First-principle investigation of displacive response in complex solids

In this work, we discuss two approaches to calculate phonon spectra of crystals within the all-electron full-potential linearized augmented-plane-wave (FLAPW) method. This method is one of the most precise implementations of Kohn-Sham (KS) density functional theory(DFT) due to the inclusion of all electrons into the calculation and the use of the full potential, i.e., no shape approximations are applied to the potential. The calculation of phonons requires the force-constant matrix (FCM). The FCM is the second-order derivative of the KS total energy with respect to two atomic displacements. Its Fourier transform yields the dynamical matrix (DM). The eigenvalues of the DM are thesquares of the phonon frequencies. Its eigenvectors are the polarization vectors. The first approach to calculate phonons is the nite-displacement (FD) method. In this method, the FCM is obtained from displacing one atom at a time from equilibrium, calculating the forces on all atoms, and dividing by the displacement amplitude. This is repeated for each atom and for each spatial direction. In practice, the number of calculations reduces signicantly by exploiting the symmetry of the crystal lattice. The FCM is transformed to the DM and the phonon energies and polarization vectors are extracted. A drawback of this approach is given by the necessity to use supercells. The phonon frequencies are only correct for phonons whose wave vector $\textit{q}$ is commensurable with the lattice. Hence, to correctly calculate phonon frequencies at small wave vectors, large super cells are needed, because the displacement pattern of such phonons repeats only after many instances of the primitive unit cell. Since the FD procedure relies on an analytical derivation of the total energy followed by a numerical one, precise forces are necessary. Otherwise, the FCM is not symmetric, forexample. We present a reformulation of the FLAPW force formalism which includes the whole unit cell into the calculation of the atomic force contribution from the core states and which incorporates additional terms to deal with the slight discontinuity of the LAPW basis functions and the quantities derived from them. The improvement of the force precision is demonstrated by the study of different criteria. We then present phonon spectra for Al, MgO, GaAs, and EuTiO$_{3}$ obtained by the FD method from forces calculated in the FLAPW approach using our reformulation. The second approach to calculate phonon spectra is density functional perturbation theory(DFPT). In DFPT, the second-order derivative of the KS total energy is directly calculated via perturbation theory. DFPT allows the determination of phonon frequencies at arbitrary wave vectors $\textit{q}$ from calculations involving the primitive unit cell, only, by treating a phonon of this wave vector as the perturbation. The first-order changes of the basis functions, the electronic density, and the potential have to be obtained from the Sternheimer equation, which is the linearized Schrodinger equation. Additionally, the second-order changes of the external potential, the ion-ion energy, and the LAPW basis functions are required. We provide formulas which explicitly include adjustments of the general DFPT approach when used in conjunction with the FLAPW method. These adjustments include Pulay terms, the correct treatment of the core state contribution, and surface terms which are analogous to those within the force formalism