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:
2
ZT =S2�T
Seebeck coefficient Electrical conductivity
Temperature of device Thermal conductivity (electrons + phonons)
Amato, Michele, et al. Chemical reviews 114.2 (2013)
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
Ø Semiconductors:
• Heat conduction primarily due to phonons
3 *Kim, Hyoungjoon, et al. Applied Physics Letters 96.23 (2010)
ZT =S2�T
⇡ lattice
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
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
Heat transferred across temperature gradient
Cold reservoir Hot reservoir
Example of “Direct Method” Implementation!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
Ø Steady state:
Ø Nanowires: huge temperature gradients are created! Ø Fourier’s law not rigorously proved for
microscopic Hamiltonian*
5
Jz =�✏
2A�t
Typical temperature profile
Nonlinear effects
Linear region: Get T gradient
Jµ = �X
⌫
µ⌫@T
@x⌫
Schelling, Patrick K., Simon R. Phillpot, and Pawel Keblinski. Physical Review B 65.14 (2002) *Amato, Michele, et al. Chemical reviews 114.2 (2013)
EMD: Green-Kubo!q Benefit: Linear response regime q Drawback: Very long simulation times needed
Ø Including longer times in integral introduces significant noise
q Definition of heat current
q For 3-body interaction (such as Tersoff*) we define the potential as:
6
µ⌫(⌧m) =1
V kBT 2
Z ⌧m
0hJµ(⌧)J⌫(0)id⌧
2-body force on atom i due to its neighbor j
3-body force
Heat current autocorrelation function (HCACF)
Schelling, Patrick K., Simon R. Phillpot, and Pawel Keblinski. Physical Review B 65.14 (2002) *Tersoff, J. Physical Review B 38.14 (1988)
J =d
dt
X
i
ri(t)"i(t)
J =X
i
vi"i +1
2
X
ij,i 6=j
rij (F ij · vi) +1
6
X
ijk
(rij + rik) (F ijk · vi)
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
7
HCACF
Fourier transform of raw heat current
F�1⇣F(J(t))(⌫)F(J(t))(⌫)
⌘(t)
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
8
*Plimpton, Steve. Journal of computational physics 117.1 (1995) **Chan, M. K. Y., et al. Physical Review B 81.17 (2010)
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
9
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!
We use the method from Ref. [*]:
F (t) ⌘�����(cor(t))
E(cor(t))
����
Numerical noise takes over
*J. Chen, G. Zhang, and B. Li. Physics Letters A 374.23 (2010)
Decay is exponential (shown in log)
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)
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
10
Length: 2 nm Surface: Pristine
Length: 50 nm Surface: Rough
~220 atoms >5500 atoms
*M. Chan et al. Physical Review B 81.17 (2010)
Preparing Nanowire for LAMMPS!
11
q Nanowire for LAMMPS (visualized in OVITO*)
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
Parse with LAMMPS
*Stukowski, Alexander. Modelling and Simulation in Materials Science and Engineering 18.1 (2010)
Solving Green-Kubo with LAMMPS!
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)
12
µ⌫(⌧m) =1
V kBT 2
Z ⌧m
0hJµ(⌧)J⌫(0)id⌧
*Amato, Michele, et al. Chemical reviews 114.2 (2013)
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
13
Annealing scheme (not to scale)
Time
Tem
pera
ture
(K) 1000
300
100 K In our work Annealing essential to good results. Other possibility: Langevin thermostat or variants thereof (not explored in depth).
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
q 40 moving averages of various window sizes (50 to 200 ps) Ø Convergence: Minimum standard deviation time gives upper limit
14 *McGaughey, Alan JH, and M. Kaviany. Advances in Heat Transfer 39 (2006)
(Figure from Ref. [*])
µ⌫(⌧m) =1
V kBT 2
Z ⌧m
0hJµ(⌧)J⌫(0)id⌧
Verify Nanowire LAMMPS Implementation!
15
q Compare to Ref. [*]
Wm
-1K
-1!
W/m/K Pure Si wire PPG wire (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)
Nanowires of Random Si-Ge Concentration!q Data set of 145 wires with random Si-Ge concentrations
Ø The “random wire (RW) data set”
q Fit data with surrogate model Ø Use ATAT with ghost lattice method*
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Distributed as expected
*Kristensen, Jesper, and Nicholas J. Zabaras. Physical Review B 91.5 (2015)
Fitting Thermal Conductivities!q Employing the fit with the CE-GLM we find
q Explore configuration space:
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MD noise is large (but as expected*)
CE
-GLM
(W
/m.K
)
Molecular dynamics (W/m.K)
CE
-GLM
(W
/m.K
)
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
(a)
(b)
0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.6
0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
RW trainRW testSPPGPPG
*M. Chan et al. Physical Review B 81.17 (2010) 18 SPPG wires
LAMMPS (MD)
q We find the PPG to have lowest thermal conductivity
18
CE
-GLM
(W
/m.K
)
Molecular dynamics (W/m.K)
CE
-GLM
(W
/m.K
)
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
(a)
(b)
0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.6
0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
RW trainRW testSPPGPPG
SPPGs generally lower than RW train and test sets as expected
Lowest-Thermal-Conductivity Structure!
19
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
*M. Chan et al. Physical Review B 81.17 (2010)
(this image of the PPG wire is from Ref. [*])
Great Comparison with Literature!
Questions?!
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Kristensen, Jesper, and Nicholas J. Zabaras
"Predicting low-thermal-conductivity Si-Ge nanowires with a modified cluster expansion method.”
Physical Review B (2015)