# Optical properties of a semi-Dirac material

## TL;DR

This paper calculates the optical conductivity of a 2D semi-Dirac material using the Kubo formalism, analyzing intraband and interband contributions, anisotropy, effects of an energy gap, impurity scattering, and extensions to related models.

## Contribution

It provides analytical formulas for optical conductivity in semi-Dirac materials, explores anisotropic effects, and generalizes the model to include higher-order dispersions and flat bands.

## Key findings

- Analytic formulas for intraband and interband conductivity components.
- Sumrule between Drude weight and interband transitions in the relativistic direction.
- Comparison of semi-Dirac optical properties with graphene.

## Abstract

Within a Kubo formalism, we calculate the absorptive part of the dynamic longitudinal conductivity $\sigma(\Omega)$ of a 2D semi-Dirac material. In the clean limit, we provide separate analytic formulas for intraband (Drude) and interband contributions for $\sigma(\Omega)$ in both the relativistic and nonrelativistic directions. At finite doping, in the relativistic direction, a sumrule holds between the increase in optical spectral weight in the Drude component and that lost in the interband optical transitions. For the nonrelativistic direction, no such sumrule applies. Results are also presented when an energy gap opens in the energy dispersion. Numerical results due to finite residual scattering are provided and analytic results for the dc limit are derived. Energy dependence and possible anisotropy in the impurity scattering rate is considered. Throughout, we provide comparison of our results for $\sqrt{\sigma_{xx}\sigma_{yy}}$ with the corresponding results for graphene. A generalization of the 2D Hamiltonian to include powers of higher order than quadratic (nonrelativistic) and linear (relativistic) is considered. We also discuss the modifications introduced when an additional flat band is included via a semi-Dirac version of the $\alpha$-${\cal T}_3$ model, for which an $\alpha$ parameter tunes between the 2D semi-Dirac (graphene-like) limit and the semi-Dirac version of the dice or ${\cal T}_3$ lattice.