Tunable in-plane bihyperbolicity in bismuth monolayer

The distinctive hyperbolic properties of natural two-dimensional (2D) materials have garnered considerable attention in recent years due to their potential to surpass the limitations of metahyperbolic surfaces. It is essential to control hyperbolic regions and the sign of optical conductivities. This study introduces the concept of “bihyperbolicity” and establishes a critical connection between the semiconducting characteristics of 2D materials and their hyperbolic attributes. Through first-principles calculations, we illustrate the applicability of this strategy to materials such as the recently synthesized bismuth monolayer. The computations revealed that altering the type ($n$ type or $p$ type) of semiconducting bismuth monolayers can lead to a reversal of conductivity signs along two orthogonal directions, consequently enabling the precise regulation of hyperbolicity. The intriguing interplay between hyperbolicity and semiconductivity lays the foundation for crafting in-plane hyperbolic heterostructures using well-established semiconductor technologies. These heterostructures unlock a plethora of exotic optical phenomena, including negative refraction and negative reflection, thereby opening new horizons in optical engineering and device design.

Figure 1

Schematic representation of bihyperbolicity in 2D materials. (a) Plasmon dispersions in 2D materials given by Eq. (5). The two dashed lines represent the asymptotes of the two dispersions. (b) Schematic electronic band profiles of bihyperbolic 2D materials. $E_F^n$ and $E_F^p$ indicate the positions of Fermi levels for the $n$-type and $p$-type cases, respectively. (c) and (d) The correlation between the anisotropic Fermi surface (left panel) and the plasmon dispersions (right panel) for the $n$-type and $p$-type doping cases. The basis vectors of the reciprocal lattice are indicated by ${\mathit{b}}_1$ and ${\mathit{b}}_2$. The Brillouin zone is represented by the square. The yellow areas in the diagram indicate regions of electron occupation in the Brillouin zone.

Figure 2

Atomic configuration and electronic structure of bismuth monolayer. (a) Atomic configuration of bismuth monolayer consisting of two buckled rectangular sublayers. (b) Orbital-resolved electronic band structures of bismuth monolayer. The energy of the VBM indicated by the dashed line was set to zero. (c), (d) The isovalue profiles of the Kohn-Sham electron wave functions at the VBM and CBM, respectively. (e), (f) The anisotropic energy contours of electronic states near the VBM and CBM within the 1/4 Brillouin zone, respectively.

Figure 3

The plasmonic properties and bihyperbolicity of bismuth monolayer. (a), (b) The electron energy loss spectra (EELS) of $n$-type and $p$-type bismuth monolayers at the doping concentration of $n_e=1.15\times {10}^{13}\mathrm{c}{\mathrm{m}}^{-2}$ and $n_h=0.72\times {10}^{13}\mathrm{c}{\mathrm{m}}^{-2}$, respectively. The yellow dots denote the specific plasmon modes obtained from first-principles calculations. The red lines represent the results of fitting these data using Eq. (6). The black dashed lines depict the boundaries of the intraband single-particle excitation regions. (c), (d) The optical conductivity of electron-doped and hole-doped bismuth monolayers. The Fermi-surface contours of the two cases are presented as the insets of these figures. (e) The variation of the hyperbolic regions of the hole-doped (left panel) and electron-doped (right panel) bismuth monolayer at different doping levels.

Figure 4

The SPPs in bismuth monolayer. (a), (b) The isofrequency contours of bismuth monolayers at the doping concentration of $n_e=1.15\times {10}^{13}\mathrm{c}{\mathrm{m}}^{-2}$ and $n_h=0.72\times {10}^{13}\mathrm{c}{\mathrm{m}}^{-2}$, respectively. The dashed lines indicate the asymptotic lines. (c), (d) The spatial distribution of electric field E of the surface plasmons at $\omega =140$ meV for the heterostructures with different interface configurations, respectively. The interface is marked by the red line. (e), (f) The corresponding isofrequency contours of the heterostructures. The Poynting vectors for the incident beam (${\mathbf{S}}_{\mathbf{i}}$), negative refraction (${\mathbf{S}}_{\mathbf{NRR}}$), negative reflection (${\mathbf{S}}_{\mathbf{NRL}}$), and positive reflection (${\mathbf{S}}_{\mathbf{PRL}}$) beams that are normal to the isofrequency contours are denoted.