Revisiting the theory of crystal polarization: The downside of employing the periodic boundary conditions

Periodic boundary condition (PBC) is a standard approximation for calculating crystalline materials properties. However, a PBC crystal is not the same as the real macroscopic crystal, therefore, if applied indiscriminately, it can lead to erroneous conclusions. For example, unlike other extensive observables such as total energy, the polarization of a macroscopic crystal, cannot always be described by a PBC model, because polarization is inherently nonlocal and strongly dependent on surface terminations, irrespective of crystal size, and moreover, the symmetry of the macroscopic crystal can be altered when the PBC is applied to a macroscopic crystal. We demonstrate in this paper that the polarization of a macroscopic crystal receives contributions from both the repeating bulk units and the crystal surfaces, which must be treated on an equal footing. When the combined system of the bulk and its surfaces are taken into account, materials traditionally classified as nonpolar can, in fact, admit polar symmetry, thus explaining why experimentalists have observed polarization in some nominally “nonpolar” systems. Our study, thus, clarifies that polarization can only exist in polar group systems and that apparent violations of the Neumann’s principle reported in some recent works originate from misinterpreting bulk PBC crystal as intrinsic macroscopic crystal, ignoring the contribution from the surfaces. We demonstrate that when the full bulk-plus-surface system is considered, the crystal polarization and symmetry is fully consistent with Neumann’s principle.