Slow spin-lattice relaxation dynamics in YbVO4 revealed by extended thermal impedance spectroscopy from AC susceptibility and AC magnetocaloric measurements

Abstract

Alternating (AC) magnetic fields can induce not only an alternating magnetization in materials, but also an alternating temperature via the magnetocaloric effect. The latter effect is typically neglected when performing AC susceptibility measurements, but consideration of both effects on an equal footing is necessary in order to reliably distinguish between internal and external causes of magnetic response and accurately extract quantitative information about relaxation processes. In order to address this, we have developed a method to measure the AC magnetocaloric effect that is compatible with AC susceptibility measurements, and also a framework to analyze these data in combination. We demonstrate the efficacy of this approach using YbVO4, a material for which strong single-ion anisotropy leads to slow spin-lattice relaxation at low temperatures via a phonon bottleneck effect. We report AC magnetic susceptibility and AC magnetocaloric effect measurements for this material as a function of field and frequency at a temperature of 3 K. We analyze the data using a discretized thermal model, and extract the field-dependence of the intrinsic spin-lattice relaxation rate. This demonstration experiment illustrates a general approach to quantitatively address multiple measured quantities in driven systems using a unified thermal circuit analysis. The thermal analysis methods presented in this report can be extended to study other magnetic, dielectric, and elastic materials exhibiting a complex response to an external driving field in the presence of internal and external relaxation, particularly when an energy dissipation process is within an accessible frequency regime.

Figures (12)

• Figure 1: (a) Schematic diagram illustrating the CEF levels and indirect transition corresponding to Yb^3+ ion embedded in the YbVO4 crystal field environment. The lower two levels represent the ground state Kramers doublet, which are split when an external magnetic field is oriented along the c-axis. The higher energy states correspond to the excited states of the crystal field levels. We consider a scenario where the temperature is much less than the Debye temperature. (b) Schematic diagram illustrating the components of the discretized thermal circuit model considered in this report. The coefficients and time constants are defined in the main text. (c,d) Schematic diagrams illustrating solutions of the equivalent thermal circuit, and using parameters characteristic of YbVO$_4$ at 3K. In the left panels, the normalized response is plotted to illustrate the phase difference between the time-dependent solution and the signals. In the middle panels, solid lines represent the real components ($\chi_{AC}'$ and $\Gamma_{AC}'$), and dashed lines represent the imaginary components ($\chi_{AC}"$ and $\Gamma_{AC}"$). In the right panels, the imaginary components of each solution are plotted against their respective real parts in a Cole-Cole plot. This representation maintains a fixed aspect ratio of 1:1 for the $x$ and $y$ axes.
• Figure 2: Data illustrating how sample mounting configurations can affect AC magnetic relaxation. Panel (a-c) illustrates three different sample mounting configurations, described in the main text. Colored bars represent the sample that is to be measured, together with its dimensions. The same sample of YbVO4 is used for all three configurations to enable direct comparison. Grey blocks indicate quartz platforms that serve as heat baths, or at least thermal connection to a larger heat bath. The sample is oriented with the magnetic field aligned along the long c-axis of the crystal (horizontal in the schematic diagrams). AC susceptibility measurements were made at 3 K, 0.1 T, using an AC field of 3 Oe. Panels (d) and (e) show the real ($\chi_{AC}'$) and imaginary parts ($\chi_{AC}"$) normalized by $\chi_T$, which was determined from fits to the Debye model. Panel (f) shows the associated Cole-Cole plot in which frequency is an implicit variable. Data are shown for the three mounting configurations, using the same colors to differentiate the three configurations. Solid lines in (d,e,f) are fit results based on Eq. \ref{['Debye']}. The encapsulated configuration (yellow data points) yields data that are closest to the idealized Debye relaxation conditions.
• Figure 3: Experimental results for YbVO4 showing the frequency dependence of the real (upper panels) and imaginary (lower panels) parts of the response functions $\chi_{AC}(\omega)$ (in panels (a,b)) and $\Gamma_{AC}(\omega)$ (in panels (c,d)). Data are shown for representative DC magnetic fields from 0 T to 1 T. All data were taken at a temperature of 3 K.
• Figure 4: Fitting of the two measured response functions $\chi_{AC}(\omega)$ and $\Gamma_{AC}(\omega)$ for YbVO4 using Eqn.s \ref{['fit_equation_chi']} and \ref{['fit_equation_MCE']}. All data were measured at 3K. (a) A Cole-Cole plot showing the imaginary ($\chi_{AC}"$) against the real ($\chi_{AC}'$) part of the dynamical susceptibility. (b) A similar plot for the AC MCE response, $\Gamma_{AC}(\omega)$. (c) The characteristic relaxation times ($\tau_\chi$, $\tau_{int}$ and $\tau_{ext}$) obtained from the fits, as described in the main text. See Appendix \ref{['app:best fit params']} for other fitted parameters.
• Figure 5: (a) Photograph of an AC MCE probe assembly, with inset showing a magnified image of the thermal sensor ($R_x$; the small black chip in the image) mounted on the crystal of YbVO4 (yellow color prism) which is mounted on a flattened section of a thin quartz rod (clear). The sample is located at the end of the quartz rod, which is attached to the ETO probe head. Before measurement, the center of the sample is aligned with the center of the applied magnetic coils following standard procedures. (b) Circuit diagram of the AC MCE measurement. Here, $R_x$ represents the thermal resistor. $R_1$, $R_2$, and $R_3$ are bridge resistors. The raw temperature oscillation signal is obtained from the bridge voltage $V_b$ via a lock-in amplifier in a dual mode with its internal reference, and external reference from the magnetic coil ($V_h$). A second lock-in amplifier measures $V_x$ to obtain the temperature profile of the thermal resistors. A Voltage Controlled Current Source converts the $1\ \text{V}$ voltage output of the lock-in amplifier to an excitation current of $100 \ \mu\text{A}$ to the Wheatstone Resistor Bridge.