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Predicting and understanding phase stability using lattice-dynamics modelling

Jonathan Skelton
February 06, 2023

Predicting and understanding phase stability using lattice-dynamics modelling

Presented at the Vienna Ab-initio Simulation Package (VASP) Ecosystem on 6th February 2023.

Jonathan Skelton

February 06, 2023
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  1. Dr Jonathan Skelton
    Department of Chemistry, University of Manchester
    ([email protected])
    Predicting and understanding phase stability
    using lattice-dynamics modelling

    View Slide

  2. Lattice dynamics
    VASP Workshop, 6th Feb 2023 | Slide 2
    Dr Jonathan Skelton
    Consider the Taylor expansion of the crystal potential energy:
    The second-order force constants 𝚽!,!! can be used to derive the phonon modes
    within the harmonic approximation
    𝜑 𝒖 = Φ#
    + '
    !
    '
    $
    Φ!
    $𝑢!
    $ +
    1
    2
    '
    !,!!
    '
    $,%
    Φ
    !,!!
    $% 𝑢!
    $𝑢
    !!
    % +
    1
    3!
    '
    !,!!,!!!
    '
    $,%,&
    Φ
    !,!!,!!!
    $%& 𝑢!
    $𝑢
    !!
    % 𝑢
    !!!
    & + ⋯
    Third- and higher-order force constants e.g. 𝚽!,!!,!!! capture various forms of
    anharmonicity and can be used to build on the basic HA e.g. for a perturbative
    treatment of phonon lifetimes
    Lattice energy
    𝑈'())
    Atomic forces (vanish
    at equilibrium)
    Harmonic approx. Anharmonicity

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  3. Phonons in solids
    VASP Workshop, 6th Feb 2023 | Slide 3
    Dr Jonathan Skelton

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  4. Phonons in solids
    VASP Workshop, 6th Feb 2023 | Slide 4
    Dr Jonathan Skelton

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  5. Phonons and phase stability
    VASP Workshop, 6th Feb 2023 | Slide 5
    Dr Jonathan Skelton
    Geometry
    Optimisation
    Energy/Volume
    EoS 𝐸(𝑉)
    Athermal Energy
    𝐸"
    Helmholtz
    Energy 𝐴(𝑇)
    Helmholtz
    Energy 𝐴(𝑉, 𝑇)
    Gibbs Energy
    𝐺(𝑇, 𝑝)
    Dynamical
    Stability
    Phonons

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  6. Example: the tin sulphides
    Rocksalt
    Pnma Cmcm 𝜋-cubic
    SnS2
    Sn2
    S3
    Zincblende
    VASP Workshop, 6th Feb 2023 | Slide 6
    Dr Jonathan Skelton

    View Slide

  7. Energetics I: convex hull
    J. M. Skelton et al., J. Phys. Chem. C 121 (12), 6446 (2017)
    VASP Workshop, 6th Feb 2023 | Slide 7
    Dr Jonathan Skelton

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  8. Thermodynamics I
    Using the harmonic approximation, we can calculate the Helmholtz free energy
    𝐴(𝑇):
    𝐴 𝑇 = 𝑈'()) + 𝐴*+, 𝑇 = 𝑈'()) + 𝑈*+,(𝑇) − 𝑇𝑆*+,(𝑇)
    The 𝐴*+, 𝑇 term is calculated using the bridge relation from the partition function
    𝑍*+, 𝑇 :
    𝑍*+,
    𝑇 = 6
    𝐪.
    exp[− ⁄
    ℏ𝜔𝐪. 2𝑘/𝑇]
    1 − exp[− ⁄
    ℏ𝜔𝐪. 𝑘/𝑇]
    𝐴*+,
    𝑇 = −
    1
    𝑁
    𝑘/
    𝑇ln 𝑍*+,
    𝑇 =
    1
    𝑁
    1
    2
    '
    𝐪.
    ℏ𝜔𝐪.
    + 𝑘/
    𝑇 '
    𝐪.
    ln 1 − exp − ⁄
    ℏ𝜔𝐪.
    𝑘/
    𝑇
    In typical DFT calculations the 𝑈'())
    is temperature independent - the phonon
    frequencies allows the temperature-dependent Helmholtz energy to be
    calculated
    VASP Workshop, 6th Feb 2023 | Slide 8
    Dr Jonathan Skelton

    View Slide

  9. Energetics II: Helmholtz energy
    J. M. Skelton et al., J. Phys. Chem. C 121 (12), 6446 (2017)
    VASP Workshop, 6th Feb 2023 | Slide 9
    Dr Jonathan Skelton

    View Slide

  10. Thermodynamics II
    Using the harmonic approximation, we can calculate the Helmholtz free energy
    𝐴(𝑇):
    𝐴(𝑇) = 𝑈'()) + 𝑈*+,(𝑇) − 𝑇𝑆*+,(𝑇)
    If we also take into account the volume dependence of 𝑈'())
    and the phonon
    frequencies, we can calculate the Gibbs free energy 𝐺(𝑇) (the quasi-harmonic
    approximation):
    𝐺 𝑇 = min
    0
    𝐴 𝑇; 𝑉 + 𝑝𝑉 = min
    0
    𝑈'())
    (𝑉) + 𝑈*+,
    (𝑇; 𝑉) − 𝑇𝑆*+,
    (𝑇; 𝑉) + 𝑝𝑉
    This is typically achieved by minimising a free-energy equation of state, which
    yields other properties such as 𝑉(𝑇) and 𝐵(𝑇) alongside 𝐺(𝑇)
    (𝐺 is arguably a more experimentally-relevant quantity, and we can also explore
    the effect of pressure through the 𝑝𝑉 term.)
    VASP Workshop, 6th Feb 2023 | Slide 10
    Dr Jonathan Skelton

    View Slide

  11. Energetics III: Gibbs vs Helmholtz
    I. Pallikara and J. M. Skelton, Phys. Chem. Chem. Phys. 23, 19219 (2021)
    VASP Workshop, 6th Feb 2023 | Slide 11
    Dr Jonathan Skelton

    View Slide

  12. Imaginary modes I: Cmcm SnSe
    J. M. Skelton et al., Phys. Rev. Lett 117, 075502 (2016)
    VASP Workshop, 6th Feb 2023 | Slide 12
    Dr Jonathan Skelton

    View Slide

  13. Dynamical stability
    U(Q)
    Q
    Real PES HA
    U(Q)
    Q
    Real PES HA
    𝑈 𝑄 =
    1
    2
    𝜇𝜔1𝑄1
    VASP Workshop, 6th Feb 2023 | Slide 13
    Dr Jonathan Skelton

    View Slide

  14. Imaginary mode PES mapping
    J. M. Skelton et al., Phys. Rev. Lett 117, 075502 (2016)
    VASP Workshop, 6th Feb 2023 | Slide 14
    Dr Jonathan Skelton

    View Slide

  15. Imaginary modes I: Cmcm SnSe
    VASP Workshop, 6th Feb 2023 | Slide 15
    Dr Jonathan Skelton
    Low 𝑇:
    Pnma
    High 𝑇: Cmcm
    (Average structure)
    I. Pallikara and J. M. Skelton, Phys. Chem. Chem. Phys. 23, 19219 (2021)

    View Slide

  16. Pressure and dynamical stability
    I. Pallikara and J. M. Skelton, Phys. Chem. Chem. Phys. 23, 19219 (2021)
    VASP Workshop, 6th Feb 2023 | Slide 16
    Dr Jonathan Skelton

    View Slide

  17. Energetics IV: p/T phase diagram
    I. Pallikara and J. M. Skelton, Phys. Chem. Chem. Phys. 23, 19219 (2021)
    VASP Workshop, 6th Feb 2023 | Slide 17
    Dr Jonathan Skelton

    View Slide

  18. Hybrid perovskites: (CH3
    NH3
    )PbI3
    Orthorhombic
    (𝑇 < 165 K)
    Tetragonal
    (𝑇 =165-327 K)
    Cubic
    (𝑇 > 327 K)
    VASP Workshop, 6th Feb 2023 | Slide 18
    Dr Jonathan Skelton
    A. N. Beecher et al., ACS Energy Lett. 1 (4), 880 (2016)

    View Slide

  19. Imaginary modes II: MAPbI3
    c-MAPbI3
    VASP Workshop, 6th Feb 2023 | Slide 19
    Dr Jonathan Skelton
    L. D. Whalley et al., Phys. Rev. B 94, 220301(R) (2016)

    View Slide

  20. VASP Workshop, 6th Feb 2023 | Slide 20
    Dr Jonathan Skelton
    Imaginary modes II: MAPbI3
    A. N. Beecher et al., ACS Energy Lett. 1 (4), 880 (2016)

    View Slide

  21. VASP Workshop, 6th Feb 2023 | Slide 21
    Dr Jonathan Skelton
    Imaginary modes II: MAPbI3
    A. N. Beecher et al., ACS Energy Lett. 1 (4), 880 (2016)

    View Slide

  22. VASP Workshop, 6th Feb 2023 | Slide 22
    Dr Jonathan Skelton
    Imaginary modes III: Bi2
    Sn2
    O7
    W. Rahim et al., Chem. Sci. 11, 7904 (2020)

    View Slide

  23. VASP Workshop, 6th Feb 2023 | Slide 23
    Dr Jonathan Skelton
    Imaginary modes III: Bi2
    Sn2
    O7
    W. Rahim et al., Chem. Sci. 11, 7904 (2020)

    View Slide

  24. Phonon spectroscopy
    VASP Workshop, 6th Feb 2023 | Slide 24
    Dr Jonathan Skelton
    B. Wei et al., Molecules 27 (19), 6431 (2022)

    View Slide

  25. Phonon spectroscopy
    VASP Workshop, 6th Feb 2023 | Slide 25
    Dr Jonathan Skelton
    B. Wei et al., Molecules 27 (19), 6431 (2022)

    View Slide

  26. Phonon spectroscopy
    VASP Workshop, 6th Feb 2023 | Slide 26
    Dr Jonathan Skelton
    𝒗 [cm-1] Ir. Rep. 𝑰!" 𝑰"#$#%
    Γ [cm-1] 𝒗 [cm-1] Ir. Rep. 𝑰!" 𝑰"#$#%
    Γ [cm-1]
    0.00 B2u - - - 397.53 Ag 0.000 1.702 0.61
    0.00 B1u - - - 407.82 B2u 0.029 0.000 1.20
    0.00 B3u - - - 415.81 B1u 0.000 0.000 1.57
    54.73 B1u 0.000 0.000 0.08 421.80 B3g 0.000 0.042 1.12
    62.88 Au 0.000 0.000 0.12 426.94 B2u 0.005 0.000 1.56
    80.03 Au 0.000 0.000 0.68 440.20 Ag 0.000 2.783 2.22
    83.26 B3u 0.002 0.000 0.50 443.30 B2g 0.000 1.001 1.95
    109.78 B2g 0.000 0.004 0.44 444.08 B1g 0.000 0.019 2.51
    116.62 B1g 0.000 0.015 0.30 454.09 B3g 0.000 0.470 2.97
    151.83 Ag 0.000 0.081 0.23 469.23 B3u 0.004 0.000 2.04
    155.94 B2u 0.001 0.000 0.52 470.48 Ag 0.000 5.650 3.36
    159.48 B3g 0.000 0.178 0.30 475.32 Au 0.000 0.000 2.43
    164.25 B2u 0.000 0.000 0.33 479.41 B1g 0.000 1.092 2.40
    183.06 B1u 0.007 0.000 0.56 491.32 B2g 0.000 0.925 2.26
    244.23 B3g 0.000 0.010 1.40 496.49 B1u 0.008 0.000 1.25
    290.37 B1u 0.168 0.000 2.74 504.58 B3g 0.000 0.001 2.73
    365.10 Ag 0.000 1.394 1.86 524.36 Ag 0.000 4.379 3.58
    386.98 B3g 0.000 0.067 1.19 528.69 B2u 0.002 0.000 1.67
    * Intensity units: 𝐼45
    - e2 amu-1; 𝐼5(6(7
    - 103 Å4 amu-1
    B. Wei et al., Molecules 27 (19), 6431 (2022)

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  27. Summary
    Phonon calculations are a computationally-tractable and powerful means for
    assessing the phase stability of materials
    A standard DFT total-energy calculation only provides the “athermal” total energy
    𝐸#
    and does not account for the effect of temperature
    The harmonic approximation provides access to the temperature-dependent
    Helmholtz free energy 𝐴(𝑇)
    The quasi-harmonic approximation provides access to the temperature- and
    pressure-dependent Gibbs free energy 𝐺(𝑇, 𝑝)
    The presence of imaginary modes in the harmonic spectrum indicate dynamical
    instabilities and can provide information about the nature of phase transitions
    and/or a means to explore the structural potential-energy surface
    Phonons calculations can also be used to simulate the infrared (IR) and Raman
    spectra, which provide a very good point of comparison to experiments
    VASP Workshop, 6th Feb 2023 | Slide 27
    Dr Jonathan Skelton

    View Slide

  28. Software + resources
    Phonopy - https://phonopy.github.io/phonopy/
    Implements the HA and QHA and interfaces very easily with VASP
    ModeMap - https://github.com/JMSkelton/ModeMap
    Add-on to Phonopy for mapping imaginary harmonic modes
    Phonopy-Spectroscopy - https://github.com/JMSkelton/Phonopy-Spectroscopy
    Add-on to Phonopy for simulating IR and Raman spectra
    Phonopy “Pro Tips” - https://www.slideshare.net/jmskelton/phonons-phonopy-pro-
    tips-2015
    Tutorial covering various aspects of Phonopy calculations
    VASP Workshop, 6th Feb 2023 | Slide 28
    Dr Jonathan Skelton

    View Slide

  29. Acknowledgements
    VASP Workshop, 6th Feb 2023 | Slide 29
    Dr Jonathan Skelton

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  30. https://bit.ly/3X0ViT0
    These slides are available on Speaker Deck:

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