Macroscopic Polarization from Nonlinear Gradient Couplings

We show that a lattice mode of arbitrary symmetry induces a well-defined macroscopic polarization at first order in the momentum and second order in the amplitude. We identify a symmetric flexoelectric-like contribution, which is sensitive to both the electrical and mechanical boundary conditions, and an antisymmetric Dzialoshinskii-Moriya-like term, which is unaffected by either. We develop the first-principles methodology to compute the relevant coupling tensors in an arbitrary crystal, which we illustrate with the example of the antiferrodistortive order parameter in ${\mathrm{SrTiO}}_3$.

Figure 1

Comparison between the atomic forces extracted from a direct calculation of an AFD cycloid (symbols) and the predictions of the macroscopic model (solid curves). Forces on Sr (black circles), Ti (red squares), and the “radial” (blue triangles) and “tangential” (green diamonds) oxygen modes of $T_{1u}$ symmetry are shown. Dashed lines show the uniform DM-like forces. (a) No elastic relaxation; (b) mechanical equilibrium. Lower insets show the sublattice sum of the forces: first-principles (triangles) and model (solid curve). Upper insets schematically illustrate the portion (shaded gray area) of the fixed- or relaxed-strain cycloidal pattern shown in the main panel.