Tunable bulk polaritons of graphene-based hyperbolic metamaterials

1. Introduction

Graphene is a two-dimension substance composed of a single layer carbon atom arranged in a honeycomb lattice and it is actually a gapless semiconductor [1]. It has attracted intensive scientific interest owing to its incredible physical properties, such as optical transparency, flexibility and extraordinary electrical properties [2–4]. With the recent developments in the fabrication of graphene with large lateral dimensions [5,6], there have been numerous graphene applications at optical, infrared and terahertz frequencies as tunable waveguiding interconnects, polarizer, optical modulator, cloaking and strong light–matter interaction platform [7–11]. Because the imaginary part of graphene conductivity can attain positive values in a certain frequency, a graphene layer can effectively behave as a “metal” layer capable of supporting a p-polarization surface plasmon polariton [3,7]. Although graphene sheet shows excellent metal-like characteristics under electric/magnetic biasing or chemical doping, it is generally believed that graphene does not conduct well enough to replace metals, since it is too “thin” to sustain an intense resonance [12]. While a hyperbolic metamaterial (HMM) can be realized by employing a multilayer graphene-dielectric composite [13–17], where the graphene sheets are substituted to mimic essential metallic behavior. HMM is a uniaxial anisotropic medium, the diagonal elements of the permittivity tensor have different signs (e.g., ${\epsilon }_x{}_x={\epsilon }_{yy}<0$, ${\epsilon }_{zz}>0$), which causes the dispersion relation to become hyperbolic rather than elliptical [18]. It can be realized at optical frequencies using metal-dielectric multilayers, where the unit cells are much smaller than the operation wavelength [19]. But the performance of metals is hampered by the difficulty in tuning their permittivity functions and the existence of optical losses. Unlike the metals, the electronic and optical properties of graphene can be tuned through electrical or chemical modification of the charge carrier density [20,21]. And the conductivity of graphene can be readily tuned not only at THz frequencies but also at optical frequencies. The loss of graphene can be very low in the frequency region of interest, it seems to be a good candidate for designing the tunable HMM operates in both THz, infrared and optical frequency ranges.

The recent study about graphene-based HMMs composed of stacked graphene sheets separated by thin dielectric layers have demonstrated that such HMMs can be used in negative refraction [13], tunable broadband hyperlens [16], tunable infrared plasmonic devices [17], and so on. In addition, such HMM structure can also support surface (Bloch) waves, which have been theoretically studied in Ref [18,22]. In this paper, we investigate the bulk polaritons of the HMM based on the graphene-dielectric layered structure. It is found that such graphene structures support controllable bulk polaritons, which can be excited by attenuated total reflection (ATR). Furthermore, the excitation of the bulk polaritons can be used in tunable optical reflection modulation.

2. Model and theory

A graphene monolayer is electrically characterized by its surface conductivity$\sigma (\omega ,E_F)$, which is highly dependent on the working frequency and Fermi energy. Within the random-phase approximation and without external magnetic field, the graphene is isotropic and the surface conductivity $\sigma$can be written as a sum of the intraband ${\sigma }_{\mathrm{int}ra}$and the interband term ${\sigma }_{\mathrm{inter}}$ [23],

Consider the periodic stack of graphene-dielectric layers as depicted in Fig. 1(a), where graphene sheets are separated by dielectric layers with thickness$t_d$and relative permittivity${\epsilon }_d$. It is assumed that the thickness of the dielectric layers are deep subwavelength but thick enough to avoid the interaction between graphene layers (e.g., interlayer transitions). Hence the graphene’s effective permittivity${\epsilon }_g$is written as ${\epsilon }_g=1+i\sigma /(\omega {\epsilon }_0{t}_g)$ [13,22], where ${\epsilon }_0$is the permittivity in vacuum,$t_g$is the effective graphene thickness. In the calculations,$t_g$is taken to be 0.5nm similar to that used in [13,26]. For simplicity, we use a constant value of relaxation time τ = 1 ps, which is similar to that in experiment [27]. Under the subwavelength limit, the graphene-dielectric composite (GDC) can be modeled as a homogeneous uniaxial anisotropic medium. And the effective medium theory (EMT) is valid to study the wave propagation in the GDC, the effective relative permittivity tensor${\epsilon }_{eff}$is a diagonal matrix in Cartesian coordinates,${\epsilon }_{xx}={\epsilon }_{yy}={\epsilon }_p$,${\epsilon }_{zz}={\epsilon }_t$,which are approximated as follows [13,19]:

 

Fig. 1 (a) Schematic of graphene-based HMM consisting of alternating graphene sheets and dielectric layers, the layers are infinite in x-y plane. (b) Illustration of the graphene-based HMM (medium B) with thickness of d sandwiched by media A and C.

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Since the thickness of a graphene sheet is negligible compared with that of the dielectric layer, the dielectric tensor in the perpendicular direction can be considered equal to the permittivity of dielectric layer (${\epsilon }_t\approx {\epsilon }_d$). However, the dielectric tensor in the parallel direction (xy-plane) varies with frequency. As an example, in Fig. 2(a) we plot the real and imaginary parts of effective permittivities ${\epsilon }_p$ and${\epsilon }_t$ with frequency. Here, we assume that the Fermi energy of graphene E_F = 0.2 eV, and T = 300 K. Silica is selected as the dielectric layer with relative permittivity${\epsilon }_d$ = 2.2 and thickness $t_d$ = 50nm, where the absorption loss of dielectric has been neglected. According to Fig. 2(a), the hyperbolic dispersion can be obtained for certain frequencies of interest in which the real part of ${\epsilon }_p$and${\epsilon }_t$ are negative and positive respectively. Figure 2(b) is the calculated equi-frequency contours (EFCs) of a graphene-based HMM at 8THz, 9THz and 10THz respectively. The solid (short dotted and the short dashed) line is calculated from the EMT, and the hollow circles represent the results based on Bloch’s theorem, which agree well with each other and indicate the validity of EMT. It is evident that the graphene-dielectric composite possesses hyperbolic dispersion relation, hence the name of effective graphene-based HMM. Note that under the same kx, kz decreases with the frequency, kz is not zero between −1.48k₀ and 1.48 k₀ (k₀ is the wavenumber in free space) because of the loss in the graphene sheets which decreases with frequency as shown in Fig. 2(a).

 

Fig. 2 (a) Effective permittivity of the graphene-based HMM. (b) The EFCs of a graphene-based HMM at 8THz, 9THz and 10THz respectively, the solid (dotted and dashed) line is calculated from EMT, the hollow circles are the results based on Bloch’s theorem.

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In the lossless case, hyperbolic dispersion only occurs when ${\epsilon }_p<0$, and the HMM allows for propagation of extraordinary waves with a large transverse wavenumber with $kx>\sqrt{{\epsilon }_t}{k}_0$. So the graphene-based HMM will support highly confined bulk plasmon modes in addition to the long-and short-range surface plasmon modes for p polarization like that in metal-dielectric based HMMs [28]. Now, we present the derivation of the polariton dispersion relations for a graphene-based HMM slab with thickness of d sandwiched by two semi-infinite media (i.e., ABC), as shown in Fig. 1(b). The effective anisotropic permittivity tensor of HMM are given by the approximate model of Eq. (2), where the slab B has the same thickness as the layered metamaterial as shown in Fig. 1(a). P-polarized wave is characterized by the existence of a Hy component of the magnetic field together with Ex and Ez components of the electric field. Here, z is the direction normal to the HMM, and x, y are the directions parallel to the slab. According to Maxwell equations, the expressions of electric and magnetic fields are given as follows:

3. Numerical results and discussions

From Eqs. (6) and (7), the effective permittivity ${\epsilon }_p$is one important parameter which determines the existence of surface and bulk polaritons for p polarization wave in HMM structure as shown in Fig. 1(b). Fortunately, the condition for ${\epsilon }_p<0$ can be controlled by the Fermi energy of the graphene sheet, the filling ratio of the graphene sheet and dielectric layer, and the number of graphene sheets Ns in one unit, that’s the graphene-based HMMs are tunable. Figure 3 depicts a regime map in a kx−f space, illustrating the regions where surface or bulk polaritons may exist. Here, HMM layer (i.e., medium B) is sandwiched by air, for the HMM layer, it is realized by the graphene-dielectric multilayer and characteristic with the effective permittivity tensor, that’s ${\epsilon }_p$ and${\epsilon }_t$. Note that no surface polaritons can be excited when ${\epsilon }_p>0$, where the graphene-dielectric composite behaves like a pure dielectric with an elliptical EFC. Two dotted curves (I) and (II) separate three different regions. Curves (I) and (II) correspond to$k_z=0$ for air and HMM, respectively. In the left region of curve (I), $k_{A/Cz}^2>0$, and in the right region of curve (II), $k_{Bz}^2>0$, no surface modes exist. In regions between curves (I) and (II), evanescent waves emerge in both air and HMM, hence, surface polaritons can exist at dual boundaries of HMM based on the positive ${\epsilon }_{A/C}$ and negative${\epsilon }_p$. While at the right region of curve (II), bulk polaritons can also exist since $k_{A/Cz}^2<0$ and $k_{Bz}^2>0$, here we focus on the bulk polaritons in this paper.

 

Fig. 3 The dependences of dispersion relations of p-polarized surface and bulk polaritons on (a) the Fermi energy E_F, (b) the number of graphene sheets (i.e., Ns). and (c) the period number of the graphene-based HMM (i.e., N), where Ns = 1, N = 6, t_d = 50nm in (a), N = 6, t_d = 50nm, E_F = 0.2eV in (b), and Ns = 1, t_d = 50nm, E_F = 0.2eV in (c). Curves (I), (II) and (III) correspond to kz = 0 in air, HMM and prism, respectively, curve (IV) in (a) denotes $k_x=n_p{k}_0\sin (\theta )$ with incident angle $\theta ={68.55}^0$ and $n_p=4$.

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The surface and bulk polaritons dispersion relations for p polarization as a function of the Fermi energy E_F at different frequency is shown in Fig. 3(a), where Ns = 1, N = 6. It is clear that the bulk polaritons exist at the right region of curve (II), they extend from line (II) to the frequency where ${\epsilon }_p=0$. The polariton dispersions strongly depend on the Fermi energy, as it can be clearly seen in Fig. 3(a), in which we calculate and plot the dispersion relations based on the effective anisotropic permittivity tensor and the slab has the same thickness as the layered metamaterial. It should be noted that the kx decreases as E_F increases with the same frequency, the existence region of the bulk polariton extends to high frequency region as E_F increases, i.e., a higher Fermi energy increases the conductivity of graphene, thereby increasing the effective plasmon frequency of graphene. Furthermore, dramatic tuning of the existence of the bulk polariton spectrum from THz to optical frequency is possible through electrical or chemical modification of the charge carrier density in graphene and a Fermi energy as high as E_F = 1-2eV has been recently realized [31]. The dash-dotted line (III) is also drawn to facilitate the discussion in the next session on the optical properties in an attenuated total reflection (ATR) setup shown in the inset of Fig. 4. Line (III) is$k_x=n_p{k}_0$, $n_p$is the refractive index of the prism used in ATR. The dotted line (IV), between lines (II) and (III), denotes $k_x=n_p{k}_0\sin (\theta )$ with incident angle$\theta ={68.55}^0$, which can excite the bulk polariton in HMM.

 

Fig. 4 The two-dimensional map of reflectance of the Otto configuration (inset), where Ns = 1, N = 6, t_d = 50nm and E_F = 0.2eV, dielectric A is air with h = 1um.

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Except for E_F, the bulk polariton properties are also dependent on the fill factions of dielectric layer and graphene sheet. If change the thickness of dielectric layer in one unit, ${\epsilon }_p$ will vary with $t_d$ remarkedly as shown in Ref.13. Here we keep the thickness of $t_d$ = 50nm, and increase the thickness of graphene by controlling the number of layers of graphene sheets. We can estimate the conductivity following the approximation $\sigma \text{'}=Ns\cdot \sigma$ for a few layer graphene, where Ns is the number of layers (Ns <6) in one period [32], $t_g^{\text{'}}=Ns\cdot t_g$. Figure 3 (b) shows the dependences of dispersion relations of p-polarized surface polaritons and bulk polaritons on the number of graphene sheets (i.e., Ns), where N = 6, $t_d$ = 50nm and E_F = 0.2eV respectively. As observed, the existence region obviously increases to high frequency band with the number of the sheet Ns. Furthermore, the bulk polariton dispersion relations are affected by the thickness of the HMM (i.e., the period number, N). The polariton dispersion relations are shown in Fig. 3(c) with different N, N = 5, N = 6, N = 7 and E_F = 0.2eV respectively. It is seen that the polaritons curve regions decrease to lower frequency with N. On the whole, the polariton dispersions exhibit an obvious tunability.

Since HMM allows for propagation of extraordinary waves with a large transverse wavenumber $kx>\sqrt{{\epsilon }_t}{k}_0$ with ${\epsilon }_p$< 0, the bulk polaritons do not interact directly with an incoming electromagnetic plane wave because of the momentum mismatch. In order to study the bulk polaritons in the graphene-based HMM, ATR technique (Otto configuration) can be employed [33] as shown in the inset of Fig. 4. HMM is placed at a distance of h from the semi-cylindrical germanium prism with$n_p=4$. The prism is used to access the wide range of effective indices and to match the wave vector of light in free space to the wave vector of the bulk polaritons. When the incidence angle $\theta$is greater than the critical angle${\theta }_c=\mathrm{arc}\sin \sqrt{{\epsilon }_A{\mu }_A/{\epsilon }_{prism}{\mu }_{prism}}$, k_Az becomes purely imaginary and an evanescent wave emerges in the gap (i.e., medium A shown in Fig. 1(b)). If the gap width h approaches infinity, the evanescent wave will decay exponentially away from the prism and total internal reflection will occur. For a finite h, however the incident waves can be coupled to the surface polaritons or bulk polaritons, resulting in a reduction in the reflectance, especially near the resonance frequencies. Transfer matrix method has been used to calculate the angular reflectance spectra for different frequency with E_F = 0.2eV. The calculated two-dimensional map of reflectance of the Otto configuration is shown in Fig. 4. The parameters used in the calculation are ε_A = ε_C = 1, h = 1um, the HMM is designed to consist of six pairs of silica/graphene sheet. As seen from the figure, the locations of minimum reflected intensity directly indicate the excitation of the bulk plasmon modes through the ATR method. When the frequency increases, the excitation of the bulk polariton needs larger incident angle, where the bulk polariton possesses nearly the same kz. The results are accordant with those in Fig. 2 (a). It can be seen from the map that the designed structure supports a bulk plasmon mode at incident angle of 68.55°for 10 THz. While the frequency of reflectance dip has somewhat deviation compared with the across between curve (IV) and the dispersion relation (dash-dotted line) shown in Fig. 3(a), because the electromagnetic fields in the medium A and the graphene-based HMM layer have been perturbed by the prism. Despite the deviations, the dispersion relation can guide the resonance frequency of bulk plasmon mode well.

Because the bulk plasmons can be effectively controlled by E_F of the graphene, we investigate the behavior of bulk plasmon modes in the HMM at different E_F under a certain incident angle and excitation frequency. Figure 5(a) shows the reflectance spectrum of the Otto configuration as a function of incident angles, the excitation frequency is 10THz. When E_F increases or decrease from 0.2 eV, the amplitude in the reflectance spectrum at the same incident angle increases significantly and the dip deviates 68.55°. The reflectance amplitude at 68.55°vs the Fermi energy are plotted as shown the Fig. 5 (b) (case 1). This tuning property of the considered HMM structure can be used in optical reflection modulation [26,34]. As observed, the reflectance can be modulated from 0.02 to 1 when E_F deviates $\pm 0.1eV$around 0.2eV, near 100% modulation depth, which is higher than 64% reported in [35]. Due to the low propagation loss, the insertion loss for an ATR modulator is much lower than that in a waveguide system [36].

 

Fig. 5 (a) Reflectance of the Otto configuration with HMM vs incident angles at different Fermi energy E_F. (b) Reflectance vs E_F, the frequency of the incident wave is 10THz (11THz), $\theta ={68.55}^0$(51.86°) for case 1 (2).

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In the Otto configuration, the medium A and the separation h are employed as an additional degree of freedom to tune the bulk plasmon dispersion and it’s excitation. In practice, the optical reflection modulator can be realized by the following procedures: the MF₂ dielectric layer with ${\epsilon }_A$ = 1.9, h = 1.5um is first coated on a glass substrate, and then the high quality multilayer is realized by stacking of many graphene/dielectric layers on the MF₂ substrate. Figure 6(a) is the corresponding two-dimensional map of reflectance for p-polarization wave, where E_F = 0.3eV, N = 7. It is seen that by changing the parameters such as the thickness h of medium A, E_F and N, the optimal reflectance dip can be obtained at different frequencies. We further study the excitations of the bulk polaritons with different Fermi energy. As shown in Fig. 6(b), the reflectance spectra became narrower when the excitation frequency is increased from 4 THz to 20 THz, where the incident angle is 51.86°. It is also possible to observe the reflectance minima for higher frequencies when the E_F increases, which is in accordance with the theoretical analyses of the bulk polaritons as shown in Fig. 3(a). This is due to the increase in negative group index of such HMM at higher Fermi energy [13]. Furthermore, the reflectance amplitudes of 11THz at 51.86° vs the E_F are also plotted as shown in Fig. 5 (b) (case 2). This tuning property of the considered HMM structure can be realized at different frequency, which indicates that large modulation bandwidth can be obtained by using the ATR modulator, such property is important in tunable reflectance modulation.

 

Fig. 6 (a)The two-dimensional map of reflectance of the Otto configuration at different frequency and incident angles, (b) at different frequency and Fermi energy, where the dielectric A is MF₂ with${\epsilon }_A$ = 1.9, h = 1.5um.

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4. Conclusion

In conclusion, we have theoretically studied the electromagnetic properties of HMM that comprises graphene/dielectric multilayers based on the permittivity homogenization model. The p-polarized polariton dispersion relations for surface and bulk polaritons in a HMM layer surrounded by different media are obtained. We have quantitatively shown that the capability of tuning the polaritons properties via Fermi energy, the thickness of HMM, the layer number of graphene sheets. The calculated reflectance spectra in an ATR configuration involving an HMM shows that the bulk polaritons can be efficiently excited at the desired frequency by choosing appropriate parameters. The modifications of the optical reflectance properties in the Otto configuration with HMM may have practical applications in tunable optical reflection modulator.