J. M. Skelton, S. K. Guillemot, I. Pallikara, J. M. Skelton and M. Zhang Department of Chemistry, University of Manchester ([email protected]) Ab initio prediction of the thermoelectric figure of merit 𝑍𝑇: application to the Sn chalcogenides
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
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)
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
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
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)
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)
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
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)
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