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Stability and Electronic Properties of Bismuth Nanotubes Bertold Rasche, Gotthard Seifert,* and Andrey Enyashin Physical Chemistry, Technical UniVersity Dresden, 01062 Dresden, Germany ReceiVed: August 27, 2010; ReVised Manuscript ReceiVed: October 29, 2010
Recently fabricated double-walled bismuth nanotubes were the first elemental nanotubes with clearly expressed layered structure analogous to carbon nanotubes. Here the results from quantum-mechanical density-functional tight-binding (DFTB) calculations are presented for Bi nanotubes of experimentally observed diameters. The calculations uncover the nature of the experimentally observed shape of bismuth nanotubes as an interplay between relatively low strain energies of Bi nanotubes and rather strong van der Waals interaction between the layers. They evidence the stability of the hexagonal prismatic morphology for double-walled zigzag nanotubes. Band structure calculations reveal that the Bi nanotubes should be semiconducting with the band gap around 0.5 eV. Introduction Carbon nanotubes have attracted attention in many fields of research due to their specific electronic, thermal, and mechanical properties, giving a great expectation for their successful applications in the synthesis of materials for micro- and nanoelectronics as well as reinforced composites.1 Soon after the discovery of carbon nanotubes, numerous attempts were performed in order to fabricate noncarbon (inorganic) nanotubes. Despite much expectation for the successful synthesis of graphene-related BN nanotubes, the first inorganic nanotubes were synthesized from sulfides with a layered structure as WS2 and MoS2.2,3 One of the first theoretical studies predicting the stability and electronic properties of elemental noncarbon nanotubes included single-walled graphene-like boron and silicon nanotubes.4-6 However, until now, there has been no report of the successful synthesis of the corresponding nanotubes. It is widely believed that carbon easily forms fullerenes and nanotubes by the formation of strong π-bonds through sp2 hybridization, a property almost missing in the case of silicon. Eventually, hexagonal networks of silicon were synthesized as buckled layers by a chemical stabilization of the sp3-state, e.g., by saturation of the free valences, for example, by hydrogen atoms, as shown for silane SiH5 and similar germane GeH6 monolayers and nanotubulenes. The most stable allotropes of boron are based on covalently bound networks of boron polyhedra (e.g., B12),7 whereas the formation of coordinatively unsaturated graphene-like structures is unlikely.8 However, there are elements that appear in layered allotropes. These elements are P, As, Sb, and Bi. They can form hexagonal atomic networks like in graphite.9 In contrast to graphite, the structure of these layers is puckered due to the presence of “lone pairs”, responsible for a weak van der Waals interaction between the layers. First theoretical calculations of the nanotubes based on the elements of Group 15 (phosphorus) indicated the stability of such nanostructures.10 However, there have been no reports about the successful preparation of P or As nanotubes. Only antimony nanotubes were prepared as a set of tubular-like structures as hollow nanosized polycrystallites.11,12 * Corresponding author. E-mail:
[email protected].
More successful were the attempts for the synthesis of bismuth nanotubes.13-16 The first experiments yielded in small amounts poorly crystallized multilayered Bi nanotubes13,14 or nanotubulenes with walls composed of polycrystallites.15,16 Recently, a new route for a high-yield synthesis of very uniform (with radii R ∼ 25-27 Å) double-walled bismuth nanotubes was presented.17 Interestingly, TEM images have revealed a tendency of these nanotubes for a faceting of the walls. However, the resolution of the TEM images was not sufficient to characterize the details of their atomic structure. Meanwhile, results of quantum-mechanical studies on the stability and electronic properties of small single-walled and ultrasmall double-walled bismuth nanotubes have been published.18,19 As a result of these calculations, these bismuth nanotubes can be characterized as semiconductors. Furthermore, it was found that their strain energies are inversely proportional to R2. In this paper, the first results of quantum-mechanical calculations for double-walled bismuth nanotubes with diameters corresponding to the experimental ones are presented. Their stability and electronic properties are compared with those of the bulk and low-dimensional allotropes of bismuth, such as mono- and bilayer, single-walled nanotubes and single- and double-walled nanostripes. We observe a spontaneous faceting of the double-walled nanotubes after geometry optimization and explain this phenomenon by a fine balance between relatively low strain energies of the bismuth nanotubes and a quite strong van der Waals interaction between the bismuth layers. Hexagonal prismatic nanotubes of zigzag chirality with R ≈ 11-30 Å are found to be the most stable nanostructures, which agrees with experimental findings.17 Finally, the electronic structure of bismuth nanotubes and layers is discussed, and a good agreement is found with the results of previous calculations,19 supporting the validity and applicability of the DFTB approach for studies of bismuth nanostructures. Method and Models All calculations of the electronic structure and geometry optimization were performed by means of the density-functional tight-binding method (DFTB)20,21 as released in DFTB+ and DyLax software packages.22 The calculations were performed within a scalar relativistic approach.23 The Kohn-Sham orbitals for bismuth were described in a 6s6p6d atomic orbital basis.
10.1021/jp1081565 2010 American Chemical Society Published on Web 11/24/2010
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TABLE 1: Bond Lengths dBi-Bi, Lattice Parameters a, and Valence Angles rBi-Bi-Bi for the Hexagonal Bulk and Some Two-Dimensional Allotropes of Bismuth As Calculated by the DFTB Method system bulk bilayer monolayer a
dBi-Bi dBi-Bi (intralayer), Å (interlayer), Å 3.072 3.071a 3.052 3.036
3.504 3.529a 3.626 -
a, Å
RBi-Bi-Bi, deg
4.546 4.542a 4.520 4.450
95.34 95.48a 95.53 94.25
Experimental values.
The study of bulk Bi has shown the necessity of the inclusion of d-orbitals, since a 6s6p-basis failed in a correct description of the lattice parameters of bulk Bi. We optimized all atomic positions as well as the lattice parameters for all structures employing periodic boundary conditions and using the conjugated-gradient method until the residual force was below 10-5 Hartree/Å. The validity and accuracy of the approach have been proved by the comparison of the calculated lattice parameters and band structure for bulk Bi with corresponding experimental and theoretical data from the literature.19,24 Only a small error was found for the lattice parameters and bond lengths within the DFTB approach, which does not exceed 0.7%, compared with experimental data (see Table 1). The largest deviation was observed in the estimation of the interlayer distance. An analysis of the band structure using DFTB for the bulk bismuth (Figure 1) shows only small differences in comparison to results of scalar relativistic calculations of the band structure, taken from the literature.19,24 We consider the stability of the bismuth nanostructures, constructed by “cutting” of hexagonal Bi monolayers into nanostripes and by rolling them to cylindrical nanotubes. Like for carbon, the bismuth cylindrical nanotubes may be described in terms of the primitive two-dimensional hexagonal lattice vectors a1 and a2 with |a1| ) |a2| ) a and two integer indices n and m creating rolling vector B ) na1 + ma2, and divided into three groups: n ) m “armchair”, n * 0, m ) 0 “zigzag”, and n * m “chiral” nanotubes. In the present study the nanotubes with n until 40 were considered, which allows a direct comparison in the size range with the synthesized nanotubes.17 Results and Discussion Structure. The calculated structural parameters of the Bi monolayer, bilayer, and the bulk are summarized in comparison
Figure 2. Optimized structures for some single-walled bismuth nanotubes: (a) with zigzag chirality and stable circular cross-sections, (b) with zigzag chirality and metastable square-like cross-sections, and (c) with armchair chirality.
with the experimental data for bulk Bi in Table 1. As mentioned above, the calculated data for the bulk material agree very well with the experimental results. The comparison of the geometries between bulk, bilayer, and monolayer shows a shortening of the Bi-Bi bonds within layers and a slight increase of the interlayer distance, going from the bulk to the bilayer structure. This may be viewed as evidence for stronger covalent bonds within the Bi monolayer than in the bilayer or bulk. The results of the geometry optimization show that the geometry of single-walled bismuth nanotubes depends considerably on their chirality (Figure 2). While all armchair singlewalled nanotubes preserve circular cross-sections, zigzag singlewalled nanotubes can be found either also with circular crosssections or with square-like cross-sections. Armchair and zigzag nanotubes with circular cross-sections show similar trends for the geometry as a function of size. With increasing radius, the bond lengths and bond angles approach those of the monolayer. However, for prismatic zigzag nanotubes, the bond lengths and angles within the facets are already for the smallest bismuth nanotubes almost equal to those of the
Figure 1. Band structures of the bulk, mono- and double layer of bismuth as calculated using the DFTB method.
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Figure 3. Optimized structures for some double-walled bismuth nanotubes: (a) with zigzag chirality, and (b) with armchair chirality.
monolayer, whereas along the corners the deviations from the monolayer are clearly established. For double-walled bismuth nanotubes, a spontaneous faceting during the geometry optimization was found for armchair as well as for zigzag nanotubes (Figure 3). However, zigzag nanotubes of small diameter have a more circular-like crosssection, whereas for large diameters, they adopt a hexagonal cross-section. All armchair nanotubes are found with rectangular cross-sections after geometry optimization. Both kinds of faceted nanotubes are characterized by Bi-Bi bond lengths within the nanotubes, which are close to that for the bilayer. The interlayer distance between bismuth walls dBi-Bi(interlayer) is also quite close to that in the Bi bilayer, but there is a variation along the circumference of the nanotubes. The interlayer distance (dBi-Bi(interlayer)) varies for zigzag and armchair nanotubes between 3.12 and 3.96 Å, and 3.11 and 4.00 Å, respectively, in comparison to dBi-Bi(interlayer) ) 3.626 Å for the bilayer. These rather short interlayer distances together with the “faceting phenomenon” reflect a quite strong interlayer interaction in the double-walled Bi nanotubes.
Rasche et al. Our finding of the faceting of double-walled Bi nanotubes could be proved by experimental data.17 However, it is faced with the problem of the low melting point of bismuth, which essentially hinders the observation of a direct image by high resolution transmission electron microscopy (TEM) due to disintegration of the bismuth nanotubes. Stability. Mono- and bilayered nanostripes are formed together with the double-walled nanotubes by the currently used synthetic routes of bismuth nanotubes.13,17 Therefore, we considered in our stability discussion the tubular structures as well as the nanostripes. While a nanotube is characterized by a strain energy compared to the plane layer, a nanostripe suffers from the dangling bonds along the edges. Double-walled nanotubes and nanostripes gain energy due to the interlayer interaction, i.e., the energies of nanotubes ENT and nanostripes ENS can be written as:
ENT ) ε∞ + εstr +
k-1 ε k vdW
(1)
ENS ) ε∞ + εx +
k-1 ε , k vdW
(2)
where ε∞, εvdW, εstr, and εx are the energy of an infinite Bi monolayer, the terms responsible for the van der Waals interaction between layers in k-layered nanostructures, the strain energy in nanotubes, and the energy due to dangling bonds in nanostripes, respectively. Such a model was already successfully applied for the study of the stability of other noncarbon (e.g., MoS2, TiO2) nanotubes, nanostripes, and nanorolls.25,26 The energy of interlayer interaction εvdW ) -0.1905 eV/atom was obtained by the calculation of the energy for two separated Bi monolayers and the Bi double layer (Table 1). It is remarkable that the strength of the interlayer interaction for bismuth is clearly larger than that for carbon in graphite (εvdW ) -0.0435 eV/atom).27
Figure 4. Energies of the bismuth nanostructures as a function of the radii (R, nanotubes) or width (R, nanostripes). Single-walled zigzag (2) and armchair cylindrical nanotubes (1), single-walled zigzag prismatic nanotubes (*), double-walled zigzag (() and armchair prismatic nanotubes (9), single- (b) and bilayered nanostripes (+).
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Figure 5. Band structures of bismuth nanotubes with small and large radii for different chiralities and number of the walls.
The strain energy per atom εstr can be obtained from the energies of a set of Bi nanotubes, which preserve their circular cross-sections. From the theory of elasticity and numerous studies of carbon and noncarbon nanotubes, this term is inversely proportional to the square of the tube radii R as εstr ) s/R2.28 The calculated values of strain energies of both armchair and zigzag single-walled Bi nanotubes with circular cross-sections also follow such behavior (see Figure 4) with s ) 4.08 eV · Å2/ atom. This result is in agreement with previous calculations of single-walled Bi nanotubes.19 The value of the slope factor for Bi nanotubes is twice as large as that for carbon nanotubes (s ) 2.18 eV · Å2/atom)29 and almost ten times smaller than that for MoS2 nanotubes (s ) 27.2 eV · Å2/atom).30 These ratios correlate well with the thickness of the corresponding monolayers, since a folding of sandwiched triatomic S-Mo-S monolayer should be more hindered than for monatomic graphene or bismuth layers. Additionally, it is identical to the calculated value of s ) 4.1 eV · Å2/atom for the hypothetical
single-walled phosphorus nanotubes.31 The stability for the faceted single-walled nanotubes can also be approximated by a 1/R2 dependence, but with a twice higher slope factor s ) 8.14 eV · Å2/atom. At R f ∞, the strain energies of all types of singlewalled nanotubes approach the energy of the infinite monolayer. The calculated energies for all single-walled nanotubes and nanostripes are summarized together with the curves using eqs 1 and 2 in Figure 4. It is seen that such nanotubes should be less stable than the other structures for R > 6 Å. For R < 6 Å, the strain energies of single-walled nanotubes are very high, and they are less stable than the corresponding nanostripes. The single-walled zigzag nanotubes with square-like cross-sections have energies higher than those for single-walled nanotubes with circular cross-sections, which demonstrates their low stability, and evidence for the synthesis of faceted single-walled nanotubes will be even less possible than for cylindrical ones. All optimized double-walled nanotubes were found only with a faceted morphology (Figure 3). The formation of polygonal
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cross-sections can be explained by the influence of interlayer interaction in bismuth, which is five times stronger than that in graphite (see above). A stronger interaction between the walls in a double-walled nanotube supports a prismatic-like crosssection. Despite the high strain energy of the correspondingly occurring edges of such faceted nanotubes, the strain energy in the flat regions is close to zero. Thus, in total, the strain energy of a faceted nanotube becomes smaller than that of a nanotube with a circular cross-section. Due to the quadratic cross-sections, the armchair double-walled nanotubes have a slope factor, s ) 9.5 eV · Å2/atom, much larger than that of zigzag nanotubes with hexagonal cross-sections, s ) 7.1 eV · Å2/atom. However, both types of nanotubes become more stable than the Bi monolayer, with R > 10 Å and approaching the stability of the Bi bilayer. The term εx in eq 2 depends on the perimeter (R) of a nanostripe and the number of dangling bonds (t) per unit cell: εstr ) t∆ε/R, where ∆ε is the difference between energies of the atoms in an infinite monolayer and the energies of the atoms at the edge of a nanostripe. We determined ∆ε ) 1.48 eV/ atom. For double-layered nanostripes, the term responsible for van der Waals interlayer interaction is the same as for doublewalled nanotubes, and their energies can be fitted according eq 2. The calculated and fitted energies of the single- and doublelayered Bi nanostripes are also shown in Figure 4. They illustrate an increase in the stability of the nanostripes with an increasing width and number of layers k. In contrast to nanotubes, the energy function for both zigzag and armchair types of the nanostripes do not differ but depends on k only. Since the functions of the energies for nanotubes and nanostripes have a different dependence on R (ENT ∼ 1/R2 and ENS ∼ 1/R), crossing points may occur that favor one or the other of these nanostructures, which is illustrated in Figure 4. In the case of single-walled nanostructures, the nanotubes are the most stable at R > 4 Å. Most interesting is the relative stability of double-walled nanotubes and nanostripes. According to the results of the calculations, there should be a region of R between ∼11 and 30 Å where the stability of zigzag nanotubes and nanostripes is almost equal. Outside of this area the nanostripes are the most stable nanostructures. However, the energy difference between tubular structures and layered structures for R > 30 Å is quite small. Thus, our estimations agree with the experimental results17 that nanotubes with R ≈ 25 Å can be synthesized as the most energetically favored structures. Electronic Properties. DFTB calculations of the band structure for the bismuth mono- and bilayers indicate an essential difference of their electronic structure around the Fermi energy in comparison to the bulk (Figure 1). The mono- and bilayers are semiconductors with band gaps around 1-2 eV, whereas bulk Bi shows metallic-like character. Single-walled bismuth nanotubes are also semiconductors with direct band gaps for zigzag tubes and indirect band gaps for armchair tubes of about 2 eV nearly independent of their chirality and radii (Figure 5). This band gap is slightly smaller than that of the monolayer. Double-walled Bi nanotubes retain semiconducting character with slightly smaller band gaps of about 0.5-1.0 eV for armchair and 0.5 eV for zigzag nanotubes (Figure 5). The band gaps of zigzag nanotubes show a less pronounced dependence on the radii than for the armchair tubes. Conclusions At present, bismuth nanotubes are the only known elemental nanotubes as morphologically identical analogues of carbon nanotubes.17 The DFTB approach allowed the study of Bi
Rasche et al. nanotubes with experimentally observed sizes and number of the walls. The results of our simulations uncover the nature of the faceted shape of bismuth nanotubes, which was also indicated in TEM experiments and is a phenomenon not yet observed for known nanotubes of other layered compounds. The geometry optimization and energy estimations evidence the spontaneous formation of a prismatic morphology of doublewalled bismuth nanotubes as more favored relative to the cylindrical nanotubes with a circular cross-section, i.e., the formation of faceted bismuth nanotubes is a thermodynamically gainful process. Further, zigzag double-walled nanotubes with hexagonal cross-sections were found to be more stable than nanotubes of armchair chirality and all single-walled nanotubes because of the relatively strong van der Waals interaction between the bismuth layers. The stability of these nanotubes with radii between 11 and 30 Å is comparable with the stability of double-walled nanostripes, which can be obtained at any size region and are the most stable bismuth nanostructures among the considered structures. Concerning the electronic structure of bismuth nanotubes and layers, it was shown that a consideration of only 6sBi- and 6pBiorbitals is not sufficient to describe both structural and electronic properties of the bulk bismuth. Extending the valence basis set with 6dBi-orbitals improves considerably the picture of the band structure and results in densities of states, which are in reasonable agreement with those of previous calculations of the Bi band structure.19,24 We have found that experimentally observed double-walled nanotubes should behave as semiconductors as opposed to the metallic character of the bulk bismuth. It opens new perspectives for the application of bismuth, which in the bulk state shows an exceptionally large Hall effect, for the preparation of sensitive nanodetectors of magnetic fields. Since the magnetoresistance measurements have already been performed for carbon nanotubes,32 similar work for bismuth nanotubes is very promising. Acknowledgment. The authors are grateful for the support from European Commission (ERC grant INTIF 226639). We thank Daniel Ko¨hler for interesting discussions. References and Notes (1) Jorio, A.; Dreselhaus, G.; Dresselhaus, M. S., Eds. Carbon Nanotubes. In Topics in Applied Physics; Springer-Verlag: Berlin; Vol. 111, 2008. (2) Tenne, R.; Margulis, L.; Genut, M.; Hodes, G. Nature 1992, 360, 444–446. (3) Margulis, L.; Salitre, G.; Tenne, R.; Talianker, M. Nature 1993, 365, 113–114. (4) Fagan, S. B.; Baierle, R. J.; Mota, R.; da Silva, A. J. R.; Fazzio, A. Phys. ReV. B 2000, 61, 9994–9996. (5) Seifert, G.; Ko¨hler, T.; Urbassek, H. M.; Herna´ndez, E.; Frauenheim, T. Phys. ReV. B 2001, 63, 193409. (6) Seifert, G.; Ko¨hler, T.; Hajnal, Z.; Frauenheim, T. Solid State Commun. 2001, 119, 653–657. (7) Boustani, I.; Quandt, A.; Kramer, P. Europhys. Lett. 1996, 36, 583– 588. (8) Ivanovskaya, V. V.; Enyashin, A. N.; Sofronov, A. A.; Makurin, Yu. N.; Medvedeva, N. I.; Ivanovskii, A. L. J. Mol. Struct. (THEOCHEM) 2003, 625, 9–16. (9) Norman, N. C. Chemistry of Arsenic, Antimony and Bismuth; Thomson Science: London, 1998. (10) Seifert, G.; Herna´ndez, E. Chem. Phys. Lett. 2000, 318, 355–356. (11) Hu, H.; Mo, M.; Yang, B.; Shao, M.; Zhang, S.; Li, Q.; Qian, Y. New J. Chem. 2003, 27, 1161–1163. (12) Wang, D.; Yu, D.; Peng, Y.; Meng, Z.; Zhang, S.; Qian, Y. Nanotechnology 2003, 14, 748–751. (13) Li, Y.; Wang, J.; Deng, Z.; Wu, Y.; Sun, X.; Yu, D.; Yang, P. J. Am. Chem. Soc. 2001, 123, 9904–9905. (14) Wang, J.; Li, Y. AdV. Mater. 2003, 15, 445–447. (15) Li, L.; Yang, Y. W.; Huang, X. H.; Li, G. H.; Ang, R.; Zhang, L. D. Appl. Phys. Lett. 2006, 88, 103119.
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