Communication
Bandgap oscillation in quasiperiodic (
BN C
)
x y
nanotubes
A. Freitas
a,n, C.G. Bezerra
a, S. Azevedo
b, L.D. Machado
a, D.O. Pedreira
aa
Universidade Federal do Rio Grande do Norte, Departamento de Física, Caixa Postal 1641, 59078-900 Natal, RN, Brazil
b
Universidade Federal da Paraíba, Departamento de Física,CCEN, Caixa Postal 5008, 58051-970 João Pessoa, PB, Brazil
a r t i c l e i n f o
Article history: Received 2 August 2016 Received in revised form 5 September 2016 Accepted 8 September 2016 Available online 13 September 2016
a b s t r a c t
In the present contribution, we applyfirst-principles calculations to study the effects of quasiperiodic disorder on the physical properties of BN and C nanotubes. We take BN nanotubes (BNNTs) and C na-notubes (CNTs) as building blocks and construct quasiperiodic BNxCynanotubes according to the
Fibo-nacci sequence. We studied armchair and zigzag nanotubes of varying diameters. Our results demon-strate that the energy gap oscillates as a function of the n-generation index of the Fibonacci sequence. Moreover, we show that the choice of the BNNTs and CNTs may lead to a quasiperiodic BNxCynanotube
presenting an adjustable energy gap. We obtained a variety of quasiperiodic nanotubes with energy gaps ranging from 0.29 eV to 1.06 eV, which may be of interest for specific technological applications. Finally, it is also demonstrated that the specific heat of the quasiperiodic zigzag and armchair nanotubes pre-sents an oscillatory behavior in the low temperature regime, and that this behavior depends on the curvature of the nanotube.
& 2016 Elsevier Ltd. All rights reserved.
1. Introduction
Since the pioneering work of Iijima [1], carbon nanotubes (CNTs) have received increasing interest due to their novel prop-erties and potential applications in several fields, such as optics [2], electronics[3]and spintronics[4]. Carbon nanotubes can be metallic or semiconducting, depending on their diameter and chirality[5–7]. Regarding their mechanical properties, simulations predicted and experiments confirmed that CNTs present high Young's modulus[8,9]andflexibility[10,11]. CNT analogues from boron nitride have also been synthesized[12]and investigated in a variety of contexts [13–16]. Despite their structural similarity, boron nitride nanotubes (BNNTs) and CNTs present rather differ-ent properties. BNNTs are always wide gap semiconductors, in spite of diameter, chirality and number of walls [17]. They also have a high degree of radial flexibility, Young modulus [18,19], thermal stability and resistance to oxidation [20]. It has been shown that BNNTs are stable up to 700 C° in air, with oxidation occurring at800 C, while CNTs are found to be oxidized at° 400 C° [21]. It is worth to point out that the properties of CNTs and BNNTs are different due to the partially ionic character of the B–N cova-lent bond, due to charge transfer[22], which is not present in the carbon material. Moreover, a variety of defects can occur in na-notubes, including torsions, junctions, vacancies and substitutional
impurities [23–25,22]. These defects usually result from the synthesis process of nanotubes, but they can also be intentionally induced in order to modify the mechanical, chemical and elec-tronic properties of such structures.
Nanotubes with compositions other than carbon and BN have also been extensively studied in recent years. For example, we can mention B C Nx y zhybrid nanotubes with different stoichiometries (x, y, z), such as BCN, BC N2 , and BC N4 [15,26–29]. These types of
na-notubes are produced through several methods, such as arc dis-charge, laser ablation and others [30,31]. It is known from the literature that well-separated BN and C regions (the so-called is-lands) are formed in B C Nx y znanotubes. The explanation for such phenomenon comes from the fact that B–N and C–C bonds are more stable than B–C and N–C ones.[33–35]. This behavior is also observed in other types of B C Nx y zhybrid nanostructures, such as monolayers[36]and graphene nanoribbons[37]. Theoretical and experimental studies reveal that B C Nx y z hybrid systems present small energy band gaps ( <2 eV), so that these materials are semiconductors. This characteristic is associated to the number of C–B and C–N bonds present in the structure, which introduce donor and acceptor states in the band-gap region, respectively. Thus, it is possible to control the gap by varying the number of C–B and C–N bonds. Such peculiarity is considered one of the most important features of B C Nx y z hybrid structures. Therefore, it is expected that these semiconductor materials present a tunable energy band gap according to the relative percentage of the B C Nx y z components. Therefore, B C Nx y zstructures could be quite promising in applications for nanoscale electronic devices, such as electron Contents lists available atScienceDirect
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Solid State Communications
http://dx.doi.org/10.1016/j.ssc.2016.09.004
0038-1098/& 2016 Elsevier Ltd. All rights reserved.
nCorresponding author.
E-mail addresses:alilianefisica@yahoo.com.br(A. Freitas),
cbezerra@dfte.ufrn.br(C.G. Bezerra),sazevedo@fisica.ufpb.br(S. Azevedo),
field emitters[41], rectifying diodes[42]and super-capacitors[43]. In the context of the B C Nx y zstructures, we point out the study done by Pedreira et al.[44], which dealt with the effect of quasi-periodic disorder on the properties of nanoribbons composed of BN and C, where such structures were arranged according to the Fibonacci sequence. The authors found that the electronic energy band gap oscillates as a function of the Fibonacci generation index n and converges to a stationary value as the index n increases. The authors explained that such oscillations are associated to the number of C–N and C–B bonds in the quasiperiodic nanoribbon. Additionally, these nanoribbons present rather complex multi-fractal energy spectra, which is a common feature of quasiperiodic systems. Undoubtedly, one of the hottest topics of current physical interest is the study of quasiperiodic structures arranged according to a given substitutional sequence (Fibonacci [45–47], Rudin–
Shapiro [48,49], Thue-Morse [50,51] and others). The most im-portant reason for the interest in these systems lies in the ex-perimental discovery of quasicrystals in 1984 [52]. Quasicrystals 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 prop-erties exhibit an erratic-like appearance)[53]. Another motivation for the study of quasiperiodic structures comes from the fact that they have been observed experimentally, so that they are not mere academic examples of quasicrystals[54].
Since (BN C)x y quasiperiodic nanoribbons present a tunable energy band gap, we can wonder whether this behavior is also observed in structures with different geometries, such as zigzag and armchair nanotubes. To the best of our knowledge, there is no in the current literature theoretical investigation about structural Fig. 1. Illustration of Fibonacci (6,6) armchair nanotubes: (a) model-I and (b) model-II. Note that an operation that exchanges Boron and Nitrogen atoms in the S1and S0of
model-I and model-II, respectively, does not lead to new building block configurations.
Fig. 2. Illustration of Fibonacci (9,0) zigzag nanotubes: (a) model-I and (b) model-II.
and electronic properties of the quasiperiodic (BN C)x y nanotubes, constructed following the Fibonacci sequence or any other type of substitutional sequence. Thus, in the present work, we apply first-principles calculations to investigate the effect of quasiperiodicity on the properties of zigzag and armchair nanotubes, composed by building blocks of C and BN arranged according to the Fibonacci quasiperiodic sequence. This paper is organized as follows. In Section 2we discuss the theoretical approach, with emphasis on the description of the quasiperiodic (BN C)x ynanotubes. InSection 3discuss the density of states, specific heat and energy band gap of the structures. Finally, ourfindings are summarized inSection 4.
2. Theoretical approach
The Fibonacci sequence is, by definition, an infinite sequence of integer numbers which satisfy the relation Fn=Fn−1+Fn−2(n≥ )2, with F0=F1=1, where Fndenotes the nth Fibonacci number. This sequence has an interesting feature: as n approaches infinity, the ratio between two consecutive terms (Fn/Fn1) approaches an
irrational number known as golden ratio, namely τ =( +1 25). At the same time, the Fibonacci sequence is an example of quasiperiodic sequence which has been widely used to simulate quasiperiodic arrays in superlattices and multilayers [45–47]. A Fibonacci su-perlattice Sncan be experimentally obtained by the junction of two building blocks A and B[54], and by the application of the sub-stitution rules (or inflation rules): A→AB, B→A. It follows that the nth stage of the Fibonacci superlattice is given by the recurrence relation Sn=Sn−1Sn−2, where n≥2, with S0 ¼ B and S1¼A. The initial generations are S0¼[B], S1¼[A], S2¼[AB], S3¼[ABA], S4¼[ABAAB], and so on, so that the n-generation hasN=Fnletters. For the quasiperiodic carbon and boron nitride nanotubes studied in this paper, we may choose carbon as the building block A and BN as the building block B, and vice versa, resulting in two models which we refer to as model-I and model-II, respectively. Such procedure is similar to the one employed by Pedreira et al. in the study of quasiperiodic C and BN nanoribbons[44].Figs. 1 and2 show schematically some generations of the Fibonacci quasiper-iodic nanotubes considered in this work. On the one hand,Figs. 1 (a) and (b) show quasiperiodic armchair nanotubes in the model-I Fig. 3. Specific heat (in units of kB) versus temperature (in units of kBT) of zigzag nanotubes. At (A), (B) and (C) we have the specific heat of (6,0) in model-I, (9,0) in model-I
and model-II, respectively. On the other hand, Figs. 2(a) and (b) show quasiperiodic zigzag nanotubes in the model-I and model-II, respectively.
In this work, each generation Sn of the Fibonacci sequence is regarded as a nanotube of length L. We consider the nanotube length until the eighth generation (S8) of the Fibonacci sequence, which corresponds to a nanotube with 21 C (21 BN) and 13 BN (13 C) building blocks in the model-I (model-II). We choose two na-notubes with armchair chirality, namely (4,4) and (6,6), and two nanotubes with zigzag chirality, namely (6,0) and (9,0). It can be seen inFigs. 1(a) and2(a) that the initial Fibonacci generations S0 and S1 are pure carbon (CNTs) and boron nitride (BNNTs) nano-tubes, respectively, while inFigs. 1(b) and2(b) is observed just the opposite. However, for n≥2, the Fibonacci generations Sn are hybrid BN and C nanotubes (BNCNTs), with the structural formula
BN Cx y, where the values of x and y depend on the diameter and chirality of the corresponding nanotube. Additionally, the values of the calculated diameters for the tubes (4,4), (6,6), (6,0), and (9,0) are ≈4.21 , 6.88 , 4.95Å Å Å, and 6.57Å, respectively, which are essentially the same for CNTs, BNNTs and BNCNTs. It is worth to point out that the diameter of the nanotubes was taken as the
average distance between diametrically opposite (or antipodal) atoms.
In order to obtain the density of states of the Fibonacci quasi-periodic nanotubes described above, we usefirst-principles calcula-tions based on the density functional theory (DFT) [55] as im-plemented in the SIESTA code[56]. It was used a norm-conserving Troullier-Martins pseudopotentials [57]in the Kleinmann–Bylander factorized form[58]and a double-
ζ
basis set, composed of numerical atomic orbitals offinite range. Polarization orbitals are included for nitrogen, boron and carbon atoms, applying the generalized gradient approximation for the exchange-correlation potential[59]. The pre-sent calculations use a grid cutoff of 21 Å, with the Brillouin zone sampled by1×1×34 k-points grid and by1×1×20 k-points grid for armchair and zigzag nanotubes, respectively. The mesh cutoff and the energy shift was chosen as 150 Ry and 0.01 Ry, respectively. All geometries were fully relaxed, with residual forces smaller than 0.1 eV/Å. For comparison, additional calculations were carried out using 0.05 eV/Å, providing insignificant changes in the calculation results, e.g. of about 0.01 eV for the gap energy. Additionally, it was assumed a convergence criterion where the self-consistency is achieved when the maximum difference between the output and the input of each element of the density matrix, in a self-consistent cycle, Fig. 4. Specific heat (in units of kB) versus temperature (in units of kBT) of armchair nanotubes. At (A), (B) and (C) we have the specific heat of (4,4) in I, (6,6) inmodel-I and (6,6) in model-model-Imodel-I, respectively.
is smaller than 104. Moreover, all calculations are based on periodic boundary conditions, where the supercell is repeated indefinitely along all directions. Therefore, the periodicity along the structure axis was chosen to be compatible with the unit cell length to form an infinitely long nanotube. For the other two directions such periodi-city is chosen to be large enough to prevent the interaction between parallel structures.
3. Results and discussion 3.1. Specific heat spectra
Recent studies [60–67] have shown that the specific heat spectra of nanostructured systems, modeled by the Fibonacci
quasiperiodic sequence, present an oscillatory behavior in the low temperature regime. Such behavior is a consequence of the mul-tifractal energy spectra of these systems. It is known the specific heat is a physical observable which can be easily measured in experiments, making possible a comparison between the experi-mental and theoretical results in order to test the descriptive and predictive accuracy of a given physical model. Bearing this in mind, we calculate the specific heat for the quasiperiodic carbon and BN nanotubes considered in this work. From now on we adopt the same procedure used by Moreira et al.[63]. Let us give a short description of such procedure. InFig. 6 is shown the density of states for each n-generation of the Fibonacci zigzag and armchair nanotubes. The partition function for a particular n-generation, using the Maxwell–Boltzmann statistics, is given by
Fig. 5. At (a), (b) and (c) we have the semi-log plot of the specific heat (in units of kB) versus temperature (in units of kBT) showing the log-periodic behavior of Fibonacci
zigzag nanotubes: (6,0) in model-I, (9,0) in model-I and (9,0) in model-II, respectively. At (d), (e) and (f) we have the semi-log plot of the specific heat showing the log-periodic behavior of Fibonacci armchair nanotubes: (4,4) in model-I, (6,6) in model-I and (6,6) in model-II, respectively.
∫
ρ ε ε= ( ) −βε ( )
Zn e d , 1
whereρ ε( )is the density of states ande−βεis the usual Boltzmann factor. Once we know the partition function, it is possible to cal-culate the specific heat by using
⎡ ⎣⎢ ⎤ ⎦⎥ ( ) = ∂ ∂ ∂ ∂ ( ) C T T T Z T ln , 2 n 2 n
where we have used β =1/T and set the Boltzmann's constant equal to unity. Moreover, 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. A more detailed discussion of such topics is given by Vallejos et al.[63]and Oliveira et al.[65].
Figs. 3and4show, respectively, the specific heat spectra for the Fibonacci zigzag and armchair nanotubes as a function of
temperature, for several n-generations. Regardless of the nanotube chirality (zig–zag or armchair), at very high temperatures ( ⟶∞T ), the specific heat for all n-generations converges and decays as T2. This asymptotic behavior is mainly due to the fact that we have considered our system bounded. On the other hand, our results re-veal that the specific heat behavior, in the low temperature region, depends on the nanotube diameter and symmetry (armchair or zigzag), and slightly on the n-generation index. In the case of zigzag nanotubes, at very low temperatures (T⟶0), the specific heat ex-hibits a non-harmonic oscillating behavior, as shown in the insets of Fig. 3. These oscillations can be seen as a superposition of Schottky anomalies. From the insets inFig. 3, we can observe that the specific heat curve for the (6,0) nanotube in model-I presents the most evi-dent oscillatory behavior. Hence, one can say that the nanotube curvature influences on the oscillatory behavior of C(T), i.e., the greater the curvature, the greater the number of oscillations of C(T). Also, at low temperatures, the specific heat reaches a maximum Fig. 6. Density of states of Fibonacci nanotubes. At (a), (b) and (c) we have the density of states of armchair nanotubes: (4,4) in model-I, (6,6) in model-I and (6,6) in model-II, respectively. At (d), (e) and (f) we have the density of states of zigzag nanotubes: (6,0) in model-I, (9,0) in model-I and (9,0) in model-II, respectively. The Fermi energy Efis
indicated by the dotted vertical line.
value whose corresponding temperature depends on the n-genera-tion. Additionally, the specific heat spectrum of the zigzag nanotubes studied in this paper has a pattern similar to the one of carbon-BN nanoribbons and DNA chains as reported in[44]and[63], respec-tively, both modeled by a Fibonacci quasiperiodic sequence.
On the other hand, a different scenario appears when we ex-amine the specific heat spectra of the quasiperiodic armchair na-notubes in the low-temperature region (seeFig. 4). In this case, as the temperature decreases, the specific heat reaches a maximum value and then begins to decrease in the low-temperature region. It is possible to observe oscillations in this region for the (4,4) nanotube in model-I. However, no oscillation is observed for the (6,6) nanotube in model-II. Those behaviors are quite clear from the insets inFig. 4. It is important to note that there is a small ripple in the curve for n¼3 in model-II (Fig. 4(C)), when compared to other n-generations, although this is not enough to assign an oscillatory behavior. We remark that this result reinforces the fact that the nanotube curvature has influence on the oscillatory be-havior of the specific heat.
Fig. 5 shows the semi-log plots of the specific heat versus temperature of the quasiperiodic nanotubes. The oscillations of the quasiperiodic zigzag and (4,4) armchair nanotubes are better il-lustrated. Clearly, it is seen a log-periodic behavior, i.e.,
α
( ) = ( )
C Tn ACn T , where A is a constant and
α
is an arbitrary numberfor both armchair and zigzag configurations. Moreover, our results show a slight difference between the periodic case (n¼2) and the quasiperiodic case (n¼3). Finally, it is not possible to infer a log-periodic behavior for the (6,6) armchair nanotubes.
Table 1
Energy band gap (Eg) and factorγnof all Fibonacci nanotubes investigated
corre-sponding to each one of the n-generation index. Armchair
n (4,4) Model-I (6,6) Model-I (6,6) Model-II Eg(eV) γn Eg(eV) γn Eg(eV) γn
0 0.00 0.00 0.00 0.00 4.37 1.00 1 4.21 1.00 4.37 1.00 0.00 0.00 2 0.55 0.50 0.72 0.50 0.72 0.50 3 1.06 0.67 0.98 0.67 0.54 0.33 4 0.74 0.60 0.81 0.60 0.66 0.40 5 0.80 0.63 0.85 0.63 0.43 0.38 6 0.77 0.62 0.83 0.62 0.46 0.39 7 0.79 0.63 0.84 0.63 0.44 0.38 8 0.77 0.63 0.84 0.63 0.44 0.38 Zig–Zag
n (6,0) Model-I (9,0) Model-I (9,0) Model-II
Eg(eV) γn Eg(eV) γn Eg(eV) γn
0 0.19 0.00 0.59 0.00 3.50 1.00 1 2.62 1.00 3.50 1.00 0.59 0.00 2 0.56 0.50 0.88 0.50 0.88 0.50 3 0.60 0.67 0.92 0.67 0.39 0.33 4 0.40 0.60 0.57 0.60 0.43 0.40 5 0.43 0.63 0.67 0.63 0.29 0.38 6 0.41 0.62 0.57 0.62 0.33 0.39 7 0.58 0.63 0.29 0.38 8 0.55 0.63
Fig. 7. Plot of the energy band gap versus Fibonacci generation index n for armchair nanotubes: (4,4) in model-I, (6,6) in model-I and (6,6) in model-II. The lines serve only to guide the eye.
Note also that the methodology employed determines only the electronic contribution to the specific heat. While the phonon contribution should be dominant except near T¼0 K[68], our re-sults provide a qualitative description of the specific heat for the investigated quasiperiodic structures.
3.2. Energy band gap
Fig. 6shows the density of states (DOS) of the Fibonacci na-notubes: armchair (Figs. 6(a)–(c)) and zigzag (Figs. 6(d)–(f)), from S2to S8. From the DOS, one can determine the allowed and for-bidden energy states for all n-generation index. The values of the energy band-gap (Eg) calculated for all n-generation index are displayed inTable 1. For model-I it is worth noting that the energy gap is, for n¼0 and n¼1, equal to the energy gap of the pure C (S0¼B) and BN (S1¼A) nanotubes, respectively. For model-II, as expected, we have exactly the opposite behavior. It is known that BNNTs are insulators regardless of their diameter and chirality, while CNTs are conductors or semiconductors[69–72]. From Ta-ble 1 we can infer that all quasiperiodic BNCNTs are semi-conductors forn≥2, independently of their diameter and chirality – we found Egvalues ranging from 0.29 to 1.06 eV. Materials with band gaps in this range are suitable for applications in optoelec-tronics, such as light emitting devices and solar cells[73,74].
Figs. 7and 8show the energy band-gap versus n-generation index of the Fibonacci sequence, for the Fibonacci armchair and zigzag nanotubes, respectively, in the model-I and model-II. For both models, the energy gap exhibits an interesting oscillatory
behavior, which depends on the length of the Fibonacci generation Sn. We can also note a slight dependence of Egon the diameter and symmetry (zigzag or armchair) of the nanotube. As the n-index increases, the oscillation amplitude decreases and Egconverges to a stationary value, namely: (i) Eg∼0.77 and 0.84 eV, for (4,4) and (6,6), respectively, in model-I; (ii) Eg∼0.41 and 0.57 eV, for (6,0) and (9,0), respectively, in model-I; and (iii) Eg∼0.49 and 0.30 eV, for (6,6) and (9,0), respectively, in model-II. In fact, Pedreira et al. [44]found similar results for Fibonacci quasiperiodic nanoribbons composed of BN and C, with both zigzag and armchair symmetries, where the energy gap also oscillates and converges to a stationary value. Therefore, we can conclude that the presence of the qua-siperiodic disorder in the atomic arrangement of the nanotubes, which are composed of BN and C, is responsible for the oscillatory behavior of the energy band gap.
On the other hand, from the projected density of states (PDOS) shown inFig. 9, for some selected Fibonacci nanotubes, it can be seen that the contribution of the carbon atoms is, in general, more expressive for the electronic states near the Fermi energy EF. The contribution of the boron atoms is dominant for the electronic states in the conduction band, while the contribution of the ni-trogen atoms is more expressive in the valence band. InTable 1, it is noticeable that, forn≥3, the energy gap in the model-II is lower than in the model-I, for both armchair and zigzag nanotubes. This result is linked to the fact that the number of carbon atoms in the model-II is greater than in the model-I.
Let us take a look at the origin of the oscillatory behavior of the energy gap from a qualitative point of view. These oscillations are Fig. 8. Plot of the energy band gap versus Fibonacci generation index n for zigzag nanotubes: (6,0) in model-I, (9,0) in model-I and (9,0) in model-II. The lines serve only to guide the eye.
associated with the BN and C concentrations present in the qua-siperiodic (BN C)x ynanotubes, as can be seen inFigs. 1and2. It is noticeable inTable 1that the energy gap values for the quasiper-iodic case (n≥2) are between the energy gap values of the CNTs and BNNTs, as expected. Moreover, depending on the BN and C concentrations, resulting from the quasiperiodic disorder, the (BN C)x y nanotubes gaps are closer to the gaps of the CNTs or BNNTs. InFigs. 1and2, we can see that BN and C concentrations depend on the specific Fibonacci n-generation. This observation motivates us to define a factor
γ
nfor each Fibonacci n-generation as: γ = ϵ ϵ + ϵ , ( )3 n BN BN Cwhere
ϵ
BNandϵ
Care the values of the BN and C concentrations, respectively. The values ofγ
nare shown inTable 1. First, we note that, for all cases investigated, the values ofγ
nexhibit an oscilla-tory behavior as the n-index increases. Furthermore, for a parti-cular n-generation, the value ofγ
nchanges from I to model-II, and it does not depend on the diameter and symmetry (zigzag or armchair) of the nanotube. Indeed, it is possible to conclude that the values ofγ
nare strongly associated with the two possi-bilities of choice of the (BN C)x ynanotubes building blocks (model-I and model-II) of the quasiperiodic Fibonacci sequence. In parti-cular, it is noticeable that the higher values of the gaps (1.06, 0.98 and 0.92 eV) occur for n¼3, whereγ
nreaches the maximum value of 0.67. Thus, it is possible to state that the higher the BN con-centration, the higher the energy gap of the nanotube. In model-I, Fig. 9. (Color online) Calculated projected density of states (PDOS) of selected Fibonacci nanotubes. At (A) and (B) we have the PDOS of (4,4) in model-I and (9,0) in model-I, respectively, from S2to S4. Black solid lines indicate the contribution due to boron, blue dotted lines to nitrogen and brown solid lines to carbon atoms. The Fermi energy Efisfor both zigzag and armchair nanotubes, regardless of their dia-meter, the higher values of
γ
n occur for odd n, while the lower values occur for even n. Due to this fact, Egincreases for n odd and decreases for n even, leading to the oscillations observed in Figs. 7and8. However, the opposite takes place in model-II: the higher (lower) values ofγ
nare observed for n even (odd). There-fore, Egincreases for n even and decreases for n odd, resulting also in the oscillatory behavior. Additionally, as n increases, the ratioγ γ
[n n 1/ −] →1and the difference γn−1−γn tends to zero. As a con-sequence, the oscillation amplitude decreases, and Egreaches a stationary value. From our results, it is clear that the convergent value of Egof the quasiperiodic nanoribbons depends on the en-ergy band gap of their building blocks (BN and C). It is also known that a prerequisite for future technological applications of nanor-ibbons is the ability tofine-tune the energy band gap.
4. Conclusions
In the present contribution,first-principles calculations based on density functional theory were applied to investigate the ef-fects of quasiperiodic disorder on the physical properties of zigzag and armchair nanotubes, composed by BN and C, and constructed according to the Fibonacci sequence. We also studied the influence of the tube diameter on its properties, by investigating two dia-meters for both the armchair and the zigzag case. We assumed CNTs and BNNTs as the building blocks A and B of the resulting (BN C)x y quasiperiodic structure, respectively, as illustrated in Figs. 1and 2. We also considered two building models (model-I and model-II). The density of states of the nanoribbons were ob-tained by usingfirst-principles calculations. The behavior of the energy band gap were investigated as a function of the n-gen-eration index of the Fibonacci sequence. For both armchair and zigzag nanotubes, in the model-I and model-II, the energy gap oscillates and, as the n-index increases, the oscillation amplitude decreases and Egconverges to a stationary value. The oscillatory behavior of the energy gap is associated to the BN and C con-centrations present in the nanotubes. Such concon-centrations derive from the quasiperiodic disorder. In general, we realized that the combinations of CNTs and BNNTs should lead to (BN C)x ynanotubes with tunable electronic structure. It is noteworthy that the in-crease of these states with the inin-crease of the n-generation also results in an increase or decrease of Eg, leading to the oscillations. We have also performed a theoretical study of the specific heat behavior of the quasiperiodic Fibonacci nanotubes. Our numerical results show that the specific heat for all cases, in the high-tem-perature limit ( ⟶∞T ), converges and decays as (T2). In the low temperature region, the specific heat starts to present non-har-monic small oscillation behavior. This is consequence of super-positions of Schottky anomalies. We also found that the nanotube curvature has influence on the oscillatory behavior of C(T). Our results suggest that the choice of the CNTs and BNNTs as the building blocks of a quasiperiodic Fibonacci sequence may lead to a nanotube with a direct, adjustable band gap, with energies ranging from 0.29 to 1.06 eV, suitable for applications in optoe-lectronics devices such as light emitting materials and solar cells. We hope that ourfindings can provide valuable insights regarding technological applications which involve nanodevices based on BN and C.
Acknowledgments
We would like to thank the Brazilian Research Agencies CNPq, PNPD/CAPES and INCT of Space Studies forfinancial support. We
are also grateful to the CLIMA cluster– UFRN and the Laboratory of Computational Physics– UFPB, where the numerical calculations were performed.
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