Microscopic theory of the inverse spin galvanic effect in anisotropic Rashba models

TL;DR

This work addresses the ISGE in 2D electron gases with reduced in-plane rotational symmetry by modeling anisotropic Rashba SOC under $C_{2v}$ and $C_{3v}$ and including disorder effects beyond the constant broadening approximation. Using both the Kubo-Streda formalism with vertex-corrected Bethe-Salpeter equations and a quantum kinetic theory approach on the Fermi surface, the authors show that vertex corrections are crucial for accurately capturing ISGE. They find that mass anisotropy alone yields no net change to ISGE once vertex corrections are accounted for, while Rashba anisotropy causes the spin polarization to acquire a directional dependence and tilt away from strict perpendicularity to the applied field; in $C_{3v}$ systems with warping, the spin polarization remains perpendicular but the magnitude diminishes with increasing warping strength. These results highlight the necessity of going beyond the constant broadening approximation to correctly predict spin-charge interconversion in anisotropic Rashba systems and provide a framework to extract Rashba anisotropies from purely electrical measurements. The work advances the microscopic understanding of spin-orbit phenomena in realistic, symmetry-lowered materials and informs experimental strategies for spin manipulation via anisotropy engineering.

Abstract

The Rashba spin-orbit coupling (SOC) is a well-known mechanism for the spin-charge interconversion via the inverse and direct spin galvanic effects. The lack of a full inversion symmetry allows the coupling of the charge current and spin density. In this paper we investigate this phenomenon when the in-plane rotational symmetry is lowered to the $C_{2v}$ and $C_{3v}$ symmetry groups, whereby the electron spectrum becomes anisotropic. We find that in the $C_{2v}$ case, depending on the ratio between the Rashba SOC strengths along the principal axes, the non-equilibrium spin density deviates notably from the $90^o$ degrees rotation, with respect to the applied electric field, familiar in the isotropic case. In the $C_{3v}$ case, when a warping cubic-in-momentum term is present, whereas the standard $90^o$ degrees rotation of the spin density remains, the spin-charge interconversion depends on the intensity of the warping itself. The microscopic theory takes into account disorder including vertex corrections, both via the diagrammatic implementation of the Kubo formula and via the quantum kinetic theory. We show that vertex corrections are crucial to capture the details of the inverse spin galvanic effect in contrast to previous treatments based on the constant broadening approximation.

Figures (5)

• Figure 1: Fermi surfaces obtained from Eq. (\ref{['Ham']}) in regime II with anisotropic mass (a) and spin (b) terms. The lower and upper bands, corresponding to the outer (red) and inner (blue) curves, are labeled $\varepsilon_-$ and $\varepsilon_+$, respectively. The direction of the arrows indicate the spin expectation values. Mass anisotropy is characterized by the ratio $m_y/m_x=2$, while the Rashba coupling anisotropy is defined by $\alpha_y/\alpha_x = 5$. The reference system is the surface state of Au(1 1 1), with effective mass $m=0.27\,m_{\mathrm{e}}$, with $\mathrm{m_e}$ being the electron mass, $\alpha = 0.33\,\mathrm{eV}\,\text{\AA}$, and Fermi energy $\varepsilon_{\mathrm{f}}=0.475$$\mathrm{eV}$Cercellier_2006. For a better representation of the Fermi surfaces, we increased the Rashba coupling by a factor of 2. The Rashba splitting is $\sim 0.12\,\mathrm{eV}$.
• Figure 2: Diagrammatic representation of the spin-current response function. The impurity-averaged bubble in (a) depicts in the Kubo formula, where blue and red solid lines with arrows represent the advanced and retarded GF sectors, respectively. The dressed vertex is indicated by orange shading. Its skeleton expansion is shown in (b), where the dashed green lines denote the averaged interaction with a single impurity, represented by a black dot. The bare current vertex is identified by a black square. (c) Expansion of the GF within the FBA, where thin lines denote bare GFs and thick lines denote dressed GFs.
• Figure 3: Inclination of the spin density as a function of the applied electric field angle $\psi$ and the ratio $\alpha_{\mathrm{r}}=\alpha_y/\alpha_x$ . Each curve corresponds to a different value of $\alpha_{\mathrm{r}}$, as indicated in the legend. The plots are computed numerically and are in perfect agreement with the analytical result given in Eq. (\ref{['angle_law']}).
• Figure 4: Fermi surfaces of a 2DEG with $C_{3v}$ symmetry in the presence of a warping term, Eq. (\ref{['warping']}). The figure is representative of Bi/Cu(1 1 1) systems Frantzeskakis_2011, with $m=-0.29\,m_{\mathrm{e}}$, $\alpha=0.85\,\mathrm{eV}\,\text{\AA}$, $\lambda = 12\,\mathrm{eV}\,\text{\AA}^3$, and $\varepsilon_{\mathrm{f}}=-0.215\,\mathrm{eV}$. The Rashba splitting is $\sim0.1\,\mathrm{eV}.$
• Figure 5: Rashba coupling dependence of the current-induced spin susceptibility for different values of the warping constant $\lambda$, listed in the legend. The inset illustrates the spin density inclination against the applied electric field angle, revealing perfect perpendicularity for any value of $\lambda$, alike the EE. The prototypical studied system is again Bi/Cu(1 1 1), introduced in Fig. (\ref{['fermi_disks_warp']}), where $v=5\times10^5\,\mathrm{m/s}$ is the Fermi velocity. The other parameters are $n_i=10^{16}\,\mathrm{m^{-2}}$ and $u_0=0.1\,\mathrm{eV}\,\mathrm{nm^2}.$