Mesoscopic Nonlinear Electrodynamic Response of the Two Band Superconductor MgB$_{2}$*

Multi-gap superconductors are expected to have exotic physics associated with internal coupling between the multiple order parameters. 10 years after its discovery, MgB$_{2}$ is a two gap superconductor that still attracts much attention for both fundamental and practical reasons. A microwave measurement of the temperature dependent third harmonic response P$_{3f}$ (T) is performed by applying a strong and localized (mesoscopic) RF magnetic field on high quality MgB$_{2}$ films. Comparing to the P$_{3f}$ (T) result on Nb thin films, the low temperature nonlinearity of MgB$_{2}$ shows a complete harmonic cancelation at a temperature near the transition temperature of the proximity-induced $\pi $ band. One possibility is that this phenomenon is due to an additional intrinsic nonlinearity arising from Josephson coupling between the $\sigma $ and $\pi $ bands. This nonlinear response then interferes with other nonlinear mechanisms coming from moving vortices and the intrinsic nonlinear Meissner effect of the two-gap system. To investigate further, we measure the magnitude and phase of the nonlinear response and compare to models of the three nonlinearity mechanisms.