FIRST-PRINCIPLES CALCULATIONS OF VIBRATIONAL LIFETIMES IN SILICON DAMIEN WEST, B.S., M.S. A DISSERTATION PHYSICS

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1 FIRST-PRINCIPLES CALCULATIONS OF VIBRATIONAL LIFETIMES IN SILICON by DAMIEN WEST, B.S., M.S. A DISSERTATION IN PHYSICS Submitted to the Graduate Faculty of Texas Tech University in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY Approved Stefan Estreicher Chairperson of the Committee Roger Lichti Mark Holtz Bill Hase Accepted John Borrelli Dean of the Graduate School August, 2006

2 Copyright 2006, Damien West

3 ACKNOWLEDGEMENTS I would like to thank my research advisor Dr. Stefan Estreicher for his guidance through the course of this research and the preparation of this dissertation. I would also like to thank Drs. Roger Lichti, Mark Holtz, and Bill Hase for serving as my committee members. I am also grateful for the support I have received from the R.A. Welch Foundation, the National Renewable Energy Laboratory, and the Department of Physics at Texas Tech. Finally, I would like to thank the Texas Tech High Performance Computing Center for more than my fair share of computing time. ii

4 TABLE OF CONTENTS ACKNOWLEDGEMENTS... ii ABSTRACT... iv LIST OF FIGURES... v LIST OF TABLES... vii I. INTRODUCTION Practical Importance Experimental Situation This Work II. THEORY Density Functional Theory Linear Response Theory SIESTA Vibrational lifetimes III. RESULTS H 2 * and isotopes H + BC and D + BC VH i HV and isotopes IH 2 and ID Intersitial Oxygen and isotopes IV. CONCLUSIONS V. REFERENCES iii

5 ABSTRACT Time-resolved infrared absorption spectroscopy of the local vibrational modes (LVMs) of light impurities in crystalline silicon reveal that the vibrational lifetimes of almost identical LVMs sometimes differ by up to two orders of magnitude. Indeed, at low temperatures, the lifetimes of the 2062cm -1 mode of H 2 *, the 1998cm -1 mode of H + BC, and the 2072cm -1 mode of V 2 H 2 are 4,8, and 295ps, respectively. Since the optical phonon of Si is about 530cm -1, these decays should all involve at least four phonons and have long and comparable lifetimes. More surprising still, the measured lifetime of the asymmetric stretch of interstitial O in Si changes by almost an order of magnitude with the isotope of O or one of its Si neighbors. In this work, ab-initio molecular dynamic simulations in periodic supercells are used to calculate the temperature dependence of vibrational lifetimes. The theoretical approach developed for these calculations can be applied to the calculation of any vibrational lifetime in any crystal. The calculations predict accurate lifetimes of the various defects in the range 50<T<200K and provide critical insight into the decay processes. iv

6 LIST OF FIGURES 1. Trunicated harmonic oscillator representing the vibrational energy levels of the Si-H bond at the Si/SiO x interface Set-up of a transient bleaching spectroscopy experiment Transient bleaching signal(s b ) vs. time for H 2 *(left) and V 2 H 2 (right) at low temperature Low temperature wag mode lifetimes. [30] Temperature dependence of the lifetime of H * 2 (left) and HV VH (right) Measured temperature dependence of the lifetime of H + BC, along with three possible fits to the data The host crystal is represented by a periodic supecell of 64 Si atoms (red box) containing a defect ps MD simulation of Si 64 at T=50K, 200K, and 500K with a Nose-Hoover thermostat (green) and the procedure outlined above (black) Structure of the H 2 * defect Localization Plots of Σα,i(esα,i)2 (where α=h s shown in black and the two Si atoms bound to them shown in red) vs. frequency Six different realizations (different initial conditions) of the decay of H 2 * at 100K Exponential fit to the average of the six decays of H 2 * at 100K, yielding a lifetime of 2.6ps decays of D 2 * at 100K Average of the three caluclated decays of D 2 * at 100K, giving a fit of 18ps Temperature dependence of H 2 *, calculated (red triangles) and experimental (solid points)...34 v

7 16. Representitive decay of H 2 * at 100K. a) H BC stretch at 2126cm -1 b)h AB stretch at 1860cm -1 c) and d) H AB wags at 853cm Scaled energies of the H AB stretch at 1860cm -1 (red curve) and the symmetric stretch of the Si about the H BC at 272cm -1 (green curve) Structure of the H + BC defect Locatization of vibrational modes of H+ BC (top) and D+ BC (bottom) Triangles (circles) show the calculated (measured[2]) lifetimes of H + BC Energies of all the normal modes in the supercell vs. time during the decay of the 2014cm -1 mode of H + BC at T=75K VH i HV defect, H's are shown in red the yellow mark the sites with Si missing Decay of VH i HV at 200K, the red and green curves are the energies of the degenerate stretch modes, the black curve is the sum of these two curves Decay of the stretch modes of VH i HV at 200K Vibrational spectra of VHHV(top), VDHV(middle), and VDDV(bottom) IH 2 defect The sum of the squares of the magnitude of the eigenvectors on the two H/D(black) and the two nearest neighbor Si(red) The decay of the high frequency stretch of IH 2 (a), along with the sum of the energies of the proposed accepting modes (b), (c)=(a)+(b) Interstitial oxygen Calculated energies of the three most active modes in the decay of v 3, labeled (a) Frequencies (for the different isotopic combinations) of the symmetric and asymmetric stretch of O i, along with the difference of these two frequencies, Δω Frequency gap law for the wag modes of H (top), the lifetimes of the stretch modes with decay order taken to be lowest possible order phonon process (bottom)...56 vi

8 LIST OF TABLES 1. Low temperature lifetimes of the stretch modes of H related defects in Si, those marked with asterisks were performed in the time domain Calculated and observed frequencies of LVMs of H 2 * isotopes Calculated lifetimes of the different isotopes of H 2 * for several temperatures O i frequencies and lifetimes, theory indicates this work...50 vii

9 CHAPTER I INTRODUCTION The topic of this dissertation is the first-principles calculation of the decay of local vibrational modes (LVMs) in crystalline Si. Light impurities in crystalline solids oscillate at frequencies much higher than the characteristic oscillations of the crystal. Since the crystal cannot oscillate at such high frequencies such vibrations are localized at the defect[1]. In recent years, the lifetimes of the LVMs of several H-related stretch modes in Si have been measured by transient bleaching spectroscopy[2,3]. The time scale of the decays vary, but are typically in the range of hundreds of femtoseconds to hundreds picoseconds[4,5,6,7,8,9]. Dynamics at this time scale are achievable using abinitio molecular dynamics (MD) simulations. The relaxation dynamics of impurity-related LVMs in crystal is unexpectedly complex. Indeed, the measured lifetimes of modes with nearly identical frequencies (eg. 2062cm -1 and 2072cm -1 ) can differ by nearly two orders of magnitude (4 and 295ps, respectively)[3]. Since the optical phonon of Si is about 530cm -1, quantum perturbation theory suggests that these decays should involve at least four phonons and have long and comparable lifetimes. In order to determine the number of phonons involved in the decay process, recent work has concentrated on the temperature dependence of the measured lifetimes. While a general theory exists[10] which allows for the fitting of the temperature dependence of the lifetimes, these fits suffer from several problems. Firstly, the fits to the data are 1

10 generally not unique; and there may be multiple possible decay channels which fit the data equally well. Furthermore adding to the confusion, using this fitting process it has been found that the 2072cm -1 mode which lives for 295ps is a five phonon process, but the 2062cm -1 mode which lives for 4ps is a 6 phonon process. Additionally, the fitting lacks predictive power and fails to explain fundamental questions about the nature of the coupling. The remainder of this Introduction covers these issues in more detail. First, we discuss the practical importance of vibrational lifetimes, particularly the impact on device lifetimes. Second, we summarize the experimental situation; including some of the very peculiar data which has been obtained for the lifetimes of stretch modes of different defect related LVMs. And finally, an overview of the present work is provided, outlining the contributions which have been made on advancing the methodology of calculating vibrational lifetimes as well as furthering the understanding of the current experimental situation. We also summarize calculations that were performed and highlight the major results. 1.1 Practical Importance The importance of vibrational lifetimes resides not only with the general problem of heat dissipation in solids, but it also has a very practical impact on devices. The prime example of this impact is the huge isotope effect observed for the transconductance lifetimes of silicon metal-oxide-semiconductor field-effect transistors (MOSFETs).[11]. Lattice mismatch at the Si/SiO x interface leads to dangling Si bonds at the interface.[12] 2

11 This defect, known as the P b center, is a well known and well characterized trap which lowers the device transconductance[12,13,14,15]. In practice these dangling bonds at the Si/SiO x interface are passivated with H. Under operation, high current densities exist directly beneath this interface. Over time hot-electron scattering events cause a degradation of the interface by the desorption of H, leading to the recovery of Si dangling bonds[16,17]. Nearly a decade ago, it was discovered that if the Si/SiO x interface is treated with D instead of H, the transconductance lifetime increases by a factor of [11] This phenomenon has been explained using a truncated harmonic oscillator model[18,19], wherein the authors have derived an expression for the desorption rate of H or D from the interface, given by ( I / e) f R ( I / e) fin + in 1 T 1 N, (1.1) which is critically dependent on the vibrational lifetime, T 1. What is of key importance in understanding this relation is that it is not a single hot electron that is responsible for the dissociation of the Si-H or Si-D bond, but instead a series of electron scattering events is required to give the LVM enough vibrational energy to break the bond. In this manner, the vibrational states of the LVM constitute a series of steps which must be climbed in order to reach dissociation, which occurs at energies above the N th step. For the Si-H bond N~12, a qualitative picture of the truncated harmonic oscillator representing the vibrational levels of the Si-H at the interface is shown in Figure 1. 3

12 Figure 1: Truncated harmonic oscillator representing the vibrational energy levels of the Si-H bond at the Si/SiO x interface. So, if an LVM has a short vibrational lifetime (T 1 < the time interval between scattering events), climbing this ladder is very difficult. If the LVM is vibrationally excited by the inelastic scattering of a hot electron, then before a second electron comes by it has already vibrationally decayed back to its ground state. Therefore, short-lived LVMs are much more resilient to electron degradation. On the other hand, if a very long lived LVM is vibrationally excited by the inelastic scattering of an electron, then relaxation to the ground state will not be completed before a second electron scatters from it. Consequently the long-lived LVM is further excited up the ladder. This process continues with more and more scattering until it has enough energy to dissociate. 4

13 1.2 Experimental Situation Vibrational lifetime observations fall into two categories, direct and indirect. The indirect measurements are obtained from line-width measurements. The vibrational lifetime, T 1, is related to the homogeneous linewidth by, 1 T1 = 2πcΓ, (1.2) where Γ 0 is the full width at half maximum. Measurements of the lifetime in this manner have to be corrected for both instrumental and inhomogeneous line broadening. Such measurements require very high resolution and very low defect concentrations.[20] Due to these difficulties, the vibrational lifetime usually cannot be obtained from the linewidth.[21,22] Recent advances in ultra-short laser pulse technology have made it possible to measure vibrational lifetimes in the time domain via transient bleaching spectroscopy[23,24,25,26]. A typical setup for such an experiment is shown in Figure 2. 0 Figure 2: Set-up of a transient bleaching spectroscopy experiment It consists of a pump beam, shown in red, and a much weaker probe beam, shown in blue. The pump beam excites the 0 1 vibrational transition of the LVM of interest and is 5

14 of long enough duration to cause bleaching, ie. half of the defects are in the 0 state and half in the 1 state. After some time delay, the probe pulse then hits the sample and from how much of the probe beam is absorbed it can be determined how much of the vibrationally excited population have decayed back to their ground state. By varying the time delay between the pump and probe pulses, the transient bleaching signal can be plotted as a function of time and fit to an exponential decay. Figure 3 shows the low temperature transient bleaching signal of the high frequency stretch modes of two different H related defects in Si, along with exponential fit. Figure 3: transient bleaching signal(s b ) vs. time for H 2 *(left) and V 2 H 2 (right) at low temperature The two defects shown in Figure 3 are the most striking examples of the disparity in the observed vibrational lifetimes, the stretch modes of the H 2 * pair (left) and of the divacancy di-hydrogen complex (VH HV), right. While these two vibrational modes have 6

15 nearly identical frequencies (2062cm -1 for H * 2 and 2072cm -1 for VH HV), the low temperature lifetimes are observed to be 4ps and 291ps, respectively. These two defects exhibit the largest difference in measured lifetimes, but this lifetime separation is not the only peculiarity in the experimental observations which have been made. Table 1 lists the direct and indirect lifetime measurements of stretch modes of H related defects in Si which have been published to date. Strangely, although there is good reason to expect that the deuterated defects would have shorter lifetimes (they have lower frequencies and consequently they can generally decay through a lower order phonon process) only VH HV exhibits this expected isotope shift; one out of eight. Defect No D substitution D substitution Ref. ω (cm -1 ) T 1 (ps) Ω (cm -1 ) T 1 (ps) H 2 * ±* [3] H + BC ±.2* [2] IH [27] IH [27] VH [27] VH [27] VH [27] VH HV ± 6* [3] Table 1: Low temperature lifetimes of the stretch modes of H related defects in Si, those marked with asterisks were performed in the time domain. 7

16 In addition to the H and D related data, several isotopes of interstitial oxygen have been studied. The vibrational lifetime of the asymmetric stretch of interstitial oxygen shows a very strong isotopic dependence. Although the LVMs of the different isotopic combinations ( 28 Si- 17 O- 28 Si, 28 Si- 16 O- 28 Si, 28 Si- 16 O- 29 Si, and 28 Si- 16 O- 30 Si) only have frequency shifts on the order of 1-2%, the measured lifetimes differ by a factor of 7 (4ps, 11ps, 20ps, and 28ps, respectively)[28,29]. This apparent independence of the lifetimes from the frequencies of the LVM of a single defect stands in sharp contrast to the traditional understanding of vibrational relaxation based on quantum perturbation theory. While the stretch mode data does not lend itself to a straightforward interpretation, the behavior of the wag modes is closer to what one might expect. In a recent paper[30], Sun et al. measured the lifetimes of the wag modes of H 2 * and D 2 * and compiled the observed lifetimes of H/D related wag modes in various semiconductors. They observed that the wag mode data follows a frequency-gap law, where the lifetime of the LVM increases exponentially with the number of phonons required in order to conserve energy. This minimum number of phonons is referred to as the decay order. Figure 4 is taken from [30] and shows the noted exponential increase with decay order of the observed lifetimes of wag modes in various semiconductors. 8

17 Figure 4: Low temperature wag mode lifetimes. [30] In addition to the low temperature measurements, the temperature dependence of the lifetimes of several defects have been measured using transient bleaching spectroscopy. The vibrational lifetimes exhibit strong temperature dependence reminiscent of peak frequency shifts, staying nearly constant in the low temperature range but falling off rapidly at higher temperatures. Figure 5 shows the temperature dependence of the previously mentioned H * 2 (left) and VH HV(right) defects. 9

18 Figure 5: Temperature dependence of the lifetime of H * 2 (left) and HV VH (right). The solid points with error bars are the data and the solid lines are the fits to the data using the model of Nitzan and Jortner[10]. This model predicts the temperature dependence of the LVM lifetime given a set of accepting modes and coupling coefficients, G i, = ω /. (1.3) ( e 1 ) ω / kt B 1 2 e 1 2π Gi Ni j kt B T1 i j= 1 In (1.3) the sum is over all possible decay channels, this being all sets of modes {ω j } whose frequencies sum to the frequency of the LVM, ω. In practice, however, one assumes only one primary decay channel and the data are fit to the lowest order phonon process with accepting modes which reproduce the observed temperature dependence. Given only one decay channel, the coupling coefficient is reduced to an overall scaling 10

19 factor and is chosen to give the correct lifetime at low temperature. While the NJ theory successfully fits the temperature dependence of the vibrational lifetimes, without a method for calculating the coupling coefficients it cannot be used to predict the vibrational lifetime of an arbitrary unmeasured defect. Further, it does not explain why a six phonon process should be 100 times faster than a five phonon process. The temperature dependence of H + BC is shown in Figure 6. The experimental data in the figure illustrates some of the difficulty in identifying the receiving modes Figure 6: Measured temperature dependence of the lifetime of H + BC, along with three possible fits to the data experimentally. While it seems that a four phonon process can be ruled out completely because it is clearly unable to approximate the temperature dependence, the temperature 11

20 dependence is represented equally well by 3 modes at 150cm -1 and 3modes at 516cm -1 as has been reported[2] or by 2 modes at 114cm -1, 2 modes at 385cm -1, and two modes at 500cm -1. The experimental situation leaves a number of fundamental questions unanswered. Why do LVMs decay via high order phonon process (5 for VHHV, 6 for H 2 *, and 6 for H bc ) when they could all decay via 4 phonons (the Γ phonon is at ~530cm - 1 )? Furthermore, why do some LVMs couple very strongly to the phonon bath (H * 2) while others do not (VH HV)? Finally, why does H * 2 (fit to a six phonon process) decay orders of magnitude faster than VH HV (fit to a five phonon process)? 1.3 This Work The primary purpose of this dissertation is to explore the feasibility of calculating vibrational lifetimes at finite temperatures using first-principles MD simulations. The method proposed herein involves using the eigenvectors of the dynamical matrix to set the initial excitation of the vibrational mode of interest as well as preparing the supercell in thermal equilibrium. MD simulations are then performed without the use of a thermostat, whereby the normal modes of the system act as the thermal bath into which the excitation decays. The energies of each normal mode are then monitored throughout the course of the simulation, in order to identify the receiving modes. To test this methodology, the lifetimes of several defects (whose lifetimes have been measured) were calculated. We started with defects whose temperature dependence have been accurately measured using transient bleaching spectroscopy: H 2 *, H + BC, and 12

21 VHHV. We found that at temperatures above 50K our calculations yield results that are in quite good agreement with the experimental data. Furthermore our calculations allowed us to unambiguously identify the decay channel of H 2 * as a two phonon process instead of the previously reported six phonon process.[3] In addition to the H related defects, we investigated several isotopes of interstitial oxygen in Si, whose lifetimes have been found to be very sensitive to isotopic substitution. We find the same sensitivity in our calculations and predict the correct ordering of the lifetimes with isotope. Furthermore, investigation of the decay channels revealed the crucial role played by all the vibrational modes localized at the defect: local, pseudo-local, and resonant. 13

22 CHAPTER II THEORY Our calculations are first-principles calculations. In the present context, this means that there are no parameters in the calculations which are fit to reproduce experimental data. The calculations are performed in a supercell of 64 Si atoms and periodic boundary conditions are imposed at the cell edge; this eliminates surface effects. In this manner, any defects studied are represented periodically in space as shown in Figure 7. In order to minimize unphysical defect-defect interactions one desires as large a supercell as possible, but this must be balanced with the increased computational requirements of dealing with ever larger supercells. Figure 7: The host crystal is represented by a periodic supecell of 64 Si atoms (red box) containing a defect. 14

23 The Born-Oppenheimer[31] approximation is used to decouple the nuclear and electronic problems. The nuclei are treated classically. The electronic problem is split into two regions, the core region and the valence region. The core region is handled using ab initio pseudopotentials. The pseudo-wavefunction is nodeless and smooth varying within a certain core radius, r c. At the cutoff radius, the logarithmic derivatives of the core and valence wavefunctions are matched. The valence region is treated using densityfunctional theory (DFT) and is solved for self-consistently. A brief overview of DFT will be given in the next section. To perform molecular dynamics (MD) simulations the forces on the nuclei need to be calculated. The nuclei are treated classically and the forces on the nuclei are obtained from the Hellmann-Feynman theorem[32], F E = = m a i, α α i, α xα, (1.4) which gives the ith component of the force on the αth atom. Newton s laws of motion are integrated to find the velocity and position of each nucleus after the time interval Δt. The nuclei are then moved to their new positions and assigned their new velocities. Once the nuclei are at their new positions the electronic problem needs to be solved selfconsistently in order to determine the new forces and then the cycle begins again. Solving for the electron density is by far the most time consuming step in this process. Note that in this approach, the electrons have zero temperature (ground state) and any quantum behavior of the nuclei (zero point energy, tunneling) is ignored. Typical MD time steps are of the order Δt=10-15 s=1fs. All of the MD simulations in the present work were 15

24 performed using the SIESTA implementation of DFT, the details of which are also given later in this chapter. 2.1 Density Functional Theory We know, of course, that if we could solve the Schrödinger equation HΨ = EΨ (2.1) for the ground-state of a crystal, we could, in principle, obtain from the solution all of its ground-state properties. However, this is a matter of staggering complexity, and is completely intractable for any but the smallest systems. Fortunately, such a brute force approach is not the only strategy for solving the quantum many body problem. DFT allows for the calculation of the ground state energy of a system without solving for the many electron wavefunction, ( r,..., r ). This section provides a short theoretical Ψ 1 N review of DFT, since this is the theoretical foundation upon which all of the energy calculations of SIESTA are based. In terms of the normalized ground-state wave function ( r,..., r ) of the N- electron assembly, the electron density, n(r) is given explicitly by Ψ 1 N * 3 3 ( ) = Ψ ( 1,..., N) Ψ( 1,..., N) 2... N nr N r r r r dr dr. (2.2) It was recognized by Thomas[33] and independently by Fermi[34] that the ground state energy in an inhomogeneous electron gas could be written in terms of the ground state density n(r). However, it was almost 40 years after Thomas pioneering work that it was proved that, for a non-degenerate state, the energy E of an N-electron system was indeed 16

25 a unique functional E[n] of the density n(r). This theorem, due to Hohenberg and Kohn[35], formally completed the Thomas-Fermi theory. The proof of this theorem for a non-degenerate ground state goes as follows. Consider N electrons moving under the influence of an external potential energy V ext (r) and their mutual Coulomb interaction. The many electron ground state wave function is denoted by Ψ. The first step in the proof is to demonstrate that, apart from an unimportant additive constant, V ext (r) is a unique functional of n(r). Let us assume first that another external potential, V ext (r), with ground state wave function Ψ, leads to the same density n(r) as V ext (r). Now, clearly, unless V ext (r)- V ext (r) = constant, the two potentials are unequal Ψ Ψ because they satisfy different Schrödinger equations. Hence, if we denote the Hamiltonians and corresponding ground state energies associated with Ψ and Ψ by H, H and E, E, we have by the minimal property of the ground-state energy that ext ext. (2.3) * * E = Ψ H Ψ dτ < Ψ H Ψ dτ = Ψ * ( H+ V V ) Ψdτ Thus, it follows that, since <Ψ H Ψ>= E, r. (2.4) 3 E < E+ ( Vext Vext ) n( ) d r Repeating the argument for E is equivalent to interchanging primed and unprimed quantities, consequently we obtain r, (2.5) 3 E < E + ( Vext Vext ) n( ) d r where we have made the assumption that the same n(r) is generated by the two different potentials. Adding (2.4) and (2.5) yields 17

26 E + E < E+ E, (2.6) which is inconsistent. Thus two different external potentials cannot generate, the same density, and therefore, within a constant, V ext (r) is a unique functional of n(r). Since, in turn V ext (r) determines the Hamiltonian H, it follows that the many electron ground state energy is a unique functional of n(r). Exploiting the fact that the energy is a unique functional of the electron density, Kohn and Sham[36] split the energy functional into a single-particle kinetic energy T s, a Hartree potential energy, and an exchange and correlation energy E xc [n]. The resulting Euler equation follows from the variational principle δ[ E μn] = 0 (2.7) as an equation for the constant chemical potential μ, paralleling the arguments of the Thomas-Fermi and Thomas-Fermi-Dirac theories: δts δ Exc μ = + VHartree() r + δ n() r δ n() r. (2.8) Kohn and Sham[36] emphasized that (2.8) is equivalent to solving a set of single-particle Schrödinger equations 1 2 V [ n; r ] ε (2.9) 2 + eff Ψ i = iψi where the potential energy V eff (r) is given by δ Exc Veff () r = VHartree() r + Vxc[;] n r = VHartree() r + δ n() r (2.10) together with 18

27 r N = Ψi 2 (2.11) i= 1 n() where the sum runs over the lowest N eigenvalues (that is, the occupied states). It is important to note that the Ψ i s from (2.9) and (2.11) do not correspond to the wavefunctions of actual electrons but of non-interacting particles. However, the ground state density of these effective particles matches the electronic one. While, of course, in any practical implementation of this procedure E must be approximated, there is no need to make an approximation in the single-particle kinetic energy functional. The price to be paid is that we have to go back to solving single particle Schrödinger equations with the above potential energy, V. It is then easy to show that the total energy is given by the following expression[36]: 1 n( r) n( r') E = εi d rd r' Vxc[ n; ] n( ) d r+ Exc[ n] 2 ' r r r r i= 1 (2.12) which is a very convenient tool for practical computations. In principle, if the exact, E xc [n] were used in these equations, the resulting selfconsistent density n(r) and energy, E, would be exact, including all many body effects. Of course, in practical calculations one must content oneself with an approximation for E xc and consequently the quality of the resulting calculation rests on how well this energy is approximated. The simplest and very practical approximation which has become a benchmark against which all others are compared is the local density approximation, LDA. In this approximation the non-uniform system is treated as a certain set of small boxes each 19

28 containing a uniform interacting electron gas. The total exchange correlation energy is the sum of the contributions of all boxes: LDA 3 Exc [ n( )] = ε xc[ n( )] n( ) d r r r r (2.13) where ε xc is the well known exchange-correlation energy per electron of a uniform electron gas of density n. The corresponding V xc is, by (2.10), given by ε xc Vxc = n() r + ε xc[()] n n r. (2.14) r n= n( ) The LDA is, by definition, exact for a homogeneous system, and arbitrarily accurate for a system of sufficiently slowly varying density. 2.2 Linear Response Theory In addition to MD simulations and ground state energy calculations, many phonon calculations were performed throughout the present research. These calculations were performed with a DFT code based on Linear Response Theory (LRT). This allows for the calculation of the eigenfrequencies and eigenvectors of all 3n normal modes (where n is the number of atoms) of the supercell via diagonalization of the dynamical matrix. The force constants, and subsequently the dynamical matrix, are calculated within the harmonic approximation without recourse to the physical displacement of any atoms. This is different from the common method, in which an atom is displaced from equilibrium by some small, and arbitrary, distance. Such treatments are not fully consistent in that the calculations of force constants imply the harmonic approximation, but the finite displacements of atoms involve anharmonic contributions to δe. 20

29 Furthermore, the resulting frequencies are dependant on how large such displacements are chosen to be, which explains why many authors predict much more accurately LVMs that have been measured than those yet unobserved. The application of LRT to DFT within the present context enables one to obtain the dynamical matrix analytically from gradients of the density relative to atomic displacement. This is achieved by solving the first order perturbation expansion of the Schrödinger equation[37], δ Hˆψ + Hˆδψ = δεψ + ε δψ. (2.15) 0, i i i 0, i 0, i i Expanding the ground state wave function, ψ 0,i on the μ th ion yields, in terms of the atomic orbitals centered ψ () r = [ c ϕ ( r R ) + c ϕ ( r R )]. (2.16) α i α iμ μ μ iμ α μ μ μ The change in the coefficients, c α iμ, are obtained from (2.15) thereby allowing for the computation of αψ and hence the change in the electronic density, occ αρ αψψ i i ψi αψi i= 1 () r = [ + ]. (2.17) This allows for the computation of the dynamical matrix, by an explicit derivation of the forces on all the atoms in the system due to an infinitesimal displacement of one of them, 2 1/2 α β D αβ = = α F β Rα Rβ ( M M ) E. (2.18) 21

30 2.3 SIESTA Our results are obtained using the SIESTA [38,39] implementation of MD simulations based on local DFT. The exchange-correlation potential of Ceperley and Alder [40] was parameterized by Perdew and Zunger [41]. Norm-conserving pseudopotentials in the Kleinman-Bylander form [42] are used to remove the core regions from the calculations. The basis sets for the valence states are linear combinations of (numerical) atomic orbitals of the Sankey type [43,44,45], generalized to be arbitrarily complete with the inclusion of multiple-zeta orbitals and polarization states. The n th zeta orbital is given by the product of a numerical radial function which is strictly zero beyond some cutoff radius, r c, and the corresponding spherical harmonic. Here, the number of zeta orbitals correspond to the number of orbitals in the basis of the same angular momentum state but different radial cutoffs. These higher-zeta orbitals are constructed by a split-valence method where the continuity and smoothly varying nature of the function is enforced at the split radius. [46] In the present calculations, we use double-zeta (DZ) basis sets. The charge density is projected on a real-space grid with equivalent cutoffs of 80Ry to calculate the exchange-correlation and Hartree potentials. The eigenvectors and eigenvalues of the dynamical matrix are calculated using linear-response theory [47,48,49]. While using pseudopotentials are not strictly necessary for DFT based selfconsistent MD simulations, they greatly reduce the amount of computation required, thereby allowing for calculations in much larger supercells. This becomes almost 22

31 mandatory in the study of defects where it is necessary to use rather large supercells to minimize defect self-interaction induced by the periodic boundary conditions. 2.4 Vibrational lifetimes There are several issues specific to the calculation of vibrational lifetimes that require special consideration. Finite temperature MD runs are typically performed with the use of a thermostat. The equations of motion are altered such that the resulting system maintains thermal equilibrium at a specified temperature. While with a proper choice of initial conditions, one could excite an LVM of interest and observe its decay to thermal equilibrium; the rate at which this occurs would be a function of the details of the thermostat chosen and would not nessesarily be indicative of the relaxation rate in the real crystal. In this work, no thermostat is used, so that the other modes of the crystal act as the thermal bath into which the initial excitation decays. We work in the microconanical ensemble, where the temperature changes but the energy is fixed. In the decay process the energy of the initial excitation is redistributed to the bulk modes of the supercell leading to small overall increase in the temperature of the supercell ~15K. Our calculations of vibrational lifetimes begin with the dynamical matrices of the supercells containing the defects. The eigenvectors of this matrix are used to fix the equilibrium background temperature of the cell and set the initial excitation of the LVM of interest. Then, classical MD simluations are performed with a time step of 0.3fs and the 3N Cartesian coordinates of the N atoms in the supercell are written as linear 23

32 combinations of the 3N normal-mode coordinates of the cell thus allowing us to monitor the energy of each normal mode as a function of time at the chosen temperature. The calculation of the dynamical matrix plays a crucial role. The eigenvectors of this matrix are used to fix the background temperature of the cell and set the initial excitation of the LVM of interest. In order to set the background temperature, the energies and phases of the bulk modes of the system are chosen such that the supercell is in thermal equilibrium. The energy of each normal mode is randomly picked from a Maxwell-Boltzmann distribution about k B T, E s β E βe de = ξs Es = kbtln(1 ξs), (19) 0 where E s is the energy of the s th normal mode and ξ s is a random number between 0 and 1. Additionally, the phases of each normal mode are randomized ( 0 ϕ < 2π ), so that the initial conditions in terms of the normal coordinates become, s q s q = 2kT B ln(1 ξs) cos( ϕs ) ω s = 2k Tln(1 ξ ) sin( ϕ ) s B s s (20) These are then resolved in Cartesian coordinates in the standard way[50], u 1 s αi qseα i mα s =, (21) where u α i is the displacement of the α th atom in the i th direction and of the s th mode. Thus, the initial displacements and velocities are given by s e is the eigenvector 24

33 2kT ln(1 ξs ) B uα = cos( ω t+ ϕ ) e m ω u t s i s s αi α s s αi 2kT B = ln(1 ξ ) sin( ω t+ ϕ ) e m α s s s s s αi (22) This procedure is similar to that described in Ref.[51]. Instead of randomly assigning the energy of the LVM of interest, this mode is initially given the an energy corresponding to the first excited vibrational state,3 ω /2. Since the initial conditions in eqn (22) assume harmonicity, the greater the excitation the poorer the accuracy. While this error is small for the modes which have only the energy associated with thermal equilibrium, it can become quite large for the mode which is initially excited. For this reason, the initial energy of the excited mode is chosen to be entirely kinetic. This choice insures that the excited mode receives the correct amount of energy even if we are well within the anharmonic region of the potential. This method not only allows us to independently choose a background temperature and excitation energy, but it also provides very good thermalization without the use of a thermostat. Figure 8 shows tests conducted on pure Si in a 64 atom supercell, at 50, 200, and 500K. The black curves are the temperature of the cell throughout a 10ps MD simulation with the initial conditions outlined above, but without any thermostat. The green curves show the temperature throughout the simulation using the standard method of assigning kinetic energies to each atom from a Maxwell-Boltzman distribution and using a Nose-Hoover thermostat[52,53]. 25

34 Figure 8: 10ps MD simulation of Si 64 at T=50K, 200K, and 500K with a Nose-Hoover thermostat (green) and the procedure outlined above (black). Allowing the other normal modes of the system to act as a heat bath has other advantages as well. Most notably, it is possible to monitor to which bulk modes the energy of the excited mode goes. By inverting equation (21) the 3N Cartesian coordinates describing the positions of all the atoms of the system can be cast in terms of normal coordinates, q = m u e. (2.23) ( s) s k kα kα s Once the normal coordinates of the system are known for each time step, the energy of each normal mode can be calculated. In the harmonic approximation, the energy in each normal mode is given by the familiar harmonic oscillator energy equation, 26

35 1 ( s = s +ωs s ) E q q. (2.24) 2 While this expression for the energy is satisfactory for modes with low energy, modes with a significant amount of energy (like the excited LVM or sometimes the receiving modes) often extend into the anharmonic region of the potential and are subsequently not well described by (2.24). This leads to large fluctuations in the calculated energy; and these fluctuations have the same frequency as the mode whose energy is being calculated. In order to overcome this difficulty, the energy of each normal mode is only calculated twice per period, when the oscillation amplitude crosses zero. This not only eliminates error based on the calculation of ω s but when q s =0 even higher order terms like qubic or quatric vanish from the energy. 27

36 CHAPTER IV RESULTS 3.1 H 2 * and isotopes The structure of the H 2 * defect is shown in Figure 9. It contains one bondcentered H (H BC ) and one anti-bonding H (H AB ), aligned along the same trigonal axis. The first step in calculating the vibrational lifetimes is calculating the dynamical matrices. This was done for each isotopic combination: H 2 *, D 2 * (D at both sites), HD* (H at the AB site and D at the BC site, and DH* (D at the AB site and H at the BC site). H AB H BC Figure 9: Structure of the H 2 * defect Table 2 shows the calculated frequencies of the of the H(D) LVMs and their observed values (where applicable). The frequencies agree well with experiment, with an error of about 3% or less. In addition to stretch and wag modes, the presence of the 28

37 impurity can introduce pseudo-localized modes. Pseudolocal modes(plvms) are modes that while localized at the defect, have a frequency which is below the Γ phonon of the crystal(~525cm -1 for Si) and are therefore in the phonon continuum of the crystal. The Si- H BC wag is an example of such a mode. Si-H BC wag Si-H AB wag Si-H AB stretch Si-H BC stretch Calc. Expr. Calc. Expr. Calc. Expr. Calc. Expr. H 2 * [54] [3] D 2 * HD* DH* Table 2: Calculated and observed frequencies of LVMs of H 2 * isotopes In order to identify the plvms, the eigenvectors of the dynamical matrix were analyzed. The orthonormal eigenvectors of the dynamical matrix give the relative displacement of each atom in the supercell for a given normal mode. We have used the sum of the magnitudes of the relative displacement of the atoms constituting the defect has been used to define a degree of localization.[55] Figure 10 shows the localization about the two H(D)s in black and the localization about the two Si s to which they are bonded in red. In this figure the LVMs show up as almost entirely localized, nearly 1. Representations like this make it easier to see which modes may play crucial roles in the decay process as well as quickly showing the differences the isotopic substitutions have on the possible decay channels. 29

38 Figure 10: Localization Plots of Σα,i(esα,i)2 (where α=h s shown in black and the two Si atoms bound to them shown in red) vs. frequency The vibrational lifetime of the Si-H/D BC stretch mode of each isotopic combination were calculated at various temperatures (50K, 100K, and 150K for H 2 * and 100K and 150K with D isotopes). Due to the sensitivity of the decay process on the randomly chosen initial conditions multiple MD runs were performed for each isotope at each temperature, 6 for H 2 *, 3 for D 2 *, 4 for HD*, and 4 for DH*. More MD simulations were performed for H 2 * than the other isotopes because the sensitivity of the decay to the initial conditions is most pronounced for the shortest lived decay processes. To illustrate this, Figure 11 shows the energy of the Si-H BC stretch mode as a function of time for the six different MD runs performed for H 2 * at 100K. Although each realization of the decay is far from exponential, when the energies of the various 30

39 runs are averaged at each time, as shown in Figure 12, the resulting curve can be nicely fit to an exponential, and for this case the exponential fit gives a decay constant of 2.6ps Figure 11: Six different realizations (different initial conditions) of the decay of H 2 * at 100K 31

40 Figure 12: Exponential fit to the average of the six decays of H 2 * at 100K, yielding a lifetime of 2.6ps. In contrast, the longer lived D 2 *, shown in Figure 13, reveals very similar decays for each of the three different numerical realizations. Figure 13: 3 decays of D 2 * at 100K 32

41 Figure 14: Average of the three caluclated decays of D 2 * at 100K, giving a fit of 18ps The results of the lifetime calculations for each istotope are presented in Table 3. While the lifetimes of both H 2 * and D 2 * have been observed, only the lifetime of H 2 * has been measured by transient bleaching spectroscopy. For H 2 * our calculated lifetime of 3.7ps at 50K, agrees reasonably well with the low temperature experimental value. Additionally, the calculated temperature dependence of H 2 * agrees reasonably well with the measured one, Figure 15 shows our calculated results (red triangles) and the experimental data (solid points). Defect 50K 100K 150K Expr. (low T) H 2 * 3.7ps 2.6ps 1.8ps 4.2*ps [3] D 2 * 18ps 14ps <4.8ps [27] HD* 38ps 12.8ps -- DH* 24ps 11.6ps -- Table 3: Calculated lifetimes of the different isotopes of H 2 * for several temperatures 33

42 Since the lifetime of D 2 * was measured via linewidth measurements, the cited lifetime constitutes a lower bound. Although the calculated lifetime of D 2 * is unexpectedly high, it is not inconsistent with the experimental data. Nevertheless, we believe this to be an error in the calculation due to the finite size of the supercell, this will be discussed at the end of this section. Figure 15: Temperature dependence of H 2 *, calculated (red triangles) and experimental [3] (solid points) For the case of H 2 *, the primary decay channel could be unambiguously determined. Figure 16 shows a representative decay of the Si-H BC stretch mode(labeled A) of H 2 *, where it can be seen that one of the primary accepting modes is the Si-H AB stretch(labeled B). 34

43 Figure 16: Representitive decay of H 2 * at 100K. a) H BC stretch at 2126cm -1 b)h AB stretch at 1860cm -1 c) and d) H AB wags at 853cm -1 While later one can see the H AB wag modes pick up energy (labeled C and D), this only happens after the H AB stretch mode picks up energy. This indicates that while the H AB stretch mode is a receiving mode of the H BC stretch decay, the H AB wag modes are receiving modes of the secondary decay of the H AB stretch and not part of the H BC stretch decay. In order to conserve energy, the frequency difference between the H BC stretch and H AB stretch of 266cm -1 (2126cm cm -1 ) must be made up by another phonon or group of phonons. Further analysis reveals that, indeed a mode near this frequency, at 271cm -1, picks up energy along with the mode at 1860cm -1. Since the 271cm -1 has a lower frequency it receives less energy than the 1860 mode, but if each mode is divided 35

44 by it s frequency they show up on the same scale. Figure 17 shows the 1860cm -1 mode (red curve) and the 271cm -1 mode (green curve) scaled by their frequencies. Figure 17: Scaled energies of the H AB stretch at 1860cm -1 (red curve) and the symmetric stretch of the Si about the H BC at 272cm -1 (green curve). These calculations strongly suggest that the decay of the Si-H BC stretch mode of H 2 * is not a six phonon process as has been suggested by other investigators, but is a two phonon process where the Si-H BC stretch mode decays to the Si-H AB stretch mode and the symmetric stretch of the two Si about H BC. We also note that as a two phonon process the short decay of H 2 * fits on the frequency-gap law curve given in Figure 4. One might expect for D 2 * to decay quickly via a two phonon process 1528cm cm cm -1, in much the same way as H 2 *. Unfortunatly, the finite size of the supercell leaves a large gap in the phonon density of states (there are no modes between 165 and 210cm -1 ) and the would be receiving mode at 190cm -1 falls almost directly in the middle of this gap. 36

45 3.2 H + BC and D + BC Bond-centered hydrogen(h BC ) exists is the 0 and + charge state and is the lowest energy configuration for isolated H in Si. While H 0 BC is seen by electron paramagnetic resonance, only H + BC is seen by FTIR.[56] The reason is that the first excited vibrational state of H 0 BC is in the conduction band and H 0 BC self-ionizes when exposed to IR radiation at the appropriate frequency. The structure of the defect is shown in Figure 18. The calculated frequency of the asymmetric stretch of H + BC (measured: 1998cm -1 )[2] and D + BC are 2014cm -1 and 1439cm -1, respectively. Figure 18: Structure of the H + BC defect The localization of H + BC is shown in Figure 19(top), from which it can be seen that H BC has no LVMs other than the asymmetric stretch at 2014 cm -1. Therefore, the LVM decay cannot involve fewer than four phonons and its lifetime does not fit on the frequency-gap law. The modes associated with H + BC are the asymmetric stretch mode at 37

46 2014 cm -1, the two wag modes at 261 cm -1, and the modes at 209cm -1 and 410 cm -1 associated with the two Si neighbors of H. Except for the LVM at 2014 cm -1, all are pseudolocal modes [27] (plvms) which readily couple to bulk phonons. Similarly, for D BC only the 1439cm -1 asymmetric stretch is an LVM, while the wag modes shift down to 203cm -1 ; the Si related plvms remain virtually unchanged and are at 209cm -1 and 409cm -1. Figure 19: Locatization of vibrational modes of H+ BC (top) and D+ BC (bottom) The vibrational lifetimes of the asymmetric stretch of H + BC were calculated as 8.7ps at 75K, 5.4ps at 100K, and 2.8ps at 150K. These values are close to the observed temperature dependence[2] and are shown in Figure 20. D + BC lifetimes are similar to H + BC and found to be 3.9ps at 100K and 2.4ps at 150K. 38

47 Figure 20: Triangles (circles) show the calculated (measured[2]) lifetimes of H + BC As T 0, the classical oscillation amplitudes vanish, instead of approaching the zero-point amplitudes commensurate with the quantum mechanical ground state. In the real crystal, the amplitudes, anharmonic couplings, and, therefore, lifetimes become constant. In classical MD simulations, the amplitudes of the receiving modes and, therefore, the anharmonic couplings go to zero and the lifetimes become very long. We 39

48 attempted to calculate the decay of H BC at 0 K and indeed failed to observe a decay. The constant lifetimes observed below 40 K are zero point oscillation effects which cannot be reproduced in our calculations. The energies of all the normal modes of the supercell during a realization of the decay of H BC are shown in Figure 21. The modes whose energies peak sharply and repeatedly throughout the simulation are the 410 cm -1 (B) and 261 cm -1 (C and D) modes. We performed four MD simulations, each at T =50, 100, and 150 K. The decay process is always very much the same. The energy in the asymmetric stretch is absorbed by the 410 and 209 cm -1 modes, which themselves decay almost immediately into bulk phonon modes. This is apparent when blowing up the inset in Fig. 3 (the inset contains 12,000 time steps) to uncover the inner structure of the B and C peaks. 40

49 Figure 21: Energies of all the normal modes in the supercell vs. time during the decay of the 2014cm -1 mode of H + BC at T=75K. The first 3.5ps are enlarged in the inset (see text). 3.3 VH i HV and isotopes The divacancy-dihydrogen complex was the most computationally challenging defect. The low temperature vibrational lifetime has been observed[3] to be ~300ps. Given the time step of our MD simulations is 0.3fs, this requires of the order of a million MD time steps. Due to the computationally intensive nature of the calculation, only one run was performed for each isotope; we expect little variation of the calculated lifetimes for different runs of very long lived decays. The calculated lowest energy structure of the defect is shown in Figure 22. It has C 2v (orthorhombic) symmetry and the Si bonds 41

50 reconstruct perfectly. The defect is of particular importance because it is believed to have similar lifetimes to Si-H surface defects.[3] The calculated LVMs are a doublet of stretch modes at 2092cm -1 and a quadruplet of wag modes at 597cm -1. Figure 22: VH i HV defect, H's are shown in red the yellow mark the sites with Si missing 42

51 Since the excited mode is degenerate, energy readily exchanges between the two stretch modes at 2092cm -1, occurring on a time scale much shorter than the decay process. Figure 23 shows the decay of VH i HV at 200K, the energies of the two stretch modes are shown in green and red. The inset is a blow up of a 5ps region, where it can be seen that energy is rapidly and periodically exchanged between the two stretch modes with a period of about 500fs. Figure 23: Decay of VH i HV at 200K, the red and green curves are the energies of the degenerate stretch modes, the black curve is the sum of these two curves The black curve is the sum of the energies of the two stretch modes, and is indicative of the decay process. Figure 24 shows the sum of the energies of the two degenerate stretch 43

52 modes for the decay of VH i HV at 200K, which yields a lifetime of 187ps; this is quite close to the measured value at this temperature of 210ps.[3] Figure 24: Decay of the stretch modes of VH i HV at 200K Analysis of the energy of the other modes in the system did not give any indication of the decay channels. Finally, the VH i HV defect has no LVMs or plvms it can couple to. Coupling to the four wag modes, just above the Γ phonon, would involve a cumbersome 3.5 -phonon process. The decay of the stretch mode occurs by coupling to multiple bulk phonons, as discussed in the experimental work, and fits on the frequencygap law. The primary reason for this is that the decay is so slow that the other modes of the system essentially remain in equilibrium throughout the entire decay process. 44

53 Figure 25: Vibrational spectra of VHHV(top), VDHV(middle), and VDDV(bottom) In addition to VH i HV, the lifetimes of VD i HV and VD i DV were calculated for T=200K. The calculated lifetimes are 117ps and 106ps, respectively. While no experimental data exists for VD i HV, the indirectly observed low temperature lifetime of VD i DV is 90ps. 3.4 IH 2 and ID 2 The IH 2 and ID 2 complexes were investigated because of experimental evidence of a strange isotope effect. Namely, the low temperature lifetimes of IH 2 and ID 2 have been indirectly observed to be, 11ps and 18ps, respectively. So, with a frequency drop of more than 500cm -1, the lifetime nearly doubles. The structure of the defect is shown in Figure 26, and consitists of an interstitial Si and 2H s. 45

54 Figure 26: IH 2 defect The lifetime of the highest frequency stretch of IH 2 and ID 2 were calculated for T=125K, yielding lifetimes of 9.2ps and 29ps, respectively. While the IH 2 lifetime is very close to the experimental data, ID 2 is slightly larger, but shift in the correct direction. Figure 27: The sum of the squares of the magnitude of the eigenvectors on the two H/D(black) and the two nearest neighbor Si(red). 46

55 The observed and calculated isotope effect can be explained by an analysis of the accepting modes for this decay. Figure 28 shows a representative decay of IH 2, from which one can see several modes strongly pick up energy. Figure 28: Energies of all the normal modes in the supercell vs. time during a realization of the decay of the 2065cm -1 stretch mode of IH 2 at T=125K. These modes are all LVMs and consist of the wag modes at 736cm -1, further wag modes at 702cm -1, and two modes at 582cm -1 and 592cm -1 which are both localized to the two Si to which the H s are bonded. The extent to which these are the accepting modes can be seen in Figure 29, where the green curve is the excited stretch mode, the red curve is the sum of the six aforementioned modes, and the black curve is the sum of the red and green curve. The fact that the black curve is almost completely constant until well after 47

56 the stretch mode has decayed indicates that nearly all of the energy from the decay goes into these six modes. Figure 29: The decay of the high frequency stretch of IH 2 (a), along with the sum of the energies of the proposed accepting modes (b), (c)=(a)+(b) Furthermore, these six modes are actually a triplet of doublets whose frequencies nearly sum to that of the 2065cm -1 mode, this allows us to identify the decay as a three phonon process where 2065(stretch) 736(wag) + 702(wag) + 587(Si related LVM). This helps explain why ID 2 has a longer lifetime, although it has a lower frequency. The ID 2 stretch frequency drops to 1481cm -1 and the wag modes drop below the gamma phonon and become delocalized. The only LVM for ID 2 to couple to is the Si related one at 587cm -1, but it can not exclusively decay into this mode and conserve energy since it 48

57 would require 2.5 phonons. Therefore, ID 2 must couple to bulk modes whereas IH 2 couples exclusively to localized modes. 3.5 Intersitial Oxygen and isotopes In addition to the hydrogen related defects, the low temperature lifetimes of a number of isotopes of interstitial oxygen have recently been observed by transient bleaching spectroscopy[28,29]. The observed lifetimes are given in column 6 of Table 4, and surprisingly show that even though the structures are identical that isotopic substitutions that change the frequency by as little as 0.3% can change the lifetime by more than a factor of 2. Furthermore, 30 Si- 16 O- 28 Si has a longer lifetime than 29 Si- 16 O- 28 Si which is in turn longer than 28 Si- 17 O- 28 Si even though in this direction the 3 phonon density of states increases.[29] Interstitial oxygen is at the bond centered site and has on average D 3d symmetry. The calculated lowest energy structure, shown in Figure 30, is slightly puckered with an Si-O-Si angle of nearly six degrees. The calculated LVMs for the various isotopes are given in column 4 of Table 4. In order to calculate the lifetimes, several MD calculations were performed for each isotope: five for the shorter-lived LVMs ( 28 Si- 17 O- 28 Si and 28 Si- 16 O- 28 Si) and three for the longer-lived ones ( 28 Si- 16 O- 29 Si and 28 Si- 16 O- 30 Si). The resulting calculated lifetimes, column 7 of Table 4, are close to the measured values and change correctly with isotope. 49

58 Figure 30: Interstitial oxygen Label Species Frequency (cm -1 ) ν 3 - ν 1 Lifetime (ps) References Exp. Theory Exp. Theory ν ν : [57] ν : [58] ν ν : [59] ν τ : [28] ν ν τ : [28] ν ν : [59] ν ν ν : [59] ν ν ν : [60] ν Table 4: O i frequencies and lifetimes, theory indicates this work In this instance much insight into the physics behind the isotope shifts can be obtained by analyzing the accepting modes. Figure 31 shows a single example of the decay of the v 3 mode of 28 Si- 16 O- 28 Si in order to illustrate the decay process. As ν 3 50