  We theoretically investigate the resonance of third harmonic generation (THG)
that has been observed at frequency being half of the superconducting gap in a
multiband disordered superconductor NbN. The central question is whether the
dominant contribution to the THG resonance comes from the Higgs mode (the
collective amplitude mode of the superconducting order parameter) or
quasiparticle excitations. To resolve this issue, we analyze a realistic
three-band model with effective intraband and interband phonon-mediated
interactions estimated from first-principles together with nonmagnetic impurity
scatterings. Using the estimated pairing interactions with multiband impurity
scattering rates being varied from clean to dirty regimes, we calculate the THG
susceptibility for NbN in a channel-resolved manner by means of the BCS and
self-consistent Born approximations. In the dirty regime, which is close to the
experimental situation, the leading contribution is given by the paramagnetic
channel of the Higgs mode having almost no polarization-angle dependence, while
the second leading contribution comes from the paramagnetic channel of
quasiparticles generally showing significant polarization-angle dependence. The
result is consistent with the recent experimental observation of no
polarization-angle dependence of THG, giving firm evidence that the Higgs mode
dominantly contributes to the THG resonance in NbN superconductors.
