Dielectric Properties of Single Crystal Calcium Tungstate

TL;DR

This work addresses the need for precise dielectric properties of CaWO$_4$ at microwave frequencies across temperatures relevant to quantum technologies. It employs microwave whispering gallery modes on a single-crystal CaWO$_4$ cylinder and uses COMSOL-based mode matching to extract the dielectric tensor components $ε_{\parallel}$ and $ε_{\perp}$, along with loss tangents, from 295 K down to 4 K. Key findings include $ε_{\parallel}=9.029\pm0.009$ and $ε_{\perp}=10.761\pm0.011$ at 295 K, and $ε_{\parallel}=8.797\pm0.009$ and $ε_{\perp}=10.442\pm0.01$ at 4 K, with room-temperature and cryogenic loss tangents of $\tan\delta_{\parallel}$ and $\tan\delta_{\perp}$ in the $10^{-5}$ and $10^{-7}$ ranges, respectively. A cryogenic loss peak near 10.5 GHz points to an unidentified paramagnetic impurity, underscoring the role of material purity. The results provide essential parameters for designing CaWO$_4$-based microwave circuits and sensing devices, and demonstrate the utility of WGM spectroscopy as a sensitive probe of lattice dynamics and spin-related loss channels.

Abstract

This investigation employed microwave whispering gallery mode (WGM) analysis to characterize the dielectric properties of a cylindrical, single-crystal sample of calcium tungstate (CaWO$_4$). Through investigation of quasi-transverse\hyp{}magnetic and quasi-transverse\hyp{}electric mode families, we can assess loss mechanisms and relative permittivity from room temperature down to cryogenic conditions. We report the biaxial permittivity values of $ε_{||} = 9.029 \pm 0.009$ and $ε_{\perp} = 10.761 \pm 0.01$ at $295$ K, and $ε_{||} = 8.797 \pm 0.009$ and $ε_{\perp} = 10.442 \pm 0.01$ at $4$ K. Components are denoted with respect to the c\hyp{}axis of the crystal unit cell. The parallel component agrees well with the published literature at MHz frequencies; however, the perpendicular component is $4.8$\% lower. The WGM technique offers greater precision, with accuracy limited primarily by the uncertainty in the crystal's dimensions. WGMs also serve as sensitive probes of lattice dynamics, enabling monitoring of temperature-dependent loss mechanisms. At room temperature, the measured loss tangents were $\tanδ_{||}^{295,\mathrm{K}} = (4.1 \pm 1.4) \times 10^{-5}$ and $\tanδ_{\perp}^{295,\mathrm{K}} = (3.64 \pm 0.92) \times 10^{-5}$. Upon cooling to 4 K, the loss tangents improved by approximately two orders of magnitude, reaching $\tanδ_{||}^{4,\mathrm{K}} = (1.56 \pm 0.52) \times 10^{-7}$ and $\tanδ_{\perp}^{4,\mathrm{K}} = (2.05 \pm 0.79) \times 10^{-7}$. These cryogenic values are higher than those reported in prior studies, likely due to a magnetic loss channel associated with an unidentified paramagnetic spin ensemble. These findings have implications for the use of CaWO$_4$ in applications such as spin-based quantum systems and cryogenic bolometry, highlighting the potential of WGMs for novel sensing applications.

Figures (8)

• Figure 1: Experimental setup for electromagnetic mode identification in the cylindrical crystal. The radiative fields around the unshielded crystal were measured in transmission using coaxial loop probes along the axial and azimuthal directions to confirm WGM mode numbers and polarisation. The orientation of the probes and their distance from the crystal surface dictated the strength of the coupling to microwave resonances of distinct mode families.
• Figure 2: Measured and simulated electromagnetic field variations for the WGE$_{7,1,1}$ mode ($f_{0}=7.965$ GHz). A Colour density plot of the measured transmission as a function of probe position near the surface of the unshielded dielectric, in the azimuthal direction (left), and the axial direction (right). B The magnetic field distribution within the CaWO$_4$ cylindrical dielectric crystal, as simulated by COMSOL.
• Figure 3: Simulated room-temperature (295 K) characteristics of the fundamental whispering gallery mode families, WGH$_{m,1,1}$ (orange) and WGE$_{m,1,1}$ (purple), plotted as a function of azimuthal mode number, $m$. The vertical, grey, dashed line indicates the threshold above which microwave resonances were considered sufficiently WGM-like for use in subsequent analysis. A Fundamental mode family frequencies. The linear regression is fitted for $m\geq6$, where $m$ is large enough to present a linear frequency dependence. The fit gives an FSR of 0.897 GHz for WGH$_{m,1,1}$ modes and 0.830 for WGE$_{m,1,1}$ modes. Using Eq. \ref{['eq:fsr']} and assuming the modes are pure TM and TE, we obtain an average radius of propagation of $r=17.9$ and $17.8$ mm, respectively. B Electric filling factors are theoretically predicted and employed for the calculation of the temperature coefficients.
• Figure 4: Schematic of resonator setup in dilution refrigerator.
• Figure 5: The WGH$_{9,1,1}$ (blue) and WGE$_{9,1,1}$ (green) were used to match eigenfrequency solutions from simulations to the experimental values. The solid line follows the measured resonance, and the lighter bands represent theoretical predictions and $0.02\%$ error encapsulating residuals. Error here is associated with the combination of simulation error and random fluctuations in measured frequencies. The spectrum on the right illustrates a temperature 'slice' of the colour density plot with the WGH$_{9,1,1}$ (blue) and WGE$_{9,1,1}$ (green) resonances marked with dashed lines.