1. science
2. physics

PREDICTING!
LOW-THERMAL-CONDUCTIVITY
SI-GE NANOWIRES!
Jesper Kristensen, !
(joint work with Prof. N. Zabaras)!
!
Applied and Engineering Physics &!
Materials Process Design & Control Laboratory!
Cornell University!
271 Clark Hall, Ithaca, NY 14853-3501 !
and!
Warwick Centre for Predictive Modelling!
University of Warwick, Coventry, CV4 7AL, UK!
The Si-Ge Nanowire!
q  One of most rapidly developing research activities in materials
science
q  Advanced applications:
Ø  High performance nanoelectronics (FETs and interconnections)
•  40 % increase in mobility compared to pure Si nanowire
Ø  Thermoelectrics
q  We will be interested in thermoelectric applications
Ø  Convert heat to electrical energy and vice versa
Ø  Figure of merit captures thermoelectric efficiency:
Electrical conductivity
Seebeck coefficient
Temperature of device
Thermal conductivity
S2 T
(electrons + phonons)
ZT =

Amato, Michele, et al. Chemical reviews 114.2 (2013)
2
The Si-Ge Nanowire as Thermoelectric Device!
q  Problem:
Ø  Electrical and thermal conductivities are highly interconnected quantities
q  Approximate:
Ø  Freeze the electronic degrees of freedom
q  Goal:
Ø  Alloy scattering is main source of thermal conductivity reduction*
Ø  Alloy Si nanowire with Ge until minimum in phonon thermal conductivity
S2 T
ZT =

Ø  Semiconductors:
•  Heat conduction primarily due to phonons
 ⇡ lattice
*Kim, Hyoungjoon, et al. Applied Physics Letters 96.23 (2010)
3
Computational Methods!
q  Computing the thermal conductivity
Ø  Non-equilibrium method
Ø  Equilibrium method
q  Non-equilibrium molecular dynamics (NEMD)
Ø  “Direct method”
Ø  Analogous to experiments
Hot reservoir
Cold reservoir
Heat transferred across temperature gradient
q  Equilibrium molecular dynamics (EMD)
Ø  Green-Kubo
•  Fluctuation-Dissipation theorem:
Relate current fluctuations to thermal conductivity (no reservoirs)
•  Benefit: Entire κ tensor computed in a single simulation
4
Example of “Direct Method” Implementation!
Typical temperature profile
q  Direct method implementation:
Ø  At each time step:
•  Add heat Δε to slab at –Lz/4
•  Subtract heat Δε from slab at Lz/4
Nonlinear
effects
Ø  Steady state:
Jz =
✏
2A t
Jµ =
X
⌫
@T
µ⌫
@x⌫
Linear region:
Get T gradient
Ø  Nanowires: huge temperature gradients are created!
Ø  Fourier’s law not rigorously proved for
microscopic Hamiltonian*
Schelling, Patrick K., Simon R. Phillpot, and Pawel Keblinski. Physical Review B 65.14 (2002)
*Amato, Michele, et al. Chemical reviews 114.2 (2013)
5
EMD: Green-Kubo!
q  Benefit: Linear response regime
q  Drawback: Very long simulation times needed
Ø  Including longer times in integral introduces significant noise
1
µ⌫ (⌧m ) =
V kB T 2
Z
⌧m
0
hJµ (⌧ )J⌫ (0)id⌧
Heat current
autocorrelation function
(HCACF)
q  Definition of heat current
J=
d X
r i (t)"i (t)
dt i
q  For 3-body interaction (such as Tersoff*) we define the potential as:
2-body force on atom i
due to its neighbor j
J=
X
i
3-body force
1 X
1X
v i "i +
r ij (F ij · v i ) +
(r ij + r ik ) (F ijk · v i )
2
6
ij,i6=j
ijk
Schelling, Patrick K., Simon R. Phillpot, and Pawel Keblinski. Physical Review B 65.14 (2002)
*Tersoff, J. Physical Review B 38.14 (1988)
6
Notes on HCACF!
q  Computing the HCACF was done as follows
Ø  Take 2n MD steps (n=24 in our case)
Ø  Use Wiener-Khinchin-Einstein theorem:
•  Autocorrelation related to Fourier transformed heat current vector
HCACF
F
1
⇣
⌘
F(J (t))(⌫)F(J (t))(⌫) (t)
Fourier transform
of raw heat current
7
Molecular Dynamics!
q  Use molecular dynamics (MD) to obtain the thermal conductivity
Ø  The large-scale atomic/molecular massively parallel simulator
(LAMMPS*)
Ø  Alternative: Ref. [**] used XMD
q  MD: Integrate Newton’s laws of motion
Ø  Give atoms initial positions and velocities
Ø  Repeat:
•  Obtain forces from interaction potential chosen
–  In our case this was Tersoff
•  Obtain accelerations
•  Update positions and velocities
*Plimpton, Steve. Journal of computational physics 117.1 (1995)
**Chan, M. K. Y., et al. Physical Review B 81.17 (2010)
8
Verify Green-Kubo Implementation in LAMMPS!
q  Bulk Si and Ge structures with Tersoff potential
Ø  Time step: 0.8 fs
Ø  Temperature 300 K
We use the method from Ref. [*]:
(cor(t))
F (t) ⌘
E(cor(t))
Decay is
exponential
(shown in log)
Numerical noise
takes over
We predict 170 W/m.K for Silicon.
Experimental value = 150 W/m.K.
We predict 90 W/m.K for Germanium.
Experimental value is 60 W/m.K.
Tersoff potential known to overshoot.
Great agreement!
*J. Chen, G. Zhang, and B. Li. Physics Letters A 374.23 (2010)
9
Creating the Nanowire!
q  In this work, we wish to model 50 nm long Si nanowires
Ø  Roughened surface
q  Ref. [*]: evidence of this equivalence (good enough for our purpose)
>5500 atoms
Length: 50 nm
Surface: Rough
~220 atoms
Length: 2 nm
Surface: Pristine
q  Similar phonon behavior
Ø  Why? Roughening scatters/excludes phonons.
Shortening the wire has a similar effect (wavelengths don’t “fit”
anymore).
q  Computational benefits of smaller system
Ø  Easier to create and implement
Ø  Faster to run
*M. Chan et al. Physical Review B 81.17 (2010)
10
Preparing Nanowire for LAMMPS!
q  Nanowire for LAMMPS (visualized in OVITO*)
Parse with LAMMPS
q  Simplification: not passivating the wires
Ø  Experimental wires passivated with, e.g., hydrogen from HF treatment
Ø  Hydrogen passivation can stabilize the system
•  Removes dangling bonds
*Stukowski, Alexander. Modelling and Simulation in Materials Science and Engineering 18.1 (2010)
11
Solving Green-Kubo with LAMMPS!
1
µ⌫ (⌧m ) =
V kB T 2
Z
⌧m
0
hJµ (⌧ )J⌫ (0)id⌧
q  Our case: µ=ν=x; so compute Jx only
q  We solved the above integral with LAMMPS as follows:
Ø  MD time step = 1 fs
Ø  Initialize atomic coordinates (minimum (local) energy)
Ø  Annealing process to deal with surface
•  After this process we were in a 300 K NVT ensemble
Ø  Nanowire axis: pressurize to 1 bar in an NPT ensemble
•  Axial strain was ~500 bar before this due to lattice mismatch
between Si and Ge of ~4.2 %* (large value)
Ø  After NPT, switched back to NVT for 1 ns
Ø  Switched to an NVE ensemble for 16 ns. Collected J in integrand.
Ø  Integrated autocorrelation of J (integrand)
*Amato, Michele, et al. Chemical reviews 114.2 (2013)
12
Annealing Scheme for Nanowire Surface!
q  Problem: Surface atoms far from equilibrium (dangling bonds)
q  Solution: The following annealing procedure was successful:
Ø  Start at T=1000 K; run for 500 ps
Ø  Lower T 100 K at a time over 10 ps
•  Each T: run for 100 ps
Temperature (K)
Annealing scheme (not to scale)
1000
100 K
In our work
Annealing essential to good results.
Other possibility:
Langevin thermostat or variants
thereof (not explored in depth).
300
Time
13
Convergence Issues with the HCACF!
q  Bulk HCACF: Predictable exponential decay
q  Nanowire HCACF: No known analytical form
Ø  Some wires: No clear convergence à Due to MD noise
q  Ref. [*]: How to integrate the HCACF
Ø  We implemented an automatic way of identifying convergence
(Figure from Ref. [*])
q  40 moving averages of various window sizes (50 to 200 ps)
Ø  Convergence: Minimum standard deviation time gives upper limit
1
µ⌫ (⌧m ) =
V kB T 2
Z
⌧m
0
hJµ (⌧ )J⌫ (0)id⌧
*McGaughey, Alan JH, and M. Kaviany. Advances in Heat Transfer 39 (2006)
14
Verify Nanowire LAMMPS Implementation!
q  Compare to Ref. [*]
W/m/K
Pure Si wire
PPG wire
Wm-1K-1!
(defined later)
Our work
(LAMMPS)
4.1 +/- 0.4
0.12 +/- 0.03
Ref. [*]
(XMD)
4.1 +/- 0.3
0.23 +/- 0.05
q  Great agreement
Ø  Main sources of discrepancy
•  Thermalization techniques
–  Surface treatment
•  MD software
•  Thermalization times
*M. Chan et al. Physical Review B 81.17 (2010)
15
Nanowires of Random Si-Ge Concentration!
q  Data set of 145 wires with random Si-Ge concentrations
Ø  The “random wire (RW) data set”
Distributed as expected
q  Fit data with surrogate model
Ø  Use ATAT with ghost lattice method*
*Kristensen, Jesper, and Nicholas J. Zabaras. Physical Review B 91.5 (2015)
16
Fitting Thermal Conductivities!
q  Employing the fit with the CE-GLM we find
CE-GLM (W/m.K)
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
(a)
MD noise is large
(but as expected*)
0
1.6
LAMMPS (MD)
RW train
RW test
SPPG
PPG
1.4
CE-GLM (W/m.K)
q  Explore configuration
space:
1.2
1.0
0.8
0.6
0.4
0.2
0
0
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
(b)
Molecular dynamics (W/m.K)
18 SPPG wires
*M. Chan et al. Physical Review B 81.17 (2010)
17
Lowest-Thermal-Conductivity Structure!
CE-GLM (W/m.K)
1.6
1.4
1.2
1.0
0.8
0.6
0.4
RW train
RW test
SPPG
PPG
0.2
(a)
0
1.6
CE-GLM (W/m.K)
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0
(b)
q  We find the PPG to have
lowest thermal conductivity
0
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
Molecular dynamics (W/m.K)
SPPGs generally lower than
RW train and test sets as expected
18
Great Comparison with Literature!
From Ref. [*] on the same problem
(using a different surrogate model and MD software)
They found as well that the PPG wire has lowest κ
(this image of the PPG wire is from Ref. [*])
*M. Chan et al. Physical Review B 81.17 (2010)
19
Questions?!
Kristensen, Jesper, and Nicholas J. Zabaras
"Predicting low-thermal-conductivity Si-Ge nanowires with a
modified cluster expansion method.”
Physical Review B (2015)
20