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

Jonathan Skelton
September 06, 2022

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

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

Jonathan Skelton

September 06, 2022
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  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: