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Abstract
Integrating gate-tunable graphene with plasmonic nanostructures or metamaterials offers a great potential in achieving dynamic control of plasmonic response. While remarkable progress has been made in realizing efficient graphene-induced modulations of plasmon resonances, a full picture of graphene-plasmon interactions and the consequent deep understanding on graphene-enabled tuning mechanism remain largely unexplored. Here, we theoretically identify, for the first time, two distinct modulation effects that can coexist in graphene-based plasmonic nanostructure: graphene can influence the plasmon resonances by either acting as equivalent nanocircuit elements or effectively altering their excitation environment, leading to totally different tuning behaviors. A general dependency of tuning features on the graphene-induced impedance, irrespective of structure geometries, is established when graphene serves as nanocircuit elements. We demonstrate that these two modulation effects can be dynamically controlled by appropriately integrating graphene with plasmonic nanostructures, which provide an active window for efficient modulation of surface plasmons. Our findings may pave the way towards realizing dynamic control of plasmonic response, which holds great potential applications in graphene-based active nanoplasmonic devices.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Surface plasmons, collective oscillations of electrons at the interface of metallic nanostructures, have attracted tremendous interests due to their unique capability of confining and manipulating light on the nanoscale [1–5]. In the applications of active plasmonics, it is highly desirable to realize dynamic modulation of plasmon resonances [6–8]. A promising way of realizing dynamical control of plasmons is to combine functional materials, such as, organic molecules [9] and inorganic quantum dots [10] with metallic nanostructures to form hybrid plasmonic devices. However, these materials require demanding syntheses or cryogenic environments. In this context, graphene, a 2-dimensional zero-band gap semiconductor, has emerged as a promising candidate for dynamical plasmon modulations due to its excellent optical properties that can be easily tuned via electrostatic gating [11–13]. The integration of graphene with plasmonic nanostructure offers opportunities for the development of highly integrated active devices, such as optical modulators [14, 15], waveguides [16], photodetectors [17, 18], and sensors [19–21].
Electrically controllable plasmon resonances have been demonstrated by integrating graphene with a variety of plasmonic nanostructures [14, 18, 19, 22–26] and metamaterials [27–31]. Efficient graphene-induced modulations of localized surface plasmon (LSP) has been observed in graphene-integrated nanoantenna array [23]. There, the hybrid system is considered as an equivalent nanocircuit, and the graphene monolayer behaves as input nanocircuit elements. The relevance of tunable LSP resonance to graphene charge carrier concentrations has been successfully explained by the “nanocircuit” effect of graphene in the frame work of equivalent nanocircuit method [32]. Whereas, the “nanocircuit” effect on the modulations of other LSP aspects including spectral amplitude and width has not been addressed [33, 34]. Efficient graphene-induced LSP modulations have also been observed in single gold nanorod covered with a graphene monolayer [19]. The dependence of plasmon frequency and width increase on the graphene dielectric permittivity is phenomenologically described by simple linear equations. The physics behind this linear dependency has not been revealed. Although, a lot work has been devoted to realizing electrical tuning in a variety of graphene-based hybrid plasmonic systems, a full picture of graphene-plasmon interactions, and the consequent deep understanding on graphene-enabled tuning mechanism is still lacking.
Here, we focus on the graphene-plasmon interaction and explore the underlying mechanism of graphene-induced modulation of surface plasmons. We theoretically demonstrate that the tuning features of surface plasmons can be directly related to the graphene load impedance under the circumstance of plasmon response with much smaller resonance shift, irrespective of specific structure geometries. We identify, for the first time, two distinct modulation effects that can coexist in graphene plasmonic nanostructure: it can influence the plasmon resonances by either acting as an equivalent nanocircuit element [32, 35, 36] or effectively altering their excitation environment, leading to totally different tuning behaviors. We reveal that both tuning effects actually originate from the Pauli blocking effect of graphene, resulting in an active window for effiecient modulation of surface plasmons. Our findings provide a powerful approach to realizing electrical control of surface plasmons, which is expected to facilitate the design of novel graphene-based active plasmonic devices with high-efficiency and desired tunability.
2. Results and discussion
2.1. Modulation effect of nanocircuit element
We first discuss the modulation effect with graphene acting as nanocircuit elements. As graphene is embedded inside the first hBN layer, lying far away from gold gratings, it affects the plasmonic system only as absorptive medium and interacts with plasmons perturbatively [37]. Although the effect of the gate-tunable graphene on the plasmon resonances in this case can be estimated by perturbation theory, considering the composites of the hybrid system as equivalent “nanocircuit” elements can dramatically facilitate the quantitative description of graphene-induced modulations on plasmons, thus giving a direct deep insight into the modulation mechanism. We extend the well-established nanocircuit method to give an analytical description for the general relevance of graphene-induced modulation behavior to the input graphene impedance.
From the perspective of nanocircuit model [23, 32, 38], a (nano-sized) piece of monolayer graphene, when integrated to a nanostructure, as depicted in Fig. 1(a), will introduce an additional impedance ZG to a nanocircuit [23]. According to the Ampere’s law, the electromagnetic field in graphene satisfies:
Here, σG denotes the graphene bulk conductivity, which is dependent on its gate-tunable Fermi energy Ef. ω is the angular frequency. The total current JT consists of two parts: the free carrier current Jf = σGE and the displacement current Jd = −iω∊0E. Under quasi-static approximation, the voltage across the graphene monolayer and the current over length d can be expressed as UG = tG|E| and IT = ld|JT| respectively. Here, tG is the thickness of the graphene. Considering the expression of JT in Eq. (1), the graphene impedance can be calculated as: Here, RG and XG are the graphene-induced resistance and reactance respectively. With the complex graphene permittivity defined as the analytical expression for ZG represented by the real (∊Gr) and imaginary (∊Gi) part of graphene permittivity can be written as: Here, ξ = tG/(ldω∊0) is defined as frequency-dependent prefactor, which is governed by geometrical parameters of the graphene sheet involved in the nanocircuit.
Fig. 1 (a) Schematic of a graphene sheet connecting to a nanostructure with length l and width d. It introduces an additional impedance ZG with a voltage UG across the graphene sheet. (b) Graphene-induced reactance XG (solid) and resistance RG (dash) as a function of graphene Fermi energy at LSP peak wavelength of 2400 nm.
In the framework of nanocircuit method, when combined with a plasmonic nanostructure, graphene, as a circuit element (resistor,inductor or capacitor), can be connected into a nanocircuit either in parallel or in series. Assume that the real part of graphene permittivity ∊Gr > 0, graphene behaves as a capacitor [32]. The total capacitance of the hybrid system CT can be expressed as either 1/CT = 1/CG + 1/C0 for series combination or CT = CG + C0 for parallel combination. Here, C0 denotes the capacitance introduced by the bare plasmonic nanostructure. Apparently, CT shows positive correlation to CG for both cases. Consider the resonance wavelength of the hybrid system , we find that λR is positively correlated to CG with an assumption that the total inductance of the hybrid system LT is unchanged as the graphene Fermi energy Ef varies. Because the graphene-induced reactance is also positively correlated to CG due to the relation of XG = −1/(ωCG) provided that the angular frequency ω is fixed or slowly varying with Ef. Therefore, we can finally conclude that the wavelength λR shows positive correlation to XG. For ∊Gr < 0, graphene serves as an inductor, graphene-induced reactance becomes XG = ωLG. In this case, the same relationship between λR and XG can be obtained using similar analysis approach.
The real part of graphene-induced impedance ZG, i.e., the graphene-induced resistance RG plays an important role in modulating the spectral amplitude of plasmon resonances [23,34,38], particularly the absorption spectra. Under illumination of external electromagnetic field, the power dissipation , which is directly related to the absorption intensity of the hybrid system, is determined by the total voltage UT loaded on the whole system and the total resistance RT. Similarly, RT can also be separated into two parts: R0 introduced by the bare plasmonic nanostructure and the graphene-induced RG. The positive correlation of RT to RG always holds for both series and parallel connecting configurations. If we assume the total voltage UT is stable with varying Ef, we can find that the power dissipation, or the absorption amplitude is negatively correlated to RG.
Figure 1(b) plots XG(ω0, Ef) (solid line) and RG(ω0, Ef) (dashed line) as a function of graphene Fermi energy for a fixed resonance frequency ω0 = 2πc/λ0 with λ0 = 2400 nm. Note that (i) these correlations are always valid since Eq. (4) is independent of the geometrical configuration of the involved nanostructures, (ii) we only consider graphene in Fig. 1(a) as a nanocircuit element without considering its role in modifying the excitation environment, and (iii) the modulation depth may differ, depending on the intensity of plasmon field that determines the graphene-light interactions [23,39], which will be discussed in details in later sections.
To verify the validity of the above conclusion, we employed a one-dimensional (1D) plasmonic nanostructure as a test model, which consists of a multi-layered metal-dielectric stack covered with a period array of metallic bar pair, as depicted in Fig. 2(a). Here, gold and hexagonal boron nitride (hBN) are used as the metallic and dielectric components. One feature of this model system is that it can support extremely narrow LSP resonances due to Wood anomalies [40,41]. The other feature lies in the fact that the first thin hBN layer (10 nm) is used to form a Fabry-Pérot microcavity [42–48] such that the electric field can be strongly localized within the layer to facilitate the graphene-plasmon interaction. The super narrow LSP resonances with strong field confinement make this plasmonic nanostructure a good candidate for investigating graphene-plasmon interactions and the graphene-induced modulation behaviors.The proposed heterostructure can be practically realized in the following processes: First the hBN layer with various thicknesses can be mechanically exfoliated onto the bottom gold plate. Then the thin gold layer can be subsequently deposited on the hBN layer by magnetron sputtering. The following hBN-graphene-hBN stack can be assembled by the polymer-free van der Waals assembly technique [49, 50]. The gold pair array is then fabricated on the top by e-beam lithography.
Fig. 2 (a) 3-dimentional schematic illustration of the test plasmonic nanostructure with a monolayer graphene embedded in the first layer of hBN. (b) The simulated absorptance spectrum for the test structure in absence of graphene with the bar-pair period p = 2400 nm, illustrating the first-order LSP centred at λ0 with a super narrow spectral width of 3 nm. (c) The simulated spatial distribution of the electric field amplitude |E| at the resonance wavelength of λ0.
The LSP wavelength of the plasmonic nanostructure can be described by the Rayleigh cutoff wavelength given by [26,40,51]:
Here, p is the array period and m is an integer, n is the substrate or superstrate refractive index corresponding to the excited environment and θ is the angle of incidence. We performed full wave simulations to obtain the optical property of LSP by using the commercial finite element solver COMSOL Multiphysics.The graphene monolayer is modeled as a thin sheet of thickness tG = 0.5 nm with a complex sheet conductivity derived from the random-phase approximation of Kubo formula [52]:
Here, ħ is the reduced Planck’s constant, e is the charge of an electron, T = 300K is the temperature, kB is the Boltzmann constant, τ−1 represents the scattering rate, and Ef is the Fermi energy, which is dependent on the graphene charge carrier concentration. ∊0 is the vacuum permittivity. The complex sheet conductivity σGs, with its real (σ1) and imaginary (σ2) part, includes two contributions from the intraband σintra (Eq. (7)) and interband σinter (Eq. (8)) transitions respectively. For the simulation of the plasmonic nanostrcuture, we consider the gold film at the bottom as a perfect reflecting layer. A 170 nm thick layer of the hexagonal boron nitride (hBN) covers on the bottom gold reflector. The thickness of the thin gold film is taken as 25 nm. Then another thin hBN layer (10 nm) is situated on the top of the thin gold layer. The gold bar pair is composed of two identical (30 nm-thick and 460 nm-wide) rectangular gold nanobars separated by a distance of 20 nm. Here, instead of using a pure gold layer right below the thin hBN layer, we insert a 170 nm-thick hBN layer between the two gold films. The permittivity of gold are given by Drude model [53] with plasma frequency ωp = 1.37 × 1016s−1. Although hBN shows strong plasmonic property due to its hyperbolic dispersion in mid-infrared frequencies [54, 55], in near infrared region discussed in our case, it can be considered as a normal dielectric material with the refractive index nBN = 1.78 [56]. The main aim of the embedded hBN layer is to obtain a larger tuning range of LSP wavelength. Compared to the reactance introducted by a pure gold layer, the presence of hBN layer gives a smaller reactance X0 (C0 and L0), which gives rise to a larger ratio of graphene-induced XG to the total reactance XT. This improves the graphene-induced tunability on LSP wavelength.We calculated the absorptance spectrum for the bare nanostructure without graphene under the illumination of a TM polarization from the top at normal incidence, that is θ = 0. In our case, the period of p = 2400 nm is used for the bar-pair array. As shown in Fig. 2(b), a strong and sharp absorptance peak occurs at λ0 = 2407.1 nm (ħω0 = 0.52 eV) with a super narrow spectral width ∼ 3 nm, corresponding to the excitation of the first-order (m = 1) LSP mode at the “air-gold” interface (superstrate, n = 1). Here, an asymmetry in the high-energy side of the spectral line is generated from the asymmetry between the substrate and the superstrate [57]. Figure 2(c) shows the distribution of the electric field [47, 58] with strong field confinement [23] within the first hBN layer. We will show in later section that this strong field localization plays an important role in explaining the graphene-induced modulation features.
We first simulated the optical response of the hybrid nanostructure with the graphene sheet embedded in the first hBN layer, as shown in Fig. 2(a), in which graphene can strongly interact with LSP due to the strong field confinement. In this case, graphene is connected into the nanostructure only as a nanocircuit element since it is not involved in LSP excitations at the air-gold interface. We simulated the absorptance spectra of the hybrid nanostructure as a function of graphene Fermi energy. The optical properties of LSP, including the resonance wavelength λR, the absorptance αR (obtained at each λR) and the spectral width Δλ (in unit of nm), are extracted and plotted (solid blue) in Figs. 3(a), 3(c), and 3(d) respectively. Several features are apparent. (i) The LSP wavelength λR exactly follows the same trend as the line shape of the graphene reactance XG (solid curve, Fig. 1(b)). (ii) The absorptance αR and the spectral width Δλ show a negative and positive correlation with the graphene resistance RG (dashed curve, Fig. 1(b)), respectively. (iii) There exists an “active” area with efficient modulations, i.e., the shaded region II around graphene Fermi energy 2|Ef| = ħω0, as shown in Figs. 3(a)–3(d) [19].
Fig. 3 (a) The LSP peak wavelength as a function of graphene Fermi energy obtained by numerical simulations (blue) and nanocircuit model (red). (b) The real part of the 2D graphene sheet conductivity with a vertical green line denoting the graphene Fermi energy of 2 |Ef| = ħω0 ≈ 0.52 eV. (c) The simulated (solid blue) and modelled (solid red) absorptance αR as a function of graphene Fermi energy obtained at the peak wavelength λR. The dashed blue line represents the simulated αR ≈ 0.84 in absence of graphene. (d) The simulated spectral widths Δλ (solid blue) of the absorptance spectra together with the graphene-induced resistance RG(Ef, λR) (red). The dashed blue line represents the simulated Δλ in absence of graphene. (e) Equivalent nanocircuit for the test hybrid nanostructure. (f) Illustration of three types of optical transition processes (I, II and III), corresponding to three modulating regions separated by the shaded area in (a–d).
To gain a deep insight into these interesting features and reveal the underlying modulation mechanism, we employed the well-established equivalent nanocircuit model [23,32,34] to give a quantitative description of the simulated tuning behavior. The calculated λR and αR are given by the red curves in Figs. 3(a) and 3(c) respectively. A very good match to the simulations (blue curves) confirms the validity of our theoretical predictions. Note that the red curve in Fig. 3(a) is vertically shifted up by 6 nm, which may result from the assumption of unchanged reactance of the dielectric and gold components with varying graphene Fermi energy.
In Fig. 3(a), the presence of graphene tunes the plasmon wavelength within a very limited range of 1.5 nm. This is attributed to the graphene-induced reactance XG that is two order of magnitude smaller than the reactance introduced by other dielectric and metallic components, resulting in the limited tunability on LSP wavelength. This small wavelength shift is actually a prerequisite for the validity of our theoretical analysis. For studying LSP tuning feature and its dependency on the graphene-induced impedance ZG, it is useful to decorrelate the effect of the plasmon frequency ωR and the graphene Fermi energy Ef appearing in Eq. (4). By assuming a small resonance shift with ωR ≈ ω0, we are able to obtain the explicit dependency of ZG purely on Ef as given in Fig. 1(b). Although, this 1.5-nm modulation range is very small in terms of its absolute value, it is still on the order of the LSP spectral width (Δλ0 ≈ 3 nm). Such a narrow LSP with this wavelength shift is sufficient to realize graphene-based plasmonic sensor with high figure of merit.
In contrast, the modulation effect on αR in Fig. 3(c) is more prominent: a modulation depth of over 40% in absorptance is obtained in near-infrared (NIR) region around 2407 nm as the graphene Fermi energy increases by less than 0.1 eV. This means that one can realize a high-efficiency NIR modulator by just applying a much smaller amount of gating voltage. In addition, the presence of graphene, irrespective of its Fermi energy, dramatically reduces the LSP absorptance αR by about 17% with respect to the absorptance in absence of graphene (dashed blue), as demonstrated by the vertical blue arrow in Fig. 3(c). This strong graphene-induced decrease in LSP absorption may originate from the screening effect of graphene which has been observed in graphene-coated dielectric substrate [59]. Graphene, even intrinsic or undoped is featured by its semimetallic electronic band structure. It offers charge carriers (electrons and holes) that can be spatially rearranged to screen the external electric fields [47,59–61], leading to a significant reduction of absorption at plasmon resonance.
Accordingly, the spectral width in Fig. 3(d) experiences an inverse tuning feature with respect to the modulation of αR in Fig. 3(c). This tuning feature of Δλ is easy to understand: graphene, acting as a resistor, provides an effective dissipation channel and modifies the damping rate of LSP. Note that Δλ also has an over-all 1-nm broadening compared with LSP spectral width in absence of graphene (dashed blue). This broadening can not be explained by the effect of graphene-induced resistance since RG goes to almost zero in Region I and III. Here, graphene is sufficiently interacting with the strongly confined plasmon field inside the thin hBN layer, the increased LSP damping can be attributed to the elastic carrier scattering processes of graphene due to its high carrier concentration [62]. To understand the third feature, i.e., the “active” Region II, we recall Eq. (4), where we can apparently see that RG and XG are directly related to the gate-dependent graphene conductivity. Figure 3(b) plots the real part of the sheet conductivity σGs, where a typical “step-like” line shape can be divided into three parts. These three parts, exactly corresponding to Region I, II and III in Fig. 1(b), are governed by three different transition processes [13,63,64]. For smaller Fermi energy with 2|Ef| much less than the LSP energy, the interband transitions are allowed, as depicted in Fig. 3(f) (left). This corresponds to Region I. As the Fermi energy increases such that 2|Ef| is much larger than the LSP energy, the effect of “Pauli blocking” dominates in Region III: the interband transitions are totally forbidden because of the empty initial states (or filled final states) [12,13], as denoted by the right part of Fig. 3(f). While in Region II with graphene Fermi energy comparable to the LSP energy, the process of disorder-mediated transitions are dominant due to the presence of disorder effects and electron-phonon coupling [13], leading to the broadening of the “step-like” line shape of σ1 around 2|Ef| = ħω0 ≈ 0.52 eV (dashed green line in Fig. 3(b)). Therefore, the microscopic origin of the active region is the disorder-mediated optical transitions of graphene.
Fig. 4 (a) Schematic of the nanostructure with graphene on top, and the distribution of the electric field at LSP resonance. The simulated (blue) and fitted (red) results of graphene-induced LSP wavelength shift (b), the corresponding absorptance (c) at LSP peak wavelength and spectral width (d) as a function of 2|Ef|. The purple lines in (b) and (c) give λR and αR obtained by nanocircuit model. (e) The real (blue) and imaginary (red) part of graphene permittivity. The blue dashed lines in (c) and (d) represent the absorptance and spectral width of LSP resonances in the bare nanostructure.
2.2. Coexistence of two modulation effects
We will show in this section that graphene can not only behave as a nanocircuit element, as discussed in Fig. 3, it may also serve as an effective medium to influence the plasmon excitations, leading to distinctly different modulation features of plasmon resonances. In this sense, the present nanostructure is an ideal test model to study the graphene-induced modulations caused by latter effect.
Using the same plasmonic nanostructure, we move the graphene sheet from the first hBN layer to the top of the nanostructure, as depicted in Fig. 4(a). Since graphene is connecting to the circuit in series for both cases, shifting graphene to the top is equivalent to “shift” the graphene-induced ZG to the top without changing the connecting configuration of the rest circuit. Therefore, one would have expected modulation behaviors that are same as those for the former case (graphene inside hBN layer) but only with reduced modulation depth because the field confinement is much weaker for the latter case (graphene on top, Fig. 2(c)). We performed simulations with the same geometrical parameters as those used in previous case. Figure 4(a) gives the distribution of electric field at LSP resonance(bottom). The simulated LSP peak positions λR, the absorptance at λR and the spectral width Δλ as a function of graphene Fermi energy are plotted in Figs. 4(b)–4(d), respectively.
Surprisingly, compared with the previous case, the LSP tuning behaviors exhibit totally different features: (i) the LSP wavelength scales linearly with the real part of graphene permittivity ∊Gr, which shows a maximum at exactly the same Fermi energy 2|Ef| = ħω0 ≈ 0.52 eV (dashed green lines in Figs. 4(b) and 4(e)), (ii) the absorptance and the spectral width show “step-like” line shape, which also show linear dependence on the imaginary part of graphene permittivity ∊Gi (red, Fig. 4(e)), and interestingly (iii) obvious variations occurring at 2|Ef| ≈ 0.6 eV are visible in Figs. 4(b)–4(d), as marked by the dashed black lines. Obviously, the first two features do not follow the link between the modulation effects and the graphene-induced impedance by considering graphene as nanocircuit element, which cannot be explained in the framework of nanocircuit model.
Here, we claim that the mono-layered graphene, can modify the excitation environment (the air) as an effective medium [26,40,51]. Essentially, in contrast to the first modulation behavior where graphene influences the plasmon resonances in a perturbative way, a non-pertubative interaction between graphene and plasmons dominates when graphene situates on top of the plasmonic nanostructure and is directly involved in the excitation environment of plasmons, which leads to distinctly different modulation behaviors. To verify this claim, we introduce an effective refractive index for the air in presence of graphene monolayer, which can be expressed as . Here, ∊air = 1 denotes the relative permittivity of air. ∊Gr(Ef, λ0) is taken at the resonance of bare nanostructure at λ0 = 2407.1 nm. A free parameter f is introduced here to represent the weight of graphene in the effective refractive index neff. Based on Eq. (5), the LSP wavelength λR can be given as
Here, A = 1.003 is a prefactor, which is determined by the LSP mode without graphene. By fitting the simulated λR to Eq. (9), we reproduce the main feature of the graphene-induced LSP wavelength shift very nicely, as shown in Fig. 4(b) (red). The fitted free parameter f gives a very small value of 1.5 × 10−5, indicating that the existence of graphene on top of the nanostructure has very little impact on the excitation medium due to relatively weak plasmon field confinement. This accounts for the negligible ±0.2 nm wavelength changes around λ0.The modulation of graphene on LSP absorptance shows distinctly different “step-like” feature with respect to that in the previous case (Fig. 3(c)). This phenomenon can be well explained by considering the graphene sheet as a “lossy” medium. The total absorption of the hybrid nanostructure has contribution from two parts: the graphene overlayer and the underneath gold components including the bar pairs and the gold films. The total absorptance at LSP peak wavelength can be expressed as the sum of graphene and gold part:
Here, α1 is an optimized parameter representing the absorptance of bare nanostructure in absence of graphene. κ is the ratio of graphene absorptance to the sum of the reflection of the hybrid structure and the absorptance of graphene. Taking β as a fitting parameter and h = 0.5 nm for the thickness of graphene sheet, we fitted the simulated absorptance to Eq. (10). The fitted result, as displayed by the red curve in Fig. 4(c), matches the main feature of the simulated αR very well. The “step-like” decreasing in the spectral width (Δλ), as shown by the solid blue curve in Fig. 4(d), originates from the Pauli blocking effect: as |Ef| increases, the efficient power dissipation channel provided by the interband transitions is switched off, thus leading to narrowed absorption spectra with higher quality factor [19].For the interesting variations in Figs. 4(b)–4(d), we note that, compared with the efficient modulations shown in Fig. 3, the variations in present case happen “exactly” at the same graphene Fermi energy around 2|Ef| ≈ 0.6 eV and keep the same tuning features, but with reduced modulation effects. This is consistent with our expectation that graphene, when moved to the top, still acts as nanocircuit elements but with reduced modulation effects. To prove this, we applied nanocircuit method to quantitatively describe the modulation behavior. The calculated results are given by the purple lines in Figs. 4(b) and 4(c), respectively. Obviously, the calculated variations with reduced modulation effect, follow exactly the same line shapes and positions as those shown in Figs. 3(a) and 3(c) (solid blue lines). This reduced modulation effect, as expected, results from the weak graphene-light interaction due to relatively weak field enhancement at the air-gold interface. This weak field enhancement also leads to the diminishing of the over-all reduction in αR and broadening in Δλ, i.e., the absorptance and the spectral width return back to those for the bare nanostructure due to the reduced screening effect and elastic carrier scattering of graphene respectively [65], as clearly shown by the dashed blue lines in Figs. 4(c) and 4(d). Note particularly in the second case that the two effects can be activated only in Region I and II where the interband and the disorder-mediated transitions are allowed. The graphene-induced modulations are deactivated for higher Fermi energy (Region III) due to the totally blocking of graphene interband transitions.
Our analysis can be used to readily explain the graphene-induced tuning behaivor of surface plasmons observed in graphene-nanorod hybrid system [19]. The linear dependence of the plasmon resonance energy and width on the graphene permittivity is similar to the simulated results in our second case (Figs. 4(c) and 4(d)). This means that the monolayer of graphene covered on top of the nanorod modifies the excitation environment of surface plasmons. In that case, the effect of “nanocircuit elements” is not observed, which may due to the fact that the connecting configuration in the hybrid system give negligible graphene-induced reactance and resistance. Therefore, with appropriately integrating graphene with plasmonic nanostructures, the modulation behavior can be dynamically controlled by employing the coexistence of the two effects. For example, one can totally avoid the effect of excitation environment by keeping graphene away from the plasmon excitation interface, like the first case discussed here. In contrast, one can also enlarge both effects by placing graphene at the excitation interface with strong field confinement. In principle, these two distinct modulation effects can be experimentally verified since the hybrid nanostructure proposed here is practically realizable. For example, stacked hBN-graphene-hBN structure integrated with metal strips has been successfully fabricated and investigated very recently [66]. Our findings pave the way towards designing graphene-based active plasmonic devices, such as high-efficiency modulators with desirable tunability.
3. Conclusions
Therefore, we can conclude that the graphene-induced LSP modulation can be understood by identifying two types of modulation effects: (i) the effect of “nanocircuit elements” which can be described by nanocircuit model, and (ii) the effect of excitation environment medium. In the first case, where graphene is embedded inside the hBN layer, the modulation behavior is determined only by the first effect, leading to the “active region” with efficient LSP modulations that can be directly related to the graphene-induced reactance and resistance. Whereas, in the second case with the graphene on top, both effects coexist and the second effect are dominant in determining the tuning features. We have shown that these two modulation effects can be dynamically controlled by appropriately integrating graphene with plasmonic nanostructures, thus opening an active window for efficient modulation of surface plasmons. The identification of the two modulation effects is expected to help in deep understanding of graphene-plasmon interactions, which may pave the way towards realizing dynamic control of plasmonic response and hold great potential applications in graphene-based active nanoplasmonic devices.
Funding
National Natural Science Foundation of China (NSFC) (No.61675139,11474207); National Key R&D Program of China (2017YFA0303600).
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