Supplementary Material for
Limit for thermal transport reduction in Si nanowires with nanoengineered
corrugations
Sean Sullivan†, Keng-Hua Lin, Stanislav Avdoshenko, Alejandro Strachan*
School of Materials Engineering and Birck Nanotechnology Center,
Purdue University, West Lafayette, Indiana 47907, USA
†Present
address: Materials Science and Engineering Program,
University of Texas at Austin, Austin, Texas 78712, USA
*Corresponding author
Email: strachan@purdue.edu
Simulation Details
Structures were generated in LAMMPS [1] and then cut to the appropriate
geometries using VMD [2]. All MD simulations were performed with LAMMPS using the
Stillinger-Weber potential.
After cutting to the desired geometry the structures were annealed from T = 10 K to
T = 300 K using isothermal/isobaric (NPT) MD simulations for 1 ns. Following this, the
wires were further thermalized at 300 K for an additional 1 ns of NPT MD, allowing the
simulation cell length along the (periodic) longitudinal direction of the wire to adjust to
zero stress. We used a Nosé-Hoover thermostat (damping parameter = 0.1 ps) and
barostat (damping parameter = 1 ps) throughout NPT anneal/relaxation.
After the relaxation, the wires were again thermalized at 300 K for an additional 100
ps at constant volume with a periodic cell length set to the average length of the last 100 ps
of the NPT run. Figure S1(a) displays the total, potential, and shifted kinetic energies of a
wire during the NPT anneal/relaxation and Figure S1(b) shows the wire length as a
function of time.
Figure S1: (a) Total, kinetic, and potential energies of a wire during NPT relaxation and
thermalization. The kinetic energies are shifted by -1.16x108 kcal/mol in order to compare
the fluctuations with the total and potential energies. System is annealed from 10 K to 300
K during the first 1000 ps and stabilized at 300 K for the second 1000 ps.
Figure S1 (continued): (b) Length change of a Si nanowire during NPT relaxation and
thermalization. The average longitudinal length during the last 100 ps is used to determine
the box size for future runs.
Figure S2 displays the post-thermalization longitudinal temperature distribution in
a wire. It is important to ensure the wires are properly thermalized before initiating the
Müller-Plathe velocity exchange algorithm to study thermal transport.
Figure S2: The average temperature of each longitudinal bin after 100 ps of thermalization
at 300 K. Deviations from the average are only ± 1 K.
Thermal conduction calculations
Heat flux generation via the Müller-Plathe algorithm was achieved using the fix
thermal/conductivity command with 80 bins in the longitudinal direction. Atomic
momenta were exchanged every 10 MD time steps (20 fs). In order to verify that this
swapping period is within the linear response regime, heat flux versus temperature
gradient was plotted for a swapping period of 10 MD time steps and 2 MD time steps
(Figure S3). Linear regressions show that, despite having only two data points per line, the
y-intercepts tends toward zero, indicating the simulations are in the linear response regime
between heat flux and temperature gradient.
Figure S3: Heat flux versus temperature gradient at two different swapping periods. The
upper three points correspond to velocity exchanges occurring every 4 fs, while the bottom
three points correspond to velocity exchanges every 20 fs. All three linear regressions
predict a y-intercept within 1x10-12 W*Å-2 of zero.
Since our simulations were very long (in excess of 8 ns) in order to converge local
quantities, the thermal conductivity calculation was performed within the NVT ensemble
with a weakly coupled Nosé-Hoover thermostat (with a coupling timescale Tdamp = 10 ps).
This is done in order to prevent potential energy drifts that can occur when performing
simulations with the NVE ensemble over long time periods due to accumulated errors in
the time integration.
To ensure the system has reached steady state, we analyze temperature evolution in
time (Figure S4) for several locations along the length of the wire. Temperatures stabilize
very quickly.
Figure S4: Temporal temperature evolution during velocity exchange in three locations
along the wire. Temperatures converge quickly, indicating steady state.
Meshing for temperature profiles
Taking advantage of the cylindrical symmetry of the nanowires, two-dimensional
temperature profiles were obtained by using a concentric cylindrical mesh. Along with the
longitudinal bins, an array of concentric cylinder bins was used in the radial direction, thus
resulting in individual concentric annuli and cylinders along the primary axis. These radial
bins have equal thickness and encapsulate the entire radial range of the nanowire, ranging
from radius 0 to 3 Å outside the nanowire surface. In general, 16 concentric cylinders
were used in addition to the 80 longitudinal bins, resulting in 1200 annuli and 80 cylinders
about the longitudinal axis. The 16 cylinders were averaged into 4 groups for clearer
viewing. Temporal averaging was identical to that for the one-dimensional profiles,
computing the average of 20 ps frames for a total of 4 ns. Average temperature per bin was
computed and mapped onto a two-dimensional profile. Arrows represent local
temperature gradients, as computed by numpy.quiver in Python.
Atomic Heat Flux Calculation
Typically, the heat flux for a group of atoms is defined as
𝐽=
1
[∑ 𝐾𝑖 𝑉𝑖 𝑣𝑖 − ∑ 𝑆𝑖 𝑣𝑖 ]
𝒱
𝑖
𝑖
where 𝒱 is the system volume, Ki and Vi are the atomic kinetic and potential energies, vi is
the velocity vector, and Si is the six component virial stress tensor. Computing only the
summands of this equation yields the per-atom flux vector; however, an approximation of
the atomic volume must be made. To avoid such approximations, the true atomic heat flux
was not calculated. Instead, a value proportional to the flux by 𝒱 was computed. This heat
flux length product (henceforth: quasi-atomic heat current or “heat current”) has units of
W*m instead of W*m-2. For N atoms, the local heat current is given by
𝑄̇𝑙𝑜𝑐𝑎𝑙 =
1
[∑ 𝐾𝑖 𝑉𝑖 𝑣𝑖 − ∑ 𝑆𝑖 𝑣𝑖 ]
𝑁
𝑖
𝑖
Temporal averages were again performed on a 20 ps basis for a total of 4 ns.
Spatial averaging was performed using the same annular meshing scheme discussed
for the temperature profiles, generating vector quantities on a quasi-atomic basis. With
vector quantities, it is necessary to perform a projection of non-longitudinal heat current
vectors onto the perpendicular radial vector:
Figure S5: Schematic illustrating the radial heat current vector projections used in the
annular binning.
For a given position along z (let z1=z2), the projection of heat current vector
[𝑄̇𝑥 , 𝑄̇𝑦 , 𝑄̇𝑧 ] onto the z-perpendicular radial vector [𝑥2 − 𝑥1 , 𝑦2 − 𝑦1 , 0] is
[Qx,Qy,Qz] •[x2 - x1, y2 - y1, z2 - z1 ] Qx * x2 + Qy * y2
=
2
2
2
(x2 - x1 ) + (y2 - y1 ) + (z2 - z1 )
x22 - y22
with inward-pointing defined as negative and outward-pointing as positive. Performing
this projection allows the three-dimensional description to be folded into a twodimensional one, while preserving the two heat current vectors in the transverse directions
within the wires.
The streamlines used to visualize the atomic heat currents were formed using a 4th
order Runge-Kutta algorithm implemented in the code streamplot.py as part of the
Matplotlib package [3]. This algorithm computes local derivatives to heat current
(velocity) vectors at each grid point and attempts to construct a linking path between
adjacent vectors without crossing other streamlines. Three-dimensional streamlines were
computed using a cubic - as opposed to cylindrical - grid in the visualization software
Mayavi [4].
As discussed briefly in the Letter, the methods used here attempt to minimize the
number of approximations made when describing thermal transport. The only
approximations made in molecular dynamics simulations are the use of classical mechanics
to describe system dynamics and the use of an interatomic potential to describe atomic
interactions. The Stillinger-Weber potential has successfully predicted thermo-mechanical
behavior of crystalline Si [1, 5-7] and classical mechanics suffices to model solids in this
temperature regime [8]. Additionally, molecular dynamics simulations adequately predict
anharmonity in solids and describe phonons naturalistically, without the need for
additional fitting parameters.
References
1.
2.
3.
4.
5.
6.
7.
8.
Stillinger, F.H. and T.A. Weber, Computer-Simulation of Local Order in Condensed
Phases of Silicon. Physical Review B, 1985. 31(8): p. 5262-5271.
Humphrey, W., A. Dalke, and K. Schulten, VMD: visual molecular dynamics. J Mol
Graph, 1996. 14(1): p. 33-8, 27-8.
Hunter, J.D., Matplotlib: A 2D graphics environment. Computing in Science &
Engineering, 2007. 9(3): p. 90-95.
Ramachandran, P. and G. Varoquaux, Mayavi: 3D Visualization of Scientific Data.
Computing in Science & Engineering, 2011. 13(2): p. 40-50.
Volz, S.G. and G. Chen, Molecular dynamics simulation of thermal conductivity of
silicon nanowires. Applied Physics Letters, 1999. 75(14): p. 2056-2058.
Volz, S.G. and G. Chen, Molecular-dynamics simulation of thermal conductivity of
silicon crystals. Physical Review B, 2000. 61(4): p. 2651-2656.
Lin, K.-H. and A. Strachan, Thermal transport in SiGe superlattice thin films and
nanowires: Effects of specimen and periodic lengths. Physical Review B, 2013. 87(11).
Rapaport, D.C., The art of molecular dynamics simulation. 1995, Cambridge ; New
York: Cambridge University Press. xiv, 400 p.