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
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
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
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)
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
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