How does lattice thermal conductivity “work”? Insights from first-principles calculations

Presented at the 42nd Collaborative Computational Chemistry No. 5 (CCP5) Annual General Meeting.

Transcript

1. J. M. Skelton, J. Cen, J. M. Flitcroft, M. Molinari, S. Moxon,
   I. Pallikara, J. Tang, J. Tse and B. Wei
   Department of Chemistry, University of Manchester
   (jonathan.skelton@manchester.ac.uk)
   How does lattice thermal conductivity “work”?
   Insights from first-principles calculations
   

   View Slide

2. Motivation: thermoelectrics
   CCP5 42nd AGM, 6th Sept 2022 | Slide 2
   Dr Jonathan M. Skelton
   𝑍𝑇 =
   𝑆!𝜎
   𝜅"#" + 𝜅#$%
   𝑇
   𝑆 - Seebeck coefficient
   𝜎 - electrical conductivity
   𝜅!"!
   - electronic thermal conductivity
   𝜅"#$
   - lattice thermal conductivity
   G. Tan et al., Chem. Rev. 116 (19), 12123 (2016)
   

   View Slide

3. Modelling thermal conductivity
   Dr Jonathan M. Skelton
   A. Togo et al., Phys. Rev. B 91, 094306 (2015)
   𝜿#$%% (𝑇) =
   1
   𝑁𝑉&
   .
   '
   𝜿'(𝑇)
   1
   𝑁𝑉&
   .
   '
   𝐶'(𝑇)𝒗' ⊗ 𝒗'𝜏'(𝑇)
   The simplest model for 𝜅"#$$
   is the single-mode relaxation time approximation (SM-RTA) - a
   closed solution to the phonon Boltzmann transport equations
   Modal heat capacity
   Mode group velocity
   𝜕𝜔%
   𝜕𝐪
   Average over phonon
   modes λ
   Phonon MFP
   Mode lifetime
   𝜏%
   =
   1
   2Γ%
   𝚲&
   𝑇 = 𝒗&
   𝜏&
   𝑇
   CCP5 42nd AGM, 6th Sept 2022 | Slide 3
   

   View Slide

4. Modelling thermal conductivity
   Dr Jonathan M. Skelton
   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
   CCP5 42nd AGM, 6th Sept 2022 | Slide 4
   

   View Slide

5. The RTA model: modal properties
   Dr Jonathan M. Skelton
   J. Tang and J. M. Skelton, J. Phys.: Condens. Matter 33 (16), 164002 (2021)
   CCP5 42nd AGM, 6th Sept 2022 | Slide 5
   

   View Slide

6. The RTA model: modal properties
   Dr Jonathan M. Skelton
   CoSb3
   CoSb3
   J. Tang and J. M. Skelton, J. Phys.: Condens. Matter 33 (16), 164002 (2021)
   CCP5 42nd AGM, 6th Sept 2022 | Slide 6
   

   View Slide

7. The RTA model: modal properties
   Dr Jonathan M. Skelton
   A. Gold-Parker et al., PNAS 115 (47), 11905 (2018)
   GaAs MAPbI3
   CCP5 42nd AGM, 6th Sept 2022 | Slide 7
   

   View Slide

8. 𝒗3
   vs. 𝜏3
   : the CRTA model
   Dr Jonathan M. Skelton
   Consider again the SM-RTA model:
   𝜿"#$$
   =
   1
   𝑁𝑉'
   3
   &
   𝜿&
   =
   1
   𝑁𝑉'
   3
   &
   𝐶&
   𝒗&
   ⊗ 𝒗&
   𝜏&
   Replace the 𝜏&
   with a constant lifetime (relaxation time) 𝜏()*+ defined as follows:
   𝜿"#$$
   𝜏()*+
   =
   1
   𝑁𝑉'
   3
   &
   𝜿&
   𝜏&
   =
   1
   𝑁𝑉'
   3
   &
   𝐶&
   𝒗&
   ⊗ 𝒗&
   𝜿"#$$
   ≈
   1
   𝑁𝑉'
   3
   &
   𝐶&
   𝒗&
   ⊗ 𝒗&
   ×𝜏()*+
   HA
   AH
   HA AH
   J. Tang and J. M. Skelton, J. Phys.: Condens. Matter 33 (16), 164002 (2021)
   CCP5 42nd AGM, 6th Sept 2022 | Slide 8
   

   View Slide

9. 𝒗3
   vs. 𝜏3
   : Si clathrates
   Dr Jonathan M. Skelton
   B. Wei et al., submitted
   CCP5 42nd AGM, 6th Sept 2022 | Slide 9
   

   View Slide

10. 𝒗3
    vs. 𝜏3
    : Si clathrates
    Dr Jonathan M. Skelton
    B. Wei et al., submitted
    𝜿!"##
    ≈
    1
    𝑁𝑉$
    &
    %
    𝐶%
    𝒗%
    ⊗ 𝒗%
    ×𝜏&'()
    CCP5 42nd AGM, 6th Sept 2022 | Slide 10
    

    View Slide

11. ⁄
    𝜿 𝜏4567: Si clathrates
    Dr Jonathan M. Skelton
    ⁄
    𝜿 𝝉𝐂𝐑𝐓𝐀
    (W m-1 K-1 ps-1) 𝒏𝐚
    Spacegroup
    d-Si 5.002 2 𝐹𝑑0
    3𝑚
    oC24 2.295 12 𝐶𝑚𝑐𝑚
    K-II / C-I 0.829 46 𝑃𝑚0
    3𝑚
    K-V / C-VI 0.815 40 𝐶𝑚𝑚𝑚
    K-VII / C-V 0.770 68 𝑃6&/𝑚𝑚𝑐
    C-II 0.458 34 𝐹𝑑0
    3𝑚
    With the exception of the Clathrate-II structure, the harmonic ⁄
    𝜿 𝜏!"#$ term correlates with:
    (1) the size of the primitive cell (𝑛%
    ); and
    (2) the spacegroup (crystal symmetry)
    Indicates that low group velocities are favoured by complex structures with large primitive
    cells and/or low symmetry
    B. Wei et al., submitted
    CCP5 42nd AGM, 6th Sept 2022 | Slide 11
    

    View Slide

12. Analysing 𝜏4567: phonon linewidths
    Dr Jonathan M. Skelton
    Γ%
    (𝑇) = &
    %&%&&
    Φ8%%&%&&
    9×{
    𝑛%&(𝑇) − 𝑛%&&(𝑇) 𝛿 𝜔 + 𝜔%& − 𝜔%&& − 𝛿 𝜔 − 𝜔%& + 𝜔%&& +
    𝑛%&(𝑇) + 𝑛%&&(𝑇) + 1 𝛿 𝜔 − 𝜔%& − 𝜔%&&
    }
    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)
    CCP5 42nd AGM, 6th Sept 2022 | Slide 12
    

    View Slide

13. Dr Jonathan M. Skelton
    A. Togo et al., Phys. Rev. B 91, 094306 (2015)
    Approximate expression for Γ!
    :
    With:
    Γ%
    (𝑇) ≈
    18𝜋
    ℏ9
    <
    𝑃𝑁9
    (𝒒%
    , 𝜔%
    , 𝑇)
    𝑁9 𝒒%, 𝜔%, 𝑇 = 𝑁9
    (;) 𝒒%, 𝜔%, 𝑇 + 𝑁9
    (9) 𝒒%, 𝜔%, 𝑇
    𝑁9
    (;) 𝒒%
    , 𝜔%
    , 𝑇 =
    1
    𝑁
    &
    %&%&&
    ∆(−𝒒%
    + 𝒒%& + 𝒒%&&) 𝑛%&(𝑇) − 𝑛%&&(𝑇) ×
    𝛿 𝜔 + 𝜔%& − 𝜔%&& − 𝛿 𝜔 − 𝜔%& + 𝜔%&&
    𝑁9
    (9) 𝒒%
    , 𝜔%
    , 𝑇 =
    1
    𝑁
    &
    %&%&&
    ∆(−𝒒%
    + 𝒒%& + 𝒒%&&) 𝑛%&(𝑇) + 𝑛%&&(𝑇) + 1 𝛿 𝜔 − 𝜔%& − 𝜔%&&
    CCP5 42nd AGM, 6th Sept 2022 | Slide 13
    Analysing 𝜏4567: phonon linewidths
    

    View Slide

14. Analysing 𝜏3
    Dr Jonathan M. Skelton
    B. Wei et al., submitted
    Γ%(𝑇) ≈
    18𝜋
    ℏ9
    <
    𝑃𝑁9(𝒒%, 𝜔%, 𝑇)
    CCP5 42nd AGM, 6th Sept 2022 | Slide 14
    

    View Slide

15. Summary
    Dr Jonathan M. Skelton
    𝜿!"##
    ⁄
    𝜿 𝜏&'() 𝜏&'()
    B
    𝑁9
    <
    𝑃
    CCP5 42nd AGM, 6th Sept 2022 | Slide 15
    RTA model gives good results for most
    systems and provides microscopic
    detail at the level of individual phonon
    modes
    Allows differences in 𝜿!"##
    to be
    attributed to differences in group
    velocities and phonon lifetimes
    Allows differences in lifetimes to be
    attributed to selection rules and
    (anharmonic) phonon interaction
    strengths
    

    View Slide

16. CRTA analysis: other TEs
    Dr Jonathan M. Skelton
    𝜅 [W m-1 K-1]
    ⁄
    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., submitted
    CCP5 42nd AGM, 6th Sept 2022 | Slide 16
    

    View Slide

17. Approximating Γ3
    Dr Jonathan M. Skelton CCP5 42nd AGM, 6th Sept 2022 | Slide 17
    S. Moxon et al., J. Mater. Chem. A 10, 1861 (2022)
    Γ%(𝑇) ≈
    18𝜋
    ℏ9
    <
    𝑃𝑁9(𝒒%, 𝜔%, 𝑇)
    

    View Slide

18. Acknowledgements
    Dr Jonathan M. Skelton CCP5 42nd AGM, 6th Sept 2022 | Slide 18
    B. Wei
    

    View Slide

19. https://bit.ly/3RohkNG
    These slides are on Speaker Deck:
    

    View Slide