Second harmonic generation in Dirac/Weyl semimetals with broken tilt inversion symmetry

We theoretically investigate the second harmonic generation in tilted Dirac/Weyl semimetals with broken tilt inversion symmetry in the absence of an external magnetic field using quantum theory. An analytical formula for the second harmonic conductivity tensor is derived, and it does not depend on the chirality of Weyl node. There are two contributions to the conductivity in the low-frequency region, one coming from the intraband transitions and describing by Drude-like effects, and the other from the interband-intraband transitions due to the linear energy dispersion of Dirac/Weyl semimetals near the Dirac/Weyl points. In the high-frequency region, the appearance of prominent resonant peaks in the nonlinear conductance originates from the two-photon absorption process. It is found that Dirac/Weyl semimetals have a very high nonlinear susceptibility, and an optimal tilt of the Dirac/Weyl node for the maximum nonlinear susceptibility has been found.