Nonlocal magnetolectric effects in diffusive conductors with spatially inhomogeneous spin-orbit coupling

Cristina Sanz-Fernández, Juan Borge, Ilya V. Tokatly, and F. Sebastián Bergeret

Phys. Rev. B 100, 195406 – Published 5 November 2019

Abstract

We present a theoretical study of nonlocal magnetoelectric effects in diffusive hybrid structures with an intrinsic linear-in-momentum spin-orbit coupling (SOC) which is assumed to be spatially inhomogeneous. Our analysis is based on the SU(2)-covariant drift-diffusion equations from which we derive boundary conditions at hybrid interfaces for SOC of any kind. Within this formulation, the spin current is covariantly conserved when the spin relaxation is only due to the intrinsic SOC. This conservation leads to the absence of spin Hall (SH) currents in homogeneous systems. If, however, extrinsic sources of spin relaxation (ESR), such as magnetic impurities and/or a random SOC at nonmagnetic impurities, are present the spin is no longer covariantly conserved, and SH currents appear. We apply our model to describe nonlocal transport in a two-dimensional system with an interface separating two regions: one normal region without intrinsic SOC and one with a Rashba SOC. We first explore the inverse spin-galvanic effect, i.e., a spin polarization induced by an electric field. We demonstrate how the spatial behavior of such spin density depends on both the direction of the electric field and the strength of the ESR rate. We also study the spin-to-charge conversion, and compute the charge current and the distribution of electrochemical potential in the whole system when a spin current is injected into the normal region. In systems with an inhomogeneous SOC varying in one spatial direction, we find an interesting nonlocal reciprocity between the spin density induced by a charge current at a given point in space, and the spatially integrated current induced by a spin density injected at the same point.

Received 26 July 2019
Revised 21 October 2019

DOI:https://doi.org/10.1103/PhysRevB.100.195406

Physics Subject Headings (PhySH)

Spin dynamics Spin-orbit coupling Spintronics

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Cristina Sanz-Fernández ^1,*, Juan Borge², Ilya V. Tokatly ^2,3,4, and F. Sebastián Bergeret^1,4,†

¹Centro de Física de Materiales (CFM-MPC), Centro Mixto CSIC-UPV/EHU, 20018 Donostia-San Sebastián, Spain
²Nano-Bio Spectroscopy Group, Departamento de Física de Materiales, Universidad del País Vasco (UPV/EHU), 20018 Donostia-San Sebastián, Spain
³IKERBASQUE, Basque Foundation for Science, 48011 Bilbao, Spain
⁴Donostia International Physics Center (DIPC), 20018 Donostia-San Sebastián, Spain

^*cristina_sanz001@ehu.eus
^†sebastian_bergeret@ehu.eus

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Issue

Vol. 100, Iss. 19 — 15 November 2019

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Images

Figure 1
Schematical view of different setups considered in this work. (a) Heterostructure with a localized SOC at the interface (blue region) between two different materials, in this case a normal conductor (gray bottom region) and an insulator (transparent top region). (b) Hybrid lateral structure for nonlocal transport measurements. The gray region is a normal conductor without SOC connected to two electrodes. At the contact with the blue electrode it is assumed a sizable SOC, whereas the orange electrode is a ferromagnet which may serve as a spin injector or detector. (c) The system under consideration to study the nonlocal inverse SGE. An electric field is applied parallel or perpendicular to the interface, between a normal conductor without SOC (gray region) and a conductor with intrinsic SOC (blue region). A spin density is induced in the latter. We are interested in the spin density at a distance $L$ away from the interface in the normal conductor. (d) Same setup as panel (c) but now a spin is injected into the normal conductor at a distance $L$ from the interface. We are interested in the charge current induced in the blue region as a consequence of the SGE.
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Figure 2
Spatial dependence of the spin density induced by applying an electric field [see Fig. 1] for different values of $r_{ext}$ . We distinguish two possible directions of the electric field: (a), (b) parallel, and (c) perpendicular to the interface. In all figures it is assumed that $λ_{s} = λ_{ext}$ , and the calculated spin density is normalized by the corresponding bulk value $S_{EE}^{a}$ of Eq. (24).
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Figure 3
(a) Spin density induced at $x = - L$ by an electric field [see Fig. 1] as a function of $r_{ext}$ . We distinguish between an electric field applied parallel and perpendicular to the interface. (b) Integrated charge current induced by a spin density injection at $x = - L$ [see Fig. 1] as a function of $r_{ext}$ . The charge current flows in different directions depending on the polarization of the injected spin density. All curves are calculated in units of $γ / (D λ_{α})$ , for $L / λ_{s} = 0.01$ , $λ_{s} = λ_{ext}$ , and normalized according to Eq. (43).
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Figure 4
(a) Spatial dependence of the charge current density $j_{1 y}$ induced by the Rashba SOC when the spin density injected at $x = - L$ is $x$ polarized. The interfacial and bulk contributions can be distinguished. (b) Redistribution of the electrochemical potential $μ$ due to the insulating boundaries placed at $y = \pm W / 2$ . The vector field lines correspond to the total charge current densities $j = - σ_{D} \nabla μ + j_{1 y} {\hat{e}}_{y}$ . Both plots are shown for pure Rashba SOC, $r_{ext} = 0$ , with $λ_{α} / λ_{s} = 0.1$ , and $L / λ_{s} = 0.1$ .
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