# Difference frequency generation in topological semimetals

## TL;DR

This paper derives a universal expression for difference frequency generation (DFG) in topological semimetals, revealing quantized responses under circular polarization and proposing experimental detection methods for topological photocurrents in metals.

## Contribution

It provides the first general derivation of DFG in metals, showing quantization in topological semimetals and identifying additional free-carrier contributions beyond semiclassical theory.

## Key findings

- DFG in chiral topological semimetals is quantized in units of e^3/h^2.
- Quantized DFG response is independent of material parameters when Δω ≫ τ^{-1}.
- Additional free-carrier contributions to DFG oscillate with frequency and phase, beyond semiclassical predictions.

## Abstract

When two lasers are applied to a non-centrosymmetric material, light can be generated at the difference of the incoming frequencies $\Delta\omega$, 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 $\Delta\omega \gg \tau^{-1}$. 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 non-chiral 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, non-interacting limit, and highlights a plausible experiment to measure topologically quantizated photocurrents in metals.