#### Abstract

The optical properties of (8,0) BC_{3} and B_{3}C single-wall carbon nanotubes (SWCNTs) are computed using *ab initio* density functional theory (DFT). The electronic band structure reveals that the Fermi energy of B_{3}C system is reduced compared to BC_{3}. The static dielectric constant in the long wavelength limit for B_{3}C system is 9 times larger than that of BC_{3} in unpolarized electromagnetic field. Within 10 eV frequency (energy) range, the absorption coefficient of B_{3}C is higher compared to BC_{3}, while, above 10 eV, it is less than that of BC_{3}. In parallel polarization, the peak of the loss function for B_{3}C is shifted to higher frequency (energy) region with significantly six orders of magnitude compared to BC_{3} system. The analysis of this study indicates that the optical anisotropies can be gained easily in these boron-doped systems by appropriately choosing the direction of the polarization of the electromagnetic field. Besides, the results of the loss functions may throw some light on the nature of collective excitations of these two systems.

#### 1. Introduction

From its very discovery in 1991, carbon nanotubes (CNTs), both single wall as well as multiwall, have attracted the attention of theoretical and experimental research groups [1, 2]. Due to their unique one-dimensional structure and unusual electronic properties, they can be regarded as one of the natural building blocks of molecular electronics. Single-wall carbon nanotubes (SWCNTs) consist of a single graphene layer rolled on a cylinder with diameters between 1 and 2 nm having lengths of the order of few hundred micrometers. The calculations reveal that their electronic properties can be semiconducting, metallic, or quasimetallic depending on their geometrical structure dictated by a quantity known as the chirality. The optical properties of 4 Å diameter pure SWCNT have been investigated [3–5] recently by first principles calculation to explain the experimental results. Most recently, *ab initio* calculations of the linear and nonlinear optical properties of pure CNTs have shown that the dielectric function depends essentially on chirality, diameter and the nature of polarizations of incident electromagnetic field [6].

The electronic properties of single-wall carbon nanotubes (SWCNTs) can be tailored [7, 8] by substituting carbon atom(s) by heteroatom (s) such as boron or nitrogen. It is well known that pure CNTs are unable to detect highly toxic gases, water molecules, and biomolecules [9]. To improve the nanosensor reliability and quality, the importance of substitution alloying of impurity atoms such as boron and nitrogen has been discussed [10]. In fact, a calculation on the chemical interaction reveals that the boron doped CNT can act as a novel sensor [11] for formaldehyde. The synthesis of composite B* _{x}*C

*N*

_{y}*tubes has been performed and their energy loss spectroscopy have been reported [12, 13]. In general, through the reaction of B*

_{z}_{2}O

_{3}with CNTs under an Ar atmosphere [14], B atom(s) can be substituted for the carbon atom(s) of SWNT. In the literature, the synthesis and electronic properties of B-substituted SWCNT have been discussed [15, 16]. A quantum chemical calculation [17] has been employed to investigate the larger mobility (i.e., electronic conductivity) of B- and N- doped CNTs. Recently, the electronic structure and optical properties of B- doped single-wall carbon nanotubes (SWCNTs) have been studied in detail, and it is found that boron is in sp

^{2}configuration [18]. It has also been shown recently that even a small amount dopant can significantly change the mesoscopic conductivity [19] of chemically B-doped CNTs. The electron current distribution in B- and N-doped armchair CNT has been investigated [20] using Density Functional Theory [DFT] and Green’s function to show a chiral flow of current. All the above examples eventually indicate the importance of the study of B-doped CNTs and invite further investigation about the optical properties of the doped system as a function of B concentration. Since this doping can alter the band structure of SWCNTs (such as band gap, the density of states near Fermi energy) considerably [21, 22], we would naturally expect some dramatic changes in the response of the B

*C*

_{x}*nanotubes under an electromagnetic field. As a matter of fact, in recent years the spectroscopic studies of BC*

_{y}_{3}SWNTs have been interpreted in terms of

*ab initio*band structure calculation [23, 24].

With this motivation, we are interested in computing the optical properties of these two boron-doped systems in different polarizations of the electromagnetic field. In particular, in this paper, we study the optical response of (8,0) BC_{3} and B_{3}C SWCNTs under the action of a uniform electric field with various polarizations direction through relaxed C-C bond length *ab initio* (DFT) calculations in the long wavelength limit. The geometrical structures of these systems were built by replacing some of the carbon atom(s) in the hexagonal ring by B atoms. The preferred boron sites were chosen having lowest total energy.

#### 2. Computational Methods

The numerical methods are employed using first-principles (DFT) with Generalized Gradient Approximation (GGA) as implemented in CASTEP code [25] in the materials studio and simulation software package. Using this method, we calculate the dielectric constant and other related optical properties such as absorption coefficients, reflectivity, and the loss function. The last three quantities are however experimentally accessible. An extensive thorough account of the numerical computation dealing with B-alloyed system has been described in our earlier work [26–28].

The optical properties of any system are generally studied by the complex dielectric function defined by . However, and are not independent of each other. Within the framework of time dependent perturbation theory along with simple dipole approximation used in CASTEP code [25], the imaginary part is given by
where and represent, respectively, the volume of the supercell and the dielectric constant of the free space. It is easy to notice that the expression in (1) is similar to Fermi’s Golden rule for time-dependent perturbation in quantum theory. Physically speaking, can be visualized as a parameter concerning the real transitions between the occupied valence band (VB) and unoccupied electronic states. The vector defines the state of polarization of the incident electric field while is the position vector. The sum over is a crucial point in such numerical calculations. It actually samples the whole region of Brillouin zone (BZ) in the space by taking a symmetrized Monkhorst-Pack [29] grid and smearing each energy level with a Gaussian spread function. The other two sums take care of the contribution of the unoccupied conduction band (CB) and occupied valence band (VB). In computing the above dielectric function, typically [(1*⁄*2) (total number of electrons +4)] number of bands were taken. No phonon contribution was taken here. Moreover, in this formulation, the local field effect and the excitonic effect have been neglected. The atomic positions are relaxed until the forces on the atoms are less than 0.01 eV/Å. The typical convergence was achieved till the tolerance in the Fermi energy is eV. After checking the convergence, for sampling the irreducible part of the Brillouin Zone (BZ), we have used () Monkhorst and Pack mesh having 6 k points with a cut-off energy of 470 eV. The smearing broadening in computing the optical properties was kept fixed at 0.5 eV.

The matrix element described in (1) for the electronic transition is computed between the single-electron energy eigenstates. To determine the wave functions in (1), we perform the first-principles spin unpolarized density functional theory using plane wave pseudopotential methods [30, 31]. Like any *ab initio* calculation, the self-consistent Kohn Sham (KS) equation has been employed here to compute the eigen function. For the exchange and correlation term, the generalized gradient approximation (GGA) as proposed by Perdew-Berke-Ernzerhof [32] is adopted. The standard norm-conserving pseudo-potential in reciprocal space is invoked for the optical calculation. Because of using of nonlocal potentials in this software, the matrix elements are modified as
In comparison with the standard local density approximation (LDA) (with appropriate modifications) used mostly in electronic band structure calculation, the optical properties of the system are normally standardized by *spin unpolarized* GGA.

The typical computational supercell used here is the 3D triclinic crystal ( Å, Å, Å, and angles ) having symmetry P1. The supercells are built by taking four units of CNT. The energy cut-off, -point sampling, geometry, GGA/norm-conserving pseudopotential are the same in these two systems as well as the pristine one. The anisotropic behavior in the optical properties can be investigated by taking into account the polarization vector of the electromagnetic field in (1). Thus, the dielectric constant can be evaluated for three separate cases: one can choose in the direction of (i) of electric field vector for the light at normal incidence (polarized); (ii) choosing in the direction of propagation of incident light at the normal incidence but the electric field vector is considered as an average over the plane perpendicular to this direction (unpolarized); (iii) choosing not in specified direction while the electric field vectors are taken as full isotropic average (polycrystalline). The directions of the field ( wave vector) have been chosen with respect to axis of the B-doped carbon nanotubes. The parallel polarization refers to (0,0,1) while perpendicular one (1,0,0).

#### 3. Numerical Results and Discussions

##### 3.1. Study of Band Structure of B_{3}C and BC_{3} Systems

Before we discuss the optical properties, we show in Figure 1 the typical ball and stick model of (8,0) BC_{3} and B_{3}C systems.

**(a)**

**(b)**

All the results presented in this paper have the same set of parameters as indicated in earlier section. For pure (8,0) we find the Fermi energy 6.028 eV with band gap at point (most symmetric point in the BZ) as 0.48 eV. However, alloying with boron atoms in (8,0) nanotubes such as in BC_{3} system, the Fermi energy reduces to 4.256 eV. With increasing more number of boron atoms in SWNTs, we find, for (8,0) B_{3}C nanotubes, a further reduction of the Fermi energy to 3.614 eV. More interestingly, we note a significant increase of the overlapping of valence and conduction band compared to pure as well as BC_{3} SWNT. This is understood simply from the fact that the electronic configuration of B atom is 1s^{2} 2s^{2}2p^{1}.

Therefore, doping by B atom always reduces the total number of electrons in the system which on the other hand implies the decrease of Fermi energy with B doping. This has also been observed in boron doped multiwalled carbon nanotubes [33, 34]. In Figure 2, we show the typical number of self-consistent field (SCF) iterations and the convergence of the typical energy per atom of B_{3}C system.

In Figure 3, we schematically show the band structure of B_{3}C system, respectively. All the energies shown in the diagram have been measured with respect to the Fermi energy (shown as the dashed line in the band diagram). The most symmetric point is known as point and is denoted by G in the energy band diagram. The band gap in BC_{3} system turns out as 0.43 eV at point, which is smaller than the pure one.

However, for B_{3}C system, the band gap turns out as 0.58 eV at point, which is larger than the pure one. This engineering of band gap at the most symmetric point in BZ may be useful in device and sensor applications. These observations are required later on to understand some of the features of the optical properties of the doped CNT systems. The band structure shown in Figure 3 reveals that near the bottom of the valence band, there is a gap in energy for all values of points in BZ. With increase of B doping, this feature is seen to be an integral part of the dispersion relation.

Apart from the difference of the Fermi energy in the two cases (see Table 1), the striking difference lies in the energy ranges between −12 eV and −14 eV. There exist energy states in BC_{3} system while those are absent in the energy spectra in the VB for all values of . Besides, it is clear from the two energy band diagrams (Figures 3 and 4) that the band width is higher in BC_{3} compared to B_{3}C one.

The partial density of states (PDOS) of B atoms of (8,0) BC_{3} and B_{3}C carbon nanotubes shown in Figure 5 indicates a series of spikes in the whole spectrum of band energy and these are basically the van Hove singularity typical characteristics of low-dimensional condensed matter systems. The low temperature scanning tunneling spectroscopy (STS) measurement can be used to verify the position of the spikes. It is seen that, in both pure and doped case, the contribution of p electrons in valence band is higher compared to its counterpart s electrons. However, the contribution of s electrons in both of the cases in the conduction band is meagre. In B_{3}C case, the contribution of p electrons at the Fermi level have been increased substantially compared to pure case. In fact, the higher value of DOS at the Fermi level signifies the metallicity character of B_{3}C. The DOS at the Fermi level is a measure of available free charge carriers. Thus, the increase of the DOS at the Fermi levels is a signature of more metallic character of B_{3}C system compared to BC_{3} one.

**(a)**

**(b)**

The direction with relatively flat dispersionless bands at various -points seems to contribute significantly to the optical absorption and hence allows one to explain the anisotropy of the optical properties. This is due to the fact that at those values the group velocity of the electronic states vanishes with increase of DOS. Hence, an increase of the value of is expected.

##### 3.2. Study of Dielectric Constant of BC_{3} and B_{3}C Systems

We compute the imaginary part of the dielectric constant within the specified frequency range. The polarizations of the electromagnetic field play an important role in computing the imaginary part of the dielectric constant. We compute the dielectric constant for parallel, perpendicular polarization and unpolarized one with normal incidence (1,0,0). The parallel polarization refers here to the direction of light parallel to the axis of respective SWCNTs. In Figure 5, we schematically show the dielectric constant (real) for both pure (8,0), BC_{3}- and B_{3}C-doped systems as a function of frequency for unpolarized scheme. It is seen in the numerical computation (not shown in the figure) that, in both cases, the imaginary part of the dielectric constant is always positive throughout the range of frequency. This can be understood very simply from (1) used for the numerical simulation study. The square of the matrix element and the even functional nature of the energy conserving delta function ensure the positivity of . This property of serves as one of the crosschecks in our numerical computation.

However, as evident from the figure itself, such a restriction is not obeyed by the real part of the dielectric constant . We also note that the static value (strictly speaking but in our numerical computation Hz.) of the dielectric constants for both pure and doped systems is always positive. This observation is further satisfied by a theorem in continuous media stating that the static electric dielectric constant is always positive [35] for any material in thermal equilibrium. The variation of static dielectric constant with concentration of B has been reported recently [36] to show that a small concentration is enough to change the value drastically from the pure (8,0) SWCNT. It is evident from Figure 6 that the static value of the dielectric constant (real as well as imaginary) of BC_{3} system is higher compared to pure one. It has been observed that the static dielectric constant of B_{3}C is higher than that of BC_{3} for any type of polarizations (see Table 1).

In case of semiconducting SWCNT, an *ab initio* tight-binding calculation [37–39] relates the static value of the dielectric constant with the energy band gap as
Here is the plasma frequency and is the energy band gap. Our numerical calculations are in qualitative agreement with the higher value of the dielectric constant B_{3}C compared to BC_{3} one. In particular, the static dielectric constant in the long wave length limit for B_{3}C system is 9 times larger than that of BC_{3} in unpolarized electromagnetic field with normal incidence (1,0,0).

##### 3.3. Study of the Absorption Spectra of the Doped System

The absorption coefficient is related to the imaginary part of the dielectric constant as where and are the refractive index and the speed of light, respectively. The absorption spectra depend critically on the nature of CNT as well as the direction of polarization. The absorption spectra are limited to UV region only.

The existence of peaks in the spectra indicates the maximum absorption at that particular energy. With doping by B atom(s), both the magnitude of the peaks and its position change significantly. We depict, in Figure 7, a comparative study of the absorption coefficient of BC_{3} and B_{3}C systems as a function of frequency in the unpolarized case with normal incidence (1,0,0). We notice that in contrast to B_{3}C, there exist several peaks in the absorption spectra. The existence of these rich absorption peaks in the overall frequency range is consistent with the theoretical tight-binding calculations made on (3,3) and (6,0) BC_{3} nanotubes [40]. It is also evident from the figure that up to 10 eV or so (frequency measured in units of energy), the absorption coefficient of B_{3}C is always higher than that of BC_{3}. However, above 10 eV, the reverse is true.

##### 3.4. Study of the Reflectivity at Normal Incidence of the Doped System

The reflectivity of any media at normal incidence is calculated from the refractive indexes via the relations (as implemented in CASTEP [25]) given by
It is clearly evident from the definition that the reflectivity is always positive in the scheduled range of the frequency and is dimensionless. is sometimes regarded as the index of refraction as a function of wavelength of light used. In Figure 8, we show the variation of reflectivity of BC_{3} and B_{3}C systems for parallel polarization as a function of frequency. It is clear from the figure that the reflectivity of B_{3}C system is always higher than that of BC_{3} in the whole range of frequency. This could be helpful in designing optical devices involving B-doped SWCNT.

##### 3.5. Study of the Loss Function of the Doped System

In this time-dependent calculation of the ground state electronic states, the interaction is between the photon and electrons. The transitions between the occupied and unoccupied states are caused by the electric field of the photon. When these excitations are collective in nature, they are termed as plasmons. The loss function, which is a direct measure of the collective excitations of the systems, is defined as . Since we are taking limit in our calculation, therefore, we are considering the loss function behavior under the long wavelength limit. The peak position of this loss function determines the typical energy of the plasmons in the system. High-resolution transmission electron microscopy (HRTEM) and nanoelectron energy loss spectroscopy (nano-EELS) can provide information about the systematic and atomic structural defects of B-doped SWCNTs [41–43]. The spectra resulting from these collective excitations can be alternatively understood as a joint DOS between VB and CB weighted by appropriate matrix elements. In terms of the real and imaginary dielectric constants, a straightforward algebra reveals that . At the plasma frequency, the above expression attains the higher value when and .

In Figures 9 and 10, we depict the anisotropic signature of BC_{3} and B_{3}C systems for unpolarized case and parallel polarization, respectively. For unpolarized case, we notice that within the range of frequency (7–12 eV), the loss function of BC_{3} is smaller than that of B_{3}C. However, above 12 eV, the loss function of BC_{3} is higher than that of B_{3}C. Besides, the main peak of BC_{3} at 7.24 eV is seen to shift to 9.89 eV with significantly higher magnitude. While in parallel polarization as evident from Figure 10, the single peak of B_{3}C at 8.39 eV is shifted to 8.81 eV for BC_{3} system with significantly six orders of magnitude. In Table 1, we summarize the main results of the two systems.

#### 4. Conclusions and Perspectives

From the first-principles relaxed C–C bond length DFT calculation of the optical property of BC_{3} and B_{3}C (8,0) SWNT systems, we have observed significant changes in the optical behavior for different polarizations. The behavior of the static dielectric constant of B-doped system depends on the flavor (nature) of the CNTs. The anisotropy signatures of the dielectric constants noticed in these systems are due to the confined geometry of the CNTs. The electronic band structure reveals that the Fermi energy of B_{3}C system is reduced compared to BC_{3}. The static dielectric constant in the long wavelength limit for B_{3}C system is 9 times larger than that of BC_{3} in unpolarized electromagnetic field with normal incidence (1,0,0). Within 10 eV frequency (energy) range, the absorption coefficient of B_{3}C is higher compared to BC_{3}, while, above 10 eV, it is less than that of BC_{3}. In parallel polarization, the peak of the loss function for B_{3}C is shifted to higher-frequency (energy) region with significantly six orders of magnitude compared to BC_{3} system. All these facts about these two systems may throw some light on the nature of collective excitations and nanoscale optical devices.

#### Acknowledgment

One of the authors (D. Jana) would like to thank the National Science Council (NSC) of Republic of China (R.O.C) for financially supporting him as a visiting researcher under Contract no. NSC 099-2912-I-002-160.