Slow-phonon control of spin Edelstein effect in Rashba $d$-wave altermagnets

TL;DR

This work develops a minimal continuum model for a 2D Rashba altermagnet with slow, Holstein-type electron-phonon coupling to study the spin Edelstein effect. Using a self-consistent static-Holstein treatment and Kubo linear response, it shows that increasing EPC progressively suppresses the induced spin polarization, culminating in a depolarization transition tied to phonon-driven band renormalization and Fermi-surface collapse. Altermagnetism introduces pronounced anisotropy and breaks the usual antisymmetry of spin susceptibilities, enabling directionally selective spin responses that are tunable by Rashba strength and doping. The results point to phonon engineering as a viable route to controllable spin polarization in spintronic devices, with potential applications in spin logic and reconfigurable spin-orbit torque systems.

Abstract

Altermagnets have zero net magnetization yet feature spin-split bands that host spin-polarized states. Here, we investigate how slow lattice vibrations (phonons) influence both the intrinsic and externally induced spin polarizations in two-dimensional $d$-wave altermagnets. For the induced spin polarizations, we employ a Rashba continuum model with electron-phonon coupling (EPC) treated at the static-Holstein level and analyze the spin Edelstein effect using the Kubo linear-response formalism. We find that moderate-to-strong EPC progressively suppresses the induced polarization via both intraband and interband channels, with a critical coupling marking the onset of complete spin Edelstein depolarization. The depolarization transition arises from a phonon-induced energy renormalization that pushes the spin-split bands anisotropically above the chemical potential, leading to a complete collapse of the Fermi surface. While (de)polarization can occur even in the Rashba non-altermagnetic phase, it remains isotropic. The presence of altermagnetism makes it anisotropic and breaks the conventional antisymmetry between spin susceptibilities that occurs with pure spin-orbit coupling, rendering the effect highly relevant for spintronic applications. We further investigate how the phonon coupling to the altermagnetic order, Rashba spin-orbit strength, and carrier doping collectively tune the depolarization transition. Our findings demonstrate that phonon scattering (e.g., through various substrates) offers a powerful means for on-demand control of spin polarization, enabling reversible switching between spin-polarized and depolarized states -- a key functionality for advancing spin logic architectures and optimizing next-generation spintronic devices.

Figures (8)

• Figure 1: (a) Real-space representation of the staggered spin configuration on a square lattice of a pristine $d_{x^2-y^2}$-wave altermagnet with phonons included and without gating, where the spin-dependent orbital texture alternates between sublattices. Sublattice states are linearly coupled to slow Holstein phonons, depicted as shaded (vibrating) sites. (b) Corresponding band structure along high-symmetry paths near the $\Gamma$-point, displaying spin-split dispersions for spin-up and spin-down channels.
• Figure 2: (a) Components of the spin Edelstein susceptibility tensor, $\chi_{\ell j}/\chi_{0}$, as a function of the EPC strength $g$ for fixed altermagnetic strength $\beta = 0.5$, RSOC strength $\lambda_{\rm R} = 0.3$ eV$\cdot$Å, and chemical potential $\mu = 0$ at Fermi energy. The susceptibilities start at a nearly constant value for weak coupling, reflecting robustness of the spin response in the perturbative regime, and then decrease monotonically with increasing $g$, eventually approaching zero at moderate-to-strong coupling, marking the onset of spin Edelstein depolarization. (b) Decomposition of $-\chi_{xy}/\chi_{0}$ into its intraband and interband contributions, along with its total response. The intraband processes provide an increasingly significant contribution, and together with the interband effects, they ultimately lead to the complete suppression of the spin Edelstein polarization at $g_{\rm c} \approx 0.25$ eV/Å.
• Figure 3: Minimum of the lower spin-split band, $\min[\mathcal{E}_{\vec{k},-}]$, as a function of the EPC $g$ for two parameter variations. (a) Variation of the altermagnetic anisotropy $\beta$ and (b) variation of the RSOC strength $\lambda_{\rm R}$. The horizontal dashed line marks the zero Fermi energy $\mu$. Open circles indicate the critical EPC $\tilde{g}_{\rm c}$ at which the lower band touches $\mu$, marking the onset of spin Edelstein depolarization at $g_{\rm c} \gtrsim\tilde{g}_{\rm c}$.
• Figure 4: DOS resolved along the $X$ and $Y$ directions for $\beta=0.5$, $\lambda_{\rm R}=0.3$ eV$\cdot$Å, $\mu=0$, and at different EPCs: (a) $g=0.04$ eV/Å, (b) $g=0.12$ eV/Å, (c) $g\approx0.18$ eV/Å, and (d) $g=0.28$ eV/Å. Insets show the corresponding band dispersions along the $\Gamma \to X$ and $\Gamma \to Y$ directions, highlighting the anisotropic reshaping of the bands with increasing $g$. When the spin bands shift away from the Fermi energy, where the chemical potential is located, the Fermi states are no longer involved due to the absence of available spin states at that energy. As a result, the intraband and interband contributions to the spin Edelstein susceptibility vanish, leading to depolarization. Moreover, the bands become anisotropic with $g$, as reflected in the DOS along different directions.
• Figure 5: (a) Transverse spin Edelstein susceptibility $-\chi_{xy}/\chi_{0}$ as a function of EPC strength $g$ for several values of the staggered lattice potential parameter or altermagnetic order $\beta$. The parameters are set to $\lambda_{\rm R} = 0.3$ eV$\cdot$Å and $\mu = 0$. Increasing $\beta$ slightly shifts the depolarization transition to larger $g$, demonstrating strong sensitivity of spin Edelstein polarization to lattice asymmetry. (b) Electronic density of states $\mathcal{D}(\mathcal{E})$ at $\beta=0$ and $g = 0.2$ eV/Å, with the inset showing the corresponding band structure $\mathcal{E}_{\vec{k}}$ along the high-symmetry directions $Y \leftarrow \Gamma \to X$. Since DOS remains identical along both directions at $\beta = 0$, the resulting finite induced spin (de)polarization is isotropic, which is generally unfavorable for spintronic applications.