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Physica E 30 (2005) 164–168 www.elsevier.com/locate/physe
Structural and electronic properties of the TiC nanotubes: Density functional-based tight binding calculations A.N. Enyashin, A.L. Ivanovskii Institute of Solid State Chemistry, Ural Division of the Russian Academy of Sciences, 620219 Ekaterinburg, Russia Received 19 May 2005; received in revised form 22 July 2005; accepted 28 August 2005 Available online 5 October 2005
Abstract Atomic models of the hypothetical single- and multi-walled cylindrical- and prismatic-like TiC nanotubes have been constructed and their structural and electronic properties have been studied by means of density functional-based tight binding (DFTB) method. The electronic bands, densities of states and binding energies are analyzed as a function of the TiC tubes sizes. Our calculations showed that TiC nanotubes are semiconducting, in contrary to the metallic-like crystalline TiC, and the band gaps tend to vanish as the number of tube walls increase. r 2005 Elsevier B.V. All rights reserved. PACS: 61.46; 73.22; 79.70.+q Keywords: Titanium carbide; Nanotubes; Electronic properties; Density functional theory
1. Introduction The discovery of carbon fullerenes [1] and nanotubes (NTs) [2] with unique properties [3,4] has stimulated the search of other inorganic cage-like structures as a promising materials for a various technological applications. The basic strategy was the search of suitable candidates among the layered inorganic materials, which may be converted into various cage-like nanoforms. Similar to the graphite sheets acting as a ‘‘precursor’’ of mentioned carbon nanostructures, the fragments of monolayers of mentioned phases can form the shells (or walls) for the new hollow clusters (or nanotubes) constructed from them with desirable structures. At present a lot of nanotubes and fullerene-like clusters of the inorganic layered materials (BN, BxCyNz, CNx, binary or ternary metal dichalcogenides, halides, some oxides, hydroxides, etc.) have been successfully predicted and synthesized, see [5–7]. On the other hand, the quite unexpected tubular or fullerene-like forms of other inorganic species are obtained Corresponding author. Tel.: +343 3745331; fax: +343 374495.
E-mail address:
[email protected] (A.N. Enyashin). 1386-9477/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2005.08.004
recently: for example, the nanotubes of the noble metals, MgO, SiC or various semiconducting materials (Si/Ge, Cr/ Si/SiGe) etc., see [5–7]. The metal carbides, especially d-metal monocarbides, MC (M ¼ Ti, Zr, Hf, V, Nb, Ta) are also very far from the layered materials and possess a high-symmetrical cubic (B1 type) structure and an isotropic system of highly covalent M–C bonds. These materials fall into the family of the refractory phases and exhibit high chemical inertness together with ultrahardness, excellent wear resistance and oxidation stability [8–10]. Therefore many efforts are made to produce d-metal monocarbides as nano-sized non-oxide ceramic powder materials [11]. The important progress was achieved also in producing and modeling of the new families of the d-metal carbide nanoforms. For example, TiC nanocrystals [12,13] as well as quasi-one-dimensional (1D) nanostructures—extended nanowires and nanorods [14–18] were prepared. Additionally various heterostructures combined from carbon nanotubes and MC nanorods are reported [19]. In 1992 Castleman et al. [20] have discovered the cage-like nanoclusters named metallocarbohedrenes (Ti8C12, Zr8C12, etc., see review [21]). Further, the TiC hollow spherical particles [22] and polyhedrons [23]
ARTICLE IN PRESS A.N. Enyashin, A.L. Ivanovskii / Physica E 30 (2005) 164–168
were obtained, as well as various types of the hybrid structures based on carbon fullerenes or NTs filled or coated with transition metals are reported [24–31], and their electronic [26–30], magnetic properties [30] and hydrogen storage capability [27,31] are predicted. Quite recently NbC nanotubes have been successfully synthesized [32]. Thus, nowadays the cage-like nanocarbides are no longer hypothetical structures and it is quite real to expect their fabrication with controllable size and diameter. In this paper, the atomic models for the hypothetical TiC nanotubes are proposed and their stability and morphological differences depending from tubes sizes are discussed. The band structure calculations were performed also in order to analyze their electronic properties. 2. Structural models and computational details In a standard fashion the atomic models of nanotubes of layered materials may be obtained as a simple rolling of the atomic sheets into seamless cylinders. In the case of TiC with the cubic lattice a realization of this way leads to the simple construction of a family of the cylindrical singlewalled tubes based on TiC (1 0 0) planar sheets. As alternative model, we have considered the ‘‘crystalline’’—like nanotubes of the prismatic morphology with square-like cross sections as depicted in Fig. 1. Note, that such morphology is known also for some inorganic nanotubes; in particular the numerous polygonized carbon nanotubes are predicted and observed [33–35]. Further we shall classify these 1D nanoforms using number of the inter-atomic Ti–C distances (n) on a square side of the starting structures. Then, for example, the single-walled TiC tube with square-like cross section presented on Fig. 1, may be labeled as 6-NT. In this way the multi-walled prismatic-like nanotubes may be classified as n @ m, n @ m @ l, etc.; the example of the double-walled 4 @ 6-NT is depicted also on the Fig. 1.
Fig. 1. The cross sections of the prismatic 1—single- (6) and 2—doublewalled (4 @ 6) TiC nanotubes before (a), after geometry optimization (b) (at T ¼ 0 K) and after MD simulation at T ¼ 300 K (c). Ti (J) and carbon () atoms are shown.
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Table 1 The average radii (/RS, A˚) for the TiC nanotubes considered Nanotubesa 2 3 4 5 6 7 8 9 10 11 12 13 14 15
/RS 2.79 4.20 5.50 7.00 8.29 9.80 11.09 12.80 14.08 15.38 16.90 18.25 19.49 20.73
a
(2.78) (4.18) (5.60) (6.97) (8.40) (9.85) (11.26) (12.66) (14.07) (15.48) (16.88) (18.29) (19.70) (21.10)
Nanotubes
/RS
16 17 18 2 3 4 5 6 7 8 2 3 4
22.45 (22.51) 23.79 (23.92) 25.18 (25.32) 4.40 5.96 7.38 8.96 10.38 11.96 13.38 5.99 7.54 9.01
@ @ @ @ @ @ @ @ @ @
4 5 6 7 8 9 10 4@6 5@7 6@8
a
The average radii for the prismatic-like nanotubes are determined as /RS ¼ P/2pk, where P is the perimeter of NT cross-section, and k—the number of the tube walls; for the cylindrical-like tubes R are given in parenthesis.
Fig. 2. The binding energies (Eb) as a function of the average radii /RS for TiC nanotubes. The data for the single-walled cylindrical (J) as well as prismatic single- (&), double- (m) and triple-walled (.) TiC NTs are presented.
In this work we have analyzed two series of infinite-long single-walled n TiC NTs (with starting square-like and circle cross sections) as a function of n from 2 to 18, and also some ‘‘crystalline’’-like (prismatic) multi-walled nanotubes with the averaged radii /RS as presented in Table 1. Our predictions were achieved using the atomistic simulations within a density functional-based tight binding scheme DFTB [36,37]. All 1D TiC nanostructures have been treated in supercell geometry, and all the atomic positions were optimized under the conjugate gradient relaxation. The initial Ti–C bond lengths were chosen the same as in bulk TiC (2.16 A˚ [9]). Additionally the electronic bands, total and partial densities of states (DOSs) of these nanostructures are obtained and analyzed.
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3. Results and discussion Fig. 1 illustrates typical initial and optimized atomic geometries of the single- and double-walled prismatic TiC nanotubes. First of all our calculations reveal, that these structures preserve the hollow configurations and indicate the possibility of the existence of the TiC tubular forms. For cylinder-like single-walled TiC NTs only small atomic rearrangements from the ideal cylindrical morphology are observed. Much pronounced relaxation effects are indicative for the single-walled ‘‘crystalline’’-like nanoforms; whereas multi-walled prismatic structures preserve as a whole the initial morphology, see Fig. 1. In addition, to proof the thermal stability of the investigated structures at least at room temperature, the molecular dynamic (MD) simulation was performed under constant volume and temperature conditions (NVT
ensemble). All MD simulations were carried out during 1000 iterations with a time step 2 fs. The examples of such structures at T ¼ 300 K (Fig. 1) demonstrate that the above mentioned morphology preserve also at the account of thermal effects. Thus, all these tests have provided strong evidence that the proposed TiC tubular forms are stable. Let us analyze the relative stability of the titanium carbide 1D nanostructures more detailed. For this purpose the binding energies (Eb) have been found as: Eb ¼ {[Eat(Ti)+Eat(C)]Etot(1D-TiC)}; here Eat(Ti,C) are the energies of individual Ti, C atoms, and Etot(1D-TiC) are the total energies of the optimized TiC 1D nanostructures. The results obtained (Fig. 2) show that the binding energies for the single-walled cylinder-like TiC NTs decreases monotonically with increasing of the tube radii, going nearly to Eb of TiC (1 0 0) planar sheet (about
Fig. 3. Electronic bands (dispersion along G-X direction of the Brillouin zone) and densities of states of the prismatic: 1— single- (8), 2— double- (6 @ 8) and 3— triple-walled (4 @ 6 @ 8) TiC NTs. The energies is given relative to E F ¼ 0. Total (full lines) and l-projected Ti 3d (-.-.-), 4p (- - -), 4 s (y) and C 2p (- - -), C2 s (y.) densities of states are presented.
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4. Conclusions
Fig. 4. The band gaps (BG) as a function of the average radii /RS for TiC nanotubes. The data for the prismatic single- (&), double- (m) and triple-walled (.) TiC NTs are presented.
7.19 eV/atom), whereas the stability of the 1D ‘‘crystalline’’ structures increase slightly with the tube radii growth. Much more significant stabilization is achieved with growth the number of tube walls, Fig. 2. This is an expected result, since it reflects the peculiarities in the bonding capabilities of Ti and C, when six-coordinated carbon tends to utilize all of three valence p orbitals, resulting in hybridization with Ti d orbitals, see [10]. Let us consider the electronic properties of TiC NTs. Fig. 3 shows the dispersion E(k) of the electronic bands (along G-X direction of the Brillouin zone, i.e. along the tube axis) as well as total and partial DOSs for some typical single- (8), double- (6 @ 8) and triple-walled (4 @ 6 @ 8 @) TiC nanotubes. For all TiC NTs the calculations provide similar band pictures. The lowest bands, with energy in the range from about 10 to 8 eV (the Fermi level E F ¼ 0 eV) are mainly composed by C 2s states. There is a gap of about 4.5 eV between the C 2s states and the bottom of the valence band. In turn the valence band is mainly composed by C 2p states. However, the density of Ti 3d states is high all over the valence band, which indicates a strong covalent interaction between Ti and C. The bottom of the conduction band is mainly composed by Ti 3d states. The top of the valence band and the bottom of the conduction band are placed in the G point of the Brillouin zone. Therefore, the considered TiC nanotubes are direct-gap semiconductors in contrary to the metalliclike crystalline TiC [10]. The main interesting differences in the electronic properties depending from the tube sizes (radii /RS and wall numbers) are connected with the near-Fermi bands. Fig. 4 illustrates the calculated behavior of the band gaps. All single-walled NTs remain semiconductors, however a nonuniform BG dependence with increasing /RS occur. Besides, the gap values with the growth of the wall number decrease rapidly. For example, the band gap values are about 0.8, 0.19 and 0.15 eV for 8, 6 @ 8 and 4 @ 6 @ 8 NTs, respectively.
In summary, the atomic models of the hypothetical TiC nanotubes are proposed and their structural, electronic properties and stability have been investigated for the first time using the DFTB band structure method. Our calculations showed that the TiC nanotubes in contrary to the metallic-like crystalline TiC are semiconducting and their band gaps tend to vanish, whereas the stability increases significantly with the increasing of the number of the tube walls. There are still numerous issues of interest for future studies. An important problem is the effect of polygonal cross-sections geometries on the electronic properties of the hollow (tubular) structures. Moreover, the most important properties of the bulk metal monocarbides are their extreme hardness and high melting points. Therefore the simulations of the mechanical and thermal properties of the 1D nanocarbides receive great interest. Acknowledgement Financial support of the RFBR (Grants 04-03-32111 and 04-03-96117) is gratefully acknowledged. A.E. thanks Prof. Dr. G. Seifert gratefully for the providing of the software and the TB parameters used in this work. References [1] H.W. Kroto, J.H. Heath, S.C. O’Brien, R.F. Curl, R.E. Smalley, Nature (London) 318 (1985) 162. [2] S. Iijima, Nature (London) 354 (1991) 56. [3] R. Saito, G. Dresselhaus, M.S. Dresselhaus, Physical Properties of Carbon Nanotubes, Imperial College Press, London, 1998. [4] P.J.F. Harris, Carbon Nanotubes and Related Structures: New Materials for the Twenty-first Century, Cambridge University Press, Cambridge, 1999. [5] R. Tenne, C.N.R. Rao, Phil. Trans. R. Soc. Lond. A 362 (2004) 2099. [6] M. Remskar, Adv. Mater. 16 (2004) 1497. [7] G.S. Zacharova, V.L. Volkov, V.V. Ivanovskaya, A.L. Ivanovskii, Nanotubes and Related Nanostructures of Metal Oxides, Ural Branch of RAS, Ekaterinburg, 2005. [8] H. Goldschmidt, Interstitial Alloys, Butterworths, London, 1967. [9] L.E. Toth, Transition Metal Carbides and Nitrides, Academic Press, NY, London, 1971. [10] V.A. Gubanov, A.L. Ivanovskii, V.P. Zhukov, Electronic Structure of Refractory Transition-Metal Carbides and Nitrides, Cambridge University Press, Cambridge, 1994. [11] A.I. Gusev, A.A. Rempel, Nanocrystalline Materials, Cambridge International Science Publishing, Cambridge, 2004. [12] A. Fukunaga, S. Chu, M.E. McHenry, J. Mater. Sci. Lett. 18 (1999) 431. [13] L. Shi, Y. Gu, L. Chen, Z. Yang, J. Ma, Y. Qian, Chem. Lett. 33 (2004) 56. [14] E.W. Wong, B.W. Maynor, L.D. Burns, C.M. Lieber, Chem. Mater. 8 (1996) 2041. [15] S.R. Qi, X.T. Huang, Z.W. Gan, X.X. Ding, Y. Cheng, J. Crystal Growth 219 (2000) 485. [16] C.H. Liang, G.W. Meng, W. Chen, Y.W. Wang, D.L. Zhang, J. Crystal Growth 220 (2000) 296. [17] X. Wang, J. Lu, P. Gou, Y. Xie, Chem. Lett. 31 (2002) 820.
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