Ab initio workflow for predicting the figure of merit of thermoelectric materials

Presented at the Indo-UK Workshop on Critical Materials for Energy and Sustainability at the University of Manchester on 1st October 2024.

Transcript

1. Dr Jonathan Skelton Department of Chemistry, University of Manchester (jonathan.skelton@manchester.ac.uk) Ab initio workflow for predicting the figure of merit of thermoelectric materials
2. Thermoelectrics: motivation 31 % 23 % 20 % 19 % 3 % Provisional UK greenhouse gas emissions national statistics (published March 2022) Tan et al., Chem. Rev. 116 (19), 12123 (2016) Dr J. M. Skelton Indo-UK Workshop on Energy, 1st October 2024 | Slide 2 𝑍𝑇 = 𝑆2𝜎 𝜅el + 𝜅latt 𝑇
3. An ab initio modelling workflow Crystal structure Convergence testing Geometry optimisation Phonon calculation 𝜅latt Electronic structure 𝑆, 𝜎, 𝜅el 𝑍𝑇 = 𝑆2σ 𝜅el + 𝜅latt 𝑇 Scattering rates: DP, 𝜔po , 𝜀∞ , 𝑍∗, 𝐶, 𝑒(0) Dr J. M. Skelton Indo-UK Workshop on Energy, 1st October 2024 | Slide 3
4. Oxychalcogenides: Bi2 ChO2 𝒂 [Å] 𝒃 [Å] 𝒄 [Å] 𝑽 [Å3] Bi2 SO2 3.81 3.81 11.90 173 Expt 3.87 3.84 11.92 177 Bi2 SeO2 3.87 3.87 12.12 182 Expt 3.88 3.88 12.21 184 Bi2 TeO2 3.96 3.96 12.68 199 Expt 3.98 3.98 12.70 201 Koyama et al., Acta Cryst. B 40, 105 (1984) Zhan et al., J. Am. Ceram. Soc. 98, 2465 (2015) Luu and Vaqueiro, J. Solid State Chem. 226, 219 (2015) Dr J. M. Skelton Indo-UK Workshop on Energy, 1st October 2024 | Slide 4
5. Modelling thermal conductivity The simplest model for 𝜅latt is the single-mode relaxation time approximation (SM-RTA) - a closed solution to the phonon Boltzmann transport equations 𝜿latt (𝑇) = 1 𝑁𝒒 𝑉 ෍ 𝒒𝑗 𝐶𝒒𝑗 (𝑇)𝒗𝒒𝑗 ⊗ 𝒗𝒒𝑗 𝜏𝒒𝑗 (𝑇) 𝐶𝒒𝑗 - phonon heat capacities 𝒗𝒒𝑗 - phonon group velocities 𝜏𝒒𝑗 - phonon lifetimes (inverse linewidths Γ𝒒𝑗 ) 𝑁𝒒 - number of 𝒒 in summation 𝑉 - unit cell volume Togo et al., Phys. Rev. B 91, 094306 (2015) Tang and Skelton, J. Phys: Condens. Matter 33 (16), 164002 (2020) Dr J. M. Skelton Indo-UK Workshop on Energy, 1st October 2024 | Slide 5
6. Lattice thermal conductivity 𝑇 [K] 𝜅(Calc.) [Wm-1K-1] 𝜅(Expt.) [Wm-1K-1] Bi2 SO2 300 2.62 2.9 Bi2 SeO2 800 0.97 0.71 Bi2 TeO2 300 0.95 0.91 Flitcroft et al., J. Phys.: Energy 6, 025011 (2024) Zhang et al., J. Mater. Chem. C 7, 14986 (2019) Pan et al., Nano Energy 69, 104394 (2020) Luu and Vaqueiro, J. Solid State Chen. 226, 219 (2015) Dr J. M. Skelton Indo-UK Workshop on Energy, 1st October 2024 | Slide 6
7. Electronic structure Flitcroft et al., J. Phys.: Energy 6, 025011 (2024) Pacquette et al., J. Photochem. Photobiol. A 277, 27 (2014) Tan et al., J. Am. Ceram. Soc. 101, 326 (2018) Luu and Vaqueiro, J. Solid State Chem. 226, 219 (2015) Bi2 SO2 : 𝐸g (Calc.) = 1.46 eV 𝐸g (Expt) = 1.5 eV Bi2 SeO2 : 𝐸g (Calc.) = 1.1 eV 𝐸g (Expt) = 1.77 eV Bi2 TeO2 : 𝐸g (Calc.) = 0.33 eV 𝐸g (Expt) = 0.23 eV Dr J. M. Skelton Indo-UK Workshop on Energy, 1st October 2024 | Slide 7
8. Modelling electrical properties Ganose et al., Nature Comm. 12, 2222 (2021) We first define the spectral conductivity tensor: Σ𝛼𝛽 𝜖, 𝑇 = 1 8𝜋3 ෍ 𝑗 න𝑣𝒌𝑗,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 Dr J. M. Skelton Indo-UK Workshop on Energy, 1st October 2024 | Slide 8
9. Modelling electrical properties The 𝓛𝑛(𝜖F , 𝑇) are determined from a band structure, a model for the 𝜏𝑗𝒌 , and a specified 𝑛/𝑇: ℒ𝛼𝛽 𝑛 𝜖F , 𝑇 = නΣ𝛼𝛽 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) ℒ𝛼𝛽 0 (𝜖F , 𝑇) 𝜅el,𝛼𝛽 (𝜖F , 𝑇) = 1 𝑒2𝑇 ℒ𝛼𝛽 1 (𝜖F , 𝑇) 2 ℒ𝛼𝛽 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) Dr J. M. Skelton Indo-UK Workshop on Energy, 1st October 2024 | Slide 9
10. Modelling electrical properties Flitcroft et al., Solids 3 (1), 155 (2022) Fixed 𝑇 = 800 K Fixed 𝑛ℎ = 1019 cm-3 Dr J. M. Skelton Indo-UK Workshop on Energy, 1st October 2024 | Slide 10
11. Electrical transport Flitcroft et al., J. Phys.: Energy 6, 025011 (2024) Dr J. M. Skelton Indo-UK Workshop on Energy, 1st October 2024 | Slide 11
12. Comparison to experiments Flitcroft et al., J. Phys.: Energy 6, 025011 (2024) 𝑇 = 300 K 𝑇 = 800 K Dr J. M. Skelton Indo-UK Workshop on Energy, 1st October 2024 | Slide 12
13. Predicted 𝒁𝑻 Flitcroft et al., J. Phys.: Energy 6, 025011 (2024) Dr J. M. Skelton Indo-UK Workshop on Energy, 1st October 2024 | Slide 13
14. Predicted 𝒁𝑻 𝒁𝑻 𝒏 [cm-3] 𝑻 [K] 𝝈 [S cm-1] 𝑺 [µV K-1] 𝑺𝟐𝝈 [mW m-1 K-2] 𝜿𝐞𝐥 [W m-1 K-1] 𝜿𝐥𝐚𝐭𝐭 [W m-1 K-1] 𝜿𝐭𝐨𝐭 [W m-1 K-1] Bi2 SO2 (n) 0.33 2.5×1019 900 120 -186 0.41 0.23 0.9 1.13 Bi2 SO2 (p) 0.72 4×1019 900 24.7 545 0.73 2.63×10-2 0.92 2.53 8×1020 900 495 287 4.08 0.55 1.45 Bi2 SeO2 (n) 0.45 2.5×1019 900 193 -180 0.62 0.39 0.87 1.25 Bi2 SeO2 (p) 1.12 5×1019 900 44.4 512 1.16 6.56 ×10-2 0.93 2.62 5×1020 900 436 318 4.41 0.65 1.51 Bi2 TeO2 (n) 1.05 5×1019 900 554 -184 1.87 1.28 0.33 1.61 Bi2 TeO2 (p) 1.36 5×1019 540 340 250 2.13 0.31 0.54 0.85 1.51 1020 640 538 213 2.45 0.58 0.46 1.04 𝑍𝑇max = 0.38 reported for n-type (Bi1.9 Ta0.1 )SeO2 @ 𝑛 = 2.1×1019 + 𝑇 = 773 K 𝑍𝑇max = 0.13 reported for n-type Bi2 TeO2 @ 𝑛 = 1.1×1019 cm-3 + 𝑇 = 573 K Tan et al., Adv. Energy Mater. 9, 1900354 (2019) Luu and Vaqueiro, J. Solid State Chem. 226, 219 (2015) Flitcroft et al., J. Phys.: Energy 6, 025011 (2024) Dr J. M. Skelton Indo-UK Workshop on Energy, 1st October 2024 | Slide 14
15. Summary High-performance thermoelectrics require a balance of a 𝑆 and 𝜎 and low 𝜅 = 𝜅latt + 𝜅el The 𝜅latt can be modelled using the single-mode relaxation-time approximation: o Provides microscopic insight at the level of individual phonon modes The 𝑆, 𝜎 and 𝜅el calculated from electronic-structure calculations and approximate models for the 𝜏el : o Can reproduce experiments reasonably well, taking into account sample variation o Can be used to explore p- and n-type doping over a wide range of carrier concentrations and “untangle” the interdependence of the 𝑆, 𝜎, 𝜅el and 𝑛 Microscopic insight from the models, and useful predictive accuracy, allow this approach to be used to identify and characterise novel TEs Dr J. M. Skelton Indo-UK Workshop on Energy, 1st October 2024 | Slide 15
16. Acknowledgements ... plus other students, mentors and collaborators too numerous to mention Dr J. M. Skelton Indo-UK Workshop on Energy, 1st October 2024 | Slide 16
17. These slides are available on Speaker Deck: https://bit.ly/3Bs718l