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)

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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

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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

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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)

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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

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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)

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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)

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