Broadband type-I hyperbolicity independent of carrier density in $\mathrm{RuOC}{\mathrm{l}}_2$ crystals

The design and fabrication of hyperbolic metamaterials require precise control over the arrangement and dimensions of constituent components, such as nanosheets or nanowires, to achieve the desired hyperbolic dispersion. However, the ongoing challenge lies in further downsizing these components to broaden the hyperbolic regime and enhance the maximal wave vector. In this study, we propose a linear quasi-one-dimensional electron gas array model as a promising category of type-I natural hyperbolic materials (NHMs). The hyperbolic properties in these NHMs remain insensitive to the position of the Fermi level, owing to the linear dispersion relation near the Fermi level. Through first-principles calculations, we have identified a highly promising candidate material, the $\mathrm{RuOC}{\mathrm{l}}_2$ crystal, for this model. Our research demonstrates that the $\mathrm{RuOC}{\mathrm{l}}_2$ crystal exhibits a broad type-I hyperbolic region that spans from the infrared to the entire visible spectrum, and this property is nearly independent of the Fermi level's position, or equivalently, the carrier density. We also investigate the exceptional negative refraction effects and directional-propagating surface plasmon polaritons that arises from the hyperbolic equifrequency contour. These findings offer a universal approach to designing type-I NHMs, as well as a compelling foundation for development of cutting-edge optoelectronic devices.

Figure 1

Schematic represention of Q1DEGA model. (a) Left: Schematic plot of the Fermi surface for Q1DEGA system. Right: The linear dependence of energy $E$ with $k_y$, where the slope $v_{\mathrm{F}}$ is a constant. (b) The real part of the permittivity for Q1DEGA systems. The shaded region stands for the hyperbolic region with $\mathrm{Re}{\varepsilon }_{xx}>0,\mathrm{Re}{\varepsilon }_{zz}>0,\mathrm{Re}{\varepsilon }_{yy}<0$. (c) Equifrequency contours (EFC) for Q1DEGA systems for $\omega <{\omega }_{p,y}$. (d) Dependencies of plasmon frequency on the carrier density in the long-wavelength limit for parabolic and linear Q1DEGA systems.

Figure 2

Lattice and electronic structures of bulk $\mathrm{RuOC}{\mathrm{l}}_2$ crystal. (a) Schematic views of the primitive cell (left panel) and conventional cell (right panel) of bulk $\mathrm{RuOC}{\mathrm{l}}_2$. (b) Atom-resolved electronic band structure of bulk $\mathrm{RuOC}{\mathrm{l}}_2$. (c) Fermi surface of bulk $\mathrm{RuOC}{\mathrm{l}}_2$, with the color representing the magnitude of the Fermi velocity. (d) Schematic diagram of the Brillouin zone and the high symmetry points of bulk $\mathrm{RuOC}{\mathrm{l}}_2$.

Figure 3

Dielectric properties of bulk $\mathrm{RuOC}{\mathrm{l}}_2$ crystal. (a) Real and imaginary parts of the permittivity along three principal directions of pristine $\mathrm{RuOC}{\mathrm{l}}_2$ crystal. (b) The contributions of electron interband transitions to the real and imaginary parts of the permittivity of pristine $\mathrm{RuOC}{\mathrm{l}}_2$ crystal. (c) Real and imaginary parts of the permittivity along three principal directions for bulk $\mathrm{RuOC}{\mathrm{l}}_2$ at the electron doping concentration of 6.6 × ${10}^{21}\mathrm{c}{\mathrm{m}}^{-3}$. (d) Variation of the plasmon frequency (green dots), squared Fermi velocity (orange dots), and hyperbolic interval (dark blue bars) of bulk $\mathrm{RuOC}{\mathrm{l}}_2$ as a function of electron doping concentration. The shaded region in (a) and (c) shows the hyperbolic frequency window.

Figure 4

Negative refraction in $\mathrm{RuOC}{\mathrm{l}}_2$. (a) Left: Schematic representation of negative refraction of a TM-polarized light incident from air to $\mathrm{RuOC}{\mathrm{l}}_2$. Right: The EFC projected onto the $k_x$-$k_y$ plane. The refracted wave vectors and Poynting vectors are indicated by the solid blue and red arrows, respectively. The nonphysical solutions are represented by the dashed arrows. (b) Real part [Re($k$)] and imaginary part [Im($k$)] of wave vector in the wavelength of 911.7 nm. (c) The electric field distribution in the x-y plane for the TM light with the wavelength of 911.7 nm and an incident angle of ${30}^{\circ }$. The Poynting vectors are marked by red arrows.

Figure 5

The spatial distribution of electric field E (log scale) of the SPPs at the frequency of $\hslash \omega =$ 1.36 eV (wavelength of 911.7 nm) in the planes of (a) $z=0$, (b) $y=0$, and (c) $x=0$. These SSPs are excited by a vertically polarized dipole located just 1 nm above the $\mathrm{RuOC}{\mathrm{l}}_2$ surface.