- Nano Express
- Open Access

# Mode manipulation and near-THz absorptions in binary grating-graphene layer structures

- Haiming Yuan
^{1, 2}, - Hanning Yang
^{1, 2}, - Pengzi Liu
^{1, 2}, - Xiangqian Jiang
^{1, 2}and - Xiudong Sun
^{1, 2}Email author

**9**:90

https://doi.org/10.1186/1556-276X-9-90

© Yuan et al.; licensee Springer. 2014

**Received:**13 December 2013**Accepted:**28 January 2014**Published:**21 February 2014

## Abstract

The excitation and absorption properties of grating coupled graphene surface plasmons were studied. It was found that whether a mode can be excited is mainly determined by the frequency of incident light and the duty ratio of gratings. In the structure consisting graphene bilayer, a blueshift of the excitation frequency existed when the distance between neighbor graphene layer were decreased gradually. In graphene-grating multilayer structures, a strong absorption (approximately 90% at maximum) was found in near-THz range.

## Keywords

- Graphene surface plasmons
- Excitation condition
- Multilayer structure absorption

## Background

Recently, a lot of work has been done based on graphene due to its unique properties in electric, magnetic, thermal, etc.[1–3]. Graphene is carbon atoms arranged in a two-dimensional honeycomb lattice, in which the electrons behave like massless Dirac fermions with linear dispersion[4, 5]. Graphene has strong plasmonic effects which can be modified by gating, by doping, and so on[2]. A controllable optical absorption was also found in structured graphene[6, 7].

Up to date, the graphene is modeled usually to be an extremely thin film with a conductivity *σ*, which consists of both intraband and interband from Kubo formula[7–9]. The intraband conductivity with Drude type plays a leading role when ℏ*ω*/*μ*_{c} was small[10]. Both transverse electric (TE) and transverse magnetic (TM) have the dispersion relations at monolayer graphene with dielectric materials on two sides[10–12]. In other words, the charge carriers coupling to electromagnetic waves will produce a new surface wave, namely graphene surface plasmons (GSPs).

In the previous works, many numerical approaches were used to study the structured graphene, for example the finite element method (FEM)[13], finite difference time domain (FDTD)[14], and others[6, 15]. A strong plasmonic response of graphene has been demonstrated in a square-wave grating with a flat graphene on top[15]. In which, the graphene-based plasmon response lead to a 45% optical absorption. In a periodic array of graphene ribbons, remarkably large GSPs result in prominent optical absorption peaks[13]. In multilayer graphene, the absorption spectrum can be decomposed into subcomponents[6], which is helpful in understanding the behavior of GSP coupling.

In this paper, we studied the binary grating bounded by graphene on both sides. The rigorous coupled wave analysis (RCWA)[16, 17] was used the first time as we know to characterize the graphene-containing periodic structures. The excitation condition and excitation intensity seemed to be influenced by the grating constant, duty ratio and the distance between the graphene layers. When introducing more graphene layers into the structure periodically, a strong absorption band was found in the near-THz range.

## Methods

### Electromagnetic mode of binary grating-graphene

*ω*≾

*μ*

_{c}(see[10]). Then the intraband conductivity can be expressed as[23]

*μ*

_{c}is the chemical potential, relating to the electron density. Equation 1 became a Drude type when

*μ*

_{c}/

*k*

_{B}

*T*≫ 1, i.e.

*ε*

_{1}and

*ε*

_{2}, respectively. The electric induced current on graphene layer resulted in a magnetic field difference, which led to the coupled GSP on graphene layer. Using Maxwell equation and boundary condition, GSP modes were proved to existed for both TE and TM polarization[12, 23–25]. For TE mode, the dispersion relation was as follows:

Because the imaginary part of conductivity (2) was positive, no solution of Equation 3 was found in real, which meant the TE mode GSP could not be excited.

*n*

_{eff}=

*β*/

*k*

_{0}=

*βc*/

*ω*as the effective index of GSP. After making a transformation of (

*ω*,

*n*

_{eff}) → (

*ω*,

*β*), the dispersion relations were obtained and plotted in Figure 1. The wave vector was normalized by

*k*

_{Λ0}= 2

*π*/

*λ*

_{0},

*λ*

_{0}= 1 μm. As a local mode, GSP modes were same as the surface plasmon polaritons (SPPs). They cannot be excited directly from the air. And in our work, gratings were used to provide an external wave vector to match the phase condition.

### Rigorous coupled wave analysis in graphene-containing structures

*h*to be the depth of grating (thickness of gratings). The

*h*was also the distance between two graphene layers. In multilayer structures of Figure 2b, 2

*h*was the longitudinal period. The structures were designed to only contain two kinds of interfaces.

In common, the conventional RCWA based on the Floquet's theorem[26] was unable to be used for the graphene-containing structures as the electric field will induce a current with current density **J** = *σ* **E**, while graphene was included.

**J**can also be expended to the sum of spatial harmonics with different wave vector components. To obtain the reflection, transmission, absorption, field distribution, and other optical properties of such structures as shown in Figure 2, a nonzero item must be included in the boundary condition of

*H*

_{y}field considering the induced current,

*H*

_{y}will also be continuous at the interface if each spatial harmonics subcomponent satisfied the boundary conditions independently,

in which *n* was the order, ± in subscripts represented approaching to *y*_{0} from two different directions. After the modification on the RCWA program, we can utilize it to deal with the graphene-containing structures.

## Results and discussion

### Phase matching condition

*ε*

_{1}-graphene-

*ε*

_{1}and

*ε*

_{1}-graphene-

*ε*

_{2}. When GSP is propagating along the interfaces, the phase shifts were

*φ*

_{1}and

*φ*

_{2}for the two kinds of interfaces, respectively.

*δ*was the total phase loss considering two abrupt phase changes when GSP propagates across the joints between the two kinds of interfaces in a grating period. At the excitation frequency, the phase change in a grating period should satisfy the relation

*N*is the integer and can be rewritten as

*f*

_{2}=

*L*/

*Λ*and

*f*

_{1}= 1 -

*f*

_{2},

*β*

_{1}and

*β*

_{2}were the wave vectors of GSP on two kinds of interfaces, respectively. When

*N*was a nonnegative integer, the GSP mode could be excited, and

*N*can be defined as the order of surface modes. The resonant frequencies can be obtained both from absorption spectrum in Figure 3 and theoretically from Equation 10 (

*δ*= 0). They were given in Table 1 and agreed well for high order modes. But for low order modes, some deviations existed between numerical and theoretical caused by the coupling of GSPs on two graphene layers.

**The resonant frequency of different orders**

Order of GSP (N) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | … |
---|---|---|---|---|---|---|---|---|

| 11.9 | 16.7 | 20.5 | 23.7 | 26.3 | 28.9 | 31.1 | … |

| 11.70 | 16.61 | 20.38 | 23.55 | 26.34 | 28.86 | 31.18 | … |

### Duty ratio and stand wave interference

*f*, where

*f*=

*φ*

_{1}/(

*φ*

_{1}+

*φ*

_{2}),

*φ*

_{1},

*φ*

_{2}had the same meaning as Equation 9. From the discussion above, when the phase match conditions were satisfied, GSPs could be coupled and absorption peaks should appear. But some break points appeared as shown in Figure 5, which meant that GSPs cannot be excited at this frequency and this duty ratio and it was reasonable to believe that there must be other process related to

*f*of controlling the excitation of GSP.

First two terms were GSP excited by one set of points (A in Figure 6) with two propagating directions (blue and green) and the last two terms were that from another set of points (B in Figure 6), where *x*_{0} is the distance of A and B in the form of light path (*k*_{0}*x*_{0} = *L*_{1}β_{1} = *φ*_{1} = (*φ*_{1 +}*φ*_{2})*f* = 2*πNf*). Because in real space, different interfaces (*ε*_{1}/*ε*_{1} and *ε*_{1}/*ε*_{2}) had different propagating constants, the expression might be complex. Here, the light path of *x* was used. It is found that scatting points A and B had a phase difference of *π*. This was caused by the different geometric symmetries. From Equation 11, when sin(*k*_{0}*x*_{0}/2) = 0, i.e., *f = m/N* ( *m* = 0, 1, …, *N*), field amplitude *F* would always be 0, which meant that the field cannot be excited. It was a cancelation process of two sets of standing waves that are coherent. So, for GSP mode of *N*, *N* + 1 of none absorption points appeared.

### Coupling of GSPs on different graphene layers and resonant frequency shift

*h*. A blueshift of absorption peaks was found when the grating became thin. The oscillator model is used to describe this phenomenon of spectrum blueshift[30, 31].

In Equation 12, κ(*n*, *h*, ∆θ) was the coupling coefficient and *n*, *h*, and ∆*θ* were order of GSP mode, thickness of grating, and phase difference of GSPs on two graphene layers, respectively. Essentially, the GSPs were surface waves so they interact with each other via evanescent interactions, and the coupling intensity decayed exponentially with *h* increasing. For fixed *h*, the lower order modes had larger skin depth (stronger coupling intensity) than the higher orders; then, the stronger coupling resulted in a large spectra shift. The phase difference of ∆*θ* also had affection to the absorption frequencies. However, in our case, the wavelength (15 meV ~ 82.8 μm) was much larger than the thickness of grating layer (*h* = 10 μm), it is reasonable to assume ∆*θ* is approximately 0. This can also be obtained clearly from the field distribution in Figure 4 that the electric fields on upper and lower graphene layers oscillated synchronously. This conclusion can still hold in multilayer graphene-grating structures. Finally, *κ*(*n*, *h*, ∆*θ*) ∝ *e*^{-hq(n)}, where$\mathit{q}\left(\mathit{n}\right)={\left[{\mathit{\beta}}^{2}\left(\mathit{n}\right)-{\mathit{\epsilon}}_{1,2}{\mathit{k}}_{0}^{2}\right]}^{1/2}$.

*x*

_{up}=

*x*

_{down}=

*x*

_{0}

*e*

^{-iωt}(no phase difference between GSP on neighbor layers), it is found that the resonant frequency became

When *h* was small (*h* < 4 μm), the larger *κ*(*n*, *h*, ∆*θ*) ∝ *e*^{
-h
} was the larger shift of resonant frequency would be. And obviously, *κ*(*n*, *h*, ∆*θ*) was approaching 0 rapidly when *h* was large enough, which meant that the resonant frequency became a stable value of${\left({\mathit{\omega}}_{0}^{2}-{\mathit{\gamma}}^{2}/2\right)}^{1/2}$. Otherwise, *κ*(*n*, *h*, ∆*θ*) was also related to the order of GSP. The high order mode had a small skin deep with weak coupling intensity and less blueshift. When *h* tends to be 0, the grating became too thin to excite the surface mode. This was why the absorption disappeared when *h* = 0 in Figure 7.

### Strong absorption in grating-graphene multilayers

## Conclusions

On conclusion, the rigorous coupled wave analysis was modified to compute the excitation of graphene surface plasmon in graphene-containing binary gratings structures. Under the phase matching conditions, the excitation of the graphene surface plasmonics was determined by the distance between graphene layers and duty ratio of gratings, and the mode suppression can be realized by modifying the grating constant and duty ratio. A blueshift of the excitation frequency was obtained for enhanced coupling between GSP of neighbor graphene layers. Increasing the number of graphene layers had almost no effect on the excitation frequency of GSP but would lead to a high absorption with negligible reflection in near-THz range. Finally, the resonant frequency and absorptions can be easily modified by manipulating the structure parameter, including grating constant, duty ratio, and distance between the graphene layers and number of grating, and graphene-containing grating might become potential applications of THz region, such as optical absorption devices, optical nonlinear, optical enhancement, and so on.

## Declarations

### Acknowledgements

This project was supported by the National Basic Research Program of China (no. 2013CB328702) and by the National Natural Science Foundation of China (no. 11374074).

## Authors’ Affiliations

## References

- Geim AK, Novoselov KS: The rise of graphene.
*Nat Mater*2007, 6: 183–191. 10.1038/nmat1849View ArticleGoogle Scholar - Grigorenko A, Polini M, Novoselov K: Graphene plasmonics.
*Nat Photonics*2012, 6: 749–758. 10.1038/nphoton.2012.262View ArticleGoogle Scholar - Bonaccorso F, Sun Z, Hasan T, Ferrari A: Graphene photonics and optoelectronics.
*Nat Photonics*2010, 4: 611–622. 10.1038/nphoton.2010.186View ArticleGoogle Scholar - Novoselov K, Geim AK, Morozov S, Jiang D, Grigorieva MKI, Dubonos S, Firsov A: Two-dimensional gas of massless Dirac fermions in graphene.
*Nature*2005, 438: 197–200. 10.1038/nature04233View ArticleGoogle Scholar - Ju L, Geng B, Horng J, Girit C, Martin M, Hao Z, Bechtel HA, Liang X, Zettl A, Shen YR: Graphene plasmonics for tunable terahertz metamaterials.
*Nat Nanotechnol*2011, 6: 630–634. 10.1038/nnano.2011.146View ArticleGoogle Scholar - Koshino M, Ando T: Magneto-optical properties of multilayer graphene.
*Phys Rev B*2008, 77: 115313.View ArticleGoogle Scholar - Gusynin V, Sharapov S, Carbotte J: Magneto-optical conductivity in graphene.
*J Phys Condens Matter*2007, 19: 026222. 10.1088/0953-8984/19/2/026222View ArticleGoogle Scholar - Dressel M:
*Electrodynamics of Solids: Optical Properties of Electrons in Matter*. Cambridge: Cambridge University Press; 2002.View ArticleGoogle Scholar - Falkovsky L, Pershoguba S: Optical far-infrared properties of a graphene monolayer and multilayer.
*Phys Rev B*2007, 76: 153410.View ArticleGoogle Scholar - Mikhailov SA, Ziegler K: New electromagnetic mode in graphene.
*Phys Rev Lett*2007, 99: 016803.View ArticleGoogle Scholar - Stern F: Polarizability of a two-dimensional electron gas.
*Phys Rev Lett*1967, 18: 546–548. 10.1103/PhysRevLett.18.546View ArticleGoogle Scholar - Jablan M, Buljan H, Soljačić M: Plasmonics in graphene at infrared frequencies.
*Phys Rev B*2009, 80: 245435.View ArticleGoogle Scholar - Nikitin AY, Guinea F, Garcia-Vidal FJ, Martin-Moreno L: Surface plasmon enhanced absorption and suppressed transmission in periodic arrays of graphene ribbons.
*Phys Rev B*2012, 85: 081405.View ArticleGoogle Scholar - Nayyeri V, Soleimani M, Ramahi OM: Modeling graphene in the finite-difference time-domain method using a surface boundary condition. In
*IEEE Transactions on Antennas and Propagation*. Piscataway: IEEE; 2013.Google Scholar - Peres N, Bludov YV, Ferreira A, Vasilevskiy MI: Exact solution for square-wave grating covered with graphene: surface plasmon-polaritons in the THz range. J. Phys. Condens. Matter 25: 125303. arXiv preprint arXiv:12116358 2012 arXiv preprint arXiv:12116358 2012Google Scholar
- Moharam M, Grann EB, Pommet DA, Gaylord T: Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings.
*JOSA A*1995, 12: 1068–1076. 10.1364/JOSAA.12.001068View ArticleGoogle Scholar - Moharam M, Pommet DA, Grann EB, Gaylord T: Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach.
*JOSA A*1995, 12: 1077–1086. 10.1364/JOSAA.12.001077View ArticleGoogle Scholar - Neto AC, Guinea F, Peres N, Novoselov KS, Geim AK: The electronic properties of graphene.
*Rev Mod Phys*2009, 81: 109. 10.1103/RevModPhys.81.109View ArticleGoogle Scholar - Ziegler K: Robust transport properties in graphene.
*Phys Rev Lett*2006, 97: 266802.View ArticleGoogle Scholar - Gusynin V, Sharapov S, Carbotte J: Unusual microwave response of Dirac quasiparticles in graphene.
*Phys Rev Lett*2006, 96: 256802.View ArticleGoogle Scholar - Falkovsky L, Varlamov A: Space-time dispersion of graphene conductivity.
*Eur Phys J B*2007, 56: 281–284. 10.1140/epjb/e2007-00142-3View ArticleGoogle Scholar - Falkovsky L: Optical properties of graphene.
*Phys. Conf. Ser*2008, 129(1):012004.View ArticleGoogle Scholar - Hanson GW: Quasi-transverse electromagnetic modes supported by a graphene parallel-plate waveguide.
*J Appl Phys*2008, 104(8):084314–084314–5. 10.1063/1.3005881View ArticleGoogle Scholar - Mikhailov S, Ziegler K: A new electromagnetic mode in graphene.
*Phys. Rev. Lett.*2007, 99: 016803.View ArticleGoogle Scholar - Economou EN: Surface plasmons in thin films.
*Phys Rev*1969, 182: 539. 10.1103/PhysRev.182.539View ArticleGoogle Scholar - Petit R:
*Electromagnetic Theory of Gratings*. Heidelberg: Springer Berlin; 1980.View ArticleGoogle Scholar - Liu H, Lalanne P: Microscopic theory of the extraordinary optical transmission.
*Nature*2008, 452: 728–731. 10.1038/nature06762View ArticleGoogle Scholar - van Beijnum F, Rétif C, Smiet CB, Liu H, Lalanne P, van Exter MP: Quasi-cylindrical wave contribution in experiments on extraordinary optical transmission.
*Nature*2012, 492: 411–414. 10.1038/nature11669View ArticleGoogle Scholar - Lalanne P, Hugonin J, Rodier J: Theory of surface plasmon generation at nanoslit apertures.
*Phys Rev Lett*2005, 95: 263902.View ArticleGoogle Scholar - Liu N, Langguth L, Weiss T, Kästel J, Fleischhauer M, Pfau T, Giessen H: Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit.
*Nat Mater*2009, 8: 758–762. 10.1038/nmat2495View ArticleGoogle Scholar - Haus HA:
*Waves and Fields in Optoelectronics*. Englewood Cliffs, NJ: Prentice-Hall; 1984.Google Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.