Role of non-linear effects and standing waves in microwave spectroscopy: Corbino measurements on superconductors and VO$_2$.

Broadband microwave spectroscopy can probe material properties in wide spectral and temperature ranges, including superconductors at cryogenic temperatures. The quality of such measurements crucially depends on the calibration, which also removes from the obtained spectra signatures of standing waves. Here we consider low-temperature reflection measurements in Corbino geometry, and we show that the non-linear response of superconducting samples close to the critical temperature can lead to strong signatures of standing waves even in a well-calibrated Corbino spectrometer. We demonstrate our findings with microwave measurements as a function of frequency and temperature for a variety of superconducting samples and for different length of the microwave transmission line. Finally we show that such non-linear effects extend beyond the case of superconductors by probing a VO$_2$ thin film at the insulator-metal transition.