Third order optical nonlinearity of three dimensional massless Dirac fermions

TL;DR

This paper derives analytical formulas for the linear and third order optical conductivities of three-dimensional massless Dirac fermions, revealing their unique optical features influenced by chemical potential and dimensionality.

Contribution

It provides the first detailed analytical expressions for third order optical nonlinearity in 3D Dirac fermions, extending understanding beyond 2D systems like graphene.

Findings

Imaginary part of linear conductivity diverges logarithmically with cutoff energy.

Real part of linear conductivity is linear in photon frequency for  > 2||.

Third order conductivity amplitude is two orders of magnitude smaller than in 2D Dirac systems.

Abstract

We present analytic expressions for the electronic contributions to the linear conductivity $\sigma^{(1)}_{3d}(\omega)$ and the third order optical conductivity $\sigma^{(3)}_{3d}(\omega_1,\omega_2,\omega_3)$ of three dimensional massless Dirac fermions, the quasi-particles relevant for the low energy excitation of topological Dirac semimetals and Weyl semimetals. Although there is no gap for massless Dirac fermions, a finite chemical potential $\mu$ can lead to an effective gap parameter, which plays an important role in the qualitative features of interband optical transitions. For gapless linear dispersion in three dimension, the imaginary part of the linear conductivity diverges as a logarithmic function of the cutoff energy, while the real part is linear with photon frequency $\omega$ as $\hbar\omega>2|\mu|$. The third order conductivity exhibits features very similar to those of two dimensional Dirac fermions, i.e., graphene, but with the amplitude for a single Dirac cone generally two orders of magnitude smaller in three dimension than in two dimension. There are many resonances associated with the chemical potential induced gap parameters, and divergences associated with the intraband transitions. The details of the third order conductivity are discussed for third harmonic generation, the Kerr effect and two-photon carrier injection, parametric frequency conversion, and two-color coherent current injection. Although the expressions we derive are limited to the clean limit at zero temperature, the generalization to include phenomenological relaxation processes at finite temperature is straightforward and is presented.