Current-induced second harmonic generation in inversion-symmetric Dirac and Weyl semimetals

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