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Investigations of the intrinsic electromagnetic nonlinearity ofsuperconductors 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-fieldmicrowave microscope on homogeneous YBa$_2$Cu$_3$O$_{7-\delta}$ (YBCO) thinfilms in a temperature range close to the critical temperature T$_c$. Astemperature is increased from below T$_c$, the harmonic magnitude exhibits amaximum, while the phase, $\pi/2$ in the superconducting state, goes through aminimum. It is found that samples with doping ranges from near optimal($\delta=0.16$) to underdoped ($\delta=0.47$) exhibit different behavior interms of both the harmonic magnitude and phase. In optimally-doped samples, theharmonic magnitude reaches its maximum at a temperature $T_M$ slightly lowerthan that associated with the minimum of phase $T_m$ and drops into thenoisefloor as soon as $T_m$ is exceeded. In underdoped samples $T_M$ is shiftedtoward lower temperatures with respect to $T_m$ and the harmonic voltagemagnitude decreases slower with temperature than in the case of optimally-dopedsamples. A field-based analytical model of $\tilde{u}_{3f}$ is presented, wherethe nonlinear behavior is introduced as corrections to the low-field,linear-response complex conductivity. The model reproduces the low-temperatureregime where the $\sigma_2$ nonlinearity dominates, in agreement with publishedtheoretical and experimental results. Additionally the model identifies $T_m$as the temperature where the order parameter relaxation time becomes comparableto the microwave probing period and reproduces semi-quantitatively theexperimental data.

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