Spin Hall effect and irreversible thermodynamics: Center-to-edge transverse current-induced voltage

Abstract

We examine the Dyakonov and Perel theory of the Spin Hall Effect (SHE) from the viewpoint of irreversible thermodynamics, which is more constraining than the symmetry arguments of pure phenomenology. As thermodynamic driving forces we include the thermal gradient, the gradient of the electrochemical potential (rather than the potential gradient and density gradient separately), and the "internal" magnetic field that is thermodynamically conjugate to the magnetization. In turn, we obtain the form of bulk transport coefficients relating the fluxes to the thermodynamic forces. Relative to Dyakonov and Perel, in addition to the new terms due to thermal gradients, the Onsager relations require three new (non-linear) terms in the current density, and minor revisions in the current density and spin current density. The center-to-edge transverse voltage difference, due both to the −β P × E term of of the number current density q and to one of the new current density terms, is calculated for the first time. An ac capacitative probe likely would not significantly disturb this effect. An Appendix explicitly relates the Anomalous Hall Effect (AHE) to the term in the (vector) number flux that is Onsager-related to the SHE term in the (tensor) spin flux.

Magnetoresistance generated from charge-spin conversion by anomalous Hall effect in metallic ferromagnetic/nonmagnetic bilayers

“…(20), represents the contribution from the charge-spin conversion caused by the anomalous Hall effect. Equations (17) and (20) indicate that the transverse resistivity shows the angular dependence that is the same as with the planar Hall effect in ferromagnets [37]. However, the dependences of Eqs.…”

“…The first one is the spin Hall magnetoresistance [6] described by the first term of Eq. (17). This term arises purely from the spin Hall effect, and is finite even when the ferromagnet does not show the anomalous Hall effect.…”

“…However, the dependences of Eqs. (17) and (20) on the thicknesses d N and d F are scaled by the spin diffusion lengths, and thus these effects can be separated from the conventional planar Hall effect by measuring the thickness dependence of total transverse resistivity. The terms in Eqs.…”

“…The terms in Eqs. (17) and (20) are classified into three contributions. The first one is the spin Hall magnetoresistance [6] described by the first term of Eq.…”

“…This new type of magnetoresistance effect, called spin Hall magnetoresistance, was explained theoretically by using the diffusive spin transport theory in the nonmagnet [6], despite the existence of other contributions being implied [7][8][9]. The key idea of the spin Hall magnetoresistance was the charge-spin conversion [10][11][12][13][14][15][16][17] caused by the spin Hall effect [18][19][20] originating from the spin-orbit interaction in nonmagnetic heavy metals. The spin Hall magnetoresistance was observed also in a metallic ferromagnetic/nonmagnetic bilayer [21][22][23][24].…”