Spin relaxation, diffusion, and Edelstein effect in chiral metal surface

We study electron spin transport at the spin-splitting surface of chiral-crystalline-structured metals and the Edelstein effect at the interface, by using the Boltzmann transport equation beyond the relaxation time approximation. We first define spin relaxation time and spin diffusion length for two-dimensional systems with anisotropic spin-orbit coupling through the spectrum of the integral kernel in the collision integral. We then explicitly take account of the interface between the chiral metal and a nonmagnetic metal with finite thickness. For this composite system, we derive analytical expressions for efficiency of the charge current–spin current interconversion as well as other coefficients found in the Edelstein effect. We also develop the Onsager reciprocity in the Edelstein effect along with experiments so that it relates local input and output, which are respectively defined in the regions separated by the interface. We finally provide a transfer matrix corresponding to the Edelstein effect through the interface, with which we can easily represent the Onsager reciprocity as well as the charge-spin conversion efficiencies we have obtained. We confirm the validity of the Boltzmann transport equation in the present system starting from the Keldysh formalism in supplemental material. Our formulation also applies to the Rashba model and other spin-splitting systems.

Figure 1

Schematic pictures of temporal spin relaxation and spatial spin diffusion [(a)] at the chiral metal surface, direct Edelstein effect [(b)], inverse Edelstein effect [(c)] across the interface, and radial spin texture of the Fermi contours at the surface [(d)]. Arrows denote spin polarization. (d) The spin-orbit coupling is set to be isotropic (left) and anisotropic (right, $\delta =\pi /64$) with the same strength of SOC $\alpha /v_{\mathrm{F}}\hslash =0.6$.

Figure 2

Eigenmode analysis of the Boltzmann transport equation for relaxation. (a) Inverse of the relaxation time $\tau$ plotted for all modes and for various anisotropy ${\alpha }_{\perp }/{\alpha }_{\parallel }=tan\delta$. All eigenmodes but two are degenerate in $\tau ={\tau }_{\text{p}}$. (b) Deviation from equilibrium in the distribution function around the Fermi contours ${\phi }_{\tau }(\theta ,\gamma )$ that has the longest relaxation time when $\delta =\pi /64$. The amplitude is put in arbitrary units. (c) Schematic illustration of the deviation of the electron distribution (b), drawn as a shift of the Fermi contours.

Figure 3

Eigenmode analysis of Boltzmann transport equation for diffusion. (a) Eigenvalues corresponding to inverse of the diffusion length $\ell >0$ arranged in descending order. The red arrow points to an isolated slowly decaying mode. The total number of components reflects the 358 grids in the azimuthal direction around the Fermi contour used in the numerical calculation. (b) Inverse of the diffusion length $\ell$ plotted for the isolated slowly decaying modes labeled (A)–(D) and for different anisotropy $\delta$. Both diffusion lengths in the $z$ direction and that in the $x$ direction are shown. (c) Deviation from equilibrium in distribution function around the Fermi contours ${\phi }_{\ell }(\theta ,\gamma )$ that has the longest diffusion length when diffusing in the $z$ direction. The amplitude is put in arbitrary units.

Figure 4

The deviation from the equilibrium distribution function around the two Fermi contours for direct Edelstein effect [(a)] and inverse Edelstein effect [(b)], and associated charge-spin conversion efficiencies at the interface $q_{\text{EE}}$ [(c)] and ${\lambda }_{\text{IEE}}$ [(d)]. The parameters are chosen as $(\alpha /v_{\mathrm{F}}\hslash ,\delta ,L/{\ell }_{\text{sf}})=(0.1,\pi /64,1)$. [(c), (d)] The colored lines show different spin diffusion rates in the three-dimensional nonmagnetic metal [defined in Eq. (28)]. Those lines collapse onto a single curve in (d).

Figure 5

Linear response specified by the reciprocal relationship of the Edelstein effect ${\lambda }_{\text{recip}}$ for varying the parameters of the interface and 3D metal. The colored lines show different spin diffusion rates in the three-dimensional nonmagnetic metal. The dashed black line shows the peak location of the response with respect to the transmission rate ${\tau }_{\text{p}}/{\tau }_{\text{t}}$. The parameters are chosen as $(\alpha /v_{\mathrm{F}}\hslash ,\delta ,L/{\ell }_{\text{sf}})=(0.1,\pi /64,1)$.