Broadband microwave spectroscopy can probe material properties in wide spectral and temperature ranges. The quality of such measurements crucially depends on the calibration, which also removes from the obtained spectra signatures of standing waves. Here we consider cryogenic Corbino-type reflection measurements on superconductors close to the critical temperature. We show that the non-linear sample response, which relates to sample heating, 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, power, and temperature and for different lengths of the microwave transmission line. Finally, we note such non-linear effects beyond the case of superconductors by probing a VO2 thin film at the insulator-metal transition.
Flexible Pasternack Precision Cables, 095 Series PE35611. The used cables have a physical length of 16 cm, 31 cm, 46 cm, and 77 cm.
From independent measurements, we know that Tc of the studied NbTiN sample is several K higher than suggested by the data in Fig. 3. This is due to the distance between the sample and the temperature sensor and the large temperature gradients in the probe of setup #4 when operated at such low temperatures.
Manufacturer specifications are for room temperature, and we neglect the temperature dependence of Lel.
We estimate that the strongest observed oscillations correspond to a temperature swing of 2 mK for the VO2 case and some tens of mK for the superconducting cases.
This argument only holds if the sample impedance is lower than the coax characteristic impedance of 50 , and all our cases fall well into this regime.