Precision microwave dielectric and magnetic susceptibility measurements of correlated electronic materials using superconducting cavities

We analyze microwave cavity perturbation methods, and show that the technique is an excellent, precision method to study the dynamic magnetic and dielectric response in the GHz frequency range. Using superconducting cavities, we obtain exceptionally high precision and sensitivity for measurements of relative changes. A dynamic electromagnetic susceptibility ${\mathrm{\zeta ̃(T)=\zeta }}^{\prime }{\mathrm{+i\zeta }}^{″}$ is introduced, which is obtained from the measured parameters: the shift of cavity resonant frequency $\mathrm{\delta f}$ and quality factor Q. We focus on the case of a spherical sample placed at the center of a cylindrical cavity resonant in the ${\mathrm{TE}}_{011}$ mode. Depending on the sample characteristics, the magnetic permeability $\mathrm{\mu ̃,}$ the dielectric permittivity ${\mathrm{\epsilon ̃=\epsilon }}^{\prime }{\mathrm{+i\epsilon }}^{″},$ and the complex conductivity ${\mathrm{\sigma ̃=\sigma }}^{\prime }{\mathrm{+i\sigma }}^{″}$ can be extracted from ${\mathrm{\zeta ̃}}_H.$ A full spherical wave analysis of the cavity perturbation indicates that: (i) In highly insulating samples with dielectric constant ε^′∼1, the measured ${\mathrm{\zeta ̃}}_H{\mathrm{\sim \chi ̃}}_M,$ enabling direct measurement of the magnetic susceptibility. The sensitivity of the method equals or surpasses that of dc superconducting quantum interference device measurements for the relative changes in magnetic susceptibility. (ii) For moderate $\mathrm{\epsilon ̃}$ and conductivity σ̃, ${\mathrm{\zeta ̃}}_H{\mathrm{\propto \epsilon ̃+i\omega \sigma ̃/\epsilon }}_0\text{−1,}$ thus enabling direct measurement of the sample dielectric constant $\mathrm{\epsilon ̃,}$ even though the sample is placed in a microwave magnetic field. (iii) For large σ we recover the surface impedance limit. (iv) Expressions are provided for the general case of a lossy dielectric represented by ${\mathrm{\epsilon ̃+i\omega \sigma ̃/\epsilon }}_0.$ We show that an inversion procedure can be used to obtain ${\mathrm{\epsilon ̃+i\omega \sigma ̃/\epsilon }}_0$ in a wide range of parameter values. This analysis has led to the observation of new phenomena in novel low-dimensional materials. We discuss results on magneto dynamics of the three-dimensional (3D) antiferromagnetic state of the spin chain compound ${\mathrm{Sr}}_{\mathrm{2}}{\mathrm{CuO}}_3.$ In dielectric susceptibility measurements in ${\mathrm{Sr}}_{\mathrm{14}}{\mathrm{Cu}}_{\mathrm{24}}{\mathrm{O}}_{41},$ we directly observe a dielectric loss peak. Dimensional resonances in the paraelectric material ${\mathrm{SrTiO}}_3$ are shown to occur due to the rapid increase of dielectric constant with decreasing temperature. The cavity perturbation methods are thus an extremely sensitive probe of charge and spin dynamics in electronic materials.