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Formation and electronic properties of double-walled boron nitride nanotubes
Solid State Communications 134 (2005) 397–402 www.elsevier.com/locate/ssc
Formation and electronic properties of double-walled boron nitride nanotubes Seung-Hoon Jhia,*, David J. Roundya,b, Steven G. Louiea, Marvin L. Cohena a
Materials Science Division, Department of Physics, Lawrence Berkeley National Laboratory, University of California at Berkeley, Le Conte 7300, Berkeley, CA 94720, USA b Department of Physics, Cornell University, Ithaca, NY 14853, USA Received 9 December 2004; accepted 4 February 2005 by P. Sheng
Abstract The electronic and structural properties of double-walled boron nitride (BN) nanotubes are studied using the first principle pseudopotential density functional method. It is shown that zigzag-type tubes have a larger formation energy for the doublewalled configuration than the armchair-type structure, and that interwall stacking plays a significant role for intertube interactions and in the formation of multiwall BN nanotubes. The fundamental energy gap of double-walled BN nanotubes was found to be smaller than that of single-walled tubes mostly due to band shift. It is shown that the electronic properties of doublewalled BN tubes exhibit a slight but noticeable difference for the zigzag and armchair type tubes studied. q 2005 Elsevier Ltd. All rights reserved. PACS: 61.48.Cc; 62.40.Ci; 71.20.Tx Keywords: A. Boron nitride; A. Nanotubes; D. Interwall; D. Electronic structure
1. Introduction The discovery of carbon nanotubes has stimulated the search for and synthesis of similar tubular nanostructures of non-carbon materials. Several inorganic fullerene-like structures have been suggested and some of them are actually synthesized (for review, see Ref. [1,2]). Hexagonal boron–nitride (hBN), for example, has a similar layered structure to graphite, and its tubular forms were first realized theoretically as a possible structure and later successfully * Corresponding author. Now at: Physics Department, Pohang University of Science and Technology, San 31, Hyojadong, Namgu, Pohang, Gyungbuk 790–784, Korea. Tel.: C82 54 279 2094; fax: C82 52 279 3099. E-mail addresses: [email protected] (S.-H. Jhi), [email protected] (D.J. Roundy), sglouie@uclink. berkeley.edu (S.G. Louie), [email protected] (M.L. Cohen).
0038-1098/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2005.02.007
produced [3,4]. Unlike carbon nanotubes, single-walled BN nanotubes were predicted to be wide gap semiconductors and have similar electronic structures independent of their chirality and diameter [5]. For that reason, BN nanotubes may be more useful than carbon nanotubes for certain applications in electronic devices or as nanoscaled insulating materials. The ionic B–N bonding in BN nanotubes can provide richer, but more complex structural properties than those of carbon nanotubes. The non-conducting property of BN may also be a disadvantage in the direct use of synthesis methods available for carbon nanotubes [6]. A recent molecular dynamics calculation showed that unmatched bonding between boron and nitrogen (called the bond frustration) leads to the closing of the nanotubes, indicating that BN nanotubes have more stringent growth conditions than carbon nanotubes [7]. Several synthesis methods have proven successful for bulk production of BN nanotubes: laser heating method [8], non-equilibrium plasma-arc method [6], and substitutional
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reaction method using carbon multiwall tubes [9–11] are found in the recent literature. Grown BN nanotubes are mostly multiwall nanotubes, and with some synthesis methods, the majority of the tubes were found to have zigzag atomic arrangements [12]. This observation seems inconsistent with a recent molecular dynamics simulation that showed a preference for armchair over zigzag structure for BN growth [7]. Because the simulation assumed singlewalled BN nanotubes, it is possible that interwall interactions are advantageous for the growth of zigzag-type tubes. Interestingly, a recent experiment using the plasmaarc method [6] showed that most of the tubes synthesized had a double-walled structure and demonstrated that it is possible to control the growth of BN nanotubes and produce them in bulk amounts under certain conditions. Although it is yet unclear, due to highly non-equilibrium synthesis conditions, whether the formation of double-walled structure is a generic phenomenon, the high yield of doublewalled BN nanotubes suggests the possibility that their growth may be related to their intrinsic properties. Another interesting issue is the effect of interwall interactions on the electronic and structural properties of multiwall BN nanotubes. It has been known that the stacking of basal planes of hBN can have a significant effect on their structural and electronic properties [15]. The BN nanotubes, which cannot have a perfect commensurating stacking of walls because of their tubular structure, may exhibit different aspects of interwall interactions from those of hBN. In this paper, we studied the electronic and structural properties of single- and double-walled BN nanotubes to investigate the effect of interwall interactions in the multiwalled tubes using the ab initio pseudopotential density functional method. We chose two representative types of BN nanotubes, namely zigzag and armchair types, to study the stacking dependence of interwall interactions.
2. Computational method Calculations were carried out with using the ab initio total energy method based on the pseudopotential density functional theory within the local density approximation (LDA). A plane-wave basis set was used with a cutoff energy for the plane-wave expansion of 60 Ry in these calculations. To study BN nanotubes, we used a supercell with a triangular lattice with the shortest distance between ˚ or larger (note that the walls of adjacent tubes of about 5.8 A ˚ ). Atomic inter-layer distance in hBN is about 3.3 A positions were fully relaxed until the Hellmann–Feynman forces on each atom was less than 1 mRy/aB with aB being the Bohr radius. We studied two different types of nanotubes, (n,0) zigzag and (n,n) armchair types. For double-walled BN-nanotubes, we put (5,5) inside (10,10) and (8,0) inside (16,0) denoted as (5,5)C(10,10) and (8,0)C(16,0), respectively. Fully relaxed structures of individual single-walled BN tubes
were first obtained, and then the tubes were put together coaxially to form double-walled BN tubes. It has been known that the atomic arrangement in neighboring layers (or stacking) is important for both the electronic and structural properties of hBN [15]. The preferential stacking in hBN is achieved by putting boron atoms on top of nitrogen atoms while avoiding the overlap between the same type of atoms in neighboring layers [16]. We chose a similar atomic arrangement for the double-walled tubes so that boron atoms on the inner tube make the least overlap along the radial direction with boron atoms on the outer tube (also similarly for nitrogen atoms). Fig. 1 shows a schematic diagram of how the atoms were arranged in zigzag tubes in our calculations. Some BN hexagon rings in inner and outer tubes have the same stacking as hBN, but they become gradually non-commensurate for stacking along the circumferential direction. However, there is still no overlap of the same type of atoms along the radial direction in this configuration. We note that such an arrangement is not possible for armchair tubes, which inevitably have mixed stacking of B–B, B–N, and N–N atoms along the radial direction regardless of axial rotation of inner or outer tubes.
3. Results We first studied the single-walled BN nanotube and also single and double layers of hexagonal graphitic BN sheets. The B–N bond-length in a graphitic single BN-sheet was ˚ which is slightly less than the calculated to be about 1.440 A ˚ in hBN (Ref. [15] and experimental value of 1.443 A references therein). The optimal inter-layer distance in double layer graphitic BN sheet was obtained to be about ˚ , which is quite close to that in hBN of about 3.33 A ˚. 3.201 A The interlayer binding energy was calculated to be about 0.01 eV/atom. Fully relaxed single-walled BN tubes exhibited several characteristic features compared to a sheet of hexagonal BN. It is known that the curvature of tubular structure causes a buckling of B–N bonds to incorporate sp3 bonding characteristics. The lone-pair electrons of nitrogen
Fig. 1. A schematic diagram of atomic arrangements in inner and outer tubes of the zigzag double-walled BN tube. Only single BN hexagons from the inner and outer tubes are shown for illustrations. Empty circles denote boron atoms and gray ones the nitrogen atoms, respectively.
S.-H. Jhi et al. / Solid State Communications 134 (2005) 397–402
energetically prefer configurations where nitrogen atoms move outward. The buckling of the B–N bonds hence leads to a formation of a dipole layer of an inner B-atom shell and an outer N-atom shell. It was found that the buckling increases as the tube diameter decreases with a maximum separation of B-atom shell and N-atom shell of about ˚ for the (8,0) tube in our calculations. This 0.078 A observation is consistent with previous theoretical reports [5,13]. The buckling occurs in a slightly different way for zigzag and armchair type tubes at smaller diameter, which is related to inequivalent local arrangements of the B–N bondings. The bond length of the two inequivalent B–N bonds in the armchair tubes is much different from each other compared to those in zigzag tubes. According to our calculations, the B–N bonds perpendicular to the tube axis have a longer ˚ than those oblique to the bond-length by as much as 0.025 A tube axis in the (10,10) armchair tube. The differences are ˚ for the not as significant for the zigzag tubes (about 0.006 A (16,0) tube). This is related to how B–N bonds are relaxed to incorporate the sp3 bonding character in tubular structures. The buckling reduces to a bond rotation in zigzag tubes while it leads to an increase of bond length along the circumferential direction for armchair tubes [18]. However, this effect becomes insignificant when the tube diameter is large enough. After obtaining the relaxed geometry for each singlewalled BN tubes, we put two single-walled tubes coaxially to make a double-walled tube and performed the structural relaxation again. An apparent change in structure occurs in the interwall spacing. Our calculated interlayer distance of ˚ as double-layer graphitic BN sheets was about 3.201 A mentioned above. The relaxed double-walled BN tubes show that the interwall distance increases for (8,0)C(16,0) zigzag tube while it decreases for (5,5)C(10,10) armchair tube. The change of the interwall spacing is mainly driven by a force to maintain the optimal spacing between inner and outer BN tubes. The difference in diameter between each single-walled tube here is smaller (larger) for zigzag tube (armchair tube) than the optimal interlayer distance of graphitic BN sheets. For example, the difference in averaged diameter between (8,0) and (16,0) single-walled BN tubes is ˚ , but it increases to 3.147 A ˚ when they form a 3.055 A double-walled tube. The outer (16,0) tube expands and the inner (8,0) tube shrinks radially in the double-walled
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structure. Calculated average radii of BN nanotubes and buckling of B–N bonds are summarized in Table 1. According to our calculations, no apparent change in the buckling of B–N bonds was observed when BN nanotubes form double-walled structures. This suggests that their chemical reactivity, which can be related to the degree of B– N buckling in the outer tube, does not change when the tubes make a double-walled structure. The experimental finding of double-walled BN nanotubes [6] may be originated from specific growth conditions such as catalysts structures or gas flows rather than from any change in the intrinsic properties of the BN nanotubes such as reduced chemical reactivity for double-walled tubes. From the calculation of the total energies of doublewalled and each single-walled tubes, we can estimate the formation energy of double-walled tubes to be about 0.08 and 0.01 eV/atom for the zigzag (8,0)C(16,0) and the armchair (5,5)C(10,10) tube, respectively. The formation energy of zigzag-type double-walled tubes was significantly larger than that of armchair type tubes and also the interlayer binding energy of graphitic BN sheets. In a previous theoretical study [14], the formation energy of the zigzag type double-walled tubes [the (7,0)C(15,0) tube, for example] was calculated to be about w0.01 eV, which is much smaller than our calculated value. The difference may be due to the different sizes of the tubes or the different stacking of atoms along the walls from ours. The interwall stacking was not clearly indicated in the paper of Okada et al. to be compared to ours [14]. The larger formation energy of zigzag tubes can be attributed to the better matched stacking of atoms compared to that in armchair tubes. The larger interwall spacing in armchair tubes may give a slightly smaller formation energy, but its effect cannot lead to a decrease of the formation energy of more than 10 meV/atom. It was shown that the interlayer binding energy can vary as much as several tens meV per atom depending on the stacking of the layers for hBN [15,16]. For zigzag tubes, it is possible to make an interwall stacking that does not have an overlap between the same type of atoms on the inner and outer tubes regardless of the tube diameter as mentioned above. However, the armchair tubes have a mixed overlap of B– B, N–N, and B–N atoms in our configurations. The larger formation energy for zigzag type tubes suggests that the multiwall BN tubes can grow preferentially in zigzag
Table 1 ˚ unit) of B-atom shell (B in subscript) and N-atom shell (N in subscript) of boron nitride nanotubes in single-walled (s in Average radius (R in A superscript) and double-walled (d in superscript) configurations
(8,0) (16,0) (5,5) (10,10)
RdB
RdN
dRdBN
RsB
RsN
dRsBN
3.192 6.354 3.491 6.948
3.252 6.386 3.555 6.980
0.06 0.032 0.064 0.032
3.269 6.349 3.490 6.982
3.347 6.379 3.555 7.013
0.078 0.030 0.065 0.031
˚ unit). The buckling of B–N bonds is also given by the difference of the shell radii in individual tubes (dR in A
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patterns due to the interwall stacking constraint, which supports the experimental observations of preferred growth of BN nanotubes in zigzag atomic arrangement [12]. The interwall stacking can also give different mechanical properties of multiwall BN nanotubes depending on their chirality. Because the energy barrier for the outer tube to slide over inner tube (or vice versa) is estimated to reach several tens meV per atom, the outer tube in zigzag multiwall BN nanotubes is expected to hardly slide along the tube axis. On the contrary, the armchair or chiral tubes may move relatively smoothly as observed in carbon nanotubes telescopic motions [19]. In order to investigate the effect of intertube interactions on electronic structure, we calculated band structures of the single- and double-walled BN nanotubes. For zigzag tubes, the outer tube [(16,0)] was found to have an LDA KohnSham energy-gap of 4.3 eV, which is almost the same as that of hBN. For the inner tube [(8,0)], the (s*–p* hybridization reduces the gap to about 3.4 eV in our LDA calculations. It was found that the energy-gap of the zigzag double-walled BN tube was reduced slightly but in a measurable amount (about 0.1 eV) as shown in Fig. 2. The reduction of the gap is mostly due to the band shift of inner and outer tubes, which is in turn due to the difference in s–p hybridization for the tubes. The energy bands of the inner tube are shifted down relative to those of the outer tube. The interwall interactions break the degeneracy of the valence band maximum (VBM) states (at the G point in Fig. 2), which also contributes to the reduction of the energy gap. It was found that the gap reduction due to band shift was more significant for armchair type tubes as shown in Fig. 3. The band gap of the armchair double-walled tube is reduced by about 0.4 eV in our calculations. Within LDA, the armchair type tubes have an indirect bandgap of about 4.4, 4.6, and 4.0 eV for (5,5), (10,10), and (5,5)C(10,10) tubes, respectively, from
Fig. 2. The band structure of zigzag BN nanotubes near the Fermi level. (a) (8,0), (b) (16,0), and (c) (8,0)C(16,0) double-walled BN nanotubes along tube axis, respectively. The top of the valence band is set at the zero of energy for all three plots. The energy gap for double-wall tube is slightly smaller than that of (16,0) tube. The valence band top states are mostly derived from the outer tubes, while the minimum of conduction bands comes from the states of the inner tube. The optical transition would thus occur through interwall hopping of electrons or holes.
Fig. 3. Similar plot of the band structure of armchair BN nanotubes near the Fermi level as in Fig. 2. (a) (5,5), (b) (10,10), and (c) (5,5)C(10,10) double-walled BN tubes along tube axis, respectively. The zero of energy is set to the level of the valence band maximum states for all three plots. The armchair tubes have an indirect gap from D to G points of the Brillouin zone. The energy gap for double-walled tube in our calculation is reduced by as much as 0.4 eV, which is significant compared to that for zigzag doublewalled BN tubes (Fig. 2).
the D to G points of the Brillouin zone. The reason for different band shift for zigzag and armchair double-walled BN nanotubes is unclear yet. This may originate from the different interwall stacking or interwall spacing of the tubes. It is noteworthy to mention that the conduction band minimum (CBM) states of the two tubes exhibit quite different characteristics as shown in Fig. 4. A signature of band gap reduction in multiwalled BN nanotubes was also reported in a recent EELS experiment that observed a downward shift of the energy loss peak by 0.6 eV with respect to the value in hBN [17]. The shift in the experiment is believed to be driven by the curvature as well as the interwall interactions in the tubes. More comprehensive studies need to be done to clearly distinguish the effect of the curvature and the interwall interactions in multiwalled BN nanotubes. For large diameter multiwall tubes, on the other hand, the energy-gap is expected to be close to that of hBN because the B–N bond buckling decreases and the interwall interaction will converge to that in hBN. Fig. 4 shows the squared wave-functions of the conduction band minimum states of the (8,0)C(16,0) zigzag (a,b) and the (5,5)C(10,1,0) armchair (c,d) doublewalled tubes in a plane perpendicular to the tube axis (a,c) and a plane along the tube axis (b,d) of the tubes. The plots of the squared wave functions reveal the characteristic symmetry of the corresponding states. The distribution of the wave functions is quite different for zigzag and armchair tubes. While the state at bottom of the conduction band of the armchair tube exhibits a signature of nearly-freeelectron-like character, the corresponding state of the zigzag tube shows a mixture of p and sp3 bonding character, and is mostly distributed in the inner tube. A weak anti-bondinglike interaction between inner and outer tubes is also seen in the conduction band minimum state of the zigzag tube
S.-H. Jhi et al. / Solid State Communications 134 (2005) 397–402
Fig. 4. Squared wave functions of the CBM states of the (8,0)C (16,0) zigzag (a,b) and the (5,5)C(10,10) armchair (c,d) doublewalled BN nanotubes in a plane perpendicular to the tube axis (a,c) and also in a plane along the tube axis (b,d). The edge of the tube is indicated by dashed lines. The CBM states in the armchair tube shows a nearly-free-electron-like character while the corresponding states in the zigzag tubes have a typical p* bonding hybridized with sp3 bonding, especially in the inner tube. We also observe that there is a weak anti-bonding-like coupling between inner and outer tubes in zigzag nanotubes.
[Fig. 4(c)]. The unlike characters of the states at bottom of the conduction band in zigzag and armchair tubes may contribute to the different band shift when the double-walled nanotubes are formed. The states at top of the valence band, on the other hand, do not show so much difference as opposed to the conduction band minimum states. Weak interwall interactions split the degenerate states at top of the valence band at G point as mentioned above. Fig. 5 shows a similar plot as in Fig. 4 of one of the degenerated states of the zigzag nanotubes. Another state has an almost identical density distribution except a rotation by 908 along the tube axis (not shown here). Because the density of the VBM and CBM states is either distributed mostly in the inner and outer tubes, respectively, the optical transition from the VBM to CBM states or vice versa should occur through the interwall hoping of electrons or holes. A recent Raman spectroscopy study [20] observed significantly different decay processes of photoluminescence in multiwalled BN tubes compared to those in hBN, which was attributed to the
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Fig. 5. Squared wave function of the valence band maximum states of the (8,0)C(16,0) tube (a) in a plane perpendicular to the tube axis and (b) in a plane along the tube axis. The edge of the tube is indicated by dashed lines. There is a weak coupling between atoms in inner and outer layers. The almost-degenerate state looks similar but rotated by 908 along the tube axis (not shown here).
spatial separation of electron and hole wave functions. Possible applications for laser devices based on this property was suggested by Okada et al. [14].
4. Summary In this paper, we discussed the electronic and structural properties of double-walled BN nanotubes. It was, found that the formation energy of double-walled tubes can depend on the stacking structure of the tubes. According to our calculations, the interwall stacking gives a preference for the growth of zigzag nanotubes, which is supported by the experimental results. The energy band gap also exhibits a measurable difference for the zigzag and armchair type tubes chosen in this study. The indirect gap of the (5,5)C (10,10) double-walled armchair tubes has a smaller gap by 0.4 eV than the single-walled tubes, whereas the reduction of the band gap in the (8,0)C(16,0) zigzag tube is about 0.1 eV. The conduction band minimum states show quite different bonding character in zigzag and armchair tubes studied.
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Acknowledgements SHJ acknowledges the helpful discussion with C.-H. Park. This work was supported by the NSF under Grant No. DMR-04-39768, and by the Office of Energy Research, Office of Basic Energy Science, Material Sciences Division of the US Department of Energy under Contract No. DEAC03-76SF00098. Supercomputer time was provided by NCSA and NERSC.
References [1] R. Tenne, Angew. Chem., Int. Ed. 42 (2003) 5124. [2] R. Tenne, A. Zettl, Top. Appl. Phys. 80 (2001) 81. [3] A. Rubio, J.L. Corkill, S.G. Louie, M.L. Cohen, Phys. Rev. B 49 (1994) 5081. [4] N.G. Chopra, R.J. Luyken, K. Cherrey, V.H. Crespi, M.L. Cohen, S.G. Louie, A. Zettl, Science 269 (1995) 966. [5] X. Blase, A. Rubio, S.G. Louie, M.L. Cohen, Europhys. Lett. 28 (1994) 335. [6] J. Cumings, A. Zettl, Chem. Phys. Lett. 316 (2000) 211. [7] X. Blase, A. De Vita, J.-C. Charlier, R. Car, Phys. Rev. Lett. 80 (1998) 1666.
[8] T. Laude, Y. Matsui, A. Marraud, B. Jouffrey, Appl. Phys. Lett. 76 (2000) 3239. [9] W.Q. Han, Y. Bando, K. Kurashima, T. Sato, Appl. Phys. Lett. 73 (1998) 3085. [10] D. Golberg, Y. Bando, K. Kurashima, T. Sato, Solid State Commun. 116 (2000) 1. [11] D. Golberg, Y. Bando, K. Kurahima, T. Sato, Chem. Phys. Lett. 323 (2000) 185. [12] D. Golberg, Y. Bando, L. Burgeois, K. Kurashima, T. Sato, Appl. Phys. Lett. 77 (2000) 1979. [13] E. Hernn´dez, C. Goze, P. Bernier, A. Rubio, Phys. Rev. Lett. 80 (1998) 4502. [14] S. Okada, S. Salto, A. Oshiyama, Phys. Rev. B 65 (2002) 165410. [15] L. Liu, Y.P. Feng, Z.X. Shen, Phys. Rev. B 68 (2003) 104102. [16] Cheol-Hwan Park, private communications. [17] G.G. Fuentes, E. Borowiak-Palen, T. Pichler, X. Liu, A. Graff, G. Behr, R.J. Kalenczuk, M. Knupfer, J. Fink, Phys. Rev. B 67 (2003) 035429. [18] M. Menon, D. Srivastava, Chem. Phys. Lett. 307 (1999) 407. [19] J. Cumings, A. Zettl, Science 289 (2000) 602. [20] J. Wu, W.-Q. Han, W. Walukiewicz, J.W. Ager, W. Shan, E.E. Haller, A. Zettl, Nano Lett. 4 (2004) 647.
Formation and electronic properties of double-walled boron nitride nanotubes Seung-Hoon Jhia,*, David J. Roundya,b, Steven G. Louiea, Marvin L. Cohena a
Materials Science Division, Department of Physics, Lawrence Berkeley National Laboratory, University of California at Berkeley, Le Conte 7300, Berkeley, CA 94720, USA b Department of Physics, Cornell University, Ithaca, NY 14853, USA Received 9 December 2004; accepted 4 February 2005 by P. Sheng
Abstract The electronic and structural properties of double-walled boron nitride (BN) nanotubes are studied using the first principle pseudopotential density functional method. It is shown that zigzag-type tubes have a larger formation energy for the doublewalled configuration than the armchair-type structure, and that interwall stacking plays a significant role for intertube interactions and in the formation of multiwall BN nanotubes. The fundamental energy gap of double-walled BN nanotubes was found to be smaller than that of single-walled tubes mostly due to band shift. It is shown that the electronic properties of doublewalled BN tubes exhibit a slight but noticeable difference for the zigzag and armchair type tubes studied. q 2005 Elsevier Ltd. All rights reserved. PACS: 61.48.Cc; 62.40.Ci; 71.20.Tx Keywords: A. Boron nitride; A. Nanotubes; D. Interwall; D. Electronic structure
1. Introduction The discovery of carbon nanotubes has stimulated the search for and synthesis of similar tubular nanostructures of non-carbon materials. Several inorganic fullerene-like structures have been suggested and some of them are actually synthesized (for review, see Ref. [1,2]). Hexagonal boron–nitride (hBN), for example, has a similar layered structure to graphite, and its tubular forms were first realized theoretically as a possible structure and later successfully * Corresponding author. Now at: Physics Department, Pohang University of Science and Technology, San 31, Hyojadong, Namgu, Pohang, Gyungbuk 790–784, Korea. Tel.: C82 54 279 2094; fax: C82 52 279 3099. E-mail addresses: [email protected] (S.-H. Jhi), [email protected] (D.J. Roundy), sglouie@uclink. berkeley.edu (S.G. Louie), [email protected] (M.L. Cohen).
0038-1098/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2005.02.007
produced [3,4]. Unlike carbon nanotubes, single-walled BN nanotubes were predicted to be wide gap semiconductors and have similar electronic structures independent of their chirality and diameter [5]. For that reason, BN nanotubes may be more useful than carbon nanotubes for certain applications in electronic devices or as nanoscaled insulating materials. The ionic B–N bonding in BN nanotubes can provide richer, but more complex structural properties than those of carbon nanotubes. The non-conducting property of BN may also be a disadvantage in the direct use of synthesis methods available for carbon nanotubes [6]. A recent molecular dynamics calculation showed that unmatched bonding between boron and nitrogen (called the bond frustration) leads to the closing of the nanotubes, indicating that BN nanotubes have more stringent growth conditions than carbon nanotubes [7]. Several synthesis methods have proven successful for bulk production of BN nanotubes: laser heating method [8], non-equilibrium plasma-arc method [6], and substitutional
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reaction method using carbon multiwall tubes [9–11] are found in the recent literature. Grown BN nanotubes are mostly multiwall nanotubes, and with some synthesis methods, the majority of the tubes were found to have zigzag atomic arrangements [12]. This observation seems inconsistent with a recent molecular dynamics simulation that showed a preference for armchair over zigzag structure for BN growth [7]. Because the simulation assumed singlewalled BN nanotubes, it is possible that interwall interactions are advantageous for the growth of zigzag-type tubes. Interestingly, a recent experiment using the plasmaarc method [6] showed that most of the tubes synthesized had a double-walled structure and demonstrated that it is possible to control the growth of BN nanotubes and produce them in bulk amounts under certain conditions. Although it is yet unclear, due to highly non-equilibrium synthesis conditions, whether the formation of double-walled structure is a generic phenomenon, the high yield of doublewalled BN nanotubes suggests the possibility that their growth may be related to their intrinsic properties. Another interesting issue is the effect of interwall interactions on the electronic and structural properties of multiwall BN nanotubes. It has been known that the stacking of basal planes of hBN can have a significant effect on their structural and electronic properties [15]. The BN nanotubes, which cannot have a perfect commensurating stacking of walls because of their tubular structure, may exhibit different aspects of interwall interactions from those of hBN. In this paper, we studied the electronic and structural properties of single- and double-walled BN nanotubes to investigate the effect of interwall interactions in the multiwalled tubes using the ab initio pseudopotential density functional method. We chose two representative types of BN nanotubes, namely zigzag and armchair types, to study the stacking dependence of interwall interactions.
2. Computational method Calculations were carried out with using the ab initio total energy method based on the pseudopotential density functional theory within the local density approximation (LDA). A plane-wave basis set was used with a cutoff energy for the plane-wave expansion of 60 Ry in these calculations. To study BN nanotubes, we used a supercell with a triangular lattice with the shortest distance between ˚ or larger (note that the walls of adjacent tubes of about 5.8 A ˚ ). Atomic inter-layer distance in hBN is about 3.3 A positions were fully relaxed until the Hellmann–Feynman forces on each atom was less than 1 mRy/aB with aB being the Bohr radius. We studied two different types of nanotubes, (n,0) zigzag and (n,n) armchair types. For double-walled BN-nanotubes, we put (5,5) inside (10,10) and (8,0) inside (16,0) denoted as (5,5)C(10,10) and (8,0)C(16,0), respectively. Fully relaxed structures of individual single-walled BN tubes
were first obtained, and then the tubes were put together coaxially to form double-walled BN tubes. It has been known that the atomic arrangement in neighboring layers (or stacking) is important for both the electronic and structural properties of hBN [15]. The preferential stacking in hBN is achieved by putting boron atoms on top of nitrogen atoms while avoiding the overlap between the same type of atoms in neighboring layers [16]. We chose a similar atomic arrangement for the double-walled tubes so that boron atoms on the inner tube make the least overlap along the radial direction with boron atoms on the outer tube (also similarly for nitrogen atoms). Fig. 1 shows a schematic diagram of how the atoms were arranged in zigzag tubes in our calculations. Some BN hexagon rings in inner and outer tubes have the same stacking as hBN, but they become gradually non-commensurate for stacking along the circumferential direction. However, there is still no overlap of the same type of atoms along the radial direction in this configuration. We note that such an arrangement is not possible for armchair tubes, which inevitably have mixed stacking of B–B, B–N, and N–N atoms along the radial direction regardless of axial rotation of inner or outer tubes.
3. Results We first studied the single-walled BN nanotube and also single and double layers of hexagonal graphitic BN sheets. The B–N bond-length in a graphitic single BN-sheet was ˚ which is slightly less than the calculated to be about 1.440 A ˚ in hBN (Ref. [15] and experimental value of 1.443 A references therein). The optimal inter-layer distance in double layer graphitic BN sheet was obtained to be about ˚ , which is quite close to that in hBN of about 3.33 A ˚. 3.201 A The interlayer binding energy was calculated to be about 0.01 eV/atom. Fully relaxed single-walled BN tubes exhibited several characteristic features compared to a sheet of hexagonal BN. It is known that the curvature of tubular structure causes a buckling of B–N bonds to incorporate sp3 bonding characteristics. The lone-pair electrons of nitrogen
Fig. 1. A schematic diagram of atomic arrangements in inner and outer tubes of the zigzag double-walled BN tube. Only single BN hexagons from the inner and outer tubes are shown for illustrations. Empty circles denote boron atoms and gray ones the nitrogen atoms, respectively.
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energetically prefer configurations where nitrogen atoms move outward. The buckling of the B–N bonds hence leads to a formation of a dipole layer of an inner B-atom shell and an outer N-atom shell. It was found that the buckling increases as the tube diameter decreases with a maximum separation of B-atom shell and N-atom shell of about ˚ for the (8,0) tube in our calculations. This 0.078 A observation is consistent with previous theoretical reports [5,13]. The buckling occurs in a slightly different way for zigzag and armchair type tubes at smaller diameter, which is related to inequivalent local arrangements of the B–N bondings. The bond length of the two inequivalent B–N bonds in the armchair tubes is much different from each other compared to those in zigzag tubes. According to our calculations, the B–N bonds perpendicular to the tube axis have a longer ˚ than those oblique to the bond-length by as much as 0.025 A tube axis in the (10,10) armchair tube. The differences are ˚ for the not as significant for the zigzag tubes (about 0.006 A (16,0) tube). This is related to how B–N bonds are relaxed to incorporate the sp3 bonding character in tubular structures. The buckling reduces to a bond rotation in zigzag tubes while it leads to an increase of bond length along the circumferential direction for armchair tubes [18]. However, this effect becomes insignificant when the tube diameter is large enough. After obtaining the relaxed geometry for each singlewalled BN tubes, we put two single-walled tubes coaxially to make a double-walled tube and performed the structural relaxation again. An apparent change in structure occurs in the interwall spacing. Our calculated interlayer distance of ˚ as double-layer graphitic BN sheets was about 3.201 A mentioned above. The relaxed double-walled BN tubes show that the interwall distance increases for (8,0)C(16,0) zigzag tube while it decreases for (5,5)C(10,10) armchair tube. The change of the interwall spacing is mainly driven by a force to maintain the optimal spacing between inner and outer BN tubes. The difference in diameter between each single-walled tube here is smaller (larger) for zigzag tube (armchair tube) than the optimal interlayer distance of graphitic BN sheets. For example, the difference in averaged diameter between (8,0) and (16,0) single-walled BN tubes is ˚ , but it increases to 3.147 A ˚ when they form a 3.055 A double-walled tube. The outer (16,0) tube expands and the inner (8,0) tube shrinks radially in the double-walled
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structure. Calculated average radii of BN nanotubes and buckling of B–N bonds are summarized in Table 1. According to our calculations, no apparent change in the buckling of B–N bonds was observed when BN nanotubes form double-walled structures. This suggests that their chemical reactivity, which can be related to the degree of B– N buckling in the outer tube, does not change when the tubes make a double-walled structure. The experimental finding of double-walled BN nanotubes [6] may be originated from specific growth conditions such as catalysts structures or gas flows rather than from any change in the intrinsic properties of the BN nanotubes such as reduced chemical reactivity for double-walled tubes. From the calculation of the total energies of doublewalled and each single-walled tubes, we can estimate the formation energy of double-walled tubes to be about 0.08 and 0.01 eV/atom for the zigzag (8,0)C(16,0) and the armchair (5,5)C(10,10) tube, respectively. The formation energy of zigzag-type double-walled tubes was significantly larger than that of armchair type tubes and also the interlayer binding energy of graphitic BN sheets. In a previous theoretical study [14], the formation energy of the zigzag type double-walled tubes [the (7,0)C(15,0) tube, for example] was calculated to be about w0.01 eV, which is much smaller than our calculated value. The difference may be due to the different sizes of the tubes or the different stacking of atoms along the walls from ours. The interwall stacking was not clearly indicated in the paper of Okada et al. to be compared to ours [14]. The larger formation energy of zigzag tubes can be attributed to the better matched stacking of atoms compared to that in armchair tubes. The larger interwall spacing in armchair tubes may give a slightly smaller formation energy, but its effect cannot lead to a decrease of the formation energy of more than 10 meV/atom. It was shown that the interlayer binding energy can vary as much as several tens meV per atom depending on the stacking of the layers for hBN [15,16]. For zigzag tubes, it is possible to make an interwall stacking that does not have an overlap between the same type of atoms on the inner and outer tubes regardless of the tube diameter as mentioned above. However, the armchair tubes have a mixed overlap of B– B, N–N, and B–N atoms in our configurations. The larger formation energy for zigzag type tubes suggests that the multiwall BN tubes can grow preferentially in zigzag
Table 1 ˚ unit) of B-atom shell (B in subscript) and N-atom shell (N in subscript) of boron nitride nanotubes in single-walled (s in Average radius (R in A superscript) and double-walled (d in superscript) configurations
(8,0) (16,0) (5,5) (10,10)
RdB
RdN
dRdBN
RsB
RsN
dRsBN
3.192 6.354 3.491 6.948
3.252 6.386 3.555 6.980
0.06 0.032 0.064 0.032
3.269 6.349 3.490 6.982
3.347 6.379 3.555 7.013
0.078 0.030 0.065 0.031
˚ unit). The buckling of B–N bonds is also given by the difference of the shell radii in individual tubes (dR in A
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patterns due to the interwall stacking constraint, which supports the experimental observations of preferred growth of BN nanotubes in zigzag atomic arrangement [12]. The interwall stacking can also give different mechanical properties of multiwall BN nanotubes depending on their chirality. Because the energy barrier for the outer tube to slide over inner tube (or vice versa) is estimated to reach several tens meV per atom, the outer tube in zigzag multiwall BN nanotubes is expected to hardly slide along the tube axis. On the contrary, the armchair or chiral tubes may move relatively smoothly as observed in carbon nanotubes telescopic motions [19]. In order to investigate the effect of intertube interactions on electronic structure, we calculated band structures of the single- and double-walled BN nanotubes. For zigzag tubes, the outer tube [(16,0)] was found to have an LDA KohnSham energy-gap of 4.3 eV, which is almost the same as that of hBN. For the inner tube [(8,0)], the (s*–p* hybridization reduces the gap to about 3.4 eV in our LDA calculations. It was found that the energy-gap of the zigzag double-walled BN tube was reduced slightly but in a measurable amount (about 0.1 eV) as shown in Fig. 2. The reduction of the gap is mostly due to the band shift of inner and outer tubes, which is in turn due to the difference in s–p hybridization for the tubes. The energy bands of the inner tube are shifted down relative to those of the outer tube. The interwall interactions break the degeneracy of the valence band maximum (VBM) states (at the G point in Fig. 2), which also contributes to the reduction of the energy gap. It was found that the gap reduction due to band shift was more significant for armchair type tubes as shown in Fig. 3. The band gap of the armchair double-walled tube is reduced by about 0.4 eV in our calculations. Within LDA, the armchair type tubes have an indirect bandgap of about 4.4, 4.6, and 4.0 eV for (5,5), (10,10), and (5,5)C(10,10) tubes, respectively, from
Fig. 2. The band structure of zigzag BN nanotubes near the Fermi level. (a) (8,0), (b) (16,0), and (c) (8,0)C(16,0) double-walled BN nanotubes along tube axis, respectively. The top of the valence band is set at the zero of energy for all three plots. The energy gap for double-wall tube is slightly smaller than that of (16,0) tube. The valence band top states are mostly derived from the outer tubes, while the minimum of conduction bands comes from the states of the inner tube. The optical transition would thus occur through interwall hopping of electrons or holes.
Fig. 3. Similar plot of the band structure of armchair BN nanotubes near the Fermi level as in Fig. 2. (a) (5,5), (b) (10,10), and (c) (5,5)C(10,10) double-walled BN tubes along tube axis, respectively. The zero of energy is set to the level of the valence band maximum states for all three plots. The armchair tubes have an indirect gap from D to G points of the Brillouin zone. The energy gap for double-walled tube in our calculation is reduced by as much as 0.4 eV, which is significant compared to that for zigzag doublewalled BN tubes (Fig. 2).
the D to G points of the Brillouin zone. The reason for different band shift for zigzag and armchair double-walled BN nanotubes is unclear yet. This may originate from the different interwall stacking or interwall spacing of the tubes. It is noteworthy to mention that the conduction band minimum (CBM) states of the two tubes exhibit quite different characteristics as shown in Fig. 4. A signature of band gap reduction in multiwalled BN nanotubes was also reported in a recent EELS experiment that observed a downward shift of the energy loss peak by 0.6 eV with respect to the value in hBN [17]. The shift in the experiment is believed to be driven by the curvature as well as the interwall interactions in the tubes. More comprehensive studies need to be done to clearly distinguish the effect of the curvature and the interwall interactions in multiwalled BN nanotubes. For large diameter multiwall tubes, on the other hand, the energy-gap is expected to be close to that of hBN because the B–N bond buckling decreases and the interwall interaction will converge to that in hBN. Fig. 4 shows the squared wave-functions of the conduction band minimum states of the (8,0)C(16,0) zigzag (a,b) and the (5,5)C(10,1,0) armchair (c,d) doublewalled tubes in a plane perpendicular to the tube axis (a,c) and a plane along the tube axis (b,d) of the tubes. The plots of the squared wave functions reveal the characteristic symmetry of the corresponding states. The distribution of the wave functions is quite different for zigzag and armchair tubes. While the state at bottom of the conduction band of the armchair tube exhibits a signature of nearly-freeelectron-like character, the corresponding state of the zigzag tube shows a mixture of p and sp3 bonding character, and is mostly distributed in the inner tube. A weak anti-bondinglike interaction between inner and outer tubes is also seen in the conduction band minimum state of the zigzag tube
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Fig. 4. Squared wave functions of the CBM states of the (8,0)C (16,0) zigzag (a,b) and the (5,5)C(10,10) armchair (c,d) doublewalled BN nanotubes in a plane perpendicular to the tube axis (a,c) and also in a plane along the tube axis (b,d). The edge of the tube is indicated by dashed lines. The CBM states in the armchair tube shows a nearly-free-electron-like character while the corresponding states in the zigzag tubes have a typical p* bonding hybridized with sp3 bonding, especially in the inner tube. We also observe that there is a weak anti-bonding-like coupling between inner and outer tubes in zigzag nanotubes.
[Fig. 4(c)]. The unlike characters of the states at bottom of the conduction band in zigzag and armchair tubes may contribute to the different band shift when the double-walled nanotubes are formed. The states at top of the valence band, on the other hand, do not show so much difference as opposed to the conduction band minimum states. Weak interwall interactions split the degenerate states at top of the valence band at G point as mentioned above. Fig. 5 shows a similar plot as in Fig. 4 of one of the degenerated states of the zigzag nanotubes. Another state has an almost identical density distribution except a rotation by 908 along the tube axis (not shown here). Because the density of the VBM and CBM states is either distributed mostly in the inner and outer tubes, respectively, the optical transition from the VBM to CBM states or vice versa should occur through the interwall hoping of electrons or holes. A recent Raman spectroscopy study [20] observed significantly different decay processes of photoluminescence in multiwalled BN tubes compared to those in hBN, which was attributed to the
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Fig. 5. Squared wave function of the valence band maximum states of the (8,0)C(16,0) tube (a) in a plane perpendicular to the tube axis and (b) in a plane along the tube axis. The edge of the tube is indicated by dashed lines. There is a weak coupling between atoms in inner and outer layers. The almost-degenerate state looks similar but rotated by 908 along the tube axis (not shown here).
spatial separation of electron and hole wave functions. Possible applications for laser devices based on this property was suggested by Okada et al. [14].
4. Summary In this paper, we discussed the electronic and structural properties of double-walled BN nanotubes. It was, found that the formation energy of double-walled tubes can depend on the stacking structure of the tubes. According to our calculations, the interwall stacking gives a preference for the growth of zigzag nanotubes, which is supported by the experimental results. The energy band gap also exhibits a measurable difference for the zigzag and armchair type tubes chosen in this study. The indirect gap of the (5,5)C (10,10) double-walled armchair tubes has a smaller gap by 0.4 eV than the single-walled tubes, whereas the reduction of the band gap in the (8,0)C(16,0) zigzag tube is about 0.1 eV. The conduction band minimum states show quite different bonding character in zigzag and armchair tubes studied.
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Acknowledgements SHJ acknowledges the helpful discussion with C.-H. Park. This work was supported by the NSF under Grant No. DMR-04-39768, and by the Office of Energy Research, Office of Basic Energy Science, Material Sciences Division of the US Department of Energy under Contract No. DEAC03-76SF00098. Supercomputer time was provided by NCSA and NERSC.
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