Ultra‑Compact Terahertz Plasmonic Waveguide with Enhanced Confinement Using a Bulk Dirac Semimetal‑Insulator‑Metal Structure

Abstract

A subwavelength terahertz plasmonic waveguide built on a bulk Dirac semimetal (BDS)–insulator–metal (BIM) stack is investigated. The structure delivers an optimized frequency band where the surface plasmon polariton (SPP) mode is tightly confined—up to a confinement factor of ½·λ0/15—while the propagation loss remains low at 1.0 dB/λ0. Introducing two silicon ribbons into the BIM waveguide creates a dynamically tunable Fabry–Pérot resonator that can shape terahertz SPPs in a deep‑subwavelength scale, offering a pathway toward ultra‑compact, tunable THz plasmonic devices and optical filters.

Background

Terahertz (THz) radiation has emerged as a versatile tool in imaging, biochemical sensing, and wireless communications – demanding ever tighter confinement, higher resolution, and greater integration – to harness its full potential – [1,2,3]. Achieving subwavelength confinement of THz waves is therefore critical, yet conventional surface plasmon polaritons (SPPs) in noble metals suffer from poor confinement, high intrinsic loss, and passive tunability – limitations that have hampered practical applications – [4,5,6].

Graphene plasmons have shown promise due to their low loss, dynamic tunability, and extreme confinement, enabling high‑resolution THz devices – examples include gate‑tunable graphene heterostructures, discrete Talbot effects, and plasmon‑induced transparency waveguides – [11–18]. These advances underline the pivotal role of extreme SPP confinement for manipulating THz radiation at deep‑subwavelength scales.

Bulk Dirac semimetals (BDS), often dubbed “3‑D graphene”, exhibit ultrahigh carrier mobilities (up to 9×10⁶ cm² V⁻¹ s⁻¹) that far exceed the best graphene (2×10⁵ cm² V⁻¹ s⁻¹) – a key advantage for low‑loss plasmons – [19]. Their dielectric response is also tunable via Fermi‑energy modulation, and materials such as Na₃Bi, Cd₃As₂, and AlCuFe quasicrystals are easier to process and more stable than graphene – positioning BDS as a promising next‑generation plasmonic material – [20,21]. However, the SPP mode confinement at a BDS‑insulator interface alone is still insufficient for deep‑subwavelength manipulation.

In this study, we present a BIM waveguide that markedly improves SPP confinement while maintaining low loss and enabling dynamic tunability. The dispersion, loss characteristics, and filtering capabilities of this highly confined mode are systematically investigated, revealing an optimized frequency window with both enhanced confinement and reduced propagation loss – a behavior rarely observed in traditional metal‑based SPPs. Moreover, the BIM mode can propagate through slits narrower than λ₀/2000, and by embedding silicon ribbons as mirrors, we realize a tunable band‑pass filter whose resonance can be shifted by adjusting the BDS Fermi energy.

Theory and Simulation

The BIM waveguide consists of a monolayer BDS film (thickness 0.2 µm) separated from a silver substrate by a dielectric spacer (permittivity εᵣ). In the THz regime, the silver behaves as a perfect electric conductor. For TM‑polarized excitation, the SPP mode propagates along x with wavevector k_SPP and decays along y into free space. Applying boundary conditions yields the dispersion relation:

$$-\frac{\varepsilon_r\sqrt{k_{\mathrm{SPP}}^2-k_0^2}}{\varepsilon_0\sqrt{k_{\mathrm{SPP}}^2-\frac{\varepsilon_r k_0^2}{\varepsilon_0}}}=\left(1+\frac{i\sigma \sqrt{k_{\mathrm{SPP}}^2-k_0^2}}{{\omega \varepsilon}_0}\right)\tanh\left(g\sqrt{k_{\mathrm{SPP}}^2-\frac{\varepsilon_r k_0^2}{\varepsilon_0}}\right),$$

where k₀ is the free‑space wavevector and σ is the complex conductivity of the BDS film (Eq. 3–4). Solving Eq. (1) gives the effective refractive index n_eff = k_SPP/k₀ = Re(n_eff) + i Im(n_eff). Re(n_eff) quantifies mode confinement, while Im(n_eff) directly relates to propagation loss.

For sufficiently large gaps (tanh[…]≈1), Eq. (1) simplifies to the single‑layer BDS dispersion (Eq. 2), highlighting the added benefit of the metal substrate in the BIM configuration.

Schematic of the BIM plasmonic waveguide: a monolayer BDS film is positioned at a gap g from a silver substrate, separated by a dielectric spacer with permittivity εᵣ. The TM‑polarized SPP mode propagates along x and decays along y. The Eₓ field distribution is shown by the red line.

Results and Discussion

We first examine how the BIM gap g and BDS Fermi energy E_F affect mode confinement and loss. With E_F=70 meV, the effective refractive indices for various g values are plotted in Fig. 2a (Re(n_eff)) and Fig. 2b (Im(n_eff)). For g≥10 µm, the curves converge above 0.05 THz, indicating that the SPP field is already confined within ~10 µm and the silver contribution is negligible. Reducing g below 1 µm markedly increases confinement and reduces loss, yielding an optimized frequency band where confinement peaks while loss diminishes—a feature uncommon in conventional metal‑based SPPs.

Figure 2c,d illustrate the dependence on E_F (g=1 µm). Increasing E_F lowers both confinement and loss, attributable to enhanced metallicity and longer carrier relaxation times. For instance, at 2.5 THz, a confinement of λ₀/15 and a loss of 1.0 dB/λ₀ are achieved with g=10 nm and E_F=70 meV.

Re(n_eff) and Im(n_eff) versus (a) gap g and (b) Fermi energy E_F for E_F=70 meV (g variable) and g=1 µm (E_F variable).

Transmission simulations (Fig. 3) confirm the reduced loss: at 1.56 THz, the BIM waveguide exhibits a transmission of 0.97, surpassing the monolayer BDS case. The corresponding magnetic field distributions (Fig. 3b,c) show a higher Re(n_eff)=2.45 for BIM versus 1.002 for monolayer, with the field tightly localized within slits <λ₀/2000.

Transmission spectra (a) and magnetic field distributions (b,c) for BIM (red) and monolayer (blue) waveguides at E_F=70 meV, g=50 µm, and 1.56 THz.

To harness the highly confined mode, we integrate two silicon ribbons (n_Si=3.4) into the dielectric spacer, forming reflective mirrors (Fig. 4a). The resulting Fabry–Pérot cavity supports standing‑wave resonances, yielding two transmission peaks at 1.56 and 2.22 THz with FWHM of 0.12 and 0.09 THz, respectively. Magnetic field maps (Fig. 4c,e) confirm first‑ and second‑order cavity modes. Outside resonance, standing waves cannot form, and the input is reflected (Fig. 4d).

The transmission can be analytically described by coupled‑mode theory (CMT):

$$T(\omega)=\frac{\kappa_w^2}{(\omega-\omega_0)^2-(\kappa_w+\kappa_i)^2},$$

where κ_w and κ_i are the waveguide coupling and intrinsic loss rates. The quality factors Q_t=ω₀/FWHM and Q_oi=−Re(n_eff)/(2Im(n_eff)) enable extraction of κ_w and κ_i, with the analytical results matching the numerical spectra (Fig. 4b).

(a) BIM waveguide with silicon ribbons (width d, separation L). (b) Numerical (blue) and CMT‑fitted (red) transmission spectra for g=1 µm, d=5 µm, L=120 µm. (c–e) Magnetic field distributions at 1.56, 1.90, and 2.22 THz.

Figure 5 explores the dependence of resonant frequency on cavity length L. The first and second resonances red‑shift with increasing L, consistent with the standing‑wave condition 2k_SPP(ω_r)L+θ=2mπ. Similarly, increasing g blue‑shifts the resonances (Fig. 6a) due to reduced Re(n_eff), while increasing E_F also blue‑shifts the peaks (Fig. 6b) and narrows the bandwidth, reflecting lower Im(n_eff). These tunability mechanisms allow dynamic control of the filter without redesigning the structure.

(a) Transmission spectra for varying L. (b) Resonant frequencies of modes 1 and 2 versus L (g=1 µm, d=5 µm, E_F=70 meV).

Transmission spectra versus (a) g and (b) E_F (other parameters fixed). (c) Re(n_eff) vs. E_F and g. (d) Im(n_eff) vs. E_F and g.

Conclusions

The BIM waveguide delivers an ultra‑compact terahertz plasmonic mode with λ₀/15 confinement and 1.0 dB/λ₀ loss, achievable in slits narrower than λ₀/2000. By incorporating silicon ribbons, we realize a dynamically tunable band‑pass filter whose resonant frequency shifts with the BDS Fermi energy, enabling THz switching and filtering without structural re‑optimization.

Methods

Numerical simulations employed a 2‑D finite‑difference time‑domain (FDTD) solver with perfectly matched layers in the x and y directions. The BDS mesh was set to 1 µm × 0.02 µm for convergence. The frequency‑dependent conductivity of BDS follows the Kubo formula under the random phase approximation – Eqs. (3) and (4) describe Reσ(Ω) and Imσ(Ω). Parameters for AlCuFe were: t=40, ε_c=3, μ=3×10⁴ cm² V⁻¹ s⁻¹, and E_F=70 meV.

No human participants, data, or animals were involved in this research.

Abbreviations

BDS:

Bulk Dirac semimetals

BIM:

BDS‑insulator‑metal

CMT:

Coupled mode theory

FDTD:

Finite‑difference time‑domain

FWHM:

Full width at half maximum

SPPs:

Surface plasmon polaritons