Post on 11-Apr-2015
Katholieke Universiteit Leuven 1425
FACULTEIT WETENSCHAPPEN DEPARTEMENT NATUURKUNDE
Transport Properties of Underdoped Cuprates
in High Magnetic Fields Promotoren: Prof. Dr. V.V. Moshchalkov Dr. J. Vanacken
Proefschrift ingediend tot het behalen van de graad van Doctor in de Wetenschappen door Lieven Trappeniers
2000
Dit werk is tot stand gekomen dankzij de intensieve samenwerking met een
aantal collega's die ik dan ook expliciet wens te bedanken. In de eerste plaats
mijn promotoren Victor Moshchalkov en Johan Vanacken. Victor, bedankt
voor de frisse ideeën, de hulp wanneer de modellen me boven het hoofd
dreigden te groeien en het nakijken van dit manuscript.
De weg die Johan en ik aflegden om supergeleiders te bestuderen in pulsvelden
was lang en voerde ons van het opbouwen van de apparatuur via
resistiviteitsmetingen op supergeleiders naar magnetisatiemetingen. Na een
aantal jaar zetten we terug de stap naar transportmetingen, wat uiteindelijk
resulteerde in deze thesis. Onderweg werden we meer vrienden dan collega's.
Bedankt Johan voor de motiverende samenwerking en het luisterend oor.
De vier cryostaten, vijf magneten en een 8 tal meetprobes die we onderweg
nodig hadden werden voor een groot stuk mogelijk dankzij de hulp van Luc
Grammet, Freddy Gentens, Johan Morren, Philippe Mispelter en Jef Haesaerts.
Jean-Pierre Locquet (IBM), bedankt voor de samples en de interessante
discussies.
De leden en ex-leden van het supergeleiders-team: Johan, Patrick, Gerd, Kris en
Liesbet zijn bedankt voor de goede sfeer en de hulp. Met natuurlijk een
speciale vermelding voor Johan en Gerd die me in de laatste rush om
experimentele data bijstonden om "in ploegen" te meten. Bedankt Patrick, voor
de samples, het nalezen van de thesis en de gesprekjes tussendoor. Bedankt ook
de goede collega's die ik onderweg mocht tegenkomen: Igne, Paul, "Mr. Li",
Alexei, Alexander, Willy, Fritz, Tony en Manus.
Ik ben ook Professor Yvan Bruynseraede erkentelijk voor de financiële steun
tijdens het moeizame begin (en einde) van mijn IWT mandaat en voor de
mogelijkheid om elk jaar een prima conferentie te kunnen bijwonen. Uiteraard
ben ik ook het IWT erkentelijk voor de vier jaar financiële steun.
Mijn dank gaat ook uit naar mijn vrienden en familieleden die gedurende de
voorbije jaren interesse toonden in mijn werk. Bedankt moeke, vake en Koen,
voor de steun.
Ik apprecieer uiteraard ook de uitzonderlijke steun die ik krijg van Els en ik
besef ten volle dat het niet evident is om in het labo mee pulsvelden-
experimenten te komen uitvoeren de avond voor je bevalling. Ik denk niet dat
veel vrouwen "zo zot" zouden zijn.
Bedankt ook Hanne, om mij te doen beseffen dat de moeilijkheden bij het
krijgen van nieuwe tandjes en het zetten van je eerste stapjes minstens even
belangrijk zijn als de thesis-perikelen van je vake.
Introduction
The origin of high-temperature superconductivity in cuprate
materials is one of the biggest puzzles in physics,
but the behaviour of these materials when they are not
superconducting is an even bigger mystery. [Batlogg2000]
The discovery of superconductivity in 1986 [Bednorz86] in a layered copper-
oxide compound with a critical temperature Tc above 30 K was the beginning of
new area of research: high-Tc superconductivity. After this initial discovery, a
whole family of high-Tc's was discovered, all having a layered structure with
CuO2 planes and intermediate building blocks. Generally, it is accepted that the
CuO2 planes are important for the superconducting properties while the
intermediate blocks act as charge reservoirs.
The temperature versus doping T-p phase diagram of the high-Tc cuprates
contains a rich variety of phase lines and crossover lines (figure 1 below).
Apart from the superconducting phase, showing up at low temperatures in a
certain range of doping, these copper oxides are antiferromagnetic (AF) Mott
insulators at low doping. At a certain level of hole doping, the long-range AF
correlations of the Cu2+ spins in the CuO2 planes are destroyed, leaving only
short range AF correlations in the material. Experimental evidence on
underdoped cuprates indicates (i) the existence of short range
antiferromagnetic fluctuations [Shirane87, Birgenau88, Rossat91, 93 & 94],
(ii) the opening of a pseudo-gap in the density of states [Ong96, Hanaguri99,
Timusk99] at temperatures far above the superconducting critical temperature
Tc and (iii) the formation of stripes in the CuO2 planes (charge stripes
intercalated by a hole free antiferromagnetic Mott insulator) [Thurston89,
Cheong91, Mason92 & 94, Yamada97, Tranquada97 & 97b, Hunt99, Aeppli97,
Dai98, Kao99, Arai99].
The occurrence of superconductivity in materials showing such exotic magnetic
and electronic properties is still the subject of intense scientific debate.
INTRODUCTION
Moreover, any model explaining the normal-state properties or the possibility of
superconductivity must account for these recent observations. The
understanding of mechanisms responsible for the appearance of these normal-
state properties may provide key information about the nature of the
superconducting mechanism itself.
This conclusion has motivated scientists to study the normal-state properties
extensively and the recent observation of the pseudo-gap and the possibility of
the presence of charge stripes can be regarded as a consequence of this renewed
interest.
Figure 1: The properties of the high-Tc cuprates vary with temperature (vertical axis) and doping of the CuO2 planes [Batlogg2000].
One physical property that has attracted much attention is the normal-state charge transport, both the resistivity and the Hall-effect. The temperature
dependence of the zero-field resistivity ρ(T) for the optimally doped cuprates
was shown to exhibit a robust linearity down to the critical temperature. When
these compounds are underdoped, a remarkable super-linear ρ(T) curve
INTRODUCTION
develops at intermediate temperatures while at higher temperatures (T > T*) the
linear ρ(T) dependence persists. It was demonstrated that, even in rather
strongly underdoped samples, showing an insulator-like ρ(T) behaviour at low
temperatures, the occurrence of superconductivity is possible. The Hall-
coefficient was found to be dependent upon doping and temperature. A
systematic study of the zero-field normal-state transport properties of
underdoped YBa2Cu3Ox thin films at T > Tc, revealed that for all samples both
ρ(T) and the Hall-data can be scaled using the same scaling temperature
[Wuyts94 & 96]. Since these two properties are closely related to scattering of
charge carriers, this suggest that scattering in these materials has a common
origin and that only the energy scale differs.
Although widely studied, the transport properties of the underdoped cuprates
still retain some mysterious features that remain to be solved. What are the
microscopic scattering mechanisms responsible for the robust linear ρ(T)
behaviour ? What is the origin of the S-shaped super-linear ρ(T) curve in
underdoped samples ? Can it be related to the opening of a pseudo (spin) gap or
the occurrence of charge stripes intercalating hole-free AF regions ? What is
the influence of hole doping on these features ? It is thus clear that the
completion of the temperature versus doping T-p phase diagram needs more
experimental investigations to be carried out.
Oddly enough, the study of the normal ground-state properties is hindered by the superconducting phase itself, since the normal-state at T < Tc is hidden
behind the zero sample resistance in the superconducting state !
Therefore, inspired by earlier work on La2-xSrxCuO4 and Bi2Sr2CuOy [Ando95,
96, 96b & 96c], in this work a dual-track approach is chosen. In a first stage,
underdoped cuprates with varying levels of doping were prepared:
YBa2Cu3Ox and (Y0.6Pr0.4)Ba2Cu3Ox epitaxial thin films [Wagner99] and
strained La1.9Sr0.1CuO4 epitaxial thin films [Locquet98]. Doing so, the whole
underdoped region of the phase diagram can be covered while the reduced
critical temperatures allow easier access to the normal-state transport properties.
In a second stage, in order to gain access to the normal-state properties below
Tc, the transport measurements were carried out in pulsed high magnetic fields up to 50 T to suppress superconductivity in a reversible way.
INTRODUCTION
Chapter 1 reviews the most important physical properties of the cuprate
superconductors, which are essential to understand the discussion made in this
work. Furthermore, it gives a review of the state-of-the-art knowledge of the
high-temperature superconductors.
In chapter 2, the employed experimental techniques and their modifications
carried out during this work are reviewed. The experimental setup and
procedure for measuring the magnetoresistivity and Hall-effect in pulsed
magnetic fields are discussed.
Chapter 3 reports our measurements of the zero-field and high-field resistivity and magnetoresistance of YBa2Cu3Ox and (Y0.6Pr0.4)Ba2Cu3Ox
epitaxial thin films [Wagner99] and strained La1.9Sr0.1CuO4 epitaxial thin films
[Locquet98]. It will be checked whether the scaling, reported for the normal
state properties above Tc [Wuyts94 & 96], also holds for the high field transport
data. By measuring the resistivity of these cuprates at very high magnetic
fields, the normal state ρ(T) curve will be constructed, even at T < Tc. Doing so,
a statement about whether the ground state (i.e. in the absence of superconductivity) is metallic or insulating will be made. These findings will
be confronted with available models for charge transport and superconductivity
in the CuO2 planes and an experimental T(x) phase diagram for YBa2Cu3Ox and
(Y0.6Pr0.4)Ba2Cu3Ox will be constructed.
In chapter 4, the findings of chapter 3 will be further substantiated with the
results of high-field Hall-effect measurements on the same thin films. Also
for the Hall-effect, the use of very high magnetic fields is essential in order to
suppress superconductivity. We will report Hall-effect measurements on
YBa2Cu3Ox and Y0.6Pr0.4Ba2Cu3Ox thin films at temperatures extending to below the critical temperature Tc. From the measurements of the high-field Hall-
resistivity ρyx(H), the Hall coefficient RH(T) at fixed field will be calculated.
The combination of these RH(T) and ρab(T) curves then allows the derivation of
the Hall-angle. Finally, the carrier density nH that can be extracted from our
Hall-data will enable us to construct a generic T(p) phase diagram for the
YBa2Cu3Ox and Y0.6Pr0.4Ba2Cu3Ox compounds.
INTRODUCTION
The data discussed in chapters 3 and 4 will be used in chapter 5 to investigate
the effect of the short-range antiferromagnetic fluctuations and possible stripe formation in the CuO2 plane on the normal state transport properties. In this
charge-stripe picture [Emery97b & 99b], dynamic metallic [Ichikawa99,
Noda99, Tajima99] stripes are thought to dominate the transport properties. To
check this idea, an existing model [Moshchalkov93, 98b], describing transport
both in the 2D Heisenberg regime (above T*) as in the 1D striped regime (below
T*) where the pseudo gap develops, will be used as a framework for the
interpretation of our high-field normal-state transport data. Finally, a generic temperature versus hole doping T(p) phase diagram will be constructed.
Contents Chapter 1 Cuprate Superconductors ....................................................1
1.1 Introduction ......................................................................................................1 1.2 Structural properties .........................................................................................1 1.3 Doping of high-temperature superconductors ..................................................4 1.4 Evolution of physical properties with doping ................................................12 1.5 Generic T(p) phase diagram ...........................................................................26
Chapter 2 Experimenting in pulsed high magnetic fields..................29 2.1 Generation of pulsed magnetic fields .............................................................29 2.2 Cryogenics .....................................................................................................31 2.3 Transport measurements ................................................................................33
Chapter 3 Normal-state resistivity of YBa2Cu3Ox, (Y1-yPry)Ba2Cu3Ox and (La1.9Sr0.1)CuO4 ...........................................................43
3.1 Introduction ....................................................................................................43 3.2 Zero-field resistivity at T > Tc ........................................................................46 3.3 Suppression of superconductivity by high magnetic fields ............................59 3.4 Normal-state resistivity at T < Tc in high magnetic fields ..............................68 3.5 Comparison with the La2-xSrxCuO4 system ....................................................81 3.6 Localisation effects at T → 0 in the YBa2Cu3Ox, (Y0.6Pr0.4)Ba2Cu3Ox and
La1.9Sr0.1CuO4 samples...................................................................................85 3.7 Conclusions ....................................................................................................94
Chapter 4 Hall-effect in YBa2Cu3Ox and (Y0.6Pr0.4)Ba2Cu3Ox thin films ...................................................................................97
4.1 Introduction ....................................................................................................97 4.2 Hall-effect in the normal state below Tc .........................................................99 4.3 Phase diagram ..............................................................................................113 4.4 Conclusions ..................................................................................................115
Chapter 5 Effect of stripe formation on the transport properties of underdoped cuprates........................................................119
5.1 Charge ordering revisited - Stripes...............................................................120 5.2 Spin ladders, a magnetic structure between 1D & 2D..................................123 5.3 Quantum transport in doped 1D and 2D Heisenberg systems......................125 5.4 Stripe ordering at low temperatures .............................................................139 5.5 Construction of the T(p) phase diagram .......................................................143 5.6 Conclusions ..................................................................................................146
Summary .................................................................................151 Appendices .................................................................................161 Bibliography .................................................................................173 Nederlandstalige samenvatting ..........................................................185 Publicatielijst .................................................................................195
Symbols
Crystallographic a, b, c Crystallographic axes / lattice parameters
d Separation of the CuO2 planes
P, Px Hydrostatic and uniaxial pressure (x = a, b or c)
εx Strain (x = a, b or c)
Temperatures and energy scales Tc, Tc,onset, Tc,mid, Tc,offset critical temperature, at the onset, middle and offset of
the transition
TN Neèl temperature for an antiferromagnet
TRVB, TB Spinon and holon pairing temperatures in the RVB model
To Temperature below which local AF correlations appear
T* Temperature below which the spin-gap opens
TMI Boundary between the metallic and insulating ρ(T)
θD Debye temperature for the phonon spectrum
∆, ∆s, ∆p, ∆SC Gap, spin-gap, pseudo-gap, superconducting gap
Ec1, Ec2 Mobility edges
EF Fermi energy
Electronic properties g(E), g(EF) Density of states (DOS), at the Fermi level
Cel, γ = Cel/T Electronic specific heat, -coefficient
n, nH Total density of charge carriers, determined from the Hall effect
p Hole concentration in the CuO2 plane
q Elementary charge, either positive or negative
µ, µH Mobility of the charge carriers, determined by the Hall-effect
Magnetic properties
J Exchange coupling
t Hopping amplitude in the t-J model
1/(T1T) Normalised spin relaxation rate (NMR)
KS Knight shift
SYMBOLS
nc Number of chains constituting a spin-ladder
σ, σ1D, σ2D Conductivity, in 1 and 2 dimensions
H, µoH Applied magnetic field expressed in respectively Gauss & Tesla
B Magnetic induction
Electric properties J (or I), Ji Current, along the i-direction
j Current density
Ei Electric field along the i-direction
ρij Element i,j of the resistivity tensor
σij Element i,j of the conductivity tensor
ρo Residual resistivity
RH, θH Hall coefficient, Hall angle
R Sheet resistance, also called resistance per square
Length scales
eBm
h=λ Magnetic length
λ Penetration depth in a superconductor
ξ Coherence length in a superconductor
ξm, ξm1D, ξm2D Magnetic correlation length, in 1D and 2D
l Mean free path for scattering of charge carriers
Lφ Inelastic length for scattering of charge carriers
CHAPTER 1 Cuprate Superconductors
1
Chapter 1
Cuprate Superconductors
1.1 Introduction
The discovery of superconductivity in 1986 [Bednorz86] in a layered copper-oxide compound came as a great surprise, not only because of the record transition temperatures Tc, but also because these materials are relatively poor conductors in the "normal" (i.e. non-superconducting) state. Indeed, these superconductors are obtained by doping parent compounds that are antiferromagnetic (AF) Mott insulators, materials in which both the antiferromagnetism and the insulating behaviour are the result of strong electron-electron interactions. At a certain level of hole doping, long-range AF correlations of the Cu2+ spins are destroyed, leaving only short range AF correlations in the material. Even now, almost fifteen years after the start of cuprate-superconductivity research, the influence of the AF correlations and the gradual doping with holes on the transport properties and the occurrence of superconductivity remains a hot topic of research.
1.2 Structural properties
All cuprate superconductors known up to now are layered perovskites
containing CuO2 layers alternated by intermediate building blocks.
Superconductivity takes place in the doped CuO2 layers while the other layers
act as charge reservoirs. One possible way of classifying these cuprates is by
the number of CuO2 planes and the nature of the intermediate building blocks.
The La2-xAxCuO4 type of cuprate superconductors (with A being Ba, Sr,
Ca, …), also known as the "2-1-4" group, contains 2 crystallographic CuO2
CHAPTER 1 Cuprate Superconductors
2
planes per unit cell, sandwiched with La-O layers. In this system, doping is
done by partly substituting La3+ by A2+ in the La-O layers, leading to a
maximum bulk Tc of 40 K. The RBa2Cu3Ox group (R being a rare-earth Y, Gd,
Eu, Tm, …), also known as the "1-2-3" group, also contains two CuO2 planes
per crystallographic unit cell, alternated with building blocks of Ba-O, R and
CuO-chain layers. In this system, doping is mostly done by changing the
oxygen content in the Cu-O layers, leading to a maximum Tc of 92 K at the
optimal doping of x = 6.95. The (Bi, Tl, Hg)mM2Can-1CunOm+2n+2 materials (M
being Ba or Sr), with m = 1 or 2 and n = 1, 2 or 3 can reach Tc values of about
133 K. The most common members of this family are "2-2-0-1" (containing 1
CuO2 layer), "2-2-1-2" (with 2 CuO2 layers) and "2-2-2-3" (having 3 CuO2
layers), but the highest Tc of 133 K is reached in HgBa2Ca2Cu3O8+δ (1-2-2-3),
without any external pressure. For obvious reasons they are sometimes called
telephone-book compounds. Here, the block layers responsible for doping are
the (Bi, Tl or Hg)-O layers. This work will focus on the YBa2Cu3Ox and
La2-xSrxCuO4 cuprate superconductors.
YBa2Cu3O7
CuO
BaO
CuO2
Y
CuO2
BaO
CuO
O(1)
Cu(1)
O(5)
O(4)
O(3)
Cu(2)
O(2)
BaCuO2
YCuO2
BaCuO3
La2-xSrxCuO4
CuO2
LaO
LaO
CuO2
LaO
LaO
CuO2
Y
Ba/La/Sr
O
Cu
O vacancy
ab
c
YBa2Cu3O7
CuO
BaO
CuO2
Y
CuO2
BaO
CuO
O(1)
Cu(1)
O(5)
O(4)
O(3)
Cu(2)
O(2)
BaCuO2
YCuO2
BaCuO3
La2-xSrxCuO4
CuO2
LaO
LaO
CuO2
LaO
LaO
CuO2
La2-xSrxCuO4
CuO2
LaO
LaO
CuO2
LaO
LaO
CuO2
Y
Ba/La/Sr
O
Cu
O vacancy
ab
c
Figure 1.1: Unit cell of YBa2Cu3O7 (left) and La2-xSrxCuO4 (right) with the identification of the non-equivalent crystallographic sites for oxygen and copper atoms in the case of YBa2Cu3O7.
Neutron [Jorgensen87, 88 & 90, Hinks87] and X-ray [Schuller87, Grant87]
diffraction measurements revealed the unit cell of the YBa2Cu3Ox system to be
CHAPTER 1 Cuprate Superconductors
3
built of three stacked perovskite blocks stacked on top of each other (BaCuO3,
YCuO2 and BaCuO2) as shown in Figure 1.1. The central Y atom is
sandwiched between two CuO2 planes. These planes are separated from the top
and bottom (basal) CuOy layers by a BaO block. In the unit cell, the Cu-atoms
in the basal CuOy layers are named Cu(1) while the Cu in the CuO2 layers is
denoted as Cu(2). Five non equivalent positions could be identified
[Jorgensen87] for the oxygen atoms. The oxygen atoms in the CuO2 planes are
referred to as O(2) and O(3), whereas the O atoms in the BaO layer are labelled
O(4). In the basal CuOy plane, two positions were found: O(1) along the b-axis
and O(5) along the a-axis. The equivalence of these two positions in the
tetragonal phase is destroyed in the orthorhombic case where oxygen atoms are
preferably located in the O(1) positions. The basal CuOy layers have therefore a
partial filling with oxygen and the total number of oxygen atoms in a unit cell
varies from 6 to 7. For x = 6, the basal planes contain no oxygen and the
structure is tetragonal. For higher oxygen contents, the basal plane is gradually
filled with oxygen, resulting in a structural phase transition from tetragonal to
orthorhombic at x ~ 6.4. Fully oxygenated compounds with x = 7 have an
orthorhombic unit cell and the material is metallic. The onset of metallic
behaviour coincides approximately with the concentration corresponding to the
structural phase transition. The evolution of the lattice parameters of bulk
YBa2Cu3Ox with varying oxygen content is summarised in figure 1.2. Because
of the presence of ab-twinning in this compound, only high resolution neutron
or X-ray techniques or the use of untwinned single crystals yield accurate
estimates for the lattice parameters.
The La2-xSrxCuO4 system was also shown [Radaelli94, Tarascon87, Takagi87
& 89, Fleming87] to have a layered structure composed of two CuO2
crystallographic layers per unit cell (figure 1.1), in this case alternated by La-O
layers. However, in this system the layers are not aligned in the c-direction as
observed in the "123" system. Here, the O atoms of successive CuO2 layers are
stacked above each other whereas the Cu atoms alternate between (0,0,0) and
(1/2,1/2,1/2) sites in adjacent layers. Moreover, although from a
crystallographers point of view there are two CuO2 layers in a unit cell, they are
separated by two LaO layers and electronically, the La214 system is considered
being a 1-layer compound. No Cu-O chains are present in this system. Doping
CHAPTER 1 Cuprate Superconductors
4
is done by partly substituting the trivalent La by the bivalent Sr (x ranging from
0 to 0.4), resulting in a structural phase transition from orthorhombic (at x = 0)
to tetragonal (at x = 0.4) symmetry. The evolution of the lattice parameters of
bulk La2-xSrxCuO4 with changing Sr content is summarised in figure 1.2.
3.75
3.80
3.85
13.013.113.213.313.413.513.6
3.75
3.80
3.85
11.5
11.6
11.7
11.811.9
12.0
0.0 0.1 0.2 0.3 0.46.0 6.2 6.4 6.6 6.8 7.0x x
a
b
c
a
b
c
YBa2Cu3Ox La2-xSrxCuO4
a,b,
c (Å
)
RT
10 Kb
a
Figure 1.2: Lattice parameters for bulk YBa2Cu3Ox (left) [Jorgensen90] and bulk La2-xSrxCuO4 (right) [Takagi89, Radaelli94], at different doping levels, determined by neutron diffraction and X-ray diffraction. All data were taken at room-temperature, except the 10 K data for La2-xSrxCuO4.
1.3 Doping of high-temperature superconductors
The undoped parent compounds of the cuprate superconductors are
antiferromagnetic insulators. The introduction of holes in the CuO2 planes
transforms the 3d9 Cu2+ state (that is fourfold co-ordinated with the O(3) oxygen
atoms) into effective S = 0 Cu3+ sites (bound state of a S = 1/2 Cu2+ with a hole
residing mainly on the 4 surrounding O 2p orbitals) thus destroying long range
AF order and providing hole-type charge carriers which, for large hole
concentrations, are mobile in the plane (by transfer of electrons). The doping of
the CuO2 plane is generally realised in one of the following ways: by changing
the oxygen content, by chemical substitution of the cations or by the application
of strain.
CHAPTER 1 Cuprate Superconductors
5
1.3.1 Doping by varying the oxygen content
This method of doping is mainly used in the YBa2Cu3Ox system where the
oxygen content can be systematically changed from x = 6 to x = 7, resulting in a
change from an AF insulator to an underdoped superconductor at x ~ 6.3 and
gradually approaching the optimally doped Tc = 92 K case at x = 6.95. The
oxygen content of the YBa2Cu3Ox samples is set to the desired value starting
from a sample with optimal oxygen content. This sample is then kept at fixed
temperature and oxygen pressure as to yield the desired oxygen content. After
that, a slow cooldown at controlled temperature and oxygen pressure is
initiated, carefully guided by a "constant oxygen-content line" in the p-T phase
diagram [Tetenbaum89]. Details of the technical realisation of the different
oxygen contents are described in appendix B.
As introduced above, the variable part in the oxygen content in the YBCO unit
cell is entirely located in the basal CuOy plane (CuO chains). At x = 6, no
oxygen is present in the basal planes, the Cu(1) atoms have a twofold co-
ordination (by O(4)) and they are in the Cu1+ state [Rushan90]. When oxygen is
added to this insulating YBa2Cu3O6 compound, the O-atoms have an equal
probability to locate themselves in the O(1) or the O(5) positions. At x ~ 6.4
however, a structural phase transition to the orthorhombic unit cell is induced
by the addition of extra oxygen and the added oxygen is located preferably in
the O(1) sites, thus forming Cu-O-Cu chain fragments along the
crystallographic b-axis.
A simplified picture of the creation of the Cu-O fragments can be described as
follows: when a neutral O-atom is introduced into the basal plane, it needs to
extract two electrons from the surrounding structure to convert into O2-. These
electrons are provided by the 2 adjacent Cu1+ ions by transforming into 3-fold
coordinated Cu2+ [Rushan90] (Cu1+-.-Cu1+ + O → Cu2+-O2--Cu2+). No holes are
introduced in the structure, so far, by creating such Cu-O-Cu fragments. At
higher doping levels, the Cu-O-Cu fragments are lengthened by adding extra O-
atoms. When binding to the added O-atom, the 3-fold coordinated Cu2+
transforms in a 4-fold coordinated Cu2+ and one hole is created in order to
provide the second electron for the O2- and thus keep the charge balance
constant (Cu2+-O2--Cu2+-.-Cu1+ + O → Cu2+-O2--Cu2+-O2--Cu2+ + hole). If the
CHAPTER 1 Cuprate Superconductors
6
added O-atom connects two Cu-O-Cu-… fragments, 2 holes are created because
the Cu-atoms already have a +2 valence (…-Cu2+-O2--Cu2+-.-Cu2+-O2--Cu2+-… +
O → …-Cu2+-O2--Cu2+-O2--Cu2+-O2--Cu2+-… + 2 holes) [Rushan90, Veal91].
The precise amount of holes created during the filling of the basal plane with
oxygen is thus strongly dependent upon the distribution of the O-atom over the
three different processes described above. At low doping levels (6.0 < x
< 6.25), the introduced O-atoms mostly create Cu-O-Cu monomers in the CuOy
basal planes (both along the a and the b axis) and almost no holes are produced.
At intermediate oxygen contents (6.25 < x < 6.8), the oxygen atoms start to
enlarge the length of the …-Cu-O-Cu-… fragments (preferably along the b-
axis) thus reducing the number of monovalent Cu(1) atoms and creating holes.
6.0 6.2 6.4 6.6 6.8 7.00.0
0.2
0.4
0.6
0.8
1.0
hole
s / u
nit c
ell
x
YBa2Cu3Ox
Figure 1.3: The number of holes per unit cell in the YBa2Cu3Ox system [Veal91].
At near optimum doping (6.5 < x ≤ 6.95), isolated oxygen vacancies are further
filled, releasing additional holes. As oxygen is added, the decreasing
probability of O(5) occupation leads to the formation of distinct equilibrium
phases for the oxygen ordering in the basal CuO plane. At low oxygen
concentrations and moderate temperatures, an imperfect tetragonal oxygen
ordering is present, while at higher concentrations orthorhombic oxygen
ordering is established by the formation of Cu-O chains along the b-axis with a
superstructure period of one (ortho I phase), two (ortho II phase) or more a-axis
CHAPTER 1 Cuprate Superconductors
7
lattice spacings, as observed by electron microscopy [Cava90, Reyes89,
Andersen99]. Based on structural calculations [deFontaine90] and assuming a
double-cell superstructure (ortho II) of the Cu-O chains at x = 6.5, the (non-
monotonic) increase of the number of holes per unit cell by the addition of
oxygen was calculated, as shown in figure 1.3 [Veal91].
The holes that are thus created in the basal CuOy planes ( by adding oxygen and
changing the valence of Cu(1)) are moved to the CuO2 planes by the transfer of
electrons (charge transfer model) to the Cu-O chains [Cava90]. The mean
valence of the Cu(2) atoms is thus increased above 2, by creating a finite
fraction of Cu ions with a formal valence of 3+ [Cava90, Brown90].
1.3.2 Doping by chemical substitution of cations
Changing the hole concentration in the CuO2 planes can also be realised by
performing a chemical substitution of the cations Y3+ and La3+ in the
YBa2Cu3Ox ("123") or La2CuO4 ("214") unit cell. If we take a closer look at
these unit cells (figure 1.1) , we can choose between substitutions in the CuO2
planes, the CuO chains (in YBa2Cu3Ox) or at the La, Ba or the Y sites.
Depending on the nature of the substituting atom and its position in the unit cell
the influence of the chemical substitution on the normal-state and
superconducting properties will differ substantially.
In the YBa2Cu3Ox system, chemical doping is generally done by substituting
Y3+ by Pr4+,3+,.. or Ca2+ or Cu by a metal like Co, Fe, Al or Zn. Co, Fe and Al
will be mainly located at the basal Cu(1) locations (the chains) while (for small
concentrations) Zn is known to locate itself on the Cu(2) sites in the CuO2 plane
[Walker95, Tarascon88, Xiao88] giving rise to a rapid reduction of Tc.
Substitution of the trivalent Y by the bivalent Ca is known to introduce extra
holes, resulting in overdoping [Neumeier89]. An interesting case, however, is
the Y1-yPry substitution, being an exception to the general observation that
almost all members of the RBa2Cu3O7 family exhibit about the same critical
temperature Tc ~ 90 K; regardless of the type of rare-earth ion R that is used.
Y1-yPryBa2Cu3Ox is widely reported to show a gradual deterioration of its
superconducting and normal state properties as the Pr content y is increased,
becoming an insulator at y > 0.55 [Dalichaouch88, Neumeier89, Xu92,
CHAPTER 1 Cuprate Superconductors
8
François96, Jiang97, Tang99]. To complicate things even further,
superconductivity has been predicted [Blackstead95] and reported
[Blackstead96] in the PrBa2Cu3O7 compound; with Pr at the rare-earth position
between the CuO2 planes !
In order to elucidate the origin of these conflicting observations, we will
elaborate on the role of the location of the Pr ions in the YBa2Cu3O7 unit cell
for the mechanisms of destroying or enabling superconductivity.
Several reasons for the destruction of superconductivity by the presence of Pr-
atoms in the YBa2Cu3O7 unit cell are reported in literature. Often, hole filling
by the (supposed tetravalent) Pr-ion is used to explain the destruction of
superconductivity while others claim magnetic pair-breaking through an
exchange interaction between the spin of the charge carriers and the spin of the
paramagnetic Pr-ion to be important. Recent investigations [Tang99,
Dalichaouch88, Xu92, Neumeier89] on Y1-yPryBa2Cu3Ox, Y1-2yPryCayBa2Cu3Ox
and YBa2-yPryCu3O7 showed that Pr at Y sites suppresses superconductivity by
both hole filling and magnetic pair-breaking. The Y-Pr sheets then act as
"internal" charge reservoirs, inside the CuO2 bilayer, with the magnetic moment
of the paramagnetic Pr atom interacting with the carriers in the CuO2 planes.
When Pr is located at the Ba sites, only hole filling occurs, which is plausible
since the distance between the CuO2 planes and the Ba site is longer than to the
R-sites in between the planes and thus the magnetic interaction should be
weaker.
In general, it is observed that PrBa2Cu3O7 has a substantial solubility for Pr at
the Ba sites and the Pr/Ba occupancy strongly depends on the substrate
temperature during thin film deposition [Tang99, Blackstead95]. Thus, in many
cases, a finite Ba/Pr mixture in the Y1-yPryBa2Cu3Ox system can therefore not be
excluded. Blackstead and Dow attribute the often-claimed absence of
superconductivity in the PrBa2Cu3O7 compound to this finite Pr/Ba substitution
[Blackstead95 & 96]; a material with Pr only located at the Y sites, in between
the planes, should then be superconducting.
Apart from the possible hole filling and magnetic pair breaking effects, the
introduction of Pr in the unit cell, substituting Y and possibly also Ba, will
CHAPTER 1 Cuprate Superconductors
9
induce appreciable disorder in the crystal (the atomic radius of Pr lies in
between those of Y and Ba).
In the La2CuO4 system, chemical doping is generally achieved by partly
substituting the trivalent La3+ by the divalent Sr2+ (or Ba2+) thus obtaining
La2-xSrxCuO4; with x = 0 to 0.4. In the undoped case, the LaO plane has one
electron to donate to the surrounding structure since only two electrons of the
three, supplied in the formation of La3+, are required in the La3+O2- plane. The
two electrons, donated by two adjacent LaO layers, are used for the charge
balance of the intermediate Cu2+(O2-)2 plane. When a fraction of the La3+ is
substituted by Sr2+, a shortage of electrons is created. The resulting reduced
transfer of electrons to the CuO2 planes leads to the addition of holes into the
CuO2 plane by increasing the mean valence of copper from +2 (x = 0) to
Cu Cu12 3−+ +
x x (x > 0).
1.3.3 Influence of epitaxial strain
The copper oxide superconductors exhibit a remarkable range of pressure
dependencies of the superconducting transition temperature (dTc/dP). The
observed values for dTc/dP can be very high (up to +7 K/GPa), zero or negative,
depending on the doping level and the precise member of the cuprate family.
The pressure and strain dependencies of Tc contain both in-plane (ab) and out-
of-plane (c) contributions and can (for a tetragonal unit cell) be written as
equation 1.1 with ε = (dbulk - dstrained)/dbulk the strain.
cc
cab
ab
cc
c
c
ab
ccc
TTT
PP
TP
P
TTT
εε∂
∂ε
ε∂∂
∂∂
∂∂
++=
++=
2)0(
2)0(
(1.1)
Thus, the dTc/dP, as observed under hydrostatic conditions, is the result of a
subtle balance between the in-plane and out-of-plane derivatives, which often
have opposite signs [Locquet98 & 98b, Fietz 96]. Since the effect of applied
pressure on the superconducting properties depends on the crystal structure and
the pressure induced deformations, it does not come as a surprise that
YBa2Cu3O7 and La2-xSrxCuO4 show a very different dTc/dP behaviour. The
CHAPTER 1 Cuprate Superconductors
10
influence of applied pressure on the critical temperature Tc of the YBa2Cu3Ox
system, which by itself has a tetragonal to orthorhombic transition at x ~ 6.4, is
shown in figure 1.4. The interpretation of this plot is still the subject of
discussion and must account for oxygen ordering and pressure effects in the
CuO chains, charge redistribution in the CuO2 plane and for the fact that in the
orthorhombic phase there is an anisotropic dTc/dP for the a and b-axis [Fietz96,
Benischke92, Kraut93, Welp92, Welp94, Meingast91, Jorgensen90b, Pickett97
and 97b]. In any case, it is clear from figure 1.4 that the non-monotonic, and
qualitatively different behaviour of dTc/dP in the a and b directions makes
epitaxial ab strain not suitable to dope this material. Even a substrate like
SrLaAlO4, with a lattice parameter of approximately 3.76 Å at room
temperature (see also figure 1.5), will not necessarily yield a higher critical
temperature because of the compressive strain in both a and b directions.
6.5 6.6 6.7 6.8 6.9 7.0-4
-2
0
2
4
6
8
10
a
b
c
hydrostatic
YBa2Cu3Ox
x
dTc/
dP (
K/G
Pa)
Figure 1.4: The variation of dTc/dP versus doping level x for hydrostatic pressure [Beniscke92] and a, b and c-axis uniaxial pressure [Kraut93, Welp94]. All data were taken below 105 K to reduce the influence of oxygen ordering.
The bulk La2-xSrxCuO4 system however, has an orthorhombic structure over
almost the whole range of doping (x < 0.2); as long as the temperature is of the
order of Tc (see figures 1.2 and 1.10). As such, no structural phase transitions
will complicate dTc/dP, although the transition is close. Moreover, no CuO
CHAPTER 1 Cuprate Superconductors
11
chains are present in this compound, taking away the problem of oxygen
reordering and pressure effects in these chains. The problem of the different
dTc/dP response in a and b directions for the orthorhombic phase - for doping
by the application of epitaxial ab strain - was in the La2-xSrxCuO4 compound
successfully solved by the use of strained ultra-thin films [Locquet96, 98 & 98b,
Sato97]. It was shown [Locquet98b & 96b] that when the thickness of a thin-
film does not exceed 500 Å, the square symmetry of the unit cell of the
substrate can be imposed onto the film.
tensile
compressive
SrTiO3
SrLaAlO4
La1.9Sr0.1CuO4
3.76
3.78
3.80
3.82
3.92
3.94
0 200 400 600 800 1000
Temperature (ºC)
Lat
tice
para
met
er (
Å)
growthtemperature
Figure 1.5: Temperature dependence of the ab-plane lattice parameters of La1.9Sr0.1CuO4, SrTiO3 and SrLaAlO4.
Locquet and co-workers have prepared strained La1.9Sr0.1CuO4 ultra-thin films
of typical thickness of 100 Å by molecular beam epitaxy with block-by-block
deposition [Locquet94]. The choice of the substrate - SrLaAlO4 (SLAO) or
SrTiO3 (STO) - enabled them to induce compressive (SLAO) or tensile (STO)
strain in the ab-plane (see figure 1.5). This deformation - which essentially
keeps the volume of the unit cell constant - increases (compressive) or decreases
(tensile) the critical temperature significantly. A doubling of the critical
temperature has been achieved [Locquet98]. Although the precise mechanism
responsible for this strong enhancement of Tc is still under discussion, it was
suggested [Locquet98b] that a reduced charge transfer to the CuO2 planes
CHAPTER 1 Cuprate Superconductors
12
(which are more separated by the compressive ab-plane stress and volume
conservation) leads to a self-doping of the plane by an increased Cu valence.
However, tentative Hall measurements [Locquet00] seem to indicate that not
only the charge carrier density but also scattering effects might play an
important role for the critical temperature Tc. Moreover, the increased (or
decreased) orbital overlap between the CuO2-plane Cu and O atoms upon the
application of compressive (or tensile) strain might also play a relevant role that
goes beyond a simple doping picture.
1.4 Evolution of physical properties with doping
1.4.1 Electronic properties
Changing the carrier concentration in the copper oxide materials will
profoundly influence the normal state electronic properties and the occurrence
of the superconducting state.
The electronic properties of a superconductor - and more specifically the
electron density of states versus energy g(E) - can be looked upon in two ways:
the semiconductor view and the bosonic approach. In the semiconductor representation (figure 1.6, left), a superconductor below its critical temperature
Tc is considered to have for simple quasiparticles an energy gap of 2∆ between a
filled lower band and an upper band which is empty at T = 0. Excitations at
finite temperature can create quasi-particles by increasing the energy of an
electron with 2∆, thus creating a hole in the valence band. This single-electron
picture does not take into account electron pairing leading to the formation of
bosons (Cooper pairs). The bosonic representation (figure 1.6, right) includes
the bose condensation in such that at T = 0 a single energy level of paired
electrons (bosons) is separated from an empty band by a gap of energy ∆. At
finite temperatures, electron pairs can break up and go to the upper band, both
gaining an energy ∆ (since the total binding energy 2∆ is divided over the two
electrons).
CHAPTER 1 Cuprate Superconductors
13
E E
EF
EF+∆
EF+∆
T = 0 T = 0
Figure 1.6: Schematic view of the superconducting gap in the semiconductor representation (left) and the boson-condensation approach with Cooper pairs (right)
In order to gain knowledge about the normal-state electronic properties of the
copper-oxide superconductors and their evolution with doping, band structure
calculations [Pickett87 & 90] and extensive studies of the optical properties of
these materials have been performed [Yu93, Kircher91]. One can represent the
normal-state density of states (DOS) versus energy schematically for different
levels of doping (figure 1.7). The empty upper band (upper "Hubbard band") is formed by non-occupied Cu- 22 yx
3d−
states while the filled lower band is
created by a hybridisation of the O- yx,p2 band and the Cu- 22 yx3d
− lower
Hubbard band [Yu93]. The undoped system is a Mott insulator, a system with
all except one (which is half-filled) orbitals in the (Cu) 3d-shell filled, which is
insulating rather than metallic by virtue of the strong electron-electron
repulsion. Doping the material (adding holes to the CuO2 planes) can be
thought of as introducing holes into the valence band and thus lowering the
Fermi energy to within that band.
However, an alternative view [Moshchalkov88 & 90, Quitmann92], inspired by
the physics of lightly doped semiconductors, considers the dopants as impurities
that create an electron-acceptor level close to the valence band (figure 1.7a).
This impurity band grows as more (randomly distributed) dopant atoms are
added - the width being determined by the mean distance between the impurities
CHAPTER 1 Cuprate Superconductors
14
- and the Fermi-level enters the band (figure 1.7a). The electrons in this band
remain localised since the impurities are still at a large distance.
At intermediate doping levels, the wave functions of the impurities have an
increasing overlap and a narrow band with metallic conductivity appears
(figure 1.7b). At the same time, due to the disorder introduced by the
impurities, the tails of the band (below and above the two so-called mobility-
edges Ec1 and Ec2) correspond to the localised states [Anderson58] and only
hopping can take place. At higher doping levels, the Fermi-level EF crosses the
upper mobility edge Ec2 and an insulator to metal transition (IMT) sets in
(figure 1.7c). From that point on, the localised (hopping) tails coexist with -and
are shunted by- the central band in which the delocalised states show metallic
conduction.
occupied
localised
extendedE
g(E)
Ec1 Ec2
EF
Ec1 Ec2
EF
(c)
(b)
(a)
E
g(E)
E
g(E)
EF
dopi
ng
Figure 1.7: Schematic representation of the density of states g(E) of a high-temperature superconductor for light doping, in the insulating regime (a), intermediate doping (b) and heavy doping where metallic transport sets in .
From a certain level of hole doping on, a superconducting gap 2∆ between
single-electron bands opens around the Fermi energy when the material is
cooled below the critical temperature Tc (figure 1.6). As a result, the density of
states g(E) will increase at EF-∆ and EF+∆. This was microscopically described
by the famous theory of Bardeen, Cooper and Schrieffer (BCS [Bardeen57])
who considered a net attractive potential V appearing between the electrons due
CHAPTER 1 Cuprate Superconductors
15
to the electron phonon interaction. The critical temperature Tc and the gap ∆
both depend exponentially on the coupling constant V and the density of states
g(EF) at the Fermi-level (with ωD the Debye frequency for phonons), leading to
a constant ratio between ∆ and Tc (equation 1.2); when assuming a not too
strong coupling (the weak coupling limit).
53.3~)0(2
13.1
2
)(
1
)(
1
cBVEgDcB
VEgD
TkeTk
e
F
F ∆
=
=∆−
−
ω
ω
h
h (1.2)
This relation between the critical temperature Tc, the potential V and the DOS
g(EF), provides a framework for a possible explanation for the very high critical
temperatures observed in the novel cuprate superconductors. One way is to turn
to a very strong coupling V between the electrons while others focus on the
enhancement of g(EF) by postulating either novel coupling interactions (e.g.
bipolarons [Alexandrov88, Mott90]) or an inhomogeneous distribution of
charge carriers within the CuO2 planes [Bianconi96, 97 & 98, Valetta97]. Only
the latter scenario seems to escape the problem of the increasing ratio 2∆/kBTc
(beyond 3.53) and the onset of lattice instabilities as the coupling gets stronger.
The opening of the superconducting gap ∆ exactly at Tc was not observed in the
novel cuprate superconductors. In these compounds a partial (pseudo) gap was
found to open at temperatures far above the onset of the macroscopic
superconducting state as some kind of precursor. This pseudo-gap will be
discussed in more detail later on in this chapter.
For various high-Tc compounds, the evolution of the superconducting critical
temperature Tc with doping was shown to obey a universal curve for the
dependence on the hole concentration p in the CuO2 plane [Tallon90, 93 & 95].
This concentration (expressed as a fraction of holes per Cu atom in the plane) is
in La2-xSrxCuO4 directly given by the strontium content x whereas in the
YBa2Cu3Ox system it is the result of bond valence calculations. The universal
Tc(p) curve can be described by the parabolic-like empirical relation
Tc(p)/Tc,max=1 - 82.6 (p - 0.16)2 (figure 1.8).
CHAPTER 1 Cuprate Superconductors
16
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.350.0
0.2
0.4
0.6
0.8
1.0
1.2
Tc/
Tc,
max
p, hole concentration
YBa2Cu3Ox
Y1-yCayBa2Cu3O6.96
Y1-yCayBa2Cu3O6
Y0.9Ca0.1Ba2Cu3Ox
La2-xSrxCuO4
Figure 1.8: Reduced critical temperature versus hole concentration (fraction of holes per Cu-atom in the CuO2 plane) for different high-Tc compounds, after [Tallon95, Cava90].
1.4.2 Magnetic structure
During the past few years, the evolution of the magnetic structure of the high Tc
cuprates upon hole doping has been studied extensively using inelastic neutron scattering (INS) (YBa2Cu3Ox [Rossat91, 93 & 94, Tranquada97, Dai98,
Arai99, Kao99], La2-xSrxCuO4 [Shirane87, Birgenau88, Endoh88, Thurston89,
Cheong91, Mason92 & 94, Yamada97, Tranquada97]) and nuclear magnetic resonance (NMR) [Alloul89, Kitaoka91, Yasuoka94 & 97, Berthier97,
Carretta99]. It was found that, at low levels of doping, these oxides are 3D
antiferromagnets (AF) with a Neèl temperature of about 410 K for the undoped
YBa2Cu3O6 compound and TN ~ 250-300 K for La2CuO4. In the undoped
compounds, the CuO2 planes are built up from S = ½ 3d9 Cu2+ and S = 0 2p6 O2-
states (figure 1.9). This results in an AF ordering of the Cu spins, mediated by
the electrons forming the S = 0 2p O-orbital (super-exchange). At temperatures
TN < T < J ~ 1700 K, the 3D AF order is destroyed but 2D AF correlations
persist. The magnetic properties of the CuO2 planes can thus be described as a
2D Heisenberg system on a square lattice by the Hamiltonian
∑ ⋅=ji
ji SSJH,
rr (1.3)
CHAPTER 1 Cuprate Superconductors
17
with i and j labelling lattice sites, iSr
are spin ½ operators and <i,j> denotes
nearest neighbours sites. J (>0), the AF exchange coupling determining the
energy scale of the problem, was estimated to be JCu-Cu ~ 125-170 meV
(previous references and [Mizuno98]).
S=1/2 Cu2+
S=1/2 Cu2+
S=0 Cu3+
(formal)
AF
disturbed AF order
Slightlydoped
Undoped
Figure 1.9: Schematic view of the CuO2 plane and the influence of hole doping on the antiferromagnetic (AF) order in the plane.
When holes are introduced into de CuO2 plane, they go to the oxygen orbitals (the O yx,p2 band lies slightly above the Cu 22 yx
3d−
lower Hubbard band) and
a formal S = 0 Cu3+ state is formed (figure 1.9). This so-called Zhang-Rice
singlet [Zhang88, Tjeng97] is a bound state of a S=1/2 Cu2+ with a hole residing
mainly on the 4 surrounding O 2p orbitals. The introduction of these effective
S = 0 sites represents a significant disturbance of the AF ordering and as a
result, the Neèl temperature is drastically decreased upon doping (figure 1.10).
At a concentration of p ~ 0.02 holes/Cu-atom, the long-range 3D AF state is
completely suppressed; although short range AF correlations still persist for
higher doping levels. In the La2-xSrxCuO4 system, these AF correlations are
purely 2D and have a correlation length of ~ 50 Å for the undoped plane and
8 Å for the CuO2 plane near optimum doping. In YBa2Cu3Ox, the neighbouring
CuO2 planes remain coupled for these AF fluctuations, even up to optimal
doping [Rossat93].
CHAPTER 1 Cuprate Superconductors
18
La2-xSrxCuO4
0 0.10 0.20 0.30
Sr content x
0
20
40
0
50
150
200
250
100
TN (K) Tc (K)
Antiferrom
agnetic
Orthorhombic Tetragonal
Superconducting
I M
Ortho Tetra
0
50
100
oxygen content x6 6.2 6.4 6.6 6.8 7
0
100
200
300
400
TN (K) Tc (K)
OrthorhombicMetallic
TetragonalInsulating
Antiferro-magnetic Superconducting
YBa2Cu3Ox
Figure 1.10: Schematic phase diagram for the magnetic, superconducting and structural properties of bulk La2-xSrxCuO4 and YBa2Cu3Ox as a function of Sr or oxygen content x [Rossat93, Fleming87, Takagi89].
Once holes are present in the plane, transfer of electrons allows a S = 1/2 Cu2+
and S = 0 Cu3+ to exchange positions and the Mott-insulator state is destroyed at
a certain critical level of doping (figure 1.10). The model describing the motion
of the S = 0 Cu3+ ions in a background of Heisenberg coupled S = 1/2 Cu2+ ions
is known as the t-J model, represented by its Hamiltonian:
( )∑∑ ++ +−⋅=σ
σσσσ,,
,,,,, ji
ijjiji
ji cccctSSJHrr
(1.4)
CHAPTER 1 Cuprate Superconductors
19
with t > 0 a "hopping amplitude" and +σ,ic and σ,ic the electron creation and
annihilation operators respectively; no double occupancy is allowed. The
exchange interaction was estimated JCu-Cu ~ 125-170 meV (see above) whereas
the hopping amplitude was calculated to be tCu-O ~ 1 - 1.24 eV [Mizuno98].
This picture is complicated by two fairly recent observations: the existence of a
pseudo spin-gap at temperatures T > Tc (for a review see [Timusk99]) and the
observation of incommensurate spin fluctuations in both La2-xSrxCuO4
[Thurston89, Cheong91, Mason92 & 94, Yamada97, Tranquada97, Hunt99] and
YBa2Cu3Ox [Aeppli97, Tranquada97, Dai98, Kao99, Arai99] pointing towards
the formation of dynamic stripes [Bianconi96, 97 &98, Emery97b & 99b,
Zaanen99, Valetta97].
1.4.2.1 The pseudo gap
The pseudo-gap is observed as a gradual and partial decrease in the density of
states near the Fermi level, at temperatures by far exceeding the
superconducting critical temperature [Ong96, Hanaguri99, Timusk99]. This
pseudo-gap has been found in underdoped La2-xSrxCuO4 as well as in
YBa2Cu3Ox as reflected in (i) a reduction of the spin-lattice relaxation rate 1/T1T
and Knight-shift KS in NMR [Alloul89, Yasuoka94 & 97, Berthier97], (ii) the
development of a gap around EF in the dynamic spin susceptibility in inelastic
neutron scattering (INS) [Rossat93, Thurston89], (iii) a decrease in the
electronic specific heat coefficient γ = Cel/T ~ g(EF) [Loram93 & 98,
Momono99], (iv) a loss of spectral weight in Raman spectroscopy
measurements [Chen97, Naeini99 & 99b], (v) the development of a gap in angle
resolved photo emission spectroscopy (ARPES) [Loeser96, Ino98 &99], (vi)
tunnelling spectroscopy [Oda98] and (vii) resistivity (see paragraph 1.4.3).
These measurements all show the opening of a partial energy gap ∆ in the DOS
below a temperature T*. ARPES measurements [Loeser96, Ino98 &99,
Timusk99] showed this gap to be consistent with a d-wave symmetry, showing
the development of a gap in the (π,0) directions in reciprocal space with arcs of
a gapless Fermi surface in the (π,π) directions in underdoped samples.
In the scientific literature, several crossover temperatures T* (To, Tco, Ts, Tsg) and
various gaps ∆ (∆s, ∆p, ∆o, …) have been used. In La2-xSrxCuO4 two crossover
CHAPTER 1 Cuprate Superconductors
20
temperatures were found: To marks the onset of AF correlations whereas below
T* the pseudo-gap opens (see figure 1.11). In YBa2Cu3Ox, the observed pseudo-
gap in the DOS was shown to be very sensitive to specific physical property
used to define the pseudo gap. Probing charge excitations (like ARPES) is
reported to yield a gap ∆p, approximately twice as large as the spin-excitation
gap ∆s as observed in NMR and INS experiments [Nakano98, Mihailovic99].
This points in the direction of two possible (complementary) mechanisms, both
leading to the opening of a depleted region in the DOS near EF. Numerous
models were proposed to address the origin of such gaps; the precise relation
between these two gaps and the superconducting gap is, however, not yet
cleared out.
p
T
3DAF
TN
Tc
SC
To
T*
spingap
AF correlations
Figure 1.11: Schematic and simplified diagram showing the superconducting and 3D antiferromagnetic (AF) phases, together with the To and T* crossover lines for the onset of AF fluctuations and pseudo gap behaviour, respectively.
One possible paradigm for explaining the opening of the spin-gap is the
resonating valence bond (RVB) model [Anderson73 & 88, Suzumura88,
Nagaosa90] in which the attributes of the electrons -spin and charge- are
separated into two types of quasiparticles carrying spin (spinons with zero
charge) and charge (holons without spin). The formation of the
superconducting state then requires both the spinons (fermions) and holons
(bosons) to be paired into Cooper pairs. In the RVB model, below a
temperature TRVB (= T*), due to the strong AF correlations, the spinons are
CHAPTER 1 Cuprate Superconductors
21
paired into so-called Zhang-Rice singlets [Zhang88, Tjeng97] and the holons
are responsible for the (non-superconducting) transport of electric current. At a
temperature TB < TRVB, also the holons undergo Bose-condensation and the
superconducting state is established (Tc = TB). The phase line for the pairing of
the spinons shows a decreasing TRVB as the level of hole doping is increased,
whereas the temperature TB for the bose-condensation of the holons increases.
The prediction by the RVB model of the existence of two separate mechanisms
for the opening of a gap is in a nice agreement with the different pseudo gaps
reported when probing charge- or spin-excitations [Nakano98, Mihailovic99].
The spin-bag mechanism [Schrieffer88] is, like the RVB model, also based on
the notion of spin-charge separation. In this model, the introduction of a hole in
an AF correlated region disturbs the AF order and a bag (a sort of a spin-
polarised cloud or an AF spin-polaron) is created in the AF in which the hole is
trapped. When another hole is added to the structure, another hole-bag is
created and the two holes are attracted to share a common bag. The resulting
pairing interaction then leads to the opening of a superconducting gap.
A model that uses similar concepts as the RVB model and spin-bag model is the
precursor-pairing model [Emery97, 97b & 99, Randeira97] combined with the
concept of charge stripes [Cheong91, Tranquada97, Arai99, Bianconi96, 97
&98, Zaanen99,]. Below a temperature To, AF correlations set in and holes are
expelled from the AF regions [Schrieffer88] when cooling down further. In that
way hole-free AF Mott-insulating regions are created, intercalated by hole-rich
metallic charge-stripes (topological doping [Kivelson96]). At this point, no
superconductivity is established yet (in contrast to the spin-bag model described
above). When lowering the temperature even further, the reduced
dimensionality resulting from the confinement of the AF correlations, allows a
spin-gap to be established [Dagotto96] at temperatures far above the
superconducting critical temperature Tc. This energy gap is then imposed upon
the metallic stripes by pair hopping of holes between the stripe and the AF
surroundings (magnetic proximity effect). The superconducting state is
recovered when the spin-gaps, that are formed locally, acquire macroscopic
coherence by Josephson coupling between the stripes. However, the precise
CHAPTER 1 Cuprate Superconductors
22
link between the J (125-170 meV) scale local physics and the Tc (10 to 20 meV)
scale long-range physics of superconductivity still remains unclear.
1.4.2.2 Formation of stripes
The above mentioned incommensurate fluctuations in the microscopic spin
ordering have been observed both in La2-xSrxCuO4 [Thurston89, Cheong91,
Mason92 & 94, Yamada97, Tranquada97, Hunt99] and YBa2Cu3Ox [Aeppli97,
Tranquada97, Dai98, Kao99, Arai99], at temperatures above the
superconducting critical temperature Tc. They show up in the structure factor
derived from inelastic neutron scattering experiments as intensity peaks that are
incommensurate with the crystal lattice spacing. These observations can be
interpreted as an argument in favour of a separation of the CuO2 planes into AF
hole-free regions (a few lattice spacings wide), alternated with metallic stripes
of holes (a superlattice of quantum stripes [Bianconi96, 97 &98]). This
interpretation is supported by 63Cu nuclear quadrupole resonance (NQR)
measurements which are sensitive to the local gradient of the electric field and
are wiped out by the strong gradients in the alternating hole free and hole rich
stripes. The observation of incommensurate fluctuations in the microscopic
magnetic ordering forms a strong experimental support for the models based on
stripe formation [Bianconi96, 97 &98, Emery97b & 99b, Zaanen99].
1.4.3 Transport properties
In a free electron system, the electrical resistivity can be written as [Ashcroft76]
τ
ρ2
*
ne
m= (1.5)
with m* the effective mass, n the density and τ the scattering time of the
electrons. Although being a naïve single electron picture, it already provides
the intimate connection between the electrical resistivity and the density and the
microscopic scattering of the electrons. This indicates that the study of the
transport properties of a material can give key information about the nature of
the charge carrier scattering.
Scattering of electrons can be due to lattice imperfections, magnetic and non-
magnetic impurities or the interactions with thermal lattice vibrations
CHAPTER 1 Cuprate Superconductors
23
(phonons). In conventional metals, the transport properties are governed by this
electron-phonon scattering. There are two relevant temperature regions relative
to the Debye temperature θD. For high temperatures T >> θD, all phonon modes
are excited and the resistivity ρ varies linearly with temperature. However, as
the temperature decreases the resistivity drops faster with temperature and
obeys a T 5 law. This is due to the fact that below θD an increasing number of
phonon modes with high excitation energy starts to freeze out, leading to a
reduced electron-phonon scattering [Ashcroft76].
>∝<<∝
+=−
−− )(
)(5
Dphel
Dphelphelo TT
TT
θρθρ
ρρρ (1.6)
The residual resistivity ρo remaining at low temperatures is then a measure for
the amount of impurities in the crystal lattice. The Debye temperature θD for
the high-temperature superconductors lies between 300 K and 450 K and the
deviation from linear resistivity should certainly be expected to show up at
T ≤ 0.2 θD [Poole95].
In the cuprate high-temperature superconducting compounds, an enormous
effort has been put in the exploration of the resistivity ρ(T) at various levels of
hole doping and different geometrical configurations between the transport
current and the crystallographic axes. It was found that these materials exhibit a
very unusual behaviour of the resistivity when doping is changed from
underdoped to optimally doped and overdoped samples [Takagi92, Kimura92,
Ito93, Batlogg94, Ando96b, Wuyts94 & 96].
At optimal doping, the in-plane resistivity ρab(T) shows a robust linear
behaviour that extends from the critical temperature Tc up to very high
temperatures; although Tc is significantly lower than the Debye temperature θD.
The absence of an excess conductivity indicates that another scattering
mechanism, besides electron-phonon scattering, may play a prominent role in
these materials. When the concentration of holes is lowered, the resistance
increases and the linear ρab(T) behaviour transforms into a super-linear
behaviour at low temperatures T < T* (figure 1.12); the so-called S-shape
behaviour. This excess conductivity was studied extensively in YBa2Cu3Ox
[Ito93, Wuyts94 & 96] and La2-xSrxCuO4 [Suzuki91, Kimura92, Takagi92,
CHAPTER 1 Cuprate Superconductors
24
Batlogg94] and was (by comparison with magnetic measurements like for
instance NMR) attributed to the opening of the spin pseudo-gap below a
temperature T*. At even lower levels of hole doping, the conductivity decreases
even further and at low temperatures, the slope dρ/dT becomes negative, a sign
for the onset of insulating (or semiconducting) behaviour (see also the phase
diagram in figure 1.13). The temperature at which dρab/dT changes sign to an
insulating behaviour decreases as the doping level is increased. The
temperature range, where the in-plane resistivity shows an insulating behaviour,
thus shrinks as doping is elevated and it is therefore camouflaged by the onset
of the superconducting phase, as can be seen in the phase diagram in
figure 1.13.
underdopednear optim
aloverdoped
ρab
Tρab
Tρab
T
ρc
T
ρc
T
ρc
T
II
Figure 1.12: Schematic overview of the transport properties of the high-Tc cuprates in the underdoped (top), optimally doped (middle) and overdoped (bottom) regime with the transport current in the ab plane (left) and parallel to the c axis (right).
So one can ask then whether the traditional description of the metal to insulator
(MI) transition by a simple vertical line in the T(p) diagram (figure 1.10) is
physically correct. This question becomes of particular relevance when
considering the fact that the out-of-plane c-axis resistance ρc(T) exhibits an
insulating alike behaviour (dρ/dT <0) in the whole underdoped to optimally
doped region of the phase diagram (figure 1.13). Only in the overdoped regime
CHAPTER 1 Cuprate Superconductors
25
a Fermi-liquid alike regime is entered, characterised by an approach of the ρ ~
T 2 behaviour, a sign of electron-electron scattering becoming dominant in
charge-carrier scattering.
To address this issue, the experiments, initiated by Ando to measure the normal-
state transport properties of Bi2Sr2CuOy (Bi2201) [Ando96c, 97, 97b] and
La2-xSrxCu4 (La214) [Ando95, 96, 96b, 97, 97b, Boebinger96] below Tc, are of
great importance. Ando and co-workers were able to suppress the masking
superconducting phase by applying very high magnetic fields and thus to reveal
the normal-state properties below Tc. In that way, it was shown that in La214
and Bi2201 both the ab-plane and c-axis transport show a diverging behaviour
of ρ(T) at low temperatures. The traditional MI phase line of figure 1.10 at a
certain critical concentration of doped holes, was therefore nuanced to yield the
phase diagram shown in figure 1.13, accounting for the results of Ando and co-
workers and the observed anisotropy.
The consequences of this puzzling behaviour for the microscopic mechanism
for conduction and scattering of the charge-carriers are still the subject of
intense debate and might be connected with the opening of the (spin) pseudo-
gap at T* and the presence of charge stripes in the CuO2 planes.
p
T
3DAF
Fermi Liquid
TN
Tc
Insulating
Metallic ρab
Insulating ρc
SC
Figure 1.13: Schematic phase diagram for the transport properties of the high-Tc cuprates. The notions metallic and insulating are defined by dρ/dT > 0 and dρ/dT <0 respectively.
CHAPTER 1 Cuprate Superconductors
26
1.5 Generic T(p) phase diagram
The copper-oxide high-temperature superconductors form a class of materials
showing unusual features in their electronic and magnetic properties which are
as fascinating as uncomprehended. Following the discussion in the previous
paragraphs we will construct a generic T(p) phase diagram for this class of
layered CuO2 materials and focus on specific points where knowledge is still
lacking. In figure 1.14 the major regimes for the electronic and magnetic
properties are sketched.
At low levels of hole doping the 3D AF phase on the T(p) diagram is rapidly
shrinking as the doping is increased and only short range AF correlations
persist, when cooling below a temperature To. Below a temperature T* a partial
gap in the DOS is formed around EF. Although being still under discussion,
experimental evidence exists for this pseudo gap to be a spin gap
(paragraph 1.4.2.1, and for a review see [Timusk99]). As revealed by
measurements in high magnetic fields on La2-xSrxCuO4 and Bi2Sr2CuOy, at low
temperatures (mostly below Tc), both the in plane and the out of plane c-axis
resistivity in the underdoped compounds show a diverging resistivity
(dρ/dT < 0), pointing to an insulating or semiconducting ground-state. Thus, in
these compounds the metal to insulator (MI) transition line was shown to
penetrate into the superconducting phase rather than being a vertical line in the
T(p) phase diagram (figure 1.14). The superconducting phase -showing up
below Tc in a very limited range of doping levels- is superimposed with this
normal-state behaviour and masks some of its anomalous features. The critical
temperature Tc has a parabolic dependence on the concentration of holes in the
CuO2 plane. At near optimum doping and above Tc, the high-Tc's are anomalous
metals showing a linear ρ(T) behaviour that is not in agreement with the
transition from ρ(T) ~ T 5 to ρ(T) ~ T at T ~ θD in conventional metals.
This phase diagram is however not complete and evokes some questions.
What happens in the two layer compound YBa2Cu3Ox ? Does it also exhibit a
MI phase line entering into the superconducting phase rather than being a
vertical line at some critical concentration of holes ?
CHAPTER 1 Cuprate Superconductors
27
p
T
3DAF
SC
spingap
anomalous metal
Fermi liquid
I M
TN
Tc
T*
AF correlations
To
MI
p
T
3DAF
SC
spingap
anomalous metal
Fermi liquid
I M
TN
Tc
T*
AF correlations
To
MI
Figure 1.14: Schematic overview of the T(p) phase diagram for the high-Tc cuprates, showing the 3D antiferromagnetic (AF) and superconducting (SC) region. The additional phase lines are explained in the text.
What can we extract from the S-shape behaviour of ρ(T) below T* in the pseudo
spin-gap regime and can we make the experimental and theoretical correlation
between these two phenomena more specific ? Can it be compared to what
happens in other techniques in the pseudo gap regime ? Are these phenomena
purely dependent upon the charge carrier density or is the situation more
complicated ? To answer these important question, high field transport
experiments (both magneto-resistivity and Hall-effect) were initiated on thin
films of YBa2Cu3Ox and (Y0.6Pr0.4)Ba2Cu3Ox with varying oxygen content and
on ultra thin films of La1.9Sr0.1CuO4 under epitaxial strain.
CHAPTER 2 Experimenting in pulsed high magnetic fields
29
Chapter 2
Experimenting in pulsed high
magnetic fields
Experimenting in high pulsed magnetic fields is not an easy task because of the
necessary miniaturisation and the transient signals in the experiment. The large
sweep rates of the magnetic field produce unwanted voltages in the wiring and
induce eddy currents in the metallic parts of the set-up. The main experimental
techniques and their modifications carried out during this work are reviewed in
this chapter. The experimental set-up and procedure for measuring the
magnetoresistivity and Hall-effect in pulsed magnetic fields are discussed.
2.1 Generation of pulsed magnetic fields
The generation of a magnetic field can be achieved by simply putting a current
through some conducting windings. Evidently, the more windings and the
higher the current, the higher the magnetic field will be. Constraints of heating,
mechanical stability and the need for enormous electric power block this
roadmap for DC fields at around 36 Tesla, achieved at the hybrid magnet of
NRIM in Tsukuba [Herlach95].
Some of these constraints can be relieved by turning to pulsed magnetic fields
using non-destructive, liquid nitrogen cooled, coils. The transient nature of
such field pulses reduces the thermal load and takes away the need for a
constant feed of a high electric current. The mechanical stability of the coil
however is in this case more complicated and the forces are far above what can
be sustained by wires with a reasonable conductivity (like copper). Therefore,
CHAPTER 2 Experimenting in pulsed high magnetic fields
30
at the K.U.Leuven, the choice was made to turn to solenoids with a non-uniform
density of windings - in order to distribute the stress over all layers -, alternated
by glass- or carbon-fibre reinforcement thus providing the necessary strength
[Li98, Herlach89 & 96]. Figure 2.1 shows a schematic drawing of a Leuven-
type coil.
0
10
20
30
40
50
0 5 10 15 20-10
0
10
20
30
40
50
µ oH
(T
)
t (ms)
µo dH
/dt (kT/s)
MeL83Ucap = 4750 VC = 24 mF
epoxycontacts
Stainless steelcylinder
carbon-fibrecomposite
glass -fibrecomposite
conductor
studding
8 - 17 mm
Figure 2.1: Schematic drawing of a coil for the generation of pulsed magnetic fields. (plot) Magnetic field and sweep rate versus time for MeL83.
The generation of the necessary high pulsed currents is achieved by the
discharge of a high-voltage capacitor bank. The switching gear that is
necessary to permit such a discharge and the diagnosis of the coil before and
after the shot is presented in figure 2.2. Four coils (g) can be powered
sequentially, with a charging time of at maximum 2 minutes in between. This
discharge is done either by a mechanical switch (f) or a stack of thyristors (f).
The switching set-up also contains safety discharge switches (d) allowing a
quick and controlled disposal of the energy in case of an emergency. The
maximum energy is 475 kJ, provided by a 38 mF capacitor bank charged to
5000 V, although the default value for daily experiments is set to 24 mF. The
CHAPTER 2 Experimenting in pulsed high magnetic fields
31
oscillations in the LRC circuit that is formed by the coil (g), the wiring and the
capacitor bank (c) are strongly damped by the 0.08 Ω crowbar circuit (e). A
typical field profile (using coil MeL83) is shown in figure 2.1. This 60 T-class
coil, used to produce the experimental results in this work, is capable of
generating 57 T with a pulse duration of ~ 17 ms and was the first user-coil in
Leuven to have carbon-fibre reinforcement between the layers of CuAg wire. It
has a bore diameter of 17.4 mm and the design constitutes of 10 layers, each
with 18 windings of CuAg-wire, resulting in L = 512 µH and R = 46 mΩ at
liquid nitrogen temperature. These values are checked before and after the field
pulse to ensure that no deterioration of the mechanical or geometrical properties
of the coil has occurred.
(a) (c) (d) (f)(e)
(g)
(h) L/R measurement
(b)
Figure 2.2: Simplified electrical scheme of the high-voltage switching-gear that enables the discharge of the capacitor bank in a chosen coil, permitting a pre- and post-shot diagnosis of the coil.
The whole pulsed field facility is controlled by a central measuring computer
that operates the high-voltage bank, performs the discharge, diagnoses the coils,
makes stringent safety checks, collects experimental data, controls the
temperature of the experiments and allows to perform a first analysis of the
data. It is evident that for safety reasons all connections of this computer to the
high-voltage switching gear or the experimental equipment are made by optical
fibre. Safety measures also include strict grounding prescriptions and the use of
shielded Faraday cages containing the experimental equipment.
2.2 Cryogenics
The miniaturisation imposed by the constraints during the design of the pulsed
field coils and the pulsed nature of the magnetic field have severe implications
CHAPTER 2 Experimenting in pulsed high magnetic fields
32
on the cryogenic technology. The tail of the cryostat (figure 2.3) can have a
maximum outer diameter of about 17 mm while leaving enough free space
inside to allow physical experiments to be carried out. Moreover, this tail is
preferably constructed out of a non-conducting (plastic, ceramic) or poorly
conducting (stainless steel) material as to reduce eddy-currents. At the
K.U.Leuven pulsed field facility it is now possible to perform measurements
from room temperature down to 360 mK [Li98, Herlach89 & 96].
vacuumspace
vacuumpump
He flowout
Cu-block+ heater
Cu-block
He contact-gas inletsample room
vacuum
sam
ple
room
15.38mm
He flow
in
flow cryostat
vibrationinsulation
N2 vessel
60 teslamagnet
magnetmountingrack
cryostatmountingplate
Figure 2.3: Schematic view of the 4He cryostat (top panel) and the magnet support with vibration insulation (bottom panel).
CHAPTER 2 Experimenting in pulsed high magnetic fields
33
The home-made 4He flow cryostat that was used to establish the data in this
work has a tail of outer diameter 16 mm and consists of 4 coaxial stainless-steel
tubes (figure 2.3, top panel). The gas flow is first conducted through a heated
copper block, then guided through the tail after which it passes another copper
block that thermally isolates the inner tube from the surroundings. This cryostat
allows experiments from room-temperature down to 4.2 Kelvin.
Another requirement for fruitful experiments in pulsed fields is an effective
insulation for mechanical vibrations between the cryostat and the magnet. This
is achieved by rigidly attaching the coil to a fixed support system. The cryostat
itself is then mounted on top of this rack using cylinders of elastic silicone
rubber (figure 2.3, bottom panel) for damping. This construction enables
experimenting in an environment with a vibrational noise level that is an order
of magnitude lower than in the absence of vibration insulation.
2.3 Transport measurements
The presence of parasitic effects like mechanical vibrations, transient voltages
induced by the pulsed magnetic fields and a high-voltage discharge in the
neighbourhood of the experimental equipment also asks for specific counter
measures to be taken.
As a first precaution, the home-made measuring probe (based on a 1 meter glass
tube of diameter 6 mm) is attached to the cryostat using vibration insulation.
Damping of the mechanical vibrations is necessary since any movement of the
measuring probe in the, in reality non-uniform, magnetic field will induce
unwanted voltages. Simulations show the inhomogeneity of the magnetic field
to be less than 0.01 % at 1 mm from the centre of the magnet, yielding parasitic
voltages in the circuit that are an order of magnitude smaller than the "direct"
µodH/dt voltages induced in the wires. However, since they are essentially
random, they cannot be compensated completely and efficient damping of
vibrations remains important.
Secondly, the metallic cryostat, together with the shielding of the connecting
measuring-cable and the shielded measuring box forms one single Faraday cage
CHAPTER 2 Experimenting in pulsed high magnetic fields
34
(figure 2.4). This provides efficient screening of spurious transient
electromagnetic effects. The various signals of the physical measurement are
led into the box containing the instrumentation devices by separate coaxial
cables.
The actual set-up for the measurement of the magneto-resistance and the Hall
effect is depicted in figure 2.4. The thin film samples are patterned in a 1 mm
long 50 µm strip as to generate a high signal even at low currents. Electric
contacts are realised by means of gold wire, attached with silver paint to thin
(< 1000 Å) gold pads deposited and annealed on the film. The constant current
through the sample is generated using a battery-operated current source,
independent of the rest of the set-up, with a series resistor of about 600 kΩ to
limit the variation of the current (caused by a varying resistance of the sample)
to less than 0.5 %. The signal, generated by the sample then is amplified by a
home-made instrumentation amplifier (based on the Burr-Brown INA103 and
INA110 low noise operational amplifiers) with a bandwidth ranging from DC
up to 150 kHz. No electronic filtering is performed. In this amplifier, the
physical signal is mixed with the µodH/dt voltage that is induced in a small
pick-up coil in the vicinity of the sample. This permits to subtract spurious
signals that are induced in the wiring and to obtain the useful physical signal.
The magnetic field is measured by monitoring the induced voltage in a second
pick-up coil with a calibrated area (S = 66.83 mm2 for the measurements in this
work). The different voltages, generated during the 17 ms field pulse, are
measured with a 4 channel, 12 bit, transient recorder (BE256 Bakker
Electronics) operating at 1 MHz.
The temperature is monitored by means of a silicon miniature diode temperature sensor (SMDT v3.0, Institute of Cryogenics, Southampton), which,
with a constant current of 10 µA applied, gives signals from 0.56 V at room
temperature up to 1.71 V at 4.2 Kelvin. This voltage is then measured using a
digital voltmeter (DVM, Keithley 2000).
All data taken by this voltmeter and the transient recorder are transferred to the
central measuring computer by means of an optical-fibre IEEE-488 data bus,
using special bus extension devices (Hewlett-Packard 37204 HP-IB extender).
CHAPTER 2 Experimenting in pulsed high magnetic fields
35
Cryostat Shielded measuring box
x1
I
SMDT
dB/dt
comp
Shieldedcable
x1
230 V
filter
IEEE
TransientRecorder
optical
Hall
MR
I
DVM
Cryostat Shielded measuring box
x1
I
SMDT
dB/dt
comp
Shieldedcable
x1x1
230 V
filter
IEEE
TransientRecorder
optical
Hall
MR
I
DVM
Figure 2.4: Schematic overview of the set-up used for transport measurements in pulsed magnetic fields. The cryostat (left) is connected to the shielded measuring box (right) by means of a shielded cable. The parts with grey background depict the wiring for measurements of respectively the magnetoresistance (MR) and the Hall-effect.
All measurements, reported in this work, were taken in the in-plane transverse
configuration (figure 2.5), with the applied current in the ab-plane and the
magnetic field applied perpendicular to this current and the ab-plane (H // c).
H
H
H
HH
I
I
I
I
I
a
b
c
in-planeI // ab
out-of-planeI // c
H // c H // ab
transverse
transverse
transverse
longitudinal
longitudinal
Figure 2.5: Geometrical configurations for transport measurements on layered materials, with I the transport current, H the external magnetic field and a, b and c the crystallographic axes.
CHAPTER 2 Experimenting in pulsed high magnetic fields
36
Although the voltage versus time traces that are transferred to the computer are
almost completely compensated for induced voltages µodH/dt, the combination
of multiple field pulses is required in order to obtain the real physical signal.
2.3.1 Magnetoresistance
For measurements of the magnetoresistance, two shots of different field
polarity and positive current are averaged (equation 2.1). Since µodH/dt is an
odd function of the magnetic field (see figure 2.11), this removes induced
µodH/dt influences that have survived the hardware compensation in the
amplifier.
)0(2
1 00 >+
== <> IVV
II
VR HH (2.1)
This experimental procedure can be illustrated with a real experiment on a
1770 Å thick YBa2Cu3O6.7 epitaxial thin film, data that will be presented in
detail in chapter 3. Figure 2.6 presents the voltage, generated in the 1 mm x
50 µm strip at 4.2 K (top) and 70 K (bottom), during a 50 T field pulse of
positive (VH>0) and negative (VH<0) polarity. The applied current was 146 µA,
resulting in a current density of approximately 1.6· 107 A/m2 (much smaller than
the critical current density of the order 109 A/m2 and the depairing current
~ 1013 A/m2) and the signal was amplified 200 times. The relevant physical
signal that is plotted in figure 2.6 after amplification corresponds to a voltage
over the sample that lies in the range 1 to 100 mV (depending on the sample and
the current). From these plots the equivalent level of noise at the sample (both
of vibrational and electronic origin), can be estimated at 70 µV at low fields and
1 mV at 50 T (the width of the noise band). The two shots with opposite
polarity of the magnetic field show a significant hysteresis at low magnetic
fields where the asymmetry in the sweep rate µodH/dt is high (figure 2.6). This
imperfect compensation is removed by combining the VH>0 and VH<0 traces
using equation 2.1, resulting in the third trace on the two figures. Both below
and above Tc the resulting traces are of a remarkable quality, showing no
significant sign of voltages induced due to mechanical vibrations. Moreover,
although the sweep rate of the magnetic field varies from 15 kT/s at rising field
to -5 kT/s at lowering magnetic field, no hysteresis was observed in these traces.
CHAPTER 2 Experimenting in pulsed high magnetic fields
37
This proves that the heating of the metallic parts (sample, contacts and cryostat) is of minimal influence on the physical measurement. Furthermore, the perfect reproducibility of all measurements reported in this
work by smaller field pulses (typically between 15 and 20 T) shows that the
transport properties, derived from pulsed field measurements are not
significantly influenced by the high sweep rate.
0 10 20 30 40 507
8
9
10
11
µoH (T)
VH > 0
VH < 0
YBa2Cu3O6.7T = 70 K
V (
x200
, Vol
t)
0 10 20 30 40 50
0
2
4
6
8
µoH (T)
V (
x200
, Vol
t)
VH > 0
VH < 0
YBa2Cu3O6.7T = 4.2 K
Figure 2.6: Transport measurements in an YBa2Cu3O6.7 epitaxial thin film at T = 4.2 K << Tc (top) and T = 70 K > Tc (bottom). The two shots VH>0 and VH<0 are shifted up by one unit with respect to the final trace.
The final traces, as presented in figure 2.6, can then be recalculated to the true
resistivity ρ by dividing by the applied current and the amplification factor and
taking into account the thickness and width of the strip and the distance between
the voltage probes. The resulting ρ(H) plots at various temperatures are
presented as one summarising plot (figure 2.7) in which the data were smoothed
by adjacent averaging over 20 points. Taking into account the 3.3 µs sampling
CHAPTER 2 Experimenting in pulsed high magnetic fields
38
rate this results in an averaging time window of 66 µs thus reducing the
influence of frequency components above 16 kHz. This frequency is still much
higher than the approximate 50 Hz at which the physical properties of interest
vary during the field pulse. From figure 2.7 it can be noted that no additional
hysteresis is induced by the low-pass filtering procedure. During the past few
years, it was shown that such an excellent quality of transport measurements
can only be achieved by a careful implementation of the precautions discussed
above.
The collection of one ρ(H) trace, including the reproducing low-field shots,
takes about 2.5 hours (depending on the temperature stability) resulting in 3 to 4
ρ(H) traces per day. Thus the construction of the summarising plot in figure 2.7
takes as a minimum one week of intensive experimenting.
0 10 20 30 40 500
100
200
300
400
500
µoH (T)
ρ (µ
Ωcm
)
4.2 K
127 KT
YBa2Cu3O6.7
Figure 2.7: Resistivity ρ versus applied field up to 50 T (up and down field-sweeps) for an epitaxial thin film of YBa2Cu3O6.7 at temperatures T = 4.2 K, 11.4 K, 19.4 K, 30.1 K, 38.8 K, 46.8 K, 52.4 K, 62.6 K, 70 K, 81 K, 100.8 K and 127 K. Note the absence of hysteresis effects and induced voltages due to mechanical vibrations.
2.3.2 Hall effect
The transverse electric field arising from the Hall-effect is measured using a
standard 5-terminal configuration where a virtual contact pad can be moved by
varying a 100 kΩ resistor, placed between the two contacts at the same side of
CHAPTER 2 Experimenting in pulsed high magnetic fields
39
the strip (figures 2.4 and 2.8). This allows the virtual contact to be placed at
exactly the opposite side of the 3rd contact pad (figure 2.8). The variable
resistor has to be of high resistance such that no significant current is flowing
through it. Doubling the value of this variable resistor was shown not to
improve the quality of the measurements but rather to complicate offset
adjustments of the amplifiers. A proper adjustment of the compensation of the
system involves three successive steps, which have to be iterated until the
desired level of compensation is obtained. As a first step, the zero current
offsets of the four operational amplifiers have to be set to zero. Step two
involves the alignment of the virtual Hall contact pad, with the current set to the
value used in the later experiments. In the third step the overall compensation
of the system for the induced voltages µodH/dt is optimised using a number of
small (6 T) field pulses. Since all three steps are interdependent, at least one
additional iteration is necessary.
VHI I
Figure 2.8: Electric scheme for a 5 terminal measurement of the Hall effect.
A proper measurement of the Hall-voltage requires four field pulses to be
combined and averaged. From figure 2.11 it can be seen that the Hall-voltage
can be obtained in two independent ways (VHall,a and VHall,b in equation 2.2), both
removing the parasitic µodH/dt and magnetoresistive signals.
+−=
+=+
=><<>
<<>>
2
22 0,00,0
,
0,00,0,,,
IHIHbHall
IHIHaHall
bHallaHallHall VV
V
VVVVV
V (2.2)
Also here, the experimental procedure can very well be illustrated with a real
experiment on a 850 Å thick Y0.6Pr0.4Ba2Cu3O6.7 epitaxial thin film, data that
will be presented in detail below in chapter 4. Figure 2.9 shows four voltage
CHAPTER 2 Experimenting in pulsed high magnetic fields
40
traces (VH>0, I>0, VH<0, I<0, VH<0, I>0 and VH>0, I<0), generated in the 1 mm x 50 µm
Hall pattern at 84.6 K during a 17 T field pulse with the polarity of both the
field H and the current I changing in between the pulses. These traces were
taken after amplification by a factor 1250 and with an applied current 600 µA,
resulting in a current density of approximately 7· 107 A/m2 (much smaller than
the critical current density (of the order 109 A/m2) and the depairing current
~ 1013 A/m2).
-3
-2
-1
0
1
2
3
V (
x125
0, V
olt)
µoH (T)
VH I> >0 0,
VH I< <0 0,
VV V
Hall aH I H I
,, ,=
+> > < <0 0 0 0
2
Y0.6Pr0.4Ba2Cu3O6.7T = 84.6 K
0 5 10 15 20
-3
-2
-1
0
1
2
3V
V VHall b
H I H I,
, ,= −+> < < >0 0 0 0
2
µoH (T)
VH I> <0 0,
VH I< >0 0,Y0.6Pr0.4Ba2Cu3O6.7T = 84.6 K
0 5 10 15 20
V (
x125
0, V
olt)
Figure 2.9: Measurements of the voltage over a Hall pattern in a Y0.6Pr0.4Ba2Cu3O6.7 epitaxial thin film at 84.6 K in pulsed fields up to 17 T: (top) VHall,a as an average of VH>0, I>0 and VH<0, I<0 and (bottom) VHall,b as an average of VH<0, I>0 and VH>0, I<0. The four shots VH>0, I>0, VH<0, I<0, VH<0, I>0 and VH>0, I<0 are shifted down by one unit with respect to the final trace. No filtering or numerical smoothing was applied.
The relevant physical signal that is plotted in figure 2.9 after amplification
corresponds to a Hall voltage across sample that reaches at maximum a few
100 µV (depending on the sample and the current). Also here, the raw traces
CHAPTER 2 Experimenting in pulsed high magnetic fields
41
VH>0, I>0, VH<0, I<0, VH<0, I>0 and VH>0, I<0 show an appreciable hysteresis over the
whole field region. Just as in the case of the magnetoresistance experiments this
imperfect compensation is removed by combining these traces in order to give 2/)( 0,00,0, <<>> += IHIHaHall VVV and 2/)( 0,00,0, ><<> +−= IHIHbHall VVV using
equation 2.2 (figure 2.9). These two traces almost perfectly overlap and show a
linear behaviour. However, when comparing these two traces in detail, small
symmetric deviations from the 4-pulse-averaged VHall were observed. This
might be due to an effect that is an even function of both field- and current
polarity; the precise origin of this effect is still under investigation. In any case
this artefact is completely removed by performing the average over four pulses
(figure 2.10).
0 5 10 15 200.0
0.5
1.0
1.5
µoH (T)
Y0.6Pr0.4Ba2Cu3O6.7T = 84.6 K
VH (
Vol
t)
VV V
Hall
Hall a Hall b=+, ,
2
0.00
0.02
0.04
0.06
0.08
0.10
0 5 10 15 20
µoH (T)
Y0.6Pr0.4Ba2Cu3O6.7
T = 84.6 K
VH/µ
oH (
V/T
)
Figure 2.10: The Hall voltage VHall in a Y0.6Pr0.4Ba2Cu3O6.7 epitaxial thin film at 84.6 K in fields up to 17 T as an average of VHall,a and VHall,b (top). The ratio VH/µoH which is proportional to the Hall coefficient RH (bottom).
CHAPTER 2 Experimenting in pulsed high magnetic fields
42
The final VH curve in the upper panel of figure 2.10 is of a remarkable quality,
showing no significant sign of mechanical vibrations or hysteresis; this curve
was smoothed with a 20-point adjacent averaging (16 kHz). Also here the high
field shots (40 to 50 T) were reproduced by smaller (10 to 20 T) pulses and the
underlying physics seems not to be significantly influenced by the sweep rate of
the magnetic field.
From this Hall voltage versus applied field, the ratio VH/µoH (proportional to the
Hall coefficient RH) can be calculated (lower panel of figure 2.10). This gives
(above Tc) an approximately constant value over the applied field range. From this ratio, we can calculate the Hall coefficient zxyH HjER ≈ by taking into
account the value of the applied current and the width of the strip.
The collection of one VH(H) trace, including the reproducing low-field shots,
takes about 4 to 5 hours (depending on the temperature stability) yielding in 2 to
3 traces/day. A reasonable characterisation of the Hall coefficient of a sample
thus takes at least 2 weeks of intensive experimenting.
µoH (T)
Induced µodH/dt
µoH (T)
Magnetoresistance
I > 0
I < 0
µoH (T)
Hall-voltage
I > 0
I < 0
µoH (T)
Induced µodH/dt
µoH (T)
Induced µodH/dt
µoH (T)
Magnetoresistance
I > 0
I < 0
µoH (T)
Magnetoresistance
I > 0
I < 0
µoH (T)
Hall-voltage
I > 0
I < 0
µoH (T)
Hall-voltage
I > 0
I < 0
Figure 2.11: Schematic view of the magnetoresistance, the Hall voltage and the parasitic µodH/dt voltage versus applied magnetic field. The full line is for positive currents while the dashed line represents the situation for negative measuring currents.
CHAPTER 3 Normal-state resistivity of Y123, YPr123 & La214
43
Chapter 3
Normal-state resistivity of
YBa2Cu3Ox, (Y1-yPry)Ba2Cu3Ox
and (La1.9Sr0.1)CuO4
3.1 Introduction
Although widely studied, the transport properties of the underdoped cuprates
still retain some mysterious features that remain to be solved. What are the
microscopic scattering mechanisms responsible for the robust linear ρ(T)
behaviour ? What is the origin of the S-shaped ρ(T) curve below T* ? Can it be
related to the opening of a pseudo (spin) gap or the occurrence of charge stripes
intercalating hole-free AF regions ? What is the influence of hole doping on
these features ? Where in the temperature (T) versus hole doping (p) phase
diagram should the boundary between the metallic and insulator-like behaviour
be drawn ? It is clear that, as discussed also in chapter 1, the completion of the
T(p) phase diagram needs more experimental investigations to be carried out.
Recently, it was shown that the zero-field normal-state ρ(T) curves for
YBa2Cu3Ox at various doping levels exhibit an almost perfect scaling onto a
universal curve by linearly scaling both temperature and resistivity [Wuyts94 &
96]. The linear part of this universal curve was interpreted in terms of
scattering of charge carriers in a 2D antiferromagnet (AF) [Moshchalkov93,
Wuyts96]. The super-linear behaviour at lower temperatures was not
understood yet.
CHAPTER 3 Normal-state resistivity of Y123, YPr123 & La214
44
This work [Moshchalkov93, Wuyts94 & 96] on the normal-state resistivity
above Tc can in a natural way be extended to the region below the critical
temperature Tc by suppressing the "unwanted" superconducting phase by the use
of strong magnetic fields (cf. the work of Ando and co-workers on La214 and
Bi2201 cited earlier). In our work we will do this on the "123" system in which
the doping level is changed by varying the oxygen content x (YBa2Cu3Ox) and
chemically substituting Pr for Y ((Y0.6Pr0.4)Ba2Cu3Ox). This dual-track
approach - lowering Tc by changing the hole content p on one side and using
very high magnetic fields on the other side - allows us to cover the whole
underdoped to optimally doped region of the T-p phase diagram and thus
address the questions raised above.
x t
(Å)
Tc,mid
(K)
∆∆T
(K)
ρρo
(µΩµΩcm)
R
(kΩΩ)
ρρ290 K
(µΩµΩcm)
ρρ290K/ρρo
#1Y 6.95 1770 92.2 1.1 48 0.8 579 ~12
#2Y 6.8 1300 73.7 3.5 35 0.6 426 ~12
#1Y 6.7 1770 58.2 2.7 209 3.6 1171 ~5.6
#2Y 6.5 1300 52.9 2.2 95 1.6 835 ~8.7
#1Y 6.45 1770 41.7 7.7 607 10.4 2237 ~3.7
#3Y 6.4 2300 17.9 13.2 1900 32 4910 ~2.6
#4YPr 6.95 850 41.4 5.3 278 4.8 658 ~2.4
#4YPr 6.85 850 31.8 6.7 459 7.8 1205 ~2.6
#4YPr 6.7 850 22.3 12.7 805 13.8 1904 ~2.4
Table 3.1: Overview of the YBa2Cu3Ox and (Y0.6Pr0.4)Ba2Cu3Ox thin films selected for this work with their thickness t, critical temperature Tc, width of the transition ∆Tc, residual resistivity ρo, sheet resistance per square R (calculated from the residual resistivity and to be compared to the quantum of resistance h/4e2 ≈ 6.45 kΩ), resistivity at 290 K ρ290 K and the ratio ρ290 K/ρo. The method to determine ρo will be discussed in paragraph 3.2.2.
CHAPTER 3 Normal-state resistivity of Y123, YPr123 & La214
45
For this purpose, a total of approximately 70 epitaxial thin films was deposited
by sputtering on SrTiO3 substrates [Wagner99]. A selection of these films was
patterned by wet chemical etching after which the oxygen content was adjusted
to the desired nominal value. The appendices give more details about the
sample preparation (A, B & C) and characterisation (D). For the YBa2Cu3Ox
compound, two films were selected for experiments in pulsed magnetic fields
(labelled #1 and #2). They were subsequently made oxygen deficient (see
table 3.1 and appendix B). Film #3 was not used for experiments in pulsed
fields, and thus only the zero-field data for this x = 6.4 sample will be presented.
For the (Y0.6Pr0.4)Ba2Cu3Ox compound, one film was selected and the oxygen
content was consecutively set to nominal values x = 6.95, x = 6.85 and x = 6.7.
The main physical properties such as thickness, critical temperature Tc,
resistivity at 290 K and residual resistivity ρo of the samples are summarised in
table 3.1.
Experimenting in high DC magnetic fields is possible up to 36 tesla (at the
NRIM in Tsukuba); when higher fields are necessary one has to turn to the
technique of pulsed magnetic fields [Herlach95]. The physical measurements
reported here, were carried out at the pulsed magnetic fields facility of the
K.U.Leuven [Herlach89 & 96] that allows measurements in magnetic fields up
to 60 T at temperatures from room temperature down to 360 mK. The transport
measurements (both magnetoresistivity and Hall-effect) were performed in a
home-made cryostat and using a home-made high-field coil. All measurements
reported in this work were performed in the transverse geometry (H ⊥ I) with
the magnetic field perpendicular to the film (H // c) and the current sent along
the ab-plane (I // ab). A detailed discussion of the experimental procedure and
the special issues arising when experimenting in transient magnetic fields was
given in chapter 2.
In this chapter we will report the normal-state resistivity of YBa2Cu3Ox and
(Y0.6Pr0.4)Ba2Cu3Ox thin films at various oxygen contents and La1.9Sr0.1CuO4
ultra thin films under epitaxial strain, above and below their critical temperature
Tc. For the YBa2Cu3Ox and (Y0.6Pr0.4)Ba2Cu3Ox compounds, we will make a
scaling analysis of both the zero-field and the high magnetic field data. The
low-temperature divergence of the resistivity will be examined.
CHAPTER 3 Normal-state resistivity of Y123, YPr123 & La214
46
3.2 Zero-field resistivity at T > Tc
The zero-field resistivity of the samples was measured using a standard 4 point
measuring technique on a 1000 x 50 µm strip patterned onto the thin film. This
well defined geometry allows to obtain a high signal and to recalculate it to the
resistivity. The precise experimental procedure is described in chapter 2.
3.2.1 Temperature dependence of the zero-field resistivity
As explained on the previous pages, the temperature dependence of the in-plane
resistivity ρab(T) was measured on two YBa2Cu3Ox and one (Y0.6Pr0.4)Ba2Cu3Ox
epitaxial thin films at various levels of oxygen content. The results for the x =
6.45, 6.5, 6.7, 6.8 and x = 6.95 YBa2Cu3Ox films are summarised in figure 3.2
while the data for the (Y0.6Pr0.4)Ba2Cu3Ox x = 6.7, 6.85 and x = 6.95 films are
presented in figure 3.3.
From these plots, the resistivity at 290 K can be extracted, yielding values
between 420 and 4900 µΩcm that increase while lowering the oxygen content,
for each sample individually. The ρ(290 K) values are summarised in table 3.1
and they are in good agreement with the values reported in literature, ranging
from 300 µΩcm up to 104 µΩcm. Thus, the high-Tc superconductors are
located somewhere between the very good metallic conductors (e.g. like Cu)
with ρ(290 K) ~ 1 - 10 µΩcm and the poor conductors ρ290 K ~ 10 - 100 µΩcm
on one side and the semiconductors with ρ290 K ~ 104 to 1015 µΩcm on the other
side. The resistivity of film #1 is slightly higher than film #2, possibly because
of the uncertainty in the measurement of the thickness (see appendix C) or
possible over-etching (same appendix). However, the smaller value of the ratio
ρ290 K/ρo (see below) for film #1, compared to film #2, indicates that film #1
apparently contains more disorder than film #2.
These resistivities can be interpreted in term of the sheet resistance (or
resistance per square) R = (ρa)/(ad) = ρ/d, with a the side of the square and d
the thickness of the sheet. This resistance per square can, for the Y123
compounds, be estimated by considering them as a stacking of thin conducting
layers (e.g. the CuO2 planes) and setting d = c/2 ~ 5.85 Å. However, since the
two CuO2 planes are lying very close together in the Y123 unit cell, setting
CHAPTER 3 Normal-state resistivity of Y123, YPr123 & La214
47
d = c would also be an acceptable choice. This apparent uncertainty in R is no
problem, since the resistance per square can be regarded as an estimate for the
maximum sheet resistance for a metallic system. The estimated sheet resistance
can be compared to the quantum of resistance h/4e2 ≈ 6.45 kΩ. (equation 3.1).
⇒<<⇒>>
==insulator1
metal14
quantum econductanc
e/squareconductanc
21
2
c
ehlk
abF ρ
(3.1)
This comparison is preferably made on the basis of the low-temperature
resistivity (residual resistivity) where temperature dependent scattering (like the
phonon contribution) is small. Although this is just a rough estimate, the values
for the sheet resistance in table 3.1 already show that the samples with the
lowest doping approach the 6.45 kΩ limit, a strong hint towards an insulating
behaviour at low temperatures.
A remarkable observation concerning the Pr doped YBa2Cu3Ox is that, although
the ρ290 K values are equivalent with the undoped compound, the residual
resistivities ρo are significantly higher. This is probably related with the fact
that, as discussed in paragraph 1.3.2, the introduction of Pr in the YBa2Cu3Ox
unit cell causes, apart from hole filling and magnetic pair breaking effects,
appreciable disorder in the crystal. The relatively low values of ρ290 K/ρo for
these films (~ 2.5 for (Y0.6Pr0.4)Ba2Cu3Ox and 3 to 12 for YBa2Cu3Ox) indicate
their relatively high impurity concentration compared to normal metals where
ρ290 K/ρo ~ 1000. The method used for obtaining a reliable estimate for ρo will
be discussed in paragraph 1.2.2.
The plots in figure 3.2 and 3.3 reveal a temperature dependency of the
resistivity that contains a rich variety of features that, moreover, are
pronouncedly sensitive to the doping level. At optimal doping (YBa2Cu3O6.95)
the ρ(T) exhibits a remarkable linearity that extends down to just above the
critical temperature Tc. At lower levels of hole doping, the regime of linear ρ(T)
shifts to higher temperatures and an S-shape, super-linear, behaviour emerges
between Tc and T* (T* is commonly used to denote the temperature at which the
linear ρ(T) transforms into the super-linear behaviour). This development of a
region T < T* with a reduced resistivity is accompanied by an increase of the
slope in the region of linear ρ(T) and an overall increase of the resistivity (both
CHAPTER 3 Normal-state resistivity of Y123, YPr123 & La214
48
ρ290 K and ρo, see also table 3.1). This is because the in-plane transport is
determined by the number of charge carriers in the plane (and hence the doping
level). The stronger increase of ρo is an indication that reducing the oxygen
content or substituting Pr for Y, apart from lowering the charge carrier density,
also induces appreciable disorder (in Pr-doped YBa2Cu3Ox, there are at least
three effects playing: hole-filling changes the density of charge carriers,
magnetic scattering by the paramagnetic Pr ions and disorder).
0
200
400
600
800
1000
ρ ab
(µΩ
cm)
0 50 100 150 200 250 3000
500
1000
1500
2000
T (K)
ρab (µΩ
cm)
YBa2Cu3Ox
#1
#2
#2
#1
#1
6.456.56.76.86.95
x
Figure 3.2: The zero-field resistivity ρ(T) for the YBa2Cu3Ox films selected for this work.
These qualitative changes in the behaviour of ρ(T) for YBa2Cu3Ox,
(Y1-yPry)Ba2Cu3Ox and also La2-xSrxCuO4 are by now widely documented in the
literature for both thin films and single crystals [Almasan97, Batlogg94,
Boebinger96, Jiang97, Levin97, Takagi92, Wuyts94 & 96]. It was also shown
that in strongly underdoped samples, at low temperatures, an increase of the
resistivity occurs upon decreasing temperature. This insulating ρ(T) behaviour
is found to be compatible with the onset of superconductivity.
The critical temperature Tc at which superconductivity sets in is also closely
correlated with the hole doping (e.g. the oxygen content or the level of Y/Pr
substitution). When reducing the oxygen content of the thin films, the critical
temperature decreases, both in YBa2Cu3Ox (figure 3.2) and (Y0.6Pr0.4)Ba2Cu3Ox
(figure 3.3).
CHAPTER 3 Normal-state resistivity of Y123, YPr123 & La214
49
0 50 100 150 200 250 3000
500
1000
1500
2000
T (K)
ρ ab
(µΩ
cm)
Y0.6Pr0.4Ba2Cu3Ox
6.76.856.95
x
Figure 3.3: The zero-field resistivity ρ(T) for the (Y0.6Pr0.4)Ba2Cu3Ox thin films selected for this work.
In figure 3.4, the Tc(x) values for the YBa2Cu3Ox thin films studied in this work
are plotted together with the values obtained for bulk samples [Beyers89,
Cava90] for which the oxygen content was accurately determined by chemical
and thermogravimetric analyses. As already reported extensively in the
literature, the Tc(x) data exhibit a systematic decrease with decreasing oxygen
content x with a plateau near Tc ~ 60 K. From this plot, a good agreement
between our thin film samples and the bulk Tc(x) data is demonstrated.
A simple experimental phase diagram is obtained by plotting the Tc(x) data for
the YBa2Cu3Ox and (Y0.6Pr0.4)Ba2Cu3Ox thin films in one graph (figure 3.5). As
noted above, both systems show a decreasing Tc(x) as x is reduced. In the
(Y1-yPry)Ba2Cu3Ox system however the substitution of 40 % of the Y atom by Pr
already suppresses Tc to about 41 K and a further reduction of the charge carrier
density by oxygen desorption induces an additional reduction of Tc. From this
phase-diagram, it is clear that the oxygen content x is not an appropriate
parameter for constructing a phase diagram for both YBa2Cu3Ox and
(Y0.6Pr0.4)Ba2Cu3Ox, since (Y0.6Pr0.4)Ba2Cu3Ox is already underdoped by the
40 % Y/Pr substitution. The precise mechanism of doping the 123 system by an
Y/Pr substitution is not yet clear (see paragraph 1.3.2). However, the solubility
for Pr at the Ba sites suggests a finite Ba/Pr substitution in the Y1-yPryBa2Cu3Ox
CHAPTER 3 Normal-state resistivity of Y123, YPr123 & La214
50
system to be possible. The suppression of superconductivity by adding Pr is
most probably a mixture of both hole filling and magnetic pair-breaking
[Tang99, Dalichaouch88, Xu92, Neumeier89] (see paragraph 1.3.2).
6.2 6.4 6.6 6.8 7.00
20
40
60
80
100
Bulk, Beyers et al. Bulk, Cava et al.
Thin Films, this work
x
Tc (
K)
YBa2Cu3Ox
Figure 3.4: Critical temperature Tc for YBa2Cu3Ox in the form of bulk material [Beyers89, Cava90] and thin films [this work].
6.0 6.2 6.4 6.6 6.8 7.00
100
200
300
400
500
T (
K)
x
YBa2Cu3Ox
AF
Y0.6Pr0.4Ba2Cu3Ox
SC
Tc
Tc
Tc
Figure 3.5: Experimental phase diagram for the YBa2Cu3Ox and the (Y0.6Pr0.4)Ba2Cu3Ox system with the critical temperature Tc, mid. The antiferromagnetic region is only indicative.
CHAPTER 3 Normal-state resistivity of Y123, YPr123 & La214
51
3.2.2 Scaling of the zero-field resistivity for different doping levels
The distinct features in the temperature dependence of the zero-field resistivity
of YBa2Cu3Ox and (Y0.6Pr0.4)Ba2Cu3Ox discussed above, i.e. a linear ρ(T) at
temperatures above, a super-linear ρ(T) behaviour below T* and a saturating or
increasing ρ(T) at low temperatures are universal for all the curves reported in
figure 3.2 and 3.3 (except for some samples the low temperature saturation).
The only difference is the temperature at which these features set in. This
observation has motivated scientists to make a successful scaling analysis of the
transport properties of YBa2Cu3Ox [Wuyts94 & 96].
A similar scaling analysis can be applied to our ρ(T) data of figures 3.2 and 3.3.
In figure 3.6 the scaled ρ(T) curve is plotted for the YBa2Cu3Ox thin films for
which the temperature is rescaled with a parameter ∆ and the resistivity is given
by (ρ-ρo)/(ρ∆-ρo). The parameter ∆ then defines the energy scale controlling the
linear and super-linear behaviour. The residual resistivity ρo is subtracted from
ρ and the resulting resistivity is then divided by ρ∆-ρo, with ρ∆ the resistivity at
T = ∆. This procedure and the nomenclature differ from the approach used by
Wuyts et al. who did not subtract the residual resistivity and used a crossover
temperature To rather than T*.
Similar to the previous reports [Wuyts94 & 96], a very nice scaling is obtained
for all YBa2Cu3Ox samples with oxygen contents from x = 6.4 up to the optimal
value of 6.95 (figure 3.6). The scaling reported in figure 3.6 is of an even better
quality than the previous attempts, due to the subtraction the residual resistivity.
This near-perfect quality of the scaling of the ρ(T) curves is most convincingly
demonstrated by the good overlap of the derivatives as illustrated in figure 3.7.
Within the noise level, the overlap is perfect; the upward deviations at low
temperatures mark the onset of superconductivity.
CHAPTER 3 Normal-state resistivity of Y123, YPr123 & La214
52
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
T/∆
ρ ρρ ρ
−−
o
o∆
x = 6.4 / 6.45 / 6.5 / 6.7 / 6.8 / 6.95
YBa2Cu3Ox
Figure 3.6: Scaling of the zero field ρ(T) data for the YBa2Cu3Ox thin films with x = 6.4, 6.45, 6.5, 6.7, 6.8 and x = 6.95. The temperature is rescaled with a parameter ∆ (an energy scale) and the resistivity is given by (ρ-ρo)/(ρ∆-ρo) in which the residual resistivity ρo is subtracted and ρ∆ is the resistivity at T = ∆.
0 1 2 30
1
2
3
4
5
∆
T/∆
dρ/d
T
YBa2Cu3Ox
x = 6.4 / 6.45 / 6.5 / 6.7 / 6.8 / 6.95
Figure 3.7: Derivative dρ/dT of the scaled resistivity of the YBa2Cu3Ox thin films shown in figure 3.6.
At this point, the question arises whether a similar scaling analysis works
equally well for the temperature dependence of the resistivity for the
(Y0.6Pr0.4)Ba2Cu3Ox thin films. This seems to be probable since qualitatively the
CHAPTER 3 Normal-state resistivity of Y123, YPr123 & La214
53
same features were observed in these films (a linear regime shifting to higher
temperatures and an S-shape region developing as the oxygen content is
decreased). The result of this scaling analysis is shown in figure 3.8. Also here,
a very good scaling of the ρ(T) data is obtained by subtracting the residual
resistivity and then linearly scaling the temperature with ∆ and the resistivity
with ρ∆. Moreover, the (Y0.6Pr0.4)Ba2Cu3Ox data collapse perfectly with the
YBa2Cu3Ox data shown in figure 3.6. Also here, the temperature derivatives of
the resistivity are in good agreement (figure 3.9).
0.0 0.5 1.0 1.5 2.0 2.5 3.0-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
ρ ρρ ρ
−−
o
o∆
T/∆
Y0.6Pr0.4Ba2Cu3Ox
x = 6.7 / 6.85 / 6.95
Figure 3.8: Scaling of the zero field ρ(T) data for the (Y0.6Pr0.4)Ba2Cu3Ox thin films with x = 6.7, 6.85 and x = 6.95. The temperature is rescaled with a parameter ∆ (an energy scale) and the resistivity is given by (ρ-ρo)/(ρ∆-ρo) in which the residual resistivity ρo is subtracted and ρ∆ is the resistivity at T = ∆.
A remarkable observation from these rescaled ρ(T) data is however that,
although the (Y0.6Pr0.4)Ba2Cu3Ox and the YBa2Cu3Ox data fall on the same
universal curve, the energy scale ∆ in the two systems differs substantially. For
the same oxygen content, the scaling parameter ∆ that is needed to obtain the
universal curve is significantly higher for the (Y0.6Pr0.4)Ba2Cu3Ox films,
although the critical temperature is about a factor of 2 lower (see table 3.10).
This apparent higher energy scale ∆ in the (Y0.6Pr0.4)Ba2Cu3Ox system is further
illustrated by the rescaled ρ(T) data in figure 3.8 for which the linear regime is
CHAPTER 3 Normal-state resistivity of Y123, YPr123 & La214
54
only reached at room temperature (whereas for the YBa2Cu3Ox system the linear
regime is entered easily for the samples with the highest oxygen content).
0 1 2 3-1
0
1
2
3
4
T/∆
dρ/d
T
Y0.6Pr0.4Ba2Cu3Ox
x = 6.7 / 6.85 / 6.95
Figure 3.9: Derivative dρ/dT of the scaled resistivity of the (Y0.6Pr0.4)Ba2Cu3Ox thin films shown in figure 3.8.
composition x Tc, mid
(K)
∆∆
(K)
T*
(K)
ρρo
(µΩµΩcm)
YBa2Cu3Ox 6.95 92.2 86.5 173 48
6.8 73.7 126.9 254 34.9
6.7 58.2 174.3 349 209
6.5 52.9 198.6 397 95.3
6.45 41.7 215.1 430 607
6.4 17.9 248.5 497 1900
(Y0.6Pr0.4)Ba2Cu3Ox 6.95 41.4 158.7 317 277.7
6.85 31.8 185.9 372 458.5
6.7 22.3 184.5 369 804.8
Table 3.10: Oxygen content x, critical temperature Tc, scaling parameter ∆, crossover temperature T* ≈ 2∆ and residual resistivity ρo for the studied thin films. The energy scale ∆ and the residual resistivity ρo are scaling parameters used in figure 3.6 and 3.8.
CHAPTER 3 Normal-state resistivity of Y123, YPr123 & La214
55
Having established the perfect scaling of the zero field ρ(T) data of both the
YBa2Cu3Ox and the (Y0.6Pr0.4)Ba2Cu3Ox system, a more detailed discussion is
appropriate. In figures 3.11 and 3.12, the rescaled data are plotted again, but
now with some extra indications to facilitate the discussion.
At high temperatures T > T*, a linear regime is observed. This regime is
universal for all samples, at least for those were T* is within our experimental
temperature interval (< 300 K). For the samples with the lowest oxygen content
this is not the case. For YBa2Cu3Ox, the temperature T* is approximately equal
to 2∆, the energy scale of the conduction process (see table 3.10). The
crossover temperature T* increases as the doping level is decreased. These
general observations are valid for both the YBa2Cu3Ox and the
(Y0.6Pr0.4)Ba2Cu3Ox system and the region of linear ρ(T) is labelled I in
figure 3.11. Since in (Y0.6Pr0.4)Ba2Cu3Ox we were not able to enter this region
unambiguously, no label was introduced there, and the temperature T* is defined as T* ≈≈ 2∆∆ (as observed experimentally for YBa2Cu3Ox).
At lower temperatures T < T*, the ρ(T) curves deviate from their linearity and
the curved, super-linear, regime sets in. Although already observed in the
individual ρ(T) curves, our scaling analysis implies this enhanced conductivity
to be a universal feature for the conduction (and hence the scattering process) in
the high-Tc samples. This region is labelled II in figures 3.11 and 3.12. The
increased conductivity is often explained by the opening of a pseudo spin-gap
below T* (for a review see [Timusk99]) and for that reason we have adopted this
nomenclature. The implications of our scaling analysis however are far more
general and are not restricted to any specific theoretical model.
At very low temperatures T < 0.25∆ for YBa2Cu3Ox and T < 0.35∆ for the
(Y0.6Pr0.4)Ba2Cu3Ox system, a saturation of the ρ(T) curves sets in, with a small
increase of ρ for the samples with the lowest oxygen content. This is the onset
of a region (labelled III) were the ρ(T) curves have a negative slope (see also
the derivatives in figure 3.7 and 3.9) and that might tentatively be identified
with the onset of localisation.
CHAPTER 3 Normal-state resistivity of Y123, YPr123 & La214
56
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
T*
∆
T/∆
ρ ρρ ρ
−−
o
o∆
x = 6.4 / 6.45 / 6.5 / 6.7 / 6.8 / 6.95
YBa2Cu3Ox
III II I
Figure 3.11: Scaling of the zero-field ρ(T) data for the YBa2Cu3Ox thin films with x = 6.4, 6.45, 6.5, 6.7, 6.8 and x = 6.95. The temperature is rescaled with ∆ (an energy scale) and the resistivity is given by (ρ-ρo)/(ρ∆-ρo) in which the residual resistivity ρo is subtracted and ρ∆ is the resistivity at T = ∆. Three regions of different ρ(T) behaviour are indicated together with the energy scale ∆ and the crossover temperature T*.
0.0 0.5 1.0 1.5 2.0 2.5 3.0-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
ρ ρρ ρ
−−
o
o∆
T/∆
Y0.6Pr0.4Ba2Cu3Ox
III II
∆
x = 6.7 / 6.85 / 6.95
Figure 3.12: Scaling of the zero field ρ(T) data for the (Y0.6Pr0.4)Ba2Cu3Ox thin films with x = 6.7, 6.85 and x = 6.95. The temperature is rescaled with a parameter ∆ (an energy scale) and the resistivity is given by (ρ-ρo)/(ρ∆-ρo) in which the residual resistivity ρo is subtracted and ρ∆ is the resistivity at T = ∆. The regions II and II of different ρ(T) behaviour are indicated, together with the energy scale ∆.
CHAPTER 3 Normal-state resistivity of Y123, YPr123 & La214
57
From the scaling, presented in figures 3.11 and 3.12, another parameter, the
residual resistivity ρρo, can be extracted. This property, that is otherwise
hardly accessible (unless by turning to some risky extrapolation), can be easily
obtained here since it's just a scaling parameter used in the construction of
figures 3.11 and 3.12. The residual resistivities ρo are summarised in table 3.10.
The perfect scaling of the metallic resistivity in at least region I and region II of
the ρ(T) plot in figures 3.11 and 3.12 is a strong indication that in all high-Tc
samples, the transport properties are dominated by the same underlying
scattering mechanism; from the strongly underdoped to the near optimally
doped samples. The only thing that changes when lowering the doping level, is
the energy scale on which this process is applicable. Since this scaling also
exhibits a good agreement with the pseudo spin-gap crossover temperature T*,
one is tended to pinpoint the origin of this dominant scattering mechanism to be
magnetic. Although this idea will be further elaborated in chapter 5, a first
phase diagram can already be constructed (figure 3.13).
6.0 6.2 6.4 6.6 6.8 7.00
100
200
300
400
500
T (
K)
x
II
I
AF
SCTc
T*Y0.6Pr0.4Ba2Cu3Ox
Tc T*
YBa2Cu3Ox
Tc T*
Figure 3.13: Experimental phase diagram for the YBa2Cu3Ox and the (Y0.6Pr0.4)Ba2Cu3Ox system with the critical temperature Tc, mid and the crossover temperature T*. The antiferromagnetic region is indicative.
In figure 3.13, the experimental phase diagram for the YBa2Cu3Ox and the
(Y0.6Pr0.4)Ba2Cu3Ox systems is now complemented with the crossover
temperature T* that marks the transition from the linear (region I) to the super-
CHAPTER 3 Normal-state resistivity of Y123, YPr123 & La214
58
linear (region II) ρ(T) behaviour on cooling down. Since the tendency of
localisation at low temperatures is still weak, region III is omitted in this phase
diagram.
From this experimental T(x) diagram it is clear that a substantial difference
exists between the YBa2Cu3Ox and the (Y0.6Pr0.4)Ba2Cu3Ox systems. This is
probably related with the fact that the introduction of Pr in the YBa2Cu3Ox unit
cell possibly causes hole filling as well as magnetic pair breaking (and disorder
in the crystal). In the (Y1-yPry)Ba2Cu3Ox system, the substitution of 40 % of the
Y atom by Pr suppresses Tc to about 41 K but elevates ∆ and hence the
crossover temperature T*. This augmentation of T* is of such an order that it
exceeds the T* observed in YBa2Cu3Ox samples with the same oxygen content,
although the critical temperature of (Y0.6Pr0.4)Ba2Cu3Ox is about a factor of two
lower.
Therefore, when checking the phase diagram of figure 3.13 more carefully, it
seems that the oxygen content x is not a suitable parameter for constructing a
generic phase diagram but one should better turn to the "real" density of charge
carriers (accounting for possible hole-filling effects). From these data, it is not
clear whether the shift of the (Y0.6Pr0.4)Ba2Cu3Ox data to lower temperature in
the T(p) diagram, due to the reduction of the charge carrier density by the Y/Pr
substitution, suffices to obtain a reasonable overlap of the boundary lines in this
T(p) diagram. At least the Tc(p) boundary for (Y0.6Pr0.4)Ba2Cu3Ox is influenced
not only by the density of charge-carriers but also by additional magnetic pair-
breaking. This question -the existence of a single T(p) phase diagram for
YBa2Cu3Ox and (Y0.6Pr0.4)Ba2Cu3Ox- will therefore be addressed later in this
work in the context of the Hall-effect data (chapter 4).
Another important question is what happens below the critical temperature Tc,
when superconductivity is suppressed by a high magnetic field ? Do all
samples show a saturating or insulating ρ(T) behaviour at low temperatures ? Is
it possible to construct a metal to insulator boundary ? Will the scaling of the
ρ(T) data still hold for the normal state below Tc ? These questions will be
addressed within the next paragraphs.
CHAPTER 3 Normal-state resistivity of Y123, YPr123 & La214
59
3.3 Suppression of superconductivity by high magnetic fields
In order to make more precise statements about the normal-state transport
properties below Tc it is necessary to suppress the superconducting state. The
application of a very high magnetic field is well-suited for this task since
experiments can be carried out during the process of the suppression of
superconductivity and, moreover, it has the advantage of being a fully reversible
process.
H
T
Hc2
Hc3
Hc1
Hirr
Tc
Hc2(0)
Figure 3.14: Schematic field versus temperature (H-T) phase diagram for type II superconductors.
The relation between type II superconductivity and externally applied magnetic
fields is quite complicated (see figure 3.14). At very small fields, all magnetic
flux is expelled and a perfect diamagnetic response is established (Meissner-
Ochsenfeld effect). Above the first critical field Hc1 however, flux lines enter a
type II superconductor individually and distribute themselves by minimising
their energy with respect to their mutual repulsion and their attraction to defects
in the superconductor. This results in a magnetic response that deviates
substantially from the ideal diamagnetic case but is of technological importance
in that a finite critical current can be accommodated due to the magnetic
gradient sustained by the flux lines that are pinned on the defects. Above the
so-called irreversibility field Hirr the flux lines become unpinned. This liquid of
individual flux lines penetrating the superconductor persists up to the second
critical field Hc2 that is significantly higher than Hc1 (see figure 3.14). Above
CHAPTER 3 Normal-state resistivity of Y123, YPr123 & La214
60
Hc2, bulk superconductivity is destroyed and only surface superconductivity
survives up to Hc3, at surfaces that are parallel to the applied field and that have
a roughness smaller than the coherence length ξ.
The suppression of bulk superconductivity at Hc2 can be attributed to two main
mechanisms. The first mechanism, known as orbital pair-breaking, is
important not too far from the critical temperature Tc (and hence low magnetic
fields) and the description using the linearised Ginzburg-Landau equations
yields a linear Hc2(T). The second mechanism, known as spin pair-breaking, is
active at lower temperatures and breaks up pairs of electrons or holes by simple
Zeeman splitting. This mechanism of breaking up the Cooper pairs is of
particular relevance in the type II superconductors in which above Hc1 the
magnetic field penetrates, couples with the spins and modifies the shape of the
Hc2(T) curve [Werthamer66] to bend down (figure 3.14). The upper limit of the
second critical field Hc2(0), the so-called Clogston-Chandrasekhar or Pauli
paramagnetic limit, in a spin-singlet superconductor can then be estimated by
comparing the energy of the magnetic Zeeman splitting µBgB with the energy
gap of the superconductor 3.53 kBTc (the BCS value), with µB the Bohr
magneton. When taking into account an extra factor 2 introduced by more
detailed calculations [Clogston62, Chandrasekhar62], the Pauli paramagnetic
limit equals to µoHP/Tc ~ 1.84 T/K. The Pauli pair-breaking will obviously
become an important issue at very high magnetic fields.
Experimentally, the Hc2(T) curve was found to have a negative curvature at high
temperatures. The Hc2(T) line transforms to an upward curvature at lower
temperatures and does not exhibit any saturation at low temperatures
[Osofsky93, Affronte94, Mackenzie94, Ando99]. These findings are in
disagreement with the conventional Werthamer-Helfand-Hohenberg (WHH)
theory [Werthamer66] (a refinement of the BCS electron-phonon theories by
including impurity scattering, electron spin and spin-orbit effects) that predicts a
negative curvature over the whole temperature range with a saturation at low
temperatures. This WHH Hc2(T) curve is plotted in figure 3.14. A lot of
theoretical effort has been put in obtaining a description of the Hc2(T) curve that
does agree with the experimental data for high-Tc's. Among the proposed
models are strong [Schossmann86] and very strong [Marsiglio87] coupling
CHAPTER 3 Normal-state resistivity of Y123, YPr123 & La214
61
models, (bi-) polaron models [Alexandrov87, 92 & 94] and theoretical work
[Tešanovic91, Rasolt92] predicting re-entrant superconductivity at high
magnetic fields H > Hc∞ (at which all charge carriers are in the lowest Landau
level) and low temperatures in very pure systems with a low density of charge
carriers.
The effect of the suppression of superconductivity, by applying a high magnetic
field, on the transport properties can not be described accurately by one single
mechanism or physical regime. At low magnetic fields (H < Hirr) all vortices in
the superconductor are pinned and a zero voltage drop occurs over the sample
(figure 3.15). When this irreversibility field is exceeded, vortices are de-pinned
and become mobile, thus giving rise to a finite dissipation. Hence, when a
small measuring current is applied, a finite voltage drop will appear due this so-
called flux creep (FC). At higher magnetic fields, the flux lines will move
freely and the Bardeen-Stephen flux flow model [Bardeen65] predicts the
resistivity to increase linearly with field (figure 3.15).
ρ
Flow
TAFF
H
FC
Hc2Hirr
ρ
Flow
TAFF
H
FC
Hc2Hirr
Figure 3.15: Schematic view of the ρ(H) transition below Tc from a irreversible superconductor to the normal state, traversing the regimes of thermally assisted flux flow (TAFF), flux creep (FC) and flux flow. These regimes are explained in the text.
At even higher fields H > Hc2 bulk superconductivity is destroyed and the
normal-state resistivity is recovered. At finite temperatures, the changes at Hirr
and Hc2 are smoothed over a finite range of magnetic fields, resulting in a
rounding of the corners of the superconducting to normal-state transition. At
Hirr, thermally assisted flux flow (TAFF) will allow flux lines to jump to
CHAPTER 3 Normal-state resistivity of Y123, YPr123 & La214
62
neighbouring potential wells in the landscape of pinning sites and thus give rise
to a finite dissipation, even below Hirr. At Hc2 the transition from the
superconducting to the normal state is smoothed by the existence of
superconducting fluctuations, even above Hc2 (figure 3.15).
The experimental resistivity versus field curves for the YBa2Cu3Ox and the
(Y0.6Pr0.4)Ba2Cu3Ox thin films are presented in figures 3.16 to 3.23. All curves
were taken in the transverse H // c and I // ab configuration at the pulsed
magnetic fields facility at the K.U.Leuven. The applied measuring current was
in the range 100-200 µA, resulting in a current density of approximately 1 to
2· 107 A/m2, much smaller than the critical current density (of the order
109 A/m2) and the depairing current 1013 A/m2. The signal was amplified 200
times and two traces with opposite field polarity were combined in order to
remove spurious inductive effects due to the high sweep rate of the magnetic
field. A description of the experimental procedure was given in chapter 2.
Figures 3.16 to 3.20 present the ρab(H) curves measured at temperatures varying
from T >> Tc down to 4.2 K for the YBa2Cu3Ox films with x = 6.95, 6.8, 6.7, 6.5
and 6.45.
0
100
200
300
400
ρ ab
(µΩ
cm)
µοH (T)
41.4 K
121.4 K
51 K
102.6 K
90.9 K
186 K
60.3 K55 K
YBa2Cu3O6.95
0 10 20 30 40 50
Figure 3.16: Field dependence of the in-plane resistivity ρab(H) for an epitaxial YBa2Cu3O6.95 thin film at temperatures 41.4, 51, 55, 60.3, 65.8, 70, 75.9, 79, 81.5, 84, 86.6, 88.8, 90.9, 94.9, 102.6, 111, 121.4, 139.3, 157.9 and 186 K. Only the data taken at rising magnetic field are shown, the data were smoothed over 20 points.
CHAPTER 3 Normal-state resistivity of Y123, YPr123 & La214
63
In the optimally doped case (figure 3.16), above Tc a small positive
magnetoresistivity is present when sweeping the magnetic field up to 50 T. For
clarity, the data presented in figure 3.16 were smoothed over 20 points and only
show the sample response during rising magnetic field. When lowering the
temperature, even just above Tc a small low-field excess-conductivity develops
that is suppressed by high magnetic fields. The metallic tendency of decreasing
resistivity with lowering temperature seems to continue itself below Tc, when
considering the ρ(µoH = 50 T) data. This is in agreement with the resistance per
square, estimated in table 3.1 and paragraph 3.2.1 to be 0.8 kΩ, far below the
limit of h/4e2 ≈ 6.45 kΩ. When reducing the temperature to below Tc, a
superconducting transition develops in the ρ(H) curves. While close to Tc the
ρ(H) transition is narrow, it broadens significantly when lowering the
temperature. This indicates that the irreversibility field Hirr(T) and the second
critical field Hc2(T) separate from each other when going to lower temperature
in the H-T phase diagram (schematically drawn in figure 3.14). Note that below
40 K the 50 tesla pulsed magnetic field does not suffices to recover the normal,
non-superconducting, state.
The YBa2Cu3O6.8 data are presented in figure 3.17, no smoothing was applied
and the data taken during both rising and lowering magnetic field are shown.
Although exhibiting more noise, no important distortion of the data can be
observed and the two branches of the ρab(H) curves coincide almost perfectly.
Only at certain temperatures a small hysteresis can be observed, in most cases
due to a small drift of the electronic equipment between the field pulses. To
illustrate the fine quality of the measurements in pulsed magnetic fields, all
ρab(H) data presented below in this work will display the data-sets taken during
both rising and lowering magnetic field. From the YBa2Cu3O6.8 data, presented
in figure 3.17, similar observations can be made as for the optimally doped case.
Also here a small magnetoresistivity effect is present above Tc and the metallic
tendency for the ρ(µoH = 50 T ) data also seems to continue below Tc. When
reducing the temperature to below Tc, a superconducting behaviour develops in
the ρ(H) curves. However, while close to Tc the ρ(H) transition is narrow and it
significantly broadens when lowering the temperature, at the lowest
temperatures the transition steepens again ! This indicates the irreversibility
line Hirr(T) and the second critical field Hc2(T) to come closer again after their
CHAPTER 3 Normal-state resistivity of Y123, YPr123 & La214
64
initial separation at intermediate temperatures (see the schematic H-T phase
diagram in figure 3.14).
0
50
100
150
200
250
ρ ab
(µΩ
cm)
µοH (T)
0 10 20 30 40 50
4.2 K
153.7 K
141.5 K
119.5 K
100.3 K91.4 K78.4 K
YBa2Cu3O6.8
Figure 3.17: Field dependence of the in-plane resistivity ρab(H) for an epitaxial YBa2Cu3O6.8 thin film at temperatures 4.2, 15, 20, 25, 30.2, 35.2, 40.5, 45.2, 51.9, 60.2, 65, 70, 78.4, 91.4, 100.3, 119.5, 141.5 and 153.7 K. The data taken during both rising and falling magnetic field are shown, no smoothing was performed.
0 10 20 30 40 500
100
200
300
400
500
ρ ab
(µΩ
cm)
µοH (T)
YBa2Cu3O6.7
4.2 K
126.9 K
100.8 K
81 K
11.4 K
Figure 3.18: Field dependence of the in-plane resistivity ρab(H) for an epitaxial YBa2Cu3O6.7 thin film at temperatures 4.2, 11.4, 19.4, 30, 38.8, 46.8, 52.4, 62.6, 69.9, 81, 100.8 and 126.9 K. The data taken during both rising and falling magnetic field are shown, a 20 point smoothing was performed.
CHAPTER 3 Normal-state resistivity of Y123, YPr123 & La214
65
Although at first sight the YBa2Cu3O6.7 data, presented in figure 3.18, are
similar to the x = 6.95 and x = 6.8 data (small magneto-resistive effect above Tc,
the transition width increasing and decreasing again as temperature is lowered)
an important difference becomes clear when taking a closer look. The obvious
metallic tendency (decreasing resistivity upon lowering temperature) above Tc is
for the YBa2Cu3O6.7 sample not unambiguously extended into the normal state
below Tc. On the contrary, according to the ρ(µoH = 50 T ) data, a saturation
(or even a mild increase) is present at around 50 K. From this one might
anticipate the YBa2Cu3O6.7 sample to be on the limit of having a localised
ground state at low temperatures.
0 10 20 30 40 500
50
100
150
ρ ab
(µΩ
cm)
µοH (T)
YBa2Cu3O6.5
73 K
4.2 K
8.9 K
14.2 K
66.9 K
Figure 3.19: Field dependence of the resistivity ρab(H) for an epitaxial YBa2Cu3O6.5 thin film at T = 4.2, 8.9, 14.2, 17.6, 22.6, 26.3, 30.5, 33.1, 36.8, 41.3, 45.9, 50.3, 54, 58.7, 62, 66.9 and 73 K. The data taken during both rising and falling magnetic field are also shown for certain temperatures, a 20 point smoothing was performed.
The ρab(H) data for the YBa2Cu3O6.5 sample at various temperatures are shown
in figure 3.19. Also in this x = 6.5 case, a ρab(H) transition from the
superconducting to the normal state develops below Tc, exhibiting an initial
broadening on cooling down, with a sharpening at the lowest temperatures.
However, this ρab(H) picture is very different from the data at higher oxygen
content in that at low temperatures the high field ρab(H) data show a strong
increase causing a crossing of the ρab(H) curves. This is a true sign of the
CHAPTER 3 Normal-state resistivity of Y123, YPr123 & La214
66
normal state resistivity at 50 T increasing strongly with decreasing temperature.
The initial tendency of metallic ρab(T) behaviour at high temperatures thus
transforms into a more insulating behaviour at low temperatures. For the
YBa2Cu3Ox sample with the lowest oxygen content, x = 6.45, this tendency is
even more pronounced (figure 3.20).
0 10 20 30 40 500
200
400
600
800
1000
98.7 K
4.2 K
ρ ab
(µΩ
cm)
µοH (T)
YBa2Cu3O6.45
8.7 K
124.7 K
73 K
13.7 K
Figure 3.20: Field dependence of the in-plane resistivity ρab(H) for an epitaxial YBa2Cu3O6.45 thin film at temperatures 4.2, 8.7, 13.7, 15.8, 20.7, 24.7, 28.7, 32.1, 40.5, 43.8, 55.3, 73, 98.7 and 124.7 K. The data taken during both rising and falling magnetic field are shown without smoothing.
In the (Y1-yPry)Ba2Cu3Ox system, the substitution of 40 % of the Y atom by Pr
already suppresses Tc to about 41 K and a further reduction of the charge carrier
density by oxygen desorption induces an additional decrease of Tc. The
disorder introduced by the Y/Pr substitution reflects in the ρab(T) curves shown
earlier in figure 3.3: a higher residual resistivity is obtained, accompanied by a
tendency for localisation at low temperatures (except for the x = 6.95 sample).
The question is what will happen below Tc when suppressing
superconductivity ? Figures 3.21, 3.22 and 3.23 present the ρab(H) curves at
temperatures varying from T >> Tc down to 4.2 K for the (Y0.6Pr0.4)Ba2Cu3Ox
films with x = 6.95, 6.85 and 6.7. In all three cases an increasing ρab(50 T) with
lowering temperature is observed. For the x = 6.95 sample this is a striking
observation since the zero-field ρab(T) curve shows no such tendency.
CHAPTER 3 Normal-state resistivity of Y123, YPr123 & La214
67
Y0.6Pr0.4Ba2Cu3O6.95
0 10 20 30 40 500
100
200
300
400
500
4.2 K8.1 K
12.7 K19.8 K
90.5 K
ρ ab
(µΩ
cm)
µοH (T)
Figure 3.21: Field dependence of the in-plane resistivity ρab(H) for an epitaxial (Y0.6Pr0.4)Ba2Cu3O6.95 thin film at temperatures 4.2, 8.1, 12.7, 19.8, 23.4, 27, 32.8, 35.5, 40.5, 45.1, 49.9, 55.5, 59.8, 70, 80.9 and 90.5 K. The data taken during both rising and falling magnetic field are shown without smoothing.
0 10 20 30 40 500
200
400
600
800
10004.2 K4.75 K
9.8 K
80 K
12.5 K16.1 K18.6 K
Y0.6Pr0.4Ba2Cu3O6.85
ρ ab
(µΩ
cm)
µοH (T)
Figure 3.22: Field dependence of the in-plane resistivity ρab(H) for an epitaxial (Y0.6Pr0.4)Ba2Cu3O6.85 thin film at temperatures 4.2, 4.75, 9.8, 12.5,16.1, 18.6, 21.2, 24.8, 27.9, 32.1, 37.6, 42.8, 51.6, 63.5, 68.7 and 80 K. The data taken during both rising and falling magnetic field are shown without smoothing.
CHAPTER 3 Normal-state resistivity of Y123, YPr123 & La214
68
0 10 20 30 40 500
500
1000
1500 Y0.6Pr0.4Ba2Cu3O6.7
ρ ab
(µΩ
cm)
µοH (T)
4.2 K
7.7 K
12.0 K
18.3 K23 K28 K36.7 K
Figure 3.23: Field dependence of the in-plane resistivity ρab(H) for an epitaxial (Y0.6Pr0.4)Ba2Cu3O6.7 thin film at temperatures 4.2, 7.7, 12, 18.3, 23, 28, 36.7, 42, 53.9 and 69.6 K. The data taken during both rising and falling magnetic field are shown, no smoothing was performed.
Thus, even at optimal oxygen content, the 40 % Y/Pr substitution introduces
enough disorder to induce a low temperature localisation. This observation of
an insulating ground state is in agreement with the sheet resistances calculated
from the residual resistivities, reported in table 3.1, that are of the order of the
limiting value of h/4e2 ≈ 6.45 kΩ for a metallic system.
3.4 Normal-state resistivity at T < Tc in high magnetic fields
On the basis of the high field ρab(H) curves for the YBa2Cu3Ox and the
(Y0.6Pr0.4)Ba2Cu3Ox thin films, presented in figures 3.16 to 3.23, it is possible to
gain access to the normal-state resistivity below Tc. However it is not
immediately clear what criterion to take in order to obtain a ρab(T) that is a true
reflection of the underlying normal-state.
Above the critical temperature Tc only a small (but nevertheless finite)
magneto-resistive effect is observed, as illustrated in figure 3.24 for the Tc,mid =
73.7 K YBa2Cu3O6.8 sample. When approaching the critical temperature,
superconducting fluctuations create an excess conductivity that can only be
CHAPTER 3 Normal-state resistivity of Y123, YPr123 & La214
69
removed by applying a high magnetic field. In this T > Tc case one can thus
choose for the normal state resistivity between the ρab(0 T) and the ρab(50 T)
data. The ρab(0 T) data have the advantage of coinciding with the zero field
ρab(T) curve but contain an excess conductivity due to superconducting
fluctuations. The ρab(50 T) data are less susceptible to the influence of the
excess conductivity but contain a small magneto resistive contribution and are
therefore slightly overestimated. This can be taken into account by truly referring to these data as the "50 T normal-state transport properties" and not simply the normal state properties.
0 10 20 30 40 50
80
100
120
140
160
180
200
220
240
78.4 K
91.4 K
100.3 K
119.5 K
141.5 K
153.7 K
YBa2Cu3O6.8
ρ ab
(µΩ
cm)
µοH (T)
Figure 3.24: Illustration of the various possible methods for obtaining the normal-state resistivity above Tc from the ρab(H) curves. The data are for an epitaxial YBa2Cu3O6.8 thin film at temperatures 78.4, 91.4, 100.3, 119.5, 141.5 and 153.7 K.
Below the critical temperature Tc, the picture is blurred by the existence of a
superconducting transition at fields H < Hc2 and superconducting fluctuations
even above this second critical field (figure 3.25). Here one does not have
simple access to the normal-state ρab(0 T) data and one is tempted into turning
to an extrapolation to zero magnetic field to obtain the "true" normal-state
resistivity. A simple linear extrapolation is easy to perform reproducibly by
using the quasi-linear regime at high magnetic fields. However, as can be seen
in both figure 3.24 and 3.25, such an extrapolation is bound to fail since it
depends heavily upon the precise nature of the real magnetoresistivity in this
CHAPTER 3 Normal-state resistivity of Y123, YPr123 & La214
70
region. From figure 3.24 it is clear that the normal state magnetoresistivity has
rather a quadratic-like behaviour than a simple linear one and a linear
extrapolation would yield values that would be significantly underestimated.
0
50
100
150
200
250
ρ ab
(µΩ
cm)
µοH (T)
0 10 20 30 40 50
4.2 K
153.7 K
141.5 K
119.5 K
100.3 K91.4 K78.4 K
YBa2Cu3O6.8
Figure 3.25: Illustration of the various possible methods for obtaining the normal-state resistivity below Tc from the ρab(H) curves. The data are for an epitaxial YBa2Cu3O6.8 thin film at temperatures 4.2, 15, 20, 25, 30.2, 35.2, 40.5, 45.2, 51.9, 60.2, 65, 70, 78.4, 91.4, 100.3, 119.5, 141.5 and 153.7 K.
This becomes particularly clear from figure 3.26 were the linear extrapolation is
shown to suffer from the excess conductivity at T ~ Tc and the systematic
underestimation at T < Tc. A quadratic extrapolation, however, is not a good
alternative due to the inherent uncertainty during its construction.
Therefore, in this work, the choice was made to take the 50 T ρab data as a true
reflection of the normal-state transport properties. This has the additional
advantage that by similarly gathering the ρab data at 0, 10, 20, 30, 35, 40, 45 and
50 T, a smooth transition from a superconductor to a metal or insulator can be
depicted. Moreover, by comparing the 45 T and the 50 T data it is possible to
see whether the ρab(T) curves at 45 and 50 T separate or not. A near-
coincidence (because of a finite magnetoresistivity) of these curves indicates
that the normal-state is fully reached and that the high field data reflect the "true" normal-state.
CHAPTER 3 Normal-state resistivity of Y123, YPr123 & La214
71
0
100
200
300
400YBa2Cu3O6.8
T (K)
0 50 100 150 200 250 300
ρ ab
(µΩ
cm)
@ 50 T
0 T linear extrapolation
H
I a
bc
Figure 3.26: Influence of the criterion on the "normal state" resistivity below Tc for the YBa2Cu3O6.8 thin film.
The experimental ρab(T) resistivity versus temperature curves for the
YBa2Cu3Ox and the (Y0.6Pr0.4)Ba2Cu3Ox thin films are shown in figures 3.27 to
3.34. These plots contain the zero field ρab(T) combined with the high field 10,
20, 30, 35, 40, 45 and 50 tesla ρab(T) curves, constructed as described above,
and for the (Y0.6Pr0.4)Ba2Cu3Ox thin films, the value of the sheet resistance
corresponding to R = h/4e2 ≈ 6.45 kΩ is indicated on the plots. All high-field
curves were taken in the transverse H // c and I // ab configuration.
Figures 3.27 to 3.31 present the ρab(T) curves for the YBa2Cu3Ox films with x =
6.95, 6.8, 6.7, 6.5 and 6.45. In the optimally doped x = 6.95 case (figure 3.27),
the metallic behaviour (dρ/dT > 0) is extended to below Tc and no saturation is
present. Moreover, below 50 K the high magnetic fields do not manage to
destroy superconductivity and the superconducting state is recovered. The
magnetic field here simply broadens the ρab(T) transition.
When the oxygen content in the YBa2Cu3Ox samples is lowered (figures 3.28 to
3.31), the ρab(T) transition is broadened even further and the reduced energy
scale (lower Tc and hence lower critical fields) causes the high field ρab(T) data
to reflect the normal-state more closely. Indeed, the 45 T ρab(T) curve follows
the 50 T trace down to low temperatures for the samples with x ≤ 6.8.
CHAPTER 3 Normal-state resistivity of Y123, YPr123 & La214
72
T (K)
0 50 100 150 200 250 3000
100
200
300
400
500
600ρ a
b (µ
Ωcm
)
50 / 45 / 40 / 35 / 30 / 20 / 10 / 0 T
YBa2Cu3O6.95
Figure 3.27: Temperature dependence of the in-plane resistivity ρab(T) for an epitaxial YBa2Cu3O6.95 thin film at 0, 10, 20, 30, 35, 40, 45 and 50 T.
0
100
200
300
400
50 / 45 / 40 / 35 / 30 / 20 / 10 / 0 T
T (K)
0 50 100 150 200 250 300
ρ ab
(µΩ
cm)
YBa2Cu3O6.8
Figure 3.28: Temperature dependence of the in-plane resistivity ρab(T) for an epitaxial YBa2Cu3O6.8 thin film at 0, 10, 20, 30, 35, 40, 45 and 50 T.
As anticipated in the ρab(H) curves in figure 3.18, at an oxygen content x = 6.7 a
plateau in the ρab(T) curve starts to develop (figure 3.29). At even lower oxygen
contents x = 6.5 (figure 3.30) and x = 6.45 (figure 3.31), the plateau transforms
into an increasingly insulator-like behaviour (dρ/dT < 0) that diverges at
x = 6.45, corresponding to the strong tendency of crossing ρab(H) curves in
CHAPTER 3 Normal-state resistivity of Y123, YPr123 & La214
73
figures 3.19 and 3.20. This is a strong indication that, even in the region of the
YBa2Cu3Ox T(x) phase diagram were superconductivity shows up, the ground
state for x ≤ 6.7 has an insulating or semiconducting nature. This observation
agrees with our estimates above (table 3.1) for the sheet resistance of these
materials that become of the order of the limiting value of h/4e2 ≈ 6.45 kΩ.
0
200
400
600
800
1000
1200
50 / 45 / 40 / 35 / 30 / 20 / 10 / 0 T
T (K)
0 50 100 150 200 250 300
ρ ab
(µΩ
cm)
YBa2Cu3O6.7
Figure 3.29: Temperature dependence of the in-plane resistivity ρab(T) for an epitaxial YBa2Cu3O6.7 thin film at 0, 10, 20, 30, 35, 40, 45 and 50 T.
0
200
400
600
800
50 / 45 / 40 / 35 / 30 / 20 / 10 / 0 T
T (K)
0 50 100 150 200 250 300
ρ ab
(µΩ
cm)
YBa2Cu3O6.5
Figure 3.30: Temperature dependence of the in-plane resistivity ρab(T) for an epitaxial YBa2Cu3O6.5 thin film at 0, 10, 20, 30, 35, 40, 45 and 50 T.
CHAPTER 3 Normal-state resistivity of Y123, YPr123 & La214
74
0
500
1000
1500
2000
50 / 45 / 40 / 35 / 30 / 20 / 10 / 0 T
T (K)
0 50 100 150 200 250 300
ρ ab
(µΩ
cm)
YBa2Cu3O6.45
Figure 3.31: Temperature dependence of the in-plane resistivity ρab(T) for an epitaxial YBa2Cu3O6.45 thin film at 0, 10, 20, 30, 35, 40, 45 and 50 T.
In the (Y1-yPry)Ba2Cu3Ox system, the substitution of 40 % of the Y atoms by Pr
suppresses Tc to about 41 K for an optimal content of oxygen. The appreciable
disorder introduced by the Y/Pr substitution is reflected in the low ρ290 K/ρ0
values: 2.4 for the (Y0.6Pr0.4)Ba2Cu3O6.95 (optimally oxygenated) compared to
3.4 and 12 for the x = 6.45 and x = 6.95 YBa2Cu3Ox samples respectively. This
is especially remarkable since the (Y0.6Pr0.4)Ba2Cu3O6.95 (ρ290 K/ρ0 ~ 2.4) and the
YBa2Cu3O6.45 (ρ290 K/ρ0 ~ 3.4) samples exhibit almost identical critical
temperatures Tc,mid of respectively 41.4 K and 41.7 K.
Figures 3.32 to 3.34 present the ρab(T) curves for the (Y0.6Pr0.4)Ba2Cu3Ox films
with x = 6.95, 6.85 and 6.7. Already in the fully oxygenated sample
(figure 3.32), the metallic behaviour (dρ/dT > 0) above Tc transforms into an
insulating behaviour (dρ/dT < 0) when suppressing the superconducting state
and its fluctuations. This tendency is stronger in the oxygen deficient x = 6.85
and x = 6.7 samples (figures 3.33 and 3.34). In these last two cases the normal
state is attained easily (even at 40 tesla) and the low temperature state of
insulating (diverging !) ρab(T) is clearly demonstrated in all these
(Y0.6Pr0.4)Ba2Cu3Ox samples, although no sign of a saturating ρab(T) is present
above Tc in the x = 6.95 sample.
CHAPTER 3 Normal-state resistivity of Y123, YPr123 & La214
75
Also for the (Y0.6Pr0.4)Ba2Cu3Ox films, the insulating behaviour agrees with the
rough estimates made in table 3.1 for the sheet resistance of these materials. On
the plots (3.32 to 3.34), the limiting value of R = h/4e2 ≈ 6.45 kΩ is indicated.
0
100
200
300
400
500
600
700
50 / 45 / 40 / 35 / 30 / 20 / 10 / 0 T
T (K)
0 50 100 150 200 250 300
ρ ab
(µΩ
cm)
Y0.6Pr0.4Ba2Cu3O6.95
Rÿ = 6.45 kΩ
Figure 3.32: Temperature dependence of the in-plane resistivity ρab(T) for an epitaxial (Y0.6Pr0.4)Ba2Cu3O6.95 thin film at 0, 10, 20, 30, 35, 40, 45 and 50 T. The value of the sheet resistance corresponding to R = h/4e2 ≈ 6.45 kΩ is indicated on the plot.
0
200
400
600
800
1000
1200Y0.6Pr0.4Ba2Cu3O6.85
50 / 45 / 40 / 35 / 30 / 20 / 10 / 0 T
T (K)
0 50 100 150 200 250 300
ρ ab
(µΩ
cm)
Rÿ = 6.45 kΩ
Figure 3.33: Temperature dependence of the in-plane resistivity ρab(T) for an epitaxial (Y0.6Pr0.4)Ba2Cu3O6.85 thin film at 0, 10, 20, 30, 35, 40, 45 and 50 T. The value of the sheet resistance corresponding to R = 6.45 kΩ is indicated on the plot.
CHAPTER 3 Normal-state resistivity of Y123, YPr123 & La214
76
45 / 40 / 35 / 30 / 20 / 10 / 0 T
T (K)
0
500
1000
1500
2000
0 50 100 150 200 250 300
ρ ab
(µΩ
cm)
Y0.6Pr0.4Ba2Cu3O6.7
Rÿ = 6.45 kΩ
Figure 3.34: Temperature dependence of the in-plane resistivity ρab(T) for an epitaxial (Y0.6Pr0.4)Ba2Cu3O6.7 thin film at 0, 10, 20, 30, 35, 40 and 45 T. The value of the sheet resistance corresponding to R = 6.45 kΩ is indicated on the plot.
3.4.1 Scaling of the metallic normal-state resistivity
Now that the normal-state transport properties below Tc for the YBa2Cu3Ox and
the (Y0.6Pr0.4)Ba2Cu3Ox samples have been determined experimentally, it is
good to take a second look at the zero-field scaling and to check whether it is
also valid for the high-field normal-state below Tc.
In figure 3.35, the scaled ρab(T) is given for the YBa2Cu3Ox; x = 6.4, 6.45, 6.5,
6.7, 6.8 and the x = 6.95 samples. For all these samples (except the x = 6.4
sample) the ρab(50 T) data were added, after being scaled by the same
parameters ∆, ρo and ρ∆ (see paragraph 3.2.2). For the optimally doped sample,
the ρab(50 T) curve follows the zero field ρab(T) accurately, until below T/∆ ~ 1
the fields necessary to suppress the superconducting state go beyond the 50 T
we can access experimentally. Therefore, below this temperature the ρab(50 T)
curve starts to deviate from the zero field ρab(T) by showing the onset of the
superconducting state. The YBa2Cu3O6.8 and YBa2Cu3O6.7 samples demonstrate
a ρab(50 T) behaviour that follows the universal curve, until at a certain
temperature there is a clear tendency towards a saturating ρab(T). Also the
YBa2Cu3O6.5 and YBa2Cu3O6.45 samples show this saturating behaviour and
CHAPTER 3 Normal-state resistivity of Y123, YPr123 & La214
77
even exhibit a diverging ρab(T) at the lowest temperatures. The data for these
last two samples (x = 6.45 and 6.5) however seem to follow the universal curve
to lower temperatures and a resistance minimum is observed at a temperature
TMI with TMI/∆ ~ 0.25. For the samples with x = 6.7 and x = 6.8, this simple
ratio is a lower limit, since the tendency for ρab(T) to saturate is already present
slightly above T ~ 0.25 ∆.
x = 6.45x = 6.5x = 6.7x = 6.8x = 6.95
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
T/∆
ρ ρρ ρ
−−
o
o∆
T*
∆
x = 6.4 / 6.45 / 6.5 / 6.7 / 6.8 / 6.95
YBa2Cu3Ox
III II I
x = 6.45x = 6.5x = 6.7x = 6.8x = 6.95
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
T/∆
ρ ρρ ρ
−−
o
o∆
T*
∆
x = 6.4 / 6.45 / 6.5 / 6.7 / 6.8 / 6.95
YBa2Cu3Ox
III II I
Figure 3.35: Scaling of the zero field and 50 tesla ρ(T) data for the YBa2Cu3Ox thin films with x = 6.4, 6.45, 6.5, 6.7, 6.8 and x = 6.95. The temperature is rescaled with a parameter ∆ (an energy scale) and the resistivity is given by (ρ-ρo)/(ρ∆-ρo) in which the residual resistivity ρo is subtracted and ρ∆ is the resistivity at T = ∆. The three regions of different ρ(T) behaviour are indicated as well as the energy scale ∆ and the crossover temperature T*.
We can conclude that the resistivity versus temperature data for the YBa2Cu3Ox
system scale - in regions I and II - perfectly onto a universal curve. For lower
temperatures, i.e. region III, the samples with the lowest oxygen content deviate
from the universal behaviour.
In figure 3.36, the scaled ρab(T) is shown for the (Y0.6Pr0.4)Ba2Cu3Ox sample
with x = 6.7, 6.85 and x = 6.95. The high field data were added after scaling
them with the same ∆, ρo and ρ∆ parameters employed for the zero-field scaling
(see paragraph 3.2.2).
CHAPTER 3 Normal-state resistivity of Y123, YPr123 & La214
78
0.0 0.5 1.0 1.5 2.0 2.5 3.0-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
ρ ρρ ρ
−−
o
o∆
T/∆
Y0.6Pr0.4Ba2Cu3Ox
III II
∆
x = 6.7 / 6.85 / 6.95
x = 6.7x = 6.85x = 6.95
0.0 0.5 1.0 1.5 2.0 2.5 3.0-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
ρ ρρ ρ
−−
o
o∆
T/∆
Y0.6Pr0.4Ba2Cu3Ox
III II
∆
x = 6.7 / 6.85 / 6.95
x = 6.7x = 6.85x = 6.95
Figure 3.36: Scaling of the zero field and 50 tesla ρ(T) data for the (Y0.6Pr0.4)Ba2Cu3Ox thin films with x = 6.85 and x = 6.95. For the x = 6.7 sample, only the 45 T data were experimentally accessible. The temperature is rescaled with a parameter ∆ (an energy scale) and the resistivity is given by (ρ-ρo)/(ρ∆-ρo) in which the residual resistivity ρo is subtracted and ρ∆ is the resistivity at T = ∆. Two of the three regions of different ρ(T) behaviour are indicated as well as the energy scale ∆.
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
-4
-2
0
2
T/∆
dρ/d
T
Y0.6Pr0.4Ba2Cu3Ox
x = 6.7 / 6.85 / 6.95
Figure 3.37: Derivative dρ/dT of the scaled resistivity of the (Y0.6Pr0.4)Ba2Cu3Ox thin films shown in figure 3.36, also including the high field data.
CHAPTER 3 Normal-state resistivity of Y123, YPr123 & La214
79
From figure 3.36, it is clear that in the (Y0.6Pr0.4)Ba2Cu3Ox case a reasonable
scaling was obtained for the low temperature diverging resistivity in region III
(apart from the already discussed scaling in regimes I and II). This is illustrated
by the good overlap of the derivatives dρ/dT of the scaled resistivity
(figure 3.37). In the (Y0.6Pr0.4)Ba2Cu3Ox case, the resistance minimum was
observed at T/∆ ~ 0.35 (see also the derivatives in figure 3.37), somewhat
higher than the ratio 0.25 found for YBa2Cu3Ox.
It is not a priori clear whether such a universal scaling should be fulfilled for the
low temperature regime in region III. The behaviour of the resistivity in this
region is dominated by the localisation of charge carriers due to the disorder. It
would rather come as a surprise if the energy scale ∆ describing the scattering
process in the metallic regions I and II would also be able to give a good
description of these insulator-like features in regime III.
The reasonable scaling of the high-field data of the (Y0.6Pr0.4)Ba2Cu3Ox samples
can be understood in view of the high energy-scale ∆ for these samples at fixed
oxygen content (table 3.10 and figure 3.36) which, combined with the higher
value T/∆ ~ 0.35 for the resistance minimum, yields a temperature TMI for the
onset of localisation that lies significantly above the TMI values for YBa2Cu3Ox
(with a lower ∆ and T/∆ ~ 0.25). This implies that the Y/Pr substitution seems
to enhance disorder considerably, resulting in a rather high TMI. The variations
in ρ(T) at low temperatures, due to the additional disorder arising from the
oxygen desorption, are made less clear by the high value of ∆, squeezing the
low-temperature data together. In the YBa2Cu3Ox system these deviations are
neither obscured by other disorder-contributions, nor compressed by the T/∆
rescaling and so significant deviations clearly show up at low temperatures.
The substantial discrepancy between the YBa2Cu3O6.7 and YBa2Cu3O6.8 samples
has an additional explanation in that the normal state magnetoresistivity ρ(H)
exhibits a strong field dependence at around 40 to 50 Kelvin (figures 3.17 and
3.18). This can also be seen from the ρ(T) plots in figures 3.28 and 3.29 where
the 40 T and 50 T separate slightly. This additional high-field effect,
superimposed onto the normal-state transport properties hinders a proper
analysis.
CHAPTER 3 Normal-state resistivity of Y123, YPr123 & La214
80
In the remaining part of this paragraph the focus will be on the metallic part of
the universal ρ(T) curve, where the scaling works fine i.e. region I and II. The
low-temperature regime of insulating behaviour (region III) will be discussed
later in this chapter.
The linear temperature dependence of the resistivity ρ(T), as observed in region
I (at high temperatures) has puzzled the high-Tc community for the last decade.
Numerous models were proposed to explain this robust linear resistivity. The
models based on the electron-phonon coupling, however, are bound to fail since
for the optimally doped compounds the absence of any deviation from linearity
down to Tc implies a coupling factor that is unable to explain the high values for
Tc. Other models involve real-space paring of polarons [Alexandrov87 & 88,
Mott90], the existence of a narrow, metallic, impurity band [Moshchalkov88 &
90, Quitmann92] or a spin-charge separation into spinons and holons in the
Resonating Valence Bond (RVB) model [Nagaosa90, Suzumura88,
Anderson88] (see also Chapter 1). However, any model trying to accurately
describe the normal-state transport properties of the cuprate superconductors
should also account for the S-shaped super-linear behaviour observed in
region II. Recent experimental evidence [Ito93, Wuyts94 & 96, Batlogg94]
shows the super-linear resistivity in this regime to be closely connected to the
antiferromagnetic (AF) fluctuations in the underdoped cuprates.
Recent models therefore include these AF fluctuations explicitly [Pines90 & 97,
Hlubina95, Chubukov96, Li99, Yanase99] and also take into account the
presence of stripes [Emery97, 97b & 99, Randeira97, Moshchalkov93, 98, 99,
99b & 99c] that are observed both in La2-xSrxCuO4 and YBa2Cu3Ox
([Thurston89, Cheong91, Mason92 & 94, Yamada97, Tranquada97, Hunt99]
and [Aeppli97, Tranquada97, Dai98, Kao99, Arai99] respectively).
The nearly antiferromagnetic Fermi-liquid model (NAFLM) for cuprates
[Hlubina95, Chubukov96, Pines97, Li99] assumes no spin-charge separation
but considers the interactions between Fermionic charge carriers and the
localised spins. Calculations in the framework of this NAFLM yield a ρ(T) ~ T
at high temperatures and an approximate ρ(T) ~ T 2 behaviour in the low
temperature region. The precise position of the crossover between these two
regimes is not yet established and the present degree of refinement of the NAFL
CHAPTER 3 Normal-state resistivity of Y123, YPr123 & La214
81
model does not yet provide a sufficient agreement with experimental data to be
conclusive [Li99]. Along the same tracks, Moshchalkov proposed a model for
quantum transport in 2 dimensional (2D) Heisenberg [Moshchalkov93] and
1 dimensional (1D) systems [Moshchalkov98b], which was successfully applied
for YBa2Cu3Ox and YBa2Cu4O8 [Moshchalkov99, 99b & 99c, Trappeniers99].
This model implicitly accounts for the recently established striped structure of
the CuO2 planes and contains the spin pseudo-gap as a simple fitting parameter.
The model yields an excellent fit of the available experimental data. In
chapter 5, the model and its applicability will be discussed in more detail.
3.5 Comparison with the La2-xSrxCuO4 system
It is interesting to compare these findings on YBa2Cu3Ox and
(Y0.6Pr0.4)Ba2Cu3Ox thin films with the results from resistivity experiments on
ultra thin La2-xSrxCuO4 layers in which Tc was additionally changed by applying
in-plane strain instead of chemical doping.
For these experiments, at the IBM research labs in Zürich, three ultra-thin
La1.9Sr0.1CuO4 films of fixed stoichiometry were prepared by molecular beam
epitaxy with block-by-block deposition [Locquet94 & 98]. The choice of the
substrate - SrLaAlO4 or SrTiO3 - determines the presence of compressive
(SrLaAlO4) or tensile (SrTiO3) epitaxial strain in the ab-plane (see figure 1.5 in
paragraph 1.3.3).
Figure 3.38 presents the ρab(T) curves for these three strained thin films. The
zero-field ρab(T) data show similar features upon the application of epitaxial
strain as were observed in the YBa2Cu3Ox system upon doping by changing the
oxygen content (or in La2-xSrxCuO4 itself upon changing Sr-content x
[Takagi92]). All three samples have a xSr = 0.1 stoichiometry which is -
ignoring strain effects- on the underdoped side of the phase diagram. Bulk
material of this composition is free of strain effects and has a Tc of about 25 K.
Samples A and B were deposited on SrLaAlO4 and have an effective thickness t
of 125 Å and 150 Å respectively. The compressive strain in these films results
in Tc values of respectively 47.1 K and 35.4 K that are significantly higher than
CHAPTER 3 Normal-state resistivity of Y123, YPr123 & La214
82
the bulk value and they even exceed the value of 38 K of the optimally doped
samples (table 3.39). The application of compressive epitaxial strain induces an
almost linear ρab(T), contrasting with the more S-shaped ρab(T) in unstrained
material [Takagi92]. Further inspection also reveals a trace of a cross-over T*
around 250 K for the sample with the largest compressive strain (sample A),
contrasting with T* ~ 400 K for the unstrained material [Takagi92]. These
observations (higher Tc, linear ρab(T) and reduced T*) suggest that the
compressive strain possibly leads to an additional doping of the CuO2 planes.
0 50 100 150 200 250 3000
200
400
600
800
T (K)
La1.9Sr0.1CuO4
ρ (µ
Ωcm
)
sample C, 150 Å on SrTiO3
sample B, 150 Å on SrLaAlO4
sample A, 125 Å on SrLaAlO4
Figure 3.38: Zero-field and 50 T resistivity versus temperature ρab(T) for the ultra-thin La1.9Sr0.1CuO4 films with tensile strain (sample C, up triangles) and compressive strain (sample B, circles and A, down triangles).
Name Substr. t
(Å)
Tc, mid
(K)
∆∆T
(K)
LC416 SrTiO3 150 14.1 7.2
LC391 SrLaAlO4 150 35.4 1.7
LC438 SrLaAlO4 125 47.1 1.7
Bulk ~ 25
Table 3.39: Overview of the ultra-thin La1.9Sr0.1CuO4 films with the substrate, thickness t and critical temperature Tc.
CHAPTER 3 Normal-state resistivity of Y123, YPr123 & La214
83
Sample C consists of 15 unit cells (~ 150 Å), deposited on SrTiO3 and the
induced tensile strain results in a critical temperature Tc ~ 14.1 K, significantly
lower than the bulk value (see table 3.39). The ρab(T) curve for this sample is
slightly curved and already around Tc a tendency towards insulating behaviour
is present. A comparison with the results on polycrystalline and thin film
La2-xSrxCuO4 [Takagi92] without strain-effects suggests that tensile strain
possibly causes an effective underdoping of the samples.
No scaling behaviour of the temperature dependence of the resistivity was
found for these three strained samples since the features in region I and II
(linear, super-linear) are not sufficiently pronounced.
Tentative results from Hall measurements on ultra-thin films under epitaxial
strain from the same research group [Locquet98] indicate a possible trend of a
decreasing carrier concentration in the CuO2 plane (lower doping) as
compressive strain is applied [Locquet2000]. If these findings are confirmed,
the origin of the marked changes in the superconducting and normal-state
properties upon compressive or tensile strain might be due to in-plane processes
like a changing orbital overlap, magnetic coupling or altered scattering
processes.
0 10 20 30 40 500
50
100
150
ρ (µ
Ωcm
)
125 Å La1.9Sr0.1CuO4 deposited on SrLaAlO4
µoH (T)
72.7 K
61.6 K
46.5 K
14.7 K
Figure 3.40: Field dependence of the in-plane resistivity ρab(H) for the 125 Å La1.9Sr0.1CuO4 film on SrLaAlO4, with compressive strain (sample A) at temperatures 14.7, 17, 20, 25, 30.2, 34.4, 39, 42, 46.5, 61.6, 72.7 K. For some traces the branch at lowering magnetic field was omitted, the data were smoothed over 20 points.
CHAPTER 3 Normal-state resistivity of Y123, YPr123 & La214
84
In order to gain knowledge upon the normal-state transport properties of these
strained ultra-thin films at temperatures below their critical temperature Tc,
transport measurements in pulsed magnetic fields were performed. The ρ(H)
traces at various temperatures above and below Tc are summarised in
figures 3.40 (sample A), 3.41 (sample B) and 3.42 (sample C). Whereas samples A and B exhibit metallic features (decreasing resistivity upon lowering temperature), sample C (tensile strained) shows a strong tendency towards an insulating ρρ(T), with a crossing of the ρ(H) traces as temperature is
lowered (figure 3.42).
0 10 20 30 40 500
50
100
150
200 150 Å La1.9Sr0.1CuO4 deposited on SrLaAlO4
4.2 K
62.6 K
43.7 K
ρ (µ
Ωcm
)
µoH (T)
Figure 3.41: Field dependence of the in-plane resistivity ρab(H) for the 150 Å La1.9Sr0.1CuO4 film with compressive strain, deposited on SrLaAlO4 (sample B) at temperatures 4.2, 13, 20.2, 30, 35.8, 43.7 and 62.6 K. The traces during both rising and lowering field are shown, no smoothing was performed.
From these ρ(H) plots, the temperature dependence of the resistivity in the
normal state (the normal state at 50 T) can be reconstructed at temperatures
below Tc. This normal state resistivity is, for all three samples, given in
figure 3.38. For the samples that show a metallic ρ(T) above Tc (samples A and
B) this behaviour is simply continued below Tc by the high field data. For
sample C, that already demonstrated a slight tendency towards a saturating
resistivity, a strong divergence of the low-temperature resistivity is seen
(figure 3.38). These observations further support that, apart from possibly
CHAPTER 3 Normal-state resistivity of Y123, YPr123 & La214
85
modifying the effective doping of the CuO2 plane, adding epitaxial strain also
causes disorder-induced scattering becoming important in materials with tensile
in-plane strain.
0
100
200
300
400
500
0 10 20 30 40 50
150 Å La1.9Sr0.1CuO4 deposited on SrTiO3
4.2 K
ρ (µ
Ωcm
)
µoH (T)
53.4 K62.5 K
Figure 3.42: Field dependence of the in-plane resistivity ρab(H) at temperatures 4.2, 7.6, 10.1, 13.7, 20.3, 26, 32, 40, 46.6, 53.4 and 62.5 K for the 150 Å La1.9Sr0.1CuO4 film with tensile strain, deposited on SrTiO3 (sample C). Only the traces during rising field are shown, the data were smoothed over 20 points.
3.6 Localisation effects at T →→ 0 in the YBa2Cu3Ox, (Y0.6Pr0.4)Ba2Cu3Ox and La1.9Sr0.1CuO4 samples
The low-temperature in-plane resistivity of the YBa2Cu3Ox and the
(Y0.6Pr0.4)Ba2Cu3Ox samples strongly increases as temperature is lowered. This
diverging ρ(T) behaviour (region III) was in the previous paragraph shown to
deviate from the universal scaling in the metallic regime (region I and II).
The ρ(T) curves for the strongly underdoped samples exhibiting such an
insulating tendency are plotted in figure 3.43 (for the x = 6.45 and 6.5
YBa2Cu3Ox films), figure 3.44 (for the x = 6.7, 6.85 and 6.95
(Y0.6Pr0.4)Ba2Cu3Ox thin film) and figure 3.38 (for sample C, the tensile strained
La1.9Sr0.1CuO4 ultra-thin film). For the two YBa2Cu3Ox samples with the lowest
oxygen content a diverging ρ(T) is seen; although the onset of the
superconducting state becomes visible at the lowest temperatures. Due to their
CHAPTER 3 Normal-state resistivity of Y123, YPr123 & La214
86
significantly lower critical temperatures, the tensile strained La1.9Sr0.1CuO4
ultra-thin film and the (Y0.6Pr0.4)Ba2Cu3Ox film (at all levels of oxygen content)
show a strongly diverging ρ(T) down to the lowest experimentally accessible
temperatures (4.2 K).
0
500
1000
1500
0 50 100 150 2000
200
400
600
T (K)
ρ ab
(µΩ
cm) YBa2Cu3Ox
x = 6.45
x = 6.5
ρab (µΩ
cm)
Figure 3.43: Temperature dependence of the in-plane resistivity ρab(T) for an epitaxial YBa2Cu3Ox thin film without an external magnetic field and at 50 tesla with x = 6.45 and 6.5.
0
500
1000
1500
2000
x = 6.95
x = 6.85
x = 6.7
T (K)
0 50 100 150 200 250 300
ρ ab
(µΩ
cm)
Y0.6Pr0.4Ba2Cu3Ox
Figure 3.44: Temperature dependence of the in-plane resistivity ρab(T) for an epitaxial (Y0.6Pr0.4)Ba2Cu3Ox thin film without an external magnetic field and at 45 tesla (for x = 6.7) and 50 tesla (for x = 6.85 and 6.95).
CHAPTER 3 Normal-state resistivity of Y123, YPr123 & La214
87
The origin of such a low temperature increase of the resistivity can be attributed
to several mechanisms. At low temperatures, a simple thermally activated
process yields a diverging ρ(T) ~ exp(1/T) behaviour. In many cases, however,
a weaker dependency is observed. The general equation 3.2 describes the so-
called hopping of charge carriers between two localised states, with To a
characteristic temperature and α a constant that is determined by the precise
conditions of this hopping process. The simple case of Mott variable range
hopping [Mott79] yields a power α = 1/3 or 1/4 in two- and three dimensions,
respectively, by assuming an energy-independent density of states near the
Fermi level (n = 0 in equation 3.3). The 3D α = 1/4 case transforms into
α = 1/3 in very high magnetic fields when the magnetic length λm becomes of
the order of the localisation radius. At low carrier concentrations, due to poor
screening, a soft Coulomb-gap can develop around the Fermi-level (0 < n ≤ 2,
n = 2 for electron-electron repulsion) thus yielding α = 1/2 in 3 dimensions
[Shklovskii84]. For arbitrary n (nature of the gap) and D (dimension of the
problem), the power α is given by equation 3.4.
=α
ρρT
TT o
o exp)( (3.2)
n
FEEEg −~)( (3.3)
1
1
+++
=Dn
nα (3.4)
To check the validity of such activated or hopping type of charge transport, all
low-temperature data with an insulating tendency (figures 3.43, 3.44 and 3.38)
were analysed. Figure 3.45 shows such an analysis for the low-temperature part
of the 50 T ρab(T) curve of the YBa2Cu3O6.45 sample; this picture is
representative for all other data. In this plot, the resistivity is plotted
logarithmically versus T -α with α ranging from 0.1 up to 0.5. Any behaviour
obeying equation 3.2 should then give a linear plot at some value of α. From
this figure, it is clear that no good correspondence is obtained for any value of
α; only in the α < 0.1 case some agreement is found. This tendency is observed
in all the low-temperature resistivity data.
CHAPTER 3 Normal-state resistivity of Y123, YPr123 & La214
88
YBa2Cu3O6.45
0.0 0.2 0.4 0.6 0.8
T-α
103
ρ ab
(µΩ
cm)
α = 0.5 0.4 0.33 0.25 0.2 0.1
50 T
high T low T
Figure 3.45: Low-temperature part of the ρab(T) curve, plotted logarithmically versus T -α for an epitaxial YBa2Cu3O6.45 thin film at 50 tesla.
The fact itself, that no correspondence is found with the known α = 1/4, 1/3 or
1/2 coefficients, is not sufficient to rule out the possibility of a hopping process
being relevant; as the precise value of this parameter is determined by assuming
a rather soft g(E) around the Fermi-level (equation 3.3). However, the linearity
observed only at α < 0.1 (by squeezing the data together) implies the physics to
obey a severely modified form of equation 3.2. This limit of α < 0.1 yields a
much smoother ρ(T) than the generic hopping-transport and our data can thus
not be accurately described by equation 3.2.
Recently, high field experiments by Ando and co-workers on the normal-state
transport properties of Bi2Sr2CuOy (Bi2201) [Ando96c, 97, 97b] and
La2-xSrxCuO4 (La214) [Ando95, 96, 96b, 97, 97b, Boebinger96] showed both
the in-plane ab as the out-of-plane c-axis high-field resistivity to diverge as
~ ln(1/T) at low-temperatures. This is indeed a much smoother divergence than
proposed by the hopping models. To check the validity of this observation for
our data on the YBa2Cu3Ox and the (Y0.6Pr0.4)Ba2Cu3Ox samples, in figures 3.46
and 3.47 the in-plane resistivities were re-plotted versus ln(T). From these two
plots, a nice linear behaviour ρ(ln(T)) can be observed thus confirming the
ln(1/T) divergence for the YBa2Cu3Ox and the (Y0.6Pr0.4)Ba2Cu3Ox samples.
Also in the tensile strained La1.9Sr0.1CuO4 ultra-thin film, a good agreement of
CHAPTER 3 Normal-state resistivity of Y123, YPr123 & La214
89
the high-field data with the ln(1/T) divergence was found. This agreement of
the high-field ρ(T) data with the ln(1/T) divergence is better than the
correspondence with a simple power law T-α [Hao00].
0
500
1000
1500
ln(T)
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.00
200
400
600
x = 6.45
x = 6.5
ρ ab
(µΩ
cm)
YBa2Cu3Ox
ρab (µΩ
cm)
50 T
Figure 3.46: Temperature dependence of the in-plane resistivity ρab versus ln(T) for an epitaxial YBa2Cu3Ox thin film without an external magnetic field and at 50 tesla with x = 6.45 and 6.5.
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.50
500
1000
1500
2000
ln(T)
ρ ab
(µΩ
cm)
Y0.6Pr0.4Ba2Cu3Ox
x = 6.95
x = 6.85
x = 6.750 T
Figure 3.47: Temperature dependence of the in-plane resistivity ρab versus ln(T) for an epitaxial (Y0.6Pr0.4)Ba2Cu3Ox thin film without external field, at 45 tesla (for x = 6.7) and at 50 tesla (for x = 6.85 and 6.95).
CHAPTER 3 Normal-state resistivity of Y123, YPr123 & La214
90
Although the low-temperature (high-field) ln(1/T) divergence is by now an
experimental fact for both the ab-plane and the c-axis resistivities in Bi2201
[Ando96c, 97, 97b] and La214 [Ando95, 96, 96b, 97, 97b, Boebinger96] and at
least the ab-plane resistivity in Y123 and YPr123 [this work and
Trappeniers99], no consensus exists on the nature of this "localisation" of the
charge carriers. A whole variety of models can provide us with a divergence of
the resistivity at low temperatures.
However, some predictions for the resistivity of the high-Tc cuprates violate the
available experimental data [Ando, Boebinger96, Trappeniers99, this work] in
the low-temperature limit and their relevance for these low-temperature normal-
state transport properties of the high-Tc cuprates can thus be questioned.
The ln(1/T) dependence of the resistivity as predicted from spin-flip scattering
in the Kondo framework fails at low temperatures where µBgB > kBT and the
"impurity" spins are aligned by the external magnetic field. For a 50 T
magnetic field, the spin-flip process starts to diminish at around T < 50 K and
the resistivity should saturate. This contradicts the experimental findings.
The model described earlier in this chapter, explaining the linear ρ(T) at high
temperatures by assuming the development of a narrow metallic impurity band
with localised edges upon doping [Moshchalkov88 & 90, Quitmann92] also
predicts a diverging resistivity at low temperatures. The disagreement of the
proposed ρ(T) ~ exp(1/T) with the experimental high-field data might be due to
the simplification by approximating the impurity band by a square density of
states g(E).
Some models cannot be ruled out immediately on the basis of experimental data
at this point.
Without being complete, it is worthwhile to mention the phenomenological c-
axis resistivity model [Zha96] predicting a divergent ρc and another model
accounting for electron interactions in 2D disordered systems [Altshuler80]
predicting a ln(1/T) dependence in both ab and c directions. Both models rely
on the suppression of the 2D in-plane density of states as measured in Knight
shift experiments. Also the bipolaron model gives a logarithmically diverging
resistivity by assuming a temperature dependent scattering time for the
CHAPTER 3 Normal-state resistivity of Y123, YPr123 & La214
91
bipolarons [Alexandrov97]. The 2D Luttinger liquid model [Anderson91,
Clarke95] predicts a resistivity that diverges as a power law, both in ρab(T) as in
ρc(T), by assuming the existence of a confined spin-charge separated liquid with
incoherent hopping between the planes. The agreement of our high-field data
with such a power law is less convincing than the correspondence with the
ln(1/T) behaviour.
In the 2D Anderson weak localisation theory, a charge carrier is localised due to
the interference with its coherently backscattered wave function. This model
was shown to yield an ln(T) correction to the conductivity [Lee85]. This
correction then transforms into an ln(1/T) correction for the resistivity (inverse
conductivity). However, coherent backscattering can be frustrated by an
external magnetic field and at 50 T might possibly result in a negative
magnetoresistivity.
A model that is of particular interest is the charge-stripe picture [Cheong91,
Tranquada97, Arai99]. The dynamic incommensurate stripes, observed also in
YBa2Cu3Ox [Aeppli97, Tranquada97, Dai98, Kao99, Arai99], are shown to
yield metallic transport properties [Noda99, Ichikawa99, Tajima99,
Moshchalkov99 & 99c]. When the 1D charge stripes are pinned and the stripe
order increases, the movement of the charge carriers is restricted to these fixed
1D paths and the resistivity is reported to increase [Noda99, Lavrov99]. These
findings for the insulating regime (region III) complement the model for
quantum transport in 1D and 2D Heisenberg systems [Moshchalkov93, 98, 99,
99b & 99c, Trappeniers99] that already gives a proper description for the
metallic (striped) regime (region II) and 2D Heisenberg regime (region I). The
possible relevance of the stripe pinning to the low temperature divergence of the
resistivity will be discussed in detail in chapter 5.
In any case, whatever the origin of the insulating-alike behaviour, we can
complement the phase diagram (figure 3.13) with the temperature TMI at which
the metallic dρ/dT > 0 transforms into an insulating dρ/dT < 0 behaviour
(figure 3.48). This boundary TMI was (arbitrarily) taken at the resistance
minimum and it separates region II and III, introduced earlier. For the lowest
oxygen contents in YBa2Cu3Ox this yields TMI ~ 0.25 ∆. As discussed above,
the insulating resistivity part of the samples with x = 6.7, 6.8 and 6.95 does not
CHAPTER 3 Normal-state resistivity of Y123, YPr123 & La214
92
scale very well onto the universal curve and therefore has no such simple
expression for TMI. Moreover, in these samples, the resistance minimum was
not observed unambiguously and a tendency to a saturating ρ(T) starts already
above TMI. Therefore, the TMI ~ 0.25 ∆ data for the x = 6.7, 6.8 and 6.95
samples, plotted in figure 3.48, represent a lower limit and must be considered with some reservations. In contrast to this, the resistivity data of the x = 6.7,
x = 6.85 and x = 6.95 (Y0.6Pr0.4)Ba2Cu3Ox samples do fall onto each other and
the resistance minimum can be approximated by TMI ~ 0.35 ∆.
6.0 6.2 6.4 6.6 6.8 7.0
0
100
200
300
400
500
T (
K)
x
AF
SCTc
T*
TMI
TN Y0.6Pr0.4Ba2Cu3Ox
YBa2Cu3Ox
Tc T* TMI
Tc T* TMI
0
100TMI
I
II
IIIII
III
SC
6.0 6.2 6.4 6.6 6.8 7.06.0 6.2 6.4 6.6 6.8 7.0
0
100
200
300
400
500
0
100
200
300
400
500
T (
K)
x
AF
SCTc
T*
TMI
TN Y0.6Pr0.4Ba2Cu3Ox
YBa2Cu3Ox
Tc T* TMITc T* TMI
Tc T* TMITc T* TMI
0
100
0
100TMI
II
IIII
IIIIIIIIII
IIIIII
SC
Figure 3.48: Experimental T(x) phase diagram for the YBa2Cu3Ox and the (Y0.6Pr0.4)Ba2Cu3Ox system, with the superconducting critical temperature Tc, mid and the crossover temperature T* between the linear and super-linear ρ(T). At this point, the temperature of the resistance minimum, indicating a transition from a metal to an insulator-like behaviour TMI is added. The antiferromagnetic region is only indicative. The Tc, mid(x) and TMI(x) lines for (Y0.6Pr0.4)Ba2Cu3Ox were shifted down by 100 K.
When checking this phase diagram more closely, it is clear that the MI
boundary between the metallic region II and the insulating region III for both
compounds is a decreasing function of the doping. Both TMI(x) lines penetrate
the underlying superconducting region below Tc(x) that can thus be thought of
as masking the underlying insulating ground state.
CHAPTER 3 Normal-state resistivity of Y123, YPr123 & La214
93
The TMI(x) line for the YBa2Cu3Ox compound seems to vanish at optimal
doping. This observation however strongly relies on the assumption that the
TMI ~ 0.25 ∆ relation holds also for the higher levels of oxygen content (x = 6.7,
6.8 and 6.95) in YBa2Cu3Ox. In any case, for the YBa2Cu3Ox compound the
TMI(x) line is below Tc(x) for all oxygen concentrations above x = 6.5, meaning
that the zero field ρ(T) will not show any tendency of an insulating behaviour.
It is only at high magnetic fields, by suppressing the superconductivity below
Tc(x), that the underlying insulating ground-state can be accessed.
The TMI(x) line for the (Y0.6Pr0.4)Ba2Cu3Ox compound does not vanish at optimal
oxygen content since these cuprates are already strongly underdoped by the
40 % Y/Pr substitution. What is truly remarkable for this compound is that the
TMI(x) line is above Tc(x) for all oxygen contents. Thus, the tendency towards
an insulating behaviour (saturation or divergence) of the resistivity is already
visible above Tc and is only enhanced by the application of a high external
magnetic field.
A final remark about this phase diagram was already made in the previous
paragraphs. From this phase diagram, it is clear that the oxygen content is not a
good parameter to construct a generic phase diagram for the YBa2Cu3Ox and
the (Y0.6Pr0.4)Ba2Cu3Ox compounds. A better parameter would be the true
charge carrier density, accessible by measurements of the Hall-effect. This
would result in a shift to the left of the Tc, T* and TMI boundaries for
(Y0.6Pr0.4)Ba2Cu3Ox with respect to the line for YBa2Cu3Ox, since at least the
supposed hole-filling effect of the Y/Pr substitution would then be taken into
account. It is not clear beforehand what will be the role of the claimed magnetic
pair-breaking in the (Y/Pr)Ba2Cu3Ox compound.
In the next chapter an attempt will be made to construct the generic T(p) phase
diagram for the YBa2Cu3Ox and (Y0.6Pr0.4)Ba2Cu3Ox compounds based on Hall-
effect measurements in very high pulsed magnetic fields.
CHAPTER 3 Normal-state resistivity of Y123, YPr123 & La214
94
3.7 Conclusions
The temperature dependent resistivity of a set of YBa2Cu3Ox, (Y0.6Pr0.4)Ba2Cu3Ox and strained La1.9Sr0.1CuO4 epitaxial thin films was
measured in zero field and at very high pulsed magnetic fields. The zero-
field data show that these samples are rather poor conductors having relatively
low ratios ρ290 K/ρo (ρ290 K/ρo ~ 2.5 for (Y0.6Pr0.4)Ba2Cu3Ox and 2.6 to 12.2 for
YBa2Cu3Ox) compared to pure normal metals where ρ290 K/ρo ~ 1000. This
agrees with previous findings on all superconducting high-Tc cuprates and
indicates their relative impurity.
Furthermore, it was shown that the zero-field normal-state resistivity above Tc
for various levels of hole doping -both for the YBa2Cu3Ox and the
(Y0.6Pr0.4)Ba2Cu3Ox members of the cuprate superconductors- can be scaled
onto one single universal curve. An energy scale ∆, the resistivity ρ∆ and the
residual resistivity ρo are used as scaling parameters. The universal curve
exhibits a region (labelled I) of linear ρ(T) at high temperatures T > T*, a
super-linear ρ(T) at intermediate temperatures T <T* (region II) and a low
temperature insulating ρ(T) regime (labelled III) at T < TMI. This low-
temperature regime is masked by the onset of superconductivity at T = Tc. The
distinct features in the temperature dependence of the metallic zero-field
resistivity of YBa2Cu3Ox and (Y0.6Pr0.4)Ba2Cu3Ox in regions I and II are
universal for all the reported zero-field curves, the only difference is the
temperature at which these features set in. The existence of a universal metallic
ρ(T) curve was interpreted as a strong indication of one single mechanism
dominating the scattering of the charge carriers in these materials. This
mechanism then determines the energy scale ∆.
In order to gain access to the low-temperature T < Tc part of the normal-state
transport properties (regions II & III), experiments in very high magnetic fields
were performed. These experiments allowed to suppress the superconducting
state thus retaining the normal-conducting high field normal-state properties.
These experiments revealed the ground state of YBa2Cu3Ox (for x ≤ 6.8),
(Y0.6Pr0.4)Ba2Cu3Ox (for all levels of oxygen content) and the tensile strained
CHAPTER 3 Normal-state resistivity of Y123, YPr123 & La214
95
La1.9Sr0.1CuO4 ultra-thin film to be of an insulating nature with the resistivity
increasing as temperature is lowered.
Performing the same scaling analysis as in the zero-field case, it was shown that
for the YBa2Cu3Ox system the high-field ρρ(T) data for region III -the insulating regime- do not scale, whereas they more or less do scale in the
(Y0.6Pr0.4)Ba2Cu3Ox system (by squeezing the data together). It was argued that
such a scaling is not very likely since it would imply that the same energy scale
∆ controls both the metallic (region I and II) and the insulating (region III) part
of the ρ(T) curves; the insulating regime most likely being dominated by
disorder.
The insulating tendency in region III was shown not to obey a simple activated
or hopping mechanism. It was argued that, most probably, spin-flip scattering
in the framework of the Kondo model or the existence of a narrow metallic
impurity band with localised edges do not play an important role in producing a
divergent resistivity upon lowering temperatures. Our low-temperature ρ(T)
data were shown to obey the ln(1/T) divergence in high magnetic fields, as
was first observed in the cuprates by Ando.
The comparison of the transport data on strained La1.9Sr0.1CuO4 ultra-thin films
with data on samples without epitaxial strain and our YBa2Cu3Ox and
(Y0.6Pr0.4)Ba2Cu3Ox samples demonstrated that epitaxial strain probably not only
influences the doping of the CuO2 planes but also modifies the disorder
scattering.
In the final section of this chapter, an experimental T(x) phase diagram was
constructed including the superconducting critical temperature Tc(x), the
temperature T*(x) marking the opening of the pseudo spin-gap and also TMI(x),
the boundary between the metallic regime (regions I and II) and the insulating
regime (region III). It was argued that a generic T(p) phase diagram (with p
the number of holes per CuO2 plane) can only be obtained when combining
these data with Hall-effect measurements. This will be realised in the next
chapter.
CHAPTER 4 Hall-effect in YBa2Cu3Ox and (Y0.6Pr0.4)Ba2Cu3Ox thin films
97
Chapter 4
Hall-effect in YBa2Cu3Ox and
(Y0.6Pr0.4)Ba2Cu3Ox thin films
4.1 Introduction
For a 3 dimensional system, the current density jr
and the electric field Er
are
related by Ejrr
⋅= σ (and by jErr
⋅= ρ ) with σ and ρ the conductivity and
resistivity tensors, respectively. In a magnetic field along the z-axis, this can be
written in more detail (equation 4.1).
=
z
y
x
zz
yyyx
xyxx
z
y
x
E
E
E
j
j
j
σσσσσ
00
0
0
(4.1)
The Onsager relations for the symmetry of the kinetic coefficients ( )()( HH jiij −= σσ ) yield yyxx σσ = and xyyx σσ −= for the diagonal and off-
diagonal components of σ . The resistivity tensor ρ is the inverse of the
conductivity tensor σ and is given by equation 4.2:
−
+=
xxxy
xyxx
xyxxyyyx
xyxx
σσσσ
σσρρρρ
22
1 and
zzzz σ
ρ1
= (4.2)
The diagonal resistivity tensor components ρxx and ρyy represent the
magnetoresistance in the transverse (H ⊥ j) configuration whereas the third
diagonal component ρzz describes the longiudinal (H // J) configuration.
The off-diagonal elements ρxy and ρyx describe the Hall effect [Hall1879] which
occurs when a current flowing through a conductor is subject to a magnetic field
CHAPTER 4 Hall-effect in YBa2Cu3Ox and (Y0.6Pr0.4)Ba2Cu3Ox thin films
98
with a component perpendicular to it (H ⊥ J) (figure 4.1). This perpendicular
component of the magnetic field tends to deflect the charge carriers by means of
the Lorentz force. In the steady state, the accumulating charges at the border of
the wire give rise to a finite electric field opposing their motion so that the lines
of current flow are straightened. This transverse electric field EH is known as
the Hall field. The relation between the applied current density jx and the
transverse electric field Ey = EH is defined by the off-diagonal element of the
resistivity tensor ρyx = Ey/jx. Since the transverse electric field EH exactly
opposes the Lorentz force it is expected (and in most cases also observed) to
vary linearly with the magnetic induction B. This leads in a natural way to the
definition of the quantity z
yx
zx
yH BBj
ER
ρ== as the Hall coefficient.
In a free electron system, this coefficient is related to the charge carrier density n by the simple nqRH 1= relation, were q is the elementary charge. In this
simple picture, the Hall coefficient is directly determined by the density of
charge carriers, whereas its sign is set by the their nature (positive or negative).
This is a surprising result since in general the Hall coefficient is dependent on
temperature as well as on magnetic field and one would expect a more
complicated expression. This more elaborate description is provided by the
semi-classical model for electron dynamics [Ashcroft76]. The description reproduces the single electron nqRH 1= result in the more general high-field
limit of clean samples in which the entire current is carried by charge carriers
from a single band.
H
jx+ + + + + + + + + + + + + + + + +
- - - - - - - - - - - - - - - - - -EyEx
xy
z
vx
-|e|v x B
t w
Figure 4.1: Schematic view of the configuration of the current density j and the applied magnetic field H in which the Lorentz force induces a transverse electric field EH known as the Hall field. The thickness t and width w of the specimen are indicated.
CHAPTER 4 Hall-effect in YBa2Cu3Ox and (Y0.6Pr0.4)Ba2Cu3Ox thin films
99
The Hall-effect in high-temperature superconductors has been reported
extensively in the literature for the regime T >> Tc [Harris92, Xiong93,
Almasan94, Wuyts94 & 96] where there it is not affected by superconducting
fluctuations close to Tc or magnetic vortices in the mixed state [Hagen93,
Kopnin99].
In this chapter, we will report Hall-effect measurements on YBa2Cu3Ox and
Y0.6Pr0.4Ba2Cu3Ox thin films at temperatures extending to below the critical
temperature Tc. From the measurements of the high-field Hall-resistivity
ρyx(H), the Hall coefficient RH(T) at fixed field will be calculated. The
combination of these RH(T) and ρab(T) curves then allows the derivation of the
Hall-angle. Finally, the carrier density nH, that can be extracted from our Hall
data, enables us to construct a generic T(p) phase diagram for the YBa2Cu3Ox
and Y0.6Pr0.4Ba2Cu3Ox compounds.
4.2 Hall-effect in the normal state below Tc
Commonly, the Hall resistivity ρyx, arising in the presence of a magnetic field,
is measured in magnetic fields of the order of 1 to 10 tesla. Such an approach
suffers from evident complications in the vicinity of the superconducting
transition. Even above the critical temperature Tc, superconducting fluctuations
change the electronic properties of the high-Tc materials. Below Tc, in the
presence of magnetic fields H < Hc2, a type II superconductor is in the so-called
mixed state with magnetic vortices penetrating the material. The presence of
these vortices strongly influences the response of both ρxx(H) and ρxy(H)
[Bardeen65, Vinokur93, Kopnin99, Hagen93]. Thus, it is clear that, in order to
gain access to the normal-state Hall effect, it is of crucial importance to use
very high magnetic fields H > Hc2(T).
In this work, the Hall-effect was measured on the same thin films whose in-
plane transverse magnetoresistance ρxx(T,H) data were presented in Chapter 3.
In the next paragraphs we will present the Hall-effect data for the YBa2Cu3Ox
thin films with x = 6.45, 6.5 and 6.95 and the Y0.6Pr0.4Ba2Cu3Ox films with
x = 6.7, 6.85 and 6.95.
CHAPTER 4 Hall-effect in YBa2Cu3Ox and (Y0.6Pr0.4)Ba2Cu3Ox thin films
100
The measurements of the Hall-effect were performed in magnetic fields up to
50 T at the pulsed fields facility of the K.U. Leuven. All measurements were
performed in the transverse (H ⊥ I) in-plane (I // ab) configuration (H // c), with
a standard 5-terminal configuration over a Hall pattern of length 1 mm, and
width w = 50 µm; the thickness t of the thin films was of the order of 1000 to
2000 Å (table 3.1 in chapter 3). The Hall signal was amplified 1250 times and
the applied current was set such that the current density did not exceed
~ 7 107 A/m2. This yields Hall-voltages across the sample that are at maximum
a few 100 µV in amplitude. This is three orders of magnitude smaller than the
typical signal in the magnetoresistivity experiments (see chapter 2).
The meticulous implementation of vibration insulation and shielding for electric
interference, combined with a low contact resistance (below 1 Ω) allowed these
measurements to be carried out with large-bandwidth amplifiers and without
any electronic filtering. The Hall-voltage reported here is the result of the
combination of four field pulses with a changing polarity of field and current
used to eliminate spurious induced µodH/dt voltages and unwanted
contributions of the magnetoresistance. A more detailed discussion of the
experimental issues coming into play when experimenting in pulsed magnetic
fields is given in chapter 2.
In figures 4.2 and 4.3, the off-diagonal Hall resistivity ρyx(H) is given for an
YBa2Cu3O6.45 film of thickness t ≅ 1770 Å. This resistivity was calculated with
equation 4.3 from the Hall voltage VH across the sample (a four-pulse average)
by taking into account the applied current Jx and the thickness t of the film. The
width of the sample does not enter the equation directly but, however, strongly
determines the acceptable current that can be applied in order to obtain a
reasonable value for the current density jx. The ρyx(H) curves presented in
figures 4.2 and 4.3 were obtained without any electronic filtering, only a 20
point adjacent averaging was performed, reducing the influence of frequency
components above 16 kHz.
tJ
V
twJ
wV
j
E
x
H
x
H
x
yyx =
==ρ (4.3)
CHAPTER 4 Hall-effect in YBa2Cu3Ox and (Y0.6Pr0.4)Ba2Cu3Ox thin films
101
0 5 10 15 20
0
5
10
15
µoH (T)
YBa2Cu3O6.45ρ y
x (µ
Ωcm
)
4.2 K
20 K
25 K
29 K
35 K
79.4 K53.3 K
t ≈ 1770 Å
Figure 4.2: The off-diagonal resistivity ρyx versus applied magnetic field for a thin YBa2Cu3O6.45 film of thickness 1770 Å at various temperatures. The magnetic field was swept to about 17 T.
0 5 10 15 20 25 30 35 40 45 50
0
10
20
30
40
µoH (T)
YBa2Cu3O6.45
ρ yx
(µΩ
cm) t ≈ 1770 Å
4.2 K
12.9 K
20 K
64.9 K
35 K47.6 K
Figure 4.3: The off-diagonal resistivity ρyx versus applied magnetic field for a thin YBa2Cu3O6.45 film of thickness 1770 Å at various temperatures. The magnetic field was swept to about 42 T.
The low- and high-field ρyx(H) curves for the YBa2Cu3O6.45 film at temperatures
ranging from T << Tc up to T >> Tc as presented in figures 4.2 and 4.3 show a
field-induced transition from ρyx ~ 0 at low temperatures and fields to a regime
CHAPTER 4 Hall-effect in YBa2Cu3Ox and (Y0.6Pr0.4)Ba2Cu3Ox thin films
102
with a finite ρyx ~ H. It is only at high magnetic fields H > Hc2 or high temperatures T > Tc that a Hyx ∝ρ (expected from the balance of the Lorentz
force with the force arising from the transverse electric field) is reached. This
further illustrates the need for high magnetic fields when probing the Hall-effect
around or below Tc.
The quality of these experiments is illustrated by taking a closer look at the
ρyx(H) curve at 4.2 K. It is only at around 35 T that a finite ρyx is induced in the
sample. Below this field, within the high-frequency noise band, the ρyx(H)
curve exhibits an almost perfect ρyx = 0.
Although in this work the focus is on the normal state (e.g. T > Tc or H > Hc2)
Hall effect, it is worthwhile to mention that the fact that the ρyx(H) curves
exhibit a transition, resembling the ρxx(H) magnetoresistive transition, has
inspired scientists to look for an eventual scaling between these two transport
properties [Hagen93, Vinokur93, Wang94, Casaca99, Kopnin99] in the mixed
state. The scaling can be written in the general form of equation 4.4
[Vinokur93, Hagen93].
Bo
xxxy Φ
=βρ
αρ (4.4)
In this equation, α(T,B) is a microscopic parameter that is predicted both to be
independent [Vinokur93] or dependent [Wang94] on flux pinning. In both
models, the exponent β is set to 2. However, recently, Kopnin and Vinokur
[Kopnin99] revisited the previous analysis of Vinokur [Vinokur93] and showed
the scaling exponent β not to be universal and to vary from β ≈ 2 to β ≈ 1
depending on the magnetic field and the concentration of defects. The β ≈ 2
case is then recovered in the limit of weak pinning. To complicate things even
further, when pinning becomes important (e.g. at low temperatures) the
parameter α can become proportional to the applied field [Vinokur93],
neutralising the 1/B in equation 4.4 and thus yielding 2xxxy ρρ ∝ . However,
since experimental evidence on this subject is conflicting, this issue is still
under debate.
CHAPTER 4 Hall-effect in YBa2Cu3Ox and (Y0.6Pr0.4)Ba2Cu3Ox thin films
103
We have used equation 4.4 in an attempt to scale our ρyx(H) and ρxx(H) curves.
The main problem in such an analysis is that it can only be performed in the
narrow field range where both ρyx and ρxx are finite and that one should not
leave the mixed state (e.g. H < Hc2). This narrow field range (~ 10 T) makes it
difficult to distinguish between β ≈ 1 or 2, or to comment upon the eventual 1/B
dependency in equation 4.4. Our data, suffering also from minor temperature
differences between ρyx(H) and ρxx(H) experiments, allow no conclusive
statement about this scaling.
Experimentally, a sign reversal of the Hall resistivity the ρyx can be observed in
moderate magnetic fields and not too far from Tc. It is not yet established
whether this is an intrinsic effect or it is just a consequence of flux pinning. In
our experiments, this sign reversal was not observed.
4.2.1 Hall coefficient
From the ρyx(H) traces introduced in the previous paragraph, the Hall-
coefficient RH can be calculated by applying equation 4.5. This coefficient is
obtained by simply dividing the off-diagonal resistivity by the magnetic
induction. In analysing our data, we have taken the external magnetic µoH to
approximate the induction B, a common assumption that is however only valid
above the Hc2(T) line where the diamagnetic response of the material vanishes.
Since, in this work, emphasis is put on the normal-state properties, this is an
acceptable procedure.
tHJ
V
Bt
J
V
BBj
ER
ox
H
zx
H
z
yx
zx
yH µ
ρ≈
===
1 (4.5)
Figure 4.4 shows the RH versus field curves at various temperatures for the same
thin YBa2Cu3O6.45 film for which the ρyx(H) traces were shown in the previous
paragraph. Also here, a field-induced transition from the superconducting to the
normal-state response is observed. The 29.1 K curve for example starts at
RH ~ 0, rises with magnetic field and reaches an approximately constant value at
high fields (where the ρyx(H) curves become linear (figure 4.3)). It is this
regime of constant RH(H) at H > Hc2 that is of importance for our normal-state
analysis.
CHAPTER 4 Hall-effect in YBa2Cu3Ox and (Y0.6Pr0.4)Ba2Cu3Ox thin films
104
0 5 10 15 20 25 30 35 40 45 50
0
5
10
µoH (T)
YBa2Cu3O6.45
t ≈ 1770 Å
RH (
10-9
m3 /
C)
4.2 K12.9 K
20 K29.1 K
35 K
53.3 K41.9 K
Figure 4.4: The Hall coefficient RH versus applied magnetic field for a thin YBa2Cu3O6.45 film of thickness 1770 Å at various temperatures.
From the RH(H) curves, the temperature dependence of the Hall-coefficient at
fixed magnetic field can be extracted by applying the same procedure as used
for the normal-state resistivity in the previous chapter.
The RH(T) curves at 10, 20, 30, 40 and 50 T were constructed for the
YBa2Cu3Ox thin films with x = 6.45, 6.5 and 6.95 and the Y0.6Pr0.4Ba2Cu3Ox
films with x = 6.7, 6.85 and 6.95. These RH(T) curves are plotted together with
their ρxx(T) equivalent (= ρab(T) in our transverse, in-plane, configuration) in
figures 4.5 to 4.10. In this work, our goal was to access the Hall-effect in the
normal state at T < Tc and the experimental efforts were therefore concentrated
on the low-temperature part of the RH(T) curve.
From these combined RH(T) and ρab(T) plots, the gradual transition from a
superconductor to the normal-state, induced by applying a high magnetic field,
can be seen. The changes upon varying hole doping of the temperature
dependence of ρab(T) at low temperatures, were discussed in the previous
chapter. The direct comparison between the ρab(T) and RH(T) curves allows us
to describe the region where the magnetic field is able to suppress the
superconducting state and where the normal-state RH(T) is fully attained.
CHAPTER 4 Hall-effect in YBa2Cu3Ox and (Y0.6Pr0.4)Ba2Cu3Ox thin films
105
0 50 100 150 200 250 3000
5
10
T (K)
0
500
1000
1500
2000
2500
10 T 20 T 30 T 40 T 50 T
RH (
10-9
m3 /
C)
ρ ab
(µΩ
cm)
YBa2Cu3O6.45
Figure 4.5: The in-plane resistivity ρab (upper panel) and the Hall coefficient RH (lower panel) versus temperature for a thin YBa2Cu3O6.45 film. The data were taken at 10, 20, 30, 40 and 50 T.
A general observation that can be made from these RH(T) plots is that, when
plotted on the same scale, strongly underdoped samples seem to show a weaker
temperature dependence of RH(T) than compounds with a higher doping level.
This observation agrees with earlier work on YBa2Cu3Ox and Y1-yPryBa2Cu3Ox
thin films [Wuyts94 & 96, Xiong93] where, however, it was shown that this is
just a consequence of "squeezing" the data together. When plotted in separate
plots, at every level of hole doping a significant temperature dependence of RH
remains, in the temperature range from Tc up to room temperature.
The RH(T) curves for the YBa2Cu3O6.45 sample show a transition to the normal-
state value of RH as temperature is increased (figure 4.5). This transition shifts
to lower temperatures as the field is increased to 40 tesla. At these high fields,
the low-temperature normal-state RH(T) is recovered, exhibiting little structure
on changing temperature. From the comparison with the in-field ρab(T) curves,
it is clear that at 40 T the normal state is entered unambiguously down to 25 K.
Below this temperature, the ρab(T) curves at 40 T and 50 T separate slightly,
indicating the onset of superconductivity. Thus, by using high pulsed magnetic
fields, we were able to extend the knowledge of the Hall-effect of this
Tc,mid = 41.7 K sample to a temperature T ~ 25 K << Tc.
CHAPTER 4 Hall-effect in YBa2Cu3Ox and (Y0.6Pr0.4)Ba2Cu3Ox thin films
106
0
200
400
600
800
0 50 100 150 200 250 3000
5
10
T (K)
10 T 20 T 30 T 40 T 50 T
RH (
10-9
m3 /
C)
ρ ab
(µΩ
cm)
YBa2Cu3O6.5
Figure 4.6: The in-plane resistivity ρab (upper panel) and the Hall coefficient RH (lower panel) versus temperature for a thin YBa2Cu3O6.5 film. The data were taken at 10, 20, 30, 40 and 50 T.
0
200
400
600
0 50 100 150 200 250 3000
5
10
T (K)
10 T 20 T 30 T 40 T 50 T
RH (
10-9
m3 /
C)
ρ ab
(µΩ
cm)
YBa2Cu3O6.95
Figure 4.7: The in-plane resistivity ρab (upper panel) and the Hall coefficient RH (lower panel) versus temperature for a thin YBa2Cu3O6.95 film. The data were taken at 10, 20, 30, 40 and 50 T.
Similar observations can be made for all the presented RH(T) plots. However, it
is worthwhile to focus for a while on the RH(T) curves for the two samples with
the lowest critical temperature Tc, Y0.6Pr0.4Ba2Cu3O6.7 and Y0.6Pr0.4Ba2Cu3O6.85,
CHAPTER 4 Hall-effect in YBa2Cu3Ox and (Y0.6Pr0.4)Ba2Cu3Ox thin films
107
presented in figures 4.8 and 4.9. For these two compounds, the 50 T magnetic
field suffices largely to suppress the superconducting state over the entire range
of temperatures that was explored experimentally. For these plots then, the
entire 50 T RH(T) curve represents the normal-state response.
0
500
1000
1500
2000
0 50 100 150 200 250 3000
5
10
T (K)
10 T 20 T 30 T 40 T 50 T
RH (
10-9
m3 /
C)
ρ ab
(µΩ
cm)
Y0.6Pr0.4Ba2Cu3O6.7
Figure 4.8: The in-plane resistivity ρab (upper panel) and the Hall coefficient RH (lower panel) versus temperature for a thin Y0.6Pr0.4Ba2Cu3O6.7 film. The data were taken at 10, 20, 30, 40 and 50 T.
0
500
1000
0 50 100 150 200 250 3000
5
10
T (K)
10 T 20 T 30 T 40 T 50 T
RH (
10-9
m3 /
C)
ρ ab
(µΩ
cm)
Y0.6Pr0.4Ba2Cu3O6.85
Figure 4.9: The in-plane resistivity ρab (upper panel) and the Hall coefficient RH (lower panel) versus temperature for a thin Y0.6Pr0.4Ba2Cu3O6.85 film. The data were taken at 10, 20, 30, 40 and 50 T.
CHAPTER 4 Hall-effect in YBa2Cu3Ox and (Y0.6Pr0.4)Ba2Cu3Ox thin films
108
0
200
400
600
0 50 100 150 200 250 3000
5
10
T (K)
10 T 20 T 30 T 40 T 50 T
RH (
10-9
m3 /
C)
ρ ab
(µΩ
cm)
Y0.6Pr0.4Ba2Cu3O6.95
Figure 4.10: The in-plane resistivity ρab (upper panel) and the Hall coefficient RH (lower panel) versus temperature for a thin Y0.6Pr0.4Ba2Cu3O6.95 film. The data were taken at 10, 20, 30, 40 and 50 T.
Although not in our experimental window, it can be remarked here that the high
temperature Hall coefficient was shown to exhibit a change in slope at about the
temperature T*, marking the crossover in the resistivity [Ilonca93, Wuyts94].
For the samples with the lowest level of hole doping, the relative insensitivity to temperature variations of the low-temperature Hall-coefficient RH is in sharp contrast with the strongly divergent resistivity ρρab(T) found in these compounds. This nearly temperature independent Hall
coefficient, also reported for Bi2201 crystals and La214 thin films [Ando97], is
an important challenge to the existing models already explaining a low-
temperature divergence of ρab(T).
Among these models, some indeed predict a constant RH at low temperatures.
The 2D Luttinger model predicts a constant RH [Anderson96b] but its predicted
power-law divergence does not agree as good with the high-field ρab(T) data as
does the ln(1/T). The conventional 2D weak localisation also yields a
temperature independent RH [Lee85] and was already shown to reproduce the
ln(1T) divergence of ρab(T) (chapter 3, paragraph 3.6).
CHAPTER 4 Hall-effect in YBa2Cu3Ox and (Y0.6Pr0.4)Ba2Cu3Ox thin films
109
4.2.2 Hall angle
A property that is closely related to the Hall-coefficient, is the Hall-angle. This
quantity can be computed by combining the Hall-coefficient with the results
from magnetoresistivity experiments by applying equation 4.6.
HRBRE
E
oH
ab
zH
ab
yx
xx
y
xH µ
ρρρρ
θ ≈===cot (4.6)
The Hall-angle thus basically is the ratio of the longitudinal and transverse
electric fields and hence also the ration between the diagonal and off-diagonal
elements of the resistivity tensor (equation 4.2); also in this equation, the
Bz ≈ µoH approximation was made. The Hall-angle is related to the Hall-
mobility by:
Hzxx
HH B
R
θρµ
cot
1== (4.7)
The importance of this Hall-angle for the high-Tc cuprates was pointed out by
Anderson [Anderson91] as a means of accounting for the deviations from the
linear temperature dependencies for ρ and RH upon changing doping level. He
argued that the Hall-angle should depend quadratically upon temperature
CTH += 2cot αθ .
This prediction comes about in the so-called 2D Luttinger liquid model, in
which spin and charge are separated and are described by spinon and holon
quasiparticles. The quasiparticle excitations are then assumed to obey two
different relaxation times. The longitudinal charge transport (the resistivity) is
determined by the relaxation time τtr, whereas the Hall-resistivity is determined
by both τtr and a transverse scattering time τH. The transverse relaxation rate
(τH)-1 is governed by spinon-spinon scattering and is proportional to T 2
[Anderson91]. Computing the Hall-angle then yields a T 2 dependence
(equation 4.8) where the constant C is introduced by a temperature independent
scattering process at magnetic impurities by the simple application of
Matthiessen's rule.
CTimpuritiesHtr
tr
yx
xxH +∝+∝= 21
cot ατττ
τρρ
θ (4.8)
CHAPTER 4 Hall-effect in YBa2Cu3Ox and (Y0.6Pr0.4)Ba2Cu3Ox thin films
110
The quadratic temperature dependence of the Hall-angle was confirmed
experimentally in a wide variety of compounds (among these are YBa2Cu3Ox
[Chien91, Wuyts94 & 96, Xiong93], (YPr)Ba2Cu3Ox [Almasan94, Xiong93]
and La2-xSrxCuO4 [Ando97]) and the correlation of the constant C in
equation 4.8 with the presence of magnetic impurities was indeed confirmed
(Zn [Chien91], Pr [Xiong93]). Wuyts et al. demonstrated a nice scaling
behaviour of the Hall-angle when using the same scaling temperature as for the
longitudinal transport data [Wuyts94 & 96] and found indications for deviations
from the T2 dependence. They also showed the slope α to be closely related to
the density of charge carriers in the plane.
The Hall-angle versus temperature curves were constructed using the high-field
RH(T) and ρab(T) data for the YBa2Cu3Ox thin films with x = 6.45, 6.5 and 6.95
and the Y0.6Pr0.4Ba2Cu3Ox films with x = 6.7, 6.85 and 6.95 and are plotted
versus T 2 in figure 4.11. From this plot, it is clear that we are able to extend the
existing studies at T > Tc to temperatures T << Tc. The region of a diverging
Hall-angle (mostly seen around T ~ Tc) was significantly shifted to lower
temperatures in the high field limit. Since our experimental effort was however
mainly concentrated on the transport properties at these low temperatures
(T < Tc), the temperature window is rather limited in comparison to most other
studies, where it extends up to T 2 ~ 10⋅104 K2. Nevertheless, a tentative
agreement with the often cited quadratic temperature dependence can be
observed, although our data do not allow a reliable determination of the slope α or to distinguish between a possible scaling or a mere shift of Hθcot .
A statement can however be made about the zero-temperature intercept C of
Hθcot . The C values estimated from figure 4.11, and the estimates for the
error, are summarised in table 4.12. The low value for the YBa2Cu3O6.5 thin
film is representative for YBa2Cu3Ox films that are not too strongly underdoped
[Wuyts94, Xiong93]. For the lowest oxygen contents (x = 6.45 in our case), the
parameter C was shown to increase, supposedly due to scattering by the
antiferromagnetic fluctuations showing up in these samples (see the discussion
in chapter 1). This is confirmed by our high-field data on the x = 6.45 and 6.5
samples. This temperature independent contribution C, arising from magnetic
scattering, was also shown to increase markedly upon a gradual substitution of
CHAPTER 4 Hall-effect in YBa2Cu3Ox and (Y0.6Pr0.4)Ba2Cu3Ox thin films
111
Y by the magnetic Pr [Xiong93]. The high-field Hθcot data for the
Y0.6Pr0.4Ba2Cu3Ox films studied in this work (table 4.12) indeed show a
significantly higher value for the intercept C when comparing to YBa2Cu3Ox
samples with a similar oxygen content.
This argument suggests that the presence of the magnetic Pr atoms, apart from
the reduction of the critical temperature Tc, causes a significant temperature-
independent contribution to the scattering of the charge carriers. This is in
agreement with the high residual resistivity ρo and the small ρ290 K/ρo resistivity
ratio, reported in chapter 3.
0
20
40
60
0.0 0.5 1.0 1.5 2.0 2.50
20
40
cot θ
H
T2 (104 K2)
Y0.6Pr0.4Ba2Cu3O6.95
Y0.6Pr0.4Ba2Cu3O6.7
Y0.6Pr0.4Ba2Cu3O6.85
YBa2Cu3O6.5
YBa2Cu3O6.45
Figure 4.11: The Hall angle plotted versus T2 for thin films of YBa2Cu3O6.45, YBa2Cu3O6.5, Y0.6Pr0.4Ba2Cu3O6.7, Y0.6Pr0.4Ba2Cu3O6.85 and Y0.6Pr0.4Ba2Cu3O6.95. The Y123 data were taken at 40 T whereas the YPr123 data are at 50 T. The arrows indicate the critical temperature Tc; the dashed lines were constructed by linearly extrapolating (the weakly temperature dependent) RH.
4.2.3 Carrier density
From the Hall-coefficient RH, the simple qnR HH 1= relation gives an estimate
for the charge carrier density. However, the marked temperature dependence of
this property over the temperature range Tc → 300 K [Xiong93, Wuyts94 & 96]
in the high-Tc's, introduces some doubt at which temperature to define nH and
about the validity of this simple approach.
CHAPTER 4 Hall-effect in YBa2Cu3Ox and (Y0.6Pr0.4)Ba2Cu3Ox thin films
112
However, in the case of our high-field Hall data, we were able to construct the
RH(T) curve down to temperatures T << Tc were the temperature dependence of
the Hall-coefficient is weak. Moreover, the use of very high magnetic fields
(50 T) at temperatures T << Tc fulfils at least two out of three conditions for the simple qnR HH 1= limit of the semi-classical model for electron dynamics
[Ashcroft76] to be valid. The demand for clean samples remains of course a
problematic issue in the high-Tc compounds (see the discussion of the residual
resistivity ρo and the resistance ratio ρ290 K/ρo in chapter 3).
In view of this, for these high-field data, the qnR HH 1= relation can be taken
as a reasonable approximation. Indeed, in this limit, the low temperature RH(T)
shows only little temperature dependence (figures 4.5 to 4.10), as also observed
in Bi2201 crystals and La214 thin films [Ando97]. Therefore it seems
appropriate to take nH(T = Tc) as an approximation for the charge carrier
density. These values are summarised in table 4.12; the error bars account for
the small temperature dependence of RH.
From these nH estimates, p, the fraction of holes per Cu-atom in the CuO2 plane,
can be calculated by simply multiplying nH by the volume of the unit cell
a· b· c ≈ 173.24 Å3 and divide it by 2, the number of Cu atoms per unit cell in the
CuO2 plane. These values are also reported in table 4.12.
Tc,mid
(K)
nH
(1027 m-3)
p
(#/Cu)
C
YBa2Cu3O6.95 92.2 2.36 ± 0.15 0.210 ± 0.01 N/A
YBa2Cu3O6.5 52.9 0.83 ± 0.03 0.073 ± 0.003 2.1 ± 2
YBa2Cu3O6.45 45.5 0.71 ± 0.02 0.062 ± 0.002 15.4 ± 2
Y0.6Pr0.4Ba2Cu3O6.95 41.4 1.24 ± 0.01 0.108 ± 0.001 7 ± 2
Y0.6Pr0.4Ba2Cu3O6.85 31.8 0.98 ± 0.01 0.086 ± 0.001 13 ± 6
Y0.6Pr0.4Ba2Cu3O6.7 22.3 0.93 ± 0.01 0.081 ± 0.001 24 ± 2
Table 4.12: Superconducting critical temperature Tc,mid, charge carrier density nH from the Hall data at T = Tc, the fraction p of holes per Cu-atom in the CuO2 plane and the zero-temperature intercept C of the quadratic part of the Hall-angle.
CHAPTER 4 Hall-effect in YBa2Cu3Ox and (Y0.6Pr0.4)Ba2Cu3Ox thin films
113
4.3 Phase diagram
It was discussed in chapter 3 that the oxygen content x is not a good parameter
to construct a phase diagram that is valid for both YBa2Cu3Ox and
Y0.6Pr0.4Ba2Cu3Ox (a so-called generic phase-diagram). It was shown that none
of the three characteristic boundary lines Tc(x), T*(x) or TMI(x) coincided for the
two systems. Also, it was argued that turning to a charge carrier density, as
obtained from the Hall data, would at least account for the supposed hole-filling
effect of the Y/Pr substitution and shift the boundary lines of the
Y0.6Pr0.4Ba2Cu3Ox system to the left in the T(x) plane with respect to the
YBa2Cu3Ox lines. The influence of the claimed magnetic pair-breaking in the
(Y/Pr)Ba2Cu3Ox compound would then become clear by a non-coincidence of
the Tc(x) lines.
The values for the hole fraction in the CuO2 planes p, as calculated from the
high-field Hall data, enable us then to construct such a generic T(p) phase
diagram for both the YBa2Cu3Ox and the Y0.6Pr0.4Ba2Cu3Ox systems. In
figure 4.13, the critical temperatures Tc,mid and the boundary TMI between the
metallic and insulating ρ(T) regimes are plotted versus p. It is clear from this
plot that the Tc(p) lines for the two compounds do not coincide but that they
show a qualitatively similar behaviour, shifted to a lower Tc for the
Y0.6Pr0.4Ba2Cu3Ox system. The fact that Tc(p)YPr < Tc(p)Y is a strong indication
that the magnetic pair-breaking in the (Y/Pr)Ba2Cu3Ox compound might play an
important role in the reduction of the superconducting critical temperature Tc.
As far as the boundary TMI(p) between the metallic and insulating ρ(T) regimes
is concerned, the discussion in paragraph 3.6 concerning the T(x) phase diagram
(figure 3.48) is still valid for this T(p) plot. For the YBa2Cu3Ox system, this
TMI(p) line is below the Tc(p) line for almost all levels of hole doping. In this
system, the insulating state is thus effectively masked by the onset of
superconductivity and is only made visible by high-field transport
measurements. For the Y0.6Pr0.4Ba2Cu3Ox compound (dotted lines in
figure 4.13), the TMI(p) line is above the Tc(p) line for all levels of oxygen
content and the insulating tendency can therefore already be seen in the zero-
field ρ(T) measurements above Tc. This strong tendency towards an insulating
CHAPTER 4 Hall-effect in YBa2Cu3Ox and (Y0.6Pr0.4)Ba2Cu3Ox thin films
114
low-temperature resistivity behaviour is in agreement with the additional
disorder introduced by the Y/Pr substitution and large residual resistivity ρo and
small resistance ratio ρ290 K/ρo for this Y0.6Pr0.4Ba2Cu3Ox compound.
0
50
100
0 0.05 0.10 0.15 0.20 0.25
T (
K)
p (holes/Cu-atom in the plane)
TMI
TMI
YBa2Cu3Ox
Y0.6Pr0.4Ba2Cu3Ox Tc TMI
Tc TMI
Tc
Tc
Figure 4.13: Superconducting critical temperature Tc (open symbols) and the boundary TMI (filled symbols) between the metallic and the insulating regimes for ρ(T), plotted versus the fraction of holes per Cu-atom in the CuO2 plane, for the YBa2Cu3Ox (diamonds) and the Y0.6Pr0.4Ba2Cu3Ox (circles) thin films. The arrow indicates the TMI = Tc point for YBa2Cu3Ox.
When we complement this T(p) plot with the crossover line T*(p) between the
linear ρ(T) at T > T* and the super-linear ρ(T) at T < T*, we obtain the phase
diagram as shown in figure 4.14. A striking observation from this plot is that
the crossover temperature T*(p)YPr ~ T*(p)Y, in contrast to the Tc(p)YPr < Tc(p)Y
discussed above. It seems that the crossover at T* is fully determined by the
density of charge carriers in the CuO2 plane, whereas for the critical
temperature Tc, additional effects come into play (for example magnetic pair
breaking).
CHAPTER 4 Hall-effect in YBa2Cu3Ox and (Y0.6Pr0.4)Ba2Cu3Ox thin films
115
0 0.5 1.0 1.5 2.0 2.5
0 0.05 0.10 0.15 0.20 0.25
0
100
200
300
400
T (
K)
p (holes/Cu-atom in the plane)
nH (1027 m-3)
T*
YBa2Cu3Ox
Y0.6Pr0.4Ba2Cu3Ox
Tc TMIT*
TcTMI
TMI
0
100
Figure 4.14: Generic T(p) phase diagram for the YBa2Cu3Ox (diamonds, solid line) and the Y0.6Pr0.4Ba2Cu3Ox (circles, dotted line) thin films. Indicated are the crossover temperature T* (filled symbols), the superconducting critical temperature Tc (open symbols) and the boundary TMI between the metallic and the insulating-alike regimes for ρ(T). All are plotted versus the fraction of holes per Cu-atom in the CuO2 plane (bottom axis). The equivalent density of charge carriers, obtained from the Hall data is indicated on the upper axis. The Tc(x) and TMI(x) boundaries for Y0.6Pr0.4Ba2Cu3Ox were shifted down by 100 K.
4.4 Conclusions
In this chapter, Hall-effect measurements on YBa2Cu3Ox and Y0.6Pr0.4Ba2Cu3Ox
thin films at temperatures extending to below the critical temperature Tc are
reported. These measurements were performed at very high pulsed magnetic
fields in order to fully access the normal state (H > Hc2).
These Hall measurements yield a signal that is three orders of magnitude
smaller than in standard magnetoresistivity experiments. The successful
accomplishment of these measurements was possible by a careful
CHAPTER 4 Hall-effect in YBa2Cu3Ox and (Y0.6Pr0.4)Ba2Cu3Ox thin films
116
implementation of vibration insulation and shielding for electric interference,
combined with a low contact resistance.
From the measurements of the high-field Hall-resistivity ρyx(H), the field
dependence of the Hall coefficient RH(H) at various temperatures was
calculated. By concentrating on the normal-state part of the data (H > Hc2), the
Hall coefficient RH(T) curves at fixed (high) field were constructed. These RH(T)
curves show a transition to the normal-state value of RH as temperature is
increased. This transition shifts to lower temperatures in high fields. At these
high fields, the low-temperature normal-state RH(T) is recovered, exhibiting
only a weak temperature dependence. By combining these RH(T) curves with
the high-field ρab(T) curves for the same samples, it is clear that at these high
fields the normal state is entered unambiguously down to temperatures T << Tc.
Thus, by using pulsed high magnetic fields, we were able to extend the
temperature range for the observation of the Hall-effect of the YBa2Cu3Ox and
Y0.6Pr0.4Ba2Cu3Ox thin films to temperatures T << Tc.
The insensitivity of the low-temperature Hall-coefficient RH to temperature variations is in sharp contrast with the strongly divergent ρρab(T) observed in these compounds. The approximately temperature-independent Hall
coefficient, is an important test for the existing models already explaining a low-
temperature divergence of ρab(T). Among the models still applicable for the
description of both ρab(T) and RH(T) is the weak localisation-model, which
explains both the constant RH at low temperatures and the ln(1/T) divergence of the high-field ρρab(T) data.
A further combination of the Hall data with the ρab(T) resistivity curves then
allows the derivation of the Hall-angle. The Hall-angle is predicted, and widely
observed, to obey a quadratic temperature dependence up to room temperature,
with a constant offset that is introduced by temperature independent scattering
at magnetic impurities. Also here, the existing studies at T > Tc are extended to
temperatures T << Tc by adding our high-field data. The region of diverging
Hall-angle (mostly observed around T ~ Tc) was significantly shifted to lower
temperatures in this high field limit. Although our experimental temperature
window is limited, a tentative agreement with the often cited quadratic
temperature dependence was observed. However, our data do not allow a
CHAPTER 4 Hall-effect in YBa2Cu3Ox and (Y0.6Pr0.4)Ba2Cu3Ox thin films
117
reliable determination of the slope or to distinguish between a possible scaling or a simple shift of Hθcot . The zero-temperature intercept of Hθcot , giving
information about the presence of magnetic scatterers, was, in the samples with
the lowest oxygen content, shown to be elevated, supposedly due to
antiferromagnetic fluctuations.
Also in the Y/Pr substituted samples, a higher offset of the Hall-angle was
observed. This suggests that the presence of the magnetic Pr atoms, apart from
the reduction of the critical temperature Tc, causes a significant temperature-
independent contribution to the scattering of the charge carriers. This is in
agreement with the high residual resistivity ρo and the small ρ290 K/ρo resistivity
ratio, reported in chapter 3.
Finally, the carrier density nH (and thus also p, the fraction of holes per Cu-atom
in the CuO2 plane) enables us to construct a generic T(p) phase diagram for the
YBa2Cu3Ox and Y0.6Pr0.4Ba2Cu3Ox compounds. Turning to this hole fraction p
accounts for the expected hole-filling effect of the Y/Pr substitution and shifts
the boundary lines of the Y0.6Pr0.4Ba2Cu3Ox system on the T-p plane to the left
with respect to the YBa2Cu3Ox lines.
The critical temperature Tc,mid(p) and the boundary TMI between the metallic and
insulating-alike ρ(T) regimes do not coincide for the YBa2Cu3Ox and
Y0.6Pr0.4Ba2Cu3Ox compounds; although demonstrating a qualitatively similar
behaviour, shifted to a lower Tc for the Y0.6Pr0.4Ba2Cu3Ox system. The fact that
Tc(p)YPr < Tc(p)Y is an indication of the magnetic pair-breaking in the
(Y/Pr)Ba2Cu3Ox compound, which might play an important role in the reduction
of the superconducting critical temperature Tc.
For the YBa2Cu3Ox system, the TMI(p) line is below the Tc(p) line for almost all
levels of hole doping and thus the insulating state at zero-field is effectively
masked by the onset of superconductivity. For the Y0.6Pr0.4Ba2Cu3Ox
compound, the TMI(p) line is above the Tc(p) line for all levels of oxygen content
and the insulating tendency can therefore already be seen in the zero-field ρ(T)
measurements above Tc. This tendency towards an insulating low-temperature
resistivity behaviour is in agreement with the additional disorder introduced by
CHAPTER 4 Hall-effect in YBa2Cu3Ox and (Y0.6Pr0.4)Ba2Cu3Ox thin films
118
the Y/Pr substitution and the large residual resistivity ρo and small resistance
ratio ρ290 K/ρo for this Y0.6Pr0.4Ba2Cu3Ox compound.
The T*(p) lines, for the crossover between the linear ρ(T) at T > T* and the
super-linear ρ(T) at T < T*, coincide for the YBa2Cu3Ox and Y0.6Pr0.4Ba2Cu3Ox
thin films. It seems that the crossover at T* is fully determined by the density of charge carriers in the CuO2 plane, whereas for the critical temperature Tc, additional effects come into play (for example magnetic pair breaking).
CHAPTER 5 Stripe formation and transport properties of underdoped cuprates
119
Chapter 5
Effect of stripe formation on
the transport properties of
underdoped cuprates
"The problem with the spin-gap is
that there are too many right ways to understand it …, not too few.
When one realises what is going on, it seems all too obvious
in several ways that one should have known all along.
(P.W. Anderson [Anderson96])
The universal ρ(T) behaviour in the underdoped YBa2Cu3Ox and
Y0.4Pr0.4Ba2Cu3Ox thin films, reported in chapter 3, strongly points into the
direction of a single scattering mechanism being dominant over the whole
underdoped regime in the Y123 system. Bearing in mind the complex magnetic
phase diagram discussed in chapter 1 with short-range AF correlations and a
pseudo gap showing up at temperatures far above the superconducting critical
temperature Tc and reminding the strong indications of stripe formation in the
CuO2 planes, it is tempting to assign the origin of this dominant scattering
mechanism to the microscopic magnetic and charge ordering.
CHAPTER 5 Stripe formation and transport properties of underdoped cuprates
120
5.1 Charge ordering revisited - Stripes
The undoped HTS are Mott insulators (with a half-filled upper orbital in the Cu
3d-shell) that are insulating rather than metallic by virtue of the strong electron-
electron repulsion. Doping the material (adding holes to the CuO2 planes)
results in the local destruction of the AF order. In the absence of long-range
Coulomb repulsion, a moderately doped AF tends to expel holes and the
material exhibits phase separation into hole-rich (metallic) regions and hole free
AF areas [Emery90]. This tendency towards local phase separation into hole-
free and hole-rich phases is frustrated when the Coulomb repulsion between the
holes is taken into account [Kivelson96]; a matter of particular relevance in the
underdoped HTS that are poor metals and in which screening is thus not
guaranteed.
The competition between the long-range Coulomb repulsion and the short-range
magnetic dipole interaction and their influence on the ordering of the holes in
the AF were studied by molecular dynamics simulations [Stojkovic99]. The
main result of these simulations is shown in figure 5.1 where the various phases
are drawn as a function of the magnetic dipole interaction (~ J) and the doping
level of the CuO2 plane. At weak dipole interactions, the Coulomb interaction
pushes the system to form a Wigner crystal. At moderate dipole interactions,
the Wigner phase is frustrated and stripes of holes (diagonal and vertical-
horizontal) show up at higher levels of hole doping. At very strong magnetic
interactions the holes forms a glassy clumped phase. The introduction of
thermal or crystallographic disorder changes this phase diagram and gives the
Wigner-phase a more glassy nature. The striped phases however retain their
main features although the stripes become of finite length (due to
crystallographic disorder) and are dynamic (at high enough temperatures).
Other groups predict exotic striped phases, including liquids crystals and glass
phases [Kivelson98].
These simulations confirm earlier experimental work showing the formation of
magnetic domains, incommensurate with the crystal lattice, both in
La2-xSrxCuO4 [Thurston89, Cheong91, Mason92 & 94, Yamada97,
Tranquada97, Hunt99] and YBa2Cu3Ox [Aeppli97, Tranquada97 & 97b, Dai98,
CHAPTER 5 Stripe formation and transport properties of underdoped cuprates
121
Kao99, Arai99]. This shows that, in reality, charge might be neither
homogeneously spread over the CuO2 planes, nor phase-separated into 2 phases
but that some complicated 1D-like structure is more probable.
diagonalstripes
clumpsWignercrystal
Dop
ing
Magnetic dipole interaction ~ J
geometricphase
Figure 5.1: Schematic phase diagram of the arrangement of holes, doped into a planar AF, in the presence of a long-range Coulomb interaction. As the magnetic interaction is becoming more important, the Wigner crystal becomes unstable and other phases show up (adapted from [Stojkovic99]).
In this charge-stripe picture [Emery97b & 99b, Moshchalkov93, 98b & 99 and
refs. above], dynamic metallic [Ichikawa99, Noda99, Tajima99] stripes are
thought to dominate the transport properties. Below a certain temperature the
tendency towards local phase separation results in the formation of dynamic
charge stripes, acting as domain walls for the antiferromagnetic (AF)
surroundings. The confinement of the AF regions leads to the development of a
spin-gap that is transferred to the holes in the stripe by hopping of electron pairs
perpendicular to the stripe, or, from an alternative point of view, by the coherent
transverse fluctuation of the charge stripe thus interacting with the underlying
AF [Emery97b & 99b]. This might result in a local pairing (with a gap equal to
the spin-gap) of charges, the so-called pre-formed pairs. Thus, the "magnetic
proximity effect" [Emery97b & 99b] then imposes the spin-gap onto the holes
in the charge-stripes. At lower temperatures, T = Tc, it is predicted that the local
pairs could acquire macroscopic coherence, an imperative condition for the
onset of bulk superconductivity [Emery97b & 99b].
CHAPTER 5 Stripe formation and transport properties of underdoped cuprates
122
Figure 5.2 presents a schematic view on the formation of dynamic charge stripes
in the underdoped CuO2 plane. On a short length scale (top) the straight, 1D,
metallic charge stripes form a domain wall for the surrounding antiferromagnet.
At intermediate length scales (middle), the holes recover a 2D motion (stripe
meandering), while at the macroscopic level (i.e. the sample size) the effective
1D conduction is recovered. Indeed, in this macroscopic view, the metallic
wires retain their 1D character as the meandering takes place on a much smaller
length scale.
Figure 5.2: A schematic view on the formation of dynamic charge stripes in the underdoped CuO2 plane (after [Zaanen99]), on a short length scale (top), an intermediate length scale (middle) and the macroscopic level (sample size).
CHAPTER 5 Stripe formation and transport properties of underdoped cuprates
123
Neutron diffraction experiments on La2-xSrxCuO4 revealed the inverse stripe
distance 1/ds to increase approximately linearly with Sr doping [Tranquada97b,
Yamada98] and start to saturate at around xSr = 0.12 where the inter-stripe
distance ds equals 4a, a being the crystallographic lattice parameter (figure 5.3 a
and b). This suggests the increasing doping to cause the stripes to be packed
denser, essentially keeping the same doping level within the stripe. In the
underdoped region, a linear correspondence between the inverse stripe distance
1/ds and the critical temperature Tc was found [Yamada98]. In the overdoped
regime (c and d in figure 5.3), the distance between the stripes was shown to be
almost constant upon doping, suggesting the holes to enter the regions between
the stripes and the doping contrast of the stripes with respect to their
surroundings to decrease.
p
1
ds
(a)
(c)
(b)
(d)
Figure 5.3: (a)-(d) A schematic view on the formation of charge stripes in the CuO2 plane upon doping (on a short length scale) (after [Moshchalkov99d]). The curve represents the inverse distance between the stripes versus the doping level of the plane.
5.2 Spin ladders, a magnetic structure between 1D & 2D
Low-dimensional quantum Heisenberg antiferromagnets are reported to exhibit
fascinating properties [Dagotto96 & 99]. The simple 1D spin ½ nearest
neighbour Heisenberg chain does not show an ordered ground state, due to
quantum fluctuations [Bethe31]. Instead, this system shows power-law
CHAPTER 5 Stripe formation and transport properties of underdoped cuprates
124
correlations with gapless excitations. The 2D Heisenberg spin ½ AF does
exhibits long range order at low temperature and they too accommodate a
gapless creation of an S = 1 excitation.
Spin-ladders (SL) are arrays of coupled chains and thus present structures that
interpolate between the 1D and the 2D case. By assembling a varying number
of these 1D chains into a SL, the influence of the dimensionality of the AF
region on the correlation length and spin-gap can be studied.
Numerical calculations revealed this crossover to be not as smooth as expected.
Ladders made of an even number of chains have so-called spin-liquid ground
states with purely short-range spin correlations that have an exponential decay.
In this even-chain SL, a finite gap exist for the creation of spin S = 1 excitations
[Dagotto92]. A ladder with an odd number of legs exhibits quite opposite
features and seems to retain some of the properties of the purely 1D single
chain, namely gapless spin excitations and a power-law decay of the AF
correlations.
Figure 5.4: Schematic drawing of the Cu2O3 sheet of SrCu2O3 (left) and the Cu3O5 sheet of Sr2Cu3O5 (right) containing a 2-leg (left) and three leg ladder (right) (adapted from [Azuma94]).
One class of materials, known to show intrinsically a SL behaviour, is formed
by the cuprates with a modified (and mixed) coordination of the CuO4 squares
in the CuO2 planes. The alternating edge- and corner-sharing between the CuO4
squares in these materials such as Srx-1Cux+1O2x, SrCu2O3, Sr2Cu3O5 and
CHAPTER 5 Stripe formation and transport properties of underdoped cuprates
125
(SrCa)14Cu24O41 [Azuma94, Dagotto96, Uehara96, Nagata97, Takano97] results
in the formation of n-leg spin ladders with nc = 1,2 or 3 (figure 5.4). The spin-
gap, predicted by the numerical work to exist in even-chain SL, was observed
experimentally whereas the 3-leg SL were shown to be gapless. Moreover,
doping 2-leg ladders by chemical substitution and external pressure induces a
crossover to more metallic transport properties, creates a super linear ρ(T)
behaviour similar to that observed in the HTS and even produces
superconductivity [Uehara96, Nagata97].
5.3 Quantum transport in doped 1D and 2D Heisenberg systems
The discussion in the previous paragraphs makes clear that the transport
properties of the high-Tc cuprates should be extremely sensitive to the
underlying microscopic magnetic structure. In order to account for the possible
inhomogeneous intercalation of AF insulating regions and metallic hole-rich
stripes, a physical model was developed [Moshchalkov93, 98, 98b, 99, 99b &
99c]. The model describes the transport both in the 2D Heisenberg regime
(above T*) as in the 1D striped regime (below T*) where the pseudo gap
develops.
A rapidly growing experimental evidence ([Dagotto96, Emery97, Tranquada97]
and previous paragraphs) indicates that the 1D scenario might be also relevant
for the description of the underdoped high-Tc cuprates where 1D stripes can be
eventually formed. Since mobile carriers in this case are expelled from the
surrounding Mott-insulator phase into the stripes, the latter then provide the
lowest resistance paths. This makes the transport properties very sensitive to
the formation of the stripes, both static and dynamic.
5.3.1 The model
The importance of the CuO2 planes for the transport properties is a widely
documented feature of the high-Tc cuprates. The confinement of the charge
carriers in these planes reduces the dimensionality for charge transport to 2
dimensions (or less) and makes the conductivity σ in such 2D metallic system
CHAPTER 5 Stripe formation and transport properties of underdoped cuprates
126
to be dominated by quantum transport. At moderate temperatures T* < T < To,
AF correlations will develop in the CuO2 planes, yielding a 2D Heisenberg
regime. At lower temperatures T < T*, local charge inhomogeneities (1D charge
stripes) will confine the AF regions, resulting in the formation of a pseudo spin-
gap. Any model trying to explain the unusual transport properties should
account for these features (i.e. the magnetic structure and the dimensionality).
The proposed approach [Moshchalkov93, 98b & 99] is based on three basic
assumptions:
1. the dominant scattering mechanism in HTS in the whole temperature
range is of magnetic origin;
2. the specific temperature dependence of the resistivity ρ(T) can be
described by the inverse quantum conductivity σ-1 with the inelastic
length Lφ being fully controlled, (via a strong interaction of holes
with Cu2+ spins, due to the magnetic proximity effect [Emery97b &
99b]) by the magnetic correlation length ξm, and, finally,
3. the proper 1D or 2D expressions should be used for calculating the
quantum conductivity with Lφ ~ ξm.
The quantum conductivity in 2D is proportional to ln(Lφ) whereas the quantum
conductivity of a single 1D wire is a linear function of the inelastic length Lφ
[Abrikosov88]. They are represented by
=−
lLe
bTT DD
φσρ ln1
~)()(2
21
2h
(5.1)
φσρ Le
bTT DD
h
2
211
11
~)()( =− (5.2)
with l the elastic length and b the thickness of the 2D layer or the diameter of
the 1D wire. These expressions for the resistivity of the 1D wires and 2D layers
can be detailed by assuming that the dominant scattering mechanism is of
magnetic origin and thus enforce the Lφ ~ ξm condition (ξm being the magnetic
correlation length). Equations 5.1 and 5.2 for the 2D and 1D resistivities can
then be modified as to give:
CHAPTER 5 Stripe formation and transport properties of underdoped cuprates
127
=
=−
lTe
bTT Dm
LDDm
)(ln1
~)()( 22
21
2ξσρ
ξφ h (5.3)
)(1
~)()( 1
2
211
1 Te
bTT DmLDD
mξσρ
ξφ h=− = (5.4)
The determination of the precise behaviour of the resistivity in the 2D
Heisenberg (T > T*) and the 1D striped (T < T*) regimes thus requires the
calculation of the magnetic correlation lengths in a 1D (ξm1D) and a 2D (ξm2D)
antiferromagnetic structure.
In the 2D Heisenberg case, the temperature dependence of the correlation length
ξm2D was calculated for the isotropic case [Hasenfratz91] and is expressed as
( )
⋅−
⋅=
T
F
F
T
F
ceTDm
2
222
2exp
221
28
πππ
ξh
(5.5)
with c being the spin velocity and F 2 a parameter that can be directly related to
the exchange interaction J. For simplicity we adopt the JF =22π relation that
is valid over a wide temperature range. For the 1D striped phase, the similarity
with the 1D even-chain Heisenberg AF spin-ladder compounds can be
employed and the spin-correlation length found by Monte Carlo simulations
[Greven96] can be taken for ξm1D :
∆−
∆+=∆ −
T
TADm exp
2)( 1
1 πξ (5.6)
where A ≈ 1.7 and ∆ is the spin-gap.
The combination of these expressions for the 1D and 2D spin correlation
lengths with the proper expression for the quantum resistance then gives the
temperature dependence of the resistivity. For temperatures T > T*, in the 2D
Heisenberg regime, surprisingly, the resistivity is a linear function of
temperature [Moshchalkov93] due to the mutual cancellation in the limit T <<
2J between the logarithmic ρ(ξm) dependency and the exponential temperature
dependence of ξm. For T < T* the striped 1D phase yields equation 5.8, with J//
the intra-chain coupling and a the spacing between the 1D wires (J// comes in to
recalculate the theorist units) [Moshchalkov98b & 99].
CHAPTER 5 Stripe formation and transport properties of underdoped cuprates
128
[ ] ( )[ ]J
T
e
b
T
JTT mDD 2
1
1122 ~expln~ln~)()(
h−
−−
= ξσρ (5.7)
[ ]
∆
−+∆
== −
TJ
TA
Jae
bTT DD exp
2)()(
////2
21
11 πσρ
h (5.8)
To verify the validity of the proposed 1D spin ladder model, a crucial test is its
application to the resistivity data obtained on the even-chain spin-ladder
compound Sr2.5Ca11.5Cu24O41 described above [Nagata97]. This compound, due
to its specific crystalline structure, definitely contains a two-leg (nc = 2) Cu2O3
ladder and therefore its resistivity along the ladder direction should indeed obey
the 1D expression 5.8. The results of the ρ(T) fit with equation 5.8 are shown in
figure 5.5. This fit demonstrates a remarkable quality over the whole
temperature range T ~ 25-300 K, except for the lowest temperatures where the
onset of the localisation effects, not considered here, is clearly visible in the
experiment. Moreover, the used fitting parameters ρo, C and ∆ all show very
reasonable values.
ρ ρ(T) CTT
= + −
0
exp∆
0 50 100 150 200 250 3000
1
2
3
4
ρ (1
0-4 Ω
cm)
T (K)
0 2 4 6 8200
220
240
∆ (K
)
pressure (GPa)
4.5 GPa
8 GPa
Sr2.5Ca11.5Cu24O41+δ
Figure 5.5: Temperature dependence of the resistivity for a Sr2.5Ca11.5Cu24O41 even-chain spin-ladder single crystal at 4.5 GPa and 8 GPa (experimental data points after [Nagata97]). The solid line represents a fit using equation 5.8 describing transport in 1D SL's.
CHAPTER 5 Stripe formation and transport properties of underdoped cuprates
129
The expected residual resistance )()2( 2//
20 aeJb πρ h∆= for b ~ 2a ~7.6 Å,
∆ ~ 200 K and J// ~ 1400 K (the normal value for the CuO2 planes) is
ρo ~ 0.5· 10-4 Ωcm which is in good agreement with ρo ~ 0.83· 10-4 Ωcm found
from the fit. The fitted gap ∆ ~ 216 K (at 8 GPa) (figure 5.5) is close to
∆ ~ 320 K determined for the undoped SL SrCu2O3 from inelastic neutron
scattering experiments [Takano97]. In doped systems it is natural to expect a
reduction of the spin gap upon doping. Therefore the difference between the
fitted value (216 K) and the one measured in an undoped system (320 K) seems
to be quite fair. Finally the calculated fitting parameter 0103.0)2()( =∆= oAC ρπ (in units of 10-4 Ωcm/K) is to be compared with
C = 0.013 (from the 8 GPa fit on figure 5.5). Using the fitting procedure for the
two pressures 4.5 GPa (∆ ~ 219 K) and 8 GPa (∆ ~ 216 K), we have obtained a
weak suppression of the spin-gap under pressure d∆/dp ~ -1 K/GPa.
The super linear ρ(T) behaviour observed in this doped even-chain SL under
external pressure indicates, by its similarity with the S-shaped ρ(T) in
underdoped HTS, that the picture of 1D transport might be relevant to the HTS.
To investigate the possibility of using the 1D scenario for describing transport
properties of the 2D CuO2 planes of the high-Tc superconductors, it is
appropriate to compare the temperature dependency of the resistivity of a
typical underdoped high-Tc material YBa2Cu4O8 with that of the even-chain SL
compound Sr2.5Ca11.5Cu24O41. The crystal structure of the YBa2Cu4O8
compound ('124') differs substantially from that of the more common
YBa2Cu3O7 ('123'), since 124 contains double CuO chains stacked along the c-
axis and shifted by b/2 along the b axis [Karpinski88]. These chains are
believed to act as charge reservoirs, therefore they may have a strong influence
on the transport in the CuO2 planes themselves. In the 124 case, the 1D features
of this double CuO chain can be expected to induce an intrinsic doping
inhomogeneity in the neighbouring CuO2 planes thus enhancing in a natural
way the formation of the 1D stripes. A weak coupling of 1D chains to 2D
planes might be sufficient to reduce the effective dimensionality by
preferentially orienting the stripes in the CuO2 planes along the chains. But
even in pure 2D planes, without their coupling to the 1D structural elements, the
formation of the 1D stripes is possible. Using a simple scaling parameter To, a
CHAPTER 5 Stripe formation and transport properties of underdoped cuprates
130
perfect overlap of the two sets of data was found: (ρ-ρo)/ρ(To) versus T/To (with
ρo being the residual resistance) for YBa2Cu4O8 and Sr2.5Ca11.5Cu24O41
(figure 5.6). Note that ρo should be subtracted from ρ(T) since ρo may contain
contributions from several scattering mechanisms depending on the sample
quality.
0.0 0.5 1.0 1.50.0
0.5
1.0
1.5
2.0
T/To
YBa2Cu4O8
∆ = (224 ± 5) K
Sr2.5Ca11.5Cu24O41+δ at 8 GPa
∆ = (216 ± 5) K
(ρ-ρ
o)/ρ
(To)
Figure 5.6: Scaling analysis on the temperature dependence of the resistivity of the underdoped high-Tc superconductor YBa2Cu4O8 and the even-leg spin-ladder Sr2.5Ca11.5Cu24O41.
This perfect scaling of the ρ(T) data of an underdoped HTS on one side and an
even-leg spin-ladder on the other side has severe implications for the nature of
the charge transport and the scattering in the high-Tc cuprates' CuO2 layers. It
convincingly demonstrates that resistivity vs. temperature dependencies of
underdoped cuprates in the pseudo-gap regime at T < T* and even-chain SL
with a spin-gap ∆ are governed by the same underlying 1D (magnetic)
mechanism.
Early experiments on twinned high-Tc samples however, created an illusion that
all planar Cu sites in the CuO2 planes are equivalent. Recent experiments on
perfect untwinned single crystals have strongly nuanced this belief. A very
large anisotropy in the ab-plane of twin-free samples has been reported for
resistivity (ρa/ρb(YBa2Cu2O7) = 2.2 [Gagnon94, Friedman90] and
ρa/ρb(YBa2Cu4O8) = 3.0 [Bucher95]), for thermal conductivity
CHAPTER 5 Stripe formation and transport properties of underdoped cuprates
131
(κa/κb(YBa2Cu4O8) = 3-4 [Cohn98]), for superfluid density [Yu92, Wang98]
and for optical conductivity [Wang98, Tajima96]. In all these experiments,
much better metallic properties have been clearly seen along the direction of the
chains (b-axis). And what is truly remarkable, that this in-plane anisotropy can
be partly suppressed by a small (only 0.4 %) amount of Zn [Wang98], which is
known to replace copper, at least for Zn concentrations up to 4 %, only in the
CuO2 planes [Tarascon88, Xiao88] ! The latter suggests that the ab-anisotropy
can not only be explained just by assuming the existence of highly conducting
CuO-chains. Instead, the observation of the anisotropy in the transport
properties in the ab-plane for YBa2Cu4O8 [Bucher95] and YBa2Cu3O7
[Gagnon94], interpreted as a large contribution of strongly metallic Cu-O chains
ρchain(T), might be re-interpreted taking into account the fact that the in-plane
anisotropy is caused by certain processes in the CuO2 planes themselves, where
the substitution of Cu by Zn takes place. In this situation we may expect that
the chains are actually imposing certain directions in the CuO2 planes for the
formation of 1D stripes.
However, inelastic neutron scattering experiments on YBa2Cu3O7 [Dai98,
Kao99, Arai99] show evidence for the existence of rather dynamic stripes and
the observation of 1D features in the transport properties should therefore not be
limited to the Cu-O chain-direction only. Moreover, although the 1D stripes are
dynamic, no averaging of the transport properties will occur, since, even for
dynamic stripes, the charge will automatically follow the most conducing paths,
i.e. stripes, even if they are moving fast. Fitting the 1D quantum transport
model [Moshchalkov93, 98b & 99] to the in-plane ρ(T) curve for YBa2Cu4O8
(equation 5.8) results in a very nice fit [Moshchalkov98b, 99, 99b & 99c], yielding a spin-gap ( )K5224 ±=∆ (figure 5.7) were the slope of ln[(ρ-ρo)/T]
versus 1/T (see inset in figure 5.7) defines the spin gap value. Therefore, we
can conclude that the resistivity of underdoped cuprates below T* (see inset in
figure 5.7) simply reflects the temperature dependence of the magnetic correlation length DDm 11 /1 ρξ = in the even-chain SL's and the pseudo-gap is
the spin-gap formed in the 1D stripes.
CHAPTER 5 Stripe formation and transport properties of underdoped cuprates
132
4 6 8 10 12 146
7
8
9
10
11
1/T (10-3 K-1)
−−
lnρ ρ
o
T
slope ∆ = 224 K
0.0
0.1
0.2
0.3
0.4ρ
(10-4
Ωcm
)
T (K)
0 50 100 150 200 250 300
ρ ρ(T) C TT
= + −
0 exp
∆
0 200 400 600 8000246810
T (K)
ρ (1
00 µ
Ωcm
)
T *
Figure 5.7: Temperature dependence of the resistivity of a YBa2Cu4O8 single crystal (open circles); the solid line represents the fit using equation 5.8. The fit parameters were ρo = 0.024 10-4 Ωcm, C = 0.00242 10-4 Ωcm/K and ∆ = 224 K. The high-temperature data taken on another crystal [Bucher94], shown in the inset, illustrate the crossover at T* to 2D (linear behaviour). Insert (upper left): fit of experimental data using equation 5.8.
0 100 200 300 400 5000
5
10
15
20
25
30
T (K)
Ksp
in (
10-2
%)
K T K K TTD( ) ( ) exp= + −
−0 1
12
∆
YBa2Cu4O8
Figure 5.8: Knight shift data KS for the YBa2Cu4O8 system [Bucher94] fitted with equation 5.9 for 2-leg spin-ladders [Troyer94]. The resulting fitting parameters are K(0) = (0.6 ± 2) 10-2 %, K1D = (870 ± 40) 10-2 % and ∆ = (222 ± 20) K [Moshchalkov99 & 99c].
CHAPTER 5 Stripe formation and transport properties of underdoped cuprates
133
In order to substantiate these observations, we can use similar ideas in the
analysis of other physical properties. Since in underdoped cuprates the scaling
temperature To, used in the scaling of ρ(T), works equally well for resistivity as
for Knight-shift data KS [Wuyts96], these KS data can also be used for fitting
with the expressions derived from the 1D SL models. For a 2-leg SL, the
temperature dependence of the Knight shift KS should obey the following
expression [Troyer94]:
( )TTTKS∆−− exp~)( 2
1
(5.9)
Fitting the KS(T) data [Bucher94] for YBa2Cu4O8 with this expression gives an excellent result (figure 5.8) with a spin-gap ( )K20222 ±=∆ which is very
close to the value ( )K5224 ±=∆ derived from the resistivity data (see above).
Therefore, for the underdoped HTS, we have related the linear ρ(T) behaviour
above T* with quantum transport in a 2D AF Heisenberg system and the S-
shaped super-linear behaviour below T* with a 1D quantum transport model for
even-chain spin ladders (the striped phase). In the next paragraphs we will
identify the universal ρ(T) behaviour reported in chapter 3 with this 1D/2D
model, extract the spin gap ∆ and construct an experimental generic T(p) phase
diagram.
5.3.2 Application to the data
In chapter 3, the in plane resistivity ρab(T) of underdoped YBa2Cu3Ox,
(Y0.6Pr0.4)Ba2Cu3Ox and La2-xSrxCuO4 thin films was shown to exhibit a linear
ρ(T) dependence at high temperatures T > T*, a super-linear behaviour at T < T*
and an insulating resistivity at the lowest temperatures for strongly underdoped
samples. This insulating behaviour was made evident by the application of very
high magnetic fields in order to suppress superconductivity. Doping the high-Tc
materials reduces the tendency towards insulating behaviour and lowers the
crossover temperature T* such that the super linear ρab(T) makes way for the
linear region, extending to lower temperatures. These general observations are,
for the YBa2Cu3Ox compound, nicely illustrated by the composite ρab(T) plots
given in figure 5.9.
CHAPTER 5 Stripe formation and transport properties of underdoped cuprates
134
0
100
200
300
400
500
600
0
200
400 0
100
200
300
0
200
400
0 50 100 150 200 250 3000
500
1000
T (K)
ρ ( µ
Ωcm
)
x = 6.5
50 / 45 / 40 / 35 / 30 / 20 / 10 / 0 T
x = 6.7
x = 6.8
x = 6.95
x = 6.45
YBa2Cu3Ox
Figure 5.9: Resistivity versus temperature for the YBa2Cu3Ox thin films studied in this work at zero-field and at 10, 20, 30, 35, 40, 45 and 50 T.
These in-plane resistivities were, both for the YBa2Cu3Ox and the
(Y0.6Pr0.4)Ba2Cu3Ox compound, shown to scale onto one universal curve
(figures 5.10 and 5.11). From these plots, a perfect scaling in regimes I (linear
part) and II (curved, super-linear ρ(T)) was observed for the zero-field curves.
In the insulating regime (III), the scaling is of less good quality.
The perfect scaling of the metallic in-plane resistivities for these compounds is a
strong indication that one scattering mechanism is dominant for the strongly
underdoped up to the near-optimally doped samples. Only the energy scale (the
scaling parameter ∆ and the crossover temperature T* ≈ 2∆) varies upon
doping.
CHAPTER 5 Stripe formation and transport properties of underdoped cuprates
135
x = 6.45x = 6.5x = 6.7x = 6.8x = 6.95
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
T/∆
ρ ρρ ρ
−−
o
o∆
T*
∆
x = 6.4 / 6.45 / 6.5 / 6.7 / 6.8 / 6.95
YBa2Cu3Ox
III II I
Figure 5.10: Scaled zero field and 50 T ρ(T) for the YBa2Cu3Ox films (x = 6.4, 6.45, 6.5, 6.7, 6.8 and x = 6.95). The regions of different ρ(T) behaviour are indicated as well as the energy scale ∆ and the crossover temperature T* ≈ 2∆; ρo is the residual resistivity and ρ∆ is the resistivity at T = ∆. Two additional lines represent the predictions from the 1D/2D quantum transport model for T < T* (eq. 5.8) and T > T* (eq. 5.7).
0.0 0.5 1.0 1.5 2.0 2.5 3.0-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
ρ ρρ ρ
−−
o
o∆
T/∆
Y0.6Pr0.4Ba2Cu3Ox
III II
∆
x = 6.7 / 6.85 / 6.95
x = 6.7x = 6.85x = 6.95
Figure 5.11: Scaled zero field and 50 T ρ(T) data for the (Y0.6Pr0.4)Ba2Cu3Ox thin films (x = 6.85 and 6.95). For the x = 6.7 sample, only the 45 T data were experimentally accessible. The regions of different ρ(T) behaviour are indicated as well as the energy scale ∆; ρo is the residual resistivity and ρ∆ is the resistivity at T = ∆. The additional line is the prediction from the 1D/2D quantum transport model for T < T* (eq 5.8).
CHAPTER 5 Stripe formation and transport properties of underdoped cuprates
136
In the framework of the previous paragraph, it is almost evident to try to
correlate this "dominant process" with the magnetic scattering mechanisms in
1D and 2D, introduced there.
For T > T*, where short range AF fluctuations are seen in inelastic neutron
scattering experiments, the resistivity is observed to have a linear temperature
dependency (region I). This regime is thus perfectly described by equation 5.7
for quantum transport in a 2D Heisenberg system with the inelastic length
determined by the magnetic (2D) correlation length.
For T < T* ≈≈ 2∆∆, were charge stripes can develop, the resistivity is observed to
have a super-linear temperature dependence (region II). This regime should
then be accurately described by equation 5.8, describing quantum transport in a
1D striped material with again the inelastic length determined by the magnetic
(1D) correlation length. To check this, the ρ(T) curve described by this
expression for 1D conduction (equation 5.8) was plotted together with the data
in figures 5.10 and 5.11. A perfect overlap with the data is established up to
slightly above T/∆ = 1. The scaling of the data was performed such that the data
fall onto the universal ρ(T) = ρo+ CTexp(-∆/T) curve with C = exp(1) =
2.7183… . In that way, the scaling parameters necessary to obtain the collapsing ρρab(T) traces directly yield estimates for the spin pseudo-gap ∆∆ within this model for transport in a 1D striped case.
Also for the tensile strained La1.9Sr0.1CuO4 ultra-thin film (150 Å deposited on
SrTiO3, sample C), showing an S-shaped in-plane resistivity ρab(T), this ρab(T)
can be fitted with expression 5.8 for 1D quantum transport (figure 5.12). The
agreement is quite reasonable for temperatures TMI < T < ∆ (as was observed
also on the YBa2Cu3Ox and the (Y0.6Pr0.4)Ba2Cu3Ox compounds) and a spin gap
of ∆ = (125 ± 20) K was obtained.
In figure 5.13, the estimates for the spin pseudo-gap ∆ and the crossover
temperature T* ≈ 2∆ are, for the YBa2Cu3Ox system, plotted versus the oxygen
content x. Like T*, the spin-gap decreases upon doping, approaching the critical
temperature Tc near the optimally doped case. This is a well documented trend
for the pseudo-gap and is not restricted to the YBa2Cu3Ox or the
(Y0.6Pr0.4)Ba2Cu3Ox compounds (for a review, see [Timusk99]).
CHAPTER 5 Stripe formation and transport properties of underdoped cuprates
137
0 50 100 150 200 250 3000
200
400
600
800
T (K)
La1.9Sr0.1CuO4
ρ (µ
Ωcm
)
tensile strained150 Å on SrTiO3
∆ = (125 ± 20) K
Figure 5.12: Temperature dependence of the in-plane resistivity ρab(T) for a tensile-strained 150 Å La1.9Sr0.1CuO4 ultra-thin film fitted with the expression for 1D quantum transport. The arrow indicated the gap ∆.
6.2 6.4 6.6 6.8 7.00
100
200
300
400
500
T (
K)
x
YBa2Cu3Ox
∆ from ρab(T)
T*≈2∆ from ρab(T)T*
∆
Figure 5.13: Spin gap ∆ and crossover temperature T* ≈ 2∆ for the for the YBa2Cu3Ox thin films, as derived from the scaling of their in-plane resistivities ρab(T) with the curve for 1D quantum transport.
CHAPTER 5 Stripe formation and transport properties of underdoped cuprates
138
6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 7.00
100
200
300
∆(K
)
x
YBa2Cu3Ox
ρab(T) on thin filmsthis work Wuyts et. al
ρab(T) on crystalstwinned detwinned
KS on 17O on 63Cu
6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 7.00
100
200
300
∆(K
)
x
YBa2Cu3Ox
ρab(T) on thin filmsthis work Wuyts et. al
ρab(T) on crystalstwinned detwinned
KS on 17O on 63Cu
Figure 5.14: The spin gap ∆ of YBa2Cu3Ox versus oxygen content x, from the scaling of the ρab(T) data with the curve for 1D quantum transport for the thin films in this work (open diamonds) and a direct fit on the films from [Wuyts94 & 96] (down triangles), twinned crystals [Ito93] (up triangles) and de-twinned crystals [Gagnon94] (squares). The spin gap obtained from a fit of the Knight-shift on 17O [Martindale96] (filled diamonds) and on 63Cu and 17O [Takigawa91] (circles) is also added.
A last, crucial, check for the 1D conductivity model [Moshchalkov98b & 99] is
the direct comparison of our values for the pseudo-gap with estimates from the
literature. In figure 5.14, we have re-plotted our ∆(x) data on thin films (open
diamonds) together with estimates from resistive measurements on other
YBa2Cu3Ox thin films [Wuyts94 &96], twinned [Ito93] and de-twinned
[Gagnon94] single crystals. Within the error bars, these data agree well.
Additionally, we have plotted estimates of the pseudo-gap as derived from
CuO2-plane 17O and 63Cu Knight-shift measurements on aligned powders
[Takigawa91, Martindale96]. Also these data, although obtained with a totally
different technique, yield estimates for the spin-gap that are in good agreement
with our ∆(x) data. This proves that the 1D quantum transport model
[Moshchalkov98b & 99], used to describe the transport in underdoped cuprates
at T < T* in a 1D striped manner, not only agrees qualitatively, but also yields
values for the pseudo spin-gap ∆ that agree well with independent estimates.
Although this correspondence is quite convincing, it should be mentioned that
experimental techniques probing charge excitations (like ARPES, quasi-particle
CHAPTER 5 Stripe formation and transport properties of underdoped cuprates
139
relaxation measurements and tunnelling experiments) yield values of the
pseudo-gap ∆p, that are significantly higher (about a factor 2) than the spin-
excitation gap ∆s as observed in NMR and INS experiments [Timusk99,
Mihailovic99]. In the 1D quantum transport model, where the inelastic length
is presumed to be dominated by the magnetic correlation length, the agreement
of our data with the gap-value determined from NMR experiments then seems to
be natural.
The only dissonance in this discussion comes from the often-cited 89Y NMR
data on underdoped YBa2Cu3Ox reported by Alloul and co-workers [Alloul88 &
89]. These Knight-shift data were shown earlier to scale very well, using the
same scaling temperature To that was derived from the scaling of ρab(T)
[Wuyts96]. This was interpreted as a strong indication that the magnetic
structure and the opening of the spin-gap are relevant also for transport
measurements, motivating the development of the 1D/2D quantum transport
model [Moshchalkov93, 98b, 99]. This argument still stands. However, when
fitting the expression for the Knight shift KS(T) (as in figure 5.8) to these data,
the resulting values for the pseudo-gap are about a factor 2 higher than the gap
values determined from resistivity measurements or data on in-plane 17O and 63Cu Knight-shift measurements on aligned powders [Takigawa91,
Martindale96]. The origin of this deviation is not clear but could be due to the
use of non-aligned powders [Alloul88 & 89] or possible differences between
NMR measurements probing inter-plane 89Y on one side and in-plane 17O and 63Cu on the other side.
5.4 Stripe ordering at low temperatures
At low temperatures, T < TMI, the metallic behaviour of the resistivity in regions
I and II transforms into an insulating, diverging, ρ(T) (region III). The
diverging high-field ρ(T) data were shown to agree better with the ln(1/T)
divergence than with a simple power law T-α. Although the origin of such a
logarithmic divergence is still strongly debated (see chapter 3), it is interesting
to analyse our data for the normal-state resistivity and Hall-effect within the
framework of the model considering stripe formation in the CuO2 plane.
CHAPTER 5 Stripe formation and transport properties of underdoped cuprates
140
In this charge-stripe picture [Cheong91, Tranquada97, Arai99, Emery97b,
Moshchalkov98b & 99], dynamic metallic [Noda99, Ichikawa99, Tajima99]
stripes are thought to dominate the transport properties. So, within this model,
one expects a strong influence on the transport properties when, for some
reason, the 1D charge stripes are fragmented or pinned. Then the movement of
the charge carriers is restricted to these fixed 1D paths (in contrast to the
dynamic stripes) and the resistivity is reported to increase [Noda99, Ichikawa99,
Lavrov99]. Moreover, in the presence of stripe fragmentation, charge carriers
have to hop to another metallic stripe, passing the intercalating Mott-insulator,
also resulting in an increased resistivity. The occurrence of fragmentation or
pinning of the otherwise dynamic stripes can be understood when one realises
that any scattering process (or local scattering centre) yielding an inelastic
length smaller than Lφ~ξm1D will destroy the fragile regime of a striped CuO2
plane. By inserting the temperature dependency of this Lφ into the conductivity
expression with the proper dimensionality, one is then, in principle, able to
describe the low-temperature ln(1/T) divergence of the high-field resistivity (see
below).
One possible type of pinning centre is the crystallographic disorder in the CuO2
plane, in the form of dislocations. These dislocations will also alter the local
electronic and magnetic structure in the plane and at low temperatures, when the
stripes are less mobile, they can be expected to pin the magnetic domain walls
formed by the charge stripes (figure 5.2). Moreover, in the case of strong
pinning, stripe fragmentation is predicted to occur [Kivelson98].
Experimentally, the pinning of charge stripes has been seen by neutron
diffraction experiments on Nd-doped and pure La2-xSrxCuO4 [Tranquada97 &
97b]. The striking observation from these data is that, although the
incommensurate features (i.e. the stripes) are almost identical, the scattering in
the pure, near optimally doped, (La2-xSrx)CuO4 system is inelastic (dynamic
stripes) whereas in the (La1.6-xNd0.4Srx)CuO4 system elastic scattering is
observed, corresponding to static stripes. In general, pinning of these stripes is
correlated with the onset of an increasing resistivity [Noda99], although stripe
pinning has been found in underdoped samples that are metallic (but close to the
CHAPTER 5 Stripe formation and transport properties of underdoped cuprates
141
metal-insulator transition) [Ichikawa99], suggesting stripe fragmentation to be
as important as pinning for the creation of an insulating state.
So, for dynamic stripes, the resistivity will be quasi 1D metallic and the Hall
response in a magnetic field will be finite since dynamic charge stripes are able
to respond to the transverse electric field acting on the charge carriers. For
pinned stripes that are not fragmented, the resistivity can be expected to
remain essentially metallic since the 1D metallic wires remain. However, such
a reduced mobility of the stripes can be expected to have a noticeable influence
on the Hall effect. When the stripes are pinned, they cannot properly react to
the Lorentz force on the charge carriers and only a reduced Hall field (and thus
Hall resistivity ρyx) is built up. However, in the presence of stripe fragmentation or inter-stripe hopping, also an effect will be present due to the
charge inter-stripe hopping across the Mott-insulator phase. This will result in
an insulating longitudinal resistivity and a small but finite Hall effect.
Recently, based on Hall effect and X-ray measurements on Nd-doped
La2-xSrxCuO4 crystals [Noda99], it was argued that the Hall conductivity σxy
(equation 5.10) is related to the inverse stripe order.
( )
( )H
HRBRH
ab
oH
ab
zH
xx
yx
xy
yxxy
xx
22222 ρ
µ
ρρ
ρ
ρρ
ρσ ≈=≈
+=
(5.10)
In order to check this idea, we have combined our high-field ρab(T) and RH(T)
data above and below Tc to calculate the Hall conductivity σxy using
equation 5.10. The results are summarised in figure 5.15, for the samples
showing a pronounced divergence of the low-temperature resistivity.
From the plots in figure 5.15, it is clear that, once the resistivity starts increasing
on lowering temperature (at T < TMI), also the Hall conductivity goes down
rapidly and hence, according to the analysis made in [Noda99], stripe order in
these underdoped YBa2Cu3Ox and Y0.6Pr0.4Ba2Cu3Ox samples increases.
However, a significant difference with the data published on the Nd-doped
La2-xSrxCuO4 crystals [Noda99] becomes clear when comparing them with our
high-field ρab(T) and RH(T) data (figures 4.5 to 4.10 in chapter 4). In our data,
the decreasing Hall conductivity σxy is almost completely due to the strongly
CHAPTER 5 Stripe formation and transport properties of underdoped cuprates
142
diverging longitudinal resistivity ρab(T) whereas the Hall response RH(T)
remains finite (and approximately constant) down to the lowest temperatures in
our experiments.
T (K)
σ xy
(arb
. uni
ts)
Y0.6Pr0.4Ba2Cu3O6.7
Y0.6Pr0.4Ba2Cu3O6.85 Y0.6Pr0.4Ba2Cu3O6.95
YBa2Cu3O6.45 YBa2Cu3O6.5
0 50 100 150
( )( )
σµ
ρxy
o H
ab H
HH R
≈2
0 50 100 150
TMI
TMITMI
TMI TMI
Figure 5.15: The off-diagonal conductivity σxy, calculated by combining the Hall coefficient RH and the in-plane resistivity ρab at 40 tesla (equation 5.10). The σxy,(T) data are presented for the YBa2Cu3O6.45, YBa2Cu3O6.5, Y0.6Pr0.4Ba2Cu3O6.7, Y0.6Pr0.4Ba2Cu3O6.85 and Y0.6Pr0.4Ba2Cu3O6.95 thin films. The arrows indicate TMI where the resistivity starts increasing on lowering temperature and the x-axis is drawn at σxy = 0.
When combining this result with the discussion about dynamic versus pinned
stripes, it becomes clear that, at low temperatures, the charge stripe picture can
only be brought in agreement with our normal-state transport data by assuming
stripe fragmentation or inter-stripe hopping effects. This causes an effective
recovery of the two-dimensional (2D) regime. By inserting the temperature
dependence of the inelastic length Lφ, of the scattering mechanisms working in
the intercalating insulating phase, into the conductivity expression for 2D
quantum transport (equation 5.7), one can calculate the low-temperature ln(1/T)
CHAPTER 5 Stripe formation and transport properties of underdoped cuprates
143
divergence of the high-field resistivity. For example, the inelastic length for
electron-electron or electron-phonon scattering, Lφ ~ 1/T€α [Abrikosov88],
combined with equation 5.7 for 2D quantum transport gives an ln(1/T)
correction to the low-temperature resistivity. Also electron interference effects
in the 2D weak localisation theory predict an ln(1/T) behaviour. Moreover, this
2D weak localisation model also agrees with our finding of a constant Hall
coefficient RH(T) at low temperatures.
5.5 Construction of the T(p) phase diagram
The construction of a so-called generic T(p) phase diagram, describing the
superconducting and normal-state transport properties of both the YBa2Cu3Ox
and the (Y0.6Pr0.4)Ba2Cu3Ox compounds, requires the combination of our high-
field transport data (chapter 3) and the estimates for the carrier concentration
from the Hall effect (chapter 4). This experimental phase diagram, already
constructed in chapter 4, can now be re-investigated in the framework of the
1D/2D quantum transport model [Moshchalkov93, 98b & 99]. Of course,
regardless of this interpretation, the experimental T(p) phase diagram, including its crossover lines remains valid.
In figure 5.16, the spin-gap ∆ as derived by applying the equation for 1D
quantum transport is plotted versus p, the fraction of holes per Cu atom in the
CuO2 plane. From this plot, it is clear that the ∆(p) data for YBa2Cu3Ox and
(Y0.6Pr0.4)Ba2Cu3Ox coincide very well, contrasting with the previous
disagreement of the T*(x) ≈ 2∆(x) crossover line for the two compounds. Thus,
the energy scale ∆(p) (for the 1D quantum transport) is well described by the
carrier density in the CuO2 plane. Moreover, since within the quantum transport
model [Moshchalkov93, 98b & 99], applied here for YBa2Cu3Ox and
(Y0.6Pr0.4)Ba2Cu3Ox, this energy scale equals the pseudo spin-gap, also this
pseudo spin-gap ∆ is well described by the carrier density in the CuO2 plane.
CHAPTER 5 Stripe formation and transport properties of underdoped cuprates
144
0
5
10
15
20
0
50
100
150
200
250
∆(K
) ∆(m
eV)
0 0.05 0.10 0.15 0.20 0.25
p (holes/Cu-atom in the plane)
YBa2Cu3Ox
Y0.6Pr0.4Ba2Cu3Ox
0
5
10
15
20
0
5
10
15
20
0
50
100
150
200
250
0
50
100
150
200
250
∆(K
) ∆(m
eV)
0 0.05 0.10 0.15 0.20 0.250 0.05 0.10 0.15 0.20 0.25
p (holes/Cu-atom in the plane)
YBa2Cu3Ox
Y0.6Pr0.4Ba2Cu3Ox
YBa2Cu3Ox
Y0.6Pr0.4Ba2Cu3Ox
Figure 5.16: The pseudo-gap ∆ for the YBa2Cu3Ox and the Y0.6Pr0.4Ba2Cu3Ox thin films, derived from scaling the ρab(T) data to the expression for 1D quantum transport, plotted versus the fraction of holes per Cu atom in the CuO2 plane.
Having established the relevance of the carrier density as a suitable parameter
for creating a phase diagram for the normal-state properties within the 1D/2D
quantum transport model, the experimental T(p) phase diagram can now be re-
interpreted in the framework of this model. In figure 5.17, the T(p) phase
diagram is re-plotted with some extra indications to facilitate the discussion.
Region I, where a metallic linear temperature dependence of the resistivity is
observed (T > T*), is accurately described by the expression for a 2D Heisenberg system where short range AF fluctuations are observed in inelastic
neutron scattering experiments. When an underdoped high-Tc cuprate is cooled
below T*, an S-shaped ρ(T) develops, that can be scaled onto a single universal
curve for both compounds. This curve is accurately described by the model for
transport in a 1D striped regime (region II) and yields values for the spin-gap
that agree well with estimates in literature. This gap is well described as a
function of the carrier density.
CHAPTER 5 Stripe formation and transport properties of underdoped cuprates
145
0 0.10 0.15 0.20 0.250.05
T (
K)
p (#/in-plane Cu)
metallic2D Heisenberg
metallic1D striped
T*
200
300
400
0
100 TcTMI
0
100
I
II
IIISC
SC
≈ ≈TMI
Tc
II
AF
YBa2Cu3Ox
Y0.6Pr0.4Ba2Cu3Ox
III
0 0.10 0.15 0.20 0.250.050 0.10 0.15 0.20 0.250.05
T (
K)
p (#/in-plane Cu)
metallic2D Heisenberg
metallic1D striped
T*
200
300
400
0
100
200
300
400
0
100 TcTMI
0
100
0
100
II
IIII
IIIIIISC
SC
≈ ≈TMI
Tc
IIII
AF
YBa2Cu3Ox
Y0.6Pr0.4Ba2Cu3Ox
IIIIII
Figure 5.17: Generic T(p) phase diagram for the YBa2Cu3Ox (diamonds, solid line) and the Y0.6Pr0.4Ba2Cu3Ox (circles, dotted line) thin films. Indicated are the 2D/1D crossover temperature T* (filled symbols), the superconducting critical temperature Tc (open symbols) and the boundary TMI between the metallic and the insulating-alike regimes for ρ(T). All are plotted versus the fraction of holes per Cu-atom in the CuO2 plane. The data for Y0.6Pr0.4Ba2Cu3Ox were shifted down by 100 K
The 1D striped regime is defined by 4 boundaries in the T(p) diagram. At low doping levels, the bulk antiferromagnetic order is recovered and the stripes
disappear. At high doping levels, the distance between stripes is expected to
decrease, charges leak into the Mott insulator phase between the stripes and as a
result, the charge stripes collapse completely (see also figure 5.3). At high temperatures, stripe meandering is expected to destroy the 1D regime,
recovering the 2D regime with antiferromagnetic fluctuations. At low
CHAPTER 5 Stripe formation and transport properties of underdoped cuprates
146
temperatures T < TMI, stripe pinning, fragmentation and inter-stripe hopping
effects establish a 2D insulating regime (region III). In the T(p) diagram, the
onset of this insulating-alike regime is indicated by TMI, below which the
resistivity increases with lowering temperature. The TMI(p) line for the
Y0.6Pr0.4Ba2Cu3Ox compound lies significantly higher than its equivalent for the
YBa2Cu3Ox samples (figure 5.17). This might, within the stripe scenario, be
due to the presence of additional disorder caused by the Y/Pr substitution,
resulting in stronger stripe pinning and fragmentation effects. At low temperatures T < Tc, the onset of macroscopic coherence between the so-
called pre-formed pairs [Emery97b & 99b] is predicted to result in the recovery
of bulk superconductivity (in the absence of high magnetic fields).
5.6 Conclusions
The universal ρ(T) behaviour in the underdoped YBa2Cu3Ox and
Y0.4Pr0.4Ba2Cu3Ox thin films, reported in chapter 3, is a strong indication of one
single scattering mechanism being dominant over the whole underdoped regime
in the Y123 system. Only the energy scale (the scaling parameter ∆ and the
crossover temperature T* ≈ 2∆) varies upon doping.
Any model trying to explain the extraordinary features of the normal-state
transport properties of the high-Tc's (linear ρ(T) at high temperatures, S-shaped
ρ(T) at intermediate temperatures and logarithmically diverging ρ(T) and
constant Hall coefficient RH at low temperatures) should also account for the
complex magnetic phase diagram for these high-Tc cuprates. In the underdoped
region of this diagram, at moderate temperatures T* < T < To, short-range
antiferromagnetic correlations develop in the CuO2 planes. Moreover, an
increasing amount of experimental and theoretical indications are in favour of
the existence of dynamic one-dimensional charge stripes in the CuO2 planes at
T < T*, acting as domain walls for the antiferromagnetic fluctuations. These
local charge inhomogeneities (1D charge stripes) will confine the AF regions,
resulting in the formation of a pseudo spin-gap at temperatures far above the
superconducting critical temperature Tc.
CHAPTER 5 Stripe formation and transport properties of underdoped cuprates
147
It is then tempting to assign the origin of the dominant scattering mechanism for
charge transport to the microscopic magnetic ordering in the planes of the high-
Tc cuprates. The importance of the CuO2 planes for the transport properties is a
widely documented feature of the high-Tc cuprates. The confinement of the
charge carriers in these planes reduces the dimensionality for charge transport to
2 dimensions (or less) and makes the conductivity σ in such 2D metallic system
to be dominated by quantum transport. In this case the approach based on the
following three basic assumptions [Moshchalkov93, 98b & 99] can be used: (i)
the dominant scattering mechanism in HTS in the whole temperature range is of
magnetic origin; (ii) the specific temperature dependence of the resistivity ρ(T)
can be described by the inverse quantum conductivity σ-1 with the inelastic
length Lφ being fully controlled by the magnetic correlation length ξm, and
finally, (iii) the proper 1D or 2D expressions should be used for calculating the
quantum conductivity with Lφ ~ ξm.
At high temperatures T* < T < To, in the 2D Heisenberg regime, the
combination of the expressions for the 2D spin correlation length with the
proper expression for the quantum resistance gives a linear temperature
dependence of the resistivity, due to the mutual cancellation between the
logarithmic ρ(ξm) dependence and the exponential temperature dependence of
ξm. This result is in perfect agreement with our finding of a linear ρ(T) at high
temperatures for all measured underdoped YBa2Cu3Ox and Y0.4Pr0.4Ba2Cu3Ox
thin films.
At intermediate temperatures TMI < T < T*, in the 1D striped regime, inelastic
neutron scattering experiments show evidence for the existence of dynamic
stripes and the observation of 1D features in the transport properties should
therefore not be limited to the Cu-O chain-direction only. Moreover, although
the 1D stripes are dynamic, no averaging of the transport properties will occur,
since, even for dynamic stripes, the charge will automatically follow the most
conducing paths, i.e. stripes, even if they are moving fast. So, in transport
experiments the magnetic correlation length ξm1D of a dynamic insulating stripe
permanently imposes the constraint Lφ~ξm1D, thus providing a persistent 1D
character of the charge transport in underdoped cuprates. Inserting this inelastic
length in the expression for 1D quantum conductivity yields an S-shaped ρ(T)
CHAPTER 5 Stripe formation and transport properties of underdoped cuprates
148
that perfectly describes the resistivity data obtained on the even-chain spin-
ladder compound Sr2.5Ca11.5Cu24O41. This compound, due to its specific
crystalline structure, definitely contains a 1D spin-ladder and therefore its
resistivity along the ladder direction should indeed obey the expression for 1D
quantum transport.
As a next step, a convincing scaling was pointed out between the resistivity of
this 1D spin-ladder compound and a typical underdoped high-Tc material,
YBa2Cu4O8, demonstrating that the resistivity versus temperature dependencies
of underdoped cuprates in the pseudo-gap regime at T < T* and even-chain SL
with a spin-gap ∆ are governed by the same underlying 1D (magnetic)
mechanism. This magnetic origin of the scattering of the charge carriers is
further confirmed by the fact that the scaling temperature To, used in the scaling
of ρ(T), works equally well for resistivity as for Knight-shift data KS and by the
nice fit of these KS data with the expressions derived from the 1D SL models.
The ρ(T) data of a La1.9Sr0.1CuO4 thin film under tensile epitaxial strain and
YBa2Cu3Ox and (Y0.6Pr0.4)Ba2Cu3Ox thin films with varying oxygen content,
scaled onto one universal curve, are all perfectly described by the expression for
1D quantum transport. The values of the spin-gap ∆, estimated from this fitting,
are in agreement with an independent determination of ∆ from resistive
measurements on other YBa2Cu3Ox thin films, twinned and de-twinned single
crystals. Moreover, they agree with estimates of the pseudo-gap as derived
from CuO2-plane 17O and 63Cu Knight-shift measurements on aligned powders.
In the 1D quantum transport model, where the inelastic length is presumed to be
dominated by the magnetic correlation length, the agreement of our data with
the gap-value determined from NMR experiments seems only natural. This
proves that our analysis, describing the transport in underdoped cuprates at
T < T* by taking into account the presence of 1D stripes, not only agrees
qualitatively, but also yields values for the pseudo spin-gap ∆ that agree well
with independent estimates.
At low temperatures T < TMI, the metallic behaviour of the resistivity at high
temperatures transforms into an insulating-alike, diverging, ρ(T) that was shown
to agree with a ln(1/T) divergence. In this chapter, our data for the normal-state
resistivity and Hall-effect was confronted with the possibility of stripe
CHAPTER 5 Stripe formation and transport properties of underdoped cuprates
149
formation in the CuO2 plane. In this charge-stripe picture, dynamic, metallic
stripes are thought to dominate the transport properties. So, within this model,
one expects a strong influence on the transport properties when, for some
reason, the 1D charge stripes are fragmented or pinned. Such a fragmentation
or pinning of the else dynamic stripes will be caused by any scattering process
(or local scattering centre) yielding an inelastic length smaller than Lφ ~ ξm1D
(disorder). It was found that, at low temperatures, the charge stripe picture can
only be brought in agreement with our normal-state transport data by assuming stripe fragmentation or inter-stripe hopping effects.
These processes invoke a strong influence of the intercalating Mott insulator
phase on the charge transport, yielding a 2D insulating resistivity and a finite Hall response. By inserting the temperature dependence of the inelastic length
Lφ, of the scattering mechanisms working in the intercalating insulating phase,
into the conductivity expression for 2D quantum transport (equation 5.7), one
can calculate the low-temperature ln(1/T) divergence of the high-field
resistivity. For example, the inelastic length for electron-electron or electron-
phonon scattering, Lφ ~ 1/T€α [Abrikosov88], combined with the expression for
2D quantum transport gives an ln(1/T) correction to the low-temperature
resistivity. Also electron interference effects in the 2D weak localisation theory
predict an ln(1/T) behaviour. Moreover, this 2D weak localisation model also
agrees with our finding of a constant Hall coefficient RH(T) at low temperatures.
The construction of a so-called generic T(p) phase diagram, describing the
superconducting and normal-state transport properties of both the YBa2Cu3Ox
and the (Y0.6Pr0.4)Ba2Cu3Ox compounds was possible by combining our high-
field transport data with the estimates for the carrier concentration from the Hall
effect. This experimental phase diagram, was in this chapter re-investigated in
the framework of the 1D/2D quantum transport model. It was shown that the
energy scale ∆(p) (for the 1D quantum transport) is well described by the carrier
density in the CuO2 plane. Moreover, since within the quantum transport model
this energy scale equals the pseudo spin-gap, also this pseudo spin-gap ∆ is well
described by the carrier density in the CuO2 plane.
The 1D striped regime is defined by four boundaries in the T(p) diagram. At
low doping levels, the bulk antiferromagnetic order is recovered and the stripes
CHAPTER 5 Stripe formation and transport properties of underdoped cuprates
150
disappear. At high doping levels, the distance between stripes is expected to
decrease and the Mott insulator phase between stripes collapses. At high temperatures, stripe meandering is expected to destroy the 1D regime,
recovering the 2D regime with antiferromagnetic fluctuations. At low temperatures T < TMI, stripe pinning, fragmentation and inter-stripe hopping
effects establish a 2D insulating regime. In the T(p) diagram, the onset of this
insulating regime is indicated by TMI, below which the resistivity increases with
lowering temperature. The TMI(p) line for the Y0.6Pr0.4Ba2Cu3Ox compound lies
significantly higher than its equivalent for the YBa2Cu3Ox samples , which
might, within the stripe picture, be due to the presence of additional disorder
due to the Y/Pr substitution, resulting in stronger stripe pinning and
fragmentation effects. At low temperatures T < Tc, the onset of macroscopic
coherence between the so-called pre-formed pairs [Emery97b & 99b] is
predicted to result in the recovery of bulk superconductivity.
SUMMARY
151
Summary
The normal-state transport properties of a set of YBa2Cu3Ox and
(Y0.6Pr0.4)Ba2Cu3Ox thin films with various oxygen concentrations and strained
La1.9Sr0.1CuO4 epitaxial thin films were measured in zero field and at very high pulsed magnetic fields. This dual-track approach - lowering the critical
temperature Tc by changing the hole content p on one side and using very high
magnetic fields on the other side - allows us to cover the whole underdoped to
optimally doped region of the T-p (temperature versus hole content in the CuO2
plane) phase diagram (figure 1) and thus explore the normal-state transport
properties for these compounds -usually studied at temperatures above the
critical temperature Tc- for the first time at temperatures T < Tc.
Figure 1: The properties of the high-Tc cuprates vary with temperature (vertical axis) and doping of the CuO2 planes [Batlogg2000].
SUMMARY
152
As a first step, inspired by earlier work [Wuyts94 & 96], it was shown that the
metallic zero-field normal-state resistivity ρρ(T) above Tc for various levels of
hole doping -both for the YBa2Cu3Ox and the (Y0.6Pr0.4)Ba2Cu3Ox members of
the cuprate superconductors- can be scaled onto one single universal curve.
An energy scale ∆, the resistivity ρ∆ at T = ∆ and the residual resistivity ρo are
used as scaling parameters. The inclusion of the residual resistivity ρo in this
scaling allowed a scaling that is of a better quality than the earlier analysis
[Wuyts94 & 96]. The universal curve exhibits three regions with a qualitatively
different ρ(T) behaviour:
(I) a region of linear ρ(T) at high temperatures T > T*,
(II) a super-linear ρ(T) at intermediate temperatures T < T*
(III) a low temperature insulating-alike ρ(T) regime at T < TMI.
The low-temperature regime (III) is to a large extent masked by the onset of
superconductivity at T = Tc. The distinct features in the temperature
dependence of the metallic zero-field resistivity ρ(T) of YBa2Cu3Ox and
(Y0.6Pr0.4)Ba2Cu3Ox in regions I and II are universal for all the reported curves,
the only difference is the temperature scale ∆∆ at which these features occur.
As a second step, the magnetoresistivity of these YBa2Cu3Ox and
(Y0.6Pr0.4)Ba2Cu3Ox epitaxial thin films was measured in pulsed high magnetic fields up to 50 T. This allowed us to investigate the normal-state ρρ(T) behaviour at temperatures below the critical temperature Tc. It was shown
that these high field data in the metallic regime scale reasonably good with the
universal ρ(T/∆) curve. However, the ρ(T) data in the low-temperature
insulating regime (of diverging resistivity) do not scale satisfactorily when
using the same scaling parameters as in the zero-field scaling.
The existence of a universal ρ(T/∆) curve for the metallic normal-state
resistivity was interpreted as a strong indication that one single mechanism
dominates the scattering of the charge carriers in these materials. This
mechanism then determines the energy scale ∆, which is dependent upon
doping.
SUMMARY
153
These experiments revealed the ground state at T < Tc of YBa2Cu3Ox (for
x ≤ 6.8), (Y0.6Pr0.4)Ba2Cu3Ox (for all levels of oxygen content) and the tensile
strained La1.9Sr0.1CuO4 ultra-thin film to be insulating, which leads to an increase of the low-temperature resistivity upon decreasing temperature.
The insulating-alike ρ(T) behaviour at low temperatures was shown not to obey
a simple activated or hopping mechanism. It was argued that, most probably,
spin-flip scattering in the framework of the Kondo model or the existence of a
narrow metallic impurity band with localised edges do not play an important
role in producing a divergent resistivity upon lowering temperatures. Our low-
temperature data were shown to have a ln(1/T) divergence in high magnetic fields, as was first observed in the cuprate La2-xSrxCuO4 by Ando [Ando95].
Models predicting a divergent resistivity, which are not conflicting with our
experimental data, include the phenomenological c-axis resistivity model
[Zha96], a model accounting for electron interactions in 2D disordered systems
[Altshuler80], the bipolaron model [Alexandrov97], the 2D Luttinger model
[Anderson91, Clarke95], the 2D Anderson weak localisation theory [Lee85] or
the pinning of the dynamic charge stripes [Kivelson98, Noda99, Ichikawa99,
Tranquada97 & 97b].
The comparison of the transport data on epitaxially strained La1.9Sr0.1CuO4
ultra-thin films with data on samples without epitaxial strain and our
YBa2Cu3Ox and (Y0.6Pr0.4)Ba2Cu3Ox samples demonstrated that epitaxial strain
probably influences not only the doping of the CuO2 planes but also the disorder
scattering.
In order to facilitate further interpretation of the normal-state resistivity data,
Hall-effect measurements on the same YBa2Cu3Ox and Y0.6Pr0.4Ba2Cu3Ox thin
films at temperatures extending to below the critical temperature Tc were
reported. These measurements were performed at very high pulsed magnetic
fields in order to fully access the normal state (H > Hc2).
The Hall measurements yield a signal that is three orders of magnitude smaller
than in standard magnetoresistivity experiments. The successful
accomplishment of these measurements was possible by a careful
SUMMARY
154
implementation of vibration insulation and shielding for electric interference,
combined with a low contact resistance to the sample.
From the measurements of the high-field Hall-resistivity ρyx(H), the field
dependence of the Hall coefficient RH(H) at various temperatures was
calculated. By concentrating on the normal-state part of the data (H > Hc2), the
Hall coefficient RH(T) curves at fixed (high) field were constructed. These RH(T)
curves show a transition to the normal-state value of RH as temperature is
increased. This transition shifts to lower temperatures in high fields. At these
high fields, the low-temperature normal-state RH(T) is recovered, exhibiting only a very weak temperature dependence. By combining the RH(T) curves
with the high-field ρab(T) curves for the same samples, it is clear that at these
high fields the normal state is entered unambiguously down to temperatures
T << Tc. Thus, by using very high magnetic fields, we were able to access the
normal-state Hall-effect in the YBa2Cu3Ox and Y0.6Pr0.4Ba2Cu3Ox thin films in
the temperature range down to temperatures T << Tc.
The quasi temperature independence of the low-temperature Hall-coefficient RH is in sharp contrast with the strongly divergent ρρab(T) observed in these compounds. The approximately temperature-independent
Hall coefficient, is an important test for the existing models already explaining a
low-temperature divergence of ρab(T). Among the models, still applicable for
the description of both ρab(T) and RH(T), the weak localisation-model explains
both the constant RH at low temperatures and the ln(1/T) divergence of the high-
field ρab(T) data.
A further combination of the Hall data with the ρab(T) resistivity curves then
allows the derivation of the Hall-angle. The Hall-angle is predicted, and widely
observed, to obey a quadratic dependence upon temperature, in the temperature
range from Tc up to room temperature, with a constant offset that is related to
the temperature independent scattering by magnetic impurities. Also here, the
existing studies at T > Tc are extended to temperatures T << Tc by our high-field
data. The region of diverging Hall-angle (mostly observed around T ~ Tc) was
significantly shifted to lower temperatures in this high field limit. Although our
experimental temperature window is limited, a tentative agreement with the
often cited quadratic temperature dependence was seen. The zero-temperature
SUMMARY
155
intercept of the Hall-angle cotθH , giving information about the presence of
magnetic scatterers, was, in the samples with the lowest oxygen content, shown
to be elevated, supposedly due to antiferromagnetic fluctuations. Also in the
Y/Pr substituted samples, a higher offset of the Hall-angle was observed. This
suggests that the presence of the magnetic Pr atoms, apart from the reduction of
the critical temperature Tc, causes a significant temperature-independent
contribution to the scattering of the charge carriers. This is in agreement with
the high residual resistivity ρo and the small ρ290 K/ρo resistivity ratio.
The carrier density nH (and thus also p, the fraction of holes per Cu-atom in
the CuO2 plane) calculated from the Hall-data enabled us to construct a generic T(p) phase diagram for the YBa2Cu3Ox and Y0.6Pr0.4Ba2Cu3Ox compounds.
The universal ρρ(T) behaviour in the underdoped YBa2Cu3Ox and
Y0.4Pr0.4Ba2Cu3Ox thin films, reported in chapter 3, strongly supports the idea
that a single scattering mechanism is dominant over the whole underdoped
regime in the Y123 system. Bearing in mind the complex magnetic phase
diagram, with short-range AF correlation and a pseudo gap showing up at
temperatures far above the superconducting critical temperature Tc and
reminding the strong indications of stripe formation in the CuO2 planes (an
inhomogeneous intercalation of AF insulating regions and metallic hole-rich
stripes), it is tempting to assign the origin of this dominant scattering mechanism to the microscopic magnetic and charge ordering.
Therefore, our high-field transport data (both the resistivity and the Hall-effect)
were used to investigate the effect of the short-range antiferromagnetic
fluctuations and possible stripe formation in the CuO2 plane on the normal
state transport properties. In this charge-stripe scenario [Emery97b & 99b],
dynamic metallic [Ichikawa99, Noda99, Tajima99] stripes are thought to
dominate the transport properties. Since mobile carriers in this case are
expelled from the surrounding Mott-insulator phase into the stripes, the latter
then provide the lowest resistance paths. This makes the transport properties
very sensitive to the formation of the stripes, both static and dynamic. To check
this idea, an existing model [Moshchalkov93, 98b], describing transport both in the 2D Heisenberg regime (above T*) as in the 1D striped regime
SUMMARY
156
(below T*) where the pseudo gap develops, was used as a framework for the
interpretation of our high-field normal-state transport data.
The proposed approach [Moshchalkov93, 98b] is based on three basic
assumptions:
1. the dominant scattering mechanism in HTS in the whole
temperature range is of magnetic origin;
2. the specific temperature dependence of the resistivity ρ(T)
can be described by the inverse quantum conductivity σ-1
with the inelastic length Lφ being fully controlled, (via a
strong interaction of holes with Cu2+ spins) by the magnetic
correlation length ξm, and, finally,
3. the proper 1D or 2D expressions should be used for
calculating the quantum conductivity with Lφ ~ ξm.
At high temperatures T* < T < To, in the 2D Heisenberg regime, the
combination of the expressions for the 2D spin correlation length with the
proper expression for the quantum resistance gives a linear temperature dependence of the resistivity, due to the mutual cancellation between the
logarithmic ρ(ξm) dependence and the exponential temperature dependence of
ξm. This result is in perfect agreement with our finding of a linear ρ(T) at high
temperatures for all measured underdoped YBa2Cu3Ox and Y0.4Pr0.4Ba2Cu3Ox
thin films.
At intermediate temperatures Tc < T < T*, in the 1D striped regime, although
the 1D stripes are dynamic, no averaging of the transport properties will occur,
since, even for dynamic stripes, the charge will automatically follow the most
conducing paths, i.e. stripes, even if they are moving fast. So, in transport
experiments the magnetic correlation length ξm1D of a dynamic insulating stripe
permanently imposes the constraint Lφ~ξm1D, thus providing a persistent 1D
character of the charge transport in underdoped cuprates. Inserting this inelastic
length in the expression for 1D quantum conductivity yields an S-shaped ρρ(T) that perfectly describes the resistivity data obtained on the even-chain spin-
ladder compound Sr2.5Ca11.5Cu24O41. This compound, due to its specific
crystalline structure, definitely contains a 1D spin-ladder and therefore its
SUMMARY
157
resistivity along the ladder direction should indeed obey the expression for 1D
quantum transport.
As a next step, a convincing scaling was found between the resistivity of this
1D spin-ladder compound and a typical underdoped high-Tc material,
YBa2Cu4O8, demonstrating that the resistivity versus temperature dependencies of underdoped cuprates in the pseudo-gap regime at T < T* and even-chain SL with a spin-gap ∆∆ are governed by the same underlying 1D (magnetic) mechanism. This magnetic origin of the scattering of the
charge carriers is further confirmed by the fact that the scaling temperature To,
used in the scaling of ρ(T), works equally well both for resistivity and for
Knight-shift data KS. A good fit of these KS data is achieved by using the
expressions derived from the 1D spin-ladder models.
The ρ(T) data of a La1.9Sr0.1CuO4 thin film under tensile epitaxial strain and
YBa2Cu3Ox and (Y0.6Pr0.4)Ba2Cu3Ox thin films with varying oxygen content,
scaled onto one universal curve, are all perfectly described by the expression for 1D quantum transport. The values of the spin-gap, found from this
fitting, are in agreement with independent estimates from resistive
measurements on other YBa2Cu3Ox thin films, twinned and de-twinned single
crystals. Moreover, they agree with estimates of the pseudo-gap as derived
from CuO2-plane 17O and 63Cu Knight-shift measurements on aligned powders.
In the 1D quantum transport model, where the inelastic length is presumed to be
dominated by the magnetic correlation length, the agreement of our data with
the gap-value determined from NMR experiments seems to be quite natural.
This proves that our approach, describing the transport in underdoped cuprates at T < T* in a 1D striped manner, not only agrees qualitatively, but also yields values for the pseudo spin-gap ∆∆ that agree well with independent estimates.
In the final chapter of this work, our data for the normal-state resistivity and
Hall-effect were analysed by considering the possibility of stripe pinning by
disorder. The metallic behaviour of the resistivity at high temperatures,
transforms at low temperatures T < TMI into an insulating-alike, diverging, ρ(T)
that was shown to agree with a ln(1/T) law, whereas the Hall-coefficient is
temperature independent at low temperatures. Within the charge-stripe picture,
SUMMARY
158
one expects a strong influence on the transport properties when, for some
reason, the 1D charge stripes are fragmented or pinned. Such a fragmentation
or pinning of the else dynamic stripes will be caused by any scattering process
(or local scattering centre) yielding an inelastic length smaller than Lφ ~ ξm1D. It
was found that, at low temperatures, the charge stripe picture can only be
brought into agreement with our normal-state transport data by assuming stripe fragmentation, promoting inter-stripe hopping effects. These processes
invoke a strong influence of the intercalating Mott insulator phase on the charge
transport, yielding a (2D) insulating resistivity and a finite Hall response.
By inserting the temperature dependence of the inelastic length Lφ, of the
scattering mechanisms working in the intercalating insulating phase, into the
conductivity expression for 2D quantum transport, one can calculate the low-
temperature ln(1/T) divergence of the high-field resistivity. For example, the
inelastic length for electron-electron or electron-phonon scattering, Lφ ~ 1/T€α
[Abrikosov88], combined with the expression for 2D quantum transport gives
an ln(1/T) correction to the low-temperature resistivity. Also electron
interference effects in the 2D weak localisation theory predict an ln(1/T)
behaviour. Moreover, this 2D weak localisation model also agrees with our
finding of a constant Hall coefficient RH(T) at low temperatures.
The construction of a so-called experimental generic T(p) phase diagram,
describing the superconducting and normal-state transport properties of both the
YBa2Cu3Ox and the (Y0.6Pr0.4)Ba2Cu3Ox compounds was possible by combining
our high-field transport data with the estimates for the carrier concentration
from the Hall effect. This experimental phase diagram (see figure 2 below, to
be compared with the schematic phase diagram on figure 1 above) was
investigated in the framework of the 1D/2D quantum transport model
[Moshchalkov93, 98b]. It was shown that the energy scale ∆(p) (for the 1D
quantum transport) is well described by the carrier density in the CuO2 plane.
Moreover, since within the quantum transport model this energy scale equals
the pseudo spin-gap, also this pseudo spin-gap ∆ is well described by the
carrier density in the CuO2 plane.
The 1D striped regime is defined by four boundaries in the T(p) diagram.
At low doping levels, the bulk antiferromagnetic order is recovered and the
SUMMARY
159
stripes disappear. At high doping levels, the distance between stripes is
expected to decrease and the Mott insulator phase between stripes collapses. At
high temperatures, stripe meandering results in the destruction of the 1D
regime, thus recovering the 2D regime with antiferromagnetic fluctuations. At
low temperatures T < TMI, stripe pinning, fragmentation and inter-stripe
hopping effects establish a 2D insulating regime. In the T(p) diagram, the onset
of this insulating-alike regime is indicated by TMI, below which the resistivity
increases with lowering temperature. The TMI(p) line for the Y0.6Pr0.4Ba2Cu3Ox
compound lies significantly higher than its equivalent for the YBa2Cu3Ox
samples , which might, within the stripe picture, be due to the presence of
additional disorder caused by the Y/Pr substitution, which enhances stripe
pinning and fragmentation effects. For the YBa2Cu3Ox system, the TMI(p) line
is below the Tc(p) line for almost all levels of hole doping and thus the
insulating state at zero-field is effectively masked by the onset of
superconductivity. For the Y0.6Pr0.4Ba2Cu3Ox compound, the TMI(p) line is
above the Tc(p) line for all levels of oxygen content and the insulating tendency
can therefore already be seen in the zero-field ρ(T) measurements above Tc. At temperatures T < Tc, the onset of macroscopic coherence between the so-
called pre-formed pairs [Emery97b & 99b] is predicted to result in the recovery
of bulk superconductivity.
The critical temperature Tc,mid(p) and the boundary TMI between the metallic and
insulating-alike ρ(T) regimes do not coincide for the YBa2Cu3Ox and
Y0.6Pr0.4Ba2Cu3Ox compounds; although showing a qualitatively similar
behaviour, shifted to a lower Tc for the Y0.6Pr0.4Ba2Cu3Ox system. The fact that,
for the same hole concentration p, the critical temperature Tc is lower in the
Y0.6Pr0.4Ba2Cu3Ox system in comparison with YBa2Cu3Ox, is an indication that
the magnetic pair-breaking in the (Y/Pr)Ba2Cu3Ox compound might play an
important role in the reduction of the superconducting critical temperature Tc.
So, by performing magnetoresistivity and Hall effect measurements in very high
magnetic fields on a selection of high-Tc superconducting thin films with
varying level of hole doping, we were able to complement the schematic T-p phase diagram in figure 1 with three experimental phase boundaries: T*(p)
describing a 2D to 1D crossover, the superconducting critical temperature Tc(p)
SUMMARY
160
and TMI(p) marking the onset of an insulating-alike behaviour. The normal-state generic phase diagram was discussed in terms of a 2D metallic
Heisenberg regime, a metallic 1D stripe region where the pseudo gap develops
and a low-temperature insulating regime.
0 0.10 0.15 0.20 0.250.05
T (
K)
p (#/in-plane Cu)
metallic2D Heisenberg
metallic1D striped
T*
200
300
400
0
100 TcTMI
0
100
I
II
IIISC
SC
≈ ≈TMI
Tc
II
AF
YBa2Cu3Ox
Y0.6Pr0.4Ba2Cu3Ox
III
0 0.10 0.15 0.20 0.250.050 0.10 0.15 0.20 0.250.05
T (
K)
p (#/in-plane Cu)
metallic2D Heisenberg
metallic1D striped
T*
200
300
400
0
100
200
300
400
0
100 TcTMI
0
100
0
100
II
IIII
IIIIIISC
SC
≈ ≈TMI
Tc
IIII
AF
YBa2Cu3Ox
Y0.6Pr0.4Ba2Cu3Ox
IIIIII
Figure 2: Generic T(p) phase diagram for the YBa2Cu3Ox (diamonds, solid line) and the Y0.6Pr0.4Ba2Cu3Ox (circles, dotted line) thin films. Indicated are the 2D/1D crossover temperature T* (filled symbols), the superconducting critical temperature Tc (open symbols) and the boundary TMI between the metallic and the insulating-alike regimes for ρ(T). All are plotted versus the fraction of holes per Cu-atom in the CuO2 plane. The data for the Y0.6Pr0.4Ba2Cu3Ox system are shifted down by 100 K.
APPENDICES
161
Appendices
Sample preparation and characterisation
Any proper characterisation of the normal state transport properties of high-Tc
superconductors -as is the aim of this work- requires the use of high-quality
single crystalline samples. This leaves the choice to either single-crystals or
epitaxial thin films. From the viewpoint of magnetoresistance and Hall effect
measurements, the use of epitaxial thin films is desirable for reasons of a higher
resistance and Hall-signal and the possibility of patterning the films in a well-
defined geometry.
In this appendix, the preparation of these thin films by high-pressure DC-sputtering will be discussed in appendix A. The procedure for changing the oxygen content of these thin films to a certain nominal value is discussed in
appendix B, while the patterning and contacting of the films is treated in
appendix C. Finally, the techniques employed for the characterisation of the
structural and superconducting properties are introduced in appendix D.
Appendix A: Thin film deposition by sputtering
The preparation of high quality thin films of YBa2Cu3Ox requires two
constraints to be fulfilled. The first constraint involves the stability of the
YBa2Cu3Ox compound during deposition. From the thermodynamic phase diagram (oxygen pressure
2OP versus temperature T), presented in figure A.1, it
is clear that the YBa2Cu3Ox compound is stable only in a narrow T-2OP
window. Combining this with the need for relatively high deposition
temperatures, in order for the deposited atoms to have enough mobility to eventually form a crystalline film, oxygen partial pressures of a few hPa are
required.
APPENDICES
162
This short discussion shows that ultra high vacuum (UHV) techniques like
molecular beam epitaxy (MBE) are only suited to deposit thin films of these
copper oxides when using differential pumping or a local flux of oxygen and
accurate control of the different sources [Locquet94]. Locquet and co-workers
have prepared the strained La1.9Sr0.1CuO4 ultra-thin films (of typical
thickness of 100 Å) used in this work by molecular beam epitaxy with block-by-block deposition [Locquet94]. The choice of the substrate - SrLaAlO4
(SLAO) or SrTiO3 (STO) - enabled them to induce compressive (SLAO) or
tensile (STO) strain in the ab-plane [Locquet96, 98 & 98b, Sato97], see also
paragraph 1.3.3.
0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5
900 800 700 600 500 400103
102
101
100
10-1
10-2
P O2 (
Pa)
1000/T (K-1)
T (oC)
YBa2Cu3OxY2BaCuO5
+BaCuO2
+CuO2
x = 6.0
x = 6.5x = 6.9deposition
Figure A.1: Thermodynamic phase diagram (oxygen partial pressure 2OP versus
temperature T), indicating the stability region for the YBa2Cu3Ox compound [Hammond89]. The shaded ellipse indicates the region of oxygen pressure and substrate temperature where the actual deposition takes place.
For the preparation of the YBa2Cu3Ox and (YPr)Ba2Cu3Ox thin films studied
in this work, it was chosen to use the high-pressure DC-sputtering technique.
For this purpose, a dedicated setup was developed, implementing the necessary
precautions to perform a high temperature deposition in an oxygen atmosphere
APPENDICES
163
[Wagner99, Teniers99]. Sputtering involves the creation of a plasma in the
sputtering gas (in our case 2.5 hPa of oxygen) by a glow-discharge maintained
by an electric field (280 V over a distance of 2 cm) between the substrate and
the target material, a water-cooled ceramic disk of stoichiometric YBa2Cu3Ox
or Y0.6Pr0.4Ba2Cu3Ox (99.99 %, 50 mm diameter, 3 mm thickness,
Superconductive Components Inc.). In this plasma, ionised +2O molecules are
created and accelerated to the target material -kept at a negative potential- and
their collision with the target releases neutral atoms from the stoichiometric
disk. These atoms condense on the substrate (heated to a temperature of
approximately 840 oC on a grounded holder) and form a thin film at a
deposition rate of 20 Å/min.
To avoid the deposition of off-stoichiometric films due to the different
sputtering yield of the four constituents of the YBa2Cu3Ox compound, all new
targets were pre-sputtered 24 hours before the first use and 45 minutes before
each deposition. This procedure creates an "altered" layer that is depleted of the
atoms with the highest sputtering yield. As such, the sputtered atoms will
regain the desired stoichiometry. The possible back-sputtering of the
deposited film by negatively charged ions is prevented by using a very high
oxygen partial pressure (2.5 hPa) that neutralises and thermalises these ions.
An important parameter in the deposition of thin films is the choice of the
substrate. In table A.2 the lattice parameters and thermal expansion
coefficients are summarised for YBa2Cu3Ox and commonly used substrates. In
this work SrTiO3, cut perpendicular to the c-axis, was chosen as a substrate
because of the good matching of both the lattice parameters and the thermal
expansion coefficient. Moreover, this substrate material does result in
interdiffusion like MgO. For reasons of thermal homogeneity (and hence the
homogeneity of the thin film) rather thick substrates (1.2 mm), surrounded by
platelets of Al2O3, were glued to the heater with silver epoxy.
APPENDICES
164
Material ab-plane symmetry
Lattice parameters
(Å)
Thermal expansion coefficient
(10-6/oC)
YBa2Cu3Ox orthorhombic a = 3.823 b = 3.886 c = 11.65
12
SrTiO3 (100) square a = 3.905 11
MgO (100) square a = 4.2 13
LaAlO3 trigonal a = 5.375
SrLaAlO4 square a = 3.76
Table A.2: Lattice parameters and thermal expansion coefficients for YBa2Cu3Ox and various substrate materials. Also the symmetry in the ab-plane of the materials is indicated.
The deposition procedure starts with 45 minutes of pre-sputtering at 2.5 hPa
oxygen pressure with the sample shielded. The actual deposition at ~ 20 Å/min
is also carried out at 2.5 hPa with a substrate temperature of 840 oC. After this,
an oxygen annealing is performed at 700 hPa and 700 oC for 15 minutes, then at
600 oC for another 15 minutes after which a slow cooldown (30 minutes) to
room temperature is performed, still keeping an oxygen pressure of 700 hPa. In
this way, an epitaxial thin film, optimally loaded with oxygen, is obtained.
Appendix B: Procedure to vary the oxygen content
As can already be seen on the thermodynamic phase diagram of figure A.1, the
YBa2Cu3Ox compound exists with variable oxygen contents (6 < x < 7),
depending on temperature and the partial pressure of oxygen in the
environment. In this diagram of oxygen partial pressure versus temperature,
phase lines of constant oxygen content for the YBa2Cu3Ox compound can be
drawn (figure B.1). In principle, at room temperature, this compound has no
stable oxygen content since these ambient conditions are far above the x = 6.9
line and samples with a reduced oxygen content will take up oxygen. The
dynamics is however so slow that at a time scale of weeks one has an effective
constant oxygen concentration.
APPENDICES
165
A simple method for changing the oxygen content in the YBa2Cu3Ox compound
was already introduced above (App. A): a slow cooldown at a constant oxygen
pressure, crossing the P(T) lines of constant oxygen content. When the desired
P(T) line is reached, the temperature is kept constant for a certain time after
which a fast quenching of the sample to room temperature is performed. Since,
during this quenching procedure, multiple P(T) lines have to be crossed, the
homogeneity of the oxygen distribution in the sample is strongly dependent
upon this quenching step.
In this work a refined method was used [Maenhoudt95], in which the P(T)
diagram is traversed as indicated in figure B.1.
108
106
102
100
10-2
10-4
104
0 200 400 600 800 1000
6.1
6.5
6.86.6
6.9
6.2
6.7
6.36.4
P O2 (
Pa)
T (oC)
Figure B.1: Phase diagram (oxygen partial pressure 2OP versus temperature T),
adapted from [Gallagher87], indicating the P(T) phase lines of constant oxygen content for the YBa2Cu3Ox compound.
As a first step, the optimally oxygenated thin film sample -covered in an
optimally oxygenated bulk YBa2Cu3Ox container- is put in a quartz tube that is
evacuated to a pressure of 10-6 mbar and consecutively put at approximately
13 hPa of oxygen pressure and heated up to the P(T) line corresponding to the
desired nominal oxygen content (x = 6.6 in figure B.1). After keeping the
system at these conditions for a few hours, a slow, computer controlled,
cooldown (1 oC/min) is initiated, carefully following the P(T) phase line of
APPENDICES
166
constant oxygen concentration (figure B.1). At low temperatures and pressure,
when the dynamics of oxygen desorption is slow, the system is quenched to
room temperature (figure B.1).
Several P(T) phase diagrams were proposed in literature, all qualitatively in
agreement with the one shown in figure B.1 [Gallagher87]. In this work it was chosen to use the
2OP -T diagram of Tetenbaum and co-workers [Tetenbaum89]
which, after comparison with other phase diagrams [Maenhoudt95], gives a
good correspondence between the critical temperatures of thin film samples
prepared according to this scheme and bulk samples [Maenhoudt95] (see
figure 3.4 for the comparison of our thin films with published data on bulk
samples). Therefore, the values for the oxygen content indicated for our thin
film samples is the nominal value from the oxygen desorption procedure.
Appendix C: Patterning and contacting of thin films
In order to obtain a well defined geometry for the magnetoresistivity and Hall-
effect measurements on the YBa2Cu3Ox and Y0.6Pr0.4Ba2Cu3Ox thin films, they
were etched into the pattern shown in figure C.1. The dimensions of the bridges
were chosen in a way to ensure sufficiently large signals.
Before the actual etching, gold contacts were evaporated (thickness ~ 500 Å)
and annealed at 500oC in an oxygen flow. This resulted in contacts with a
resistance below 1 Ω; no influence of this procedure on the superconducting
transition was observed.
As a first step in the etching procedure, a thin layer of photo resist (OLIN
HPR504) is spinned onto the film, which is then hardened by baking at 90oC for
about 12 minutes. While covered with the proper chromium-on-glass mask, the
film is then illuminated by UV light during 10 seconds. After that, the resist
layer is developed (OLIN HPRD407) and the film is rinsed with distilled water
and dried with nitrogen gas. The etching itself is performed with HNO3 during
10 to 30 seconds, depending on the thickness of the film. After checking
whether the etching was successful, the resist layer is removed with acetone.
APPENDICES
167
1 mm
11
22
33 44 55
Figure C.1: The pattern etched onto the YBa2Cu3Ox and Y0.6Pr0.4Ba2Cu3Ox thin films. The dimensions of the bridges are (1) 3000 x 300 µm, (2) 1000 x 50 µm, (3) 100 x 10 µm, (4) 50 x 5 µm and (5) 30 x 5 µm.
For the measurements in pulsed magnetic fields, bridge 2 was selected
(figure C.1) and cut out of the film using a diamond wire-saw. Due to its length
to width ratio (l/w = 20), this narrow 1000 x 50 µm strip has the advantage of
giving a high signal in transport experiments (important for measurements in
pulsed fields), even when a small current is applied.
The disadvantage of these dimensions is that even a small over-etching can
result in deviations of the transport properties. However, for all films studied in
this work, no broadening of the resistive superconducting transition, a sign of
local degradation, was observed after etching.
Finally, the small 3 x 3 mm piece of the film, containing strip 2, was mounted
on a plexi-glass holder and contacted with gold wire (0.1 mm diameter) using
silver paint.
The thickness of the films was determined by atomic force microscopy (AFM)
on the patterned strips (see table C.2). The edge of the strip was traversed
several times at different positions along the strip. The indicated thickness t is
the average of these measurements and the error was estimated to be
approximately 100 Å.
APPENDICES
168
Sample composition t (Å)
#1 YBa2Cu3Ox 1770
#2 YBa2Cu3Ox 1300
#3 YBa2Cu3Ox 2300
#4 Y0.6Pr0.4Ba2Cu3Ox 850
Figure C.2: The thickness of the films studied in this work as determined by AFM measurements. More information on these samples is given in paragraph 3.1
Appendix D: Characterisation
D.1 Structural characterisation by X-ray diffraction (XRD)
The structural quality of the thin films was checked by x-ray diffraction (XRD)
measurements. This allows the characterisation of the texturisation of the thin
films. Since the wavelength of the x-rays is of the same order as the lattice
parameters, they are diffracted by the lattice and the condition for constructive interference is given by Bragg's law λθ nd =sin2 with d being the interlayer
spacing, θ the angle of incidence, λ the wavelength of the x-rays and n an
integer number.
θ
2θX-raysource
Detector
Thin film
Figure D.1: The Bragg-Brentano XRD geometry with fixed and equal distance between the sample and the detector and the sample and the source. The sample has an angular degree of freedom (θ) while the detector moves in a circle (2θ.)
The x-ray diffraction measurements in this work were carried out in the Bragg-
Brentano geometry (figure D.1) using a Rigaku 12 kW rotating anode x-ray
diffractometer using Cu-Kα radiation (wavelength 1.5408 Å, mostly used at
7.5 kW). Figures D.2 and D.3 show a θθ-2θθ scan for a YBa2Cu3Ox and a
APPENDICES
169
Y0.6Pr0.4Ba2Cu3Ox thin film, in which the sample rotates with an angle θ while
the detector performs a 2θ rotation. In that way, all directions are scanned in
which constructive interference can give rise to intensity peaks (Bragg's law).
0 10 20 30 40 50 60 700
5
10
15
20
25
(007
)
(200
) SrT
iO3 (
006)
(005
)
(004
)
(100
) SrT
iO3 (
003)
(002
)
(001
)
(008
)
2θ (degree)
Inte
nsity
(a.
u.)
0 20 40 60
log(
inte
nsity
)
2θ (degree)
YBa2Cu3O7 on SrTiO3
(007
)
(200
) SrT
iO3 (
006)
(005
)
(004
)
(100
) SrT
iO3 (
003)
(002
)
(001
)
(008
)
Figure D.2: XRD spectrum of a YBa2Cu3Ox thin film on SrTiO3, measured during a θ€-2θ scan. The inset shows the same data, with a logarithmic intensity scale.
0 10 20 30 40 50 60 700
5
10
15
20
25
30
(007
)
(200
) SrT
iO3 (
006)
(005
)
(004
)
(100
) SrT
iO3 (
003)
(002
)
(001
)
Inte
nsity
(a.
u.)
2θ (degree)
Y0.6Pr0.4Ba2Cu3O7 on SrTiO3
0 20 40 60
2θ (degree)
log(
inte
nsity
)
(007
)
(200
) SrT
iO3 (
006)
(0
05)
(004
)
(100
) SrT
iO3 (
003)
(0
02)
(001
)
(008
)
Figure D.3: XRD spectrum of a Y0.6Pr0.4Ba2Cu3Ox thin film on SrTiO3, measured during a θ -2θ scan. The inset shows the same data, with a logarithmic intensity scale.
APPENDICES
170
From these figures (D.2 and D.3) it is clear that mainly (00l) peaks are visible,
indicating a strong c-axis orientation of the material. The absence of any other
significant peaks proves the single phase character of the films. The small
peaks in between the (00l) peaks (three orders of magnitude smaller than the
(00l) peaks) were shown to originate either from the substrate or the sample
holder (aluminium & plasticine), since they are observed also in measuring the
XRD pattern of a virgin substrate.
The distribution of the crystallographic directions around the c-axis is
characterised by a rocking-scan in which the detector is kept fixed at an
intensity peak while the sample is moved around the original angle θ. The
rocking-curves for the same YBa2Cu3Ox and Y0.6Pr0.4Ba2Cu3Ox thin films are
shown in figures D.4 and D.5, respectively. The crystallographic orientations in
the material have a narrow distribution around the c-axis with a full width at
half maximum (FWHM) of 0.22o for the YBa2Cu3Ox and 0.24o for the
Y0.6Pr0.4Ba2Cu3Ox thin films. This indicates that our YBa2Cu3Ox and
Y0.6Pr0.4Ba2Cu3Ox thin films are epitaxial, apart from possible ab-twinning
effects due to the small a-b difference.
18.5 19.0 19.5 20.00
5
10
15
20(005)-peak rocking curve2θ = 38.56o
FWHM = 0.22o
θ (degree)
Inte
nsity
(a.
u.)
YBa2Cu3O7 on SrTiO3
Figure D.4: Rocking curve around the (005) line for the YBa2Cu3Ox thin film of figure D.2.
APPENDICES
171
18.5 19.0 19.5 20.00
2
4
6
8
10(005)-peak rocking curve2θ = 38.61o
FWHM = 0.24o
Y0.6Pr0.4Ba2Cu3O7 on SrTiO3
θ (degree)
Inte
nsity
(a.
u.)
Figure D.5: Rocking curve around the (005) line for the Y0.6Pr0.4Ba2Cu3Ox thin film of figure D.3.
Also the La1.9Sr0.1CuO4 ultra thin films, in which epitaxial strain was induced by
depositing them on SrLaAlO4 (compressive strain) or SrTiO4 (tensile strain),
were studied by XRD [Locquet94, 96, 96b, 98 & 98b]. The θ -2θ scans for
these films only show (00l) peaks -indicating their good epitaxy and c-axis
orientation- and finite size peaks -suggesting a film roughness of ± 1 unit cell.
Structural refinements allow to estimate the lattice parameters and to show that
the tensile and compressive strain basically keep the volume of the unit cell
constant.
D.2 Superconducting properties
The quality of the as-deposited, fully oxygenated, YBa2Cu3Ox and
Y0.6Pr0.4Ba2Cu3Ox thin films was checked by measuring the superconducting
transition by AC susceptibility measurements. This was done with a 77.7 Hz
AC field of amplitude 3.6 G generated in an excitation coil coupled to a
compensation coil separated from the detection coil by the (unpatterned) thin
film. Any diamagnetic response to this small magnetic field changes the
coupling between these coils. The difference between the (now unbalanced)
compensation and detection coils is proportional to the AC susceptibility. This
APPENDICES
172
method involves a much stricter quality check than resistive R(T) measurements
since it involves the whole film whereas R(T) measurements only probe the first
superconducting path (percolation). For our optimally oxygenated
YBa2Cu3O6.95 films a χAC transition width of less than 1.5 K is typical (see
figure D.6).
90 91 92 93 94 95
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
T (K)
χ ac / |
χ ac(
90 K
)|
YBa2Cu3Ox thin film3.6 G @ 77.7 Hz
χ′′
χ′
Figure D.6: The AC susceptibility as a function of temperature for a YBa2Cu3O6.95 thin film. The measurement was carried out with an excitation of 3.6 G at a frequency of 77.7 Hz; χac' and χac'' denote the in-phase and out-of-phase signal.
After patterning and depositing gold contacts, the superconducting transition
was checked by measuring the resistive R(T) transition. For the samples
discussed in this work, these R(T) data and the resulting values for Tc are
summarised in paragraphs 3.1 and 3.2.
BIBLIOGRAPHY
173
Bibliography
A A.A. Abrikosov, "Fundamentals of the theory of metals", (North-Holland Amsterdam) 1988.
G. Aeppli et al., Science 278 (1997) 1432.
M. Affronte, J. Marcus, C. Escribe-Filippini, A. Sulpice, H. Rakoto, J.M. Broto, J.C. Ousset, S. Askenazy and A.G.M. Jansen, Phys. Rev. B 49 (1994) 3502.
A.S. Alexandrov, Pis'ma Zh. Eksp. Teor. Fiz. 46 Prilozh. (1987) 128.
A.S. Alexandrov, Phys. Rev. B 38 (1988) 925.
A.S. Alexandrov and A.B. Krebs, Sov. Phys. Usp. 35 (1992) 345.
A.S. Alexandrov, A.M. Bratkovsky and N.F. Mott, Phys. Rev. Lett. 72 (1994) 1734.
A.S. Alexandrov, Physica C 282-287 (1997) 269.
H. Alloul, P. Mendels, G. Collin and P. Monod, Phys. Rev. Lett. 61 (1988) 746.
H. Alloul, T. Ohno and P. Mendels, Phys. Rev. Lett. 63 (1989) 1700.
C.C. Almasan, S.H. Han, K. Yoshiara, M. Buchgeister, B.W. Lee, L.M. Paulius, D.A. Gajewski and M.B. Maple, Physica B 199-200 (1994) 288.
C.C. Almasan, G.A. Levin, C.N. Jiang, T. Stein, D.A. Gajewski, S.H. Han and M.B. Maple, Physica C 282-287 (1997) 1129.
B.L. Altshuler and A.G. Aronov, Phys. Rev. Lett. 44 (1980) 1288.
N.H. Andersen, M. von Zimmermann, T. Frello, M. Käll, D. Mønster, P.-A. Lindgård, J. Madsen, T. Niemöller, H.F. Poulsen, O. Schmidt, J.R. Schneider, Th. Wolf, P. Dosanjh, R. Liang and W.N. Hardy, Physica C 317 (1999) 259.
P.W. Anderson, Phys. Rev. 109 (1958) 1492.
P.W. Anderson, Mater. Res. Bull. 8 (1973) 153.
P.W. Anderson and Z. Zou, Phys. Rev. Lett. 60 (1988) 132.
P.W. Anderson, Phys. Rev. Lett. 67 (1991) 2092.
P.W. Anderson, J. Phys.: Cond. Matter 8 (1996) 10083.
P.W. Anderson, T.V. Ramakrishnan, S. Strong and D.G. Clarke, Phys. Rev. Lett. 77 (1996) 4241.
Y. Ando, G.S. Boebinger, A. Passner, T. Kimura and K. Kishio, Phys. Rev. Lett. 75 (1995) 4662.
BIBLIOGRAPHY
174
Y. Ando, G.S. Boebinger, A. Passner, R.J. Cava, T. Kimura, J. Shimoyama and K. Kishio, Proc. 10th Anniversary HTS Workshop (1996) Houston.
Y. Ando, G.S. Boebinger, A. Passner, K. Tamasaku, N. Ichikawa, S. Uchida, M. Okuya, T. Kimura, J. Shimoyama and K. Kishio, J. Low Temp. Phys. 105 (1996) 867.
Y. Ando. G.S. Boebinger, A. Passner, N.L. Wang, C. Geibel and F. Steglich, Phys. Rev. Lett. 77 (1996) 2065.
Y. Ando, G.S. Boebinger, A.Passner, N.L. Wang, C. Geibel, F. Steglich, I.E. Trofimov and F.F. Balakirev, Phys. Rev. B 56 (1997) R8530.
Y. Ando, G.S. Boebinger, A. Passner, N.L. Wang, C. Geibel, F. Steglich, T. Kimura, M. Okuya, J. Shimoyama, K. Kishio, K. Tamasaku, N. Ichikawa and S. Uchida, Physica C 282-287 (1997) 240.
Y. Ando, G.S. Boebinger, A. Passner, L.F. Schneemeyer, T. Kimura, M. Okuya, S. Watauchi, J. Shimoyama, K. Kishio, K. Tamasaku, N. Ichikawa and S. Uchida, cond-mat/9908190.
M. Arai, T. Nishijima, Y. Endoh, T. Egami, S. Tajima, K. Tomimoto, Y. Shiohara, M. Takahashi, A. Garret and S. M. Bennington, Phys. Rev. Lett. 83 (1999) 608.
N.W. Ashcroft and N.D. Mermin, "Solid State Physics", Saunders Company (Philadelphia) 1976.
M. Azuma, Z. Hiroi, M. Takano, K. Ishida and Y. Kitaoka, Phys. Rev. Lett. 73 (1994) 3463.
B J. Bardeen, L.N. Cooper and J.R. Schrieffer, Phys. Rev. 108 (1957) 1175.
J. Bardeen and M.J. Stephen, Phys. Rev. 140A (1965) 1197.
B. Batlogg, H.Y. Hwang, H. Takagi, R.J. Cava, H.L. Kao and J. Kwo, Physica C 235-240 (1994) 130.
B. Batlogg and Ch. Varma, Phys. World 13 (2000) 33.
J.G. Bednorz and K.A. Müller, Z. Phys. B 64 (1986) 188.
R. Benischke, T. Weber, W.H. Fietz, J. Metzger, K. Grube, T. Wolf and H.Wühl, Physica C 203 (1992) 293.
C. Berthier, M.-H. Julien, O. Bakharev, M. Horvatic and P. Ségransan, Physica C 282-287 (1997) 227.
H.A. Bethe, Z. Phys. 71 (1931) 205.
R. Beyers and T.M. Shaw, Sol. St. Phys. 42 (1989) 135.
A. Bianconi, N.L. Saini, T. Rossetti, A. Lanzara, A. Perali, M. Missori, H. Oyanagi, H. Yamaguchi, Y. Nishihara and D.H. Ha, Phys. Rev. B 54 (1996) 12018.
BIBLIOGRAPHY
175
A. Bianconi, A. Valletta, A. Perali and N.L. Saini, Sol. St. Comm. 102 (1997) 369.
A. Bianconi, A. Valletta, A. Perali and N.L. Saini, Physica C 296 (1998) 269.
R.J. Birgenau, D.R. Gabbe, H.P. Jenssen, M.A. Kastner, P.J. Picone, T.R. Thurston, G. Shirane, Y. Endoh, M. Sato, K. Yamada, Y. Hidaka, M. Oda, Y. Enomoto, M. Suzuki and T. Murakami, Phys. Rev. B 38 (1988) 6614.
H.A. Blackstead and J.D Dow, Phys. Rev. B 51 (1995) 11830.
H.A. Blackstead, J.D. Dow, D.B. Chrisey, J.S. Horwitz, M.A. Black, P.J. McGinn, A.E. Klunzinger and D.B. Pulling, Phys. Rev. B 54 (1996) 6122.
G.S. Boebinger, Y. Ando, A. Passner, T. Kimura, M. Okuya, J. Shimoyama, K. Kishio, K. Tamasaku, N. Ichikawa and S. Uchida, Phys. Rev. Lett. 77 (1996) 5417.
I.D. Brown, Physica C 169 (1990) 105.
B. Bucher, P. Steiner and P. Wachter, Physica B 199-200 (1994) 268.
B. Bucher and P. Wachter, Phys. Rev. B 51 (1995) 3309.
C P. Carretta, F. Tedoldi, A. Rigamonti, F. Galli, F. Borsa, J.H. Cho and D.C. Johnston, Eur. Phys. J. B 10 (1999) 233.
A. Casaca, G. Bonfait, C. Dubourdieu, F. Weiss and J.P. Sénateur, Phys. Rev. B 59 (1999) 1538.
R.J. Cava, A.W. Hewat, E.A. Hewat, B. Batlogg, M. Marezio, K.M. Rabe, J.J. Krajewski, W.F. Peck Jr. and L.W. Rupp Jr., Physica C 165 (1990) 419.
B.S. Chandrasekhar, Appl. Phys. Lett. 1 (1962) 7.
X.K. Chen, J.G. Naeini, K.C. Hewitt, J.C. Irwin, R. Liang and W.N. Hardy, Phys. Rev. B 56 (1997) R513.
S.-W. Cheong, G. Aeppli, T.E. Mason, H. Mook, S.M. Hayden, P.C. Canfield, Z. Fisk, K.N. Clausen and J.L. Martinez, Phys. Rev. Lett. 67 (1991) 1791.
T.R. Chien, Z.Z. Wang and N.P. Ong, Phys. Rev. Lett. 67 (1991) 2088.
A.V. Chubukov, D. Pines and B. Stojkovic, J. Phys.: Cond. Matter 8 (1996) 10017.
D.G. Clarke, S.P. Strong and P.W. Anderson, Phys. Rev. Lett. 74 (1995) 4499.
A.M. Clogston, Phys. Rev. Lett. 9 (1962) 266.
J.L Cohn and J. Karpinski, cond-mat/9810152.
D E. Dagotto, J. Riera and D.J. Scalapino, Phys. Rev. B 45 (1992) 5744.
E. Dagotto and T.M. Rice, Science 271 (1996) 618.
BIBLIOGRAPHY
176
E. Dagotto, Rep. Progr. Phys. 62 (1999) 1525.
P. Dai, H.A. Mook and F. Dogan, Phys. Rev. Lett. 80 (1998) 1738.
Y. Dalichaouch, M.S. Torikachvili, E.A. Early, B.W. Lee, C.L. Seaman, K.N. Yang, H. Zhou and M.B. Maple, Solid State Commun. 65 (1988) 1001.
D. de Fontaine, G. Ceder and M. Asta, J. Less Common Metals 164-165 (1990) 108.
E V.J. Emery, S.A. Kivelson and H.Q. Lin, Phys. Rev. Lett. 64 (1990) 475.
V.J. Emery, S.A. Kivelson and O. Zachar, Physica C 282-287 (1997) 174.
V.J. Emery, S.A. Kivelson and O. Zachar, Phys. Rev. B 56 (1997) 6120.
V.J. Emery and S.A. Kivelson, cond-mat/9902179
V.J. Emery, S.A. Kivelson and J.M. Tranquada, Proc. Natl. Acad. Sci. USA 96 (1999) 8814.; cond-mat/9907228.
Y. Endoh, K. Yamada, R.J. Birgenau, D.R. Grabbe, H.P. Jenssen, M.A. Kastner, C.J. Peters, P.J. Picone, T.R. Thurston, J.M. Tranquada, G. Shirane, Y. Hidaka, M. Oda, Y. Enomoto, M. Suzuki and T. Murakami, Phys. Rev. B 37 (1988) 7443.
F W.H. Fietz, R. Quenzel, H.A. Ludwig, K. Grube, S.I. Schlachter, F.W Hornung, T. Wolf, A. Erb, M. Kläser and G. Müller-Vogt, Physica C 270 (1996) 258.
R.M. Fleming, B. Batlogg, R.J. Cava and E.A. Rietman, Phys. Rev. B 35 (1987) 7191.
I. François, "High Frequency Properties of YBaCuO Films: from Fundamentals to Applications ", PhD Thesis (1996) K.U.Leuven.
T.A. Friedmann, M.W. Rabin, J. Giapintzakis, J.P. Rice and D.M. Ginsberg, Phys. Rev. B 42 (1990) 6217.
G R. Gagnon, Ch. Lupien and L. Taillefer, Phys. Rev. B 50 (1994) 3458.
P.K. Gallagher, Adv. Ceram. Mat. 2 (1987) 632.
P.M. Grant, R. Beyers, E. Engler, G. Lim, S. Parkin, M. Ramirez, V. Lee, A. Nazzal, J. Vasquez and R. Savoy, Phys. Rev. B 35 (1987) 7242.
M. Greven, R.J. Birgenau and U.-J. Wiese, Phys. Rev. Lett. 77 (1996) 1865.
H S.J. Hagen, A.W. Smith, M. Rajeswari, J.L. Peng, Z.Y. Li, R.L. Greene, S.N. Mao, X.X. Xi, S. Bhattacharya, Q. Li and C.J. Lobb, Phys. Rev. B 47 (1993) 1064.
BIBLIOGRAPHY
177
E.H. Hall, Am. J. Math. 2 (1879) 287.
R.H. Hammond and R. Bormann, Physica C 162-164 (1989) 703.
T. Hanaguri, M. Naito, K. Kitazawa, Physica C 317-318 (1999) 345.
Z. Hao, B.R. Zhao, B.Y. Zhu, L. Trappeniers, J. Vanacken and V.V. Moshchalkov, submitted to Phys. Rev. Lett. (2000)
J.M. Harris, Y.F. Yan and N.P. Ong, Phys. Rev.B 46 (1992) 14293.
P. Hasenfratz and F. Niedermayer, Phys. Lett. B 268 (1991) 231.
F. Herlach, L. Van Bockstal, M. van de Burgt and G. Heremans, Physica B 155 (1989) 61.
F. Herlach and J.A.A.J. Perenboom, Physica B 211 (1995) 1.
F. Herlach, Ch. Agosta, R. Bogaerts, W. Boon, I. Deckers, A. De Keyser, N. Harrison, A. Lagutin, L. Li, L. Trappeniers, J. Vanacken, L. Van Bockstal and Ann Van Esch, Physica B 216 (1996) 161.
D.G. Hinks, L. Soderholm, D.W. Capone, J.D. Jorgensen, I.K. Schuller, C.U. Segre, K. Zhang and J.D Grace, Appl. Phys. Lett. 50 (1987) 1688.
R. Hlubina and T.M. Rice, Phys. Rev. B 51 (1995) 9253.
A.W. Hunt, P.M. Singer, K.R. Thurber and T. Imai, Phys. Rev. Lett. 82 (1999) 4300.
I N. Ichikawa, S. Uchida, J.M. Tranquada, T. Niemoller, P.M. Gehring, S.H. Lee and J.R. Schneider, cond-mat/9910037.
G. Ilonca, M. Mehbod, A. Lanckbeen and R. Deltour, Phys. Rev. B 47 (1993) 15265.
A. Ino, T. Mizokawa, K. Kobayashi, A. Fujimori, T. Sasagawa, T. Kimura, K. Kishio, K. Tamasaku, H. Eisaki and S. Uchida, cond-mat/9806341.
A. Ino, Ch. Kim, T. Mizokawa, Z.-X. Shen, A. Fujimori, M. Takaba, K. Tamasaku, H. Eisaka and S. Uchida, J. Phys. Soc. Jpn. 68 (1999) 1496.
T. Ito, K. Takenaka and S. Uchida, Phys. Rev. Lett. 70 (1993) 3995.
J C.N. Jiang, A.R. Baldwin, G.A. Levin, T. Stein, C.C. Almasan, D.A. Gajewski, S.H. Han and M.B. Maple, Phys. Rev. B 55 (1997) R3390.
J.D. Jorgensen, M.A. Beno, D.G. Hinks, L. Soderholm, K.J. Volin, R.L. Hitterman, J.D. Grace, I.K. Schuller, C.U. Segre, K. Zhang and M.S. Kleefisch, Phys. Rev. B 36 (1987) 3608.
J.D. Jorgensen, H. Shaked, D.G. Hinks, B. Dabrowski, B.W. Veal, A.P. Paulikas, L.J. Nowicki, G.W. Crabtree, W.K. Kwok and L.H. Nunez, Physica C 153-155 (1988) 578.
BIBLIOGRAPHY
178
J.D. Jorgensen, B.W. Veal, A.P. Paulikas, L.J. Nowicki, G.W. Crabtree, H. Claus and W.K. Kwok, Phys. Rev. B 41 (1990) 1863.
J.D. Jorgensen, Sh. Pei, P. Lightfoot, D.G. Hinks, B.W. Veal, B. Dabrowski, A.P. Paulikas, R. Kleb and I.D. Brown, Physica C 171 (1990) 93.
K Y.-J. Kao, Q. Si and K. Levin, cond-mat/9908302.
J. Karpinski, E. Kaldis, E. Jilek, S. Rusiecki and B. Bucher, Nature 336 (1988) 660.
T. Kimura, K. Kishio, T. Kobayashi, Y. Nakayama, N. Motohira, K. Kitazawa and K. Yamafuji, Physica C 192 (1992) 247.
J. Kircher, M.K. Kelly, S. Rashkeev, M. Alouani, D. Fuchs and M. Cardona, Phys. Rev. B 44 (1991) 217.
Y. Kitaoka, K. Ishida, S. Oshugi, K. Fujiwara and K. Asayama, Physica C 185-189 (1991) 98.
S.A. Kivelson and V.J. Emery, Synth. Met. 80 (1996) 151.
S.A. Kivelson, E. Fradkin and V.J. Emery, Nature 393 (1998) 550.
N.B. Kopnin and V.M. Vinokur, Phys. Rev. Lett. 83 (1999) 4864.
O. Kraut, C. Meingast, G. Bräuchle, H. Claus, A. Erb, G. Müller-Vogt and H.Wühl, Physica C 205 (1993) 139.
L A.N. Lavrov, Y. Ando, K. Segawa and J. Takeya, Phys. Rev. Lett. 83 (1999) 1419.
P.A. Lee and T.V. Ramakrishnan, Rev. Mod. Phys. 57 (1985) 287.
G.A. Levin, T. Stein, C.N. Jiang, C.C. Almasan, D.A. Gajewski, S.H. Han and M.B. Maple, Physica C 282-287 (1997) 1127.
L. Li, "High Performance Pulsed Magnets: Theory, Design and Construction", PhD Thesis (1998) K.U.Leuven.
J.X. Li, W.C. Wu and T.K. Lee, Phys. Rev. B 60 (1999) 3629.
J.-P. Locquet, A. Catana, E. Mächler, C. Gerber and J.G. Bednorz, Appl. Phys. Lett. 64 (1994) 372.
J.-P. Locquet and E.J. Williams, Acta Physica Polonica A 92 (1996) 69.
J.-P. Locquet, Y. Jaccard, A. Cretton, E.J. Williams, F. Arrouy, E. Mächler, T. Schneider, Ø. Fischer and P. Martinoli, Phys. Rev. B 54 (1996) 7481.
J.-P. Locquet, J.Perret, J. Fompeyrine, E. Mächler, J.W. Seo and G. Van Tendeloo, Nature 394 (1998) 453.
J.-P. Locquet, J.Perret, J.W. Seo and J. Fompeyrine, SPIE proceedings 1998.
BIBLIOGRAPHY
179
J.-P. Locquet, private communication (2000).
A.G. Loeser, Z.-X. Shen, D.S. Dessau, D.S. Marshall, C.H. Park, P. Fournier, A. Kapitulnik, Science 273 (1996) 325.
J.W. Loram, K.A. Mirza, J.R. Cooper and W.Y. Liang, Phys. Rev. Lett. 71 (1993) 1740.
J.W. Loram, K.A. Mirza, J.R. Cooper and J.L. Tallon, J. Phys. Chem. Solids 59 (1998) 2091.
M A.P. Mackenzie, S.R. Julian, A. Carrington, G.G. Lonzarich, D.J.C. Walker, J.R. Cooper and D.C. Sinclair, Physica C 235-240 (1994) 233.
M. Maenhoudt, "Persistent photoinduced phenomena in oxygen deficient YBCO thin films", PhD thesis K.U.Leuven (1995).
F. Marsiglio and J.P. Carbotte, Phys. Rev. B 36 (1987) 3633.
T.E. Mason, G. Aeppli and H.A Mook, Phys. Rev. Lett. 68 (1992) 1414.
T.E. Mason, G. Aeppli, S.M. Hayden, A.P. Ramirez and H.A. Mook, Physica B 199-200 (1994) 284.
J.A. Martindale and P.C. Hammel, Phil. Mag. B 74 (1996) 573.
C. Meingast, O. Kraut, T. Wolf, H.Wühl, A. Erb and G. Müller-Vogt, Phys. Rev. Lett. 67 (1991) 1634.
D. Mihailovic, V.V. Kabanov, K. Zagar and J. Demsar, Phys. Rev. B 60 (1999) R6995.
Y. Mizuno, T. Tohyama and S. Maekawa, Phys. Rev. B 58 (1998) R14713.
N. Momono, T. Matsuzaki, T. Nagata, M. Oda and M. Ido, to appear in J. of Low Temp. Phys. (1999).
N. Momono, R. Dipasupil, H. Ishiguro, S. Saigo, T. Nakano, M. Oda and M. Ido, Physica C 317-318 (1999) 603.
V.V. Moshchalkov, Physica C 156 (1988) 473.
V.V. Moshchalkov, Physica B 163 (1990) 59.
V.V. Moshchalkov, Sol. St Comm. 86 (1993) 715.
V.V. Moshchalkov, L. Trappeniers, J. Vanacken, Proc. ARW "Symmetry and pairing in superconductors", april 28 - may 2 (Yalta) 1998.
V.V. Moshchalkov, cond-mat/9802281.
V.V. Moshchalkov, L. Trappeniers, J. Vanacken, Europhys. Lett. 46 (1999) 75.
V.V. Moshchalkov, L. Trappeniers, J. Vanacken, Physica C 318 (1999) 361.
V.V. Moshchalkov, L. Trappeniers, J. Vanacken, J. Low Temp. Phys. 117 (1999) 1283.
BIBLIOGRAPHY
180
V.V. Moshchalkov and V.A. Ivanov, cond-mat/9912091.
N.F. Mott and E.A. Davis, Electronic Processes in Non-Crystalline Materials, 2nd ed. (Clarendon, Oxford) 1979.
N.F. Mott, Physica A 168 (1990) 221.
N J.G. Naeini, X.K. Chen, J.C. Irwin, M. Okuya, T. Kimura and K. Kishio, Phys. Rev. B 59 (1999) 9642.
J.G. Naeini, J.C. Irwin, T. Sasagawa, Y. Togawa and K. Kishio, cond-mat/9909342.
N. Nagaosa and P.A. Lee, Phys. Rev. Lett. 64 (1990) 2450.
T. Nagata, M. Uehara, J. Goto, N. Komiya, J. Akimitsu, N. Motoyama, H. Eisaki, S. Uchida, H. Takahashi, T. Nakanishi and N. Mori, Physica C 282-287 (1997) 153.
T. Nakano, N. Momono, M. Oda and M. Ido, J. Phys. Soc. Jpn. 67 (1998) 2622.
J.J. Neumeier, T. Bjornholm, M.B. Maple and I.K. Schuller, Phys. Rev. Lett. 63 (1989) 2516.
T. Noda, H. Eisaki and S. Uchida, Science 286 (1999) 265.
O M. Oda, K. Hoya, N. Abe, M. Yokoyama, N. Momono, T. Nakano, T. Nagata and M. Ido, Int. J. of Mod. Phys B 12 (1998) 3179.
N.P. Ong, Science 273 (1996) 321.
M.S. Osofsky, R.J. Soulen, S.A. Wolf, J.M. Broto, H. Rakoto, J.C. Ousset, G. Coffe, S. Askenazy, P. Pari, I. Bozovic, J.N. Eckstein and G.F. Virshup, Phys. Rev. Lett. 71 (1993) 2315.
P W.E. Pickett, H. Krakauer, D.A. Papaconstantopoulos and L.L. Boyer, Phys. Rev. B 35 (1987) 7252.
W.E. Pickett, R.E. Cohen and H. Krakauer, Phys. Rev. B 42 (1990) 8764.
W.E. Pickett, Physica C 289 (1997) 51.
W.E. Pickett, Phys. Rev. Lett. 78 (1997) 1960.
D. Pines, Physica B 163 (1990) 78.
D. Pines, Physica C 282-287 (1997) 273.
Ch.P. Poole, H.A. Farach, R.J. Creswick, "Superconductivity" Academic Press (California) 1995.
BIBLIOGRAPHY
181
Q C. Quitmann, D. Andrich, C. Jarchow, M. Fleuster, B. Beschoten, G. Güntherodt, V.V. Moshchalkov, G. Mante and R. Manzke, Phys. Rev.B 46 (1992) 11813.
R P.G. Radaelli, D.G. Hinks, A.W. Mitchell, B.A. Hunter, J.L. Wagner, B. Dabrowski, K.G. Vandervoort, H.K. Viswanathan and J.D. Jorgensen, Phys. Rev. B 49 (1994) 4163.
M. Randeira, cond-mat/9710223
M. Rasolt, Z. Tešanovic, Rev. Mod. Phys. 64 (1992) 709.
J. Reyes-Gasga, T. Krekels, G. Van Tendeloo, J. Van Landuyt, S. Amelinckx, W.H.M. Bruggink and H. Verweij, Physica C 159 (1989) 831.
J. Rossat-Mignod, L.P. Regnault, C. Vettier, P. Bourges, P. Burlet, J. Bossy, J.Y. Henry and G. Lapertot, Physica C 185-189 (1991) 86.
J. Rossat-Mignod, L.P. Regnault, P. Bourges, P. Burlet, C. Vettier and J.Y. Henry, Physica B 192 (1993) 109.
J. Rossat-Mignod, P. Bourges, F. Onufrieva, L.P. Regnault, J.Y. Henry, P. Burlet and C. Vettier, Physica B 199-200 (1994) 281.
H. Rushan, G. Zizhao, Y. Daole and L. Qing, Phys. Rev. B 41 (1990) 6683.
S H. Sato, H. Yamamoto and M. Naito, Physica C 274 (1997) 221.
M. Schossmann and E. Schachinger, Phys. Rev. B 33 (1986) 6123.
J.R. Schrieffer, X.-G. Wen and S.-C. Zhang, Phys. Rev. Lett. 60 (1988) 944.
I.K. Schuller, D.G. Hinks, M.A. Beno, D.W. Capone II, L. Soderholm, J.-P. Locquet, Y. Bruynseraede, C.U. Segre, and K. Zhang, Solid State Commun. 63 (1987) 385.
G. Shirane, Y. Endoh, R.J. Birgenau, M.A. Kastner, Y. Hidaka, M. Oda, M. Suzuki and T. Murakami, Phys. Rev. Lett. 59 (1987) 1613.
B.I. Shklovskii and A.L. Efros, in Electronic Properties of Doped Semiconductors, Ed. by M. Cardona, H.J. Queisser, Springer series in Solid-State Sciences (Berlin) 1984.
B.P. Stojkovic, Z.G. Yu, A.R. Bishop, A.H. Castro Neto and N. Gronbech-Jensen, Phys. Rev. Lett. 82 (1999) 4679.
N, Suzuki and M. Hikita, Phys. Rev. B 44 (1991) 249.
Y. Suzumura, Y. Hasegawa and H. Fukuyama, J. Phys. Soc. Jpn. 57 (1988) L401.
BIBLIOGRAPHY
182
T S. Tajima, R. Hauff, W.-J. Jang, A. Rykov, Y. Sato and I. Terasaki, Journ. Low Temp. Phys. 105 (1996) 743.
S. Tajima, N.L. Wang, N. Ichikawa, H. Eisaki, S. Uchida, H. Kitano, T. Hanaguri and A. Maeda, Europhys. Lett. 47 (1999) 715.
H. Takagi, S. Uchida, K. Kitazawa and S. Tanaka, Jpn. J. Appl. Phys. Pt2 26 (1987) L123.
H. Takagi, T. Ido, S. Ishibashi, M. Uota, S. Uchida and Y. Tokura, Phys. Rev. B 40 (1989) 2254.
H. Takagi, B. Batlogg, H.L. Kao, J. Kwo, R.J. Cava, J.J. Krajewski and W.F. Peck, Phys. Rev. Lett. 69 (1992) 2975.
M. Takano, M. Azuma, Y. Fujishiro, M. Nohara, H. Takagi, M. Fujiwara, H. Yasuoka, S. Ohsugi, Y. Kitaoka and R.S. Eccleston, Physica C 282-287 (1997) 149.
M. Takigawa, A.P. Reyes, P.C. Hammel, J.D. Thompson, R.H. Heffner, Z. Fisk and K.C. Ott, Phys. Rev. B 43 (1991) 247.
J.L. Tallon, Physica C 168 (1990) 85.
J.L. Tallon and N.E. Flower, Physica C 204 (1993) 237.
J.L. Tallon, C. Bernhard, H. Shaked, R.L. Hitterman and J.D. Jorgensen, Phys. Rev. B 51 (1995) 12911.
W.H. Tang and J. Gao, Physica C 315 (1999) 59.
J.M. Tarascon, L.H. Greene, W.R. McKinnon, G.W. Hull and H. Geballe, Science 235 (1987) 1373.
J.M. Tarascon, P. Barboux, P.F. Miceli, L.H. Greene and G.W. Hull, Phys. Rev. B 37 (1988) 7458.
G. Teniers in "Studie van de normaalgeleidende transporteigenschappen van dunne YBCO films beneden de kritische temperatuur", licentiaatthesis K.U.Leuven (1999).
Z. Tešanovic, Int. J. Mod. Phys. B 5 (1991) 353.
M. Tetenbaum, L.A. Curtiss, B. Tani, B. Czech and M. Blander, Physica C 158 (1989) 371.
T.R. Thurston, R.J. Birgenau, M.A. Karstner, N.W. Preyer, G. Shirane, Y. Fujii, K. Yamada, Y. Endoh, K. Kakurai, M. Matsuda, Y. Hidaka and T Murakami, Phys. Rev. B 40 (1989) 4585.
T. Timusk and B. Statt, Rep. Progr. Phys. 62 (1999) 61.
L.H. Tjeng, B. Sinkovic, N.B. Brookes, J.B. Goedkoop, R. Hesper, E. Pellegrin, F.M.F de Groot, S. Altieri, S.L. Hulbert, E. Shekel and G.A. Sawatzky, Phys. Rev. Lett. 78 (1997) 1126.
BIBLIOGRAPHY
183
J.M. Tranquada, Physica C 282-287 (1997) 166.
J.M. Tranquada, Phys. Rev. Lett. 78 (1997) 338.
L. Trappeniers, J. Vanacken, P. Wagner, G. Teniers, S. Curras, J. Perret, P. Martinoli, J.-P. Locquet, V.V. Moshchalkov and Y. Bruynseraede, J. Low Temp. Phys. 117 (1999) 681.
M. Troyer, H. Tsunetsugu and D. Würtz, Phys. Rev. B 50 (1994) 13515.
U-V M. Uehara, T. Nagata, J. Akimitsu, H. Takahashi, N. Mori and K. Kinoshita, J. Phys. Soc. Jpn. 65 (1996) 2764.
A. Valetta, G. Bardelloni, M. Brunelli, A. Lanzara, A. Bianconi and N.L. Saini, J. of Supercond. 10 (1997) 383.
B.W. Veal and A.P. Paulikas, Physica C 184 (1991) 321.
V.M. Vinokur, V.B. Geshkenbein, M.V. Feigel'man and G. Blatter, Phys. Rev. Lett. 71 (1993) 1242.
W P. Wagner, private communication (1999).
D.J.C. Walker, O. Laborde, A.P. Mackenzie, S.R. Julian, A. Carrington, J.W. Loram and J.R. Cooper, Phys. Rev. B 51 (1995) 9375.
Z.D. Wang, J. Dong and C.S. Ting, Phys. Rev. Lett. 72 (1994) 3875.
N.L. Wang, S. Tajima, A.I. Rykov and K. Tomimoto, Phys. Rev. B 57 (1998) R11081.
U. Welp, M. Grimsditch, S. Fleshler, W. Nessler, J. Downey, G.W. Crabtree and J. Guimpel, Phys. Rev. Lett. 69 (1992) 2130.
U. Welp, M. Grimsditch, S. Fleshler, W. Nessler, B. Veal and G.W. Crabtree, J. of Supercond. 7 (1994) 159.
N.R. Werthamer, E. Helfand and P.C. Hohenberg, Phys. Rev. 147 (1966) 295.
B. Wuyts, "Normal-state transport properties of metallic oxygen deficient YBa2Cu3Ox films", PhD thesis (1994) K.U.Leuven.
B. Wuyts, V.V. Moshchalkov and Y. Bruynseraede, Phys. Rev. B 53 (1996) 9418.
X-Y-Z G. Xiao, M.Z. Cieplak, D. Musser, A. Gavrin, F.H. Streitz, C.L. Chien, J.J. Rhyne and J.A. Gotaas, Nature 332 (1988) 238.
P. Xiong, G. Xiao and X.D. Wu, Phys. Rev. B 47 (1993) 5516.
Y. Xu and W. Guan, Phys. Rev. B 45 (1992) 3176.
BIBLIOGRAPHY
184
K. Yamada, C.H. Lee, Y. Endoh. G. Shirane, R.J. Birgenau and M.A. Kastner, Physica C 282-287 (1997) 85.
K. Yamada, C.H. Lee, K. Kurahashi, J. Wada, S. Wakimoto, S. Ueki, H. Kimura, Y. Endoh. S. Hosoya, G. Shirane, R.J. Birgenau, M. Greven, M.A. Kastner and Y.J. Kim, Phys. Rev. B 57 (1998) 6165.
Y. Yanase and K. Yamada, J. Phys. Soc. Jpn. 68 (1999) 548.
H. Yasuoka, S. Kambe, Y. Itoh and T. Machi, Physica B 199-200 (1994) 278.
H. Yasuoka, Physica C 282-287 (1997) 119.
R.C. Yu, M.B. Salamon, J.P. Lu, W.C. Lee, Phys. Rev. Lett. 69 (1992) 1431.
G. Yu, C.H. Lee, D. Mihailovic, A.J. Heeger, C. Fincher, N. Herron and E.M. McCarron, Phys. Rev. B 48 (1993) 7545.
J. Zaanen, Science 286 (1999) 251.
Y. Zha, S.L. Cooper and D. Pines, Phys. Rev. B 53 (1996) 8253.
F.C. Zhang and T.M. Rice, Phys. Rev. B 37 (1988) 3759.
NEDERLANDSTALIGE SAMENVATTING
185
Nederlandstalige samenvatting
De eigenschappen van het elektrisch transport in een set van YBa2Cu3Ox en
(Y0.6Pr0.4)Ba2Cu3Ox dunne filmen met verschillende zuurstofinhoud en
La1.9Sr0.1CuO4 ultra dunne filmen onder epitaxiale spanning werden bestudeerd
in de normaal-geleidende toestand. Hiertoe werden de transport
eigenschappen (weerstand en Hall-effect) opgemeten in nulveld en in gepulste
hoge magneetvelden. De keuze voor een tweevoudige aanpak - enerzijds het
verlagen van de kritische temperatuur Tc door de ladingsdragers-concentratie p
te veranderen en anderzijds het gebruik van hoge magneetvelden - laat toe het
gehele ondergedopeerde gebied van het T-p (temperatuur versus gaten-
concentratie in het CuO2 vlak) fasediagram (figuur 1) te bestrijken. Het wordt
dan mogelijk om de transporteigenschappen van deze materialen in de normale
toestand (d.w.z. niet supergeleidend) te bestuderen beneden de kritische temperatuur Tc.
Figuur 1: De eigenschappen van de hoge-Tc koperoxides veranderen sterk met veranderende temperatuur (vertikale as) en dopering van het CuO2 vlak [Batlogg2000].
NEDERLANDSTALIGE SAMENVATTING
186
In een eerste stap (geïnspireerd door vroeger werk [Wuyts94 & 96]) werd
aangetoond dat de nulveld normaal-geleidende resistiviteit ρρ(T) boven Tc
schaalt op een universele curve voor verschillende zuurstofinhouden x, zowel
voor YBa2Cu3Ox als voor (Y0.6Pr0.4)Ba2Cu3Ox. Deze schaling werd uitgevoerd
met drie schalingsparameters: een energieschaal ∆, de resistiviteit ρ∆ bij T = ∆
en de residuele resistiviteit ρo. Het expliciet inbrengen van de residuele
resistiviteit ρo in deze schaling zorgde voor een betere schaling dan de vroegere
analyse [Wuyts94 & 96]. De universele curve vertoont drie gebieden met een
kwalitatief verschillend ρ(T) gedrag:
(I) een gebied van lineaire ρ(T) bij hoge temperaturen T > T*,
(II) een super-lineaire ρ(T) bij temperaturen TMI < T < T*
(III) een isolerend ρ(T) regime bij T < TMI.
Het lage-temperatuur regime (III) zit voor een groot gedeelte verstopt achter de
supergeleidende fase die optreedt bij T < Tc. De karakteristieken in de
temperatuursafhankelijkheid van de metallische nulveld resistiviteit ρ(T) van
YBa2Cu3Ox en (Y0.6Pr0.4)Ba2Cu3Ox in gebieden I en II zijn universeel voor alle
gemeten curves. Het enige verschil is de temperatuursschaal ∆∆ bij dewelke de
regimes optreden.
In een tweede stap werd de magnetoresistiviteit van deze YBa2Cu3Ox en
(Y0.6Pr0.4)Ba2Cu3Ox epitaxiale dunne filmen gemeten in gepulste hoge magneetvelden tot 50 T. Dit liet ons toe het ρρ(T) gedrag in de normale toestand bij temperaturen onder de kritische temperatuur Tc te meten.
Deze hoge veld gegevens in het metallische regime schalen goed met de
universele ρ(T/∆) curve. Alleen de ρ(T) data in het isolerende regime bij lage
temperaturen schalen niet bevredigend wanneer dezelfde parameters gebruikt
worden als in de schaling van de nulveld gegevens.
Het bestaan van een universele ρ(T/∆) curve voor de metallische resistiviteit in
de normale toestand werd geïnterpreteerd als een sterke indicatie dat in deze
materialen één enkel mechanisme de verstrooiing van de ladingsdragers
domineert. Het is dit mechanisme dat de energieschaal ∆ (die afhankelijk is van
de dopering) bepaalt.
NEDERLANDSTALIGE SAMENVATTING
187
Deze experimenten toonden aan dat de grondtoestand bij T < Tc van
YBa2Cu3Ox (voor x ≤ 6.8), (Y0.6Pr0.4)Ba2Cu3Ox (voor alle niveaus van
zuurstofinhoud) en epitaxiaal uitgerekt La1.9Sr0.1CuO4 eerder isolerend is. De
divergentie van de resistiviteit bij lage temperaturen volgt een ln(1/T) wet in hoge magneetvelden en de implicaties voor de theoretische modellen werden
besproken.
De vergelijking van de transportgegevens voor epitaxiaal gespannen
La1.9Sr0.1CuO4 ultra-dunne filmen met gegevens op monsters zonder deze
spanning en met onze meetgegevens op YBa2Cu3Ox en (Y0.6Pr0.4)Ba2Cu3Ox
monsters wijst er op dat epitaxiale spanning waarschijnlijk niet alleen de
dopering van de CuO2 vlakken maar ook de verstrooiing aan wanorde
beïnvloedt.
Ten einde een verdere interpretatie van de resistiviteitsgegevens in de normale
toestand mogelijk te maken, werden Hall-effect metingen onder de kritische
temperatuur Tc uitgevoerd op dezelfde YBa2Cu3Ox en Y0.6Pr0.4Ba2Cu3Ox dunne
filmen. Deze metingen werden in zeer hoge gepulste magneetvelden uitgevoerd
om zo de normale-toestand (d.w.z. niet-supergeleidend) te bestuderen (H > Hc2).
De Hall metingen geven een signaal dat drie grootte-ordes kleiner is dan in
gewone magnetoresistiviteitsexperimenten. De succesvolle uitvoering van
dergelijke experimenten was slechts mogelijk door een zorgvuldige
implementatie van vibratie-isolatie, afscherming voor elektromagnetische
interferentie en het realiseren van een lage contactweerstand met het
meetmonster.
Uit de metingen van de hoge-veld Hall-resistiviteit ρyx(H) werd de
veldafhankelijkheid van de Hall-coëfficiënt RH(H) bij verschillende
temperaturen berekend. Door ons te concentreren op het gedeelte in de normale
toestand (H > Hc2), werd de temperatuursafhankelijkheid van de Hall coëfficiënt
RH(T) bij een vast (hoog) magneetveld geconstrueerd. Bij deze hoge velden
wordt de Hall-coëffciënt RH(T) in de normale toestand gemeten, zelfs bij
T < Tc. Deze RH(T) vertoont slechts een zeer zwakke temperatuursafhankelijkheid bij lage temperaturen. Door de RH(T) curves te
combineren met de hoge-veld ρab(T) curves voor de zelfde samples wordt
NEDERLANDSTALIGE SAMENVATTING
188
duidelijk dat bij deze hoge velden de normale toestand volledig wordt bereikt,
zelfs bij temperaturen T << Tc.
De temperatuurs-onafhankelijkheid van de lage-temperatuur Hall-coëfficiënt RH staat in scherp contrast met de sterk divergerende ρρab(T) in deze materialen. Dit is een belangrijke test voor de bestaande modellen die
reeds een lage-temperatuur divergentie van ρab(T) voorspellen. Van deze
modellen verklaart het zwakke lokalisatie-model zowel de constante RH bij lage
temperaturen als de ln(1/T) divergentie van de hoge-veld ρab(T) gegevens.
De ladingsdragersdichtheid nH (en aldus ook p, de fractie van gaten per Cu-
atoom in de CuO2 vlakken), berekend uit de Hall-data, liet toe een generisch T(p) fasediagram op te stellen voor de YBa2Cu3Ox en Y0.6Pr0.4Ba2Cu3Ox materialen.
Het universele ρρ(T) gedrag in de ondergedopeerde YBa2Cu3Ox en
Y0.4Pr0.4Ba2Cu3Ox dunne filmen is een sterk argument voor het idee dat één
enkel verstrooiingsmechanisme het hele ondergedopeerde regime in het Y123
systeem domineert. Met het complexe magnetische fasediagram in het
achterhoofd (korte-afstands AF correlaties en een pseudo energiekloof bij
temperaturen ver boven de supergeleidende kritische temperatuur Tc) en
geconfronteerd met de sterke aanwijzingen voor de vorming van ladingsstrepen
in de CuO2 vlakken is het redelijk om de oorsprong van dit dominant verstrooiingsmechanisme toe te wijzen aan de microscopische ordening van magnetisme en ladingen.
Daarom gebruikten we onze hoge-veld transport gegevens (zowel de resistiviteit
als het Hall-effect) om het effect op de transport-eigenschappen in de normale
toestand van de korte-afstands antiferromagnetische fluctuaties en de mogelijke
vorming van ladingsstrepen in het CuO2 vlak na te gaan. In dit ladingsstreep
scenario [Emery97b & 99b], worden dynamische, metallische [Ichikawa99,
Noda99, Tajima99] strepen geacht de transporteigenschappen te bepalen.
Vermits mobiele ladingsdragers in dit geval uit de Mott isolator naar de strepen
gestoten worden, vormen deze strepen de paden met de laagste weerstand.
Daardoor zijn de transporteigenschappen zeer gevoelig voor de vorming van
zowel statische als dynamische strepen. Om deze ideeën te verifiëren werd een
NEDERLANDSTALIGE SAMENVATTING
189
bestaand model [Moshchalkov93, 98b & 99] gebruikt. Dit model beschrijft het
elektrisch transport zowel in het tweedimensionale (2D) Heisenberg regime (boven T*) als in het ééndimensionale (1D) regime (onder T*) met ladingsstrepen, waar zich de pseudo energiekloof ontwikkelt.
Deze benadering [Moshchalkov93, 98b & 99] is gebaseerd op drie aannames:
1. het dominante verstrooiingsmechanisme in hoge-Tc materialen in
het hele temperatuursgebied is van magnetische oorsprong;
2. de specifieke temperatuursafhankelijkheid van de resistiviteit ρ(T)
kan worden beschreven door de inverse kwantumgeleidbaarheid
σ -1 met de inelastische lengte Lφ bepaald door de magnetische
correlatielengte ξm (via een sterke interactie van gaten met Cu2+
spins);
3. de gepaste 1D of 2D uitdrukkingen moeten worden gebruikt om de
kwantum geleidbaarheid uit te rekenen met Lφ ~ ξm.
Bij hoge temperaturen T* < T < To, in het 2D Heisenberg regime, geeft de
combinatie van de uitdrukkingen voor de 2D spin correlatielengte met de
gepaste uitdrukking voor de kwantumweerstand een lineaire temperatuursafhankelijkheid van de resistiviteit. Dit resultaat is in perfecte
overeenstemming met de lineaire ρ(T) bij hoge temperaturen voor alle gemeten
ondergedopeerde YBa2Cu3Ox en Y0.4Pr0.4Ba2Cu3Ox dunne filmen.
Bij matige temperaturen Tc < T < T*, in het 1D gestreept regime, geeft het
invullen van de 1D magnetische correlatielengte in de uitdrukking voor 1D
kwantumgeleidbaarheid een gebogen, super-lineaire ρρ(T) die de resistiviteit
van Sr2.5Ca11.5Cu24O41, een spin-ladder met een even aantal benen, perfect
beschrijft. Dit materiaal bevat door zijn specifieke kristalstructuur zeker een 1D
spin-ladder en daarom zou de resistiviteit volgens de richting van de ladder
zeker de uitdrukking voor 1D kwantumtransport moeten volgen.
In een volgende stap werd een overtuigende schaling aangetoond tussen de
resistiviteit van deze 1D spin-ladder en een typisch ondergedopeerd hoge-Tc
materiaal, YBa2Cu4O8. Dit wijst erop dat de temperatuursafhankelijkheden van de resistiviteit van ondergedopeerde koperoxides (in het regime met
NEDERLANDSTALIGE SAMENVATTING
190
een pseudo-energiekloof bij T < T*) enerzijds en spin ladders met een even aantal benen met een spin-energiekloof ∆∆ anderzijds, bepaald worden door hetzelfde onderliggende 1D (magnetisch) mechanisme. De magnetische
oorsprong van de verstrooiing van de ladingsdragers wordt verder bevestigd
door het feit dat de schalingstemperatuur, gebruikt in de schaling van ρ(T), even
goed werkt voor resistiviteits- als voor Knight-shift gegevens KS. Deze KS
gegevens kunnen goed gefit worden met de uitdrukkingen die afgeleid werden
in de 1D spin-ladder modellen.
De ρ(T) gegevens van een La1.9Sr0.1CuO4 dunne film onder epitaxiale
trekspanning en YBa2Cu3Ox en (Y0.6Pr0.4)Ba2Cu3Ox dunne filmen met
verschillende zuurstof inhoud (geschaald op een universele curve) zijn alle
perfect beschreven door de uitdrukking voor 1D kwantum transport. De
waardes van de spin-energiekloof, die gevonden werden uit deze fitting, zijn in
overeenstemming met onafhankelijke afschattingen op basis van resistieve
metingen op andere YBa2Cu3Ox dunne filmen, "getwinde" en ge-"detwinde"
éénkristallen. Bovendien komen ze overeen met afschattingen van de pseudo-
energiekloof uit 17O en 63Cu Knight-shift metingen op opgelijnde poeders. In
het 1D kwantumtransport model, waar de inelastische lengte bepaald wordt
door de magnetische correlatielengte, lijkt de overeenstemming van onze data
met de energiekloof bepaald uit NMR experimenten eerder evident. Dit bewijst
dat onze aanpak (het beschrijven van het transport in ondergedopeerde koperoxides bij T < T* in een 1D gestreepte manier) niet alleen kwalitatief overeenkomt maar ook waardes geeft voor de pseudo spin-energiekloof ∆∆ die goed overeenkomen met onafhankelijke afschattingen.
In het laatste hoofdstuk van dit werk werden onze data voor de resistiviteit en
het Hall-effect in de normale toestand geconfronteerd met de mogelijkheid van
de hechting van de ladingsstrepen ("stripe pinning") door wanorde. Het
metallische gedrag van de resistiviteit bij hoge temperaturen gaat bij lage
temperaturen T < TMI over in een meer isolerend, divergerend, ρ(T) gedrag dat
overeenkomt met een ln(1/T) wet. De Hall-coëfficiënt daarentegen is
temperatuursonafhankelijk bij lage temperaturen. Binnen het ladingsstreep
beeld kan men een sterke invloed verwachten op de transport eigenschappen
wanneer de 1D ladingsstrepen gefragmenteerd zijn of vastgehecht worden.
NEDERLANDSTALIGE SAMENVATTING
191
Zulke fragmentatie of hechting van de ladingsstrepen kan worden veroorzaakt
door elk verstrooiingsproces dat een inelastische lengte oplegt die kleiner is dan
Lφ ~ ξm1D.
Bij lage temperaturen, kan het ladingsstreep beeld alleen in overeenstemming worden gebracht met onze transportgegevens in de normale toestand door aan te nemen dat er streep-fragmentatie optreedt, zodat inter-streep hopping effecten belangrijk worden. Deze processen veroorzaken een sterke invloed
van de tussenliggende Mott isolator fase op het ladingstransport wat resulteert
in een (2D) isolerende resistiviteit en een klein Hall signaal. Door de
inelastische lengte van het proces dat werkzaam is in de tussenliggende AF in te
vullen in de uitdrukking voor 2D kwantumtransport zou dan het geobserveerde
ln(1/T) gedrag bekomen moeten worden. Ook elektron interferentie effecten in
het kader van 2D zwakke lokalisatie geven een ln(1/T) divergentie voor de
resistiviteit en een constante Hall-coëfficiënt RH(T) bij lage temperaturen, in
overeenstemming met onze resultaten.
De constructie van een zogenaamd experimenteel generisch T(p) fase diagram,
dat de transport eigenschappen in de normale toestand en de supergeleidende
eigenschappen van zowel de YBa2Cu3Ox en de (Y0.6Pr0.4)Ba2Cu3Ox materialen
beschrijft, was mogelijk door onze hoge-veld transportgegevens te combineren
met de afschattingen voor de ladingsdragersdichtheid uit het Hall effect. Dit
experimentele fasediagram (zie figuur 2 hieronder, te vergelijken met het
schematische fasediagram op figuur 1 hierboven) werd onderzocht in het kader
van het 1D/2D kwantumtransport model [Moshchalkov93, 98b & 99]. De
energie schaal ∆(p) (voor 1D kwantum transport) is goed beschreven door de
ladingsdragersdichtheid in de CuO2 vlak. Vermits deze energieschaal binnen
het kwantumtransport model precies de pseudo spin-energiekloof is, is ook deze
pseudo spin-energiekloof ∆ goed beschreven door de ladingsdragersdichtheid
in het CuO2 vlak.
Het 1D gestreept regime is gedefinieerd door vier grenzen in het T(p) diagram. Bij lage dopering wordt de bulk antiferromagnetische orde
herwonnen en verdwijnen de strepen. Bij sterke dopering zal de afstand tussen
de strepen verminderen en zal de tussenliggende Mott isolator fase verdwijnen.
Bij hoge temperaturen, kunnen de ladingsstrepen meanderen zodat het 1D
NEDERLANDSTALIGE SAMENVATTING
192
regime vernietigd wordt en een 2D regime met antiferromagnetische fluctuaties
optreedt. Bij lage temperaturen T < TMI, kunnen de ladingsstrepen
vastgehecht ("pinned") of gefragmenteerd worden en inter-streep hopping
effecten veroorzaken dan een 2D isolerend regime. In het T(p) diagram wordt
dit isolerend gebied aangeduid met TMI, onder dewelke de resistiviteit toeneemt
met afnemende temperatuur. De TMI(p) lijn voor het Y0.6Pr0.4Ba2Cu3Ox
materiaal ligt beduidend hoger dan de TMI(p) lijn voor de YBa2Cu3Ox samples.
Dit kan binnen het ladingsstrepen-model verklaard worden door de bijkomende
wanorde veroorzaakt door de Y/Pr substitutie, waardoor de hechting van de
strepen en fragmentatie effecten versterkt worden. Voor het YBa2Cu3Ox
systeem ligt de TMI(p) lijn onder de Tc(p) lijn voor bijna alle niveaus van
gatendopering en aldus wordt de isolerende toestand bij nulveld effectief
gemaskeerd door het optreden van supergeleiding. Voor het Y0.6Pr0.4Ba2Cu3Ox
materiaal ligt TMI(p) lijn boven de Tc(p) lijn voor alle niveaus van
zuurstofinhoud en de isolerende tendens kan daarom reeds worden gezien in de
nulveld ρ(T) metingen boven Tc. Bij temperaturen T < Tc, kan het optreden
van macroscopische coherentie tussen de zogenaamde voorgevormde paren
[Emery97b & 99b] resulteren in bulk supergeleiding.
Alhoewel ze een kwalitatief gelijkaardig gedrag vertonen, vallen de kritische
temperatuur Tc,mid(p) en de grens TMI tussen de metallische en meer isolerende
ρ(T) regimes niet samen voor de YBa2Cu3Ox en Y0.6Pr0.4Ba2Cu3Ox materialen.
Het feit dat voor dezelfde gatenconcentratie p, de kritische temperatuur Tc lager
is in het Y0.6Pr0.4Ba2Cu3Ox systeem in vergelijking met YBa2Cu3Ox, is een
indicatie dat naast wanorde ook magnetische paar-breking in het
(Y/Pr)Ba2Cu3Ox materiaal een belangrijke rol zou kunnen spelen in de reductie
van de supergeleidende kritische temperatuur Tc.
Door magnetoresistiviteitsmetingen en Hall-effect metingen uit te voeren in
zeer hoge magnetische velden op een selectie van hoge-Tc supergeleidende
dunne filmen met verscheidene niveaus van gatendopering, konden we het
schematisch T-p fase diagram uit figuur 1 complementeren met drie
experimentele faselijnen: de T*(p) lijn die een 2D naar 1D overgang beschrijft,
de supergeleidende kritische temperatuur Tc(p) en TMI(p) voor het begin van een
isolerend gedrag. Het generisch fasediagram voor de normale toestand werd
NEDERLANDSTALIGE SAMENVATTING
193
besproken in het kader van een 2D metallisch Heisenberg regime, een
metallisch gebied met 1D ladingsstrepen waar de pseudo energiekloof zich
ontwikkelt en een lage-temperatuur isolerend regime.
0 0.10 0.15 0.20 0.250.05
T (
K)
p (#/in-plane Cu)
metallic2D Heisenberg
metallic1D striped
T*
200
300
400
0
100 TcTMI
0
100
I
II
IIISC
SC
≈ ≈TMI
Tc
II
AF
YBa2Cu3Ox
Y0.6Pr0.4Ba2Cu3Ox
III
0 0.10 0.15 0.20 0.250.050 0.10 0.15 0.20 0.250.05
T (
K)
p (#/in-plane Cu)
metallic2D Heisenberg
metallic1D striped
T*
200
300
400
0
100
200
300
400
0
100 TcTMI
0
100
0
100
II
IIII
IIIIIISC
SC
≈ ≈TMI
Tc
IIII
AF
YBa2Cu3Ox
Y0.6Pr0.4Ba2Cu3Ox
IIIIII
Figuur 2: Generisch T(p) fasediagram voor de YBa2Cu3Ox (ruiten, volle lijn) en de Y0.6Pr0.4Ba2Cu3Ox (cirkels, puntlijn) dunne filmen. De 2D/1D overgangstemperatuur T* (volle symbolen), de supergeleidende kritische temperatuur Tc (open symbolen) en de grens TMI tussen de metallische en de meer isolerende regimes voor ρ(T) zijn aangeduid op de figuur. Alle gegevens zijn uitgezet ten opzichte van de fractie van gaten per Cu-atoom in het CuO2 vlak. De gegevens voor het Y0.6Pr0.4Ba2Cu3Ox systeem zijn 100 K verschoven.
PUBLICATIELIJST
195
Publicatielijst
Lieven Trappeniers (1994-1999)
Publicaties in internationale tijdschriften Study of High-Temperature Superconducting Thin Films in Magnetic Fields up to 50 Tesla, J. Vanacken, L. Trappeniers, A.S. Lagutin, P. Wagner, U. Frey, H. Adrian, F. Herlach, V.V. Moshchalkov en Y. Bruynseraede, Institute of Physics Conference Series 148 (1995) 971-974.
Strong Pinning in Melt-textured YBa2Cu3O7 with non superconducting Y2BaCuO5 inclusions, K. Rosseel, D. Dierickx, J. Lapin, V.V. Metlushko, W. Boon, L. Trappeniers, J. Vanacken, F. Herlach, V.V. Moshchalkov, Y. Bruynseraede and O. Vanderbiest, Institute of Physics Conference Series 148 (1995) 279-282.
Hopping conductivity of magnetic polarons in epitaxial Pr0.5Sr0.5MnO3 films, P. Wagner, V. Metlushko, J. Vanacken, L. Trappeniers, V.V. Moshchalkov, Y. Bruynseraede, Czechoslovak Journal of Physics 46 (1996), 2004-2006.
Magnetic Transitions and Magneto-Transport in Epitaxial Pr0.5Sr0.5MnO3 Thin Films, P. Wagner, V. Metlushko, M. Van Bael, R.J.M. Vullers, L. Trappeniers, A. Vantomme, J. Vanacken, G. Kido, V.V. Moshchalkov and Y. Bruynseraede, Journal de Physique IV 6 (1996) C3-309-343.
Observation of Magneto-Thermal Instabilities in YxTm1-xBa2Cu3O7 Single Crystal in Pulsed Magnetic Field Magnetization Measurements, L. Trappeniers, J. Vanacken, I.N. Goncharov, K. Rosseel, A.S. Lagutin, W. Boon, F. Herlach, V.V. Moshchalkov and Y. Bruynseraede, Journal of Low Temperature Physics 105 (1996) 1029-1034.
Experimental Techniques for Pulsed Magnetic Fields, F. Herlach, C.C. Agosta, R. Bogaerts, W. Boon, I. Deckers, A. De Keyser, N. Harrison, A. Lagutin, L. Li, L. Trappeniers, J. Vanacken, L. Van Bockstal and A. Van Esch, Physica B 216 (1996) 161-165.
Magnets, semiconductors and organic conductors at the Leuven pulsed field laboratory, F. Herlach, R. Bogaerts, W. Boon, I. Deckers, A. De Keyser, N. Harrison, L. Li, P. Reinders, K. Rosseel, L. Trappeniers, J. Vanacken, L. Van Bockstal, A. Van Esch, A. House, J. Singleton, M. Kartsovnik, J. De Boeck, G. Borghs, "High Magnetic Fields on the Physics of Semiconductors II", G. Landwehr, W. Ossau (eds.) II, 905-914 (1997) ISBN 981-02-3076-1
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Magneto-Transport in Epitaxial Thin Films of the Magnetic Perovskite Pr0.5Sr0.5MnO3, P.H. Wagner, V. Metlushko, L. Trappeniers, A. Vantomme, J. Vanacken, G. Kido, V.V. Moshchalkov and Y. Bruynseraede, Phys. Rev.B 55 (1997) 3699.
Anomalous Hall effect in thin films of Pr0.5Sr0.5MnO3, P. Wagner, D. Mazilu, L. Trappeniers, V.V. Moshchalkov and Y. Bruynseraede, Phys. Rev. B 55 (1997) 14721-14724.
Influence of Processing Parameters on Critical Currents and Irreversibility Fields of Fast Melt Processed YBa2Cu3O7 with Y2BaCuO5 inclusions, K. Rosseel, D. Dierickx, J. Vanacken, L. Trappeniers, W. Boon, F. Herlach, V.V. Moshchalkov and Y. Bruynseraede, Inst. Phys. Conf. Ser. 158 (1997) 1587-1590.
Construction of the Current-Voltage Characteristic in a 12 Decade Voltage Window using Magnetisation Measurements, J. Vanacken, K. Rosseel, L. Trappeniers, M. Van Bael, A.S. Lagutin, D. Dierickx, W. Boon, F. Herlach, V.V. Moshchalkov and Y. Bruynseraede, Inst. Phys. Conf. Ser. 158 (1997) 985-988.
Influence of Columnar Defects on Critical Currents and Irreversibility Fields in (YxTm1-x)Ba2Cu3O7 Single Crystals, L. Trappeniers, J. Vanacken, K. Rosseel, A.Yu. Didyk, I.N. Goncharov, L.I. Leonyuk, W. Boon, F. Herlach, V.V. Moshchalkov, and Y. Bruynseraede, Inst. Phys. Conf. Ser. 158 (1997) 1591-1594.
Irreversible Magnetic Properties of Ceramic HgBa2Ca2Cu3O8 at Very High Pulsed Magnetic Fields, L. Trappeniers, J. Vanacken, K. Rosseel, S. Reich, W. Boon, F. Herlach, V.V. Moshchalkov and Y. Bruynseraede, Journal of Superconductivity 11 (1998) 35.
Scaling Behavior of the Normal State Properties of the Underdoped High Tc Cuprates, V.V. Moshchalkov, L. Trappeniers and J. Vanacken, “Superconducting and related oxides: Physics and nano-engineering III”, Edited by D. Pavuna and I. Bozovic, SPIE proc. 3481 (SPIE, Bellingham, 1998) 10-16.
Spin Dependent Hopping and Colossal Negative Magnetoresistance in Epitaxial Nd0.52Sr0.48MnO3 Films in Fields up to 50 T, P. Wagner, I. Gordon, L. Trappeniers, J. Vanacken, F. Herlach, V.V. Moshchalkov and Y. Bruynseraede, Phys. Rev. Lett. 81 (1998) 3980.
Magnetic Phase Diagram of the Spin-Peierls Compound CuGeO3 doped with Al and Sn, S.V. Demishev, L. Weckhuysen, J. Vanacken, L. Trappeniers, F. Herlach, Y. Bruynseraede, V.V. Moshchalkov, A.A. Pronin, N.E. Sluchanko, N.A. Samarin, J. Meersschaut en L.I. Leonyuk, Phys. Rev. B 58 (1998) 6321-6329.
Critical Currents, Pinning Forces and Irreversibility Fields in (Tm1-xYx)Ba2Cu3O7 Single Crystals with Columnar Defects in Fields up to 50 T, L. Trappeniers, J. Vanacken, L. Weckhuysen, K. Rosseel, A.Yu. Didyk, I.N. Goncharov, L.I. Leonyuk, W. Boon, F. Herlach, V.V. Moshchalkov and Y. Bruynseraede, Physica C 313 (1999) 1-10.
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1D Quantum Transport in the Even-Chain Spin-Ladder Compound Sr2.5Ca11.5Cu24O21 and YBa2Cu4O8 , V.V. Moshchalkov, L. Trappeniers and J. Vanacken, Europhys. Lett. 46 (1999) 75-81.
Influence of Columnar Defects on Critical Currents, Pinning Forces and Irreversibility Fields in (Tm1-xYx)Ba2Cu3O7 Single Crystals up to 50 T, L. Trappeniers, J. Vanacken, L. Weckhuysen, K. Rosseel, W. Boon, F. Herlach, V.V. Moshchalkov, Y. Bruynseraede, A. Didyk, I. Goncharov and L. Leonyuk, “Physics and Material Science of Vortex States, Flux Pinning and Dynamics” NATO ASI series 345-355, Ed. R. Kossowsky.
Simulation and Calibration of an Open Inductive Sensor for Pulsed Field Magnetization Measurements, L. Weckhuysen, J. Vanacken, L. Trappeniers, M.J. Van Bael, W. Boon, K. Rosseel, F. Herlach, V.V. Moshchalkov, Y. Bruynseraede, Rev. Sci. Instr. 70 (1999) 2708-2710.
Normal State Resistivity of Underdoped YBa2Cu3Ox thin films and La2-xSrxCuO4 ultra-thin films under epitaxial strain, L. Trappeniers, J. Vanacken, P. Wagner, G. Teniers, S. Curras, J. Perret, P. Martinoli, J.-P. Locquet, V.V. Moshchalkov and Y. Bruynseraede, J. Low Temp. Phys. 117 (1999) 681-685.
Doped CuO2 planes in High-Tc Cuprates: 2D or not 2D ?, V.V. Moshchalkov, L. Trappeniers and J. Vanacken, J. Low Temp. Phys. 117 (1999) 1283-1287.
Ter perse 1D Quantum Transport in the Even-Chain Spin-Ladder Compound Sr2.5Ca11.5Cu24O21 and YBa2Cu4O8 , V.V. Moshchalkov, L. Trappeniers and J. Vanacken, Proc. of the second int. conf. on “Stripes and High Tc Superconductors”, Rome, June 2-6, 1998 to be published in Journal of Superconductivity
Pseudo-Gap and Crossover from the 2D Heisenberg to the even - leg Spin Ladder regime in underdoped cuprates, V.V. Moshchalkov, L. Trappeniers and J. Vanacken, Proc. of the NATO ARW on “Symmetry and Pairing in Superconductors”, Yalta , April 28-May 2, 1998.
Magnetic Field Induced Localisation and Current driven Delocalisation in Y0.5Pr0.5Ba2Cu3Ox Films, Z. Hao, B.R. Zhao, B.Y. Zhu, L. Trappeniers, J. Vanacken and V.V. Moshchalkov, submitted to Phys. Rev. Lett.
Stripe formation and disorder induced stripe fragmentation, V.V. Moshchalkov, L. Trappeniers, G. Teniers, J. Vanacken, P. Wagner and Y. Bruynseraede, submitted to Physica C.
Stripes and dimensional crossovers in high-Tc cuprates, V.V. Moshchalkov, L. Trappeniers and J. Vanacken, submitted to Physica C.
Normal State Resistivity of Underdoped YBa2Cu3O7-d Films and La2-XSrXCuO4 Ultra -Thin Films in Fields up to 60 T, L. Trappeniers, J. Vanacken, P. Wagner,
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S. Curras, G. Teniers, J. Perret, J.-P. Locquet, V.V. Moshchalkov and Y. Bruynseraede, submitted to Physica C.
Zonder leescomité Critical Parameters of High Tc Cuprates at 60 T, V.V. Moshchalkov, J. Vanacken, L. Trappeniers, K. Rosseel, D. Dierickx, P. Wagner, W. Boon, I.N. Goncharov, A. Yu Didyk, A.S. Lagutin, L.I. Leonyuk, N. Harrison, F. Herlach and Y. Bruynseraede, Physicalia 19 (1997) 205-218.
The K.U.Leuven pulsed magnet facility, L. Van Bockstal, R. Bogaerts, W. Boon, A. De Keyser, M. Hayne, L. Liang, L. Trappeniers, P. Reinders, K. Rosseel, J. Vanacken, M. Van Cleemput and A. Van Esch, Physicalia 19 (1997) 165-175
Publicaties op internationale conferenties Influence of columnar pinning defects on the critical currents in (YxTm1-x)Ba2Cu3O7 single crystals, L. Trappeniers, J. Vanacken, I.N. Goncharov, W. Boon, F. Herlach, V.V. Moshchalkov and Y. Bruynseraede, Paper presented at NATO Advanced Study Institute "Materials Aspects of High Tc superconductivity: 10 years after the discovery", Delphi, Griekenland (1996).
Experimental techniques for pulsed magnetic fields in the 60 T/20 ms range, L. Van Bockstal, R. Bogaerts, W. Boon, I. Deckers, A. De Keyser, N. Harrison, A. Lagutin, L. Li, L. Trappeniers, J. Vanacken, A. Van Esch, F. Herlach, Proceedings of Physical Phenomena at High Magnetic Fields - II, ed. Z. Fisk, L. Gor'kov, D. Meltzer and R. Schrieffer,(Conference Paper) World Scientific, Singapore (1996) 755-760. 505