Enhanced plasmonic light absorption engineering of graphene: simulation by boundary-integral spectral element method

Jun Niu; Ma Luo; Jinfeng Zhu; Qing Huo Liu

doi:10.1364/OE.23.004539

1. Introduction

Graphene has drawn a surge of interest in both academia and industry due to its unique properties. Because of its exceptional electromagnetic and mechanical properties, such as the extremely high quantum efficiency for light-matter interactions, controllable inter-band transition and saturable nonlinear absorption, graphene is considered as a promising alternative for silicon in the next generation advanced optoelectronic devices [1, 2]. Meanwhile, over the past few years, a considerable number of graphene-based optoelectronic devices have been demonstrated and studied. One representative device is the graphene-based photodetector [3, 4]. However, despite the outstanding properties of graphene including fast carrier mobility and high quantum efficiency, some fundamental limitations still need to be overcome. Typically, the optical absorption of floated single atom layer graphene is essentially poor for a satisfying photoresponsivity. Another limitation is that graphene also shows a considerably flat absorption in the visible to near-infrared spectra, i.e., in this spectra, graphene does not provide a frequency selectivity.

One promising approach to overcome these two drawbacks is to utilize the localized surface plasmon resonance (LSPR) of periodically patterned noble metal nanoparticless (NPs). With a properly designed plasmonic antenna, the photo-responsivity can be dramatically improved with significant enhancement of optical response, while simultaneously generating a frequency selectivity for the device [4–6]. In addition, the optimal operation frequency of the device can be effectively tuned by varying plasmonic antenna’s parameters. Previous research reports that gold NP arrays have been demonstrated as a promising plasmonic nano-antenna for graphene-based photodetectors with effective near field excitation and hot electron contribution [8]. With gold plasmonic antennae, we are able to induce strong photovoltage and photocurrent in graphene-based photodetectors through enhanced light absorption in visible and near-infrared spectra.

Another critical aspect of graphene-based nano-scale photonic device design is the accuracy and efficiency of numerical techniques. Despite the promising role of graphene-based photodetectors with plasmonic antennae, its specific design is very challenging without a powerful numerical simulation technique. Currently, the prevailing numerical approaches in this area are concentrated on the finite-domain time-difference (FDTD) method and the classic finite element method (FEM). Despite the flexibility of the FDTD and the FEM, these methods suffer from a slow convergence because of their lower-order approximations. In order to guarantee satisfying accuracy, these methods usually require a sampling density of over 20 points per wavelength (PPW). Meanwhile, even though the time domain method can theoretically simulate a relatively wide spectral band, this strategy may be time- and resource-consuming in specific nano-photonic cases. For instance, when the concerned spectra is very large, the electromagnetic simulation covers both the electrically large and electrically small problems according specific frequency components. A corresponding challenge is that for electrically small problems, a large number of time steps is required to obtain the steady state results; while for electrically large problems, the computation resources required by the classical time domain methods may be prohibitively large due to the required high sampling density. In addition, under some circumstances, the frequency-dependent refractive index of various materials in the visible and near-infrared spectra cannot be described or accurately approximated as functions of frequency in analytical forms. In nano-photonic simulations, an accurate wide bandwidth solution is hence difficult to be obtained.

This paper focuses on light absorption engineering of graphene around visible spectrum by LSPR of gold NPs with the boundary-integral spectral element method (BI-SEM). In previous studies, graphene’s strong light absorption is mainly achieved in the infrared spectrum [7]. However, its performance around the visible spectrum is of great significance in practical engineering designs. The research by J. Zhu et al. [4] theoretically explores the performance and discipline of uniformly placed cuboid Au NPs by the FDTD method, and reaches an optimal light absorption of 30.3 %. Here, with the proposed BI-SEM, we further study the absorption enhancement of graphene with boosted plasmonic effects. Under the objective of improving the performance of graphene-based photodetectors, the advanced light absorption enhancement of graphene is theoretically explored with BI-SEM, which shows an outstanding accuracy and excellent efficiency. Our study shows that with the light concentration by the LSPR and the electromagnetic coupling of gold nano-antennae, the tunable selective optical absorption enhancement of graphene can be achieved in the simulated spectra of 300 nm to 1000 nm. By adjusting the gold plasmonic antennae, a peak graphene absorption rate of 67.54% is obtained in the simulation frequency band. This absorption is enhanced by 30 times compared with that of the floating graphene layer, and 2.2 times with the optimal value of the research by reference [4]. To our knowledge, it is also the highest reported absorption rate achieved inside graphene layer in the concerned spectra.

It is worth noting that the efficient analysis of the optical absorption largely depends on the high performance of the BI-SEM. Compared with the classical FEM and the FDTD, the BI-SEM can efficiently analyze the electromagnetic scattering problem with spectral accuracy, thus enabling fast numerical analysis of large scale graphene-based photodetectors. Considering the possible large number of unknowns when solving the assembled linear equation, an iterative solver can show a poor convergence while a direct solver can be prohibitively expensive. Thus, throughout all the simulation cases, the block Thomas algorithm is utilized to further boost the computational efficiency [21].

2. Simulation of the localized surface plasmon resonance

2.1. Graphene’s refractive index and optical absorption

Based on the Fresnel coefficients approach [15], the complex refractive index of single atom layer graphene can be described by the following equations

ñ (ω) = n (ω) + j k (ω)

\begin{matrix} n (ω) = 3.0, & k (ω) = - \frac{c}{2 ω d n} \end{matrix} \ln (1 - π α)

where α = 1/137 is the fine structure constant and c is the speed of light in vacuum; ω is the angular frequency; d is the thickness of the graphene layer; n(ω) is real part of the complex-value refractive index, and k(ω) is the extinction coefficient which determines the optical absorption in graphene.

The ultrathin graphene is modeled as a 2-dimensional surface characterized by the appropriate boundary conditions in many researches. However, in this research, the key for graphene’s optical absorption engineering is to confine the majority of the incident energy inside the graphene layer with the LSPR. To emphasis this physical process, the single layer graphene is considered as an electromagnetic dielectric medium with a thickness of 0.34 nm in the following study. With BI-SEM numerical analyses, the 3-dimensional bulk description of graphene becomes even more rewarding when the researchers aims to further calculate the amount of energy confined inside the graphene or to analysis the corresponding optical field distribution.

The optical absorption of the graphene layer at an angular frequency is calculated as the ratio between the absorbed power in the graphene and the incident power, as is defined by

A (ω) = \frac{P_{d}}{P_{i n c}} = \frac{\int \int \int_{V} ω n k ε_{0} | E |^{2} d V}{\int \int_{S} S_{i n c} \cdot d S}

where S_inc denotes the Poynting vector of the incident field impinging upon the entire structure; V is the volume of the graphene layer, and the surface integral area S is the upper intersection area of the graphene layer.

Based on above equations, the absorption engineering of the single layer graphene can be analyzed with numerical electromagnetic simulations. Theoretically, the optical absorption of a floating single layer graphene is 2.3% with an optical transmission of about 97.7% [3, 24]. In this research, our goal is to design nano-structures with a high-efficiency frequency domain computational electromagnetic technique, so that plasmonic effects can be utilized for tunable graphene absorption enhancement.

2.2. Computation method

In this work, the frequency domain BI-SEM with block-Thomas acceleration is used to efficiently analyze the absorption engineering of graphene. Under the scheme of BI-SEM, the interior simulation domain is computed by the spectral element method (SEM), while the radiation boundary is modeled by a set of boundary integral equations. For periodic nanostructures simulations, the Bloch periodic boundary conditions are applied to the lateral boundaries, while the top and bottom open boundaries are truncated by the boundary integral solved by method of moments (MoM) [18].

Under the scheme of BI-SEM, the scattering problems reduce to solving the following equations, which combines the weak-form vector Helmholtz, the electric field integral equation, and the magnetic field integral equation:

\begin{array}{r} \int_{V} [{(μ_{r}^{- 1} \nabla \times Φ_{i})}^{T} \cdot (\nabla \times E) - k_{0}^{2} \cdot {(ε_{r} Φ_{i})}^{T} \cdot E] d V \\ - 2 j k_{0} \int_{S} Φ_{i}^{T} \cdot [K ({\tilde{J}}_{s}) - ε_{r} \cdot L (M_{s})] d S^{'} = 2 j k_{0} \int_{S} Φ_{i}^{T} \cdot (\hat{n} \times {\bar{H}}^{i n c}) d S^{'} \end{array}

j k_{0} \int_{S} Φ_{i}^{T} \cdot [μ_{r} L ({\tilde{J}}_{s}) + \frac{1}{2} \hat{n} \times E + K (M_{s})] d S^{'} = j k_{0} \int_{S} Φ_{i}^{T} \cdot (\hat{n} \times E^{i n c}) d S^{'}

After obtaining the linear equation system discretized from Eq. (4)–(5), the block-Thomas algorithm is used to accelerate the solving process. The block Thomas method is designed for block tridiagonal linear systems. For our problem, the block Thomas algorithm can be used level by level for partial acceleration. Readers can refer to the reference by Chen et al. [21], Luo et al. [17], Meurant et al. [16] and references therein for detailed discussions and pseudo codes about this algorithm.

When simulating the cylindrical or ring geometry with conventional FEM or FDTD method, a very dense mesh must be required around the curve boundary. This is because when using the 1st order geometrical element, such as tetrahedron or rectilinear hexahedron, an especially high sampling density is needed to reduce the meshing error around the curve boundaries. If this requirement is not met, the error generated in the meshing process will inevitably compromise the overall simulation accuracy. Meanwhile, if high-order geometrical elements (curve-edge elements) are used in the mesh process for conventional FEM, it is not appropriate to expect an improvement in either the efficiency of the numerical technique or the final accuracy. This is because even though high-order geometrical elements can precisely model the complex geometry, the covariant and contravariant mapping between its physical domain and reference domain requires extra computation, which is not negligible for a relative large number of elements. Furthermore, when applying high-order geometrical elements, the associated numerical quadratures all require a higher sampling density to achieve a reasonable accuracy. For instance, the accurate result of Gauss-Lobatto-Legendre (GLL) quadrature can no longer be obtained under this circumstance. Instead, the number of sampling points must be increased according to specific cases. This requirement, in fact, brings considerable extra computation cost, and severely raises the time cost. The most common attempt to reduce the overall computing load is using higher order but larger geometrical element in the meshing process. When the high-order geometrical elements are utilized in combination with a high-order method, an acceptable improvement of the overall numerical scheme can be achieved.

Fortunately, BI-SEM shows a spectral accuracy inside the computation domain, by which the advantage of high-order geometrical discretization can be realized. With GLL approximation, p-refinement is much more efficient than h-refinement. Based on above facts, the relatively sparse high-order geometrical discretization becomes meaningful for BI-SEM. When simulating the curved edge (such as arches) structures, the mapping of the coordinates between the reference domain and the physical domain is accomplished with the interpolation of second order scalar basis functions. By decreasing the number of unknowns, the computation cost of this mapping becomes secondary and fully acceptable. With this strategy, not only the meshing error can be minimized, but also the performance of the overall numerical scheme can be boosted.

3. Enhancement structures and numerical results

The BI-SEM’s high efficiency makes it an effective tool for practical designs and optimizations. With this method, a set of localized surface plasmon resonance enhancement gold nano antennae with the design of Bragg reflector are studied for graphene’s optical response improvement within the spectra of 300 nm to 1000 nm. A geometrical parameters are determined in combination of systematic numerical simulations and reference researches. Additionally, the dispersive parameters of the materials are carefully taken into consideration in the numerical simulations to guarantee trustworthy results. In all the shown cases, the numerical technique is accelerated by a five-level block Thomas solver. Compared with previous study, our research shows that the optical absorption of graphene can be further enhanced with frequency selectivities.

3.1. Floating single layer graphene

A simple case is first studied in order to validate the proposed numerical analysis method. As is well known, theoretically and experimentally, the floating single layer graphene has a spectral non-selective optical absorption of 2.3% with almost zero reflection in the concerned spectral range of 300 nm to 1000 nm. Here the BI-SEM with block-Thomas acceleration is first used to calculate the optical absorption of the floating single layer graphene model.

The numerical results as well as the experimental data from reference [19] are given in Fig. 1. The result is in excellent agreement with the theoretical value, and also reasonably matches the experimental data.

Fig. 1 Light Transmission for Floating Single Layer Graphene

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3.2. Light absorption engineering with nanoparticles

Considering the thickness of a single layer graphene is only about 0.34 nm, the 2.3% optical absorption of the floating single layer graphene is already very impressive. However, when designing graphene-based nano-scale photonic devices, such as the graphene-based photodetectors, this light absorption is still too poor to guarantee a satisfying photoresponsivity. In this section, we propose a set of design dedicated to confine spectral selective energy inside the graphene layer by appropriate plasmon resonance excitation.

The key for effectively shaping the light absorption capability of graphene as well as the corresponding spectra is to effectively engineer the plasmon resonance generated by plasmon nano-antennae. The plasmon resonance is largely determined by the parameters of each single particle as well as the inter-particle electromagnetic coupling.

The reference by Zhu et al. [4] theoretically explores the optical absorption enhancement with uniformly placed square gold NPs, which shows that with appropriate choice of the parameters, graphene can achieve an optimal absorption of 30.3%. Here we further study the possibility of further boosting the optical absorption of graphene with plasmonic phenomena.

A schematic drawing of a graphene-based photodetector is given in Fig. 2(a). The reflector layer generates a phase shift of the wave reflected by the substrate. By changing the thickness and structure of the reflector layer, we are able to tune the phase difference between reflective optical wave from graphene and that from the substrate. This phase difference, in fact, directly contributes to the interferometric property of the photodetector. Additionally, the noble metal NPs can act as plasmonic antennae which generate certain degree of plasmon resonance. In order to achieve optimally enhanced absorption, both the effects of optical interferences and plasmon resonances should be taken into consideration. When the intrinsic interferometric absorption peak coincides with the plasmon resonant frequency, it is expected that the enhanced near field has a good contribution to the optical loss in graphene.

Fig. 2 Sketch of (a) the schematic graphene-based photodetector; (b) cuboid cluster NP with Bragg reflector; (c) cylindrical NP with Bragg reflector; (d) ring NP with Bragg reflector.

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3.3. Cuboid Au NP cluster with Bragg reflector

We first study the performance of cuboid shape Au NP clusters in comparison with the conventional uniformly distributed NPs. As mentioned above, both the parameter of single NP and the inter-particle electromagnetic coupling contribute to the graphene layer’s final behavior. Here the influences of Au NP clusters inner gap, their interval, and the contribution of a specially designed Bragg reflector are all studied comprehensively.

This category of plasmonic enhancement strategy is shown in Fig. 2(b). The periodic Au NP clusters are incorporated into the photodetectors directly above the graphene layer. Keeping the total area of the top noble metal surface as 2500 nm², each NP cluster consists of 4 identical cuboid Au NPs with the side length of 25 nm. In addition to the Au antenna cluster, the classical silica reflector is replaced by a specially designed Bragg reflector, which is placed below graphene. In this model, the Bragg reflector consists of 6 Si/SiO₂ bilayers, where the thickness of each bilayer is 100 nm+100 nm. Below the Bragg reflector is the silicon substrate. In numerical simulation, the substrate is appropriate to be considered as infinitely thick. Thus the substrate can be truncated by the EFIEs and MFIEs at its top surface, making the computation domain much smaller than the conventional FDTD method.

As another validation of BI-SEM code, a comparison of the numerical results calculated from BI-SEM and FEM are provided in Fig. 3. With a 4-th generation Intel i-7 CPU, FEM requires 354 min and 9.8 GB memory to obtain the numerical results of one single design, while the computation cost of BI-SEM is 59 min and 3 GB, respectively.

Fig. 3 Simulation results from BI-SEM and FEM.

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In all the following systematic simulations, normal incident optical fields are applied to the graphene-based model, which is also the most common situations in practical engineering. However, considering the Bragg reflector behaves sensitively under oblique incidence, its contribution becomes an interesting and meaningful topic when coupling with the LSPR. Figure 4 provides the light absorption rate of graphene under various incident polar angles. In this model, the distance between adjacent cuboid Au NP clusters is chosen as 30 nm, and the inner gap is chosen as 2 nm. Despite the proposed structure’s performance shows a strong correlation with the incident angle, it has a relatively large tolerance to the incident offset. Within the incident angle of −40 degree to +40 degree, a significant light absorption enhancement is observed in the graphene layer.

Fig. 4 Cuboid NPs with Bragg reflector, incident polar angle varies.

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Both the gap inside each NP cluster and the interval between adjacent NP clusters are able to, and in fact are designed to, control the inter-particle electromagnetic coupling strength. Conventionally, the noble metal NPs are placed in a uniform pattern. However, even though this strategy is capable of achieving a strong surface plasmon resonance, the conflict of interparticle coupling enhancement and effective plasmon surface area optimization always exists. Achieving a strong inter-particle electromagnetic coupling usually requires a close spatial interval between adjacent NPs; on the other hand, over-intensive metal coverages of the graphene inevitably weaken the utilization efficiency of plasmon resonance. In this section, we numerically study the strategy of split-NP clusters with BI-SEM. By keeping the percentage of gold-covered area unchanged, the contribution of the inner gap of each NP cluster is first studied. As a comparison, for a fixed inner gap value, the influence of distance between adjacent NP clusters is further systemically investigated.

Figure 5(a) shows the influence of the inner gap of each NP cluster, where the distance between adjacent clusters is supposed to be 70 nm. In Fig. 5(b), we focus on the influence of the gap between adjacent clusters, where the gap in each cluster is typically chosen as 3 nm. Numerical results show that the split-NP cluster structure enhances the strength of the plasmonic phenomenon, and boost the maximum graphene absorption by 110% compared with the uniform NP array. As a typical instance, here the electric field distribution on the interface between graphene and the Au NPs are compared in Fig. 6 between conventional NP arrays and cluster NP arrays. It can be seen inside the gap, the electromagnetic near field distributes with more uniformly enhancement, which improves the plasmonic light concentration with higher energy efficiency. On the contrary, for non-split Au NP model (gap=0 nm), the near electric field mainly concentrates on the 4 singular corners, which not only produces unnecessary energy losses, but also generates a less efficient plasmon resonance.

Fig. 5 Structure with Bragg reflector (a) cluster interval=70 nm, inner gap varies; (b) inner gap=3 nm, cluster interval varies.

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Fig. 6 Under 870nm TM_y normal incidence, electric field distribution for (a) inner gap=3 nm; (b) inner gap=0 nm.

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The numerical study shows that the combination of NP cluster arrays and the Bragg reflector is able to boost the light absorption of graphene in the visible spectra and near infrared spectra. Compared with the conventional non-split NP arrays, an essentially stronger surface plasmon resonance are excited inside the gap of each NP cluster, while the energy loss caused by NPs sharp corners are reduced to some degree. Due to this improved NP structure and well matched plasmon resonance frequency and intrinsic interferometirc peak, a peak absorption rate of 67.54% is observed at the wavelength 860 nm. This strong light absorption is achieved by choosing the gap inside each NP cluster as 3 nm, while separating adjacent clusters by 30 nm.

3.4. Cylindrical and ring NP with/without Bragg reflector

Cylindrical and ring noble metal NPs are widely used in nano-scale photonic structures to generate effective resonances. The rectangular corners of cuboid metal NPs may become a drawback for nano-devices under certain circumstances. It is well known that these sharp corners can be singular points for electromagnetic fields and undesirably confine a large portion of energy, while at the same time negatively affect the surface plasmon resonance with the accompanying energy losses. With these concerns, it is reasonable to expect that the undesired power loss caused by the geometry can be reduced by removing the sharp corners of the cuboid Au NPs. In addition, since the strong near field in graphene is highly concentrated surrounding the bottom edges of gold NPs, appropriate improvement of the interface between gold NPs and graphene serves as an alternative strategy for each NP’s plasmonic efficiency enhancement. In this section, we theoretically study the graphene light absorption contribution of a set of cylindrical and ring category Au NPs sketched in Fig. 2(c) – 2(d). Without loss of generality, for all the ring NP models, the inner radius is typically chosen as half of its outer radius.

We first study the influence of solid cylindrical NPs with varying diameters and adjacent distance. Compared with the uniform metal NP pattern, this design does not explicitly improve the inter-particle coupling strength, but reduces the unnecessary energy loss and improves the plasmon resonance efficiency by removing the four singular corners. Next, to further boost the plasmonic efficiency of each noble metal NP, we consider a set of ring gold NPs for LSPR manipulation.

In previous studies, graphene’s optimal light absorption in the visible spectrum is reported as 30.3% with classic silica reflector and cuboid NPs [4]. Here we first investigate the performance of ring and cylindrical with the SiO₂ reflector. Referring to previous research reported by Zhu et al. [4], a typically optimal thickness of 300 nm is chosen for the SiO₂ reflector. In order to make this structure comparable with the cuboid gold NP case, here the height of each NP is designed identical to that of the cuboid NP cluster. Similarly, we first try to find an appropriate diameter of the gold NP. Then, with this design of gold NP obtained, different distances between adjacent NPs are simulated. By varying the distance between adjacent cylindrical NPs, the inter-particle coupling strength can be controlled in a predicable way.

Numerical results in Fig. 7 – 8 show that even with classic silica reflectors, a peak graphene light absorption can be achieve as 41% by the ring NP structure at the incident wavelength of 690 nm. With this improvement on metal NP design, the optimal graphene absorption rate is over 35% higher than the maximum value reported by Zhu et al. [4], where cuboid Au NPs are systematically studied. Compared with cylindrical NPs, ring-NP-based models show a better frequency selectivity manipulation and a slightly higher graphene absorption. Moreover, by comparing Fig. 7(b) and 8(b), the ring NP structure also shows a stronger and more sensitive inter-particle coupling, whose strength can be sharply tunned by changing the distance between adjacent gold NPs.

Fig. 7 Cylindrical NP with classic SiO₂ reflector (a) adjacent distance=40 nm, radius varies; (b) $radius = 30 / \sqrt{π} nm$ , adjacent distance varies.

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Fig. 8 Ring NP with classic SiO₂ reflector (a) adjacent distance=40 nm, radius varies; (b) $radius = 30 / \sqrt{π} nm$ , adjacent distance varies.

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In general, by changing the outer radius of the ring NP, we can significantly change the local resonant frequency and consequently manipulate frequency selective absorption curve of graphene layer. Meanwhile, when varying the distance between adjacent NPs, the coupling strength of NPs are effectively controlled. This property can serve as a strategy to tune the strength of graphenes absorption without severely affecting the optimal absorption frequency.

As a comparable case with the cuboid cluster Au NP models, the same Bragg reflector design is built into the model for further graphene absorption enhancement. Fig. 9 – 10 provide the BI-SEM results for the cylindrical NP models and the ring NP models, respectively. Similar to the cuboid cluster NP case, the absorption of graphene can be enhanced dramatically in the simulated spectra. For cylindrical Au NPs, a peak graphene light absorption of 58.64% is observed under 860 nm incidence, while a peak graphene light absorption of 40.27% is observed under 850 nm incidence for ring Au NPs.

Fig. 9 Cylindrical NP with Bragg reflector (a) adjacent distance=40 nm, radius varies; (b) $radius = 30 / \sqrt{π} nm$ , distance between adjacent NPs varies.

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Fig. 10 Ring NP with Bragg reflector (a) distance=40 nm, radius varies; (b) $radius = 30 / \sqrt{π} nm$ , distance between adjacent NPs varies.

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4. Conclusions

In this work, the BI-SEM, which combines the boundary integral equations with the spectral element method, is utilized for numerical study of the light absorption of graphene. Benefiting from the spectral accuracy of the SEM and the efficiency of the surface integral equations used to truncate the open boundary, the proposed hybrid numerical technique shows excellent efficiency and accuracy.

Based on BI-SEM, the periodic cuboid Au NP clusters as well as the cylindrical/ring clusters of Au NP are studied as the nano antenna for graphene light absorption engineering. Numerical results confirm that these designs are able to essentially boost the light absorption of graphene with a frequency selective property. To further enhance the plasmon resonance strength, a Bragg reflector is also designed for the proposed structures, which dramatically assists the enhancement and tuning of graphene light absorption.

Through plasmonic near-field engineering by controlling the type of reflector, the pitch and category of plasmonic antenna, the light absorption of graphene can be efficiently manipulated. A peak graphene light absorption rate of 41% is reported in the visible spectra, while 67.54% is reported in the whole spectra studied in this work.

Acknowledgments

This work was supported by National Science Foundation under grant ECCS-1102109.

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Enhanced plasmonic light absorption engineering of graphene: simulation by boundary-integral spectral element method

Abstract

1. Introduction

2. Simulation of the localized surface plasmon resonance

2.1. Graphene’s refractive index and optical absorption

2.2. Computation method

3. Enhancement structures and numerical results

3.1. Floating single layer graphene

3.2. Light absorption engineering with nanoparticles

3.3. Cuboid Au NP cluster with Bragg reflector

3.4. Cylindrical and ring NP with/without Bragg reflector

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (10)

Equations (5)

Optics Express