Phase-sensitive Harmonic Measurements of Microwave Nonlinearities in Cuprate Thin Films

Investigations of the intrinsic electromagnetic nonlinearity of superconductors give insight into the fundamental physics of these materials. Phase-sensitive third-order harmonic voltage data $\tilde{u}_{3f}=|u_{3f}|exp(i\phi_{3f})$ are acquired with a near-field microwave microscope on homogeneous YBa$_2$Cu$_3$O$_{7-\delta}$ (YBCO) thin films in a temperature range close to the critical temperature T$_c$. As temperature is increased from below T$_c$, the harmonic magnitude exhibits a maximum, while the phase, $\pi/2$ in the superconducting state, goes through a minimum. It is found that samples with doping ranges from near optimal ($\delta=0.16$) to underdoped ($\delta=0.47$) exhibit different behavior in terms of both the harmonic magnitude and phase. In optimally-doped samples, the harmonic magnitude reaches its maximum at a temperature $T_M$ slightly lower than that associated with the minimum of phase $T_m$ and drops into the noisefloor as soon as $T_m$ is exceeded. In underdoped samples $T_M$ is shifted toward lower temperatures with respect to $T_m$ and the harmonic voltage magnitude decreases slower with temperature than in the case of optimally-doped samples. A field-based analytical model of $\tilde{u}_{3f}$ is presented, where the nonlinear behavior is introduced as corrections to the low-field, linear-response complex conductivity. The model reproduces the low-temperature regime where the $\sigma_2$ nonlinearity dominates, in agreement with published theoretical and experimental results. Additionally the model identifies $T_m$ as the temperature where the order parameter relaxation time becomes comparable to the microwave probing period and reproduces semi-quantitatively the experimental data.