A first principles study of the optical properties of BxCy single wall nanotubes

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A ﬁrst principles study of the optical properties

of B

x

C

y

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 coefficient 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 field. The peak of the loss func-tion in parallel polarizafunc-tion and unpolarized cases shifts to a lower frequency with increasing concentrafunc-tion up to 50% but then shifts to a higher frequency. The non-linear fits 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.

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 impor-tance of substitutional alloying of impurity atoms such as boron, nitrogen have been discussed [6]. In fact, a

calculation on the chemical interaction reveals that the boron doped CNT can act as a novel sensor[7]for formal-dehyde. The synthesis of composite BxCyNztubes 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 litera-ture, the synthesis and electronic properties of B-substi-tuted 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 sp2conﬁguration[14]. It has also been shown recently that even a small amount dopant can signiﬁcantly 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

*

Corresponding authors. Fax: +91 (0) 3323509755 (D. Jana); fax: +886 2 23655405 (L.-C. Chen).

E-mail addresses:[email protected](D. Jana),[email protected]

(L.-C. Chen).

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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, diame-ter and the nature of polarizations of incident electromag-netic field. In this paper, we study the response of the B alloyed CNTs (pure metallic, semiconductor and quasi-metallic) under the action of a uniform electric field with various polarizations direction through relaxed C–C bond length ab initio density functional theory (DFT) calcula-tions. 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 first principles computa-tion, we have only one fixed 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 nat-ure of its variation with chemical doping of B concentra-tion in a semiconductor (8, 0) nanotube.

2. Numerical methods

The optical properties of any system are generally studied by the com-plex dielectric function deﬁned by ~_{DðxÞ ¼ eðxÞ~}_{EðxÞ ¼ ½e}₁_ðxÞþ ie2ðxÞ~EðxÞ. 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 ﬁrst-order time dependent perturbation the-ory. In the simple dipole approximation used in CASTEP code[25], the imaginary part is given by

e2ðq ! 0~u; hxÞ ¼ 2e2_p Xe0 X k;V ;C jhwC kj~u~rjw V kij 2 dðEC k E V k EÞ ð1Þ

Xand e0represent respectively the volume of the super-cell and the dielec-tric constant of the free space. The sum over k is a crucial point in numer-ical 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, ~u;~rrespectively represent the polarization vector of the incident electric ﬁeld 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 ﬁeld ef-fect 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

computing the optical properties was kept ﬁxed 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· 106_{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 BxCysystem

Before we discuss the optical properties, we show in

Fig. 1 the typical ball and stick model of pure (8, 0) and

BC3system. All the results presented in this numerical

cal-culation have same set of parameters as indicated in earlier section. InFig. 2, we schematically show the band structure of BC3system respectively. All the energies shown in the

diagram have been measured with respect to the Fermi energy. For pure (8, 0) we ﬁnd 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) nano-tubes, the Fermi energy reduces to 4.256 eV with overlap-ping 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 ﬁnd for (8, 0) B3C

nanotubes, a further reduction of the Fermi energy to 3.614 eV. More interestingly, we note that a signiﬁcant increase of the overlapping of valence and conduction band in compared to pure as well as BC3SWNT. 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 figure) 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 flatness of the band at various k-points seems to contribute signifi-cantly to the optical absorption. The partial density of states (PDOS) of (8, 0) B3C carbon nanotubes shown in

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of band energy and these are basically the van Hove singu-larity 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 com-pared to pure case. In fact, the higher value of DOS at the Fermi level signiﬁes 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,

Fig. 1. Ball and stick model of (a) pure (8, 0) and (b) BC3tube in 3d triclinic structure.

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

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the Fermi Energy decreases exponentially. This is under-stood simply from the fact that the electronic conﬁguration of B atom is 1s2 2s22p1. Therefore, doping by B atom always reduces the total no. of electrons N0in 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. InFig. 4b, we show the

sche-matic variation of the band gap at C point of the BZ with B doping. A polynomial ﬁt to the data obtained from the band structure calculations reveals that the minimum of the band gap is obtained at some critical doping concentra-tion (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.

-1 0 1 2 3 4 5 6 -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 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.

0.0 0.2 0.4 0.6 0.8 1.0 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Band Gap at Γ (eV)

Boron Doping Concentration Simulation Data Polynomial Fit 0.0 0.2 0.4 0.6 0.8 1.0 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 Simulation Data Exponential Fit Ferm i Energy EF (eV)

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-oﬀ, k-point sampling, geometry, GGA/norm-conserving pseudo-potential has been kept constant.

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3.2. Study of dielectric constant of BxCysystem

We compute the imaginary part of the dielectric con-stant within the speciﬁed frequency range for three types of CNT namely pure metallic, semiconductor and quasi-metallic. 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. InFig. 5, we schematically show the dielectric constant (real as well as imaginary) for both pure (8, 0) and BC3doped system as a function of frequency. It is

evi-dent that in both cases, the imaginary part of the dielectric constant is always positive throughout the range of fre-quency. This can be understood very simply from Eq.(1)

used for this simulation study. The square of the matrix ele-ment and the even functional nature of the energy conserv-ing 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 ﬁgure 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 satisﬁed 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 fromFig. 5that 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 ﬂavors of CNT and electro magnetic ﬁeld is summarized inTables 1 and 2in 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 con-stant depends on both polarizations as well as on the nat-ure of CNT. The parallel polarization refers to the direction of light parallel to the axis of the CNT. For exam-ple, 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) semiconduc-tor 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

-10 0 10 20 30 40 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 Dielectric Function 1

ε

2 1

ε

2 Frequency (eV) -10 0 10 20 30 40 0 2 4 6 8 10 12 14 16 18 20 22 24 26 Dielectric Function Frequency (eV)

Fig. 5. Typical variation of dielectric constants of (a) pure (8, 0) and (b) BC3nanotube in parallel polarization as a function of frequency (x). Table 1

Un-polarized light with incidence direction (1, 0, 0)

Nature of CNT/diameter (nm) Static e1(x) Static e2(x) Pure Doped Pure Doped

(5, 5)/0.6783 7.301 8.588 0.059 0.083

(6, 6)/0.8140 7.852 10.140 0.229 0.326

(7, 0)/0.5483 43.408 28.968 6.278 4.698 (10, 0)/0.7833 43.408 26.094 6.790 4.065 (9, 0)/0.7049 11.760 72.039 1.006 21.365

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e1ð0Þ ¼ 1 þ

ðhxpÞ2

ð5:4EgÞ

2 ð2Þ

Here xpis the plasma frequency and Egis 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

e1ð0Þ < 5 ð3Þ

However, in our numerical calculation, the static (real) dielectric constants of all the various CNTs having diame-ter less than 1 nm violate the above inequality as noticed

fromTables 1 and 2. This has been also noticed in another

first principles calculation[21] of the optical properties of CNTs having diameter 4 A˚ . Moreover, Eq. (3) indicates infinite static dielectric constants for pure and quasi-metal-lic CNT. The infinite value is intuitively expected in view of the conducting nature[34]of the available free electrons in CNT. However, we get finite positive values for pure and quasi-metallic CNT along with their doped counterparts. We believe that the finiteness of the static values arises due to non-zero positive values of band gaps of all flavors of CNT and the small diameter of the CNT. For quasi-metallic tubes such as (9, 0) replacing one of the carbon atoms in hexagonal network by one B atom always en-hances 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 rea-sonably with the values for parallel polarization inTable 2. The similar calculation of the (real) static dielectric con-stant 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 inTable 2. We will

quasi-metallic case is remarkable in all polarization direc-tions 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 reflectivity, refractive index [38] and absorption coefficient can be obtained from the dielectric constant. Below we present the variation of the absorption coefficient and the Loss function that suggests the typical nature of collective exci-tations of the system.

3.3. Study of the absorption spectra of the doped system The absorption coeﬃcient a is related to the imaginary part of the dielectric constant as

a¼e2x

nc ð4Þ

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 maxi-mum absorption at that particular energy. With doping by B atom(s), both the magnitude of the peaks and its po-sition change signiﬁcantly. The appearance of several peaks in the absorption spectra in perpendicular and unpolarized one makes the analysis little bit complicated. For our con-venience, we concentrate on the maximum value of the absorption coeﬃcient in all three cases. We depict in

Fig. 6a the dramatic variation in absorption coeﬃcient

with doping concentration. A simple polynomial ﬁt for parallel polarization suggests a parabolic type of depen-dence with doping having a minimum of magnitude 6.48· 106m1 at a concentration of 0.40. In other words, for electromagnetic light parallel to axis of (8, 0) CNTs, be-low 0.40 concentration amaxdecreases with doping

concen-tration while above this (0.40) it increases with concentration. We also observe the same type of behavior as indicated respectively inFig. 6a and b for perpendicular

(m1) concentration

Parallel 6.48· 106 _0.400 Perpendicular 8.53· 106 0.290 Unpolarized 7.95· 106 0.400

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polarization and un-polarized light. We notice in Fig. 6a the maximum value of the absorption coeﬃcient amax

ranges from 1.65· 107

m1 to 2.17· 107

m1. These val-ues, however, vary for perpendicular as well as unpolarized light. The values of the B doping concentration at which the minimum of amax occurs however diﬀer very slightly

from parallel polarization case as shown inTable 3. 3.4. Study of the loss function of the doped system

The loss function, which is a direct measure of the col-lective excitations of the systems, is deﬁned as Im[1/ e(q, x)]. Since we are taking the q! 0 limit in our calcula-tion, therefore we are considering the loss function behav-ior under the long wavelength limit. The peak position of this loss function determines the typical energy of the plas-mons in the system. High-resolution transmission electron microscopy (HRTEM) and nano-electron energy loss spec-troscopy (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 inFig. 7the typical variation of Loss function for parallel polarization in case of pure and doped semi-conductor (8, 0) SWNT. Interestingly, we note the appear-ance of single peak in this pure as well as doped CNT in contrast to multiple peaks in metallic pure and doped sys-tem. This represents a unique collective mode of excitation in parallel polarization only. It is also evident from the ﬁg-ure that the magnitude of the peak of the loss function and its position is modiﬁed 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 fre-quency can be attributed to the reduction of plasma

0.0 0.2 0.4 0.6 0.8 1.0 6.0x106 8.0x106 1.0x107 1.2x107 1.4x107 1.6x107 1.8x107 2.0x107 αma x (m -1 ) αma x (m -1 )

Boron Doping Concentration Boron Doping Concentration

Unpolarized Light Polynomial fit of data

0.0 0.2 0.4 0.6 0.8 1.0 6.0x106 9.0x106 1.2x107 1.5x107 1.8x107 2.1x107 2.4x107 Parallel Polarization Perpendicular Polarization Polynomial Fit of data Polynomial Fit of data

a

b

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

0 1 2 3 4 5 6 7 8 9 10 11 12 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 Frequency (eV) Loss Function 0 1 2 3 4 5 6 7 8 9 10 11 12 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 Frequency (eV) Loss Function

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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 res-onance frequency and the band gap at C point, we ﬁnd the static dielectric constant for pure and doped case as 21.29 and 92.42, respectively. These values are reasonable with the values shown inTable 1. Thus, we conclude that even a small percentage (3.125%) of B doping can signiﬁcantly modify the collective excitations of the pure system under various polarization directions.

We depict inFig. 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 ﬁrst 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 coeﬃcient as well as the plasmon fre-quency 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 shoul-der. In such a situation, we have concentrated on the main

bered 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 ﬁgure, we note that Eq. (3) predicts the largest value for pure (8, 0) SWCNT while the simulation suggests

10 12 14 16 18 20 Plasma Frequency p (eV) Perpendicular Polarization Linear Fit of data

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 6 7 8 9 10 11 12 ω Plasma Frequency p (eV) ω

Boron Doping Concentration Boron Doping Concentration

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

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. 0.0 0.2 0.4 0.6 0.8 1.0 0 400 800 1200 ε1 (0)

Boron Doping Concentration

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

0.0 0.2 0.4 0.6 0.8 1.0

4 8 12 16

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.

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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 speciﬁc 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 ﬁrst principles relaxed C–C bond length DFT calculation of the optical property of BxCySWNT systems,

we have observed significant changes in the optical behav-ior for different CNT systems (radius < 1 nm) with 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. In all three cases of the incident electromag-netic wave (i.e. parallel polarization, perpendicular polari-zation and unpolarized light) the maximum value of the absorption coefficient varies significantly with B doping concentration indicating a unique minimum (0.40 for par-allel 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 fits to the plasma resonance frequency variation with B con-centration indicate the existence of a unique minimum. Acknowledgements

One of the authors (DJ) would like to thank the Na-tional Science Council (NSC) of the Republic of China for ﬁnancially supporting him as a visiting researcher under Contract No. NSC93-2811-M-002-034. Thanks to Chun-Liang 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.

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