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Understanding and controlling the heat transport in thermoelectric materials

Understanding and controlling the heat transport in thermoelectric materials

Presented at the Thomas Young Centre (TYC) Symposium on "Modelling Phonons in Materials" on 26th January 2023.

Jonathan Skelton

January 26, 2023
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  1. Dr Jonathan Skelton
    Department of Chemistry, University of Manchester
    ([email protected])
    Understanding and controlling
    the heat transport in thermoelectric materials

    View Slide

  2. Acknowledgements
    TYC Seminar, 26th Jan 2023 | Slide 2
    Dr Jonathan Skelton
    ... plus other students, mentors and
    collaborators too numerous to mention

    View Slide

  3. Overview
    TYC Seminar, 26th Jan 2023 | Slide 3
    Dr Jonathan Skelton
    o Thermoelectric power and the global
    energy challenge
    o Modelling lattice thermal conductivity
    o Models for understanding 𝜅latt
    :
    • CRTA model - 𝒗λ
    vs. 𝜏λ
    • Constant 𝑃λ
    model - ഥ
    𝑁2
    (𝜔) vs. ෨
    𝑃
    o Strategies for controlling 𝜅latt
    :
    • Reducing 𝒗λ
    - alloying and doping
    • Reducing 𝜏λ
    - “rattler” TEs
    o Modelling the thermoelectric figure of
    merit
    o Recent highlights and current work
    http://bit.ly/3H3ys7x

    View Slide

  4. The global energy challenge
    34 %
    26 %
    19 %
    18 %
    3 %
    1000 MW nuclear power plant:
    o 650 MW waste heat
    o 3 % ≈ 20 MW ≈ 50,000 homes
    300-500 W from exhaust gases:
    o 2 % lower fuel consumption
    o 2.4 Mt reduction in CO2
    Thermoelectric generators allow waste
    heat to be recovered as electricity
    TEGs with ~3 % energy recovery (𝑍𝑇 = 1) are
    considered industrially viable
    1. Provisional UK greenhouse gas emissions national statistics (published June 2020)
    2. EPSRC Thermoelectric Network Roadmap (2018)
    TYC Seminar, 26th Jan 2023 | Slide 4
    Dr Jonathan Skelton

    View Slide

  5. Thermoelectric materials
    𝑍𝑇 =
    𝑆2𝜎
    𝜅ele
    + 𝜅lat
    𝑇
    𝑆 - Seebeck coefficient
    𝜎 - electrical conductivity
    𝜅ele
    - electronic thermal conductivity
    𝜅lat
    - lattice thermal conductivity
    G. Tan et al., Chem. Rev. 116 (19), 12123 (2016)
    TYC Seminar, 26th Jan 2023 | Slide 5
    Dr Jonathan Skelton

    View Slide

  6. Modelling thermal conductivity
    A. Togo et al., Phys. Rev. B 91, 094306 (2015)
    𝜿latt
    𝑇 =
    1
    𝑁𝑉0

    𝜆
    𝜿𝜆
    𝑇 =
    1
    𝑁𝑉0

    𝜆
    𝐶𝜆
    (𝑇)𝒗𝜆
    ⊗ 𝒗𝜆
    𝜏𝜆
    (𝑇)
    The simplest model for 𝜅latt
    is the single-mode relaxation time approximation (SM-RTA) - a
    closed solution to the phonon Boltzmann transport equations
    Modal heat capacity
    Mode group velocity
    𝜕𝜔λ
    𝜕𝐪
    Average over phonon
    modes λ
    Phonon MFP
    Mode lifetime
    𝜏λ
    =
    1
    2Γλ
    𝚲𝜆
    𝑇 = 𝒗𝜆
    𝜏𝜆
    𝑇
    TYC Seminar, 26th Jan 2023 | Slide 6
    Dr Jonathan Skelton

    View Slide

  7. Modelling thermal conductivity
    TYC Seminar, 26th Jan 2023 | Slide 7
    Dr Jonathan Skelton

    View Slide

  8. Modelling thermal conductivity
    A. Togo et al., Phys. Rev. B 91, 094306 (2015)
    J. Tang and J. M. Skelton, J. Phys.: Condens. Matter 33 (16), 164002 (2021)
    CoSb3
    TYC Seminar, 26th Jan 2023 | Slide 8
    Dr Jonathan Skelton

    View Slide

  9. Modelling thermal conductivity
    J. Tang and J. M. Skelton, J. Phys.: Condens. Matter 33 (16), 164002 (2021)
    TYC Seminar, 26th Jan 2023 | Slide 9
    Dr Jonathan Skelton

    View Slide

  10. Modelling thermal conductivity
    A. Gold-Parker et al., PNAS 115 (47), 11905 (2018)
    GaAs (CH3
    NH3
    )PbI3
    TYC Seminar, 26th Jan 2023 | Slide 10
    Dr Jonathan Skelton

    View Slide

  11. 𝒗𝜆
    vs. 𝜏𝜆
    : the CRTA model
    Consider again the SM-RTA model:
    𝜿latt
    =
    1
    𝑁𝑉0

    𝜆
    𝜿𝜆
    =
    1
    𝑁𝑉0

    𝜆
    𝐶𝜆
    𝒗𝜆
    ⊗ 𝒗𝜆
    𝜏𝜆
    Replace the 𝜏𝜆
    with a constant lifetime (relaxation time) 𝜏CRTA defined as follows:
    𝜿latt
    𝜏CRTA
    =
    1
    𝑁𝑉0

    𝜆
    𝜿𝜆
    𝜏𝜆
    =
    1
    𝑁𝑉0

    𝜆
    𝐶𝜆
    𝒗𝜆
    ⊗ 𝒗𝜆
    𝜿latt

    1
    𝑁𝑉0

    𝜆
    𝐶𝜆
    𝒗𝜆
    ⊗ 𝒗𝜆
    × 𝜏CRTA
    HA
    AH
    HA AH
    J. Tang and J. M. Skelton, J. Phys.: Condens. Matter 33 (16), 164002 (2021)
    TYC Seminar, 26th Jan 2023 | Slide 11
    Dr Jonathan Skelton

    View Slide

  12. J. M. Skelton, J. Mater. Chem. C (2021), DOI: 10.1039/D1TC02026A
    𝒗𝜆
    vs. 𝜏𝜆
    : SnS/SnSe
    TYC Seminar, 26th Jan 2023 | Slide 12
    Dr Jonathan Skelton
    SnS
    SnSe

    View Slide

  13. 𝒗𝜆
    vs. 𝜏𝜆
    : other TEs
    𝜅 [W m-1 K-1]
    Τ
    𝜅 𝝉𝐂𝐑𝐓𝐀
    [W m-1 K-1 ps-1] 𝝉𝐂𝐑𝐓𝐀 [ps]
    Si 136.24 5.002 27.2
    SnS 2.15 0.718 3.00
    SnSe 1.58 0.372 4.23
    CoSb3
    9.98 0.273 36.6
    Bi2
    S3
    (Pnma) 0.90 0.423 2.14
    Bi2
    Se3
    (R-3m) 1.82 0.293 6.20
    Bi2
    Te3
    (R-3m) 0.87 0.199 4.41
    J. M. Skelton, J. Mater. Chem. C 9, 11772 (2021)
    J. Tang and J. M. Skelton, J. Phys.: Condens. Matter 33 (16), 164002 (2021)
    J. Cen, I. Pallikara and J. M. Skelton, Chem. Mater. 33 (21), 8404 (2021)
    B. Wei et al., Molecules 27 (19), 6431 (2022)
    TYC Seminar, 26th Jan 2023 | Slide 13
    Dr Jonathan Skelton

    View Slide

  14. 𝒗𝜆
    vs. 𝜏𝜆
    : Si clathrates
    TYC Seminar, 26th Jan 2023 | Slide 14
    Dr Jonathan Skelton
    B. Wei et al., Molecules 27 (19), 6431 (2022)

    View Slide

  15. 𝒗𝜆
    vs. 𝜏𝜆
    : Si clathrates
    TYC Seminar, 26th Jan 2023 | Slide 15
    Dr Jonathan Skelton
    𝜿latt

    1
    𝑁𝑉0

    𝜆
    𝐶𝜆
    𝒗𝜆
    ⊗ 𝒗𝜆
    × 𝜏CRTA
    B. Wei et al., Molecules 27 (19), 6431 (2022)

    View Slide

  16. 𝒗𝜆
    vs. 𝜏𝜆
    : Si clathrates
    𝜿 (W m-1 K-1)
    Τ
    𝜿 𝝉𝐂𝐑𝐓𝐀
    (W m-1 K-1 ps-1) 𝝉𝐂𝐑𝐓𝐀 (ps)
    d-Si 136.24 5.002 27.24
    oC24 40.92 2.295 17.83
    K-II / C-I 43.54 0.829 52.52
    K-V / C-VI 36.29 0.815 44.53
    K-VII / C-V 31.16 0.770 40.45
    C-II 6.33 0.458 13.81
    Spacegroup 𝒏𝐚
    𝐹𝑑ത
    3𝑚 2
    𝐶𝑚𝑐𝑚 12
    𝑃𝑚ത
    3𝑚 46
    𝐶𝑚𝑚𝑚 40
    𝑃63
    /𝑚𝑚𝑐 68
    𝐹𝑑ത
    3𝑚 34
    With the exception of the Clathrate-II structure, the harmonic Τ
    𝜿 𝜏CRTA term correlates with:
    1) the size of the primitive cell (𝑛a
    ); and
    2) the spacegroup (crystal symmetry)
    Implies low group velocities are favoured by complex structures with large primitive cells
    and/or low symmetry
    TYC Seminar, 26th Jan 2023 | Slide 16
    Dr Jonathan Skelton
    B. Wei et al., Molecules 27 (19), 6431 (2022)

    View Slide

  17. Analysing 𝜏𝜆
    : phonon linewidths
    Γ𝜆
    (𝑇) = ෍
    𝜆′𝜆′′
    Φ−𝜆𝜆′𝜆′′
    2 × {
    𝑛𝜆′
    (𝑇) − 𝑛𝜆′′
    (𝑇) 𝛿 𝜔 + 𝜔𝜆′
    − 𝜔𝜆′′
    − 𝛿 𝜔 − 𝜔𝜆′
    + 𝜔𝜆′′ +
    𝑛𝜆′
    (𝑇) + 𝑛𝜆′′
    (𝑇) + 1 𝛿 𝜔 − 𝜔𝜆′
    − 𝜔𝜆′′
    }
    Collision
    Decay
    Three-phonon interaction strength - includes
    conservation of momentum(“anharmonicity”)
    Conservation of energy
    (“selection rules”)
    A. Togo et al., Phys. Rev. B 91, 094306 (2015)
    TYC Seminar, 26th Jan 2023 | Slide 17
    Dr Jonathan Skelton

    View Slide

  18. Analysing 𝜏𝜆
    : phonon linewidths
    A. Togo et al., Phys. Rev. B 91, 094306 (2015)
    Approximate expression for Γ𝜆
    :
    With:
    Γ𝜆
    (𝑇) ≈
    18𝜋
    ℏ2

    𝑃𝑁2
    (𝒒𝜆
    , 𝜔𝜆
    , 𝑇)
    𝑁2
    𝒒𝜆
    , 𝜔𝜆
    , 𝑇 = 𝑁
    2
    (1) 𝒒𝜆
    , 𝜔𝜆
    , 𝑇 + 𝑁
    2
    (2) 𝒒𝜆
    , 𝜔𝜆
    , 𝑇
    𝑁
    2
    (1) 𝒒𝜆
    , 𝜔𝜆
    , 𝑇 =
    1
    𝑁

    𝜆′𝜆′′
    ∆(−𝒒𝜆
    + 𝒒𝜆′
    + 𝒒𝜆′′
    ) 𝑛𝜆′
    (𝑇) − 𝑛𝜆′′
    (𝑇) ×
    𝛿 𝜔 + 𝜔𝜆′
    − 𝜔𝜆′′
    − 𝛿 𝜔 − 𝜔𝜆′
    + 𝜔𝜆′′
    𝑁
    2
    (2) 𝒒𝜆
    , 𝜔𝜆
    , 𝑇 =
    1
    𝑁

    𝜆′𝜆′′
    ∆(−𝒒𝜆
    + 𝒒𝜆′
    + 𝒒𝜆′′
    ) 𝑛𝜆′
    (𝑇) + 𝑛𝜆′′
    (𝑇) + 1 𝛿 𝜔 − 𝜔𝜆′
    − 𝜔𝜆′′
    TYC Seminar, 26th Jan 2023 | Slide 18
    Dr Jonathan Skelton

    View Slide

  19. Analysing 𝜏𝜆
    : phonon linewidths
    Nanoscale Energy Harvesting, 24th Aug 2022 | Slide 19
    Dr Jonathan M. Skelton
    Γ𝜆
    (𝑇) ≈
    18𝜋
    ℏ2

    𝑃𝑁2
    (𝒒𝜆
    , 𝜔𝜆
    , 𝑇)
    B. Wei et al., Molecules 27 (19), 6431 (2022)

    View Slide

  20. Analysing 𝜏𝜆
    Γ𝜆
    (𝑇) ≈
    18𝜋
    ℏ2

    𝑃𝑁2
    (𝒒𝜆
    , 𝜔𝜆
    , 𝑇)
    TYC Seminar, 26th Jan 2023 | Slide 20
    Dr Jonathan Skelton
    B. Wei et al., Molecules 27 (19), 6431 (2022)

    View Slide

  21. Workflow
    𝜿latt
    (𝑇)
    Τ
    𝜿 𝜏CRTA 𝜏CRTA

    𝑁2

    𝑃
    Phonopy + Phono3py
    A. Togo and I. Tanka, Scr. Mater. 108, 1 (2015)
    A. Togo et al., Phys. Rev. B 91, 094306 (2015)
    TYC Seminar, 26th Jan 2023 | Slide 21
    Dr Jonathan Skelton

    View Slide

  22. Reducing 𝒗𝜆
    I: alloying
    C.-C. Lin et al., Chem. Mater. 29 (12), 5344 (2017)
    SnSe
    15-20 % S
    TYC Seminar, 26th Jan 2023 | Slide 22
    Dr Jonathan Skelton

    View Slide

  23. Reducing 𝒗𝜆
    I: alloying
    54.2 % ↓
    Sn(S0.1875
    Se0.8125
    )
    SnSe
    J. M. Skelton, J. Mater. Chem. C 9, 11772 (2021)
    TYC Seminar, 26th Jan 2023 | Slide 23
    Dr Jonathan Skelton

    View Slide

  24. Reducing 𝒗𝜆
    II: discordant doping
    H. Xie et al., J. Am. Chem. Soc. 141 (47), 18900 (2019)
    TYC Seminar, 26th Jan 2023 | Slide 24
    Dr Jonathan Skelton

    View Slide

  25. Reducing 𝜏𝜆
    I: “rattler” TEs
    J. W. Schwartz and C. T. Walker, Phys. Rev. B 155, 959 (1967)
    E. S. Toberer et al., J. Mater. Chem. 21, 15843 (2011)
    “One phonon” model for resonant
    scattering:
    𝜏−1 = ෍
    𝑖
    𝑐𝑖
    𝜔2𝑇2
    𝜔𝑖
    2 − 𝜔2 2
    + 𝛾𝑖
    𝜔𝑖
    2𝜔2
    TYC Seminar, 26th Jan 2023 | Slide 25
    Dr Jonathan Skelton

    View Slide

  26. Reducing 𝜏𝜆
    I: “rattler” TEs
    TYC Seminar, 26th Jan 2023 | Slide 26
    Dr Jonathan Skelton
    J. Tang and J. M. Skelton, J. Phys.: Condens. Matter 33 (16), 164002 (2021)
    Filler 𝒎𝐗
    [amu] 𝒓𝐗
    [pm]
    He 4.0026 31
    Ne 20.180 38
    Ar 39.948 71
    Kr 83.798 88
    Xe 131.29 108
    Noble gases are chemically inert (closed shell, unlikely to reduce/oxidise host framework) and
    are likely closest it is possible to get to a “hard sphere” filler

    View Slide

  27. Reducing 𝜏𝜆
    I: “rattler” TEs
    TYC Seminar, 26th Jan 2023 | Slide 27
    Dr Jonathan Skelton
    J. Tang and J. M. Skelton, J. Phys.: Condens. Matter 33 (16), 164002 (2021)
    We can define a rattling frequency ሚ
    𝑓𝑥
    for the noble gas fillers X based on the 𝑫 XX, 𝐪 = Γ :
    𝑫 XX, 𝐪 = Γ =
    1
    𝑚X

    𝑙′
    𝚽 X0, X𝑙′
    What happens to 𝜅latt
    if we artificially change the 𝑚X
    while keeping the 𝚽 fixed?

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  28. Reducing 𝜏𝜆
    II: hybrid TEs (?)
    A. Gold-Parker et al., PNAS 115 (47), 11905 (2018)
    TYC Seminar, 26th Jan 2023 | Slide 28
    Dr Jonathan Skelton

    View Slide

  29. Workflow
    𝜿latt
    (𝑇)
    Τ
    𝜿 𝜏CRTA 𝜏CRTA

    𝑁2

    𝑃
    𝑺(𝑛, 𝑇) 𝝈(𝑛, 𝑇) 𝜿el
    (𝑛, 𝑇)
    Phonopy + Phono3py AMSET
    𝑍𝑇(𝑛, 𝑇)
    A. Togo and I. Tanka, Scr. Mater. 108, 1 (2015)
    A. Togo et al., Phys. Rev. B 91, 094306 (2015)
    A. M. Ganose et al., Nature Comm. 12, 2222 (2021)
    TYC Seminar, 26th Jan 2023 | Slide 29
    Dr Jonathan Skelton

    View Slide

  30. Predicting 𝒁𝑻
    J. M. Flitcroft et al., Solids 3 (1), 155 (2022)
    TYC Seminar, 26th Jan 2023 | Slide 30
    Dr Jonathan Skelton

    View Slide

  31. High-performance oxide TEs
    W. Rahim et al., J. Mater. Chem. A 8, 16405 (2020)
    W. Rahim et al., J. Mater. Chem. A 9, 20417 (2021)
    K. Brlec et al., J. Mater. Chem. A 10, 16813 (2022)
    𝛼-Bi2
    Sn2
    O7
    𝑛 = 1.73 × 1019 cm-3
    𝑍𝑇 = 0.36 (385 K)
    Ca4
    Sb2
    O / Ca4
    Bi2
    O
    𝑝 = 4.64 / 2.15 × 1019 cm-3
    𝑍𝑇 = 1.58 / 2.14 (1000 K)
    Y2
    Ti2
    O5
    S2
    𝑛 = 2.37 × 1020 cm-3
    𝑍𝑇 = 1.18 (1000 K)
    TYC Seminar, 26th Jan 2023 | Slide 31
    Dr Jonathan Skelton

    View Slide

  32. 𝝅-cubic SnS/SnSe
    TYC Seminar, 26th Jan 2023 | Slide 32
    Dr Jonathan Skelton
    𝒁𝑻 𝐦𝐚𝐱
    𝑺𝟐𝝈
    [mW m-1 K-2]
    𝜿𝐭𝐨𝐭
    [W m-1 K-1]
    SnS (Pnma) 1.75 1.87 1.07
    SnSe (Pnma) 2.81 2.62 0.93
    SnSe (RS) 2.60 10.90 3.02
    R. E. Abutbul et al., CrystEngComm 18, 5188 (2016)
    J. M. Flitcroft et al., Solids 3 (1), 155 (2022)

    View Slide

  33. “Hybrid” TEs
    TYC Seminar, 26th Jan 2023 | Slide 33
    Dr Jonathan Skelton

    View Slide

  34. Summary
    o The SM-RTA can give quantitative predictions of the 𝜅latt
    of a wide range of materials
    o The contributions of individual phonon modes can be used to obtain microscopic insight
    into how the 𝜅latt
    “works”:
    • CRTA model: 𝒗λ
    vs. 𝜏λ
    • Constant 𝑃λ
    model: ഥ
    𝑁2
    vs. ෨
    𝑃
    o Modelling on Si allotropes shows that low 𝜅latt
    is favoured by:
    1) Large primitive cells
    2) Lower crystal symmetry
    o With reference to existing TEs, the CRTA model can be used to suggest strategies for
    reducing the 𝜅latt
    :
    • 𝒗λ
    - alloying and discordant-atom doping
    • 𝜏λ
    - introducing “ratters”, small molecules may be particularly effective
    o These ideas are being explored in our current work -- watch this space!
    TYC Seminar, 26th Jan 2023 | Slide 34
    Dr Jonathan Skelton

    View Slide

  35. Phono3py-Power-Tools
    TYC Seminar, 26th Jan 2023 | Slide 35
    Dr Jonathan Skelton
    https://github.com/skelton-group/Phono3py-Power-Tools

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  36. Thankyou for listening!
    Any questions?

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