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

Presented at the 6th UK Materials Chemistry Consortium (MCC) Conference on 5th July 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 MCC Conference, 5th July 2024 | Slide 2 𝑍𝑇 = 𝑆!𝜎 𝜅"# + 𝜅#$%% 𝑇
3. An ab initio modelling workflow Crystal structure Convergence testing Geometry optimisation Phonon calculation 𝜅!"## Electronic structure 𝑆, 𝜎, 𝜅$! 𝑍𝑇 = 𝑆%σ 𝜅$! + 𝜅!"## 𝑇 Scattering rates: DP, 𝜔&' , 𝜀( , 𝑍∗, 𝐶, 𝑒(+) Dr J. M. Skelton MCC Conference, 5th July 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 MCC Conference, 5th July 2024 | Slide 4
5. Modelling thermal conductivity The simplest model for 𝜅!"## is the single-mode relaxation time approximation (SM-RTA) - a closed solution to the phonon Boltzmann transport equations 𝜿!"## (𝑇) = 1 𝑁𝑉 ) $ 𝐶$ (𝑇)𝒗$ ⊗ 𝒗$ 𝜏$ (𝑇) 𝐶$ - phonon heat capacities 𝒗$ - phonon group velocities 𝜏$ - phonon lifetimes (inverse linewidths Γ$ ) 𝑁 - 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 MCC Conference, 5th July 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 MCC Conference, 5th July 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 : 𝐸% (Calc.) = 1.46 eV 𝐸% (Expt) = 1.5 eV Bi2 SeO2 : 𝐸% (Calc.) = 1.1 eV 𝐸% (Expt) = 1.77 eV Bi2 TeO2 : 𝐸% (Calc.) = 0.33 eV 𝐸% (Expt) = 0.23 eV Dr J. M. Skelton MCC Conference, 5th July 2024 | Slide 7
8. Modelling electrical properties Ganose et al., Nature Comm. 12, 2222 (2021) We first define the spectral conductivity tensor: Σ&' 𝜖, 𝑇 = 1 8𝜋( 6 ) 7 𝑣𝒌),& 𝑣𝒌),' 𝜏𝒌) 𝑇 𝛿 𝜖 − 𝜖𝒌) 𝑑𝒌 This is used to calculate the 𝑛th-order moments of the generalised transport coefficients: ℒ&' , 𝜖- , 𝑇 = 7 Σ&' 𝜖, 𝑇 𝜖 − 𝜖- , − 𝜕𝑓 𝜖, 𝜖- , 𝑇 𝜕𝜖 𝜕𝜖 𝑓 𝜖, 𝜖- , 𝑇 = 1 exp ⁄ 𝜖 − 𝜖- 𝑘. 𝑇 + 1 Where: o The 𝒗𝒌) are obtained from a high-quality band structure o The 𝜏𝒌) can be: treated as a constant 𝜏/! ; approximated by model equations for different scattering processes; or calculated from the electron-phonon coupling o The 𝜖- (= 𝜇) is set by the DoS and a specified extrinsic carrier concentration 𝑛 Dr J. M. Skelton MCC Conference, 5th July 2024 | Slide 8
9. Modelling electrical properties The 𝓛,(𝜖- , 𝑇) are determined from a band structure, a model for the 𝜏)𝒌 , and a specified 𝑛/𝑇: ℒ&' , 𝜖- , 𝑇 = 7 Σ&' 𝜖, 𝑇 𝜖 − 𝜖- , − 𝜕𝑓 𝜖, 𝜖- , 𝑇 𝜕𝜖 𝜕𝜖 The electrical transport coefficients can be determined from the 𝓛,(𝜖- , 𝑇) as: 𝜎&' (𝜖- , 𝑇) = ℒ&' 0 (𝜖- , 𝑇) 𝑆&' (𝜖- , 𝑇) = 1 𝑒𝑇 ℒ&' 1 (𝜖- , 𝑇) ℒ&' 0 (𝜖- , 𝑇) 𝜅/!,&' (𝜖- , 𝑇) = 1 𝑒2𝑇 ℒ&' 1 (𝜖- , 𝑇) 2 ℒ&' 0 (𝜖- , 𝑇) − ℒ&' 2 (𝜖- , 𝑇) Note that when using the CRTA (i.e. 𝜏𝒌) → 𝜏/! ): o The 𝑺 are the ratio of two 𝓛, and the 𝜏/! cancel o The 𝝈 and 𝜿/! are obtained with respect to 𝜏/! (𝜏/! ~ 10-14 s) Ganose et al., Nature Comm. 12, 2222 (2021) Dr J. M. Skelton MCC Conference, 5th July 2024 | Slide 9
10. Modelling electrical properties Flitcroft et al., Solids 3 (1), 155 (2022) Fixed 𝑇 = 800 K Fixed 𝑛3 = 1019 cm-3 Dr J. M. Skelton MCC Conference, 5th July 2024 | Slide 10
11. Electrical transport Flitcroft et al., J. Phys.: Energy 6, 025011 (2024) Dr J. M. Skelton MCC Conference, 5th July 2024 | Slide 11
12. Comparison to experiments Dr J. M. Skelton MCC Conference, 5th July 2024 | Slide 12
13. Comparison to experiments Flitcroft et al., J. Phys.: Energy 6, 025011 (2024) Dr J. M. Skelton MCC Conference, 5th July 2024 | Slide 13
14. Comparison to experiments Flitcroft et al., J. Phys.: Energy 6, 025011 (2024) 𝑇 = 300 K 𝑇 = 800 K Dr J. M. Skelton MCC Conference, 5th July 2024 | Slide 14
15. Predicted 𝒁𝑻 Flitcroft et al., J. Phys.: Energy 6, 025011 (2024) Dr J. M. Skelton MCC Conference, 5th July 2024 | Slide 15
16. 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 𝑍𝑇4"5 = 0.38 reported for n-type (Bi1.9 Ta0.1 )SeO2 @ 𝑛 = 2.1×1019 + 𝑇 = 773 K 𝑍𝑇4"5 = 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 MCC Conference, 5th July 2024 | Slide 16
17. Some “pro tips” (?) Problem: Accurate 𝐸% and 𝜀6 typically require hybrid functionals – can be expensive, especially if convergence w.r.t. 𝑘-point sampling is slow Solution: Try non-self-consistent hybrid calculations Skelton et al., J. Chem. Theor. Comput. 16 (6), 3543 (2020) Dr J. M. Skelton MCC Conference, 5th July 2024 | Slide 17
18. Some “pro tips” (?) Problem: 𝜅!"## calculations on doped/alloy systems are prohibitively expensive Solution: The 𝜅!"## can be very well approximated from a harmonic phonon calculation and averaged three-phonon interaction strengths from a pristine/endpoint structure Skelton, J. Mater. Chem. C 9 (35), 11772 (2021) Liu et al., J. Mater. Chem. C 12 (2), 508 (2024) Dr J. M. Skelton MCC Conference, 5th July 2024 | Slide 18
19. Some “pro tips” (?) Problem: 𝜅!"## calculations on doped/alloy systems are prohibitively expensive Solution: The 𝜅!"## can be very well approximated from a harmonic phonon calculation and averaged three-phonon interaction strengths from a pristine/endpoint structure Problem: To calculate 𝜔78 and/or 𝜀98:9; we sometimes need to combine GGA phonon calculations with hybrid Born charges 𝑍∗ Solution: Implemented calculating 𝜔78 /𝜀98:9; from Phonopy calculations Problem: 𝜅!"## calculations on doped/alloy systems are prohibitively expensive Solution: The 𝜅!"## can be very well approximated from a harmonic phonon calculation and averaged three-phonon interaction strengths from a pristine/endpoint structure Dr J. M. Skelton MCC Conference, 5th July 2024 | Slide 19
20. Summary High-performance thermoelectrics require a balance of a 𝑆 and 𝜎 and low 𝜅 = 𝜅!"## + 𝜅/! The 𝜅!"## 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 𝜅/! calculated from electronic-structure calculations and approximate models for the 𝜏/! : 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, 𝜅/! 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 MCC Conference, 5th July 2024 | Slide 20
21. Acknowledgements ... plus other students, mentors and collaborators too numerous to mention Dr J. M. Skelton MCC Conference, 5th July 2024 | Slide 21
22. These slides are available on Speaker Deck: https://bit.ly/3RUtVdG