Predicting Surface Impedance of Metasurface–Graphene Hybrid Structures in the Terahertz Regime

Introduction

Artificial impedance metasurfaces have recently attracted significant attention due to their unique ability to manipulate electromagnetic waves, enabling applications such as holography, high‑resolution imaging, carpet cloaking, and broadband absorption. However, the inherently dispersive nature of conventional metasurfaces limits their performance to narrowband operation. Graphene, with its voltage‑tunable conductivity, offers a dynamic platform to overcome this limitation, allowing real‑time control of plasmonic and metasurface characteristics across the THz and optical spectra. Consequently, hybrid metasurface‑graphene structures have emerged as promising candidates for tunable, compact THz devices.

Analytical models for the equivalent surface impedance of both metasurfaces and graphene sheets have been developed in the literature, yet a unified model that accurately captures the interaction between a square‑patch metasurface and a graphene sheet remains elusive. In this study, we fill this gap by deriving a comprehensive analytical framework and validating it against full‑wave simulations.

Methods

Impedances for Square Patches and Graphene Sheets

The canonical metasurface‑graphene absorber consists of a thin patterned layer above a metallic ground plane, separated by a dielectric spacer of thickness h. The unit cell is a square patch of side D with a narrow slot of width w (Fig. 1a). Under the condition D ≪ λ, a transmission‑line equivalent circuit can be constructed (Fig. 1b). The input impedance of the unit cell is expressed as

\(\displaystyle \frac{1}{Z_{\text{in}}}=\frac{1}{Z_{1}}+\frac{1}{Z_{\text{mg}}}=\frac{1}{jZ_{h}\tan(k_{zh}h)}+\frac{1}{Z_{\text{mg}}}\) (1)

where Z_h and k_zh are the substrate impedance and propagation constant, respectively. The overall absorptivity at normal incidence follows from the reflection coefficient S₁₁:

\(\displaystyle A(\omega)=1-|S_{11}|^{2}=1-\left|\frac{Z_{\text{in}}-120\pi}{Z_{\text{in}}+120\pi}\right|^{2}\) (2)

Equation (1) enables extraction of the hybrid surface impedance Z_mg from simulated S-parameters, revealing the interplay between patch geometry and graphene conductivity.

Impedance of Square Patches

For an electrically dense array of perfectly conducting patches, the grid impedance is given by

\(\displaystyle Z_{m}=\frac{D}{w}Z_{s}-j\frac{\eta_{\text{eff}}}{2\alpha}\) (3)

with the effective wave impedance η_eff = √(μ₀/ε₀ε_eff) and the geometrical factor

\(\displaystyle \alpha=\frac{k_{\text{eff}}D}{\pi}\ln\left[\frac{1}{\sin(\pi w/2D)}\right]\) (4)

where ε_eff ≈ (ε_r+1)/2 and k_eff = k₀√ε_eff. These expressions hold when the patch period is much smaller than the operating wavelength.

To validate the analytical model, we performed full‑wave simulations in Ansoft HFSS with a resistive sheet (Z_s = 35 Ω/sq), spacer thickness 20 µm, and relative permittivity ε_r = 3.2(1−j0.045). The extracted patch impedance from the simulated input impedance matched the analytical prediction closely, confirming the model’s accuracy.

Impedance of Graphene Sheets

Graphene is treated as an infinitesimally thin conductive surface. Its surface conductivity is described by the Kubo formula:

\(\displaystyle \sigma_{g}=\frac{j e^{2}k_{B}T}{\pi\hbar^{2}(\omega+j/\tau)}\left[\frac{\mu_{c}}{k_{B}T}+2\ln\left(e^{-\mu_{c}/k_{B}T}+1\right)\right]+\frac{j e^{2}}{4\pi\hbar}\ln\left[\frac{2|\mu_{c}|-\hbar(\omega+j/\tau)}{2|\mu_{c}|+\hbar(\omega+j/\tau)}\right]\) (5)

Assuming room temperature (T = 300 K) and a relaxation time τ = 0.1 ps, the sheet impedance is Z_g = 1/σ_g = R_g + jX_g. Figure 4 shows that both resistance and reactance decrease monotonically with increasing chemical potential μ_c, while remaining nearly flat across the 0.2–6 THz band for a fixed μ_c.

Results and Discussion

Unlike the conventional assumption that the hybrid impedance is simply the parallel combination of patch and graphene impedances, our simulations reveal a more complex interaction. Figure 5 compares analytical and simulated values of the real and imaginary parts of Z_mg for various chemical potentials (w = 19 µm). While the analytical trend captures the overall frequency dependence, the absolute values differ significantly, especially at low μ_c where graphene’s impedance dominates.

To further elucidate the size dependence, we varied the patch width from 17 to 19.5 µm and extracted Z_mg for a fixed μ_c = 0.4 eV (Fig. 7). The real part of the impedance decreases with increasing w up to 0.31 THz, then rises, reflecting the balance between capacitive coupling and graphene’s sheet resistance. The imaginary part follows a similar trend, confirming the interplay between geometry and material response.

Current density maps at 3 THz (Fig. 8) illustrate that larger patches support lower surface currents, consistent with the inverse relationship between patch length and current magnitude given by Eq. (11). Integrated current values from HFSS corroborate this trend, reinforcing the physical interpretation of the impedance variations.

Conclusions

We have developed and validated a comprehensive analytical framework for predicting the surface impedance of metasurface–graphene hybrids in the THz regime. The derived formulas for square‑patch impedance, combined with the Kubo‑based graphene sheet impedance, enable rapid design of absorbers, antennas, and other devices without exhaustive full‑wave simulations. The study also clarifies how patch size and graphene chemical potential jointly influence the effective impedance, offering clear design levers for tunable THz components.

Abbreviations

HFSS:

High‑frequency structure simulation

TEM:

Transverse electromagnetic

THz:

Terahertz