Unified ab initio formulation of flexoelectricity and strain-gradient elasticity

The theory of flexoelectricity and that of nonlocal elasticity are closely related, and are often considered together when modeling strain-gradient effects in solids. Here I show, based on a first-principles lattice-dynamical analysis, that their relationship is much more intimate than previously thought, and their consistent simultaneous treatment is crucial for obtaining correct physical answers. In particular, I identify a gauge invariance in the theory, whereby the energies associated to strain-gradient elasticity and flexoelectrically induced electric fields are individually reference dependent, and only when summed up they yield a well-defined result. To illustrate this, I construct a minimal thermodynamic functional incorporating strain-gradient effects, and establish a formal link between the continuum description and ab initio phonon dispersion curves to calculate the relevant tensor quantities. As a practical demonstration, I apply such a formalism to bulk ${\mathrm{SrTiO}}_3$, where I find an unusually strong contribution of nonlocal elasticity, mediated by the interaction between the ferroelectric soft mode and the transverse acoustic branches. These results have important implications towards the construction of well-defined thermodynamic theories where flexoelectricity and ferroelectricity coexist. More generally, they open exciting new avenues for the implementation of hierarchical multiscale concepts in the first-principles simulation of crystalline insulators.

Figure 1

Convergence of various quantities with the real-space cutoff of the interatomic force constants along a given direction. (a), (b) Flexocoupling coefficients along [100] and [110]; solid and dashed lines refer to longitudinal and transverse modes, respectively; the reported values are the deviations with respect to the $n=10$ point. (c) Electronic octupolar moments; the converged values are ${\tilde{O}}_{100}=-109.1$ a.u. and ${\tilde{O}}_{110}=-115.4$ a.u. (to be compared with ${\tilde{O}}_{100}=-108.8$ a.u. and ${\tilde{O}}_{110}=-115.3$ a.u., obtained in Ref. [11]). (d) Electronic contribution to the fourth-order dispersion (${\tilde{h}}^{\mathrm{el}}$).

Figure 2

Dispersion relation of a linear chain (black curve) as approximated by local elasticity (green dotted-dashed line) and SGE (red dashed line).

Figure 3

(a) Lattice-dynamical analysis of the continuum strain-gradient theory of Sec. 2a, applied to the ${\mathrm{SrTiO}}_3$ case. Black solid curves represent the dispersion of the Fourier-interpolated ab initio phonon frequencies (thin and thick correspond, respectively, to longitudinal and transverse branches); green dotted-dashed lines correspond to the linear dispersion given by classical elasticity; red dashed curves correspond to the continuum model. The inset indicates the paths in the 2D Brillouin zone that correspond to the reported phonon branches. (b) Revised continuum functional of Eq. (105), either with (dotted curves) or without (dashed curves) the “background” SGE term; only transverse modes are shown.