Spin transport at interfaces with spin-orbit coupling: Phenomenology

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Spin transport at interfaces with spin-orbit coupling: Phenomenology

V. P. Amin and M. D. Stiles

Phys. Rev. B 94, 104420 – Published 16 September 2016

Abstract

This paper presents the boundary conditions needed for drift-diffusion models to treat interfaces with spin-orbit coupling. Using these boundary conditions for heavy-metal/ferromagnet bilayers, solutions of the drift-diffusion equations agree with solutions of the spin-dependent Boltzmann equation and allow for a much simpler interpretation of the results. A key feature of these boundary conditions is their ability to capture the role that in-plane electric fields have in the generation of spin currents that flow perpendicularly to the interface. The generation of these spin currents is a direct consequence of the effect of interfacial spin-orbit coupling on interfacial scattering. In heavy-metal/ferromagnet bilayers, these spin currents provide an important mechanism for the creation of dampinglike and fieldlike torques; they also lead to possible reinterpretations of experiments in which interfacial contributions to spin torques are thought to be suppressed.

Received 23 June 2016

DOI:https://doi.org/10.1103/PhysRevB.94.104420

Physics Subject Headings (PhySH)

Spin polarization Spin-orbit coupling Spintronics

Nanostructures

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

V. P. Amin^1,2,* and M. D. Stiles²

¹Maryland NanoCenter, University of Maryland, College Park, Maryland 20742, USA
²Center for Nanoscale Science and Technology, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA

^*vivek.amin@nist.gov

Spin transport at interfaces with spin-orbit coupling: Formalism

V. P. Amin and M. D. Stiles

Phys. Rev. B 94, 104419 (2016)

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Vol. 94, Iss. 10 — 1 September 2016

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Images

Figure 1
Spin and particle currents created by in-plane electric fields in heavy-metal/ferromagnet bilayers. (a) illustrates some scattering processes that give rise to spin currents in the bulk. (b) depicts these currents within the bilayer as they would be represented in a drift-diffusion approach. White arrows represent particle currents while the colored arrows represent spin currents. The direction of these arrows indicates the direction of flow while the color (blue/red) indicates the direction of spin polarization (positive/negative) along the $y$ axis. (c) shows slices of the nonequilibrium distribution functions that correspond to the currents in (b), as described by the Boltzmann equation. The $k$ -space plots each contain the Fermi surface (black circle) and the nonequilibrium distribution functions. Distortions of the distribution functions outside (inside) the Fermi surface indicate an excess (deficit) of carriers at each $k$ point. The width of the curves shows the degree of spin polarization and the colors indicate the direction of spin polarization as before. For example, carriers in the heavy metal moving towards or away the interface have opposite spin polarization; thus the spin Hall current represents a flux of angular momentum (spin current) but no net spin accumulation. (d) illustrates scattering processes that give rise to spin currents near the interface due to interfacial spin-orbit coupling. (e) depicts those currents analogously to (b), and shows that in-plane charge currents give rise to out-of-plane spin currents that are generated at the interface due to interfacial spin-orbit coupling. (f) shows the nonequilibrium distribution function corresponding to these currents. The distribution functions demonstrate that carriers carry a net spin current and exhibit a net spin accumulation (unlike the case with the spin Hall effect). The spin accumulation exerts a torque on the magnetization at the interface via the exchange interaction, while the spin currents exert torques on the neighboring ferromagnet layer via the spin-transfer mechanism.
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Figure 2
Dampinglike $(τ_{d}^{tot})$ and fieldlike $(τ_{f}^{tot})$ total spin torques plotted versus the heavy-metal thickness $(t)$ . Here $l_{sf} = 2.57$ nm as in Ref. [11]. The spin torques shown originate from either the bulk spin Hall effect or the interfacial Rashba effect. The ratio of the dampinglike and fieldlike components of the spin Hall torque roughly match the ratio between the real and imaginary parts of the mixing conductance. Additionally, the spin Hall torque saturates at thicknesses roughly twice that of the Rashba torque. This thickness-related suppression provides one possible mechanism for Rashba torques to surpass spin Hall torques in thin bilayer systems.
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Figure 3
Spin torques in a Co/Pt bilayer plotted versus the interfacial exchange strength $u_{ex}$ (with no spin Hall effect). The solid curves give the Boltzmann solution, while the dashed curves and the circles give the drift-diffusion/generalized circuit theory solution. The circles are based on the conductivity and torkivity tensors computed in Appendix pp2, while the dashed curves use more accurately computed tensors outlined in Appendix pp4. (a)–(d) The dampinglike $(d)$ and fieldlike $(f)$ components of the total spin torque, shown for various $u_{R}$ . As $u_{ex}$ increases, the total spin torque becomes mostly fieldlike. (e)–(f) Breakdown of the total spin torque into its interfacial (red) and bulk (blue) parts. For weak $u_{ex}$ the bulk spin torque dominates, while for strong $u_{ex}$ the interfacial spin torque dominates. The spin current density $j_{d}^{E} (0^{+})$ (which causes a spin torque by flowing into the ferromagnet) significantly contributes to the total dampinglike spin torque for weak $u_{ex}$ . However, as $u_{ex}$ increases the interfacial spin torque must increase as well; eventually its fieldlike component exceeds all other contributions.
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Figure 4
Comparison of the Boltzmann approach (solid curves) and drift-diffusion/generalized circuit theory approach (circles), with the latter using the boundary parameters computed in Appendix pp2. The panels display spin torques in a Co/Pt bilayer in the absence of the spin Hall effect, plotted versus Rashba parameter $(u_{R})$ for various exchange parameter values $(u_{ex})$ . (a) and (b) represent the total spin torque, while (c)/(d) and (e)/(f) represent the interfacial and bulk contributions respectively. Both approaches quantitatively agree on the parametrization of the each spin torque, although the drift-diffusion/circuit theory approach slightly underestimates the value of the dampinglike spin torque. The conductivity $(σ)$ and torkivity $(γ)$ parameters enable the drift-diffusion equations to describe Rashba spin-orbit torques by capturing the $k$ -dependent spin-orbit scattering present in the Boltzmann equation. Without the inclusion of these parameters, no such drift-diffusion solution exists.
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