  Second-order nonlinear optical responses, including photogalvanic effect
(PGE) and second harmonic generation (SHG), are important physical phenomena in
nonlinear optics. The PGE (SHG) related to linearly and circularly polarized
light are called the linear and circular PGE (LPGE and CPGE) [linear and
circular SHG (LSHG and CSHG)], respectively. In this work, we use the quantum
kinetics under relaxation time approximation to study the dependence of
second-order nonlinear optical responses on Fermi level and frequency under
different out-of-plane electric fields in WTe$_2$ monolayer from radio to
infrared region. We find that the maximum frequency at which the Berry
curvature dipole mechanism for the nonlinear Hall effect plays a major role is
about 1 THz. In radio and microwave regions, two large peaks of nonlinear
conductivities occur when the Fermi level is equal to the energy corresponding
to gap-opening points. In terms of frequency, in radio region, LPGE and SHG
conductivities maintain a large constant while the CPGE conductivity
disappears. In microwave region, LPGE and SHG start to decrease with increasing
frequency while the CPGE is large. In 125-300 THz region and in y direction,
the presence of DC current without the disturbance of second harmonic current
under circularly polarized light may be useful for fabricating new
optoelectronic devices. Moreover, we illustrate that when calculating the
nonlinear optical responses of practical materials, the theories in the clean
limit fail and it is necessary to use a theory that considers scattering
effects. We also point out that for materials with femtosecond-scale relaxation
times and complex energy band structures, the quantum kinetics is more accurate
than the semi-classical Boltzmann equation method. Besides, phenomenological
expressions of PGE and SHG are provided.
