Gated 2D Plasmons and Tunable Resonances

• Gated 2D Plasmons are collective electron oscillations in 2D systems modulated by nearby metal gates that tune screening and carrier density.
• Gating introduces hybridization of plasmon modes, leading to quantized subbands, localized defect states, and enhanced near-field confinement.
• Electrostatic tuning via gate bias enables dynamic control of plasmon resonances, fostering advances in THz modulators and nano-optoelectronic devices.

Gated two-dimensional (2D) plasmons refer to collective oscillations of conduction electrons in 2D electron systems (2DES) subject to electrostatic gating, which modifies the electron density, screening, and boundaries in a spatially controlled manner. Ribbon, disk, finger, or grating-shaped metal gates introduce hybridization between distinct plasmonic modes, allow precise electrostatic tuning, facilitate coupling to external electromagnetic radiation in the THz to mid-IR regime, and provide access to phenomena not present in ungated systems—such as deep subwavelength field confinement, near-field enhancement, acousto-plasmonic effects, and plasmon instabilities. The interplay of gate geometry, local carrier density, boundary conditions, and external bias enables a rich set of phenomena essential to both plasmonic device engineering and fundamental condensed matter electrodynamics.

1. Fundamental Principles of Gated 2D Plasmons

A 2D electron system, such as a quantum well or graphene layer, supports plasmon modes whose electrostatic character and dispersion relations are highly sensitive to boundary conditions and screening. Applying a gate—typically a nearby metal electrode—screens the Coulomb interactions under the gate, modifies the carrier density when biased, and imposes distinct modal geometry (see, e.g., a periodic array of metal fingers on a 2DEG (Davoyan et al., 2011), back-gated graphene (Fei et al., 2012), or partial-gate disk geometry (Zabolotnykh et al., 2018, Zabolotnykh et al., 2021)).

• Infinite Gate (Fully Gated 2DES): Screening by a large top gate replaces the unscreened, square-root plasmon dispersion $\omega_p(q) = \sqrt{2\pi e^2 n q / (\kappa m)}$ with a linear, "acoustic" branch $\omega_g(q) = q V_p$, $V_p = \sqrt{4\pi e^2 n d / m\kappa}$.
• Partial or Patterned Gating: Finite-width gates or gratings create a spatial hybridization of "gated" (screened) and "ungated" (unscreened) plasmon segments (Zabolotnykh et al., 2018, Nikonov et al., 18 Jun 2024), giving rise to quantized subbands, hybrid "near-gate" modes, and analytically tractable solutions for modal frequencies and field localization.
• Electrostatic Tuning: Biasing the gate modifies the local carrier density, allowing the tunable shift of plasmon resonance frequency at fixed wavenumber (central in graphene (Fei et al., 2012, Jin et al., 2013), quantum wells, and hybrid plasmonic THz devices (Davoyan et al., 2011, Sai et al., 2023)).

This gating control is the backbone for dynamic modulation, field enhancement, and modal engineering in 2D plasmonic systems.

2. Modal Structure and Dispersion Engineering

The presence and geometry of a gate fundamentally alter the modal structure of plasmons in a 2DES:

• Band Structures in Periodic Gating:
  • Periodic stripes or gratings create 1D plasmonic crystals. The modal spectrum folds into Brillouin zones, resulting in bands and gaps determined by Kronig-Penney-type relations (Dyer et al., 2016, Nikonov et al., 18 Jun 2024, Sai et al., 2023). For example, for a grating with gated width $W$ and period $a$, the matching of solutions under and between gates with different screening leads to bulk plasmonic bands.
• Localized States—Defect and Tamm Plasmons:
  • Gates with intentionally different widths (e.g., a single "defect" finger (Davoyan et al., 2011), an edge stripe in a grating (Nikonov et al., 18 Jun 2024), or a disk-shaped top gate (Zabolotnykh et al., 2021)) induce localized modes either within spectral gaps (Tamm plasmons) or as Fano-like quasi-bound states coupled to the continuum.
• Fundamental Gapless and Hybrid Modes:
  • In finite strip gating, a new $N=0$ mode appears which is gapless and exhibits a hybridized square-root dispersion, with the charge oscillation and fields concentrated near the gate edge (Zabolotnykh et al., 2018). The corresponding mathematical form arises from boundary value solutions to the inhomogeneous Poisson equation with appropriate matching at the gate edge.

The modal landscape thus comprises:

• Gated (acoustic) modes;
• Ungated (square-root) modes;
• Hybrid edge/near-gate modes;
• Tamm/defect states in bandgaps.

3. Near-field Concentration and Field Enhancement

Gated 2D plasmon systems enable deep subwavelength confinement and strong near-field enhancement, surpassing free-space field amplitudes by orders of magnitude. The key mechanisms are:

• Cavity Pumping by Lattice Plasmons: In a periodically gated 2DEG with a centrally biased gate (defect), the delocalized lattice plasmons can resonantly drive the cavity modes, leading to deep-subwavelength concentration and enhancement factors $|E/E_0| \sim 10^4$ (Davoyan et al., 2011). The excitation factor

$$F = \frac{1}{2a}\,\frac{\Re \int_{\text{defect}} \sigma(\omega) |E_x|^2 dx}{P_\text{in}}$$

quantifies local absorption and field build-up.

• Edge Launching/Field "Hot Spots": At the boundary of a gate (e.g., metal stripe edge on 2DEG), the incident p-polarized electromagnetic field is strongly enhanced due to the singularity of the screening field, leading to efficient plasmon launching (Moiseenko et al., 7 May 2025). The magnitude of the launched field can exceed the incoming wave, maximized for optimal ratios of 2DES reactance and gate-channel separation.
• Van der Waals Heterostructure Resonances: In finite disk top gates over graphene or other 2D layers, "near-gate" modes form quantized quasi-bound states whose spectral position and lifetime are tunable through geometry and external fields. They manifest as Fano-type resonances in scattering cross-sections and are directly accessible via near-field imaging (Zabolotnykh et al., 2021).

Field control at the nanometric scale underpins applications in THz nonlinear optics and detector sensitivity.

4. Tunability and Electrically Induced Phase Transitions

Gate voltage and geometry provide exceptional tunability for 2D plasmons:

• Carrier Density Modulation: In both quantum wells and graphene, the plasmon frequency scales with local carrier density, $n$, as $\omega_p \propto \sqrt{n q}$ (ungated) or linearly in $q$ and $\sqrt{n}$ (gated) (Fei et al., 2012, Pisarra et al., 2013). Gate bias allows both amplitude and wavelength of the plasmon to be electrically controlled in real time.
• Plasmonic Crystal Phase Transition: In grating-gated heterostructures, continuous tuning of the modulation degree induces a transition between a delocalized phase (where both gated and ungated regions participate collectively) and a localized phase (where only the ungated regions support well-defined resonances) (Sai et al., 2023). The transition is smooth in the modulation parameter $\rho = (n_O-n_G)/(n_O+n_G)$.
• Edge Gating: Even after the depletion of carriers in a gated region, lateral depletion under the gate extends into the ungated region, causing "edge gating"—even ungated plasmon frequencies remain sensitive to the external bias (Sai et al., 2023), a critical consideration for device design.

This tunability extends the operational range for THz modulators and dynamic plasmonic circuits.

5. Hybridization, Coupling, and Instabilities

Plasmons in gated 2D systems can be engineered to exhibit strong coupling, hybridization, and even electrical instabilities:

• Mode Hybridization and Coherent Effects: Coupling between multiple voltage-tuned resonators or between extended and localized modes (e.g., defect modes, crystal edge/Tamm states) leads to coherent phenomena such as electromagnetically induced transparency (EIT) analogues and Fano resonances (Dyer et al., 2016, Zabolotnykh et al., 2021, Nikonov et al., 18 Jun 2024). The spectral splitting (vacuum Rabi analog, EIT dip) is determined by the coupling strength, losses, and device geometry.
• Plasmon Instabilities: In periodically gated plasmonic crystals under applied dc bias, amplified-reflection Dyakonov–Shur-style instabilities yield traveling plasmon modes with gain exceeding the collision-damping rate (Petrov et al., 2016). The threshold drift velocity can be tuned below the plasmon phase and carrier saturation velocities by geometric and dielectric engineering, opening prospects for THz emission.
• Relativistic Regime: For high-conductivity 2DES, electrodynamic (relativistic) corrections become nontrivial (when $2\pi\sigma_{2D} > c$), supporting robust, weakly damped, hybrid plasmon–photon modes surviving to room temperature (Muravev et al., 2015). The polaritonic nature and unusual magneto-dispersion are leveraged for terahertz device operation.

6. Experimental Manifestations and Applications

Gated 2D plasmons underpin a diverse set of applications and measurement techniques:

Mechanism	Main Application Domains	Example Papers
Field Focusing/Near-field Pump	THz nonlinear optoelectronics, detectors	(Davoyan et al., 2011)
Gate Tunable Propagation	IR/THz waveguides, optical switches, sensors	(Fei et al., 2012, Jin et al., 2013)
Diffraction/Edge Launching	Integrated THz sources, photon drag detection	(Moiseenko et al., 7 May 2025, Moiseenko et al., 18 Oct 2025)
Hybridization/Crystals	Planar THz metamaterials, slow-light devices	(Dyer et al., 2016, Nikonov et al., 18 Jun 2024)
Fano/Tamm/Near-gate Resonances	Spectroscopy, optical antennas, feedback design	(Zabolotnykh et al., 2021, Cheremisin, 2023)

Gate-defined plasmonic nanostructures enable reconfigurable, field-enhanced THz sources, detectors, subwavelength imaging, and quantum information architectures, as well as providing platforms for fundamental exploration of carrier dynamics, plasmon–phonon coupling, and non-local effects in the condensed matter context.

7. Mathematical Frameworks and Key Formulas

The theoretical analysis of gated 2D plasmons employs a suite of analytic and semi-analytic models:

• Poisson/Continuity + Local Conductivity: Starting from the Poisson equation and including the gate as boundary condition or source [e.g., $\nabla^2 \phi(x,z) +$ source terms], with current governed by a local Drude (or Dirac) conductivity.
• Dispersion Relations:
  • Ungated: $\omega_p(q) = \sqrt{2\pi e^2 n q / (\kappa m)}$
  • Gated (infinite gate): $\omega_g(q) = q V_p$, $V_p = \sqrt{4\pi e^2 n d / m\kappa}$
  • Finite gate hybrid: $$\omega_0(q_y) = \sqrt{(8\pi e^2 n d)/(m\kappa)\,|q_y|/L_x}$$ (Zabolotnykh et al., 2018)
• Boundary-Value Problems: For partial or patterned gates, one solves differential equations with matching conditions at gate edges or defects (example: Bessel function approach in disk geometry (Cheremisin, 2023, Zabolotnykh et al., 2021)).
• Scattering Solutions: For diffraction and launching at partial gate edges, the Wiener–Hopf technique yields exact field and amplitude expressions, including the enhancement at $k_0 z_0 \ll 1$ and the separation-based impedance matching (Moiseenko et al., 7 May 2025).
• Non-trivial Angular Dependence: Reflection/transmission coefficients for plasmonic refraction at interfaces exhibit nontrivial Brewster-type no-reflection conditions and nontrivial phases (Svintsov et al., 2023).

These frameworks permit both rigorous spectral calculations and device-oriented performance predictions.

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Gated 2D plasmons thus encapsulate an interplay between electron confinement, electromagnetic screening, gate-induced spatial potential engineering, and external field control. They serve as a foundation for tunable, high-field, and deeply subwavelength phenomena essential for integrated THz photonics, sensing, and future nanoscale quantum devices. The field is marked by analytic tractability enabling direct connection between theoretical models and experimental data, as well as robust applicability across a wide range of material platforms and operating frequencies.