cSRR Terahertz Circuits: Design & Applications

• cSRRs are planar meta-atoms defined by subwavelength complementary split ring apertures that enable engineered LC resonance for precise THz modulation and sensing.
• The integration with dynamic materials like VO₂ leverages phase transitions to induce significant resonance shifts and enhance transmission control in THz applications.
• Design trade-offs involving gap dimensions, array density, and substrate choice are critical for tuning the quality factor, operational bandwidth, and overall device performance.

Terahertz (THz) circuit-based complementary split ring resonators (cSRRs) are planar electromagnetic meta-atoms defined by subwavelength slot apertures in a metallic surface, whose engineered LC resonance enables strong field confinement and highly tunable THz responses. cSRRs are central elements in metamaterial and metasurface architectures for THz modulation, filtering, sensing, and quantum light–matter interaction. Their planar, complementary topology (where metal is replaced by apertures) offers distinct operational characteristics compared to conventional SRRs, particularly in transmission-based device engineering and active hybridization with dynamic quantum materials.

1. Geometry, Materials, and Equivalent Circuit Representation

The canonical cSRR geometry consists of an aperture shaped as a split ring (square, circular, or rectangular), with dimensional parameters set by application requirements such as resonance frequency and electromagnetic mode confinement. For example, in vanadium dioxide (VO₂)-integrated cSRRs, a square aperture with outer side 45 μm, central gap width of 3 μm, and metal line width 5 μm is patterned in a 150 nm Au film with a 10 nm Ti adhesion underlayer, on a 100 nm VO₂ film grown on ~500 μm sapphire (Huang et al., 2022). For high-Q THz meta-atoms on silicon membranes, dual-ring cSRRs with outer footprint 10 μm × 25.5 μm, 500 nm gap, and multilayer Cu/Au are employed (Keller et al., 2019). In gate-tunable single-meta-atom light–matter coupling platforms, circular cSRRs (D ≈ 7.2 μm, gap 1 μm) are defined by etching into a 40 nm Au layer (Cr adhesor), atop a GaAs/AlGaAs 2DEG (Jöchl et al., 5 Nov 2025).

The electromagnetic response of a cSRR is accurately captured with a lumped-element LC circuit model. The inductance L arises from the current path encircling the perimeter (with $L\sim\mu_0\,l_\mathrm{eff}$), and the capacitance C is dominated by the fields across the split gap in the surrounding dielectric (substrate or active film):

• $L\simeq\mu_0 l_\mathrm{eff}$, where $l_\mathrm{eff}$ is twice the average ring radius.
• $$C\simeq\epsilon_0\epsilon_\mathrm{eff} A_\mathrm{gap}/g$$, with $A_\mathrm{gap}$ set by the gap-facing area, $g$ the gap width, and $\epsilon_\mathrm{eff}$ the local permittivity.

The resonance occurs at $f_0 = (2\pi \sqrt{LC})^{-1}$, with strong sensitivity to $\epsilon_\mathrm{eff}$—a property key to integrating active materials. Parameter choices and substrate properties are summarized in the following table:

Platform	Primary Metal / Thickness	Substrate / Film	Gap Width	Typical $f_0$
VO₂-cSRR metasurface	Au, 150 nm (+10 nm Ti)	Sapphire/VO₂, 100 nm	3 μm	0.47 THz
Si-membrane cSRR array	Cu 250 nm + Au 20 nm	10 μm Si-membrane	500 nm	1.12 THz
2DEG gate-tunable cSRR	Au 40 nm (+4 nm Cr), etched	GaAs/AlGaAs 2DEG	1 μm	255 GHz

2. Resonance Linewidth, Quality Factor, and Collective Effects

The THz cSRR’s resonance characteristics—especially the linewidth and quality factor $Q=f_0/\Gamma_\mathrm{tot}$—are governed by both intrinsic (Ohmic/material) and extrinsic (radiative) losses. In dense arrays on thin Si membranes, the total linewidth scales as $\Gamma(N)=\Gamma_0+\alpha N$, where $N=1/a^2$ is the meta-atom areal density (Keller et al., 2019). The linear dependence evidences a collective superradiant effect: in-phase gap oscillations (“bright mode”) enhance radiative damping with increasing density, resulting in homogeneous Lorentzian broadening.

Experimental results show:

• Decreasing array period $a$ (higher density): $Q$ drops from 17.9 (for $a=180\,\mu$m) to 6.6 ($a=40\,\mu$m), while linewidth increases from 63 GHz to 170 GHz ($f_0=1.12$ THz).
• The superradiant slope $$\alpha\sim2\times10^5\,\text{GHz}\cdot\mu\text{m}^2$$ for the modified cSRR design.
• Use of thin ($<10\,\mu$m) Si membranes suppresses surface plasmon polariton (SPP) leakage, preserving high-Q operation by shifting SPP modes out of the LC-mode’s frequency.

The trade-off between fill factor (interaction strength) and $Q$ is a key design parameter. For single-cSRRs intended for ultrastrong-light–matter coupling, the $Q$ factor is primarily set by Ohmic and radiation losses and can be kept high by metal choice, trace width, and gap engineering (Jöchl et al., 5 Nov 2025).

3. Integration with Dynamic Quantum Materials

A defining advancement in terahertz cSRR design is hybridization with materials whose dielectric function can be externally modulated—exemplified by vanadium dioxide (VO₂) heterostructures exhibiting a sharp, thermally driven insulator–metal transition (IMT) near 340 K.

The cSRR–VO₂ system demonstrates:

• The effective dielectric permittivity of the active film ($\epsilon_\mathrm{eff}$) is described by Maxwell–Garnett effective medium theory (MG-EMT):

$$\epsilon_\mathrm{eff}(p) = \epsilon_\mathrm{i} + \frac{p\,(\epsilon_\mathrm{m}-\epsilon_\mathrm{i})}{1-p\,(\epsilon_\mathrm{m}-\epsilon_\mathrm{i})/(\epsilon_\mathrm{m}+2\epsilon_\mathrm{i})}$$

where $p$ is the metallic volume fraction, $\epsilon_\mathrm{m}$ ($\approx -90$ at 0.5 THz) and $\epsilon_\mathrm{i}$ ($\approx 10$) are the metallic and insulating VO₂ permittivities.

• Upon crossing the percolation threshold $p_\mathrm{t}\approx0.62$, $\epsilon_\mathrm{eff}$ spikes, drastically enlarging $C$ and producing a marked resonance redshift—a shift of 90 GHz ($0.47\rightarrow0.38$ THz) is observed for a few K temperature sweep through the IMT (Huang et al., 2022).
• The modulation amplitude of THz transmission through the structure ($\Delta T$) is enhanced from $42\%$ (pristine VO₂) to $68.3\%$ (with cSRR), a $62.4\%$ improvement, due to resonance structure and $\epsilon_\mathrm{eff}$-dependent capacitive modulation.

MG-EMT, when coupled to equivalent circuit modeling, enables accurate prediction of these effects, confirming the cSRR’s function as a “spectral lever” on the underlying active material’s phase response.

4. Implementation of Gate-Tunable cSRRs and Single-Atom Ultrastrong Coupling

Single cSRR meta-atoms can be engineered for gate-tunable THz light–matter coupling by integrating them with a two-dimensional electron gas (2DEG) and incorporating an electrostatic gate (Jöchl et al., 5 Nov 2025). Key operational features include:

• A cSRR etched in a Schottky metal on a GaAs/AlGaAs 2DEG enables DC gating, which laterally confines electrons, reducing the electron channel width $d$ from $0.92\,\mu$m ($V_G=0.75$ V) to $0.41\,\mu$m ($V_G=4.0$ V), thus decreasing the number of THz-coupled electrons ($N_e$).
• The normalized vacuum Rabi splitting $\eta=g/\omega_\mathrm{res}$ is tuned in-situ from $0.456$ to $0.184$, corresponding to $\sim7900$ to $1300$ coupled electrons. The Rabi coupling $g\propto\sqrt{N_e}$ is directly adjusted via $V_G$.
• The system is modeled using the multi-mode Hopfield Hamiltonian, with measured polariton branches and lineshapes fitting the full quantum model with diamagnetic $A^2$ term included.

Standing plasma (M1) modes emerge under strong in-plane confinement, visible as blueshifting features in transmission maps under increasing negative gate bias. The ability to dynamically modulate $\eta$ and field profiles at the single-cSRR level enables studies of ultrastrong and deep-strong coupling physics, dynamic quantum optics, and highly sensitive THz sensing.

5. Design Strategies and Practical Guidelines

Designing cSRRs for optimal THz circuit functionality involves material platform selection, geometric fine-tuning, and loss-channel engineering as dictated by intended application:

• For maximal amplitude modulation and frequency tunability, employ cSRR geometries that invert the LC resonance feature (transmission dip to peak) and enlarge gap-facing area ($A_\mathrm{gap}$) under the active layer.
• Gap width ($g$) should be minimized to enhance $C$ and resonance sensitivity but balanced to maintain manageable insertion loss and fabrication feasibility; typical gaps are $300$–$500$ nm (Keller et al., 2019).
• Active material film thickness and crystalline quality must facilitate a well-defined, percolative phase transition (e.g., $d_\mathrm{VO2}\sim100$ nm, hybrid MBE growth for VO₂).
• Substrates with low-loss and low-$\epsilon_r$, such as thin high-resistivity Si or sapphire, reduce parasitic modes and support subwavelength mode confinement.
• Array periodicity $a$ should remain $>50\,\mu$m ($\sim\lambda_0/5$ at $f_0$) to avoid diffraction, but small enough for significant field enhancement and device interaction strength; large $a$ yields higher $Q$ but lower fill factor.

A plausible implication is that subwavelength cSRR meta-atoms, integrated with quantum materials or active devices, provide a versatile toolbox for dynamic THz photonics, with design continuously guided by circuit modeling, full-wave simulation (e.g., CST with Drude permittivities), and effective medium analysis.

6. Applications and Implications for Terahertz Circuit Engineering

THz cSRRs support a spectrum of functionalities:

• As metasurface modulators, cSRR–VO₂ structures yield high-depth, thermally induced THz transmission control, with large, sharp spectral shifts suited for dynamic filtering, amplitude modulation, and switching at cryogenic or ambient $T$ (Huang et al., 2022).
• In the collective regime (high-density cSRR arrays) on thin Si membranes, field-enhanced high-Q resonances are exploited for on-chip THz filtering, sensing, and platform development for nonlinear spectroscopy (Keller et al., 2019).
• Electrically gated single-cSRR platforms bridge toward quantum optics and polariton condensate control. Gate-tuned $\eta$ enables tailored ultrastrong- and deep-strong coupling regimes, with applications in vacuum field engineering and subwavelength THz detection (Jöchl et al., 5 Nov 2025).

The integration route—combining circuit-derived resonance engineering, dynamic material platforms, and carefully designed electromagnetic environment—positions terahertz cSRRs as fundamental building blocks for next-generation active, tunable, and quantum-enabled THz technologies.