Plasmonic Characteristics of the Graphene-Photonic Crystal Composite Structure in the IR Regime

In this work, a graphene-photonic crystal composite structure is proposed, in which graphene nanoribbons are placed on top of a photonic crystal that is composed of two alternating dielectric materials. The optical properties of the composite structure are investigated in the infrared regime, by varying the Fermi energy and the layer number of the graphene, as well as the polarization angle of the incident light. Plasmonic resonance effects are revealed and the resonance wavelength, the peak width, and the peak intensity are determined. Quantitatively, the polarization-angle-dependent peak intensity is well fitted to a trigonometric function. It is demonstrated that the resonance peak is quite sensitive to the dielectric material that is adjacent to the graphene. It is also shown that the structure has a high sensitivity of 7660 nm/RIU to the refractive index of the surrounding environment, with a figure of merit of 6.38. Based on the results, the graphene-photonic crystal composite structure proposed in this work may have potential applications in designing of plasmonic devices, such as ones that can be used to detect subtle variations in the refractive index of surrounding environments.


Introduction
Conventional plasmonic materials have a relatively low performance such as slow response and limited tunability [1][2][3]. Compared with plasmonic metals, graphene surface plasmons are dynamically tunable. The characteristic wavelength can be adjusted in the fixed structure, and hence its optical properties can be modified; this makes up for the difficulty of changing the optical properties of metallic materials whose structure is fixed. In addition, the propagation length of graphene surface plasmons can reach as long as the micron scale, which greatly broadens its application area. Thanks to its unique electronic properties, graphene, the single layer of graphite, that consists of hexagonally arranged carbon atoms, however, possesses great potential for the design of plasmonic devices with fast response and broad tunability, including optical switches [4,5], filters [6,7], and metasurfaces [8,9]. To date, a variety of structures that are based on graphene plasmonics have been proposed and intensively investigated in the infrared regime. For example, tunable plasmonic devices using graphene insulator stacks in the near-infrared band were experimentally demonstrated [10]. A monolayer graphene structure with tunable gates in the mid-infrared range was proposed to support the fabrication of planar nano-integrated plasmonic circuits [11]. Simulations were performed to achieve perfect plasmonic absorption in the far-infrared band using a metal-grapheneinsulator-metal structure [12]. Two-dimensional plasmonic crystals composed of graphene nanodisks were proposed, and the isotropic distribution of the negative/positive effective permeability and permittivity was numerically demonstrated [13]. Besides, the plasmonic properties of bilayer graphene nanostructures were numerically simulated and the associated plasmonic resonance effects were reported [14]. Recently, several graphene square nano-resonator-based structures were also designed, which may be used as basic modules for optical calculations and signal processing in the mid-infrared band [15].
In addition to simple graphene nanostructures, composite structures that are constituted by graphene and other optical functional materials become more and more attractive, since several unique properties have been reported to the community. Among them, photonic crystals (PCs), the artificial materials that are made of alternating compositions of dielectric materials with different refractive indices, have drawn much attention in the past decades. So far, composite structures that consist of graphene and PCs have shown a variety of new properties, via both theories and experiments. For instance, a highly sensitive dual-core photonic crystal fiber for surface plasmon resonance (PCF-SPR) biosensors with a silvergraphene layer was investigated, demonstrating that the spectra of the sensors could be optimized by adjusting the structural parameters, which could be widely used in biological and biochemical detection [16]. A PC structure embedded in graphene was proposed to achieve broadband terahertz absorption of graphene and improve the absorption efficiency [17]. The absorption properties of a metal-graphene PC-metal composite structure in the visible region were also investigated, providing a useful reference for the development of graphene-based multiband optical absorption materials [18]. Based on the experiments with one-dimensional strain-tunable PCs, it was demonstrated that 1D graphene dielectric PCs with controllable photonic band gaps that were achieved by uniaxially straining graphene were capable of showing contraction or broadening along with different chain crystal directions [19]. In addition, the magneto-optical properties of one-dimensional PCs with graphene defect layers were also investigated, and the results showed that the steady-state behavior of the medium depended entirely on the strength of the magnetic and coherent coupling fields [20].
In this work, a composite structure consisting of graphene nanoribbons and photonic crystals is proposed. In the infrared wavelength range from 10 to 60 m, the plasmonic properties of the composite structure are investigated, with varying the parameters including the Fermi energy, the layer number of the graphene, and the polarization angle of the incident light. The electric field distribution at the graphene plane is presented, and tuning effects of the resonance wavelength, the peak width, and the peak intensity are also demonstrated. Besides, the sensitivity of the composite structure to the surrounding dielectric environment is  Fig. 1 Schematic of the graphene-PC composite structure. A set of graphene nanoribbons was placed on top of the PC that consisted of two alternating dielectric materials (labeled A and B, respectively). In the figure, d g is the width of the graphene nanoribbons, and h 1 and h 2 represent the thicknesses of materials A and B, respectively. A plane-wave light was normally incident along the -z-axis (not shown for brevity), and the light's polarization ( p ) was defined as the angle between the electric vector of light with respect to the x-axis. The graphene nanoribbons and the PC were all infinite in the x-y plane addressed, with the sensitivity and the figure of merit (FOM) being determined. Note that the proposed structure is only studied by numerical simulations in this work; in practice, similar systems based on graphene and photonic crystals have already been experimentally fabricated, for example, a photonic crystal fiber based on graphene was synthesized by chemical vapor deposition in ref. [21].

Structure and Method
As indicated in Fig. 1, two dielectric materials were alternately layered to form a photonic crystal structure, and a set of graphene nanoribbons was placed on top of the PC. The width of the graphene nanoribbons is labeled d g . The thicknesses of dielectric A and B are indicated in h 1 and h 2 , respectively. In all calculations, the alternating dielectric materials A and B were repeated 9 times, which was found to be sufficient to capture the key features of the optical properties of the PC. To investigate the optical properties of the graphene-PC composite structure, the Fermi energy of the graphene (labeled c ) and the light's polarization angle ( p ) were systematically varied.
In this work, the finite-difference time-domain (FDTD) method was used for all simulations [22]. A plane-wave light was normally incident along the -z-axis in the wavelength range of 10-60 m. The perfect match layer (PML) boundary condition was used in the z direction, and periodic boundary conditions were employed in the x − y plane. To avoid the possible artifacts that might be induced by the simulation method, the mesh size was always kept less than 1/10 of the shortest wavelength studied in the simulation region of non plasmon-carrying media. In this paper, a plane monitor measuring the light transmittance (located in the x − y plane) was placed below the composite structure, and the electric field distribution of the structure was also collected utilizing a linear monitor placed along the z direction (not shown in Fig. 1 for clarity). Note that the values for the thicknesses of the dielectric materials were carefully chosen so that the bandgap of the PC fit in the wavelength region that was studied in this work.
For single-layer graphene, the optical constant may be derived from its surface conductivity, , by the following relationship where intra and inter are the intraband and interband terms, respectively, which may be given by the following equations [23] In Eq. 2, e is the electron charge, k B is the Boltzmann's constant, ℏ is the reduced Planck's constant, and T is the temperature (T = 300K was used in this work). is the angular frequency of the incident light, and c is the Fermi energy of graphene. Γ is the scattering rate, which is normally determined by experiments. In this work, the phenomenological scattering rate was assumed to be Γ = 0.11meV, this number was based on the typical values of the carrier mobility, and a similar value was also used in the literature [24][25][26]. Note that the interband term has no analytical solutions, and yet an approximation can be made when k B T ≪ c , ℏ [27]. In this work, the above condition was held; thus, the interband term was approximately expressed by Eq. 3.

Results and Discussions
The light transmittance of the PC that was composed of dielectric materials with different refractive indices was first studied, and the resultant spectra in the range of 20-60 m are shown in Fig. 2. In the figures, the graphene's Fermi energy was c = 0.20eV, the layer number was N = 1, and the width of the nanoribbons was d g = 300nm. In each figure of Fig. 2, the refractive index of dielectric B was varied while that of dielectric A was fixed. Taking Fig. 2b as an example, the refractive index of dielectric B was changed from 2.2 to 3.3, whereas dielectric A was kept 2.0. It is obvious that a sharp peak in the transmittance spectrum is observed for all cases, which is a clear manifestation of the plasmonic resonance induced in the composite nanostructure under the irradiation of the incident light. By examining all figures in Fig. 2, at a given value for the refractive index of dielectric A, the position of the resonance peak has few variations upon the change in dielectric B's refractive index. This reveals that the resonance wavelength of the composite structure is almost independent of the refractive index of dielectric B. However, by comparing the results through Fig. 2a to c, it is evident that the resonance wavelength was increased from 25.8 to 44.2 m as the refractive index of dielectric A was raised from 1.46 to 2.9. This demonstrates that the resonance peak is influenced by the dielectric material that is in close proximity to the graphene nanoribbons.
To further verify the above observations, the plasmonic properties of the graphene-PC composite structure are quantitatively analyzed, by varying the value for the refractive index of dielectric A. The plasmonic resonance-related parameters are tabulated in Table 1, including the resonance wavelength (labeled 0 ), the full width at half maximum (FWHM, labeled 0 ), and the relative peak intensity (denoted I 0 ). In this work the relative peak intensity was defined by I 0 = (1 − T r )×100% , where T r is the transmittance at resonance. For example, when the transmittance at resonance is 0.049, its relative peak intensity is 95.1% . In Table 1, dielectric B was kept to be TiO2 with a refractive index of 2.9. The dielectric A's refractive index was changed among 1.46, 1.8, 2.0, 2.2, and 2.4, representing the materials of SiO 2 , Al 2 O 3 , Si 3 N 4 , Sb 2 O 3 , and ZnS, respectively. h 1 was 290nm, and h 2 was 500, 420, 362.5, 340, and 310nm, respectively. Regarding Table 1, with increasing the value of the refractive index of the dielectric material that is in close proximity to the graphene, it is observed that the resonance wavelength is increased from 25.8 to 37.5 m, correlating to the redshift trend that is addressed in Fig. 2. It is obvious that Fig. 4 Electric field distribution along the z-axis for the cases shown in Fig. 3. The Fermi energy of the graphene was c = a 0.20eV, b 0.25eV, c 0.30eV, d 0.35eV, and e 0.40eV, respectively Table 2 The values of the resonance wavelength ( 0 ), the FWHM ( 0 ), and the relative peak intensity ( I 0 ), determined from the resonance peaks shown in Fig. 3  the FWHM is also broadened from 0.87 to 2.88 m, while the relative peak intensity is nearly unchanged with a value of around 95% . It is worthwhile noting when the refractive index was less than 1.46, the obtained pattern was similar, and no examples are given in this paper for brevity. According to Eqs. 1 through 3, it is known that the surface conductivity of graphene can be tuned by changing its Fermi energy, which in turn affects the optical properties of graphene due to its surface plasmon excitations, and this effect was demonstrated in a previous work [28]. In this work, the effect of the graphene's Fermi energy on the optical charateristics of the graphene-PC composite structure was also investigated. For simplicity, the dielectric materials were set to be Si 3 N 4 and TiO 2 , respectively. The layer number of graphene was N = 1. The transmittance of the composite structure was calculated by varying the Fermi energy of the graphene in the wavelength range of 20-45 m, and the corresponding results are shown in Fig. 3. In the figure, a blue-shift trend of the resonance peak is witnessed in the transmittance spectrum, with increasing the Fermi energy from 0.20 to 0.40eV. In order to probe the resonance properties, the electric field of the composite structure was further determined, and the resultant distributions are presented in Fig. 4.
Regarding Fig. 4, it is clear that the "hot region" of the electric field in the wavelength domain strongly depends on the value of graphene's Fermi energy. As the Fermi energy is raised from 0.20 to 0.40eV, the electric field "hot region" is gradually shifted from the greater wavelength towards the smaller wavelength, which coincides with the blue-shift trend that is indicated in Fig. 3. Furthermore, the resonance properties of the composite structure were quantitatively derived from the spectra given in Fig. 3, and the corresponding values are presented in Table 2. It is clear from Table 2 that with increasing the Fermi energy, the resonance wavelength is decreased (i.e., from 32.5 to 23.0 m), and the FWHM is gradually broadened from 1.38 to 1.60 m. The relative peak intensity appears subtle variations between 96.6 and 98.2%.
In addition to the Fermi energy, the layer number is another parameter that may affect the surface conductivity and hence the optical properties of graphene-based systems. To study the effects of graphene's layer number on the plasmonic properties of the composite structure, another set of simulations were also performed, where the layer number was changed from N = 1 to 5. The simulated transmittance curves are shown in Fig. 5 in which c = 0.20eV and the dielectric materials were set to be Si 3 N 4 and TiO 2 , respectively. It is clear from Fig. 5 that the resonance peak is blueshifted with increasing graphene's layer number, revealing an increase in the energy that is associated with the plasmonic mode in the graphene since the surface conductivity is enhanced. Similar effects were also reported and addressed in previous work [29]. Besides, the electric field distribution of the composite system was also calculated, and the results are shown in Fig. 6. In Fig. 6, the electric field "hot region" is gradually tuned from the greater wavelength to the shorter wavelength; this is consistent with the blue-shift fashion of the plasmonic resonance wavelength that is revealed in Fig. 5. This observation illustrates the tunability of the plasmonic resonance of the graphene-PC composite structure, by adjusting the layer number of the graphene. Similar to Table 2, the values for 0 , 0 , and I 0 were also measured from Fig. 5, and the results are given in Table 3. Referring to Table 3, the resonance wavelength is significantly decreased as the layer number is increased. However, the FWHM and the relative peak intensity are barely changed. Note that the anomaly in 0 for N = 3 is accounted for by the fact that the plasmonic resonance peak induced in the graphene with N = 3 (i.e., at 19.0 m) coincides with the badgap region that is associated with the PC (i.e., around 17-22 m).
Apart from the structural parameters studied above, it is also known to the community that surface plasmons may be affected by the polarization of the incident light, especially for graphene nanoribbon-based structures, since the plasmonic wave that is induced in the nanoribbons depends on the relative angle of the light's electric vector with respect to the ribbon axis. To probe this polarization effect, the composite structure was further studied by varying the polarization angle (labeled p ) from 0 to 90 o . The resultant transmittance curves are shown in Fig. 7, and the electric field distributions are given in Fig. 8, where the dielectric materials were set to be Si 3 N 4 and TiO 2 , respectively. The layer number of graphene was N = 1. Regarding Fig. 7, it is evident that the resonance wavelength appears constant in the transmittance curve as the polarization angle is changed from 0 to 90 o . Referring to Fig. 8, with changing the polarization angle from 0 to 90 o , it is observed that the field intensity is gradually weakened, while the spectral position of the "hot region" remains unchanged (i.e., at 32.5 m). Similar effects were also observed and addressed in ref. [30]. Referring to Fig. 7 again, the intensity of the plasmonic resonance peak is decreased with increasing the polarization angle. In order to study this polarization effect quantitatively, the relative peak intensities were determined and their values are indicated as dots in Fig. 9. The values were then fitted to a trigonometric function as follows, Fig. 6 Electric field distribution along the z-axis for the cases shown in Fig. 5. The layer number was N = a 1, b 2, c 3, d 4, and e 5, respectively. Note that the intensity of the color bar is different in each figure Table 3 The values of the resonance wavelength ( 0 ), the FWHM ( 0 ), and the relative peak intensity ( I 0 ), determined from the resonance peaks shown in Fig ig. 8 Electric field distribution along the z-axis for the cases shown in Fig. 7. The polarization of the incident light was p = a 0 • , b 30 • , c 60 • , and d 90 • where I A is an amplitude that captures the variation of the relative intensity ( I 0 ), and I off is an offset constant; they are both in the same units as I 0 . off is an offset angle that is in the units of degrees. is a coefficient that characterizes the "frequency" of the variation. The best fit is also presented as (4) I 0 = I A cos( × p + off ) + I off a solid curve in Fig. 9, and the best fitting parameters were derived to be I A = 41.1% , I off = 55.5% , off = 0.15 o , and = 2.06. By examining Fig. 9, a good agreement between the simulated data and the best fit is evident, and this is a clear manifestation of the effect associated with the polarization of the incident light that is irradiated on the plasmonic system. Furthermore, it is known to the community that changing the refractive index of the surrounding medium also has an effect on the characteristics of plasmonic systems. Therefore, the equipartition excitonic response of the graphene-PC composite structure was also investigated, with a perturbation of the refractive index of the surrounding medium. The corresponding transmittance spectra are shown in Fig. 10. In the simulations, the graphene nanoribbons were c = 0.20eV and N = 1; dielectric materials A and B were Si 3 N 4 and TiO 2 , respectively. It is observed from Fig. 10 that the resonance wavelength is red-shifted with increasing the refractive index of the surrounding medium. This is consistent with the known fact that the plasmonic resonance properties also dependent on the surrounding medium in which the plasmonic structures are embedded. For the graphene-PC composite structure proposed in this work, the resonance peak wavelength is shifted from 32.5 to 36.3 m, as the refractive index is increased from 1.0 to 1.5.
As a possible sensor, the sensitivity (S) of the composite structure may be estimated by using the following relationship,  Fig. 9 The variation of the relative peak intensity as a function of the light's polarization angle, determined from the resonance peaks shown in Fig. 7. The solid curve is a best fit to Eq. 4 where Δ is the wavelength shift of the resonance peak, and Δ n is the change in the refractive index units (RIU) [31]. Based on Eq. 5, the sensitivity of the proposed structure was determined to be 7660nm/RIU. Usually, the figure of merit (FOM) can be used to evaluate the sensor performance, which is another important parameter to measure sensor performance. The FOM in sensors is usually defined as the ratio between sensitivity and FWHM of the peak (FOM = S/FWHM). A FOM of 6.38 is achieved in this work. The structures and sensitivities of several refractive index sensors are listed in Table 4, as a comparison with the structure presented in this work. A plasmon refractive index sensor based on a ring-type pentagonal was found to have a maximum sensitivity of 1500nm/RIU [31,32]. A metal-insulatormetal waveguide refractive index sensor structure based on a crossed structure embedded in a toroidal resonant cavity was designed to obtain a maximum sensitivity of 2325nm/ RIU [31,32]. A D-shaped optical refractive index sensor with graphene-gold deposited platform was proposed, which can reach a maximum sensitivity of 4391nm/RIU [33]. In this work, a sensitivity of as great as 7660 nm/RIU can be achieved in the proposed graphene-PC structure.

Conclusions
In this work, a graphene nanoribbon-photonic crystal composite structure has been proposed. It has been found that the optical properties of the photonic crystals depend only on the dielectric material that is in close proximity to the graphene. The transmittance curves and the electric field distributions of the composite structure have been calculated in the infrared range, by varying the Fermi energy and the layer number of the graphene, respectively. The resonance wavelength, the FWHM, and the relative peak intensity have been determined, according to the resonance peaks. The incident light's polarization effects on the plasmonic resonance have also been addressed, and the relative peak intensity as a function of polarization angle has been found to be well fitted by employing a trigonometric function. It has been demonstrated that the plasmonic resonance properties of the graphene-PC composite structure can be systematically tuned, and the proposed structure has a potential to be used as a sensor. The sensitivity and FOM have been estimated to be 7660nm/RIU and 6.38, respectively. Based on the results, the graphene-PC composite structure proposed in this work may be of significance in designs of surface plasmon-based optoelectronic devices that can operate in the infrared regime.