Chirality-dependent spin polarization in metals: linear and quadratic responses

Abstract

We study spin polarization induced by locally injected electric currents in a metal whose spin--orbit coupling reflects its structural chirality. We reveal both spin polarization in the bulk in the linear response and antiparallel spin polarization near the interface in the quadratic response to external electric currents, and reproduce the experimentally observed correlation between the chirality of the metal and the direction of spin polarization. In particular, we elucidate that the sign of the spin polarization in the quadratic response is opposite to that expected from the bulk spin current. This sign discrepancy originates from spin polarization induced by dipole-like charge distribution appearing in the quadratic response.

Figures (4)

• Figure 1: Schematics for the earlier studies on the linear response against the DC driving current Inui2020Nabei2020Shiota2021Shishido2021Shishido2023 and those on the quadratic response to the AC driving current nakajima2023Nakajima_Thesis. They show the chirality-dependent spin polarization, and the antiparallel spin near interfaces in the quadratic response. The dotted arrows in the right panel indicate the flow direction of carriers with rightward polarized spin.
• Figure 2: Schematics for the deviation of the distribution function around the Fermi surfaces of the spin-splitting bands in (a) linear and (b) quadratic responses, for $q=-e$ and $\alpha > 0$. The shaded regions in orange (gray) denote the excess (deficit) distributions. These excess distributions stem from differences in the density of states between the spin-splitting band, which, for instance in the linear response, leads to a net spin density. The arrows represent the direction of spin in each Bloch state. These figures account for the bulk responses shown in Eqs. \ref{['eq:bulk-response-s1']} and \ref{['eq:bulk-response-j2']}.
• Figure 3: (a, b) Schematics of the setup to measure the linear and quadratic responses to the locally uniform electric field. Red symbols and blue arrows roughly illustrate the spatial profiles of charge and spin, respectively, for $q=-e$ and $\alpha > 0$. (c, d) The spatial dependence of the excess charge density in the linear and quadratic responses, $\rho^{(1)}$ and $\rho^{(2)}$. (e, f) Spatial dependence of spin polarization in the linear and quadratic responses. The inset shows the enlarged view of $\mathcal{S}^{(2)}(z)$ (blue line). The green, purple and orange solid lines show the three components $(-\tau_0)\partial j_{s;zz}^{(2)}/\partial z$, $\beta_1(z)E^{(1)}(z)$ and $\beta_2E^{(2)}(z)$ scaled by $(q\mathcal{E}_0\tau_0)^2m\alpha/4\pi^2$, respectively, whereas the black dashed line shows $s_z^{(2)}(z)$ obtained by the conventional relaxation time approximation neglecting the contribution of the excess electron density in Eq. \ref{['eq: St-constant-W']}. For clarity, we set $\lambda_{\mathrm{TF}} / \ell = 0.1$ and $L / \ell = 50$ in panels (c--f). (g) The monotonic behavior of $\Pi_{zz}(z)$ near the interface $z = 0$ in the limit of $\lambda_{\mathrm{TF}}/\ell \to 0$.
• Figure 4: Enlarged view of spatial variations in electric quantities and the spin polarization near the interface for $q=-e$ and $\alpha>0$. (a) The excess charge density $\rho^{(2)}(z)$ shown for $\lambda_{\rm TF} / \ell = 0.1$. (b) The distributions of the electrochemical potential gradient of the order of $j_0$, $\mathcal{E}_{p,r}^{(1)}(z)$, are shown. Ohm's law as a local relation does not hold near the surface, within a distance on the order of the mean free path $\ell$. For clarity, the cases $p=1, r=0$ (black curve, where Ohm's law as a local relation holds throughout the system) and $p=2, r=0$ (red curve) are shown. Results for other combinations of $\{p,r\}$ are qualitatively similar to those for $\{p=2,r=0\}$. (c) The spin polarization $s_z^{(1)}(z)$ and $s_z^{(2)}(z)$ under condition (i) $J_{\bm{k}\gamma}=J_{p,\bm{k}\gamma}$ for $p=1\text{--}4$ and (ii) $J_{\bm{k}\gamma}=rJ_{p=1,\bm{k}\gamma}+(1-r)J_{p=2,\bm{k}\gamma}$ for $r=0,1/3,2/3,1$.