Low-Temperature Properties of Whispering-Gallery Modes in Isotopically Pure Silicon-28

Whispering-gallery-(WG) mode resonators are machined from a boule of single-crystal isotopically pure silicon-28. Before machining, the as-grown rod is measured in a cavity, with the best Bragg confined modes exhibiting microwave $Q$ factors on the order of a million for frequencies between 10 and 15 GHz. After machining the rod into smaller cylindrical WG-mode resonators, the frequencies of the fundamental mode families are used to determine the relative permittivity of the material to be $11.488\pm 0.024$ near 4 K, with the precision limited only by the dimensional accuracy of the resonator. However, the machining degraded the $Q$ factors to below $4\times {10}^4$. Raman spectroscopy is used to optimize postmachining surface treatments to restore high-$Q$-factors. This is an enabling step for the use of such resonators for hybrid quantum systems and frequency-conversion applications, as silicon-28 also has very low phonon losses, can host very narrow linewidth spin ensembles, and is a material commonly used in optical applications.

Figure 1

(a) Photograph of the ${}^{28}\mathrm{Si}$ rod of 55 mm in length and approximately 15 mm diameter before machining. (b) Three of the four WG-mode ${}^{28}\mathrm{Si}$ cylindrical resonators cut and machined from the rod.

Figure 2

Graphical representation of the ${}^{28}\mathrm{Si}$ WG-mode resonator-loaded cavity. The magnitude of the real part of the axial electric field component $\left|E_z\right|$ of the 19.535 GHz ${\mathrm{WGH}}_{7,1,1}$ mode, calculated by finite-element analysis using comsol, is shown overlaid on the resonator. (Left) Cross section through in the $r$-$\theta$ plane at $z=0$. (Right) Cross section through the $r$-$z$ plane. The coaxial coupling probes used for excitation and measurement are shown, which couple to the evanescent field of the resonator.

Figure 3

Measured and calculated mode frequencies of the fundamental WGH and WGE mode frequencies with ${ϵ}_r=11.488$.

Figure 4

Raman spectra showing the first-order peak of silicon at 520 ${\mathrm{cm}}^{-1}$ for one of the WG-mode resonators pictured in Fig. 1. (a) Spectra measured at several locations on the side and top surfaces of the resonator after machining, and (b) after undergoing the acid treatment and annealing procedure. Surface strain is evident in the linewidth broadening and shifting of the Raman peak.

Figure 5

(a) Measured $Q$ factors of the resonator before (gray) and after (coloured) treatment, ensuring enough incident power to obtain maximum $Q$ factor. Additionally, potential $Q$ factors assuming dominant limiting loss factors are plotted for cavity losses; $Q_c$, assuming a cavity wall conductivity of ${\sigma }_c={10}^7$ S/m, hopping conductivity; plotted for $p_e{\sigma }_{\mathrm{hop}}={10}^{-5}$ S/m, as well as pure dielectric loss assuming a loss tangent filling factor product $p_e\tan \delta ={10}^{-6}$. (b) Transmission spectrum of the ${\mathrm{WGH}}_{7,1,1}$ mode; the highest $Q$-factor mode measured.

Figure 6

Measured $Q$ factors categorized into mode families, plotted proportion of normal $E$ field of the curved surface of the cylinder as defined by Eq. (4). The apparent inverse relationship explains the consistent hierarchy of $Q$ factors for each mode family.