Theory of spin torque due to spin-orbit coupling

The combined effect of spin-orbit coupling and exchange interaction in a single ferromagnetic layer is investigated. It is shown that, in nonequilibrium regime, the spin-orbit interaction (SOI) gives rise to a transverse spin density that exerts a torque on the local magnetization. The spin torque depends on the symmetry properties of the SOI. For the inversion-symmetry-preserved SOI such as the impurity SOI and the Luttinger spin-orbit band, the spin torque is a high-order effect too small to lead to a reasonable critical switching current density. For the inversion-symmetry-broken SOI, e.g., Rashba and Dresselhaus SOIs, the torque is on the first order of the SOI parameter and can be effectively used to control the magnetization direction using critical switching current densities as low as ${10}^4–{10}^6\text{A}/{\text{cm}}^2$. We also address the relation between the spin torque and the anisotropic magnetoresistance. Finally, a number of systems are proposed for the experimental observation of the SOI-induced torque.

Figure 1

Schematics of the Fermi surface for majority (top) and minority (bottom) electrons (a) in a ferromagnet without Rashba SOI, (b) in a layer with Rashba SOI and without exchange interaction, and (c) in a ferromagnet with small Rashba SOI.