Real-space representation of uniaxial magnetic anisotropy of ferromagnetic metals with non-periodic structures

A real-space representation of magnetic anisotropy (MA) in metallic ferromagnets is formulated in a d-orbital tight-binding model. By adopting the second-order perturbation for the spin–orbit interaction (SOI), which takes into account the direction of magnetisation, the lowest order of the uniaxial MA constant $K_{\text{u}}$ is expressed in terms of non-local Green's functions and the matrix elements of the SOI. The non-local Green's functions are calculated using a symmetry-conserving recursive method. The validity of the method is examined by comparing the results obtained with those calculated by the first-principles method. The method is applied to calculate layer- or site-resolved $K_{\text{u}}$ near body-centred cubic (bcc) Fe and face-centred cubic Ni surfaces with various surface structures. We find that surface resonant states contribute considerably to the uniaxial MA of bcc Fe thin films. Moreover, it is shown that the uniaxial MA is determined by rather short-range atomic configurations. (paper)