Graphene-based plasmonic structures feature large tunability, high spatial confinement, and potentially low loss, and are therefore an emerging technology for unconventional manipulation of light. In this paper, we demonstrate electrically tunable terahertz plasmonic crystals consisting of square-lattice graphene periodic anti-dot arrays on a substrate. Transmission spectroscopy reveals multiple distinct resonances arising from excitations of graphene surface-plasmon–polariton (SPP) modes on different branches of the SPP dispersion curves inherent to the periodic structures. The resonance frequencies are readily tuned electrostatically with the Si back-gate and exhibit the dependency on the carrier density unique to SPP in graphene. Simulations show excellent agreement with the experiments and further illustrate the symmetry-based selection rule for the excited graphene SPP modes. Such graphene plasmonic crystals may lead to a broad range of applications including plasmonic waveguide and transformation optics. Exploiting higher-order graphene SPP modes is an effective way to further facilitate field localization and enhancement.
© 2015 Optical Society of America
Among other extraordinary electronic and optical properties of graphene , its ability to support highly confined surface-plasmon-polariton (SPP) waves in the mid-infrared to terahertz (THz) frequency range with large tunability and potentially low loss has motivated considerable research efforts in recent years [2–15], in light of a broad range of potential applications such as tunable plasmonic waveguides , reconfigurable metasurfaces , transformation optics , optical modulators , photo-detectors , chemical sensors , and so on. Similarly to SPP waves at a metal-dielectric interface , as a result of their high spatial confinement, SPP waves in an intact graphene sheet do not couple to free-space electro-magnetic (EM) waves due to a large momentum mismatch. Various approaches to bridge the momentum gap and achieve excitation of graphene SPP with free-space EM waves have been theoretically studied and/or experimentally demonstrated, including patterning graphene to form lower-dimensional structures (e.g., ribbons and disks) as cavities for localized SPP modes [6–10], or exciting SPP waves in continuous graphene with a near-field high-momentum components of light scattered by sub-wavelength structures such as a sharp metal tip [12,13] or a grating . Alternatively, efficient coupling can be accomplished with periodically modulated graphene such as a graphene anti-dot array which possesses modified dispersion relations of SPP waves compared to that of pristine graphene . In a fashion analogous to photonic crystal slabs [24,25], multiple branches of graphene SPP dispersion curves emerge and form a band-structure in the first Brillouin zone of the anti-dot array as a result of the Bragg scattering from the periodic lattice, thus allowing for direct coupling of free-space EM waves to selected SPP modes above the light line. In addition to the large tunability intrinsic to graphene-based structures, the plasmonic band-structure of a graphene anti-dot array can be tailored by its structural details, making such structures promising versatile building blocks for more complex devices in different applications. Periodic anti-dot arrays have previously been studied in the context of two-dimensional electron gas (2DEG) in semiconductor hetero-structures [26–29] where intriguing collective carrier excitations (plasmonic and magneto-plasmonic modes) brought forth by the anti-dot arrays were observed; however, those investigations were limited at cryogenic temperatures. Plasmonic crystals employing graphene anti-dot arrays operating at room temperature have been demonstrated recently [30,31] where resonance features in the transmission spectra attributed to graphene SPP modes were observed in hexagonal-lattice graphene anti-dot arrays. However, neither electrostatic tuning of these resonances nor resolved multi-band resonances inherent to the periodic structures was reported.
In this paper we demonstrate various large-area electrically tunable THz plasmonic crystals consisting of graphene periodic anti-dot arrays in the square-lattice configuration. Transmission characterizations evidently reveal multiple resonances, corresponding to modes on different branches of the graphene SPP dispersion curves. Carrier density () dependent measurements are conducted by employing electrostatic doping with the Si back-gate, which confirm that the resonance frequencies () shift according to the scaling law unique to SPP waves in monolayer graphene (). The experimentally observed resonances show excellent agreement with the full-wave simulation results which further reveal the symmetry-based selection rule for the excited SPP modes.
2. DESIGNS OF THE GRAPHENE ANTI-DOT ARRAY THz PLASMONIC CRYSTALS
We focus on square-lattice anti-dot array structures with round apertures in part due to their isotropic nature, high symmetry, and structural simplicity [only two structural parameters, i.e., the lattice constant and the aperture diameter as illustrated in Fig. 1(a)]. Another key motivation for studying such square-lattice structures is that in general half of the higher-order dispersion curves in the associated SPP band-structure host modes that can be efficiently excited by free-space EM waves (referred to as coupled SPP modes later), whereas this ratio is 1/3 for hexagonal-lattice structures. This is a consequence of the symmetry-based selection rule for the coupled SPP modes in such 2D crystal type structures . Similarly to photonic crystal slab structures [24,25], the SPP band-structure of a graphene periodic anti-dot array  is a combined result of the zone-folding of the single SPP dispersion curve of an intact graphene sheet into the first Brillouin zone by different orders of lattice-induced Bragg scatterings, and the mixing and degeneracy-lifting due to the finite-sized apertures. As a first-order estimate, such a band-structure can also be analytically obtained with the empty lattice approximation , and an example is presented in Fig. S1 (Supplement 1). Near the -point (center) of the first Brillouin zone, each SPP dispersion curve except for the lowest branch lies above the light line and hence it is possible for the associated SPP modes to couple to free-space EM waves. Such a coupling would be manifested as enhanced absorption and reflection, and thus reduced transmission in the form of a resonance at the frequency of the SPP mode. However, since the -point is a high-symmetry point, the field profile (or equivalently the charge distribution) of any SPP mode in the vicinity of the -point also preserves certain symmetry properties of the lattice structure, which belong to the point group for the case of a square lattice . Besides the momentum-matching requirement, the symmetry property of a SPP mode also has to be compatible with that of the incident EM wave of a specific polarization to achieve efficient coupling. More specifically, for the square-lattice structures studied in this work, the profiles of the -direction electric field of coupled SPP modes in the plane of the lattice structure (–plane) should be symmetric under one mirror reflection while anti-symmetric under the orthogonal mirror reflection. SPP modes above the light line but with different symmetry properties are consequently uncoupled modes. Such a symmetry-based selection rule for coupled SPP modes can be more intuitively perceived from the simulation results in Fig. 1. The simulated transmission, reflection and absorption spectra of a graphene anti-dot array on a substrate are plotted in Fig. 1(b). Two sharp resonances associated with two different graphene SPP modes are present in the frequency range concerned, with their electric field profiles shown in Fig. 1(c). Both field profiles, under mirror reflection operations, are symmetric with respect to the plane and anti-symmetric with respect to the plane, indeed matching the symmetry of the incident EM wave which is linearly polarized in the -direction. It is worth noticing that the higher-order mode in Fig. 1(b) is considerably stronger than the simulated higher-order mode of a hexagonal-lattice graphene anti-dot array with a similar filling factor of the apertures in Ref. . From the perspective that different branches of the SPP dispersion curves are due to Bragg scatterings associated with different reciprocal lattice vectors [23,29], the reciprocal lattice vectors corresponding to the field profiles in Fig. 1(c) are [0,1] (top) and [0,2] (bottom), respectively. However, as the relatively large apertures introduce strong mixing of different orders of Bragg scatterings, both modes also contain significant contributions from the reciprocal lattice vector [1,1] as is markedly revealed by the 2D FFT of the field profiles in Fig. 1(c). Since the spatial confinement factor of graphene SPP ( being the SPP wavelength) decreases with the free-space wavelength , exploiting higher-order modes in the THz frequency range is an effective way to achieve high spatial confinement comparable to that in the mid-infrared [12,13] and significantly higher than that of noble-metal-based plasmonics .
3. SAMPLE FABRICATION AND BASIC CHARACTERIZATIONS
Square-lattice graphene periodic anti-dot arrays with various dimensions are designed and implemented on large-area monolayer graphene on a substrate. The details of the investigated structures are listed in Table 1. These structures are designed to ensure that the dominant coupled graphene SPP modes are in the THz frequency range and well below the frequency of the surface optical phonon mode in the substrate at (14.5 THz) . The graphene sheet is patterned with deep-UV photolithography and plasma etching (fabrication details in Supplement 1). Each graphene anti-dot array structure has a dimension of . An SEM image detailing the microscopic structure of a fabricated sample is shown in Fig. 2(a). The fabrication process unavoidably introduces small discrepancies between the diameters of the etched apertures (also listed in Table 1) and their designed values, which are taken into account when analyzing the experimental data. Metal electrodes are deposited to form contacts with the graphene structure in a field-effect transistor-type configuration with the Si substrate as the back-gate, facilitating electrostatic tuning of the carrier density and characterizations of the electrical properties of the graphene sheet. Several unpatterned graphene samples are prepared in a similar configuration, and electrical characterizations show that the graphene sheet is naturally p-doped with an average hole mobility of (Fig. S2, Supplement 1). Characterizations on the patterned graphene samples suggest that the carrier mobility is not significantly affected by the fabrication process. Raman spectroscopy is performed on several graphene anti-dot arrays, and signature Raman spectra of monolayer graphene are observed [Fig. 2(b)].
4. EXPERIMENTAL DEMONSTRATION OF TUNABLE MULTI-BAND SPP EXCITATIONS
To probe the coupled SPP modes of the graphene anti-dot arrays, transmission spectroscopy is carried out with a Fourier-transform infrared (FTIR) spectrometer (Bruker Vertex 80v). The samples are placed in the vacuum compartment of the FTIR, and all the measurements are conducted at room temperature. Due to the isotropic nature of the square-lattice anti-dot arrays with round apertures, the transmission should not depend on the polarization of the incident illumination (experimentally verified); therefore, unpolarized broadband emission from a Globar is employed as the illumination beam in a nearly normal-incident (slightly focusing) scheme for the measurements. Following the conventional procedure , the transmission spectra of each anti-dot array are measured at various back-gate voltages and hence different carrier densities including the charge neutrality point (CNP). Transmission extinction spectra at biases away from the CNP defined as 1-() are extracted, which clearly reveal the carrier-density-dependent spectral features of the graphene-based structure by reliably eliminating the background spectral information due to the substrate and the measurement setup. Figure 3 shows the extinction spectra of four different anti-dot arrays (SQ1–SQ4) in comparison with those of an unpatterned graphene sample, whereas the spectra for the other investigated structures (SQ5–SQ7) are presented in Fig. S3 (Supplement 1). In sharp contrast to the extinction spectra of the unpatterned graphene sample [Fig. 3(e)], the only feature of which is a Drude-type tail progressing with increasing carrier density, distinct peaks with rising prominences are observed in the extinction spectra of all the graphene anti-dot arrays [Figs. 3(a)–3(d)]. Moreover, in each set of spectra shown in Figs. 3(b)–3(d), two peaks are clearly visible despite their relatively broad profiles. The positions of all the peaks shift towards the high-frequency side with increasing carrier density, and further quantitative analyses show that the frequency shift is consistent with the carrier-density-dependent frequency scaling law , where is the Fermi energy, unique to SPP waves in monolayer graphene [Fig. 3(f)], confirming the graphene SPP origin of all the observed resonance peaks. The separated resonances in each of Figs. 3(b)–3(d) originate from SPP modes on different branches of the SPP dispersion curves associated with the corresponding graphene anti-dot array, which are further analyzed and confirmed by the simulation results presented in a subsequent paragraph. To the best of our knowledge, such direct experimental observation of the multi-band nature of SPP dispersion curves inherent to the periodic structures of graphene anti-dot arrays has not been reported yet. Higher-order SPP modes were also rarely observed in semiconductor-based 2DEG anti-dot arrays despite their much higher carrier mobility at cryogenic temperatures .
5. SIMULATED PROFILES OF THE EXCITED SPP MODES
Full-wave simulations of the transmission spectra are performed for all the studied graphene anti-dot arrays using finite-element frequency domain methods (details in Supplement 1). For the simulation results presented, the Fermi energy is set to which roughly corresponds to the highest carrier (hole) density achieved experimentally (), and two different values of carrier relaxation time , 1 and 0.1 ps, are assumed where the latter is relatively close to that of the graphene material used. Figure 4 shows the simulated extinction spectra in comparison with the experimental results of the same four structures in Fig. 3, whereas results for the other structures are included in Fig. S4 (Supplement 1). The simulated spectra corresponding to (red curves) show excellent agreement with the experimental results in terms of the number of observable peaks and their positions, although the resonance peaks are more prominent in the simulated spectra, suggesting that the carrier relaxation time of the real samples is even shorter. Scatterings due to the etched graphene edges  and the surface optical phonon [8,10] may have significant impact on the plasmon damping rate. Furthermore, the spectra corresponding to (blue curves) unveil more details on the origins of the observed broader higher-frequency peaks in Figs. 4(c) and 4(d) as well as Figs. 3(c) and 3(d), i.e., they do not stem from a single SPP resonance but contain contributions from multiple SPP resonances on different branches of SPP dispersion curves. These individual resonances are to be clearly distinguished only in structures with much higher carrier mobility.
To further gain insight of the individual coupled SPP modes, electric field profiles are calculated for each SPP resonance in all the structures. Figure 5 shows the -direction electric field profiles (in an –plane right above the graphene sheet) associated with the four prominent resonances in Fig. 4(d). Two distinctive features are noticeable from the field profiles of all the investigated structures. The first feature is that all of the field profiles have the same symmetry properties under mirror reflection operations as those in Fig. 1(c), indeed complying with the symmetry property of the linearly polarized (in the –direction) normal-incident plane wave, as required by the previously mentioned symmetry-based selection rule. The second feature is that each field profile has certain dominant spatial frequencies corresponding to specific reciprocal lattice vectors, which are clearly revealed in the 2D FFT of the field profiles (see the insets). In general, modes with higher energy are associated with higher-order reciprocal lattice vectors and have more enhanced spatial confinement. Due to the large ratio of the aperture diameter to the lattice constant (), the mixing of Bragg scatterings of different orders is significant and hence each mode is associated with multiple dominant reciprocal lattice vectors.
In summary, various large-area graphene periodic anti-dot arrays in the square-lattice configuration were investigated, and multiple electrically tunable resonances in the transmission extinction spectra (originating from excitations of graphene SPP modes on different bands of SPP dispersion curves inherent to the periodic structures) are evidently resolved at room temperature. Numerical simulations confirm the experimental observations and further reveal the symmetry-based selection rule for the excited SPP modes. The presented experimental and simulation results further illustrate the high flexibility of band-structure engineering of such graphene-based plasmonic crystals, demonstrate their advantages over noble-metal-based plasmonics, and can be utilized as design guidelines for more complex structures and devices such as tunable plasmonic waveguides, reconfigurable metasurfaces, and transformation optics. The excitation of higher-order SPP modes observed in such structures is of significant technological interest due to the more enhanced mode spatial confinement and field localization, and also provides an interesting platform for further study of higher-order magneto-plasmonic responses in periodic anti-dot arrays.
European Commission (EC) through project GOSFEL; Swiss National Science Foundation (SNSF) through NCCR QSIT.
The authors thank Sergey Mikhailov for valuable discussions and Christofer Hierold for granting access to a micro-Raman spectrometer. The cleanroom facility FIRST at ETH Zurich is also acknowledged.
See Supplement 1 for supporting content.
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