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Plasmon induced transparency in graphene based terahertz meta-materials

Koijam Monika Devi1 (a), Maidul Islam1, Dibakar Roy Chowdhury2, Amarendra K. Sarma1 and GaganKumar1

1 Department of Physics, Indian Institute of Technology Guwahati, Guwahati 781039, Assam, India2 Mahindra Ecole Centrale, Jeedimetla, Hyderabad 500043, Telangana, India

PACS 73.20.Mf – Collective excitations (including excitons, polarons, plasmons and other charge-density excitations)

PACS 78.67.Pt – MetamaterialsPACS 78.67.Wj – Optical properties of graphene

Abstract – Plasmon induced transparency (PIT) effect in a terahertz graphene metamaterialis numerically and theoretically analyzed. The proposed metamaterial comprises of a pair ofgraphene split ring resonators placed alternately on both sides of a graphene strip of nanometerscale. The PIT effect in the graphene metamaterial is studied for different vertical and horizontalconfigurations. Our results reveal that there is no PIT effect in the graphene metamaterial whenthe centers of both the split ring resonators and the graphene strip are collinear to each other.This is a noteworthy feature, as the PIT effect does not vanish for similar configuration in ametal-based metamaterial structure. We have further shown that the PIT effect can be tuned byvarying the Fermi energy of graphene layer. A theoretical model using the three level plasmonicsystem is established in order to validate the numerical results. Our studies could be significantin designing graphene based frequency agile ultra-thin devices for terahertz applications.

Introduction. – Electromagnetically induced trans-parency (EIT) is an interference effect that occurs in athree level atomic system. In EIT, an atom that absorbsa particular light signal is rendered transparent by shininganother light signal having nearly the same resonance fre-quency [1]. In 2008, a novel study done by Zhang and hisco-workers revealed that an EIT like phenomenon can oc-cur in plasmonic metamaterials (MMs) [2]. The plasmonicanalogue of this EIT effect is known as the Plasmon In-duced Transparency (PIT) effect in MMs [2–5]. PIT usu-ally occurs as a result of interference between the brightand dark plasmonic modes. The bright mode strongly cou-ples with the incident light while the dark mode couplesweakly with the incident light [6]. For the PIT effect tooccur, both the bright and the dark modes should havesimilar resonant frequencies [2, 7]. Then, the destructiveinterference of these modes induces a narrow transparencyregion in the otherwise absorptive spectrum. Since thedetection of the PIT effect, a lot of theoretical as well asexperimental studies have been done revealing its poten-tial in various applications such as ultrafast sensing [8–10],

(a)E-mail: [email protected]

switching applications [10], slow light phenomena [11–14],terahertz modulations [15] etc.

For most of the metal based MMs, tuning of the PITeffect is achieved by altering the geometrical parameters.Recently, active tuning of the PIT effect has also beenachieved by incorporating nonlinear media or semicon-ductor material in the MM structure [16, 17]. However,only limited tuning could be achieved in case of the metalstructures. The quest for a better and efficient tuningof the PIT effect, led to the study of this effect usinggraphene based MMs. Graphene provides extreme fieldconfinement and low propagation losses. Its Fermi energycan be varied easily by applying an external gate voltageor through chemical doping [18–22]. This has kindled alot of interest in exploring the efficient dynamic tunabiltyof the PIT effect in the graphene MMs in the mid-infrared[23–30] as well as terahertz frequency regions. Tuningof slow light through PIT effect in graphene MM struc-tures has been studied through the variation of Fermi en-ergy [25–27]. Metal split ring resonators (SRRs) coupledto a graphene strip has also been found to exhibit thePIT effect [29]. Comb like structure [23, 31] as well as

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dipole-dipole coupling [32], a graphene ring coupled to agraphene strip [33], intergrated graphene waveguides orarrays [34, 35] are some configurations that has been in-vestigated in recent years. In spite of these investigations,none of the study has been focused in exploring the PITeffect in graphene based terahertz MM structures to thebest of our knowledge. The field of graphene and tera-hertz MMs has grown significantly in last few years. Con-sequently, there is ample scope to explore and understandthe PIT effect in graphene based MMs and optimize theirperformance to actualize the construction of terahertz de-vices.

In this article, we report a novel graphene MM com-prising of a pair of graphene SRRs placed alternately onboth sides of the graphene strip. The PIT effect in a ter-ahertz graphene MM structure is numerically and theo-retically analyzed. The transmission characteristics of theproposed graphene MM structure are studied for differentvertical configurations. As opposed to similar metal MMstructure, the PIT effect in the graphene MM vanisheswhen the centers of both graphene SRRs and the graphenestrip are collinear to each other. We have also thoroughlyexamined the PIT effect in the graphene MM when theSRRs are displaced horizontally w.r.t. the graphene strip.Further, the PIT effect is studied for different values ofFermi energy of the graphene material. Finally, a theo-retical model based on the three level plasmonic systemis established in order to validate our numerical observa-tions.

SubstrateGraphene

TransmittedTHz

x

yz

t

Incident THz

(a)

0.2

0.4

0.6

0.8

1.0

Tran

smitt

ance

f (THz)0.5 0.7 0.9 1.1 1.3 1.5

PIT

Strip SRRs

(b)

aLw

a

d

pw

2

1

ps

Fig. 1: (a) Schematic diagram of the graphene MM struc-ture, (b) Individual transmission profile versus frequency forthe graphene strip, graphene SRRs and the graphene PIT MM.

Design of graphene terahertz metamaterial. –The schematic diagram of the graphene MM structure isshown in fig. 1(a). The transmission characteristics of thegraphene MM structure are investigated using finite ele-ment method based frequency domain solver in the CSTMicrowave Studio. The MM structure is composed of astrip and a pair of SRRs made of graphene material hav-ing a thickness of 1 nm placed on a substrate of thickness

t = 10 µm, having a dielectric permittivity of 2.4. Thegraphene strip and the graphene SRRs are designed suchthat they have similar resonance frequency, with very lit-tle deviation, upon excitation by the incident light. Thelength and width of the graphene strip is L = 19 µmand w1 = 1.1 µm, respectively. The outer dimensionsof each graphene SRR is a × a = 6.7 µm × 6.7 µm andwidth is w2 = 0.9 µm. ′d′ is the distance between thegraphene strip and the graphene SRRs while ′s′ is the dis-tance between the centers of the two graphene SRRs. Theperiodicity of the structure is taken as p = 24 µm. TheMM structure is simulated under unit cell boundary con-ditions in the x-y plane. An adaptive mesh size of theorder of λ/10, where λ is the wavelength of the incidentradiation, is employed. Open boundary conditions is setalong the direction of light propagation. The conductivityof graphene in the terahertz frequency range is describedby a simplified Drude expression [18]:

σ(ω) =e2Ef

(π~)2i

ω + iτ−1(1)

where e is the electronic charge and Ef is the Fermi en-ergy and τ is the intrinsic relaxation time of the graphenematerial. The transmission results of the graphene strip,graphene SRRs and the hybrid graphene MM structuresare shown in fig. 1(b). The red line represents the trans-mittance of the graphene strip while the black traces repre-sent the transmittance of the graphene SRRs. It is evidentfrom the figure that the graphene strip is directly excitedby the incident radiation whereas the fundamental modeof the pair of graphene SRRs is not excited by the incidentradiation. Hence, the graphene strip behaves as the brightmode while the graphene SRRs behave as the dark mode.When these graphene SRRs are kept in the vicinity of thegraphene strip, they couple through the induced electricfield of the graphene strip. This coupling causes a PITeffect in the graphene MM structure, which is representedby the blue line in fig. 1(b).

Plasmon induced transparency: Numerical sim-ulations. – The transmission characteristics of the pro-posed MM structure are studied for different vertical andhorizontal configurations. Fig. 2(a) represents the trans-mittance of the graphene MM structure for different vari-ations in ′s′ while d = 1.5 µm is fixed. The red linerepresents the transmittance of the MM structure whens = 0 µm. The blue line represents transmittance fors = 4 µm while the cyan traces signify the transmittancefor s = 6 µm. The green and the orange traces representthe transmittance of the MM structure for s = 8 µm ands = 11 µm, respectively. Simulations reveal that there isno PIT effect in the graphene MM when both the SRRsare collinear to each other i.e. s = 0 µm. However, as thegraphene SRRs are displaced away from each other in thevertical direction, the graphene MM starts exhibiting thePIT effect. Next, the transmittance of the graphene MMstructure for different values of ′d′ is shown in fig. 2(b). In

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this case, we have assumed ′s′ to be fixed. The orange linerepresents the transmittance of the MM structure whend = 0.5 µm. The green line represents the transmittancefor d = 1 µm while the cyan traces signify the transmit-tance for d = 1.5 µm. The blue and the red traces repre-sent the transmittance for d = 2.5 µm and d = 3.5 µm,respectively. It may be noted that, as d increases, the PITwindow becomes narrower due to the reduced coupling be-tween the graphene strip and the graphene SRRs.

0 μm4 μm6 μm8 μm11 μm0.2

0.4

0.6

0.8

1.0

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f (THz)0.5 0.7 0.9 1.1 1.3 1.5

(a)

vertical displacement

0.2

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ance

f (THz)0.5 0.7 0.9 1.1 1.3 1.5

(b)

0.5 μm1.0 μm1.5 μm2.5 μm3.5 μm

horizontal displacement

Fig. 2: Numerically simulated transmittance of the grapheneMM structure for (a) different values of vertical distance ′s′

by fixing the horizontal distance, d = 1.5 µm and (b) differentvalues of the horizontal distance ′d′ for a fixed value of verticaldistance, s = 11 µm.

In order to get insight about the PIT effect in thegraphene MM, we study the electric field profiles for var-ious vertical and horizontal configurations. Fig. 3(a) to3(c) represents the electric field profiles of the MM struc-ture for s = 0 µm, 6 µm and 11 µm. It is evident fromfig. 3(a), that the coupling between the graphene strip andthe graphene SRRs is negligible for s = 0 µm i.e. when thecenters of the graphene SRRs and the graphene strip arecollinear. This leads to the vanishing of the PIT effect inthe MM structure. This is a very interesting feature as thePIT effect does not vanish for the similar configuration ina metal based MM structure. In metals, when the SRRsare shifted away from the center in the vertical direction,the PIT effect occurs due to inductive coupling via themagnetic field as well as capacitive coupling via the elec-tric field of the middle strip. When the SRRs placed close

to the center of the middle strip, the dark mode excita-tion occurs due to the magnetic field of the strip whereaswhen the SRRs are close to the edges of the strip, theelectric field is responsible for the excitation of dark moderesonance [36]. We believe that in graphene based MM,the dark mode is neither excited by the electric field northe magnetic field of the graphene strip. However, as theSRRs are displaced away from the center in the oppositedirection, the graphene MM structure starts to exhibit thePIT effect.The electric field profile for s = 6 µm is shownin fig. 3(b). As the SRRs are further displaced towardsthe edge of the strip, the PIT window becomes wider dueto an increased coupling between the graphene strip andthe graphene SRRs. Fig. 3(c) represents the electric fieldprofile for s = 11 µm. In this case, the graphene SRRsis close to the edge of the graphene strip and hence thedark mode excitation takes place through the electric fieldof the graphene strip. The electric field profiles for differ-ent horizontal displacements i.e. d = 0.5 µm, 2.5 µm and3.5 µm are also shown in figs. 3(d), 3(e) and 3(f), respec-tively. It is clearly evident from the fig. 3(d) to 3(f) that

s = 0 μm

Min

Max

(a)

xy

s = 11 μm(c)

xy

s = 6 μm(b)

xy

d = 0.5 μm

(d)

xy

(f)

d = 3.5 μm

xy

(e)

d = 2.5 μm

xy

Fig. 3: Electric field profile of the graphene MM structure fordifferent vertical configurations (a) s = 0 µm, (b) s = 6 µm,(c) s = 11 µm and different horizontal configurations (d) d =0.5 µm, (e) d = 2.5 µm and (f) d = 3.5 µm. The incidentelectric field is along the direction of the green arrow.

the coupling between the graphene strip and the grapheneSRRs is the maximum for d = 0.5 µm. This configurationcorresponds to a wider PIT window (see fig. 2(b)). As thegraphene SRRs are displaced away from the graphene stripin the horizontal direction, the coupling between them re-duces. When d = 3.5 µm, the graphene strip and thegraphene SRRs are almost uncoupled. This results in thenarrowing of the PIT window and fading of the PIT effect

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in the MM structure.

Min

Max

0.5 0.7 0.9 1.1 1.30.4

0.5

0.6

0.7Fe

rmi e

nerg

y (e

V)

f (THz)1.5

Fig. 4: Contour plot of the transmittance of the graphene MMby varying Ef of graphene layer from 0.4 eV to 0.7 eV.

Further, to investigate the tunability of the PIT effect inour proposed near field coupled graphene MM, the trans-mission characteristics were studied by varying the Fermienergy ′Ef

′ of the graphene layer. Fig. 4 represents thecontour plot of the transmittance of the graphene MMstructure for different values of Ef . When Ef = 0.4 eV,the PIT peak frequency is observed at fR = 0.87 THz.As the Fermi energy is increased gradually from 0.4 eV to0.7 eV, the PIT peak frequency increases, finally reachinga value of fr = 1.2 THz. It is evident from the figurethat the PIT window exhibits a blue shift with the in-crease of the Fermi energy. It may be worthwhile to men-tion that the resonance frequency of the graphene layerstrongly depends upon the Fermi energy by the relation,fr ∝

√Ef [18–21]. This dependence of the peak frequency

on the Fermi energy is clearly observed in our numericalresults. Hence, an efficient tuning of the PIT effect of theMM structure, without altering the geometrical parame-ters, can be achieved by tuning the Fermi energy of thegraphene layer. This opens up the possibility of realiz-ing reconfigurable or active MMs [37] based on graphenestructures.

Analytical model for PIT effect. – In order tobetter understand the physics of the PIT effect in thegraphene MMs, the three level plasmonic model [2,25,26]is employed. The model consists of a bright plasmonicstate, which strongly couples with the incident field and adark plasmonic state which does not couple to the incidentTHz beam. We assume the bright mode and dark modeto have nearly same resonances i.e ω1 and ω2, respectively.The field amplitude of the bright and dark modes can beexpressed as

(ω − ω1 + iγ1)a1 + κa2 = −g1E(ω − ω2 + iγ2)a2 + κa1 = 0

(2)

where γ1, γ2 are the damping factors of the bright andthe dark modes, respectively. ω is the incident frequency,κ is the coupling coefficient between the bright and the

dark modes while g1 is a parameter describing the couplingbetween the incident light and the bright resonator. Bysolving the eq. (2) we obtain

a1 =g1E(ω − ω2 + iγ2)

κ2 − (ω − ω1 + iγ1)(ω − ω2 + iγ2)(3)

Then, the transmission of the graphene MM can be ex-pressed using [25] t(ω) = 1− |a1/E|2 as follows

t(ω) = 1− | g1(ω − ω2 + iγ2)

κ2 − (ω − ω1 + iγ1)(ω − ω2 + iγ2)|2 (4)

The theoretical fitting of the square of the transmissionrepresented by eq. (4) for different values of ′s′ and ′d′ areshown in figs. 5(a) and 5(b), respectively. The parametersfor the theoretical fitting for different values of ′s′ and ′d′

are given in table 1 and in table 2 , respectively.

0.2

0.4

0.6

0.8

1.0

Tran

smitt

ance

f (THz)0.5 0.7 0.9 1.1 1.3 1.5

(a)

0 μm4 μm6 μm8 μm

11 μmvertical

displacement

0.2

0.4

0.6

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smitt

ance

f (THz)0.5 0.7 0.9 1.1 1.3 1.5

(b)

0.5 μm1.0 μm1.5 μm2.5 μm3.5 μm

horizontal displacement

Fig. 5: Theoretically fitted transmittance of the graphene MMstructure for (a) different values of ′s′ and (b) different valuesof ′d′.

Table 1: Parameters for the theoretical fit for various ′s′.

s (µm) g1 (THz) γ1 (THz) γ2 (THz) κ (THz)0 0.2 0.28 0.15 0.054 0.2 0.205 0.35 0.256 0.2 0.15 0.36 0.318 0.2 0.12 0.37 0.3611 0.2 0.105 0.38 0.42

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Table 2: Parameters for the theoretical fit for various ′d′.

d (µm) g1 (THz) γ1 (THz) γ2 (THz) κ (THz)0.5 0.21 0.012 0.45 0.71 0.21 0.045 0.43 0.54

1.5 0.21 0.11 0.36 0.422.5 0.21 0.145 0.34 0.343.5 0.21 0.185 0.33 0.26

The geometrical parameter g1 = 0.2 or 0.21 is same forall the values of ′s′ and ′d′, representing a constant freespace coupling between the bright mode resonator andelectric field of the incident linearly polarized THz beam.γ1 gradually decreases for variation of s from 0 µm to11 µm whereas it increases for variation of d from 0.5 µmto 3.5 µm. γ2 increases for variation of s from 0 µm to11 µm and decreases for variation of d from 0.5 µm to3.5 µm. As ′s′ increases from 0 µm to 11 µm, κ graduallyincreases indicating an increase in the coupling betweenthe bright and the dark mode of the graphene MM, whichresults in the widening of the PIT window (fig. 5(a)). κreduces steadily as ′d′ increases indicating a reduction ofcoupling between the bright and the dark mode resultingin the narrowing of the PIT window (fig. 5(b)). The the-oretical transmission results are in good agreement withour numerical results.

Conclusion. – In summary, the plasmon inducedtransparency effect in a near field coupled metamaterialcomprising of a graphene strip and a pair of graphene splitring resonators is numerically and theoretically analyzed.The plasmon induced transparency effect is modulated byvarying the vertical displacement as well as horizontal dis-placement of the split ring resonators w.r.t. the graphenestrip. As the graphene SRRs are displaced in the verti-cal direction, the PIT effect vanishes when the SRRs arecollinear to each other. When the graphene SRRs aredisplaced in the horizontal direction, the PIT window be-comes narrower due to reduced coupling between the darkand bright modes. Further, the PIT effect is studied fordifferent values of Fermi energy of the graphene material.The PIT window exhibits a blue shift as the Fermi en-ergy gradually increases from 0.4 eV to 0.7 eV. Finally, atheoretical model based on the three level plasmonic sys-tem is provided in order to validate our numerical findings.The theoretical results are in good agreement with the nu-merical results. Our studies may be significant in design-ing frequency tunable, active and reconfigurable terahertzmetamaterials in near future.

∗ ∗ ∗

The author, GK gratefully acknowledges the finan-cial support from the Board of Research in NuclearSciences (BRNS), India (34/20/17/2015/BRNS). AuthorDRC gratefully acknowledges the financial support fromthe SERB, Department of Science and Technology, India

(EMR/2015/001339). Author KMD would like to thankMHRD, Government of India for a reasearch fellowship.

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