Perpendicular magnetic anisotropy of two-dimensional Rashba ferromagnets

We compute the magnetocrystalline anisotropy energy within two-dimensional Rashba models. For a ferromagnetic free-electron Rashba model, the magnetic anisotropy is exactly zero regardless of the strength of the Rashba coupling, unless only the lowest band is occupied. For this latter case, the model predicts in-plane anisotropy. For a more realistic Rashba model with finite band width, the magnetic anisotropy evolves from in-plane to perpendicular and back to in-plane as bands are progressively filled. This evolution agrees with first-principles calculations on the interfacial anisotropy, suggesting that the Rashba model captures energetics leading to anisotropy originating from the interface provided that the model takes account of the finite Brillouin zone. The results show that the electron density modulation by doping or an external voltage is more important for voltage-controlled magnetic anisotropy than the modulation of the Rashba parameter.