Torque method for the theoretical determination of magnetocrystalline anisotropy

We propose a torque method for the theoretical determination of the magnetocrystalline anisotropy (MCA) energy for systems with uniaxial symmetry. While the dependence of the total energy on the angle between the magnetization and the normal axis (θ) can be expressed as E(θ)=E0+K2sin2(θ)+K 4sin4(θ), we show that the MCA energy [defined as EMCA=E(θ=90°)-E(θ=0°)=K2+K 4] can be easily evaluated through the expectation value of the angular derivative of the spin-orbit coupling Hamiltonian (torque) at an angle of θ=45°. Unlike other procedures, the proposed method is independent of the validity of the MCA force theorem, or of the absolute accuracy of two total energy calculations. Calculated MCA energies for the free Fe monolayer with different lattice constants are analyzed and compared with results of other ab initio calculations, especially those obtained with our previously reported state tracking method.