Domain boundaries in Luttinger-Tisza ordered dipole lattices

Motivated by the recent interest in the possible ordering of the CH$_3$NH$_3$ dipoles in the material CH$_3$NH$_3$PbI$_3$, we investigate the properties of domain boundaries in a simple cubic lattice of dipoles. We perform numerical simulations in which we set the boundary conditions such that the dipoles at opposite sides of the simulated sample are ordered in different directions, hence simulating a domain boundary. We calculate the lowest energy configuration under this constraint. We find that if we consider only dipole-dipole interactions the dipole orientations tend to gradually transform between the two orientations at the two opposite ends of the sample. When we take into consideration the finite spatial size of the CH$_3$NH$_3$ molecules and go beyond the point dipole approximation, we find that the domain boundary becomes sharper. For the parameters of CH$_3$NH$_3$PbI$_3$, our results indicate that the optimal energy structure has a boundary region of a width on the order of a single unit cell.