The Einstein–de Haas (EdH) effect is a fundamental, mechanical consequence of any temporal change of magnetism in an object. EdH torque results from conserving the object's total angular momentum: The angular momenta of all the specimen's magnetic moments, together with its mechanical angular momentum. Although the EdH effect is usually small and difficult to observe, it increases in magnitude with detection frequency. We explore the frequency dependence of EdH torque for a thin film permalloy microstructure by employing a ladder of flexural beam modes (with five distinct resonance frequencies spanning from 3 to 208 MHz) within a nanocavity optomechanical torque sensor via magnetic hysteresis curves measured at mechanical resonances. At low dc fields, the gyrotropic resonance of a magnetic vortex spin texture overlaps the 208 MHz mechanical mode. The massive EdH mechanical torques arising from this coresonance yield a fingerprint of vortex core pinning and depinning in the sample. The experimental results are discussed in relation to mechanical torques predicted from both macrospin (at high dc magnetic field) and finite-difference solutions to the Landau-Lifshitz-Gilbert (LLG) equation. A global fit of the LLG solutions to the frequency-dependent data reveals a statistically significant discrepancy between the experimentally observed and simulated torque phase behaviours at spin-texture transitions that can be reduced through the addition of a time constant to the conversion between magnetic cross-product torque and mechanical torque, constrained by experiment to be in the range of 0.5–4 ns.

• Received 27 October 2023
• Revised 15 January 2024
• Accepted 18 January 2024

DOI:https://doi.org/10.1103/PhysRevB.109.064404