Introduction

Hyperbolic metamaterials are engineered materials specifically designed to have opposite signs of permittivity tensors along two crystallographic directions, resulting in a hyperbolic dispersion relation. These materials have the unique capability to support surface plasmon polaritons (SPPs) that exhibit highly confined electromagnetic fields. SPPs are collective oscillations of electrons (plasmons) coupled to photons at the interface of a metal and a dielectric material. Within hyperbolic materials, SPPs can be tightly confined, allowing for subwavelength waveguiding and significantly enhancing light-matter interactions at the nanoscale. These properties have profound implications for a wide range of applications, including sensing, nonlinear optics, and integrated photonic circuits1,2,3,4.

The SPPs with hyperbolic dispersion have primarily been a topic of discussion within the realm of artificially engineered metamaterials. These engineered materials encompass a diverse range of configurations, such as metasurfaces and nanostructured van der Waals materials, providing unique opportunities for manipulating light at subwavelength scales2,3,4,5,6. The emergence of two-dimensional (2D) materials presents an additional avenue for hosting SPPs owning to their distinct plasmonic characteristics, such as low damping rate, high confinement, and exceptional tunability7,8,9. Natural hyperbolic 2D materials possess inherent advantages over metasurfaces, as their atomic scale periodicity enables the generation of large wavevectors and eliminates the need for intricate surface patterning. The plasmon frequency in conventional electron gas systems is generally strongly influenced by the carrier density n, following an n1/2 dependence for long-wavelength plasmons10, while for Dirac plasmons in graphene, it follows an n1/4 dependence11,12. This dependence on carrier density allows for control of plasmon polaritons by altering the electron density through chemical doping and gating13,14, but it also makes them susceptible to environmental disturbances15. Furthermore, the plasmons in anisotropic 2D materials offer a versatile platform for manipulating polariton propagation and light-matter interactions8,16. Of particular interest is the hyperbolic isofrequency contours, which enable large wave-vector propagation and high photonic density of states (DOS)2,17. While hyperbolic plasmons have mostly been realized in artificially engineered metamaterials18, there is a growing interest in utilizing 2D materials with intrinsic anisotropic electronic states as natural platforms for hosting hyperbolic SPPs, benefiting from their merits of low-loss and tunability7,19,20. Black phosphorus (BP)19,21, α-MoO322,23, and WTe224,25 and several multilayer transition metal dichalcogenides (TDMs)26 have been identified as capable of hosting anisotropic 2D plasmons. However, finding a viable method to enhance the anisotropy of plasmon modes and expand the intervals of hyperbolic response in 2D materials continues to pose challenges. These advancements are crucial to meet the specific requirements of applications that involve the directional transport of SPPs.

In this study, we present a promising design principle for hyperbolic natural 2D materials, characterized by a linear energy-momentum relation along the y-direction in the electronic band structure, which mimics a quasi-one-dimensional electron gas (Q1DEG) system. Our research demonstrates the immense potential of this model in hosting highly directional plasmons where the plasmon frequency remains independent of carrier density. Through first-principles calculations, we identify a promising candidate material, RuOCl2 monolayer to realize this design. Our calculations unveil that the RuOCl2 monolayer can exclusively support plasmons along the y-direction, achieving frequencies as high as 1.75 eV, while lacking plasmon excitation along the x-direction. Furthermore, these plasmons are insensitive to variation in the Fermi level. Additionally, the RuOCl2 monolayer also displays a broad and low-loss hyperbolic region spanning from 0 to 4.45 eV, making it highly capable of supporting directional hyperbolic SPPs. The natural 2D hyperbolic material designed from this principle encompasses multiple benefits, such as carrier-density-independent frequency, high directionality, and low loss. These findings open exciting prospects for the development of high-performance directional plasmon transmission devices.

Results

A general formula of 2D anisotropic plasmon dispersion

Plasmon dispersion is determined by the zeros of the dielectric function \(\varepsilon ({{{{{\boldsymbol{q}}}}}},\omega )\). Assuming the intraband transitions of electrons dominate the dielectric function, we have \({F}_{ll{\prime} }({{{{{\boldsymbol{k}}}}}},{{{{{\boldsymbol{q}}}}}})={\delta }_{ll{\prime} }\). In this case, the polarization function \(\varPi ({{{{{\boldsymbol{q}}}}}},\omega )\) can be written in the Lindhard expression,

$$\varPi ({{{{{\boldsymbol{q}}}}}},\omega )=\frac{{g}_{s}}{{(2\pi )}^{2}}\mathop{\sum}\limits _{l}\int d{{{{{\boldsymbol{k}}}}}}\frac{f({E}_{{{{{{\boldsymbol{k}}}}}},l})-f({E}_{{{{{{\boldsymbol{k}}}}}}+{{{{{\boldsymbol{q}}}}}},l})}{\omega +{E}_{{{{{{\boldsymbol{k}}}}}},l}-{E}_{{{{{{\boldsymbol{k}}}}}}+{{{{{\boldsymbol{q}}}}}},l}}.$$
(1)

In the long-wavelength approximation (q→0), we have \({E}_{{{{{{\boldsymbol{k}}}}}},l}-{E}_{{{{{{\boldsymbol{k}}}}}}+{{{{{\boldsymbol{q}}}}}},l}\approx -{\nabla }_{{{{{{\boldsymbol{k}}}}}}}{E}_{{{{{{\boldsymbol{k}}}}}},l}\cdot {{{{{\boldsymbol{q}}}}}}\) and \(f({E}_{{{{{{\boldsymbol{k}}}}}},l})-f({E}_{{{{{{\boldsymbol{k}}}}}}+{{{{{\boldsymbol{q}}}}}},l})\approx \big(-\frac{\partial f}{\partial E}\big){\nabla }_{{{{{{\boldsymbol{k}}}}}}}{E}_{{{{{{\boldsymbol{k}}}}}},l}\cdot {{{{{\boldsymbol{q}}}}}}\). The polarization function is written as,

$$\varPi ({{{{{\boldsymbol{q}}}}}},\omega )\approx \frac{{g}_{s}}{{(2\pi )}^{2}}\mathop{\sum}\limits _{l}\int d{{{{{\boldsymbol{k}}}}}}\frac{{\nabla }_{{{{{{\boldsymbol{k}}}}}}}{E}_{{{{{{\boldsymbol{k}}}}}},l}\cdot {{{{{\boldsymbol{q}}}}}}}{\omega -{\nabla }_{{{{{{\boldsymbol{k}}}}}}}{E}_{{{{{{\boldsymbol{k}}}}}},l}\cdot {{{{{\boldsymbol{q}}}}}}}\left(-\frac{\partial f}{\partial E}\right)$$
(2)

Replacing \({\nabla }_{{{{{{\boldsymbol{k}}}}}}}{E}_{{{{{{\boldsymbol{k}}}}}},l}\) with electron velocity, \({{{{{{\boldsymbol{\upsilon }}}}}}}_{{{{{{\boldsymbol{k}}}}}}{,}{l}}={\nabla }_{{{{{{\boldsymbol{k}}}}}}}{E}_{{{{{{\boldsymbol{k}}}}}},l}={\upsilon }_{x}{{{{{\boldsymbol{i}}}}}}+{\upsilon }_{y}{{{{{\boldsymbol{j}}}}}}\) and taking T = 0 K, we have,

$$\varPi ({{{{{\boldsymbol{q}}}}}},\omega )\approx \frac{{g}_{s}}{{(2\pi )}^{2}}\mathop{\sum}\limits _{l}\int d{{{{{\boldsymbol{k}}}}}}\frac{{{{{{{\boldsymbol{\upsilon }}}}}}}_{{{{{{\boldsymbol{k}}}}}},l}\cdot {{{{{\boldsymbol{q}}}}}}}{\omega }\left(1+\frac{{{{{{{\boldsymbol{\upsilon }}}}}}}_{{{{{{\boldsymbol{k}}}}}},l}\cdot {{{{{\boldsymbol{q}}}}}}}{\omega }\right)\delta ({{{\rm E}}}_{{{{{{\boldsymbol{k}}}}}},l}-{E}_{F}),$$
(3)

where EF is the Fermi level. Defining the electron DOS at the Fermi level \(\rho ({E}_{F})=\frac{{g}_{s}}{{(2\pi )}^{2}}\mathop{\sum}\limits _{l}\int {d}^{2}{{{{{\boldsymbol{k}}}}}}\delta ({E}_{{{{{{\rm{k}}}}}},l}-{E}_{F})\) and the averaged square Fermi velocities \(\langle {\upsilon }_{x,y}^{2}\rangle =\mathop{\sum}\limits _{l}\int d{{{{{\boldsymbol{k}}}}}}{{v}}_{x,y}^{2}\delta ({E}_{{{{{{\boldsymbol{k}}}}}},l}-{E}_{F})/\mathop{\sum}\limits _{l}\int {d}^{2}{{{{{\boldsymbol{k}}}}}}\delta ({E}_{{{{{{\boldsymbol{k}}}}}},l}-{E}_{F})\), we have a simple formula of the polarization function,

$$\prod ({{{{{\boldsymbol{q}}}}}},\omega )=\rho ({E}_{F})(\langle {v}_{x}^{2}\rangle {\cos }^{2}\theta +\langle {v}_{y}^{2}\rangle {\sin }^{2}\theta )\frac{{q}^{2}}{{\omega }^{2}}.$$
(4)

In this expression, θ represents the angle between q and the x direction. By solving the zeros of the dielectric function, we get a general formula of anisotropic plasmon dispersion in the long-wavelength limit,

$$\omega ({{{{{\boldsymbol{q}}}}}})={({\omega }_{x}^{2}(q){\cos }^{2}\theta +{\omega }_{y}^{2}(q){\sin }^{2}\theta )}^{1/2}$$
(5)

with\({\omega }_{x/y}(q)={\alpha }_{x/y}{q}^{1/2}\), \({\alpha }_{x}={(2\pi {e}^{2}\rho ({E}_{F})\langle {\upsilon }_{x}^{2}\rangle /{\varepsilon }_{r})}^{1/2}\) and \({\alpha }_{y}={(2\pi {e}^{2}\rho ({E}_{F})\langle {\upsilon }_{y}^{2}\rangle /{\varepsilon }_{r})}^{1/2}\). The anisotropy of the plasmon dispersion can be featured by the ratio of \({\alpha }_{y}/{\alpha }_{x}=\sqrt{\langle {\upsilon }_{y}^{2}\rangle /\langle {\upsilon }_{x}^{2}\rangle }\). \({\alpha }_{x}={\alpha }_{y}\) corresponds to an isotropic plasmon dispersion, as shown in Fig. 1a. As a limiting case of anisotropy, \({\alpha }_{x}=0\) and \({\alpha }_{y}\ne 0\), the plasmon dispersion is depicted in Fig. 1b, where hyperbolic-like isofrequency curves in this anisotropic plasmon dispersion are evident.

Fig. 1: Schematic diagrams of plasmon dispersions.
Fig. 1: Schematic diagrams of plasmon dispersions.
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a The isotropic plasmon with \({\alpha }_{x}={\alpha }_{y}\). b The anisotropic plasmon dispersion with \({\alpha }_{x}=0\) and \({\alpha }_{y}\ne 0\). c The electronic band structure of a quasi-one-dimensional electron gas (Q1DEG) system with a linear energy-momentum dispersion relation of \(E({{{{{\boldsymbol{k}}}}}})=\hslash {v}_{F}|{k}_{y}|-{E}_{F}\). d Schematic representation of the distinct carrier density-dependent plasmon frequency for conventional electron gas (EG) (blue curve) and Q1DEG systems (pink line).

Anisotropic plasmon dispersion of quasi-1DEG systems

The linear energy-momentum dispersion (also named Dirac cones) in the electronic band structure of graphene leads to unusual electron transport properties27. In this work, we consider a highly anisotropic 2D lattice with a linear energy-momentum relation along one direction (taken as y-direction), \(E({{{{{\boldsymbol{k}}}}}})=\hslash {\upsilon }_{F}|{k}_{y}|-{E}_{F}\), while being insulating along the x-direction. This model emulates a Q1DEG system, where the electrons are restricted to a 1D geometry while the Coulomb interaction between them assumes a 2D or 3D form28,29. This highly anisotropic electronic band gives rise to an opened Fermi surface consisting of two parallel lines oriented in the x-direction, as shown in Fig. 1c.

According to Eq. (5), this anisotropic linear energy-momentum relation gives the plasmon dispersion of the Q1DEG system as

$$\omega ({{{{{\boldsymbol{q}}}}}})={\alpha }_{y}|\,\sin \theta |{q}^{1/2},$$
(6)

with \({\alpha }_{y}={({g}_{s}{e}^{2}{\upsilon }_{F}G/{\varepsilon }_{r}\pi )}^{1/2}\) and \({\alpha }_{x}=0\), where G represents the length of the first Brillouin zone along the y-direction. Equation (6) provides valuable insights into the dispersion behavior of plasmons in this Q1DEG system. The plasmon mode exhibits distinct properties depending on the direction of momentum transfer q, as depicted in Fig. 1b. For q//kx (θ = 0) the plasmon mode is completely forbidden, while for q//ky (θ = 90°), the plasmon energy reaches its maximum. Moreover, the hyperbolic-like isofrequency curves allow for the manipulation of plasmon polaritons by changing the polarization of the incident light. One such phenomenon is the hyperbolic plasmon polariton, commonly observed in artificially engineered metamaterials18,30. Additionally, one intriguing property of plasmons in Q1DEG systems is that according to Eq. (6), their frequency \(\omega\) remains constant regardless of carrier density n or Fermi level EF. This is in contrast to the behavior of plasmons in conventional EG systems in all dimensions of \(\omega \sim {n}^{1/2}\)10,31, as shown in Fig. 1d.

Plasmon dispersion of RuOCl2 monolayer

To implement the aforementioned model, we focused on studying a monolayer of RuOCl2 oxide dichloride. It is worth noting that the bulk form of this material has already been successfully synthesized32, providing a suitable foundation for our investigation. First-principles calculations revealed that the exfoliation energy of the RuOCl2 monolayer from its bulk material is determined to be 14.9 meV/Å2. This value is lower than the exfoliation energies of graphene (21.8 meV/Å2)33 and phosphorene (23.0 meV/Å2)34 from their respective bulk counterparts, indicating that it can be readily exfoliated and isolated as a 2D material. The monolayer has a rectangular lattice structure with a space group of Pmmm, as depicted in Fig. 2a. The lattice constants along the two orthogonal basis vectors are respectively a = 3.55 Å and b = 3.70 Å (The structure parameters shown in the ‘Supplementary Data 1’). Each Ru atom is coordinated by two O atoms and four Cl atoms with the Ru-O plane sandwiched by two Cl planes. The anisotropic atomic arrangements along the two basis vectors (taken as x- and y-directions, respectively) are clearly visible with Ru-O-Ru molecular chains orientated along the y direction and bridged by Cl atoms along the x-direction.

Fig. 2: Crystal and electronic structures of RuOCl2 monolayer.
Fig. 2: Crystal and electronic structures of RuOCl2 monolayer.
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a Lattice structure with top view in the up panel and lateral view in the down panel. b The electron localization function (ELF). The iso-values of 0 and 1 indicate complete electron delocalization and perfect electron localization respectively. c The Fermi surfaces, with the arrows indicating the electron velocities at the Fermi surfaces. d The electronic band structure and density of states (DOS). The energy at the Fermi level is set to zero. The symbol θ in the inset represents the angle between q and the x direction.

Electron Localization Function (ELF) quantifies the extent of electron localization. ELF values range between 0 and 1, with ELF = 0 signifying complete electron delocalization, and ELF = 1 denoting perfect electron localization. The ELF of the RuOCl2 monolayer in Fig. 2b shows the electron gas features with ELF = 0.5 along the Ru-O-Ru chains, implying the anisotropic electronic properties of the RuOCl2 monolayer.

The Fermi surface and electronic band structure of the RuOCl2 monolayer are depicted in Fig. 2c, d, where the metallic nature is quite evident. Along the Γ−Y direction (y-direction), there exist two energy bands across the Fermi level, as shown in Fig. 2d. The two bands exhibit nearly linear energy-momentum dispersions in region of the Fermi level, consistent with the above model of Q1DEG. The electronic states near the Fermi level arise mainly from dxy and dyz orbitals of Ru atoms, with minor contribution from px and pz orbitals of O atoms, which form π-bond along the y-direction and highly dispersive energy band along the Γ−Y direction. Along the Γ−X direction, however, a pseudo gap appears near the Fermi level. The pronounced anisotropic behavior observed in the RuOCl2 system can be attributed to the unique atomic arrangements along its two primary vectors. Specifically, the presence of the Ru-O-Ru chain along the y-direction induces substantial hybridization between the Ru 4d orbitals and the O 2p orbitals, resulting in the formation of dispersive energy bands. Conversely, along the x-direction, the hybridization between the Ru 4d orbitals and the Cl 3p orbitals in the Ru-Cl-Ru chains is relatively weak due to the substantial energy difference between them, leading to the emergence of a pseudo gap. Moreover, the electron DOS of RuOCl2 monolayer shown in Fig. 2d is nearly constant near the Fermi level, indicating linear dispersion of the band in this region. The Q1DEG nature of the RuOCl2 monolayer can be visualized more vividly in the Fermi surface shown in Fig. 2c. The Fermi surface contains flat lines orientated along the Γ−X direction, consistent with the Q1DEG model. It is noteworthy that the carriers in the bands move predominantly along the Γ−Y direction with a Fermi velocity of up to 7.5 × 105 m/s. This value is comparable to the Fermi velocity of graphene (~8.5 × 105 m/s)35,36 and confirms the Q1DEG features of the RuOCl2 monolayer. These results demonstrate that the electronic band structure of the RuOCl2 monolayer satisfies the requirement of the Q1DEG model proposed in this study.

We then calculated the electron energy loss spectrum (EELS) of RuOCl2 monolayer along Γ−X (x-) and Γ−Y (y-) directions using first-principles calculations, as shown in Fig. 3a. At small q for both directions, the plasmon energy is proportional to q1/2, which is consistent with Eq. (5) and similar to the plasmon behavior in conventional 2DEG37. As q increases, the plasmon dispersion gradually deviates from q1/2 and becomes flatten. Compared to Γ−Y direction, the plasmon of Γ−X direction shows extremely weak intensity and saturates at around 0.08 eV, while the plasmon along Γ−Y can reach energy up to 1.75 eV before decaying into electron-hole pairs. To quantify the anisotropy of plasmon, we fitted the plasmon dispersions along the Γ−X and Γ−Y directions using Eq. (5). The fitting data with αy = 12.06 eV Å1/2 and αx = 0.29 eV Å1/2 are indicated by the dotted lines in Fig. 3a. The large \({\alpha }_{y}/{\alpha }_{x}\) ratio (~42) corresponds to a significant effective mass anisotropy of \({m}_{x}^{\ast }/{m}_{y}^{\ast }\approx 1764\), which is considerably larger than those observed in other 2D anisotropic plasmon materials such as monolayer BP (~4.2)20, borophene (~3.7)38 and MoOCl2 (~23.6)39. The anisotropic plasmons also exhibit hyperbolic-like isofrequency curves, as depicted in Fig. 3b, consistent with the Q1DEG model. This feature is distinctly distinct from conventional plasmons which are characterized by elliptical or circular isofrequency curves (IFC). Electromagnetic waves with hyperbolic IFC in hyperbolic media exhibit anomalous properties, including negative refraction, super-resolution imaging, and nonlinear optical effects18,40,41,42,43. Consequently, the anisotropic plasmons in the RuOCl2 monolayer hold great promise for a range of applications.

Fig. 3: The plasmonic properties of RuOCl2 monolayer.
Fig. 3: The plasmonic properties of RuOCl2 monolayer.
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a The electron energy loss spectrum (EELS) along the Γ-X and Γ-Y direction. The plasmon dispersions calculated by Eq. (5) are represented by white solid lines. The enlarged view of the plasmons dispersion along the Γ-X is presented in the left panel. b The isofrequency curves of the plasmon dispersion of RuOCl2 monolayer. c The variation of α along the Γ-X and Γ-Y direction as a function of energy relative to the Fermi level at qy = 0.0034 Å−1.

Both \({\alpha }_{x}\) and \({\alpha }_{y}\) of the RuOCl2 monolayer are insensitive to the position of the Fermi level (EF) in the range of |E − EF| < 0.2 eV, as shown in Fig. 3c. Since the carrier density (n) of a material is determined by the Fermi level, the anisotropic plasmons in the RuOCl2 monolayer thus exhibits a n-independent feature, consistent with the Q1DEG model. This property can be attributed to the fact that in Q1DEG systems, the electrons are confined to one dimension and the screening is reduced, leading to a weaker dependence of plasmon frequency on the carrier density.

Hyperbolic regions of RuOCl2 monolayer

In 2D materials, the optical conductivity describes the dielectric properties44, where the real part \({{{{\mathrm{Re}}}}}\sigma\) characterizes the energy loss and the imaginary determines the hyperbolic interval through \({{\mbox{Im}}}[{\sigma }_{xx}(\omega )]\times {{\mbox{Im}}}[{\sigma }_{yy}(\omega )] < 0\). The optical conductivities of RuOCl2 monolayer along x and y directions are plotted in Fig. 4a. A broad hyperbolic regime with \({{\mbox{Im}}}{\sigma }_{xx} < 0\) and \({{\mbox{Im}}}{\sigma }_{yy} > 0\) is observed from 0 eV to 4.45 eV along with a significant suppression of energy loss within this range. To investigate the origin of the hyperbolic response, we separate the optical conductivity into two distinct components9:

$${\sigma }_{jj}(\omega )=\frac{i}{\pi }\frac{{D}_{jj}}{(\omega +i\varGamma )}+\frac{i}{\pi }\frac{\omega {S}_{jj}}{{\omega }^{2}-{\omega }_{b}^{2}+i\omega \eta }$$
(7)
Fig. 4: The conductivity spectra of RuOCl2 monolayer.
Fig. 4: The conductivity spectra of RuOCl2 monolayer.
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a The real and imaginary components of the optical conductivity. The inset shows a partial enlargement of the band structure with the blue arrow marking the onset energy of interband transitions. The contribution of b interband transitions and c intraband transitions of electrons to the conductivity.

The first term represents the intraband contributions, in which Djj is the Drude weight. The second term is interband contributions. Sjj is the spectral weight and \({\omega }_{b}\) is the frequency of interband transitions. Γ and η are the damping rates for intraband and interband transitions, respectively. The decomposed optical conductivities of RuOCl2 monolayer are presented in Fig. 4b, c. The effect of Q1DEG behavior of RuOCl2 monolayer on \(\sigma (\omega )\) is clear, where the intraband transitions dominate the \({\sigma }_{yy}(\omega )\) at low energy region, but have no contribution to the \({\sigma }_{xx}(\omega )\). When the energy reaches around 2 eV, the interband transitions begin to contribute to \({\sigma }_{yy}\), resulting in a weak broad absorption peak in the y direction, as shown in Fig. 4c. Moreover, the energy loss due to light adsorption that is determined by the real part of conductivity (\({{{{\mathrm{Re}}}}}{\sigma }_{xx}\) and \({{{{\mathrm{Re}}}}}{\sigma }_{yy}\)) is quite low in the energy range of 0–3.0 eV, which is promising for supporting hyperbolic plasmons in this region.

Lastly, we examine the directionality of surface plasmons on the RuOCl2 monolayer, which emerges from the coupling between the Q1DEG within the monolayer and the electromagnetic field of the incident light. Taking into account only the eigenmodes confined to monolayer (x–y plane), specifically \({e}^{i({q}_{x}x+{q}_{y}y)}{e}^{-pz}\) (for z > 0) and \({e}^{i({q}_{x}x+{q}_{y}y)}{e}^{pz}\) for (z < 0), we can derive the dispersion of the surface plasmons20,

$$({q}_{x}^{2}-{k}_{0}^{2}){\sigma }_{xx}+({q}_{y}^{2}-{k}_{0}^{2}){\sigma }_{yy}=2ip\omega ({\varepsilon }_{0}+\frac{{\mu }_{0}{\sigma }_{xx}{\sigma }_{yy}}{4})$$
(8)

In this expression, \({\varepsilon }_{0}\), \({\mu }_{0}\) and \({k}_{0}=\omega \sqrt{{\varepsilon }_{0}{\mu }_{0}}\) represent the permittivity, permeability and wave number in vacuum, \(p=\sqrt{{q}_{x}^{2}+{q}_{y}^{2}-{k}_{0}^{2}}\). For \({{\mbox{Im}}}\sigma > > {{{{\mathrm{Re}}}}}\sigma\), the isofrequency curve of ω is approximately determined by the equation,

$$\frac{({q}_{x}^{2}/{k}_{0}^{2}-1)+\zeta ({q}_{y}^{2}/{k}_{0}^{2}-1)}{{({q}_{x}^{2}/{k}_{0}^{2}+{q}_{y}^{2}/{k}_{0}^{2}-1)}^{1/2}}=\gamma ,$$
(9)

with\(\gamma =2(\frac{{\varepsilon }_{0}}{{\mu }_{0}Im{\sigma }_{xx}\times Im{\sigma }_{yy}}-\frac{1}{4})Im{\sigma }_{yy}{(\frac{{\varepsilon }_{0}}{{\mu }_{0}})}^{-1/2}\) and \(\zeta =Im{\sigma }_{yy}/Im{\sigma }_{xx}\). In the case of \(\zeta < 0\), Eq. (9) is reduced to hyperbola with the asymptotic equation of \({q}_{x}^{2}+\zeta {q}_{y}^{2}=0\) for \(|{q}_{x}| > > {k}_{0}\) and \(|{q}_{y}| > > {k}_{0}\). The group velocity of surface plasmons normal to the hyperbolic asymptotes determines the direction of surface plasmon beams. Therefore, the propagation direction of plasmon beams is described by \(y=\pm x{|\zeta |}^{1/2}\) with the angle of \(\varphi =\pm {\tan }^{-1}{|\zeta |}^{1/2}\) relative to the x-direction.

To illustrate the discussed concepts, we performed numerical simulations on the propagation of surface plasmons in a monolayer of RuOCl2. Our simulation considered a circular RuOCl2 monolayer with a radius of 100 nm and a thickness of 1 nm, surrounded by a vacuum. We excited surface waves by placing a z-directionally polarized dipole 1 nm above the sheet. To obtain the dispersion relation and spatial distribution of the surface plasmon electric field, we numerically solve Maxwell’s equation using a commercial finite-difference time-domain method. The conductivities of the RuOCl2 sheet used in our simulations were obtained from the first-principles calculations mentioned earlier. We select three frequencies, namely ω = 0.5, 1.5, and 2.0 eV within the hyperbolic regions. At these frequencies, the conductivity tensors were determined as follows: σxx = 0.0016–0.20i and σyy = 3.16 + 15.56i, σxx = 0.043–0.7i and σyy = 0.42 + 4.93i, σxx = 0.173–1.42i and σyy = 0.80 + 3.08i, respectively. The isofrequency contours corresponding to these frequencies obtained from Eq. (8) are depicted in Fig. 5a, which verify the hyperbolic nature of the surface plasmons. Figure 5b–d presents the electric field distribution obtained by solving Maxwell’s equations using the finite-element method (FEM). The spatial distribution of the electric field on the RuOCl2 sheet clearly demonstrates that the energy of the surface plasmons propagates as narrow beams, confirming the directional propagation features of the surface plasmons in the RuOCl2 sheet. Moreover, the angles of the surface plasmon beams relative to the x-direction, which was measured as 87.6°, 68.3°, and 48.7°, are consistent with the theoretical predictions based on \(\varphi =\pm {\tan }^{-1}{|\zeta |}^{1/2}\). The propagation lengths of hyperbolic plasmon modes can be determined by calculating the reciprocal of the imaginary part of the vector \({l}_{pl}=1/{{\mbox{Im}}}k\). For the three specific frequencies considered, the propagation lengths are ~3.6 μm, 0.88 μm, and 0.14 μm, respectively. These values are approximately 0.78, 1.87, and 0.61 times the plasmon wavelength, providing evidence of the substantial losses exhibited by plasmons. Nevertheless, such directional surface plasmons hold great promise in diverse fields, such as photonic circuitry, sensing, nanophotonic devices, offering opportunities for advancements in technology and scientific research.

Fig. 5: The isofrequency contours and the spatial electric field distribution.
Fig. 5: The isofrequency contours and the spatial electric field distribution.
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a The isofrequency contours of the surface plasmons on the RuOCl2 monolayer corresponding to the frequencies of 0.5, 1.5, and 2.0 eV. b–d The spatial distribution of electric field E of the surface plasmons excited by a z-directionally polarized dipole in a circular RuOCl2 monolayer with a radius of 100 nm and a thickness of 1 nm at \(\omega =\)0.5, 1.5, and 2.0 eV, respectively.

Conclusion

In summary, we introduce a model of strongly anisotropic 2D materials that emulate a Q1DEG system, capable of supporting highly directional plasmons that are immune to changes in carrier density (or Fermi level). Using first-principles calculations, we identify a promising candidate 2D material, the RuOCl2 monolayer, which can be easily stripped from bulk material, to realize this model. The RuOCl2 monolayer exhibits anisotropic electronic band structures with a metallic nature along the Γ−Y direction while being insulating along the Γ−X direction, in agreement with the Q1DEG model. This leads to distinct plasmons along the x- and y-directions. Plasmons along the y-direction have a frequency of up to 1.75 eV and exhibit minimal damping, while those along the x-direction are negligible. Furthermore, our first-principles calculations show that the RuOCl2 monolayer exhibits a broadband and low-loss hyperbolic interval due to Q1DEG feature. The methodology proposed in this study is also relevant for studying plasmon behavior in other Q1DEG systems with similar electronic characteristics, such as OsOCl2 monolayer45.

Furthermore, our research suggests that to achieve robustly anisotropic plasmons, as well as highly directional hyperbolic surface plasmons in anisotropic materials, it is essential for the electronic properties to demonstrate an ideal Q1DEG behavior. This behavior entails the material exhibiting metallic characteristics with highly dispersive electronic bands along one particular direction, while simultaneously displaying insulating properties with a band gap along the orthogonal direction. This distinct anisotropic electronic structure is pivotal in facilitating the unidirectional propagation of plasmons along the conducting axis. Furthermore, our objective is to attain a higher onset energy for interband transitions, as this parameter indicates that plasmons can attain higher frequencies without experiencing significant attenuation. This paves the way for the development of promising plasmonic applications and offers a promising platform for directional polariton devices.

Method

Linear response theory for plasmons

The plasmonic properties were studied using linear response theory with random phase approximation (RPA) which has been rigorously validated in various other 2D materials46,47,48,49. The frequency and wave-vector dependent polarization function for a 2D system are calculated within time-dependent density functional theory (TDDFT) formalism using the RPA50,51,

$$\varPi ({{{{{\boldsymbol{q}}}}}},\omega )=\frac{{g}_{s}}{{(2\pi )}^{2}}\mathop{\sum}\limits _{l,l^{\prime} }\int d{{{{{\boldsymbol{k}}}}}}\frac{f({E}_{{{{{{\boldsymbol{k}}}}}},l})-f({E}_{{{{{{\boldsymbol{k}}}}}}+{{{{{\boldsymbol{q}}}}}},l^{\prime} })}{\omega +{E}_{{{{{{\boldsymbol{k}}}}}},l}-{E}_{{{{{{\boldsymbol{k}}}}}}+{{{{{\boldsymbol{q}}}}}},l^{\prime} }+i\eta }{F}_{l,l^{\prime} }({{{{{\boldsymbol{k}}}}}},{{{{{\boldsymbol{q}}}}}}),$$
(10)

where gs = 2 is the spin degeneracy. The broadening parameter \(\eta\) is taken to be 0.05 eV. The Fermi-Dirac distribution function denoted by \(f(E)\) acts as a step function at \(T=0\). \({E}_{{{{{{\bf{k}}}}}}l}\) and \(|{{{{{\boldsymbol{k}}}}}},l\rangle\) are the eigenvalues and eigenstates of the ground state of Hamiltonian matrix, where \(l\) represents the band indices. The dynamic dielectric function \(\varepsilon ({{{{{\boldsymbol{q}}}}}},\omega )\) is determined from the polarization function using the equation:

$$\varepsilon ({{{{{\boldsymbol{q}}}}}},\omega )=1-{V}_{2D}\varPi ({{{{{\boldsymbol{q}}}}}},\omega )$$
(11)

where \({V}_{2D}=2\pi {e}^{2}/{\varepsilon }_{r}q\) is the Fourier transform of the Coulomb potential of 2D systems52 with \({\varepsilon }_{r}\) being the background dielectric constant. The EELS, calculated from the inverse of \(\varepsilon ({{{{{\boldsymbol{q}}}}}},\omega )\) using the expression

$$L({{{{{\boldsymbol{q}}}}}},\omega )=-{{\mbox{Im}}}(1/\varepsilon ({{{{{\boldsymbol{q}}}}}},\omega )).$$
(12)

The plasmons are indicated by the peaks or local maxima of the EELS.

First-principles calculations

Our first-principles calculations were conducted using density-functional theory, which was implemented through the Vienna ab simulation package (VASP)53 and GPAW codes54. Both codes employ the projected augmented-wave method to model interactions between electrons and ions55. The exchange-correlation function was treated self-consistently within the generalized gradient approximation (GGA) using the Perdew-Burke-Ernzerhof (PBE) functional56. The cutoff energy was set to 500 eV. To account for the strong Coulomb interaction between the partially filled 4d-shells of Ru, we employed the GGA + U method57 with the Hubbard interaction parameter of Ueff = 2 eV (where U = 2 eV and J = 0 eV). The VASP code was used to calculate the structure relaxation and electronic properties of the RuOCl2 monolayer with the k-point mesh of 14 × 13 × 1. A vacuum space of 20 Å was used along the z-direction to exclude the interaction between neighboring images. The lattice constants and the atomic positions were fully relaxed until the atomic forces on the atoms were below than 0.01 eV/Å and the total energy change was under 10−5 eV. The band structures of the RuOCl2 monolayer were constructed using the maximally localized Wannier functions were constructed using the WANNIER90 package58, enabling the acquisition of energy eigenvalues and eigenstates for subsequent polarization function calculations.

The in-plane optical conductivity tensor of the RuOCl2 monolayer was determined by calculating the density-density response function using the RPA within the GPAW code59. For improving accuracy, a denser k mesh of 40 × 39 × 1 was utilized for the monolayer RuOCl2 to converge the optical conductivity calculations, with up to 40 empty bands taken into account to describe the response function.