Spin-orbit coupling and the Edelstein effect at conducting ferroelectric domain walls

Head-to-head ferroelectric domain walls are intrinsically charged, and are typically compensated by a mix of oppositely charged defects and free electrons. The free electrons form a two-dimensional electron gas (2DEG) along the domain wall. In many cases, inversion symmetry is broken at the wall, which implies that the 2DEG is subject to nontrivial spin-orbit coupling. Here, we use symmetry arguments to construct a generic six-band tight-binding electronic Hamiltonian for a $90^\circ$ head-to-head ferroelectric domain wall. The model, which includes spin-orbit physics and has a multi-orbital $t_{2g}$ band structure that is common to transition-metal perovskites, is applied to BaTiO$_3$. We find that the 2DEG develops an Ising spin texture, with spins aligned perpendicular to the domain wall. We contrast this with the Rashba spin texture that should emerge at weakly conducting $90^\circ$ head-to-tail domain walls. We then show that the head-to-head domain walls should have a measurable Edelstein effect (that is, a current-induced magnetization), even in the dilute limit and at room temperature, and describe a simple experiment to measure it.