Quantum Saturation of Magnetoelectric Coupling in Fe$_3$O$_4$ Nanoparticles

Abstract

We report magnetoelectric coupling in nanoparticle assemblies that persists to temperatures over 300 times lower than in previous studies. The ME response saturates at low temperature, revealing a quantum plateau of the coupling. A field dependence analysis shows a crossover from quadratic to linear behavior, captured by a phenomenological expansion $C(B,T) \simeq C_0(T) + a_1(T) B + a_2(T) B^2$. The magnitude of the extracted quadratic coefficient, $|a_2(T)|$, follows a power law $|a_2(T)| \sim T^{-\alpha}$ with $\alpha \approx 1.15$, indicating proximity to a quantum critical regime. The observed saturation reflects a new intrinsic energy scale, distinct from finite-size or extrinsic effects. These results establish nanoparticle assemblies as a new platform for studying quantum magnetoelectric phenomena.