Hyperbolic Plasmons in Anisotropic Media

Updated 4 August 2025

Hyperbolic plasmons are electromagnetic excitations whose hyperbolic dispersion arises from strong anisotropy in the permittivity or conductivity tensor, enabling unbounded in-plane momentum.
They leverage both natural and engineered materials—such as metamaterials, van der Waals systems, and topological semimetals—to achieve extreme spatial confinement and directional energy transport.
Their tunability via factors like gating, strain, and patterning underpins applications in sub-diffraction imaging, sensing, nanophotonics, and quantum light modulation.

Hyperbolic plasmons are electromagnetic collective excitations—plasmon polaritons—whose dispersion in momentum space becomes hyperbolic due to strong in-plane anisotropy of the optical response. This property emerges in natural and artificial systems where the permittivity or conductivity tensor exhibits principal components with opposite signs or large magnitude differences, enabling wave propagation with unbounded in-plane momenta, extreme spatial confinement, directional canalization, and new regimes of light–matter interaction. The concept of hyperbolic plasmons underpins a broad class of phenomena across metamaterials, van der Waals systems, semimetals, topological materials, engineered metasurfaces, and correlated metals, with implications for sub-diffractional optics, imaging, sensing, and quantum nanophotonics.

1. Physical Principles of Hyperbolic Plasmons

The fundamental prerequisite for hyperbolic plasmon behavior is a highly anisotropic dielectric (permittivity) or conductivity tensor in the relevant dimensions, such that the in-plane optical response can be represented as

$\epsilon = \begin{pmatrix} \epsilon_{xx} & 0 \ 0 & \epsilon_{yy} \end{pmatrix}$

or, equivalently in 2D systems, through a conductivity tensor

$\sigma = \begin{pmatrix} \sigma_{xx} & 0 \ 0 & \sigma_{yy} \end{pmatrix}$

The "hyperbolic regime" is realized when the real parts (or, for optical frequencies, the imaginary parts relevant for losses) of these components have opposite signs, i.e.,

$\mathrm{Im}[\sigma_{xx}(\omega)] \cdot \mathrm{Im}[\sigma_{yy}(\omega)] < 0$

or, in bulk:

$\mathrm{Re}[\epsilon_{xx}(\omega)] \cdot \mathrm{Re}[\epsilon_{yy}(\omega)] < 0$

Under these conditions, the isofrequency surface for the plasmonic (or polaritonic) modes in the $\bf{k}$ -space—the set of wavevectors $\bf{k}$ for which a given frequency $\omega$ is allowed—takes the form of a hyperbola rather than an ellipse. For a typical two-dimensional sheet, the long-wavelength plasmon dispersion is modified to:

$q_y = \pm q_x \sqrt{|\mathrm{Im}[\sigma_{xx}]/\mathrm{Im}[\sigma_{yy}]|}$

(Nemilentsau et al., 2015, Espinosa-Champo et al., 30 Jul 2025, Yin et al., 2020, Gangaraj et al., 2015)

The group velocity, being perpendicular to the isofrequency contour, channels energy along directions set by the optical anisotropy.

2. Hyperbolic Plasmons in Metamaterials and Multilayers

Artificial hyperbolic metamaterials (HMMs) are engineered composites such that their effective permittivities are of opposite sign along orthogonal axes. The canonical HMM consists of alternating metal and dielectric layers (or metallic nanorods in a dielectric host):

$\epsilon = \mathrm{diag}(\epsilon_1, \epsilon_2, \epsilon_2)$

with $\epsilon_1 < 0$ , $\epsilon_2 > 0$ (Type I) or vice versa (Type II).

HMMs exhibit a continuum of bulk high- $k$ modes—volume plasmon polaritons (VPPs)—arising from the hybridization of elementary excitations localized in each unit cell (e.g., short-range SPPs in metallic layers). The existence criterion for a broad VPP (i.e., hyperbolic plasmon) band is quantified via a normalized pole parameter $|\xi|\gtrsim 0.1$ –$1$, where $\xi$ describes the resonance strength in the unit cell’s scattering response (Zhukovsky et al., 2014). The Bloch condition for the metamaterial stack becomes:

$\cos(k_B D) = \frac{1}{2}\left[(\beta-1) A(\beta) - (\beta + 1) A^{-1}(\beta)\right],\qquad \beta = \kappa / \kappa_p$

where $A(\beta)$ encodes the unit cell properties and $D$ is the period.

In graphene–dielectric multilayers, analogous hybridization of high-Im( $\sigma$ ) transverse magnetic (TM) modes produces broadband hyperbolic bands, with gate-tunable frequency ranges (Zhukovsky et al., 2014). In contrast, the long-range SPPs and transverse electric (TE) plasmons do not give rise to robust hyperbolic behavior, as their pole parameter $\xi\to 0$ restricts the high- $k$ bandwidth.

3. Surface and Interface Hyperbolic Plasmons

Surface states such as surface plasmon polaritons (SPPs) at metal–dielectric or uniaxial–isotropic interfaces exhibit additional richness. With uniaxial anisotropy, the SPP dispersion can become hyperbolic, supporting Dyakonov plasmons—hybrid modes with mixed TE/TM polarization, whose isofrequency contours are hyperbolae as opposed to the circles of conventional SPPs (Takayama et al., 2015). Their existence and propagation direction are determined by eigenvalue equations that depend on the permittivities and propagation angle. For certain signs of the metrics $(\epsilon_o, \epsilon_e \mid \epsilon_c)$ , two entirely new classes of hyperbolic surface waves exist—distinguished from Dyakonov plasmons—which do not require a hyperbolic bulk.

Directional propagation and strong field localization—manifested in diverging effective refractive index at cut-off angles—enable steering, routing, and switching of surface waves via material parameter control (Takayama et al., 2015). In 2D materials, such as graphene metasurfaces or black phosphorus, anisotropic optical conductivity gives rise to highly directive beam propagation and hybridized eigenmodes, with direct control over directionality, spatial confinement, and polarization via geometry, stacking, or gating (Gangaraj et al., 2015, Veen et al., 2018, Nemilentsau et al., 2015).

4. Realization in Natural Anisotropic Crystals and Correlated Metals

In several van der Waals, topological, and semimetallic materials, hyperbolic plasmons arise due to natural in-plane anisotropy and distinct band topology. Black phosphorus, with pronounced armchair–zigzag symmetry breaking, hosts broad-band hyperbolic regions whose plasmonic character, spectral window, and damping are tunable via thickness, strain, carrier doping, and optical gain (Veen et al., 2018). Borophene phases with tilted Dirac cones combine intrinsic high carrier density and strong anisotropy, supporting visible- and mid-IR hyperbolic plasmon polaritons with low losses and highly tunable confinement (Torbatian et al., 2021).

In correlated metals such as MoOCl $_2$ , orbital-selective Peierls phases and Fermi surface reconstructions yield extreme dielectric anisotropy, but collective hyperbolic plasmon polaritons (HPPs) persist and even become long-lived despite strong incoherence and renormalized plasma frequency (Ruta et al., 9 Jun 2024). The broad, low-loss HPP bands reflect combined effects of orbital selectivity, strong electronic interactions, and band-structure engineered hyperbolicity—a scenario similarly identified in nodal-line systems such as ZrSiSe, where van Hove singularities suppress interband losses and boost the plasmonic response, yielding bulk-guided hyperbolic wave propagation (Shao et al., 2022).

5. Nonlocal and Moiré Effects, Topological Transitions, and Meta-Gratings

Hyperbolic plasmon phenomena are further enriched in composite, moiré, and nonlocal systems:

At high-quality metal–dielectric interfaces, nonlocal electron dynamics produce a thin "hyperbolic layer" that supports hyper-plasmons: surface waves with both strong field confinement and long propagation length, existing even beyond the conventional plasmon resonance. The "hyperbolic blockade" can suppress traditional SPPs while enabling new broadband surface modes (Narimanov, 2017).
Moiré physics arises when two hyperbolic metasurfaces (e.g., crossed graphene nanoribbon arrays), rotated relative to one another, produce topological transitions from hyperbolic to elliptical band structures at magic angles governed by the angle between the principal axes. Field canalization, anti-crossing points, and plasmon spin-Hall effects—chirality-dependent, unidirectional propagation controllable by polarization—can be engineered by adjusting the relative orientation and spacing (Hu et al., 2020).
In hyperbolic meta-gratings, a plasmonic diffraction grating atop a type II HMM enables phase-matching and far-field excitation of high- $k$ Bloch plasmon polariton modes. The grating injects the necessary in-plane momentum, overcoming the mismatch and facilitating tunable, strongly confined, and slow-light plasmonic bands, validated by spectroscopic and near-field imaging (Maccaferri et al., 2020).

6. Tunability, Applications, and Probing Electronic Order

Hyperbolic plasmons are inherently tunable via both extrinsic and intrinsic means, with the plasmonic bandwidth, directionality, and confinement engineered by:

Gate doping (varies carrier concentration and thus optical anisotropy in 2D materials) (Nemilentsau et al., 2015, Jiang et al., 2018)
Mechanical strain and stacking (in black phosphorus, borophene, ReS $_2$ ) (Veen et al., 2018, Torbatian et al., 2021, Kiran et al., 2023)
Patterning (holey graphene, periodic metasurface design) (Espinosa-Champo et al., 30 Jul 2025, Yin et al., 2020)
Population inversion or optical gain (hyperbolic regions in BP and related systems) (Veen et al., 2018)

Applications include:

Nano-imaging and super-resolution: Hyperlenses composed of HMMs or metasurfaces transfer evanescent, subwavelength details to the far field via high- $k$ hyperbolic modes (Guo et al., 2012, Yin et al., 2020).
Sensing: High photonic density of states and strong field enhancement increase sensitivity to refractive index and environmental changes (Maccaferri et al., 2020, Shao et al., 2022).
Light emission and modulation: Hyperbolic substrates modify the radiative decay of emitters, boosting Purcell factors, enabling directional spontaneous emission, and enhancing or suppressing emission rates (Guo et al., 2012, Veen et al., 2018).
Integrated photonics and on-chip routing: Broadband, low-loss, and highly directional propagation benefits nanophotonic circuitry, beam steering, channel multiplexing, and active device switching (Hu et al., 2020, Nemilentsau et al., 2015, Jiang et al., 2018).
Probing electronic correlations: HPPs in bad metals serve as self-referencing measures of the intraband dynamics, coherence, and Fermi surface properties (Ruta et al., 9 Jun 2024). Likewise, in topological and nodal-line semimetals, they probe key spectral features (Shao et al., 2022, Wang et al., 2021).

7. Theoretical Formalism and Design Rules

The hyperbolic plasmon regime is analyzed through a combination of:

Effective medium theory (for multilayers/artificials), yielding bulk or interface dispersion relations of the form

$\frac{k_x^2}{\epsilon_{yy}} + \frac{k_y^2}{\epsilon_{xx}} = \left(\frac{\omega}{c}\right)^2$

Green’s function and Sommerfeld integral approaches (for 2D anisotropic surfaces), where the pole structure and branch cuts in the complex plane dictate the SPP/Plasmon propagation (Gangaraj et al., 2015)
Analytical and numerical methods (Bloch waves, transfer/scattering matrices, many-body perturbation theory, kinetic equation with boundary conditions), tailored to both macroscopic and ab initio descriptions (Zhukovsky et al., 2014, Narimanov, 2017, Torbatian et al., 2021)
Isotropy criteria and propagation angle expressions—e.g., for 2D materials, the hyperbola’s opening angle (direction of energy propagation) is

$\theta = \arctan\sqrt{|\mathrm{Im}[\sigma_{yy}]/\mathrm{Im}[\sigma_{xx}]|}$

(Nemilentsau et al., 2015)

Design rules specify that to achieve broadband, strongly confined, and robust hyperbolic plasmonic behavior, one needs: (i) a strong resonance pole in the elementary excitation/unit cell, (ii) deeply subwavelength dielectric/mathematical cell sizes, (iii) minimal dissipative losses (high figure-of-merit Im( $\sigma$ )/Re( $\sigma$ )), and (iv) tunable anisotropy—potentially with dynamical control (Zhukovsky et al., 2014, Nemilentsau et al., 2015).

Hyperbolic plasmons unify themes from classical metamaterials to quantum-correlated and topological matter, offering a rich, versatile, and dynamically tunable platform for advancing nanophotonics, optoelectronics, and spectroscopy. Their paper intertwines fundamental condensed matter physics, advanced materials design, and the quest for probing and manipulating light–matter interactions at the ultimate subwavelength and quantum limits.