Phonon-phonon interactions: First principles...
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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)
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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)
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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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