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*

16

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:

17

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?!

20

Kristensen, Jesper, and Nicholas J. Zabaras

"Predicting low-thermal-conductivity Si-Ge nanowires with a modified cluster expansion method.”

Physical Review B (2015)