Graphene Plasmon Cavities Made with Silicon Carbide.

We propose a simple way to create tunable plasmonic cavities in the infrared (IR) range using graphene films suspended upon a silicon carbide (SiC) grating and present a numerical investigation, using the finite element method, on the absorption properties and field distributions of such resonant structures. We find at certain frequencies within the SiC reststrahlen band that the structured SiC substrate acts as a perfect reflector, providing a cavity effect by establishing graphene plasmon standing waves. We also provide clear evidence of strong coupling phenomena between the localized surface phonon polariton resonances in the SiC grating with the graphene surface plasmon cavity modes, which is revealed by a Rabi splitting in the absorption spectrum. This paves the way to build simple plasmonic structures, using well-known materials and experimental techniques, that can be used to excite graphene plasmons efficiently, even at normal incidence, as well as explore cavity quantum electrodynamics and potential applications in IR spectroscopy.

Introduction

Graphene is a two-dimensional (2D) single-atom thick layer of carbon, and after the experimental proof of existence in 2004,[1] its remarkable physical properties have led to one of the most prolific areas of research in nanoscience. In particular, its extreme 2D geometry and distinct band structure lead to novel interaction with light and have resulted in immense interest from the plasmonics and nano-optics communities.[2] Doped graphene supports collective oscillations of the free carriers known as surface plasmon polaritons (SPPs) that can be either propagating or localized[2,3] and has been recognized as a suitable alternative to noble metals in the field of plasmonics. These plasmonic modes are characterized by large field confinement [plasmon wavelengths, a 100 times smaller than the free-space wavelength, have been demonstrated at mid-infrared (IR) frequencies[4]] and enhancements and are particularly appealing in graphene because of the tunability of the Fermi energy via chemical doping or electrical gating. Optical gaps of up to 2 eV (which corresponds to EF ≈ 1 eV) can be created.[5] This paves the way for cheap, reliable, and ultrafast optical modulators. At terahertz and IR frequencies, graphene plasmons have been conclusively demonstrated by a number of different experimental techniques including scanning near-field microscopy,[6−8] optical far-field spectroscopy,[9] and electron energy loss spectroscopy.[10]

Silicon carbide (SiC) is a polar dielectric that supports subdiffraction confinement of light using the excitation of surface phonon polaritons (SPhPs) that result from the coupling of a photon with the collective oscillation of the polar lattice atoms.[11] Because of the energy scale of optical phonons, the reststrahlen band [a narrow and material-dependent frequency range bounded by the longitudinal optical (LO) and transverse optical (TO) phonon frequencies] inherently occurs within the IR to terahertz regime, where polar dielectrics act as an optical metal, that is, they exhibit high reflectivity and negative real part of the dielectric function. The long phonon lifetimes (picoseconds), as compared to electron scattering times in metals (10s of femtoseconds), means that the imaginary part of the dielectric function can be orders of magnitude smaller than that for noble metals, leading to low-loss systems with high quality factors and narrow line widths.[11] As previously shown, nanostructuring SiC results in localized surface phonon polaritons (LSPhPs) with a high Q and strong field confinement.[12,13] SiC also has a large band gap and high thermal, mechanical, and chemical stability, making it highly suitable for a number of applications.

In this work, we explore the coupling between SiC LSPhPs and graphene SPPs and explore the unique properties of the hybrid modes,[14] which combine positive characteristics of the graphene SPPs (large light confinement and tunable) and SiC LSPhPs (low loss). We consider a single-layer graphene sheet on the top of a SiC cavity. Structuring of SiC[12,13] leads to the excitation of both the graphene SPP and a LSPhP in SiC. This is convenient as the difficulty in nanofabrication of graphene devices, as well as the large momentum mismatch between graphene SPPs and the incident light, means that the creation of graphene-based resonators that can be excited at normal incidence, is challenging. A numerical investigation, using finite element analysis method from COMSOL Multiphysics, on the absorption properties and field distributions of this system in the IR range is reported. At frequencies in the reststrahlen band, but away from the LSPhP resonances, SiC acts as a “perfect conductor”, and at certain lengths of the grating groove, a plasmon standing wave is set up. Here, by a perfect conductor, we simply mean that the real part of the permittivity is extremely large and negative (small field penetration), and we assume that SiC is undoped and hence is not conductive. This setup is a different way to excite localized graphene plasmons and achieve strong resonant absorption, without using nanoribbons. Note that although we consider the cavity as part of a periodic structure (i.e., a grating that is convenient for experiments and numerical simulations), we do not consider the excitation of modes using diffraction. The necessary momentum is provided by the abrupt breaking of the symmetry of the cavity, not by the periodicity.a

There is a lot of interest on the coupling of surface phonons and graphene plasmons, which becomes important at the mid-IR range for typical polar dielectric substrates such as SiC[15,16] and SiO2.[6,17,18] It has also been demonstrated in graphene/hexagonal boron nitride (h-BN) heterostructures[4,19−22] and thin layers of surface-absorbed polymers.[23] Recently, the interaction of localized plasmons in a gold antenna with a SiO2 coating and the substrate has been explored, and it was found that strong coupling between the modes leads to a transparency window.[24] We find, near the LO phonon frequency, a strong interaction between the LSPhPs and the graphene plasmons, leading to Rabi splitting, signaling the coherent energy transfer between the two coupled modes.[25,26] The high quality factor Q and the small effective volume V of the hybrid modes make this system a suitable system for cavity quantum electrodynamics (QED)[27] where one is interested in the strong mixing of light and individual emitters. In photonic cavities, the modal volume is on the order of λ3, but by using localized surface plasmons (LSPs) and/or LSPhPs, one can confine light fields evanescently and achieve subdiffraction modal volume. Using hybrid modes to tailor the spectral and spatial profile of localized plasmons to aid strong coupling to emitters has been demonstrated.[28] By picking suitable constituent modes that have complimentary characteristics, the Q/V ratio, which is proportional to the Purcell factor, can be increased to enhance strong light–matter interactions with emitters. Our work provides a potential method to efficiently excite unique hybrid modes for confining and manipulating light and could pave the way for applications in tunable broadband molecular spectroscopy in the terahertz and IR range.[29] Furthermore, the device potentially provides a new system to study cavity QED. The high quality factor and the small modal volume make this system a very suitable platform for inducing and exploring strong light–matter interactions.

Results and Discussion

In Figure , we show the system to be studied. It is composed of a monolayer graphene sheet deposited on a SiC grating, and we assume that the two are in physical contact with one another. The graphene sheet is located on the x–z plane with the grating period in the x direction. The incident light propagates in the y direction from above and is normal to the graphene surface. It is TM so that SPPs can be excited above the grating in the graphene layer. The height, period, ridge width, and groove width of the grating are, respectively, represented by the letters h, Λ, w, and L (see Figure b). The graphene thickness in the simulations is 0.5 nm, which we have checked is converged for all results. The cavity supporting the graphene SPPs is given by the zone where the graphene is suspended (see Figure c).

Figure 1

Schematics of the SiC grating and graphene plasmonic device. (a) Three-dimensional view of the whole structure and (b) cross-sectional view (the incident light is transverse magnetic (TM) polarized normal incident in the air, where Λ is the grating period, w is the width of the grating ridge, L is the width of the grating groove, and h is the grating height). (c) Sketch of the standing waves of graphene SPP resonance in a grating cavity.

We model the optical properties of graphene using the local random phase approximation model that is known to work well.[2,30] We also use the zero-temperature limit that is a good approximation even for finite temperatures. The conductivity is given bywhere τ is the carrier relaxation time and is linked to the carrier mobility μ via the equation , θ denotes a step function, and EF is the Fermi energy. The first term describes intraband transitions and is Drude-like. This will dominate the response over the second term, which describes interband transitions, at the frequencies and levels of doping we consider. In Figure a,b, we plot the corresponding dielectric function for EF = 0.64 eV and μ = 10 000 cm2/(V s) (which is a typical maximum value for the mobility of chemical-vapor-deposition-grown graphene), where it can be seen that graphene, for a single-layer structure, has a remarkably large absorption at visible wavelengths.

Figure 2

Material properties of single-layer graphene in air (a,b) and bulk SiC (c,d). The chemical potential of graphene is EF = 0.64 eV, and carrier mobility is μ = 10 000 cm2/(V s) in all our simulations unless otherwise specified. (a) Absorption of graphene: from the inset, it is shown that graphene has remarkably high absorption at visible light range. (b) Permittivity of graphene. (c) SiC absorption and its reststrahlen band. (d) Permittivity of SiC.

The dielectric function for polar dielectrics such as SiC can be described by a Lorentz oscillator modelwhich has a pole at the TO phonon frequency ωTO and a zero-crossing point at θLO. For SiC,[31] ϵ∞ = 6.56, ωLO = 973 cm–1, ωTO = 797 cm–1, and γ = 4.76 cm–1; in Figure c,d, we show the absorption and permittivity of SiC. The SPhPs can only be supported in a narrow spectral bandwidth (in the case of SiC around 176 cm–1), but crucially they correspond to frequencies at which graphene plasmons can be excited, paving the way for interactions between the two types of excitations.

If the momentum mismatch is overcome, graphene SPPs can be excited by the normal incident light. Assuming that interband transitions can be ignored, the dispersion relation for the plasmon waves in a free-standing graphene sheet is given by[30]where kSPP is the in-plane wavevector of the plasmon and ϵ1 and ϵ2 are the relative permittivity of the materials in the infinite half-spaces above and below the graphene sheet, respectively. The dispersion is plotted in Figure a for a graphene sheet in vacuum for different levels of doping; the dispersion curve indicates that the plasmon wavelength is on the order of 10–100 times smaller than the incident wavelength, revealing the large confinement and the consequent momentum mismatch that must be overcome for efficient coupling. Unless otherwise specified, we use a Fermi energy of 0.64 eV throughout this work.

Figure 3

(a) Graphene SPP wavelength for different values of the chemical potential. (b) Total absorption (PabsNG) vs grating period (Λ) and incident wavelength (λ) for the SiC grating (with no graphene) structure for w = h = 1 μm. (c) Total absorption (PabsT) vs grating period and incident wavelength for the SiC grating and graphene structure for w = h = 1 μm. (d) Absorption of the graphene layer (PabsG) vs grating period and incident wavelength for the SiC grating and graphene structure for w = h = 1 μm. The blue lines indicate a fit to a Fabry–Pérot model for a phase shift of −π (see eq ).

To investigate our system, we begin by calculating the absorption of a SiC grating with no graphene sheet present. We assume that SiC is infinitely thick below the grating to switch off the transmission channel and simplify the analysis. We consider wavelengths and grating periods in the range 10–14 and 1.1–5 μm, respectively, with the grating height and the ridge width fixed at 1 μm. The results are shown in Figure b. We see that within the reststrahlen band there is now a number of peaks as compared to the plain SiC case (Figure c); this is due to the interaction of the LO phonon and the grating, which leads to the excitation of LSPhPs.[13] There is a region of large absorption at the LSPhP resonances, but elsewhere in the band the absorption is close to 0% as SiC acts as a perfect reflector. Outside of the reststrahlen band, for frequencies less than ωLO and larger than ωTO, the absorption is very similar to that for a plain SiC surface shown in Figure c.

Next, we add graphene on the top of the grating, with the results shown in Figure c,d. We consider two quantities: PabsT, which represents the total absorption (sum of the absorption of the SiC grating and the graphene layer), and PabsG, which is the absorption in the graphene. The most striking difference is the appearance of strong absorption peaks in the low absorption region of the reststrahlen band for certain grating period lengths. This can be understood by looking at the near field of the y component of the electric field for each peak, as shown in Figure a. The field is concentrated in the graphene layer and has a fixed number of wavelengths in the region directly over air. The plasmon modes are antisymmetric, which is a consequence of the electric field boundary conditions. The normal incident field excites SPPs at the edge, which are out of phase and will only constructively interfere for odd modes.[32] In the parts of the graphene, which are located on the top of SiC, there exists a very weak field and consequently there is low absorption in these locations. This can be understood by noting in this spectral region that SiC acts, to a good approximation, as a perfect electrical conductor and screens out any external field very effectively. In other words, the field is zero outside of the cavity region and leads to plasmon standing waves being established. To confirm this, in Figure a, we show the absorption varying with the cavity length (for a fixed wavelength λ = 12 μm). We find, at integer multiples of L/λSPP, absorption peaks that correspond to the plasmon standing waves. This is shown pictorially in Figure c. For non-normal incident light, it is also possible to excite symmetric plasmon standing waves as the symmetry of the system is broken. Note that there is also a small peak located near L/λSPP = 0, which is a LSPhP mode that is visible without graphene (see Figure b); it is present for small cavities because of near-field coupling between the two SiC slabs. Our findings have some similarities with earlier work on the plasmonic coupling between graphene SPPs and magnetic polaritons in silver gratings.[33] Note that the cavities used in that work are very deep (a width of 300 nm and a height of 2 μm) as compared to that of our system, where the cavity height and the width studied are always equal or comparable, which makes the experimental realization potentially much simpler. The highest field enhancement in our system is located on the graphene layer, which is very convenient for potential applications in molecular sensors. Note that the fact that the peak position does not change for varying Λ (for a fixed cavity length) is a confirmation that these are localized cavity modes and are not modes excited by diffraction effects owing to the grating periodicity.[34] In an experiment, it would be more natural to explore these standing wave modes by scanning the excitation light wavelength for fixed geometric parameters; in Figure b, we show this for various chemical potentials for L = 1.06 μm, w = 2.0 μm, and h = 1 μm. We find that as the Fermi energy is increased the absorption peak red shifts (as the SPP wavelength changes with EF) and grows in intensity. This illustrates that our system can act as a tunable cavity, thanks to the inclusion of the graphene sheet. In Figure c, we show how the grating height impacts the absorption of graphene; we find that the absorption intensity is strongly dependent on the height. When the height is changed, the graphene SPP wavelength does not change; this is because when h is large (h > 500 nm here), then eq applies and λSPP depends only on the material directly above and below the suspended graphene (in this case, air).

Figure 4

(a) Absorption of the graphene layer at different geometric parameters L/λSPP of the SiC grating. The peaks at integer numbers of L/λSPP correspond to the standing waves of the graphene SPPs as indicated by the above near-field plot of Ey for each peak. The electric field (here, normalized to the incident field magnitude) goes to zero at the boundaries, meaning that the plasmon modes must be antisymmetric. (b) Absorption of the graphene layer against excitation wavelength for fixed geometric parameters: L = 1.06 μm, w = 2.0 μm, and h = 1 μm (shown for different Fermi energies). (c) Absorption of the graphene layer at different geometric parameters for different grating heights h and EF = 0.64 eV.

To confirm our intuition about these standing wave modes, we use the graphene dispersion in eq along with a Fabry–Pérot modelwhere δϕ is the phase shift. Taking a phase shift of −π, we find a good agreement with the simulation results (see the blue lines in Figure d) in the region near the TO energy where the grating acts as a perfect electric conductor (i.e., plasmon standing waves in the cavity). At frequencies away from this, the simple model breaks down as the phase shift is frequency-dependent and will differ from −π. This is due to the grating not acting as a perfect reflector any more and allowing some field penetration into the SiC. To quantify the suitability of the cavity as an experimental system for strong light–matter interactions, we consider the ratio Q/V. To calculate the volume, we consider a square cavity (with sides of length L, we use w = 1 μm and L is calculated using a period Λ that sets up a plasmon standing wave) with a plasmon standing wave setup in both directions; the dimension perpendicular to the graphene sheet is given by the out-of-plane decay of the plasmon: [kSPP(ω)][35] (as kSPP ≫ 2π/λ, then kz ≈ ikSPP, and for 12 μm incident wavelength, this distance corresponds to 0.17 μm). The quality factor can be estimated using the equation Q ≈ ω/Δω, where we obtain the line width Δω using a Lorentzian fit. At a wavelength of 12 μm, we find a value of Q/V = 8 × 105/λ03, which is comparable to the values found for other graphene nanoresonators.[4]

Another interesting feature, revealed in Figure d (and a zoom-in is provided in Figure a,b), is the presence of an anticrossing point in the absorption spectrum close to the LO phonon frequency; this is a manifestation of strong coupling between the two modes, the graphene plasmon and a localized cavity SPhP,[13] and can be intuitively understood by considering two coupled harmonic oscillators.[26] The shape of the Rabi splitting shown in Figure a,b is a consequence of the dispersive graphene SPP (given by eq ) and the LSPhP at a fixed wavelength of 10.6 μm. The strong coupling regime is very dependent on the level of the damping of the system; thus, the small line widths of graphene plasmons and SiC SPhPs are favorable for the observation of Rabi splitting. It should be noted that at frequencies close to the LO phonon energy graphene SPPs can become lossy because of substrate phonons.[4] In our cavity-grating system, a substantial amount of the graphene is free-standing and not in contact with the substrate, which should negate this.

Figure 5

(a) Zoom-in of the absorption in the graphene layers (for w = h = 1 μm) shown in Figure d between the wavelengths 10.2 and 10.8 μm, highlighting the Rabi splitting for different modes. (b) Zoom-in for m = 2 mode (the purple dashed line indicates the LSPhP wavelength and the cyan dashed line shows the graphene SPP dispersion using the Fabry–Pérot model with a phase shift of −1.35π). The blue line shows the resulting splitting of the hybrid mode.

In Figure , we show a slice of the absorption in the graphene, apply a sum of two Lorentzian functions to fit the curve, and quantify the strength of the coupling. We find the peak separation (i.e., the Rabi splitting) to be ωR = 1 meV. This value can then be used with a coupled harmonic oscillator model to describe the anticrossing[36]where we take the LSPhP energy to be fixed at a value of 0.12 eV, the graphene SPP dispersion is given by eqs and 4, and we use the phase shift as a fitting parameter as it can no longer be expected to be −π at these wavelengths. The result for the fit to the m = 2 mode is shown in Figure b, where a good fit is obtained using a phase shift of δϕ = −1.35π; this indicates that at these frequencies SiC does not act as a perfect reflector.

Figure 6

Spectra shown correspond to L = 1.45 μm, which means that Λ = 2.45 μm. The summation of two Lorentz equations is used to fit these peaks. The full widths at half maximum of these two peaks are 0.004 rad/μm (W1) and 0.005 rad/μm (W2). The peak distance is D = 0.005 rad/μm. The ratio D/W = 1 quantifies the strength of the coupling and shows that this is the strong coupling regime.

Also shown in Figure are two near-field plots showing the different nature of the hybrid mode depending on the frequency; at lower frequencies, the mode is more phonon-polariton-like and the absorption is situated at the bottom of the cavity. At larger frequencies, the mode is more graphene-plasmon-like and the absorption is concentrated in the graphene layer on the top. Thus, over small frequency range, one has a lot of spatial control of where absorption takes place in this system, which is a consequence of strong coupling between the modes. Further control of the relative contribution of each mode is given by the cavity width and the doping level. To quantify the coupling, we compare the line width of the constituent modes (W1 and W2) to the splitting (D), for Rabi splitting to be experimentally observable D ≳ W.[26] Taking the largest line width, we get the ratio of D/W = 1, which justifies considering this interaction of the two modes as strong coupling.

Conclusions

We have studied a graphene and SiC cavity system and have shown its suitability as an experimentally realizable, tunable cavity that could be useful for cavity QED and molecular sensing in the mid-IR range. We have shown that the inclusion of a graphene monolayer on a SiC grating leads to a number of new modes in the low absorption part of the reststrahlen band. These modes can be explained as antisymmetric plasmon standing waves, set up by the SiC grating that acts as a perfect conductor at certain frequencies, which are well-explained by a simple Fabry–Pérot model. By altering the chemical potential of the graphene or changing the cavity width, one can change the spectral position of these peaks, meaning the system acts as a very tunable plasmonic cavity. Near the LO phonon energy, we find evidence of strong coupling between the graphene SPP and the LSPhP, leading to a hybrid mode[14] that exhibits characteristics of both constituent modes. Our results also offer promise that a similar coupling with local trace level quantities of molecules could be possible. We find that over a small frequency range the spatial profile of the hybrid modes can change significantly. The tunability, high quality factor, and low modal volume make this system an ideal setup for molecular sensing and to study cavity QED.