Interband optical conductivity in two-dimensional semi-Dirac bands tilting along the quadratic dispersion

TL;DR

This work analyzes interband optical response in two-dimensional semi-Dirac materials with a quadratic dispersion along one direction and a linear dispersion along the other, focusing on tilts along the quadratic direction and classifying Lifshitz phases by the tilt parameter $t_x$ defined as $t_x=\frac{v_{tx}}{2\sqrt{a|\mu|}}$. Using Kubo linear response, the interband conductivity decouples into direction-specific scaling factors $S_{jj}^{(\mathrm{IB})}(\omega)$ and dimensionless spectral functions $\Gamma_{jj}^{(\mathrm{IB})}(\omega,\mu,t_x)$, revealing anisotropic power-law behavior in $S_{xx}$ and $S_{yy}$ and a tilt-insensitive high-frequency background. A key result is a robust fixed point at $\omega=2\mu$ in the interband conductivity for $0<t_x\le 1$, explained by a corresponding fixed point in the joint density of states (JDOS); for $t_x>1$ four separate optical-transition thresholds emerge due to two Fermi pockets, highlighting Lifshitz-phase–dependent spectral fingerprints. These findings distinguish tilted 2D SDBs from tilts along the linear dispersion and Dirac cases, offering theoretical benchmarks for experimental identification and guiding future tilt-engineering of anisotropic electronic systems.

Abstract

Two-dimensional (2D) semi-Dirac materials feature a unique anisotropic band structure characterized by quadratic dispersion along one spatial direction and linear dispersion along the other, effectively hybridizing ordinary and Dirac fermions. The anisotropy of energy dispersion can be further modulated through band tilting along either spatial direction of the wave vector. We propose a new definition of tilt parameter to characterize Lifshitz phases in 2D semi-Dirac bands tilting along the quadratically dispersing direction. Using linear response theory, we theoretically investigate the interband optical conductivity of 2D tilted semi-Dirac bands. Our analytical zero-temperature results reveal pronounced distinctions from Dirac and semi-Dirac systems tilting along the linearly dispersing direction. Notably, we find that spectral fixed point emerges in the optical conductivity over a specific range of the tilt parameter, a phenomenon explained by the corresponding behavior of the joint density of states. These findings provide a robust theoretical framework for identifying and characterizing 2D tilted semi-Dirac materials and establish clear spectral fingerprints that distinguish different kinds of 2D semi-Dirac bands and Dirac bands. Our predictions can guide future experimental studies of anisotropic band engineering and tilt-dependent phenomena.

Figures (5)

• Figure 1: Schematic diagrams for energy bands, Fermi surfaces, and corresponding critical frequencies $\omega_{\lambda}^{\pm}$ of interband optical transitions in 2D SDBs tilting along the quadratic dispersion for $n$-doped case.
• Figure 2: Dimensionless auxiliary functions $\Gamma_{xx}^{(\mathrm{IB})}(\omega,\mu,t_{x})$, $\Gamma_{yy}^{(\mathrm{IB})}(\omega,\mu,t_{x})$ , and their geometric mean $\bar{\Gamma}^{(\mathrm{IB})}(\omega,\mu,t_{x})$ are shown in panels (a)-(c). Interband LOCs $\mathrm{Re}~\sigma_{xx}^{(\mathrm{IB})}(\omega,\mu,t_{x})$, $\mathrm{Re}~\sigma_{yy}^{(\mathrm{IB})}(\omega,\mu,t_{x})$, and their geometric mean $\mathrm{Re}~\bar{\sigma}^{(\mathrm{IB})}(\omega,\mu,t_{x})$ are shown in panels (d)-(f). The fixed points are denoted by the solid black dots. The critical frequencies $\omega_{\lambda}^{\pm}$ in the interband LOCs are indicated in panels (d)-(f). The legends for all panels are shown in panel (a).
• Figure 3: Joint density of state. The fixed point are denoted by the solid black dots. The critical frequencies $\omega_{\lambda}^{\pm}$ are indicated. Two critical frequencies $\omega_{+}^{\pm}$ contributed by electron pocket ($\lambda=+$) appear in all tilted phases, but other two critical frequencies $\omega_{-}^{\pm}$ contributed by hole pocket ($\lambda=-$) are present only in the type-II phase.
• Figure 4: The states involved in the interband optical transition with $\omega=2\mu$ are gathered around a cyan stripe, and the Fermi surfaces are denoted by the red and blue lines. The location of states occupying below them are meshed with light-blue and light-red slashes, and the crossed regions represent that the interband optical transition therein is forbidden by Pauli exclusion principle, rendering the corresponding states represented by the black dashed lines do not contribute to the interband optical transition.
• Figure 5: Interband LOCs $\mathrm{Re}~\sigma_{jj}^{(\mathrm{IB})}(\omega,\mu,t_{i})$ with $i,j$ covering both $x$ and $y$. The analytical results adopted to plot this figure are based on the present work and Refs. PRBTan2022PRBHou2023PRBYan2023. In panels (c), (d), (g) and (h), $\mathrm{Re}~\sigma_{xx}^{(\mathrm{IB})}(\omega,\mu,t_{x})=\mathrm{Re}~\sigma_{yy}^{(\mathrm{IB})}(\omega,\mu,t_{y})$ and $\mathrm{Re}~\sigma_{xx}^{(\mathrm{IB})}(\omega,\mu,t_{y})=\mathrm{Re}~\sigma_{yy}^{(\mathrm{IB})}(\omega,\mu,t_{x})$. Each column represents a distinct group of panels.