Molecular Simulations of Thermal
Conductivity of Si/SiGe
Superlattice Nanowires(Course Project for ME 346)
Yongxing Shen*, Annica Heyman**, and Ningdong Huang**Departments of Materials Science and Engineering* and Applied Physics**
Instructor: Wei CaiStanford University
December 2004
Heat
117
Abstract
We have studied the thermal conductivity of Si/SiGe nanowires using molecular dynamics
simulations. Wires of Si, SiGe, and Si/SiGe superlattice structures were prepared in order to
compare the effect of phonon scattering due to Ge imperfections and Si-SiGe interfaces.
Starting from the bulk Si [111] direction, wires were prepared by randomly replacing Si with
Ge and then relaxing at 0 K followed by equilibration at room temperature. The molecular
dynamics software used was modified to extract the heat density during an NVE run. From
the heat density we expected to calculate the thermal conductivity using the fluctuation-
dissipation theorem. Unfortunately, we discovered that the software did not conserve the total
energy and we were therefore never able to extract any physical relevant data on the thermal
conductivity. At the writing of this report, the program bug has not yet been found.
217
Introduction
Si/SixGe1-x (0<x<1) superlattice nanowires have been attracting more and more attention
because of their potential application as thermoelectric materials and thus a future key role
in thermoelectric refrigeration [1], [2]. This comes from their low thermal conductivity [3]
as reported, which would enhance the thermoelectric figure-of-merit (ZT), which
characterizes the performance of a certain kind of thermoelectric material. ZT is defined as
!
" TSZT
2
= (1)
where ! is the electrical conductivity, " the thermal conductivity, T the absolute
temperature and S the thermal power that comes from the Seebeck effect:
TSE != (2)
which connects the applied temperature gradient and the subsequent electric field.
Superlattice nanowire is a kind of nanowire in which two or more kinds of materials alternate
periodically along the axis, as in Figure 1. Here the two materials are Si and SixGe1-x alloy.
Figure 1. Si/SixGe1-x superlattice nanowire, Si and SixGe1-x alternate along the nanowire axis.
The thermal conductivity of such structures is dominated by the scattering of phonons caused
by the deviation from perfect lattice. The phonon scattering centers in the materials include
external surface, interfaces and impurity (Ge atoms).
The aim of the present study is to examine the Si-SixGe1-x interfacial effect contributed to the
low thermal conductivity of the superlattice nanowire by comparing the thermal
conductivities of this structure and corresponding homogeneous nanowires – Si nanowires
and SixGe1-x nanowires. The structures involved are shown in Figure 2. Molecular dynamics
simulation is used since it is the motion of the atoms rather than the electrons that dominates
to the thermal conductivity of these semiconducting materials.
Si
SixGe1-x
317
Figure 2. Different types of nanowires to be examined. The cross-sectional diameter is 15.4 Å or 23.0 Å (as in
pure Si before relaxation); the length is 4 periods or 37.6 Å (as in pure Si before relaxation). The structures are:
(a) Pure Si; (b) SixGe1-x alloy; (c) Superlattice nanowire where Si and SixGe1-x alternate by two periods (d)
Superlattice nanowire where Si and SixGe1-x alternate by one period.
In the present report, we will first introduce the background physics of thermal conductivity,
then go through the simulation details, and finally present the results and discussion.
BackgroundEarlier results:
The thermal conductivity of pure Si nanowires has already been studied theoretically with
MD simulation [5] and experimentally [4]. The observed thermal conductivity of Si
nanowires is much lower than the bulk value, suggesting that phonon–boundary scattering
controls thermal transport in Si nanowires. The MD results also reveal a one to two orders of
magnitude reduction in the thermal conductivity of nanowires compared to those of
corresponding bulk Si crystals. A solution of the Boltzmann transport equation matches the
MD results quite well and indicates that diffuse boundary scattering causes the thermal
conductivity drop as observed in the MD simulation.
However, the mechanism may be quite different in the Si/SiGe supperlattice nanowires.
Previous experiments show that the thermal conductivities of individual Si/SixGe1-x
superlattice nanowires are much lower than those of the pure Si ones of similar diameters (see
Figure 3), which leads to the conclusion that alloy scattering of phonons in the Si–Ge
segments and/or Si-SixGe1-x interfacial scattering is the dominant scattering mechanism in
these superlattice nanowires [3]. In addition, the result that the thermal conductivity of the
Si/SiGe nanowire has similar temperature dependence to that of 2D superlattice film but much
lower values (shown in Figure 3(a)) also suggests that nanowire boundary scattering also
contributes to thermal conductivity reduction. In the reference, they also give out a possible
explanation: while short-wavelength acoustic phonons are effectively scattered by atomic
Si
SixGe1-x
(b)
(a)
(c)
(d)
417
scale point imperfections in the SiGe alloy segments, long-wavelength acoustic phonons are
scattered by the nanowire boundary.
Figure 3. (a) Thermal conductivities of 58 and 83 nm diameter single crystalline Si/SixGe1-x superlattice
nanowires. The value of x is ~ 0.9– 0.95 and the superlattice period is 100–150 nm. Thermal conductivities of a
30 nm period 2D Si/Si0.7Ge0.3 superlattice film and Si0.9Ge0.1 alloy film (3.5 mm thick) are also shown. (b)
Thermal conductivities of single crystalline pure Si nanowires. The number besides each curve denotes the
corresponding wire diameter [5].
Theoretical analysis using incoherent particle model shows that the thermal resistivity of
Si/Ge superlattice nanowires is due entirely to the diameter effect and SiGe alloy scattering
[6]. Below ~30K, alloy scattering is less important due to the lower characteristic thermal
frequencies, and the superlattice nanowire behaves almost like a pure Si nanowire limited by
diameter scattering. Above ~30K, on the other hand, alloy scattering in the SiGe segments
begins to play an increasingly important role, and by 300K is comparable to the diameter
effect.
However in both either [3] or [6], the interfacial effect of Si-SixGe1-x has been overlooked and
should be seperated from the scattering within the SiGe segment. In order to do this, we carry
out our simulation on both alloyed SixGe1-x nanowire and Si/SixGe1-x superlattice nanowires in
Figure 2.
Theoretical background for thermal conductivity:
Thermal conductivity is the physical property of materials that expresses the heat current that
flows in a material in response to a temperature gradient. Mathematically this means
!
J = "#$T (3)
where J is the heat current, T the temperature, and " the thermal conductivity. (Note that the
thermal conductivity is formally a tensor but the off-diagonal elements will always be zero
and for cubic or isotropic materials the diagonal elements will all be equal). The heat current
can be calculated as
517
!
J = 1
V
d
dtriEi
i
"#
$ %
&
' ( (4)
where the summation runs over all atoms in the volume V and each atom has the position r
and the total energy E, i.e. kinetic plus potential energy. Using the SI system, " will be given
in units of W/(mK).
The thermal conductivity belongs to a category of coefficients related to transport phenomena.
In this case the transport is associated with heat and occurs since a temperature gradient is
present in the material. As with all transport properties, the definition of thermal conductivity
is intimately connected with a non-equilibrium situation. In an MD simulation this can be
implemented by introducing so called phantom atoms at each end of the sample and forcing
them to vibrate according to specified temperatures, T1 in one end and T2 (<T1) in the other
end of the sample. The temperature gradient is then simply approximated as (T1 - T2)/L, where
L is the length of the sample and the heat flow through the material is measured using eq. (4).
There exists however an alternative route to calculate transport properties that only require
equilibrium simulations, and therefore is often preferred over a non-equilibrium calculation.
This approach is made possible by the Fluctuation-Dissipation Theorem (FDT) which was
fully developed by R. Kubo (see for example [7]) in the 1950s. The FDT provides a general
relationship between the response of a system to an external disturbance and the internal
fluctuations in a system in the absence of any disturbance. The former being characterized by
a response function and by definition describes a non-equilibrium situation whereas the latter
is related to an equilibrium situation and where fluctuations are expressed using correlation
functions. The derivation of the FDT is quite complex and involves keeping only linear terms
in the perturbation part of the Hamiltonian (hence the alternative name: linear response
theory). In the classical limit for transport problems the theorem takes the form of an infinite
time integral of an equilibrium time correlation function:
!"
=0
)0()(d AtAt &&# (5)
where # is the transport coefficient in question and A(t) the dynamical quantity conjugate to
the applied force that causes the perturbation. The < > bracket indicates an ensemble average.
Associated with any expression of the above kind is a so called Einstein relation:
!
2t" = A(t) # A(0)( )2
(6)
which holds for large times t. The integral expression in eq. (5) can exactly be obtained by
differentiating eq. (6), but at large times the approximate Einstein form will also be valid.
Probably the most familiar application of the FDT to transport problems is the formula for
calculating the diffusion coefficient, D:
617
!
D = 1
3dt v
i(t)v
i(0)
0
"
# or, equivalently,
!
2tD = 1
3ri(t) " r
i(0)( )
2. (7)
For thermal conductivity the FDT stipulates that
!
" = V
kBT
2dt J# (t)J# (0)
0
$
% or, equivalently,
!
2t" =V
kBT2
qi# (t) $ qi# (0)( )2
(8)
where $ = x, y and z, and
!
q" = 1
Vri" Ei # Ei( )
i
$ is the heat density.
As will be apparent in the next section, the second form in eq. (8) is much easier implemented
for a Tersoff potential and therefore became our choice for calculating ".
To conclude this section it is probably worth pointing out a few things. First, the formulas
above hold only for a microcanonical ensemble, i.e. they should only be calculated when
using an NVE ensemble. Second, obviously we cannot take an infinite time integral when
using MD as stated above but need to limit ourselves to finite times. As correlation functions
necessarily decays to zero after some time, this is formally not a problem but one needs to be
aware of the fact that using a too short time-interval may miss important contributions to the
correlation function. For calculating thermal conductivity in Si nanowires 20 ps has been
reported as a reasonable time interval [5]. For application of the second form in eq. (8) it may
be necessary to use an even longer time interval. Third, the bracket notation indicates an
ensemble average which in MD is usually treated as a time average. Thus " can be calculated
using
!
q" (t) # q" (0)( )2
= 1
nmax
q" (n$ + t) # q" (n$ )( )2
n=1
nmax
% (9)
where % is an appropriately chosen time-interval. Note that this requires the calculational time
to include not only the full correlation time but also enough time to accurately sample the
distribution function. Last, it is probably wise to raise a word of caution. All the above
formulas for transport coefficients are classical limits of corresponding quantum mechanical
relations. For temperatures much higher than the Debye temperature, the classical expressions
are automatically valid, but the Debye temperature in Si for example is 640K which is
significantly higher than room temperature. It is possible to apply certain quantum corrections
to the classical result in order to get a more accurate result. These include correcting for the
zero-point energies and the fact that classically the discreteness of the energy levels is always
neglected. However, such schemes will always be more or less ad-hoc. For example, Li et al.
[10] employed a correction scheme for the thermal conductivity of crystalline &-SiC based on
the hypothesis that it is possible to establish a one-to-one correspondence between the real
quantum system and the classical MD simulation such that all the physical variables are the
same. Their corrected thermal conductivity at elevated temperatures was better than the
uncorrected, but at room temperature they were both as good (or bad).
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It is however true that in the harmonic approximation of the interatomic potential the quantum
and the classical expression coincide (of course one should probably also take into account
the correctness of the interatomic potential in the quantum limit as well). So under the
assumption that the main features of the anharmonic effects are captures accurately by the
MD simulation it is possible to carry out only classical calculations of the correlation function
with no additional quantum corrections. This is often the approach taken and will also be
applied here.
Simulation Details:
To address the problem described above we have used MD simulations with a classic
interatomic Tersoff potential [8] for Si and Ge. The particular software package we used was,
and partially still is, being developed in Prof. K.J. Cho’s group by Byeongchan Lee. It has
capability for, among other things, 0K relaxations, NVTL ensembles, and NVE ensembles.
The remainder of this section will in detail describe the various steps in our approach to
calculate the thermal conductivity of Si/Ge nanowires. First we will motivate the choices of
the particular structures we have studied, followed by the specifics on how these were
generated. Then we will discuss how to extract the heat density, q$, from a MD calculation,
starting with a discussing on how to define the potential energy per atom and continuing with
a short explanation of the important parts of the program and the changes we have
implemented in it. This is followed by a short paragraph on how we have calculated the
thermal conductivity with the heat density as input. Finally we will summarize the whole
simulation procedure. All results will be reported in the next section.
Initialization of the structures
Both bulk Si and Ge are diamond structure, and the atoms in the alloy SixGe1-x have the same
coordination numbers and similar bond natures as the diamond structure except the lattice is
distorted due to the difference in sizes of both species, which can be simulated by energy
minimization. We start with Si structure and randomly replace the Si atoms with Ge according
to the composition x. We use x = 0.75, the same as recently synthesized [9].
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Figure 4. Cross-sectional view of the atoms in non-relaxed Si nanowires (as in bulk). The circles (o), the
asterisks (*) and the dots (.) denote the atoms in A, B and C layers respectively.
Since Si and Ge grow preferably on the {111} plane [12], we set the nanowire axis as [111]
direction. The stacking sequence of (111) plane in diamond structure is
A AB BC CA AB BC CA….
where the meaning of A, B and C sites is shown in Figure 4 and AABBCC forms a period
along the axis. The interplanar spacings of AA, BB and CC layers are 4/3a (a is the lattice
constant and equals to 5.432Å), while those of AB, BC and CA layers are 12/3a .
We set the x and y dimensions of the simulation cell large enough (200 times a) to make sure
the nanowires in the periodic 2D array don’t interact each other, and periodic boundary
condition along the z direction.
The simulation cells we use have 4 or 8 periods along the z direction. The cross sectional
radius before relaxation is twice or three times 2a , which is the unit vector in the cross
sectional triangular lattice.
Energy minimization
An NVT ensemble at T=0 K is used. Since the code is not capable of using Parrinello-
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Rahman method, we have to iterate in order to find the optimized nanowire length. The base
is that at energy minimum, the virial pressure should also be zero since the nanowire would be
unstressed.
When we have calculated two pairs of length and pressure, we can linearly interpolate or
extrapolate to find the equilibrium length, and go on to iterate to get the desired accuracy.
Equilibration
The NVT ensemble with Langevin thermostat is employed to find the equilibrium
configuration at room temperature. In Langevin thermostat, each particle receives a friction
and a random force so as to maintain the temperature.
Extraction of the heat density:
To calculate the thermal conductivity we need to at the very least extract the time-dependent
heat density during a relatively long run, using an NVE ensemble. While the software
package we have used does output e.g. positions, total energies and virial pressure at a
specified update rate, it was not set up for calculating thermal conductivities. We decided to
extract the total heat density (actually heat density times volume) in each direction (x, y, and
z) from the MD run and then afterwards calculate the thermal conductivity from these data.
Since the heat density consists of position multiplied with the total energy of each atom we
first of all need to calculate the total energy of each atom. The kinetic energy is trivially found
for each atom but unless a simple pair potential is used the potential energy cannot uniquely
be partitioned between the individual atoms. As a simple pair potential only accurately
describes the noble gases, the potential energy per atom will not be a unique function for most
relevant systems. For Si and Ge it is important to not only include the distances between
atoms but also the local environment since these elements prefer to sit in diamond-like
structures. The classic Tersoff potential is modeled as a pair potential,
!
Vij = fC (rij ) fR (rij ) + bij fA (rij )[ ] (10)
but the coefficient (bij) of the attractive term is dependent on a third atom, k ' i, j, and its
distances and angles to atoms i and j, effectively making the Tersoff potential into a many-
body potential. However, since it is defined as a pair-potential there is a very natural, though
not unique, way of defining the potential energy per atom as simply Vij/2. This partitioning of
course ignores the neighboring atoms k. Using this simple partitioning Li et al. [10] calculated
the thermal conductivity of crystalline &-SiC using the same Tersoff potential. It has also been
suggested that for short-range interactions, the details of the partitioning is not of importance
since the temperature gradient varies over a macroscopic scale [11]. We will therefore adopt
Vij/2 as the potential energy of atom i. It is also apparent here why the second form of the FDT
for the thermal conducitivity (eq. (8)) is to prefer, at least implementation wise, since the
potential energy per atom is easily obtained whereas the derivative of the potential energy
with respect to atoms i, j, and k is quite complex.
1017
In order to describe how we extracted the heat density from the MD run we need to briefly
explain the software structure. The program is written in C++ and for a MD simulation an
object known as the MDExecutionManager manages the overall work with reading in input
files, initializing, advancing in time, and writing output files. All the data of the physical
system is stored in a System object which, most importantly, manages a list of Particle
objects. The Particle object knows about its position, velocity and acceleration. Integration
algorithms and interatomic potentials are both separate objects set up according to the
specifications in the input files.
To calculate the heat density we added the member variable eigenpotential to the Particle
object and in the interatomic potential object Tersoff we added functions that set the
eigenpotential in each Particle to Vij/2. Then, from the System object we accessed the
eigenpotential, velocity, position, and mass of each Particle to calculate the heat density which
was stored as a member variable in the System. Finally, the heat density variable was printed
to file from the MDExecutionManager at fixed time intervals.
Calculating the thermal conductivity:
With the heat density data written to file we were able to read this data into Matlab and using
eqs. (8)-(9) we calculated the thermal conductivity in each direction using the short script
found in Appendix 1.
Simulation procedure:
We conclude this section by summarizing the simulation procedure:
1) Generating initial structures: Finding T = 0K minimum configuration
2) NVTL equilibration
3) NVE runs to extract heat density
4) Post-processing of heat density data
Results and Discussion:Optimized nanowire length
The nanowires expand with respect to their lattice constant in bulk materials because of the
presence of free surface.
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Virial Pressure vs Nanowire Length
-150
-100
-50
0
50
100
150
200
37.4 37.6 37.8 38 38.2 38.4 38.6
Nanowire Length (Angstrom)
Vir
ial
Pre
ssu
re (
eV/A
ng
stro
m^3
)(a)
(b)
(c)
(d)
Figure 5. Finding the relaxed lengths of the nanowires. The cross-section diameter is 23.0 Å (as pure Si, before
relaxation); the length is 4 periods (as pure Si, 37.6 Å before relaxation). The structures are: (a) Pure Si; (b)
SixGe1-x alloy; (c) Superlattice nanowire where Si and SixGe1-x alternate by two periods each (d) Superlattice
nanowire where Si and SixGe1-x alternate by one period each.
Thermal conductivity
To calculate the thermal conductivity we preformed long (200 ps) runs using an NVE
ensemble with our slightly modified program. However, we ran into some serious problems.
Some of the structures we run suffered badly from total energy conservation problems; the
total energy increased with, in some cases, many eV during a run. With a closer look on all
our data, we found that the total energy in almost all runs, suffered from the same problem,
even though often limited to a few tenth of an eV. To rule out the possibility of us using a too
large a time step, we went from using 0.5 fs to 0.05 fs but the same overall energy
conservation problem persisted even though the instantaneous energy fluctuations decreased.
Figure 6 shows the total energy plotted versus simulation time for (a), with zoom-in in (b), 0.5
fs and (c) 0.05 fs time step for the 2a structure (our smallest radius and the (b) structure in
Figure 2). Both the energy conservation problem and the instantaneous fluctuations are clearly
visible.
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Figure 6. Total energy vs. simulation time for the 2B structure. (a) using a 0.5 fs time step. (b) zoom-in of (a). (c)
using a 0.05 fs time step.
This energy conservation problem is not a bug due to our modification but unfortunately
appears in the original code as well and it is present when using both pure Si and Si/Ge alloy.
At the writing of this report, it was not clear what is causing this problem. A short discussion
of our debugging effort can however be found in Appendix 2.
With this energy conservation bug in the code causing the total energy to increase, the validity
of using our heat density results is of course highly questionable. Not only since the
derivation of the expressions for the thermal conductivity using the FDT explicitly requires an
NVE ensemble and but even more importantly, that the bug is probably very likely to also
generate the wrong trajectory, and thus the program will not accurately simulate the true
atomic system. Note especially that even for runs where the energy is approximately
conserved this bug may result in unphysical data.
With this said we still chose to report some data on the thermal conductivity in Figures 7 and
8 as function of the time t in eq. (8). Figure 7 shows the thermal conductivity for structures 2a,
2b, 2c, and 2d respectively (the 2 refers to the smaller radius, the letter corresponds to the
structures shown in Figure 2 and the periodic length was 4 unit cells) using a 0.5 fs time step.
Figure 8 (a)-(d) show the same structures but with a 0.05 fs time step. Figure 8 (e) shows the
2a structure with a periodic length of 8 unit cells. The energy conservation problem were
worst for the (b) and (e) graphs in both figures. However, we will refrain from any
interpretation of the data in Figures 7 and 8 as we do not want to make any predictions based
on presumably faulty data but only conclude that there are several things that seem
questionable in the graphs below. Among these are the inconsistency of the thermal
conductivity in the x and y directions between different structures and the seemingly higher
value of " for SiGe. Also the thermal conductivity seems to not have converged with respect
to t in many of the graphs. If this is a real effect or due to the energy conservation problem is
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hard to estimate. On a positive note is, however, that all thermal conductivities are of the
anticipated order of magnitude, i.e. 1 W/mK or less.
(a)
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
50 55 60 65 70 75 80 85 90 95 100105110115120125130135140145150
t(ps)
the
rma
l c
on
du
cti
vit
y(W
/mK
)
kappaxx
kappayy
kappazz
(b)
0
0.05
0.1
0.15
0.2
0.25
50 55 60 65 70 75 80 85 90 95 100105110115120125130135140145150
t(ps)
the
rma
l c
on
du
cti
vit
y(W
/mK
)
kapaaxx
kappayy
kappazz
(c)
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
50 55 60 65 70 75 80 85 90 95 100105110115120125130135140145150
t(ps)
the
rma
l c
on
du
cti
vit
y(W
/mK
)
kappaxx
kappayy
kappazz
(d)
0
0.05
0.1
0.15
0.2
0.25
50 55 60 65 70 75 80 85 90 95 100105110115120125130135140145150
t(ps)
the
rma
l c
on
du
cti
vit
y(W
/mK
)
kappaxx
kappayy
kappazz
Figure 7. Thermal conductivity as function of the time t in eq. (8). (a) Structure 2a, (b) structure 2b, (c) structure
2c, (d) structure 2d. Time step 0.5 fs.
(a)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
5
5.5 6
6.5 7
7.5 8
8.5 9
9.5 10
10.5 11
11.5 12
12.5 13
13.5 14
14.5 15
t(ps)
the
rma
l c
on
du
cti
vit
y(W
/mK
)
kappaxx
kappayy
kappazz
(b)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
5
5.5 6
6.5 7
7.5 8
8.5 9
9.5 10
10.5 11
11.5 12
12.5 13
13.5 14
14.5 15
t(ps)
therm
al co
nd
ucti
vit
y(W
/mK
)
kappaxx
kappayy
kappazz
(c)
0
0.1
0.2
0.3
0.4
0.5
0.6
5
5.5 6
6.5 7
7.5 8
8.5 9
9.5 10
10.5 11
11.5 12
12.5 13
13.5 14
14.5 15
t(ps)
the
rma
l c
on
du
cti
vit
y(W
/mK
)
kappaxx
kappayh
kappazz
(d)
0
0.1
0.2
0.3
0.4
0.5
0.6
5
5.5 6
6.5 7
7.5 8
8.5 9
9.5 10
10.5 11
11.5 12
12.5 13
13.5 14
14.5 15
t(ps)
the
rma
l c
on
du
cti
vit
y(W
/mK
)
kappaxx
kappyy
kappazz
(e)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
5
5.5 6
6.5 7
7.5 8
8.5 9
9.5 10
10.5 11
11.5 12
12.5 13
13.5 14
14.5 15
t(ps)
the
rma
l c
on
du
cti
vit
y(W
/mK
)
kappaxx
kappayy
kappazz
Figure 8. Thermal conductivity as function of the time t in eq. (8). (a) Structure 2a, (b) structure 2b, (c) structure
2c, (d) structure 2d, (e) Structure 2a but double length (8 unit cells). Time step 0.05 fs.
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Future work suggestion
1) Find out the underlying reason why we could not get the correct conserved energy and
correct the code to get results for valid thermal conductivity
2) Examine the contribution of the quantum effect
3) Simulate the electrical conductivity and thermal power in order to have the complete
picture of thermoelectricity
4) Find the optimized parameters to get the best thermoelectric figure of merit (ZT). These
parameters include the ratio of both phases within one period, the compositions in each
phase and the cross-sectional diameter.
Acknowledgements
We would like to thank the instructor of the course ME 346, Wei Cai, for his offer of this
opportunity for us to practice much of the skills and techniques talked about in the course.
And we would also like to thank Byeongchan Lee for debugging his code that we used for this
project. Without his help, we would not have come up with these results though we have not
actually reached the valid results.
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References
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[6] C. Dames et al, J. Appl. Phys 95, 682 (2004)
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Appendix 1: Matlab script to calculate thermal conductivity
% constants and variableskB = 8.621e-5; %in eV/K (Boltzmann's constant)T = 0.027021/kB; % temperature(in units of K)volume = 6.9772e3; % the volume of the supercell, in units of A^3(unrelaxed)N = 40000; % number of sets of input datadT = 10*0.5; % time interval of input data, in units of fstmin = 10000; % the minimum time interval to calculate thethermal conductivity, in units of dTtmax = 30000; % the maximum time interval to calculate thethermal conductivity, in units of dT% nmax = N - t; % maximum index to calculate the time average ofdelta epsilon multipling volumeep_V = zeros(N,3); % place to save data read from inputkappa = zeros(1,3); % thermal conductivityfactor = 1.602E-19/1.0e-10/1.0e-15;
fid = fopen('/2D.txt');if(fid == -1) fprintf('could not open file\n');else for i=1:N, ep_V(i,:) = fscanf(fid,'%f',[1,3]); endendfclose(fid);
number_t = floor((tmax - tmin)/1000);for k=0:number_t t = tmin+k*1000; nmax = N - t;% maximum index to calculate the ensemble (time) average kappa = zeros(1,3); for i=1:nmax, kappa = kappa + (ep_V(i+t,:) - ep_V(i,:)).^2; end kappa = kappa/nmax/kB/T^2/(2*t*dT)/volume*factor; fprintf('%d %f %f %f\n', t, kappa);end
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Appendix 2: Discussion on energy conservation bug in the NVE ensemble
Since the discovery of the energy conservation problem we have tried to find the cause of this
problem. Byeongchan Lee, the current “owner” of the code, has been involved in this process
as well. We have not been able to find any errors when going through the algorithm file and
the neighbor list file, which seemed to be likely candidates for such problems we are
experiencing. The strongest indication we have so far in fact points to problems in the Tersoff
potential file. This is because systems using other potentials, such as different versions of
EAM (Embedded Atom Method) seem to run a NVE ensemble properly. Of course, the
program is quite complex and involves several hierarchies and it is possible the problem is to
be found at a higher level.
