How does lattice thermal conductivity “work”? Insights from first-principles calculations

Presented at the 42nd Collaborative Computational Chemistry No. 5 (CCP5) Annual General Meeting.

Transcript

1. J. M. Skelton, J. Cen, J. M. Flitcroft, M. Molinari, S. Moxon, I. Pallikara, J. Tang, J. Tse and B. Wei Department of Chemistry, University of Manchester (jonathan.skelton@manchester.ac.uk) How does lattice thermal conductivity “work”? Insights from first-principles calculations
2. Motivation: thermoelectrics CCP5 42nd AGM, 6th Sept 2022 | Slide 2 Dr Jonathan M. Skelton 𝑍𝑇 = 𝑆!𝜎 𝜅"#" + 𝜅#$% 𝑇 𝑆 - Seebeck coefficient 𝜎 - electrical conductivity 𝜅!"! - electronic thermal conductivity 𝜅"#$ - lattice thermal conductivity G. Tan et al., Chem. Rev. 116 (19), 12123 (2016)
3. Modelling thermal conductivity Dr Jonathan M. Skelton A. Togo et al., Phys. Rev. B 91, 094306 (2015) 𝜿#$%% (𝑇) = 1 𝑁𝑉& . ' 𝜿'(𝑇) 1 𝑁𝑉& . ' 𝐶'(𝑇)𝒗' ⊗ 𝒗'𝜏'(𝑇) The simplest model for 𝜅"#$$ is the single-mode relaxation time approximation (SM-RTA) - a closed solution to the phonon Boltzmann transport equations Modal heat capacity Mode group velocity 𝜕𝜔% 𝜕𝐪 Average over phonon modes λ Phonon MFP Mode lifetime 𝜏% = 1 2Γ% 𝚲& 𝑇 = 𝒗& 𝜏& 𝑇 CCP5 42nd AGM, 6th Sept 2022 | Slide 3
4. Modelling thermal conductivity Dr Jonathan M. Skelton A. Togo et al., Phys. Rev. B 91, 094306 (2015) J. Tang and J. M. Skelton, J. Phys.: Condens. Matter 33 (16), 164002 (2021) CoSb3 CCP5 42nd AGM, 6th Sept 2022 | Slide 4
5. The RTA model: modal properties Dr Jonathan M. Skelton J. Tang and J. M. Skelton, J. Phys.: Condens. Matter 33 (16), 164002 (2021) CCP5 42nd AGM, 6th Sept 2022 | Slide 5
6. The RTA model: modal properties Dr Jonathan M. Skelton CoSb3 CoSb3 J. Tang and J. M. Skelton, J. Phys.: Condens. Matter 33 (16), 164002 (2021) CCP5 42nd AGM, 6th Sept 2022 | Slide 6
7. The RTA model: modal properties Dr Jonathan M. Skelton A. Gold-Parker et al., PNAS 115 (47), 11905 (2018) GaAs MAPbI3 CCP5 42nd AGM, 6th Sept 2022 | Slide 7
8. 𝒗3 vs. 𝜏3 : the CRTA model Dr Jonathan M. Skelton Consider again the SM-RTA model: 𝜿"#$$ = 1 𝑁𝑉' 3 & 𝜿& = 1 𝑁𝑉' 3 & 𝐶& 𝒗& ⊗ 𝒗& 𝜏& Replace the 𝜏& with a constant lifetime (relaxation time) 𝜏()*+ defined as follows: 𝜿"#$$ 𝜏()*+ = 1 𝑁𝑉' 3 & 𝜿& 𝜏& = 1 𝑁𝑉' 3 & 𝐶& 𝒗& ⊗ 𝒗& 𝜿"#$$ ≈ 1 𝑁𝑉' 3 & 𝐶& 𝒗& ⊗ 𝒗& ×𝜏()*+ HA AH HA AH J. Tang and J. M. Skelton, J. Phys.: Condens. Matter 33 (16), 164002 (2021) CCP5 42nd AGM, 6th Sept 2022 | Slide 8
9. 𝒗3 vs. 𝜏3 : Si clathrates Dr Jonathan M. Skelton B. Wei et al., submitted CCP5 42nd AGM, 6th Sept 2022 | Slide 9
10. 𝒗3 vs. 𝜏3 : Si clathrates Dr Jonathan M. Skelton B. Wei et al., submitted 𝜿!"## ≈ 1 𝑁𝑉$ & % 𝐶% 𝒗% ⊗ 𝒗% ×𝜏&'() CCP5 42nd AGM, 6th Sept 2022 | Slide 10
11. ⁄ 𝜿 𝜏4567: Si clathrates Dr Jonathan M. Skelton ⁄ 𝜿 𝝉𝐂𝐑𝐓𝐀 (W m-1 K-1 ps-1) 𝒏𝐚 Spacegroup d-Si 5.002 2 𝐹𝑑0 3𝑚 oC24 2.295 12 𝐶𝑚𝑐𝑚 K-II / C-I 0.829 46 𝑃𝑚0 3𝑚 K-V / C-VI 0.815 40 𝐶𝑚𝑚𝑚 K-VII / C-V 0.770 68 𝑃6&/𝑚𝑚𝑐 C-II 0.458 34 𝐹𝑑0 3𝑚 With the exception of the Clathrate-II structure, the harmonic ⁄ 𝜿 𝜏!"#$ term correlates with: (1) the size of the primitive cell (𝑛% ); and (2) the spacegroup (crystal symmetry) Indicates that low group velocities are favoured by complex structures with large primitive cells and/or low symmetry B. Wei et al., submitted CCP5 42nd AGM, 6th Sept 2022 | Slide 11
12. Analysing 𝜏4567: phonon linewidths Dr Jonathan M. Skelton Γ% (𝑇) = & %&%&& Φ8%%&%&& 9×{ 𝑛%&(𝑇) − 𝑛%&&(𝑇) 𝛿 𝜔 + 𝜔%& − 𝜔%&& − 𝛿 𝜔 − 𝜔%& + 𝜔%&& + 𝑛%&(𝑇) + 𝑛%&&(𝑇) + 1 𝛿 𝜔 − 𝜔%& − 𝜔%&& } Collision Decay Three-phonon interaction strength - includes conservation of momentum (“anharmonicity”) Conservation of energy (“selection rules”) A. Togo et al., Phys. Rev. B 91, 094306 (2015) CCP5 42nd AGM, 6th Sept 2022 | Slide 12
13. Dr Jonathan M. Skelton A. Togo et al., Phys. Rev. B 91, 094306 (2015) Approximate expression for Γ! : With: Γ% (𝑇) ≈ 18𝜋 ℏ9 < 𝑃𝑁9 (𝒒% , 𝜔% , 𝑇) 𝑁9 𝒒%, 𝜔%, 𝑇 = 𝑁9 (;) 𝒒%, 𝜔%, 𝑇 + 𝑁9 (9) 𝒒%, 𝜔%, 𝑇 𝑁9 (;) 𝒒% , 𝜔% , 𝑇 = 1 𝑁 & %&%&& ∆(−𝒒% + 𝒒%& + 𝒒%&&) 𝑛%&(𝑇) − 𝑛%&&(𝑇) × 𝛿 𝜔 + 𝜔%& − 𝜔%&& − 𝛿 𝜔 − 𝜔%& + 𝜔%&& 𝑁9 (9) 𝒒% , 𝜔% , 𝑇 = 1 𝑁 & %&%&& ∆(−𝒒% + 𝒒%& + 𝒒%&&) 𝑛%&(𝑇) + 𝑛%&&(𝑇) + 1 𝛿 𝜔 − 𝜔%& − 𝜔%&& CCP5 42nd AGM, 6th Sept 2022 | Slide 13 Analysing 𝜏4567: phonon linewidths
14. Analysing 𝜏3 Dr Jonathan M. Skelton B. Wei et al., submitted Γ%(𝑇) ≈ 18𝜋 ℏ9 < 𝑃𝑁9(𝒒%, 𝜔%, 𝑇) CCP5 42nd AGM, 6th Sept 2022 | Slide 14
15. Summary Dr Jonathan M. Skelton 𝜿!"## ⁄ 𝜿 𝜏&'() 𝜏&'() B 𝑁9 < 𝑃 CCP5 42nd AGM, 6th Sept 2022 | Slide 15 RTA model gives good results for most systems and provides microscopic detail at the level of individual phonon modes Allows differences in 𝜿!"## to be attributed to differences in group velocities and phonon lifetimes Allows differences in lifetimes to be attributed to selection rules and (anharmonic) phonon interaction strengths
16. CRTA analysis: other TEs Dr Jonathan M. Skelton 𝜅 [W m-1 K-1] ⁄ 𝜅 𝝉𝐂𝐑𝐓𝐀 [W m-1 K-1 ps-1] 𝝉𝐂𝐑𝐓𝐀 [ps] Si 136.24 5.002 27.2 SnS 2.15 0.718 3.00 SnSe 1.58 0.372 4.23 CoSb3 9.98 0.273 36.6 Bi2 S3 (Pnma) 0.90 0.423 2.14 Bi2 Se3 (R-3m) 1.82 0.293 6.20 Bi2 Te3 (R-3m) 0.87 0.199 4.41 J. M. Skelton, J. Mater. Chem. C 9, 11772 (2021) J. Tang and J. M. Skelton, J. Phys.: Condens. Matter 33 (16), 164002 (2021) J. Cen, I. Pallikara and J. M. Skelton, Chem. Mater. 33 (21), 8404 (2021) B. Wei et al., submitted CCP5 42nd AGM, 6th Sept 2022 | Slide 16
17. Approximating Γ3 Dr Jonathan M. Skelton CCP5 42nd AGM, 6th Sept 2022 | Slide 17 S. Moxon et al., J. Mater. Chem. A 10, 1861 (2022) Γ%(𝑇) ≈ 18𝜋 ℏ9 < 𝑃𝑁9(𝒒%, 𝜔%, 𝑇)
18. Acknowledgements Dr Jonathan M. Skelton CCP5 42nd AGM, 6th Sept 2022 | Slide 18 B. Wei
19. https://bit.ly/3RohkNG These slides are on Speaker Deck: