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Understanding and controlling the heat transport in thermoelectric materials

Understanding and controlling the heat transport in thermoelectric materials

Presented at the Thomas Young Centre (TYC) Symposium on "Modelling Phonons in Materials" on 26th January 2023.

Jonathan Skelton

January 23, 2023
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  1. Dr Jonathan Skelton Department of Chemistry, University of Manchester ([email protected])

    Understanding and controlling the heat transport in thermoelectric materials
  2. Acknowledgements TYC Seminar, 26th Jan 2023 | Slide 2 Dr

    Jonathan Skelton ... plus other students, mentors and collaborators too numerous to mention
  3. Overview TYC Seminar, 26th Jan 2023 | Slide 3 Dr

    Jonathan Skelton o Thermoelectric power and the global energy challenge o Modelling lattice thermal conductivity o Models for understanding 𝜅latt : • CRTA model - 𝒗λ vs. 𝜏λ • Constant 𝑃λ model - ഥ 𝑁2 (𝜔) vs. ෨ 𝑃 o Strategies for controlling 𝜅latt : • Reducing 𝒗λ - alloying and doping • Reducing 𝜏λ - “rattler” TEs o Modelling the thermoelectric figure of merit o Recent highlights and current work http://bit.ly/3H3ys7x
  4. The global energy challenge 34 % 26 % 19 %

    18 % 3 % 1000 MW nuclear power plant: o 650 MW waste heat o 3 % ≈ 20 MW ≈ 50,000 homes 300-500 W from exhaust gases: o 2 % lower fuel consumption o 2.4 Mt reduction in CO2 Thermoelectric generators allow waste heat to be recovered as electricity TEGs with ~3 % energy recovery (𝑍𝑇 = 1) are considered industrially viable 1. Provisional UK greenhouse gas emissions national statistics (published June 2020) 2. EPSRC Thermoelectric Network Roadmap (2018) TYC Seminar, 26th Jan 2023 | Slide 4 Dr Jonathan Skelton
  5. Thermoelectric materials 𝑍𝑇 = 𝑆2𝜎 𝜅ele + 𝜅lat 𝑇 𝑆

    - Seebeck coefficient 𝜎 - electrical conductivity 𝜅ele - electronic thermal conductivity 𝜅lat - lattice thermal conductivity G. Tan et al., Chem. Rev. 116 (19), 12123 (2016) TYC Seminar, 26th Jan 2023 | Slide 5 Dr Jonathan Skelton
  6. Modelling thermal conductivity A. Togo et al., Phys. Rev. B

    91, 094306 (2015) 𝜿latt 𝑇 = 1 𝑁𝑉0 ෍ 𝜆 𝜿𝜆 𝑇 = 1 𝑁𝑉0 ෍ 𝜆 𝐶𝜆 (𝑇)𝒗𝜆 ⊗ 𝒗𝜆 𝜏𝜆 (𝑇) The simplest model for 𝜅latt is the single-mode relaxation time approximation (SM-RTA) - a closed solution to the phonon Boltzmann transport equations Modal heat capacity Mode group velocity 𝜕𝜔λ 𝜕𝐪 Average over phonon modes λ Phonon MFP Mode lifetime 𝜏λ = 1 2Γλ 𝚲𝜆 𝑇 = 𝒗𝜆 𝜏𝜆 𝑇 TYC Seminar, 26th Jan 2023 | Slide 6 Dr Jonathan Skelton
  7. Modelling thermal conductivity TYC Seminar, 26th Jan 2023 | Slide

    7 Dr Jonathan Skelton
  8. Modelling thermal conductivity 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 TYC Seminar, 26th Jan 2023 | Slide 8 Dr Jonathan Skelton
  9. Modelling thermal conductivity J. Tang and J. M. Skelton, J.

    Phys.: Condens. Matter 33 (16), 164002 (2021) TYC Seminar, 26th Jan 2023 | Slide 9 Dr Jonathan Skelton
  10. Modelling thermal conductivity A. Gold-Parker et al., PNAS 115 (47),

    11905 (2018) GaAs (CH3 NH3 )PbI3 TYC Seminar, 26th Jan 2023 | Slide 10 Dr Jonathan Skelton
  11. 𝒗𝜆 vs. 𝜏𝜆 : the CRTA model Consider again the

    SM-RTA model: 𝜿latt = 1 𝑁𝑉0 ෍ 𝜆 𝜿𝜆 = 1 𝑁𝑉0 ෍ 𝜆 𝐶𝜆 𝒗𝜆 ⊗ 𝒗𝜆 𝜏𝜆 Replace the 𝜏𝜆 with a constant lifetime (relaxation time) 𝜏CRTA defined as follows: 𝜿latt 𝜏CRTA = 1 𝑁𝑉0 ෍ 𝜆 𝜿𝜆 𝜏𝜆 = 1 𝑁𝑉0 ෍ 𝜆 𝐶𝜆 𝒗𝜆 ⊗ 𝒗𝜆 𝜿latt ≈ 1 𝑁𝑉0 ෍ 𝜆 𝐶𝜆 𝒗𝜆 ⊗ 𝒗𝜆 × 𝜏CRTA HA AH HA AH J. Tang and J. M. Skelton, J. Phys.: Condens. Matter 33 (16), 164002 (2021) TYC Seminar, 26th Jan 2023 | Slide 11 Dr Jonathan Skelton
  12. J. M. Skelton, J. Mater. Chem. C (2021), DOI: 10.1039/D1TC02026A

    𝒗𝜆 vs. 𝜏𝜆 : SnS/SnSe TYC Seminar, 26th Jan 2023 | Slide 12 Dr Jonathan Skelton SnS SnSe
  13. 𝒗𝜆 vs. 𝜏𝜆 : other TEs 𝜅 [W m-1 K-1]

    Τ 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., Molecules 27 (19), 6431 (2022) TYC Seminar, 26th Jan 2023 | Slide 13 Dr Jonathan Skelton
  14. 𝒗𝜆 vs. 𝜏𝜆 : Si clathrates TYC Seminar, 26th Jan

    2023 | Slide 14 Dr Jonathan Skelton B. Wei et al., Molecules 27 (19), 6431 (2022)
  15. 𝒗𝜆 vs. 𝜏𝜆 : Si clathrates TYC Seminar, 26th Jan

    2023 | Slide 15 Dr Jonathan Skelton 𝜿latt ≈ 1 𝑁𝑉0 ෍ 𝜆 𝐶𝜆 𝒗𝜆 ⊗ 𝒗𝜆 × 𝜏CRTA B. Wei et al., Molecules 27 (19), 6431 (2022)
  16. 𝒗𝜆 vs. 𝜏𝜆 : Si clathrates 𝜿 (W m-1 K-1)

    Τ 𝜿 𝝉𝐂𝐑𝐓𝐀 (W m-1 K-1 ps-1) 𝝉𝐂𝐑𝐓𝐀 (ps) d-Si 136.24 5.002 27.24 oC24 40.92 2.295 17.83 K-II / C-I 43.54 0.829 52.52 K-V / C-VI 36.29 0.815 44.53 K-VII / C-V 31.16 0.770 40.45 C-II 6.33 0.458 13.81 Spacegroup 𝒏𝐚 𝐹𝑑ത 3𝑚 2 𝐶𝑚𝑐𝑚 12 𝑃𝑚ത 3𝑚 46 𝐶𝑚𝑚𝑚 40 𝑃63 /𝑚𝑚𝑐 68 𝐹𝑑ത 3𝑚 34 With the exception of the Clathrate-II structure, the harmonic Τ 𝜿 𝜏CRTA term correlates with: 1) the size of the primitive cell (𝑛a ); and 2) the spacegroup (crystal symmetry) Implies low group velocities are favoured by complex structures with large primitive cells and/or low symmetry TYC Seminar, 26th Jan 2023 | Slide 16 Dr Jonathan Skelton B. Wei et al., Molecules 27 (19), 6431 (2022)
  17. Analysing 𝜏𝜆 : phonon linewidths Γ𝜆 (𝑇) = ෍ 𝜆′𝜆′′

    Φ−𝜆𝜆′𝜆′′ 2 × { 𝑛𝜆′ (𝑇) − 𝑛𝜆′′ (𝑇) 𝛿 𝜔 + 𝜔𝜆′ − 𝜔𝜆′′ − 𝛿 𝜔 − 𝜔𝜆′ + 𝜔𝜆′′ + 𝑛𝜆′ (𝑇) + 𝑛𝜆′′ (𝑇) + 1 𝛿 𝜔 − 𝜔𝜆′ − 𝜔𝜆′′ } 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) TYC Seminar, 26th Jan 2023 | Slide 17 Dr Jonathan Skelton
  18. Analysing 𝜏𝜆 : phonon linewidths A. Togo et al., Phys.

    Rev. B 91, 094306 (2015) Approximate expression for Γ𝜆 : With: Γ𝜆 (𝑇) ≈ 18𝜋 ℏ2 ෨ 𝑃𝑁2 (𝒒𝜆 , 𝜔𝜆 , 𝑇) 𝑁2 𝒒𝜆 , 𝜔𝜆 , 𝑇 = 𝑁 2 (1) 𝒒𝜆 , 𝜔𝜆 , 𝑇 + 𝑁 2 (2) 𝒒𝜆 , 𝜔𝜆 , 𝑇 𝑁 2 (1) 𝒒𝜆 , 𝜔𝜆 , 𝑇 = 1 𝑁 ෍ 𝜆′𝜆′′ ∆(−𝒒𝜆 + 𝒒𝜆′ + 𝒒𝜆′′ ) 𝑛𝜆′ (𝑇) − 𝑛𝜆′′ (𝑇) × 𝛿 𝜔 + 𝜔𝜆′ − 𝜔𝜆′′ − 𝛿 𝜔 − 𝜔𝜆′ + 𝜔𝜆′′ 𝑁 2 (2) 𝒒𝜆 , 𝜔𝜆 , 𝑇 = 1 𝑁 ෍ 𝜆′𝜆′′ ∆(−𝒒𝜆 + 𝒒𝜆′ + 𝒒𝜆′′ ) 𝑛𝜆′ (𝑇) + 𝑛𝜆′′ (𝑇) + 1 𝛿 𝜔 − 𝜔𝜆′ − 𝜔𝜆′′ TYC Seminar, 26th Jan 2023 | Slide 18 Dr Jonathan Skelton
  19. Analysing 𝜏𝜆 : phonon linewidths Nanoscale Energy Harvesting, 24th Aug

    2022 | Slide 19 Dr Jonathan M. Skelton Γ𝜆 (𝑇) ≈ 18𝜋 ℏ2 ෨ 𝑃𝑁2 (𝒒𝜆 , 𝜔𝜆 , 𝑇) B. Wei et al., Molecules 27 (19), 6431 (2022)
  20. Analysing 𝜏𝜆 Γ𝜆 (𝑇) ≈ 18𝜋 ℏ2 ෨ 𝑃𝑁2 (𝒒𝜆

    , 𝜔𝜆 , 𝑇) TYC Seminar, 26th Jan 2023 | Slide 20 Dr Jonathan Skelton B. Wei et al., Molecules 27 (19), 6431 (2022)
  21. Workflow 𝜿latt (𝑇) Τ 𝜿 𝜏CRTA 𝜏CRTA ഥ 𝑁2 ෨

    𝑃 Phonopy + Phono3py A. Togo and I. Tanka, Scr. Mater. 108, 1 (2015) A. Togo et al., Phys. Rev. B 91, 094306 (2015) TYC Seminar, 26th Jan 2023 | Slide 21 Dr Jonathan Skelton
  22. Reducing 𝒗𝜆 I: alloying C.-C. Lin et al., Chem. Mater.

    29 (12), 5344 (2017) SnSe 15-20 % S TYC Seminar, 26th Jan 2023 | Slide 22 Dr Jonathan Skelton
  23. Reducing 𝒗𝜆 I: alloying 54.2 % ↓ Sn(S0.1875 Se0.8125 )

    SnSe J. M. Skelton, J. Mater. Chem. C 9, 11772 (2021) TYC Seminar, 26th Jan 2023 | Slide 23 Dr Jonathan Skelton
  24. Reducing 𝒗𝜆 II: discordant doping H. Xie et al., J.

    Am. Chem. Soc. 141 (47), 18900 (2019) TYC Seminar, 26th Jan 2023 | Slide 24 Dr Jonathan Skelton
  25. Reducing 𝜏𝜆 I: “rattler” TEs J. W. Schwartz and C.

    T. Walker, Phys. Rev. B 155, 959 (1967) E. S. Toberer et al., J. Mater. Chem. 21, 15843 (2011) “One phonon” model for resonant scattering: 𝜏−1 = ෍ 𝑖 𝑐𝑖 𝜔2𝑇2 𝜔𝑖 2 − 𝜔2 2 + 𝛾𝑖 𝜔𝑖 2𝜔2 TYC Seminar, 26th Jan 2023 | Slide 25 Dr Jonathan Skelton
  26. Reducing 𝜏𝜆 I: “rattler” TEs TYC Seminar, 26th Jan 2023

    | Slide 26 Dr Jonathan Skelton J. Tang and J. M. Skelton, J. Phys.: Condens. Matter 33 (16), 164002 (2021) Filler 𝒎𝐗 [amu] 𝒓𝐗 [pm] He 4.0026 31 Ne 20.180 38 Ar 39.948 71 Kr 83.798 88 Xe 131.29 108 Noble gases are chemically inert (closed shell, unlikely to reduce/oxidise host framework) and are likely closest it is possible to get to a “hard sphere” filler
  27. Reducing 𝜏𝜆 I: “rattler” TEs TYC Seminar, 26th Jan 2023

    | Slide 27 Dr Jonathan Skelton J. Tang and J. M. Skelton, J. Phys.: Condens. Matter 33 (16), 164002 (2021) We can define a rattling frequency ሚ 𝑓𝑥 for the noble gas fillers X based on the 𝑫 XX, 𝐪 = Γ : 𝑫 XX, 𝐪 = Γ = 1 𝑚X ෍ 𝑙′ 𝚽 X0, X𝑙′ What happens to 𝜅latt if we artificially change the 𝑚X while keeping the 𝚽 fixed?
  28. Reducing 𝜏𝜆 II: hybrid TEs (?) A. Gold-Parker et al.,

    PNAS 115 (47), 11905 (2018) TYC Seminar, 26th Jan 2023 | Slide 28 Dr Jonathan Skelton
  29. Workflow 𝜿latt (𝑇) Τ 𝜿 𝜏CRTA 𝜏CRTA ഥ 𝑁2 ෨

    𝑃 𝑺(𝑛, 𝑇) 𝝈(𝑛, 𝑇) 𝜿el (𝑛, 𝑇) Phonopy + Phono3py AMSET 𝑍𝑇(𝑛, 𝑇) A. Togo and I. Tanka, Scr. Mater. 108, 1 (2015) A. Togo et al., Phys. Rev. B 91, 094306 (2015) A. M. Ganose et al., Nature Comm. 12, 2222 (2021) TYC Seminar, 26th Jan 2023 | Slide 29 Dr Jonathan Skelton
  30. Predicting 𝒁𝑻 J. M. Flitcroft et al., Solids 3 (1),

    155 (2022) TYC Seminar, 26th Jan 2023 | Slide 30 Dr Jonathan Skelton
  31. High-performance oxide TEs W. Rahim et al., J. Mater. Chem.

    A 8, 16405 (2020) W. Rahim et al., J. Mater. Chem. A 9, 20417 (2021) K. Brlec et al., J. Mater. Chem. A 10, 16813 (2022) 𝛼-Bi2 Sn2 O7 𝑛 = 1.73 × 1019 cm-3 𝑍𝑇 = 0.36 (385 K) Ca4 Sb2 O / Ca4 Bi2 O 𝑝 = 4.64 / 2.15 × 1019 cm-3 𝑍𝑇 = 1.58 / 2.14 (1000 K) Y2 Ti2 O5 S2 𝑛 = 2.37 × 1020 cm-3 𝑍𝑇 = 1.18 (1000 K) TYC Seminar, 26th Jan 2023 | Slide 31 Dr Jonathan Skelton
  32. 𝝅-cubic SnS/SnSe TYC Seminar, 26th Jan 2023 | Slide 32

    Dr Jonathan Skelton 𝒁𝑻 𝐦𝐚𝐱 𝑺𝟐𝝈 [mW m-1 K-2] 𝜿𝐭𝐨𝐭 [W m-1 K-1] SnS (Pnma) 1.75 1.87 1.07 SnSe (Pnma) 2.81 2.62 0.93 SnSe (RS) 2.60 10.90 3.02 R. E. Abutbul et al., CrystEngComm 18, 5188 (2016) J. M. Flitcroft et al., Solids 3 (1), 155 (2022)
  33. “Hybrid” TEs TYC Seminar, 26th Jan 2023 | Slide 33

    Dr Jonathan Skelton
  34. Summary o The SM-RTA can give quantitative predictions of the

    𝜅latt of a wide range of materials o The contributions of individual phonon modes can be used to obtain microscopic insight into how the 𝜅latt “works”: • CRTA model: 𝒗λ vs. 𝜏λ • Constant 𝑃λ model: ഥ 𝑁2 vs. ෨ 𝑃 o Modelling on Si allotropes shows that low 𝜅latt is favoured by: 1) Large primitive cells 2) Lower crystal symmetry o With reference to existing TEs, the CRTA model can be used to suggest strategies for reducing the 𝜅latt : • 𝒗λ - alloying and discordant-atom doping • 𝜏λ - introducing “ratters”, small molecules may be particularly effective o These ideas are being explored in our current work -- watch this space! TYC Seminar, 26th Jan 2023 | Slide 34 Dr Jonathan Skelton
  35. Phono3py-Power-Tools TYC Seminar, 26th Jan 2023 | Slide 35 Dr

    Jonathan Skelton https://github.com/skelton-group/Phono3py-Power-Tools
  36. Thankyou for listening! Any questions?