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