Difference frequency generation in topological semimetals

When two lasers are applied to a noncentrosymmetric material, it can generate light at the difference of the incoming frequencies $\mathrm{\Delta }\omega$, a phenomenon known as difference frequency generation (DFG), well characterized in semiconductors. In this work, we derive a general expression for DFG in metals, which we use to show that the DFG in chiral topological semimetals under circular polarized light is quantized in units of $e^3/h^2$ and independent of material parameters, including the scattering time $\tau$, when $\mathrm{\Delta }\omega \gg {\tau }^{-1}$. In this regime, DFG provides a simpler alternative to measure a quantized response in metals compared to previous proposals based on single frequency experiments. Our general derivation unmasks, in addition, a free-carrier contribution to the circular DFG beyond the semiclassical one. This contribution can be written as a Fermi surface integral, features strong frequency dependence, and oscillates with a $\pi /2$ shift with respect to the quantized contribution. We make predictions for the circular DFG of chiral and nonchiral materials using generic effective models, and ab initio calculations for TaAs and RhSi. Our work provides a complete picture of the DFG in the length gauge approach, in the clean, noninteracting limit, and highlights a plausible experiment to measure topologically quantized photocurrents in metals.

Figure 1

Trace and traceless parts of CPGE tensors for a Weyl node with chirality $\chi =-1$, with tilt $\tilde{u}=0$ (green dashed) and $\tilde{u}=0.4$ (blue solid). (a) Trace of injection tensor $\beta$, where a quantized plateau is observed. (b) Traceless part of injection ${\beta }_F$. (c) Trace of free-carrier tensor $\gamma$. Note the $1/\omega$ divergence stemming from ${\gamma }_2$ which always cancels after summing over all nodes in the Brillouin zone. (d) Traceless part of free-carrier ${\gamma }_F$. (a), (b) and (c), (d) are in units of ${\beta }_0=i\pi e^3/h^2$ and ${\beta }_0/\mu$, respectively. Vertical dashed lines mark the frequencies $\hslash {\omega }_{\pm }=2\mu /(1\pm \tilde{u})$.

Figure 2

[(a) and (b)] Diagonal parts of the circular DFG, $\beta$ (blue) and $\gamma$ (green) for a chiral Weyl semimetal with two Weyl nodes at energies ${\mu }_L$ and ${\mu }_R$, with ${\mu }_L/{\mu }_R=0.25$, and tilts ${\tilde{u}}_L=0.2{\tilde{u}}_R=0.25$. [(c) and (d)] Same quantities for the linear approximation of a multifold material in space group 198, with a threefold at ${\mu }_{3f}$ and a double Weyl fourfold fermion at ${\mu }_{4f}$ with ${\mu }_{3f}/{\mu }_{4f}=0.25$.

Figure 3

First-principles prediction for circular DFG. a) and b) show ${\beta }_{xy}$ and ${\gamma }_{xy}$ for TaAs with spin-orbit coupling (SOC). The activation frequencies of the W2 and W1 pairs of nodes at 25 and 45 meV (dashed vertical lines) mark the peaks in (a) and (b), in qualitative agreement with Figs. 1 and 1. In (c) and (d), we show the trace $\beta$ (with and without SOC) and ${\gamma }_{xx}$ with SOC for RhSi. The activation frequencies of the threefold at $\mathrm{\Gamma }$ and fourfold fermion at $R$, 0.1 and 0.625 eV, respectively, are marked by dashed vertical lines. The horizontal dashed line in (b) marks the effective monopole charge of 4 corresponding to $4{\beta }_0=0.318e^3/{\hslash }^2$. For (b) and (d), the total free-carrier contribution $\gamma$ (green solid) is composed of the semiclassical (${\gamma }_2$, red dashed) and the novel free-carrier contribution (${\gamma }_1$, orange dotted), which qualitatively agree with Fig. 2.