Author's Accepted Manuscript

Optical properties of one-dimensional photo-nic crystals containing graphene sheets

Amir Madani, Samad Roshan Entezar

PII: S0921-4526(13)00513-9DOI: http://dx.doi.org/10.1016/j.physb.2013.08.041Reference: PHYSB307861

To appear in: Physica B

Received date: 17 June 2013Revised date: 21 August 2013Accepted date: 27 August 2013

Cite this article as: Amir Madani, Samad Roshan Entezar, Optical properties ofone-dimensional photonic crystals containing graphene sheets, Physica B, http://dx.doi.org/10.1016/j.physb.2013.08.041

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Optical properties of one-dimensional photonic crystals

containing graphene sheets

Amir Madania,b,∗, Samad Roshan Entezara

aFaculty of Physics, University of Tabriz, Tabriz, IranbDepartment of laser and optical engineering, University of Bonab, Bonab, Iran

Abstract

The transmission properties of a one-dimensional photonic crystal containinggraphene monolayers are investigated using transfer matrix method. It isshown that the structure has a new type of the photonic band gap in the THzregion which is almost omnidirectional and insensitive to the polarization.The results show that the characteristic properties of this band gap dependon the optical parameters of the graphene sheets and can be controlled viaa gate voltage. The difference between this gap and the structural Bragggaps is investigated by plotting the electromagnetic field profiles inside theone-dimensional photonic crystal for some critical frequencies.

Keywords: Photonic crystal, Graphene, Omnidirectional, Tunability.PACS: 42.70.Qs, 81.05.ue, 78.67.Wj.

1. Introduction

During the past two decades, photonic crystals (PC) which are artificiallyfabricated materials with periodic modulation in dielectric properties, havegained much attention due to their ability to create a range of forbiddenfrequencies known as photonic band gap (PBG) [1]. PBGs have many in-teresting and attractive applications in optical reflectors [2], localization ofphoton [3], control of spontaneous emission from atoms [4, 5], fabrication ofPC waveguides [6] and etc. Beside the geometrical parameters of the PCs,

∗Corresponding author: A. MadaniEmail address: a-madani@tabrizu.ac.irTel: ++984113393349 Fax: ++984113341244

Preprint submitted to Elsevier August 30, 2013

the type of the constituent materials used in the PC fabrication has greatsignificance in their band structure. PBGs in conventional dielectric PCs de-pend strongly on the polarization and incidence angle of the electromagneticwaves while this may be different in other types of PCs. Recently, the opticalproperties of the PCs containing various kinds of materials including metals,semiconductors and metamaterials have been investigated [7, 8, 9, 10]. Forexample, the band structure of the one-dimensional photonic crystals (1DPC) containing single-negative or double-negative metamaterials have beenstudied [11, 12, 13]. It is shown that this kind of PC can show omnidirec-tional band gap which its central frequency and width can be invariant uponthe change of scaling and it is insensitive to disorders, incident angles andpolarizations.

On the other hand, tunability of the PBGs is an essential feature in thedynamic controlling of the transmittance spectrum of a PC. This can bepossible by controlling the optical properties of the constituent materials.So, it is highly appropriate to use the materials with externally controllableoptical properties. Liquid crystals (LCs) as a kind of uniaxial dielectricmaterials with tunable optical properties have been used extensively in thisregard [14, 15, 16].

Graphene, an allotrope of carbon atoms arranged in one atom thickhoney-comb lattice, can be introduced as another alternative with control-lable optical properties [17]. Beside the general features such as high mo-bility of carriers, flexibility, robustness and environmental stability [18, 19],graphene has some properties which can make it a suitable option in de-signing photonic devices. At the frequency ranges of THz and far-IR, thedissipative losses of graphene is less than the usual metals and its electronicand also optical response is described by the surface conductivity which isrelated to its chemical potential and can be controlled and tuned by voltage[20, 21]. Due to these unique characteristics, people have been motivatedto study the multilayer structures containing graphene sheets. For example,THz optical properties and plasmonics of graphene multilayers have beeninvestigated in the recent years [22, 23, 24]. Also, THz hyperbolic metama-terials and hyperlenses made of graphene multilayers have been studied bydifferent groups [25, 26].

In the present paper, we are interested in studying the photonic bandstructure of a 1D PC in which the graphene sheets are embedded betweenadjacent dielectric layers as a controlling elements. We employ the trans-fer matrix method [27] to analyze the PBGs of the 1D PC for both of the

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TE and TM polarizations. The results are compared with the PBGs of asimilar structure without graphene sheets. It is found that the presence ofthe graphene sheets results in the creation of a new type of PBG in the THzregion which is omnidirectional and also insensitive to the polarization. Also,the tunability of the PBGs is investigated by analyzing the influence of thechemical potential of the graphene sheets on the photonic band structure ofthe 1D PC. Finally, the behavior of the electromagnetic fields inside the 1DPC at some critical frequencies is investigated by plotting the electric fieldprofiles.

2. Theoretical model

We consider a 1D PC with a periodic structure of (AB)N in air as shownin Fig.1. Here, A and B represent two isotropic dielectric materials with thepermittivity of εA and εB and thicknesses of dA and dB, respectively. Weconsider that the layers are nonmagnetic and their permeability are µA =µB = 1. N is the period number and a plane wave is incident at an angleθ upon the 1D PC from air. The interfaces of the layers are parallel to the(x − y) plane and the z axis is normal to the structure. It is assumed thatthe graphene monolayers are embedded between adjacent dielectric layers andσg(ω) represents the surface conductivity of the graphene. σg(ω) is governedby the Kubo formula [28] including the intraband and interband transitioncontributions as σg(ω) = σintra

g (ω) + σinterg (ω), where

σintrag (ω) =

e2

4~i

2π{16 kBT

~ωln(2 cosh(

µc

2kBT))},

σinterg (ω) =

e2

4~{12+

1

πarctan

~ω − 2µc

2kBT− i

2πln

(~ω + 2µc)2

(~ω − 2µc)2 + (2kBT )2}. (1)

Here, e is the charge of an electron, kB is the Boltzmann constant, T is thetemperature in K and µc is the chemical potential which can be controlledby the gate voltage. These expressions show that the interband contributionplays the leading role around the absorption threshold, ω ≈ 2µ, while theintraband contribution is important at relatively low frequencies, ω < µ.

We use the well-known transfer matrix method to obtain the transmissionof the structure as

T = | 2q0(qtT11 + q0T22)− (T21 + q0qtT12)

|2. (2)

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T11, T12, T21 and T22 are the elements of the total transfer matrix T =[MA(dA, ω)MB(dB, ω)]

N where Mj(dj, ω) (j = A,B) is obtained as

Mj(dj, ω) =

(cos(kzjdj) (i/qj)sin(kzjdj)

σgcos(kzjdj) + iqjsin(kzjdj) (iσg/qj)sin(kzjdj) + cos(kzjdj)

)(3)

for the TE waves and

Mj(dj, ω) =

(cos(kzjdj)− iσgqjsin(kzjdj) (i/qj)sin(kzjdj)− σgcos(kzjdj)

(iqj)sin(kzjdj) cos(kzjdj)

)(4)

for the TM waves. Here, qj = (−kzj/ωµ0µj) for TE waves and qj = (+kzj/ωε0εj)for TM waves and q0 and qt are defined as the corresponding q parametersof the incidence and exit media which are chosen as air.

3. Results and discussion

In our numerical calculations, we take the optical and geometrical pa-rameters of the system as follows: dA = 10µm, dB = 10µm and N = 15.We limit our calculations only to the far-IR and THz region at the frequencyrange of 0− 8 THz due to the low loss of the graphene sheets in this regime.Besides, the dispersion of the usual dielectric materials is not noticeable inthis region [29]. So, we consider the relative permittivity of the layers A andB as εA = 5 + iγ and εB = 2.5 + iγ respectively, where γ shows the lossof the dielectric materials. Here, we assume that the surface conductivity ofgraphene sheets is given by Eq.1 with µc = 0.2 eV and T = 300K.

At first, we study the transmission properties of our structure and com-pare the results with the structure without graphene sheets. So, we plotthe transmission spectrum of the structure for the normal incidence of thewaves in Fig.2(a,b) for the structure without and with the graphene sheets,respectively. Here, the solid lines show the transmission for the case of loss-less dielectric materials (γ = 0) while the dashed (γ = 0.01) and dotted(γ = 0.1) lines represent the results for the case of lossy materials. As onecan see from the Fig.2(a), the system has only one frequency gap for the caseof σg = 0 in the defined frequency range. This gap is the structural Bragggap of the 1D PC and is located around 3.5− 4.5 THz. On the other hand,Fig.2(b) represents an additional PBG in the lower frequencies for the 1DPC containing the graphene sheets which may be due to the presence of thegraphene. So, we call it graphene induced photonic band gap, GIPBG. The

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existence of the GIPBG for different values of γ reveals that this band gap issolely due to the existence of the graphene sheets. As a result, we considerγ = 0 in the rest of paper.

To see the effect of the graphene monolayers on the PBGs of the 1D PC,we consider the dispersion relation, cos[K(dA + dB)] = λ1 + λ2, which onecan obtain using Eqs. 3 and 4 where

λ1 = cos[kzAdA]cos[kzBdB]−1

2(q̃Aq̃B

+q̃Bq̃A

)sin[kzAdA]sin[kzBdB]

λ2 = −iσg q̃Asin[kzAdA]cos[kzBdB]− iσg q̃Bsin[kzBdB]cos[kzAdA]

−1

2σ2g q̃Aq̃Bsin[kzAdA]sin[kzBdB]. (5)

Here, K is the Bloch wave number and q̃j (j = A,B) is a parameter which isdefined as q̃j = −1/qj and q̃j = qj for the TE and TM waves, respectively. λ1

depends only on the optical and geometrical parameters of the 1D PC whileλ2 is related to the graphene sheets and disappears when σg = 0. Since, σg

strongly depends on the frequency at the THz region (see Fig.3(a)), signifi-cantly affects the propagation properties of the structure. At the frequencyrange of interest, the intraband transition of electrons dominates and thesurface conductivity of the graphene sheets is given by σintra

g which is animaginary number. So, λ2 can be expressed as

λ2 = |σg|q̃Asin[kzAdA]cos[kzBdB] + |σg|q̃Bsin[kzBdB]cos[kzAdA]

+1

2|σg|2q̃Aq̃Bsin[kzAdA]sin[kzBdB]. (6)

To observe the contributions of λ1 and λ2 in the creation of the PBGs, weplot λ1, λ2 and λ1+λ2 as functions of the frequency for the normal incidenceof the waves (Fig.3(b)). As it is clear from the figure, the band gap condition(|λ1+λ2| > 1) is fulfilled only at the frequency range of 3.5− 4.5 THz whenwe neglect the contribution of λ2, i.e. when σg = 0. For σg ̸= 0, we haveto consider the contribution of λ2. As a result, the additional GIPBG isappeared at the lower frequency interval (0 − 1.35 THz). It is evident thatthese results are in good agreement with the results of Fig.2 and one cansay that the GIPBG in Fig.2(b) is the direct consequence of the graphenesheets. However, the effect of the graphene is not remarkable in the secondgap which may be due to the negligible values of σg in the correspondingfrequencies (see Fig.2(a)).

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To study the dependence of the PBGs on the incidence angle, we plot thephotonic band structure of the system as a function of θ for σg ̸= 0 in Fig.4.Here, the gaps are represented by the dark areas while the bright areas showthe allowed bands for the TE and TM polarizations. It is clear from Fig.4(a)that the PBG around 5 THz is polarization sensitive and depends stronglyon the incidence angle which is the characteristic of the conventional Bragggap. On the other hand, we see that the PBG around 1 THz (GIPBG) isomnidirectional and also polarization insensitive. This band gap which iscompletely different from the usual Bragg gaps is remarkably suitable forthe designing of filters in the THz region. Since the GIPBG is due to theexitance of the graphene sheets, this band gap can be created when thegraphene sheets are embedded in a uniform dielectric material (see Fig.4(b)).Moreover, we see that the band structure of the uniform dielectric materialcontains a Bragg gap which is due to the the exitance of the graphene sheets.

The advantage of this structure comparing with the similar PC structureswith metallic elements is the tunability of the PBGs due to the controllableelectronic and optical characteristics of the graphene. As mentioned before,these characteristics depend on the surface conductivity of the graphene, σg,which can be controlled by a gate voltage via tuning the chemical potential,µc. To show this, we plot the PBGs of the 1D PC as a function of µc for thenormal incidence of the waves in Fig.5. The 1D PC shows almost the samebehavior for both the TE and TM polarizations but the effect of the µc onthe GIPBG is more considerable than the second one. It is evident from thefigure that one can adjust the filtering bandwidth of the structure via tuningof the µc.

In order to investigate other difference between the GIPBG and Bragggap, we plot the transverse electric field inside the structure for the frequencyat or around the first and second band gaps in Fig.6. Here, we consider nor-mal incidence with four different frequencies, (a) f = 0.67 THz at the midof the first PBG , (b) f = 1.34 THz at the upper edge of the first PBG, (c)f = 4.2 THz at the mid of the second PBG and (d) f = 4.66 THz at the up-per edge of the second PBG (which are marked as a, b, c and d in Fig.2(b)).The profiles show the variation of the electric field of the waves inside thestructure. Fig.6 reveals that the transverse electric field does not show theoscillatory behavior for the frequency at the GIPBG which is in contrast tothe second PBG (see Fig.6(a,c)). Moreover, the transverse electric field hasits maximum at the mid of the layers A and B with minimums at the inter-faces (see Fig.6(b)). While, the transverse electric field has its maximums at

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the layers B with its minimums at the layers A for the frequency near thesecond PBG (see Fig.6(d)).

4. Conclusion

In summary, the transmission properties of a 1D PC containing graphenemonolayers are investigated theoretically using transfer matrix method forthe TE and TM polarization. The graphene monolayers are embedded be-tween adjacent dielectric layers. It is shown that the presence of the graphenesheets results in the creation of a new type of PBG in the THz region whichis omnidirectional and insensitive to the polarization. The width of this gapstrongly depends on the surface conductivity of the graphene and can betuned via a gate voltage, so it can be used in the designing of the tunablefar-IR filters or switches. Also, we plotted the electric field profiles inside the1D PC for some critical frequencies and explained that the properties of thegraphene based new type gap is different from the conventional Bragg gaps.

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Figure 1: (color online) Structure of the 1D PC consisting of alternate layers of the type(A) and (B). The graphene monolayers are embedded between dielectric layers.

Figure 2: (Color online) The transmission spectrum of the 1D PC for the cases of a) σg = 0and b) σg ̸= 0 for the normal incidence of the waves. Temperature and chemical potentialare chosen as µc = 0.2 eV and T = 300K. Here, γ = 0, γ = 0.01 and γ = 0.1 are used inthe plotting of the solid, dashed and dotted lines, respectively.

Figure 3: (Color online) a) Surface conductivity of the graphene as a function of frequency.b) λ1, λ2 and λ1 + λ2 as functions of the frequency for the normal incidence of the waves.The other parameters are the same as Fig.2.

Figure 4: (Color online) a) PBGs of the 1D PC as a function of θ for the structure withgraphene sheets. b) PBGs of the graphene multilayer structure which is embedded in auniform medium. The other parameters are the same as Fig.2.

Figure 5: (Color online) PBGs of the 1D PC as a function of µc for the normal incidenceof light and T = 300K. Here the dark and the bright areas show the gaps and allowedbands, respectively.

Figure 6: (Color online) The electromagnetic fields inside the 1D PC for the frequenciesa) f = 0.67 THz at the mid of the first gap, b) f = 1.34 THz at the upper edge of thefirst gap, c) f = 4.2 THz at the mid of the second gap and d) f = 4.66 THz at the upperedge of the second gap. The waves are incident normal to the interfaces and the otherparameters are the same as Fig.2.

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Figure 1

11

Figure 2

12

Figure 3

13

Figure 4

14

Figure 5

15

Figure 6

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