First-Principles Theory of Spatial Dispersion: Dynamical Quadrupoles and Flexoelectricity

Abstract

Density-functional perturbation theory (DFPT) is nowadays the method of choice for the accurate computation of linear and non-linear response properties of materials from first principles. A notable advantage of DFPT over alternative approaches is the possibility of treating incommensurate lattice distortions with an arbitrary wavevector, q, at essentially the same computational cost as the latticeperiodic case. Here we show that q can be formally treated as a perturbation parameter, and used in conjunction with established results of perturbation theory (e.g. the "2n + 1" theorem) to perform a long-wave expansion of an arbitrary response function in powers of the wavevector components. This provides a powerful, general framework to accessing a wide range of spatial dispersion effects that were formerly difficult to calculate by means of first-principles electronic-structure methods. In particular, the physical response to the spatial gradient of any external field can now be calculated at negligible cost, by using the response functions to uniform perturbations (electric, magnetic or strain fields) as the only input. We demonstrate our method by calculating the flexoelectric and dynamical quadrupole tensors of selected crystalline insulators and model systems.

ABINIT: Overview and focus on selected capabilities

“…The ABINIT implementation of the long-wave DFPT has been used to compute the clamped-ion FxE tensor of Si and SrTiO 3 , obtaining an excellent agreement with the existing results in the literature. 109 The computed spatial dispersion properties accurately reproduce the sum rules predicted in Refs. 110 and 102 that relate them with specific quantities from uniform perturbation theories.…”

“…En route toward a practical implementation, additional technical and formal issues were addressed, regarding the generalization of the uniform strain response to a "metric-wave" perturbation 106,107 and the proper definition of the current-density operator in the presence of nonlocal pseudopotentials. 108 These efforts culminated with the present long-wave method, 109 which paves the way toward the computation of many spatial dispersion properties with a computational burden that is comparable to conventional linear-response calculations.…”

Long-range quadrupole electron-phonon interaction from first principles

“…The electron distribution changes in response to a displacement of an atom from its equilibrium position. The cell-integrated charge response to a displacement of atom κ due to a phonon with wave-vector q → 0 can be written as a multipole expansion [21]: where summation over the Cartesian indices β and γ is implied. This polarization response defines the Born effective charge Z κ , a rank-2 tensor associated with the dipole term, and the dynamical quadrupole Q κ , the rank-3 tensor in the quadrupole term; both tensors can be computed in the DFPT framework [15,19].…”

“…We use the dynamical quadrupole tensors computed in Refs. [21,32]. The phonon dispersions and e-ph perturbation potentials on coarse qpoint grids are computed with DFPT [15].…”

“…The open question is how one can carry out the eph matrix element interpolation in the region near q = 0 using analytical expressions for the long-range dipole and quadrupole terms. These expressions have been obtained by Vogl [18], but need to be rewritten in the ab initio formalism and computed with first-principles quantities such as the atomic dynamical dipoles [15] and quadrupoles [19][20][21] induced by lattice vibrations, which can be computed with DFPT. For each atom κ, one can obtain Born charge (Z κ ) and dynamical quadrupole (Q κ ) tensors, which, once contracted with the phonon eigenvector, give the atomic contributions to the dipole and quadrupole e-ph interactions.…”