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

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

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

(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

(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

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

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

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

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

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