Resonant torsion magnetometry in anisotropic quantum materials

Unusual behavior of quantum materials commonly arises from their effective low-dimensional physics, which reflects the underlying anisotropy in the spin and charge degrees of freedom. Torque magnetometry is a highly sensitive technique to directly quantify the anisotropy in quantum materials, such as the layered high-T\(_c\) superconductors, anisotropic quantum spin-liquids, and the surface states of topological insulators. Here we introduce the magnetotropic coefficient \(k=\partial^2 F/\partial \theta^2\), the second derivative of the free energy F with respect to the angle \(\theta\) between the sample and the applied magnetic field, and report a simple and effective method to experimentally detect it. A sub-\(\mu\)g crystallite is placed at the tip of a commercially available atomic force microscopy cantilever, and we show that \(k\) can be quantitatively inferred from a shift in the resonant frequency under magnetic field. While related to the magnetic torque \(\tau=\partial F/\partial \theta\), \(k\) takes the role of torque susceptibility, and thus provides distinct insights into anisotropic materials akin to the difference between magnetization and magnetic susceptibility. The thermodynamic coefficient \(k\) is discontinuous at second-order phase transitions and subject to Ehrenfest relations with the specific heat and magnetic susceptibility. We apply this simple yet quantitative method on the exemplary cases of the Weyl-semimetal NbP and the spin-liquid candidate RuCl\(_3\), yet it is broadly applicable in quantum materials research.