**A general formula of 2D anisotropic plasmon dispersion.** Plasmon dispersion is determined by the zeros of the dielectric function \(\varepsilon ({\text{q}},\omega )\). Assuming the intraband transitions of electrons dominate the dielectric function, we have \({F_{ll^{\prime}}}({{k}},{{q}})={\delta _{ll^{\prime}}}\). In this case, the polarization function \(\Pi ({{q}},\omega )\) can be written in the Linhard expression,

$$\Pi ({{q}},\omega )=\frac{{{g_s}}}{{{{(2\pi )}^2}}}\sum\limits_{l} {\int d } {{k}}\frac{{f({E_{{{k}},l}}) - f({E_{{{k}}+{{q}},l}})}}{{\omega +{E_{{{k}},l}} - {E_{{{k}}+{{q}},l}}}}$$

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

$$\Pi ({{q}},\omega ) \approx \frac{{{g_s}}}{{{{(2\pi )}^2}}}\sum\limits_{l} {\int d } {{k}}\frac{{{\nabla _{{k}}}{E_{{{k}},l}} \cdot {\text{ }}{{q}}}}{{\omega - {\text{ }}{\nabla _{{k}}}{E_{{{k}},l}} \cdot {\text{ }}{{q}}}}\left( { - \frac{{\partial f}}{{\partial E}}} \right)$$

2

Replacing \({\nabla _{{k}}}{E_{{{k}},l}}\) with electron velocity, \({{\upsilon}_{{{k}},l}}={\nabla _{{k}}}{E_{{{k}},l}}={\upsilon _x}{{i}}+{\upsilon _y}{{j}}\) and taking T = 0 K, we have,

$$\Pi ({{q}},\omega ) \approx \frac{{{g_s}}}{{{{(2\pi )}^2}}}\sum\limits_{l} {\int {d{{k}}\frac{{{{\upsilon}_{{{k}},l}} \cdot {\text{ }}{{q}}}}{\omega }\left( {1+\frac{{{{\upsilon}_{{{k}},l}} \cdot {\text{ }}{{q}}}}{\omega }} \right)\delta ({{\rm E}_{{{k}},l}} - {E_F})} }$$

3,

where *E**F* is the Fermi level. Defining the electron density of states (DOS) at the Fermi level \(\rho ({E_F})=\frac{{{g_s}}}{{{{(2\pi )}^2}}}\sum\limits_{l} {\int {{d^2}{{k}}} } \delta ({E_{{\text{k}},l}} - {E_F})\) and the averaged square Fermi velocities \(\left\langle {\upsilon _{{x,y}}^{2}} \right\rangle =\sum\limits_{l} {\int {d{{k}}v_{{x,y}}^{2}} } \delta ({E_{{{k}},l}} - {E_F})/\sum\limits_{l} {\int {{d^2}{{k}}} } \delta ({E_{{{k}},l}} - {E_F})\), we have a simple formula of the polarization function,

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

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

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with\({\omega _{x/y}}(q)={\alpha _{x/y}}{q^{1/2}}{\text{ }}\), \({\alpha _x}={\text{ }}{\left( {2\pi {e^2}\rho ({E_F})\left\langle {\upsilon _{x}^{2}} \right\rangle /{\varepsilon _r}} \right)^{1/2}}\) and \({\alpha _y}={\text{ }}{\left( {2\pi {e^2}\rho ({E_F})\left\langle {\upsilon _{y}^{2}} \right\rangle /{\varepsilon _r}} \right)^{1/2}}\). The anisotropy of the plasmon dispersion can be featured by the ratio of \({\alpha _y}/{\alpha _x}=\sqrt {\left\langle {\upsilon _{y}^{2}} \right\rangle /\left\langle {\upsilon _{x}^{2}} \right\rangle }\). \({\alpha _x}={\alpha _y}\)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.

**Anisotropic plasmon dispersion of quasi-1DEG systems.** The linear energy-momentum dispersion (also named as Dirac cones) in the electronic band structure of graphene leads to unusual electron transport properties 25. In this work, we consider a highly-anisotropic 2D lattice with a linear energy-momentum relation along one direction (taken as *y-*direction), \(E({{k}})=\hbar {\upsilon _F}|{k_y}| - {E_F}\), 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 form 26, 27. 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

$$\omega ({{q}})={\alpha _y}|\sin \theta |{q^{1/2}}$$

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with \({\alpha _y}={\left( {{g_s}{e^2}{\upsilon _F}G/{\varepsilon _r}\pi } \right)^{1/2}}\) and \({\alpha _x}=0\), where *G* represents the length of the first Brillouin zone along the *y*-direction. Eq. (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** // *k**x* (*θ* = 0) the plasmon mode is completely forbidden, while for **q** // *k**y* (*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 metamaterials 17, 28. Additionally, one intriguing properties of plasmons in Q1DEG systems is that according to Eq. (6), their frequency\(\omega\)remains constant regardless of carrier density *n* or Fermi level *E**F*. This is in contrast to the behavior of plasmons in conventional EG systems in all dimensions of \(\omega \sim {n^{1/2}}\) 8, 29, as shown in Fig. 1d.

**Plasmon dispersion of RuOCl** **2** **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 synthesized 30, 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) 31 and phosphorene (23.0 meV/Å2) 32 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. Each Ru atom is coordinated by two O atoms and four Cl atoms with the RuO 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. The electron localization function (ELF) 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 and 2d, 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 the Fermi level, in consistent with the above model of Q1DEG. The electronic states near the Fermi level arise mainly from *d**xy* and *d**yz* orbitals of Ru atoms, with minor contribution from *p**x* and *p**z* 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. Moreover, the electron density of states (DOS) of RuOCl2 monolayer shown in Fig. 2d are 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, in 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 up to 7.5×105 m/s. This value is comparable to the Fermi velocity of graphene (~ 8.5×105 m/s) 33, 34 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 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 *q*1/2, which is consistent with Eq. (5) and similar to the plasmon behavior in conventional 2DEG 35. As *q* increases, the plasmon dispersion gradually deviates from *q*1/2 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 the energy up to 1.75 eV. 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 \({\alpha _y}\)= 12.06 eV Å1/2 and \({\alpha _x}\)= 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 \(m_{x}^{*}/m_{y}^{*} \approx 1764\), which is considerably larger than those observed in other 2D anisotropic plasmon materials 18, 36, 37, 38. The anisotropic plasmons also exhibit hyperbolic-like isofrequency curves, as depicted in Fig. 3b, in consistent with the Q1DEG model. This feature is remarkably distinct from conventional plasmons which are characterized by elliptical or circular isofrequency curves. Electromagnetic waves with hyperbolic isofrequency curves in hyperbolic mediums exhibit anomalous properties, including negative refraction, super-resolution imaging, and nonlinear optical effects. Consequently, the anisotropic plasmons in the RuOCl2 monolayer hold great promise for a range of applications.

Notably, 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, in 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 RuOCl** **2** **monolayer.** In 2D materials, the optical conductivity describes the dielectric properties 39, where the real part \(\operatorname{Re} \sigma\) characterizes the energy loss and the imaginary determines the hyperbolic interval through \(\operatorname{Im} [{\sigma _{xx}}(\omega )] \times \operatorname{Im} [{\sigma _{yy}}(\omega )]<0\). The optical conductivities of RuOCl2 monolayer along *x* and *y* directions are plotted in Fig. 4a. Notably, a broad hyperbolic regime with \(\operatorname{Im} {\sigma _{xx}}<0\) and \(\operatorname{Im} {\sigma _{yy}}>0\) 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 components7:

$${\sigma _{jj}}(\omega )=\frac{i}{\pi }\frac{{{D_{jj}}}}{{(\omega +i\Gamma )}}+\frac{i}{\pi }\frac{{\omega {S_{jj}}}}{{{\omega ^2} - \omega _{b}^{2}+i\omega \eta }}$$

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The first term represents the intraband contributions, in which \({D_{jj}}\) is the Drude weight. The second term is interband contributions. *S**jj* 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 and 4c. 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. Notably, the energy loss due to light adsorption that is determined by the real part of conductivity (\(\operatorname{Re} {\sigma _{xx}}\) and\(\operatorname{Re} {\sigma _{yy}}\) ) 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 (xy 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 plasmons 20,

$$\left( {q_{x}^{2} - k_{0}^{2}} \right){\sigma _{xx}}+\left( {q_{y}^{2} - k_{0}^{2}} \right){\sigma _{yy}}=2ip\omega \left( {{\varepsilon _0}+\frac{{{\mu _0}{\sigma _{xx}}{\sigma _{yy}}}}{4}} \right)$$

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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 \(\operatorname{Im} \sigma >>\operatorname{Re} \sigma\), the isofrequency curve of \(\omega\) is approximately determined by the equation,

$$\frac{{\left( {q_{x}^{2}/k_{0}^{2} - 1} \right)+\zeta \left( {q_{y}^{2}/k_{0}^{2} - 1} \right)}}{{{{\left( {q_{x}^{2}/k_{0}^{2}+q_{y}^{2}/k_{0}^{2} - 1} \right)}^{1/2}}}}=\gamma$$

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with\(\gamma =2\left( {\frac{{{\varepsilon _0}}}{{{\mu _0}\text{I}\text{m}{\sigma _{xx}} \times \text{I}\text{m}{\sigma _{yy}}}} - \frac{1}{4}} \right)\text{I}\text{m}{\sigma _{yy}}{\left( {\frac{{{\varepsilon _0}}}{{{\mu _0}}}} \right)^{ - 1/2}}\) and \(\zeta =\text{I}\text{m}{\sigma _{yy}}/\text{I}\text{m}{\sigma _{xx}}\). 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 \(\left| {{q_x}} \right|>>{k_0}\) and \(\left| {{q_y}} \right|>>{k_0}\). 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 \(y= \pm x{\left| \zeta \right|^{1/2}}\)with the angle of \(\varphi = \pm {\tan ^{ - 1}}{\left| \zeta \right|^{1/2}}\) 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 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 surface plasmon electric field, we numerically solve the 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: *σ**xx* = 0.0016–0.20i and *σ**yy* = 3.16 + 15.56i, *σ**xx* = 0.043–0.7i and *σ**yy* = 0.42 + 4.93i, *σ**xx* = 0.173–1.42i and *σ**yy* = 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. Figures 5b to 5d present 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 were measured as 87.6°, 68.3° and 48.7°, are consistent with the theoretical predictions based on \(\varphi = \pm {\tan ^{ - 1}}{\left| \zeta \right|^{1/2}}\). 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.