$30 off During Our Annual Pro Sale. View Details »

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

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.

Jonathan Skelton

November 16, 2023
Tweet

More Decks by Jonathan Skelton

Other Decks in Science

Transcript

  1. 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

    View Slide

  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)

    View Slide

  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

    View Slide

  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)

    𝛼𝛽
    0 (𝜖F
    , 𝑇)
    𝜅el,𝛼𝛽
    (𝜖F
    , 𝑇) =
    1
    𝑒2𝑇
    ℒ𝛼𝛽
    1 (𝜖F
    , 𝑇) 2

    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)

    View Slide

  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

    View Slide

  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

    View Slide

  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)

    View Slide

  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)

    View Slide

  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)

    View Slide

  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)

    View Slide

  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

    View Slide

  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

    View Slide

  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)

    View Slide

  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)

    View Slide

  15. 𝑷𝒏𝒎𝒂 vs 𝝅 SnSe
    EPSRC TE Network Meeting, 16th Nov 2023 | Slide 15
    Dr Jonathan Skelton

    View Slide

  16. Acknowledgements
    EPSRC TE Network Meeting, 16th Nov 2023 | Slide 16
    Dr Jonathan Skelton

    View Slide

  17. https://bit.ly/3MMzkAX
    These slides are on Speaker Deck:

    View Slide