Second harmonic generation (SHG) is a fundamental nonlinear optical phenomenon widely used both for experimental probes of materials and for application to optical devices. Even-order nonlinear optical responses including SHG generally require the breaking of inversion symmetry, and thus have been utilized to study noncentrosymmetric materials. Here, we study theoretically the SHG in inversion-symmetric Dirac and Weyl semimetals under a DC current which breaks the inversion symmetry by creating a nonequilibrium steady state. Based on analytic and numerical calculations, we find that Dirac and Weyl semimetals exhibit strong SHG upon application of finite current. Our experimental estimation for a Dirac semimetal ${\mathrm{Cd}}_3{\mathrm{As}}_2$ and a magnetic Weyl semimetal ${\mathrm{Co}}_3{\mathrm{Sn}}_2{\mathrm{S}}_2$ suggests that the induced susceptibility ${\chi }^{\left(2\right)}$ for practical applied current densities can reach ${10}^5\mathrm{pm}{\mathrm{V}}^{-1}$ with mid-IR or far-IR light. This value is ${10}^2$–${10}^4$ times larger than those of typical nonlinear optical materials. We also discuss experimental approaches to observe the current-induced SHG and comment on current-induced SHG in other topological semimetals in connection with recent experiments.

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• Electric field effects
• Nonlinear optical susceptibility
• Nonlinear optics
• Ultrafast optics
• Dirac semimetal
• Topological materials
• Weyl semimetal
• Boltzmann theory
• Perturbation theory