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J. M. Skelton Department of Chemistry, University of Manchester ([email protected]) Theory-Led Control of Heat Transport in Thermoelectric Materials

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The global energy challenge Nanoscale Energy Harvesting, 24th Aug 2022 | Slide 2 Dr Jonathan M. Skelton 34 % 26 % 19 % 18 % 3 % 1000 MW nuclear power plant: o 650 MW waste heat o 3 % β‰ˆ 20 MW β‰ˆ 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)

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Thermoelectric materials Nanoscale Energy Harvesting, 24th Aug 2022 | Slide 3 Dr Jonathan M. Skelton 𝑍𝑇 = 𝑆2𝜎 πœ…ele + πœ…lat 𝑇 𝑆 - Seebeck coefficient 𝜎 - electrical conductivity πœ…ele - electronic thermal conductivity πœ…lat - lattice thermal conductivity G. Tan et al., Chem. Rev. 116 (19), 12123 (2016)

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Acknowledgements CMD29, 23rd August 2022 | Slide 4 Dr Jonathan M. Skelton ... plus mentors and collaborators too numerous to mention

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Modelling thermal conductivity Nanoscale Energy Harvesting, 24th Aug 2022 | Slide 5 Dr Jonathan M. Skelton A. Togo et al., Phys. Rev. B 91, 094306 (2015) 𝜿latt (𝑇) = 1 𝑁𝑉0 ෍ πœ† πœΏπœ† (𝑇) 1 𝑁𝑉0 ෍ πœ† πΆπœ† (𝑇)π’—πœ† βŠ— π’—πœ† πœπœ† (𝑇) The simplest model for πœ…latt is the single-mode relaxation time approximation (SM-RTA) - a closed solution to the phonon Boltzmann transport equations Modal heat capacity Mode group velocity πœ•πœ”Ξ» πœ•πͺ Average over phonon modes Ξ» Phonon MFP Mode lifetime 𝜏λ = 1 2Γλ πš²πœ† 𝑇 = π’—πœ† πœπœ† 𝑇

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Modelling thermal conductivity Nanoscale Energy Harvesting, 24th Aug 2022 | Slide 6 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

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Modelling thermal conductivity Nanoscale Energy Harvesting, 24th Aug 2022 | Slide 7 Dr Jonathan M. Skelton J. Tang and J. M. Skelton, J. Phys.: Condens. Matter 33 (16), 164002 (2021)

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Modelling thermal conductivity Nanoscale Energy Harvesting, 24th Aug 2022 | Slide 8 Dr Jonathan M. Skelton A. Gold-Parker et al., PNAS 115 (47), 11905 (2018) GaAs MAPbI3

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Modelling thermal conductivity Nanoscale Energy Harvesting, 24th Aug 2022 | Slide 9 Dr Jonathan M. Skelton CoSb3 CoSb3 J. Tang and J. M. Skelton, J. Phys.: Condens. Matter 33 (16), 164002 (2021)

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Modelling thermal conductivity Nanoscale Energy Harvesting, 24th Aug 2022 | Slide 10 Dr Jonathan M. Skelton CoSb3 CoSb3 J. Tang and J. M. Skelton, J. Phys.: Condens. Matter 33 (16), 164002 (2021)

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π’—πœ† vs. πœπœ† : the CRTA model Nanoscale Energy Harvesting, 24th Aug 2022 | Slide 11 Dr Jonathan M. Skelton Consider again the SM-RTA model: 𝜿latt = 1 𝑁𝑉0 ෍ πœ† πœΏπœ† = 1 𝑁𝑉0 ෍ πœ† πΆπœ† π’—πœ† βŠ— π’—πœ† πœπœ† Replace the πœπœ† with a constant lifetime (relaxation time) 𝜏CRTA defined as follows: 𝜿latt 𝜏CRTA = 1 𝑁𝑉0 ෍ πœ† πœΏπœ† πœπœ† = 1 𝑁𝑉0 ෍ πœ† πΆπœ† π’—πœ† βŠ— π’—πœ† 𝜿latt β‰ˆ 1 𝑁𝑉0 ෍ πœ† πΆπœ† π’—πœ† βŠ— π’—πœ† Γ— 𝜏CRTA HA AH HA AH J. Tang and J. M. Skelton, J. Phys.: Condens. Matter 33 (16), 164002 (2021)

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π’—πœ† vs. πœπœ† : Si clathrates Nanoscale Energy Harvesting, 24th Aug 2022 | Slide 12 Dr Jonathan M. Skelton B. Wei et al., in preparation

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π’—πœ† vs. πœπœ† : Si clathrates Nanoscale Energy Harvesting, 24th Aug 2022 | Slide 13 Dr Jonathan M. Skelton B. Wei et al., in preparation 𝜿latt β‰ˆ 1 𝑁𝑉0 ෍ πœ† πΆπœ† π’—πœ† βŠ— π’—πœ† Γ— 𝜏CRTA

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π’—πœ† vs. πœπœ† : Si clathrates Nanoscale Energy Harvesting, 24th Aug 2022 | Slide 14 Dr Jonathan M. Skelton 𝜿 (W m-1 K-1) Ξ€ 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 π’πš 𝐹𝑑ഀ 3π‘š 2 πΆπ‘šπ‘π‘š 12 π‘ƒπ‘šΰ΄€ 3π‘š 46 πΆπ‘šπ‘šπ‘š 40 𝑃63 /π‘šπ‘šπ‘ 68 𝐹𝑑ഀ 3π‘š 34 With the exception of the Clathrate-II structure, the harmonic Ξ€ 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 B. Wei et al., in preparation

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Analysing πœπœ† : phonon linewidths Nanoscale Energy Harvesting, 24th Aug 2022 | Slide 15 Dr Jonathan M. Skelton Ξ“πœ† (𝑇) = ෍ πœ†β€²πœ†β€²β€² Ξ¦βˆ’πœ†πœ†β€²πœ†β€²β€² 2 Γ— { π‘›πœ†β€² (𝑇) βˆ’ π‘›πœ†β€²β€² (𝑇) 𝛿 πœ” + πœ”πœ†β€² βˆ’ πœ”πœ†β€²β€² βˆ’ 𝛿 πœ” βˆ’ πœ”πœ†β€² + πœ”πœ†β€²β€² + π‘›πœ†β€² (𝑇) + π‘›πœ†β€²β€² (𝑇) + 1 𝛿 πœ” βˆ’ πœ”πœ†β€² βˆ’ πœ”πœ†β€²β€² } Collision Decay Three-phonon interaction strength - includes conservation of momentum(β€œanharmonicity”) Conservation of energy (β€œselection rules”) A. Togo et al., Phys. Rev. B 91, 094306 (2015)

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Analysing πœπœ† : phonon linewidths Nanoscale Energy Harvesting, 24th Aug 2022 | Slide 16 Dr Jonathan M. Skelton A. Togo et al., Phys. Rev. B 91, 094306 (2015) Approximate expression for Ξ“πœ† : With: Ξ“πœ† (𝑇) β‰ˆ 18πœ‹ ℏ2 ΰ·¨ 𝑃𝑁2 (π’’πœ† , πœ”πœ† , 𝑇) 𝑁2 π’’πœ† , πœ”πœ† , 𝑇 = 𝑁 2 (1) π’’πœ† , πœ”πœ† , 𝑇 + 𝑁 2 (2) π’’πœ† , πœ”πœ† , 𝑇 𝑁 2 (1) π’’πœ† , πœ”πœ† , 𝑇 = 1 𝑁 ෍ πœ†β€²πœ†β€²β€² βˆ†(βˆ’π’’πœ† + π’’πœ†β€² + π’’πœ†β€²β€² ) π‘›πœ†β€² (𝑇) βˆ’ π‘›πœ†β€²β€² (𝑇) Γ— 𝛿 πœ” + πœ”πœ†β€² βˆ’ πœ”πœ†β€²β€² βˆ’ 𝛿 πœ” βˆ’ πœ”πœ†β€² + πœ”πœ†β€²β€² 𝑁 2 (2) π’’πœ† , πœ”πœ† , 𝑇 = 1 𝑁 ෍ πœ†β€²πœ†β€²β€² βˆ†(βˆ’π’’πœ† + π’’πœ†β€² + π’’πœ†β€²β€² ) π‘›πœ†β€² (𝑇) + π‘›πœ†β€²β€² (𝑇) + 1 𝛿 πœ” βˆ’ πœ”πœ†β€² βˆ’ πœ”πœ†β€²β€²

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Analysing πœπœ† Nanoscale Energy Harvesting, 24th Aug 2022 | Slide 17 Dr Jonathan M. Skelton B. Wei et al., in preparation Ξ“πœ† (𝑇) β‰ˆ 18πœ‹ ℏ2 ΰ·¨ 𝑃𝑁2 (π’’πœ† , πœ”πœ† , 𝑇)

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Workflow Nanoscale Energy Harvesting, 24th Aug 2022 | Slide 18 Dr Jonathan M. Skelton 𝜿latt (𝑇) Ξ€ 𝜿 𝜏CRTA 𝜏CRTA ΰ΄₯ 𝑁2 ΰ·¨ 𝑃 Phonopy + Phono3py A. Togo and I. Tanka, Scr. Mater. 108, 1 (2015) A. Togo et al., Phys. Rev. B 91, 094306 (2015)

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π’—πœ† vs. πœπœ† : other TEs Nanoscale Energy Harvesting, 24th Aug 2022 | Slide 19 Dr Jonathan M. Skelton πœ… [W m-1 K-1] Ξ€ πœ… 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., in preparation

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Reducing π’—πœ† I: alloying Nanoscale Energy Harvesting, 24th Aug 2022 | Slide 20 Dr Jonathan M. Skelton C.-C. Lin et al., Chem. Mater. 29 (12), 5344 (2017) SnSe 15-20 % S

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Reducing π’—πœ† I: alloying Nanoscale Energy Harvesting, 24th Aug 2022 | Slide 21 Dr Jonathan M. Skelton 54.2 % ↓ Sn(S0.1875 Se0.8125 ) SnSe J. M. Skelton, J. Mater. Chem. C 9, 11772 (2021)

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Reducing π’—πœ† II: discordant doping Nanoscale Energy Harvesting, 24th Aug 2022 | Slide 22 Dr Jonathan M. Skelton H. Xie et al., J. Am. Chem. Soc. 141 (47), 18900 (2019)

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Reducing πœπœ† I: β€œrattler” TEs Nanoscale Energy Harvesting, 24th Aug 2022 | Slide 23 Dr Jonathan M. Skelton Schwartz and Walker, Phys. Rev. B 155, 959 (1967) E. S. Toberer et al., J. Mater. Chem. 21, 15843 (2011) J. Tang and J. M. Skelton, J. Phys.: Condens. Matter 33 (16), 164002 (2021) β€œOne phonon” model for resonant scattering: πœβˆ’1 = ෍ 𝑖 𝑐𝑖 πœ”2𝑇2 πœ”π‘– 2 βˆ’ πœ”2 2 + 𝛾𝑖 πœ”π‘– 2πœ”2

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Reducing πœπœ† II: hybrid TEs (?) Nanoscale Energy Harvesting, 24th Aug 2022 | Slide 24 Dr Jonathan M. Skelton A. Gold-Parker et al., PNAS 115 (47), 11905 (2018)

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Workflow Nanoscale Energy Harvesting, 24th Aug 2022 | Slide 25 Dr Jonathan M. Skelton 𝜿latt (𝑇) Ξ€ 𝜿 𝜏CRTA 𝜏CRTA ΰ΄₯ 𝑁2 ΰ·¨ 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)

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Predicting 𝒁𝑻 Nanoscale Energy Harvesting, 24th Aug 2022 | Slide 26 Dr Jonathan M. Skelton J. M. Flitcroft et al., Solids 3 (1), 155 (2022)

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Challenges and opportunities Nanoscale Energy Harvesting, 24th Aug 2022 | Slide 27 Dr Jonathan M. Skelton D. W. Davies et al., Chem 1 (4), 617 (2016)

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Challenges and opportunities Nanoscale Energy Harvesting, 24th Aug 2022 | Slide 28 Dr Jonathan M. Skelton 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 Γ— 1019 cm-3 𝑍𝑇 = 0.36 (385 K) Ca4 Sb2 O / Ca4 Bi2 O 𝑝 = 4.64 / 2.15 Γ— 1019 cm-3 𝑍𝑇 = 1.58 / 2.14 (1000 K) Y2 Ti2 O5 S2 𝑛 = 2.37 Γ— 1020 cm-3 𝑍𝑇 = 1.18 (1000 K)

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Challenges and opportunities Nanoscale Energy Harvesting, 24th Aug 2022 | Slide 29 Dr Jonathan M. Skelton

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Challenges and opportunities Nanoscale Energy Harvesting, 24th Aug 2022 | Slide 30 Dr Jonathan M. Skelton Some questions to consider (definitely not exhaustive...): 1. Is there a β€œkiller” application of TEs that we should target? 2. If so, what are the parameters? β€’ Operating temperature and target 𝑍𝑇? β€’ Target cost per device? β€’ Constraints on elemental composition? β€’ Constraints on synthesis/device fabrication for scale up? 3. How can theory and experiment best work together? β€’ Is modelling in a position to provide actionable suggestions to improve 𝑍𝑇? β€’ Would it be possible(/useful) to use modelling to screen candidates to narrow the focus of experiments? β€’ Are existing theoretical techniques sufficient, or is more development needed (e.g. faster, cheaper, easier to use, capability to model new things ...)?

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https://bit.ly/3AjphfB These slides are available on Speaker Deck: