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Phonon-phonon interactions: First principles theoryT. M. Gibbons, M. B. Bebek, By. Kang, C. M. Stanley, and S. K. Estreicher Citation: Journal of Applied Physics 118, 085103 (2015); doi: 10.1063/1.4929452 View online: http://dx.doi.org/10.1063/1.4929452 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/118/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Electron-phonon coupling and thermal conductance at a metal-semiconductor interface: First-principles analysis J. Appl. Phys. 117, 134502 (2015); 10.1063/1.4916729 Isobaric first-principles molecular dynamics of liquid water with nonlocal van der Waals interactions J. Chem. Phys. 142, 034501 (2015); 10.1063/1.4905333 Phonon and thermal transport properties of the misfit-layered oxide thermoelectric Ca3Co4O9 from firstprinciples Appl. Phys. Lett. 104, 251910 (2014); 10.1063/1.4885389 Chirality dependence of quantum thermal transport in carbon nanotubes at low temperatures: A first-principlesstudy J. Chem. Phys. 139, 044711 (2013); 10.1063/1.4816476 Multiphonon hole trapping from first principles J. Vac. Sci. Technol. B 29, 01A201 (2011); 10.1116/1.3533269

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Phonon-phonon interactions: First principles theory

T. M. Gibbons, M. B. Bebek, By. Kang, C. M. Stanley, and S. K. Estreichera)

Physics Department, Texas Tech University, Lubbock, Texas 79409-1051, USA

(Received 24 June 2015; accepted 11 August 2015; published online 28 August 2015)

We present the details of a method to perform molecular-dynamics (MD) simulations without ther-

mostat and with very small temperature fluctuations 6DT starting with MD step 1. It involves pre-

paring the supercell at the time t¼ 0 in physically correct microstates using the eigenvectors of the

dynamical matrix. Each initial microstate corresponds to a different distribution of kinetic and poten-

tial energies for each vibrational mode (the total energy of each microstate is the same). Averaging

the MD runs over many initial microstates further reduces DT. The electronic states are obtained

using first-principles theory (density-functional theory in periodic supercells). Three applications are

discussed: the lifetime and decay of vibrational excitations, the isotope dependence of thermal con-

ductivities, and the flow of heat at an interface. VC 2015 AIP Publishing LLC.

[http://dx.doi.org/10.1063/1.4929452]

I. INTRODUCTION

Phonon-phonon interactions (the coupling between nor-

mal vibrational modes) play a central role in an atomic-level

understanding of heat flow and of the interactions between

thermal phonons and defects. Since low-frequency phonons

are often involved, excellent temperature control is required

starting with the first molecular-dynamics (MD) time step.

Indeed, large temperature fluctuations obscure (at least some

of) the interactions. Furthermore, no thermostat can be used

since the modes couple faster to it than to each other.

Finally, when defects are involved, first-principle methods

are needed. If the system contains N atoms, there are 3N nor-

mal modes (including the 6 translational and rotational

modes). In a macroscopic sample, N is far too large for a

detailed analysis but in nanostructures or—for theorists—in

periodic supercells, the details of phonon-phonon interac-

tions can be studied.

In this paper, we describe the details of a method to per-

form such non-equilibrium MD simulations. The tempera-

ture fluctuations are very small starting with MD time step 1

and remain almost perfectly constant with time. The energies

and amplitudes of all the normal vibrational modes can be

monitored as a function of time as they couple to each other.

The method involves preparing the system at the time t¼ 0

in a random distribution of kinetic and potential energies for

all the modes (initial microstate). The temperature fluctua-

tions are small and can be further reduced by averaging over

n � 20–30 initial microstates. This approach requires sub-

stantial amounts of computer time since the MD runs must

be repeated n times, but computer power increases fast and

calculations that are highly demanding today become routine

tomorrow.

The method we describe can be used in conjunction

with any electronic-structure code as long as accurate

dynamical matrices are obtained. Since the three examples

discussed here involve defects, we use first-principles theory:

density-functional theory (DFT) for the electronic-structure

in 3D- or 1D-periodic supercells.

In Sec. II, we outline the limitations of conventional

MD, which prevent the study of phonon-phonon interactions,

at least while the system is away from equilibrium or from a

steady state situation. In Sec. III, we describe how to prepare

the system in an initial microstate. In Sec. IV, we discuss the

MD simulations and the averaging over initial microstates.

Section V contains three examples of vibrational coupling,

which require a high degree of temperature control. In

Sec. VI, we comment on the limitations and strengths of the

method.

II. LIMITATIONS OF CONVENTIONALNON-EQUILIBRIUM MD SIMULATIONS

Conventional MD simulations begin (time t¼ 0) with all

the nuclei in their equilibrium configuration and a Maxwell-

Boltzmann distribution of nuclear velocities corresponding

to the temperature T0. Although this does produce the

desired temperature for the system, all the normal modes

start with zero potential energy. This is an unphysical situa-

tion (minimum entropy) since all the 3N normal modes of

the system start exactly in phase. As the MD run begins, all

the modes lose kinetic and gain potential energy simultane-

ously, and the temperature drops precipitously. The tempera-

ture fluctuates widely as the MD run proceeds, with

fluctuations 6DT initially comparable to T0. Reducing and

stabilizing DT requires a thermostat.1 The energy of the sys-

tem is controlled by the thermostat, and the system is not

truly microcanonical.

Thus, there are two reasons why conventional MD simu-

lations do not allow the study of the interactions between

normal vibrational modes. First, the large temperature fluctu-

ations—especially in the beginning of the simulation—hide

many single-mode excitations. Second, the excited modes

couple to the thermostat much faster (every few time steps)

than to each other. Conventional MD simulations are often

used to study a system after it has reached the steady-statea)[email protected]

0021-8979/2015/118(8)/085103/8/$30.00 VC 2015 AIP Publishing LLC118, 085103-1

JOURNAL OF APPLIED PHYSICS 118, 085103 (2015)

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(after thousands of time steps) rather than during the non-

equilibrium phase itself (from t¼ 0 to the steady-state).

III. SUPERCELL PREPARATION (TIME t 5 0)

A physically correct initial microstate of the system has

a random distribution of mode phases, that is a random dis-

tribution of initial kinetic and potential energies for all the

vibrational modes, corresponding to the desired total energy.

Each such distribution is just one of an infinite number of

initial microstates, all corresponding to the same macrostate.

Thus, in order to perform MD simulations at the temper-

ature T0, the velocities va and positions ra of all the nuclei ain the system corresponding to a specific microstate must be

known at the time t¼ 0. These initial velocities and positions

can be obtained from the dynamical matrix of the system.

In order to prepare the system away from equilibrium,

the first step is to prepare the entire cell in thermal equilib-

rium at T0 (Sec. III A). And then, a part of the supercell is

prepared at a different temperature T1 (Sec. III B). This

“part” may be a small region of the supercell (if a tempera-

ture gradient is desired) or simply the addition of a phonon

to a specific vibrational mode (to mimic the thermal or opti-

cal excitation of a specific mode).

A. Preparation in thermal equilibrium

This preparation involves a combination of normal

vibrational modes with random phases (i.e., random distribu-

tions of kinetic and potential energies) and a distribution of

mode energies such that the average energy per mode is

kBT, where kB is the Boltzmann constant.

Supercell preparation was first developed2,3 to calculate

vibrational lifetimes using DFT for the electronic structure

within the SIESTA package.4,5 This is still our basic approach

today. However, supercell preparation can be implemented

using any level of theory for the electronic structure (empirical

or any first-principles software package) as long as an accurate

dynamical matrix can be obtained. This matrix should have no

negative eigenvalues, implying that the system is at a true min-

imum of the potential-energy surface. Therefore, the dynami-

cal matrix is calculated at 0 K, and its eigenvalues and

eigenvectors are only approximate when T> 0.

The dimensions of the supercell must be optimized for

each charge state, basis set, and exchange-correlation poten-

tial. If the nuclear coordinates are written in units of the lattice

constant(s), the cell can be optimized with great accuracy

(maximum force component smaller than 10�5 or 10�6 eV/A).

When the nuclear coordinates are in A or aB, the optimization

is less accurate (�10�3 eV/A). When a defect is included in

the optimized cell, a conjugate gradient algorithm with a small

force tolerance is used. We normally want the maximum force

component to be smaller than 0.003 eV/A.

The next step is to calculate the dynamical matrix of the

system. The eigenvalues and eigenvectors of this matrix are

the square of the normal-mode frequencies xs2 and the rela-

tive displacements esai (i¼ x, y, z) of the nuclei a¼ 1…N for

each mode s, respectively. The 3N eigenvectors are ortho-

normal. If the system is at a minimum of the 3N-dimensional

potential-energy surface, all the eigenvalues are positive,

except for the three translational modes with x¼ 0 (there are

no rotational degrees of freedom if periodic boundary condi-

tions are applied). In practice, it is not unusual to have a few

negative eigenvalues. Additional geometry optimizations are

then needed to eliminate them. This is achieved6 by nudging

the atoms associated with the imaginary modes in the direc-

tion of the corresponding eigenvector of the dynamical ma-

trix. A new conjugate gradient is then performed and the

dynamical matrix is calculated. A few iterations usually suf-

fice to eliminate the imaginary modes or reduce their number

to just one or two (their amplitudes are set to zero in the be-

ginning of the MD run).

Once the dynamical matrix is obtained, it is used to cal-

culate the initial positions and velocities of all the nuclei cor-

responding to thermal equilibrium at the temperature T0.

Since the normal-mode frequencies slowly shift as the tem-

perature increases and therefore the eigenvectors also

change, the supercell preparation is strictly valid only at

T0¼ 0. However, it remains a very good approximation up

to at least room temperature. We verified that the magnitude

of the temperature fluctuations during MD runs at 125 K

remains almost perfectly constant for hundreds of thousands

of time steps,3 starting with MD step 1. This shows that the

system started very close to thermal equilibrium at t¼ 0.

The Cartesian coordinates rsai¼ (1/�ma)qse

sai of nucleus

a in the mode s are related to the eigenvectors esai via

qs(T0,t), the normal-mode coordinate. The qs’s are not known

but, up to moderate temperatures, they are approximately

harmonic: qs¼As(T0)cos(xstþus). This introduces a ran-

dom distribution of phases us in the range [0,2p), which

determines the potential/kinetic energy ratio of each mode s

at the time t¼ 0. Note that the harmonic assumption is onlyused to convert Cartesian-to-normal-mode coordinates and

vice versa. The MD runs involve all the anharmonic terms

since the force on each nucleus at each time step is obtained

from the total energy via the Hellman-Feynman theorem.7

The energy of mode s is Es¼ 1=2As2xs

2. We introduce a

randomized Boltzmann distribution of energies bexp{�bEs},

where b¼ 1/kBT0, which produces an average energy per

mode equal to kBT0. Using the inverse-transform method for

generating distributions, we write the cumulative distribution

function fs ¼Ð Es

0be�bEdE and invert it to get Es¼�kBT0

ln(1�fs), where fs is a random number in the interval [0,1].

Equating this to the energy, we get As2(T0)¼�2kBT0

ln(1�fs)/xs2.

A random distribution of mode phases (us) and energies

(via fs) produces the normal-mode amplitudes As, from

which we obtain8 the nuclear coordinates rsai and velocities

vsai of the microstate at t¼ 0. The result is a supercell very

close to thermal equilibrium at the temperature T0 with a

random distribution of kinetic and potential energies for the

normal modes. The subsequent MD simulations, performed

without thermostat, exhibit small and time-independent tem-

perature fluctuations. No thermalization run is needed.

B. Preparation away from thermal equilibrium

Once the cell is in thermal equilibrium at T0, one can

easily drive it away from equilibrium. We discuss two

085103-2 Gibbons et al. J. Appl. Phys. 118, 085103 (2015)

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possibilities: exciting a single normal mode above the back-

ground temperature and setting up an initial temperature

gradient.

In order to calculate the vibrational lifetime of a phonon

with frequency xs (with �hxs> kBT0), the supercell is pre-

pared at T0 except for the vibrational mode s, which is

assigned the potential energy 3�hxs/2 (zero-point energy plus

one phonon) using the appropriate eigenvector of the dynam-

ical matrix. This produces a classical oscillator with the

same initial amplitude as the quantum-mechanical one and

therefore the same anharmonic coupling. The added potential

energy increases the temperature of the cell since 3NkBTcell

¼ (3N� 1)kBT0þ 3�hxs/2, where N is the number of atoms

in the cell. In Refs. 2 and 3, the temperature at which the life-

times were calculated was reported to be T0 instead of Tcell,

a shift of �20 K in Si64 if xs� 2000 cm�1. The energy of the

excited mode is monitored during the MD run, and its decay

is fit to an exponential with time constant equal to the vibra-

tional lifetime s(Tcell). Sometimes, one (or more) receiving

mode(s) can be identified.2,3,9,10

In order to study heat flow, a small region of the cell is

prepared at a higher temperature (Thot) than the rest of the

cell (T0¼Tcold). This involves recalculating the amplitudes

of all the modes such that the temperature of the atoms in the

small region is Thot. Since the mode energies scale with As2,

this involves a factor S¼ �(Thot/Tcold) and the Cartesian posi-

tions and velocities of the nuclei in the hot region are simply:

ra ! Sra and va ! Sva. All the modes remain in phase at

the hot/cold interface.

When calculating a thermal conductivity j, the hot

region is a thin slice of an elongated supercell, and the tem-

perature vs. time is recorded in the central slice. Because of

the periodic boundary conditions, heat flows toward the cen-

tral slice from the hot slice and from the nearest image hot

slice, and the impact of the defect distribution in the entire

cell is accounted for. The temperature T(t) in the central slice

is fit to an analytic solution of the heat diffusion equation in

order to extract j in a manner similar to the experimental

laser-flash method11 (Sec. V B).

These are the two simplest examples of supercell prepa-

ration away from equilibrium. But the method is very flexi-

ble and allows more complicated configurations of hot and

cold regions to be generated at t¼ 0, corresponding to a

wide range of initial temperature profiles.

IV. MD RUNS AND AVERAGINGOVER INITIAL MICROSTATES

Once the supercell is prepared, MD runs produce fluctu-

ations DT that are typically within 4% of T0 and, more

importantly, constant with time. Figure 1 compares constant-

T MD runs at T0¼ 125 K of the Si250H40 nanowire with time

step Dt¼ 1.0 fs for 2000 time steps. The top two runs (con-

ventional MD) start with a random distribution of velocities

and the Nos�e-Hoover12 (top) or Langevin13 (middle) thermo-

stats. The bottom run uses supercell preparation (no thermo-

stat) for a single initial microstate. The two conventional MD

runs exhibit huge T fluctuations for several hundred time

steps, which then stabilize to about 69 K, corresponding here

to about 7%T0. In contrast, supercell preparation with a single

initial microstate produces smaller and time-independent

fluctuations.

Thus, supercell preparation mimics thermal equilibrium

much better than an initial distribution of velocities. But,

each initial microstate corresponds to just one of an infinite

number of possible distributions of mode phases and ener-

gies corresponding to the same macrostate. In an experimen-

tal context, the number of atoms N may be on the order of

1020 as compared to at most N� 103 for theory at the first-

principles level. Much can be gained by averaging MD runs

over many initial microstates.

Figure 2 compares the temperature fluctuations for the

same 1000 time-step MD run with supercell preparation for

FIG. 1. Constant-temperature MD runs for the Si250H40 nanowire at

T¼ 125 K (time step 1 fs) using a Maxwell-Boltzmann distribution of veloc-

ities at t¼ 0 with the Nos�e-Hoover (top, fictitious mass q¼ 100 Ryd s2) or

Langevin (middle, damping constant c¼ 100 fs�1) thermostat, compared to

a single supercell preparation run without thermostat (bottom). Note the

huge noise in the two conventional MD runs for the first 300 or 400 time

steps. There is virtually no noise with supercell preparation. The thin hori-

zontal lines show the temperature fluctuations 6DT obtained from the stand-

ard deviation (beyond 250 time steps when a thermostat was used).

FIG. 2. Total temperature T of the supercell during a non-equilibrium MD

simulation for the elongated 3D-periodic Si192 supercell prepared with a thin

slice at 180 K and the rest at 120 K. After averaging over 3000 initial micro-

states, the standard deviation DT drops to 0.15 K. No thermalization run is

needed. The total temperature (energy) of the cell oscillates around a con-

stant value. The vertical scale is different from that in Fig. 1.

085103-3 Gibbons et al. J. Appl. Phys. 118, 085103 (2015)

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a single initial microstate and after averaging over n¼ 30,

100, 1000, and 3000 initial microstates. The averaging (over

initial microstates) causes the temperature fluctuations DT to

drop to 60.91 K, 60.52 K, 60.23 K, and 60.15 K, respec-

tively. Note that what we do here differs from the averaging

sometimes performed in conventional MD runs in which

each run begins in a state with a different distribution of nu-

clear velocities, but the initial potential energy of all the

modes is always zero. In contrast, each of our initial states

involves a different random distribution of kinetic and poten-

tial energies for the modes.

Since no thermostat is used, the total energy of the

supercell remains strictly constant as expected in the micro-

canonical ensemble. Our longest run (about 240 ps, almost

106 time steps, with no averaging) involved the calculation

of a very long vibrational lifetime.3 The total energy and the

amplitude of DT remained constant with time.

Figure 3 shows DT vs. the log(n), where n is the number

of initial microstates used for averaging. DT drops to below

1%T0 after averaging over n� 30 MD runs and then only

slowly decreases for larger n. Thus, the most time-

consuming part of the calculations is not the calculation of

the dynamical matrix but the n MD runs required to average

over initial microstates. DT can be made very small indeed.

Note that the reductions in the thermal fluctuations shown

above are our best results. Typical MD runs averaged over

30 initial microstates with larger temperature gradients, at

higher temperatures, and/or with extended defects such as

interfaces produce DT in the range 1%–2%T0.

In the microcanonical ensemble,14,15 the temperature

fluctuations DT are proportional to the temperature and inver-

sely proportional to the square root of the number of atoms:

DT�T/�N. The temperature dependence is of course the

same here, but we deal with an effective N because of the

large number of initial microstates included in the calcula-

tions. The number of MD runs necessary to achieve small val-

ues of DT can be reduced by taking a number of precautions.

First, preparation at T0 refers to the average temperature

of the entire cell. This initial temperature is not necessarily

uniform throughout the cell, especially if it contains defects

that are associated with localized vibrational modes.16,17

Some initial microstates generate hot or cold spots that do not

match the physical system under study. Before starting the

MD run, we verify that T0 is approximately uniform by divid-

ing our cell into small regions and comparing the temperatures

of each region to T0. We reject the preparations that produce

temperature variations exceeding a few percent of T0. The tol-

erance depends on the geometry of the cell, the size of the

regions, and the nature of the problem investigated.

Second, random mode phases mean random distribu-

tions of potential and kinetic energies for the 3 N modes in

the system. On the average (or for very large N), the kinetic/

potential energy ratio is very close to 50/50. But for the small

number of modes considered here (a few thousand), the ki-

netic/potential energy distribution can deviate from the

desired 50/50 ratio. If a particular initial microstate generates

too much (too little) potential energy, then the temperature

of the system stabilizes at a higher (lower) temperature than

the target T0. This can occur in non-equilibrium MD runs.

We eliminate those runs by monitoring the temperature of

the cell for �100 time steps. If it increases or decreases by

more than �10% of T0, then we know that the actual kinetic/

potential ratio in the initial microstate was not close enough

to 50/50 and a new preparation is performed.

A third problem is associated with initial microstates

that result in strongly coupled modes with above-average

energies and in phase at t¼ 0 (or the opposite). In such cases,

the MD run produces unexpected behavior: an excited mode

gains energy instead of decaying, or a hot spot appears.

These occurrences are rare and do not matter if the number

of microstates is n!1. But if n� 30, such runs can have

an undesirably large weight. We eliminate those runs by

monitoring the energy of the excited mode (when calculating

lifetimes) or the temperature of the central slice (when calcu-

lating thermal conductivities) for the first 50–100 time steps.

If this energy or temperature immediately increases or

decreases by some percent of their initial energy or tempera-

ture, the run is rejected.

V. APPLICATIONS

We discuss here three applications that involve defects

(first-principles theory is desirable), a strong temperature de-

pendence (strict temperature control is required), and vibra-

tional coupling (the energies or amplitudes of individual

vibrational modes are monitored). These applications are the

lifetime and decay of vibrational excitations, the dependence

of the thermal conductivity on the isotopic composition, and

heat flow at interfaces.

A. Lifetime and decay of vibrational excitations

The temperature dependence of the vibrational lifetimes

of high-frequency local vibrational modes associated with H

(or D) impurities in Si has been measured by transient-

bleaching spectroscopy,18–20 a pump-probe infra-red (IR)

absorption technique. In these experiments, a sample initially

in thermal equilibrium at the temperature T in a cryostat is

exposed to a strong laser pulse at the precise frequency of a

Si-H (or Si-D) mode, which adds one phonon to these

FIG. 3. Temperature fluctuations DT (standard deviation) after averaging

over n initial microstates at T0¼ 125 K. DT drops to about 1% T0 for n¼ 30

and to 0.12% T0 for n¼ 3000. Only the horizontal scale is logarithmic.

085103-4 Gibbons et al. J. Appl. Phys. 118, 085103 (2015)

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oscillators. In the “bleached” state, half the oscillators are in

the first excited state and half in the ground state. After a

short time delay Dt, a much weaker probe beam at the same

frequency passes through the sample and its intensity is

measured. The absorption of the probe beam vs. Dt yields

the vibrational lifetime s(T).

Typical data show that s(T) is independent of T below

�50 or 75 K, and then drops roughly linearly with T. The ex-

istence of a temperature-independent region was unexpected.

A second feature of the data involves the large differences in

the lifetimes of vibrational modes with similar frequencies.

For example, the Si-H stretch mode frequencies of the so-

called H2* and VH�HV defects are very close to each other at

2062 and 2072 cm�1, respectively, but their low-T lifetimes

differ by almost two orders of magnitude, 4.2 and 291 ps,

respectively.18 Also, puzzling are some huge isotope effects.

For example, the low-T lifetime of the asymmetric stretch of

interstitial O in Si varies strongly with the isotope of one of

its Si nearest neighbors.9 Indeed, the isotope combinations28Si–16O–28Si, 29Si–16O–28Si, and 30Si–16O–28Si yield

almost identical frequencies (1187, 1185, and 1183 cm�1,

respectively) but very different lifetimes (11, 19, and 27 ps,

respectively). The isotope effect associated with a D substi-

tution in the CH2* defect is even larger.10 All these results

have been explained using non-equilibrium MD with our

supercell preparation.2,3,9,10

At the time t¼ 0, the supercell is prepared at the temper-

ature T0 except for the vibrational mode s, which is assigned

the potential energy 3�hxs/2. During the MD run, the energy

of all the normal modes is monitored at every time step. This

allows the vibrational lifetime of the excited mode to be cal-

culated at Tcell (Sec. III B). Sometimes, one or more of the

receiving modes can be identified, as well as the existence of

more than one decay channels.

For example, one receiving mode of the CH2* defect can

be identified.10 This defect consists of two H interstitials

trapped by substitutional carbon (Cs) in Si: one H at

the bond-centered (bc) and the other at the anti-bonding

(ab) site. The trigonal configuration of interest here is

Hab–Cs���Hbc–Si. The measured frequency and low-T life-

time of the Hab–Cs stretch mode are 2688.3 cm�1 and 2.4 ps,

respectively. The calculated frequency and lifetime of this

mode (in the Si64 periodic supercell) are 2567 cm�1 and 2.7

to 4.8 ps, respectively. Figure 4 shows the amplitudes of the

decaying Hab–Cs stretch mode and that of one of the receiv-

ing mode, the Hbc–Si stretch mode at 2183 cm�1. This is a

two-phonon decay.21 One of the receiving modes is always

Hbc–Si but the second one is a different bulk mode in MD

runs with different initial microstates.

The same method can be used to calculate the lifetime

and decay channel(s) of any normal vibrational mode in the

system. This includes low-frequency defect-related and host-

crystal (bulk) modes. Defect-related modes are localized in

space (Spatially Localized Modes or SLMs) and typically

survive for dozens or hundreds of periods of oscillation.

Bulk modes are delocalized and usually decay within less

than one period.17

Note that vibrational lifetimes cannot be calculated at

arbitrarily low temperatures because the nuclear dynamics

are classical. When T drops, the classical amplitudes become

very small, the modes become harmonic, no longer couple,

and the calculated lifetimes become infinite. In the quantum

system, the receiving modes reach their zero-point energy

state at some critical temperature Tc below which the oscilla-

tion amplitudes, anharmonic couplings, and lifetimes remain

constant. Tc is typically18–20 in the range �50 to 75 K. The

exact value depends on the frequencies of the receiving

modes. Above Tc, the individual oscillators remain quantum

objects but the IR absorption technique measures �1016 cm�3

oscillators simultaneously. The average behaves classically,

and the calculated lifetimes agree with the measured ones.2,3

B. Isotope-dependence of the thermal conductivity

It is well-known that isotopically mixed materials have

a smaller lattice thermal conductivity than the corresponding

isotopically pure ones22–24 (for reviews, see Refs. 25–27).

This effect has also been discussed at various levels of

theory.28–30 The example we discuss here involves the ab-initio theoretical laser-flash technique,31–33 which mimics

the experimental method of the same name.11 The central

point of this section is to show that the temperature fluctua-

tions are small enough to monitor changes in temperature on

the order of 1 K or less.

An elongated supercell is prepared in thermal equilib-

rium at Tcold except for a thin slice that is prepared at the

higher temperature Thot (Sec. III B). During the MD run, the

temperature T(t) is monitored in the central slice as it

increases from Tcold to Tfinal. This T(t) is fit to an analytic so-

lution to the heat transport equation T(x,t)¼Tcoldþ (Tfinal

�Tcold)Rn(�1)nexp{�n2p2at/x2}, where x is the position of

the central slice relative to the hot slice (we use n¼ 10). The

fit gives the thermal diffusivity a, from which the thermal

conductivity j¼ aqC is obtained. The density q is that of the

supercell and the specific heat C(Tfinal) is obtained from the

calculated phonon density of states.34–36 In periodic super-

cells, the nearest image hot slice causes heat to propagate to-

ward the central slice from both sides and the impact of

defects in the entire cell is accounted for. After averaging,

FIG. 4. The Hab–C stretch mode (2567 cm�1, black line) in the

Hab–C���Hbc–Si defect always decays into the nearby Hbc–Si stretch mode

(2183 cm�1, red line) plus a bulk mode near 384 cm�1. The calculations

were done at Tcell¼ 75 K and involved averaging over 30 initial microstates.

085103-5 Gibbons et al. J. Appl. Phys. 118, 085103 (2015)

129.118.29.167 On: Mon, 31 Aug 2015 18:51:25

the amplitude of the temperature fluctuations is small enough

to allow the monitoring of small temperature variations and

subtle changes in thermal conductivity.

Figure 5 shows T(t) in the central slice of the isotopi-

cally pure 28Si192 and mixed natSi192¼ 28Si17729 Si9

30Si6 3D-

periodic supercells. The isotopes are distributed randomly.

The supercell was prepared with Thot¼ 180 K and

Tcold¼ 120 K, so that Tfinal¼ 125 K. The temperature of the

central slice (where T(t) is recorded) increases from 120 to

125 K. The fit of T(t) yields j(28Si)¼ 2.1� 10�2 and j(natSi)

¼ 1.5� 10�2W/mK. These values must be compared31,32 to

nanowire37 and not bulk values.

In the case of isotopically pure semiconductors,38–40 the

net increase in the thermal conductivity of the isotopically

pure material relative to the isotopically mixed one is largest

at the temperature at which the thermal conductivity is maxi-

mum (about 200 K in Si nanowires37). The calculated 40%

increase in the thermal conductivity at 125 K is in the

expected range.

C. Heat flow at interfaces

The interactions between thermal phonons and interfaces

are an active area of research. The early descriptions were

qualitative. In the acoustic mismatch model, the coefficient of

thermal reflection and transmission at an interface are

obtained from the acoustic impedance of the two materials. In

the diffuse mismatch model, all the phonons are assumed to

undergo elastic scattering at the interface.25,26,41 Recent calcu-

lations do not include such assumptions and involve semi-

empirical non-equilibrium MD simulations25,26,42 including

wave-packet dynamics,43–45 as well as equilibrium approaches

based on the Green-Kubo formalism.46,47

Our last example involves heat flow and the SijX inter-

face, where X stands for C or Ge. The calculations are per-

formed for a d-layer (4-atomic layers thick) in the

nanowire17,49 Si225X25H40. The interface itself is a defect17

and is associated with SLMs. The initial temperature of the

nanowire is T0¼ 120 K, and all the vibrational modes up to

x0¼ kBT0/�h are thermally populated. A thin layer on the Si

side is then prepared at 520 K (in thermal equilibrium, the

system is at 140 K). Thermal phonons propagate through Si

and reach the SijX interface at the temperature T0þ dT. The

quasi-continuum of thermal phonons in the frequency range

[x0,x0þ dx] (where dx¼ kBdT/�h) resonantly excite the

interface SLMs in the same frequency range, resulting in

phonon trapping. This trapping occurs very fast (a few tens

of a picosecond) in the case of the SijGe interface since the

Si-Ge SLMs have low frequencies: these excitations involve

resonant coupling, a one-phonon process.21 In the case of the

SijC interface, the higher-frequency interface SLMs can only

be excited by much slower two-phonon processes (�8–20 ps).

This heating of the interface, described46 in terms of “stuck

modes,” has been identified as the main contribution to contact

resistance. The vibrational excitations in SLMs survive for a

few dozen periods of oscillation and then decay into lower-

frequency bulk modes. This decay depends on the availability

of receiving modes, not on the origin of the excitation.

The temperature of the d-layer vs. time is shown in

Fig. 6. In the relevant frequency range, the SijC interface has

far fewer SLMs than the SijGe one and therefore traps heat

much slower. Furthermore, Ge has many more low-

frequency receiving modes than Si, and Si many more than

C. Thus, the SijGe interface not only traps more phonons

faster but these phonons decay mostly into Ge. On the other

hand, few phonons trap at the SijC interface in the first few

ps, since one must wait some 8–20 ps for two-phonon proc-

esses to kick in and excite higher-frequency SLMs. These

trapped phonons then decay predominantly back into Si,

which has far more low-frequency receiving modes. As a

result, the temperature of the Ge layer increases rapidly,

while the C layer remains cold for a long time. More detail

about these results will be published elsewhere.48

Thus, the amount of heat that propagates through the

interface or is reflected back into Si depends on three factors:

the temperature (background T0 and heat front T0þ dT); the

number and localization of interface SLMs in that

FIG. 5. Temperature vs. time in the central slice of the isotopically pure28Si192 (upper red curves) and isotopically mixed natSi192¼ 28Si177

29Si930Si6

(lower black curves) supercells. The solid lines show the fit from which

the thermal conductivity is extracted. The MD runs (3500 time steps) were

averaged over n¼ 50 initial microstates. The plot involves a 50-time-step

running average. Note that very small changes in temperature are easily

calculated.

FIG. 6. Temperature vs. time of a Ge or C d-layer in the Si225X25H40 nano-

wire (X¼Ge or C). The temperature of the Ge layer increases much faster

than that of the C layer (see text). The MD runs (5000 time steps) were aver-

aged over n¼ 50 initial microstates. The plot involves a 50-time-step run-

ning average.

085103-6 Gibbons et al. J. Appl. Phys. 118, 085103 (2015)

129.118.29.167 On: Mon, 31 Aug 2015 18:51:25

temperature range; the number of receiving modes on both

sides of the interface.

VI. DISCUSSION

We have presented the details of an approach to MD

simulations, which uses no thermostat, produces constant

temperature fluctuations 6DT starting with MD step 1, and

allows DT to be made as small as desired by averaging over

n initial microstates. The central ingredients are supercell

preparation at t¼ 0 followed by averaging over n� 30 initial

mode phase and energy distributions. The supercell can be

prepared in a wide range of initial configurations, such as

temperature gradients or specific vibrational excitations. Any

electronic-structure method can be used as long as an accu-

rate dynamical matrix can be obtained. The examples dis-

cussed involve defects, and all the calculations are done at

the first-principles level using the SIESTA package. The

high degree of temperature control allows us to monitor T

changes smaller than 1 K and record the amplitudes or ener-

gies of all the normal modes in the system as a function of

time.

None of our calculations show the scattering of thermal

phonons by defects. Instead, we see phonon trapping in

defect-related SLMs. Note that the importance of the defect-

related degrees of freedom was first pointed out half-a-cen-

tury ago.49 At low temperatures, the trapping lasts for dozens

or hundreds of periods of oscillation, long enough for

the origin of the excitation to be forgotten. The decay of

trapped phonons depends on the availability of receiving

modes.17,48,50 The interactions between thermal phonons and

an interface depend on three factors: the temperature (back-

ground temperature T0 and temperature of the heat front

T0þ dT), the interface SLMs (number and localization of

SLMs in the frequency range dx¼ kBdT/�h), and the number

of lower-frequency receiving modes in the two materials.

A. Limitations

The dynamical matrix is calculated at T¼ 0 K, when all

the nuclei are in their equilibrium configuration. But, its

eigenvalues and eigenvectors shift as the temperature

increases. Thus, the supercell preparation and the transfor-

mation from Cartesian to normal-mode coordinates are not

exact for T> 0 K. Our experience is that the 0 K dynamical

matrix remains a very good approximation at least up to

room temperature. However, it can be tricky to determine

which mode is which at a few hundred K, especially when

many modes have very similar frequencies at 0 K.

The need to average the MD runs over n initial micro-

states renders the method CPU intensive and therefore lim-

ited to relatively small supercells at the first-principles level.

This should be much less of an issue when using empirical

potentials.

Finally, the MD runs are classical and the nuclei obey

Newton’s laws of motion. No classical MD simulation is

accurate at arbitrarily low temperatures. As discussed in

Sec. V A, the temperature-dependence of the calculated

vibrational lifetimes matches the one measured by transient-

bleaching spectroscopy above a (defect-dependent) critical

temperature Tc. Experimentally, below Tc, all the receiving

modes are in their zero-point state leading to constant anhar-

monic coupling and lifetimes. Theoretically, below Tc, the

classical MD runs produce harmonic oscillators and infinite

lifetimes. However, for T>Tc, the average behavior of the

�1016 cm�3 oscillators measured by IR absorption matches

the classical calculations.

B. Strengths

Supercell preparation allows strictly microcanonical

MD simulations to be performed: no thermostat is used and

the total energy of the supercell remains constant. The tem-

perature fluctuations DT can be made very small by averag-

ing over n initial microstates, which mimics supercells with

a much larger number of atoms.

A supercell can be prepared with just about any distribu-

tion of initial temperatures within the cell, the simplest being

a small warm region within a colder environment. The sub-

sequent MD run allows the study of the system as it returns

to equilibrium.

Since no thermostat is used, the normal-vibrational modes

of the system couple only to each other, making it possible to

study the coupling between any normal modes.

The small temperature fluctuations DT allow the moni-

toring of changes in temperature as small as 1 K and, in the

case of vibrational lifetimes, the identification of one (or

more) receiving mode(s). The energies and amplitudes of all

the normal modes can be monitored as function of time.

When used in conjunction with density-functional based

electronic-structure calculations (“ab-initio” MD), the inter-

actions between heat flow and defects can be studied at the

atomic level without the usual empirical “phonon scattering”

assumption. The results obtained so far17 show no scattering

but phonon trapping in defect-related SLMs.

Finally, an important use of non-equilibrium MD

involves setting-up and maintaining a temperature gradient

in order to calculate the thermal conductivity using Fourier’s

law. Maintaining the gradient requires two thermostats. With

conventional MD, the calculations involve substantial tem-

perature gradients because of the large noise resulting from

the unphysical initial state (all the normal modes have zero

potential energy at t¼ 0). Lengthy thermalization runs are

required to stabilize the temperature fluctuations and achieve

a steady-state situation. Our supercell preparation could not

only substantially reduce the temperature fluctuations DT

and allow the use of much smaller gradients but also elimi-

nate the need for thermalization runs.

ACKNOWLEDGMENTS

The work of S.K.E. was supported in part by the grant

D-1126 from the R. A. Welch Foundation. The authors are

thankful to Texas Tech’s High-Performance Computer

Center for substantial amounts of CPU time.

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