Band-selective third-harmonic generation in superconducting MgB$_2$: Possible evidence for Higgs amplitude mode in the dirty limit

We report on time-resolved linear and nonlinear terahertz spectroscopy of the two-band superconductor MgB$_2$ with the superconducting transition temperature $T_c \approx 36$ K. Third-harmonic generation (THG) is observed below $T_c$ by driving the system with intense narrowband THz pulses. For the pump-pulse frequencies $f=$ 0.3, 0.4, and 0.5 THz, temperature-dependent evolution of the THG signals exhibits a resonance maximum at the temperatures with the resonance conditions $2f=2\Delta_\pi(T)$ fulfilled, for the dirty-limit superconducting gap $2\Delta_\pi$. In contrast, for $f=$ 0.6 and 0.7 THz with $2f>2\Delta_\pi(T\rightarrow0)=1.03$ THz, the THG intensity increases monotonically with decreasing temperature. Moreover, for $2f<2\Delta_\pi(T\rightarrow0)$ the THG is found nearly isotropic with respect to the pump-pulse polarization. These results suggest a predominant contribution of the driven Higgs amplitude mode of the dirty-limit $\pi$-band superconducting gap, pointing to the importance of scattering for observation of the Higgs mode in superconductors.