Plasmonic cloak using graphene at infrared frequencies

Optics Communications ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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Plasmonic cloak using graphene at infrared frequencies Yan Xiu Li, Fan Min Kong n, Kang Li, Hua Wei Zhuang School of Information Science and Engineering, Shandong University, Jinan 250100, China

art ic l e i nf o

a b s t r a c t

Article history: Received 28 February 2015 Received in revised form 16 May 2015 Accepted 21 May 2015

A carpet cloak based on graphene is designed and realized by making an approximate hemisphere surface which behaves as a flat surface, and the performances of the cloak are simulated by finite element method. The cloak performs perfectly through tuning conductivity of the graphene. The incident wave can propagate on the curved surface without being disturbed, and an object under the curved surface will be cloaked. It is indicated that graphene can be a platform for “on-atom-thick” cloaks, and the proposed methods can be applied in the practical design. & 2015 Published by Elsevier B.V.

Keywords: Anisotropic optical materials Three-dimensional fabrication Invisibility cloaks Transforms

1. Introduction

2. Theoretical analyses

Electromagnetic cloaking, which was firstly designed by Pendry and his coworkers in 2006, is undoubtedly a most intriguing application in modern communication [1]. Soon, this was verified experimentally by Smith et al. in the microwave regime [2]. From then on, it has aroused great interests in exploring in various cloaks [3–5]. However, most of the cloaks require anisotropy, where some components of the permittivity and permeability tensors are singularities [6–8] or perform imperfectly [9]. These reasons limit the use of the cloaks in practical application. More recently, as a good platform for plasmonics, graphene has attracted more and more interest in potential application because of its tunability of the conductivity and strong confinement of surface plasmon polariton (SPP) waves [10]. The effective refractive index of the graphene is related to its conductivity. Through tuning graphene conductivity, the desired effective refractive index can be achieved, and then the SPP waves can be manipulated freely. A design for one-atom-thick cloak is presented in this paper. It is achieved by utilizing an approximate hemispherical geometry with graded index to create a perfect, isotropic, and omnidirectional cloak. A curved surface with given refractive index will modify the effective distance seen by a SPP wave. The conductivity profile will ensure that the propagation characteristics of a flat surface are emulated by the hemispherical one, and this makes the curvature invisible to SPP waves confined to the hemispherical surface.

Graphene's complex conductivity (σg = σg , r + iσg , i ) can be calculated according to the Kubo formula [11,12]:

n

Corresponding author. E-mail address: [email protected] (F.M. Kong).

σ (ω, μc , τ , T ) =−

ie2 (ω − i/τ ) π ℏ2

⎡ 1 ⎢ × ⎢⎣ (ω − i/τ )2 −

∫0

∞

∫0

∞

⎛ ∂fd (ξ ) ∂fd ( − ξ ) ⎞ ⎟ dξ ξ⎜ − ∂ξ ⎝ ∂ξ ⎠

fd ( − ξ ) − fd (ξ ) (ω +

i/τ )2

−

4 (ξ /ℏ)2

⎤ dξ ⎥ , ⎦

(1)

where ω is the radian frequency, τ is the relaxation time, T is the temperature, and μc is the chemical potential, e is the electron charge, ℏ = h/2π is the reduced Planck's constant, ξ is the energy, fd (ξ ) = (e(ξ − μc ) / kB T + 1)−1 is the Fermi–Dirac distribution, and kB is the Boltzmann's constant [13]. The graphene's conductivity depends on the chemical potential. The tunability of the chemical potential of the graphene can be implemented via gate voltage, electric field, magnetic field, and chemical doping [11,12,14]. The schematic diagram of the hemisphere emulating a plane is presented in Fig. 1. It is calculated by equating the optical path lengths on the plane and hemispherical surfaces for two orthogonal paths. The first is a circular path of the fixed radius (illustrated in red in Fig. 1). The second is a radial path of fixed angle (illustrated in blue in Fig. 1). The refractive index of the curvature depends on the angle. In Fig. 1, the radius of the hemisphere is a , the red lines are the circular geometric paths (c1 and c2) and the blue lines denote the radial path lengths (r1 and r2), separately in

http://dx.doi.org/10.1016/j.optcom.2015.05.047 0030-4018/& 2015 Published by Elsevier B.V.

Please cite this article as: Y.X. Li, et al., Optics Communications (2015), http://dx.doi.org/10.1016/j.optcom.2015.05.047i

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Y.X. Li et al. / Optics Communications ∎ (∎∎∎∎) ∎∎∎–∎∎∎

2

Fig. 1. The diagram illustrating the relation between the hemisphere surface and the plane. The green line, R (θ ) , which is a constant, denotes the length from the origin to the hemispherical surface. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) -4

4.0x10

-4

g,r

-4

g,i

3.5x10 3.0x10

-4

2.5x10

g (S)

1 2 3 4 5 6 7 8 9 10 11 12 Q2 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

-4

2.0x10

-4

Fig. 4. (a) The refractive index profile required by the hemispherical cloak. (b), (c) Simulations of the wave propagation on curved graphene with the appropriate graded refractive index. (d), (e) Cross-sectional plots of (b) and (c).

1.5x10

-4

1.0x10

-5

5.0x10

n1 (r ) dr = n2 (θ ) R (θ )2 + R′ (θ )2 dθ

0.0 0.0

0.1

0.2

0.3

0.4

0.5 0.6 (eV) c

0.7

0.8

0.9

1.0

Fig. 2. The relationship curve between graphene's complex conductivity and chemical potential at f = 50 THz , T = 300 K , and Γ = 0.43 eV .

when a plasmonic cloak is designed, in order to reduce reflections due to the transition between the flat and the hemispherical surfaces, the requirement must be met that n2 (π /2) = n1 (a) . According to the boundary condition, n2 (π /2) = n1 (a), as well as Eqs. (2) and (3), aftersome manipulation to find the transformed index, the required index profile n2 (θ ) for the hemispherical cloak, can be found:

n2 (θ ) = n1 (r )(csc (θ ) − cot (θ ))/ sin (θ ),

Fig. 3. Simulations of the wave propagation on graphene with hemispherical geometry, where the refractive index of the graphene is homogeneous. (a) The incident wave is a plane SPP wave. (b) The incident wave is a spherical SPP wave. (c), (d) Cross-sectional plots of (a) and (b).

flat and hemispherical surface. Here, n1 (r ) and n2 (θ ) are the relative refractive index of the flat and the hemispherical surface, respectively. The infinitesimal optical path elements of a ray traversing the two radial and circular geometric paths are respectively equated to give [15]:

n1 (r )⋅2πr = n2 (θ )⋅2πR (θ ) sin (θ ) and

(2)

(3)

(4)

According to Eq. (1), when the working frequency is set at f = 50 THz, temperature is T = 300 K , and the phenomenological scattering rate is Γ = 0.43 meV , the relationship curve between graphene's complex conductivity and chemical potential is given in Fig. 2. According to previous studies, it is known that when σg , i > 0, a graphene layer behaves as a very thin metal layer supporting highly confined transverse-magnetic (TM) SPP waves [14,16,17]. For practical application, the surface of the cloak can be divided into m parts with equal arc length for every band along the z -axis. Every part is substituted by a narrow ribbon of graphene with fixed refractive index n2, j ( j represents the j th part). Consequently, the surface constituted by graphene is approximate to a hemisphere surface. For the TM SPP wave on the graphene, its dispersion relation can be expressed as [11,14,18–20]:

⎛ 2 ⎞2 ⎟⎟ , β = k 0 1 − ⎜⎜ ⎝ σg η0 ⎠

(5)

where k0 and η0 are respectively the free space wavenumber and the intrinsic impedance of free space. From Fig. 2, it can be seen that when μc > 0.15 eV , the real part of the graphene's conductivity is much smaller than its imaginary part. As a consequence, the dispersion relationship can be approximately written as:

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n2

n2 n2

n2

n2 58.9 64.4 73.3

3

55.9

87.4

n2 110

Fig. 5. (a) The discrete refractive index profile required by the hemispherical cloak when it is divided into six bands. (b), (c) The simulations of the hemispherical cloak with discrete refractive index.

n2 56.7n2 n2 58.9 n2 62.3 n2 67.0 n2 73.3 n2 82.0 n2 93.7 n2 110

55.4

Fig. 6. (a) The discrete refractive index profile required by the hemispherical cloak when it is divided into nine bands. (b), (c) The simulations of the hemispherical cloak with discrete refractive index.

n2 56.0 n2 58.9 n2 64.4 n2 73.3 n2

n2

87.4

110

n2

55.2 n2 57.2 n2 61.3 n2 68.4

n2 n2

79.6 97.3

Fig. 7. (a) The discrete refractive index profile required by the hemispherical cloak when it is divided into 12 bands. (b), (c) The simulations of the hemispherical cloak with discrete refractive index.

Table 1 Thickness of SiO2 when the applied voltage is 30 V.

nspp

Refractive index

t (nm)

0 1 2 3 4 5 6

110 110 87.4 73.3 64.4 58.9 55.9

441 441 324 242 203 173 158

Then, the requirement can be obtained that the graded conductivity of graphene should fulfill:

Fig. 8. Schematic of uneven SiO2 underneath the graphene layer is shown.

2 , ≈ σg, i η0

Layer

σg, i =

2 , n2, j η0

(7)

(6)

where σg , i is the imaginary part of graphene conductivity. To analog the same spatial variation, the effective mode index of the SPP wave should vary spatially according to Eq. (4). For a designated part of the graphene surface, there will be n2, j corresponding to it.

3. Simulation results In the simulations, the radius of the hemisphere a is 100 nm. According to [21], the effective refractive index of graphene has

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little relationship with its thickness, for saving simulation computer resource, the graphene thickness is set as 1 nm. From Eq. (7), when the relative refractive index is set to be 110, the imaginary part of graphene conductivity is 0.048 mS and the real part of it can be neglected. And when σg 4 0.048 mS, μc and σg are in linear relationship, then relative refractive index on the flat plane and on the background graphene is set to be 110. Fig. 3(a) illustrates the Ez component of a plane TM wave incident from the left when the graphene's refractive index is homogeneous, 110. It is clear that the plane wave fronts are severely distorted by the curvature. The same results will be presented, when a point source, has been utilized for excitation of a spherical SPP wave in Fig. 3(b). When n1 (r ) = 110, the graded index of the cloak can be expressed as:

n2 (θ ) = 110 (csc (θ ) − cot (θ ))/ sin (θ )

(8)

The profile of n2 (θ ) is shown in Fig. 4(a). Fig. 4(b) and (c) illustrates the simulation results of the cloak. In these insets, it can be seen that the modified index profile may correct the phase difference and reconstruct the wave fronts. Thus the cloak would perform perfectly. In the practical fabrication, the hemisphere surface is composed of some graphene ribbons with fixed refractive index. Because the tunability of the chemical potential of the graphene, we can see that it is feasible to realize the graded refractive index by manipulating the conductivity of graphene. The discretization process is performed on the continuous distribution to give a very simple six-banded, nine-banded and twelve-banded structure to evaluate the practical performance with equal arc length for every band along the z -axis, in which band n2 (θ ) is set as fixed refractive index. The simulation results are shown in Figs. 5–7. From the results above, it can be concluded that the hemispherical surface is subdivided into an obviously larger number, the simulation results will be better. According to Eq. (8), the required conductivity of graphene can be achieved. To create required conductivity patterns along the graphene surface, the similar methods suggested in [14] is used. Since the distance between graphene and Si substrate is nonuniform, the static electric field due to the same bias voltage is not uniform too. It means that the chemical potential is different at different segments, which results in the inhomogeneous conductivity. The schematic of uneven SiO2 is shown in Fig. 8. The applied bias voltage (Vbias ) modifies graphene carrier density (ns ) as [22]:

Cox Vbias = qens ,

(9)

where Cox = εd ε0/t is the gate capacitance, ε0 and εd are the permittivity of free space and SiO2 respectively, t is the thickness of the SiO2 layer. In addition, the carrier density is related with chemical potential via the expression [22]:

ns =

2 π ℏ2v 2f

∫0

∞

ξ [fd (ξ − μc ) − fd (ξ + μc )]∂ξ ,

(10)

where vf is the Fermi velocity (~108 cm/s in graphene). The desired thickness t of the SiO2 layer can be accurately retrieved by numerically solving Eqs. (1), (9), and (10). When the bias voltage is 30 V , εd = 3.9 and the curved surface is subdivided into six bands, the thickness of SiO2 is shown in Table 1.

When the curved surface is divided into nine and 12 bands, the same method can be used to achieve the thickness of SiO2.

4. Conclusion In summary, it is demonstrated that a practical proposal of a plasmonic cloak with graphene is applicable in this article. First, in theory the equivalent refractive index of the hemispherical cloak is calculated and the performance of the cloak is accurately verified by the simulation of wave propagation in the hemisphere geometry. Then, a discretization of the index distribution is adopted and an approximate hemispherical surface is composed of some ribbons of graphene. Finally, the performance of the graphene cloak is demonstrated to be robust. It is indicated that graphene can be a platform for “on-atom-thick” cloaks, and the proposed methods can be applied in the design of antennas and practical devices with graphene.

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