Lattice vibrations

The atoms in a material undergo regular vibrational motion around their equilibrium positions, a phenomenon that is of fundamental importance for the overall behavior of the material. In crystalline solids in particular these vibrations are periodic in nature and can be described using quasi-particles named phonons that represent collective excitations of the crystal lattice.

At the first level of approximation, phonons can be obtained within the so-called harmonic approximation, which implies non-interacting quasi-particles with (accordingly) infinite lifetimes. In principle, harmonic theory allows one to obtain a wealth of information including phonon dispersions and Helmholtz free energies. Using a slight extension, known as the quasi-harmonic approximation, it is even possible to predict e.g., thermal expansion. The approach has, however, various limitations related e.g., to the fact that real materials do always exhibit some degree of anharmonicity, which is most pronounced at elevated temperatures but even at low temperatures responsible for e.g., temperature dependent frequencies, finite phonon life times, or a finite thermal conductivity. In addition there are many materials, which are mechanically unstable at zero temperature but stabilized by vibrations at finite temperature (e.g., the group-IV transition metals in BCC structure, cubic ZrO2, various cubic perovskite structures such as BaTiO3). In all these cases one must explicitly account for the anharmonicity of the lattice vibrations, which leads to phonon-phonon coupling.

Force constants

The most essential ingredient required for analyzing phonons in a material is the set of force constants (FCs). Knowledge of the latter allows one to compute the forces on the atoms solely based on their displacements. Tools such as phonopy [TogTan15] exist for obtaining and analyzing the second order FCs, whereas third order FCs can be extracted using e.g., phono3py [TogChaTan15] or shengBTE [LiCarKat14]. There are even codes such as alamode [TadGohTsu14] that enable one to obtain the fourth order FCs. While phonopy and shengBTE employ finite displacement schemes to construct FCs and typically carry out a systematic enumeration of the possible displacements up to a certain cutoff, alamode relies on forces from molecular dynamics (MD) simulations as input.

It is also possible to obtain FCs from density functional perturbation theory (DFPT) calculations as implemented e.g., in the abinit and quantum-espresso. In that case the FCs are constructed by transformation of the dynamical matrices obtained on a regular q-point mesh.

The hiPhive approach

hiPhive enables one to efficiently obtain high order FCs (e.g., of fourth or sixth order) including large and low-symmetry systems. It employs a supercell approach similar to phonopy, shengBTE, or alamode but does not rely on a specific type of input configuration (i.e. enumerated displacements or configurations from MD simulations). Rather it employs advanced optimization techniques that are designed to find sparse solutions, which in the present case reflect the short-range nature of the FCs. If the input configurations are constructed sensibly this approach allows one to obtain FCs using a much smaller number of input configurations and thus to reduce the computational effort, usually in the form of density functional theory (DFT) calculations, considerably. This approach becomes genuinely advantageous already for obtaining second order FCs in large and/or low symmetry systems (defects, interfaces, surfaces, large unit cells etc). hiPhive truly excels when it comes to higher order FCs, for which a strict enumeration scheme quickly leads to an explosion of displacement calculations.