Highly tunable hybrid metamaterials employing split-ring resonators strongly coupled to graphene surface plasmons

Metamaterials and plasmonics are powerful tools for unconventional manipulation and harnessing of light. Metamaterials can be engineered to possess intriguing properties lacking in natural materials, such as negative refractive index. Plasmonics offers capabilities of confining light in subwavelength dimensions and enhancing light–matter interactions. Recently, the technological potential of graphene-based plasmonics has been recognized as the latter features large tunability, higher field-confinement and lower loss compared with metal-based plasmonics. Here, we introduce hybrid structures comprising graphene plasmonic resonators coupled to conventional split-ring resonators, thus demonstrating a type of highly tunable metamaterial, where the interaction between the two resonances reaches the strong-coupling regime. Such hybrid metamaterials are employed as high-speed THz modulators, exhibiting ∼60% transmission modulation and operating speed in excess of 40 MHz. This device concept also provides a platform for exploring cavity-enhanced light–matter interactions and optical processes in graphene plasmonic structures for applications including sensing, photo-detection and nonlinear frequency generation.

Supplementary Note 1. Comparison between a conventional C-SRR and a modified twopiece C-SRR designs

Supplementary Note 2. Screening of the GR localized plasmons by nearby metallic structures
If the distance between the edges of a GR and the nearby metallic structures (such as the C-SRR in this work) is sufficiently small compared to the thickness of the metallic structures, the localized SP resonance frequency of the GR is reduced as a result of the screening effect from the metal. The collective oscillation of the charge carriers (i.e. the SP resonance) in a GR is a consequence of the restoring force arising due to the charge density fluctuations with the linear charge density , localized near the GR edges and indicated by the positive (+) and negative (-) signs in Supplementary Fig. 2. In the presence of nearby thick (compared to the distances d 1 and d 2 ) metallic structures which may be approximated as infinitely high metallic walls extending in both directions, this restoring force is modified by the image charges induced in the metal, As a result, the SP resonance frequency is reduced to where is the unscreened SP resonance frequency of the GR and . This calculation provides the theoretical upper limit of the screening effect induced by nearby metallic structures, while the actual reduction of the SP resonance frequency depends on the geometrical details of the structures and can be computed numerically.
Supplementary Note 3. Fitting procedure of the hybridized modes of the C-SRR-GR hybrid metamaterials The efficient coupling of the C-SRR LC resonance to the GR localized SP resonance leads to the observed hybridized modes in the transmission spectra of the C-SRR-GR hybrid metamaterials. To quantitatively analyse the carrier density dependent spectral response of these hybrid metamaterials, we have also investigated the tuning of the SP resonances in bare GR arrays (without the C-SRRs). The GR arrays are fabricated on the same SiO 2 /Si substrates used for the hybrid metamaterials, which allow for tuning of the carrier density via the applied back-gate voltage [1]. SiO 2 is a polar dielectric and as such, supports surface optical (SO) phonons, which can couple to SP resonances in the GRs via the long-range Fröhlich interaction [2,3], resulting in multiple surface-plasmon-phonon-polariton (SP3) modes [1,4]. The transmission spectrum of such an array of 400-nm-wide GRs is measured as a function of carrier density and plotted in Supplementary Fig. 3a. In the measured frequency range, two resonances are observed, labelled P 1 and P 2 , which originate from the interaction of the graphene SP and the SO phonon at ~14.5 THz [1]. As the carrier density is increased, there is a monotonic increase in the frequency of both resonances, with the peak positions plotted as a function of carrier density (n 1/4 ) in Supplementary Fig. 3b.
To derive an equation to describe the anti-crossing of the plasmon and SO phonon modes, we consider a 2D layer with the conductivity ( ) lying on the boundary z = 0 of two media with the bulk dielectric functions ( ) and ( ). The spectrum of electromagnetic waves running along this boundary in the x-direction and localized at this boundary is determined by the dispersion equation (in Gaussian units) The conductivity can be described using the Drude model, which for the case of graphene and neglecting the carrier relaxation is where is the sheet carrier density and is the Fermi velocity. We assume the first medium is vacuum, ( ) and the second has the dielectric function ( ) and consider several special cases. Firstly, the second medium is also vacuum ( ) . The dispersion equation then gives where we define as the plasma frequency for graphene in vacuum. In the second case, we assume there is no graphene, i.e.
, and the second medium is a polar semiconductor with where is the frequency of the transverse optical phonons, and and are the dielectric constants at frequencies much lower and much higher than , respectively. The dispersion equation gives and can be rewritten as (9) where is the frequency of the SO phonons. Therefore, the dispersion equation at the interface of a vacuum and a polar semiconductor can be written as Finally, for the case of graphene on the surface of a polar crystal, the dispersion equation is which leads to the following biquadratic equation, where the two solutions describe the coupled plasmon-phonon modes, To fit the experimental data shown in Supplementary Fig. 3b, we use Eq. 12 with defined according to Eq. 6 with a slight modification as where A is a fitting parameter to take into account the relation between the frequency of the localized SP resonance in a GR and the frequency of a SP wave with the wave vector |q| in a continuous graphene sheet (Eq. 6); the factor A was numerically found to be in [5], see Figure 1 there (see also [6]). The solid lines in Supplementary Fig. 3b show a fit to the experimental data using Eq. 12, with = 14.5 THz (ref 1), =3.9 and the fit parameters =2.81 and A=0.77. The lower branch, is subsequently used to fit the anti-crossing of the C-SRR-GR device HM1, employing Eq. 1 in the main text with , and with = 14.5 THz, =3.9, =2.81 as fixed parameters and A, β, -as fitting parameters. The same fitting procedure is carried out for the simulated transmission spectra of HM1 in Fig. 4 of the main text. Supplementary Table 1 lists the fit parameters of both the experimental and the simulated transmission spectra of HM1. Excellent agreement is found between the experimental results and the simulation.
It is worth pointing out that when evaluating the experimental carrier density of the 400 nm wide GRs at a given back-gate voltage, the parallel-plate capacitor model is not a good approximation in this case, as the GR width is comparable to the thickness d (~300 nm) of the dielectric layer and the edge effect is hence significant. To this end we consider a thin conducting layer of the width W (the GR) lying on a dielectric slab at a distance d from the grounded conducting substrate. The distribution of the electric potential in the free space and in the dielectric is described by the Laplace equation with the boundary conditions at the grounded substrate and at the GR. Solving this equation with the Green function technique, we reduce the problem to an integral equation for the electron density ( ) inside the GR where | |. Expanding the density ( ) in a series of Chebyshev polynomials ( ) we reduce the problem to an infinite set of algebraic equations for the expansion coefficients. Truncating this set (the convergence is checked and found to be very fast) we get the relation between the average density of electrons in the GR 〈 ( )〉 and the gate voltage. The resulting ratio of the capacitances as a function of at (SiO 2 ) is shown in Supplementary Fig. 4, where is the capacitance predicted by the parallel-plate capacitor model.

Supplementary Note 4. Equivalent circuit model of the C-SRR-GR hybrid metamaterials
We introduce an equivalent circuit model as an alternative description of the operating mechanism of the C-SRR-GR hybrid metamaterials, which also provides a more intuitive interpretation of the coupling strength characterized by the parameter in Eq. 1 in the main text. The equivalent circuit diagram for the C-SRR-GR hybrid structure is illustrated in Supplementary Fig. 5 (enclosed by the dashed square), in which both the C-SRR and the GR are modeled as LC-circuits (losses are neglected for simplicity). The GR is modeled as a parallel LC-circuit because the SP resonance corresponds to enhanced reflection, while the C-SRR is modeled as a series LC-circuit due to enhanced transmission at the resonance. Since the coupling between the C-SRR and the GR is established via the interaction between the near-field electric field of one structure and the charge distribution of the other and vice versa, such a coupling is modeled as a capacitive coupler between the two LC-circuits in the equivalent circuit diagram. Notice that the effective inductor associated with the GR SP resonance is mainly contributed by the kinetic inductance of the charge carriers [7,8].
Since the C-SRR is excited mainly by the incident electro-magnetic waves whereas the GR is driven mostly by the enhanced electric field in the capacitor gap of the C-SRR, the equivalent circuit for the entire system is constructed as shown in Supplementary Fig. 5, with and representing the impedance of the free space and the substrate, respectively. Therefore, the transmission peaks correspond to the zeros of the total impedance of the C-SRR-GR equivalent circuit measured between port 2 and port 3, which is straightforwardly derived to be The two zeros of ( ) occur at Let ( ) ( ) then Eq. 16 becomes The above Eq. 17 reproduces Eq. 1 in the main text describing the anti-crossing behavior of two strongly coupled resonances, and the physical meaning of in this equivalent circuit model is the relative strength of the capacitive coupling.
Supplementary Note 5. Comparison of the simulated near-field E-field amplitude between the GR alone and the C-SRR-GR hybrid structure Supplementary Fig. 6 shows the simulated E-field (all 3 orthogonal components included) amplitude distributions of the plasmonic resonance of a GR (Supplementary Fig. 6a) and of the two hybridized modes of a C-SRR-GR hybrid structure HM1 (Supplementary Fig. 6b,c) employing the same GR width. These field distributions correspond to the plane right below the GR or the C-SRR-GR structures. The 3 colour plots are in the same linear scale for direct comparison. The near-field enhancement associated with the C-SRR-GR is about one order of magnitude higher than that with the GR alone.
Supplementary Note 6. Simulated carrier density dependent spectral response of the C-SRR-GR hybrid metamaterials operating at ~4 THz The simulated transmission spectra of the hybrid metamaterial device HM2 at various Fermi energies (hence carrier densities) and assuming two different carrier relaxation times (50 fs and 100 fs) are shown in Supplementary Fig. 7, in comparison with that of the reference bare C-SRR array. The transmission spectra in Supplementary Fig. 7a exhibit good agreement with the experimental observations in Fig. 5a in the main text. On the other hand, the transmission spectra in Supplementary  Fig. 7b clearly reveal the double-peak feature owing to the two hybridized modes in the strong coupling regime, similar to the experimentally observed spectral responses of the 10 THz devices in Fig. 3 in the main text. From the spectra in Supplementary Fig. 7b, a coupling strength of ~0.8 THz is extracted for the coupling between the C-SRR LC resonance and the GR localized SP resonance associated with this particular structure.

Supplementary Note 7. Additional analysis of the modulation speed measurement on the C-SRR-GR hybrid metamaterials
The demonstrated modulation speed of the C-SRR-GR hybrid metamaterials is limited by the effective RC time constant, contributed by both the device itself and the external driving circuit. The total capacitance of the device mainly stems from the capacitor formed between the Si substrate backgate and the C-SRR-GR array, with the ~300 nm SiO 2 layer as the dielectric medium. Since the C-SRR-GR array covers most of the device surface area, the associated capacitance can be estimated using the parallel plate capacitor model, which yields ~115 pF for a 1 mm×1 mm device (HM2) and ~29 pF for a 0.5 mm×0.5 mm device (HM3). The effective total resistance R is further extracted from the modulation speed measurement, and the extracted values are ~80  for HM2 and ~180  for HM3, respectively, excluding the 50 input resistance of the driving voltage source. Thanks to the parallel electrical connection of all the C-SRR-GR unit cells, the resistance associated with the entire C-SRR-GR array is low (Supplementary Fig. 8) and contributes to only a small fraction of the total resistance. Hence, the major contribution to the device total resistance is from the relatively low-doped Si substrate (resistivity ~10 Ωcm). Since electrical contacts for the current devices are made at the edges of the chip, the current flows in-plane in the Si substrate when charging and discharging the capacitor, resulting in large resistance. Such substrate resistance also depends on the area of the metamaterial field and its location on the chip. Since the measured device HM3 is located closer to the chip center than device HM2 and has smaller area, the substrate resistance seen by HM3 is higher. However, this substrate resistance contribution can be significantly decreased by more than one order of magnitude if electrical contacts are made on the back side of the thin Si substrate, which would facilitate the current to flow perpendicularly instead of in-plane. Such a back contact can be realized with a properly designed wire-grid structure, imposing minimal influence on the transmission of the THz radiation. Alternatively, one can also make use of such hybrid metamaterial structures to modulate the reflection of incident THz radiation, and in this case the Si substrate can be doped much higher.