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Temperature and pressure effects on phase stability from theoretical modelling: application to the tin-sulphide phase space

Temperature and pressure effects on phase stability from theoretical modelling: application to the tin-sulphide phase space

Presented at the 29th General Conference of the Condensed Matter Division of the European Physical Society on Materials Chemistry (CMD29).

Jonathan Skelton

August 23, 2022
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  1. Ioanna Pallikara, Joseph M. Flitcroft and Jonathan M. Skelton Department

    of Chemistry, University of Manchester (jonathan.skelton@manchester.ac.uk) Temperature and pressure effects on phase stability from theoretical modelling: application to the tin-sulphide phase space
  2. The tin chalcogenides CMD29, 23rd August 2022 | Slide 2

    Dr Jonathan M. Skelton
  3. The tin sulphide phase space CMD29, 23rd August 2022 |

    Slide 3 Dr Jonathan M. Skelton Rocksalt Pnma Cmcm 𝜋-cubic SnS2 Sn2 S3 Zincblende
  4. “Cubic” tin sulphide CMD29, 23rd August 2022 | Slide 4

    Dr Jonathan M. Skelton 1967 2006 2012 2015 2016 2017 Initial identification of rocksalt SnS by epitaxial growth First report of tetrahedral zincblende nanoparticles Modelling suggests ZB SnS is energetically and dynamically unstable Electron diffraction suggests ZB nanoparticles are a new phase in the cubic P21 3 spacegroup (𝜋-SnS) Another report of ZB SnS Atomic structure of 𝜋-SnS solved by X-ray diffraction Modelling confirms 𝜋-SnS is energetically metastable and dynamically stable
  5. Energetics I: convex hull CMD29, 23rd August 2022 | Slide

    5 Dr Jonathan M. Skelton J. M. Skelton et al., J. Phys. Chem. C 121 (12), 6446 (2017)
  6. Structural dynamics of solids CMD29, 23rd August 2022 | Slide

    6 Dr Jonathan M. 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
  7. Dynamical stability CMD29, 23rd August 2022 | Slide 7 Dr

    Jonathan M. Skelton U(Q) Q Real PES HA U(Q) Q Real PES HA 𝑈 𝑄 = 1 2 𝜇𝜔*𝑄*
  8. Dynamical stability CMD29, 23rd August 2022 | Slide 8 Dr

    Jonathan M. Skelton J. M. Skelton et al., J. Phys. Chem. C 121 (12), 6446 (2017)
  9. Dynamical stability CMD29, 23rd August 2022 | Slide 9 Dr

    Jonathan M. Skelton J. M. Skelton et al., J. Phys. Chem. C 121 (12), 6446 (2017)
  10. Imaginary modes in Cmcm SnX CMD29, 23rd August 2022 |

    Slide 10 Dr Jonathan M. Skelton J. M. Skelton et al., Phys. Rev. Lett 117, 075502 (2016)
  11. Imaginary modes in Cmcm SnX CMD29, 23rd August 2022 |

    Slide 11 Dr Jonathan M. Skelton J. M. Skelton et al., Phys. Rev. Lett 117, 075502 (2016)
  12. RS and ZB SnS under pressure CMD29, 23rd August 2022

    | Slide 12 Dr Jonathan M. Skelton J. M. Skelton et al., J. Phys. Chem. C 121 (12), 6446 (2017)
  13. Statistical thermodynamics I CMD29, 23rd August 2022 | Slide 13

    Dr Jonathan M. Skelton 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 = : 𝐪/ exp[− ⁄ ℏ𝜔𝐪/ 2𝑘0𝑇] 1 − exp[− ⁄ ℏ𝜔𝐪/ 𝑘0𝑇] 𝐴+,- 𝑇 = − 1 𝑁 𝑘0 𝑇ln 𝑍+,- 𝑇 = 1 𝑁 1 2 ( 𝐪/ ℏ𝜔𝐪/ + 𝑘0 𝑇 ( 𝐪/ ln 1 − exp − ⁄ ℏ𝜔𝐪/ 𝑘0 𝑇 In typical DFT calculations the 𝑈'()) is temperature independent - the phonon frequencies allows the temperature-dependent Helmholtz energy to be calculated
  14. Energetics II: Helmholtz energy CMD29, 23rd August 2022 | Slide

    14 Dr Jonathan M. Skelton J. M. Skelton et al., J. Phys. Chem. C 121 (12), 6446 (2017)
  15. Statistical thermodynamics II CMD29, 23rd August 2022 | Slide 15

    Dr Jonathan M. Skelton 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 1 𝐴 𝑇; 𝑉 + 𝑝𝑉 = min 1 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.)
  16. Energetics III: Gibbs vs Helmholtz CMD29, 23rd August 2022 |

    Slide 16 Dr Jonathan M. Skelton I. Pallikara and J. M. Skelton, Phys. Chem. Chem. Phys. 23, 19219 (2021)
  17. Energetics IV: p/T phase diagram CMD29, 23rd August 2022 |

    Slide 17 Dr Jonathan M. Skelton I. Pallikara and J. M. Skelton, Phys. Chem. Chem. Phys. 23, 19219 (2021)
  18. Cmcm SnS under pressure CMD29, 23rd August 2022 | Slide

    18 Dr Jonathan M. Skelton I. Pallikara and J. M. Skelton, Phys. Chem. Chem. Phys. 23, 19219 (2021)
  19. RS SnS -> 𝝅 SnS transition barrier CMD29, 23rd August

    2022 | Slide 19 Dr Jonathan M. Skelton I. Pallikara and J. M. Skelton, Phys. Chem. Chem. Phys. 23, 19219 (2021)
  20. Energetics IV: p/T phase diagram CMD29, 23rd August 2022 |

    Slide 20 Dr Jonathan M. Skelton I. Pallikara and J. M. Skelton, Phys. Chem. Chem. Phys. 23, 19219 (2021)
  21. Application: thermoelectrics CMD29, 23rd August 2022 | Slide 21 Dr

    Jonathan M. Skelton G. Tan et al., Chem. Rev. 116 (19), 12123 (2016) 𝑍𝑇 = 𝑆!𝜎 𝜅"#" + 𝜅#$% 𝑇 𝑆 - Seebeck coefficient 𝜎 - electrical conductivity 𝜅"#" - electronic thermal conductivity 𝜅#$% - lattice thermal conductivity
  22. Application: thermoelectrics CMD29, 23rd August 2022 | Slide 22 Dr

    Jonathan M. Skelton J. M. Flitcroft et al., Solids 3 (1), 155 (2022)
  23. Application: thermoelectrics CMD29, 23rd August 2022 | Slide 23 Dr

    Jonathan M. Skelton n [cm-3] T [K] 𝒁𝑻 𝝈 [S cm-1] 𝑺 [𝝁V K-1] 𝑺𝟐𝝈 [mW m-1 K-2] 𝜿𝐭𝐨𝐭 [W m-1 K-1] SnS (Pnma) 4.64 × 10-19 1000 1.75 252 272 1.87 1.07 SnS (RS, Eq.) 1020 720 2.99 885 297 7.84 1.70 SnS (RS, 13 GPa) 1020 800 1.48 999 303 9.19 4.33 SnSe (Pnma) 4.64 × 10-19 1000 2.81 348 274 2.62 0.93 SnSe (RS, Eq.) 1020 800 2.60 1196 302 10.90 3.02 J. M. Flitcroft et al., Solids 3 (1), 155 (2022)
  24. Summary CMD29, 23rd August 2022 | Slide 24 Dr Jonathan

    M. Skelton The tin sulphides Snx Sy have a rich phase space with five known or proposed SnS phases plus SnS2 and Sn2 S3 Helmholtz and Gibbs free energy calculations using the HA/QHA predict the observed soft-mode phase transition between the Pnma and Cmcm phases Both RS and ZB SnS are dynamically unstable, but RS SnS can be stabilised under compression e.g. by epitaxial growth Sn2 S3 is predicted to be unstable with respect to disproportionation based on its lattice energy, but is stabilised at finite temperature by vibrational entropy The QHA 𝑝/𝑇 phase diagram of SnS features only the Pnma and Cmcm phases; pressure shifts the Pnma ↔ Cmcm transition to lower 𝑇, and above ~11-12 GPa the Cmcm phase becomes both energetically and dynamically stable If synthetically accessible, the RS phase of SnS has the potential to show superior thermoelectric properties to the Pnma phase
  25. Acknowledgements CMD29, 23rd August 2022 | Slide 25 Dr Jonathan

    M. Skelton
  26. https://bit.ly/3AAaTBk These slides are available on Speaker Deck: