Asymmetric dynamical charges in two-dimensional ferroelectrics

Ferroelectricity is commonly understood in terms of dynamical charges, which represent the dipole moments generated by atomic displacements or the forces induced by electric fields. In ferroelectrics with a high degree of symmetry, the dynamical charges are typically symmetric tensors, and can be visualized as ellipsoids. In van der Waals (vdW) materials which break centrosymmetry, a new type of ferroelectricity arises which differs greatly from conventional ferroelectrics. The polarization is purely electronic, arising from an interlayer charge transfer, and most of the polarization generated is perpendicular to atomic motion. We show that the unconventional properties of vdW ferroelectrics are manifested in their dynamical charges, which exhibit spatial modulation and intrinsic asymmetry. Dynamical charges in vdW ferroelectrics, and more generally, any strongly anisotropic ferroelectric, can be visualized as deformable, nonideal ellipsoids dependent on the atomic configuration. Furthermore, we show that, due to the mixed electrostatic boundary conditions employed for two-dimensional (2D) materials, nondiagonal dynamical charges in 2D materials are always asymmetric.