Predicting Thermoelectric Properties From First Principles

Transcription

1 Predicting Thermoelectric Properties From First Principles Paul von Allmen, Seungwon Lee, Fabiano Oyafuso Abhijit Shevade, Joey Czikmantory and Hook Hua Jet Propulsion Laboratory Markus Buehler, Haibin Su, Adri Van Duin, William Goddard III California Institute of Technology Collaboration: Art Freeman, Northwestern University Tahir Cagin, Texas U&M University Funding: DARPA, JPL, ONR

2 State-of-the-Art Thermoelectric Materials Strategy for improving ZT Alloy PbSeTe/PbTe QD Quantum Confinement Phonon-blocking/ Electron-transmitting Interface R. Venkatasubramanian, E. Siivola, Th. Colpitts and B. O Quinn, Nature 413, 597 (2001) T.C. Harman, P.J. Taylor, M.P. Walsh and B.E. LaForge, Sci. 297, 2229 (2002)

3 Computing Thermoelectric Properties First principles electronic structure Parameter fitting (Genetic Algorithm) Boltzmann Transport Equation Green Kubo formalism Tight binding calculation Figure of Merit ZT = S2 σt κ e +κ ph Electronic contribution: - Seebeck coefficient - Electrical conductivity - Thermal conductivity Lattice contribution: - Thermal conductivity First principles Force Fields Boltzmann Transport Equation Green Kubo formalism

4 Atoms Materials Design Software - EZTB StructureBuilder Optics ForceFieldKeating Electrons Absorption ForceFieldReaxFF ElasticMinimization CrystalStructure TightBinding Chi2 Chi3 χ (2) χ (3) Atomic Positions EigenSolver BandStructure Band Structure Density of States ab initio data Experimental data LanczosSolver Minimization GeneticAlgorithm Transport RelaxationTime Boltzmann ZT Completed Initiated PoissonSolver GreenKubo Caltech New NonEquilGreenFct

5 EZTB - Software Technology 1. C++ Object Oriented eases development and maintenance 2. Python and SWIG wrapping Improves portability Facilitates development of new process flows Integration into CMDF 3. XML data representation Improves portability Enables scripting instructions 4. Web Portal Middleware (JPL-WIGLAF) Simple remote browser access Job monitoring on cluster computers

6 Web Enabling Materials Design Software Example python script Read crystal Set up crystal Set up H Diagonalize H

7 Tight Binding Model Tight Binding Hamiltonian H = (ν ) ε i + t ij iν ijµν (µν ) i: neighbors (1st, 2nd, ) ν: sp 3 d 5 s*, 20 orbitals Spin-orbit term: on-site Structure Definition - Uniform Bulk: primitive vectors, atom basis - Nanostructures: atomic positions - Material: tight binding parameters fitted to ab initio and experimental data σ anti-bonding sp σ bond P x P x σ bonding π anti-bonding + = P x P x π bonding

8 Empirical Relaxation Time Approximation Linear response expressions for the charge and heat currents: Electrical conductivity: σ = L 11 j e = L 11 E + L 12 ( T ) j q = L 21 E + L 22 ( T ) Thermal conductivity (j e =0): Seebeck coefficient (j e =0): κ e = L 22 L 21 ( L 11 ) 1 L 12 S = ( L 11 ) 1 L 12 Linearized Boltzmann Transport Equation yields the linear coefficients: L 11 = Λ (0) L 21 = TL 12 = 1 e Λ(1) L 22 = 1 e 2 T Λ(2) Λ (α) mn = e 2 τ v n (k) = 1 h ε(k) k n d 3 k 4π 3 f v ε m (k)v n (k) ε(k) µ ( ) α Relaxation time fitted to experiment

9 Silicon Band Structure Empirical tight-binding method with spds* orbitals Density of States (a.u.) Energy (ev) E gap =1.12 ev

10 Relaxation time (n-si) (from fitting to experimental mobility) Computed ZT for Bulk n-si Relaxation Time (s) 2.5E E E E E-14 ZT Experiment Theory T=300K 0.0E E+14 1.E+15 1.E+16 1.E+17 1.E+18 1.E+19 1.E+20 Experiment Theory Electron Density (1/cm^3) n-ge E+14 1.E+16 1.E+18 1.E+20 Electron Density (1/cm^3) n-bi 2 Te 3 ZT E+13 1.E+15 1.E+17 1.E+19 1.E+21 Electron Density (1/cm^3)

11 n-doped Silicon Nanowires ZT figure of merit Si-NW 2x2 0.2 Si-NW 1x1 Si-NW 2x T=300K ZT E+15 1.E+17 1.E+19 Density (1/cm^3) - Semi-classical model for the electronic contribution to transport - Lattice contribution to the thermal conductivity adjusted for finite-size effects J 1.E+21

12 Boltzmann Transport Equation (BTE) f n( k) ef v n (k)+ f n( k) ε n T T v (k) = C (k) n n Beyond Relaxation Time Approximation C n (k) = V d k S 8π 3 n n (k, k ) f n n Electrical conductivity: Thermal conductivity: v n (k) = 1 ε(k) h k n { ( k) ( 1 f ( )) + S k,k) f ( k )( 1 f n ( k) )} σ αβ = e2 F β el κ αβ Seebeck coefficient: S = F T = e2 β T n n n k n n ( d 3 k 4π 3 f n (k)v α (k) n f is obtained from BTE with F along e β and with T=0 d 3 k 4π f (k)v 3 n α (k)( ε(k) µ ) For a given T, F is such that J el =0 f is obtained from BTE with T along e β and F such that J el =0

13 Targets 1st principles experiments Sensitivity Studies Inverse Problem optimization δp = C pt δt Tight-Binding Parameters Direct Model Eigenvalue problem δt = C tp δp Variation of transport coefficients Physical Quantities Conductivity Optical properties σ i (t +δt) = σ i (t)+ δt α Correlation matrices = = ( C pt ) βα

14 Silicon Band Structure Parametrized Tight- Binding method with spds* orbitals T. Boykin et al, Phys. Rev. B 69, (2004)

15 Silicon Band Structure Parametrized Tight- Binding method with spds* orbitals T. Boykin et al, Phys. Rev. B 69, (2004)

16 N Targets 1st principles experiments = Sensitivity Studies Tight-Binding Parameters Direct Model Eigenvalue problem = Inverse Problem optimization δp = C pt δt δt = C tp δp Variation of transport coefficients σ i (t +δt) = σ i (t)+ δt α Correlation matrices ( C pt ) βα Physical Quantities Conductivity Optical properties Computational process I 1. Compute Compute 4. M C pt = C tp 1 N = M = ( C tp ) αβ = ( C pt ) βα

17 N Targets 1st principles experiments = Sensitivity Studies Tight-Binding Parameters Direct Model Eigenvalue problem = Inverse Problem optimization δp = C pt δt δt = C tp δp Variation of transport coefficients σ i (t +δt) = σ i (t)+ δt α Correlation matrices ( C pt ) βα Target evaluation inaccurate Physical Quantities Conductivity Optical properties Computational process I 1. Compute Compute 4. M C pt = C tp 1 N = M = ( C tp ) αβ = ( C pt ) βα

18 N Targets 1st principles experiments p α t β Sensitivity Studies Inverse Problem optimization δp = C pt δt σ i (t +δt) = σ i (t)+ δt α Tight-Binding Parameters Direct Model Eigenvalue problem δt = C tp δp Variation of transport coefficients Correlation matrices (optimization) p α (t 1,K,t β +δt β,k,t N ) p α (t) δt β Physical Quantities Conductivity Optical properties Computational process II 1. Compute 2. Compute 3. M N M = ( C pt ) αβ = p α ( C pt ) βα t β

19 N Targets 1st principles experiments p α t β Sensitivity Studies Inverse Problem optimization δp = C pt δt σ i (t +δt) = σ i (t)+ δt α Tight-Binding Parameters Direct Model Eigenvalue problem δt = C tp δp Variation of transport coefficients Correlation matrices (optimization) Optimization inaccurate p α (t 1,K,t β +δt β,k,t N ) p α (t) δt β Physical Quantities Conductivity Optical properties Computational process II 1. Compute 2. Compute 3. M = N M ( C pt ) αβ = p α ( C pt ) βα t β

20 N Targets 1st principles experiments Sensitivity Studies Inverse Problem optimization δp = C pt δt σ i (t +δt) = σ i (t)+ δt α Tight-Binding Parameters Direct Model Eigenvalue problem δt = C tp δp Variation of transport coefficients Correlation matrices (optimization) σ i(t 1,K,t β +δt β,k,t N ) σ i (t) δt α M Physical Quantities Conductivity Optical properties Computational process III 1. Compute N M

21 N Targets 1st principles experiments Sensitivity Studies Inverse Problem optimization δp = C pt δt σ i (t +δt) = σ i (t)+ δt α Tight-Binding Parameters Direct Model Eigenvalue problem δt = C tp δp Variation of transport coefficients Correlation matrices (optimization) σ i(t 1,K,t β +δt β,k,t N ) σ i (t) δt α M Physical Quantities Conductivity Optical properties Computational process III 1. Compute N M?

22 Lattice Contribution to the Thermal Conductivity 1. Monte Carlo solution of the Boltzmann Transport Equation for phonons Anharmonicity in Force Field Phonon scattering Thermal Conductivity Monte Carlo solution: Drift in real space Scattering (Normal, Umklapp) r(t+δt) = r(t) + v.δt First Principles Force Field P NU = 1-exp(Δt/τ NU ) τ NU --1 = τ U τ N --1 If random number < P NU phonon scattered Status Developed Monte Carlo Code Validating Code by comparing with Literature calculation for Si film Using empirical scattering rates Plans Compute thermal conductivity for bulk Si, Ge, Bi 2 Te 3 and Sb 2 Te 3 Substitute empirical scattering rates with first principal ones Thermal conductivity for nanostructures

23 Lattice Contribution to the Thermal Conductivity 2. Green-Kubo formula Heat Current Auto-Correlation Molecular Dynamics Heat current Fluctuation Dissipation theorem 1 λαβ (ω ) = Vk BT 2 Thermal Conductivity dt < j α (0) jβ (t) > e iωt j= 0 d Ei ri dt i N < jα (0) jβ (t) >= jα (t i ) jβ (t i + t) Heat current auto-correlation function i=0 Status First attempt for C, Tersoff potential Slow convergence of auto correlation function (1ns time-series for convergence) Plans Validate code for thermal conductivity of C, Si, Ge Compute λ for Bi2Te3 (develop appropriate FF). Explore various nanostructure designs

24 Conclusion Status 1. Electronic contribution to ZT Computed within the empirical relaxation time approximation Good agreement with experiment Initiated BTE solution for acoustic phonon and ionized impurity scattering (deformation potentials fitted, experimental phonon energies) 2. Lattice contribution Initiated code for BTE and Green-Kubo approach Future plans Complete outlined computational framework Reduce the number of fitting parameters by using first principles results for the scattering processes Apply to other material classes such as skutterudites Compute ZT for nano-patterned materials Propose materials with large ZT