Discrete spectroscopy via Einstein-de Haas torques for nanomechanical probe of a magnetic resonance*

The Einstein-de Haas (EdH) effect is a fundamental consequence of angular momentum conservation for a magnetic system undergoing temporal change in magnetization: rotated magnetic moments from an external field require a compensating mechanical torque [1]. Although this effect is typically extremely small and difficult to measure, the resulting torque amplitude scales directly with frequency. As a result, EdH torques that are negligible at low frequencies can vastly exceed the usually dominant cross-product torque for sufficiently high driving frequency. The frequency-dependence of EdH torques is explored in a thin-film permalloy microstructure affixed to a multi-modal nanomechanical resonator [2]. The sensor supports five distinct mechanical torque modes between 3 MHz and 208 MHz which are probed through magnetic hysteresis curves read out via an optomechanical cavity. The spin texture supported by the permalloy microstructure at low bias fields manifests as a magnetic vortex. A gyrotropic resonance in this spin configuration overlaps in frequency with the 208 MHz mechanical resonance. Features present in the EdH torque signal around these overlapping resonances are suggestive of vortex pinning on imperfections in the thin film; comparison with micromagnetic simulation supports this interpretation. High-field micromagnetic simulations of the quasi-uniform spin texture at high bias fields suggests a significant discrepancy between the data and model at high detection frequency as the system nears a transition out of a quasi-uniform spin-configuration. This discrepancy is addressed through phenomenological addition of a time constant in the range of 0.5–4 ns to the calculation of cross-product torques. 

[1] A. Einstein and W. J. de Haas, in KNAW, Proceedings, Vol. 18 I (1915) p. 696.

[2] K.R. Fast, J.E.Losby, G.Hajisalem, et al., Phys. Rev. B 109, 064404 (2024).