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Infinite single-walled boron-nitride nanotubes studies by LGTO-PBC-DFT method
Diamond & Related Materials 18 (2009) 351–354
Contents lists available at ScienceDirect
Diamond & Related Materials j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / d i a m o n d
Infinite single-walled boron-nitride nanotubes studies by LGTO-PBC-DFT method Yu-Ma Chou a,⁎, Houng-Wei Wang b, Yih-Jiun Lin c, Wen-Hao Chen c, Bo-Cheng Wang c,⁎ a b c
Department of Physics, Chinese Culture University, Taipei 110, Taiwan Center for Condensed Matter Sciences, National Taiwan University, Taipei 106, Taiwan Department of Chemistry, Tamkang University, Tamsui 251, Taiwan
a r t i c l e
i n f o
Available online 5 November 2008 Keywords: BN nanotube Zigzag Armchair PBC DFT
a b s t r a c t Localized Gaussian type orbital-periodic boundary condition-density functional theory (LGTO-PBC-DFT) method was used to determine the electronic and detailed geometrical structures of (n, 0) zigzag type for n = 6 – 33 and (n, n) armchair type for n = 3 – 15 single-walled boron-nitride (BN) nanotubes with infinite tubular lengths. The calculations reveal that the calculated Eg (band gap between HOCO and LUCO) increases with increasing tubular diameter and eventually converge to 5.03 eV for BN nanotubes of larger tubular diameter. According to the calculated Egs, the BN nanotubes are semi-conductor and their conductivities are not sensitive to the tubular diameter. Theoretically, the calculated bond length decreases with increasing tubular diameter. Based on our calculations, the bond length and angle do converge to 1.45 Å and 120 degree, respectively. Thus, the structures of BN nanotubes with the infinite tubular length approach the perfect hexagonal network when the tubular diameter increases. The calculated results also indicate that zigzag BN nanotubes with the tubular diameter larger than 18 Å display 3n properties in the calculated Eg, which is also obtained for zigzag carbon nanotubes. © 2008 Elsevier B.V. All rights reserved.
1. Introduction Since the discovery of carbon fullerene C60 by Smalley et al. in 1985 and carbon nanotubes by Iijima in 1991, the unusual electronic and structural properties of carbon nanotubes have been received the increasing attention of chemists and physicists [1–4]. Being closer neighbors of carbon in the Periodic Table, boron-nitride (BN) nanotube with proper alternative substituted of carbon having similar structures and mechanical properties are viewed as ideal analogs of carbon nanotubes. Recently, the experiments have shown that BN nanotubes contain unique properties and could be applied in various branch of material science [5]. The electronic properties of carbon nanotubes vary widely with their lengths, diameter and chirality [6–8]. Carbon nanotubes can be either metal or semi-conductor and are potentially used in a wide variety of applications [9–11]. In 1994, Rubio et al. predicted the existence of BN nanotubes by theoretical computations [12,13]. According to the earlier quantum mechanical calculations, the BN nanotubes are hexagon-like with the same resemblance to carbon nanotubes; where the band gap energy is about 5.5 eV [14]. The BN nanotubes have been synthesized and shown to be the electrical insulators that are independent of their tubular length, diameter, and chirality [15]. Chopra and Ma successfully synthesized multi-layered ⁎ Corresponding authors. E-mail addresses: [email protected] (Y.-M. Chou), [email protected] (B.-C. Wang). 0925-9635/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.diamond.2008.10.026
BN nanotubes via the non-equilibrium plasma-arc and chemical vapor deposition (CVD) methods, respectively [16]. Then, TEM and HREM images indicate that multi-layered BN nanotubes display hexagonal networks with 7 to 9 concentric cylinder-like layers without fullerenelike caps [17–21]. In 2001, Bengu and Marks proposed a new method to synthesize the single-walled BN nanotubes [22]. This new synthesis technique has made it possible to bring BN nanotubes into the nanotechnology applications. In recent studies on BN nanotubes, several papers have been presented and used first-principle to investigate the small radius single-walled BN nanotubes with open end [15,23]. Charlier et al. predicted the geometric structures of BN nanotubes based on their formation mechanism [24]. Later reports explained their mechanical properties with theoretical computation methods [25]. Very recently, Zhang et al performed ab initio and DFT calculations to study the effect of chemisorption on the BN nanotubes [26,27]. Although higher level calculations have been done for the small segment of single-walled nanotubes with various diameters, they did not provide the real model of the BN nanotubes. In order to investigate the infinite system, the DFT calculation with Gaussian molecular orbital length may be used and extend it to infinite tubular length with periodic boundary condition (PBC) model. Scuseria et al. employed PBC-DFT with localized Gaussian type orbitals (LGTO) to generate the optimized structures of (5, 0) zigzag carbon nanotube [28]. Quite recently, the infinite carbon nanotubes with zigzag and armchair types have been simulated by used the LGTO-PBC-DFT calculation and the detailed geometrical structures have been reported [8]. Thus, we have
352
Y.-M. Chou et al. / Diamond & Related Materials 18 (2009) 351–354 Table 1 Calculated HOCO, LUMO and Eg of zigzag type nanotubes by VSXC/6–31G(d) calculation level with PBC-DFT method
Fig. 1. Morphological structures of (6, 0) zigzag BN nanotube with the PBC-DFT method.
confidence that the LGTO-PBC-DFT method is a suitable approach to determine the physical properties of nanotube with infinite tubular length. In this paper, the investigation of BN nanotubes is focused on the zigzag and armchair types. LGTO-PBC-DFT calculation was performed on these systems at pure DFT functional (PBE, VSXC, etc.) level with 6– 31G(d) basis sets and PBC model. The test calculations were carried out by using several functionals including hybrid function and basis sets. The optimized-geometrical structures, electronic structures and related band gap of (n, 0) zigzag (6 ≤ n ≤ 33) and (n, n) armchair (3 ≤ n ≤ 15) single-walled BN nanotubes with infinite tubular length were generated. 2. Calculation details Conveniently, BN nanotube could be presented by a (n, m) pair of numbers; where the (n, 0) and (n, n) BN nanotubes designate the zigzag and armchair types, respectively. For the zigzag type BN nanotube, n denotes the number of triazines in the circumference of the tube and the translation axis is the tubular length. For the PBC-DFT calculation of single-walled BN nanotube, we start with the single layer for the unit cell and extend it along the tubular axis to infinite length by using the PBC model (Fig. 1). The PBC-DFT method is implemented in the Gaussian 03 program revision C.02 program package [29]. The PBC model in the Gaussian 03 package is based on Gaussian type orbitals (GTOs) [30], Bloch function [31] was employed to transform GTOs into “crystalline orbitals” for calculating the periodic boundary condition systems [28,32–35]. For obtaining a high precision geometry, an extremely tight optimization convergence criteria was used, and an ultrafine was chosen for the integral grid option. In the present study, the energy gap between the energy of HOCO and that of LUCO corresponds to the band gap; we use the definition given by Eg = ELUCO − EHOCO. In order to examine the efficiency of various functionals and basis sets with the LGTO-PBC-DFT method, (9, 0) zigzag BN nanotube was optimized with various types of functionals (VSXC, BLYP, LSDA, PBEPBE, PW91PW91, and B3LYP.) by using 6–31G(d) basis set, and the corresponding total energy and band gap energies were computed. The VSXC functional calculation that gives the lowest calculated total energy and larger relative energy as compared to those calculated by BLYP, LSDA, PBEPBE, PW91PW91 and B3LYP. The calculated total energies by BLYP and B3LYP are quite close to those calculated with VSXC. The calculated Eg by BLYP, LSDA, PBEPBE and PW91PW91are very closed each other and lower
Fig. 2. Morphological structures of (5, 5) armchair BN nanotube with the PBC-DFT method.
(n, 0)
HOCO
LUCO
Eg
n=6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33
−5.9507 −6.0441 −5.9168 −5.8753 −5.9216 −5.8642 −5.8386 −5.8614 −5.8345 −5.8127 −5.8271 −5.8084 −5.7940 −5.8035 −5.7925 −5.7811 −5.7874 −5.7794 −5.7700 −5.7756 −5.7690 −5.7625 −5.7648 −5.7603 −5.7547 −5.7579 −5.7539 −5.7461
−2.9925 −2.3797 −2.0219 −1.7577 −1.5539 −1.4010 −1.2777 −1.1755 −1.0970 −1.0260 −0.9660 −0.9170 −0.8752 −0.8404 −0.8085 −0.7808 −0.7562 −0.7357 −0.7331 −0.7324 −0.7236 −0.7269 −0.7241 −0.7168 −0.7214 −0.7201 −0.7130 −0.7152
2.9582 3.6645 3.8949 4.1176 4.3677 4.4632 4.5609 4.6859 4.7375 4.7868 4.8611 4.8913 4.9188 4.9632 4.9840 5.0003 5.0312 5.0437 5.0368 5.0432 5.0454 5.0356 5.0407 5.0436 5.0333 5.0379 5.0408 5.0309
than those of VSXC and B3LYP. Also the basis set effects are considered in our calculation, the STO-3G, 3–21G, 6–31G(d), 6–311G (d), cc-pVDZ and D95 basis sets are performed with VSXC functional for the basis set dependence check. The calculated total energy and Eg are very closed while using the 6–31G(d), 6–311G(d), cc-pVDZ and D95 basis sets, where the total energy and Eg are around − 14.88 keV and 4.10 eV, respectively. In particular, the total energy shows strong basis set dependence while the band gap convergence was achieved with 6–31G(d). Unfortunately, they require a long computing time when use the basis sets 6–311G(d), cc-pVDZ and D95. Thus, considering both accuracy and computing time, we chose the 6–31G(d) basis set for the following calculation. 3. Results and discussions On the basis of the structure analysis, the prominent structural property of zigzag BN nanotubes with a small radius, as compared to carbon nanotubes, is the existence of buckling on the tubular surface, which resulted from boron and nitrogen moving slightly inward and outward of the nanotube axis, respectively. In particular, the structure of zigzag BN nanotube is different from that of armchair BN nanotube, where boron and nitrogen atoms determine the rings with different diameter along the circumference of the tube. For example, the optimized diameters of (6, 0) zigzag BN nanotube are 5.022 Å and 4.813 Å for nitrogen and boron rings, respectively; i.e., there is almost 0.2 Å diameter difference in the same tube. The calculated tubular diameters increase from 4.850 Å for the (6, 0) BN nanotube to 26.436 Å for the (33, 0) BN nanotube. Thus, the variation with tubular diameter is about 0.8 Å for each increment of n. For the calculated geometrical parameters of infinite BN nanotubes, there are three types B\N bonds (N1\B2, B2\N3 and N3\B4) and four types of B\N\B bond angles (A1, A2, A3 and A4) (Fig. 1). The bond length B2\N3 decreases while decreasing the tubular diameter. Contrarily, the N1\B2 and N3\B4 bond lengths increase with decreasing the tubular diameter. For the zigzag BN nanotubes (n ≧ 21), the calculated bond lengths N1\B2,
Y.-M. Chou et al. / Diamond & Related Materials 18 (2009) 351–354 Table 2 Calculated HOCO, LUCO and Eg of armchair type nanotubes by VSXC/6–31G(d) calculation level with PBC-DFT method (n, n)
HOCO
LUCO
Eg
n=3 4 5 6 7 8 9 10 11 12 13 14 15
− 5.9810 − 5.9211 − 5.8785 − 5.8573 − 5.8348 − 5.8194 − 5.8072 − 5.7969 − 5.7884 − 5.7811 − 5.7781 − 5.7690 − 5.7613
−1.2770 −1.1809 −0.9445 −0.8838 −0.7844 −0.7592 −0.7542 −0.7488 −0.7439 −0.7396 −0.7378 −0.7318 −0.7269
4.7040 4.7402 4.9340 4.9735 5.0504 5.0601 5.0531 5.0481 5.0445 5.0415 5.0403 5.0372 5.0344
and N3\B4 converge to almost the same magnitude 1.453 Å and B2\N3 keeps within 1.452 Å. Compared the calculated bond angle in the BN nanotube, the bond angle A4 displays a larger variation with the decreased in the tubular diameters (it increases from 110 degree for the (6, 0) BN nanotube to 119 degree for the (20, 0) BN nanotube) while the other calculated bond angles A1, A2 and A3 are around 120 degree. These effects may be explained by the hybridization and weakening of sp 2 bonds caused by the strong concaved curvature of the BN nanotube. The calculated bond length of zigzag BN nanotube along the tube circumference is slightly longer than those of along the tube axis for small radii nanotubes. Theoretically, the electrons can contribute to the hexagonal resonance due to the unpaired electron of the nitrogen atom and the nearby three boron atoms, where the electron density will be shared. This effect becomes much more obvious as the pπ resonance built in the hexagonal network of BN nanotube. Thus, nitrogen atom with a stronger electronic affinity should contribute less of its unpaired electron contributing to the hexagonal ring and produces a smaller pπ resonance in the hexagonal network. In addition, the unequal electronic distribution along the BN bonding causes variation of the bond length; BN bonds can causes the variation of the bond length. For example, BN bond along the tube axis has more electronic density than those of the BN bond along the tube circumference. Thus, the calculated Eg of BN nanotubes should be larger than those of carbon nanotubes [36,37]. The results indicate that the bond lengths N1\B2, B2\N3, and N3\B4 of armchair BN nanotubes (Fig. 2) decreased while the tubular diameter increased and the calculated bond angles A1, A2, A3, and A4 increased up to 120 degree while the tubular diameter increased. For the small radii (4, 4), (5, 5) and (6, 6) BN nanotubes, the calculated A2 values are under 118 degree which may have obvious buckling effect in the tube surface, since the sp3 distribution of nitrogen atom still produces a small bump around the nitrogen atom. Up to (11, 11) BN nanotube, the calculated bond lengths and angles indicated that these BN nanotubes have the perfect hexagonal network with the sp 2 bonding distribution and all of B\N bond lengths are kept around 1.453 Å. Thus, this structure is much closed to that of zigzag carbon nanotube with the same tubular size. Tables 1 and 2 show the calculated HOCO, LUCO and Eg for zigzag and armchair types BN nanotubes with the PBC-DFT calculation. For the calculated Eg of zigzag type BN nanotubes, the lowest calculated Eg is 2.96 eV for the (6, 0) BN nanotube and the calculated Eg increase by around 0.1 – 0.2 eV for each increment of n, and finally converge to around 5.04 eV when n ≥ 26. In particular, the zigzag BN nanotubes are shown to exhibits no obvious 3n oscillation property dependence in the calculated Eg, when the tubular diameter is less than 16.8 Å and n b 21. This observation is not the same as those of single-walled
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carbon nanotubes. By increasing the calculated diameter of BN nanotube up to 16.8 Å and n ≧ 21, the calculated Eg has the 3n oscillation property. Thus, the zigzag BN nanotube with larger diameter may have the carbon nanotube property. The trend of the calculated Eg increments is different from that of carbon nanotubes since the strong concaved curvature during the small diameter BN nanotube. The calculated Eg for armchair (n, n) increases while increasing the tubular diameter, and finally converges to 5.03 eV. Thus, the armchair type BN nanotubes are semiconductors and their calculated Eg have no obvious dependence on tubular diameters. Theoretically, the calculated Eg for both zigzag and armchair BN nanotubes converge to 5.03 eV, which is in good agreement with the experimental data of 5.5 eV [14]. 4. Conclusion The PBC-DFT calculations with the VSXC functional and 6–31G (d) basis set on the optimized-geometrical structures of BN nanotubes with infinite tubular length have revealed that there are significant differences between the BN and carbon single-walled nanotubes. The differences can be attributed to the added dimension of nitrogen and boron atoms moving outward and inward respectively on the tubular surface of BN nanotubes. The calculated Eg of the zigzag and armchair BN nanotubes were shown to converge to 5.03 eV as the tubular diameter increases. Our calculation results also agree very well with earlier experimental results that showed, for diameter about 50 nm, the energy gaps are between 4.5–4.8 eV [38]. Particularly, the BN nanotubes with infinite tubular length are semiconductors and their conductivities are independent on tubular diameters. The calculated average BN bond length and bond angle are about 1.45 Å and 120 degree for both the zigzag and armchair types. For the zigzag BN nanotubes with the tubular diameter larger than 16.8 Å and n ≧ 21, the calculated Eg shows the 3n oscillation property. The physical properties of these BN nanotubes seem to resemble those of carbon nanotubes. Acknowledgments We thank you Profs. Chhiu-Tsu Lin and Hsiu-Fu Hsu for reading the manuscript and the National Science Council of Taiwan for financial supporting this research. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
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Contents lists available at ScienceDirect
Diamond & Related Materials j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / d i a m o n d
Infinite single-walled boron-nitride nanotubes studies by LGTO-PBC-DFT method Yu-Ma Chou a,⁎, Houng-Wei Wang b, Yih-Jiun Lin c, Wen-Hao Chen c, Bo-Cheng Wang c,⁎ a b c
Department of Physics, Chinese Culture University, Taipei 110, Taiwan Center for Condensed Matter Sciences, National Taiwan University, Taipei 106, Taiwan Department of Chemistry, Tamkang University, Tamsui 251, Taiwan
a r t i c l e
i n f o
Available online 5 November 2008 Keywords: BN nanotube Zigzag Armchair PBC DFT
a b s t r a c t Localized Gaussian type orbital-periodic boundary condition-density functional theory (LGTO-PBC-DFT) method was used to determine the electronic and detailed geometrical structures of (n, 0) zigzag type for n = 6 – 33 and (n, n) armchair type for n = 3 – 15 single-walled boron-nitride (BN) nanotubes with infinite tubular lengths. The calculations reveal that the calculated Eg (band gap between HOCO and LUCO) increases with increasing tubular diameter and eventually converge to 5.03 eV for BN nanotubes of larger tubular diameter. According to the calculated Egs, the BN nanotubes are semi-conductor and their conductivities are not sensitive to the tubular diameter. Theoretically, the calculated bond length decreases with increasing tubular diameter. Based on our calculations, the bond length and angle do converge to 1.45 Å and 120 degree, respectively. Thus, the structures of BN nanotubes with the infinite tubular length approach the perfect hexagonal network when the tubular diameter increases. The calculated results also indicate that zigzag BN nanotubes with the tubular diameter larger than 18 Å display 3n properties in the calculated Eg, which is also obtained for zigzag carbon nanotubes. © 2008 Elsevier B.V. All rights reserved.
1. Introduction Since the discovery of carbon fullerene C60 by Smalley et al. in 1985 and carbon nanotubes by Iijima in 1991, the unusual electronic and structural properties of carbon nanotubes have been received the increasing attention of chemists and physicists [1–4]. Being closer neighbors of carbon in the Periodic Table, boron-nitride (BN) nanotube with proper alternative substituted of carbon having similar structures and mechanical properties are viewed as ideal analogs of carbon nanotubes. Recently, the experiments have shown that BN nanotubes contain unique properties and could be applied in various branch of material science [5]. The electronic properties of carbon nanotubes vary widely with their lengths, diameter and chirality [6–8]. Carbon nanotubes can be either metal or semi-conductor and are potentially used in a wide variety of applications [9–11]. In 1994, Rubio et al. predicted the existence of BN nanotubes by theoretical computations [12,13]. According to the earlier quantum mechanical calculations, the BN nanotubes are hexagon-like with the same resemblance to carbon nanotubes; where the band gap energy is about 5.5 eV [14]. The BN nanotubes have been synthesized and shown to be the electrical insulators that are independent of their tubular length, diameter, and chirality [15]. Chopra and Ma successfully synthesized multi-layered ⁎ Corresponding authors. E-mail addresses: [email protected] (Y.-M. Chou), [email protected] (B.-C. Wang). 0925-9635/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.diamond.2008.10.026
BN nanotubes via the non-equilibrium plasma-arc and chemical vapor deposition (CVD) methods, respectively [16]. Then, TEM and HREM images indicate that multi-layered BN nanotubes display hexagonal networks with 7 to 9 concentric cylinder-like layers without fullerenelike caps [17–21]. In 2001, Bengu and Marks proposed a new method to synthesize the single-walled BN nanotubes [22]. This new synthesis technique has made it possible to bring BN nanotubes into the nanotechnology applications. In recent studies on BN nanotubes, several papers have been presented and used first-principle to investigate the small radius single-walled BN nanotubes with open end [15,23]. Charlier et al. predicted the geometric structures of BN nanotubes based on their formation mechanism [24]. Later reports explained their mechanical properties with theoretical computation methods [25]. Very recently, Zhang et al performed ab initio and DFT calculations to study the effect of chemisorption on the BN nanotubes [26,27]. Although higher level calculations have been done for the small segment of single-walled nanotubes with various diameters, they did not provide the real model of the BN nanotubes. In order to investigate the infinite system, the DFT calculation with Gaussian molecular orbital length may be used and extend it to infinite tubular length with periodic boundary condition (PBC) model. Scuseria et al. employed PBC-DFT with localized Gaussian type orbitals (LGTO) to generate the optimized structures of (5, 0) zigzag carbon nanotube [28]. Quite recently, the infinite carbon nanotubes with zigzag and armchair types have been simulated by used the LGTO-PBC-DFT calculation and the detailed geometrical structures have been reported [8]. Thus, we have
352
Y.-M. Chou et al. / Diamond & Related Materials 18 (2009) 351–354 Table 1 Calculated HOCO, LUMO and Eg of zigzag type nanotubes by VSXC/6–31G(d) calculation level with PBC-DFT method
Fig. 1. Morphological structures of (6, 0) zigzag BN nanotube with the PBC-DFT method.
confidence that the LGTO-PBC-DFT method is a suitable approach to determine the physical properties of nanotube with infinite tubular length. In this paper, the investigation of BN nanotubes is focused on the zigzag and armchair types. LGTO-PBC-DFT calculation was performed on these systems at pure DFT functional (PBE, VSXC, etc.) level with 6– 31G(d) basis sets and PBC model. The test calculations were carried out by using several functionals including hybrid function and basis sets. The optimized-geometrical structures, electronic structures and related band gap of (n, 0) zigzag (6 ≤ n ≤ 33) and (n, n) armchair (3 ≤ n ≤ 15) single-walled BN nanotubes with infinite tubular length were generated. 2. Calculation details Conveniently, BN nanotube could be presented by a (n, m) pair of numbers; where the (n, 0) and (n, n) BN nanotubes designate the zigzag and armchair types, respectively. For the zigzag type BN nanotube, n denotes the number of triazines in the circumference of the tube and the translation axis is the tubular length. For the PBC-DFT calculation of single-walled BN nanotube, we start with the single layer for the unit cell and extend it along the tubular axis to infinite length by using the PBC model (Fig. 1). The PBC-DFT method is implemented in the Gaussian 03 program revision C.02 program package [29]. The PBC model in the Gaussian 03 package is based on Gaussian type orbitals (GTOs) [30], Bloch function [31] was employed to transform GTOs into “crystalline orbitals” for calculating the periodic boundary condition systems [28,32–35]. For obtaining a high precision geometry, an extremely tight optimization convergence criteria was used, and an ultrafine was chosen for the integral grid option. In the present study, the energy gap between the energy of HOCO and that of LUCO corresponds to the band gap; we use the definition given by Eg = ELUCO − EHOCO. In order to examine the efficiency of various functionals and basis sets with the LGTO-PBC-DFT method, (9, 0) zigzag BN nanotube was optimized with various types of functionals (VSXC, BLYP, LSDA, PBEPBE, PW91PW91, and B3LYP.) by using 6–31G(d) basis set, and the corresponding total energy and band gap energies were computed. The VSXC functional calculation that gives the lowest calculated total energy and larger relative energy as compared to those calculated by BLYP, LSDA, PBEPBE, PW91PW91 and B3LYP. The calculated total energies by BLYP and B3LYP are quite close to those calculated with VSXC. The calculated Eg by BLYP, LSDA, PBEPBE and PW91PW91are very closed each other and lower
Fig. 2. Morphological structures of (5, 5) armchair BN nanotube with the PBC-DFT method.
(n, 0)
HOCO
LUCO
Eg
n=6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33
−5.9507 −6.0441 −5.9168 −5.8753 −5.9216 −5.8642 −5.8386 −5.8614 −5.8345 −5.8127 −5.8271 −5.8084 −5.7940 −5.8035 −5.7925 −5.7811 −5.7874 −5.7794 −5.7700 −5.7756 −5.7690 −5.7625 −5.7648 −5.7603 −5.7547 −5.7579 −5.7539 −5.7461
−2.9925 −2.3797 −2.0219 −1.7577 −1.5539 −1.4010 −1.2777 −1.1755 −1.0970 −1.0260 −0.9660 −0.9170 −0.8752 −0.8404 −0.8085 −0.7808 −0.7562 −0.7357 −0.7331 −0.7324 −0.7236 −0.7269 −0.7241 −0.7168 −0.7214 −0.7201 −0.7130 −0.7152
2.9582 3.6645 3.8949 4.1176 4.3677 4.4632 4.5609 4.6859 4.7375 4.7868 4.8611 4.8913 4.9188 4.9632 4.9840 5.0003 5.0312 5.0437 5.0368 5.0432 5.0454 5.0356 5.0407 5.0436 5.0333 5.0379 5.0408 5.0309
than those of VSXC and B3LYP. Also the basis set effects are considered in our calculation, the STO-3G, 3–21G, 6–31G(d), 6–311G (d), cc-pVDZ and D95 basis sets are performed with VSXC functional for the basis set dependence check. The calculated total energy and Eg are very closed while using the 6–31G(d), 6–311G(d), cc-pVDZ and D95 basis sets, where the total energy and Eg are around − 14.88 keV and 4.10 eV, respectively. In particular, the total energy shows strong basis set dependence while the band gap convergence was achieved with 6–31G(d). Unfortunately, they require a long computing time when use the basis sets 6–311G(d), cc-pVDZ and D95. Thus, considering both accuracy and computing time, we chose the 6–31G(d) basis set for the following calculation. 3. Results and discussions On the basis of the structure analysis, the prominent structural property of zigzag BN nanotubes with a small radius, as compared to carbon nanotubes, is the existence of buckling on the tubular surface, which resulted from boron and nitrogen moving slightly inward and outward of the nanotube axis, respectively. In particular, the structure of zigzag BN nanotube is different from that of armchair BN nanotube, where boron and nitrogen atoms determine the rings with different diameter along the circumference of the tube. For example, the optimized diameters of (6, 0) zigzag BN nanotube are 5.022 Å and 4.813 Å for nitrogen and boron rings, respectively; i.e., there is almost 0.2 Å diameter difference in the same tube. The calculated tubular diameters increase from 4.850 Å for the (6, 0) BN nanotube to 26.436 Å for the (33, 0) BN nanotube. Thus, the variation with tubular diameter is about 0.8 Å for each increment of n. For the calculated geometrical parameters of infinite BN nanotubes, there are three types B\N bonds (N1\B2, B2\N3 and N3\B4) and four types of B\N\B bond angles (A1, A2, A3 and A4) (Fig. 1). The bond length B2\N3 decreases while decreasing the tubular diameter. Contrarily, the N1\B2 and N3\B4 bond lengths increase with decreasing the tubular diameter. For the zigzag BN nanotubes (n ≧ 21), the calculated bond lengths N1\B2,
Y.-M. Chou et al. / Diamond & Related Materials 18 (2009) 351–354 Table 2 Calculated HOCO, LUCO and Eg of armchair type nanotubes by VSXC/6–31G(d) calculation level with PBC-DFT method (n, n)
HOCO
LUCO
Eg
n=3 4 5 6 7 8 9 10 11 12 13 14 15
− 5.9810 − 5.9211 − 5.8785 − 5.8573 − 5.8348 − 5.8194 − 5.8072 − 5.7969 − 5.7884 − 5.7811 − 5.7781 − 5.7690 − 5.7613
−1.2770 −1.1809 −0.9445 −0.8838 −0.7844 −0.7592 −0.7542 −0.7488 −0.7439 −0.7396 −0.7378 −0.7318 −0.7269
4.7040 4.7402 4.9340 4.9735 5.0504 5.0601 5.0531 5.0481 5.0445 5.0415 5.0403 5.0372 5.0344
and N3\B4 converge to almost the same magnitude 1.453 Å and B2\N3 keeps within 1.452 Å. Compared the calculated bond angle in the BN nanotube, the bond angle A4 displays a larger variation with the decreased in the tubular diameters (it increases from 110 degree for the (6, 0) BN nanotube to 119 degree for the (20, 0) BN nanotube) while the other calculated bond angles A1, A2 and A3 are around 120 degree. These effects may be explained by the hybridization and weakening of sp 2 bonds caused by the strong concaved curvature of the BN nanotube. The calculated bond length of zigzag BN nanotube along the tube circumference is slightly longer than those of along the tube axis for small radii nanotubes. Theoretically, the electrons can contribute to the hexagonal resonance due to the unpaired electron of the nitrogen atom and the nearby three boron atoms, where the electron density will be shared. This effect becomes much more obvious as the pπ resonance built in the hexagonal network of BN nanotube. Thus, nitrogen atom with a stronger electronic affinity should contribute less of its unpaired electron contributing to the hexagonal ring and produces a smaller pπ resonance in the hexagonal network. In addition, the unequal electronic distribution along the BN bonding causes variation of the bond length; BN bonds can causes the variation of the bond length. For example, BN bond along the tube axis has more electronic density than those of the BN bond along the tube circumference. Thus, the calculated Eg of BN nanotubes should be larger than those of carbon nanotubes [36,37]. The results indicate that the bond lengths N1\B2, B2\N3, and N3\B4 of armchair BN nanotubes (Fig. 2) decreased while the tubular diameter increased and the calculated bond angles A1, A2, A3, and A4 increased up to 120 degree while the tubular diameter increased. For the small radii (4, 4), (5, 5) and (6, 6) BN nanotubes, the calculated A2 values are under 118 degree which may have obvious buckling effect in the tube surface, since the sp3 distribution of nitrogen atom still produces a small bump around the nitrogen atom. Up to (11, 11) BN nanotube, the calculated bond lengths and angles indicated that these BN nanotubes have the perfect hexagonal network with the sp 2 bonding distribution and all of B\N bond lengths are kept around 1.453 Å. Thus, this structure is much closed to that of zigzag carbon nanotube with the same tubular size. Tables 1 and 2 show the calculated HOCO, LUCO and Eg for zigzag and armchair types BN nanotubes with the PBC-DFT calculation. For the calculated Eg of zigzag type BN nanotubes, the lowest calculated Eg is 2.96 eV for the (6, 0) BN nanotube and the calculated Eg increase by around 0.1 – 0.2 eV for each increment of n, and finally converge to around 5.04 eV when n ≥ 26. In particular, the zigzag BN nanotubes are shown to exhibits no obvious 3n oscillation property dependence in the calculated Eg, when the tubular diameter is less than 16.8 Å and n b 21. This observation is not the same as those of single-walled
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carbon nanotubes. By increasing the calculated diameter of BN nanotube up to 16.8 Å and n ≧ 21, the calculated Eg has the 3n oscillation property. Thus, the zigzag BN nanotube with larger diameter may have the carbon nanotube property. The trend of the calculated Eg increments is different from that of carbon nanotubes since the strong concaved curvature during the small diameter BN nanotube. The calculated Eg for armchair (n, n) increases while increasing the tubular diameter, and finally converges to 5.03 eV. Thus, the armchair type BN nanotubes are semiconductors and their calculated Eg have no obvious dependence on tubular diameters. Theoretically, the calculated Eg for both zigzag and armchair BN nanotubes converge to 5.03 eV, which is in good agreement with the experimental data of 5.5 eV [14]. 4. Conclusion The PBC-DFT calculations with the VSXC functional and 6–31G (d) basis set on the optimized-geometrical structures of BN nanotubes with infinite tubular length have revealed that there are significant differences between the BN and carbon single-walled nanotubes. The differences can be attributed to the added dimension of nitrogen and boron atoms moving outward and inward respectively on the tubular surface of BN nanotubes. The calculated Eg of the zigzag and armchair BN nanotubes were shown to converge to 5.03 eV as the tubular diameter increases. Our calculation results also agree very well with earlier experimental results that showed, for diameter about 50 nm, the energy gaps are between 4.5–4.8 eV [38]. Particularly, the BN nanotubes with infinite tubular length are semiconductors and their conductivities are independent on tubular diameters. The calculated average BN bond length and bond angle are about 1.45 Å and 120 degree for both the zigzag and armchair types. For the zigzag BN nanotubes with the tubular diameter larger than 16.8 Å and n ≧ 21, the calculated Eg shows the 3n oscillation property. The physical properties of these BN nanotubes seem to resemble those of carbon nanotubes. Acknowledgments We thank you Profs. Chhiu-Tsu Lin and Hsiu-Fu Hsu for reading the manuscript and the National Science Council of Taiwan for financial supporting this research. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
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