Plasmons in Waveguide Structures Formed by

ISSN 00213640, JETP Letters, 2013, Vol. 97, No. 9, pp. 535–539. © Pleiades Publishing, Inc., 2013. Original Russian Text © P.I. Buslaev, I.V. Iorsh, I.V. Shadrivov, P.A. Belov, Yu.S. Kivshar, 2013, published in Pis’ma v Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 2013, Vol. 97, No. 9, pp. 619–623.

Plasmons in Waveguide Structures Formed by Two Graphene Layers P. I. Buslaeva, I. V. Iorsha, I. V. Shadrivova, b, P. A. Belova, and Yu. S. Kivsharb a

National Research University of Information Technologies, Mechanics, and Optics, St. Petersburg, 197101 Russia email: [email protected] b Nonlinear Physics Center, Research School of Physics and Engineering, Australian National University, ACT 0200 Canberra, Australia Received March 20, 2013; in final form, April 9, 2013

Plasmon modes in a waveguide formed by two parallel graphene layers with an insulator spacer layer are con sidered. The existence of TM and TE guided modes is predicted and their properties are compared with those of plasmons in metal/insulator waveguides. DOI: 10.1134/S0021364013090063

Much attention has been attracted in recent years to graphene, a material made up of a twodimensional lat tice of carbon atoms [1–4]. From the fundamental point of view, graphene is interesting because charge carriers in this material are characterized by a linear dis persion relation, which leads to such phenomena as the roomtemperature quantum Hall effect [5] and the existence of a nonzero Berry phase for charge carriers [2]. Graphene is also promising for applications owing to the high chargecarrier mobility, which is required for the new generation of electronic integrated circuits [6]. In the last few years, the interaction of graphene with an electromagnetic field has become actively studied. It was predicted that TE and TMpolarized localized surface waves can propagate along a graphene surface, the dispersion of these waves falling in the terahertz frequency range [7]. A concept for implementing twodimensional optical integrated cir cuits based on graphene was proposed in [8]. An important property of surface waves in graphene is the possibility to efficiently control their propagation and dispersion characteristics by changing the gate voltage or the magnetic field oriented perpendicularly to the graphene plane [9, 10]. Recently, it was demonstrated that an optical mod ulator can be made on the basis of a waveguide system formed by two graphene layers [11]. Absorption in sys tems of this kind was considered in [12]. The possibil ities of controlling the dispersion of waveguide modes by the magnetic field, gate voltage, or temperature were examined in [9, 13]. However, only the disper sion properties of TMpolarized surface waves, similar to guided modes in metal/insulator plasmonic waveguides, were investigated in these studies. Here, we demonstrate that a system of two graphene layers spaced by a dielectric layer with the thickness d and relative permittivity ε] (Fig. 1) sup

ports both TE and TMpolarized waveguide modes. The ranges of existence of TEpolarized modes are investigated. The characteristics of guided modes in the system under study are compared with those in metal/insulator plasmonic waveguides. In order to determine the spectrum of propagating guided modes localized in the vicinity of the two dimensional layer, we write the boundary conditions for the tangential components of the electric and mag netic fields: ( E 1 – E 2 ) × n 1–2 = 0,

(1)

4π σ ( ω, μ, γ, T )E , ( H 1 – H 2 ) × n 1–2 =  || c

(2)

where n1–2 is the unit vector along the normal oriented from region 1 to region 2 and E|| is the electric field of the wave in the xz plane, which induces current in the graphene layers.

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Fig. 1. Waveguide structure formed by two parallel graphene layers separated by a dielectric layer with the thickness d and dielectric constant ε.

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Writing the equations that describe propagating plane waves (see Fig. 2) and applying boundary condi tions (1), (2) for the waveguide consisting of two graphene layers, we obtain the equations for TM polarized waves ⎛ 1 ⎜ ⎜ 4π q ⎜ i   σ + 1 ⎜ c k0 ⎜ ⎜ 0 ⎜ ⎜ ⎝ 0

⎞ ⎟ ⎟ 0 ⎟ ⎟ ⎟ q σ + 1 ⎟ – ε q i 4π   ⎟ q' c k 0 ⎟ ⎠ –1 1 q'd

–1 –e q q q'd – ε  ε e q' q' q q'd ε  e q' –e ⎛ ⎜ ⎜ ×⎜ ⎜ ⎜ ⎝

q'd

0

2

2

q' = β – εk 0 , and β = kz is the magnitude of the wave vector component along the propagation direction. The graphene conductivity σ was calculated under various assumptions in [14–17]. We use the results obtained in [16], since losses in graphene were taken into account in that paper: 2

(3)

∞

⎧ ∂f d ( Ᏹ ) ∂f d ( – Ᏹ ) 1 × ⎨  2 Ᏹ   –  dᏱ ∂Ᏹ ∂Ᏹ ⎩ ( ω + i ⋅ 2γ ) 0

∫

(7)

∞

⎫ fd ( –Ᏹ ) – fd ( Ᏹ ) 2 dᏱ ⎬, –  2 ( ω + i ⋅ 2γ ) – 4 ( Ᏹ /ប ) ⎭ 0

∫

where

Ᏹ–μ f d ( Ᏹ ) = exp ⎛ ⎞ + 1 ⎝ kB T ⎠

and TEpolarized waves

⎛ ⎜ ⎜ ×⎜ ⎜ ⎜ ⎝

2

2

β – k0 ,

( ω + i ⋅ 2γ) σ ( ω, μ, γ, T ) = ie  2 πប

E1 ⎞ ⎟ + E2 ⎟ ⎟ = 0, – ⎟ E2 ⎟ E3 ⎠

⎛ q q'd 1 – ε q ε e 0 ⎜ q' q' ⎜ ⎜ 4π k q'd ⎜ i  0 σ – 1 1 0 e ⎜ c q ⎜ q'd 4π k ⎜ 1 i  0 σ – 1 0 e ⎜ c q ⎜ q'd ⎜ – ε q 1 0 ε qe ⎝ q' q'

Here, k0 = ω/c, c is the speed of light, q =

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(4)

H1 ⎞ ⎟ + ⎟ H2 ⎟ = 0. – H2 ⎟ ⎟ H3 ⎠

Setting the determinant of the matrices in Eqs. (3) and (4) equal to zero, we obtain the dispersion relations for electric and magnetic surface waves: q'd ⎧ 4π q qe +1  for TM 1 , ⎪ 1 + i   σ ( ω ) = – ε   c k0 q' e q'd – 1 ⎪ ⎨ q'd ⎪ 4π q qe –1   σ ( ω ) = – ε    for TM 2 , ⎪ 1 + i  c k0 q' e q'd + 1 ⎩

(5)

q'd ⎧ 4π k 0 q' e – 1  for TE 1 , ⎪ i   σ ( ω ) – 1 =   q e q'd + 1 ⎪ c q ⎨ q'd ⎪ 4π k 0 q' e + 1   σ ( ω ) – 1 =    for TE 2 . ⎪ i  q'd c q q – 1 e ⎩

(6)

–1

,

e is the elementary charge, ប is the Planck constant, kB is the Boltzmann constant, ω is the frequency of the wave, γ is the decay constant, Ᏹ is the electron energy, μ is the chemical potential, and T is the temperature. The conductivity calculated according to Eq. (7) is plotted in Fig. 3. One can see that, for certain frequen cies, the imaginary part of the conductivity becomes negative. This means that, for these frequencies, TE polarized surface waves can propagate along a single graphene layer [7]. For other frequencies, only TM polarized surface waves can propagate in singlelayer graphene. In this context, it is reasonable to suggest that waveguide systems formed by two graphene layers can also support both TE and TMpolarized propa gating waves at certain frequencies. Figure 4 shows the profiles of the waveguide modes. One can see that mode TM1 is antisymmetric and mode TM2 is symmetric. In addition, there is only one antisymmetric TE1 mode in the TE polarization. The localization of this mode is weak. Figures 5 and 6 show the dispersion curves for the two TM modes and the TE mode, respectively. According to these figures, both TE and TMpolar ized plasmon modes exist in the terahertz frequency range. Note that the TE1 mode has almost linear dis persion and small losses. This is a consequence of the small degree of localization of this mode and indicates that it is nearly equivalent to a plane wave propagating in free space. In contrast, TMpolarized modes are strongly localized and, as a consequence, are characterized by small propagation lengths and strong dispersion, which differs markedly from that of a plane electro JETP LETTERS

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Fig. 2. Waveguide system consisting of two graphene layers +

–

with a dielectric layer (ε = εs) between them; E1, E 2 , E 2 , +

–

and E3 and H1, H 2 , H 2 , and H3 are the components of the electric and magnetic field, respectively, along the propagation direction in the corresponding spatial regions; d is the separation between the graphene layers; and q' and q are introduced in the text following Eqs. (3), (4).

Fig. 3. Conductivity of a graphene layer in units of σ0 = πe2/(2h) versus the frequency for γ = 1 meV, T = 300 K, and μ = 0.2 eV.

magnetic wave in free space. The dispersion curve in Fig. 5a has a region of negative group velocity. It should be noted, however, that this region is also char acterized by high losses, so that the imaginary part of the waveguide number β becomes comparable to or larger than its real part. The absence of the symmetric TE2 mode in this system needs an additional discussion. It can easily be shown that the relationship q ≈ 0 is always satisfied for a TEpolarized mode. Analyzing the dispersion rela tion for the mode TE2 in the absence of losses, we obtain the following condition for the existence of this mode: ε < – cot ε – 1 k 0 d 4π σ''    , c ε–1 2

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the equation q =

2

2

β – ε1 k0 :

⎧ ( ε 2 – ε 1 )ε 1 2 4π ⎪ q =  dk 0 + i  k 0 σε 1 for TE 1 , c 2ε 2 ⎪ ⎨ ε ⎪ q = i 4π  k 0 σε 1 – 2 1 for TE 2 . ⎪ c d ε2 ⎩

(9)

The analysis of these expressions indicates that mode TE1 can exist if ε1/ε2 < 0.4, while the numerical calcu

(8)

which is satisfied for d > 10–4 cm. When losses are taken into account, the minimum spacing between the graphene layers required for the existence of the TE2 mode increases further. Obviously, the waveguide description given above is valid if the linear dimensions of the system along the direction of propagation are much larger than the transverse dimensions. Currently, the linear sizes of highquality graphene films obtained experimentally do not exceed a few microns. Thus, at present, the implementation of a waveguide system supporting a symmetric TEpolarized mode seems unfeasible. In addition to the spacing between the layers, the existence of TEpolarized modes depends also on the dielectric constants ε1 and ε2 of the medium outside the waveguide system and the insulator separating the graphene single layers, respectively. As was mentioned above, q ≈0 for TE modes. In the system under study, d = 10–6 cm and k0 ≈ 104 cm–1. If we use the approxi mation eq'd ≈ 1 + q'd on the righthand sides of the dis persion relations, then, discarding terms of the second JETP LETTERS

order, we can easily obtain an approximate solution to

Fig. 4. Field distribution profiles for the waveguide modes calculated for T = 300 K, μ = 0.2 eV, εs = 4, d = 10–6 cm, γ = 1 meV, and frequency បω/μ = 0.9 (TM modes) and 1.9 (TE mode). (a) TM modes are characterized by the real part of the tangential electricfield component. (b) TE mode is characterized by the real part of the tangential magneticfield component.

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Fig. 5. Dispersion curves for TMpolarized localized modes calculated for μ = 0.2 eV, T = 300 K, d = 10–6 cm, and γ = 1 meV: (a) Re(β) and (b) Im(β) versus frequency.

Fig. 6. Dispersion curves for TEpolarized localized modes calculated for μ = 0.2 eV, T = 300 K, d = 10–6 cm, and γ = 1 meV: (a) Re(β) and (b) Im(β) versus frequency.

lation yields an estimate ε1/ε2 < 0.77. For mode TE2, the solution does not satisfy the condition q ≈ 0; this fact gives additional indirect evidence of the absence of this mode. Metal/insulator plasmonic waveguides support the propagation of coupled surface plasmons similar to those in the waveguide system with two graphene lay ers considered above. For the sake of comparison, let us consider two different threelayer systems, i.e., metal/insulator/metal and insulator/metal/insulator ones. The dielectric function of a metal is given by the Drude formula

Here, i, j ∈ {1, 2}; εi is the dielectric function of the

2

ωp ε m ( ω ) = ε ∞ –  . ω ( ω + iγ )

(10)

It should be noted that only TMpolarized surface waves exist in metal/insulator structures. The disper sion relation for these systems can be obtained in the following form: ε i q j – ε j q i –qi d = ±1 .  e εi qj + εj qi

(11)

2

2

corresponding layer; qi = β – ε i k 0 ; β and k0 were introduced above; and d is the thickness of the middle layer. Comparing different types of surface waves, let us consider the following parameters: (λImβ)–1, which characterizes the plasmon propagation length, and (λReq)–1, which describes plasmon localization. The characteristics of different structures are given in the table. The table lists the parameters of a waveguide system formed by two graphene layers and of two metal/insu lator multilayer structures (metal/insulator/metal and insulator/metal/insulator ones). Furthermore, the characteristics of the graphenebased waveguide sys tem are compared for low and high frequencies. At low frequencies, the plasmon propagation length (λImβ)–1 in the system with graphene layers is somewhat larger than that in metal/insulator structures. At high fre quencies, where the imaginary part of β increases by several orders of magnitude (see Fig. 5b), plasmons hardly propagate. The plasmon localization length JETP LETTERS

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Plasmon parameters in different structures*

Propagation length Degree of localization

DLG lf

DLG hf

IMI

MIM

101–102 101–103

10–3–10–2 ~101

100–102 102–104

100–102 ~101

* The propagation length and degree of localization are given in units of plasmon wavelength λ and layer spacing d, respectively. DLG stands for the waveguide system with two graphene layers; its parameters are calculated for low frequencies (lf), corresponding to the lower parts (ω < 0.2 eV) of the curves in Fig. 5b, and high frequencies (hf), corresponding to the upper parts (ω > 0.2 eV) of the curves in Fig. 5b. MIM and IMI stand for metal/insulator/metal and insulator/metal/insulator structures, respectively.

(λReq)–1 is nearly the same for graphenebased and metal/insulator structures; however, at low frequen cies, plasmons are more strongly localized in the sys tem with graphene layers than in metal/insulator structures. Thus, at low frequencies, the characteris tics of the system formed by two graphene layers are somewhat better than those of metal/insulator struc tures. We also calculated the dependences of the propaga tion and decay constants of the TM1 mode on the chemical potential in graphene in the vicinity of the Dirac point. These dependences calculated for a fre quency of 0.1 eV are shown in Fig. 7. One can see that the guided modes of the structure are symmetric with respect to the Dirac point. Upon approaching the bandtoband absorption threshold (μ/ω = 0.5), both the degree of localization of the surface wave Reβ and the mode decay constant Imβ increase. Nevertheless, there exists a region where the wave is strongly local ized and losses are still fairly low. Thus, we have obtained the dispersion relations for plasmonic modes in a waveguide system formed by two graphene layers separated by a layer of insulator. We

Fig. 7. Propagation and decay constants of mode TM1 ver sus the chemical potential in the vicinity of the Dirac point.

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have demonstrated that TE and TMpolarized plas mon modes exist in such a system in the terahertz fre quency range. The group velocity of these waves has been analyzed and the possibility of the existence of guided modes with negative group velocity has been shown. The characteristics of guided modes in the sys tem with two graphene layers have been compared to those of plasmon modes in metal/insulator nanostruc tures. REFERENCES 1. K. S. Novoselov, A. K. Geim, S. V. Morozov, et al., Nature 485, 197 (2005). 2. Y. Zhang, Y. W. Tan, H. L. Stormer, and P. Kim, Nature 438, 201 (2006). 3. A. K. Geim and K. S. Novoselov, Nature Mater. 6, 183 (2007). 4. M. I. Katsnelson, Mater. Today 10, 20 (2007). 5. K. S. Novoselov, Z. Jiang, Y. Zhang, et al., Science 315, 1379 (2007). 6. T. Palacios, Nature Nanotechnol. 6, 464 (2011). 7. S. A. Mikhailov and K. Ziegler, Phys. Rev. Lett. 99, 016803 (2011). 8. A. Vakil and N. Engheta, Science 332, 1291 (2011). 9. C. H. Gan, H. S. Chu, and E. P. Li, Phys. Rev. B 85, 125431 (2012). 10. I. Iorsh, I. V. Shadrivov, P. A. Belov, and Yu. S. Kivshar, JETP Lett. 97, 287 (2013). 11. M. Liu, X. Yin, and X. Zhang, Nano Lett. 12, 1482 (2012). 12. A. A. Dubinov, V. Ya. Aleshkin, V. Mitin, et al., J. Phys.: Condens. Matter 23, 145302 (2011). 13. D. Svintsov, V. Vyurkov, V. Ryzhii, and T. Otsuji, arXiv:1211.3629 (2012). 14. L. A. Falkovsky and A. A. Varlamov, Eur. Phys. J. B 56, 281 (2007). 15. F. T. Vasko and V. Ryzhii, Phys. Rev. B 76, 233404 (2007). 16. G. W. Hanson, J. Appl. Phys. 103, 064302 (2008). 17. T. Stauber, N. M. R. Peres, and A. K. Geim, Phys. Rev. B 78, 085432 (2008).

Translated by M. Skorikov