We present analytic expressions for the electronic contributions to the linear conductivity σ_3d⁽¹⁾(ω) and the third-order optical conductivity σ_3d⁽³⁾(ω₁, ω₂, ω₃) 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 μ 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 ω as ℏω > 2|μ|. 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.

中文翻译：

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三维无质量狄拉克费米子的三阶光学非线性

我们本解析表达式用于向所述线性导电性的电子贡献σ _{3 d}^（1）（ω）和σ三阶光电导率_{3 d}^（3）（ω ₁，ω ₂，ω ₃）三维无质量狄拉克费米子，与拓扑狄拉克和韦尔半金属的低能激发有关的准粒子。尽管无质量狄拉克费米子没有间隙，但有限的化学势μ可以产生有效的间隙参数，这在带间光学跃迁的定性特征中起着重要的作用。对于三维的无间隙线性弥散，线性电导率的虚部作为截止能量的对数函数而发散，而实部呈线性，光子频率ω为asω> 2 |μ|。三阶电导率具有与二维狄拉克费米子非常相似的特征，即石墨烯，但单个狄拉克锥的振幅在三维上通常比二维小2个数量级。有许多与化学势诱导的间隙参数相关的共振，以及与带内跃迁相关的发散。讨论了三次谐波产生，Kerr效应和双光子载流子注入，参数频率转换以及双色相干电流注入的三阶电导率的详细信息。尽管我们得出的表达式仅限于零温度下的纯净极限，但包含有限温度下的现象学松弛过程的概化是简单明了的。与2D材料相比，承载三维Dirac费米子的材料的整体性质允许通过调整样品厚度来增强非线性信号的可能性。因此，可以设想这种材料在非线性光子器件中的广泛应用。