Extrinsic and Intrinsic Contributions to the Spin Hall Effect of Alloys

Abstract

A fully relativistic description of the spin-orbit induced spin Hall effect is presented that is based on Kubo's linear response formalism. Using an appropriate operator for the spin-current density a KuboStředa-like equation for the spin Hall conductivity (SHC) is obtained. An implementation using the Korringa-Kohn-Rostoker band structure method in combination with the coherent potential approximation allow detailed investigations on various alloy systems. A decomposition of the SHC into intrinsic and extrinsic contributions is suggested. Accompanying calculations for the skew-scattering contribution of the SHC using the Boltzmann equation demonstrate the equivalence to the Kubo formalism in the dilute alloy regime and support the suggested decomposition scheme. DOI: 10.1103/PhysRevLett.106.056601 PACS numbers: 72.25.Ba, 71.15.Rf, 75.76.+j, 85.75.Àd The emerging research field of spintronics has developed very rapidly during recent years. The reason for the broad interest in this field is based on the close connection to fundamental scientific questions as well as its impact on technology [1,2]. In this context, the spin Hall effect (SHE) is one of the most promising phenomena. It denotes the observation that a charge current flowing through a solid is accompanied by a transversal spin current. This occurs even for nonmagnetic solids as was demonstrated by experiments on pure Pt [3].Both the anomalous Hall effect (AHE) in ferromagnets and the SHE are caused by the influence of spin-orbit coupling (SOC). Accordingly, their theoretical description is quite similar [4][5][6][7][8][9][10][11][12][13][14][15]. For ideal systems an intrinsic mechanism was identified which allows the expression of the corresponding response function in terms of the Berry curvature [5,7]. On this basis, ab initio calculations for the intrinsic spin Hall conductivity (SHC) were performed [8][9][10][11]. As for the AHE, the additional extrinsic SHC in dilute and concentrated alloys is ascribed to skew and sidejump scattering caused by SOC. The role of these mechanisms for the SHE has been studied so far primarily by model calculations [12,13]. First principle calculations for the extrinsic SHC of dilute alloys on the basis of the Boltzmann formalism that account for the skew-scattering mechanism have been performed only very recently [14,15]. However, a complete description of intrinsic and extrinsic mechanisms giving rise to the SHE applicable to ideal as well as alloy systems, as it is presented below, was missing so far. As pointed out by several authors [16,17], a central issue for such an approach is an adequate definition for the spin-current density operator that accounts for SOC. This was supplied recently by Vernes et al. [18] by starting from the Bargmann-Wigner four-vector spin polarization operator T [19]. Demanding that the spin polarization is connected with the spin-current density via a corresponding continuity equation an explicit expression for the spin-current operator was given.An adequate formal basis for the di...

Anomalous and spin Hall effects in hcp cobalt from GGA+Ucalculations

“…And the calculated SHCs 32,33 are found to agree well with the corresponding experimental values in many metals such as Al 35 , Pt 36,37 , Au and Pd 38 , suggesting that the intrinsic SHE is dominant in the high resistivity regime in pure metals. Nonetheless, recent experimental 39,40 and theoretical [41][42][43][44][45] studies showed that certain impurities could also give rise to large SHE in impure metals. However, the SHE in ferromagnets remains unexplored.…”

Giant spin Nernst effect induced by resonant scattering at surfaces of metallic films

“…Owing to the fact that the SHE has been successfully observed in various types of experiments [7][8][9], it is expected that the spin Nernst effect would also be detectable. However, limitations on the magnitude of temperature gradients in metals can diminish the magnitude of the spin Nernst currents [10].Ab-initio studies [11][12][13][14] and experiments [15] suggest that the extrinsic SHE induced by the skew-scattering off impurities can be large due to a large difference in the spin-orbit coupling strength of the impurities and the host. However, this argument is not applicable to the SNE, because the thermal transport coefficients entering the expression for the spin Nernst conductivity (SNC) are determined to a first approximation by the derivative of the conductivities around the Fermi energy (E F ), and not by the their values directly at it.…”

Spin Orbit Coupling Controlled Spin Pumping and Spin Hall Magnetoresistance Effects

“…It is therefore indicated that the SHC and spin Hall angle in the NM layer can be tuned significantly via the SOC strength. As one of major issues in this field [17][18][19][20][21][22] , the mechanism of the extrinsic SHC needs further investigations.…”

“…The measured SHC should in principle consist of the intrinsic and extrinsic terms for x in the region from 0 to 1.0. The extrinsic one, caused by the asymmetric scattering at impurity sites is expected to be negligible near the ending data points 18,43 and to become prominent near x = 0.5. Clearly, for the present PdPt alloys, the intrinsic term plays a dominant role in the measured SHC and the extrinsic term can be neglected, as observed in Ir doped Pt 18 .…”

“…As an outstanding issue, however, the quantitative dependence of the SHE on the SOC strength is still unclear. Moreover, the SHE is known to be contributed by the intrinsic, the skew scattering, and the side-jump terms [17][18][19][20][21][22] . The issue of the SHE scaling law has not been understood yet although many attempts have been made.…”