Current-induced second harmonic generation of Dirac or Weyl semimetals in a strong magnetic field

We theoretically investigate the second harmonic generation (SHG) of Dirac or Weyl semimetals under parallel DC electric and strong magnetic fields using the Boltzmann equation approach. The DC current-induced SHG process originates from the optical intraband transitions of the Landau subbands. It is a remarkable fingerprint of three-dimensional Dirac or Weyl dispersions and absent in other materials. An analytical formula for the SHG tensor is derived, and it shows chirality independence. The second-order nonlinear optical susceptibility is proportional to the DC field, giving strong optical nonlinearity up to ${10}^{5}\phantom{\rule{0.28em}{0ex}}\mathrm{pm}/\mathrm{V}$ with THz light, which is several orders larger than that of the usual materials. More interesting, the second harmonic conductivity is found to exhibit periodicity in magnetic field $1/B$ oscillations, which are similar to the Shubnikov--de Haas oscillations. Thus our work proposes another approach for SHG and potential applications of Dirac or Weyl semimetals in nonlinear optics.