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.