Introduction

Because of the ability to overcome the diffraction limit of light in microchip-sized devices, SPPs are considered as one of the most promising candidates for nanophotonic components1,2,3. For this purpose, graphene has been investigated as a one-atom-thick platform for terahertz devices due to its ability to support and control surface plasmon polariton (SPP) surface waves in the terahertz regime4. Unlike those in noble metals, SPP surface waves in graphene can be tightly confined to the graphene layer because the real part of the effective index in graphene is much larger than that in free space5,6,7,8,9,10,11,12. In addition, graphene’s most unique and remarkable property is that its conductivity and permittivity can be dynamically tuned by controlling the chemical potential4, 11.

A number of schemes have been reported to create desired permittivity patterns on graphene flakes at the sub-wavelength scale13, 14. Moreover, the large effective index of graphene can efficiently constrain SPP waves around graphene sheet, which presents a variety of possibilities in the design of optical transformation devices, such as modulators15,16,17,18, filter19, cloaks20, and absorbers21,22,23,24. Additionally, transformation plasmonics has opened new avenues towards the realization of on-chip nanophotonic devices13. In particular, because of its easy fabrication, the uneven ground plane biased structure can be used to realize various terahertz devices, such as “electromagnetic black holes”25 and “beam-scanning planar lenses”26. The uneven ground plane below the dielectric spacer support can be used to easily tune the chemical potential of graphene to achieve desired permittivity patterns. However, this scheme is limited by soft-boundary effects27, which imply that the designed uneven ground plane pattern cannot be precisely mapped to the required permittivity pattern.

One way to realize graded index metasurface is to use a various ground-graphene distance approach, as reported in ref. 25. Different ground-graphene distances correspond to different chemical potentials, therefore leading to different indexes. However, this is very difficult to fabricate and suffers severe soft-boundary effects, which can lead to unexpectedly deterioration of the intended function. An alternative approach is reported in ref. 21, in which only two different ground-graphene distances are used. One distance corresponds to a background index, while another corresponds to a “dot” index, which is usually much larger than the background index. The required gradient effective index can then be realized by carefully designing the pattern with various dot sizes. In this way, the manufacturing processes can be easily controlled using planar integration techniques. However, too small dot size should be avoided since it will not only increase fabrication difficulties but lead to more apparent soft-boundary effects as well. In this paper, a new kind of Dot-Density-Renderer (DDR) quasicrystal structure is proposed to simulate gradient effective index by design density of equal size dots. This can efficiently reduce the soft-boundary effects. Moreover, due to the dynamic tuning capability of graphene, the DDR structure provides more design flexibility.

The graphene SPP platform has a finite propagation length, which will limit the device size on this platform. Within the platform, interferences between devices should be carefully controlled by constructing isolating structures. In this paper, a flexible electromagnetic (EM) fence is proposed to trap and absorb the interferences from other devices. Hence, the entire graphene SPP platform can be divided into different operation regions for different devices. Using the designed graded-index structure, both sides of the EM fence can trap and absorb EM waves. Unlike the EM black hole structure reported in ref. 28, the proposed EM fence has no core region. This can be attributed to the large imaginary part of the effective index of the dots region. Definitely, this feature enables an efficient reduction of the EM fence size.

Result

Theoretical analysis of the EM fence

In this paper, a Dot-Density-Renderer (DDR) structure is used to build a quasicrystal SPP platform. As shown in Fig. 1, the photonic crystal platform is composed of an uneven bottom ground and a substrate with a monolayer graphene sheet on top. Salient substrate dots with equal diameters and heights are periodically etched and grown inside the ground. When a static voltage bias is applied between the graphene sheet and the uneven ground, it will create two different electric fields on the graphene sheet due to the different voltage gate distances for the dot area and the background area. As a result, two different chemical potentials will be generated, thus yielding two different equivalent complex permittivities for these two different areas. The relationship between the control voltage V g and the SPP wave number β can be easily derived based on the Kubo formula, as given in Method. Notably, salient dots with a constant size are used here to relax the requirement for fabrication tolerances21 and minimize soft-boundary effects27. Clearly, the effective index will mainly depend on the dot density, which will be used to realize the graded-index materials and guided SPP waves on the graphene platform. In this paper, an Al2O3 substrate with a relative permittivity of 3.2 is selected as the dielectric material.

Figure 1
figure 1

The schematic illustration of the monolayer graphene salient dots biased structure.

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Figure 2 illustrates a concept of strip-shaped EM fence operating at terahertz bands based on the DDR structure. As shown in Fig. 2, the proposed strip EM fence has two absorbing regions, which can trap and absorb the SPP incident waves that approach from both sides. In order to ensure trapping of EM waves, the graded effective index of the EM fence n eff should satisfy the expression in Eq. (1)29,30,31.

$${n}_{eff}(r)=\{\begin{array}{cc}{n}_{0} & r > {R}_{in}\\ {n}_{0}{(\frac{{R}_{in}}{r})}^{\rho } & r\le {R}_{in}\end{array}$$
(1)

where R in is the half-width of the fence, r represents the distance from the central line of the fence, n 0 is the background index. It should be noted that the power index ρ should be greater than zero to enable trapping of EM waves towards the central line of the EM fence, as shown on the right side of Fig. 2. As discussed in ref. 29, the higher power index ρ, the better trapping capability the EM fence can achieve. Alternatively, if we choose a negative value for ρ, the SPP incident waves will rebound away from the fence, as shown on the left side of Fig. 2.

Figure 2
figure 2

The schematic illustration of the influence of ρ on trapping performance.

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The reported EM black holes29 usually require shell regions to trap the incident waves into the absorbing core region. However, because the dot regions in our DDR structure can be easily designed to exhibit a lossy characteristic, i.e. a large imaginary part of effective index, which act like the lossy core in ref. 29, SPP incident waves can then be absorbed while travelling through the dot regions of the EM fence. Therefore, the power index ρ in Eq. (1) does not need to be greater than 2 as required in refs 29,30,31. This will clearly bring us more freedom in practical designs.

The most significant feature of the proposed DDR structure is its easiness to simulate the graded index n eff by carefully designing the dot density. To guarantee penetration of EM waves into the designed EM fence, one should ensure that the operating frequency has not fallen into the forbidden bands of the 2D photonic crystal structure. Since the EM fence does not have a regular lattice distribution, its band structure cannot be easily calculated using analytical methods. Therefore, the distribution of dots should be carefully arranged in a practical design to minimize unwanted reflections at the design frequency band.

For better absorbing performance, a high imaginary part of the effective index is preferred for the proposed EM fence. For this purpose, the dots region in the DDR structure should be designed to operate at the absorption range, which corresponds to a large n dot as shown in Fig. 3a. The background region should be designed to fall in the transmission range to ensure transmission of SPP waves, corresponding to a small n 0 . However, there is a limit to the extent that n dot can be increased, as the large contrast between n dot and n 0 will result in a sparse dot distribution of the DDR structure. This will clearly increase possibilities of unwanted reflections, thus damaging the trapping capability. Another reason to avoid too large n dot is that it will experience a steep n r curve as shown in Fig. 3a, which implies a more sensitive voltage control of the DDR structure. Figure 3b gives the calculated complex index as a function of frequency for three typical chemical potentials. We can then define the cut-off frequency under which the TM mode can be supported. For example, the cut-off frequency for 0.03 eV chemical potential can be estimated as about 13.2 THz. However, strong dispersion will be experienced when the operating frequency is close to the cut-off frequency. Therefore, we will investigate the EM fence under 12 THz in this design.

Figure 3
figure 3

(a) SPP index as a function of chemical potential. The curves are calculated at f = 8 THz, τ = 0.3 ps and T = 300 K. (b) SPP index as a function of frequency at μ c  = 0.02, 0.025 and 0.03 eV, τ = 0.3 ps and T = 300 K.

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Structure and design of the EM fence

Black strip

Figure 4 shows the proposed design of a black strip based on the DDR structure. The effective index of the i-th row can be approximated by Eq. (2), where S dot is the area of a single dot, and d represents the distance between two adjacent rows. By substituting Eq. (2) into Eq. (1), the number of dots per unit length of the i-th row N dot can then be easily obtained.

$${n}_{eff}(i)=\frac{{n}_{dot}{S}_{dot}{N}_{dot}(i)+{n}_{0}(d-{S}_{dot}{N}_{dot}(i))}{d}$$
(2)
Figure 4
figure 4

(a) Illustration of index n dot and n 0 for dots region and background region, respectively. (b) Top-view of the black strip on graphene metasurface. (c) The schematic of dot distribution for the black strip.

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In this design, the chemical potentials of the dots and background are chosen to be 0.03 eV and 0.3 eV, respectively. The corresponding index can then be calculated as n dot  = 64.24 + 1.40j and n 0  = 7.67 + 0.09j at 8 THz. Other design parameters are gives as ρ = 1.45, the diameter of dots D dot  = 100 nm, the distance between two adjacent row d = 200 nm. The designed black strip is composed of 20 rows, 10 rows each side, with a total width of 60 µm. The calculated number of dots per unit length for each row and the corresponding effective index are shown in Fig. 5a. It should be noted that the resulting effective index will be slightly changed for different frequencies. However, the distribution of the effective index will be quite similar, leading to a similar trapping capability.

Figure 5
figure 5

(a) The dot density distribution of the designed black strip and the corresponding effective index. (b) Illustration of reflection and transmission with incident angle θ. (c,d) The reflectivity as a function of frequency and incident angle. (e,f) The transmissivity as a function of frequency and incident angle. (g,h) Field distributions of the black strip illuminated with a point source and a line source at 8 THz.

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To validate the absorbing performance of the designed black strip, the finite element method (FEM) software COMSOL is employed to analyze the scattering. For this purpose, a fitting method is used to describe index n dot and n 0 as function of frequency in COMSOL simulations without need of complicated numerical analysis.

The simulated transmissivity and reflectivity of the designed black strip are shown in Fig. 5c–f as a function of frequency f and incident angle θ. As shown, the reflectivity is better than −10 dB between 6 THz and 12 THz for incident angles from 5 deg to 90 deg. Apparently, the residual reflections are from the unwanted scattering effects of the edge dots of the designed black strip. The transmissivity is found to be better than −30 dB between 8 THz and 12 THz for most of incident angles. Theoretically, the absorption increases with frequencies, this is because that the imaginary part of the index for the dots region becomes larger for higher frequencies, as shown in Fig. 3b. In conclusion, the designed black strip works satisfactorily within a frequency range between 8 THz and 12 THz.

Figure 5g,h illustrates the field distributions of the designed black strip at 8 THz with two different illumination sources, i.e. a point source in Fig. 5g and a line source in Fig. 5h. It can be clearly observed in both cases that the incident SPP waves are trapped towards the central line of the black strip and get well absorbed. However, noticeable reflections can be observed with the incident angle of 45 deg in Fig. 5g. This quite agrees with our simulations in Fig. 5d, in which the reflection achieves its maximum at incident angle of 45 deg. Moreover, the difficulty in observing the transmitted waves indicates that the transmissivity is very small. This phenomenon indicates a strong absorption behavior of the designed EM fence has been achieved.

Black Ring

To demonstrate the flexibility of the EM fence, a black ring structure is introduced, in which the effective index of each ring is designed to imitate the gradient index material. Similarly, the number of dots N dot (i) can be calculated according to Eq. (2). For this purpose, an effective index distribution as function of the radius r is given in Eq. (3). To simplify the design, R in is selected as the asymptotic line of the outer ring, and R out is selected as the asymptote line of the inner ring.

$${n}_{eff}(r)=\{\begin{array}{cc}{n}_{0} & 0\le r < {R}_{in}\\ {n}_{0}{(\frac{{R}_{out}-{R}_{in}}{{R}_{out}-r})}^{\rho } & {R}_{in}\le r < \frac{{R}_{out}+{R}_{in}}{2}\\ {n}_{0}{(\frac{{R}_{out}-{R}_{in}}{r-{R}_{in}})}^{\rho } & \frac{{R}_{out}+{R}_{in}}{2}\le r < {R}_{out}\\ {n}_{0} & {R}_{out}\le r\end{array}$$
(3)

For the black ring, the chemical potential of the dot and the background are chosen to be 0.03 eV and 0.5 eV, respectively. The corresponding index can then be calculated as n dot  = 64.24 + 1.40j and n 0  = 4.66 + 0.05j. Same as the black strip, the diameter of dots is selected as D dot  = 100 nm, and the distance between two adjacent rows d = 200 nm. Other design parameters used in Eq. (3) are chosen as ρ = 2.04, R in  = 16 µm and R out  = 24 µm. The calculated number of dots per unit length and the corresponding effective index are shown in Fig. 6b.

Figure 6
figure 6

(a) Illustration of reflections and transmissions with inward and outward incident waves. (b) The dot density distribution of the designed black ring and the corresponding effective index. (c) The simulated transmissivities and reflectivities with inward and outward incident waves. (d,e) Field distributions of the black semi-ring illuminated with inward and outward incident waves at 8 THz.

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Figure 6c shows the simulated transmissivity and reflectivity of the designed semi-ring illuminated from two sides of the semi-ring, as marked in Fig. 6a. Similar to the black strip, a transmissivity of less than −30 dB is observed for incidence from both sides, when the frequency is larger than 8 THz. The corresponding reflectivity is better than −10 dB. Figure 6d,e shows the electric field distribution of the designed black-ring at 8 THz, in which good absorption performance is observed. The same conclusion can be made that good fence performance can be achieved for the designed black ring for frequencies between 8 THz to 12 THz.

Black complex

To demonstrate the performance of the designed EM fence structures, a couple of examples are given in this section. Figure 7 shows a black ring with an open end. The incident SPP wave EM-1 travels along a single-line-defect photonic crystal waveguide and then couples into the black ring. At the same time, a line source of SPP wave EM-2 illuminates the black ring from the outside. Figure 7a shows a snapshot of the electric fields simultaneously illuminated by SPP waves EM-1 and EM-2. For EM-1, a good travelling wave characteristic can be observed inside the photonic crystal waveguide. Also, a cylindrical phase front is quite obvious inside the black ring, which indicates there are very few reflections caused by the black ring. For EM-2, however, much stronger reflections can be seen from the single-line-defect photonic crystal waveguide than from the black ring. Furthermore, it can be seen that the EM-1 and EM-2 are well isolated by the designed black ring, which validates the good performance of the EM fence. This is more obvious when comparing the waves inside and outside of the black ring in Fig. 7b, which shows the amplitude distribution of the electric fields from COMSOL simulations. Also, higher contrast fringes can be observed from the photonic crystal waveguide, which indicates much stronger reflections are experienced.

Figure 7
figure 7

(a) Field distribution of black complex with two incident waves EM-1 and EM-2 at 8 THz. (b) Amplitude distribution.

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For more flexible designs, the strip and ring structures can be combined together for better performances. Figure 8 shows another example of composite structure, in which a black strip is right angle bended in the middle. As expected, appreciable reflections can be observed from the right-angle corner for both left-side incidence and right-side incidence, as shown in Fig. 8a,b. The reflections can be somewhat reduced by using a quarter length of black ring to build EM fence with a rounded corner, as shown in Fig. 8c,d.

Figure 8
figure 8

Field distributions of composite structures of the EM fence at 8 THz. (a,b) A right-angle corner composite EM fence. (c,d) A rounded-corner composite EM fence.

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Conclusion

In summary, we theoretically realize a concept of electromagnetic fence based on Dot-Density-Renderer quasi-photonic crystal structure, which is built on a monolayer graphene SPP platform with an uneven ground bias. Since all dots are in the same diameter, the soft-boundary effects can be effectively contained when an uneven ground structure is used to bias the graphene sheet. Two types of electromagnetic fences are designed, i.e. black strip and black ring, which exhibit quite favorable absorption properties over a very wide frequency band for incident waves over a large range of angles. Furthermore, the composite structures enable the realization of more flexible fence applications in the field of transformational plasmon optics.

Method

SPP waves on a graphene platform

The surface conductivity of graphene can be modeled using the Kubo formula:

$$\sigma ={\sigma }_{intra}+{\sigma }_{inter}$$
(4)

where σ intra is due to the intraband contribution, and σ inter is due to the interband contribution.

$${\sigma }_{intra}(\omega )=-j\frac{{e}^{2}{k}_{B}T}{\pi {\hslash }^{2}(\omega -j{\tau }^{-1})}[\frac{{\mu }_{c}}{{k}_{B}T}+2\,\mathrm{ln}({e}^{-{\mu }_{c}/{k}_{B}T}+1)]$$
(5)
$${\sigma }_{inter}(\omega )=\frac{-j{e}^{2}}{4\pi \hslash }\,\mathrm{ln}(\frac{2|{\mu }_{c}|-(\omega -j{\tau }^{-1})\hslash }{2|{\mu }_{c}|+(\omega -j{\tau }^{-1})\hslash })$$
(6)

where k B is the Boltzmann’s constant, e is the charge of an electron, and ћ is the reduced Planck’s constant. The Kubo formula shows that the surface conductivity of graphene depends on the frequency ω, scattering rate τ, temperature T, and chemical potential μ c . The scattering rate τ is given by ref. 32:

$$\tau =\frac{\mu \hslash \sqrt{\pi n}}{e|{\nu }_{F}|}$$
(7)

where |v F | = 1.1 × 106 m/s is the Fermi velocity, n is the carrier density, and μ is the mobility. Because the mobility μ of CVD graphene grown on hexagonal-boron nitride can easily approach a value of up to 60000 cm2/(Vs)33, a choice of τ = 0.3 ps is an appropriate assumption for our device simulation, according to Eq. (7).

In general, the surface conductivity of graphene is a complex number. According to Maxwell’s Equations, the complex conductivity of graphene can be used to obtain the equivalent complex permittivity of a very thin graphene sheet. Assuming a thickness Δ of the graphene sheet, we can obtain the following relations:

$$\varepsilon =\frac{-{\sigma }_{i}}{\omega {\rm{\Delta }}}+{\varepsilon }_{0}+j\frac{{\sigma }_{r}}{\omega {\rm{\Delta }}}.$$
(8)
$${\varepsilon }_{r}=\frac{-\,{\sigma }_{i}}{\omega {\rm{\Delta }}}+{\varepsilon }_{0}\approx \frac{-{\sigma }_{i}}{\omega {\rm{\Delta }}}.\quad \quad {\varepsilon }_{i}=j\frac{{\sigma }_{r}}{\omega {\rm{\Delta }}}.$$
(9)

Therefore, when the complex conductivity of a Δ-thick graphene sheet exhibits a positive imaginary part, it effectively behaves as a very thin “metal” sheet. A transverse-magnetic (TM) SPP surface wave can subsequently be supported on this sheet, and the guided mode wave number β can be expressed by Eq. (10) as Δ → 0.

$$\beta ={k}_{0}\sqrt{1-\frac{2}{{\eta }_{0}\sigma }}.$$
(10)

Here, η 0 is the free-space wave impedance, and k 0 is the free-space wave number. Notably, the wave number β is a complex number because the conductivity σ is complex. For SPP waves, the effective propagation length can be defined as 1/Im(β). The graded-index structure can be designed by modifying the real part of β.

For an isolated graphene sheet, the chemical potential μ c can be determined by the carrier density n using the following equation:

$$n=\frac{2}{\pi {\hslash }^{2}{v}_{F}^{2}}[{({k}_{B}T)}^{2}{\int }_{-{\mu }_{c}/{k}_{B}T}^{{\mu }_{c}/{k}_{B}T}\frac{x}{{e}^{x}+1}dx+{k}_{B}T{\mu }_{c}\,\mathrm{ln}({e}^{-{\mu }_{c}/{k}_{B}T}+1)+{k}_{B}T{\mu }_{c}\,\mathrm{ln}({e}^{{\mu }_{c}/{k}_{B}T}+1)]$$
(11)

Applying a voltage V g on the gate can modulate the carrier density of graphene through the following equation:

$${V}_{g}=\frac{\hslash |{\nu }_{F}|\sqrt{\pi n}}{e}+\frac{ne}{{C}_{g}}.$$
(12)

where C g  = ε/t is the geometric capacitance, ε is the permittivity of the dielectric material, and t is the dielectric layer thickness, which may be the key factor for controlling the electrical characteristics of graphene in practical design. It should be noted that the mode area of monolayer graphene sheet waveguides is extremely small34 (approximately 10−7λ0 2). In other words, the distance between the graphene sheet and the uneven ground is large enough to ensure that the guided mode of the SPP waves will not be affected.