Intrinsic anisotropy of degree of transport spin polarization in typical ferromagnets

A general approach is presented for investigation of the anisotropy of the degree of transport spin polarization (P) in ferromagnets both in the ballistic regime, P_1,i, and in the diffusive regime, P_2,i, as a function of crystal direction. The validity of this approach is confirmed by the benchmark calculation for the isotropic P. By this approach, we have investigated the anisotropy of P in bcc Fe, fcc Co, fcc Ni and hcp Co. For cubic structures, P_1,i shows a small but appreciable anisotropy, due to the difference in the electronic orbital extension for spin-up and spin-down conduction bands. However, P_2,i shows an isotropic feature for the cubic structure, as a result of the combination of its dependence on the square of electron velocity and the lattice symmetry. On the other hand, for hcp Co, both P_1,i and P_2,i show a very strong anisotropy. The large anisotropy of P_1,i and P_2,i in hcp Co is mainly attributed to the anisotropy of spin-down ballistic (diffusive) conductance.