Ab initio prediction of the thermoelectric figure of merit ZT: application to the Sn chalcogenides

Presented at the EPSRC Thermoelectric Network Meeting on the 16th November 2023.

Transcript

1. J. M. Skelton, S. K. Guillemot, I. Pallikara, J. M. Skelton and M. Zhang Department of Chemistry, University of Manchester (jonathan.skelton@manchester.ac.uk) Ab initio prediction of the thermoelectric figure of merit 𝑍𝑇: application to the Sn chalcogenides
2. The thermoelectric figure of merit Dr Jonathan Skelton EPSRC TE Network Meeting, 16th Nov 2023 | Slide 2 𝑍𝑇 = 𝑆2𝜎 𝜅el + 𝜅latt 𝑇 𝑆 - Seebeck coefficient 𝜎 - electrical conductivity 𝜅el - electronic thermal conductivity 𝜅latt - lattice thermal conductivity Tan et al., Chem. Rev. 116 (19), 12123 (2016)
3. Electron transport: 𝑺, 𝝈 and 𝜿𝐞𝐥 Dr Jonathan Skelton EPSRC TE Network Meeting, 16th Nov 2023 | Slide 3 Ganose et al., Nature Comm. 12, 2222 (2021) We first define the spectral conductivity tensor: Σ𝛼𝛽 𝜖, 𝑇 = 1 8𝜋3 ෍ 𝑗 න 𝑣𝒌𝑗,𝛼 𝑣𝒌𝑗,𝛽 𝜏𝒌𝑗 𝑇 𝛿 𝜖 − 𝜖𝒌𝑗 𝑑𝒌 This is used to calculate the 𝑛th-order moments of the generalised transport coefficients: ℒ𝛼𝛽 𝑛 𝜖F , 𝑇 = න Σ𝛼𝛽 𝜖, 𝑇 𝜖 − 𝜖F 𝑛 − 𝜕𝑓 𝜖, 𝜖F , 𝑇 𝜕𝜖 𝜕𝜖 𝑓 𝜖, 𝜖F , 𝑇 = 1 exp Τ 𝜖 − 𝜖F 𝑘B 𝑇 + 1 Where: o The 𝒗𝒌𝑗 are obtained from a high-quality band structure o The 𝜏𝒌𝑗 can be: treated as a constant 𝜏el ; approximated by model equations for different scattering processes; or calculated from the electron-phonon coupling o The 𝜖F (= 𝜇) is set by the DoS and a specified extrinsic carrier concentration 𝑛
4. Electron transport: 𝑺, 𝝈 and 𝜿𝐞𝐥 Dr Jonathan Skelton EPSRC TE Network Meeting, 16th Nov 2023 | Slide 4 The 𝓛𝑛(𝜖F , 𝑇) are determined from a band structure, a model for the 𝜏𝑗𝒌 , and a specified 𝑛/𝑇: ℒ𝛼𝛽 𝑛 𝜖F , 𝑇 = න Σ𝛼𝛽 𝜖, 𝑇 𝜖 − 𝜖F 𝑛 − 𝜕𝑓 𝜖, 𝜖F , 𝑇 𝜕𝜖 𝜕𝜖 The electrical transport coefficients can be determined from the 𝓛𝑛(𝜖F , 𝑇) as: 𝜎𝛼𝛽 (𝜖F , 𝑇) = ℒ𝛼𝛽 0 (𝜖F , 𝑇) 𝑆𝛼𝛽 (𝜖F , 𝑇) = 1 𝑒𝑇 ℒ𝛼𝛽 1 (𝜖F , 𝑇) ℒ 𝛼𝛽 0 (𝜖F , 𝑇) 𝜅el,𝛼𝛽 (𝜖F , 𝑇) = 1 𝑒2𝑇 ℒ𝛼𝛽 1 (𝜖F , 𝑇) 2 ℒ 𝛼𝛽 0 (𝜖F , 𝑇) − ℒ𝛼𝛽 2 (𝜖F , 𝑇) Note that when using the CRTA (i.e. 𝜏𝒌𝑗 → 𝜏el ): o The 𝑺 are the ratio of two 𝓛𝑛 and the 𝜏el cancel o The 𝝈 and 𝜿el are obtained with respect to 𝜏el (𝜏el ~ 10-14 s) Ganose et al., Nature Comm. 12, 2222 (2021)
5. Electron transport: 𝑷𝒏𝒎𝒂 SnS/Se Dr Jonathan Skelton EPSRC TE Network Meeting, 16th Nov 2023 | Slide 5 Flitcroft et al., Solids 3 (1), 155 (2022) Fixed 𝑇 = 800 K Fixed 𝑛ℎ = 1019 cm-3
6. Phonon transport: 𝜿𝐥𝐚𝐭𝐭 Dr Jonathan Skelton EPSRC TE Network Meeting, 16th Nov 2023 | Slide 6 A. Togo et al., Phys. Rev. B 91, 094306 (2015) The simplest model for 𝜅latt is the single-mode relaxation time approximation (SM-RTA) - a closed solution to the phonon Boltzmann transport equations: 𝜿latt,𝛼𝛽 (𝑇) = 1 𝑁𝒒 𝑉 ෍ 𝒒𝑗 𝜅𝒒𝑗,𝛼𝛽 (𝑇) = 1 𝑁𝒒 𝑉 ෍ 𝒒𝑗 𝐶𝒒𝑗 (𝑇)𝑣𝒒𝑗,𝛼 𝑣𝒒𝑗,𝛽 𝜏𝒒𝑗 (𝑇) Where: o 𝑁𝒒 is the number of wavevectors 𝒒 included in the summation and 𝑉 is the cell volume o The heat capacities 𝐶𝒒𝑗 and group velocities 𝒗𝒒𝑗 are determined from the phonon frequencies 𝜔𝒒𝑗 and the frequency dispersion Τ 𝜕𝜔𝒒𝑗 𝜕𝒒 o The 𝜏𝒒𝑗 are determined from the 𝜔𝒒𝑗 and eigenvectors 𝑾𝒒𝑗 plus the anharmonic third- (or higher-)order force constants
7. Phonon transport: 𝑷𝒏𝒎𝒂 SnS/Se Dr Jonathan Skelton EPSRC TE Network Meeting, 16th Nov 2023 | Slide 7 Skelton, J. Mater. Chem. C 9, 11772 (2021)
8. 𝑨𝒃 𝒊𝒏𝒊𝒕𝒊𝒐 prediction of 𝒁𝑻 Dr Jonathan Skelton EPSRC TE Network Meeting, 16th Nov 2023 | Slide 8 We can now combine our workflows for predicting the electrical and phonon transport coefficients (i.e. 𝑆/𝜎/𝜅el and 𝜅latt ) to calculate 𝑍𝑇: 𝑍𝑇𝛼𝛽 (𝑛, 𝑇) = 𝑆𝛼𝛽 𝑛, 𝑇 2 𝜎𝛼𝛽 (𝑛, 𝑇) 𝜅el,𝛼𝛽 (𝑛, 𝑇) + 𝜅latt,𝛼𝛽 (𝑇) × 𝑇 Skelton, J. Mater. Chem. C 9, 11772 (2021) Flitcroft et al., Solids 3 (1), 155 (2022)
9. 𝑨𝒃 𝒊𝒏𝒊𝒕𝒊𝒐 prediction of 𝒁𝑻 Dr Jonathan Skelton EPSRC TE Network Meeting, 16th Nov 2023 | Slide 9 𝑛 [cm-3] 𝑇 [K] 𝑍𝑇 𝝈 [S cm-1] 𝑆 [𝝁V K-1] PF [mW m-1 K-2] 𝜅𝐞𝐥 + 𝜅𝐥𝐚𝐭𝐭 = 𝜅𝐭𝐨𝐭 [W m-1 K-1] Ref. 1 1019 823 0.7-2.4 14-85 326-386 0.21-1 0.22-0.25 0.25-0.35 Calc. 1019 820 1-2.2 33-196 373-388 0.5-2.8 0.06-0.23 0.35-0.79 0.4-1 Ref. 2 4 × 1019 773 1.1-2.1 40-150 300 0.25-1.5 0.4-0.6 Calc. 4.6 × 1019 780 1.5-2.3 159-918 240-257 1-5.4 0.17-1 0.36-0.83 0.53-1.83 Ref. 3 1019 773 3.1 110 250 0.85 0.23 Calc. 1019 780 1.8 122 374 1.7 0.14 0.61 0.75 General trend for calculations to overestimate 𝜎 and PFs, under/overestimate 𝑆 in some cases, overestimate 𝜅latt , but often get reasonable “ballpark” values for 𝑍𝑇 [1] Zhao et al., Nature 508, 373 (2014) [2] Zhao et al., Science 351 (6269), 141 (2015) [3] Zhou et al., Nature Mater. 20, 1378 (2021) Skelton, J. Mater. Chem. C 9, 11772 (2021) Flitcroft et al., Solids 3 (1), 155 (2022)
10. 𝑷𝒏𝒎𝒂 SnSe: p- vs n-type Dr Jonathan Skelton EPSRC TE Network Meeting, 16th Nov 2023 | Slide 10 Flitcroft et al., Solids 3 (1), 155 (2022) Zhang et al., J. Mater. Chem. C 11, 14833 (2023)
11. Flitcroft et al., Solids 3 (1), 155 (2022) Zhang et al., J. Mater. Chem. C 11, 14833 (2023) 𝑷𝒏𝒎𝒂 SnSe: p- vs n-type Dr Jonathan Skelton EPSRC TE Network Meeting, 16th Nov 2023 | Slide 11
12. 𝑷𝒏𝒎𝒂 SnSe: p- vs n-type Dr Jonathan Skelton EPSRC TE Network Meeting, 16th Nov 2023 | Slide 12 Duong et al., Nature Comm. 7, 13713 (2016) Zhang et al., J. Alloys Compd. 910, 164900 (2022) Literature suggests n-doping can achieve 𝑍𝑇 approaching p-doped SnSe: 1. Bi-doped SnSe: 𝑛 = 2.1 × 1019 cm-3, 𝑍𝑇 = 0.4-2.2 @ 773 K 2. NdCl3 -doped SnSe: 𝑛 = 4.78 × 1015 cm-3, 𝑍𝑇 = 0.5-1.3 @ 773 K
13. Trends in 𝜿𝐥𝐚𝐭𝐭 with structure type EPSRC TE Network Meeting, 16th Nov 2023 | Slide 13 𝑃𝑛𝑚𝑎 𝐶𝑚𝑐𝑚 𝑅3𝑚 𝐹𝑚ത 3𝑚 Lower group velocities 𝒗λ Stronger anharmonicity = shorter lifetimes 𝜏λ Larger “scattering phase space” = shorter 𝜏λ Dr Jonathan Skelton Guillemot et al., ChemRxiv preprint, DOI: 10.26434/chemrxiv-2023-5q2v1 (2023)
14. Trends in 𝜿𝐥𝐚𝐭𝐭 with structure type EPSRC TE Network Meeting, 16th Nov 2023 | Slide 14 Dr Jonathan Skelton 𝐶𝑚𝑐𝑚 SnCh: ? Low symmetry  Large(-ish) cell (𝑛𝑎 = 4)  Sn constrained to a locally-symmetric environment 𝜋-cubic SnCh: ? High symmetry  (Very) large cell (𝑛𝑎 = 64)  Sn local geometry similar to 𝑃𝑛𝑚𝑎 phase Abutbul et al., CrystEngComm 18, 1918 (2016) Guillemot et al., ChemRxiv preprint (2023)
15. 𝑷𝒏𝒎𝒂 vs 𝝅 SnSe EPSRC TE Network Meeting, 16th Nov 2023 | Slide 15 Dr Jonathan Skelton
16. Acknowledgements EPSRC TE Network Meeting, 16th Nov 2023 | Slide 16 Dr Jonathan Skelton
17. https://bit.ly/3MMzkAX These slides are on Speaker Deck: