Thermal transport from first-principles DFT calculations. Keivan Esfarjani MIT. Department of Mechanical Engineering. 5/23/2012 Phonon UWM 1

Transcription

1 Thermal transport from first-principles DFT calculations Keivan Esfarjani Department of Mechanical Engineering MIT 5/23/2012 Phonon UWM 1

2 Classical MD simulations use an empirical potential fitted to reproduce lattice constant, cohesive energy, elastic properties, some phonons But are not supposed to produce correct third derivatives of the potential energy, determining phonon lifetimes They can be used to explain trends in thermal transport, but can not predict quantitatively the thermal conductivity. 5/23/2012 Phonon UWM 2

3 Goal To develop a general methodology to get accurate estimates of thermal properties using available density functional theory (DFT) tools. To make accurate / reliable predictions without fitting any parameters to experimental data. 5/23/2012 Phonon UWM 3

4 Outline New force field based on FP-DFT Approaches for thermal conductivity calculation + illustrations Lattice dynamics model MD Green s functions for coherent transport Conclusions 5/23/2012 Phonon UWM 4

5 Force Force field model 1 F E u u u i i i ij j ijk j k j, 2! jk, 1 3! displacements jkl, ijkl j k l 2 3 E E E i ij ijk ui ui u j ui u j u k 4 ijkl ui u j uk ul u u u ; ; ; E Derivatives are calculated at or near equilibrium Exact for small displacements! 5/23/2012 Phonon UWM 5

6 As the FCs are 2 nd or higher rank tensors, there are a huge number of terms in the Taylor expansion! 9 for each pair * number of pairs 27 for each triplet * number of triplets 81 for each quadruplet * number of quadruplets They need to be reduced Method to extract anharmonic force constants from first principles calculations Keivan Esfarjani and Harold T. Stokes Phys. Rev. B 77, (2008) 5/23/2012 Phonon UWM 6

7 Symmetry constraints The FCs are related by the following symmetry constraints: Permutation of the order of derivatives ij ji Global Translational invariance ij ijk j j 0 ; 0 5/23/2012 Phonon UWM 7 Method to extract anharmonic force constants from first principles calculations Keivan Esfarjani and Harold T. Stokes, Phys. Rev. B 77, (2008)

8 Symmetry constraints Global Rotational invariance Links FCs to higher-order ones R R ; R Group symmetry operation invariance ' ' ij ' ' S(i)S( j) ' ' These symmetries must be enforced for any FC model to be physically correct R S S... 5/23/2012 Phonon UWM 8

9 Methodology : FP-DFT calculations in a supercell (real space) Supercell size should ideally be larger than the range of the force constants 5/23/2012 Phonon UWM 9

10 Extraction of FCs: 1 F u u u... i i ij j ijk j k j, 2! jk, Constraints are linear in FCs Force-displacement relations also linear in FCs U Invariance constraints 0 F FP-DFT Force-displacement data Over-complete system of linear equations FCs are deduced from a SVD algorithm Method to extract anharmonic force constants from first principles calculations Keivan Esfarjani and Harold T. Stokes, Phys. Rev. B 77, (2008) 5/23/2012 Phonon UWM 10

11 How accurate are the obtained FCs? Move one Si atom along [111] 5/23/2012 Phonon UWM 11

12 More validation Use the FCs to calculate forces for arbitrary atomic displacements Denominator s(poly) s(sw) DISP=0.1 Ang DISP=0.2 Ang Error is due to choosing (5,1,1) neighbors in the model 5/23/2012 Phonon UWM 12

13 Now what? Use second derivatives to get phonon dispersion Use the third derivatives to get phonon lifetimes Use all to do MD Alternative approach to determining FCs uses DFPT (See talk by Derek Stewart) 5/23/2012 Phonon UWM 13

14 Summary of the approach FP-DFT in a supercell MD-GK Model Potential Thermal Properties Latt Dyn 5/23/2012 Phonon UWM 14

15 Phonon Dispersion For Si, Graphene, ZrCoSb N. Mingo, K. Esfarjani, D. A. Broido, and D. A. Stewart Phys. Rev. B 81, (2010) K. Esfarjani, G. Chen, and H. T. Stokes Phys. Rev. B 84, (2011) k d ln d ln V k ZrCoSb Half-heusler; By J. Shiomi, KE, G. Chen, PRB 84 (2011) 5/23/2012 Phonon UWM 15

16 Lifetimes due to 3-phonon processes G q2 q2 q2 q3 q i q i q i q1 Normal process q1 Umklapp process q1 a 4-phonon process V 1 3! ijk, ijk u i u j u k 1 4! ijkl ijkl, u i u j u k u l... Phonon creation and annihilation operators W( i, f ) 2 i V f 2 ( E i E f ) Fermi Golden Rule 5/23/2012 Phonon UWM 16

17 Thermal conductivity Use of the relaxation time approximation Relaxation time C v per mode 5/23/2012 Phonon UWM 17

18 ZrCoSb lifetimes: LD-FGR versus MD 70 Velocity autocorrelation (5x5x5 supercell=1000 Si atoms) Frequency (Thz) 5/23/2012 Phonon UWM 18 Shiomi, Esfarjani, Chen, Phys. Rev. B 84, (2011)

19 Thermal conductivity of ZrCoSb Simulations vs Experiments Impurity scattering 1/ 1/ B 0 4 Sb Zr Co Y. Xia et al., J. Appl. Phys. 88, 1952 (2000). T. Sekimoto et al., Jpn. J. Appl. Phys. 46, L673 (2007). Shiomi, Esfarjani, Chen, Phys. Rev. B 84, (2011)

20 Alloying effect on half-heusler thermal conductivity Transferability of force constants 22 Wm -1 K -1 Sb Co Zr ZrCoSb Only changing mass k (Wm -1 K -1 ) Green Kubo Calculation (300 K) Hf x Zr 1-x CoSb Hf 1-x Ti x CoSb Zr x Ti 1-x CoSb Experiment Hf 0.5 Zr 0.5 CoSb 0.8 Sn 0.2 (Yan, Nano Lett, 2011) 5 16 Wm -1 K -1 Sb Hf Co HfCoSb x Shiomi, Esfarjani, Chen, Phys. Rev. B 84, (2011)

21 Gruneisen parameter Frequency (THz) Frequency (mev) Thermal conductivity of PbTe Microscopic mechanism of low thermal conductivity in lead telluride T Shiga, J Shiomi, J Ma, O Delaire, T Radzynski, A Lusakowski, K E, and G Chen Phys. Rev. B 85, (2012) Takuma Shiga 4 This work W. Cochran, et al., (1966) 16 Why is thermal conductivity of PbTe so low (~2 Wm -1 K TO 0 0 Rocksalt PbTe X K L

22 Phonons in lead telluride Microscopic mechanism of low thermal conductivity in lead telluride T Shiga, J Shiomi, J Ma, O Delaire, T Radzynski, A Lusakowski, K E, and G Chen Phys. Rev. B 85, (2012) Takuma Shiga

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24 Summary of LD: MFP distribution 5/23/2012 Phonon UWM 24

25 MD using the polynomial force field PROS: Of accuracy similar to DFT Includes anharmonic effects to higher order CONS: Can not describe structural changes Hard to include surfaces and defects Ideal for thermal conductivity using GK 5/23/2012 Phonon UWM 25

26 Equilibrium Green-Kubo (GK) 5/23/2012 Phonon UWM 26

27 k (W/mK) Half-Heuslers 25 k xx 20 k yy k zz t (ps) 5/23/2012 Phonon UWM 27

28 Silicon 5/23/2012 Phonon UWM 28

29 Wave effects in thermal transport Quantization of thermal conductance Coherent (ballistic) phonon contribution 5/23/2012 Phonon UWM 29

30 Coherent phonon transport Landauer formula d n(t, ) G D m( ) v gm ( ) m 2 T 0 Transmission formula = 1 5/23/2012 Phonon UWM 30

31 Experimental samples 12 nm AlAs 9 periods 7 periods 5 periods 12 nm GaAs 3 periods 1 period ~100 nm Al 12 nm GaAs 12 nm AlAs 350μm GaAs 5/23/2012 Phonon UWM 31

32 GaAs-[(AlAS) 22 /(GaAs) 22 ] N=1,,9 -Al GaAs AlAs GaAs AlAs GaAs Al 1 unit= 22 layers AlAs+22 layers GaAs Period length=2*(22*5.4 Ang)=23.76 nm N-periods (N=1,3,5,7,9); 3x3 cross section for a total of 1584,,up to atoms 5/23/2012 Phonon UWM 32

33 Transmission and Conductivity (a) (b) 5/23/2012 Phonon UWM 33

34 Conclusions The developed force field can be calculated for an arbitrary crystal structure It can be used to: Do LD, get phonon spectra, lifetimes, k (fast) Do MD, get lifetimes and k including alloy scattering and anharmonic effects nonperturbatively Set up the GFs to calculate transmission & conductance LD faster and more accurate than GK-MD, but valid at low T In nanostructured materials: Wave effects are important, Heat transport can have a large ballistic component Boundary scattering can be dominant Optical modes contribution is increased 5/23/2012 Phonon UWM 34

35 Acknowledgements Gang Chen (MIT) Junichiro Shiomi (U Tokyo) ZrCoSb, PbTe Takuma Shiga (U Tokyo) PbTe Maria Luckyanova (MIT) GaAs/AlAs SLs J. Garg (MIT) GaAs/AlAs phonon lifetimes TF Luo (Notre Dame) GaAs 5/23/2012 Phonon UWM 35