Third-Order Optical Nonlinearity of Three-Dimensional Massless Dirac Fermions

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 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 dimensions, the imaginary part of the linear conductivity diverges as a logarithmic function of the cutoff energy, while the real part is linear, with a photon frequency omega as h omega > 2 vertical bar mu vertical bar. The third-order conductivity exhibits features very similar to those of two-dimensional Dirac Fermions, that is, graphene, but with the amplitude for a single Dirac cone generally 2 orders of magnitude smaller in three dimensions than in two dimensions. 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. In contrast with 2D materials, the bulk nature of materials that host three-dimensional Dirac Fermions allows for the possibility of enhancing nonlinear signals by tuning the sample thickness; thus, broad applications of such materials in nonlinear photonic devices can be envisioned.