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Optical Materials
journal homepage:www.elsevier.com/locate/optmat
Transmission spectra in graphene-based octonacci one-dimensional
photonic quasicrystals
E.F. Silva
a, C.H. Costa
b, M.S. Vasconcelos
c,∗, D.H.A.L. Anselmo
aaDepartamento de Física Teórica e Experimental, Universidade Federal do Rio Grande do Norte, 59600-900, Natal, RN, Brazil bUniversidade Federal do Ceará, Campus Avançado de Russas, 62900-000, Russas, CE, Brazil
cEscola de Ciências e Tecnologia, Universidade Federal do Rio Grande do Norte, 59072-970, Natal, RN, Brazil
A R T I C L E I N F O
Keywords: Photonic band gaps Quasiperiodic system Octonacci sequence Opticalfilters
A B S T R A C T
In this work, we have studied theoretically the optical transmissivity spectra in graphene-based quasiperiodic dielectric multilayers made of SiO2and TiO2which are juxtaposed in accordance with the Octonacci sequence, with a graphene monolayer between them. Our results show that all the optical spectra are affected, and their band gaps are slightly displaced for high frequencies. We also show that the properties of fractality and self-similarity of the spectra are maintained for high frequencies. We have also analyzed the influence of the incident angle and the chemical potential, and shown that when one changes the chemical potential, there is not an appreciable difference between TE and TM transmission spectra for angles close to the normal incidence. The absorption is also calculated as a function of the frequency, and is shown that sharp peaks will appear at the edges of the allowed band frequencies when we increase the Octonacci generation number. Our model can be used in the study of polarizingfilters based on photonic crystals, in order to perform an angular selectivity in the incident light. Also, it is possible to tune the chemical potential of graphene layers and set up an angular selective filter using these phenomena.
1. Introduction
Studies of the propagation of electromagnetic waves in multilayers have received much attention, especially since 1987 with the article by E. Yablonovitch [1]. In his study, Yablonovitch has presented a semi-conductor multilayer capable of guiding and confining electromagnetic radiation. Such artificial material has become known in the literature as “photonic crystal”. The photonic crystals have well-defined periodicity and display some properties similar to those found microscopically in the semiconductors. Nowadays, the concept of photonic crystals is more comprehensive. In a general way, they are formed by two or more types of dielectric materials spatially organized in 1D, 2D or 3D, in a periodic pattern. What makes photonic crystals extraordinary is their ability to allow or prohibit the propagation of light in their multilayer structure, and this causes the so-called photonic band gaps to occur. A very powerful theoretical technique in the study of the propagation of electromagnetic waves in photonic crystals is the transfer-matrix tech-nique [2]. By using this approach, we can, for example, calculate the optical spectrum (transmittance and reflectance) or even light absorp-tion in a multilayered system with optimum accuracy. This is useful to be compared with experimental measures [3]. In this work, we will
approach the transfer-matrix technique with more details for the pho-tonic crystals with graphene.
On the other hand, the crystals have a very important property that characterize them: the translational symmetry. However, the Nobel Prize in chemistry attributed to Dan Shechtman in 2011 was due to the study of quasi-crystals [4] made by him and his team in 1984 [5]. They reported the existence of a metallic solid which diffracts electrons like a single crystal but has an icosahedral point group symmetry which is forbidden crystallographically. If the sample is rotated through the angles of them35point group (icosahedral) symmetry, the patterns of electron diffraction in a given area clearly display the six fivefold, ten threefold, andfifteen twofold axes which are characteristics of icosa-hedral symmetry, concluding that this new icosaicosa-hedral phase has symmetries intermediate between those of a crystal and an amorphous solid [6]. Despite this peculiar characteristic, these structures have a long-range order and a well ordered atomic structure, as expected for a crystal. The growing interest in the study of these structures is mainly due to striking magnetic, electronic and structural properties that they present, which are very different from those already known for a con-ventional crystal [7,8]. From a mathematical point of view, one of the main motivations for the study of quasicrystals is that they have a
https://doi.org/10.1016/j.optmat.2018.06.031
Received 26 April 2018; Received in revised form 25 May 2018; Accepted 14 June 2018
∗Corresponding author. Escola de Ciências e Tecnologia, Universidade Federal do Rio Grande do Norte, 59072-970, Natal, RN, Brazil. E-mail address:[email protected](M.S. Vasconcelos).
0925-3467/ © 2018 Elsevier B.V. All rights reserved.
fragmented spectrum that reveals a pattern of self-similarity [9]. In fact, the energy spectra of these systems are fractals [9–13]. Photonic crys-tals can be treated as photonic quasicryscrys-tals if the spatial distribution or their unit cells and, in our case, their refractive indexes are organized quasiperiodically [11,12]. Usually, this organization is done under a mathematical substitution rule. In this work, we study quasicrystals constructed according to the substitution rule of Octonacci [14]. This sequence has a geometric origin: the octagonal Ammann-Becker tiling [15,16] is ruled by the Octonacci sequence.
Also, the search for new optical devices with different properties had a great advance recently with the development of the studies of graphenes [17], mainly after the works of Novoselov and Geim in 2004 [18]. Graphene is a two-dimensional carbon material with one-atom-thickness, composed ofsp2 hybridized carbon atoms that form a
hagonal network known as honeycomb. Several extraordinary and ex-citing properties are attributed to graphene. It is known, for example, that it has excellent transport properties, a characteristic which makes it an excellent charge and heat conductor. With regard to its mechanical properties, graphene is one of the thinnest and most resistant materials known to us. Moreover, with respect to the optical properties, it is known that the graphenes absorb little light, making them conductors practically transparent. Perhaps the most interesting property of gra-phenes is that they do not have an electronic band gap, as a conven-tional semiconductor. It was showed that it has a zero band gap in their electronic structure, so we can consider them as a null gap semi-conductor or zero band gap semisemi-conductor [19]. Also, it has been well established that the conventional band gap in one dimensional (1D) photonic crystal (or photonic quasicrystal), is modified due to an in-tercalating graphene layer between its interfaces [14,20]. Recently, we have studied the effect of thermal power spectra and the emittance spectra of the electromagnetic radiation through 1D quasiperiodic photonic crystals due to graphene layer between its interfaces [14]. We have shown that the effect of the inclusion of the graphene layer in a photonic crystal gives rise to the so-called graphene induced photonic band gap, an additional (chemical potential dependent) narrow band gap in the range of lower frequencies. Besides, the localization of the band gaps is shifted toward higher frequencies, when compared to the same structure without the graphene layers. Also, we have concluded that the presence of the graphene monolayers between the dielectric layers reduces the transmissivity on the whole range of frequency and induces a transmission gap in the low-frequency region. Indeed, these photonic band gaps are unusual because they do not come from Bragg reflections [20]. In the present work, it is considered that between the interfaces of the layers of the photonic crystal, we have a sheet of graphene characterized by the optical conductivityσg, so we can ana-lyze the effect of placing graphene sheets in multilayered systems. We restrict our study only to 1D photonic crystals defined by a multilayered dielectric system as defined below. Also, we will approach the transfer-matrix technique with more details for the photonic crystals with gra-phene.
This paper is organized as follow: in Sec.2, we describe the Octo-nacci sequence and calculate the transmission spectra of TM (p waves) and TE (s waves) in photonic quasicrystals (PQCs), where we use the transfer-matrix approach explicitly. In Sec.3, the transmittance spectra are presented, as a function of both the frequency and incident angle of the electromagnetic wave. The self-similar behavior of the transmission spectra, for waves with normal incidence, is observed. Our main con-clusions are given in Sec.4.
2. Theoretical model: Octonacci superlattice
We build our Octonacci multilayer by the juxtaposition, in a qua-siperiodic manner, of two building blocks A (SiO2) and B (TiO2). The rule of growth, as a function of the index n, of the multilayer is given iteratively bySn=Sn−1Sn−2Sn−1, forn≥3 [21], withS1=AandS2=B.
This rule of growth induces the number of the building blocks to
increase according to the Pell numberPn=2Pn−1+Pn−2 (withP1=1,
=
P2 1andn≥3). In fact, the total number of the building blocks in n-th stage is equal to Pn. Besides, the ratio between the number of B blocks and A blocks, isτ=1+ 2, in the limitn→ ∞. An alternative way to grow the Octonacci sequence is by recurring to the substitution rule, A→B, B→BAB. From a mathematical point of view, this sequence is classified as Pisot-Vijayaraghavan (PV), when one consider the negative eigenvalue of the matrix of substitution, namely, σ−=1− 2, with
≤
−
σ 1. This mean that if it is a PV irrational number, thefluctuation of the physical properties of the substitution sequence is more accentuated [22]. However, similar to the Fibonacci sequence, the ratioτ between the number of A and B building blocks is equal toσ+=1+ 2, which is a characteristic of the Fibonacci generalized sequences [23–25]. This sequence is generally confused with the Pell Sequence, or the Fibonacci Silver mean generalization. The difference between those sequences is that Octonacci sequence is symmetric with respect to its substitution rule B→BAB, while the silver mean is not symmetric, since it follows the rule B→BBA [26].
Here, we will use the transfer-matrix approach to analyze the transmittance spectra (for a review see Ref. [27]). There are reports describing other techniques to analyze these spectra in Fibonacci structures, for example, spectral analysis [28]. In our model, an elec-tromagnetic radiation of frequencyω and p-polarization (TM waves), or, s-polarization (TE waves) is incident, from a transparent medium C. The wave incides at an angleθ with respect to the perpendicular di-rection of the multilayered system (as shown inFig. 1). The multi-layered system is formed by an arrangement of blocks of materials A and B. The transmittance (T), the reflectance (R), and the absorption (A) coefficients are given, respectively, by (for details, see Ref. [14]),
= = = − − R M M T M A R T (2,1) (1,1) , 1 (1,1) , 1 , n n n 2 2 (1) Here, the M i jn( , ), with i, j=1,2, are components of the optical transfer-matrixMn, calculated for a given n-th generation, of the Oc-tonacci sequence. They link the coefficients of the electromagnetic fields in the left region <z 0 to those in the right regionz>L, where L is the size of the quasiperiodic structure.
The graphene layers, which are inserted between adjacent dielectric media, present a frequency-dependent optical conductivity σ ωg( )given by Refs. [29,30]: = + σ ωg( ) σgintra( )ω σginter( ),ω (2) with = + + + −
[
(
)]
σ ω i e π ω i μ k T e ( ) ℏ(ℏ Γ) 2 ln 1 , gintra c B K μ k T 2 ( c/B K) (3) and = ⎡ ⎣ ⎢ − + + + ⎤ ⎦ ⎥ σ ω i e π μ ω i μ ω i ( ) 4 ℏln 2 (ℏ Γ) 2 (ℏ Γ) . ginter c c 2 (4) In expressions above, e is the electronic charge, kBis the Boltzmann's constant,ℏ= h π/2 is the reduced Planck's constant,TKis the absoluteFig. 1. The geometrical schematic representation of the multilayered photonic structure considered in this work. The layers A and B have, respectively, thicknesses dAand dB, and they constitute the elementary units of the quasi-periodic structure. (For interpretation of the references to color in thisfigure legend, the reader is referred to the Web version of this article.)
temperature, Γ is the damping factor for graphene and μc is the che-mical potential. This one can be controlled, for example, by applying a gate voltage.
The optical transfer-matrices can be obtained,firstly, for the case in which we have two dielectric media A and B, with thicknessesdAand
dB, and (non-dispersive) refractive indexes nA andnB, respectively. Surrounding them, there is a transparent medium C (vacuum), with a refractive index nC(Fig. 1). Here we have considered both materials as nonmagnetic, such that the magnetic permeability of each material is equal to one,μA=μB=1. When an obliquely incident light wave tra-vels across the interfacesα→β(α β, = A, B or C), its transmission is represented by the matrix [31].
= ⎡ ⎣ ⎢ + + − − − + + − ⎤ ⎦ ⎥ → M k k σ μ ω k k σ μ ω k k σ μ ω k k σ μ ω 1 2 1 (1/ )( ) 1 (1/ )( ) 1 (1/ )( ) 1 (1/ )( ) , αTEβ zα zβ g zα zβ g zα zβ g zα zβ g 0 0 0 0 (5) where = ⎡ ⎣ ⎢⎛⎝ ⎞⎠ − ⎤ ⎦ ⎥ = k n ω c k , (α A B, orC) zα α x 2 2 1/2 (6) and = k n ω c sin ,θ x C (7) For TE or s-polarization waves, due to Snell's law, all the x-com-ponents of the wavevector are the same, i.e.,
= = =
kx kxAsinθA kxBsinθB kCsinθ, withkC=n ω cC /. Also, for p-polar-ization waves travelling through the interface α β, from mediumα to β, the transmission matrix is given by
= ⎡ ⎣ ⎢ + + − + − − + − ⎤ ⎦ ⎥ → M σ ε ω σ ε ω σ ε ω σ ε ω 1 2 1 Λ [1/Λ ( / )] 1 Λ [1/Λ ( / )] 1 Λ [1/Λ ( / )] 1 Λ [1/Λ ( / )] , αTMβ β α g β α g β α g β α g 0 0 0 0 (8)
where Λi=kzi/ ,εi i=α β, . Furthermore, the light wave travelling within a layerγ (γ=Aor B), for both TM and TE waves, is described by the propagation matrix [31].
= ⎡ ⎣ ⎢ − ⎤ ⎦ ⎥ M ik d ik d exp( ) 0 0 exp( ) , γ zγ γ zγ γ (9)
wheredγ is the thickness of the slab A or B. These matrices are de-termined through the application of Maxwell's boundary conditions, in each interface, considering a plane wave incident from a transparent medium C (seeFig. 1). Therefore, we can determine the transmission and propagation matrices along the multilayer system, as follow
⎜ ⎟ ⎜ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟= ⎛ ⎝ ⎞ ⎠ = ⎛ ⎝ ⎞ ⎠ … A A M A M A 0 0 , C C CBAB BC C N n C N 1(0) 2 (0) 1 ( ) 1 ( ) (10) whereMnis the transfer-matrix for the n-th generation, given by
= … = ⋯
Mn MCBAB BC MCBM M MB BA A M MB BC (11)
The iterative rule for the Octonacci photonic quasicrystals allows us to write the following formula for the transfer-matrices in our Octonacci multilayered system,
= ≥
Mn MCB nT MBC, (forn 3) (12)
where
= − − −
Tn Tn 1Tn 2Tn 1, (13)
whose initial conditions areM1=MCBT M1 BC,M2=MCBT M2 BC,T1=MB, andT2=M M M M MB BA A AB B. Therefore, with the knowledge of equations (12)and(13), we can calculate numerically the transmission spectra for a given range of frequency, and for any Octonacci generation. This task is the objective of the next section.
Fig. 2. Transmission spectra as a function of the reduced frequency ω ω/ 0for normally incident waves: (a)fifth generation of Octonacci sequence ( =P5 17 layers); (b) same as (a), but for the reduced range of frequency
≤ω ω ≤
0.834 / 0 1.245; in (c), (d) and (e) we have the same as (a), but for sixth (P6=41layers), seventh (P7=99layers) and eighth (P8=239layers) Octonacci sequence generations, with the frequency ranges, respectively,
≤ω ω ≤
0.965 / 0 1.123,1.013≤ω ω/ 0≤1.074, and1.032≤ω ω/ 0≤1.055.
Fig. 3. Absorption spectra as a function of reduced frequency ω ω/ 0and the generation number n considering a normal incidence forfifth (red solid line), sixth (blue solid line), seventh (green solid line) and eighth (orange solid line) generations of Octonacci sequence. As the generation increases, which corre-sponds to more graphene at interfaces, the absorption coefficient A also in-creases. (For interpretation of the references to color in thisfigure legend, the reader is referred to the Web version of this article.)
3. Numerical results
We now proceed to discuss our numerical results, for the optical transmission spectra, of the Octonacci quasiperiodic structures. The physical parameters are the same used in Refs. [20], which are suitable for titanium dioxide (B) and silicon dioxide (A); they are virtually ab-sorption-free above 400 nm [3]. We also define the individual layers as quarter-wave (QW) layers, for which the quasiperiodicity is expected to be more effective, with the central wavelength λ0=60 μm. These
requirements result in the physical thicknessdγ=(60/4nγ)μm,γ=A or B, such thatn dA A=n dB B. Their refraction indexes around the central wavelengthλ0=60μm arenA=1.45andnB=2.30, respectively, which providesdA≈10.34 μm anddB≈6.52μm. We also have chosen a re-duced frequencyΩ=ω ω/ 0=λ λ0/ [11], withω0=2πc λ/ 0≈31.4THz.
The transmission spectrum of the light wave, depicted as a function of the reduced frequency ω ω/ 0for thefifth generation of the Octonacci
sequence (hereP5=17layers), is presented inFig. 2a, for a normally incident wave, i.e.,θ=0∘. As it is expected, the transmission spectrum
Fig. 4. Transmission spectra as a function of both reduced frequency ω ω/ 0and incidence angle θ, for TE (left) and TM (right) waves for the: (a)fifth; (b) sixth; (c) seventh and (d) eighth generations of Octonacci sequence. In allfigures the white (black) color signs a transmission coefficient equal to 1 (0). The omnidirectional photonic band gaps, which are enhanced by the narrow green rectangles, emerge only for TE waves and for a small reduced frequency region0.67≤Ω≤0.71for all generations investigated. (For interpretation of the references to color in thisfigure legend, the reader is referred to the Web version of this article.)
Fig. 5. Transmission coefficient for TE (left) and TM (right) waves versus the reduced frequency Ω and the chemical potential μcforθ=25 . In (a) we have 5th, in (b)∘ we have 6th, in (c), we have 7th and,finally, in (d) we have 8th generations.
does not display a mirror symmetry around the reduced frequency at mid-gap,ω ω/ 0=1, even when the QW condition, for a periodic mul-tilayer, was taken into account [31,34,35]. The central frequency was shifted to a new value of reduced frequency around 1.043 and 1.044. Besides, one can also observe a band gap forΩ 0.287, which corre-< sponds toω<9 THz. Since the origin of this gap is related to the pre-sence of graphene at interfaces, it is called graphene induced band gap (GIBG).
The transmission spectrum also presents a scaling property, with regard to the generation number of Octonacci sequence, when one looks at a symmetrical interval around the mid-gap reduced frequency [14]. In order to illustrate this feature, inFig. 2b the transmission spectrum is presented as the same shown inFig. 2a, but for a reduced
range of frequency0.834≤ω ω/ 0≤1.245. As one can see, the rescaled spectrum resembles the one for the sixth generation of the Octonacci sequence (whereP6=41layers), respectively, displayed inFig. 2c. The
range of frequency is reduced by a scale factor, in our case, roughly equal to 2.6. InFig. 2d and e, we present the transmission spectra for seventh and eighth generations (whereP7=99 andP8=239layers),
considering the following frequency ranges, respectively, ≤ω ω ≤
1.013 / 0 1.074 and1.032≤ω ω/ 0≤1.055. This spectrum repeats
every generation, for an Octonacci 1D photonic quasicrystal. This scaling characteristic is strongly related to a self-similar behavior of the spectrum. In spite of this self-similar behavior, the transmission coef-ficient diminishes, as we increase the generation number n, and con-sequently, the number of graphene at interfaces, resulting in the Fig. 6. Same asFig. 5, but now forθ=50∘.
increase of the absorption. This behavior can be seen onFig. 3, which is a plot of the absorption coefficient, A=1−T−R, as a function of reduced frequency ω ω/ 0 and the generation number n, considering a
normally incident wave.
InFig. 4, we display the transmission coefficient versus the reduced frequency ω ω/ 0and the incident angleθ (in degrees) for 5th, 6th, 7th
and 8th generations (Fig. 4a, 4b, 4c and 4d), respectively, for TE (left) and TM (right) polarizations. For TE waves, one can notice a narrow region, with well defined “omnidirectional photonic band gaps” (OPBGs), which are independent of the incidence angle, and it is en-hanced by the thin green rectangles. The OPBGs are at the same loca-tion and have the same width, approximately0.67≤Ω≤0.71, which corresponds to a frequency range from 21 to 22.3 THz. It seems clear that the existence of these band gaps is caused by the long-range order of the positional structure of the layers in the Octonacci lattice, and this, in turn, accounts for the anomalous interference, giving rise to these OPBGs. Here, we call attention to the fact that there is some confusion in the literature about the term“omnidirectional band gap”. Some authors consider that an omnidirectional band gap is one which appears in both TE waves and TM waves within the cone of light, even though it is not at all incidence angles [36], while in other works, the authors consider that an“omnidirectional band gap” is the one which occurs for all incident angles, without taking into account the incident wave polarization [37], and in the case where the gap appears for both TE and TM polarization, they are called “complete band gaps”. We, particularly in this work, use the second definition, since the term “omnidirectional” refers to all values of the angle of incidence or di-rections of incidence of light, that is, it is a gap which exists in-dependently of the value of the angle of incidence, and has no direct relation to the type of polarization. Specifically, in the literature, there are renowned works that refer to the type of polarization in the study of optical band gaps using the terms that are,“polarization-independent band gap” [38] or “polarization-independent omnidirectional band gap” [39]. In the latter, the gap is independent of both the polarization and the angle of incidence.
InFigs. 5–7, we plot the transmission coefficient for TE (left) and TM (right) waves versus the reduced frequency Ω and the chemical potential μc for a given incident angle, namely,θ=25∘,θ=50∘and
= ∘
θ 75, respectively. For all these Figures, in (a) we have 5th, in (b) we have 6th, in (c), we have 7th and,finally, in (d) we have 8th genera-tions. Forθ=25∘,Fig. 5, we do not observe an appreciable difference
between TE and TM transmission spectra because the incidence angle is still close to the normal incidence case. As the incidence angle in-creases, for instance, takingθ=50∘, displayed onFig. 6, the expected
difference between the transmission coefficient and band gaps position for TE and TM waves becomes more pronounced, with a higher trans-mission for TM than for TE waves, but, the photonic band gaps are narrower for TM than for TE polarizations. For θ=75∘, the light
transmission spectra for TE and TM waves presented on Fig. 7, are strongly influenced by graphene: firstly, we observe that the transmis-sion coefficient is lower for TE waves than for TM ones; secondly, we observe that there are some frequency regions which correspond to band gaps for TE polarizations and allowed bands for TM waves, for example, at the spectra of reduced frequencies around 0.75 and 1.5. This is very useful to design light polarizingfilters based on photonic crystals [40,41].
4. Conclusions
In summary, in this work, we have theoretically studied the trans-mission spectra for graphene-based Octonacci one-dimensional pho-tonic quasicrystals. First, we have shown that for perpendicular (θ=0) incidence, the transmission spectrum does not have a mirror symmetry around the mid-gap reduced frequencyω ω/ 0=1due to the inclusion of the graphene at their interfaces. The central frequency was shifted to a new value of reduced frequency (seeFig. 2) when compared to the case
without graphene, and consequently giving rise to the so-called GIBG. With the exception of GIBG, the transmission spectrum keeps the scaling property, with respect to the generation number n of the Oc-tonacci sequence. Also, we have studied the absorption spectra showing that the peaks in transmissivity get lower as we increase the generation number n. The increase in generation n leads to, consequently, the in-crease in the number of graphene monolayers that are at the interfaces. Indeed, the absorption coefficient increases, generating absorption peaks (seeFig. 3) at the edges of the allowed bands [20]. Also, and only for TE waves, we could observe a narrow region with a well defined omnidirectional photonic band gaps, which are independent of the in-cident angle in the frequency range from 21 to 22.3 THz. The appear-ance of the band gaps is due to the long-range order of the stacking of layers, in the Octonacci multilayered system. By analyzing the influence of the chemical potential in the optical spectra, we have concluded that there is not an appreciable difference between TE and TM transmission spectra for angles close to the normal incidence, and, otherwise, for high angles, the transmittance is lower for TE waves than for TM ones. Polarizingfilters based on photonic crystals use phenomena like this to perform an angular selectivity in the incident light. Therefore, in our case, it is possible to tune the chemical potential of graphene layers and set up an angular selectivefilter using these phenomena [41]. Acknowledgements
The authors would like to thank CAPES and CNPq (Brazilian Science Funding Agencies) for thefinancial support.
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