Laurens W. Molenkamp

Physikalisches Institut, EP3Universität Würzburg

Overview

- HgTe/CdTe bandstructure, quantum spin Hall effect- HgTe as a Dirac system- Dirac surface states of strained bulk HgTe

band structure

D.J. Chadi et al. PRB, 3058 (1972)

fundamental energy gap

meV 30086 EE meV 30086 EE

semi-metal or semiconductor

HgTe

-1.0 -0.5 0.0 0.5 1.0k (0.01 )

-1500

-1000

-500

0

500

1000

E(m

eV) 8

6

7

-1.0 -0.5 0.0 0.5 1.0k (0.01 )

-1500

-1000

-500

0

500

1000

E(m

eV) 8

6

7

Eg

Type-III QW

VBO = 570 meV

HgCdTeHgCdTeHgTe

HgCdTe

HH1E1

QW < 63 Å

HgTe

inverted normal

band structure

conduction band

valence band

HgTe-Quantum Wells

Layer Structure

gate

insulator

cap layer

doping layer

barrier

barrierquantum well

doping layer

buffer

substrate

Au

100 nm Si N /SiO

3 4 2

25 nm CdTe

CdZnTe(001)

25 nm CdTe10 nm HgCdTe x = 0.79 nm HgCdTe with I10 nm HgCdTe x = 0.74 - 12 nm HgTe10 nm HgCdTe x = 0.7 9 nm HgCdTe with I10 nm HgCdTe x = 0.7

symmetric or asymmetricdoping

Carrier densities: ns = 1x1011 ... 2x1012 cm-2

Carrier mobilities: = 1x105 ... 1.5x106 cm2/Vs

-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 80

100

200

300

400

500

µ=1.06*106cm2(Vs)-1

nHall=4.01*1011cm-2

Q2134a_Gate

B[T]

Rxx

[]

-15000

-10000

-5000

0

5000

10000

15000

Graph2

Rxy

[]

123456

k (0.01 -1)

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

Ener

gyE(

k)(e

V)

k || (1,1)k || (1,0)k = (kx,ky)

k || (1,1)k || (1,0)k = (kx,ky)

4 nm QW 15 nm QW

normal

semiconductor

inverted

semiconductor

1 2 3 4 5 6

k (0.01 -1)

-0.20

-0.15

-0.10

-0.05

0.00

0.50

0.10

0.15

0.20

E2

H1H2

E1L1

0.6 0.8 1.0 1.2 1.4

dHgTe (100 )

E2E2

E1E1H1H1

H2H2H3H3

H4H4 H5H5

H6H6L1L1

Band Gap Engineering

Bandstructure HgTe

E

k

E1

H1

invertedgap

4.0nm 6.2 nm 7.0 nm

normalgap

H1

E1

B.A Bernevig, T.L. Hughes, S.C. Zhang, Science 314, 1757 (2006)

QSHE, Simplified Picture

normalinsulator

bulk

bulkinsulating

entire sampleinsulating

m > 0 m < 0

QSHE

-1.0 -0.5 0.0 0.5 1.0 1.5 2.0103

104

105

106

G = 2 e2/h

Rxx

/

(VGate- Vthr) / V

Observation of QSH Effect

(1 x 0.5) m2

(1 x 1) m2(2 x 1) m2

(1 x 1) m2

non-inverted

M. König et al., Science 318, 766 (2007).

1 m 2 m

1 2 3

6 5 4

1 m

1 m

5 m

1 2

34

(a) (b)

Verify helical edge state transport

Multiterminal /Non-local transport samples

Multi-Terminal Probe

210001121000012100001210000121100012

T

heIG

heIG

t

t

2

23

144

2

14

142

232

generally

22 2)1(

ehnR t

3exp4

2 t

t

RR

heG t

2

exp,4 2

Landauer-Büttiker Formalism normal conducting contacts no QSHE

Configurations would be equivalent in quantum adiabatic regime

-1 0 1 2 30

5

10

15

20

25

30

35

40

R (k

)

V* (V)

I: 1-4V: 2-3

R14,23=1/2 h/e2

R14,14=3/2 h/e2

I: 1-3V: 5-6

R13,13=4/3 h/e2

R13,54=1/3 h/e2

-1 0 1 2 3 4

V* (V)

Multi-Terminal Measurements

A. Roth et al., Science 325, 294 (2009).

0.0 0.5 1.0 1.5 2.00

5

10

15

20

25

R

(k

)

V* (V)

I: 1-4V: 2-3

1

3

2

4

R14,23=1/4 h/e2

R14,14=3/4 h/e2

Non-Local data on H-bar

A. Roth et al., Science 325, 294 (2009).

H-bar for detection of Spin-Hall-Effect

(electrical detection through inverse SHE)

E.M. Hankiewicz et al ., PRB 70, R241301 (2004)

200 nm 200 nm

-2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.00

100

200

300

400

500

0

2

4

6

8

10

12

R12

,36

()

Vg (V)

I (n

A)

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.00.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

T = 2 K

Rno

nloc

al /

kVGate / V

I / n

A

n-conductingp-conducting

insu

latin

g

– Suppress non-local QSHE using long leads or narrow wires

– Intrinsic metallic SHE only shows up for holes: larger spin-orbit

– Amplitude in agreement with modeling (E. Hankiewicz, J. Sinova)

H-bar experiments

C. Brüne et al., Nature Physics 6, 448 (2010).

-1,5 -1,0 -0,5 0,0 0,5 1,0 1,5 2,00

100

200

300

400

500

R21

,36

()

Vg* (V)

-1,5 -1,0 -0,5 0,0 0,5 1,0 1,5 2,00

100

200

300

400

500

R36

,21

()

Vg* (V)-1,5 -1,0 -0,5 0,0 0,5 1,0 1,5 2,0

0,0

0,5

1,0

1,5

2,0

R45

,21

(k

)

Vg* (V)

-1,5 -1,0 -0,5 0,0 0,5 1,0 1,5 2,00,0

0,2

0,4

0,6

0,8

1,0

1,2

1,4

1,6

R21

,45

(k

)

Vg* (V)

Q2197(a)

p-cond. insul. n-cond.

p-cond. insul. n-cond.

p-cond. insul. n-cond.

Q2198(b)

p-cond. insul. n-cond.

I

V

I

V

I

V

I

V

C. Brüne et al., Nature Physics 6, 448 (2010).

Bandstructure HgTe

E

k

E1

H1

invertedgap

4.0nm 6.2 nm 7.0 nm

normalgap

H1

E1

B.A Bernevig, T.L. Hughes, S.C. Zhang, Science 314, 1757 (2006)

Dispersion at d=dc is Dirac-like

For well thickness d=6.3 nm, the gap closes,especially the conductionband shows a linear dispersion: single Dirac cone

Zero mode dispersion

Zero mode spin splitting allows to select sample at dc.

B. Büttner et al., Nature Physics doi:10.1038/nphys1914

Large g-factor (g=55) responsible for spin slitting already at low fields.Hall quantization reflects single valley character of the band structure:a HgTe quantum well at d=6.3 nm is half-graphene.

B. Büttner et al., Nature Physics doi:10.1038/nphys1914

Quantum Hall effect shows Berry phase

Landau-fan

Color coded: gate voltage derivative of longitudinal resistivity.Fits: left – 8-band Kane model, right – Dirac Hamiltonian

B. Büttner et al., Nature Physics doi:10.1038/nphys1914

Dirac peak at B=0

Peak width and mobilities comparable with/better than free standing grapheneScattering mechanisms: probably mass fluctuations + Coulomb (fit is Kubo model)

Mobility for finite Dirac mass

B. Büttner et al., Phys. Rev. Lett. 106, 076802 (2011).

Originally increase in mobility from reduced impurity scattering, then changeover to behavior due to well width (Dirac mass) fluctuations.

Mobility for finite Dirac mass

B. Büttner et al., Phys. Rev. Lett. 106, 076802 (2011).

Modeling by Grigory Tkachov and Ewelina Hankiewicz:Mass and disorder induce backscattering of Dirac fermions.

Bulk HgTe as a 3-D Topological ‚Insulator‘

-1.0 -0.5 0.0 0.5 1.0k (0.01 )

-1500

-1000

-500

0

500

1000

E(m

eV) 8

6

7

-1.0 -0.5 0.0 0.5 1.0k (0.01 )

-1500

-1000

-500

0

500

1000

E(m

eV) 8

6

7

Bulk HgTe is semimetal,

topological surface state overlaps w/ valenceband.

k(1/a)

E-E

F(eV

)ARPES:

Yulin Chen, ZX Shen, Stanford

C. Brüne et al., Phys. Rev. Lett. 106, 126803 (2011).

70 nm layer on CdTe substrate:coherent strain opens gap

0 2 4 6 8 10 12 14 160

2000

4000

6000

8000

10000

12000

14000

16000

0

2000

4000

6000

8000

10000

12000

14000

Rxx (SdH)

Rxx

in O

hm

B in Tesla

Rxy (Hall)

Rxy

in O

hm

Bulk HgTe as a 3-D Topological ‚Insulator‘

@ 20 mK: bulk conductivity almost frozen out - Surface state mobility ca. 35000 cm2/Vs

C. Brüne et al., Phys. Rev. Lett. 106, 126803 (2011).

2 4 6 8 10 12 14 160

2

4

6

8

10

12

14

xy [e

2 /h]

B [T]

Bulk HgTe as a 3-D Topological ‚Insulator‘

@ 20 mK: same data, plotted as conductivity

3D HgTe-calculations

2 4 6 8 10 12 14 160

2000

4000

6000

8000

10000

2.73.54.45.67.69.711 33.94.96.78.510.112

experiment

Rxx

in O

hm

B in Tesla

n=3.7*1011 cm-2

n=4.85*1011 cm-2

n=(4.85+3.7)*1011 cm-2

DO

S

Red and blue lines : DOS for each of the Dirac-cones with the corresponding fixed 2D-density,Green line: the sum of the blue and red lines

C. Brüne et al., Phys. Rev. Lett. 106, 126803 (2011).

Conclusions– HgTe quantum wells: normal and inverted gap, linear (Dirac) dispersion

– First observation of Quantum Spin Hall Effect

– At d=dc, a HgTe QW is ideal model system for zero massDirac fermion physics

– Can conveniently study Dirac fermions w/ finite Dirac mass

– Strained 3D layers show QHE of topological surface statesCollaborators:Bastian Büttner, Christoph Brüne, Hartmut Buhmann, Markus König, Matthias Mühlbauer, Andreas Roth, Volkmar Hock

Theory: Alina Novik, Chaoxing Liu, Ewelina Hankiewicz , Grigory Tkachov,Patrick Recher, Björn Trauzettel (all @ Würzburg), Jairo Sinova (TAMU), Shoucheng Zhang, Xiaoliang Qi (Stanford)

Funding: DFG (SPP Spintronics, DFG-JST FG Topotronics), Humboldt Stiftung, EU-ERC AG