The Frequency-Dependent Spin Contribution to the Magnetoelectric Tensor of Cr$_2$O$_3$: A First-Principles Study
The magnetoelectric (ME) effect provides a promising pathway for controlling magnetic functionalities using electric fields. While first-principles methods for the static linear ME response are well established, comparable approaches for the frequency-dependent response remain less developed, despite experiments showing pronounced finite-frequency resonances. Here, we investigate the dynamical spin-induced linear ME response from first principles and systematically compare the independent-particle approximation (IPA), random-phase approximation (RPA), time-dependent density functional theory (TDDFT), and the Bethe-Salpeter equation (BSE). We apply these methods to the prototypical ME material Cr$_2$O$_3$ and compare the results with available experimental and theoretical studies. We find that the IPA and RPA fail to reproduce the previously reported finite static limit of the spin-induced response. Within the BSE framework, pronounced excitonic resonances emerge in the ME spectrum, in qualitative agreement with experiment. We also identify a magnon-like peak that coincides with a pole of the transverse spin susceptibility while remaining essentially dark in optical absorption, highlighting the sensitivity of the ME response to spin excitations. TDDFT places this mode closer to the expected low-energy magnonic regime and yields a sizable static spin-induced response. Our results show that these frameworks capture complementary aspects of the dynamical ME response. Low-energy collective spin excitations are required to recover the static limit, whereas electron-hole interactions are essential for reproducing the excitonic resonances.