Probing a two-dimensional soft ferromagnet Cr$_2$Ge$_2$Te$_6$ by a tuning fork resonator

Abstract

Magnetic anisotropy encodes key information about the free-energy landscape of magnetic materials, but its quantitative characterization often requires probes beyond conventional magnetometry. A quartz tuning-fork resonator provides direct access to the magnetotropic susceptibility. Here we use this technique to investigate the magnetic anisotropy of the layered ferromagnet Cr$_2$Ge$_2$Te$_6$. The temperature-, field-, and angle-dependent responses are consistently described by a quasi-two-dimensional (2D) easy-axis ferromagnetic model. In particular, the evolution of the magnetotropic susceptibility reveals how the angular profile changes from a conventional cos(2$θ$) form to a pronounced dip structure as the magnetization approaches directional saturation. These results establishCr$_2$Ge$_2$Te$_6$ as an ideal reference system for tuning-fork-based magnetotropic measurements. More broadly, they provide a useful framework for distinguishing spin-origin anisotropy from orbital magnetism, as in the case of CsV3Sb5. Our work demonstrates that tuning-fork resonators offer a sensitive thermodynamic probe of the rotational stiffness of magnetization in anisotropic low-dimensional magnets.

Figures (4)

• Figure 1: (a) Crystal structure of Cr$_2$Ge$_2$Te$_6$. (b) Schematic illustration of the tuning-fork measurement setup. The sample is mounted at the front end of the cantilever, with the rotation axis perpendicular to the $c$-axis. $\theta$ is defined as the angle between the magnetic field $H$ and the $c$-axis. (c) Temperature dependence of the magnetic susceptibility of Cr$_2$Ge$_2$Te$_6$ for $H \parallel ab$ and $H \parallel c$. The inset shows the temperature dependence of the resonant frequency shift, $\delta f=f(0.5~\mathrm{T})-f(0~\mathrm{T})$, for $H \parallel ab$. (d) Magnetization curves measured at 10 K, showing the saturation fields for $H \parallel ab$ and $H \parallel c$.
• Figure 2: (a) Angular dependence of $\Delta f$ at different temperatures for $\mu_0H=0.5$ T. (b) Forward and backward rotation measurements of $\Delta f$ at 100 K and 5 T in the paramagnetic phase. The green dashed line shows a $\cos(2\theta)$ fit. (c) Forward and backward rotation measurements of $\Delta f$ at 10 K and 0.5 T in the ferromagnetic phase. The green dashed line shows the fit to the phenomenological easy-axis ferromagnetic model. (d) Temperature dependence of $\Delta f$ at $\theta=90^\circ$, extracted from panel (a).
• Figure 3: (a) Angular dependence of the resonant frequency at different magnetic fields at 10 K. (b) Field dependence of $k_{\mathbf n}$ at selected field orientations. (c) Derivative of $k_{\mathbf n}$ with respect to $H$. The peak values are used to define the saturation field $H_{\mathrm S}$. (d) Angular dependence of $H_{\mathrm S}(\theta)$ extracted from panel (c), together with two different fits. The black dashed line is proportional to $1/\cos\theta$. The green solid line is obtained from the energy minimization based on Eqs. (\ref{['eq3']}) and (\ref{['eq4']}).
• Figure 4: Schematic illustration of the magnetic response of Cr moments in Cr$_2$Ge$_2$Te$_6$ under a rotating magnetic field at low temperature. Panels (a)-(c) show the spin configurations in three field regimes: (a) $H<H_{\mathrm{SE}}$, (b) $H_{\mathrm{SE}} \leq H < H_{\mathrm{SH}}$, and (c) $H \gg H_{\mathrm{SH}}$. Panels (d)-(f) show the corresponding measured (left panel) and simulated (right panel) angular dependence of the magnetotropic susceptibility.