Bandgap oscillation in quasiperiodic carbon-BN nanoribbons

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Bandgap oscillation in quasiperiodic carbon-BN nanoribbons

D.O. Pedreira

a

, S. Azevedo

b

, C.G. Bezerra

a,c

, A. Viol

a

, G.M. Viswanathan

a

, M.S. Ferreira

c a

Departamento de Física, Universidade Federal do Rio Grande do Norte, Natal, RN 59078-900, Brazil

b

Departamento de Física, Universidade Federal da Paraíba, CCEN, João Pessoa, PB 58051-970, Brazil

c_{School of Physics, Trinity College Dublin, Dublin 2, Ireland}

a r t i c l e i n f o

Article history:

Received 9 September 2013 Received in revised form 24 October 2013

Accepted 15 November 2013 by R. Merlin

Available online 25 November 2013 Keywords:

A. Boron carbon nitride nanoribbons B. First-principle calculations C. Speciﬁc heat

D. Quasicrystals

a b s t r a c t

In this work we address the effects of quasiperiodic disorder on the physical properties of nanoribbons, composed by BN and C, constructed according to the Fibonacci quasiperiodic sequence. We assume BN and C as the building blocks of the resulting quasiperiodic structure. The density of states and energy band gap were obtained through ab-initio calculations based on the density functional theory. We report the effects of the quasiperiodic disorder on the oscillatory behavior of the speciﬁc heat, in the low temperature regime, and on the behavior of the energy band gap. In particular, we show that the electronic energy band gap oscillates as a function of the Fibonacci generation index n. Our results suggest that the choice of the building block materials of the quasiperiodic sequence, with appropriate band gap energies, may lead to a tuneable band gap of quasiperiodic nanoribbons.

1. Introduction

In the last 10 years, carbon based nanostructures (CBN), such as graphene, nanotubes and fullerenes[1–5], have been attractive objects of research and technological interest due to their unique structural, optical and mechanical properties. Those structures provide ideal systems for studying low-dimensional electron systems, i.e., 0D full-erenes[6], 2D graphene[7], and 1D carbon nanotubes and graphene nanoribbons[8]. A carbon nanotube can be considered as a perfect graphene sheet wrapped up into a cylinder. In addition, fullerene results from a tridimensional array of pentagons and hexagons, with carbon atoms at the corners, like a soccer ball, and a graphene nanoribbon (GNR) can be formed from a single graphene sheet by ”cutting” a narrow strip from it. In all these CBNs the presence of defects/disorder can induce signiﬁcant changes, causing relevant modiﬁcations to their physical properties. This can be explored theoretically and experimentally detected indicating, for example, that CBNs are not as perfect as previously believed. Defects such as 5_–7 rings in an hexagonal honeycomb array, kinks, junctions, vacancies and substitutional impurities [9–13] may occur in CBNs and other nanoscale structures. These defects result from the synthesis process or being intentionally induced in order to modify the electronic, chemical and mechanical properties of these materials.

As a result of the growing interest in CBN in general, other structures are now in focus, among which graphene nanoribbons

(GNR) [14,15]and hexagonal boron nitride (BN) sheets. BN has a large band gap which remains essentially unaltered when a BN sheet is rolled up into a nanotube. An interesting possibility is the partial substitution of carbon by boron and nitrogen in graphene, leading to the formation of ternary layered compounds of distinct stoichiometries. Several methods have been proposed to fabricate monolayer or few-layer BN sheets[16–18]but only recently the synthesis of BN nanoribbons (BNNRs) has been reported[19]. This provides a route to the production of nanoribbons having electro-nic properties that are intermediate between those of graphite and BN, making it possible to tune the compound electronic structure to the needs of particular applications in electronic devices.

The discovery of quasicrystals in 1984 aroused a great interest, both theoretically and experimentally, in quasiperiodic systems

[20]. One of the most important reasons for that is because they can be deﬁned as an intermediate state between an ordered crystal (their deﬁnition and construction follow purely determi-nistic rules) and a disordered solid (many of their physical proper-ties exhibit an erratic-like appearance) [21]. On the theoretical side, a wide variety of particles, namely, electrons, phonons, plasmon–polaritons, magnons, etc., have been studied in quasi-periodic systems [22]. In fact, very recently the properties of Fibonacci semiconductor systems were studied [23,24]. A quite complex fractal energy spectrum, which can be considered as their basic signature, is a common feature of these systems. On the experimental side, in order to construct a quasiperiodic structure, two building blocks (A and B) are juxtaposed following a quasi-periodic sequence[25].

Contents lists available atScienceDirect

journal homepage:www.elsevier.com/locate/ssc

Solid State Communications

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of CBNs, a timely question that naturally arises is how sensitive their electronic properties are to quasiperiodic disorder. To address this question, and also motivated by the very recent experimental realization of planar carbon-BN heterostructures[26], in this work we use a toy model to study the inﬂuence of quasiperiodicity on the properties of nanoribbons composed by building blocks of carbon and boron nitride arranged according to the Fibonacci quasiperiodic sequence. This paper is organized as follows. In

Section 2we discuss the theoretical model, with emphasis on the description of the Fibonacci quasiperiodic sequence and quasiper-iodic carbon-BN nanoribbons. The density of states, speciﬁc heat and energy band gap of the structures are calculated inSection 3. Finally, ourﬁndings are summarized inSection 4.

2. Theoretical framework

The Fibonacci sequence Snis generated by appending the n2

generation to the n1 one, i.e., Sn¼ Sn 1Sn 2 (nZ2). This

con-struction algorithm requires initial conditions which are chosen to be S0¼ B and S1¼ A. This sequence can also be constructed by

repeated operations starting with an A and applying the substitu-tion rules (or inﬂation rules): A-AB, B-A. The n generation has N¼ Fnletters, where Fndenotes the nth Fibonacci number

satisfy-ing Fn¼ Fn 1þFn 2 with F0¼ F1¼ 1. Fn 1 and Fn 2 are the

numbers of letters A and B, respectively. When no1, the ratio Fn=Fn 1is given by an irrational number which is known as the

golden mean,

τ

¼ ð1þpffiffiffi5Þ=2. The initial Fibonacci generations are S0¼ ½B, S1¼ ½A, S2¼ ½AB, S3¼ ½ABA, etc. The generation S2¼ ½AB

recovers the periodic case. From an experimental perspective, a Fibonacci structure can be constructed by juxtaposing two building blocks A and B. For the quasiperiodic nanoribbons considered in this work, we choose C as the building block A and BN as the building block B. Fig. 1 shows schematically some Fibonacci generations of the quasiperiodic nanoribbons considered here.Fig. 1(a) shows zigzag quasiperiodic nanoribbons andFig. 1(b) shows armchair quasiperiodic nanoribbons.

In the present work each Fibonacci generation is adopted as the planar unit cell in order to generate the nanoribbon structure. Our calculations were done considering unit cell size up to the ninth generation (S9) of the Fibonacci sequence, which means a unit cell

with F8¼ 34 C building blocks and F7¼ 21 BN building blocks. The

obtained supercell size assures that point defects are isolated from each other, avoiding defect–defect interactions. In order to obtain the density of states of the above described quasiperiodic nanoribbons, we have used ﬁrst-principles calculations based on the density functional theory [27], as implemented in the SIESTA code [28].

potentials [29], in the Kleinmann–Bylander factorized form [30], and a linear combination of numeric atomic orbitals was used to represent a double-zeta basis set with polarized functions (DZP). In order to obtain the Hamiltonian matrix elements, we used a grid in real space, which was obtained using a mesh cutoff of 150 Ry. Integrals in k-space were performed in aﬁnite uniform grid of the Brillouin zone. For the present calculations, the k-grid cutoff was 26 Å, and the corresponding energy-shift was 0.01 Ry. Such orbitals were included for nitrogen, boron and carbon atoms, and the generalized gradient approximation (GGA) [31] was used for the exchange–correlation potential. All the geometries were fully relaxed, until residual forces become smaller than 0.06 eV/Å. Additionally, it was adopted, as a convergence criterion, that the self-consistency would be achieved when the maximum difference between the output and the input of each element of the density matrix, in a self-consistentﬁeld cycle, turns out to be equal or smaller than 104_.

3. Numerical results

3.1. Multifractal analysis of the density of states

Fig. 2shows the density of states of the Fibonacci nanoribbons for Fibonacci zigzag (Fig. 2(a)) and Fibonacci armchair (Fig. 2(b)), from S2 until S9. From Fig. 2, we can infer the forbidden and

allowed energies, for various generation indexes n. With the purpose of characterizing the density of states, we investigate its multifractal behavior which is, in general, a common property of strange attractors in nonlinear systems [32]. In order to study these objects, it is convenient to introduce the function fð

α

Þ, known as the multifractal spectrum or the spectrum of scaling indices. Loosely, one may think of the multifractal as an interwoven set of fractals of different dimensions fð

α

Þ, where

α

is a measure of their relative strength[33]. The formalism relies on the fact that highly nonuniform probability distributions arise from the nonuniformity of the system. Usually, the singularity spectrum has a parabolic-like shape, distributed in a ﬁnite range ½

α

min;

α

max, which are the

minimum and maximum singularity strengths of the measure inten-sity, respectively. They correspond also to the exponents governing the scaling behavior in the most concentrated and rareﬁed regions of the attractor. The value of

Δα

¼ ð

α

max

α

minÞ may be used as a parameter

to measure the degree of disorder of the distribution[22].

We analyze the density of states using multifractal analysis. In principle, such techniques are only useful if the underlying data are not smooth, i.e. if they have inherent randomness or non-analytic behavior. In our case, we do not expect the density of states to have non-analytic properties. Nevertheless, multifractal

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analysis can shed insight into the multiple scales seen in the density of states. The multifractal spectra, of singularity spectrum of data, contain information about n-point correlations [34], unlike the power spectrum calculated from Fourier transforms, for example. Multifractal spectra can thus provide more informa-tion compared to two-point correlainforma-tion funcinforma-tions. For a more general discussion about multifractality see for example Refs.[35,36].

Here we use the Multifractal Detrended Fluctuation Analysis (MF-DFA) method to study the multifractal spectra[37]. A differ-ent method is the Wavelet Transform Modulus Maximus (WTMM) method[38]. Both WTMM and MF-DFA avoid certain difﬁculties of managing diverging negative moments. Moreover, another advan-tage of methods such as WTMM and MF-DFA is that they are not limited to stationary, positive, and normalized data, unlike more traditional methods. We brieﬂy summarize the MF-DFA method, which is described in more detail in many other articles and books. First, we numerically “integrate” the density of states

ρ

ð

ε

Þ to create a random walk proﬁle. We discretize the energy variable as

ε

k¼ k=kmaxð

ε

max

ε

minÞþ

ε

min, where

ε

min and

ε

max are the

minimum and maximum energies being considered and kmax is

the size of the random walk. We then deﬁne yðiÞ ∑i

k¼ 1

ρ

ð

ε

kÞ,

which we can interpret as a one dimensional random walk. We then divide the series y(i) into Nssegments of size s. We use

non-overlapping segments, so the product sNsgives the total length of

the complete series. For each segment

ν

, we apply linear regres-sion (generalizable to polynomial regresregres-sion) to calculate a best

ﬁtting “trend” yνðiÞ, where i ¼ 1; 2; …; s: For each segment

ν

we can

thus deﬁne a mean square ﬂuctuation F2ð

ν

; sÞ

1 s ∑

s

i¼ 1jyðð

ν

1ÞsþiÞyνðiÞj

2_; _ð1Þ

which corresponds to a measure of the second moment for segment

ν

. We then deﬁne a qth order ﬂuctuation function FqðsÞ 1 Ns ∑ Ns ν¼ 1 F2ð

ν

; sÞq=2: ð2Þ

Positive values of q weigh large ﬂuctuations while negative moments weigh small ﬂuctuations. The fact that F2ð

ν

; sÞ always

remains positive (for non-trivial data) avoids the divergence of the negative moments. The scaling of the ﬂuctuation for moment q follows

FqðsÞ sqHðqÞ; ð3Þ

where H(q) represents a generalized Hurst exponent, with the traditional Hurst exponent given by H¼ Hð2Þ. Monofractal data series have a unique Hurst exponent HðqÞ ¼ H. In contrast, for multifractal series the value of H(q) depends on q.

Fig. 3 shows the fð

α

Þ spectra of the Fibonacci nanoribbons, Fibonacci zigzag (Fig. 3(a)) and Fibonacci armchair (Fig. 3(b)), for four generations of the sequence S5, S6, S7and S8. As we can see,

the spectra have well-deﬁned f ð

α

Þ functions, which means that the density of states of the quasiperiodic nanoribbons is a multi-fractal set. As the generations index n increases, the widths of the

-5 0 5 Energy (eV) 0 50 100 DOS 2 3 4 5 6 7 8 9 -5 0 5 Energy (eV) 0 50 100 DOS 2 3 4 5 6 7 8 9

Fig. 2. (Colour online) Density of states of Fibonacci nanoribbons, with n varying from 3 to 9, for (a) zigzag and (b) armchair.

0.8 1 1.2 1.4 1.6 α 0 0.2 0.4 0.6 0.8 1 n=5 n=6 n=7 n=8 5 6 7 8 n 0.6 0.8 1 1.2 1.4 0.7 0.9 1.1 1.3 1.5 1.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 α 0.2 0.4 0.6 0.8 1 f(α ) 0 n=5 n=6 n=7 n=8 5 6 7 8 n 0.6 0.8 1 1.2 1.4 f(α )

Fig. 3. (Colour online) Approximate estimate of the multifractal spectrum fðαÞ for Fibonacci nanoribbons for: (a) zigzag and (b) armchair. The curves shown are quadratic least squaresﬁts of the points calculated using MD-DFA. The insets show αminandαmaxversus Fibonacci generation index n.

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is obeyed for both symmetries (zigzag and armchair). In all curves the extremes

α

min and

α

max of the abscissa of the fð

α

Þ curves

represent the minimum and maximum of the singularity exponent

α

, once

α

minand

α

max characterize the scaling properties of the

most concentrated and most rariﬁed region of the measure, respectively. One can observe fromFig. 3that

Δα

of zigzag case is greater than

Δα

of armchair case. As we said previously,

Δα

¼ ð

α

max

α

minÞ may be used as a parameter to measure the

degree of disorder of the distribution[22]. Therefore, we can infer from these results that the armchair quasiperiodic nanoribbon is more ordered than the zigzag quasiperiodic nanoribbon.

3.2. Speciﬁc heat

Specific heat is an important response function in condensed matter physics and it is a physical observable whose experimental measurements are relatively easy to be performed. Therefore, it is possible to compare experimental results with the theoretical predictions so that we can test the reliability of the physical models. In particular, the specific heat of quasiperiodic systems has been studied because of the thermodynamical consequences of their fractal spectra of energy. For example, Tsallis and colla-borators[39], in a pioneering work, showed that the specific heat of such a systems exhibits a very particular behavior: it oscillates log-periodically. Many other articles about the thermodynamical properties of fractal spectra of energy, including connections with natural spectra, were produced afterward (see Ref. [40] and references therein). With this in mind, we calculate the electronic contribution to the specific heat of the quasiperiodic carbon-BN nanoribbons. At this point we should remark that, while the phononic contribution to the specific heat in graphene is known to be dominant, its electronic counterpart is simpler to calculate and yet captures all the essential features contained in the quasiperiodicity order of the system. Therefore, in order to calculate the specific heat of the quasiperiodic carbon-BN nanor-ibbons, we have followed the lines of Ref. [41]. Let us briefly describe the procedure. InFig. 2we show the density of states for each generation n. The partition function for a specific generation n is

Zn¼

Z

ρ

ð

ε

Þeβε_d

_ε

_ð4Þ

where

ρ

ð

ε

Þ is the density of states in the range d

ε

and eβεis the well known Boltzmann factor. The associate speciﬁc heat is then given by CnðTÞ ¼ ∂ ∂T T 2∂ ln Zn ∂T : ð5Þ

Here we used

β

¼ 1=T (by choosing the Boltzmann's constant kB¼ 1). We justify the use of a classical Maxwell–Boltzmann

statistics because for fermions the classical scenario survives the inclusion of a more appropriate quantum Fermi–Dirac statistics

[42,43].

In Fig. 4, we show the specific heat spectra of the Fibonacci zigzag and armchair nanoribbons as a function of the temperature, for several generations n of the Fibonacci sequence. For high temperatures (T-1), the specific heat for all generations con-verges and decays as T 2. As the temperature decreases, the specific heat increases up to a maximum value. The corresponding temperature for this maximum value depends on the generation of the Fibonacci sequence, although one can see a clear tendency for a common temperature value as n increases. After the maximum value, the specific heat falls into the low temperature region. In this region it starts to present a non-harmonic small oscillation

behavior, as shown in the insets ofFig. 4. This can be interpreted as a superposition of Schottky anomalies corresponding to the different scales of the multifractal density of states.

These oscillations are better illustrated inFigs. 5and6, which depict semi-log and log-log plots of the speciﬁc heat against the temperature. Fig. 5 shows clearly a log-periodic behavior, i.e., CnðTÞ ¼ ACnðaTÞ, where A is a constant and a an arbitrary number,

for both armchair and zigzag conﬁgurations. The mean value d around which C(T) oscillates log-periodically can be given approxi-mately by the so-called spectral dimension (the exponent of a power law ﬁt of the density of states), associated to the minimum singularity exponent

α

in the multifractal curve fð

α

Þ. It is interesting to remark that previous results on the specific heat of quasiperiodic systems show a very distinct behavior between the periodic and quasiperiodic cases and two classes of oscilla-tions, one for the even and other for the odd generations of the quasiperiodic sequence, in the plot log CðTÞ versus log T [44]. However, as illustrated inFig. 6, one cannot clearly observe such an oscillation pattern in our results. Moreover, our results show a mild difference between the periodic case (n¼2) and the quasiperiodic case (nZ3). The absence of the even-odd oscillation pattern and the mild difference between the periodic and quasi-periodic cases can be explained qualitatively as follows. The quasiperiodic nanoribbons of our toy model are not robust enough to emphasize the quasiperiodicity in the specific heat, although as illustrated inFigs. 2and3the density of states is a multifractal set. Besides, the building blocks are not strongly coupled which destroys the even-odd oscillation pattern. A similar symmetry break of the oscillation pattern was reported in the specific heat of quasiperiodic

0 15 30 T 0 1 C(T) ₂ 3 4 5 6 7 8 9 0 15 30 T 0 1 2 C(T) ₂ 3 4 5 6 7 8 9

Fig. 4. (Colour online) Speciﬁc heat (in units of kB) versus temperature (in units of

kBT) for (a) zigzag and (b) armchair. The insets show the low-temperature behavior

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magnetic superlattices with reduced coupling between adjacent building blocks[45].

3.3. Energy band gap

Before concluding, let us discuss the much more interesting behavior of the energy band gap.Fig. 7shows the energy band gap (Eg) versus the Fibonacci generation index n for zigzag (Fig. 7(a))

and armchair (Fig. 7(b)) nanoribbons, and a summary of the calculated Egresults is shown inTable 1. The band gap presents

an interesting oscillatory behavior, which depends on the size of the nanoribbon and slightly depends on the nanoribbon symmetry (zigzag or armchair). We remark that the band gap results for n¼0 and n¼1 correspond to pure BN (S0¼ B) and C (S1¼ A)

nanor-ibbons. For both symmetries, as n increases, the oscillation amplitude decreases with Eg converging to a stationary value,

i.e., Eg 2:74 eV (zigzag) and Eg 2:56 eV (armchair).

From a qualitative perspective, let us take a closer look at the origin of the band gap oscillations. Such oscillations can be associated to the number of C–N and C–B bonds in the nanoribbon, as illustrated in Fig. 1. It is well known that electronic band structure behavior of BxNyCzstructures depends on the acceptor

states due to holes (which are introduced by substitutional boron atoms that form C–B bonds) and the donor states due to electrons (which are introduced by substitutional nitrogen atoms that form C–N bonds). Therefore, the increasing or decreasing of C–N and C–B bonds means the increasing or decreasing of acceptor and donor states. It is easy to see fromFig. 1that the number of C–N and C–B bonds depends on the considered Fibonacci generation. For a given Fibonacci generation n, the number of C–N bonds is

given by Fn 2, while the number of C–B bonds is given by Fn 2

(n odd) or Fn 21 (n even). The difference between the number of

C–N and C–B bonds, Fn 2ðFn 21Þ, may be equal to 0 (n odd) or

1 (n even), leading to the oscillations observed in Fig. 7. As n increases, _½ðFn 21Þ=Fn 2-1 and the difference between the

number of C–N and C–B bonds tends to zero. As a consequence, the oscillation amplitude decreases, and Egreaches a convergent

value. From our results, it is clear that the convergent value of Egof

the quasiperiodic nanoribbons depends on the energy band gap of their building blocks (BN and C). It is also known that a pre-requisite for future technological applications of nanoribbons is the ability toﬁne-tune the energy band gap. We must bear in mind that the energy band gaps presented here are for a qualitative toy model. We believe that, although they may seem limited in values close to the convergent energy gap, in more realistic situations the available band gap values will not be limited. Our results suggest that the choice of the building block materials of the quasiperiodic sequence, with appropriate band gap energies, may lead to a tuneable band gap of the quasiperiodic nanoribbon. This opens new perspectives in technological applica-tions due to the feasibility of tuneable quasiperiodic nanodevices.

4. Conclusions

In summary, we have studied quasiperiodic Fibonacci nanor-ibbons, composed by BN and C, with both symmetries armchair and zigzag. We assume C and BN as the building blocks A and B of the resulting quasiperiodic structure, respectively, as illustrated in

Fig. 1. The density of states of the nanoribbons were obtained by usingﬁrst-principles calculations based on the density functional theory. The multifractal proﬁle of the density of states was

log T 0 1 2 C(T) 2 3 4 5 6 7 8 9 0.01 0.1 1 10 100 log T 0 1 2 C(T) 2 3 4 5 6 7 8 9 0.01 0.1 1 10 100

Fig. 5. (Colour online) Semi-log plot of the speciﬁc heat (in units of kB) versus

temperature (in units of kBT) showing the log-periodic behavior for (a) zigzag and

(b) armchair. 0.01 0.1 1 10 100 log T 0.01 0.1 1 log C(T) 2 3 4 5 6 7 8 9 log T 0.01 0.1 1 log C(T) 2 3 4 5 6 7 8 9 0.01 0.1 1 10 100

Fig. 6. (Colour online) Log-log plot of the speciﬁc heat (in units of kB) versus

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characterized, by Multifractal Detrended Fluctuation Analysis (MF-DFA), through the so-called fð

α

Þ function, which corresponds to a set of fractals of different dimensions fð

α

Þ, where

α

is a measure of their relative strength. We have also performed a theoretical study of the specific heat behavior of the quasiper-iodic Fibonacci nanoribbons. Our numerical results show that the specific heat for all cases, in the high-temperature limit (T-1), converges and decays as T 2. In the low temperature region, the specific heat starts to present non-harmonic small oscillation behavior, as shown in the insets ofFig. 4. This is consequence of superpositions of Schottky anomalies corresponding to the different scales of the multifractal density of states. We remark that, different from other quasiperiodic systems, our results do not show an

odd-interesting, however, is the behavior of the energy band gap. As illustrated inFig. 7, the energy band gap Egoscillates as a function of

the Fibonacci generation index n. These oscillations are associated to the number of C–N and C–B bonds in the quasiperiodic nanoribbon. For a given Fibonacci generation n, the number of C–N bonds is given by Fn 2, while the number of C–B bonds is

given by Fn 2 (n odd) and Fn 21 (n even). As the Fibonacci

generation index n increases, the difference between C–N and C–B bonds tends to zero and the oscillation amplitude decreases, with Eg

converging to a stationary value, i.e., Eg 2:56 eV (armchair) and

Eg 2:74 eV (zigzag). Finally, our results for the band gap energy, of

the quasiperiodic nanoribbons, may opens new perspectives in technological applications due to the feasibility of a tuneable band gap of a quasiperiodic nanodevice.

Acknowledgments

We would like to thank LFC Pereira for a critical reading of the manuscript. We also thank the Brazilian Research Agencies CNPq and FAPERN for _{ﬁnancial support. CGB acknowledges ﬁnancial} support from CAPES (Grant no. 10144-12-9) and INCT of Space Studies. SA acknowledges INCT of Carbon Nanomaterials and Nanobiotec (Capes). MSF acknowledges support from Science Foundation Ireland (Grant no. SFI 11/RFP.1/MTR/3083). We are also grateful to The International Institute of Physics (IIP) - UFRN where the numerical calculations were performed.

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Fig. 7. (Colour online) Plot of the energy band gap versus Fibonacci generation index n for (a) zigzag and (b) armchair. The lines serve only to guide the eye.

Table 1

Energy band gap (Eg) values in eV for different

Fibonacci generations n. Generation (n) Eg Zigzag Armchair 0 0.00 1.22 1 4.64 4.23 2 2.29 2.54 3 2.96 2.79 4 2.69 2.61 5 2.77 2.60 6 2.73 2.62 7 2.75 2.57 8 2.73 2.62 9 2.74 2.56

(7)

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