APPLIED PHYSICS REVIEWS—FOCUSED REVIEW Vibrational … · 2017-12-06 · theory and molecular dynamics ~MD! simulations. This ap-proach is described in the theory section ~Sec

JOURNAL OF APPLIED PHYSICS VOLUME 93, NUMBER 5 1 MARCH 2003

APPLIED PHYSICS REVIEWS—FOCUSED REVIEW

Vibrational lifetimes of hydrogen in siliconG. Lupkea)

Department of Applied Science, The College of William & Mary, Williamsburg, Virginia 23187

N. H. Tolk and L. C. FeldmanDepartment of Physics and Astronomy, Vanderbilt University, Nashville, Tennessce 37235

~Received 2 July 2002; accepted 14 August 2002!

Characterization of defect and impurity reactions, dissociation, and migration in semiconductorsrequires a detailed understanding of the rates and pathways of vibrational energy flow, of the energytransfer channels, and of the coupling mechanisms between local modes and the phonon bath of thehost material. Significant progress in reaching this goal has been accomplished in recent landmarkstudies exploring the excitation and dynamics of vibrational states associated with hydrogen insilicon. The lifetime of the Si–H stretch mode is found to be extremely dependent on the localsolid-state structure, ranging from picoseconds for interstitial-like hydrogen, hundreds ofpicoseconds for hydrogen–vacancy complexes, to several nanoseconds for hydrogen bonded to Sisurfaces—over three orders of magnitude variation. Such large variations in lifetime~transitionprobability! are extraordinarily rare in solid-state science. The level of theoretical investigation intothe vibrational lifetime of the Si–H oscillator is less advanced. This state of affairs is partly becauseof the difficulties in explicitly treating slow relaxation processes in complex systems, and partlybecause, as suggested by experiment, a highly anharmonic coupling mechanism is apparentlyresponsible for the~multiphonon! relaxation process. Even more importantly, because of the highfrequency of the Si–H stretching motion, a quantum mechanical treatment of the Si–H oscillator isrequired. A combination of Bloch–Redfield theory and molecular dynamics simulation seemspromising in describing the relaxation process of the Si–H vibrational modes. It is the aim of thisreview article to present a comprehensive overview of the recent accomplishments, currentunderstandings, and future directions in this emerging field of time-resolved vibrationalspectroscopy of point defects in solids. ©2003 American Institute of Physics.@DOI: 10.1063/1.1517166#

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TABLE OF CONTENTS

I. INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . . . . .2317II. THEORY. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2318

A. Bloch–Redfield theory. . . . . . . . . . . . . . . . . . . . .2319B. Vibrational relaxation and pure dephasing. . . . . 2320C. Multiphonon relaxation. . . . . . . . . . . . . . . . . . . . 2320D. System-field interaction. . . . . . . . . . . . . . . . . . . . 2321

III. EXPERIMENT. . . . . . . . . . . . . . . . . . . . . . . . . . . . .2322A. Pump–probe technique. . . . . . . . . . . . . . . . . . . . 2322B. Transient-grating method. . . . . . . . . . . . . . . . . . . 2323C. Time-resolved sum–frequency generation. . . . . 2323D. Infrared absorption spectroscopy. . . . . . . . . . . . . 2324

IV. CRYSTALLINE SILICON. . . . . . . . . . . . . . . . . . . . 2324A. Bond-center hydrogen. . . . . . . . . . . . . . . . . . . . .2325B. Structural dependence. . . . . . . . . . . . . . . . . . . . .2326C. Decay channels.. . . . . . . . . . . . . . . . . . . . . . . . . .2327D. Deuterium lifetimes. . . . . . . . . . . . . . . . . . . . . . .2328

V. AMORPHOUS SILICON. . . . . . . . . . . . . . . . . . . . .2329

a!Electronic mail: [email protected]

2310021-8979/2003/93(5)/2317/20/$20.00

Downloaded 24 May 2004 to 129.59.117.76. Redistribution subject to AI

A. Hydrogen stretch band. . . . . . . . . . . . . . . . . . . . .2329B. Molecular dynamics simulations. . . . . . . . . . . . . 2330

VI. SILICON SURFACES. . . . . . . . . . . . . . . . . . . . . . .2331A. Si~111!:H Surface. . . . . . . . . . . . . . . . . . . . . . . . .2331B. Si~100!:H Surface. .. . . . . . . . . . . . . . . . . . . . . . .2332C. Theoretical results. . . . . . . . . . . . . . . . . . . . . . . .2333

VII. CONCLUSIONS AND OUTLOOK. . . . . . . . . . . 2334

I. INTRODUCTION

Study of the relaxation dynamics of local vibrationmodes~LVMs! in semiconductors down to the picosecotime scale is an emerging field. Direct time-domain measuments of the lifetime of LVMs is made possible by receadvances in tunable, ultrafast, high-intensity infrared~IR!light sources, including optical parametric amplifiers~OPAs!~Refs. 1 and 2! and free-electron lasers~FELs!.3,4 Such stud-ies allow the elucidation of the nature and pathways ofergy transfer at defects and impurities in semiconductorsder dynamical conditions. The coupling between LVMs athe heat~phonon! bath is responsible for the flow of energ

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P license or copyright, see http://jap.aip.org/jap/copyright.jsp

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2318 J. Appl. Phys., Vol. 93, No. 5, 1 March 2003 Appl. Phys. Rev.: Lupke, Tolk, and Feldman

into and out of impurity and defect complexes. This is important, because when defects act as recombination ortering centers in semiconductors, the electronic excitathat is deposited at the defects is then available to promdefect migration and reactions, an area of defect physicshas an important impact on the degradation of electronicphotonic devices. Hence, a better understanding of the phcal properties of defects and impurities in semiconductcan be obtained by studying the dynamics of the elasticinelastic local interactions.

Information on the inelastic microscopic interactions cbe obtained from the lifetime of LVMs. Vibrational energrelaxation in condensed phases entails the redistributiothe excess energy into low-frequency~often vibrational! de-grees of freedom that are anharmonically coupled to thetially excited local mode. These inelastic interactions ofoscillator with the low-frequency modes result in energylaxation times that are typically hundreds of femtosecondhundreds of picoseconds in solids.5–10 The energy relaxationlifetime most often cannot be derived from the linewidththe linear absorption spectrum.11,12This is due to the fact thagenerally elastic interactions leading to dephasing of thebrational transition and the associated line broadening~puredephasing! occur on even shorter time scales.13–16 Further-more, inhomogeneous effects may contribute significantlythe observed absorption linewidth. Therefore time-domanonlinear spectroscopic experiments, such as pump–psaturation spectroscopy, are necessary to obtain informaon the vibrational population lifetime.

Light impurities in semiconductors, such as hydrogand deuterium, give rise to localized vibrational modes wfrequencies above the phonon bands of the solid.17 The vi-brational spectroscopy of these high-frequency modesbecome an important probe of defects in solids and oprovides information about defect structure and properthat cannot be obtained by other methods.18 LVM spectros-copy has been applied extensively to study isolated impties and impurity complexes in crystalline solids, includiionic crystals17 and semiconductors.18 One of the impurity-materials systems studied most extensively by LVM sptroscopy is H in semiconductors, which has attracted minterest because of hydrogen’s ability to interact with virally any lattice imperfection, including impurities, intrinsdefects, surfaces, and interfaces, thus possibly changingelectronic properties of the material.19,20 Until now, mostspectroscopic studies of H-related LVMs in semiconducthave been carried out in the frequency domain, which prothe time-averaged optical response of the modes. Coquently, very little is known about the dynamics of themodes, i.e., the time scales and mechanisms for populaand phase relaxation upon excitation. Such informationcrucial since excited vibrational states may be involvedthe dissociation of the bond between H and the lattice.instance, Persson, Avouris, Lyding, and their co-workhave reported intriguing data on the different behaviorsthe desorption of deuterium~D! and H bonded to Si.21–24

Although the chemical binding of these species is essentthe same, the difference in desorption behavior was attuted to differences in their vibrational lifetimes.25 To under-


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stand such processes, it is necessary to know the time son which excited vibrational states decay, which is giventhe lifetime of the states.

The aim of this review article is to present a comprehesive overview of the recent theoretical and experimentalsults with respect to the vibrational lifetimes of hydrogen adeuterium-related modes in silicon. Theoretical results hbeen obtained with a method that combines Bloch–Redfitheory and molecular dynamics~MD! simulations. This ap-proach is described in the theory section~Sec. II A!. A theo-retical description of vibrational relaxation and dephasprocesses is outlined in Sec. II B. Given the general imptance of multiphonon relaxation for the lifetime of LVMs,theory of relaxation rates based on a microscopic Hamtonian is presented in Sec. II C. The interaction betweensystem and the radiation field is briefly described in the ltheory section~Sec. II D!.

The various experimental techniques used to measvibrational lifetimes are presented in Sec. III. The pumprobe transient bleaching technique is the most commtime-resolved method for vibrational lifetime measureme~Sec. III A!. The transient-grating method belongs to tsame class of degenerate four-wave-mixing techniques apump–probe method, but allows for background-free dettion, which may increase the signal-to-noise ratio in lifetimmeasurements~Sec. III B!. Lifetime measurements of adsobates on surfaces require a time-resolved surface-senstechnique such as IR sum–frequency generation~SFG!. Asimple combination of SFG and transient bleaching isscribed in Sec. III C. The lifetime of LVMs can also be stuied in the frequency domain by infrared absorption spectrcopy ~IRAS! ~Sec. III D!.

The remainder of this review article consists of thrsections presenting experimental results of the lifetime ofSi–H stretch mode in different environments, including crytalline Si ~Sec. IV!, hydrogenated amorphous Si~Sec. V!,and hydrogen adsorbed on a Si surface~Sec. VI!. Section IVreviews recent lifetime measurements of H-related defectcrystalline silicon, including bond-center hydrogen~Sec.IV A !, the structural dependence for different H-related dfects~Sec. IV B!, their decay channels~Sec. IV C!, and theirrelationship to deuterium lifetimes~Sec. IV D!. Section Vreviews lifetime studies in amorphous silicon~a-Si!. The re-sults of pump–probe and transient-grating experimentspresented in Sec. V A. Section V B reports MD simulatioof vibrational lifetimes of local and extended modes ina-Si.The last experimental section~Sec. VI! reviews the vibra-tional lifetimes of the Si–H stretch mode on Si surfaceSections VI A and VI B present lifetime data from H:Si~111!and H:Si~100! surfaces, respectively. Theoretical resultsthe vibrational lifetimes of the hydrogen stretch mode onsurfaces are reported in Sec. VI C. The article concludes wa brief summary and an evaluation of future directions~Sec.VII !.

II. THEORY

The level of theoretical investigation into the vibrationlifetime of the Si–H oscillator is less advanced. This state


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2319J. Appl. Phys., Vol. 93, No. 5, 1 March 2003 Appl. Phys. Rev.: Lupke, Tolk, and Feldman

affairs is partly because of the difficulties in explicitly treaing slow relaxation processes in complex systems, and pabecause, as suggested by experiment, a highly anharmcoupling mechanism is apparently responsible for the reation process. Even more importantly, because of the hfrequency of the Si–H stretching motion, results from a clsical treatment of the Si–H oscillators may not be partilarly accurate. Bloch–Redfield theory provides the quantmechanical equations of motion of the density operator osystem~local oscillator! in contact with a thermal bath. Expressions for the vibrational relaxation and dephasing rare derived by defining an effective vibrational Hamiltonithat represents canonical averages over the bath modeexpanding the system–bath-interaction Hamiltonian to horder in the phonon coordinates an expression for the mtiphonon relaxation rate is derived. A theoretical descriptof the interaction between the system and an incoming lapulse is presented in the last part of this section.

A. Bloch–Redfield theory

Theoretical results have been obtained with a metthat combines Bloch–Redfield theory and molecular dynaics simulations.26–29 The Bloch–Redfield theory componeallows one to treat the multilevel population evolution ofhigh-frequency Si–H oscillator in a quantum mechanifashion. The molecular dynamics component, on the ohand, allows one to treat the interaction of the Si–H strewith the vibrational degrees of freedom of the bath for a hlevel of complexity.

A brief description of the Bloch–Redfield theorygiven here as it applies to the vibrational energy relaxatdynamics of the Si–H oscillator. The total Hamiltonianwritten as follows:

H~q,Q!5HS~Q!1HB~q!1HI~q,Q!, ~1!

where Q and q represent the coordinates of the quantuSi–H oscillator and the bath, respectively.HS is the Hamil-tonian for the vibrational levels of the system~Si–H bond!,HB is the Hamiltonian of the bath~phonons!, andHI is theinteraction between the LVM and the phonon bath. Typicathe system contains a few degrees of freedom of interesQ,whereas the bath consists of many degrees of freedomq.The total density operator in the joint system and bath phspace is too complex and cannot be calculated explicitly

Bloch–Redfield theory utilizes an important advantaof the density operator, namely, that it offers the possibito describe only a part of the system of interest, i.e., it pvides a reduced description. In this approach,26,30,31one as-sumes that the state of the bath can be well described atimes by a fixed, thermal equilibrium density matrix, aderives an equation for the ‘‘reduced density matrix’’ of tQ system, which involves some correlation functions of oerators mixing theQ system and the bath. Assuming the todensity matrix at t50 can be writtens(0)5r(0)rB ,where the bath density operator is given byrB

5exp@2bHB /Tr(2bHB)#, with b5(kBT)21. The densitymatrix elements of the system in interaction with the thermbath obey the equations of motion32


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rab~ t !52 ivabrab~ t !2 (a8,b8

Rab,a8b8ra8b8~ t !, ~2!

where rab(t) is the reduced density matrix element avab5(Ea2Eb)/\ is the frequency of oscillation of coherence between the eigenstatesua& and ub& of the system. Thefirst term on the right-hand side of Eq.~2! describes theunperturbed motion of the system as determined byHamiltonian HS . The matrixR describes the relaxation othe system by the statistical perturbationHI(t). It will becalled the ‘‘relaxation matrix.’’ The elements of the Redfietensor have a straightforward interpretation.Rab,a8b8 is arate constant for the processrab→ra8b8 . If a5a8 and b5b8, this is a dephasing rate for the coherencerab . If a5b and a85b8, then the term represents a populationlaxation rate between eigenstatesua& andua8& of the system.Other terms describe the coupling between populationscoherences and coherence transfer. The relaxation mRab,a8b8 is given by26–29

Rab,a8b852db8b(g

Gag,ga81

2daa8(g

Gb8g,gb2

1Gb8b,aa81

1Gb8b,aa82 , ~3!

\2Gb8b,aa81

5^b8uqub&âuqua8&

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dte2 ivaa8t^F~ t !F~0!&, ~4!

\2Gb8b,aa82

5^b8uqub&âuqua8&

3E0

`

dte2 ivb8bt^F~0!F~ t !&. ~5!

In Eqs. ~3!–~5!, âuqua8& is the transition matrix ele-ment between eigenstatesua& and ua8&, daa8 is the Kro-necker delta, and the dynamical functionF(t) is the classicalfluctuating force on the Si–H bond arising from the couplito the other modes of the system. In the derivation of Eq.~2!many assumptions have been made:30,31 ~1! The interactionHamiltonian can be written asHI(t)5qF(t), which is to becomputed with the oscillator frozen at its equilibrium postion. ~2! The thermal bath remains in thermal equilibrium~3! The relaxation times of the system under study are mlonger than the autocorrelation timetc of the force. The firstassumption is mostly of a technical nature and representproblem if the coupling is small. The second assumptionreasonable because the amount of energy transferred tobath is small and also because the many-body bath cantablish equilibrium quickly. The third assumption can bchecked self-consistently by performing the MD simulation

The correlation functions appearing in the expressiofor G6 are quantum mechanical. Replacing them by thclassical limit violates detailed balance and does not leathermodynamic equilibrium.33 To go smoothly to the classical limit one rewrites them in terms of the symmetrized crelation function, which is then replaced by the classicaltocorrelation function, which can be computed by meansMD.33 Bloch–Redfield theory seems promising in describivibrational relaxation processes.


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B. Vibrational relaxation and pure dephasing

The vibrational Hamiltonian,H(q,Q), is a multidimen-sional operator that describes the static and dynamic proties of the vibrational states. We can formally reduce it toone-dimensional effective Hamiltonian by defining a neoperator that represents a canonical average over themodes. Let

H~Q!5^H~q,Q!&, ~6!

V~q,Q!5H~q,Q!2H~Q!, ~7!

whereV(q,Q) represents fluctuations of this operator froits canonical average. ExpandingV(q,Q) to second order inthe system coordinate about the equilibrium positionD gives

V~q,Q!5V~q,D!1]V

]QUD

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2

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~8!

The zeroth-order term does not depend onQ and is thusincluded in the bath Hamiltonian. The displaced oscillastates will be denoted by Greek letters with a prime. In terof the displaced oscillator states, the operator describingsystem–bath interaction becomes

HI5 (a8b8

fa8b8Pa8b8 , ~9!

wherefa8b85â8uV(q,Q)ub8& are elements of the fluctuation matrix, andPa8b85ua8&^b8u are the vibrational operators.

Assuming that the coupling between the system coonate and the bath is sufficiently weak, the rate constGa8b8 , for population relaxation from vibrational statea8 tob8 can be written as a Fourier transform of the autocorretion function of the appropriate fluctuation matrelements:34,35

Ga8b851/T15h22E2`

`

dteiva8b8t^fa8b8~ t !fa8b8~0!&,

~10!

wherefa8b8(t)5exp(iHBt)fa8b8(0)exp(2iHBt). The correla-tion function appearing in Eq.~10! involves only the bathvariables and is assumed to decay exponentially with a tconstanttc , the bath correlation time.

Coherent excitation results in a phase coherence betwthe vibrational states,ra8b8(t), which gives rise to a macroscopic vibrational amplitude. In the absence of populatrelaxation, the phase coherence described byra8b8(t) will,in the Bloch approximation, decay exponentially with a timconstantT2* . Again, perturbation theory gives the result

Ga8b8* 51/T2* 5h22/2E

2`

`

dt^@fa8a8~ t !2fb8b8~ t !#

3@fa8a8~0!2fb8b8~0!#&. ~11!

The total dephasing rate between two levels is given by

Ga8b85(g8

Ga8g81(g8

Gb8g81Ga8b8* , ~12!


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where the summations denote the sum of all theT1 processesoriginating in statesa8 andb8. The theory outlined abovein which the bath is described by fluctuation operators, pvides a convenient way to calculate the various relaxatrates.

C. Multiphonon relaxation

If the vibrational relaxation process involves transferienergy directly to the bath modes~phonons! in a single step,and the amount of energy transferred exceeds by many tithe maximum energy of a single phonon, then many phonwill have to be created simultaneously. This processknown as multiphonon relaxation. One might reasonaimagine that the rate of such a high-order process wouldquite small. Therefore, it is remarkable that even if the nuber of phonons emitted is as high as ten or more, mtiphonon processes typically compete successfully withdiative decay, and are in fact often the dominant relaxatmechanism.36–51

Given the general importance of multiphonon relaxatioit is clearly desirable to develop a theory of relaxation rabased on a microscopic Hamiltonian. Existing theoriesvolve one of two approaches: the adiabatic~Born–Oppenheimer! or ‘‘static-coupling’’ ~crude Born–Oppenheimer! methods. The first method is usually invokewhen discussing relaxation between electronic states, butbe applied to high-frequency vibrations as well. The coornates are divided into fast~electronic or high-frequency vi-bration! and slow~phonon! components. One neglects for thmoment the kinetic energy of the phonons, and findseigenstates of the Hamiltonian for fixed phonon coordinaThe eigenvalues of this procedure generate the usual abatic potential surfaces, and transitions between thesefaces are due to the ‘‘nonadiabatic’’ coupling~phonon kineticenergy! term in the full Hamiltonian. Even when this nonadiabatic coupling is taken to lowest order in perturbatitheory, multiphonon transitions emerge. This was theproach pioneered by Kubo and Toyozawa,52 Perlin,53 Miy-akawa and Dexter,54 and others.55–60

The second approach goes by several different nabut its essence is as follows: one assumes that the Hatonian can be written as

H5HS1HB1HI . ~13!

The Hamiltonian of the local vibrational mode,

HS5\va†a, ~14!

is characterized in terms of the frequencyv, creation opera-tor a†, and annihilation operatora. The ~harmonic! Hamil-tonian for the phonon bath is given by

HB5(k

\vbk†bk , ~15!

where bk† and bk are the phonon creation and annihilatio

operators, respectively. There are at least three differentsible routes to multiphonon relaxation based on the formthe interaction HamiltonianHI .61 In the most common approachHI is strictly off-diagonal in the two-level basis of th


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local vibrational mode.61,62The off-diagonal matrix elementof HI are expanded to high order in the phonon coordinaand are treated by lowest-order time-dependent perturbatheory. The interaction Hamiltonian is then given by

HI5($n%

\~G$n%B$n%a†1G$n%

* B$n%† a!, ~16!

whereB$n%5P i 51Nn bi ,$n% , B$n%

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† , andG$n% denotesthe temperature-independent coupling strength of the chnel $n%. $n% is characterized by the set$v1

(n) ,v2(n) , ...,vNn

(n) ,%of accepting mode frequencies. Energy is conserved indecay process,\v5( j 51

Nn \v j(n) . The contribution to the de

cay rate due toHI is given by the standard result of firsorder time-dependent perturbation theory.63 The total decayrate ~inverse lifetime! is given as the sum of the rates of aof the decay channels:61,62

1

T152p(

$n%uG$n%u

2n$n%r$n% . ~17!

The functionn$n% describes the temperature-dependent polation of the receiving modes:

n$n%5exp~b\v!21

P j 51Nn @exp~b\v j

~n!!21#. ~18!

The compound spectral density of accepting states,r$n% , canbe expressed in terms of a convolution of single specdensities of states:

r$n%5E dv1~n!¯E dv~Nn21!

~n! r1~n!~v1

~n!!¯rNn

~n!~vNn

~n!!.

~19!

In the low-temperature limit (kBT!\v j(n) , for all j!, the

decay rate reflects spontaneous decay intoNn acceptingmodes andn$n%>1. At higher temperatures the decay raincreases due to stimulated emission in a fashion determby the frequencies of the accepting modes.61,62As a rule, theprocess with the least amount of phonons should be favo

D. System–field interaction

In conventional relaxation theories for excited states,assumption of fast dissipation results in dynamics thatinsensitive to the nature of the prepared state. For cawhere either population relaxation and/or pure dephasingcurs on a time scale comparable to~or longer than! the initialvibrational excitation, this is not necessarily true. In thecases one must explicitly take into account the frequencytemporal properties of the excitation pulse. The interactbetween the system and an incoming laser pulse is give

HSF~ t !52mE~ t !, ~20!

where m is the transition dipole operator andE(t) is thetime-dependent electric field, which can be written as

E~ t !5E0~ t !exp~ ivLt !, ~21!

whereE0(t) is the temporal profile of the pulse andvL thecenter frequency. Assuming the excitation pulse istransform-limited pulse from a mode-locked laser, the sptral width is determined by the Fourier transform ofE0(t).


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We assume that only the first excited stateub& is opticallyaccessible from the ground stateua& of the system, so that thetransition dipole operator has the form

m5mab$ua&^bu1ub&âu%, ~22!

where the matrix elements of the transition dipole operahave the formmab5âumub&. For simplicity, we will as-sume there is no dissipation in the system during the exction pulse. The validity of this approximation for any givepulse duration will obviously depend on the magnitude offluctuations. In the absence of dissipation, the dynamicsthe density matrix is governed by the quantum mechanLiouville equation

]r~ t !

]t52

i

\@H1HSF~ t !,r~ t !#. ~23!

In order to solve the Liouville equation, the timedependent perturbation theory with the perturbation HamtonianHSF(t) has been used to calculate thenth-order den-sity operator.64 Here, we consider the rate equatiodescription of the density matrix elements to describeevolution of the nonequilibrium state.

We assume that the frequency of the excitation puvL , is tuned in resonance with the one-quantum vibratiotransition betweenua& and ub& states, i.e.,vL5vab . Due tothe resonance enhancement, the vibrational transition fua& to the other stateug& ~Þub&! can be ignored. Also, for thesake of simplicity it is assumed that due to the anharmoniof the bth mode the transition fromub& to u2b&, whereu2b&denotes the overtone state, is negligibly small in comparito the transition probability fromua& to ub&—note that thistwo-state assumption is, however, not entirely necessardevelop a theory of the equations of motion of the noneqlibrium density operator. We next consider the equationsmotion of the density operator:65

rab52 ivabrab1i

\E~ t !mab~rbb2raa!2Gabrab ,

~24!

rba5 ivabrab2i

\E~ t !mba~rbb2raa!2Gabrba ,

~25!

raa52i

\E~ t !~mabrba2mbarab!2garaa1gbrbb ,

~26!

rbb5i

\E~ t !~mabrba2mbarab!1garaa2gbrbb ,

~27!

wherega andgb represent population relaxation (T1) ratesand Gab is the dephasing (T2) rate. In Eqs.~24!–~27!, thevibrational population transfer fromua& to ub& states is fullydescribed by the time-dependent pump–pulse-induced tsition rate,V(t), which is given as66

V~ t ![2p

\2 umabE~ t !u2gab~v!, ~28!


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where the line-shape function,gab(v), is assumed to be aLorentzian function:

gab~v!5Gab

p@~v2vab!21Gab2 #

. ~29!

The solution of Eqs.~24!–~27! can be used to describthe time evolution of the nonequilibrium density operarNeq(t). The preparation step of the nonequilibrium state wtreated nonperturbatively. However, one can use secoorder~with respect to the pump IR field! perturbation theoryto obtainrNeq(t), if the pump intensity is not too large. Nexwe consider the probe–field–matter interaction Hamiltoniwhich is given by

HSP~ t !52m•Epr~ t !, ~30!

whereEpr(t) is the electric field of the infrared probe pulswhich peaks att5t. The density operator at timet is givenby

rNeq~1! ~t !.req2

i

\ E0

t

dt@HSP~ t !,rNeq~ t !#. ~31!

Even though the pump field is assumed to be strong,probe field is weak, and therefore the perturbation approis quantitatively acceptable. In Eq.~31!, the second term corresponds to the first-order~with respect to the infrared probfield! contribution to the density operator. The nonequilrium polarization, which is defined as the dipole momeaveraged over the nonequilibrium density operator, is giby

PNeq~ t !5^mrNeq~1! ~ t !&. ~32!

Finally, the weak probe absorption signal can be callated, within the slowly varying-amplitude approximation,

Sb~t!54pvab

cn~vab!Im E

0

t

dtEpr* ~ t !PNeq~ t !Y E0

t

dtuEpr~ t !u2,

~33!

wheren(vab) is the index of refraction. The time evolutioof Sb(t),

Sb~t!} exp@2t/T1#, ~34!

provides direct information on the vibrational lifetime of thfirst excited state of the local vibrational mode.

III. EXPERIMENT

In order to observe vibrational lifetimes in semicondutors on the picosecond time scale, several ultrafast optechniques have been employed. The most common tidomain ~pump–probe! measurement techniques include t

FIG. 1. Schematic of the pump–probe transient bleaching setup.


rsd-

,

ech

-tn

-

-ale-

transient bleaching technique, the transient-grating methand a combination of sum–frequency generation and trsient bleaching. In special cases, the vibrational lifetimealso be obtained in the frequency domain by IR absorptspectroscopy. This section presents a brief descriptionthese optical techniques.

A. Pump–probe technique

Lifetimes ranging from a few picoseconds to sevenanoseconds can be measured directly by the pump–ptransient bleaching technique utilizing ultrashort pulses fra tunable infrared laser. Typically, two different tunable, ifrared lasers are used: ultrafast optical parameamplifiers1,2 or picosecond free-electron lasers.3,4 These la-sers can deliver pulses with a time duration of;1 ps, spec-tral width of ;15 cm21, and pulse energies of severalmJ.Both lasers can in principle be tuned in wavelength betw1–10 mm, thus covering the Si–H stretch vibrations. Tmajor difference between these lasers is the pulse repetfrequency~PRF!, which for OPAs is typically limited to afew kHz, whereas FELs can achieve much higher PRFs.example, the FEL at the Thomas Jefferson National Accerator Facility ~TJNAF! operates at a PRF of 18 MHz. Ahigh PRF and excellent pulse-to-pulse stability are necesto achieve a high signal-to-noise ratio in pump–probe msurements.

A schematic diagram of a typical setup for a pumpprobe transient bleaching experiment is shown in Fig. 1. Tlaser beam is split into two parts, pump and probe, carry91% and 9% of the power, respectively. The two incideultrashort laser pulses are spatially overlapped on the samThe pump excites a fraction of the Si–H bonds to the fiexcited state, which causes a transient increase in the trmission coefficient of the sample that decays over timeto the decay of the excited mode. The time evolution oftransient bleaching signalSb ,

Sb~t!}exp@2t/T1#, ~35!

is monitored by varying the time delay~t! between the pumpand probe pulses. The signalSb is detected with an amplifiedliquid-nitrogen-cooled HgCdTe or InSb detector. To improthe sensitivity of the experiment a lock-in amplifier is ulized along with an optical chopper, which modulates texcitation for phase-sensitive detection.

FIG. 2. Schematic of the transient grating experiment.


e

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i

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onve

ba

tatord,rbyit

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il-

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eakthe


B. Transient-grating method

The vibrational lifetime can also be probed by anothtime-domain technique, transient grating~TG! spectroscopy.The following typical setup is considered~Fig. 2!. At timet50 two short excitation pulses, (k1) and (k2), are superim-posed on the sample under an angleu, which causes an interference pattern that leads to a spatial modulation ofabsorption coefficient, i.e., a transient grating. The decaythe grating as a result of vibrational population relaxationmonitored by applying a probe pulse, (k3), at timet5t. Thediffracted probe intensitySb(t) decays as

Sb~t!5exp@22t/T1#. ~36!

Higher signal-to-noise ratios may be obtained with tmethod than with the pump–probe technique, sincetransient-grating signal can be observed in a background-direction. However, it is necessary to confirm that spatransport or thermal effects do not affect the decay oftransient-grating signal.

C. Time-resolved sum–frequency generation

Measurements of the vibrational lifetime of Si–H stretmodes on hydrogenated silicon surfaces requires a surfsensitive time-resolved optical technique. In the pastyears the nonlinear optical techniques of second-harmgeneration ~SHG! and sum–frequency generation haemerged as useful surface probes.66–69 In SFG, two inputlaser beams at frequenciesv1 andv2 overlap at the surfaceand generate a sum–frequency output atvs5v11v2 inboth transmitted and reflected directions~Fig. 3!. SHG is aspecial case of SFG withv15v2 . The output is highly di-rectional as required by the boundary conditions and candetected by a photomultiplier through filters and a smmonochromator.

The theory of surface SFG has been described in deelsewhere.70 We summarize here only the results relevantthe measurements. Both techniques rely on second-ononlinear-optical processes.66 In centrosymmetric mediasuch as silicon, second-order nonlinear processes are foden, but at a surface or interface, the inversion symmetrnecessarily broken. This is the basis for the surface sens

FIG. 3. Schematic of the pump–probe SFG experiment.


r

eofs

eeele

e-5ic

ell

il

er

id-isiv-

ity of SFG and SHG. As an intense electromagnetic fiirradiates a material the polarizability that varies linearly fweak fields becomes nonlinear. This leads to variousquency mixing processes. The source of SFG is the nonlinoptical polarization induced in the medium:66

P~vs!5xs~2!~vs5v11v2!:E~v1!E~v2!, ~37!

wherexs(2) is the second-order nonlinear optical susceptib

ity and E(v i) is the field atv i .A semiclassical perturbative treatment of the interact

between the molecule and the electromagnetic field isquired to obtain a complete expression of the nonlinear sceptibility. For SHG, this expression can be found in sevepublications.66,71–74Guyot-Sionnest and co-workers consiered the resonant second-order nonlinearity as a perturbaof the molecular polarizability by a normal mode excitatiosuch that101

P~ t !5S avis~1!1

]avis~1!

]qq~ t !1¯ DEvis~ t !. ~38!

The IR fieldEIR(t) drives the normal modeq(t),

q~ t !5Dre*

m*EIR~ t !

v022v IR

2 2 iv IRT221 , ~39!

where Dr is the population difference between the groustate and the excited state. The induced polarizability atsum frequency is then

a~2!~v5v IR1vvis!5Dr]avis

~1!

]q

e*

m*1

v022v IR

2 2 iv IRT221 .

~40!

From Eq.~40! it is clear that the mode has to be Ramand IR active to produce SFG. The total surface susceptity

xs~2!5Nâ~2!&1xNR

~2! , ~41!

contains the resonant contribution of the excited mowhich depends on the molecular coverageN and on the av-erage molecular orientation, and a nonresonant contribufrom the adsorbate or the substrate. The SFG signal,I (vs),is proportional to66

I ~vs!'ux~2!u2I ~vvis!I ~v IR!, ~42!

whereI (v i), i 5s, IR, vis, is the intensity of the SFG, infrared, and visible pulse. The visible beam energy densitylimited by the damage threshold of the substrate. As a csequence, it is advantageous to perform SFG experimwith short ~picosecond! pulse laser sources.

Lifetime measurements are performed by a simple cobination of SFG and transient bleaching~see Sec. III A!. Inthis scheme, the surface is first excited by an intense infrabeam. This leads to a new population differenceDr for theadsorbate vibration. The SFG from a delayed pair of a winfrared beam and a visible beam is then detected. SinceSFG signal is proportional to (Dr)2 @Eqs.~40!–~42!# it givesdirectly the ground-state recovery time.


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intry

orustra-tz-

e,idthocalti-

ionm-

ofns.


D. Infrared absorption spectroscopy

The lifetime of LVMs can also be studied in the frequency domain by IRAS. The shape of an absorption lingenerally given by the convolution of its homogeneous lshape with a function describing the inhomogeneobroadening.75 In single crystals, inhomogeneous broadenresults from strain fields induced by lattice defects presenthe sample. Since the most abundant imperfections in c

FIG. 4. FTIR spectrum of proton-implanted crystalline silicon.


isesgins-

talline samples are defects formed during crystal growthion implantation, one would expect the inhomogeneobroadening to decrease with decreasing impurity concention CI . Generally, the homogeneous line shape is Lorenian with a full width at half maximum~FWHM! ~in cm21!given by

GH51

2pcT11

1

pcT2*, ~43!

where T1 is the lifetime of population~energy! relaxationandT2* is the time of phase relaxation~pure dephasing!. Thefirst term determines the natural linewidth of the modwhereas the second term generally dominates the linewat elevated temperatures due to elastic scattering of the lmode with phonons. The vibrational lifetime is a good esmate of the natural linewidth.

IV. CRYSTALLINE SILICON

In 1975, Stein reported a pioneering infrared absorptstudy of crystalline silicon implanted with protons at rootemperature~RT!.76 After implantation more than ten absorption lines were observed in the range 1800–2250 cm21 ~seeFig. 4!. The lines shifted down in frequency by a factor1/& when deuterons were implanted instead of proto

FIG. 5. Structure of the interstitial-type defects~top!, and the vacancy-type complexes~bottom!. Large spheres are Si, whereas small spheres are H.


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er

yindmta

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onantioene

ntedge

ichto

thethe

ientn

h a

nalthesbe

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thate

-

itsho--

99he

-


Based on the observed isotopic shifts and the proximitythe lines to those of Si–H bonds in molecules, Stein assigthe lines to local vibrational stretch modes of Si–H bonassociated with hydrogen-decorated implantation-indudefects~see Fig. 5!. Since the original work by Stein, a number of additional Si–H absorption lines have been obserin proton-implanted silicon77,78 or silicon grown in a H2 am-bient and subsequently irradiated with electrons79,80 orneutrons.81 At present, more than 30 distinct absorption linhave been observed in the frequency region associatedSi–H stretch modes. In addition, a number of lines have bobserved in the region 550–900 cm21, which are ascribed toangular vibrations of Si–H bonds.77,78

A. Bond-center hydrogen

Recently, Budde and co-workers have performed fitransient bleaching experiments of the vibrational lifetimebond-center hydrogen in crystalline silicon using the FELthe TJNAF.82 The lifetime of the stretch mode of bond-centH in Si, HBC

(1) , at 1998 cm21 was measured to beT157.860.2 ps at 20 K. There are several good reasons for studHBC

1 : It is the most fundamental H-related defect in Si, aplays a key role in the reactions of H with defects and ipurities. Moreover, it has been studied in great deexperimentally76,83–86 and theoretically87 and is thus wellcharacterized. Finally, HBC

(1) gives rise to an intense absortion line at 1998 cm21 due to the excitation of the stretcmode, where the H vibrates parallel to the Si–H–Si bondaxis ~Fig. 6!.76,83,84The large absorption cross section of thmode makes it ideal for transient bleaching experiments

Experimental studies of point defects require low defconcentrations in order to reduce the effect of interactibetween defects, an additional source of inhomogenebroadening. For transient bleaching spectroscopyhydrogen-related local modes in proton-implanted silicthe practical maximum concentration is at 0.2 at. %. Impltation of higher concentrations results in a broad absorpfeature in the Si–H stretch range rather than an increasthe intensity of the sharp absorption lines of hydrogerelated point defects.88 Pump–probe studies typically requir

FIG. 6. ~a! Absorbance spectrum of proton-implanted Si showing the 1cm21 line of HBC

(1) . Solid line represents a Lorentzian fit to the data. Tspectrum was measured at;10 K on a sample withCH517 ppm. ~b!FWHM of the 1998 cm21 line vs CH . The line represents the natural linewidth G0 obtained from the lifetime of the mode and Eq.~44!.


fedsd

d

ithn

tft

ng

-il

tsusn,-nin-

area densities~doses! of hydrogen of;131017 cm22. Inorder to meet both requirements, the samples were implawith ions at a series of 28 different energies in the ran1.0–1.8 MeV, yielding a uniform H concentrationCH of1.73102 ppm formed from;15 mm to a depth of 47mm.The samples were cooled to 80 K during implantation whwas accomplished by implanting into a cryostat attachedthe end of the accelerator beam line. After implantationcryostat was detached from the beam line and moved tospectroscopic setup.

The vibrational lifetime of the 1998 cm21 mode wasmeasured using a standard pump–probe setup~Fig. 1!. Forexperimental details see Ref. 82. Figure 7 shows the transbleaching signalSb as a function of time delay betweepump and probe measured at;20 K with the FEL. The dataare well represented by a single-exponential decay wittime constantT157.860.2 ps.

The excited 1998 cm21 mode can in principle decay intophotons, electronic degrees of freedom, or other vibratiomodes. The radiative lifetime can be estimated to be oforder of milliseconds,89 which rules out radiative decay athe dominating mechanism. Electronic decay can alsoruled out because the positive charge state of HBC has nooccupied electronic levels in the band gap. Consequenonly electronic transitions from the valence band to the cduction band or directly to the donor level of HBC are pos-sible, both of which require more energy than the 1998 cm21

available. Budde and co-workers, therefore, concludedthe 1998 cm21 mode decays into vibrational modes of thsystem consisting of a Si lattice with a H atom located at abond-center site.82

The lifetime of the 1998 cm21 mode has also been studied in the frequency domain by IRAS.82 The shape of anabsorption line is generally given by the convolution ofhomogeneous line shape with a function describing the inmogeneous broadening.75 At very low temperatures the homogeneous line shape is Lorentzian with a FWHM~in cm21!given by

8

FIG. 7. Decay of the transient bleaching signal from the 1998 cm21 line ofHBC

(1) measured at;20 K with the TJNAF FEL.


icttioine

he

t-

lyal

onoan

ra-

250ch

tran-er,eir

II,

ndntal

e

es

cts


G051

2pcT1. ~44!

For LVMs in crystals, the inhomogeneous broadeningdetermined by the strain fields induced by lattice defepresent in the sample. Since the most abundant imperfecin the H-implanted Si samples are defects formed durimplantation, one would expect the inhomogeneous broading to decrease with decreasingCH .

Figure 6~a! shows the absorption line associated with tfundamental transition of the 1998 cm21 mode measuredwith IRAS at;10 K, which is well represented by a Lorenzian. The FWHM of the absorption line is shown in Fig. 6~b!as a function ofCH . The width decreases monotonicalwith decreasingCH and nearly coincides with the naturlinewidth obtained from Eq.~44! for CH;1 ppm. Thus, theresults obtained in time and frequency domains are fully csistent. Moreover, the convergence demonstrates that gestimates of the vibrational lifetime at low temperature cbe obtained from IRAS on samples with low concentratioof H and lattice defects.

FIG. 8. IR absorbance spectrum ofe-irradiated, H-doped Si, showing thlines of H2* and HV•VH(110) .

TABLE I. Lifetime T1 for a selection of stretch modes of H-related defein crystalline Si measured at;10 K.

Defect v ~cm21! T1 ~ps! Ref.

H2* 2062 4.2 96IH2 1987 12 94IH2 1990 11 94HBC

(1) 1998 7.8 82VH2 2122 60 94VH2 2145 42 94VH4 2223 56 94

HV•VH(110) 2072 295 94


ssnsgn-

-odns

B. Structural dependence

When Si implanted with protons at cryogenic tempetures is heated above;180 K, the 1998 cm21 line disappearsand a series of new lines is formed in the range 1800–2cm21 ~Fig. 4!, which have been identified as Si–H stretmodes of distinct H-related defects~see e.g. Refs. 90–93!.The average absorption cross section of these modes is;25times smaller than that of the 1998 cm21 mode,84 whichmakes direct lifetime measurements on these defects bysient bleaching spectroscopy much more difficult. Howevone can estimate the lifetimes of these modes from thspectral widths at low temperature. As discussed in Sec. I82

the lifetime of the stateu1& of a local vibrational mode isgiven by

T151

2pcG0, ~45!

wherec is the speed of light andG0 is the full width at halfmaximum of the absorption line, given in cm21, associatedwith the fundamental transition of the mode. For Eq.~45! tobe valid,G0 has to correspond to the natural linewidth, ameasured FWHMs may have to be corrected for instrumeand inhomogeneous broadening.

FIG. 9. Decay of the transient bleaching signal from the 2072.5 cm21 modeof HV•VH(110) at 10 K.

FIG. 10. Temperature dependence ofT1 of the 2062 cm21 mode of H2* ~leftside! and the 2072 cm21 mode of HV•VH(110) ~right side!. The solid linesare fits from Eqs.~17! and~18! for decays into the sets of accepting mod$165, 165, 165, 522, 522, 522% cm21 and$343, 343, 343, 521, 521% cm21 ofthe 2062 and 2072 cm21 modes.


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th

.

wi

b-ig

alm

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t o

im

t

ect

oTtw62

hea

hathetcie

esrgath

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ter-

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f-

by10

K.

ar

es,

aye

e-. Incaytureture

ngby

t-de-ting

cal-

ts


To minimize the error inT1 introduced by such corrections, Budde and co-workers performed IRAS measuremat the highest resolution possible using samples with vlow defect concentrations.94 The samples were 15 mm thicand consisted ofn-type Si (@As#5331014 cm21) grown bythe float-zone method. As described in detail in Ref. 94,samples were doped with H~Si:H! by annealing at 1250 °Cfor 30 min in quartz ampules containing H2 gas at 0.66 atmThe concentration of H was;131016 cm21.94 The pointdefects studied were created by irradiating the samples2.5 MeV electrons at RT to a total dose of 1017 cm22. Buddeand co-workers estimated the defect concentration to;1016 cm23,94 ;50 times smaller than in the protonimplanted samples studied in Ref. 82, which results in snificantly less inhomogeneous broadening.

The IR absorbance spectra of the Si:H samples revea series of absorption lines in the Si–H stretch regions, soof which are shown in Fig. 8. The lifetimes of the Si–modes were obtained in the following way: First, thFWHMs of the lines were obtained by fitting Lorentzian linshapes to the measured absorbance profiles. These FWwere then corrected for instrumental broadening, and thesult was entered into Eq.~45!. As shown in Table I, thelifetimes of Si–H stretch modes are extremely dependenthe defect structure, ranging from 4.2 ps for H2* to more than295 ps for HV•VH(110).

94 The narrowest line in the Si:Hspectrum originates from the 2072.5 cm21 mode ofHV•VH(110).

94,95 Having a width of only 0.0203 cm21, thismode has a lifetime of at least 262 ps. To obtain a reliableT1

for this defect, Budde and co-workers measured the lifetdirectly with transient bleaching spectroscopy. Figureshows a semilog plot of the transient bleaching signalSb

versus time delay measured at;10 K with the TJNAF FEL.The signal decreases exponentially with a time constanT1

529566 ps.94

Table I shows that the lifetimes of Si–H stretch moddepend strongly on the bonding configuration of the defeFor instance, the interstitial-type defects H2* , IH2, and HBC

(1)

have lifetimes of 4.2–12 ps, whereas the lifetimesvacancy–hydrogen complexes are at least 40–295 ps.most striking example of the structural dependence is theorders of magnitude difference in the lifetime of the 20and 2072 cm21 modes of H2* and HV•VH(110). The strongstructural dependence might be explained by eitpseudolocalized modes or LVMs being involved in the decprocess. Being localized around the defect, such modesmuch larger vibrational amplitudes on the atoms close toSi–H bond than, e.g., lattice phonons, which may causstrongly enhanced anharmonic coupling to the Si–H stremode. However, significant distortions of the Si–Si bondsthe vicinity of the defects are required for pseudolocalizmodes or LVMs to form,17 which is consistent with highlydistorted interstitial-type defects having shorter lifetimthan vacancy-type defects. Furthermore, to explain the ladifference in lifetimes, one should also consider the anhmonic coupling strength between the stretch mode andlow-frequency modes. In the HV•VH(110) center, it certainlyseems plausible that the relatively large open volume ofvacancies would lead to a small interaction between the


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-

ede

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drogen and the surrounding silicon atoms. This small inaction would lead to a longer vibrational lifetime.

C. Decay channels

Lupke and co-workers investigated the vibrational relaation channels of the 2062 and 2072 cm21 modes of H2* andHV•VH(110) in Si, in order to elucidate the two-orders-omagnitude difference in their lifetimes.96 The lifetimeT1 ofthese modes is measured as a function of temperaturetime-resolved transient bleaching spectroscopy. Figureshows T1 versus temperature for H2* ~left side! andHV•VH(110) ~right side!. In the case of the 2062 cm21 mode,T1 is nearly constant up to;60 K, where it starts to de-crease, reaching half of its low-temperature value at 130The solid lines in Fig. 10 are fits using Eq.~17! with thevibrational relaxation channel of the 2062 cm21 mode repre-sented by a set of six accepting modes$165, 165, 165, 522,522, 522% cm21. This set of accepting modes is very similto that of the 1998 cm21 mode of bond-center H in Si,82

which is well described by decays into six accepting mod$150, 150, 150, 516, 516, 516% cm21. In contrast, the lifetimeof the 2072 cm21 mode starts to decrease at;120 K reach-ing half of its low-temperature value at 250 K. The decchannel of the 2072 cm21 mode is well represented by fivaccepting modes with frequencies$343, 343, 343, 521, 521%cm21.

The vibrational relaxation channels are not uniquely dtermined by the temperature dependence of the lifetimesparticular, the high-frequency phonon modes of the dechannels do not have a strong influence on the temperadependence of the lifetimes. For example, the temperadependence of the 2072 cm21 mode is equally well repre-sented by a set of accepting modes of$343, 343, 343, 343,700% cm21. However, the wave number of the acceptimode with the lowest frequency is quite well determinedthe onset of the temperature dependence ofT1 , for example,;60 K for H2* and;120 K for HV•VH(110). Figure 10 alsoshows that in both cases the calculations using Eq.~17! de-viate from the measuredT1 at elevated temperatures, indicaing that additional relaxation channels may increase thecay rate at higher temperatures. The low-frequency accepmodes of 165 cm21 for H2* ~and 150 cm21 for bond-centerhydrogen! nearly coincide with transverse acoustic~TA!phonons of the undistorted Si crystal, which have beenculated in Ref. 97. Likewise, the 343 cm21 modes involved

TABLE II. Lifetime T1 for a selection of stretch modes of D-related defecin crystalline Si measured at;10 K ~Ref. 95!.

DefectSi-D

v ~cm21!Si-D

T1 ~ps!Si-H

T1 ~ps!

D2* 1500.1 4.8 4.2ID2 1446.5 20 12ID2 1448.7 18 11VD2 1547.9 70 60VD2 1565.1 55 42VD4 1617.5 143 56

DV•VD(110) 1510.4 93 295


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ysal

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ro-or,onave

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in the decay of the stretch mode of HV•VH(110) nearly coin-cide with the peak in the longitudinal acoustic~LA ! phonondensity.

The fact that these modes have very different lifetimesurprising at first glance considering that both Si–H stremodes have nearly the same frequency and symmetry,the compound density of accepting phonon states of theperturbed latticer$n% is almost identical. The critical differ-ence between these defects is their local structure, intersor vacancy like, which gives rise to a very different locvibrational mode structures, referred to as pseudolocalmodes~PLMs!. Evidently it is these local modes that govethe lifetimes, and are responsible for the observed prefetial couplings to characteristic bulk phonons. A more detaiunderstanding of the nature of these PLMs is requiredfully elucidate the vibrational energy relaxation and transchannels in solids.

D. Deuterium lifetimes

Lyding, Hess, and Kizilyalli demonstrated that H plaan active role in the degradation process of silicon metoxide–semiconductor field-effect transistors~MOSFETs! byshowing that devices treated with deuterium~D! have trans-conductance lifetimes 10–50 times longer than H-passivadevices.98 Similar ‘‘giant isotope effects’’ have been observed in other contexts.21,99,100 Dissociation of Si–H andSi–D bonds is believed to be caused by inelastic elecscattering, which excites the bond to either a dissociaelectronic state or to excited vibrational states.21–24

The vibrational excitation mechanism of hydrogerelated modes has been described by the truncated harmoscillator~THO! model.21–24This model describes the Si–Hand Si–D bonds as harmonic oscillators, and assumesthe bonds dissociate at a rate given by the rate of excitafrom the highest bound vibrational stateuNmax& to the lowestunbound stateuNmax11&. Within this model, the dissociationrate is approximately given by

FIG. 11. Comparison of the Si–H~full ! and Si–D~dashed! absorption linesof HV•VH(110) and VH4 .


shndn-

ialld

n-dor

–

d

ne

nic

atn

R;Nmax11

T1S Gexc

Gexc11/T1D Nmax11

, ~46!

whereT1 is the lifetime of the vibrational stateu1& andGexc isthe excitation rate of theu0&→u1& transition.21–24 The THOmodel is in qualitative agreement with H desorptioexperiments.23,24However, this agreement may be fortuitouFor instance, it is well known that Si–H and Si–D bonds aanharmonic, which causes the level spacing to decreaseincreasingN. This will affect the value ofNmax in Eq. ~46!,and hence, also the desorption rate predicted by the moMoreover, the assumption of a single normal mode maybe justified, since both the stretching and the bending momay play a role in the dissociation process.25 Despite thesepotentially unjustified assumptions, the THO model provida framework for discussing bond dissociation via vibrationexcitation. For instance, it is evident from Eq.~46! that Gexc

has to be on the order of the decay rate, 1/T1 , for bondbreaking to proceed via this mechanism. The model also pvides a simple explanation for the H/D isotope effect. Finstance,Nmax for Si–D is;& times greater than for Si–Hwhich would lead to a great reduction in the dissociatirate. Moreover, Si–D modes are usually believed to hshorter T1 than the corresponding Si–H modes,25 whichwould make the isotope effect even greater.

Table II shows the lifetimes of the Si–D stretch modeand allows for a direct comparison of the lifetimes of H-a

FIG. 12. FTIR spectra ofa-Si:H showing the Si–H and Si–H2 stretch bandsat 2000 and 2090 cm21 ~with permission from Ref. 111!.

FIG. 13. Decay of the transient bleaching signal from the Si–H and Si–2

stretch modes ina-Si:H ~with permission from Ref. 111!.


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i-S

ofdi

re

r-

te

uitedis

suo-

ondnsbli-icce

rn

thei-o-

ofing

Hof

plexm-tesHtion

ent

–

00at


D-related LVMs. Substitution of H with D affects the lifetime of LVMs in two ways. First, the vibrational amplitudeless for Si–D modes than for Si–H modes, and sincedecay of the LVM into other modes is an anharmonic effeSi–D modes have longer lifetimes than the correspondSi–H modes. Second, the vibrational energy of the Simode is;& less than that of the Si–H mode. Consequenthe order of the decay process, i.e., the number of modeswhich the LVM decays, can be smaller for Si–D than fSi–H. Since the strengths of anharmonic terms are expeto decrease with increasing order, this suggests that Smodes have shorter lifetimes than the corresponding Smodes. Until recently, the second contribution has beensumed to be dominant,25 which is consistent with the expermental results for Si–H and Si–D modes on H-terminatedsurfaces.101 In contrast, Table II shows that the lifetimesD-related stretch modes are longer than the corresponSi–H lifetimes for all defects except HV•VH(110), for whichthe Si–D lifetime is three times shorter than for Si–H. Figu11 shows the inverted isotope effect for HV•VH(110). Ac-cording to Eq.~46!, the inverted isotope dependence ofT1

observed for HV•VH(110), implies that this defect, and pehaps also Pb :H centers at the Si/SiO2 interface, exhibitsmuch larger differences in Si–D and Si–H dissociation rathan most other H-related defects.

No H-related centers at the Si/SiO2 interface, includingPb :H, have ever been observed directly. It is, therefore, qlikely that the giant isotope effect will have to be explainby theoretical methods. The lifetime data provide an indpensable benchmark for such theory.

V. AMORPHOUS SILICON

In hydrogenated amorphous silicon (a-Si:H), the pres-ence of hydrogen eliminates most of the defects. As a rehigh-qualitya-Si:H layers can be grown. However, the intrduction of excess carriers through moderate illuminationelectronic injection increases the density of dangling bo~DBs! to nearly 1017 cm23, even in the most degradatioresistant materials.102–106These excess carrier-induced DBare metastable; they are annealed out in a few hours at a150 °C. However, their rapid formation sharply limits appcation ofa-Si:H as an inexpensive material for photovoltaand electronic applications. The source of the carrier-indu

FIG. 14. Transient-grating signal from the 2000 cm21 stretch band ofa-Si:Htaken at room temperature~with permission from Ref. 112!.


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metastability has been elus`ve; since its discovery by Staebleand Wronski ~SW! in 1976,107 several models have beepublished, but no previous model is satisfactory.108,109Branzrecently proposed a hydrogen collision model to explainSW effect.110 In this model, DBs are created when recombnation of light-induced carriers stimulates emission of mbile H from Si–H bonds to form a mobile Si–H–Si bond-centered H configuration leaving a DB behind. A collisiontwo mobile H atoms creates a metastable complex containa pair of Si–H bonds in close proximity. The two mobileare then immobilized. Light-induced or thermal annealingthe SW effect leads to the breakup of the metastable comand the subsequent retrapping of the mobile H to DBs copletes the SW annealing. A detailed knowledge of the raand pathways of vibrational energy flow from the Si–bonds may provide some information about the degradamechanism ofa-Si:H.

A. Hydrogen stretch band

Xu and co-workers reported the first direct measuremof the vibrational lifetime of the Si–H and Si–H2 stretchingmodes ina-Si:H obtained by the equal-wavelength pumpprobe technique using the stanford FEL.111 Figure 12 showsthe Fourier transform infrared~FTIR! spectrum of thea-Si:Hfilms grown at low substrate temperature~250 °C! on ~100!c-Si wafers to eliminate most defects. The band at 20cm21 is due to the Si–H stretching mode and the band

FIG. 15. Spectral dependence of the geometric mean lifetime~circles! at 10K. The linear absorption spectrum~solid line! is shown along with thecontributions from mono- and polyhydride~dashed lines! ~with permissionfrom Ref. 112!.

FIG. 16. Temperature dependence of the mean decay rate~inverse lifetime!.Also indicated are various possible decay channels~a!–~e!. The phonondensity ofa-Si:H is shown in the inset~with permission from Ref. 113!.


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2090 cm21 to the Si–H2 stretching mode. All of the peakhave a full width at half maximum of;100 cm21.

Figure 13 shows the decay of the transient bleachsignal for the two wavelengths, 5.0 and 4.75mm, in reso-nance with the Si–H and Si–H2 stretching mode,respectively.111 The data can be fitted very well with a biexponential function using the following expression:

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wheret1 andt2 (t1,t2) are two relaxation times. The energy relaxation times of the Si–H stretch mode are;20 and;100 ps for both samples whereas the energy relaxatimes of the Si–H2 stretch mode are;13 and;50 ps.111 Xuand co-workers proposed a model in which each time cstant corresponds to a different physical process inmaterial.111 In this model, the faster decay time corresponto redistribution within the Si–H stretch vibrational banand the slower time to energy transfer from the stretch mto low-frequency accepting modes.

Rella and co-workers reported pump–probe atransient-grating lifetime measurements performed onSi–H stretch mode as a function of temperature and walength using the FELIX laser.112,113From these data the authors concluded that energy redistribution does not occursignificant extent within the Si–H stretch vibrational banand that the stretch mode decays primarily into three bquanta plus one TA-like bulk phonon. Therefore, the strevibrations are localized, and do not form a phonon-like baof states.

Rella and co-workers performed primarily transiengrating experiments, which permits lifetime measuremewith extremely high signal-to-noise ratio, since the diffractsignal is detected against ‘‘zero’’ background.112,113 Figure14 shows a typical scan taken at room temperature, showthe decay of the transient-grating signal.112 The data are fit tothe square of a double exponential, and decay times of

FIG. 17. Average kinetic energy as a function of time and frequency aexcitation of the 78.7 meV mode~with permission from Ref. 115!.


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and 74.5 ps were obtained, which are in agreement withrelaxation times obtained from transient bleaching byet al.111

Rella and co-workers performed also lifetime measuments as a function of temperature, and across the Smonohydride absorption band.112 Because the shape of thmeasured decay curves remained unchanged, it is possibdefine a geometric mean lifetimeAt fasttslow of the Si–Hstretch mode shown in Fig. 15 for a temperature of 10 K112

Since the wavelength dependence of the lifetime doeschange with temperature the authors conclude that the strmode does not form a vibrational band ina-Si:H.

Figure 16 shows the mean lifetime as a functiontemperature.113 It is clear that only the decay channel of threbending modes plus one TA phonon fits the data well. Thefore, even after the decay of the stretch mode, the vibratioenergy remains largely localized in the form of bend moexcitation, with only a small fraction of the energy dissipaing into the phonon bath of the amorphous-Si host. The rson for the strongly nonexponential decay of the vibratiolifetime is still unclear. It is likely that disorder in the material leads to a wide variation in the microscopic environmeof the hydrogen, thus leading to a wide distribution of vibrtional lifetimes as well as the observed broadening ofabsorption line. Such a view is qualitatively consistent wthe single-crystal data where large variations in lifetimeassociated with minor changes in defect configurations.vestigations are currently underway to test this model,well as to examine the possibility of applying these tecniques to a variety of problems, ranging from a detailstudy of the effects of visible light aging on the structureamorphous silicon, to a more fundamental inquiry into tnature of disordered materials.

B. Molecular dynamics simulations

Fabian and Allen114 and Bickham and Feldman115 calcu-lated vibrational lifetimes in amorphous silicon. Both grouused the same potential and structural models in their stexcept that the later group also made use of a 4096 amodel in addition to a 216 atom model. Phenomenologistructural and interatomic potential models were employBoth groups found that at moderate temperatures, lifetimare on the order of 10 ps and lifetimes of localized and

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FIG. 18. Si–H spectra from the Si~111!:H surface:~a! linear absorption,~b!SFG, and~c! SFG after transient bleaching~with permission from Ref. 101!.


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tended nonpropagating modes are comparable. BickhamFeldman calculated the vibrational lifetimes for modelsamorphous silicon using molecular dynamics simulations115

These types of numerical experiments have been usedcreasingly to study the properties of amorphous materiThey have been used to investigate the Raman spectruamorphous silicon116 and high-frequency modes in fusesilica.117 Although the use of atomistic models restricts tinterpretation to the classical regime, experimentally relevinformation about the amorphous structure and vibratiomodes has been obtained.

In these computer simulations, the instantaneousplacements, velocities, and accelerations are used as iconditions. The kinetic energy is introduced into selecmodes of vibration and monitored as a function of time. Fure 17 shows the log kinetic energies of all 648 modes o216 atom supercell as a function of time and frequencyan initial excitation of the highest-frequency mode withv0

578.7 meV. Although the excitation energy is put intosingle mode, a small portion is quickly and nonradiativetransferred to other modes by anharmonic coupling. Thetial kinetic energy spectrum, therefore, has a strong peathe perturbation frequency with secondary peaks at lofrequencies. As the simulation progresses, this nonequrium distribution evolves into a microcanonical ensemwith each mode having the same kinetic energy. The tscale for this relaxation is on the order of tens of picosonds, and similar decay times have been obtained for olocalized and extended, nonpropagating modes in theatom model.118

These MD simulations using large amorphous supercdemonstrate the ability to extract experimentally relevantformation from computer simulations. The lifetimes founby this method are in good agreement with the perturbacalculations of Fabian and Allen114 and are on the order of 1ps at low temperatures in both 216 and 4096 atom supercThe lifetimes of high-frequency localized modes and etended, nonpropagating modes with intermediate frequenare comparable. Although these initial calculations of the

FIG. 19. Decay of the transient bleaching signal from the Si–H stremode on the Si~111!:H surface. The solid line is a least-squares fit giviT15795 ps. The dashed curve indicates the temporal resolution~with per-mission from Ref. 101!.


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brational lifetimes in amorphous silicon did not include tSi–H stretch vibration, the obtained decay rates of the hifrequency localized modes are comparable. Future Msimulations on hydrogenated amorphous silicon can be uto elucidate the strongly nonexponential decay of the Sistretch mode.

VI. SILICON SURFACES

The first studies on the relaxation dynamics of adsorbvibrations used conventional linear absorption and transbleaching. To compensate for the weak surface optical dsity, poorly characterized high surface area materials habe used.119–121On the other hand, to study single-crystal sufaces, the potential of the nonlinear optical techniquessum–frequency generation was recognized early.71–74,122

Since then, data on the relaxation of adsorbatessingle-crystal metal,123,124 semiconductors,101 and insulatorsurfaces125 have been available using SFG transient refltivity or fluorescence techniques. This section surveyspioneering work done by Guyot-Sionnest and co-workersthe lifetime of the Si–H stretching vibration for thSi(111)/H:131 and Si(100)/H:231 surfaces. Theoreticaresults have been obtained by a method that combBloch–Redfield theory with MD simulations~Sec. VI C!.

A. Si „111…:H surface

Guyot-Sionnest, Dumas, Chabal, and Higashi presenthe first time-resolved measurements of the vibrational ltime of the Si–H stretch mode for an ideally H-terminatSi~111! surface.101 The hydrogenated surface is characterizby a single Si–H stretch vibration at 2083.7 cm21 polarizedpurely perpendicularly to the surface~Fig. 18!. Furthermore,in contrast to the many relaxation channels available to mecules on metals~electronic, intramolecular! with strong dy-namic dipoles~intermolecular!, the Si–H stretch mode caonly relax via anharmonic coupling to one H mode~Si–Hbend! or to the substrate motion~Si surface phonons!. Theabsence of resonant electronic excitation and the large

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FIG. 20. Temperature dependence of the lifetime of the Si–H stretch mon Si~111!:H surface~with permission from Ref. 101!.


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ergy difference between the Si–H stretch mode and the ovibrational modes@nSi–H,bend5637 cm21 ~Ref. 126! andnphonon<500 cm21] imply that a long lifetime is expectedThe temperature dependence of the Si–H stretch linewimplies, in fact, that this mode is anharmonically coupledboth the Si–H bend and a Si surface phonon at a lofrequency, leading to a well-defined~dephasing orT2)broadening.127 Furthermore, molecular-dynamics calcultions with no adjustable parameters~i.e., based onab initiocalculations! have been performed for this system.128

To measure the lifetime of the Si–H stretch on tSi~111! surface, Guyot-Sionnest and co-workers usedmethod of transient bleaching in conjunction with tsurface-specific SFG technique.101 In this scheme~Sec. III C!an intense infrared pulse resonant with the Si–H stretchbration at 2083.7 cm21 pumps the ground state into the firexcited state. After a varying delay, the difference betwethe population of the ground state and the excited vibratiostateDr is then probed by SFG from a pair of weaker infrred and visible beams. If the nonresonant contributionsmall and the anharmonicity is larger than the bandwidththe laser pulse, then the SFG signal is just proportional tosquare ofDr. Therefore, as for the transient bleaching, omeasures a ground-state recovery time.

Although the infrared cross section is small, the SFsignal is fairly strong because of a significant Raman crsection and the perfect ordering of the surface~Fig. 18!. Fig-ure 18~c! shows the SFG spectrum obtained with the pucoming 200 ps before the probe beam. The saturation ofabsorption leads to a 60% decrease of the SFG signal.ground-state recovery time is then directly obtained by vaing the delay between the pump and probe and measuringremaining saturation. The data are presented as the rar[12r 1/2'2rexc in Fig. 19 where ln(r) is plotted versus timedelay. Within the experimental accuracy a straight lineobserved with a lifetimeT150.860.1 ns of the Si–H stretchvibration.

This lifetime is very long compared to what one wouassume for chemisorbed molecules on surfaces. It illustrthe fact that the Si–H stretching mode is well decoupfrom the bending mode and the substrate phonons. In ation, unlike molecules on metals, transfer to electronic extations cannot occur since the band-gap energy is mlarger than the vibrational quantum. The lifetime as a fution of temperature is shown in Fig. 20.101 The temperaturedependence ofT1 is well represented by a decay mechaniinvolving three bending modes around 630 cm21 and onesilicon phonon around 200 cm21 ~dashed curve in Fig. 20!.This decay channel was confirmed by theoretical calculatiusing perturbation theory by Gai and Voth132 ~Sec. VI C!.

The work by Guyot-Sionnest and co-workers constitumeasurements of the lifetime of an adsorbate–substratebration. The system, ideal in many aspects, gives a csingle-exponential decay of a spectrally well-defined vibtional mode. This measurement, along with the thoroulinewidth study,127 gives a complete experimental charactization of the anharmonic coupling between the Si–H streand the lower-frequency modes. This long Si–H stretch ltime may allow various surface photophysics experiment


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be done, such as multiphoton absorption, selective destion, and chemical reaction.

B. Si „100…:H surface

For Si(100)/H:231 the lifetime, from classical molecular dynamics, was predicted to be in excess of 20 ns,126 muchlonger than the lifetime on Si(111)/H:131. Overall, the pic-ture looks a priori fairly different for both surfaces. Themeasured lifetimes at low temperatures are highly sensito the sample preparation and thus reflect more the influeof defects than an intrinsic relaxation rate. Lifetimes longthan 6 ns have been observed at 100 K.129 Si(100)/D:231shows a shorter lifetime~250 ps! and much higher

FIG. 21. Decay of theSmode on Si~100!:H surface at 100 K with the pumpresonant with theS ~open circles! and A modes~closed circles! ~with per-mission from Ref. 129!.

FIG. 22. Decay of the transient bleaching signal from the Si–D stremode on the Si~100!:D surface at 300 K~full circles! and 100 K ~opensquares!. The solid lines are single-exponential fits givingT15225 ps at 300K and T15218 ps at 100 K~with permission from Ref. 129!.


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reproducibility.129 Guyot-Sionnest and co-workers have oserved the energy transfer between the symmetric andsymmetric modes of the surface dimers at lotemperature.129

For the Si(100)/H:231 surface the coupling of the twSi–H bonds in the unit cell induces a symmetric~S! 2099cm21 mode and an antisymmetric~A! 2087 cm21 mode.130

The A mode is relatively weak in the SFG response duesmall effectivex (2) components. For IR light polarized paallel (p-pol.) to the plane of incidence, theS mode is verydominant with no detectableA mode contribution. With theIR pulse '80% vertically polarized (s-pol.), the A modedominates but it is about 30 times weaker.

To probe the population dynamics of the system, bmodes were pumped and theS mode was probed.129 Figure21 shows saturation spectra obtained at 100 K with the pu80% s-pol., where saturation of theS mode induced bypumping either modes appears clearly. Both response cuwere fitted with a fast decay componentT1,AS590615 psand a slow decay,T151.360.1 ns.129 Therefore, the energytransfer between both modes (T1,AS) is relatively slow at 100K. It is faster at higher temperatures and not noticeableroom temperature.

While the relaxation rate is fairly consistent at rootemperature to 1.2560.1 ns, the low-temperature dashowed consistentlyT1,AS590 ps butT1 decreased from 8 to1.2 ns in the course of several days and cooling cycles129

This strong dependence ofT1 on the sample history is aindication thatT1 is not the intrinsic lifetime but is due todefect-induced relaxation. A detailed study of sevesamples revealed thatT1 at 100 K can be longer than 6 nwhile at 300 K it is around 1.2 ns.129 This temperature dependence is much stronger than for Si~111! since there therelaxation varied from 0.9 to 1.4 ns in the same temperarange~Fig. 20!. Such a temperature dependence cannoexplained by a multiphonon relaxation process unless vlow-energy phonons are involved~;100 cm21!. The defectsresponsible for the enhanced relaxation rate are not kno

Si(100)/D:231 also showed a strong SFG resonanfrom theSmode though at 1530 cm21. The relaxation occursover 250 ps with little variation~30 ps! from sample tosample.129 This appears to be the intrinsic value for the liftime of the deuterium stretch mode. Measurements take

FIG. 23. Fluctuating force autocorrelation function along the Si–H bond300 K ~with permission from Ref. 132!.


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300 and 100 K also showed little difference~Fig. 22!.129 Thefaster relaxation of Si–D and its weak temperature depdence are consistent with a multiphonon relaxation procof lower order now probably involving three Si–Si phonosince the Si–D bending mode is below the Si phonon e~520 cm21!.131 The reliability of the measurement oSi(100)/D:231 is attributed to the shorter intrinsic relaxation time that prevents defects from playing an importrole.129

C. Theoretical results

Tully and co-workers performed the first theoreticstudy of the lifetime of the Si–H stretch vibration on asurface using a classical Langevin dynamics calculation128

These authors obtained an estimate for the lifetime of thv51 Si–H vibrational level on a model Si~100! surface to beT1>231028 s, which is much larger than the experimetally observed lifetime ofT1.1.531029 s.101 First attemptsto treat this problem within perturbation theory were undtaken by Gai and Voth.132 These authors calculated the vibrtional relaxation rate of an excited Si–H stretch mode onideally H-terminated H/Si(111)131 surface using a methothat combines Bloch–Redfield theory with classical Msimulations~see Sec. II A!.26–29In the calculation by Gai andVoth,132 it has been assumed that the Si–H vibrational enerelaxation dynamics can be understood by studying the qutum population evolution for only one of the Si–H oscillatoon the H/Si(111)131 surface. This local-mode-like picturneglects the mobility of the vibrational excitations betwethe equivalent Si–H oscillators on the surface. The laeffect should be more relevant to the vibrational dephasdynamics23 than to the energy relaxation process~i.e., eachof the equivalent Si–H oscillators should have an identienergy decay pathway into the bulk!.

The Redfield equations@Eqs. ~3!–~5!# were solved nu-merically for the four lowest-energy levels of the chosSi–H oscillator.132 From the slope of the logarithm of thev51 state probability versus time, the theoretical lifetimethev51 state was determined to be 1.760.1 ns.132 The cor-responding experimental value is 0.95 ns.101 Gai and Vothconsidered also the dominant damping term in the redudensity matrix equation for thev51 state probability, to elu-cidate the mechanism for the Si–H stretching relaxation132

The Fourier transform of the fluctuating force autocorrelatfunction shown in Fig. 23 provides useful information on tcoupling between the Si–H stretch and the rest of the systThe fundamental transition frequencyv1052084 cm21 isshown by the solid vertical line in Fig. 23. This transitiofrequency is somewhat to the blue of a prominent featurethe transform of the force autocorrelation function, whican be attributed to the second overtone~i.e., three quanta! ofthe Si–Si–H bending motion (vbend;640 cm21). Gai andVoth suggested that thev51 state relaxes through couplinto a combination state consisting of three Si–H bendquanta plus a phonon to make up the small frequency mmatch~as postulated in Refs. 101, 128, 133 and 134!. Thisresult was confirmed by Ermorshin and co-workers who ua quantum mechanical treatment of the highly localiz

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Si–H bending modes.135 In the approach by Gai and Voth thbending modes are treated classically as bath modes.132 Animportant result of the calculations by Gai and Voth is tfact that the coupling between the Si–H stretch and theSi–H bends is kinetic in character.132 For the simple poten-tial energy function employed in these calculations, thereno potential energy coupling terms between the Si–H streand the Si–Si–H bends. This behavior has also been idefied, e.g., in the study of vibrational energy relaxationisolated molecules.136

It is also possible to calculate the lifetime of thev52state, which is thought to be considerably shorter thanlifetime of thev51 state. The Fourier transform of the flutuating force autocorrelation function at the frequencyv5v21 is shown by the vertical dashed line in Fig. 2Clearly, thev52 state of the Si–H stretch is more closelyresonance with the bending-mode-induced peak in Fig.Correspondingly, the calculated lifetime is 0.13 ns, which;10 times shorter than that calculated for thev51 funda-mental. The level of agreement between theory and expment is more uncertain in this case because the linewidtdominated by dephasing broadening. More accurate calctions based on improved potential energy functions aresubject of future research.

VII. CONCLUSIONS AND OUTLOOK

Before closing, we would like to place the current eperimental and theoretical accomplishments in this emergfield in the proper perspective. Budde and co-workers hmeasured the lifetimes of a selection of Si–H stretch moof point defects in Si. The lifetimes of interstitial-type dfects were found to be very short~1.6–8 ps!, whereasvacancy-type defects have lifetimes up to several hundof picoseconds—two orders of magnitude variation. Tstrong dependence of the lifetime on the atomic structurethe defect suggests that pseudolocalized modes or LVMsinvolved in the vibrational relaxation of Si–H bonds of poidefects in solids. Moreover, to explain the large differencelifetimes, one should also consider the anharmonic coupstrength between the stretch mode and the low-frequemodes. It certainly seems plausible that the relatively laopen volume of the vacancies would lead to a small intertion between the hydrogen and the surrounding siliconoms. This small interaction would lead to a longer vibrtional lifetime. This interpretation is consistent with thobserved long vibrational lifetime of hydrogen bonded tosurfaces. Both the Si~111!/H and the Si(100)/H:231 sys-tems have Si–H lifetimes in excess of 1000 ps at;100K,101,129i.e., 2–3 orders of magnitude longer than for Si–stretch modes of interstitial-type point defects.

Furthermore, the measured vibrational lifetimes of tcorresponding Si–D stretch modes are key to understanthe physics of the giant isotope effect. Against conventiowisdom, lifetimes of Si–D modes are found to be typicalonger than for the corresponding Si–H modes. Hydrogterminated dangling bonds, exemplified by the HV•VH(110)

complex, have particularly long lifetimes, and the H/D istope dependencies of their lifetimes are ‘‘inverted.’’ The i


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verted isotope dependence is also found for H/D stremodes on Si surfaces. Based on this observation, it is psible that PbH: centers might be particularly susceptibledissociation by multiple vibrational excitation, and are liketo exhibit a particularly large isotope effect. No H-relatecenters at the Si/SiO2 interface, including Pb :H, have beenobserved directly. It is, therefore, quite likely that the giaisotope effect will have to be explained by theoretical meods.

Finally, it is interesting to compare the results by Budand co-workers with measurements of Si–H modes in diffent environments. The lifetime of Si–H stretch modesamorphous Si (a-Si:H) was studied by pump–probe antransient-grating spectroscopy. In this material, the popution relaxation is biexponential with a short decay time;10 ps and a long decay time of;100 ps.111,112 The fastcomponent was initially ascribed to redistribution within thSi–H stretch band and the slow component to decay ofSi–H stretch modes into Si–H bend modes or phonons111

However, more recent experiments indicate that the twocay times are due to stretching vibrations at different shaving different lifetimes.112 The results of the work byBudde and co-workers support the latest interpretation.

Modeling of the dynamics of the multiphonon relaxatioprocess has also played a key role in the understanding ovibrational lifetime of hydrogen. The results by Gai and Voand others show that the vibrational lifetime as well ascorresponding dominant decay channel can be obtained qaccurately using a perturbative calculation, involving BlocRedfield theory and MD simulations, in which the Si–H ocillator is treated quantum mechanically. Quantitative modexplaining the large structural dependence of the vibratiolifetime require detailed knowledge of the local structurethe defects and the associated local phonons. The measlifetimes provide an indispensable benchmark to devemore accurate calculations based on improved potentialergy functions.

Future directions will focus on elucidating the stronstructural dependence of the vibrational lifetime and enetransfer channels of the Si–H stretch mode. Temperatand strain-dependent lifetime measurements will shed lon the preferential coupling between the Si–H oscillator athe bulk phonons. Two-color transient bleaching experimewill allow direct measurements of the energy transfer procbetween localized modes, enlightening the role of the being mode in the decay process. Characterization and conof materials on the time scale associated with the fundamtal vibration period of solid-state atomic motion will reveinsights into energy pathways, defect reactions, and comformation. These exciting directions provide the promiseestablishing a solid-state science involving quantum-lecontrol.

ACKNOWLEDGMENTS

First and foremost, the authors wish to thank MichaBudde who, while visiting Vanderbilt University as a posdoctorate, conducted the vibrational lifetime measuremeof bond-center hydrogen in silicon, raising the art of defe


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spectroscopy to new heights. The authors are indebtemany colleagues with whom they have collaborated on vous lifetime experiments of hydrogen- and deuterium-relamodes in silicon. The groups by M. Stavola at Lehigh Uversity and A. K. Ramdas at Purdue University performhigh-resolution FTIR absorption spectroscopy in combition with samples with very low defect densities to measthe vibrational lifetime of a variety of H- and D-related dfects in Si. J. C. Tully provided stimulating discussionstheoretical calculations of vibrational lifetimes. The authoare grateful to P. Guyot-Sionnest and Y. J. Chabal for lettthem reproduce some of their exciting results obtained frhydrogenated Si surfaces. Their papers taught the autholot about vibrational energy relaxation of hydrogen onsurfaces. The authors thank M. Rella, G. A. Voth, and J.Feldman for generously contributing figures for this articThe authors are indebted to the many postdoctoratesstudents who conducted vibrational lifetime measuremeincluding X. Zhang, C. Cheney Parks, B. Sun, and A. FraThe authors gratefully acknowledge the staff at the TJNFEL for providing them with high-quality beam time and WM. Augustyniak for technical assistance. This work was sported in part by DOE through Grant No. DE-FG099ER45781~C.W.M. and V.U.!, ONR ~C.W.M. and V.U.!,NSF through Grant No. DMR-00-76027~C.W.M.!, and theThomas F. and Kate Miller Jeffress Memorial Trust throuGrant No. J-545~C.W.M.!.

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