Spin projection and spin current density within relativistic electronic-transport calculations

Abstract

A spin projection scheme is presented which allows the decomposition of the electric conductivity into two different spin channels within fully relativistic ab initio transport calculations that account for the impact of spin-orbit coupling. This is demonstrated by calculations of the spin-resolved conductivity of Fe 1−x Cr x and Co 1−x Pt x disordered alloys on the basis of the corresponding Kubo-Greenwood equation implemented using the Korringa-Kohn-Rostoker coherent-potential approximation band-structure method. In addition, results for the residual resistivity of diluted Ni-based alloys are presented that are compared to theoretical and experimental ones that rely on Mott's two-current model for spin-polarized systems. The application of the scheme to deal with the spin-orbit induced spin-Hall effect is discussed in addition. DOI: 10.1103/PhysRevB.82.140402 PACS number͑s͒: 72.25.Ba, 71.70.Ej, 72.15.Eb, 11.80.La During the last years research activities in spintronics increased very rapidly. The reason for the growing interest in this field is based on the close connection with fundamental scientific questions as well as its impact on technology. 1,2 Compared to standard electronics where only the charge of the electrons is used, spintronics uses the charge of the electrons in combination with the spin degree of freedom. One of the most exciting effects within spintronics is the spin-Hall effect ͑SHE͒. 3,4 The SHE appears when an electric current flows through a medium with spin-orbit coupling present, leading to a spin current perpendicular to the charge current. This effect is even present in nonmagnetic materials as could be demonstrated experimentally, e.g., for Pt. 5 For a theoretical investigation of effects such as the SHE it is obviously crucial to have a reliable description for the spin-dependent transport that accounts for the impact of spin-orbit coupling in a proper way. Most investigations in this field were based on the Pauli equation including spinorbit coupling explicitly as a relativistic correction term and representing the spin current density essentially by a combination of the Pauli spin matrix z with the conventional current density operator. 6 Very few investigations have been done so far on the basis of the Dirac equation using an expression for the spin current density, albeit introduced in an ad hoc manner. 7 In contrast to these approximate schemes to deal with spin-dependent transport the approach suggested by Vernes et al. 8 supplies a fully relativistic and coherent description of electronic spin polarization and the associated spin current density. This approach based on the fourcomponent polarization operator T introduced by Bargmann and Wigner 9 leads, in particular, to a corresponding set of continuity equations.In the present work we introduce spin projection operators derived from the polarization operator T. This allows a decomposition of the conductivity into contributions from each spin channel within fully relativistic transport calculations. Applications on the sp...

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