Modeling thermal conductivity: a Green-Kubo approach

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1 Modeling thermal conductivity: a Green-Kubo approach Fabiano Oyafuso, Paul von Allmen, Markus Bühler Jet Propulsion Laboratory Pasadena, CA Funding: DARPA

2 Outline Motivation -- thermoelectrics Theory Implementation Preliminary results

3 Motivation: Thermoelectrics Goal: Optimize ZT (efficiency) engineering of phonon & electronic energy dispersions Focus of this work is on lattice contribution, κ ph

4 Motivation: Optimization of ZT First principles electronic structure Figure of Merit Parameter fitting (Genetic Algorithm) Boltzmann Transport Equation Green Kubo formalism Tight binding calculation Electronic contribution: - Seebeck coefficient - Electrical conductivity - Thermal conductivity Lattice contribution: - Thermal conductivity First principles Force Fields Boltzmann Transport Equation Green Kubo formalism

5 Lattice Contribution to Thermal Conductivity Goals / General Approach: Principal goal is the optimization of thermoelectric figure of merit ZT. Validate code by computing λ for C, Si, Ge Compute λ for various bulk materials (e.g. Bi 2 Te 3 ) Generalize to simple nanostructures. Technical Approach: Can compute λ by MD or BTE One option is NEMD (non-equilibrium), but NEMD requires large temperature gradients boundary condition issues at interfaces Instead pursue Green-Kubo approach: Fluctuation-Dissipation theorem: Status: Derived analytic expression for heat flux for Tersoff potential. Implemented heat flux calculation in IMD (ITAP) code and integrated output thereof into separate code to compute thermal conductivity. Computed λ for diamond structures (C,Si,Ge). Implementation: Modified pre-existing software, ITAP: Two options: numerical/analytic differentiation compute heat flux. Modified IMD code compute j(t) and save to binary file Postprocessing Module compute heat auto correlation function integrate to compute thermal conductivity

6 Evaluation of heat current Symmetrized Tersoff:

7 Convergence of auto-correlation function Convergence requires a large number of ensembles: tens of ns required

8 Three approaches: Direct integration Fit frequency dependent thermal conductivity to single relaxation time approximation (Volz/Chen). Conductivity calculation: three approaches Fit autocorrelation function directly to a small set of relaxation times (Che et al.) Ansatz:

9 Results: λ(n)

10 Results: C, Si, Ge Material C λ (calc) 1390 λ (calc) 1200 (Brenner) λ (exp) Che/Cagin/Deng/Goddard C, natural (Brenner) 2190 Si 180 Che/Cagin/Deng/Goddard 230 (Stillinger-Weber) Anthony et al. 130 (natural) Volz/Chen Capinski/Maris/ et al Ge (Tersoff) 62 (natural) Dong/Sankey/Myles Slack/Glassbrenner * λ: units of W/m/K Good agreement with previous theoretical calculations. Reasonable agreement with experiment -- trends are correct. Ratio λ( 12 C) / λ(c natural ) = 1.4 in agreement with experiment.

11 Summary Demonstrated modification of existing software to compute thermal conductivities using a Green- Kubo approach. Evaluated several convergence schemes. Obtained preliminary results for diamond-like materials in agreement with previous theoretical models and experiment.

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