1. INTRODUCTION

Recently, much attention has been paid to graphene plasmonics, in which a collective excitation of massless Dirac fermions in graphene, namely, graphene plasmon, plays a crucial role in controlling the radiation field in the THz region [1]. The controllability is implemented by the electric gating or chemical doping, along with nano processing of graphene sheets with various shapes. For instance, a doped graphene disk, whose diameter is approximately 100 nm, acts as a high-$Q$ cavity, and exhibits the vacuum Rabi splitting of an adsorbed molecule having an electric dipole transition [2]. Further controllability is obtained in a periodic arrangement of graphene disks, where a perfect absorption is light is predicted in the one-monolayer-thick atomic membrane [3]. Washing-board-like periodic corrugation of graphene [4] and graphene antidot arrays [5,6] have also been proposed to create novel light-matter interaction. However, it is still difficult to realize these specimens with high accuracy.

Here, we propose an alternative to this route. We employ a two-dimensional diffraction grating in the vicinity of the doped graphene. Excess charge in the graphene sheet produces the electrostatic potential. It is modulated by the grating, resulting in a periodic electrostatic potential. Affected by the periodic potential, a charge redistribution takes place, forming a periodic charge-density distribution depending on the grating geometry. This implies that the Fermi energy of the graphene measured from the Dirac point becomes a function of space. In a system with the Dirac spectrum, the Fermi energy is proportional to the square root of the number density of the electrons. Moreover, the optical conductivity is proportional to the Fermi energy in the Drude approximation [7]. Therefore, if the graphene sheet exhibits a periodic charge-density modulation, we have a periodic modulation of the optical conductivity as in the graphene disk array [8]. We do not need the nanofabrication of the graphene disk array, nor periodic corrugation. We just need a diffraction grating, in which various nanofabrication technologies are available.

Besides, the diffraction grating also modulates a possible evanescent AC wave, accompanied by, for instance, the graphene plasmon. Therefore, a joint role of the diffraction grating is clarified. That is, it modulates the electrostatic potential, giving rise to a periodic modulation of the optical conductivity, and it modulates the evanescent wave. Both the modulations affect the plasmon polariton dispersion, and can yield a plasmonic band gap structure. In addition, as is familiar in the photonic crystal, point and line defects affect strongly the electromagnetic field in a band gap. Thus, many applications are available without actually processing graphene, but with processing the diffraction gratings into cavity, waveguide, and so on. Here, we study a triangular-lattice grating as a typical example.

A similar idea of using diffraction gratings was reported previously [9 –12]. However, the authors focused mainly on weak modulation of the graphene plasmon polariton having no complete band gaps. Such a grating modulation acts as merely a grating coupler for the plasmon polariton. They also neglect the electrostatic modulation of the optical conductivity. Therefore, a full consideration of both the electrostatic potential and dynamical radiation field strongly modulated by grating is yet to be explored.

This paper is organized as follows. In Section 2, we show that a periodic modulation in the optical conductivity of the doped graphene can be realized by a two-dimensional diffraction grating. In Section 3, dynamical light-scattering in the modulated graphene is presented, taking the diffraction grating also into account. Numerical results and discussions on the periodic modulation are given in Section 4; we emphasize the band engineering of the graphene plasmon polariton. Finally, we summarize the results.

2. ELECTROSTATIC MODULATION OF GRAPHENE BY GRATING

Let us consider the electrostatic potential problem in a doped graphene above a two-dimensional diffraction grating. The system under consideration is shown in Fig. 1. The graphene is regarded as a two-dimensional conducting sheet having excess charges because of an electric gating to reservoir or a chemical doping.

 

Fig. 1. Schematic illustration of the system under study. The doped (or gated) graphene sheet is placed above a diffraction grating. The electrostatic potential induced by the excess charge in the graphene is modulated by the grating, showing the self-consistent charge-density distribution in the graphene.

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The grating is characterized by a periodic arrangement of the permittivity $ϵ$:

Since the system has the lattice translational invariance, the Bloch theorem can be applied. In the electrostatic problem, Bloch momentum ${\mathit{k}}_{\parallel }$ must be zero, because the graphene should exhibit the equipotentiality. Inside the grating, the electrostatic potential $\psi$ satisfies

In the doped graphene at $z=z_c$, the potential must be constant. We take $\psi (z_c)=0$ without loss of generality. The overall potential profile in the graphene-grating system becomes

The induced charge density in the graphene is then given by

3. DYNAMICAL LIGHT SCATTERING IN MODULATED GRAPHENE

So far, we have considered the electrostatic potential in the graphene-grating system. The grating affects not only the electrostatic potential, but also the dynamical radiation field through relevant evanescent waves. Therefore, it is desirable to have a theoretical framework to deal with the dynamical light scattering there. The key quantity is the S-matrix, which describes the light scattering processes in the graphene-grating system. Once we have the S-matrices of isolated graphene and grating, the S-matrix of the combined system is readily available by the so-called layer-by-layer procedure [13]. It is well known that the S-matrix of the diffraction grating is obtained by the rigorous coupled-wave analysis (RCWA) [14 –16]. On the other hand, the S-matrix of the modulated graphene is less known. Thus, we present the basic formalism to construct the S-matrix of the modulated graphene.

Once we have a periodic charge-density modulation, optical conductivity also modulates. In the Drude approximation, the conductivity becomes [7]

 

Fig. 2. Schematic illustration of the Fermi energy and charge-density modulation.

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Since the optical conductivity is periodic in plane, it is possible to expand the conductivity as

Let us construct the S-matrix of the modulated graphene. Suppose we have an isolated, but modulated, graphene placed at $z=0$, embedded in a background medium of permittivity ${ϵ}_u$. Since the system has the lattice translational invariance, the eigenstates of AC wave are characterized by Bloch momentum ${\mathit{k}}_{\parallel }$ parallel to the graphene. The S-matrix is then defined by

By imposing the boundary condition, the S-matrix is given as

Besides, the S-matrix of the grating is obtained by a standard algorithm of the RCWA. Combining the two S-matrices, we can obtain the total S-matrix of the graphene-grating system. The combined system has various optical modes, originating from the graphene plasmon polariton and the (quasi) guided modes from the grating itself [16,17].

The graphene plasmon polariton of the uniformly doped graphene in the background medium of permittivity ${ϵ}_u$ is given by [18]

On the other hand, the (quasi) guided modes in the grating depend strongly on the grating geometry. It is roughly approximated by the Brillouin-zone folding of the guided modes in the effective homogeneous slab with spatially averaged permittivity ${ϵ}_{\mathrm{eff}}={ϵ}_b+({ϵ}_a-{ϵ}_b)f$ [17]. We should note, however, that if high-index substrate is employed for the grating, no such guided mode emerges.

The quasi-guided modes in the graphene-grating system are identified as the frequency peaks of the optical density of states, which can be calculated from the total S-matrix [19]. The true-guided modes in the graphene-grating system are obtained as the poles of the total S-matrix. Their secular equation becomes

4. NUMERICAL RESULTS AND DISCUSSION

To be specific, let us consider a two-dimensional diffraction grating composed of the triangular lattice of the circular rod with radius $r_a=0.3a$ and permittivity ${ϵ}_a=12$ embedded in air (${ϵ}_b=1$). Here, $a$ is the lattice constant, and is taken to be 1 μm. The grating has a thickness of $t=0.6a$, supported by a high-index substrate (${ϵ}_l=12$) of the semi-infinite thickness. The medium above the grating is air (${ϵ}_u=1$). The doped graphene is placed above the grating with distance $d=0.01a$, and the average Fermi energy $E_F$ is taken to be 0.4 eV measured from the Dirac point. The average charge density is then given by

Figure 3 shows the electrostatic potential profile in the $xz$ plane. We can see a periodic modulation of the electrostatic potential by the diffraction grating. If the grating is replaced by a homogeneous slab, no periodic modulation is observed. The potential has the minimum at the graphene sheet because of the negative charge of the doped electrons.

 

Fig. 3. Electrostatic potential $\psi$ in the $xz$ plane ($y=0$) for a graphene-grating system. The grating consists of the triangular array of circular rods with radius $r_a=0.3a$ and permittivity ${ϵ}_a=12$ embedded in air (${ϵ}_b=1$). The grating has a thickness of $t=0.6a$, supported by a semi-infinite high-index substrate with permittivity ${ϵ}_l=12$. The region above the grating is air (${ϵ}_u=1$). The lattice constant $a$ is taken to be 1 μm. The doped graphene, whose average Fermi energy is 0.4 eV, is placed above the grating with distance $d=0.01a$. The lower panel shows the cross-sectional profiles at $(x,y)=(\mathrm{0,0})$ (inside a rod) and $(x,y)=(0.5a,0)$ (outside the rods). The potential at the graphene sheet is taken to be zero.

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Although the modulation of the electrostatic potential is not strong, the charge density inside the graphene is strongly modulated as shown in Fig. 4. Here, the charge density is normalized by the minus sign of the average one. In the vicinity of the rods, the charge density becomes about two times larger than the average. The electrons are thus accumulated there. This charge-density modulation implies a periodic modulation of the optical conductivity as well.

 

Fig. 4. Charge density ${\rho }^{2d}$ in the graphene sheet, induced by the diffraction grating. The charge density is normalized by $|{\rho }_0|$, the absolute value of the average charge density. The same parameters as in Fig. 3 are employed.

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Obviously, the periodic charge-density modulation becomes weak with the increasing distance between the graphene and grating. We show in Fig. 5 the minimum and maximum charge densities as a function of the distance. Although a stronger charge modulation is expected for a smaller distance below $d=0.01a$, we have a convergence problem with respect to the number of reciprocal lattice vectors taken in the numerical calculation. In this way, the two-dimensional charge-density modulation emerges just by placing the doped graphene in the vicinity of the two-dimensional diffraction grating.

 

Fig. 5. Maximum and minimum charge densities in the graphene sheet, as a function of the distance $d$ between the graphene and grating. The same parameters as in Fig. 3, except for $d$ (distance between the graphene and grating), are employed.

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Next, we consider dynamical light-scattering problems, particularly the resulting photonic band structure induced by the periodic modulation of the optical conductivity. In our case of the high-index substrate for the grating, there is no guided mode localized around the grating. Thus, the resulting photonic bands of the graphene-grating system come from the graphene plasmon polariton.

Figure 6 shows the photonic band structure of the graphene-grating system with distance $d=0.01a$. We observe three photonic band gaps around $\omega a/2\pi c=0.036$, 0.05, and 0.063, owing to the periodic modulation. The first and second gaps have moderate frequency widths, and can be enlarged as shown later. The last one is very narrow and is thus fragile against disorder.

 

Fig. 6. Photonic band structure of the graphene-grating system. The same parameters as in Fig. 3 are employed. The solid and dashed lines are the light lines of air and substrate, respectively. The photonic band structure is evaluated in the dissipation-less limit $\tau \to \infty$. For comparison, the imaginary part in the eigenfrequency, caused by finite $\tau$ of $4\times {10}^{-13}\mathrm{s}$, is also indicated by the error bars, for some representative points.

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In the band calculation, we assume the dissipation-less limit $\tau \to \infty$. In actual specimens, relaxation time $\tau$ is finite, typically $4\times {10}^{-13}\mathrm{s}$ at $E_F=0.4\mathrm{eV}$ [2]. This finite relaxation time results in a nonzero imaginary part in the eigenfrequencies. We estimate the imaginary part by the full-width at half-maximum in the absorption spectrum for the incident evanescent (propagating at $\mathrm{\Gamma }$) wave from the top, having momentum ${\mathit{k}}_{\parallel }$ at the $\mathrm{\Gamma }$, M, and K points. The results are indicated as the error bars in Fig. 6. We should note that more direct evaluation of the imaginary part is available in the S-matrix formalism [16,20,21]. As for the numerical convergence, a good convergence within 1% error is achieved with $N_{\mathit{g}}=169$, where $N_{\mathit{g}}$ is the number of reciprocal lattice vectors taken in the numerical calculation.

We should remember that the band structure originates from the graphene plasmon polariton, which is $p$-polarized and has the even parity in the nonperiodic system. When the periodic modulation is added, both $p$- and $s$-polarized components mix. Moreover, it is no longer parity even, because the $z$-parity symmetry is broken by the grating. Thus, the photonic band structure under consideration is not classified according to polarization and parity. The photonic band gaps are polarization- and parity-independent.

This band structure shows a striking contrast to that in two-dimensional photonic crystal slabs [17,22,23]. There, nontrivial band structure emerges at higher frequencies, typically above $\omega a/2\pi c\sim 0.25$. In the frequency range of Fig. 6, an effective medium approximation is generally applicable for the photonic crystal slabs, and no band gaps are found. Moreover, in photonic crystal slabs, where hole arrays of the triangular lattice are often employed, photonic band gaps are usually limited in even-parity modes in $z$-symmetric structures with respect to the planes bisecting the slabs [22]. Odd-parity modes tend to have no photonic band gaps. If $z$-asymmetric structures are employed as in our case, the even- and odd-parity modes mix, and band gaps generally close.

Since there is a scale invariance in the Maxwell equation, the band structure in terms of the normalized frequency $\omega a/2\pi c$ depends on the dimensionless parameters $r_a/a$, $t/a$, $d/a$, and $E_{f}a/hc$, along with the permittivity constants ${ϵ}_a$, ${ϵ}_b$, ${ϵ}_u$, and ${ϵ}_l$. In addition, we found that change in the band structure with respect to $E_{f}a/hc$ is nearly described by the scale factor of $\sqrt{E_{f}a/hc}$. For instance, if we increase $E_{f}a/hc$ by factor 2, then the band structure in terms of $\omega a/2\pi c$ increases with factor $\sqrt{2}$ approximately. This is reasonable, in accordance with the $\sqrt{{\omega }_c}$ dependence of the graphene plasmon found in Eq. (40).

The photonic band structure does not change much if we change the normalized thickness $t/a$ and the substrate permittivity ${ϵ}_l$ of the grating. It is also reasonable that the photonic band structure comes from the graphene plasmon polariton, which is tightly localized around the graphene. However, through the electrostatic potential, the lowest two bands are affected by the thickness and the substrate permittivity.

Therefore, if we fix the permittivity (${ϵ}_a$ and ${ϵ}_b$) of the grating itself, the rest parameters are the radius $r_a/a$ of the rods and the distance $d/a$ between the graphene and grating. As naturally expected, if the distance is small enough, the charge density and band structure are strongly modulated. A large distance results in the empty-lattice-like photonic bands.

The photonic band gaps around $\omega a/2\pi c=0.036$ and 0.05 in Fig. 6 can be the platforms of various optical functions, such as defect waveguiding and the cavity as in photonic crystal slabs. A remarkable feature is the ultimate thickness of the system. That is, we can create such functions in a one-monolayer-thick atomic membrane, assisted by the grating.

To manage these functions, it is desirable to have wider band gaps. For optimization, the gap map is shown in Fig. 7, where the gap frequencies are plotted as a function of rod radius, keeping the other parameters fixed. Here, the narrow band gap found in Fig. 6 is neglected. As Fig. 7 shows, the higher (in frequency) band gap is optimized at about $r_a=0.27a$. The lower band gap is optimized at about $r_a=0.18a$.

 

Fig. 7. Gap map of the photonic band structure in the graphene-grating system, as a function of rod radius $r_a$. The same parameters as in Fig. 3, except for $r_a$, are employed.

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Concerning other grating geometries, we have examined square, honeycomb, and Lieb lattices of circular rods or holes, keeping the permittivity contrast and the other parameters fixed except for the rod or hole radius. Among them, we found that hole-type honeycomb and kagome lattices can exhibit photonic band gaps. Figure 8 shows the photonic band structure of the graphene-grating system having a hole-type honeycomb-lattice structure in the grating. Again, we can see the band gap between the third and fourth bands. This band gap is also common in the kagome-lattice system. The band gap is observed only for nearly close-packed holes, and is smaller than in the rod-type triangular lattice. Besides, there is no band gap between the first and second band. Therefore, from the viewpoint of wide band gaps, the rod-type triangular-lattice structure seems to be the best grating geometries for the modulated graphene plasmon polariton.

 

Fig. 8. Photonic band structure of the doped graphene on a diffraction grating of a hole-type honeycomb lattice. The following parameters are employed: ${ϵ}_a=1$, ${ϵ}_b=12$, ${ϵ}_u=1$, ${ϵ}_l=12$, $r_a=0.28a$, $t=0.6a$, $d=0.01a$, $E_F=0.4\mathrm{eV}$, and $a=1\mathrm{\mu m}$. The solid and dashed lines are the light lines of air and substrate, respectively.

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Although we focus on the photonic band gap, the photonic band regions are also useful. The modulated graphene plasmon polariton can couple to external light inside the light cone. The emergence of the quasi-guided modes there results in a strong modulation of the optical density of states as in the photonic crystal slab. Since the graphene plasmon polariton lies typically in the THz frequency range, the modulation can yield various phenomena, including enhanced light emission and nonlinear optics in the frequency range.

5. CONCLUSION

In conclusion, we have presented a method of creating a periodic modulation in doped graphene by a two-dimensional diffraction grating. The graphene-grating system exhibits a periodic charge-density modulation in the graphene sheet, so that the optical conductivity also modulates periodically. This modulation, along with the modulated evanescent wave because of the grating, gives rise to the photonic band structure of the graphene plasmon polariton. In particular, photonic band gaps are found in rod-type triangular-lattice and hole-type honeycomb or kagome-lattice gratings. Using the photonic band structure, various optical functions are expected to occur via graphene plasmon polaritons in one-monolayer-thick atomic membrane, supported by two-dimensional diffraction gratings.