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 ${10}^5\mathrm{pm}/\mathrm{V}$ 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.

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• Landau levels
• Nonlinear optical susceptibility
• Photoconductivity
• Semimetals
• Boltzmann theory
• Optical second-harmonic generation