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

Altermagnets have zero net magnetization yet feature spin-split bands. 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 polarization, 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 to probe EPC-induced contributions. We find that, under a specific symmetry-lowering pattern such as a piezomagnetically active strain that explicitly breaks the inherent $C_4 \mathcal{T}$ symmetry, moderate-to-strong EPC progressively suppresses the induced polarization via both intraband and interband channels, with a threshold coupling marking the onset of complete spin Edelstein depolarization. The depolarization arises from a phonon-induced energy renormalization that leads to a complete collapse of the Fermi surface. While depolarization 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. Our findings demonstrate that static phononic effects offer 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.