DYNAMICS OF VIBRATIONS IN AMORPHOUS SILICON

DYNAMICS OF VIBRATIONS

IN AMORPHOUS SILICON

DYNAMICS OF VIBRATIONS

IN AMORPHOUS SILICON

DYNAMICA VAN VIBRATIES

IN AMORF SILICIUM

(met een samenvatting in het Nederlands)

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de Universiteit Utrechtop gezag van de Rector Magnificus, prof. dr. H.O. Voorma,ingevolge het besluit van het College voor Promoties in het open-baar te verdedigen op maandag 6 december 1999 des middagste 2.30 uur

door

MARJOLEIN VAN DER VOORT

geboren op 9 februari 1974, te Leiden

Promotor: prof. dr. J.I. Dijkhuis

Faculteit Natuur- en Sterrenkunde,Debye Instituut, Universiteit Utrecht

CIP-GEGEVENS KONINKLIJKE BIBLIOTHEEK, DEN HAAG

Voort, Marjolein van der

Dynamics of Vibrations in Amorphous Silicon(Dynamica van Vibraties in Amorf Silicium)/Marjolein van der Voort. - Utrecht: Universiteit Utrecht,Faculteit Natuur- en Sterrenkunde, Debye InstituutProefschrift Universiteit Utrecht. Met samenvatting in het Nederlands.ISBN 90-393-2257-0

“Who doesn’t believe in magic will never find it.”

Roald Dahl

The work described in this thesis has been supported by the “Stichting voor Fun-damenteel Onderzoek der Materie (FOM)”, which is financially supported by the“Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO)”.

CONTENTS

1 Introduction 91.1 Thermal properties of amorphous solids . . . . . . . . . . . . . . . 101.2 Studying vibrational dynamics . . . . . . . . . . . . . . . . . . . . 111.3 Amorphous silicon . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.4 Outline of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 Pulsed anti-Stokes Raman experiments in hydrogenated amorphoussilicon 172.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 Experimental details . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.1 Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2.2 Generation and detection of vibrations . . . . . . . . . . . . . 212.2.3 Time-resolved measurements - experimental setup . . . . . . 22

2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.3.1 Pump-probe anti-Stokes Raman spectroscopy . . . . . . . . . 252.3.2 Pump-probe luminescence measurements . . . . . . . . . . . 28

2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.4.1 Non-equilibrium phonons . . . . . . . . . . . . . . . . . . . 322.4.2 Slow electronic contribution to the vibrational signals . . . . . 352.4.3 Phonon decay in amorphous silicon . . . . . . . . . . . . . . 43

2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3 Dynamics of vibrations in a mixed amorphous-nanocrystalline Sisystem 533.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.2 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.2.1 Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.2.2 Time-resolved Raman spectroscopy . . . . . . . . . . . . . . 563.2.3 Phonon-induced luminescence experiments . . . . . . . . . . 57

3.3 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . 613.3.1 Raman experiments . . . . . . . . . . . . . . . . . . . . . . . 613.3.2 Experiments with the ruby phonon detector . . . . . . . . . . 66

7

8 Contents

3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.4.1 Raman spectra of a-nc-Si:H . . . . . . . . . . . . . . . . . . 693.4.2 Frequency-dependent lifetime of TO phonons . . . . . . . . . 723.4.3 Long-lived TA phonons . . . . . . . . . . . . . . . . . . . . 783.4.4 Long-lasting phonon induced luminescence signals . . . . . . 79

3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4 Infrared Free Electron Laser experiments on SiH vibrations in a-Si:H 874.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.2 Experimental procedures . . . . . . . . . . . . . . . . . . . . . . . 89

4.2.1 Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 894.2.2 Pump-probe methods . . . . . . . . . . . . . . . . . . . . . . 90

4.3 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . 954.3.1 Temperature dependence . . . . . . . . . . . . . . . . . . . . 974.3.2 Wavelength dependence . . . . . . . . . . . . . . . . . . . . 101

4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1044.4.1 Comparison with other experiments . . . . . . . . . . . . . . 1044.4.2 Mechanisms of energy relaxation . . . . . . . . . . . . . . . 1064.4.3 Mechanisms of dephasing . . . . . . . . . . . . . . . . . . . 112

4.5 Conclusions and perspectives . . . . . . . . . . . . . . . . . . . . . 117

References 119

Achtergrond en overzicht in een andere taal 125

Tot Slot 129

Curriculum Vitae 131

CHAPTER 1

Introduction

Nature has decided to create solids in such a way, that the constituting atoms forman ordered, crystalline, or disordered, amorphous, structure. Man has always beensurrounded by both variants of condensed matter, but in spite of that, for a longtime solid-state research has been dealing almost entirely with crystalline mate-rials. A reason for this can be found in the absence of long-range order in thestructure of amorphous solids. The very existence of a periodic lattice in crystalshas allowed for the development of an analytical framework that is used to de-scribe many of the physical phenomena encountered [1]. The lack of periodicityin amorphous systems makes their theoretical description much more involved [2].

During the last three decades, however, the interest in amorphous solids hasbeen rapidly growing [3]. Indeed, experiments have shown that these materialsexhibit many features that are unique to them and not shared by their crystallinecounterparts. On the other hand, amorphous and crystalline materials have proper-ties in common (like the bandgap in semiconductors), the explanation of which inconventional solid-state physics is generally related to the periodicity of the crystallattice. Apparently, the role of long-range order in the exact nature of solids hasbeen exaggerated in the conventional descriptions. But the fact that so many as-pects of crystals can be understood if their lattices are considered periodic, demon-strates that order in itself has to be of importance. The question is what propertiesare related to order on what length scale. Therefore, a thorough understanding ofthe physics of amorphous materials will contribute to the understanding of crystalsas well. This notion, and their striking universal properties make amorphous solidsfascinating materials to study.

One of the many interesting aspects concerns the behavior of vibrations inamorphous systems. Phonons, in the way they occur as extended propagating vi-brations of the periodic lattice in crystals, in these materials only form a minor partof the vibrational spectrum. But of course, also the atoms in amorphous solids os-cillate extensively, thereby contributing for example to the conduction of heat. Aswill be discussed in somewhat more detail below, the thermal properties of amor-

9

10 Chapter 1

phous materials are anomalous compared to those of crystals, and are, especiallyfor temperatures beyond ∼ 10 K, not very well understood. The underlying prob-lem in fact is the absence of a generally accepted picture of the high-frequencyvibrational excitations that are responsible for the thermal transport at these tem-peratures. For this reason, several years ago a project was started in our groupwith the aim to gain insight into the dynamics of high-frequency vibrations inamorphous semiconductors [4]. This thesis describes our latest experimental re-sults.

1.1 Thermal properties of amorphous solids

As early as 1911, Eucken [5] recognized that the thermal properties of non-crystalline solids are quite different from those of crystalline materials. Nowa-days, it is well-established that amorphous and glassy solids exhibit a set of ther-modynamic characteristics not found in crystals [6]. Surprisingly, these so-calledanomalies have appeared to be universal, i.e. only weakly dependent on the ex-act chemical composition of the amorphous material. Distinguished examples ofthe anomalous thermal properties are the excess specific heat at low temperatures(T 100 K), the linear increase of the heat capacity with decreasing temperaturefor T 1 K, the peak in the reduced heat capacity at temperatures between 2 Kand 100 K, and the ‘plateau’ in the thermal conductivity in the same temperatureregion. In general, the thermal conductivity of amorphous materials is orders ofmagnitude lower than in their crystalline counterparts.

The ‘two-level-system’ (TLS) model [7] has turned out to be successful indescribing the thermal properties of amorphous solids at the lowest temperatures(T ≤ 1 K). For higher temperatures, this model has been extended to the ‘soft-potential’ model [8] which explains the thermal behavior from 1 K up to 100 K.A central hypothesis of both models is the existence of local low-frequency ex-citations that interact with long-wavelength propagating acoustic modes. For in-stance, in the TLS model, the low-frequency excitations are described by asymmet-ric double-well potentials (two-level-systems) that correspond to two equilibriumpositions of a group of atoms in the glassy system.

Although the two models together can explain the anomalies observed up to100 K phenomenologically, there is no general consensus about the microscopicorigin of the two-level-systems and soft potentials. Closely related with this prob-lem are the controversies concerning the nature of the higher frequency vibrationalexcitations. Indeed, it is generally accepted that vibrations in the long-wavelengthlimit, like the acoustic modes considered by the above mentioned models, act likephonons in a crystal, since on long length scales the disorder averages out, and the

Introduction 11

amorphous solid can be considered as a homogeneous elastic medium. For shorterwavelengths, however, the phonons are scattered off structural irregularities. Thescattering efficiency increases with the phonon frequency, and at some frequencythe phonon mean free path becomes of the order of the phonon wavelength. Thecross-over, also known as the Ioffe-Regel limit [9] typically occurs for frequenciesof ∼ 1 THz, which corresponds to the temperature region where the plateaus in thethermal conductivity are found. What happens with vibrations of these and higherenergies is still a matter of intense debate. It has been suggested that vibrationsabove the Ioffe-Regel limit are strongly localized [10, 11, 12, 13]. That is, thatthe Ioffe-Regel limit coincides with a mobility edge, by analogy with the acceptedideas of Anderson-localization for electrons [14]. However, other authors have re-jected the existence of a mobility edge for vibrations, and claim that also in glasses,high-frequency vibrations are propagating waves [15]. Even interpretations havebeen put forward that fall in between these extremes, suggesting that vibrationalstates above the Ioffe-Regel limit are neither propagating plane waves, nor local-ized modes, but diffusive excitations: they carry heat by means of diffusion, whena temperature gradient is present [16, 17, 18].

Not only measurements of the thermal conductivity and specific heat of glassesindicate the existence of characteristic vibrational excitations not found in crystals.In the frequency region that corresponds to the plateau in the thermal conductivity,results of Raman [19] and inelastic incoherent neutron scattering experiments [20]show the broad so-called ‘Boson peak’. The occurrence of this peak is usually as-signed to an excess of the vibrational density of states in this frequency region, andhas often been associated with the soft oscillators described by the soft-potentialmodel [8]. However, conclusions drawn from these and more recent inelastic x-ray scattering [21, 22] measurements are not decisive in determining the nature ofthese vibrational modes.

1.2 Studying vibrational dynamics

In the case of crystals, experiments on the dynamics of high-frequency phononshave proven to give a wealth of information on elementary phonon scattering pro-cesses [23]. Some examples of the many phenomena investigated include bal-listic propagation, spectral and spatial diffusion, and anharmonic decay. In non-crystalline solids, on the other hand, the behavior of non-equilibrium phonons hasonly been examined on a limited scale. Most experiments that have been reportedaddress the transport and scattering of acoustic modes for frequencies between∼ 0.1 and 1 THz [24]. The results of those studies indicate that the phonon scat-tering is much stronger than in crystals, and for a great deal inelastic, suggesting

12 Chapter 1

again an excess of the vibrational density of states of glasses compared to crystalsfor these frequencies.

As noted above, the behavior of vibrations with frequencies higher than 1 THzis extremely interesting, since for these frequencies, vibrations are scattered tosuch a degree, that the concept of a phonon with a well-defined wave vector isnot applicable any more. But what is the concept that applies instead? In spiteof such intriguing questions, the number of dynamical experiments investigatinghigh-frequency (ν 1 THz) vibrations in non-crystalline solids is even more lim-ited than for lower frequencies. In our group, pulsed Raman spectroscopy has beenused to examine high-frequency (3-17 THz) phonon-like modes in hydrogenatedand pure amorphous Si for the first time [4, 25]. The results obtained by Scholtenet al. give even after some years rise to vivid discussions. For example, vibrationalanharmonic lifetimes were determined that were up to three orders of magnitudelonger than for vibrations of the same frequency in crystals, the interpretation ofwhich is still a matter of debate [18, 26, 27]. These and other remarkable findingsthat will be referred to throughout this thesis have indicated new directions for re-search in this field. With the experiments presented in this thesis we have exploredseveral of these new directions.

To further investigate the dynamics of high-frequency phonon-like vibrationsin amorphous silicon, we have used several pulsed laser techniques, based on Ra-man spectroscopy, phonon-induced luminescence, and infra-red spectroscopy.

Raman and infrared spectroscopy are particularly suitable techniques to studyvibrations in amorphous materials. Due to the absence of periodicity, all vibra-tional eigenvectors are allowed to contribute to the spectra, and not just zero-wavevector states, like in crystals [28]. However, although the selection rulesknown in crystals are relaxed, they are not completely absent. Consequently, notall parts of the vibrational density of states show up equally strongly in the spec-tra [29]. For instance, in the case of amorphous semiconductors like a-Si, Ramanspectra show much stronger contributions from the lowest-frequency vibrationalbands than infrared absorption spectra for the same frequency region. On the otherhand, vibrations of the SiH bond in hydrogenated a-Si are much more pronouncedin the infrared absorption spectra. So depending on the type of vibration studied,either infrared or Raman spectroscopy is the more suitable technique. In the ex-periments described in this thesis, Raman spectroscopy is applied to investigatethe behavior of excitations involving the vibration of Si atoms with respect to eachother. For convenience, these vibrations are called ‘phonons’, by analogy with thevibrations that occur at the same frequencies in crystalline Si. In addition, infraredtechniques are used to study the dynamics of vibrations supported by the SiH bondof a-Si:H.

Introduction 13

Intermezzo In the remainder of this section we list some standard expressionsconcerning the vibrational density of states and occupation numbers, that are usedthroughout the next three chapters.

Since the goal of our experiments is to study the dynamics of vibrations, notthe vibrational density of states is the quantity of prime interest, but rather the occu-pation number nω of the modes of frequency ω in a certain region of the spectrum.In fact, we are interested in the temporal evolution of a non-equilibrium vibra-tional population created instantaneously at time zero. In the case of the Ramanexperiments, we use the well known fact that the first-order anti-Stokes Raman in-tensity, IAS(ω), is proportional to nω and the corresponding Stokes intensity, IS, tonω +1. So if both the Stokes and anti-Stokes spectra are measured, the occupationnumbers can be computed by

nω = IAS/(IS − IAS). (1.1)

At some occasions, however, it is useful to know the phonon density of statesρ(ω). For amorphous solids, ρ(ω) can be derived empirically from the StokesRaman spectrum measured for a phonon population in thermal equilibrium, usingthe relation [28]

IS(ω) ∝(

1− ωωL

)3 [nω + 1

ω

]C(ω)ρ(ω). (1.2)

Here, ωL corresponds to the frequency of the laser light exciting the spectrum,and C(ω) is the weighting factor, i.e. coupling parameter, between the phonondensity of states and the equilibrium Stokes Raman spectrum. Further, in thermalequilibrium nω of course refers to the Bose-Einstein phonon occupation numberfor temperature T :

nω,BE =1

exp[ωkBT

]−1

. (1.3)

In the case of amorphous silicon, the coupling parameter C(ω) has been obtainedby combining Raman and inelastic neutron scattering data [30]. A similar com-bination of experiments has shown that, except for low frequencies in the Ramanspectra (ω < 3 THz), C(ω) has a comparable frequency dependence for samplesprepared in different ways [31]. Apparently, variations in the Raman spectra ofdifferent types of a-Si(:H) are caused by variations in the vibrational density ofstates, and not by variations in C(ω). We make use of the linear relation betweenC(ω)/ω and ω calculated by Berntsen [32]: C(ω)/ω ≈ 1 ·10−3 + 0.35ω, where ωis expressed in wavenumbers.

14 Chapter 1

1.3 Amorphous silicon

Amorphous silicon was first prepared in 1824 by Berzelius. Several decades laterDeville succeeded in preparing crystalline Si [33]. However, it took until the late1940s before the technological importance of silicon was realized. At that time, theamorphous variant was believed to be useless, because of its high defect density,limiting the conductivity and preventing the material from doping.

In the late 60s it was discovered that thin amorphous silicon films can be grownby glow-discharge deposition of silane [34]. This hydrogenated material (a-Si:H)appeared to have a much lower defect density, due to the passivation of unpairedelectrons (‘dangling bonds’) by hydrogen. When Spear and LeComber discov-ered that a-Si:H can be substitutionally doped during deposition [35], the materialbecame really promising. Indeed, a-Si:H has become a low-cost solar panel ma-terial. However, wider use is impeded by its degradation under illumination, alsocalled Staebler-Wronski effect [36]. The metastability of a-Si:H has been a topicof intense research for more than twenty years now, and new preparation meth-ods are still being developed [37]. Gradually, the efforts have resulted in a deeperunderstanding of the structural, optical and electronic properties of this prototypeamorphous semiconductor [38].

As already noted, amorphous silicon has a bandgap. In several aspects thisgap is different from the bandgap of crystalline Si. Crystalline Si has an indicrectbandgap of 1.1 eV. Due to the absence of periodicity in amorphous Si, the mo-mentum conservation rules are relaxed. Consequently, a-Si(:H) acts as a directband gap semiconductor, and has a gap of ∼ 1.7 eV. The distribution of Si-Si bondangles and bond lengths in the amorphous structure gives rise to electronic statesnear the valence and conduction band edges (known as ‘tail states’). In addition,the presence of dangling bonds result in a distribution of defect states near thecenter of the band gap.

In this thesis, a-Si:H is used as a model system for examining the vibrationaldynamics of amorphous semiconductors. Scholten has compiled the available datafor the thermal properties of a-Si and a-Si:H, to investigate if amorphous siliconcan be regarded as a typical amorphous solid [4]. He concluded that most ofthe thermal properties characteristic for amorphous materials are also observed ina-Si. For a-Si:H, unfortunately, not enough data are present to draw this conclu-sion. Anyway, the limited set of data available are consistent with what one wouldexpect for an amorphous solid.

Introduction 15

1.4 Outline of this thesis

The aim of this thesis is to elucidate the dynamics of high-frequency vibrationsin amorphous silicon. For this purpose, we have employed various pulsed lasertechniques based on Raman spectroscopy, phonon-induced luminescence, andinfra-red spectroscopy, and studied the temporal evolution of non-equilibrium vi-brational populations in hydrogenated amorphous silicon. The vibrational dy-namics have been examined in two types of a-Si:H, and in a mixed amorphous-nanocrystalline Si system.

In Chapter 2, we report on investigations of the temporal evolution of non-equilibrium phonon populations in a-Si:H. We studied the dynamics of phononswith a frequency between ∼ 3 and 17 THz, both in a-Si:H layers grown byplasma enhanced chemical vapor deposition (PE) and in films prepared by hot-wire assisted chemical vapor deposition (HW). Phonons were created during therelaxation and recombination of optically excited charge carriers, and detected bymeans of anti-Stokes Raman spectroscopy. Our observations are in agreement withthe results reported by Scholten et al., and confirm that high-frequency phonons ina-Si:H can have lifetimes that are several orders of magnitude longer than in crys-tals. This result is obtained both for PE- and HW-grown a-Si:H. In addition, in thePE sample we observed an even slower contribution to the Raman signal, whichwas not present in the HW layers. We propose a model to explain this backgroundas resulting from slow electronic processes in the amorphous semiconductor. Atthe end of Chapter 2, existing theories that yield predictions of the lifetimes ofphonons in amorphous materials are reviewed and discussed. It turns out that noneof them is able to satisfactorily describe all our results, probably because the struc-ture of real a-Si:H is different than assumed in the models. We speculate that thelong phonon lifetimes are caused by the medium-range order of a-Si:H, i.e. that theanharmonic decay of the high-frequency phonons is suppressed for phonons resid-ing inside nanometer sized regions (paracrystals) that are significantly decoupledfrom their surrounding.

In Chapter 3, the effects of the presence of nanometer sized ordered regionson the phonon dynamics are investigated in more detail. The chapter describes re-sults of pulsed Raman and phonon-induced luminescence experiments on a mixedamorphous-nanocrystalline silicon system (a-nc-Si:H). With these experiments,the decay and transport of non-equilibrium phonons in a-nc-Si:H were examinedand compared with the behavior of phonons of the same frequencies in a-Si:H.From the Raman measurements, we find that phonons of the highest frequenciessupported by the crystallites (∼ 17 THz) have a frequency-dependent lifetime. Weexplain this observation as resulting from the polydispersity of the nc-Si crystal-

16 Chapter 1

lites. A simple model is presented, showing that the frequency-dependence canbe understood as caused by a suppression of the anharmonic decay of the high-frequency phonons confined in the nanometer sized regions. The suppression isthe largest in the smallest particles. Another conclusion drawn from the Ramanresults is that phonons at lower frequencies (∼ 3 to 7 THz) have a lifetime atleast one order of magnitude longer than that of phonons of the same frequency ina-Si:H. This effect is also accounted for by the model. In the phonon-induced lu-minescence experiments, the diffusion of 29-cm−1 phonons through the a-nc-Si:Hand a-Si:H material was examined. Transport through the a-nc-Si:H film appearedto be much slower than through the a-Si:H layer. We explain these results againas effects of phonon confinement on the phonon dynamics and relate them to thelong phonon lifetimes observed in a-Si:H.

In the experiments presented in Chapter 4, vibrations were excited by directconversion of infrared photons into vibrational quanta, without the intervention ofcharge carriers. The infrared radiation was produced by a free-electron laser. Bymeans of vibrational photon-echo, transient-grating, and transient-transmission ex-periments we studied the dynamics of SiH vibrations in a-Si:H. In particular, thevibrational population relaxation and phase relaxation of the SiH stretching modewas investigated for temperatures between 10 and 300 K. The results indicate thatthe vibrational energy relaxes directly into SiH bending modes and Si phonons,with a distribution of rates determined by the amorphous surrounding. Conversely,the pure dephasing appears to be single exponential, and can be modeled by de-phasing via two-phonon interactions.

CHAPTER 2

Pulsed anti-Stokes Raman experimentsin hydrogenated amorphous silicon

In this chapter, we report on investigations of the temporal evolution of non-equilibrium phonon populations in hydrogenated amorphous silicon held at 1.8 K.Phonons were created during the relaxation and recombination of optically excitedcharge carriers, and detected by means of anti-Stokes Raman spectroscopy. Timeresolution was achieved with a pump-probe configuration. In all samples studied,decay times of ∼ 70 ns were obtained for LA and TO vibrations. The TA modesappeared to have lifetimes shorter (< 10 ns) than could be resolved with the exper-imental setup. In films prepared by plasma enhanced chemical vapor deposition,we observed an additional slowly decaying ( 100 ns) contribution to the Ramansignal, which was not present in layers grown by hot-wire assisted chemical vapordeposition. We propose a model to explain this slow background as resulting fromlaser-induced fast nonradiative recombination of mobile with localized carriers.Results of pulsed luminescence experiments support this model. Finally, varioustheories will be reviewed and discussed that yield predictions of the lifetimes ofphonons in amorphous materials.

2.1 Introduction

An effective method to investigate dynamics of high-frequency phonons in amor-phous semiconductors is provided by time-resolved Raman spectroscopy. Scholtenet al. [25] employed this technique for the first time to examine transient phononpopulations in (hydrogenated) amorphous Si held at 2 K, in a frequency range of3 to 17 THz (100 to 550 cm−1). Especially interesting about this energy range is,that it corresponds to the region where the exact nature of the vibrational excita-tions (phonons) is still unclear. It is this frequency range namely, where phononsare scattered to such a degree, that the phonon mean free path approaches its wave-length (see Chapter 1).

17

18 Chapter 2

The results of the studies by Scholten et al. are still rather striking and underdebate. First, the order of magnitude of the population-decay times measured forphonons of the highest frequencies appeared to be about four orders of magnitudelonger than for phonons of the same energies in crystalline Si (having a lifetime of∼10 ps [39]). Secondly, the dependence of the lifetime on phonon energy turnedout to be opposite to what one generally observes in crystals: the measured de-cay times in a-Si, namely, increase with increasing energy. Identical results wereobtained in a-Si and a-Si:H.

To explain these remarkable findings, it was proposed that in a-Si(:H), phononswith an energy higher than a critical value are strongly localized. It was suggestedthat a crossover (mobility edge) from extended to localized vibrations occurs at aphonon frequency well below the frequencies addressed in the experiments. In thisexplanation, the reduced decay rate is due to the fact that the localized excitationsare decoupled from each other. A quantitative analysis of the data was presented interms of the fracton model, yielding good agreement between the measurementsand the theoretical predictions, with a reasonable value for the anharmonic cou-pling constant [40].

On the other hand, the outcome of computer simulations by Fabian et al. [18]disagree with the above-mentioned experimental results, and are at variance withthe idea of localization as presumed in the fracton model. Anharmonic decay rateswere calculated for vibrational states in a 216-atom model of a-Si. It was foundthat the high-frequency modes decay on picosecond time scales, and that at lowtemperatures, vibrational lifetimes decrease with increasing frequency. So in con-trast with the experimental observations, the decay is even faster than for the samemodes in c-Si, and the frequency dependence is similar to the dependence in crys-tals. Further, the existence of a mobility-edge and crossover from extended to lo-calized modes was confirmed. However, the crossover frequency was determinedto exceed the highest frequencies studied by Scholten et al. Moreover, accordingto the calculations, even the localized modes have picosecond lifetimes.

An important difference between the calculations and the experiments lies inthe influence of the electronic properties of the amorphous semiconductor on thevibrational dynamics. In the experiments, vibrations are generated during the re-laxation of optically excited charge carriers. At low temperatures, the electronicexcitations may persist for very long (ms) times [42], which complicates the in-terpretation of the observed vibrational signals. In the computer models of a-Si,electronic processes are not considered at all [27]. Another point is that both inthe fracton model and in the computer simulations, a-Si is assumed to be statis-tically homogeneous, which, as some authors claim, is not realistic [43]. As aconsequence, in both approaches the possibility is ignored that structural inhomo-

Pulsed anti-Stokes Raman experiments ... 19

geneities, like voids, affect the vibrational dynamics in a-Si(:H).In this chapter, we present results of pulsed luminescence and anti-Stokes Ra-

man experiments on two types of a-Si:H. Due to modifications of the experimentalsetup used by Scholten et al., we were able in the present study to monitor the tem-poral evolution of a vibrational population during a much longer period after itscreation and over a larger frequency range than was possible previously. Scholtenin some cases extracted lifetimes from the temporal profile of Raman signals thatlasted for ∼ 10 ns, which is up to one order of magnitude shorter than the lifetimeshe obtained [40]. Such problems were not encountered this time, because the fulldecay could be monitored.

Finally, we investigated for the first time the influence of electronic processeson the vibrational signals. The interplay of electronic and vibrational processes ina-Si:H has received much attention recently [44], because of its suspected role inthe light-induced degradation of this material, also known as the Staebler-Wronskieffect [36, 45].

2.2 Experimental details

2.2.1 Samples

For the pulsed anti-Stokes Raman experiments, device-quality intrinsic a-Si:H lay-ers were prepared in two different ways. The first type of samples (PE) was grownby means of plasma enhanced chemical vapor deposition in a system describedby Madan et al. [46]. One micrometer of a-Si:H material was deposited on a0.5-mm-thick crystalline silicon wafer held at a temperature of 320C. The layerscontained about 11 at.% hydrogen. Other samples (HW) with thicknesses of 1.0and 1.5 µm were prepared by hot-wire assisted chemical vapor deposition [47],again on c-Si substrates, held at 450 C. The hydrogen concentration of the HWmaterial amounted to 8 at.%.

The thickness of the a-Si:H samples was chosen to be much larger than thepenetration depth (∼ 100 nm) of the green light used in the measurements. Animportant advantage of c-Si as a substrate material is its large thermal conductivity.At low temperatures, c-Si acts as a heat sink for the vibrations generated in theamorphous film. Another advantage is that c-Si, different than standard substrateslike Corning glass, does not exhibit luminescence

In Fig. 2.1 typical Stokes Raman spectra of the PE and HW samples arepresented, as obtained with a standard Raman setup used for characterization ofa-Si:H and related materials. Both spectra display the typical broad lines that aregenerally observed in the Raman spectra of (hydrogenated) a-Si [48]. The bandscentered at 140, 300, and 480 cm−1, are denoted as transverse acoustic (TA), lon-

20 Chapter 2

0 200 400 600

b)

Sto

kes

Ram

an in

tens

ity

wavenumber (cm-1)

a)

Figure 2.1 Stokes Raman spectra of a) PE-grown a-Si:H and b) HW-deposited a-Si:H.The spectra were excited with the 514.5-nm line of an argon-ion laser, analyzed with aSPEX 1877 0.6-m Triple Spectrometer, and recorded by a liquid-nitrogen cooled CCDcamera (EG&G).

gitudinal acoustic (LA), and transverse optic (TO) vibrations, respectively, and theshoulder visible around 400 cm−1 as LO-like modes. It should be noted that thisnotation is only used for convenience, to indicate vibrational energies, and is basedon the analogy of these spectra with the density of phonon states in crystalline Si.No wavevectors can be assigned to high-frequency vibrations in amorphous solids,and since they also lack a clear polarization [49], the labels used are rather arbi-trary.

At first sight, the two curves of Fig. 2.1 do not exhibit large differences. Care-ful examination of the data, however, shows that in the region of the TO peak, thespectrum of the HW film is slightly narrower than that of the PE sample (HWHMof 32 ± 0.5 cm −1 as compared to 34 ± 0.5 cm−1). It has been demonstrated thatthe width of the TO band of amorphous silicon is proportional to the width of thedistribution of bond angles present in the structure, and is therefore a measure ofthe short-range order of the material [50]. Although the difference between themeasured widths seems rather small, one generally concludes from these numbersthat the material of the HW sample is less disordered than that of the PE layer [51].


This makes a comparison of the observed vibrational dynamics in the two types ofa-Si:H of particular interest.

Feenstra [52] characterized the electronic properties of a-Si:H layers preparedin the same apparatus and under the same deposition conditions as the ones weused. He determined the (Tauc) band gaps to be 1.7 eV for the HW layer, and1.78 eV for the PE film. These numbers will be used in the quantitative analysisof the experimental results. Feenstra also examined the concentration of defectstates, and found densities of 3 ·1016 cm−3 and 5 ·1015 cm−3 for as-grown HWand PE samples, respectively. Under illumination, these numbers increased to2 ·1017 cm−3 in case of HW a-Si:H and 1 ·1017 cm−3 for the PE film. The concen-tration of tail states is in both types of materials of the order of 1020 cm−3.

2.2.2 Generation and detection of vibrations

The samples were immersed in superfluid He (T = 1.8 K) to remove the thermalpopulation of vibrations in the energy range of interest. Then, a vibrational popu-lation was created by means of optical excitation, as is depicted schematically inFig. 2.2. By absorption of green photons with an energy (∼ 2.3 eV) larger than thegap, charge carriers are excited from the valence to the conduction band. These hotcarriers relax on a picosecond time scale [53] to localized states in the band tails,converting their excess energy into phonons. The relaxation process proceeds viarecombination, in which radiative and nonradiative decay channels compete, thelatter being an additional source of vibrational excitations.

Light of the same laser that generated the phonons was used to study the vi-brational population created, where we exploited the fact that light is scatteredinelastically off the phonons present in the excited sample volume. The scatteredlight was analyzed by means of a standard Raman detection scheme, yielding theStokes and anti-Stokes Raman spectra. Since the phonon occupation numbers nω

reached in these type of experiments, are generally much smaller than unity, onlythe anti-Stokes spectra contain measurable information about the excited popula-tion We recall that the intensity of the anti-Stokes signal is ∝ nω whereas the Stokesintensity is ∝ nω + 1. As long as nω 1, increasing the excitation power, P, (andthus nω) does not result in a measurable increase of nω + 1. But, of course, themore light is incident on the sample, the more light is scattered (inelastically). Inother words, the Stokes Raman intensity should increase linearly with P, whereasa quadratic dependence of the anti-Stokes intensity is expected, at least if nω scaleslinearly with P, as was observed by Scholten et al. [25].

Apart from Raman scattering, also luminescence produced by the radiativerecombination of excited carriers could be detected. Because radiative and non-radiative recombination processes are competing in the relaxation path described

22 Chapter 2

Figure 2.2 By absorption of green photons with an energy (∼ 2.3 eV) larger than thegap, charge carriers are excited from the valence to the conduction band. The hot carri-ers rapidly relax to localized states in the band tails, converting their excess energy intophonons. Other steps of the relaxation process include radiative and nonradiative recom-bination, the latter forming an additional source of vibrational excitations.

above, also luminescence experiments may provide useful information about non-radiative recombination and phonon generation, as we will show.

2.2.3 Time-resolved measurements - experimental setup

To investigate the temporal evolution of the phonon population created, a ‘pump-probe’ configuration with two pulsed lasers was used. The low repetition ratepulse-train of each laser generates an anti-Stokes Raman signal. If two trainsare used to excite and to probe a vibrational population, and pulses of one laser(probe) are delayed with respect to the pulses of the other (pump), the integratedanti-Stokes signal consists of a superposition of three contributions. A first con-tribution is proportional to the amount of phonons created with the pump pulsesand detected with the pump. The second contribution corresponds to the phononscreated and detected with light of the probe. The third part is proportional to theamount of phonons that is created with pulses of the pump, survives the delay∆ t between pump and probe, and is detected with the probe. Of course, for suffi-ciently long delays the third contribution becomes negligible. Thus, by subtractinga spectrum measured with ∆ t → ∞ from a spectrum recorded for a particular ∆ t,


Figure 2.3 Experimental setup used for the time-resolved Raman experiments. PG =pulse generator, VD = variable delay (Stanford Research Systems model DG 535 fourchannel digital delay/ pulse generator), M = mirror, BS = beam splitter, PR = polarizationrotator, L = lens, S = sample, DM = double monochromator, SG = spectrograph. a-Si:Hwas excited with the output of two frequency-doubled, Q-switched Nd:YAG lasers (Spec-tra Physics, Quanta Ray DCR-3 and GCR-130). With a diaphragm, a homogeneous part ofthe ring profile of the DCR-3 beam was selected. The sample was immersed in superfluidHe in a Magnex dynamic flow cryostat with optical access, equipped with a Lakeshoretemperature controller. Spectra were recorded with a Tek 512 CCD chip, controlled by aPrinceton Instruments ST 130 detector controller.

the spectrum of phonons that survived ∆ t is obtained. The decay of the excitedphonon population was studied by monitoring this difference spectrum versus ∆ t.

The experimental setup employed for the time-resolved Raman experimentsis depicted schematically in Fig. 2.3. Phonon generation was accomplished withthe output of two frequency-doubled, Q-switched Nd:YAG lasers (λ = 532 nm,corresponding to 2.3-eV photons, with a penetration depth in a-Si:H of ∼ 100 nm).Both lasers produced pulses with a width of ∼ 10 ns (FWHM) at a rate of 30 Hz.Pulses of the probe were electronically delayed with respect to those of the pump.The range over which the delay could be varied without causing temporal overlapof pump and probe pulses was limited at one end by the pulse length and at theother end by the repetition rate, resulting in a large range of ∼ 15 ns up to ∼ 15 ms.The average excitation powers in both laser beams typically amounted to 5 mW

24 Chapter 2

(∼ 170 µJ per pulse) at the sample. The spatial profiles of the laser beams consistedof alternating darker and intenser rings. With a diaphragm, a homogeneous partof the total profile of the first beam was selected. In case of the second beam,however, the ring structure was too fine to select a region of constant intensitywithout losing too much power. Therefore, the excitation power varied over theexcited sample volume. Additionally, because of the relatively small penetrationdepth of the green light into the a-Si:H material, the excitation density decreasedwith increasing depth into the sample.

The two laser beams were made collinear and weakly focused onto the sample,under an angle of ∼ 20 (Brewster-angle) with the sample’s surface. By means ofa polarization rotator, the polarization directions of the two laser beams were madeequal. A second polarization rotator served to minimize the specular reflections ofboth beams from the sample, thereby maximizing the amount of light absorbed.

The sample was immersed in superfluid He, at a temperature of ∼ 1.8 K, ina cryostat with optical access. In this way, not only the thermal phonon popula-tion was suppressed, but also the heat deposited by the laser pulses was carried offrapidly enough between the laser pulses to prevent excessive heating and crystal-lization of the sample. Also to reduce heat production, the beams were focusedto spots not smaller than ∼ 1 mm2. Just decreasing the excitation power wasnot an option, due to the weakness of the Raman signals. Thus, for an averageexcitation power of 5 mW, the absorbed energy density per pulse amounted to∼ 1.5 ·103 J/cm3. Fortunately, damage of the a-Si:H samples is easily recognizedby a characteristic change of the Raman spectra: a sharp Raman peak developsaround 520 cm−1, corresponding to the TO resonance of crystalline Si, and thecontribution of elastically scattered light increases drastically.

Scattered light was collected in a backscattering geometry and projected ontothe entrance slit of a grating double monochromator in subtractive mode, that de-fines the spectral window of the measurement. In this stage, the elastically scat-tered (Rayleigh) light was strongly suppressed relative to the inelastically scatteredlight. With a third grating, the light was dispersed on a liquid-nitrogen cooled CCDdetector, consisting of a two-dimensional (512 × 512) detector-array. The spectrawere recorded with a 15-minute exposure time of the CCD camera. In addition, thedetected intensity was integrated over 300 detector-channels in the direction per-pendicular to the direction of dispersion. The spectral resolution was ∼ 40 cm−1.

In the same setup, pump-probe luminescence experiments were performed un-der identical conditions as the Raman measurements. In that case, only the selectedspectral window was shifted. Limited by the range of the gratings, a window of814-857 nm ( 1.53-1.45 eV) was chosen. In a-Si:H, luminescence at these wave-lengths is caused by radiative recombination of electrons and holes trapped in the


200 400 6000

100

200

300

PE

HW

inte

nsity

(co

unts

/ 9

00 s

)

Raman shift (cm-1)

Figure 2.4 Anti-Stokes Raman spectra of the HW and PE sample (upper two curves andlower two curves, respectively), measured at 1.8 K, for ∆t = 100 ns (solid lines) and ∆t =15 ms (dashed lines).

tail states. Since the yield of the luminescence measurements was higher than theRaman signals, in this case exposure times of only 2.5 minutes appeared to besufficient.

2.3 Results

2.3.1 Pump-probe anti-Stokes Raman spectroscopy

In Fig. 2.4 time-integrated anti-Stokes Raman spectra of both the PE and HWsamples are presented, in each case for delays ∆t of 100 ns and 15 ms. Althoughthe spectra were recorded with less spectral resolution and at 1.8 K, they all exhibitthe characteristic bands corresponding to TA, LA, and TO phonons, that werealready observed with higher resolution in the room-temperature Stokes Ramanspectra of Fig. 2.1.

Since the spectrometer settings were not identical in the experiments on thetwo samples, the relative intensities of different peaks in the upper and lower twocurves of Fig. 2.4 cannot be directly compared. The suppression of elasticallyscattered light was lower in case of the measurements on the HW layer, resulting in

26 Chapter 2

the relatively high intensity in the spectral region of the TA phonons. However, thedifference in the overall intensities of the signals of the two types of a-Si:H is real.Accurate measurements of the anti-Stokes spectra of the two layers, under exactlythe same experimental conditions, namely, confirmed that the signal intensities inthe HW sample are higher. In the case of excitation with one laser beam (power∼ 3 mW), the height of the anti-Stokes spectrum of this sample at all frequenciesexceeds that of the PE layer by a factor of ∼ 1.4, even though the absorbed energydensity in the two types of a-Si:H is the same.

Another remarkable feature of Fig. 2.4 is that in the PE sample, the anti-Stokesintensity, IAS(ω), significantly drops going from ∆t = 100 ns to 15 ms, whereasIAS(ω) remains virtually constant in the HW layer. This is one of the main pointsof this chapter and will be elaborated below.

For both samples, the dependence of the Raman intensity on excitation powerwas studied (for excitation with one laser beam). As expected, the Stokes signalsIS(ω) increased linearly with P. Results for IAS(ω) are shown in Fig. 2.5. Herewe find that IAS(ω) exhibits for all frequencies an almost quadratic dependence onP. From the ratio between IS(ω) and IAS(ω), the phonon occupation numbers nω

can be computed (Eq. 1.1). For the frequencies and powers presented, nω rangedbetween ∼ 0.03 and 0.3, so that we can safely put nω ≈ IAS(ω)/IS(ω). Then, theobservations that IS(ω) ∝ P and IAS(ω) ∝ P2 imply that nω increases linearly withP, independent of ω. So we find that the shape of the phonon spectrum exciteddoes not change with P. In the discussion of the results this observation will beused to demonstrate that the phonon distribution studied is not thermalized but hasa non-equilibrium character.

To examine the temporal evolution of the generated phonon distribution, a se-ries of anti-Stokes spectra was recorded as a function of ∆t. The time-dependentcontributions to the signals were obtained by subtracting a spectrum taken with∆t = 15 ms from each measurement. (IAS(ω) measured at ∆t = 15 ms was equal tothe sum of the separate contributions of the two lasers). In Fig. 2.6.a, the normal-ized TO part of the ‘time-dependent’ anti-Stokes spectra of both types of a-Si:His plotted vs. the delay. Fig. 2.6.b shows the contributions from different parts ofthe phonon spectrum of the PE sample only. To improve signal-to-noise ratios,all intensities were integrated over a range of frequencies: 140-220 cm−1, 240-380 cm−1, and 420-550 cm−1 for the TA, LA, and TO modes, respectively. Withinthe selected spectral regions, no dependence of the decay rate on frequency wasobserved. In both type of films, a relatively fast decay time τ ∼ 70 ns is measuredfor the TO phonons. However, in the PE sample, IAS(ω) does not decay to zero, butinstead to a slowly ( 100 ns) decaying background. The level of this backgrounddepends on excitation power. For excitation with half of the power of both laser


101

102

quadratic

linear

a)

anti-

Sto

kes

inte

nsity

(ar

b. u

nits

)

10

101

102

b)

power (mW)

Figure 2.5 Dependence of the anti-Stokes Raman signals on excitation power, measuredin a) HW-deposited and b) PE-grown a-Si:H, with excitation by a single laser beam. Thefilled symbols correspond to measurements with the DCR-3, the open symbols to experi-ments performed with the GCR-130 (squares = TO, triangles = LA, diamonds = TA). Tofacilitate comparison between the results obtained at different frequencies, the intensitiesof the LA and TA signals were for one value of the excitation power (5 mW) normalized tothe TO intensity. The dotted and dashed lines indicate linear and quadratic dependences,respectively.

beams, the intensity of the ‘background’ is about half of the intensity obtainedwith full power (5 mW). The slow contribution is observed in the PE sample foras well the TO as the LA and TA modes, but seems absent in the HW sample.Yet, in measurements of the transient anti-Stokes spectra of PE a-Si:H grown in adifferent laboratory [54], it showed up again, suggesting that it is characteristic forPE-grown a-Si:H.

Next, the time-dependent part of the signals will be considered. Accordingto Fig. 2.6.b, in the LA region of the spectrum the population decays on aboutthe same time scale as the TO part. The decays are nonexponential, and can becharacterized by a mean decay time, 〈τ〉, of 70 ± 10 ns. Here, we used the relation〈 τ 〉 =

RtIAS(t)dt/

RIAS(t)dt, with IAS corrected for the background. Decay times

of the TA part of the spectrum are too short to determine with the experimental

28 Chapter 2

0.0

0.5

1.0a)

inte

nsity

(ar

b. u

nits

)

0 50 100 150 2000.0

0.5

1.0b)

delay ∆t (ns)

Figure 2.6 a) Normalized anti-Stokes signal of the TO phonons as a function of ∆t in theHW () and PE () sample. b) Normalized anti-Stokes intensities vs. ∆t measured in PEa-Si:H ( = TO, = LA and • = TA). To focus the analysis on the time-dependent contri-bution of IAS(ω), IAS(ω) recorded for ∆t = 15 ms was subtracted from all measurements.

setup. With the exception of the slow background, the spectra of the HW sampleexhibit the same behavior (not shown). In both samples, the value of the decaytimes obtained appeared not to depend on excitation power.

Similar experiments on a bare c-Si substrate did not show any transient anti-Stokes contributions, confirming that the above presented results reflect the gener-ation and decay of phonons in the a-Si:H layers. Of course, the ∼ 10 ps lifetimesof the Raman-active (TO) phonons in c-Si are far too short to show up in the pre-sented experiments.

2.3.2 Pump-probe luminescence measurements

Luminescence experiments were performed under the same experimental condi-tions as the Raman measurements. Fig. 2.7 shows how the intensity of the tailstate luminescence of the two a-Si:H layers depends on excitation power, in caseof excitation with one laser beam. Obviously, the luminescence is saturated forthe typical excitation powers (5 mW) used in the time-resolved Raman experi-


0 2 4 6 80.0

0.5

1.0

power (mW)

inte

nsity

(ar

b. u

nits

)

Figure 2.7 Dependence of the luminescence intensities excited with a single laser beamon excitation power. The filled symbols were obtained with the DCR-3; open symbolscorrespond to experiments performed with the GCR-130. Triangles and circles were mea-sured for the HW and PE sample, respectively. The arrow indicates the typical power usedfor the time-resolved Raman experiments.

ments. Probably, for these large absorbed energy densities (1.5·103 J/cm−3 perpulse), nonradiative decay channels dominate the final step in the relaxation of theexcited charge carriers (see Fig. 2.2). We note that the saturation level reached forthe PE sample is two times higher than the maximum luminescence intensity pro-duced by the HW sample (note that the anti-Stokes signals were higher in the HWlayer). This suggests a difference in the ratios of radiative and nonradiative recom-bination rates in the two types of a-Si:H. Indeed, if nonradiative recombination isslower in PE-grown a-Si:H, both the higher saturation level of the luminescence,and the lower anti-Stokes Raman signals in this material can be explained, as willbe shown later.

In case of excitation with the GCR-130, saturation occurs at a lower powerthan in measurements with the DCR-3. The fact that the profile of the beam ofthe GCR-130 is less homogeneous than the profile generated by the DCR-3, is themost probable reason for that difference: even if the average laser intensities arethe same, some parts of the studied sample volume are excited more strongly and

30 Chapter 2

10-8

10-7

10-6

10-5

10-4

10-3

10-2

0.00

0.25

0.50

0.75

1.00

delay (s)

inte

nsity

(ar

b. u

nits

)

810 820 830 840 850

400

600

800

wavelength(nm)

cou

nts

/ 1

50

s

Figure 2.8 Luminescence intensity at 840 nm of HW () and PE (•) a-Si:H as a functionof ∆t, normalized to the sum of the luminescence intensities obtained with each laserseparately ( in the inset, dotted line in the main graph). Also shown in the inset arethe luminescence spectra measured in the PE layer for ∆t = 20 ns () and ∆t = 15 ms(O). The measurements were performed under the same experimental conditions as thetime-resolved Raman experiments; for instance, the average excitation power amounted to5 mW (corresponding to an absorbed energy density of 1.5·10 3 J/cm−3 per pulse).

get saturated by the intenser parts of the GCR-130 beam.In Fig. 2.8, results of the pump-probe luminescence measurements performed

under the same conditions as the Raman experiments are presented. The lumi-nescence intensities are normalized to the sum of the luminescence intensities ob-tained with the pump and probe separately. For delays of several milliseconds, theluminescence generated in the HW layer equals this sum. In the PE sample, the‘sum value’ is just not reached for the longest delays (see the difference betweenthe ∆t = 15 ms and ‘sum’ luminescence spectra depicted in the inset of Fig. 2.8).Also shown in the inset is a spectrum measured in the PE layer for ∆t = 20 ns.As can be seen, the shape of the spectra does not change significantly with thedelay. Finally, we note that no fast contributions to the time-dependent signalsare observed. Unfortunately, for the excitation powers (5 mW) that were used forthe Raman measurements, it was not possible to record spectra for ∆t’s between


∼ 1 µs and 1 ms, due to the appearance of a ‘helium-bubble’. Optically exciteda-Si:H slightly heats up, and as a result, some liquid He at the sample surface va-porizes. If this bubble is still present when the next laser pulse arrives, the sampleis damaged, due to the reduced cooling efficiency. For excitation powers one or-der of magnitude lower, it was possible to monitor the luminescence over the fullrange of delay times. The intensity appeared to increase gradually with the delay,until a maximum was reached after ∼ 10 ms.

2.4 Discussion

Two contributions to the transient anti-Stokes Raman signals were observed (seeFig. 2.6): a relatively fast decaying part, and a slow background. Their main char-acteristics are summarized in Table. 2.1. Obviously, the two components behavequite differently. The first, the ‘fast’ decay, was studied earlier by Scholten et al.[25, 40] and interpreted in terms of elementary phonon decay processes. Resultsof their experiments are in agreement with the observations presented here. In thenext subsection, several arguments will be given that at least support the idea thatanti-Stokes Raman measurements in a-Si:H probe the decay of non-equilibriumphonons. As will become clear from that, however, the new observation of aslowly decaying contribution to IAS cannot be explained in the same manner. Itappears that long-lived electronic excitations may influence the vibrational signalsover a long time span.

‘fast’ component ‘slow’ component

sample type PE present presentHW present absent

decay time at TA < 10 ns ∼ 1 msLA ∼ 70 ns ”TO ” ”

power-dependence IAS ∝ Pq q ∼ 2 q < 2

Table 2.1 Main characteristics of the two contributions to the anti-Stokes Raman signals.It should be noted that the q < 2 power-dependence of the ‘slow’ component was notinvestigated in detail.

32 Chapter 2

2.4.1 Non-equilibrium phonons

Before conclusions about phonon lifetimes can be drawn from the measurements,we first demonstrate that the populations responsible for the anti-Stokes Ramansignals have a non-equilibrium character. In case the phonon distribution in thesample would reach thermal equilibrium sufficiently rapidly, the cooling-down ofthe sample would be measured in the time-resolved Raman experiments, instead ofphonon lifetimes. To verify that indeed non-equilibrium phonon populations pro-duce the signals, we resort to the same arguments as those presented by Scholten etal. [25], that are based on the observed excitation power-dependence of the Ramanspectrum and the frequency-dependence of the obtained decay times.

From the quadratic increase of the anti-Stokes Raman intensity with excita-tion power (Fig. 2.5), and the linear dependence of the Stokes intensity on P, onecan conclude that the shape of the phonon spectrum excited (nω) is independent ofP. However, in the case of thermal equilibrium, the phonon occupation would bedescribed by the Bose-Einstein distribution (Eq. 1.3), characterized by an equilib-rium temperature that increases with excitation power. Consequently, the shape ofthe spectrum would significantly depend on P, which is not observed.

Furthermore, if the phonon distribution in the sample would be in ther-mal equilibrium all the time, the decay of the population would correspond toa decrease in temperature in time. Then, if ∂T/∂t is negative, the decay rate−∂ ln(nω)/∂t, of a phonon population obeying Bose-Einstein statistics, is an in-creasing function of ω. Although 〈 τ 〉 was not determined for many frequencies,the experimental results do not exhibit any indications for an increase of 〈 τ 〉 withω. On the contrary, regarding the ‘fast’ contribution to the signals, the decay of theTA part of the spectrum was too fast to monitor (< 10 ns), whereas 〈 τ 〉 ∼ 70 nscould be determined for the LA and TO frequencies. Scholten et al. observed asimilar dependence. Further, the ‘slow’ component of the Raman signals seems tobe frequency-independent within the experimental error.

These two arguments lead us to the conclusion that a non-equilibrium phonondistribution was present in our experiments.

Electronic and vibrational contributions

Now that we know that the phonon population monitored via the anti-Stokes Ra-man intensity has a non-equilibrium character, the next question is what causesits decay. Since all decay times obtained are several orders of magnitude longerthan the anharmonic decay times of phonons generally observed in crystals, it isdifficult to believe that these times correspond to the bare lifetimes of phonons inamorphous silicon. Actually, it has been suggested that due to disorder, the ef-


fect of anharmonicity in a-Si is even larger than in c-Si, which would result in anincreased anharmonic decay rate in a-Si, instead of a slower decay [18].

An explanation one could think of is, that not the phonons have long lifetimes,but that the source of electronic excitations that generate the phonons is long-lived. The vibrational signals may persist much longer than the phonons live if theabsorbed energy is not transformed instantaneously into phonons, but gets accu-mulated in the electronic system of the amorphous semiconductor. Indeed, carriersthat are trapped in the tail states of a-Si:H are known to have up to millisecond life-times at low temperatures [42] (see also Fig. 2.8). However, the very occurrence ofelectronic processes that have characteristic time scales comparable with the decaytimes of the Raman signals does not necessarily explain our experimental results.Of course, also the amount of phonons created in the electronic processes shouldagree with the observed phonon occupation numbers. Some rough estimates aregiven now to illustrate that only phonon generation and decay rates in a certainrange can account for the measured occupation numbers.

Occupation numbers and lifetimes - some estimates In the excited a-Si:H vol-ume (∼ 10−7 cm3), roughly 0.75 · 1016 optical phonon modes are available. Weassume for simplicity that all these modes have a frequency that corresponds tothe TO part of the Raman spectrum. Then, the maximum number of quanta thatcan be created from the ∼ 150 µJ energy absorbed during a pulse is ∼ 1.6 · 1016.If the phonons in the excitation volume are generated at a constant rate G duringthe laser pulse, and the population decays exponentially with a time constant τ, theoccupation n of the modes in the excited volume at time t during the pulse can bewritten as

n(t) ≈ Gτ(1− e−t/τ)0.75 ·1016 . (2.1)

Below, we use n(5ns) as a measure for the average TO phonon occupation duringthe laser pulses.

One way to explain the 70-ns decay times of the Raman signals, is to saythat the phonon source decays on a 70-ns time scale. In that case, about 13%of the total amount of phonons emitted is produced during the 10-ns laser pulses(and contributes to the Raman signal). Then, G ∼ 2.1 · 1023 s−1, and n(5ns) onlyapproaches the experimentally obtained n (typically 0.15 ± 0.05 for the TO modes)if τ 10 ns, which is three orders of magnitude longer than the lifetime of TOmodes in c-Si. Even if all energy would be converted into TO modes during thepulse (G ∼ 1.6 · 1024 s−1), nanosecond TO lifetimes would be needed to accountfor the measured n (but in that case, of course we would not obtain 70-ns decay

34 Chapter 2

times). Hence, we cannot explain the results by assuming that the phonons havelifetimes comparable to the picosecond lifetime of TO modes in c-Si, and thatthe temporal shape of the Raman signals entirely originates from the decay ofthe phonon source. On the other hand, a similar simple calculation shows thatif only ∼15% of the TO population has a 70-ns lifetime, while the other 85%decays on much shorter (say 10 ps) time scales, occupation numbers of n ∼ 0.15are attainable. So the experiments do not necessarily tell us that all phonons ina-Si:H have long lifetimes.

From these considerations it also follows, that we can only account for the highphonon occupation numbers observed, if a large amount of phonons is created dur-ing a short period. A scenario not mentioned yet, that may satisfy this constraint,is that part of the energy absorbed is stored in the form of long-lived electronicexcitations, for instance by means of a tail state population. If this stored energycan be released quickly by renewed optical excitation, relatively high phonon pop-ulations could be generated with a second laser pulse, even for delays larger thanthe phonon lifetimes.

In that case, the spectral distribution of the ‘extra’ phonons that are createdwith the second pulse is not expected to depend on the delay between the pumpand probe pulses, since the energy distribution of the electronic excitations remainsunaltered in the period between two laser pulses. Indeed, the shape of the Ramanspectra measured for long delay times (∆t > 300 ns) does not depend on ∆t. Fromthis, and more arguments that follow from the analysis of section 2.4.2, we con-clude that the slow part of the anti-Stokes Raman signals has an electronic origin,and does not reflect the phonon lifetimes.

Yet, we stress that the shape of the Raman spectrum does depend on ∆t forshort delays (∆t < 300 ns). Hence, the fast part of the signals cannot have thesame origin as the slow background. In this context it is important to note that, onthe tens-of-nanosecond time scales that characterize the fast contribution, the lu-minescence intensity plotted in Fig. 2.8 only increases by a small factor comparedto the factor by which the Raman intensity decreases. If electronic (nonradiativerecombination) processes would be responsible for the fast component in the Ra-man signals, one would expect a fast component in the luminescence signals aswell, since both result from recombination events and are proportional to the den-sity of excited carriers. This leads us to the conclusion that the fast decay timesobserved do not relate to electronic processes, but correspond to phonon decay ina-Si:H. We emphasize once more that not all phonons in a-Si:H necessarily havelong lifetimes. From Fig. 2.6 it can indeed be seen that the decay is nonexponen-tial. The setup used is unfortunately not sensitive to decays shorter than ∼ 10 ns.In the case of the measurements performed by Scholten et al., temporal resolution


down to ∼ 1 ns could be achieved, which is still much longer than the expected‘shortest’ lifetimes of phonons in a-Si. In any case, the experiments show that atleast part of the phonons behave anomalously in comparison with the crystallinecase.

In the remainder of this chapter, both the electronic and vibrational contribu-tions to the temporal behavior of the anti-Stokes Raman signals will be examinedin more detail. A simple rate-equation model will be presented that appears to de-scribe the slow component of the transient Raman spectra qualitatively. It containsphenomenological parameters concerning the electronic decay processes in a-Si:H.Further, existing models will be reviewed and discussed that treat the behavior ofphonons in amorphous materials. Predictions regarding the phonon dynamics thatfollow from the models are compared with the results of the Raman experiments.

2.4.2 Slow electronic contribution to the vibrational signals

Since the slow ( 100 ns) contribution to the anti-Stokes Raman signals cannotbe accounted for by just the longevity of the phonons (τ < 100 ns), we reconsiderthe mechanisms of phonon generation. We take the point of view that part of thephonon generation occurs during the first laser pulse, synchronously with the ab-sorption of the green photons, whereas another part is triggered by the absorptionof the second laser pulse.

As mentioned before (see Fig. 2.2), phonons are produced at various stages inthe process of carrier relaxation that follows the photo-excitation of a-Si:H. Thefirst step, thermalization, takes place on a picosecond time scale [53]. Hence, itis too fast to explain the slow background. However, the second stage, recom-bination, may be slow if carriers localized in the tail states are involved. In theabsence of optical pumping, the lifetime of carriers in the tail states is known to beas long as ∼ 1 ms [42], i.e. of the right order to match the slow contribution to theRaman signals. The recombination of electrons and holes that are both trapped inthe tail states, is far too slow to account for the measured occupation numbers (seethe discussion in Sec. 2.4.1.). But the recombination of free carriers with carrierstrapped in the tail states is efficient enough to play an essential role in the phonongeneration, as will be demonstrated below.

The idea is roughly as follows: during the first laser pulse, phonons are createdboth due to relaxation of carriers to the band edges, and due to non-radiative re-combination. Of course, all phonons contribute to the Raman signal. At the sametime, tail states are filled with electrons and holes, and energy is stored. Becauseof the long lifetimes of these states, trapped carriers survive the period betweenthe first and second laser pulse (for delays up to ∼ 1 ms). Consequently, at thestart of the second laser pulse, many tail states are still occupied. The abundantly

36 Chapter 2

present free carriers freshly created by the probe quite efficiently recombine withthe trapped ones. As a result, the number of nonradiative transitions, and hence thephonon generation rate during the second laser pulse is higher than during the firstpulse. In other words, even if the total phonon population has vanished, the secondpulse may induce an intenser anti-Stokes Raman spectrum than the first. Thus, thephonon population is absent during the major part of the period in between the twopulses. Instead, after the first pulse, energy is stored in the electronic system, andis triggered to be converted to phonons by the arrival of the second pulse.

Geminate and non-geminate recombination

Recombination processes in hydrogenated amorphous silicon have been exten-sively studied [38, 55]. One distinguishes so-called ‘geminate’ and ‘non-geminate’transitions. In the case of geminate recombination, the recombining electrons andholes are correlated in the sense that they have been created in the same photonabsorption event. In general, the rate of geminate processes does not depend onthe density of excited carriers. On the other hand, non-geminate recombination(of uncorrelated electrons and holes) becomes more probable when more carri-ers are excited. At a certain excitation power, non-geminate processes start todominate the recombination. According to Stachowitz et al. [56], in a-Si:H onlynon-geminate transitions result in non-radiative recombination, whereas geminateprocesses are radiative. The results of their analysis indicate that recombinationswitches from predominantly radiative to mostly nonradiative for average carriergeneration rates Gc higher than ∼ 1019 cm−3 s−1. In that case, in our experimentsnongeminate recombination should indeed contribute significantly to the phononpopulation: typically, 5 mW of green light was incident on a sample volume of10−7 cm3, corresponding to an average Gc of ∼ 1023 absorbed phonons cm−3 s−1.The fact that luminescence was highly saturated for these powers (see Fig. 2.7)supports this conclusion.

In the model we propose, the most relevant recombination processes involvethe recombination of free with trapped (immobile) carriers. Such transitions areessentially non-geminate and, according to the results of the luminescence mea-surements, under the conditions of our experiments predominantly nonradiative.In this connection we note that it has been proposed recently that nonradiative re-combination events in a-Si:H may result in the breaking of SiH bonds [45]. Theconcomitant increase of the number of dangling bond defects is now believed toexplain the light-induced degradation of a-Si:H that was discovered by Staeblerand Wronski in 1977 [36, 45, 57] .


Model: importance of the relative rates of relaxation and recombination

To investigate if non-geminate recombination processes may explain the slow con-tribution to the anti-Stokes Raman signals, we resort to a simple rate-equationmodel that connects the concentrations of free carriers (Nf ), trapped carriers (Nt),and phonons of frequency ω (Nω). For simplicity, both the thermalization rate offree carriers, w, and the non-geminate (also called ‘bimolecular’) recombinationrate, C, are taken equal for electrons and holes. Further, radiative (geminate) de-cay is not taken into account, consistent with the fact that nonradiative processesdominate the recombination for the excitation powers used in the experiments.Also recombination of trapped holes with trapped electrons is neglected, becauseof the intrinsically low rates of such processes. Then, we arrive at the followingcoupled rate-equations for the carrier concentrations

N f = g(t)−wNf −CNf (Nf + Nt) and

Nt = wNf −CNf Nt. (2.2)

The term g(t) describes the generation of free carriers by pulsed optical excitationand is determined by P, and is taken constant during the pulse. The term wNf

relates to the trapping of free carriers at the tail states. This component is propor-tional to the phonon generation rate due to relaxation. The contributions involv-ing C correspond to the nonradiative recombination of free carriers with free andtrapped carriers, where CNf (Nf +Nt) describes the phonon generation rate causedby recombination, and the rate constants are taken equal. Because relaxation to theband edges is much faster than the other processes, this term is approximated byCNf Nt .

The total phonon generation rate, Gph(t), is equal to a weighted sum of the con-tributions originating from relaxation and recombination. The ratio of the amountof energy released by relaxation to the amount of energy released during recombi-nation, β, is determined by the difference in energy of the photons that excite thecarriers, EYAG, and the gap, Egap, of a-Si:H, so that β = (EYAG −Egap)/Egap. Thus,the total phonon generation rate is described by

Gph(t) ∝ CNf Nt + βwNf . (2.3)

If we assume, for simplicity, that phonons decay mono-exponentially with a decaytime τω that only depends on the phonon frequency, the concentration of phononsof that frequency is given by

dNω(t)dt

= Gph(t)− Nω(t)τω

. (2.4)

38 Chapter 2

Then, if the sample is excited with a laser pulse of length t1, starting at t = 0 andcausing an average carrier generation rate g, the anti-Stokes Raman intensity forphonons of frequency ω due to this pulse is

IAS ∝ gZ t1

0Nω(t)dt, (2.5)

where Nω(t) is obtained from solving equations 2.2 and 2.4.Our goal is to investigate if the presence of electronic excitations may increase

the height of the anti-Stokes Raman spectrum detected with a second laser pulseas compared to the spectrum detected with the first pulse only, even in the casethat the complete phonon population has decayed between the pulses. Therefore,the situation is considered where a second laser pulse, of the same width and in-tensity as the first pulse, succeeds the first pulse after a period ∆t much longer thanthe phonon decay time τω. At the same time, ∆t is taken much shorter than thelifetime of the tail states, so that at the start of the second pulse, Nt has the samevalue as at the end of the first. In practice, that would correspond to a ∆t between10 ns and 0.1 ms for phonons with the shortest lifetimes (TA) and to a ∆t between∼ 200 ns and 0.1 ms for the long-lived TO phonons. From the pump-probe lumi-nescence measurements shown in Fig. 2.8 we know that indeed in both samplesstudied, electronic excitations survive the delays that are relevant for the Ramanexperiments.

To simplify the calculations, two extreme cases are distinguished. The firstcase occurs when the phonon decay time is much shorter than the duration of thelaser pulse (τω t1), as is the case for TA phonons [58]. The second situationoccurs, when the phonon decay time is much longer than the pulse duration (τω t1), but still much shorter than ∆t. This is the case for TO phonons.

Short-lived (TA) phonons For phonons with a lifetime τω t1, we may as-sume that the population decay is in equilibrium with the phonon generation, sothat Nω ≈ 0, and Nω(t) = Gph(t)τω. In that case, the anti-Stokes Raman intensi-

ties detected with the first and second laser pulse, I(1)TA and I(2)

TA , respectively, areproportional to

I(i)TA ∝ gτω

Z t1

0G(i)

ph(t)dt, (2.6)

with i = 1, 2.

Long-lived (TO) phonons In the case that the phonons have lifetimes τω t1,the phonon population increases as long as phonons are generated. Then, Nω ≈


Gph and Nω(t) =R t

0 Gph(t ′)dt′. This leads to anti-Stokes Raman intensities I(i)TOdescribed by

I(i)TO ∝ g

Z t1

0

Z t

0G(i)

ph(t′)dt′ dt. (2.7)

Exact solutions of Eq. 2.2 can be obtained in terms of the inverse function off (x) = xex. With help of this solution the integrals of Eq. 2.6 and 2.7 can beevaluated. In this way, the curves presented in the following were obtained.

Application to the experiment

To apply the model to the experiments presented in this chapter, numerical valueshave to be substituted for the parameters. Of these parameters, β, C and w de-pend on the type of material. As already mentioned, the Tauc gap of the a-Si:Hlayers used is 1.7 eV for the HW sample, and 1.78 eV for the PE layer, corre-sponding to β = 0.37 and β = 0.31, respectively (EYAG = 2.33 eV). Values for Cand w are at present not available for exactly these two samples and are there-fore taken from literature. Values given for C vary between 2 ·10−9 cm3s−1 and2.3 · 10−8 cm3s−1 [44, 59, 60, 61]. For the thermalization rate w, values between1013 s−1 and 1012 s−1 have been found [60]. In the calculations, only the ratio ofC/w enters the equations. From the numbers given here, we find that the valuesfor C/w range from 2 · 10−22 cm3 to 2.3 · 10−20 cm3. Finally, g is estimated fromthe pulse energy and pulse width, assuming that all absorbed photons produce anexcited electron.

Now, the contributions of carrier recombination and relaxation to the TO anti-Stokes Raman intensities can be calculated, for different combinations of C/w andβ. Exploring calculations were performed for two values of C/w, 5 ·10−22 cm3 and5 ·10−21 cm3, and β = 0.31 taken the same in both cases. From that calculation, welearned that only for the lower of the two values taken for C/w, 5 ·10−22 cm3, thetotal intensity measured with the second pulse was higher than that measured withthe first pulse. Since that result was closest to the situation we encountered for thePE sample, and the curves for the other value of C/w appeared to agree with theexperimental results obtained for HW a-Si:H, the calculation was repeated. Duringthe second evaluation, the Raman intensities were calculated for combinations ofthe lower C/w with β = 0.31 corresponding to the PE sample, and the higher C/wwith β = 0.37 of the HW layer. The results of that calculation are shown in Fig. 2.9.

From these curves, it can be seen that it is indeed possible to obtain an ‘ex-tra’ contribution to the anti-Stokes Raman signals measured with a second pulse,as compared to the intensities obtained with a first pulse, even in the case that all

40 Chapter 2

0

10 1st pulse

a)

C/ w = 5 * 10-21

cm3

C/ w = 5 * 10-22

cm3

inte

nsity

(ar

b. u

nits

)

0

10 2nd

pulse

b)

0 2 4 6 80

10

c)

1st pulse

d)

2nd

pulse

e)

0 2 4 6 8

f)

power (mW)

Figure 2.9 calculated TO anti-Stokes Raman intensities vs. P for C/w = 5 · 10−22 cm3

& β = 0.31 [a), b) and c)] and C/w = 5 · 10−21 cm3 & β = 0.37 [d), e) and f)]. For bothcombinations of C/w and β, the contributions to the signals due to relaxation (solid lines)and nonradiative recombination (dashed lines) are given, that are produced by the first laserpulse [a) and d)] and the second laser pulse [b) and e)].The curves in c) and f) display thetotal anti-Stokes Raman intensities calculated for the first and second laser pulse (dottedlines and dash-dotted lines, respectively).

phonons have decayed during the delay time between these two pulses. A com-bination of the excitation power and material parameters determines if this extracontribution is present or not. This may explain why a ‘background’ signal wasobserved in the PE sample, and not in the HW a-Si:H layer. These graphs furthershow, that both relaxation and non-radiative recombination contribute significantlyto all obtained signals.

In Fig. 2.10.a the ratio of the total TO intensity measured with the first pulse,as predicted for the two combinations of C/w and β, is plotted as a function ofP. According to the calculation, the total intensity should be between 1.2 and1.6 times higher for the case that C/w = 5 · 10−21 cm3 and β = 0.37 (the ‘HWcase’) than for to the other combination (the ‘PE case’). This is consistent withthe result that for an excitation power of 3 mW, the anti-Stokes signal obtained


0.0

0.4

0.8

1.2

1.6a)

ratio

0 2 4 6 8

b)

(arb

. un

its)

inte

nsity

power (mW)

Figure 2.10 a) Ratio of the calculated total intensity measured with the first pulse forC/w = 5 · 10−22 cm3 and β = 0.31, divided by the total intensity obtained with the firstpulse for C/w = 5 ·10−21 cm3 and β = 0.37, as a function of P. b) Calculated intensity ofthe ‘background’ signal vs. excitation power.

in the HW sample was 1.4 times higher than that measured in the PE film. Ofcourse, when equal amounts of light are absorbed by the two a-Si:H layers, alsothe amount of energy dissipated should be the same. Apparently, the number ofphonons produced during the laser pulses in the PE sample is lower than in the HWlayer, which is in accordance with the observation that the luminescence intensityin the PE film exceeds that of the HW sample (see Fig. 2.7). If the nonradiativerecombination rate C is lower in PE a-Si:H, indeed a higher saturation level of theluminescence would be expected for the PE sample. If in addition, w is of the sameorder in both materials, C/w is lower for the PE sample, as is also suggested bythe outcome of the above calculations.

Fig 2.10.b shows how the height of the ‘background’ (i.e. the difference be-tween the curves in Fig. 2.9.c) depends on the excitation power. Although thedependence is not linear, the calculated values agree with the fact that the intensityof the ‘background’ contribution measured with an excitation power of ∼ 5 mWwas twice as high as for P ∼ 2.5 mW.

Further, we note that the curves presented in Fig. 2.9 and 2.10 have been calcu-

42 Chapter 2

lated in the long-phonon-lifetime-limit (TO phonons), but that results obtained forthe TA phonons are qualitatively the same. In other words, the slow contributionto the signals behaves similar for different phonon frequencies, as was observedexperimentally.

The same calculations that yield the curves presented in Fig. 2.9 and Fig. 2.10,also give a rough estimate for the concentrations Nt of trapped and Nf of freecarriers that have to be generated to account for the ‘background’. We find that,for the lower value of C/w (5 ·10−22 cm3), Nt and Nf reach values of 5 ·1020 cm−3

and 1 ·1016 cm−3, respectively. For the higher C/w (5 ·10−21 cm3), Nt approaches2 · 1020 cm−3 and Nf does not exceed 1 · 1016 cm−3 again. The concentrationdetermined for Nf seems reasonable, since of the order of 1023 cm−3 electronicstates are available in the conduction band of a-Si:H. On the other hand, the valuesfor Nt are quite high, compared to the 1020 cm−3 tail states that are present in bothtypes of a-Si:H. This is probably a result of the roughness of our model.

In our analysis, we assume that all absorbed energy is either present in the formof electronic excitations, or is directly released by emission of phonons, whichmay in fact be not the case. Branz for example stated that during the relaxation ofphoto-excited carriers in a-Si:H, SiH bonds may break [45], leading to danglingbonds and mobile hydrogen atoms. The mobile H diffuses through the material,and is either retrapped at a dangling bond at another location, or forms, togetherwith another excited mobile H atom, a metastable H2 complex. In both cases,energy (about 1 eV per broken bond) is temporarily (for tens of microsecondsup to tens of seconds) stored in a ‘cloud’ of mobile H atoms, which prevents thegeneration of phonons. This would mean that in the above model, we overestimateboth the number of phonons created during thermalization, and the populationformed during recombination of the excited carriers. To explain the background,however, mainly the ratio of the amounts of phonons produced during the twoprocesses is of importance. So if energy is taken up and stored by alternative kindsof excitations (for instance mobile H atoms), the background may be accountedfor with lower, realistic values of Nf and Nt .

Another notion that supports the model is that Nt should be interpreted as theconcentration of immobile carriers. Also when all tail states are occupied, the hotcharge carriers will thermalize, to end up at local minima of the conduction andvalence bands. In the present context, such immobilized carriers are expected tobehave similarly to the carriers trapped in the tail states. Hence we do not believethat the high concentration of Nt that is predicted by the calculations is a reasonto discard the above discussed model. The main point we try to make with themodel is, that a significant part of the total energy that is eventually converted intophonons is only released when the sample is excited with a second pulse, as long


as that pulse appears within the electronic lifetime of the tail states.In conclusion, the presented model qualitatively explains why a slow back-

ground contribution to the transient anti-Stokes Raman signals was observed, with-out the need to invoke extreme longevity of the phonons in a-Si:H.

2.4.3 Phonon decay in amorphous silicon

In this subsection, several models are presented that treat the anharmonic decay ofphonons in a-Si:H. Predictions regarding the phonon dynamics that follow fromthe theoretical studies are compared with the results of the Raman experiments.

The fracton model

Several years ago, a model was developed to describe the structure and dynam-ics of fractal media [11, 62], the so-called fracton model. More recently, Orbachand Jagannathan gave theoretical support to the interpretation of the striking re-sults reported by Scholten et al., which was based on the fracton model [26]. Thecalculations demonstrated that high-frequency phonons in a-Si may indeed havelifetimes as long as suggested by the outcome of the pulsed Raman experiments.Furthermore, according to the fracton model, the phonon lifetimes should increasewith increasing frequency. Both the long lifetimes and frequency dependence ofthe decay rates appear to result from a reduced spatial extent of phonons with anenergy higher than the so-called Ioffe-Regel limit. The Ioffe-Regel limit is typi-cally encountered for phonons with a frequency of ∼ 1 THz, i.e. at an energy muchlower than the lowest energy addressed by the Raman experiments. In the fractondescription, the spatial extent of phonons beyond this energy becomes more andmore limited (the phonons become more and more ‘localized’), eventually lead-ing to a decreased spatial overlap of the vibrational states, and consequently anelongation of the phonon lifetimes against anharmonic break up.

Although amorphous materials are in general not fractal, their universal ther-mal properties appear to be similar to those found for fractal networks [62]. An-other pronounced characteristic of both amorphous and fractal structures is the ab-sence of translational symmetry, which may explain similarities in physical prop-erties of the two classes of materials. Originally, the fracton model was intended todescribe the thermal properties of glasses in general, but in order to keep the cal-culations feasible they were restricted to fractals. In any case, the fracton modeldescribes the thermal characteristics of fractals and other glasses rather accurately,and one might expect it to be capable of explaining other phenomena encoun-tered in glasses. This was a reason for Scholten et al. to make the comparison ofthis model with the experimentally observed dynamical properties of phonons in

44 Chapter 2

a-Si(:H). A further justification to take this approach was that the fracton modelwas at that time the only analytical theory treating the anharmonic decay of vibra-tions above the Ioffe-Regel limit.

A crucial ingredient of the fracton model is the assumption that the Ioffe-Regellimit, ωIR coincides with a mobility-edge ωc. Below the crossover frequency, vi-brations are extended and propagating, and called ‘phonons’, in analogy with theextended and propagating lattice vibrations in crystals. Vibrations with a frequencyhigher than ωc are localized and called ‘fractons’. Their wavefunctions are char-acterized by a localization length lω, described by

lω =ξ2

(ωc

ω

)d/D. (2.8)

The characteristic crossover length, ξ, can be interpreted as the wavelength belowwhich vibrations must be considered as fractons, and is approximated by 2πvs/ωc,with vs the sound velocity. In a-Si:H vs ∼ 5 · 103 m/s. ξ is sometimes identifiedwith the length scale of medium range order in amorphous media [63]. Further,D is the fractal or Hausdorff dimension, which describes with what power of thelength, L, the mass of the system increases (for L ξ). The fracton or spectraldimension of the network is denoted by d. Only if d ≤ 2, the wavefunctions appearto be localized.

The assumption that ωIR = ωc also gives a straightforward explanation for theplateau in the heat conductivity, κ, vs. T , since the localized vibrations cannotcontribute directly to the conduction of heat. Therefore, in this description, ωc

corresponds to the temperature at which the plateau in the heat conductivity setsin. At temperatures beyond the plateau, κ(T ) rises again with T . This behavior isattributed to phonon-assisted ‘hopping’ of fractons to nearby fracton states. Ex-actly these processes form the dominant anharmonic decay channel for fractonswith ω ωc (the frequency range of the vibrations we and Scholten examined).In the fracton model, anharmonicity is introduced as a perturbation, leading to thefollowing expression for the rate at which the decay of a fracton to a fracton oflower energy by emission of a phonon takes place [26]

1τ(ω)

=16π3C2

effΩρ3v2

s ξ7ω3c

(ωωc

)4 dD +d−3

ωc exp

[−

(ωωc

)dφ/D]

. (2.9)

The relation is valid at low temperatures (kBT ωc). Ceff is the effective an-harmonic coupling constant, Ω = 2π3/2/Γ(3/2) = 12.5, and ρ is the mass density.The superlocalization exponent, dφ describes the strength of the localization. From


this equation we see that the fracton lifetime τω has a frequency dependence

τω = A

(ωωc

)−4 dD−d+3

exp

[(ωωc

)dφ/D]

, (2.10)

instead of the familiar ω−5 dependence in crystals. We find that τω is an increas-

ing function of ω if (ω/ωc)dφ/D > (4d + Dd − 3D)/dφ. In our case, where d = 2and D = 3, this means that ω 2ωc to satisfy the inequality. From the positionof the plateau in the thermal conductivity of a-Si (T ∼ 10 K [64]), we estimateωc ∼ 65 cm−1 [65], which implies that the condition is met for the whole range offrequencies examined in the anti-Stokes Raman experiments.

Scholten et al. fitted Eq. 2.10 to the data of τω vs. ω obtained from his ex-periments. The quality of the fits turned out to be particularly sensitive to theprefactor in the right hand side of Eq. 2.10, and less to the combination of ωc, dand dphi/D. From the best fit, and substitution of all known material parameters,an effective coupling constant Ceff ≈ 2 · 1011 was obtained [40], a value which isvery close to the third-order elastic constant of c-Si [66]. However, this apparentlyrealistic value is almost 5 orders of magnitude lower than the value of Ceff requiredto describe the thermal conductivity of a-Si above the plateau, at least if it resultsentirely from phonon assisted fracton hopping. Recently, Orbach presented a mod-ification of the model, where the anharmonic decay of fractons is still describedin the same way, but an additional contribution to the thermal conductivity abovethe plateau is assumed to be supplied by ‘strongly localized modes’ [67]. The Ceff

for these strongly localized states is believed to be much higher than the abovementioned value for the fractons. In the new description, fractons are ‘mesoscopi-cally localized’ and do not interact with the strongly localized states. Further, thestrongly localized modes are associated with the Boson peak that is often observedin the Raman spectra of glasses. However, no evidence exists for the occurrenceof a Boson peak in a-Si.

An important consequence of the vibrational decay as described by the fractonmodel is its inherent nonexponentiality: due to a distribution in hopping distances,and thus a distribution of hopping probabilities there is a distribution of decay ratesfor fractons of the same energy. The decays observed and presented in Fig. 2.6 areindeed clearly nonexponential.

It should be noted that the current experiments do not confirm the dependenceof τω on ω obtained by Scholten et al. quantitatively. Although 〈 τ 〉 obtainedfor the TO vibrations from the data of Fig. 2.6 is similar to the value reported byScholten in Ref. [68], our results do not indicate a gradual decrease of the lifetimeswith decreasing frequency. Rather, the mean decay rates appear to change abruptlyfrom ‘short’ to ‘long’ at a frequency of ∼ 200 cm−1.

46 Chapter 2

Recently, several objections against the interpretation of the experiments interms of the fracton model have been put forward [18, 49]. For instance, accordingto other descriptions, the Ioffe-Regel limit should not necessarily be identified witha crossover from extended to localized vibrational states. In addition, some authorshave claimed that the process of a fracton decaying into a fracton + phonon is notthe main decay channel for localized vibrations [18]. The fracton model neglectsthe decay of a fracton into two fractons of lower energy because the probability ofa process involving spatial overlap of three localized modes seems low. Accordingto Fabian et al., this would be true if the vibrational states would be dilute, likethe localized electronic band tail states, which they claim is not the case if allmodes with frequencies > ωc are localized. A simple (but may be too simple)estimate indeed shows that if say 80 % of the modes in a-Si:H is localized (thatis ∼ 1.2 · 1023 modes per cm3), their localization length should be smaller thanthe average distance between two Si atoms if the states do not overlap in space.The calculations of Fabian et al. suggest that the rate of a ‘three-localized-statetransition’ is much higher than the transition involving two fractons and a phonon,resulting in lifetimes of the localized states of the order of 1 ps.

The diffuson model

Recently, Fabian et al. reported results of numerical calculations of anharmonicdecay rates in a model system of a-Si [18, 49]. In contrast to the fracton model, inthese calculations the Ioffe-Regel limit does not necessarily coincide with a mo-bility edge. Indeed, vibrational modes that have a wavelength comparable to theirmean free path cannot be considered as waves, and no wavevectors or polarizationcan be assigned to them. Fabian et al. introduced the following nomenclature:the term ‘vibron’ refers to all vibrational eigenmodes of the amorphous systemand is an amorphous analogon for the ‘phonons’ of crystals. Vibrons can be ei-ther localized, ‘locons’, or extended (delocalized), ‘extendons’. The eigenvectorsof locons are characterized by a localization length (like the eigenvectors of frac-tons, see Eq. 2.8). Further, the extended modes with frequencies ω < ωIR possesswavevectors, wavelengths, and velocities, and are called ‘propagons’, as opposedto ‘diffusons’, that are extended but do not exhibit any of these wave properties.Of all vibrons, the extended and propagating propagons act similar to lattice vibra-tions in crystals.

To investigate the properties of vibrons in a-Si numerically, models for theatomic coordinates and the interatomic forces are required. Fabian et al. studiedrandom-network models of 216 and 1000 atoms (and occasionally of 4096 atoms)with atomic coordinates generated by the algorithm of Wooten, Weiner and Weaire[69]. The atoms are contained in cubic boxes that are continued periodically to in-


finity. Interatomic forces are represented by the Stillinger-Weber potential [70].With these ingredients, the vibrational eigenvectors of the a-Si model were calcu-lated.

One test of how realistic the model systems are was performed by evaluatingthe radial distribution function, g(r). Close agreement between the calculated andexperimental values of g(r) was obtained. However, this test may be not verystringent, since the shape of g(r) seems to depend mostly on the atom densityand nearest neighbor distances, and does not reflect that each Si atom is roughlytetrahedrally coordinated. That is, g(r) is not very sensitive to the exact localstructure.

A second comparison between the model and ‘real’ a-Si was made by evalu-ating the vibrational density of states. The calculated density of states appearedto agree qualitatively with neutron scattering data, although the high frequencies(ω 160 cm−1) are overestimated by ∼ 15 %.

A parameter that indicates if a vibron is a locon or extendon is the inverseparticipation ratio 1/P. For extended vibrations, 1/P ∼ 1/N, N being the numberof atoms in the system. For a mode localized at M atoms, 1/P ∼ 1/M. Fabianet al. calculated 1/P as a function of frequency. They found that 1/P starts toincrease abruptly at a frequency ∼ 574 cm−1, and defined this frequency as themobility edge, ωc. Corrected for the 15 % frequency overestimation, this valuebecomes 488 cm−1, which is quite close to center of the TO peak in a-Si. Ob-viously, according to these calculations ωc = ωIR. When the disorder of the a-Simodel was enhanced artificially, ωc decreased, but never reached ωIR. Apart fromthe locons with frequencies > ωc, a few modes in the low frequency (ω < ωIR)region of the vibrational spectrum and several with a frequency ∼ 260 cm−1 haveanomalously large inverse participation ratios. Still, their eigenfunctions are notcharacterized by a localization length. Fabian and Feldman argue that these statesare temporarily trapped in regions of peculiar coordination numbers, from whichthey may tunnel to extended modes [27, 71]. The number of these so-called res-onant modes decreases if the size of the a-Si model is taken larger, but Feldmanet al. believe that such excitations may be more pronounced in real a-Si, sincethe models are homogeneous on length scales 0.4 nm, whereas real glasses mayhave structural defects such as voids that would attract these resonant modes.

Because of the large difference between ωIR and ωc, the major part of thevibrons are diffusons, and are neither propagating (their group velocity is zero)nor localized. Allen and Feldman have developed a theory describing the thermaltransport in glasses [16]. They suggest that wavepackets consisting of diffusonsspread out diffusively (hence the name of these modes), resulting in thermal trans-port (diffusion). According to their model, the plateau in the heat conductivity of

48 Chapter 2

amorphous solids corresponds to a crossover from heat being transferred by ex-tended modes to predominantly by diffusons. The thermal conductivity calculatedfor models of a-Si is in agreement with the results of experiments. Still, anhar-monic decay processes were not included in their theory. Inspired by the work ofScholten et al., the treatment was extended, taking into account anharmonicity. Bymeans of second order perturbation theory, relaxation times for different modesand different temperatures of the a-Si model were evaluated. Both the process of avibron breaking up into two, and the process where a vibron absorbs a thermal vi-bron were contained in the equations. Their results suggest that vibrons in glassesdecay on picosecond time scales, independent of the exact character of the modes:locons, diffusons, and propagons have similar relaxation times. Moreover, for allvibrons decay rates appeared to increase with increasing frequency. The validityof the perturbation theory was confirmed by molecular dynamics simulations ofBickham and Feldman [27], that made use of the same a-Si model as Fabian et al.At moderate temperatures, lifetimes of ∼ 10 ps were obtained both for localizedand extended nonpropagating modes.

Both the outcome of the molecular dynamics simulations and the perturba-tive approach are in strong disagreement with our (and Scholten’s) experimentalresults. This suggests that the simulations and experiments do not measure thesame quantities on the same system. The decay rates calculated in the simula-tions are defined for single-frequency vibrational state populations, and small de-viations from equilibrium. On the other hand, in the experiments vibrations arecreated over the whole spectrum and the vibrational system is excited far out ofequilibrium. It could be [27] that upon excitation, local structural rearrangementstake place. These rearrangements, as well as the intrinsic fluctuations betweenmetastable structural states of the glass, are not taken into account by the modelsyet, although these may modify the normal-mode structure. Further, the modelsystem of a-Si used in the calculations may not be representative of ‘real’ amor-phous Si. The 216-atom large supercells that are used to build the a-Si model arerepeated with periodic boundary conditions. In this way, translational symmetry ona ∼ 1.5 nm length scale is introduced, possibly resulting in the unwanted presenceof extended low frequency modes. Inhomogeneities with larger dimensions do notoccur in such a system. Indeed, real a-Si contains structural inhomogeneities likevoids that are not included in the model [43]. In this aspect, the advantage of thefracton model is, that it explicitly describes the properties of translational invariantstructures. The characteristic crossover length ξ that is of importance in this modelis assumed to be much larger (∼ 10 nm) than the above mentioned 1.5 nm. It isnot clear yet to what extent the structural irregularities that exist in real a-Si(:H)may influence the vibrational dynamics.


Alternative explanations

Apart from the fracton and diffuson models, anharmonic decay of phonons ina-Si has as far as we know only been treated by Bottger et al. [13]. Their the-oretical investigations address among other things the behavior of nonthermallyexcited localized modes and the contribution of these models to thermal conduc-tivity. The model resembles the fracton model in several aspects, although theapproach is more general in the sense that it is not developed for fractal media,but for disordered dielectric materials. Also in this model, high-frequency (opti-cal) phonon modes are Anderson-localized due to disorder. And as in the fractondescription, the plateau in the heat conductivity is a consequence of the localizedmodes not being able to contribute directly to the thermal conduction. Again, thehigh-temperature thermal conductivity, κ, is due to thermally induced hopping ofthe localized optical modes. When anharmonicity is increased, κ decreases, but itstemperature dependence remains the same. In fact, also the model of Bottger etal. describes the universal behavior of κ in glassy materials well. Further, accord-ing to their description it is not inconceivable that the localized modes have life-times as long as suggested by our experimental results. The distribution of phononoccupation numbers they evaluated for an externally pumped collection of local-ized modes appeared to be a non-equilibrium distribution, like in the experiments.However, the same objections that have been put forward to the interpretation ofour results in terms of the fracton model also hold for the model by Bottger et al.

A difficulty that arose earlier when comparing the theoretical descriptions withthe experiments is, that it is not clear whether both are examining the same sys-tem. Apart from the absence of electronic processes in each of the mentionedmodels, an obvious uncertainty concerns the structure of amorphous Si, which isin all models assumed to be (statistically) homogeneous. However, it is known thata-Si contains structural inhomogeneities, that result for instance in the lower massdensity of a-Si as compared to c-Si [43]. Already in 1970, Brodsky et al. statedthat bulk a-Si consists of ‘building blocks’ with dimensions between 1 and 1.5 nm,and contains numerous internal microscopic surfaces [72]. More recently, electronmicroscopy studies have revealed that amorphous tetrahedral semiconductors likea-Si and a-Ge are structurally equivalent to a collection of polycrystalline grainswith diameters < 3 nm [73]. This concept is relevant from the point of view ofour experiments. The presence of building blocks and microscopic surfaces mayresult in ‘phonon interference effects’.

The concept of wavelength is not really meaningful for phonons with a fre-quency larger than the Ioffe-Regel limit, like we study. But as a crude estimate forlength scales, λ, that may be relevant for the properties of a certain acoustic mode,we may just divide the average sound velocity in a-Si (5· 103 ms−1) by the fre-

50 Chapter 2

quency of that mode. Hence, we find that λ 1 nm for frequencies 170 cm−1.Remarkably, we obtained mean decay times that were much shorter for the modeswith frequencies 200 cm−1 than for the modes with higher frequencies. Wespeculate that this may be a result of the high-frequency modes being ‘trapped’inside a-Si building blocks, rather than being localized in Anderson’s sense. Onecould imagine that if the blocks are decoupled from the surrounding, the anhar-monic decay is suppressed when it involves the break-up of a mode confined to ablock into modes that have typical wavelengths not fitting the block dimensions.And also the decay of modes with lower frequencies may be more complicated(and slower) than in homogeneous structures.

One origin for the decoupling one could think of is that a large acoustic mis-match exists at the boundaries of the blocks, for instance due to voids. Further,amorphous silicon is rather structured (‘crystalline-like’) on small (< 1 nm) lengthscales, which is expressed in relatively narrow bond-angle and bond-length distri-butions. Therefore, on the length scales of the building blocks, a ‘crystal direction’can be identified. If these directions are different for neighboring blocks, like atgrain-boundaries in polycrystalline Si, the vibrational dynamics may be affected.

Finally, we note that, in order to explain our results following this line ofreasoning, it is not essential that all building blocks are decoupled. As the ex-perimental setup used is only sensitive to long-lived phonons, it could be that thebehavior observed is not representative for all phonons. It cannot be excluded thatthe majority of the phonons even has a lifetime (much) shorter than 10 ns.

2.5 Conclusions

In summary, we present results of pulsed Raman experiments on the population de-cay of optically excited phonons in hydrogenated Si. In all samples studied, meandecay times of ∼ 70 ns were obtained for phonons with frequencies 200 cm−1.The lower energy modes appeared to have lifetimes shorter (< 10 ns) than could beresolved with the experimental setup. These observations are in qualitative agree-ment with the results presented by Scholten et al. [25, 40]. The exact reason forthe extremely long phonon lifetimes is still a matter of debate. In this chapter,several models describing the anharmonic decay of phonons in a-Si are being re-viewed. Unfortunately, none of these models can account for all our experimentalfindings. A possibility not included by the discussed models is, that structural in-homogeneities affect the vibrational dynamics in a-Si(:H). It has been suggestedthat bulk a-Si consists of ‘building blocks’ with dimensions between 1 and 1.5 nm,length scales that are likely to be relevant for the dynamics of vibrations of the fre-quencies studied. The idea that a portion of the phonons in a-Si:H have such long


lifetimes because they are confined to nanometer scale regions in the amorphousstructure is worth further exploring.

Apart from the relatively fast decays interpreted as phonon lifetimes, an addi-tional slowly ( 100 ns) decaying contribution to the Raman signals was detectedin a-Si:H films prepared by plasma enhanced chemical vapor deposition. No suchcontribution was observed in layers grown by hot-wire assisted chemical vapor de-position. A model is presented that qualitatively explains the ‘slow background’as resulting from slow electronic recombination processes. It turns out that bothrelaxation of hot charge carriers and nonradiative recombination of free carrierswith carriers localized in the tail states contribute significantly to the Raman sig-nals. The ratio between the relaxation and recombination rates depends on materialparameters and the excitation power. This may explain why, for the range of ex-citation powers used in the experiments, the slow background is present in the PEsamples and absent in case of the HW layers. This proves that the pulsed Ramantechnique is sensitive to relative rates of electronic processes.

The results presented in this chapter partially motivated the experiments dis-cussed in the following chapters. To get an impression of the importance of struc-tural inhomogeneities for the vibrational dynamics, pulsed Raman experimentswere performed on PE-grown a-Si:H samples containing a large volume fractionof small (∼ 5 nm) c-Si particles (Chapter 3). Further, because of the obvious in-fluence of electronic processes on the temporal evolution of the Raman signals,experiments not involving electronic excitation of the system appeared interesting.The pulsed infrared experiments presented in Chapter 4 show that vibrational dy-namics in a-Si:H can indeed be examined by direct excitation of vibrations, withoutintervention of charge carriers.

Finally, we note that measurements performed with an experimental setup sen-sitive to decay times much shorter than the lifetimes that Scholten and we obtainedcould shed new light on the mystery why high-frequency phonons in a-Si(:H) seemto live so long.

CHAPTER 3

Dynamics of vibrations in a mixedamorphous-nanocrystalline Si system

In this chapter, results of pulsed Raman and phonon-induced luminescence ex-periments on a mixed amorphous-nanocrystalline silicon system (a-nc-Si:H) arereported. With these experiments, the decay and transport of non-equilibriumphonons in a-nc-Si:H was examined, and compared with the behavior of phononsof the same frequencies in a-Si:H and c-Si. From the Raman measurements,we find that in the spectral region of the TO vibrations in the crystallites (505-520 cm−1), phonons have shorter decay times than the TO phonons in a-Si:H, butlonger than in c-Si. In addition, the lifetimes increase with decreasing frequency,from < 10 ns at 515 cm−1 to ∼ 30 ns at 505 cm−1. For comparison, in c-Si Ra-man active TO phonons have a frequency of 520 cm−1 and lifetime of ∼ 10 ps,while in a-Si:H, the 480-cm−1 TO modes decay on a ∼ 70-ns time scale. Further,for TA phonons in a-nc-Si:H at ∼ 150 cm−1, we measured a much longer decaytime (τ ∼ 50 ns) than for phonons of the same energy in a-Si:H (τ < 10 ns). Fi-nally, the diffusion of 29-cm−1 phonons through the a-nc-Si:H and a-Si:H materialwas examined in phonon-induced luminescence experiments. Transport throughthe a-nc-Si:H film appeared to be much slower than through the a-Si:H layer. Weexplain these results as effects of phonon confinement.

3.1 Introduction

Numerous studies have been carried out to investigate the effects of confinementon the properties of phonons in semiconductor nanoparticles [74]. In Raman scat-tering experiments, size-dependent shifts and broadening of the peaks of opticaland acoustic phonons have been observed [75, 76, 77, 78]. Yet only few examplesexist of experiments in which the dynamical aspects of confined phonons havebeen studied. In porous nanocrystalline corundum, one has measured millisecondlifetimes of size-quantized vibrations with a frequency of 20 cm−1 [79]. In small-

53

54 Chapter 3

grain polycrystalline corundum the decay of phonons with frequencies between29 and 80 cm−1 has been examined. Also in those experiments, surprisingly longphonon decay times (τ > 1 ms) have been discovered [80]. In the corundum-basedmaterials, the vibrational dynamics turned out to be rather sensitive to the ratioof the wavelength λ of the vibrational excitation studied and the typical size a ofthe micro structures that constitute the system. For λ/a 1, the observed phonondynamics is similar to that encountered in bulk crystals: the measured phonon life-times are of the same order of magnitude, and the phonon propagation appearedto be almost ballistic [81]. In contrast, for λ/a ∼ 1, vibrational dynamics resem-ble more the behavior observed in glasses: the vibrations decay on much longertime scales, and propagate diffusively [25, 82]. The transition from ‘crystalline’to ‘glassy’ dynamics appeared to be gradual. These observations have inspiredus to examine to what extent the anomalous vibrational properties of glassy andamorphous solids are related to their micro structure. In Chapter 2 we already pro-posed that the remarkable dynamical properties of phonons in a-Si:H [25, 40] mayoriginate from a-Si:H consisting of very small (∼ 1 nm), mechanically more orless decoupled ‘crystalline’ building blocks. To justify such ideas, it is of interestto study dynamics of vibrations in a mixed amorphous-nanocrystalline material,that exhibits crystalline order on nanometer length scales. Such experiments mayprovide a clue to understand the dynamics of phonons in amorphous systems.

In this chapter, results of pulsed Raman and phonon transport experiments on amixed amorphous-nanocrystalline silicon system (a-nc-Si:H) are presented. Withthese experiments we examined the decay of high-frequency phonons, and trans-port of phonons with a frequency close to the expected lowest frequency supportedby the nanoparticles in a-nc-Si:H. The results are compared with the behavior ofphonons of the same frequencies in a-Si:H.

3.2 Experimental

3.2.1 Samples

To examine the difference between phonon dynamics in pure a-Si:H and mixedamorphous-nanocrystalline Si (a-nc-Si:H) layers, two types of samples were pre-pared. Both were grown on sapphire wafers (Al2O3 containing 20 at. ppm of Cr3+

ions) by plasma-enhanced chemical vapor deposition, to a thickness of ∼ 0.4 µm.By carefully controlling the deposition parameters, one type of layer was made tocontain a large volume fraction (∼ 24 %) of Si crystallites with an average diameterof ∼ 4.5 nm. These numbers were extracted from a detailed analysis of the Ramanspectra and electrical conductivity of the film [83]. The other type of layer wasproduced in the same run, but did not contain such crystallites. This was accom-

Dynamics of vibrations in a-nc-Si:H 55

400 450 500 550

wavenumber (cm-1)

Sto

kes

Ram

an in

tens

ity

0 200 400 600

Figure 3.1 Room-temperature Stokes Raman spectra of the a-nc-Si:H sample excitedwith an argon-ion laser. The inset shows the complete first-order Raman spectrum, an-alyzed with a SPEX 1877 0.6-m Triple Spectrometer and recorded by a liquid-nitrogencooled CCD camera (EG&G). The full graph focuses on the TO-part of the spectrum,where the solid lines indicate the contributions from a-Si:H and nc-Si, centered at 480 and517 cm−1, respectively, and the dotted line is their sum. This spectrum was taken with aSPEX 14018 0.85-m Double Monochromator equipped with a Peltier-cooled EMI 9863kb/100 photo multiplier tube.

plished by depositing it with the substrate positioned at a different location insidethe deposition chamber. The samples were prepared at the A.F. Ioffe Physical-Technical Institute, St. Petersburg (Russia). For details, we refer to Ref. [83].

Fig. 3.1 shows room-temperature Stokes Raman spectra of the a-nc-Si:H sam-ple. We observe both the contributions from the a-Si:H and from the nc-Si re-gions to the Raman spectrum: a broad peak centered at 480 cm−1 correspondingto the Raman spectrum of the TO phonons in a-Si:H, and a narrower resonance at515 cm−1 caused by the TO modes in the crystallites. The exact spectral positionand width of the crystalline peak depends on the size of the crystallites, as will bediscussed in more detail in Sec. 3.4.1.

The Raman spectrum of the layer without crystallites is very similar to that ofthe PE samples used throughout this thesis (see Fig. 2.1).

56 Chapter 3

3.2.2 Time-resolved Raman spectroscopy

The setup for the time-resolved Raman experiments on a-nc-Si:H was analogous tothat of the pulsed Raman measurements on a-Si:H discussed in Chapter 2. Again,to remove the thermal population of phonon modes in the energy range of inter-est, the samples were immersed in superfluid He (T∼ 1.8 K). A non-equilibriumvibrational population was created during the relaxation and recombination of op-tically excited charge carriers (see Fig. 2.2). The phonons were detected by meansof anti-Stokes Raman spectroscopy. To investigate the temporal evolution of thephonon populations, pump-probe configurations were used, employing two pulsed,frequency-doubled Nd:YAG lasers (λ = 532 nm). Both generated a train of 10-nspulses at a repetition rate of 30 Hz. Pulses of the second laser (probe) were givenan electronically controlled delay with respect to those of the first (pump). Bymeasuring anti-Stokes intensities as a function of the delay between pump andprobe, the phonon dynamics was examined on time scales of 10 ns up to 15 ms.

A first series of the experiments was performed with the same CCD detectionscheme as used for the measurements presented in Chapter 2 (see Fig. 2.3): anti-Stokes Raman spectra were excited by focusing the laser beams to a ∼ 1 mm2

spot on the sample, analyzed by a triple monochromator, and recorded by a CCDcamera, with a 15-minute exposure time. The average power of both beams wastypically 10 mW at the position of the sample, corresponding to an absorbed en-ergy density per pulse of ∼ 3 · 103 J/cm3. A narrower slit width of the spectrom-eters was chosen than in Chapter 2, in order to enhance the spectral resolution to∼ 20 cm−1.

A second series of measurements was carried out with a higher spectral reso-lution, in order to study the dynamics of TO phonons as a function of the frequencywithin the TO peak. The aim of these experiments was to see whether differencescan be detected between the behavior of TO phonons in a-Si:H and the crystallites.This was not possible with the above-mentioned CCD detection scheme, since therequired slit-width resulted in too low signal intensities: the number of dark countsin the period between two laser pulses was higher than the amount of anti-StokesRaman scattered photons detected during a laser pulse. Therefore, the experimen-tal setup was modified, as is drawn schematically in Fig. 3.2. A larger, conven-tional double monochromator was used, equipped with a Peltier-cooled photo mul-tiplier tube (PMT), followed by photon-counting electronics. A time-to-amplitudeconverter (TAC) was used to define a gate: only photons detected during a se-lected interval in time were recorded, to reject the majority of dark counts. Thegate was set at 500 ns, long enough to capture both the pump and probe pulses.During those intervals, the signals were monitored with a temporal resolution of∼ 5 ns. The increased spectral resolution of the larger double monochromator


Figure 3.2 Experimental setup used for the time-resolved Raman measurements as afunction of the frequency within the TO peak. PG= pulse generator, VD = variable delay,FD = fixed delay (Stanford Research Systems model DG 535), M = mirror, BS = beamsplitter, PR = polarization rotator, L = lens, S = sample, DM = double monochromator(0.85-m, SPEX 14018), PMT = photo multiplier tube (EMI 9863 kb/100, Peltier-cooled)CFD = constant fraction discriminator (TENNELEC TC 453) TAC = time-to-amplitudeconverter (EG&G ORTEC 457), PHA = pulse height analyzer (the Nucleus). a-nc-Si:Hwas excited with the 10-ns pulses of two frequency-doubled, Q-switched Nd:YAG lasers(Spectra Physics, Quanta Ray DCR-3 and GCR-130). The sample was immersed in su-perfluid He in a Magnex dynamic flow cryostat with optical access, equipped with aLakeshore temperature controller.

was 12 cm−1. In contrast with the CCD-setup, Raman spectra were obtained byrecording time-traces for different spectral positions of the monochromator. Thedecay of the phonon population was studied by performing the experiments as afunction of the delay between the pump and probe pulses. Stokes measurementswere averaged over 500 s, anti-Stokes signals over 1500 s.

3.2.3 Phonon-induced luminescence experiments

All phonons studied in the Raman experiments have a frequency that is muchhigher than the Ioffe-Regel limit ωIR in a-Si:H [41]. Other experimental tech-niques have to be used to investigate dynamics of phonons with lower frequencies,closer to, or even lower than ωIR. As will be explained below, phonon-induced

58 Chapter 3

Figure 3.3 Principle of the detector for 29-cm−1 phonons in ruby, based on the energylevels of Cr3+. Following optical excitation, the metastable 2E levels are populated, re-sulting in the R1 and R2 luminescence lines. At low temperatures, the intensity of R2 is ameasure for the 29-cm−1 phonons injected from the sample into the ruby substrate.

luminescence experiments, based on the so-called ‘ruby phonon detector’, can beexploited to examine the transport of 29 cm−1 phonons in our samples. Apartfrom the fact that this frequency happens to be close to ωIR in a-Si:H, it is alsoof the order of the lowest frequency that can be supported by the nanometer sizecrystallites that are present in the a-nc-Si:H layer under investigation. These cir-cumstances make a study of the transport of 29 cm−1 phonons in the a-nc-Si:Hfilm of particular interest.

The ruby phonon detector

Advantage was taken of the fact that the layers studied in this chapter were grownon sapphire substrates, which in fact is very dilute ruby, and can be used as anoptical detector for 29-cm−1 phonons injected by the a-Si:H and a-nc-Si:H filmsinto the substrate [41]. The principle of the ruby phonon detector is based onthe energy levels of the Cr3+ ions present in the substrate, and the radiative andnonradiative transitions between them (see Fig. 3.3) [84].

Following optical excitation of the 4T1 and 4T2 bands of the Cr3+ ions, fast(∼ ps) radiationless transitions take place that result in population of the metastable2E levels. The lower of the two, the E(2E) state, decays radiatively to the 4A2

ground state, resulting in an intense luminescence line (R1). The upper of the 2Elevels, the 2A(2E) state, at low temperatures preferentially decays into the E(2E)level, under the emission of a 29-cm−1 phonon. The typical time scale of this pro-


cess (∼ 700 ps) [85] is much shorter than the lifetime of the R1 transition (∼ 4 ms),resulting in a much higher population of the E(2E) compared to the 2A(2E) levelat low temperatures. Consequently, the R2 luminescence caused by the radiativedecay of the 2A(2E) into the 4A2 ground state (also with a τ ∼ 4 ms), is muchweaker than R1, and actually even negligible at 2K. However, when a populationof 29-cm−1 phonons is injected in the excited region of the ruby substrate, tran-sitions of the E(2E) to the 2A(2E) level occur, increasing the intensity of the R2

luminescence. Hence, in the absence of a thermal 29-cm−1 phonon population,the temporal evolution of a 29-cm−1 phonon population emitted from the sam-ple layer into the ruby substrate can be monitored via the R2 luminescence. Forequal radiative decay probabilities of the 2A(2E) and E(2E) levels, the ratio of theluminescence intensities, R2/R1, is given by

R2

R1=

n29

n29 + 1, (3.1)

where n29 is the occupation number of the 29-cm−1 phonons in the optically ex-cited region.

Only when the concentration of excited Cr3+ ions in the substrate is lowenough, the 29-cm−1 phonons travel ballistically through the substrate, with a rel-atively small chance to be absorbed by a Cr3+ ion. In that case, the time-resolutionof the phonon detector is limited by the spontaneous phonon emission rate of the2A(2E) → E(2E) transition. Then, the temporal profile of the R2 luminescence re-flects the evolution of the 29-cm−1 phonon flux injected into the excited region. Incontrast, in the presence of high concentrations of excited Cr3+ ions, the phonontransport may become diffusive, due to multiple absorption and re-emission events.Under such so-called ‘phonon bottleneck’ conditions, the phonon detector may be-come much slower [86], and even inappropriate for our purpose.

Scholten et al. for the first time studied the transport of 29-cm−1 phonons in a-Si:H using the ruby phonon detector [41]. For the excitation powers they used (andidentical substrates) the phonon bottleneck effect was proven to be absent. Sincewe performed the experiments with even lower excitation powers, the transportof 29-cm−1 phonons through our substrates is definitely ballistic. Scholten et al.further noted that the temporal resolution of the detector is limited by the finitedimensions of the excited region in the substrate and the sound velocity of 29-cm−1 phonons in ruby. For an excited cylinder with a diameter of ∼100 µm thisresults in a resolution of ∼ 10 ns [41].

60 Chapter 3

Figure 3.4 Schematic representation of the two configurations used for the phonon-induced luminescence experiments.

Experimental setup

Phonons were generated in the a-nc-Si:H and a-Si:H layers during the relaxationand recombination of hot carriers excited by the absorption of green photons (seeFig. 2.2). The green light (λ = 514.5 nm, penetration depth in a-Si:H ∼ 100 nm)was emitted by a mode-locked, cavity-dumped argon-ion laser (Spectra Physicslaser model 2060-10SA, mode locker model 342, and cavity dumper model 344),producing 1-ns pulses at repetition rates of 800 kHz or 4 MHz. The beam wasfocused to a ∼100-µm spot on the sample, where the average excitation poweramounted to ∼ 4 mW. For a repetition rate of 4 MHz this corresponds to an ab-sorbed energy density of ∼ 1.3 J/cm3 per pulse. Part of the light that was not ab-sorbed by the a(-nc)-Si:H layers excited Cr3+ ions in the substrate, and activatedthe ruby phonon detector. Luminescence from the substrate was collected andprojected on the entrance slit of a double monochromator (0.85-m, SPEX 14018).The spectral slit width was set to ∼ 5 cm−1. The same photon counting electronicswas employed as for the time-resolved Raman measurements (Fig. 3.2). A Peltier-cooled photo multiplier tube (Hamamatsu R943-02) was used for detection of thered photons. The temporal evolution of the 29-cm−1 phonon population emittedby the sample was monitored with a resolution of ∼ 10 ns, limited by the speedof the ruby phonon detector, as discussed above. The samples were mounted in acontinuous-flow cryostat (Oxford Optistat) operating at 1.8 K, to remove thermalphonons.

To examine the transport of 29-cm−1 phonons through the samples, the ex-periments were carried out for two configurations (see Fig. 3.4): one in which thelaser beam was incident at the a(-nc)-Si:H side of the sample, and phonons hadto travel ∼ 300 nm from the phonon source to the detector, and a second config-


-400 -200 0 200 400

anti-Stokes (x 2.5)

Stokes

Raman shift (cm-1)

inte

nsity

(ar

b. u

nits

)

Figure 3.5 Raman spectra of a-nc-Si:H () and a-Si:H (•) as measured with the CCD forT ∼ 1.8 K. Anti-Stokes intensities have been multiplied by a factor of 2.5.

uration where the laser entered the sample from the side of the substrate, corre-sponding to an average distance between the phonon source and detector equal tothe penetration depth of the exciting light (∼ 100 nm). Throughout this chapter,the two configurations are referred to as configuration (1) and configuration (2),respectively.

3.3 Experimental results

3.3.1 Raman experiments

Figure 3.5 shows low-temperature Stokes and anti-Stokes Raman spectra of boththe a-Si:H and a-nc-Si:H samples, obtained with the CCD detection scheme. Thebroad lines typical for phonons in a-Si:H can be distinguished. From Fig. 3.1we know that the TO peak of the a-nc-Si:H layer consists of a superposition ofa broad amorphous TO peak (centered at 480 cm−1) and a 15 cm−1-wide linecentered at 515 cm−1 that corresponds to TO phonons in nc-Si. However, in caseof the measurements presented in Fig. 3.5, the resolution of the spectrometer wasnot sufficient to resolve the amorphous and nanocrystallite peaks: only a shift of

62 Chapter 3

10

10

100

power(mW)

inte

nsity

(ar

b. u

nits

)

Figure 3.6 Dependence of the TO anti-Stokes signal of the a-nc-Si:H sample on exci-tation power, for excitation with one laser. The line indicates a quadratic dependence onpower.

the total TO peak is observed. Later in this section, low-temperature spectra willbe presented taken with a higher spectral resolution.

In a second experiment, we examined for the a-nc-Si:H sample the dependenceof the Raman intensity on excitation power P. The Stokes intensities appearedto increase linearly with P, whereas the anti-Stokes signals exhibit a quadraticdependence on excitation power. Fig. 3.6 illustrates the P2-dependence of theanti-Stokes TO intensity of a-nc-Si:H. Following the same line of reasoning as inChapter 2, we conclude from these observations that the occupation numbers nω ofthe excited phonon population in a-nc-Si:H increase linearly with P, and that thepopulation studied must have a nonthermal and nonequilibrium character.

The temporal evolution of the generated non-equilibrium phonon populationwas investigated by recording a series of anti-Stokes spectra for different delays∆t between the pulses of the pump and probe laser. To extract the time-dependentcontribution to the spectra, a spectrum taken with a delay of 15 ms was subtractedfrom each measurement. It turned out that also in the samples examined in thischapter, a slowly (τ 100 ns) decaying, frequency-independent component con-tributed to the anti-Stokes intensity IAS(ω). In Chapter 2, this slow background was


0.0

0.5

1.0 a)

inte

nsity

(ar

b.un

its)

0 50 100 150 2000.0

0.5

1.0 b)

delay (ns)

Figure 3.7 a) Normalized anti-Stokes intensities as a function of the delay, measured ina-nc-Si:H in the spectral regions of TO (), LA (), and TA (•) vibrations. To extractthe time-dependent contributions to the signals, spectra taken with a delay of 15 ms anda 30% background corresponding to electronic effects (see Chapter 2) were subtracted.b) Normalized TO anti-Stokes intensities vs. delay in a-nc-Si:H () and a-Si:H (). Again,the mentioned background signals were subtracted.

investigated in more detail, and was demonstrated to be related to slow electronicprocesses taking place in the PE material under the relevant excitation conditions.Since no clear difference between the behavior of the slow contribution in a-Si:Hand a-nc-Si:H was observed, no further experiments on this effect were done. Inall cases, the corresponding background levels were subtracted from the signals,to focus the attention on the differences in phonon dynamics in the two systems.

In Fig. 3.7, we present the results as normalized anti-Stokes Raman intensitiesplotted versus ∆t. Fig. 3.7.a shows the temporal evolution of IAS(ω) in the a-nc-Si:H layer, for three spectral regions (cf. Fig. 2.6 for a-Si:H). As for the graphspresented in Chapter 2, all intensities were integrated over a sufficiently broadrange of frequencies, to improve the signal-to-noise ratios. For the TA, LA, andTO phonons, the integration was carried out over a range of 140-220 cm−1, 240-380 cm−1, and 420-550 cm−1, respectively. In Fig. 3.7.b, the behavior of TOphonons in a-Si:H and a-nc-Si:H is compared. As can be seen, the decay of theTA, LA and TO phonon population in a-nc-Si:H closely resembles the decay of LA

64 Chapter 3

100 200 300 400 500 600 7000

50

100

150

200

x 1.8

Raman shift (cm-1)

inte

nsity

(co

unts

/ 9

00 s

)

Figure 3.8 Differential anti-stokes spectra obtained by subtracting spectra measured witha delay of 200 ns from the spectra taken with a delay of 20 ns, in a-Si:H (•) and a-nc-Si:H().

and TO phonons in a-Si:H without nanocrystals, both with respect to the temporalshape of the signals as to the absolute values of the decay times. Concerning thedecay of the TA phonons, this observation is quite remarkable, since it implies thatthe TA phonon population in a-nc-Si:H decays much slower ( τ ∼ 50-70 ns) thanin pure a-Si:H (τ 1 ns). The strikingly different behavior of TA phonons in a-nc-Si:H compared to a-Si:H is also apparent in Fig. 3.8, which shows the spectrumof phonons that survive a delay of 20 ns. In a-Si:H, the differential anti-Stokesintensity rapidly decreases with decreasing frequency for ω 250 cm−1, whereasthe differential signal in a-nc-Si:H is nearly constant, signifying that phonons livelong over the full frequency region.

A second series of measurements was performed with the higher spectral reso-lution of the modified Raman setup with gated detection. In these experiments, wefocus on the dynamics of TO phonons in a-nc-Si:H. In the TO part of the spectrum,contributions to the Raman signals from phonons related to the crystallites can bedistinguished from the normal a-Si:H spectrum, which makes it feasible to explorethe differences between the dynamical properties of TO phonons in a-Si:H and thenanocrystallites.


470 480 490 500 510 520 530

0.2

0.4

0.6

0.8

1.0

x 3

wavenumber (cm-1)

inte

nsity

(co

unts

/s)

Figure 3.9 Stokes () and anti-Stokes (N) TO peaks of the a-nc-Si:H film, obtained withthe gated detection scheme. Anti-Stokes intensities are multiplied by 3. The arrows indi-cate the frequencies where time-resolved experiments with the same detection setup wereperformed (Fig. 3.10). The dotted and dash-dotted lines serve as a guide to the eye.

In Fig. 3.9 the low-temperature Stokes and anti-Stokes TO peaks of the a-nc-Si:H layer are presented. In these graphs, the different contributions of phononsin a-Si:H and nc-Si can be recognized. Remarkably, the part of the anti-Stokesspectrum that corresponds to the TO phonons in the nanocrystals appears to peakat an approximately 10-cm−1 lower energy than in the Stokes spectrum. Thissuggests that during the pulses, it is easier to maintain a phonon population atthe lower phonon frequencies (∼ 500 cm−1), i.e. closer to the maximum of theamorphous TO peak, than at the higher frequencies (∼ 515 cm−1), closer to thecenter of the crystalline TO peak. An obvious explanation for these observations isthat phonons at higher frequencies decay faster than at lower frequencies. To verifythis interpretation, we employed the pump-probe setup and studied the decay of theanti-Stokes intensity at three selected frequencies within the TO peak: 495, 505and 515 cm−1, as indicated by the arrows in Fig. 3.9. Results of these experimentsare presented in Fig. 3.10, which shows the time-dependent part of IAS measuredat the selected frequencies as a function of the delay between pump and probepulses. Apparently, the decay of phonons at the lowest frequency (495 cm−1) is

66 Chapter 3

0 40 80 120 160 200

0.0

0.2

0.4

0.6

0.8

1.0

1.2

495 cm-1

505 cm-1

515 cm-1

delay (ns)

inte

nsity

(ar

b. u

nits

)

Figure 3.10 Normalized time-dependent anti-Stokes intensities measured at three fre-quencies within the TO peak of a-nc-Si:H, as a function of the delay.

the slowest (τ ∼ 50 ns). For 505 cm−1, the decay is faster, and at a frequency of515 cm−1, it is too fast (< 10 ns) to be observed with our setup. (Recall that atzero delay a strong contribution is present at 515 cm−1).

3.3.2 Experiments with the ruby phonon detector

Results of the experiments with the ruby phonon detector are shown in Fig. 3.11.For configuration (1) (Fig. 3.11.a), the phonon-induced luminescence signalsR2(t)/R1(t) of a-Si:H and a-nc-Si:H are clearly different. The trailing edge ofR2(t)/R1(t) obtained for the a-nc-Si:H film is much longer (∼ 0.5 µs) than thatobserved for the a-Si:H layer. For the 4-MHz repetition rate at which these mea-surements are taken, the phonon induced signals in a-nc-Si:H even do not com-pletely vanish during the interval between two laser pulses, and cause the higherbaseline level of R2(t)/R1(t) in a-nc-Si:H. Experiments performed at a repetitionrate of 800 kHz showed equally low baseline levels of the phonon-induced signalsin a-nc-Si:H and a-Si:H. The rise time of R2/R1 is in both materials almost thesame (∼ 10 ns).

In configuration (2), as shown in Fig. 3.11.b, the difference in temporal shapes


a)

R2

/ R1

(arb

.uni

ts)

-20 0 20 40 60 80 100

b)

time (ns)

Figure 3.11 Temporal evolution of the normalized luminescence signals induced by theinjection of 29-cm−1 phonons generated in the a-nc-Si:H () and a-Si:H (•) into the rubysubstrate. Graphs in (a) and (b) correspond to the signals obtained for configuration (1) and(2), respectively. The difference in baselines in (a) results from an incomplete recovery ofthe phonon signal in a-nc-Si:H when experiments are performed with a repetition rate of4 MHz. For repetition rates of 800 KHz and lower the baselines in a-Si:H and a-nc-Si:Hhave the same level. All measurements presented here were performed with an excitationpower of ∼ 3.6 mW, corresponding to an absorbed energy density of ∼ 1.2 J/cm 3 perpulse.

of R2(t)/R1(t) is less pronounced than in configuration (1). Both the signals ina-nc-Si:H and a-Si:H have sharp leading edges (∼ 10 ns) and relatively fast trailingedges (40-60 ns).

Fig. 3.12 shows signals obtained in a-nc-Si:H, for two different excitationpowers. As can be seen, the temporal shape of the curve measured for configu-ration (1) remains virtually the same when the power is decreased by a factor offour. Signals obtained for configuration (2), however, appear to become signif-icantly faster when the power is decreased by the same amount. Both the timescales of the rise and the decay of R2(t)/R1 are shorter for lower power. The in-tegrated R2(t)/R1(t) intensity, however, increases linearly with excitation power,for both configurations. This directly demonstrates that no phonons get lost in thea-nc-Si:H film.

68 Chapter 3

a)

-20 0 20 40 60 80 100

0

0

b)

R2

/ R1

(arb

.uni

ts)

time (ns)

Figure 3.12 Normalized phonon-induced signals from the a-nc-Si:H layer for two ex-citation powers: 3.6 mW () and 0.9 mW (•). Graphs in (a) and (b) correspond to thesignals obtained for configuration (1) and (2), respectively. For comparison, the baselinelevels of the signals obtained in configuration (1) have been subtracted.

In the case of a-Si:H, R2/R1 exhibits a steeper rise for lower powers. On theother hand, the trailing edge decays on the same time scale for different excitationpowers. Further, in this material results of the measurements performed for config-uration (1) again resembled those for configuration (2), for each of the excitationpowers. All the observations in a-Si:H are in agreement with the results reportedby Scholten et al. [41].

3.4 Discussion

Similarities and differences appear, when the results of the pulsed Raman andphonon transport experiments obtained for a-nc-Si:H and a-Si:H are compared.Long-lived vibrational signals were observed in both types of materials. As al-ready noted, the quadratic dependence of the anti-Stokes intensity on excitationpower indicates that in a-nc-Si:H, a non-equilibrium phonon population can begenerated, as is the case in a-Si:H. The decay of this population can thus be in-vestigated via the temporal evolution of the anti-Stokes spectra. Two contributions


can be distinguished in the time-dependent anti-Stokes intensity IAS(ω, t) mea-sured in a-nc-Si:H: a relatively fast decaying part(τ 50 ns, depending on thephonon frequency), and a slowly decaying background (τ 100 ns, independentof the phonon frequency). A similar superposition of anti-Stokes contributionswas observed in the PE a-Si:H samples (see Chapter 2). Also the a-nc-Si:H layerwas deposited by PECVD, and characteristics of the two components in the Ra-man signals of a-nc-Si:H are similar to those in the PE a-Si:H layer. In Chapter2 we already demonstrated that in PE a-Si:H, the decay of the fast contributionto IAS(ω, t) corresponds to phonon decay, and that the slow background originatesfrom slow electronic processes. We conclude here that the same situation occursin a-nc-Si:H prepared by PECVD.

The most pronounced differences between the results of the Raman experi-ments in a-nc-Si:H and a-Si:H are, that the lifetime of the TO phonons in a-nc-Si:His frequency-dependent, and that TA phonons in a-nc-Si:H appear to survive overmuch longer periods than in a-Si:H. Further, experiments performed with the rubyphonon detector reveal that in a-nc-Si:H, complete recovery of the phonon-inducedluminescence signals takes more than 250 ns (compared to ∼ 70 ns in a-Si:H). Allthese issues will be addressed in the remainder of this section. But first, the char-acteristics of the equilibrium Raman spectra of a-nc-Si:H are discussed in moredetail.

3.4.1 Raman spectra of a-nc-Si:H

As is clear from Fig. 3.1, the Raman spectra of a-nc-Si:H are different both fromthe spectra of c-Si and a-Si:H. In fact, the spectra suggest that the total scatteringintensity is a sum of contributions from a-Si:H and from the crystallites. Therelative weight of these components then,depends both on the concentration ofthe crystalline particles and their sizes, since the Raman scattering cross sectiondepends strongly on the size of the crystallites [83, 87].

The crystallites themselves produce Raman lines that are shifted and broad-ened relative to the Raman line of bulk c-Si, which is centered at ω0 =521 cm−1

and has a narrow line width Γc = 3.5 cm−1 at room temperature. For small crys-tallites namely, the (q = 0)-selection rule for Raman scattering in crystalline Si isrelaxed. Consequently, in small crystallites, phonons well beyond the center ofthe Brillouin zone contribute to the Raman spectrum. Neutron scattering data ob-tained for crystalline Si tell us that the dispersion relation for the TO phonons is adecreasing function of |q| at the center of the Brillouin zone, see Fig. 3.13. There-fore, contributions of larger q in the light scattering in the nanocrystallites result ina broader Raman peak at a lower frequency than in bulk crystalline Si [75, 76].

A generally used but simplified model that describes the TO Raman spectra

70 Chapter 3

0

5

10

15

wavevector

TO

LO

LA

TA

LΓ

freq

uenc

y (T

Hz)

Figure 3.13 Phonon dispersion of crystalline Si, shown for one crystal direction (Γ−L),taken from [88]. The lines correspond to the results of model calculations; the circlesrepresent experimental values obtained from neutron diffraction measurements.

produced by nanocrystalline Si films, considers a bulk crystal in which phononpackets reside that are spatially confined to a volume corresponding to the dimen-sions of the crystallites. The phonon wavefunction of wavevector q0 is then writtenas a superposition of eigenfunctions

Ψ(q0,r) =Z

d3q C(q0,q)eiq·r, (3.2)

where C(q0,q) is determined by the shape of the confinement volume, and theboundary conditions. For a spherical volume, and Gaussian wavepackets withenvelope exp[−2r2/d2], C(q0,q) is proportional to exp[−|q0 +q|2d2/2]. The firstorder Raman contribution Inc(ω) of such a confined phonon packet is given by[75, 76]

Inc(ω) = AZ

d3q |C(0,q)|2[ω−ω(q)]2 +(Γc/2)2 , (3.3)

where only q0 = 0 phonons are assumed to contribute, by analogy with the q = 0selection rule in bulk crystals. Further, ω(q) denotes the phonon dispersion rela-tion, and A is a constant. The length of the phonon wavevector, q, is expressed


in units of 2π/a0, with a0 the lattice constant (in c-Si a0 = 5.4 A). The diameterof the crystallites, d, is also given in units of a0. To simplify the calculation, theintegral is evaluated for a spherical Brillouin zone, and an isotropic phonon disper-sion, ω(q) = ω0(1−0.18q2), that tracks the crystalline dispersion relation for low|q|. These assumptions are justified for not too small particles, where only a smallregion in the center of the Brillouin zone, contributes to the Raman integral. Ford 6× a0, these restrictions are fulfilled.

The amorphous TO Raman component Ia can be approximated by a Gaussianof height B, centered at the position of the amorphous line (480 cm−1), with theappropriate width Γa.

By fitting the sum of Inc and Ia to the measured Raman spectrum, the aver-age size of the nanocrystals can be estimated. A, B, d and Γa are used as fittingparameters. The spectral resolution of the experimental setup was included by con-voluting Ia(ω)+ Inc(ω) with a Gaussian of the appropriate width. In this way, thetwo solid lines were obtained that are displayed in Fig. 3.1. The high-frequencypeak corresponds to the Raman contribution of crystallites with a diameter of8.3 ×a0 (4.5 nm). As can be seen, the calculated curve nicely tracks the measure-ment, except in a region between 480 and 500 cm−1, and for frequencies lowerthan 450 cm−1. The first deviation suggests a dispersion of sizes of the crystallitesin our sample, which is not included in the model. Further, the extra Raman signalbelow 450 cm−1 originates from LO phonons in a-Si:H, also not accounted forby the model. Further, Iqbal et al. measured the TO peak-frequency ωnc of theRaman contribution of Si nanocrystallites as a function of d, by combining Ramanwith X-ray diffraction experiments [77], an analysis that has not been done forour samples. They found ωnc = 512 cm−1 for d = 3.5 nm, increasing linearly toωnc = 517 cm−1 for d = 7 nm. For larger nanocrystals, ωnc(d) appeared to riseless steeply, and become equal to ω0 for d ∼ 15 nm. Their results suggest thatthe model presented above underestimates the diameter of the Si particles. Similardeviations of calculated values for d were reported by Richter et al. [75]. In filmsprepared by Okada et al., even an ωnc of 480 cm−1 was measured for particleswith a diameter as large as 7 nm [89]. Indeed, it should be stressed that valuesof d resulting from the above model should not be taken too literally, since ωnc

is sensitive to the shape of the particles, the material in which the crystallites areembedded, and the method of deposition. Also the choice to describe the phononwavepackets by a Gaussian with envelope exp[−2r2/L2] is rather arbitrary.

Sometimes the values of the parameters that follow from the fit of Ia(ω) +Inc(ω) to the Raman spectrum of a film are used to determine the volume fractionsof the a-Si:H and nanocrystalline phases in the film [37]. However, also in layerswhere X-ray diffraction and transmission electron microscopy experiments indi-

72 Chapter 3

cated that the material consisted of nc-Si particles without surrounding a-Si:H, abroad contribution to the Raman spectrum, centered at 480 cm−1, was observed[89, 90]. This component has been assigned to a surface-like mode at the grainboundaries in the nc-Si films. Of course, such observations make the method ofdetermining the a-Si:H and nc-Si volume fractions from the Raman spectra ques-tionable. On the other hand, one might wonder if the broad Raman componentaround 480 cm−1 in a-Si:H has the same origin, and is caused by surface-likemodes at internal surfaces in the a-Si:H tissue [77]. This corroborates our proposalthat a-Si:H consists of nanocrystalline building blocks.

Another example of an effect not included in the simplified model discussedabove, is the inavoidable lattice expansion of nanocrystals smaller than ∼ 10 nm.This phenomenon is related to an increased contribution of the surface energy ofsmall particles, and has been examined by means of X-ray diffraction [77]. Iqbal etal. observed that the expansion increases to a maximum of ∼ 1 % for a crystallitesize of ∼ 3 nm. Of course, the increased atomic distances result in lower vibra-tional frequencies (i.e. a shift of ωnc). Further, the distribution of atomic distancesbecomes broader, which contributes to the broadening of the Raman peak.

3.4.2 Frequency-dependent lifetime of TO phonons

The time-resolved anti-Stokes Raman experiments of Fig. 3.10 demonstrate, that ina-nc-Si:H, the decay rate of the TO phonons is frequency dependent. The phononlifetimes increase with decreasing frequency, from < 10 ns at 515 cm−1 to ∼ 50 nsat 495 cm−1. Below, a possible explanation for the frequency dependence of theTO lifetimes is discussed.

Size dependent lifetimes of TO phonon in nc-Si From the room-temperatureStokes Raman spectrum of a-nc-Si:H, it became clear that the spectrum cannot befully described by the sum of an a-Si:H component and a contribution from onemonodisperse collection of nc-Si particles as given by Eq. 3.3. To improve theagreement with the measurement, additional contributions to the calculated curveare needed at frequencies between 480 and 515 cm−1. The presence of nc-Siparticles with a diameter smaller than the average d as considered in the simplifiedmodel, would explain such additional contributions, since smaller particles giverise to TO peaks at lower frequencies [77].

By the same token, the difference in Stokes and anti-Stokes TO peak fre-quencies that showed up in Fig. 3.9 may also be related to the polydispersity ofthe nc-Si crystallites in the a-nc-Si:H film. One can argue that during one laserpulse, the larger particles (d 4.5 nm) that correspond to the larger Raman shifts


(ωnc 515 cm−1), become less populated with TO phonons than small nanocrys-tals (∼ 1.5 nm) that give the smaller TO Raman shifts (ωnc ∼ 495 cm−1). In thatcase, only the small crystallites produce the non-equilibrium anti-Stokes TO in-tensity, whereas all particles contribute to the spontaneous Stokes signals. Thedifference in non-equilibrium occupation of TO modes for particles of differentsizes can in turn be explained as the result of a size-dependent decay rate of theTO phonons. In this picture namely, the results presented in Fig. 3.10 show thatphonons in the smaller particles have a long lifetime (τ ∼ 50 ns), while phononsresiding in larger crystallites decay much faster. That would suggest that the anhar-monic decay of TO vibrations is suppressed in the smaller particles. We proposethat the slowing down of the anharmonic decay is related to phonon confinementeffects, that become important as soon as the characteristic length scale, say, thewavelength of THz phonons, approaches the size of the nanocrystals. In this con-nection, we note that in the temperature region that corresponds to the frequencyof THz phonons, one has reported an excess of the phonon specific heat of leadand palladium nanoparticles of similar sizes as ours, compared to that of the bulk-crystalline counterparts [91, 92]. This behavior has been attributed to phonon con-finement effects [93].

An effect which is not taken into account by the model used to describe the StokesRaman spectrum of a-nc-Si:H, is the discreteness of the spectrum that exists inan isolated nanocrystallite [94]. Directly connected with that is the existence ofa minimum frequency, ωmin, for the acoustic phonons in an isolated nanocrystal.Inherent to the existence of an ωmin is the occurrence of a gap between the groundstate and the first vibrational level. To put it simply, not all phonons that may existin bulk crystals fit into a small particle. The isolated particle acts as a phononcavity. Consequently, physical properties that are determined by the vibrationaldensity of states, like the phonon specific heat and the anharmonic decay, candeviate significantly from those of bulk crystals. Depending on the actual spectrumof eigenstates in a specific energy range, the physical properties can be suppressedor enhanced compared to the bulk crystalline case. For example the specific heatis expected to fall exponentially below a temperature that correspond to the energyof the gap [92].

In the following, we explore if the concept of a discrete vibrational spectrummay explain the observed size-dependent anharmonic lifetimes of the TO modesin nc-Si. For this purpose, we need to know the eigenfrequency spectrum of thenc-Si crystallites for frequencies between ∼200 cm−1 and ∼300 cm−1, the rangeof frequencies that is expected to be relevant for the anharmonic break up of the∼ 500-cm−1 TO phonons. The vibrational eigenfrequencies of a homogeneous

74 Chapter 3

spherical body were calculated by Lamb in 1882 [95]. More recently, Tamura etal. further elaborated Lamb’s approach, and investigated the influence of embed-ding, clamping, and deformation of the sphere on the shape of the spectrum [94].Effects of that turned out to be mainly expressed in relatively small shifts of theeigenfrequencies of the free, perfectly spherical body. We note that both Lamb andTamura described the behavior of continuous elastic media, taking into account thematerials anisotropy.

To obtain a measure of the vibrational spectrum of nc-Si for the relevant fre-quencies, we follow a procedure based on the work of Lamb, that has been outlinedfor other nanocrystalline materials by various authors [93, 96, 97]. To make theproblem tractable, the nanocrystals are in this approach represented by homoge-neous spheres, that are considered as isotropic continuous elastic bodies. Then, theeigenfrequencies of a particle are determined by the solutions of the wave equa-tion (∇2 + q2)u = 0, with the stress-free (∇ ·u = 0) or, alternatively, displacementfree (u = 0) boundary conditions on the surface of the particle, for the free andclamped sphere, respectively. The eigenvectors qlm can be found and are given byqlm = alm/(d/2), with d the particle diameter. For a free sphere, alm is the m-thzero of the derivative of the l-th spherical Bessel function of the first kind. In thecase of a clamped surface, alm represents the m-th zero of the l-th Bessel functionitself. The degeneracy of qlm is (2l + 1). For the eigenfrequencies ωlm we simplytake ωlm = cqlm, where c is the average sound velocity, calculated from the trans-verse and longitudinal sound velocities (for Si, c ≈ 6.3 ·103 ms−1). The resultingvibrational spectra are presented in Fig. 3.14 for two particle diameters.

As expected, these graphs show that smaller particles have a smaller numberof eigenfrequencies in the relevant frequency range, and a higher lowest eigenfre-quency ωmin than larger particles. Of course, the curves do not correspond to theactual eigenfrequencies of the Si nanocrystallites in our a-nc-Si:H sample. Prob-ably, most particles are neither completely free, nor fully clamped. Further, it isunlikely that the crystallites are perfect spheres. Finally, one could argue aboutthe validity of the elastic continuum approximation, especially for particles withdiameters only several times the atomic distance of Si. Taking into account suchdetails, however, would mainly influence the exact frequencies of the eigenmodes.Therefore, we believe that the calculated spectra give a rough, but faithful mea-sure for the dependence of ωmin and the spacing between the modes, on the size ofthe Si particles. In fact, in Raman experiments on nucleated glasses a remarkablygood agreement of the calculated and measured size dependence of ωmin has beenobserved for crystallites with diameters between 10 and 40 nm [98].

We expect that the discreteness of the vibrational spectra of nanostructuredmaterials has effects on the phonon dynamics. If the modes that are required for


0

10

20

30

40 a) free 4.5 nm

mod

e de

gene

racy

0 100 200 3000

10

20

30

40 b) free 2.5 nm

wavenumber (cm-1)

c) clamped 4.5 nm

100 200 300

d) clamped 2.5 nm

Figure 3.14 Eigenfrequencies of free [a) and b)] and clamped [c) and d)] crystalline Sispheres, with a diameter of 4.5 nm [a) and c)] and 2.5 nm [b) and d)].

the regular anharmonic break-up of a phonon into two lower frequency modes arenot available, the bulk-crystalline decay channels are suppressed. Then, other, andin bulk crystals less probable transitions govern the decay, which decreases thedecay rate. The phonon experiments on porous corundum indeed showed that themodes just above ωmin have extremely long lifetimes [79, 80]. However, the TOmodes examined in our experiments have a frequency that is almost an order ofmagnitude higher than ωmin in a 4.5-nm particle, which makes the problem morecomplicated than in the case of the above mentioned measurements. To estimatethe suppression of the anharmonic decay of the TO modes, we determine the size-dependent probability that the relevant acoustic modes are available for the decay.

As a first approximation, we assume that the TO modes break up into twolower frequency modes. The TO modes have a size-dependent minimum wavevec-tor qmin, equal to the minimum wavevector of the acoustic modes derived above,qmin = q11 = a11/(d/2). In the case of a free isolated sphere, qmin ∼ 1.84/(d/2). Forexample, if d = 4.5 nm, qmin ∼ 0.14(π/a0). From the phonon dispersion relationin c-Si (Fig. 3.13) we infer that the TO modes break up into two LA modes withfrequencies between and 200 and 300 cm−1, determined by conservation of energyand momentum. In the following, the above calculated eigenfrequency spectra for

76 Chapter 3

0.01 0.1 1

0.01

0.1

1c)

linewidth (cm-1)

P

10-2

10-1

100

101

a)

nor

mal

ized

den

sity

of s

tate

s

200 250 300

10-2

10-1

100

101

b)

wavenumber (cm-1)

Figure 3.15 Normalized vibrational density of states for d = 4.5 nm (a) and d = 1.5 nm(b). The Lorentzians have a width of 0.1 cm−1. The straight lines indicate a normalizedvibrational density of states of 1/30. Figure c) gives an estimate of the chance P that fora pair of randomly chosen frequencies between 200 and 300 cm −1 the density of statesis 1/30 in order to contribute to the decay of a TO mode within 10 ns. This chance isplotted as a function of the width of the Lorentzians, for d = 4.5 nm (), d = 2.5 nm (),and d = 1.5 nm (O).

free spheres of diameter d are taken as a measure of the vibrational density ofstates in nc-Si particles of a given typical size. As the vibrational excitations havea finite lifetime, the resonances are expected to have a Lorentzian line shape witha finite width. For simplicity, the same width is assigned to all Lorentzians. Then,the quantity we wish to compute is the density of states of acoustic phonons perunit of volume, relative to that in bulk crystalline Si. The reason is that the rate ofanharmonic break up scales with the density of final states, which in our case of TOdecay is determined by the density of acoustic modes between 200 and 300 cm−1.It turns out that the average vibrational density of states per unit of volume in thisfrequency range is independent of the particle diameter, and is therefore chosenas the ‘crystalline value’, and set to unity. In this way, we composed the normal-ized spectra that are shown in Fig. 3.15.a and 3.15.b, for 4.5- and 1.5-nm particles,respectively, and in both cases a line width of 0.1 cm−1. It is clear from these spec-tra that, although at some frequencies the density of states in the small particles is


much higher than in the larger crystallites, it is much lower for the major part of thefrequencies between 200 and 300 cm−1. This implies that accidentally the anhar-monic break up of the TO modes may be speeded up compared to the crystallinecase, if both acoustic modes involved in the decay correspond to resonances of thecrystallite. However, because of the overall lower vibrational density of states, weexpect the majority of nanoparticles to exhibit a lower decay rate of the TO modesas compared to bulk c-Si, of course, to a degree depending on the natural width ofthe modes and the size of the crystallites. Below we make an attempt to quantifythis statement.

First, we recall that the time-resolution of our experimental setup is 10 ns.Hence, the shortest phonon lifetime that can be measured is about one thousandtimes longer than the 10-ps lifetime of TO modes in c-Si [66]. So knowing that theTO decay rate scales with the square of the density of states of acoustic phonons[99], we will only be able to monitor phonon decay in nanoparticles for which thedensity of states at the frequencies of the accepting acoustic modes is

√103 ≈ 30

times lower than in c-Si. Therefore, we calculate the chance, P, that for a pairof randomly chosen frequencies between 200 and 300 cm−1 the normalized den-sity of states is ≥ 1/30, using spectra like the two presented in Fig. 3.15.a and3.15.b. This calculation is performed as a function of the width of the Lorentzians.Fig. 3.15.c shows the results for particle sizes of 1.5, 2.5, and 4.5 nm. Now, torelate the TO lifetime with the crystallite size, we identify P with the probabilitythat a TO phonon has decayed after 10 ns (103× 10 ps). According to Fig. 3.15.c,for a line width of 0.1 cm−1, P is equal to 0.7, 0.34 and 0.14 for crystallites witha diameter of 4.5 nm, 2.5 nm, and 1.5 nm, respectively. For monoexponentiallydecaying TO populations, those figures would correspond to lifetimes of 10 ns,24 ns and 74 ns. For comparison, from the Raman measurements, lifetimes wereobtained that increased from < 10 ns to ∼ 50 ns, for TO frequencies decreasingfrom 515 to 495 cm−1 (corresponding to d ≈ 4.5 to 1.5 nm). Hence, the estimatespresented here suggest that the frequency dependence of the TO lifetimes can beexplained as resulting from a size-dependent suppression of the anharmonic TOdecay. We emphasize however, that this is only true if the modes to which theTO phonons decay are sufficiently narrow in frequency (∼ 0.1 cm−1). Such a linewidth corresponds to a 50-ps lifetime, suggesting that the modes oscillate ∼ 400times without being affected by their surroundings. In this connection we notethat the particles are close together, because of the high volume fraction of crys-tallites in our a-nc-Si:H samples, the particles necessarily are close together. Asa result, also the a-Si:H material in between the nanocrystallites is structured ona nanometer length scale, which again implies that not all vibrational modes aresupported. Then, a mode of a specific frequency may get ‘trapped’ inside a crys-

78 Chapter 3

tallite, if the bordering amorphous tissue and the nearest neighbor crystallites havea sufficiently low vibrational density of states in the relevant range of frequencies.In that case, 50-ps or longer lifetimes may not be unreasonable.

We conclude that the observed frequency dependence of the TO lifetimes canbe explained by taking into account the discreteness of the vibrational spectra ofthe nanocrystallites.

Additional remarks The components of the Raman spectra that are assigned tothe nc-Si particles, are in frequency exactly in the region between the TO peak ofc-Si and that of a-Si:H. The measured lifetimes of the modes in this region are alsoin between the values found for TO modes in c-Si and a-Si:H, decreasing from τ <10 ns for the frequencies close to the c-Si resonance, to τ ∼ 50 ns for frequenciesapproaching the TO peak of a-Si. The reason for the long decay times of phononsin a-Si:H without nanocrystallites is still unclear. However, if we associate theobserved longevity with the above explanation for the lifetimes measured in a-nc-Si:H, it seems reasonable to consider a-Si:H as a material consisting of a largenumber of small (∼ 1 nm) isolated regions, in which the positions of the atomsare correlated. Indeed, one finds that a-Si:H exhibits ‘medium’ range order onnanometer length scales [73]. If, in addition, vibrations in neighboring regions aresufficiently decoupled, the vibrational spectrum becomes discrete, resulting in asuppressed decay of the TO modes.

3.4.3 Long-lived TA phonons

From the time-resolved anti-Stokes Raman measurements displayed in Fig. 3.7,we concluded that in a-nc-Si:H, the modes in the spectral region of TA phononsdecay much slower than in a-Si:H.

The observed longevity of the TA modes can be explained using similar argu-ments as presented above for the long-lived TO modes. The explanation for thelongevity of the TO modes is based on the relative mode spacing of the vibrationalspectrum in the frequency range between 200 and 300 cm−1. In the case of thelower frequency TA modes, the anharmonic decay involves acoustic modes in therange between 75 and 100 cm−1. We recall that the mode spacing is directly depen-dent on the particle diameter, qlm = alm/(d/2). Hence, the relative mode spacingin the range of 75-100 cm−1 for 4.5 nm crystallites is near to the mode spacing inthe 200-300 cm−1 region for 1.5-nm particles. Therefore, it can be expected thatthe TA modes in 4.5-nm particles have lifetimes comparable to the lifetimes of theTO phonons in 1.5-nm nanocrystals. Further, in the case of the 1.5-nm particles,the frequency region of the TA modes is just above ωmin. For such small particles,


the intra-particle anharmonic break up may be virtually inhibited. Consideringthe amorphous tissue in between the crystallites, we note once more that also thea-Si:H tissue is structured on length scales of the order of 10 nm, due to the pres-ence of the crystallites. This may in turn suppress the anharmonic break up of TAvibrations in the amorphous material. In contrast, pure a-Si:H is not structured on10-nm length scales [37], consistent with the observed short lifetimes of the TAmodes in that material.

3.4.4 Long-lasting phonon induced luminescence signals

Until now, the discussion mainly concerned the results of the time-resolved Ra-man experiments determined by the dynamics of very high-frequency phonons ina-nc-Si:H. As this point we address the phonon transport measurements performedwith the ruby phonon detector operating at a much lower frequency (29 cm−1). Wefound also in these experiments that the phonon dynamics in a-nc-Si:H is markedlydifferent from that in a-Si:H. The outcome of the transport experiments on a-Si:His consistent with the results published by Scholten et al., who extensively studiedthe scattering of 29-cm−1 phonons in a-Si:H [41]. They showed that in a-Si:H, thetransport of 29-cm−1 phonons is governed by elastic spatial diffusion, with a diffu-sion coefficient of ∼ 1 cm2/s. Encouraged by their analysis, we try to explain theresults of the phonon transport measurements on a-nc-Si:H in terms of an elasticdiffusion model.

In the present experiments, 29-cm−1 phonons were generated in the a-nc-Si:Hfilm by means of optical excitation. Due to the finite penetration depth of thegreen light into the sample, not all phonons were created at the same depth, andconsequently did not have to travel the same distance to the ruby detector. Below,we first solve the diffusion equation for a phonon density n(x,x′, t), created at t=0,at a depth x′ > 0 in an a-nc-Si:H layer. Subsequently, a solution is constructedfor the situation where phonons are generated according to an excitation profilewith a certain depth into the sample. From that, the total flux Φ(t) of phononsinjected into the Al2O3 that are resonant with the ruby detector can be determined.The results for Φ(t) will be compared with the measured temporal evolution of thephonon induced luminescence intensity.

In case the transport of 29-cm−1 phonons through the a-nc-Si:H layer is en-tirely determined by elastic spatial diffusion, and characterized by a single diffu-sion coefficient D, the one-dimensional diffusion equation for n(x,x′, t) applies,that reads

∂n(x,x′, t)∂t

= D∂2n(x,x′, t)

∂x2 . (3.4)

80 Chapter 3

The initial condition is n(x,x′,0) = n0δ(x− x′), with n0 the number of resonantphonons injected per unit area, and δ(x−x′) the Dirac delta function. The positionx = 0 corresponds to the interface between the a-nc-Si:H layer and the surroundingHe gas. For an infinite medium, the solution of Eq. 3.4 is given by

n(x,x′, t) =n0√4πDt

exp

[−(x− x′)2

4Dt

], (3.5)

which corresponds to a phonon flux φ(x,x′, t) = −∂n/∂x of

φ(x,x′, t) =2πn0(x− x′)(4πDt)3/2

exp

[−(x− x′)2

4Dt

]. (3.6)

Of course, the sample is not an infinite medium. Boundary conditions areimposed by the properties of the interfaces between the sample and the substrateat one side, and between the sample and the He gas at the other side. The Al2O3

substrate is assumed to act as an ideal sink for the 29-cm−1 phonons, while the Hesurface is taken 100% reflective. Because of the first condition, the phonon densityat the interface with the substrate (x = L) should always be zero: n(L,x′, t) = 0.According to the second condition, the net flux of 29-cm−1 phonons through theHe surface has to vanish at all times. The problem is solved using the method ofimages [41], as is illustrated in Fig. 3.16. To fulfill both boundary conditions, theinitial distribution n(x,x′,0) has to be symmetric around x = 0, and antisymmetricaround x = L. This is accomplished by introducing a positive delta peak n0δ(x+x′)at x = −x′, and two negative peaks at x = 2L− x′ and x = 2L + x′. To preservesymmetry around x = 0 again two delta peaks have to be added at x = −2L + x′

and x = −2L− x′, destroying again the antisymmetry around x = L. To satisfy thetwo conditions, the procedure has to be repeated to infinity, leading to an infinitenumber of image phonon distributions (see Fig. 3.16). The resulting phonon fluxΦ(x′, t) at the interface with the Al2O3 substrate is equal to

Φ(x′, t) =∞

∑i=0

(−1)i 4πn0

(4πDt)3/2

×

[(2i+ 1)L− x′] exp

[−((2i+ 1)L− x′)2

4Dt

]

+ [(2i+ 1)L+ x′] exp

[−((2i+ 1)L+ x′)2

4Dt

].

(3.7)

If the phonons are created according to an exponentially decaying excitation pro-file, with a penetration depth 1/α into the sample, the total phonon flux Φ1(t) for


etc.etc.

x'x

n(x,x',0)

8L-8L

6L-6L

-4L 4L

-2L 2L

Figure 3.16 Initial distribution n(x,x ′,0) of the density of phonons that are resonantwith the ruby detector, as constructed with the method of images. The distribution ismade symmetric around x = 0 and antisymmetric around x = L, to account for the 100 %reflectivity of the interface between the He gas and the sample (x = 0) and the completeabsorption of the phonons by the substrate (x = L).

configuration (1) at x = L is described by

Φ1(t) ∝Z L

0dx′e−αx′Φ(x′, t). (3.8)

To calculate the phonon flux Φ2(t) for configuration (2), exp[−αx′] in Eq. 3.8 isreplaced by exp[−α(L− x′)]. When comparing Φ1,2(t) with the R2/R1 obtainedfrom the measurements, the 10-ns resolution of the ruby phonon detector has tobe taken into account. This is done by convoluting Φ1,2(t) with a Gaussian of thecorresponding width. Then, Φ1,2 can be fitted to the data, substituting L = 0.4 µm,and using 1/α and D as fitting parameters.

By fitting Φ1(t) to the data acquired in configuration (1), we obtained for thevalues of the parameters: 1/α = 150 ± 20 nm, and D = (6± 0.5) · 10−3 cm2/s. Apenetration depth of 150 nm seems quite reasonable for 514-nm photons traversinga-nc-Si:H. Note however, that the value for D is more than two orders of magnitudesmaller than that determined for 29-cm−1 phonons in a-Si:H by Scholten et al.[41]. As can be seen from Fig. 3.17.a, Φ1(t) describes the data quite well forthese values of 1/α and D. However, Fig. 3.17.b shows that Φ2(t) by no means

82 Chapter 3

a)

-20 0 20 40 60 80 1000

0

b)

R2

/ R1

(arb

.uni

ts)

time (ns)

Figure 3.17 a) Comparison of the calculated Φ1(t) (solid line) with the normalized mea-sured 29-cm−1 phonon flux in a-nc-Si:H in configuration (1). The fitting parameters 1/αand D have values of 150 nm and 6 · 10−3 cm2/s, respectively. b) Comparison of Φ2(t)with the normalized phonon flux measured in configuration (2). For α and D the same val-ues have been substituted that followed from the fit of Φ1(t) to the data for configuration(1). This Φ2(t) clearly does not describe the measurements. In both graphs, results of themeasurements are presented for excitation powers of 3.6 mW () and 0.9 mW (•).

fits the experimental traces for configuration (2), for the same values of 1/α and D.The observed temporal profile is significantly slower than predicted by the abovemodel. In addition, our model with a constant D can of course never describe theexcitation power dependence of the traces recorded for this configuration.

Scholten et al. also observed power dependences of the phonon signals in opti-cally excited a-Si:H. They demonstrated, that incorporation of inelastic scatteringof 29-cm−1 phonons off higher frequency phonons made the diffusion model toaccount for the results of the experiments. The absorption of 29-cm−1 phonons bythe higher frequency modes decelerates the rise of the R2(t) signals. This effectnaturally becomes stronger for higher excitation powers, because of the generationof more high-frequency phonons. Such effects were indeed observed by Scholtenet al. Their decay of the R2(t) intensity, however, was shown to be independentof excitation power, and characterized by a 45-ns decay time. Scholten et al. re-lated this tail to the generation of 29-cm−1 phonons by the anharmonic break up


of the long-lived higher frequency phonons. In summary, in their interpretationthe temporal shape of R2(t) in strongly excited a-Si:H is mainly determined by thedynamics of the higher frequency phonons, and not so much by the diffusion ofthe 29-cm−1 phonons themselves.

We note that the results of our experiments on a-nc-Si:H are quite differ-ent from those reported by Scholten et al. First of all, the diffusion of 29-cm−1

phonons in a-nc-Si:H appears to be much slower than in a-Si:H. Secondly, the in-tegrated R2(t)/R1(t) intensity scales linearly with excitation power in a-nc-Si:H,for both experimental configurations. This makes an explanation of the resultsin terms of inelastic scattering of the 29-cm−1 phonons quite unlikely. As to thephonon generation, we observed that in a-nc-Si:H, in contrast to a-Si:H, the decayof the R2(t) signals in configuration (2) is clearly dependent on excitation power.

Considering the low diffusion coefficient of 29-cm−1 phonons in a-nc-Si:H,we emphasize that the material is structured on length scales comparable to thewavelength of the phonons under investigation. This can lead to strong elasticscattering, resulting in slow diffusion [79, 80]. Regions may even exist that cannotsupport the 29-cm−1 phonons at all, further obstructing the propagation of suchvibrations. The diffusion coefficient is related to the transport mean free path ,and the energy velocity ve by D = ve/3 [100]. From this relation we estimatethat ve ∼ 4 · 102 m/s, assuming that roughly equals the length scale of struc-tural variation in our sample, which is ∼ 5 nm. Note that the energy velocity isabout an order of magnitude smaller than the average velocity of sound vs in botha-Si:H and c-Si. In general, ve/vs ≤ 1, because collisions take time, and close toresonances this ratio may even become exceptionally small. For scattering of lightin TiO2 powder, for example a ratio of ve/c as small as 0.16 has been observed[101], where c is the speed of light.

The mechanism of temporarily storing the vibrational energy in a resonance iscaptured by this energy velocity ve [101]. From the ratio ve/vs we can estimate theaverage storage time, or dwell time, τd, of the 29-cm−1 phonons. The average timeτs between two scattering events is determined by the sum of the time requiredto traverse the distance between two scatterers and the dwell time, τs ≡ /ve =/vs + τd. Using the previous estimates for , ve, and vs, we find that τs ∼ 10 ps.This is remarkably close to the 50-ps natural lifetime we inferred for the Lambmodes in Sec. 3.4.2, suggesting that the energy may be stored in the form of Lambmodes.

The very fact that the integrated signal intensity increases linearly with exci-tation power, indeed suggests that elastic scattering processes govern the transportof 29-cm−1 phonons in a-nc-Si:H, and that no phonons get lost. The slowing downof the phonon signal for increasing excitation powers in configuration (2) could be

84 Chapter 3

due to an enhancement of the elastic scattering efficiency in the optically excitedregion of the sample. Such phenomena are reminiscent of the phonon bottleneckeffect extensively studied for 29-cm−1 phonons in ruby [84, 102]. In that system,the phonon diffusion becomes slower, the more Cr3+ ions are excited. Multiple ab-sorption and re-emission by the 2A(2E) ↔ E(2E) resonance of the Cr3+ producesa random walk of the 29-cm−1 phonons and consequently an excitation power de-pendent diffusion coefficient. Also in these experiments, R2(t)/R1(t) increaseslinearly with P, indicative of elastic scattering.

As already mentioned in the description of the experimental setup, the bottle-neck effect is absent in the ruby detector itself, at least for the excitation powers weused. However, phenomena similar to the phonon bottleneck effect in ruby mayoccur in the a-nc-Si:H, if suitable resonances are present with an energy differenceof 29 cm−1. Such resonances may be provided by the anharmonic coupling be-tween the Lamb modes of a nanoparticle. The energy of a 29 cm−1 phonon canbe absorbed, and thus stored, in a nanoparticle if its vibrational spectrum containssets of levels with an energy difference of 29 cm−1. If the lower level of such aset is occupied, resonant absorption and subsequently re-emission of a 29 cm−1

phonon can again take place. In the period in between these two events, the energyis trapped inside the nanoparticle, and propagation is inhibited. This process ismore pronounced for higher excitation powers, when more vibrational levels areoccupied. This might explain the observed power dependence of the R2(t)/R1(t)for configuration (2). In configuration (1), this effect is less important, since thesignals are dominated by phonons that travel through lower excited regions.

We conclude that also the transport of 29-cm−1 phonons in a-nc-Si:H isstrongly influenced by phonon resonances in our nanostructured material. To drawany firm conclusions about the processes responsible for the low, power-dependentdiffusion coefficient, more experiments are required. For example, measurementsas a function of the thickness of the a-nc-Si:H layer, or for different phonon fre-quencies may provide valuable information.

3.5 Conclusions

In this chapter, we report on the results of pulsed Raman and phonon-inducedluminescence experiments on a mixed amorphous-nanocrystalline Si system(a-nc-Si:H). The techniques were used to study the dynamics of terahertz phonons,and compare it with the behavior of comparable phonons in a-Si:H. The main goalof the experiments was to investigate the influence of the presence of nanometerscale structures on the phonon dynamics.

From the Raman experiments, we find that the decay time of phonons in the


spectral region of the TO vibrations in the crystallites is frequency dependent.The lifetimes are observed to increase with decreasing frequency, from < 10 ns at515 cm−1 to ∼ 30 ns at 505 cm−1. We present a model to explain the frequencydependent TO lifetimes as resulting from a size-dependent suppression of the an-harmonic decay inside the nanoparticles: the smaller the particles are, the slowerthe anharmonic decay of the TO modes. A crucial ingredient of our model is thatthe acoustic resonances (Lamb modes) of the nanoparticles have a narrow naturalline width (∼ 0.1 cm−1). We go even further and connect this typical behavior innanocrystalline silicon with the long TO lifetimes measured in a-Si:H, and sug-gest that one may consider a-Si:H as a material containing a large amount of small(∼ 1 nm) decoupled regions, in which the decay of the TO phonons can be severelysuppressed.

Another conclusion that follows from the pulsed Raman measurements is, thatin a-nc-Si:H, the modes in the spectral region of the TA phonons decay muchslower than in a-Si:H. The same model that is used to explain the longevity ofthe TO phonons also accounts for the obtained TA lifetimes. Compared to thelength scale of the structures that are of importance for the dynamics of the TOphonons, structures of larger dimensions appear to influence the dynamics of theTA vibrations.

Also the outcome of the phonon transport measurements illustrates the effectof nanometer scale structure on the behavior of THz phonons. The results are de-scribed in terms of elastic spatial diffusion, characterized by a diffusion coefficientD∼ (6±0.5) ·10−3 cm2/s, a value that is more than two orders of magnitude lowerthan for 29-cm−1 phonons in a-Si:H. The slow diffusion is ascribed to the strongresonant scattering of phonons inside a material that is structured on length scalesequal to the wavelength of the phonons.

The diffusion of 29-cm−1 phonons is observed to become even slower (‘bottle-necked’) when more vibrations are excited. This behavior is attributed to the pres-ence of additional vibrational resonances with an energy difference of 29 cm−1,that are capable of storing the phonon energy, thereby temporarily inhibiting prop-agation.

CHAPTER 4

Infrared Free Electron Laserexperiments on SiH vibrations in a-Si:H

We present results of the first vibrational photon-echo, transient-grating, and tem-perature dependent transient-transmission experiments on a-Si:H. Using thesetechniques, and the infrared light of a free electron laser, the vibrational popula-tion relaxation and phase relaxation of the SiH stretching mode were investigatedfor temperatures between 10 and 300 K. In this chapter, the relaxation path anddephasing mechanism will be discussed. As will be shown, both reflect the amor-phicity of the system. Careful analysis of the data indicates that the vibrationalenergy relaxes directly into SiH bending modes and Si phonons, with a distri-bution of rates determined by the amorphous surrounding. Conversely, the puredephasing appears to be single exponential, and can be modeled by dephasing viatwo-phonon interactions.

4.1 Introduction

Although experiments on non-equilibrium phonons in crystals have proven to givea wealth of information on elementary phonon scattering processes [23], the num-ber of dynamical experiments investigating high-frequency (ν 1 THz) vibrationsin non-crystalline solids is quite limited. In Chapter 2, pulsed Raman spectroscopyhas been employed to examine high-frequency phonon-like modes in various typesof hydrogenated amorphous Si. Our results are in agreement with earlier resultsof Scholten et al. [25], and confirm that in a-Si and a-Si:H, high-frequency vi-brations have a lifetime several orders of magnitude longer than in crystals. How-ever, this interpretation contradicts the outcome of recent computer simulations byFabian et al. [18, 27], yielding picosecond lifetimes of phonon-like modes in a-Si,i.e. of the same order of magnitude or even shorter than the lifetimes found forc-Si. A difficulty when comparing results of the Raman measurements with thoseof the simulations is that in the simulations decay rates are calculated for single-

87

88 Chapter 4

frequency vibrational state populations, whereas in the experiments vibrations areexcited over the full spectrum. Another complication is caused by the fact thatelectronic processes are ignored in the calculations. In the experiments however,charge carriers are essential to generate the vibrations. Indeed, it is known thatelectronic excitations in amorphous hosts may have very long lifetimes [42], sothat it cannot be excluded that slow electronic processes play a role in the mea-sured vibrational signals. In fact, in Chapter 2 these type of effects were examinedin detail. It was concluded that the relatively fast part of the Raman signals cannotbe explained other than originating from vibrational dynamics. On the other hand,it was demonstrated that slow electronic processes indeed influence the Ramansignals.

Time-resolved infrared experiments provide a way to examine the propertiesof non-equilibrium vibrations in glassy materials without the above-mentioned am-biguities. With infrared radiation, namely, vibrations can be excited resonantly bydirect conversion of infrared photons into vibrational quanta, without excitation ofcharge carriers. In spite of this advantage, the number of reported infrared exper-iments on the dynamics of vibrations in disordered solids is quite limited. A fewinfrared transient-bleaching studies have been published, investigating the behav-ior of molecular vibrations in an inorganic amorphous host [103, 104, 105]. In ad-dition, Tokmakoff et al. have presented results of infrared vibrational photon-echomeasurements on the CO stretching mode of W(CO)6 in various organic glasses[106, 107]. Yet, the conclusions drawn from these studies mainly relate to the dy-namics of molecular vibrations, and not to the vibrational modes of the amorphoushost itself.

In this chapter, we present results of the first infrared vibrational photon-echoand transient-grating experiments on an inorganic glass. Using these techniquesand the infrared light pulses of the Free Electron Laser FELIX [108] (Rijnhuizen,The Netherlands), we investigated the dynamics of SiH stretching vibrations ina-Si:H. By combining the echo and grating measurements we were able to ex-amine both the population relaxation and pure dephasing of an excited stretchingpopulation. The main goal of the experiments was to elucidate the dynamics of‘phonon-like’ vibrations in amorphous systems, using the local SiH vibrations asa probe for the properties of the surrounding a-Si phonon bath. Apart from this,knowledge of the dynamical properties of the SiH bonds is of great interest fromtechnological point of view. The dynamical behavior and breaking of SiH bondsattracts a lot of attention nowadays, as it has been demonstrated that the light in-duced degradation of a-Si:H, which is one of the most tedious problems in amor-phous silicon technology, is accompanied by a change in the number of SiH bonds[45, 109].

Infrared Free Electron Laser experiments ... 89

4.2 Experimental procedures

4.2.1 Samples

For the experiments described in this chapter, two types of device-quality hy-drogenated amorphous silicon films were used, similar to the films described inChapter 2. With plasma-enhanced chemical vapor deposition (PE), a 1.0-µm thicka-Si:H layer was grown on a crystalline silicon substrate held at 320C [37, 46].This layer contained about 11 at.% hydrogen. A second a-Si:H film, with a thick-ness of 1.5 µm, was grown by hot-wire assisted chemical vapor deposition (HW),also on a c-Si substrate, held at a temperature of 450 C [37, 47]. The hydrogenconcentration of this layer amounted to 8 at.%. The c-Si substrates were polishedon both sides to reduce stray light. Crystalline Si is transparent for infrared radia-tion of the wavelengths used in the experiments. Most measurements presented inthis chapter were performed on PE a-Si:H, and only part of them (when mentioned)on HW a-Si:H.

Due to the presence of SiHx bonds, the infrared absorption spectra of thea-Si:H films exhibit several absorption lines (see Fig. 4.1). For example, the ab-sorption around 630 cm−1 corresponds to the rocking modes of SiH, SiH2 andSiH3, and is generally used to determine the hydrogen concentration of a-Si:Hfilms [110]. Since the rocking modes supported by the monohydride and differentmultihydrides have the same resonance frequencies, it is impossible to distinguishthe different contributions of different SiHx bonds to the total absorption. In case ofthe stretching mode, however, the resonance supported by the monohydride bonds(at 2000 cm−1) can be separated from the contributions of multihydrides and hy-drogen present at the surface of voids, that cause an absorption line centered at2090 cm−1. In materials science, advantage is taken from this fact to express thequality of the material in terms of the ratio between the absorption at 2090 cm−1

and the sum of the absorptions at 2090 cm−1 and 2000 cm−1. High-quality a-Si:H,namely, only contains isolated SiH bonds and consequently shows no infrared ab-sorption at 2090 cm−1. According to this criterion, the HW samples are of higherquality than the layers deposited with the PE method (see Fig. 4.1).

In most of the experiments described below, the infrared light of FELIX wastuned in resonance with the SiH stretching mode. The advantage of studying thedynamics of the SiH stretching vibrations is, that modes of only one type are ex-cited and monitored, which facilitates the interpretation of the results.

In the cases where comparison was possible, no remarkable differences be-tween the dynamics of vibrations in HW a-Si:H and PE a-Si:H were observed.

90 Chapter 4

1900 2000 2100 22000

200

400

600b)

abso

rptio

n co

effic

ient

(cm

-1)

wavenumber (cm-1)

0

300

600

900

1200a)

1000 2000

1000 2000

Figure 4.1 Infrared absorption of a) PE-grown a-Si:H and b) HW-deposited a-Si:H. Theinsets show the raw transmission spectra as measured with a Digilab FTS-40 Fourier trans-form infrared spectrometer (Bio-Rad) equipped with a liquid-nitrogen cooled HgCdTe de-tector. The plotted absorption coefficients are determined from the transmission spectraafter background subtraction. The total absorption around 2000 cm −1 can be fitted withthe sum of two Gaussians, one centered at 2000 cm−1, corresponding to the stretchingmode of SiH, and a second with a center frequency of 2090 cm −1, corresponding to thecontributions of multihydrides and hydrogen bound to the surface of voids in the material.

4.2.2 Pump-probe methods

The free-electron laser FELIX provides a tunable pulsed infrared light beam, withan average power of ∼ 5 mW. The beam consists of Fourier-limited micropulseswith a minimum duration of 0.5 ps and a repetition rate of 25 MHz. The mi-cropulses form a train (macropulse) with a duration of 2.5 µs that is repeated at arate of 5 or 10 Hz. When the wavelength of FELIX is tuned in resonance with theSiH stretching mode, about 12 % of the light is absorbed in our sample and directlyconverted into a monochromatic vibrational population. Pump-probe methods canbe used to study the dynamics of these vibrations. In the experiments presented be-low, the population relaxation was monitored with transient-transmission and moreaccurately with transient-grating techniques, and phase relaxation was examinedby means of photon-echo measurements.


Figure 4.2 Schematic representation of the transient-transmission setup. BS = beamsplitter, M = mirror, PM = gold-plated parabolic mirror, S = sample, D = detector.

As the spectral width of the FELIX pulses (≤ 15 cm−1) is much narrowerthan the width of the SiH stretching resonance in a-Si:H (FWHM ∼ 100 cm−1),the measurements could be performed as a function of the excitation wavelengthwithin the absorption line. All experiments were done with the sample mountedin a cryostat, to allow for temperature-dependent measurements in a range of 10to 300 K. A liquid-nitrogen cooled HgCdTe diode was used for detection, read outby a boxcar that integrated the electronic signals over one macropulse (or, in caseof the transient-transmission measurements, over 10 macropulses).

Transient-transmission

Fig. 4.2 gives a schematic representation of the transient-transmission setup. Forthe transient-transmission measurements, the FEL beam is split into one pump andtwo probe beams. Pulses of the probe have a 100-times lower energy than those ofthe pump, and can be given a variable delay. The strong pump excites a fractionof the SiH bonds into the first vibrational level (ν = 1). Due to anharmonicity, the(ν = 2)-resonance is shifted from the ground-state resonance , which leads to areduced absorption of the sample at the pumping wavelength [103]. By measuringthe transmitted intensity of the first probe as a function of the delay between pump

92 Chapter 4

and probe, the decay of the population excited with the pump is studied, and thecharacteristic population decay time, T1, is obtained. Pulses of the second probeare arriving at a fixed time (20 ns) after the pulses of the first. The 20-ns delayis assumed to be long enough for all vibrations to decay, so that the pulses of thesecond probe can be used for reference measurements, to correct for backgroundsignals. Special precautions are taken to compensate for drift of the electronics.

Transient-grating

An alternative way to examine T1 is provided by the so-called transient-gratingtechnique (see Fig. 4.3). Here, the FEL beam is split into two pump beams andone probe. The probe pulses have an energy of about 35% of that of the pumpingpulses. By focusing the two synchronous pump beams onto one spot in the sample,a spatial vibrational excitation grating is formed. The delayed probe beam is Braggdiffracted off this grating into the direction determined by kd = k1 ± (k2 − k3),where k1, k2, k3 and kd are the wavevectors of the probe, first pump, second pump,and diffracted signal, respectively. By measuring the diffracted signal as a functionof the delay between the pump-pair and the probe, the population decay can bemonitored, and T1 extracted. The signal intensity namely, is proportional to thesquare of the electric field of the probe beam that is diffracted off the vibrationalpopulation grating, and consequently decays with T1/2. The advantage of thisway of measuring T1 is that the diffracted signal can be detected background-free,whereas the transient-bleaching method requires the accurate detection of smallchanges (< 3%) of the probe intensity. Still, the transient-bleaching measurementsare necessary to confirm that the grating signals indeed are caused by a vibrationalpopulation grating, and not by spatial transport or thermal effects.

Vibrational photon-echo

In addition to examining the decay of a vibrational population (energy relaxation),also information on the dephasing (phase relaxation) is of interest. The vibrationalphoton-echo experiment allows us to study the dephasing of the SiH stretchingmodes and explore the effects of the surrounding temperature bath.

After coherent excitation of a vibrational population, the macroscopic coher-ence is lost at a rate 1/(2T1)+ 1/T′

2 + 1/T∗2. We now briefly discuss the different

loss terms. Firstly, when the population decays, the coherence disappears at arate 1/(2T1). Secondly, interactions between the oscillators and the surroundingtemperature bath result in phase relaxation, at a rate 1/T′2. Finally, due to the inho-mogeneous broadening of the resonance, a set of oscillators with slightly differentresonance frequencies are excited. The oscillators run out of phase after the pump


Figure 4.3 Experimental setup for the transient-grating measurements. BS=beam split-ter, M=mirror, L=BaF2 lens, S=sample, D=detector. The lower part of the figure shows thegeometry of the beams at the sample. The dark-grey, grey and white arrows represent thepump-pair, the probe beam, and the diffracted probe beams, respectively. For the photon-echo experiments, a similar setup with a single pump was used. For measurements withexcitation wavelengths longer than 11 µm, the BaF2 lenses were replaced by gold-platedparabolic mirrors.

pulse has vanished, but keep a phase relation with the exciting field. This so-called ‘inhomogeneous-broadening dephasing’ process, or free-induction decay,is characterized by T∗

2. As noted earlier, the spectral width of the resonance understudy is much broader than that of the laser line. As a result, T∗2 is determined bythe spectral width of the laser pulses, ∆ν and equals in our case (∆ν ∼ 15 cm−1),

94 Chapter 4

T∗2 = 1/(π∆ν) ∼ 0.7 ps.

Of all contributions mentioned, only the inhomogeneous-broadening dephas-ing is reversible, and actually forms, together with the saturability of the resonance,the basis for the echo experiment. After excitation with a first pulse, followed bysome time t of free dephasing, a second pulse interferes with the polarization leftin the sample, giving rise to a time-reversed dephasing process, in case of a sat-urable transition. This rephasing process produces an echo at time 2t. As the otherdephasing contributions (1/(2T1) and 1/T′

2) are irreversible, the intensity of theecho decreases with increasing time t. So by measuring Sp(t) vs. t, the irreversibledephasing processes are studied. For exponential dephasing, Sp(t) evolves in timeaccording to S0 · e−4t/T2 [111], where

1T2

=1

2T1+

1T′

2. (4.1)

Because of the anharmonicity of the stretching mode, it can be considered asa two-level system [112], and hence the interaction of the stretching populationwith resonant radiation can be described by the Bloch model [113]. In terms of theBloch model, the echo signal at t=0, S0, is related to the intensities of the incidentbeams according to [111, 113]

S0 ∝ [sin θ1 sin2 (θ2/2)]2. (4.2)

Here, the Bloch angle θi is a dimensionless quantity determined both by the in-coming electric field, Ei, of the two pulses (i = 1,2), and the coupling, κ ≡ 2µ/,between the field and the oscillators, where µ is the magnitude of the dipole mo-ment of the oscillators, i.e. θi reads

θ1,2(t) =Z

κE1,2(t ′)dt′ =| I1,2 |1/2, (4.3)

where, I is the (dimensionless) effective laser intensity. We note that θ1 is directlyrelated to the difference in occupation numbers of the excited state (ne) and theground state (ng) by

−cos θ1 = ne −ng. (4.4)

These relations are used in the discussion of the results.Because separate information on T1 is obtainable using the transient-

transmission and transient-grating methods, the echo results in principle allow usto determine the pure dephasing time T′2. Thus, interactions can be examined be-tween the SiH vibrations and their environment, other than those responsible forthe energy relaxation.


In practice, almost the same setup was used for the echo experiments as wasused for the transient-grating measurements. Compared to the scheme depicted inFig. 4.3, the second beam splitter was removed and the delay adjusted, in orderto reverse the sequence of the stronger and weaker pulses arriving at the sample.In this connection we note, that the intensity of the echo is higher if the weakerpulses arrive first (see Eq. 4.2). Further, the detector had to be repositioned, asthe direction into which the echo is emitted is given byke = 2k2 − k1, where ke,k2, and k1 are the wavevectors of the echo, the second beam, and the first beam,respectively [113].

4.3 Experimental results

The graphs depicted in Fig. 4.4 are representative of the signals we obtained withthe transient-transmission, transient-grating, and photon-echo technique (St, Sg,and Sp, respectively). In all cases, FELIX was tuned to the center of the SiHstretching line (2000 cm−1). As an example, results are shown for two temper-atures, 15 and 250 K. We checked that the signals induced originate from theamorphous Si film and not from the substrate, by comparing them with room-temperature transient-transmission measurements on a bare c-Si wafer. As ex-pected, these experiments only yielded an autocorrelation of the FELIX pulse,independent of the excitation wavelength.

It can be seen that the signal-to-noise ratios of the grating and echo mea-surements are higher than those of the transient-transmission data. As mentionedabove, this can be traced back to the background-free signal detection, which isnot possible with the transient-transmission setup. Nevertheless, the transient-transmission experiments are useful, as already noted, since they provide a checkfor the origin of the grating signals. For comparison, Fig. 4.5 shows once moreresults of the transient-transmission and transient-grating experiments at 15 and250 K, where we took the square root of the grating signals. Apart from the coher-ent artifact in the grating measurements at zero delay, the curves measured with thetwo techniques show virtually the same temporal behavior. This directly demon-strates that other mechanisms than population decay, such as heating, which alsomay cause grating signals, are not involved in the measured decay. Therefore, weconclude that the transient-grating technique is appropriate for examining the en-ergy relaxation of a population of SiH stretching vibrations in a-Si:H. It turned outthat the improved signal-to-noise ratio makes accurate T1 experiments feasible asa function of the excitation wavelength within the absorption line, and for differ-ent excitation powers. For the system under study, this is not the case with thetransient-transmission method.

96 Chapter 4

10-3

10-2

10-1

b)

Sg

10-3

10-2

10-1

a)

St

0 50 100 150 200

10-3

10-2

10-1

c)

delay (ps)

Sp

Figure 4.4 a) Transient-transmission, b) transient-grating, and c) photon-echo signals,(St, Sg and Sp, respectively) for T =15 K (•) and T = 250 K (), as a function of thedelay between pump(s) and probe. The signals are averaged over ten scans of the opticaldelay. For the transient-transmission experiments, micropulses of pump and probe havean energy of 3.5 µJ and 0.1 µJ, respectively. The grating measurements are performed withtwo pump beams consisting of 1.0 µJ pulses, and a probe of 0.35 µJ pulses. The echosignals are obtained with a pump energy of 0.3 µJ and probe energy of 1.75 µJ. All beamsare focused to a 150-170-µm-diameter spot in the sample.

A remarkable characteristic of all curves, which is not too obvious from thetransient-transmission results only, is their nonexponential shape. Similar exper-iments on SiH stretching vibrations of an H-terminated crystalline Si surface didnot show any nonexponentiality [114]. To examine if and how the nonexponen-tiality is caused by the influence of the amorphous environment on the vibrational


0 50 100 15010

-2

10-1

100

scal

ed in

tens

ity

time (ps)

Figure 4.5 Transient-grating (lines) and transient-transmission data (symbols) for twotemperatures ( and dotted line: 15 K, • and solid line: 250 K). For comparison, thesquare root of the grating intensities was taken. Apart from the coherent artifact in thegrating signals at zero delay, the curves measured with the two techniques show the sametemporal behavior.

dynamics of the local SiH stretching mode, the measurements were repeated fordifferent temperatures between 10 and 300 K, and as a function of the excitationwavelength.

4.3.1 Temperature dependence

Decay of the vibrational population

To find an explanation for the nonexponential decay, the dependence of the tem-poral shape of the T1 data on temperature is of importance. We discovered that allnormalized transient-grating signals can be merged to one universal curve (inset ofFig. 4.6), just by rescaling the time axis of each curve Sg(t) with the average time,〈Tg 〉 of that curve, defined as

〈Tg 〉 =R

tSg(t)dtRSg(t)dt

. (4.5)

We thus find that the shape of the population decay, and therewith the degree ofnonexponentiality, does not depend on temperature. It also turned out that theexcitation power does not affect the shape of Sg.

98 Chapter 4

10 100

0.008

0.010

0.012

0.014

0.016

0.018

temperature (K)

mea

n ra

te (

TH

z)

0 2 4 6 8

10-3

10-2

10-1

100

gra

ting

inte

nsi

ty

t/<T>

Figure 4.6 Temperature dependence of 1/〈T1 〉. No dependence of 1/〈T1 〉 on the excita-tion power was observed. The inset shows the mean transient-grating curve obtained fromaveraging all scaled transient-grating measurements. The decay is clearly nonexponential.

From the ratios between the average times at different temperatures, the tem-perature dependence of the mean decay rate (1/〈T1 〉 = 2/〈Tg 〉) can be obtained.To get an impression of the absolute values of the population decay times, themean decay time of the 15 K curve was determined: 〈 T1 〉15K =137 ±6 ps. To-gether with the scaling factors, this value provides all information required to plotthe mean decay rate 1/〈T1 〉 vs. temperature. The results of our exercise are shownin Fig. 4.6. As can be seen, the population decay speeds up above temperatures ofsay 80 K. In addition, since all scaled curves have the same shape, we have aver-aged them to a ‘mean transient-grating curve’, as plotted in the inset of Fig. 4.6.This curve will be instrumental to extract information on the pure phase relaxationof the stretching vibrations from the photon-echo measurements.

We checked that the transient-transmission data lead to equivalent results, ex-cept for, of course, the larger errors.

Phase relaxation and pure dephasing of the vibrational population

In Fig. 4.7, results of the echo measurements are plotted in terms of the meandephasing rate, 1/〈 T2 〉, vs. temperature. We take 〈 T2 〉 = 4 · 〈 Tp 〉, with 〈 Tp 〉


10 100

10-2

10-1

temperature (K)

mea

n de

phas

ing

rate

(T

Hz)

Figure 4.7 〈T2 〉 as a function of temperature, as obtained from the photon-echo experi-ments. The measurements were performed on PE-grown a-Si:H (circles) and HW-growna-Si:H (triangles), and for normal and double excitation power (open and filled symbols,respectively).

the average time of each echo curve Sp(t), defined similarly as in Eq. 4.5. Theexperiments were performed both with pump pulses of ∼ 0.3 µJ and probe pulsesof ∼ 1.75 µJ (normal excitation power), and with double excitation power, eachfor the two types of a-Si:H. For both samples, 〈T2 〉 is shorter for higher excitationpower. Unfortunately, absolute values of the dephasing rates measured in the twomaterials cannot be directly compared, as the experiments were done in differentruns, with inavoidable differences of the intensity of the FEL beam and of the thefocusing of the beams at the sample.

A more fundamental parameter than the total dephasing time T2 is the puredephasing time T′

2, which is generally independent of the population decay, exceptfor the specific case when interactions between the individual excitations deter-mine the dephasing. The effect of such interactions will be discussed in Sec. 4.4.4,where we conclude, that they do not play a role in our experiments.

For exponentially decaying T1 and T2 measurements, Eq. 4.1 suffices to deter-mine the pure dephasing time T′

2. In our case, however, a complication arises sincethe population decay of SiH stretching vibrations in a-Si:H is nonexponential (see

100 Chapter 4

Fig. 4.6). Although average times like 〈 T1 〉 and 〈 T2 〉 describe the results, theyobviously cannot be simply inserted into Eq. 4.1 to find T′2. In the following wepresent a route to extract the desired information on the pure dephasing. For thenonexponential decays we write

Sg(t) =Z

g1(τ1)e−2t/τ1dτ1 and

Sp(t) =Z

g2(τ2)e−4t/τ2dτ2.(4.6)

Here gi(τi) represents the distribution of ‘microscopic’ τi’s leading to 〈 Ti 〉, withi = 1,2 and t is time. We assume that g2(τ2) can be separated into two contributionscorresponding to τ1’s and τ′2’s, in such a way that g2(τ2)dτ2= g1(τ1)dτ1g(τ′2)dτ2.Then, if 1/τ2 is replaced by 1/2τ1 + 1/τ′2, by analogy with Eq. 4.1 for the ex-ponential case, we obtain Sp(t) ∝

R Rg1(τ1)g(τ′2)e

−2t/τ1 e−4t/τ′2dτ1dτ′2. From thisrelation, it is immediately clear that dividing Sp(t) by Sg(t) leads to an expressionthat only contains the desired pure dephasing contributions

Sp(t)/Sg(t) ∝Z

g(τ′2)e−4t/τ′2dτ′2. (4.7)

To improve the quality of the Sp(t)/Sg(t) data, Sp was divided by the properlyrescaled ‘mean transient curve’ of Fig. 4.6, instead of the raw Sg traces, to takefull advantage of the higher signal-to-noise ratio of the ‘mean transient-gratingmeasurement’. For this, an S′g was derived for each temperature by multiplyingthe time-axis of the ‘mean-transient-grating curve’ with the 1/〈 Tg 〉 obtained forthat temperature.

The inset of Fig. 4.8 shows the result of dividing Sp by S′g for T=100 K.

In contrast to all traces presented in Fig. 4.4, this curve appears to be single-exponential, and according to Eq. 4.7 decays with T′2/4. The same is true for allother Sp(t)/S′

g(t) obtained for temperatures of 30 K and higher. This fact justifiesthe assumption made in deriving Eq. 4.7, and, more importantly, makes it feasibleto extract T′

2 as a function of temperature (see Fig. 4.8). As was already clear fromFig. 4.7, values of T′

2 appear to depend on the excitation power for temperaturesbelow ∼ 150 K, as opposed to the behavior observed for 〈T1 〉. Up to 30 K, resultsof Sp(t)/S′

g(t) are unfortunately noisy and partly nonexponential. These T′2 dataare just plotted for completeness, but further ignored.

As we did not perform transient-grating experiments on HW-grown a-Si:H,T′

2 was only determined for the PE sample. Interesting to note in this context,however, is that room-temperature transient-transmission experiments revealed nodifferences between the two samples. Together with the results of Fig. 4.7, thissuggests that the dynamics of SiH stretching vibrations is roughly the same in thetwo types of a-Si:H-material.


10 100

10-3

10-2

10-1

2

5

temperature (K)

pure

dep

hasi

ng r

ate

(TH

z)

0 60

∆t

Figure 4.8 Dependence of 1/T ′2 on temperature. Values of 1/T ′

2 were determined fromthe single-exponential curves obtained by dividing the photon-echo by the grating signals.In the inset Sp/S′

g vs. the delay ∆t for 100 K is shown as an example, yielding T ′2 =

230± 20 ps. Results are presented for Sp(t) measured with normal power () and doubleexcitation power (•). For temperatures higher than 30 K, error bars are of the order of thesymbol size.

4.3.2 Wavelength dependence

Because of the finite spectral width of the FELIX pulses, the infrared pumpingpulse excites a packet of oscillators with a range (∼ 15 cm−1) of resonance fre-quencies, within the inhomogeneously broadened stretching line (100 cm−1). Toassess if the selected range of resonance frequencies is directly connected withthe distribution of decay times that contribute to 〈 T1 〉, the transient-gratingand photon-echo experiments were repeated for different excitation wavelengthswithin the inhomogeneous SiH stretching absorption line. Results of those mea-surements for a temperature between 10 and 15 K are collected in Fig. 4.9.

Both 〈 T1 〉 and 〈 T2 〉 appear to vary by no more than a factor of two for ex-citation energies between 1900 and 2100 cm−1. The mean dephasing rate is evenrather constant within the SiH stretching absorption, but starts to deviate for ex-citation around the resonance at 2090 cm−1. The same behavior is observed atT = 160 K and room-temperature, where the 〈T2 〉’s obtained are shorter. Room-

102 Chapter 4

0.005

0.010

0.015a)

mea

n ra

te (

TH

z)

1900 2000 21000.0

0.5

1.0

b)

nor

m. i

nten

sity

wavenumber(cm-1)

Figure 4.9 a) Mean decay 〈 T1 〉 and dephasing rates 〈 T2 〉 (open triangles and filledcircles respectively) vs. excitation wavelength, obtained from transient-grating experi-ments at 15 K and photon-echo measurements at 10 K. b) Normalized intensities of thelow-temperature transient-grating (open triangles) and photon-echo (filled circles) mea-surements as a function of the excitation wavelength. The line represents the square of thenormalized absorption spectrum shown in Fig. 4.1.a.

temperature transient-transmission measurements, though somewhat faster, exhibita similar wavelength dependence of 〈T1 〉 as the low-temperature grating experi-ments.

Fig. 4.9.b serves to show that the normalized signal intensity tracks the squareof the absorption profile measured at room temperature. We note that at these pulsepowers, the amplitudes of as well the transient-grating as the photon-echo signalswere observed to depend quadratically on the power of the FEL beam. For excita-tion energies at the higher wing of the stretching resonance, deviations are presentbetween the squared absorption profile and the intensities of the time-resolvedmeasurements, because of a decreased performance and lower output power of thefree electron laser.

All results presented until now are of experiments carried out with thewavelength of FELIX tuned in resonance with the absorption line centered at


0 10 20 30

0.0

0.1

0.2

0.3

0.4

0.5

delay (ps)

echo

inte

nsity

-10 0 10 20 30

10-3

10-2

10-1

Figure 4.10 Intensity of the photon-echo signal as a function of the delay between pulsesof the first and second FEL beam, for T = 7K and an excitation energy of 645 cm −1.Because of the presence of an H-absorption line at the center of the bending resonance,this signal is more pronounced than that measured at 630 cm−1. The inset displays thesame data on a semi-log plot. The ‘bumps’ in the signal for delays of ∼ 12 ps and ∼ 24 psare caused by reflections from the substrate.

2000 cm−1. Fig. 4.10 shows the outcome of a photon-echo measurement exploringthe SiH resonance centered at 630-cm−1 (see the insets of Fig. 4.1). The exper-iment was performed at 7 K. It can be clearly seen that this signal decays muchfaster than the low-temperature echo result depicted in Fig. 4.4. Indeed, for the630-cm−1 line, we find 〈T2 〉 ∼ 10 ps, whereas 〈T2 〉 = 140 ± 5 ps for excitation at2000 cm−1. At 630 cm−1, no dependence of 〈T2 〉 on temperature was observed,possibly because the echo decays faster than can be resolved with the experimen-tal setup. In a room-temperature reference measurement on a c-Si substrate it waschecked that the echos indeed originate from the a-Si:H layer. The ‘bumps’ in thesignal for delays of ∼ 12 ps and ∼ 24 ps are spurious reflections from the substrate.

104 Chapter 4

4.4 Discussion

4.4.1 Comparison with other experiments

As noted earlier, few time-resolved infrared experiments have been performedon the dynamics of vibrations in glassy or amorphous systems. Transient-transmission measurements have been reported by Heilweil et al. [103], Happek etal. [104] and Xu et al. [105], while Zimdars et al. and Tokmakoff et al. presentedresults of infrared vibrational photon-echo experiments [106, 107].

Heilweil et al. examined the vibrational lifetime of OH modes of hydroxylgroups embedded in amorphous fused silica over a wide temperature range. Forthe mode they studied (centered at 3660 cm−1) exponentially decaying transient-transmission signals were obtained. The results were characterized by a T1 thatdecreased from 109 to 15 ps when the temperature was increased from 100 to1450 K. Further, the temperature dependence of T1 was interpreted in terms ofa nonradiative relaxation theory, similar to the analysis we will present below toexplain the temperature dependence of 〈T1 〉. The authors concluded that the OHmode decays into four Si-OH vibrations, stretching or bending, with energies be-tween 800 and 1060 cm−1. The uniexponential behavior in fused silica is probablyrelated to the fact that all modes involved in the population decay of the OH modesin SiO2 are local, molecular vibrations. Conversely, the nonexponentiality we ob-serve is caused by the characteristics of the ‘phonon-like’ modes and the interplayof the local SiH vibrations with those Si phonons, as will be discussed in moredetail in the following sections. We finally note in connection to the silica exper-iments, that the 109-ps lifetime at 100 K is very close to the 〈 T1 〉 of 112 ps weobtained for T = 120 K at the center of the SiH stretching line.

Also in the experiments of Happek et al. (again on a different mode in adifferent glass), the 100-ps lifetime comes back. Happek et al. investigatedthe frequency dependence of the relaxation properties of SH stretching vibra-tions in hydrogen-doped As2S3 glass. The SH stretching absorption is centeredat 2485 cm−1 and has a FWHM of 100 cm−1. Relaxation rates were measured at asample temperature of 8 K and in a frequency range between 2425 and 2505 cm−1,yielding a T1 that increased from 59 ps at 2425 cm−1 to 360 ps at 2502 cm−1. Itwas suggested that the frequency dependence of the decay time is caused by a sitedependence of the coupling between the SH stretching vibrations and the vibra-tional modes of the As2S3 glass. The factor of 6 change in decay rate is muchlarger than the difference of less than a factor of 2 in 〈T1 〉’s we present in Fig. 4.9.In contrast to the case of a-Si:H, the IR absorption spectrum of As2S3 does notshow any resonances between the broad band corresponding to the phonons ofthe amorphous host and the SH stretching absorption. Further, the broad phonon


band has a sharp high-frequency cutoff around 360 cm−1. Consequently, at least7 vibrational quanta of the host are required to match the energy of the SH mode.Then, indeed small variations of the coupling between the SH vibrations and thephonons may have large effects.

Xu et al. reported the first transient-transmission experiments on SiHx stretch-ing vibrations (2000 cm−1 and 2090 cm−1) in a-Si:H, taken at room temperature.Our room-temperature measurements reproduce their results. For excitation at2090 cm−1, a little faster population decay was observed than at the center of theSiH stretching line, which is also confirmed by our measurements. Their expla-nation for the nonexponential population decay is sought in fast energy transferbetween the stretching modes. However, our much more complete set of data al-lows us to draw another conclusion, as will be discussed in the next section.

Zimdars et al. reported the first infrared vibrational photon echo experimentsconducted on a glass. They examined both T1 and T2 of the CO stretching modeof W(CO)6 dissolved in 2-methyltetrahydrofuran (2-MTHF). The CO stretchingmode of W(CO)6 is centered around 1960 cm−1 and has a FWHM of ∼ 15 cm−1.The 2-MTHF solvent exhibits a glass-transition at T= 90 K and has a melting pointof 140 K. For temperatures between 16 and 140 K, the intensities of the photonecho and lifetime measurements decayed uniexponentially, giving T1 = 44 ps andT2 = 60 ps at T = 16 K. Above 140 K (where 2-MTHF has turned into a liquid), theechos decayed faster than the 3-ps time resolution of the experimental setup. Atthe same temperature, the population decay became biexponential. Of all resultspresented, only the low-temperature T1 value is somehow comparable with the〈 T1 〉 we obtained. On the other hand, the glassy system studied by Zimdars etal. differs in so many aspects from a-Si:H, that differences between their andour observations seem not surprising. For example, a-Si:H is inorganic and hasa several times higher glass transition temperature, and therefore does not showa phase transition in the temperature range of the present experiments. Further,the SiH bonds are directly connected to, or even a part of the amorphous Si host,which is not the case for the CO bonds. Finally, the 100-cm−1 line width of the SiHstretching mode in a-Si:H is much broader than the 15-cm−1 width of the CO linestudied. This already suggests a larger influence of the amorphous environment onthe SiH modes. The experiments presented in Ref. [106] were repeated for severaldifferent organic glassy solvents [107], revealing distinct vibrational dynamics ineach glass, but always a quadratic temperature dependence of T′2.

Another interesting system to compare with, is that studied by Guyot-Sionnestet al. [114]. They performed infrared lifetime and photon echo experiments on theSiH stretching mode of an H-terminated crystalline Si surface. Other than for theSiH modes in a-Si:H, all signals were found to decay exponentially. At 120 K, a

106 Chapter 4

T1 of 1.4 ns and T2 of 35 ps were obtained. Although the value of T2 is close to the40-ps 〈T2 〉 we found at 120 K, the one-order-of-magnitude difference between T1

and 〈T1 〉 indicates that the pure dephasing time in the two materials has to be quitedifferent in order to get a comparable T2 and 〈 T2 〉. These findings suggest thatboth the population decay and dephasing mechanisms of SiH stretching vibrationsin a crystalline environment are markedly different from those in an amorphoussurrounding.

4.4.2 Mechanisms of energy relaxation

From the temperature dependence of the energy relaxation rate it is possible to de-cide which lower-energy vibrational modes govern the decay of the SiH stretchingvibrations. Theoretical models exist, that describe the relation between the pop-ulation decay rate γ = 1/T1 and the temperature T , in terms of the frequency ωof the initially excited mode and the frequencies ωi of the accepting modes [115],that is

γ(T ) = γ(0)exp(ω/kBT )−1

∏i[exp(ωi/kBT )−1]. (4.8)

The zero-temperature relaxation rate, γ(0), is determined by the density of states ofthe accepting modes and the coupling of the stretching vibrations to those modes.Restrictions on ωi are given by the energy conservation rule∑i ωi = ω. From theStokes Raman spectrum displayed in Fig. 2.1 we know the spectrum of acceptingmodes that are available in the form of a-Si phonons. Further, the SiH bendingmode at 630 cm−1 is, of course, an important potential accepting mode. In fact, itis the only local mode with an energy lower than the SiH stretching vibrations.

For each combination of ωi to which the stretching vibrations may couple,Eq. 4.8 gives a characteristic temperature dependence of γ. In Fig. 4.11 the tem-perature dependences of conceivable energy-conserving decay paths are plotted,together with the 1/〈 T1 〉 values obtained from the experiments. The parameterγ(0) is adjusted to fit the data. In general, the more vibrational quanta needed,the steeper the temperature dependence will be, because of the lower average en-ergy per quantum. It is clear that the decay channel involving three SiH bendingquanta and one TA-like a-Si mode fits the data best. No other combinations werefound that reproduce the measured temperature dependence of 1/〈T1 〉 within theexperimental error. This leads us to the conclusion that in a-Si:H one stretchingvibration decays into three SiH bending vibrations (623 cm−1) and one TA a-Siphonon (ω1 = 133 ± 5 cm−1). The value found for the bending frequency isabout 10 cm−1 lower than the center frequency of the bending resonance show-ing up in Fig. 4.1. In this respect we note that the fit is not very sensitive to the


10 100

10-3

10-2

10-1

WR(T )

a

b)

T' 2

(TH

z)

temperature (K)

10-2

a)

<T

1> (

TH

z)

Figure 4.11 a) Temperature dependence of 1/〈T 1 〉. The lines describe the decay of oneSiH stretching vibration into: 3 bending modes (623 cm−1) and 1 TA a-Si phonon (133 ±5 cm−1) (solid line), 4 TO a-Si modes (500 cm−1) (dotted line), 2 TO and 3 LA modes(333 cm−1) (dashed line), and 3 TO modes + 1 LA (325 cm−1) + 1 TA (175 cm−1) (dash-dotted line), according to Eq. 4.8.b) Dependence of T′

2 on temperature. The solid lines are described by Eq. 4.13, withWR(40 K) = (4.3± 0.3) · 10−2 GHz. For the lower curve (‘normal’ excitation power),a = 1.8± 0.1 GHz. For the upper curve (‘double’ excitation power) a is twice as high.The dashed lines indicate the contributions of WR(T ) and a.

exact energy of the bending modes. To obtain the values for the energies and theerror, Eq. 4.8 was fitted to the data by varying ω1 and taking ω2 = ω3 = ω4 =(2000 −ω1)/3. So the 623-cm−1 frequency is in fact the average frequency ofthe three SiH bending modes excited. We presume that the same SiH bond sup-ports the SiH stretching vibration, and the three bending vibrations into which thestretching energy relaxes. Then, the anharmonicity of the bending mode explainswhy the value obtained is relatively low.

In the preceding paragraph we determined to which modes the SiH stretchingvibrations decay. This identification, however, does not explain the marked non-

108 Chapter 4

exponentiality of the population decay. In the remainder of this subsection, weconsider several scenarios to elucidate the observed nonexponential decays.

Resonant dipole-dipole energy transfer A natural way to explain nonexponen-tial decay is given in terms of energy distribution between the stretching modesby resonant dipole-dipole energy transfer [116]. If an excited SiH bond is locatedclosely enough to a neighbor SiH bond, both spatially and spectrally, energy trans-fer can take place, in this case predominantly via a dipolar interaction. Then, en-ergy is carried over to neighboring resonant dipoles that in general have a differentorientation with respect to the incoming electric field than the originally excitedones. This energy transfer would result in a decrease of both the transient-gratingand the transient-transmission signals. A rough estimate of the rate γij of this trans-fer for uncorrelated individual oscillators can be made using the expression givenby Forster [117] (in C.G.S. units)

γij =(µi

n

)2 (µ j

n

)2 κ2ij

2cR6ij

Zgi(ν)gj(ν)dν. (4.9)

Here, µi, j are the dynamic dipole moments of oscillators i and j, n is the refrac-tive index of the medium surrounding the dipoles, c is the velocity of light in themedium, R the distance between the oscillators, and g(ν) is the normalized lineshape of the vibrational mode (where ν is given in cm−1). The geometrical factorκij is determined by the angle θij between the two dipole vectors, and the angles θi

and θ j between the respective dipole vector and the vector connecting both dipoles:κij = cos θij − 3cos θi cos θ j. For randomly oriented dipoles i and j, κ2

ij is equal to2/3. It is common practice to describe the homogeneous line shape g(ν) by aLorentzian, so that the maximum value of the integral in Eq. 4.9, I, is 1/(π∆hom),with ∆hom the full width at half maximum of the Lorentzian. From the values of1/〈T2 〉 obtained by the echo experiments, ∆hom can be estimated using the relation∆hom = 1/(πT2), which is exact for exponential dephasing. Thus, we find that I ∼3.2 cm at low temperature (where ∆hom ∼ 0.1 cm−1), and I ∼ 0.35 cm (∆hom ∼0.9 cm−1) at room temperature.

Values for the dynamic dipole moment are estimated from the measured ab-sorption coefficient α [113]

α(ω) ∼ 4π2Nωc

( µ

n

)2gα(ω), (4.10)

where N is the total amount of SiH modes per volume fraction (the number ofmodes excited is neglected ), and gα(ω) the normalized inhomogeneous line shape.Here, the frequency ω is expressed in rad s−1. From the absorption measurement


presented in Fig. 4.1, we find that the inhomogeneously broadened absorption linecan be described by a Gaussian of width (FWHM) ∆inhom = 100 cm−1, so thatgα(ωc) = 4.2 ·10−14 rad−1s at the center of the line. Then, by combining Eq. 4.9and 4.10, and substituting N = 5 · 1021 cm−3, α(ωc) 1200 cm−1, n 4, andωc = 3.8 · 1014 rad s−1, we arrive at an energy transfer rate between two SiHmodes at the center of the absorption line of γij 7 · 10−34 · I/R6. To estimateR, we note that of all SiH stretching modes, only a fraction ∆hom/∆inhom has a res-onance frequency ωc. Consequently, R (N ·∆hom/∆inhom)−1/3. In this way, weobtain γij 5 ·104 s−1 at low and γij 7 ·105 s−1 at room temperature. Both ratesare much too low to account for the decay times observed. Besides, in some pre-liminary polarization-dependent transient grating measurements, no dependenceof the temporal shape of the signals on the probe polarization was seen. Moreover,in order to get temperature-independent signal shapes, γij should have the samedependence on temperature as γ(T ), which is quite unlikely. We conclude thatresonant dipole-dipole energy transfer cannot explain our results.

Phonon assisted energy redistribution within the SiH stretching line Accord-ing to Xu et al. [105], the break-up of an SiH stretching vibration into lower energymodes is the second process in a two-step relaxation mechanism. In their model,the break-up is preceded by fast phonon-assisted energy redistribution within theSiH stretching band: the pumping pulse excites SiH modes within a narrow energyrange and due to interactions between stretching vibrations and low-energy phononmodes, energy is redistributed over the complete SiH band. Then, the nonexponen-tial decay in fact is approximated by a bi-exponential decay, where the fast decaytime τ1 corresponds to the mentioned energy redistribution step, and the slow timeτ2 to the breaking-up of one stretching vibration into lower lying accepting modes.They obtained τ1 ∼ 20 ps and τ2 ∼ 100 ps. To compare their results with ours, wefitted the transient-grating data taken at room-temperature to the square of a doubleexponential function (recall that the transient grating signal is proportional to thesquare of the transient population grating), yielding τ1 = 10.6 ps and τ2 =74.5 ps.These numbers nicely coincide with the values found by Xu et al. for a differentsample. Nevertheless, we do not believe that this model can explain our results.Firstly, a bi-exponential curve does not fit the grating data very well (we note thatthe quality of our transient-transmission data does not allow this conclusion). Asecond argument refers to the temperature-independent shape of the grating signal.If indeed the proposed processes would take place in succession, phonons of dif-ferent energies would be involved in the two steps, say 50 cm−1 in the first step and∼ 133 cm−1 in the second. In that case, the ratio between τ1 and τ2 must changewith temperature at some point, resulting in a change in shape of the population

110 Chapter 4

decay signal. A final reason to refute the explanation of Xu et al. is provided bythe wavelength dependence of 〈T1 〉. For phonon-assisted energy redistribution atthe red side of the SiH stretching line, absorption of a phonon must occur. But atlow temperatures, the phonons of the required energy have extremely low thermalpopulation numbers, which should lead to a drastic increase of τ1 (and 〈 T1 〉) ascompared to its room-temperature value. Again, no such effect was observed.

All these arguments provide evidence that within the lifetime of the SiHstretching vibrations, no phonon-assisted energy redistribution over the SiH bandtakes place. In other words, the SiH stretching mode does not form a vibrationalband, and the vibrational energy stays highly localized until the excitation decays.

Transition to a ‘dark state’ Another effect that is consistent with nonexponen-tially decaying T1 signals, originates from a frequency shift of the resonance stud-ied, caused by the decay of the population [118]. One can imagine that the reso-nance frequency of the stretching mode at a particular site changes when a stretch-ing vibration decays into one TA and three bending vibrations, since the SiH bondparticipates in the bending vibrations as well. Then, the resonance frequency re-covers as soon as the bending vibrations have decayed to the ground state. Inthat case, again two time constants show up in the transient-grating measurements.As mentioned already, the obtained signals do not exactly match a biexponentialcurve. Another argument against this mechanism to be operative in our case is thata first, faster decay is followed by a slower step. This would imply that the bend-ing vibrations have a longer lifetime than the stretching modes. Unfortunately, wehave not measured the T1 of the bending mode. Yet, the low-temperature photon-echo experiments on the bending vibration suggest that the dynamics of the bend-ing modes is much faster than that of the stretching vibrations. A third argumentagainst this so-called ‘dark state’ scenario is provided again by the temperatureindependence of the shape of the vibrational decay signals. Because the bendingvibrations decay into a set of bulk a-Si vibrations, the temperature dependence ofthe population decay rate γ(T )bend of that process must be different than the γ(T )that is related to the decay of the stretching modes (see Eq. 4.8). In that case, theshape of the 〈T1 〉 measurements must depend on temperature. The final argumentagainst the ‘dark state’ explanation is based on the similarity of the temporal shapeof the echo and grating signals at low temperatures (see Fig. 4.4). This similar-ity indicates that the dephasing time is governed by the population decay rate atlow T . If the decay proceeds via the dark state, all phase information is lost al-ready after the first step, which would result in single-exponential echo curves withcharacteristic times corresponding to the rate of the first step in the decay process.We conclude from Fig. 4.4 that this concept is at variance with our experimentalresults.


In conclusion, the results of the measurements indicate that the SiH stretchingvibrations decay in one step as described by Eq. 4.8. No redistribution, reorien-tation, or ‘dark-state’ transitions seem to govern the signals. Hence, the observednonexponentiality of our 〈 T1 〉 signals is not explained by the standard conceptsdescribing nonexponential decay, and other mechanisms have to be identified. Forthis reason, we now discuss the effects of the amorphous surroundings on the vi-brational relaxation.

Nonexponentiality caused by the amorphous environment As noted above,we take the view that the stretching vibrations decay in one single step, and thatthe distribution of decay times is related with that step. We suspect that SiHstretching vibrations at different sites have a different lifetime, because of the vari-ation in local environments. As the distribution of decay times is measured to betemperature-independent, we now search for temperature-independent factors thatmay cause a variation of decay times over the ensemble of SiH modes excited.

In this context, an explanation for the nonexponentiality one could think of is,that T1 depends on the oscillator frequency. Because of the finite spectral widthof the FELIX pulses, the infrared pumping pulse excites a collection of oscillatorswith different resonance frequencies, leading to nonexponential decays in casethey do not all have equal lifetimes. In Fig. 4.9, however, we find that 〈T1 〉 variesno more than a factor of two within the SiH stretching line. On the other hand, weknow that when a sum of two exponentials is fitted to the transient-grating signalsmeasured at the center of the line, the ratio of the two decay times appears to beabout 7, i.e. much larger than the overall variation of 〈T1 〉 through the line. Thisexcludes the possibility to explain the nonexponential decay by a direct relationbetween T1 and the oscillator frequency.

The nonexponential decays of 〈T1 〉 may be explained by a spatial variation inthe local density of vibrational states to which the stretching vibrations decay, andthe anharmonic coupling to those states. Both the coupling and vibrational densityof states are independent of temperature, as we wish. Particularly T1 is expectedto be sensitive to variations in the local density of states, due to the condition ofenergy conservation. Because of the absence of such restrictions in the dephasingprocess, the pure dephasing is expected to be much less sensitive to local variationsin the density of states. We recall that the pure dephasing is observed to be single-exponential.

Unfortunately, the results of our experiments do not allow us to separate theeffect of variations in the vibrational density of states or the coupling between themodes. What they do indicate, on the other hand, is that only a remarkably narrowrange of TA frequencies centered around 133 cm−1 contributes to the decay. If

112 Chapter 4

a broad range of TA modes would be involved, namely, the temporal shape ofthe decay would change with temperature, which is not the case. The narrowrange suggests a correlation between the frequencies of the decaying stretchingvibration and of the resulting bending excitations. Since each of the stretching,bending, and TA resonances has an inhomogeneous linewidth of about 100 cm−1,we would expect that a broad spectrum of TA phonons would be excited during thedecay, if the frequencies of the homogeneous stretching and bending lines wouldbe uncorrelated. In contrast, if the frequency of the bending vibrations scales withthe frequency of the stretching mode, a much narrower range of TA frequenciesis required to make up the difference in energy between the stretching vibrationand three bending excitations for each excited stretching mode. In that case, avariation of the local density of 133-cm−1 modes may explain the nonexponentialpopulation decay. The above idea can be verified experimentally by investigatingthe temperature dependence of the population decay as a function of the excitationwavelength within the inhomogeneous stretching absorption line.

4.4.3 Mechanisms of dephasing

In glasses doped with rare-earth ions, one generally observes a quadratic tem-perature dependence for the homogeneous linewidth of optical transitions in thetemperature region of our experiments [119]. Also Tokmakoff et al. [107] re-ported a T2-behavior of the vibrational dephasing of the CO stretching vibrationsin their inorganic glasses, for comparable temperatures. At lower temperatures(below ∼10 K) T1.2−1.5-behavior has been reported many times [119]. In crys-tals, the T2-dependence is characteristic for dephasing due to two-phonon Ramanprocesses [120] and occurs for T > 0.5 TD, TD being the Debye temperature. Ina two-phonon Raman collision, a phonon scatters elastically off an electronic orvibrational excitation. The phonon is absorbed, promoting the excitation into a vir-tual state, and instantaneously emitted again into a random direction. As a result,the macroscopic coherence of the collection of excitations is decreased. The rateof such a process, WR, can be calculated using Fermi’s Golden Rule

WR =2π2

Z Z|Mab|2ρ(ωa)ρ(ωb)δ(ωa −ωb)dωadωb. (4.11)

Here ρ(ωa,b) is the phonon density of states at angular frequency ωa,b of the incom-ing and emitted phonon, respectively, and Mab is the matrix element proportionalto

Mab ∝ (ωaωb)1/2[nωa(nωb + 1)]1/2. (4.12)


The proportionality depends on the type of excitation studied, and the detailed cou-plings involved, which may be frequency dependent [121]. In thermal equilibrium,nωa,b are the Bose-Einstein phonon occupation numbers (Eq. 1.3).

In the case of bulk crystalline materials, one usually takes for ρ(ω) the Debyedensity of states. Then, the processes described by Eq. 4.11 lead to a T2 temper-ature dependence of the dephasing rate for T > 0.5 TD, and a T 7 dependence inthe low temperature limit. Such dependences have been confirmed experimentally[121].

In some situations, however, phonons of a specific energy dominate thedephasing, resulting in thermally activated behavior (also in agreement withEq. 4.11). For instance, the phase relaxation of SiH stretching vibrations of anH-terminated c-Si surface exhibited an exponentially activated temperature depen-dence, that could be traced back to the coupling to a Si surface mode centered at210 cm−1 [114].

To evaluate WR for excitations in a glass, ρ(ω) (and Mab) need to be modifiedcompared to the crystalline case [122]. Different methods to perform such mod-ifications have been proposed, based on phonon [122], two-level-system [119],soft-potential [123] and fracton descriptions [124]. All approaches appear to givethe T α temperature dependence of the dephasing rate observed in glasses.

In the case of a-Si:H, however, ρ(ω) can be determined directly from Ramanmeasurements, according to Eq. 1.2. Thus, ρ(ω) of our PE sample was obtainedfrom the Raman spectrum shown in Fig. 2.1.a. When substituting this ρ(ω) intoEq. 4.11, it turned out that the dephasing data can be fitted well with a curvedescribed by

1T ′

2= a+WR(T ), (4.13)

where any frequency dependence of the proportionality constant in Eq. 4.12 hasbeen neglected. The curve clearly deviates from a T2-temperature dependence(Fig. 4.11). For the experiments performed with ‘normal’ excitation power, weobtain a = 1.8 ± 0.1 GHz and WR(40 K) = (4.3 ± 0.3) ·102 GHz. In the case ofdouble excitation power, a is twice as high, but WR(40 K) remains unaffected.

Apparently, another process contributes to the dephasing, which is temper-ature independent and speeds up when more SiH vibrations are excited. Suchpower-dependent effects have not been reported for the dephasing of optical tran-sitions. Different than in those studies, the excitations we study are vibrational.It could be, that dipolar interactions between the excited SiH modes result in de-phasing, which is an effect that should indeed scale with excitation power. Anotherpossibility is that the excitations studied eventually decay into phonons, that may

114 Chapter 4

induce dephasing. In that case, nω in Eq. 4.11 deviates from the Bose-Einsteinoccupation numbers at the given temperature, either because of heating effects, ordue to the presence of non-equilibrium phonons.

In all of these cases, the amount of energy absorbed by the sample, and theoccupation numbers of the modes excited are of importance. In the remainder ofthis subsection, estimates will be made of the occupation numbers of the relevantvibrational modes, and of the ball park of dephasing rates expected to be reachedwith each of the three proposed mechanisms.

Occupation numbers - some estimates The fraction of SiH stretching modesthat gets excited during the experiments can be estimated both from the intensityof the beams that are incident on and partly absorbed in the sample, and from thepower dependence of the photon-echo signals.

Roughly 10% of the incident ∼ 1 µJ micropulse pump is converted into stretch-ing vibrations. This corresponds to 2.5 · 1012 quanta that are distributed over theSiH modes resonant with the 15 cm−1-wide FEL pulse. Since the Gaussian inho-mogeneous SiH stretching line has a width of 100 cm−1, about 12% of all stretch-ing modes is addressed. Then, the typical number of stretching modes accessiblein the ∼ 2 · 10−8 cm−3 excited volume that has to accommodate the quanta, is∼ 1.2 ·1013. This results in an occupation number of the excited SiH stretching vi-brations of the order of n∼ 0.2. A check on this number is obtained by consideringthe power dependence of the echo signal.

In the way the echo experiments were employed, pulses of the second beamwere 5.7 times stronger than the pulses of the first beam. In that case, Eq. 4.2predicts that the intensity of the echo signal depends on the power of FELIX asplotted in Fig. 4.12. A quadratic dependence on excitation power was observed inall echo experiments. As can be seen in the graph, that may happen in a range ofeffective intensities I1 between 0.6 and 0.85 (0.24π θ1 0.29π). Using Eq. 4.4and assuming ne + ng = 1, we find from these values that ne 0.15-0.2. In thisregion, ne increases only slightly less than linear with excitation power. The occu-pation numbers obtained here are consistent with the numbers estimated directlyfrom the intensity of the FEL beam.

Dipolar interactions between excited SiH stretching vibrations Situations ex-ist [125], where dephasing is induced by resonant dipole-dipole energy transfer. Inthe previous section, the rate of such interactions was already estimated in relationto population relaxation. The obtained numbers were too low (5 · 104 s−1) to ac-count for the 〈T1 〉 measurements. Although the rate of the power dependent con-tribution to the pure dephasing that we have to account for is lower (∼ 1.8 ·106 s−1)


0.1 1

10-3

10-2

10-1

100

I1

effective intensity

echo

sig

nal (

arb.

uni

ts)

Figure 4.12 Intensity of the echo-signals as a function of the power of the FEL beamas described by Eq. 4.2 for a factor 5.7 difference between the intensities of the first andsecond beam. Quadratic and cubic dependencies are shown for comparison (solid anddotted line, respectively).

than the population decay rate, it is still more than an order of magnitude higherthan the dipolar energy transfer rate. Another dipolar dephasing mechanism iscaused by the interactions between excited SiH vibrations, in a way similar to thedephasing induced by the phonon bath. An upper limit of the rate Γ at which thatmay happen can be obtained from Eq. 4.9, when it is taken into account that allexcited modes (in a 15-cm−1-wide energy range) participate. Thus, we arrived ata Γ of the order of 107, again too low to explain the dephasing times plotted inFig. 4.11. We note that this mechanism is in any case not compatible with theexponentiality of the T′

2 curves: if the dephasing results from interactions betweenexcited modes, the rate should decrease in time, as the excited modes decay. Thatis at variance with our observations.

Dephasing caused by a slowly diffusing phonon-bath All energy that isabsorbed by the SiH stretching vibrations will eventually be transformed intophonons. When the phonons have a lifetime that is short compared to the intervalbetween two FELIX micropulses, thermal equilibrium is established during that

116 Chapter 4

interval. The question is if the occupation numbers reached are high enough toaccount for the power-dependent portion of the dephasing or not. Here, the tem-perature reached during the pulse, and the speed at which the energy diffuses outof the excited region play a decisive role.

In the temperature range where the experiments were performed, the heat ca-pacity of a-Si:H is roughly equal to 2 ·10−3 Jkg−1K−4 [126]. So in the absence ofthermal diffusion, after l pulses the temperature Tl in the excited sample volume is

Tl =[

T 4l−1 +

4Q2 ·10−3M

]1/4

. (4.14)

Here, Q is the energy absorbed during one pulse, for normal excitation powerQ ∼ 0.1 µJ, and M is the mass of the excited region, M ∼ 8 ·10−11 kg. The Bose-Einstein occupation numbers associated with Tl can be inserted in Eq. 4.11 to cal-culate the (maximum) dephasing rate reached after l pulses.

From Eq. 4.11 and 4.12 we estimate what temperature Tl would result in aWR(Tl) equal to the observed a. We find, for ‘normal’ excitation power (a =1.8 GHz) Tl ∼ 98 K, and for double excitation power (a = 3.6 GHz) T′

l ∼ 118 K.Eq. 4.14 then tells us that more than 30 pulses would be required to reach thosetemperatures when the cryostat is kept at for example 30 K.

Of course, it is not reasonable to assume that all energy absorbed is convertedinto heat and stays in the volume of the laser focus for more than 1 µs. The resultsof the ruby experiments presented in Chapter 3 show that for example 29 cm−1-vibrations travel through a 0.5 µm-thick a-Si:H film within 5 ns. That means thatdiffusion is able to drain a significant amount of absorbed energy on the time scaleof the experiment, resulting in a lower effective Q in Eq. 4.14. Consequently,an even larger l would be required to explain the 1/T′2 values obtained from theexperiments. Another point is that the measurements were averaged over ∼50micropulses. Then, Eq. 4.14 and 4.11 predict that 1/T′

2 changes over more than anorder of magnitude in the course of one measurement, quite incompatible with theobtained 1/T′

2 curves. We conclude that heating alone cannot explain the observedtemperature dependence of the pure dephasing.

Dephasing due to long-lived high-frequency phonons It is likely that the bend-ing vibrations produced in the decay of the stretching mode split up into two LASi phonons of half the energy. From the anti-Stokes Raman experiments of Chap-ter 2, we concluded that part of the LA phonons in a-Si:H is long-lived, and hasa lifetime τ of ∼ 70 ns. This time is longer than the 40-ns interval between twoFELIX micropulses. Consequently, part of the LA modes excited after one pulse


survives the interval and cause dephasing of the SiH vibrations excited with thenext pulse.

From Eqs. 4.11 and 4.12 we may estimate what occupations nLA of the LAmodes are needed to explain the observed dephasing rates of 1.8 and 3.6 GHz. Ifwe assume that mainly modes between 300 and 325 cm−1 get populated due tothe decay of the SiH bending vibrations, occupation numbers nLA,1 ∼ 0.23 andnLA,2 ∼ 0.4 over this range of modes are required to reach a = 1.8 GHz and a =3.6 GHz, respectively. The amount of energy corresponding to an occupation of0.23 of those LA modes in the excited a-Si:H volume is ∼ 0.18 µJ. An occupationof 0.4 of the same modes corresponds to an energy of ∼ 0.32 µJ. These numbersare remarkably close to the amount of energy absorbed by the sample during onemicropulse: typical 12 % of ∼ 1 µJ in case of ‘normal’ excitation power and twicethis amount for ‘double’ excitation power.

The fact that nLA,2 and nLA,1 differ by about a factor of two agrees with theconclusion made before, that in the region of excitation powers used, occupationnumbers of the stretching mode increase almost linearly with power.

Of course, other modes than those between 300 and 325 cm−1 also get pop-ulated when the SiH stretching and bending modes decay. Thus, the average oc-cupation numbers are somewhat lower than the values just mentioned, but moremodes will contribute to the dephasing, as the total amount of energy involved isthe same.

Although these estimates are very rough, they clearly show that the extra de-phasing of the SiH stretching vibrations observed is consistent with the existenceof long-lived high-frequency phonons in a-Si:H.

4.5 Conclusions and perspectives

In summary, we present results of pulsed infrared experiments on the vibrationalpopulation relaxation and phase relaxation of the SiH stretching mode in a-Si:H.Due to the amorphous environment, SiH vibrations at different sites have a dif-ferent lifetime, resulting in nonexponential population decay. The temperaturedependence of the mean decay rate indicates that each stretching vibration decaysinto three SiH bending vibrations and one TA-like a-Si mode. The temporal shapeof the population decay is temperature independent, implying that phonon-assistedenergy redistribution over the stretching modes, nor reorientation by resonant en-ergy transfer, nor dark-state transitions play a role. Further, the fact that the spec-tral dependence of the population decay is also temperature-independent confirmthat the stretching modes do not form a vibrational band, but instead are highlylocalized in amorphous Si.

118 Chapter 4

An explanation for the nonexponentiality observed could be, that the density ofstates of the accepting modes required for the decay (to fulfill energy conservation)and the anharmonic coupling to those modes varies throughout the amorphousmaterial.

We have extracted exponentially decaying pure dephasing curves from thenonexponential transient-grating and photon-echo measurements. The tempera-ture dependence of T′

2 suggests that two processes contribute to the pure dephas-ing, one of which is power-dependent but temperature-independent. The tempera-ture dependent portion of the dephasing can be described by two-phonon-Ramanprocesses, that are governed by the thermal a-Si phonon bath. We assume thatthe additional, temperature-independent contribution to the dephasing originatesfrom the presence of long-lived (τ > 40 ns) non-equilibrium phonons, generatedduring the decay of the SiH vibrations. We conclude that the dephasing is inducedby interactions between the stretching vibrations and the phonon-like modes ofa-Si:H.

The dephasing rate depends on a local average of the vibrational density ofstates and occupation numbers, that is expected not to vary substantially fromSiH bond to SiH bond. This explains why the observed pure dephasing is single-exponential.

It would be interesting to repeat the experiments on a-Si:D samples, to con-firm our conclusions concerning the vibrational decay mechanism. In addition,experiments have demonstrated that a-Si:D is more stable against light-induceddegradation than a-Si:H [127]. It has been proposed that this results from the dif-ferent vibrational and dynamical properties of the SiH and Si-D bonds. Moleculardynamics simulations support the idea [128], but experimental evidence is stilllacking. The techniques and analysis presented in this chapter may provide a wayto obtain information that is relevant in this context.

Further, in a combination with a synchronized laser in the visible, it shouldbe possible to detect the LA phonon population and check its longevity. In a two-color infrared experiment, the formation of SiH bending vibrations that followsthe decay of the stretching modes could be monitored. We hope that our measure-ments stimulate new theoretical and computational studies on the dynamics of SiHvibrations in an amorphous surrounding.

We finally note that one of the motivations to start the pulsed infrared ex-periments was to study the dynamics of phonon-like modes in amorphous silicondirectly. Unfortunately, it appeared impossible to carry out pump-probe measure-ments on the Si-Si modes directly, because of the too low absorption and satura-bility of the oscillators.

REFERENCES

[1] N.W. Ashcroft and N.D. Mermin, Solid State Physics, (Holt, Rinehart and Winston,Philadelphia, 1976).

[2] J.M. Ziman, Models of disorder, the theoretical physics of homogeneously disor-dered systems, (Cambridge University Press, Cambridge, 1979).

[3] S.R. Elliott, Physics of amorphous materials, (Longman group Limited, London,1983).

[4] A.J. Scholten, Dynamics of high-frequency phonons in amorphous silicon, PhDthesis, Utrecht (1995).

[5] A. Eucken, Ann. Physik 34, 185 (1911).[6] R.C. Zeller and R.O. Pohl, Phys. Rev. B 4, 2029 (1971); R.B. Stephens, Phys. Rev.

B 8, 2896 (1973).[7] P.W. Anderson, B.I. Halperin, and C.M. Varma, Philos. Mag. 25, 1 (1972); W.A.

Phillips, J. Low Temp. Phys. 7, 351 (1972); for a review, see W.A. Phillips, Rep.Prog. Phys. 50, 1657 (1987).

[8] V.G. Karpov, M.I. Klinger, and F.N. Ignat’ev, Zh. Eksp. Teor. Fiz. 84, 760(1983)[Sov. Phys. JETP 57, 439 (1983)]; U. Buchenau, Y.M. Galperin, V.L. Gure-vich, D.A. Parshin, M.A. Ramos, and H.R. Schober, Phys. Rev. B 46, 2798 (1992);for a review see: B. Yu, M. Galperin, V.G. Karpov, and V.I. Kozub, Adv. Phys. 38,669 (1989).

[9] A.F. Ioffe and A.R. Regel, Prog. Semicond. 4, 237 (1960).[10] J.E. Graebner, B. Golding, and L.C. Allen, Phys. Rev. B 34, 5696 (1986); S.

Alexander, C. Laermans, R. Orbach, and H.M. Rosenberg, Phys. Rev. B 28, 4615(1983).

[11] A. Jagannathan, R. Orbach and O. Entin-Wohlman, Phys. Rev. B 39, 13 465 (1989).[12] T. Nakayama, Phys. Rev. Lett. 80, 1244 (1998).[13] H. Bottger and T. Damker, Phys. Rev. B 52, 12 481 (1995).[14] N.F. Mott and E.A Davis, Electronic processes in non-crystalline materials,

(Clarendon Press, Oxford, 1971); P.W. Anderson, Phys. Rev. 109, 1492 (1958).[15] R. Dell’Anna, G. Ruocco, M. Sampoli, and G. Viliani, Phys. Rev. Lett 80, 1236

(1998).[16] P.B. Allen and J.L. Feldman, Phys. Rev. Lett. 62, 645 (1989); P.B. Allen and J.L.

Feldman, Phys. Rev. B 48, 12 581 (1993); J.L. Feldman, M. Kluge, P.B. Allen andF. Wooten, Phys. Rev. B 48 12 589 (1993).

[17] P. Sheng, M. Zhou, and Z.Q. Zhang, Phys. Rev. Lett. 72, 234 (1994); P. Sheng andM. Zhou, Science 253, 539 (1991).

[18] J. Fabian and P.B. Allen, Phys. Rev. Lett. 77, 3839 (1996).[19] V.K. Malinovsky and A.P. Sokolov, Solid State Commun. 57, 757 (1986).[20] U. Buchenau, H.M. Zhou, N. Nucker, K.S. Gilroy, and W.A. Phillips, Phys. Rev.

119

120 References

Lett. 60, 1318 (1988); U. Buchenau, M. Prager, N. Nucker, A.J. Dianoux, N. Ah-mad, and W.A. Phillips, Phys. Rev. B 34, 5665 (1986).

[21] M. Foret, E. Courtens, R. Vacher and J. Suck, Phys. Rev. Lett 77, 3831 (1996); R.Vacher, J. Pelous and E. Courtens, Phys. Rev. B, 56, R481 (1997).

[22] P. Benassi, M. Krisch, C. Mascovecchio, V. Mazzacurati, G. Monaco, G. Ruocco,F. Sette, and R. Verbeni , Phys. Rev. Lett 77, 3835 (1996); C. Mascovecchio, G.Ruocco, F. Sette, P. Benassi, A. Cunsolo, M. Krisch, V. Mazzacurati, A. Mermet,G. Monaco, and R. Verbeni, Phys. Rev. B 55, 8049 (1997).

[23] For a comprehensive review, see M.N. Wybourne and J.K. Wigmore, Rep. Prog.Phys. 51, 923 (1988).

[24] W. Dietsche and H. Kinder, Phys. Rev. Lett. 43, 1413 (1979); J. Mebert andW. Eisenmenger, Z. Phys. B 95, 231 (1994); A.V. Akimov, A.A. Kaplyanskii, J.Kocka, A.S. Moskalenko, and J. Stuchlik, Zh. Eksp. Teor. Fiz. 100, 1340 (1991)[Sov. Phys. JETP 73, 742 (1991)]; T.C. Zhu, H.J. Maris, and J. Tauc, Phys. Rev. B44, 4281 (1991).

[25] A.J. Scholten, A.V. Akimov, and J.I. Dijkhuis, Phys. Rev. B 47, 13 910 (1993); A.J.Scholten, P.A.W.E. Verleg, J.I. Dijkhuis, A.V. Akimov, R.S. Meltzer and R.Orbach,Solid State Phenom. 44-46 289 (1995).

[26] R. Orbach and A. Jagannathan, J. Phys. Chem. 98, 7411 (1994).[27] J.L. Feldman, P.B. Allen, and S.R. Bickham, Phys. Rev. B 59, 3551 (1999); S.R.

Bickham and J.L. Feldman, Phys. Rev. B 57, 12 234 (1998).[28] R. Shuker and R.W. Gammon, Phys. Rev. Lett. 25, 222 (1970).[29] R. Alben, D. Weaire, J.E. Smith Jr., and M.H. Brodsky, Phys. Rev. B 11, 2271

(1975); M.H. Brodsky, M. Cardona, and J.J. Cuomo, Phys. Rev. B 16, 3556 (1977).[30] F. Li and J.S. Lannin, Phys. Rev. B 39, 6220 (1989).[31] N. Maley and J.S. Lannin, Phys. Rev. B 35, 2456 (1987).[32] A.J.M. Berntsen, Structural disorder in pure and hydrogenated amorphous silicon,

PhD thesis, Utrecht (1993).[33] R.C. Weast, M. Astle, and W.H. Beyer (Eds.),Handbook of Chemistry and Physics,

(CRC Press Inc., Boca Raton, 1984), 64-th edition.[34] H.F. Sterling and R.C.G. Swan, Solid State Electron. 8, 653 (1965); R.C. Chittick,

J.H. Alexander, and H.F. Sterling, J. Electrochem. Soc. 116, 77 (1969).[35] W. Spear and P. LeComber, Solid State Commun. 17, 1193 (1975).[36] D.L. Staebler and C.R. Wronski, Appl. Phys. Lett. 31, 292 (1977).[37] R.E.I. Schropp and M. Zeman, Amorphous and Microcrystalline Silicon Solar

Cells: Modeling, Materials and Device Technology (Kluwer Academic Publish-ers, Boston, 1998).

[38] for a review, see R.A. Street, Hydrogenated amorphous silicon (Cambridge Uni-versity Press, 1991).

[39] J. Menendez and M. Cardona, Phys. Rev. B 29, 2051 (1984).[40] A.J. Scholten and J.I. Dijkhuis, Phys. Rev. B 53, 3837 (1996).[41] A.J. Scholten, A.V. Akimov, and J.I. Dijkhuis, Phys. Rev. B 54 12 151 (1996).[42] C. Tsang and R.A. Street, Phys. Rev. B 19, 3027 (1979).[43] R. Biswas, I. Kwon, A.M. Bouchard, C.M. Soukoulis, and G.S. Grest, Phys. Rev.

References 121

B 39, 5101 (1989).[44] I.A. Shkrob and R.A. Crowell, Phys. Rev. B 57, 12 207 (1998), and references

therein.[45] H.M. Branz, Phys. Rev. B 59, 5498 (1999); H.M. Branz, Solid State Commun. 105,

387 (1998).[46] A. Madan, P. Rava, R.E.I. Schropp and B. von Roedern, Appl. Surf. Science 70/71,

716 (1993).[47] R.E.I. Schropp, K.F. Feenstra, E.C. Molenbroek, H. Meiling and J.K. Rath, Philos.

Mag. B 76, 309 (1997).[48] J.E. Smith, M.H. Brodsky, B.L. Crowder and M.I. Nathan, Phys. Rev. Lett. 26, 642

(1971).[49] P.B. Allen, J.L. Feldman, J. Fabian, and F. Wooten, to be published in Phil. Mag.

B.[50] D. Beeman, R.Tsu, and M.F. Thorpe, Phys. Rev. B 32, 874 (1985).[51] A.M. Brockhoff, private communication.[52] K.F. Feenstra, Hot-wire chemical vapour deposition of amorphous silicon and the

application in solar cells, PhD thesis, Utrecht (1998).[53] Z. Vardeny and J. Tauc, Phys. Rev. Lett. 25, 1223 (1981).[54] Those a-Si:H layers were prepared for the experiments of chapter 3, at the A.F

Ioffe Physical-Technical Institute in St-Petersburg (Russia). The samples were alsodeposited by means of PE chemical vapor depositon, but on sapphire substrates.

[55] E.A. Schiff, J. Non-Cryst. Sol. 190, 1 (1995).[56] R. Stachowitz, M. Schubert, and W. Fuhs, J. Non-Cryst. Sol. 227, 190 (1998).[57] Proceedings of the 18-th International Conference on Amorphous and Microcrys-

talline Semiconductors Science and Technology[58] With the setup used for the experiments presented in this chapter, it cannot be ver-

ified that TA phonons have lifetimes much shorter than the pulse-width. However,Scholten et al. [40], determined TA lifetimes of the order of 1 ns.

[59] R.I. Devlen, G.S. Kanner, Z. Vardeny and J. Tauc, Solid State Commun. 78, 665(1991).

[60] P.M. Fauchet, D.Hulin, R. Vanderhaghen, A. Mourchid and W.L. Nighan Jr., J.Non-Cryst. Sol. 141, 76 (1992); A. Mourchid, R. Vanderhaghen, D. Hulin, andP.M. Fauchet, Phys. Rev. B 42, 7667 (1990); P.M. Fauchet, D. Hulin, A. Migus, A.Antonetti, J. Kolodzey, and S. Wagner, Phys. Rev. Lett. 57, 2438 (1986).

[61] A. Esser, K. Seibert, H. Kurz, G.N. Parsons, C. Wang, B.N. Davidson, G. Lucov-sky, and R.J. Nemanich, Phys. Rev. B 41, 2879 (1990).

[62] for a review, see T. Nakayama, K. Yakubo, and R.L. Orbach, Rev. Mod. Phys. 66,382 (1994).

[63] S.R. Elliott, Europhys. Lett. 19, 201 (1992).[64] D.G. Cahill, M. Katiyar, and J.R. Abelson, Phys. Rev. B 50, 6077 (1994).[65] From measurements that yielded the mean free path against elastic scattering of

29-cm−1 phonons in a-Si:H, Scholten et al. extrapolated the same value for themobility edge [40, 41].

[66] S. Tamura, Phys. Rev. B 31, 2574 (1985).

122 References

[67] T. Nakayama and R.L. Orbach, Physica B 263, 261 (1999).[68] A.J. Scholten, A.V. Akimov, P.A.W.E. Verleg, J.I. Dijkhuis, and R.S. Meltzer, J.

Non-Cryst.Sol. 164-166 923 (1993).[69] F. Wooten, K. Winer, and D. Weaire, Phys. Rev. Lett. 54, 1392 (1985).[70] F.H. Stillinger and T.A. Weber, Phys. Rev. B 31, 5262 (1985).[71] J. Fabian, Vibrations in C60 and Amorphous Silicon, Ph.D. dissertation, SUNY

Stony Brook (1997).[72] M.H. Brodsky, R.S. Title, K. Weiser and G.D. Petit, Phys. Rev. B 1, 2632 (1970).[73] M.M.J. Treacy, J.M. Gibson, and P.J. Keblinski, J. Non-Cryst. Sol. 231, 99 (1998);

M.M.J. Treacy, P.M. Voyles, and J.M. Gibson, to be published in J. Non-Cryst.Sol.; J.M. Gibson and M.M.J. Treacy, Phys. Rev. Lett. 78, 1074 (1997).

[74] J.P. Leburton, J. Pasqual, and C. Sotomayor Torres (Eds.), Phonons in Semiconduc-tor Nanostructures, Vol. 236 of Nato Science Series E: Applied Sciences, KluwerAcademic Publishers, Dordrecht (1993).

[75] H. Richter, Z.P. Wang, and L.Ley, Solid State Commun. 39, 625 (1981).[76] I.H. Campbell and P.M. Fauchet, Solid State Commun. 58, 739 (1986).[77] Z. Iqbal and S. Veprek, Journal of Physics C 15, 377 (1993).[78] B. Champagnon, B. Andriasolo, and E. Duval, J.Chem.Phys. 94, 5237 (1991); A.

Tanaka, S. Onari, and T. Arai, Phys.Rev. B 47, 1237 (1993); M. Fujii and Y. Kan-zawa, Phys.Rev.B 54, R8373 (1996).

[79] S.P. Feofilov, A.A. Kaplyanskii, and R.I. Zakharchenya, J. Lumin. 66&67, 349(1996); A.A. Kaplyanskii, S.P. Feofilov, and R.I. Zakharchenya, Opt. Spectrosc.79, 653 (1995).

[80] S.P. Feofilov, A.A. Kaplyanskii, A.B. Kulinkin, and R.I. Zakharchenya, Physica B263-264, 695 (1999).

[81] S.P. Feofilov, A.A. Kaplyanskii, and M.B. Melnikov, Physica B 219-220, 773(1996).

[82] S.A. Basun, P. Deren, S.P. Feofilov, A.A. Kaplyanskii, and W. Strek, in: S. Hun-klinger, W. Ludwig, and G. Weiss (Eds.), Phonons ’89 Vol.2, World Scientific,Singapore, 1424 (1990).

[83] V.G. Golubev, V. Yu, A.V. Medvedev, A.B. Pevtsov and N.A. Feoktistov, Phys.Solid State 39, 1197 (1997).

[84] K.F. Renk and J. Deisenhofer, Phys. Rev. Lett. 26, 764 (1971).[85] M.F.H. Overwijk, P.J. Rump, J.I. Dijkhuis and H.W. de Wijn, Phys. Rev. B 44,

4157 (1991).[86] R.J.G. Goossens, J.I. Dijkhuis, and H.W. de Wijn, Phys. Rev. B 32, 7065 (1985).[87] E. Bustarret, M.A. Hachicha, and M. Brunel, Appl. Phys. Lett. 52, 1675 (1988);

R. Tsu, J. Gonzalez-Hernandez, S.S. Chao, S.C. Lee, and K. Tanaka, Appl. Phys.Lett. 40, 534 (1982).

[88] W. Weber, Phys. Rev. B 15, 4789 (1977).[89] T. Okada, T. Iwaki, K. Yamamoto, H. Kasahara, and K. Abe, Solid State Commun.

49, 809 (1984).[90] S. Veprek, F.-A. Sarott, and Z. Iqbal, Phys. Rev. B 36, 3344 (1987).[91] V. Novotny and P.P.M. Meincke, Phys. Rev. B 8, 4186 (1975); V. Novotny, P.P.M.

References 123

Meincke, and J.H.P. Watson, Phys. Rev. Lett. 28, 901 (1972); G.H. Comsa, D.Heitkamp, and H.S. Rade, Solid State Commun. 24, 547 (1977).

[92] F. Bartolome, J. Bartolome, C. Benelli, A. Caneschi, D. Gatteschi, C. Paulsen,M.G. Pini, A. Rettori, R. Sessoli, and Y. Volokitin, Phys. Rev. Lett. 99, 382 (1996).

[93] H.P. Baltes and E.R. Hilf, Solid State Commun. 12, 369 (1973).[94] A. Tamura, K. Higeta, and T. Ichinokawa, J. Phys. C 15, 4975 (1982).[95] H. Lamb, Proc. Math. Soc. London 13, 189 (1882).[96] R. Lautenschlager, Solid State Commun. 16, 1331 (1975).[97] Y.E. Volokitin, Electrons and phonons in nanocluster materials, PhD thesis, Leiden

(1997).[98] E. Duval, A. Boukenter, and B. Champagnon, Phys. Rev. Lett. 56, 2052 (1986).[99] P.G. Klemens, Phys.Rev. 148, 845 (1966); R. Orbach, Phys. Rev. Lett. 16, 15

(1966).[100] H.P. Schriemer, M.L. Cowan, J.H. Page, P. Sheng, Z. Liu, and D.A. Weitz, Phys.

Rev. Lett. 79, 3166 (1997).[101] M.P. van Albada, B.A. van Tiggelen, A. Lagendijk, and A. Tip, Phys. Rev. Lett. 66,

3132 (1991).[102] J.I. Dijkhuis, A. van der Pol, and H.W. de Wijn, Phys. Rev. Lett. 37, 1554 (1976).[103] E.J. Heilweil, M.P. Cassassa, R.R. Cavanagh and J.C. Stephenson, Chem. Phys.

Lett. 117,185 (1985).[104] U. Happek, J.R. Engholm, A.J. Sievers, J. Lum. 58, 17 (1994); J.R. Engholm, C.W.

Rella, H.A. Schwettman, and U. Happek, Phys. Rev. Lett. 77, 1302 (1996).[105] Z. Xu, P.M. Fauchet, C.W. Rella, H.A. Schwettman, and C.C. Tsai, J. Non-Cryst.

Sol. 198-200, 11 (1996).[106] D. Zimdars, A. Tokmakoff, S. Chen, S.R. Greenfield, M.D. Fayer, T.I. Smith, and

H.A. Schwettman, Phys. Rev. Lett. 70, 2718 (1993)[107] A.Tokmakoff, D. Zimdars, R.S. Urdahl, R.S. Francis, A.S. Kwok, and M.D. Fayer,

J. Phys. Chem. 99, 13310 (1995).[108] for details on FELIX, the Free Electron Laser for Infrared eXperiments, visit the

website http://www.rijnh.nl.[109] Z. Yiping, Z. Dianlin, K. Guanglin, P. Guangqin, and L. Xianbo, Phys. Rev. Lett.

74, 558 (1995).[110] A.A. Langford, M.L. Fleet, B.P. Nelson, W.A. Lanford, and N. Maley, Phys. Rev.

B 45, 13367 (1992); J.D. Ouwens and R.E.I. Schropp, Phys. Rev. B 54, 17 759(1996).

[111] P. Meystre and M. Sargent III, Elements of Quantum Optics (Springer-Verlag,Berlin, 1991); Y.R. Shen, The principles of Nonlinear Optics, (John Wiley & SonsInc., New York, 1984).

[112] The second level of the SiH stretching mode has a center frequency of 3940 cm −1,see Y.J. Chabal and C.K.N. Patel, Phys. Rev. Lett. 53 210 (1984).

[113] L. Allen and J.H. Eberly, Optical Resonance and Two-Level Atoms (John Wiley &Sons Inc., New York, 1975).

[114] P. Guyot-Sionnest, Phys. Rev. Lett. 66, 1489 (1991); P. Guyot-Sionnest, P. Dumas,Y.J. Chabal, and G.S. Higashi, Phys. Rev. Lett. 64, 2156 (1990); P. Dumas, Y.J.

124 References

Chabal, and G.S. Higashi, Phys. Rev. Lett. 65, 1124 (1990).[115] A. Nitzan, S. Mukamel, and J. Jortner, J. Chem. Phys. 60, 3929 (1974); A. Nitzan

and R.J. Silbey, J. Chem. Phys. 60, 4070 (1974).[116] M. Tomita, K. Ohosumi, H. Ikari, Phys. Rev. B 50, 10 369 (1994).[117] M. Morin, P. Jakob, N.J. Levinos, Y.J. Chabal and A.L. Harris, J. Chem. Phys. 96,

6203 (1992); T. Forster, Discuss. Faraday Soc. 27, 7 (1959).[118] A.L Harris, L. Rothberg, L.H. Dubois, N.J. Levinos, and L. Dhar, Phys. Rev. Lett.

64, 2086 (1990); S. Woutersen, U. Emmerichs, and H.J. Bakker, J. Chem. Phys.107, 1483 (1997).

[119] R.M. Macfarlane and R.M. Shelby, J. Lum. 36, 179 (1987).[120] J. Hegarty and W.M. Yen, Phys. Rev. Lett. 43, 1126 (1979).[121] see for example B. Henderson and G.F. Imbush, Optical Spectroscopy of Inorganic

Solids (Clarendon Press, Oxford, 1989), p. 231.[122] D.L. Huber, J. Non-Cryst. Sol. 51, 241 (1982).[123] A.J. Garcıa and J. Fernandez, Phys. Rev. B 56, 579 (1997).[124] G.S. Dixon, R.C. Powell, and X. Gang, Phys. Rev.B 33, 2713 (1986).[125] A. Abragam, The Principles of Nuclear Magnetism (Oxford University Press, Lon-

don, 1961), chapter IV.[126] this number is extrapolated from the values reported by Graebner et al. and Mertig

et al.: J.E. Graebner, B. Golding, L.C. Allen, J.C. Knights, and D.K. Biegelsen,Phys. Rev. B 29, 3744 (1984); M. Mertig, G. Pompe, and E. Hegenbarth, Sol. Stat.Commun. 49, 369 (1984).

[127] S. Sugiyama, J. Yang, and S. Guha, Appl. Phys. Lett. 70, 378 (1997).[128] R. Biswas, Y.-P. Li, and B.C. Pan, Appl. Phys. Lett. 72, 3500 (1998).

DYNAMICA VAN VIBRATIES IN AMORF SILICIUM

achtergrond en overzicht in een andere taal

Vaste stoffen, waaronder silicium, komen voor in vormen die meer of minder geor-dend zijn. Kristallen, b.v. keukenzout en kwarts, bezitten de grootste orde: deatomen waaruit ze zijn opgebouwd vormen een patroon dat zichzelf eindeloos her-haalt. In een amorfe (van het Griekse amorphos = ongevormd) stof, zoals venster-glas, zijn de patronen als het ware ‘mislukt’, zodat de herhaling die in kristallenaanwezig is ontbreekt. Ten gevolge van deze wanordelijkheid zijn de eigenschap-pen van amorfe materialen moeilijker te doorgronden dan die van kristallen. Dieextra moeilijkheid heeft ervoor gezorgd dat het vaste-stofonderzoek tot zo’n 25jaar geleden voornamelijk op kristallen gericht was, en zich nauwelijks met amorfematerialen bezig hield. De meeste theorien die voor vaste stoffen zijn opgesteld be-treffen dan ook kristallen, terwijl inmiddels is gebleken dat ze voor amorfe stoffenvaak onjuist zijn. Amorfe materialen zijn juist daarom interessant voor weten-schappelijk onderzoek.

De toegenomen aandacht voor amorfe stoffen heeft niet alleen te maken metpuur wetenschappelijke interesse. Ten gevolge van technologische ontwikkelin-gen kunnen tegenwoordig veel materialen in zowel kristallijne als amorfe vormworden geproduceerd. Het is gebleken dat de productie van amorfe stoffen vaakveel goedkoper is. Daarnaast laat de amorfe structuur gemakkelijker toe dat on-derdelen in allerlei vormen en maten worden gefabriceerd. Glas zoals we er inhet dagelijks leven aan gewend zijn vormt daarvan een goed voorbeeld. Technolo-gen zijn daarom geınteresseerd om te weten op welke plaatsen kristallijn materiaaldoor een amorfe variant kan worden vervangen. Omdat het daarvoor noodzakelijkis de eigenschappen van amorfe stoffen te begrijpen, heeft het onderzoek een flinkeduw in de rug gekregen.

De wanordelijke structuur van amorfe vaste stoffen maakt het moeilijk funda-mentele vraagstukken op te lossen. Zo is ondanks het intensieve experimentele entheoretische onderzoek van de laatste tientallen jaren nog steeds niet precies be-kend hoe bijvoorbeeld warmte zich door een amorfe stof voortplant. Warmte wordtveroorzaakt door vibraties, en eigenlijk komt het erop neer dat we niet weten hoevibraties zich in zo’n ongeordende omgeving gedragen.

Dit proefschrift beschrijft experimenteel onderzoek naar het gedrag van vi-braties in amorf silicium. Silicium (Si) is een van de meest voorkomende ele-

125

126 achtergrond en overzicht in een andere taal

menten op aarde, en wordt in kristallijne vorm veelvuldig gebruikt in bijvoorbeeldde halfgeleiderindustrie, voor de productie van chips. Mede dankzij die bredetoepassing staat ook amorf Si in de belangstelling. In Utrecht is er een groep on-derzoekers bezig met de ontwikkeling van amorf Si zonnecellen en transistoren.Zij zijn in staat hoge kwaliteit amorf Si lagen te maken. Het lag daarom voor onsvoor de hand om als voorbeeld van een amorfe stof amorf Si te nemen.

Er komt nog heel wat bij kijken om in een amorf Si monster vibraties te kunnenbestuderen. Om het gedrag van de vibraties te onderzoeken is het nodig ze opeen goed gedefinieerd moment te maken, en ze daarna te volgen, liefst totdat zewegsterven. Wat lastige bijkomstigheden daarbij zijn dat er inderdaad een maniermoet zijn om ze op een gedefinieerd moment te kunnen creeren, en dat de trillingenbinnen in het monster zichtbaar moeten worden gemaakt. Verder zijn in iederevaste stof bij kamertemperatuur vanzelf al zoveel vibraties aanwezig (‘thermischevibraties’), dat de trillingen die je extra maakt er niet goed van te onderscheidenzijn. Om dat laatste probleem te verhelpen werden alle metingen uitgevoerd ineen cryostaat: een soort koelkast voor hele lage temperaturen. Daarin werden demonsters met vloeibaar helium afgekoeld tot een temperatuur van -271 C. Bijzulke lage temperaturen blijven er maar weinig ‘thermische vibraties’ over. Hetcreeren en detecteren van de extra vibraties gebeurde met behulp van laserlicht,dat door een raampje in de cryostaat naar binnen werd gestuurd.

In onze eerste experimenten (hoofdstuk 2) maakten we gebruik van groenelaserpulsen. Amorf Si is in staat om groen licht efficient te absorberen. Maardezelfde hoeveelheid energie die wordt geabsorbeerd moet ook weer wordenafgevoerd. In amorf Si gebeurt dit voor een klein deel door het uitzenden vanlicht. Het grootste deel van de geabsorbeerde energie wordt echter omgezet in vi-braties, precies wat we voor ons onderzoek nodig hebben. Niet al het groene lichtwordt door het monster geabsorbeerd. Met het niet-geabsorbeerde deel worden degecreeerde trillingen zichtbaar gemaakt. Als licht namelijk een vibratie tegen komt(‘ermee botst’), kan de energie van die vibratie door het licht worden opgenomen.Daarbij verandert de kleur van het licht een beetje, zodat het licht dat van hetamorf Si monster afkomt een andere kleur heeft dan waarmee we begonnen zijn.Door middel van een kleuropname kunnen de trillingen binnenin het materiaaldan zichtbaar gemaakt worden, een methode die ‘Raman spectroscopie’ genoemdwordt. Door op verschillende momenten na het begin van de eerste laserpuls zo’nopname te maken kan het verloop van de trillingen in de tijd gevolgd worden.

De resultaten van onze metingen laten zien dat niet alle trillingen in amorf Sizich vrij door het materiaal kunnen voortbewegen, wat in kristallen wel het gevalis. De dynamica lijkt bepaald door de aanwezigheid van kleine geordende gebied-jes, waarin trillingen opgesloten kunnen raken. Daar, geısoleerd van hun omge-

achtergrond en overzicht in een andere taal 127

ving, leiden ze een veel langer leven dan vergelijkbare vibraties in een kristal. Dezeinterpretatie hebben we niet alleen uit de zojuist beschreven metingen afgeleid.Om de invloed van kleine geordende gebiedjes op het gedrag van de trillingen tebestuderen hebben we dezelfde experimenten ook aan een ander monster gedaan,dat kleine Si kristalletjes bevatte (hoofdstuk 3). Ook die metingen geven aan dat deaanwezigheid van geordende structuurtjes de uitdoving van de trillingen sterk kanvertragen. De mate van vertraging hangt af van de afmetingen van de geordendestructuurtjes en van de frequentie van de trillingen.

Bij het uitvoeren van experimenteel onderzoek wordt altijd geprobeerd opverschillende manieren informatie uit hetzelfde systeem te halen. Er moet im-mers getest worden of de resultaten niet door de meetmethode worden bepaald.We hebben daarom ook experimenten gedaan waarin de trillingen op een anderemanier gedetecteerd werden (hoofdstuk 3), en waarin zowel de methode van cre-atie als van detectie anders was (hoofdstuk 4).

In de experimenten met aangepaste detectie maakten we gebruik van een spe-ciale detector voor vibraties, de ‘robijndetector’. Deze detector is alleen gevoeligvoor trillingen met een frequentie van ∼ 1012 Hz. De amorf Si lagen (met en zon-der kristalletjes) werden op de detector gegroeid. De metingen werden in dit gevaldwars door het monster heen gedaan: aan een kant werden de vibraties gemaakt,weer met groene laserpulsen, en aan de andere kant werden trillingen die door delaag heen waren gereisd gedetecteerd. In de aanwezigheid van groen licht is derobijndetector in staat om direct wanneer er een vibratie met een frequentie van1012 Hz arriveert een rood licht pulsje uit te zenden. Door bij te houden wanneerdat gebeurt kan de voortplanting (‘propagatie’) van de 1012 Hz trillingen wordenbestudeerd. Uit deze metingen vonden we dat de propagatie van de trillingen doorde amorf Si laag met kristalletjes veel trager is dan door de laag zonder kristal-letjes, wat bevestigt dat de dynamica van vibraties vertraagd kan worden door deaanwezigheid van kleine geordende structuren.

De experimenten die in hoofdstuk 4 worden beschreven vormen letterlijk eenhoofdstuk apart. Behalve de creatie en detectie methodes was ook het type trillin-gen dat bestudeerd werd heel anders. Het amorf Si materiaal zoals gebruikt vooralle in dit proefschrift vermelde experimenten bevat behalve Si atomen ook water-stof (H). Dat waterstof is gedeeltelijk in het amorfe systeem ingebouwd, in de vormvan Si-H bindingen. In dit hoofdstuk wordt speciaal naar de eigentrillingen van diebindingen gekeken, en naar hoe het gedrag van de Si-H vibraties door de trillin-gen en eigenschappen van het omgevende materiaal beınvloed wordt. De vibratieswerden dit keer niet met groen licht gemaakt, maar met infrarode laserpulsen.De energie van het infrarode licht komt precies overeen met de energie van deSi-H vibraties. Het licht kan daardoor direct in Si-H trillingen worden omgezet;

128 achtergrond en overzicht in een andere taal

de vibraties worden als het ware ‘aangeslingerd’. De trillende (‘aangeslagen’) Si-Hbindingen reageren anders op het inkomende licht dan de niet-aangeslagen bindin-gen. Dat is te merken als er nog een infrarode puls achter de eerste aangestuurdwordt. Als de tweede puls echter te laat komt, zijn de aangeslagen trillingen weerverdwenen, en reageert het monster weer precies als van te voren op het infrarodelicht. Door nu de tijd tussen de eerste en tweede puls te varieren, en steeds een me-ting te doen konden we er onder andere achter komen hoe lang het duurt voordatde trillingen uitgedoofd zijn. Zo vonden we dat de Si-H trillingen die wij in amorfSi bestudeerd hebben juist korter leven dan in een kristal. Uiteindelijk worden zeomgezet in vibraties van de amorfe omgeving. Ten gevolge van de wanordelijkestructuur is de omgeving voor verschillende Si-H bindingen anders. De metingengeven aan dat Si-H trillingen op verschillende plaatsen daardoor met verschillendesnelheden uitdoven. Doordat de vibraties die in de amorfe omgeving gevormdworden de Si-H trillingen weer beınvloeden konden we verder zien dat een deelvan dıe vibraties zeer lang blijft bestaan. Dat is geheel in overeenstemming met deresultaten van hoofdstuk 2.

TOT SLOT

Promoveren doe je gelukkig niet alleen. Zonder de hulp van velen was er van dıtboekje weinig terecht gekomen. Bij deze wil ik graag een aantal mensen voor hunbijdrage bedanken.

Op de eerste plaats bedank ik mijn promotor Jaap Dijkhuis. Vooral zijn en-thousiasme en creativiteit heb ik als stimulerend ervaren. Ook het met inten-sieve discussies gepaard gaande ‘opjapen’ van geschreven stukken heb ik zeergewaardeerd. I am grateful to Andrey Akimov, who participated in almost allexperiments described in this thesis. He learned me a lot about interpreting resultsand believing in what you see (magic?).

Grote bewondering heb ik voor de vakkundigheid en vindingrijkheid van FransWollenberg. Tijdens de eerste anderhalf jaar van mijn opleiding tot onderzoekerwerd in samenwerking met Rob Beelen en later ook Paul Jurrius de opstellingin OL 221 tot de grond toe afgebroken en in grotendeels vernieuwde vorm weeropgebouwd. Dankzij deze operatie zijn veel van de problemen waar mijn voor-ganger Andries Scholten mee te kampen had mij bespaard gebleven. De lasershadden er niet altijd zin in. Gelukkig was Cees de Kok immer bereid zijn kritischeblik te laten glijden over laser- en andere optische problemen, en de handen uit demouwen te steken om ze op te lossen. Harold de Wijn ben ik erkentelijk voor hetin bruikleen geven van een Nd:YAG laser, en later nog een dubbel-spectrometer.Zonder de ‘laser van beneden’ hadden we de meeste experimenten beschreven inde hoofdstukken 2 en 3 niet kunnen doen. Drie lasers en de nodige randapparatuurvan Spectra Physics vroegen af en toe om aandacht van Hans Regeer en Paul Graf.Ook hun vakkundige en klantvriendelijke werkwijze mogen wel even genoemdworden. Nico Kuipers, Johan Keijzer en Jan van Eijk bedank ik voor de aanvoervan vele vaten vloeibaar stikstof en helium.

De a-Si:H lagen gebruikt voor de metingen beschreven in de hoofdstukken 2en 4 werden in een depositie systeem van ‘the Utrecht Solar Energy Laboratory’gemaakt door Karine van der Werf. Ik kon bij haar, evenals bij Kees Feenstra,Ruud Schropp, Stefania Acco, Jeike Wallinga, Ernst Ullersma en Anke Brock-hoff, altijd terecht voor vraagjes en vragen over amorf silicium. Ernst en Ankehielpen mij bovendien de infrarood spectrometer en Raman opstelling van AGF tegebruiken. Ook Werner van der Weg bedank ik voor de interesse in het onderzoek.

Ik ben blij dat Gerard van Lingen bereid was de achterkanten van de samplesvoor de FELIX experimenten op het laatste moment nog te polijsten. Ook het

129

130 Tot Slot

door hem gemaakte ‘bootje’ voor de CCD-camera heeft zijn diensten bewezen. Iwould like to acknowledge A.B. Pevtsov, N.A. Feoktistov, and A.A. Kaplyanskiifor providing us with the a-Si:H and a-nc-Si:H samples used for the experimentspresented in chapter 3.

Ik bedank Paul Planken en Lex van der Meer voor hun inzet om met de vrijeelectronen laser FELIX signaal uit een a-Si:H laagje te halen. Op hun aandringenwerd er nog een keer bundeltijd aangevraagd. De daarop volgende vruchtbaresamenwerking met Chris Rella, die in hoog tempo opstellingen ombouwde en weeraan de praat wist te krijgen, heeft geresulteerd in hoofdstuk 4. Ook Guido Knippelswas bereid tot in de kleine uurtjes assistentie te verlenen en heeft, net als Rene vanBuuren, Peter Delmee, Nils Dobbe, Wim Mastop, Harm Pellemans en Theo Ram,bijgedragen aan een prettige en zinvolle tijd bij FELIX.

Van de studenten Otto Muskens, Gert-Jan Smit, Irene Hoeven en Bas Meijsheb ik veel geleerd. Ze hebben allen op hun eigen manier een essentiele bijdrageaan het onderzoek geleverd.

Toen ik aangenomen werd waren Paul Vledder, Paul Verleg en Michiel Petersal aardig op weg als onderzoekers. Daar heb ik dankbaar gebruik van gemaakt.Paul Vledder leerde me omgaan met de femtoseconde opstelling en de Magnexcryostaat, die nog voordat Paul vertrokken was werd overgeplaatst naar OL 221 omvoor het fonon onderzoek te worden ingezet. Paul Verleg wist zich soms crucialedetails uit zijn afstudeertijd bij Andries te herinneren, waardoor de gewenste spec-tra er toch nog kwamen. Paul Verleg en Jeroen Bakker waren fijne kamergenoten,net als Michiel ‘Noise Boys’, en gelukkig in staat tot zinvollere discussies danhun bijnaam doet vermoeden. Petra de Jongh was (tot vanavond) een gezel-lige buurvrouw. Ook onze samenwerking als duovoorzitsters heb ik als plezierigervaren. Verder bedank ik de nog niet genoemde (ex-)gecondenseerde-materie-genoten voor de cooperatieve sfeer.

‘The Amsterdam connection’, met in het bijzonder Mischa Megens en RikKop, bedank ik voor de waardevolle tips.

Mijn ouders en ‘schoon’ouders, bedankt voor de goede zorgen. Frank heeftop vele manieren bijgedragen aan (ook de inhoud van) dit proefschrift. Gelukkigzijn we net zo gek, met ieder een boekje. Laten we nu maar gaan.

Marjolein

CURRICULUM VITAE

De schrijfster van dit proefschrift werd op zaterdag 9 februari 1974 in Leiden terwereld gebracht. In juni 1991 behaalde ze aan het Bernardinuscollege te Heerlenhet Gymnasium β diploma (cum laude). In hetzelfde jaar begon ze in Utrechtaan de studie Natuurkunde. Het afstudeerwerk voor die studie werd uitgevoerdbij de werkgroep Biofysica, waarna in augustus 1995 het doctoraal examen ex-perimentele Natuurkunde werd afgelegd. Van 15 oktober 1995 tot 1 december1999 werkte ze als onderzoeker in opleiding (OIO) in dienst van de stichting voorFundamenteel Onderzoek der Materie (FOM) bij de werkgroep GecondenseerdeMaterie van het Debye Instituut, Universiteit Utrecht. Ze verrichtte daar onder-zoek naar de dynamica van vibraties in wanordelijk silicium, het onderzoek waar-van de belangrijkste resultaten in dit proefschrift worden beschreven. In het kadervan hetzelfde onderzoek bezocht ze de volgende conferenties, waar delen van hetwerk werden gepresenteerd: de ‘Gordon Research Conference on Order and Dis-order in Solids’ (New London, New Hampshire, 1996), de ‘International Confer-ence on Amorphous and Microcrystalline Semiconductors’ (Budapest, Hongarije,1997), de ‘International Conference on Phonon Scattering in Condensed Matter’(Lancaster, Engeland, 1998), de ‘International Workshop on Disordered Systems’(Molveno, Italie, 1999), en de ‘International Conference on Dynamical Processesin Excited States of Solids’ (Humacao, Puerto Rico, 1999). Naast het onder-zoek begeleidde Marjolein tweedejaars studenten bij het hoofdvak natuurkunde,en derde- en vierdejaars studenten bij het uitvoeren van experimenteel onderzoek.Verder was ze onder andere als secretaris en voorzitster van de aio-commissie ac-tief binnen het Debye Instituut.

131

DYNAMICS OF VIBRATIONS IN AMORPHOUS SILICON

Documents

Transcript of DYNAMICS OF VIBRATIONS IN AMORPHOUS SILICON