Resonant torsion magnetometry in anisotropic quantum materials

Abstract

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 $k={\partial }^{2}F/\partial \hat{I}{¸}^2$, the second derivative of the free energy F with respect to the angle $\hat{I}¸$ between the sample and the applied magnetic field, and report a simple and effective method to experimentally detect it. A sub-$\hat{I}¼$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 $\ddot{I}„=\partial F/\partial \hat{I}¸$, $k$ 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.

7 pages including 6 figures and methods section