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Carbon 45 (2007) 1482–1491 www.elsevier.com/locate/carbon

A ?rst principles study of the optical properties of BxCy single wall nanotubes

Debnarayan Jana a,*, Li-Chyong Chen b,*, Chun Wei Chen c, Surojit Chattopadhyay d, Kuei-Hsien Chen d

a

Department of Physics, University College of Science and Technology, University of Calcutta, Kolkata 700 009, West Bengal, India b Center for Condensed Matter Sciences, National Taiwan University, Taipei 106, Taiwan c Department of Material Science and Engineering, National Taiwan University, Taipei 106, Taiwan d Institute of Atomic and Molecular Sciences, Academia Sinica, Taipei 106, Taiwan Received 19 June 2006; accepted 9 March 2007 Available online 16 March 2007

Abstract The optical properties of small radius (<1 nm) single wall carbon nanotubes (SWCNTs) alloyed with boron were examined using relaxed C–C bond length ab initio calculations in the long wavelength limit. The magnitude of the static dielectric constant essentially depends on the B concentration as well as the direction of polarization. The maximum value of the absorption coe?cient is shown to strongly depend on the concentration of B in a non-linear way with a minimum at a critical concentration of 0.40 for both the parallel polarization and the un-polarized cases and of 0.29 for perpendicular polarization of the electromagnetic ?eld. The peak of the loss function in parallel polarization and unpolarized cases shifts to a lower frequency with increasing concentration up to 50% but then shifts to a higher frequency. The non-linear ?ts to the plasma resonance frequency variation with B concentration indicate the existence of a unique minimum. All these factors may shed light on the nature of collective excitations in B-alloyed SWCNTs. ? 2007 Elsevier Ltd. All rights reserved.

1. Introduction Carbon nanotubes (CNTs) because of to their unique one-dimensional structure and unusual physical, chemical and mechanical properties have attracted the attention of theoretical and experimental research groups [1,2]. The electronic properties of single wall carbon nanotubes (SWCNTs) can be tailored [3,4] by substituting carbon atom(s) by heteroatoms (s) such as boron or nitrogen. It is well known that pure CNTs are unable to detect highly toxic gases, water molecules and biomolecules [5]. To improve the nanosensor reliability and quality, the importance of substitutional alloying of impurity atoms such as boron, nitrogen have been discussed [6]. In fact, a

* Corresponding authors. Fax: +91 (0) 3323509755 (D. Jana); fax: +886 2 23655405 (L.-C. Chen). E-mail addresses: djphy@caluniv.ac.in (D. Jana), chenlc@ntu.edu.tw (L.-C. Chen).

calculation on the chemical interaction reveals that the boron doped CNT can act as a novel sensor [7] for formaldehyde. The synthesis of composite BxCyNz tubes has been performed and their energy loss spectroscopies have been reported [8,9]. In general, through the reaction with B2O3 with CNTs under an Ar atmosphere [10] B atom(s) can be substituted for the carbon atoms(s) of SWNT. In literature, the synthesis and electronic properties of B-substituted SWCNT have been discussed [11,12]. A quantum chemical calculation [13] 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 sp2 con?guration [14]. It has also been shown recently that even a small amount dopant can signi?cantly change the mesoscopic conductivity [15] of chemically B doped CNTs. The electron current distribution in B and N doped armchair CNT have been investigated [16] using

0008-6223/$ - see front matter ? 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.carbon.2007.03.024

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DFT and Green’s function to show a chiral ?ow 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 [17,18], we would naturally expect some dramatic changes in the response of the BxCy nanotubes under an electromagnetic ?eld. As a matter of fact, in recent years the spectroscopic studies of BC3 SWNTs have been interpreted in terms of ab initio band structure calculation [19,20]. ? The optical properties of 4 A diameter pure SWCNT have been investigated [21–23] recently by ?rst principles calculation to explain the experimental results. Most recently, ab initio calculations of the linear and non-linear optical properties of pure CNTs have shown that [24] the dielectric function depends essentially on chirality, diameter and the nature of polarizations of incident electromagnetic ?eld. In this paper, we study the response of the B alloyed CNTs (pure metallic, semiconductor and quasimetallic) under the action of a uniform electric ?eld with various polarizations direction through relaxed C–C bond length ab initio density functional theory (DFT) calculations. The geometrical structure of impure system was built by replacing one of the carbon atom(s) in the hexagonal ring by B atom(s). The preferred boron sites were chosen having lowest total energy. In this ?rst principles computation, we have only one ?xed parameter from the experiment namely the carbon–carbon bond length (0.142 nm) for pure SWCNT. In particular, we concentrate on the static long wavelength (x ! 0, q ! 0) optical response of the system apart from its variation with frequency and discuss the nature of its variation with chemical doping of B concentration in a semiconductor (8, 0) nanotube.

2. Numerical methods

The optical properties of any system are generally studied by the complex dielectric function de?ned by ~ D?x? ? e?x?~ E?x? ? ?e1 ?x?? ie2 ?x??~ E?x?. However, e1(x) and e2(x) are not independent of each other. In this numerical simulation, the imaginary part of the dielectric function has been computed by using ?rst-order time dependent perturbation theory. In the simple dipole approximation used in CASTEP code [25], the imaginary part is given by e2 ?q ! 0~; hx? ? u 2e2 p X jhwC j~ ?~ V ij2 d?EC ? EV ? E? k k k u rjwk Xe0 k;V ;C ?1?

?eld. No phonon contribution is taken into account here. Moreover, in this formulation, the local ?eld e?ect and the excitonic e?ect have been neglected. To determine the wave functions in Eq. (1), we perform the ?rst principles spin unpolarized density functional theory using plane wave pseudopotential methods [26,27]. Like any ab initio calculation, the self-consistent Kohn Sham (KS) equation has been used to compute the eigen function here. For the exchange and correlation term, the generalized gradient approximation (GGA) as proposed by Perdew–Berke–Ernzerhof [28] is adopted. The standard norm-conserving pseudo-potential in reciprocal space is invoked for the optical calculation. Compared to the standard local density approximation (LDA) (with appropriate modi?cations) used mostly in electronic band structure calculation, the optical properties of the system are normally standardized by spin un-polarized GGA. A cuto? energy of 470 eV for the grid integration was adopted for computing the charge density. For Brillouin zone (BZ) integration along the tube axis, we have used six Monkhorst [29] k-points. The smearing broadening in computing the optical properties was kept ?xed at 0.5 eV. The atomic posi? tions are relaxed until the forces on the atoms are less than 0.01 eV/A. The typical convergence was achieved till the tolerance in the Fermi energy is 0.1 · 10?6 eV. The typical computational super cell used here (which includes typical four units of CNT) is the 3d triclinic crystal (a = ? ? ? 18.801 A, b = 19.004 A, c = 4.219 A and angles a = b = 90°, c = 120°) having symmetry P1.

3. Results and discussion 3.1. Study of band structure of BxCy system Before we discuss the optical properties, we show in Fig. 1 the typical ball and stick model of pure (8, 0) and BC3 system. All the results presented in this numerical calculation have same set of parameters as indicated in earlier section. In Fig. 2, we schematically show the band structure of BC3 system respectively. All the energies shown in the diagram have been measured with respect to the Fermi energy. For pure (8, 0) we ?nd the Fermi energy 6.028 eV with band gap at C point (most symmetric point in the BZ) as 0.48 eV. We also notice that the Fermi energy (the dashed line) is within the valence band and conduction band. However, alloying with Boron atoms in (8, 0) nanotubes, the Fermi energy reduces to 4.256 eV with overlapping of the few energies of valence and conduction band. The band gap in this case turns out as 0.43 eV at C point, which is smaller than the pure one. With increasing more number of boron atoms in SWNTs, we ?nd for (8, 0) B3C nanotubes, a further reduction of the Fermi energy to 3.614 eV. More interestingly, we note that a signi?cant increase of the overlapping of valence and conduction band in compared to pure as well as BC3 SWNT. The band gap in this case turns out as 0.58 eV at C point, which is larger than the pure one. The band structure of B3C (not shown in ?gure) also reveals that near the bottom of the valence band, there is a gap in energy for all values of k-points in BZ. With increase of B doping, this feature is seen to be an integral part of the dispersion relation. The ?atness of the band at various k-points seems to contribute signi?cantly to the optical absorption. The partial density of states (PDOS) of (8, 0) B3C carbon nanotubes shown in Fig. 3 indicates a series of spikes in the whole spectrum

X and e0 represent respectively the volume of the super-cell and the dielectric constant of the free space. The sum over k is a crucial point in numerical calculation. It actually samples the whole region of Brillouin zone (BZ) in the k space. The other two sums take care 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] no. of bands were taken. Here, ~;~ respectively represent u r the polarization vector of the incident electric ?eld and position vector. The matrix element of this dot product of these two vectors is computed between the single electron energy eigen states. Since the magnetic ?eld effect is weaker by a factor of v/c, the transition matrix elements between the eigenstates of CB and VB have been calculated only due to the electric

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Fig. 1. Ball and stick model of (a) pure (8, 0) and (b) BC3 tube in 3d triclinic structure.

Energy (eV) 8 6 4 2 0 -2 -4 -6 -8 -10 -12 -14 -16 -18 G F Q Z G

Fig. 2. Band structure of (8, 0) BC3 SWNT. The dashed line is the Fermi energy level.

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 counter part s electrons. The con-

tribution of s electrons in both the cases in the conduction band is meagre. In B3C 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 signi?es the metallicity character of B3C. In Fig. 4a, the variation of Fermi energy with B doping is indicated. It is observed that with increase of B doing,

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Density of States (electrons/eV)

17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 -1 -18 -17 -16 -15 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

s p sum

Energy (eV)

Fig. 3. Partial density of states (PDOS) of (8, 0) B3C SWNT and the dashed line is the position of the Fermi energy level. The energies are measured with respect to this Fermi level.

6.5

1.0

6.0 Fermi Energy EF (eV) 5.5 5.0 4.5 4.0 3.5

Band Gap at Γ (eV)

Simulation Data Exponential Fit

0.9 0.8 0.7 0.6 0.5 0.4

Simulation Data Polynomial Fit

3.0 0.0 0.2 0.4 0.6 0.8 1.0 Boron Doping Concentration

0.0

0.2

0.4

0.6

0.8

1.0

Boron Doping Concentration

Fig. 4. Variation of Fermi energy with (a) B doping concentration (b) typical variation of the band gap at C point with B doping. The energy cut-o?, k-point sampling, geometry, GGA/norm-conserving pseudo-potential has been kept constant.

the Fermi Energy decreases exponentially. This is understood simply from the fact that the electronic con?guration of B atom is 1s2 2s22p1. Therefore, doping by B atom always reduces the total no. of electrons N0 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 multi-walled carbon nanotubes [30,31]. Now, we concentrate here on the band gap in the most symmetric point of the BZ. In Fig. 4b, we show the sche-

matic variation of the band gap at C point of the BZ with B doping. A polynomial ?t to the data obtained from the band structure calculations reveals that the minimum of the band gap is obtained at some critical doping concentration ($55%). 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.

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3.2. Study of dielectric constant of BxCy system We compute the imaginary part of the dielectric constant within the speci?ed frequency range for three types of CNT namely pure metallic, semiconductor and quasimetallic. It was suggested [21] that because of the presence of the density of scatterers in the super-cell, the imaginary dielectric constant needed to be renormalized. However, we do not account such renormalization in our calculation. Further work is required to justify this renormalization procedure. In Fig. 5, we schematically show the dielectric constant (real as well as imaginary) for both pure (8, 0) and BC3 doped system as a function of frequency. It is evident 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 Eq. (1) used for this simulation study. The square of the matrix element and the even functional nature of the energy conserving delta function ensure the positivity of e2. This property of e2 serves as one of the cross-checks in our numerical computation. However, as evident from the ?gure itself that such a restriction is not obeyed by the real part of the dielectric constant e1. We also note that the static value (strictly speaking x ! 0, but in our numerical computation x = 0.0150 Hz.) of the dielectric constants for both pure and doped system is always positive. This observation is satis?ed by a theorem in continuous media stating that the static electric dielectric constant is always positive [32] for any material in thermal equilibrium. The variation of static dielectric constant with concentration of B has been reported recently [33] to show that a small concentration is enough to change the value drastically from the pure (8, 0) SWCNT. It is evident from Fig. 5 that the static value of the dielectric constant (real as well as imaginary) of BC3

system is higher compared to pure one. The an-isotropic behavior with respect to various ?avors of CNT and electro magnetic ?eld is summarized in Tables 1 and 2 in the static values of the dielectric constants of pure and doped system (replacing one of the carbon atom by B atom) at 0.0150 Hz. We note that the change of value of static dielectric constant depends on both polarizations as well as on the nature of CNT. The parallel polarization refers to the direction of light parallel to the axis of the CNT. For example, in pure case, the static values for parallel polarization for (7, 0) and (9, 0) are higher than their perpendicular polarization counterparts while the reverse is true for (5, 5) and (6, 6). For pure metallic tubes (5, 5) or (6, 6), the doping increases the static dielectric constant for parallel polarization but decreases its value for perpendicular polarization. The behavior of (7, 0) and (10, 0) semiconductor CNT is intriguing with respect to doping as well as polarization. In this semiconductor case, the doping lowers the static values for parallel as well for unpolarized light while enhances the value a small amount in perpendicular situation. In case of semiconducting SWCNT, an ab initio tight-binding calculation [34,35] relates the static value of the dielectric constant with the energy band gap as

Table 1 Un-polarized light with incidence direction (1, 0, 0) Nature of CNT/diameter (nm) Static e1(x) Pure (5, 5)/0.6783 (6, 6)/0.8140 (7, 0)/0.5483 (10, 0)/0.7833 (9, 0)/0.7049 7.301 7.852 43.408 43.408 11.760 Doped 8.588 10.140 28.968 26.094 72.039 Static e2(x) Pure 0.059 0.229 6.278 6.790 1.006 Doped 0.083 0.326 4.698 4.065 21.365

40

40

Dielectric Function

30

30

Dielectric Function

ε1 ε2

20

20

ε1 ε2

10

10

0

0

-10 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

-10 0 2 4 6 8 10 12 14 16 18 20 22 24 26

Frequency (eV)

Frequency (eV)

Fig. 5. Typical variation of dielectric constants of (a) pure (8, 0) and (b) BC3 nanotube in parallel polarization as a function of frequency (x).

D. Jana et al. / Carbon 45 (2007) 1482–1491 Table 2 Parallel polarization and perpendicular polarization Nature of CNT/diameter (nm) Parallel polarization Static e1(x) Pure (5, 5)/0.6783 (6, 6)/0.8140 (7, 0)/0.5483 (10, 0)/0.7833 (9, 0)/0.7049 6.487 5.889 77.112 78.582 16.596 Doped 9.327 11.282 40.982 40.146 132.391 Static e2(x) Pure 0.021 0.016 11.948 13.321 1.686 Doped 0.081 0.340 6.677 6.737 41.321 Perpendicular polarization Static e1(x) Pure 9.313 11.471 7.752 9.802 5.571 Doped 8.705 9.836 11.901 11.947 10.341 Static e2(x) Pure 0.199 0.688 0.165 0.422 0.142

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Doped 0.109 0.356 1.097 1.164 0.978

Table 3 Study of minima of amax Polarization Parallel Perpendicular Unpolarized Minimum value of amax (m?1) 6.48 · 106 8.53 · 106 7.95 · 106 Respective boron doping concentration 0.400 0.290 0.400

e1 ?0? ? 1 ?

?xp ? h

2

?5:4Eg ?2

?2?

Here xp is the plasma frequency and Eg is the energy band gap. Based on this equation, a strong upper bound [36] to the static dielectric constant of a semiconductor SWCNT was suggested as e1 ?0? < 5 ?3?

come back to this point when we will discuss the loss function of this quasi-metallic system. A substantial order of magnitude of increase of the static dielectric constant in quasi-metallic case is remarkable in all polarization directions even with unpolarized light having incidence direction (1, 0, 0). This feature of quasi-metallic SWCNT can be used to distinguish it from semiconductor or metallic SWCNT. The reason may be due to presence of small band gap along with the increase of free charge carrier in the doped system. All the other optical quantities such as re?ectivity, refractive index [38] and absorption coe?cient can be obtained from the dielectric constant. Below we present the variation of the absorption coe?cient and the Loss function that suggests the typical nature of collective excitations of the system. 3.3. Study of the absorption spectra of the doped system The absorption coe?cient a is related to the imaginary part of the dielectric constant as a? e2 x nc ?4?

However, in our numerical calculation, the static (real) dielectric constants of all the various CNTs having diameter less than 1 nm violate the above inequality as noticed from Tables 1 and 2. This has been also noticed in another ?rst principles calculation [21] of the optical properties of ? CNTs having diameter 4 A. Moreover, Eq. (3) indicates in?nite static dielectric constants for pure and quasi-metallic CNT. The in?nite value is intuitively expected in view of the conducting nature [34] of the available free electrons in CNT. However, we get ?nite positive values for pure and quasi-metallic CNT along with their doped counterparts. We believe that the ?niteness of the static values arises due to non-zero positive values of band gaps of all ?avors of CNT and the small diameter of the CNT. For quasimetallic tubes such as (9, 0) replacing one of the carbon atoms in hexagonal network by one B atom always enhances the static dielectric constant value independent of polarization. In other words, the value of static dielectric susceptibility is increased in doped case for this particular type of CNT. A simple estimation using Eq. (3) based on the plasma frequency and the band gap at C point for (7, 0) semiconductor CNT predicts 87.22 and 51.58 for pure and doped case, respectively. These values agree quite reasonably with the values for parallel polarization in Table 2. The similar calculation of the (real) static dielectric constant for (9, 0) pure tube taking into account the band gap [37] of 0.08 eV yields 6401 that is quite high compared to the ab initio calculation value shown in Table 2. We will

where n and c are the refractive index and the speed of light, respectively. The absorption spectra depend crucially on the nature of CNT and 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 signi?cantly. The appearance of several peaks in the absorption spectra in perpendicular and unpolarized one makes the analysis little bit complicated. For our convenience, we concentrate on the maximum value of the absorption coe?cient in all three cases. We depict in Fig. 6a the dramatic variation in absorption coe?cient with doping concentration. A simple polynomial ?t for parallel polarization suggests a parabolic type of dependence with doping having a minimum of magnitude 6.48 · 106 m?1 at a concentration of 0.40. In other words, for electromagnetic light parallel to axis of (8, 0) CNTs, below 0.40 concentration amax decreases with doping concentration while above this (0.40) it increases with concentration. We also observe the same type of behavior as indicated respectively in Fig. 6a and b for perpendicular

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2.4x10 7 2.1x10

7

D. Jana et al. / Carbon 45 (2007) 1482–1491

2.0x10 1.8x10 1.6x10

7

a

Parallel Polarization Perpendicular Polarization Polynomial Fit of data Polynomial Fit of data

7

b

Unpolarized Light Polynomial fit of data

7

1.8x10 7

αmax(m )

1.5x10 7 1.2x10 7 9.0x10 6 6.0x10

6

αmax(m )

-1

-1

1.4x10 1.2x10 1.0x10 8.0x10

7

7

7

6

0.0

0.2

0.4

0.6

0.8

1.0

6.0x10

6

0.0

0.2

0.4

0.6

0.8

1.0

Boron Doping Concentration

Boron Doping Concentration

Fig. 6. Variation of the maximum value of the absorption coe?cient (amax) with B doping concentration in (a) parallel polarization as well as perpendicular polarization and (b) unpolarized light.

polarization and un-polarized light. We notice in Fig. 6a the maximum value of the absorption coe?cient amax ranges from 1.65 · 107 m?1 to 2.17 · 107 m?1. These values, however, vary for perpendicular as well as unpolarized light. The values of the B doping concentration at which the minimum of amax occurs however di?er very slightly from parallel polarization case as shown in Table 3. 3.4. Study of the loss function of the doped system The loss function, which is a direct measure of the collective excitations of the systems, is de?ned as Im[?1/ e(q, x)]. Since we are taking the q ! 0 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 nano-electron energy loss spectroscopy (nano-EELS) can provide information about the systematics and atomic structural defects of B-doped

SWCNTs [39–41]. However, here we are interested in the variation of the peak position as one replaces the carbon atom(s) by B atom(s). We show in Fig. 7 the typical variation of Loss function for parallel polarization in case of pure and doped semiconductor (8, 0) SWNT. Interestingly, we note the appearance of single peak in this pure as well as doped CNT in contrast to multiple peaks in metallic pure and doped system. This represents a unique collective mode of excitation in parallel polarization only. It is also evident from the ?gure that the magnitude of the peak of the loss function and its position is modi?ed in the doped case. The single peak at 9.73 eV, however, shifts to 9.78 eV on B doping on the CNT. This appearance of single peak (9.5–10 eV) at long wavelength limit (q ! 0) may be attributed to the typical unique collective excitation of p electrons. This value can be compared with the values [35] obtained for p plasmons at 5.2 eV peak and r + p plasmons at 21.5 eV for wave vec? tor of 0.15 A. The shifting of the peak towards lower frequency can be attributed to the reduction of plasma

12 11 10 9 8 7 6 5 4 3 2 1 0 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

12 11

Loss Function

10 9 8 7 6 5 4 3 2 1 0 0 2 4 6 8

Loss Function

10 12 14 16 18 20 22 24 26 28

Frequency (eV)

Frequency (eV)

Fig. 7. Loss function of semiconductor (8, 0) (a) pure and (b) BC3 SWNT for parallel polarization.

D. Jana et al. / Carbon 45 (2007) 1482–1491

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frequency due to the decrease of total number of electrons on doping. This is evident from the fact that the Fermi energy also decreases with B doping in the system (see Fig. 4a). In perpendicular polarization, the appearance of multiple peaks in the Loss function implies the existence of various collective excitations involving r and p electrons in the system. This can be taken as one of the characteristic features of any type of SWCNTs in perpendicular polarization. Because of the existence of several peaks, it is however di?cult to analyze systematically their behavior with doping concentration. The single peak also appears in quasimetallic (9, 0) tube in parallel polarization. For example, in case of quasi-metallic tube (9, 0), the pure peak at 9.062 eV is shifted to higher frequency of 9.575 eV upon doping. Even the magnitude of the Loss function is increased in doped case. This may be attributed to increase of free charge carriers. In fact, taking into this plasma resonance frequency and the band gap at C point, we ?nd the static dielectric constant for pure and doped case as 21.29 and 92.42, respectively. These values are reasonable with the values shown in Table 1. Thus, we conclude that even a small percentage (3.125%) of B doping can signi?cantly modify the collective excitations of the pure system under various polarization directions. We depict in Fig. 8, the typical variation of the plasma frequency of doped (8, 0) SWNT computed from the Loss function in parallel polarization as well as unpolarized light with the B concentration. Similar to amax, with increase of B doping the plasma resonance frequency ?rst decreases and then increases after some critical concentration. This implies that there exists a unique concentration ($0.44) at which the minimum value of the plasmon frequency is obtained. It may be noted that at the same concentration, both the absorption coe?cient as well as the plasmon frequency assume their respective minimum value. In case of unpolarized light, we observe several peaks up to 50% B doping, above this doping, only a single peak with a shoulder. In such a situation, we have concentrated on the main

peaks (i.e. peaks having highest magnitude of Loss function) positions up to 50% doping for calculating the plasma resonance frequency. It is to be noted that the minimum value of xp and the corresponding B doping do depend on the nature of electromagnetic ?eld. For example, we ?nd the minimum value of the plasma frequency (the corresponding doping concentration) for parallel polarization at 6.94 eV (0.44) while for un-polarized light at 6.64 eV (0.33). The study also demonstrates that the concentration at which the minimum value of xp occurs is large in parallel polarization in compared to un-polarized case. With these plasma frequencies and the band gap at C point taken from Fig. 4b, we use Eq. (3) to calculate the static real dielectric constant. However, it is to be remembered that Eq. (3) is strictly valid for pure semiconductor SWNT. These computed values are compared in Fig. 9 with simulated ab intio values for parallel polarization. From this ?gure, we note that Eq. (3) predicts the largest value for pure (8, 0) SWCNT while the simulation suggests

1200

16

Ab initio Simulation From equation (3) Polynomial fit of data

800

ε 1(0)

12

8

400

4 0.0 0.2 0.4 0.6 0.8 1.0

0 0.0 0.2 0.4 0.6 0.8 Boron Doping Concentration 1.0

Fig. 9. Comparison of the ab initio static dielectric constant values with Eq. (3). The inset is the expanded version of the values calculated from Eq. (3).

20 12

Parallel Polarization Unpolarized Light Polynomial Fit of data Polynomial Fit of data

Plasma Frequency ω p (eV)

11 10 9 8 7 6

Plasma Frequency ω p (eV)

18

Perpendicular Polarization Linear Fit of data

16

14

12

10 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Boron Doping Concentration

Boron Doping Concentration

Fig. 8. Variation of the plasma resonance frequency with B doping concentration in (a) parallel polarization as well as for unpolarized light and (b) perpendicular polarization.

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the other extreme case i.e. doped one. All the calculated values from Eq. (3) are less than that of the simulations one. Moreover, all these values violate the upper bound restriction as predicted for pure case [36]. Similar behavior has been observed for un-polarized case also. Though the results presented above are based on some speci?c choice of parameters, however, the qualitative gross features of the optical quantity remain unaltered with the change of parameters. 4. Conclusions and perspectives From the ?rst principles relaxed C–C bond length DFT calculation of the optical property of BxCy SWNT systems, we have observed signi?cant changes in the optical behavior for di?erent CNT systems (radius < 1 nm) with di?erent polarizations. The behavior of the static dielectric constant of B doped system depends on the ?avor (nature) of the CNTs. The anisotropy signatures of the dielectric constants noticed in these systems are due to the con?ned geometry of the CNTs. In all three cases of the incident electromagnetic wave (i.e. parallel polarization, perpendicular polarization and unpolarized light) the maximum value of the absorption coe?cient varies signi?cantly with B doping concentration indicating a unique minimum (0.40 for parallel as well as unpolarized while 0.29 for perpendicular one). It is observed that the peak of the loss function in parallel polarization and unpolarized cases shifts to a lower frequency with increasing concentration up to 50% but then shifts to a higher frequency. Also, the non-linear ?ts to the plasma resonance frequency variation with B concentration indicate the existence of a unique minimum. Acknowledgements One of the authors (DJ) would like to thank the National Science Council (NSC) of the Republic of China for ?nancially supporting him as a visiting researcher under Contract No. NSC93-2811-M-002-034. Thanks to ChunLiang Yeh (IAMS and NCU) for presenting this work in ICON-05 as a poster. Discussions with Dr. C.L. Sun and Dr. Yue-Lin Huang are gratefully acknowledged. References

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