SCIENCE CHINA Physics, Mechanics & Astronomy • Article •
April 2014 Vol.57 No.4: 644–651 doi: 10.1007/s11433-013-5387-8
Structural and electronic properties of chiral single-wall copper nanotubes DUAN YingNi1,2, ZHANG JianMin1* & XU KeWei3 1 College of Physics and Information Technology, Shaanxi Normal University, Xi’an 710062, China; Department of Medical Engineering and Technology, Xinjiang Medical University, Urumqi 830011, China; 3 College of Physics and Mechanical and Electronic Engineering, Xi’an University of Arts and Science, Xi’an 710065, China 2
Received August 13, 2013; accepted September 17, 2013; published online February 14, 2014
The structural, energetic and electronic properties of chiral (n, m) (3n6, n/2mn) single-wall copper nanotubes (CuNTs) have been investigated by using projector-augmented wave method based on density-functional theory. The (4, 3) CuNT is energetically stable and should be observed experimentally in both free-standing and tip-suspended conditions, whereas the (5, 5) and (6, 4) CuNTs should be observed in free-standing and tip-suspended conditions, respectively. The number of conductance channels in the CuNTs does not always correspond to the number of atomic strands comprising the nanotube. Charge density contours show that there is an enhanced interatomic interaction in CuNTs compared with Cu bulk. Current transporting states display different periods and chirality, the combined effects of which lead to weaker chiral currents on CuNTs. density-functional theory, Cu nanotube, structural property, electronic property PACS number(s): 71.15.Mb, 73.63.Fg, 64.70.Nd, 73.22.-f Citation:
Duan Y N, Zhang J M, Xu K W. Structural and electronic properties of chiral single-wall copper nanotubes. Sci China-Phys Mech Astron, 2014, 57: 644651, doi: 10.1007/s11433-013-5387-8
Metal quasi one-dimensional (1D) structures, such as nanowires and nanotubes, have attracted considerable attention due to their intriguing physical and chemical properties that are very different from bulk materials [1–4] and they are very promising building blocks for nanoelectronics [5] and nanoelectromechanical systems [6]. The recent rapid progress in experimental techniques enables us to fabricate these metal quasi 1D structures and measure their novel properties. Takayanagi and co-workers [7,8] successfully fabricated a tip-suspended gold nanowire with a diameter less than 2 nm in an ultrahigh-vacuum transmission electron microscopy (UHV-TEM). Interestingly, this very thin nanowire exhibits a helical multishell structure that has a specific “magic” size, as predicted theoretically [9]. Furthermore, they have also fabricated a single-wall helical gold *Corresponding author (email:
[email protected]) © Science China Press and Springer-Verlag Berlin Heidelberg 2014
nanotube with a small diameter of 0.4 nm in an UHV-TEM [2]. Then many experimental methods have been used to investigate the structural and functional properties of Au, Ag, Pt, Ti, Ni and Cu nanotubes [2,10–17] and nanowires [7–9,13,16]. These experimental works on nanotubes have motivated theoretical studies attempting to understand the formation, structural evolution and physical properties of the nanotubes [18–22]. Copper is a widely used commercial metal due to its availability and outstanding properties such as good strength, excellent malleability and superior corrosion resistance. Moreover, copper has excellent electrical conductivity and thermal conductivity, and therefore, quasi 1D copper nanostructures are being explored as inexpensive, electrically conducting additives to composites and transparent conducting electrodes for solar cell and flexible electronic devices. Among metal nanotubes, fewer efforts have phys.scichina.com
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been devoted to copper nanotubes (CuNTs). CuNTs with controllable inner diameter and open ends have been fabricated using electroless deposition on anodic aluminum oxide templates by Li et al. [14]. Kamalakar et al. [15] successfully fabricate the dense arrays of aligned single crystalline CuNTs using electrodeposition method in the presence of a rotating electric field. More recently, Meng et al. [16] find that the 1D anisotropic growth of Cu nanowires and nanotubes in solution is driven by axial screw dislocations. As well-characterized experiments on CuNTs remain challenging tasks, stability and formation mechanism of CuNTs determined by experiment remain elusive so far. Theoretical studies based on first-principles density-functional theory (DFT) have successfully provided comprehensive insight into the stability, electronic and magnetic properties of metallic nanotubes [18–22]. Senger et al. [18] have calculated the band structures, charge densities, and quantum ballistic conductance of many single-wall gold nanotubes and explained why the experimentally observed (5, 3) gold tube suspended between two gold tips is favored. Based upon first-principles calculations, Elizondo et al. [19] studied a variety of possible structures for single-walled silver nanotubes in 2006. Konar et al. [20] have carried out the electronic structure calculations for different kinds of achiral and chiral single-walled platinum nanotubes, double-walled platinum nanotubes, and helical multi-shell wire structures under the formalism of density functional theory. Cai et al. [22] investigated the influence of absorbates (CO molecule and O atom) and defects on the electronic and transport properties of Au (5, 3) and Au (5, 5) nanotubes via the first-principles method based on DFT. They found that both the absorbate and defect decrease the conductance of Au nanotube. But to our knowledge, no systematic studies have been done on the structural and electronic properties of single-wall CuNTs. So in this study, the structural and electronic properties of the single-wall CuNTs have been investigated by using the first-principles projector-augmented wave (PAW) potential within the DFT framework under the generalized-gradient approximation (GGA). The rest of the paper is organized as follows. In sect. 1, the calculation method is given. In sect. 2, the structures, stability and electronic properties of single-wall CuNTs are analyzed and discussed. Finally, the conclusions of the work are presented in sect. 3.
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of the systems [27]. The cutoff energy for the plane waves is chosen to be 400 eV, which is significantly higher than 273 eV provided for the copper in the PAW potential. We choose a conjugate-gradient algorithm to relax the ions into their ground states, and the energies and the forces on each ion are converged to less than 104 eV/atom and 0.02 eV/Å, respectively. The Brillouin zone integration is performed within the Gamma centered Monkhorst-Pack scheme using (1×1×15) k-points [28]. To avoid the numerical instability due to level crossing and quasi-degeneracy near the Fermi level, we use a method of Methfessel-Paxton order N (N=1) with a width of 0.2 eV. We set the periodicity of the CuNTs along the tube axis direction, and the 20 Å vacuum spaces perpendicular to the tube axis direction to eliminate the interaction between tube and its periodic images. In order to test the quality of the above parameters, the lattice parameter and binding energy for bulk Cu crystal in the face centered cubic (FCC) lattice has been calculated. The calculated lattice constant and binding energy of bulk Cu are 3.626 Å and 3.47 eV, respectively, which are in good agreement with experimental values of 3.615 Å and 3.49 eV [29]. The calculated lattice constant also agrees well with previous DFT-GGA value 3.64 Å [30].
2 Results and discussions 2.1
Stability of the structures
The starting geometries of the copper tubular structures have been obtained by simply rolling up the 2D triangular (111) lattice sheet of copper atoms and mapping the atoms onto the surface of a cylinder, comparable to rolling up a graphite sheet for a carbon nanotube, as described in Figure 1. The structure of a nanotube may be described entirely in
1 Methods The calculations are implemented within the Vienna ab-initio simulation package (VASP) [23–25] within the framework of DFT using the PAW [26] potential and a plane-wave basis set. The 3d104s1 is chosen as the valence electrons for Cu atom. To treat electron exchange and correlation, we chose the Perdew-Burke-Ernzerhof formulation of the GGA, which yields the correct ground-state structure
Figure 1 (Color online) The 2D triangular (111) lattice sheet of copper atoms forming the single-wall CuNTs. The nearest-neighbor distance between two copper atoms in (111) lattice sheet is chosen to be optimized value 2.556 Å. The chiral vector Ch is represented by Ch= na1+ma2 (n,m), where n, m are integers and a1, a2 are the unit vectors of the triangular copper sheet.
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terms of the length and chirality. The chirality and diameter are uniquely defined in terms of the magnitude of the components of the chiral vector C h na1 ma2 (n, m), where n, m are integers and a1, a2 are the unit vectors of the triangular copper sheet. Therefore, the chiral vector Ch connects two crystallographically equivalent sites on the sheet. Based on the symmetry of the triangular lattice, all possible configurations of the single-wall nanotubes can be represented by chiral vectors within the irreducible area enclosed by the two lines of symmetry, n=2m and n=m, in which the chiral angle θ ranges from 0° to 30°. A single-wall nanotube (n, m) consists of n close-packed helical strands. The axial cell dimension LZ0 of each nanotube is taken to be the minimum length to ensure the periodicity. In this manuscript, we consider the chiral (n, m) CuNTs with 3n6 and n/2mn, since (n, nm) nanotubes are equivalent to (n, m) nanotubes just with an opposite chirality. Taking three atomic strands nanotubes as examples, we only consider (3, 2) and (3, 3) tubes, since (3, 1) and (3, 0) tubes are equivalent to (3, 2) and (3, 3) tubes, respectively. Figure 2 depicts possible structures of the (n, m) CuNTs with 3n6 and n/2mn. There are two kinds of the thinnest nanotubes consisting of three atomic strands, i.e., (3, 2) tube with 14 atoms per unit cell, and (3, 3) tube with 6 atoms per unit cell. For four atomic strands nanotube, there are three kinds of nanotubes, (4, 2), (4, 3) and (4, 4) tubes with 4, 26 and 8 atoms per unit cell, respectively. Among them, (4, 2) tube is considered the thinnest crystalline nanotube with FCC like structure. The ultrathin tubes formed by three or four atomic strands also exhibit tubular structures, similar to the gold nanotubes observed in the previous experiment [2]. Three kinds of nanotubes with five atomic stands are considered for (5, m) tubes: among them,
Figure 2
(Color online) Atomic structure of (n, m) CuNTs.
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(5, 3) and (5, 4) tubes with different chirality, while (5, 5) tube with icosahedral packing. There are four kinds of nanotubes with six atomic strands, (6, 3), (6, 4), (6, 5) and (6, 6) tubes with 6, 28, 45 and 12 atoms per unit cell, respectively. First, we study the structure and stability of the fully relaxed (n, m) CuNTs, for which the initial structural parameters, optimized structural parameters and energetic properties are summarized in Table 1. There are N atoms in one unit cell of the tube with initial (optimized) periodic length LZ0 (LZ) and diameter D0 (D). The calculated binding energy Eb, string tension f and number of conductional channels G are tabled. The binding energy Eb for 2D triangular (111) lattice sheet of copper atoms is also listed for comparison. The diameter for an initial rolling up (n, m) nanotube is given by D0
d n2 m2 nm ,
(1)
where d=2.556 Å is the Cu-Cu bond length of the optimized 2D triangular Cu (111) lattice sheet. For the nanotubes, the optimized cell length LZ along the tube axis is slightly shorter than the initial cell length LZ0, while the optimized diameter D is slightly larger than the initial diameter D0 of the (n, m) tube rolled up from 2D triangular Cu (111) lattice sheet. This is because of the curvature effect, which weakens the bonds that are wrapping around the circumference of the tube. Thus, it is also found that the average bond length of fully relaxed tubes in the direction of the circumference is larger than those along the axis of tube. The increase of the diameter caused by optimization is remarkable in thin nanotubes due to their larger curvature. Consequently, the average bond length of the thin
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Table 1 Initial structural parameters, optimized structural parameters and energetic properties of (n, m) CuNTs. There are N atoms in one unit cell of the tube with initial (optimized) periodic length LZ0 (LZ) and diameter D0 (D). The calculated binding energy Eb, string tension f and number of conductional channels G are tabled. The binding energy Eb for 2D triangular (111) lattice sheet of copper atoms is also listed for comparison Structure (n, m) (3, 2) (3, 3) (4, 2) (4, 3) (4, 4) (5, 3) (5, 4) (5, 5) (6, 3) (6, 4) (6, 5) (6, 6) Sheet
N 14 6 4 26 8 38 14 10 6 28 45 12
Initial structure parameters LZ0 (Å) D0 (Å) 11.99 2.15 4.46 2.44 2.58 2.82 16.09 2.93 4.46 3.26 19.45 3.55 6.82 3.73 4.46 4.07 2.58 4.23 11.80 4.31 18.03 4.53 4.46 4.88
tube is larger than that of the thick tube. The binding energy Eb for a given tube is calculated as the difference of the energy of an individual atom Eatom and the total energy per atom Etot/N, i.e., Eb Eatom Etot /N . It is so defined that a positive value indicates a local or a global minimum of the Born-Oppenheimer surface which generally corresponds to a stable structure with respect to N-isolated Cu atoms [18]. The positive binding energy for all the nanotubes indicates that forming the nanotube is an exothermic process. Figure 3(a) shows a plot of the binding energy Eb as a function of the diameter D of the optimized CuNT. The binding energy Eb has an increasing tendency with increasing the optimized diameter D, reflecting the nanotubes with larger D are generally more stable. This is quite reasonable because the larger bending deformations are needed to obtain the smaller diameter tubes. In addition, the local maxima of the binding energy Eb for the (4, 3) and (5, 5) tubes also indicate these tubes demonstrate an enhanced stability compared to their immediate neighboring tubes. Notice that in the large diameter region, the binding energy approaches the value of 2.761 eV/atom for the binding energy of 2D Cu (111) sheet. Our calculations are done for the infinitely extended free-standing nanotubes, while the nanotubes fabricated in experiments and applied in interconnecters and nanodevices are finite and suspended between two electrodes. Therefore, it is important to know whether the free-standing but finite length CuNTs are stable. Hence, the binding energy Eb can not be taken as a criterion to decide on the long-lived metastable states of suspended nanowires or nanotubes. As discussed in a previous work [9], the string tension f is an important parameter for describing the stability of a nanowire or a nanotube in a tip-suspended condition. We calculate the string tension f of a nanotube, which is defined by considering the positive work required to draw the tube out of the bulk material. It is given by f ( F N ) / LZ . Here, is
LZ (Å) 10.91 3.95 2.46 15.14 4.15 18.37 6.30 4.09 2.43 11.11 16.84 4.19
Optimized structure parameters and energetic properties D (Å) Eb (eV/atom) f (eV/Å) G (G0) 2.56 2.422 1.40 3 2.89 2.430 1.65 3 2.95 2.454 1.73 4 3.15 2.591 1.59 3 3.47 2.590 1.78 3 3.64 2.630 1.83 5 3.88 2.640 1.95 3 4.22 2.677 2.05 5 4.46 2.655 2.13 6 4.54 2.710 2.03 5 4.55 2.734 2.09 5 4.89 2.739 2.23 5 2.761
the chemical potential of bulk copper and is chosen to be = 3.719 eV and F is the free energy of one unit cell of nanotube, which is equal to the total energy at zero temperature. Figure 3(b) shows the string tension f increases with increasing the optimized tube diameter D, except two local minima for (4, 3) and (6, 4) tubes with respect to their immediate neighboring tubes. This indicates the (4, 3) and (6, 4) tubes are two long-lived “magic” CuNTs. The “magic” (4, 3) gold tube and (6, 4) platinum tube were also obtained by Senger et al. [18] and Konar et al. [20], respectively.
Figure 3 Binding energy Eb (a) and string tension f (b) of the optimized CuNTs as a function of the nanotube diameter D. The binding energy 2.761 eV of the optimized 2D triangular Cu (111) lattice sheet is also shown with the dashed line in (a). The solid symbols indicate the local maxima in (a), but the local minima in (b).
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Electronic properties
Figure 4 shows the electronic band structure and total density of states (DOS) of the CuNTs at their equilibrium structures. For all 12 CuNTs considered, there are several bands crossing the Fermi level, which shows all the CuNTs are metallic. According to the Landauer formula [31], in the ideal situation, the quantum conductance of the system will be G M 2e2 /h MG0 , where M M ( Ef ) is the number of channels at the Fermi level. So the quantum conductance can be easily obtained by counting the number of channels at the Fermi level and the contribution of each channel to the quantum conductance is 2e2 / h independently from the band dispersion of the specific channel [32]. This method has been successfully used to determine the quantum conductance G of the Cu nanowires [33–36]. In this way, the quantum conductance of the free-standing CuNTs can be determined from the band structures by
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counting the number of the bands crossing the Fermi level. The determined quantum conductance properties of the CuNTs are also listed in the last column of Table 1. We can see that, similar to the case of the carbon nanotube, the quantum conductance of the CuNTs is sensitive to the chirality. The quantum conductance of 3G0 , 4G0 , 3G0 and 3G0 for (3, 3), (4, 2), (4, 3) and (4, 4) tubes are consistent with the previous results [33]. According to the dispersion of the bands near the Fermi level, the CuNTs can be divided into three groups. For (n, n/2) zigzag tubes, such as (4, 2) and (6, 3) tubes with straight atomic strands along the tube axis and short periodic length LZ of about 2.5 Å as well as a smaller number of the atoms per unit cell, the dispersions are large. For (n, n) staggered tubes, such as (3, 3), (4, 4), (5, 5) and (6, 6) tubes with the n atoms lying on one circumference of the tube and middle periodic length LZ of about 4.0 Å as well as middle number of the atoms per unit cell, the dispersions are middle. For (n, m) chirality tubes (3, 2),
Figure 4 (Color online) Electronic band structures and total density of states (DOS) of the optimized (n, m) CuNTs. The Fermi levels are shown with dashed lines. As one example, the partial DOS projected onto Cu 3d and 4s orbitals is shown in DOS plot of (4, 3) CuNT.
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(4, 3), (5, 3), (5, 4), (6, 4) and (6, 5) with a long periodic length LZ ranging from 6.3 Å to 18.37 Å as well as a large number of the atoms per unit cell, not only a small dispersion but also more bands are observed in the energy window. A nonzero DOS at the Fermi level also shows all CuNTs are metallic. As is shown in DOS plot of the (4, 3) tube, the partial DOS shows that the most occupied states ranging from 1 eV to 4 eV are dominant of the Cu 3d orbital which is completely filled by 10 electrons. The one Cu 4s electron contributes a less broad state across all the occupied energy window. Charge density contours of the optimized (n, m) CuNTs on the plane through at least one Cu atom and perpendicular to the tube axis are shown in Figure 5 together with those on the (110) (lower left) and (121) (lower right) planes perpendicular to the (111) plane of Cu optimized bulk
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crystal for comparison. The absence of charge in the central region of all the CuNTs indicates the hollowness of the nanotube, hence demonstrating the tubular character of the nanotube. The connected charge isodensity curves show all the CuNTs are metallic. The zigzag (n, n/2) nanotubes (staggered (n, n) nanotubes) can be obtained by rolling up partial atoms on (111) plane (solid lines on lower two panels) surrounding the [110] axis ( [121] axis). Comparing the charge density around the solid lines on the lower two panels for Cu bulk cases, we find that an enhanced interaction appears between the atoms of the CuNTs. This is because, although the atom on the tube-wall possesses half neighbors (6) of the FCC bulk Cu (12), the vanishing of the near neighbor atoms outside and inside the tube-wall makes the tube-wall atoms give their partial electrons, which are originally given to (or shared with) the vanishing neighbors to remaining neighbors.
Figure 5 (Color online) Charge density contours of the optimized (n, m) CuNTs on the plane through at least one Cu atom and perpendicular to the tube axis together with those on the (110) (lower left) and (121) (lower right) planes perpendicular to the (111) plane of Cu optimized bulk crystal for comparison. The charge isodensity curves are drawn from 0.04 e/Å3 to 4 e/Å3 with the increment of 0.04 e/Å3.
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Prediction of chiral currents in chiral nanotubes is of special interest due to the possibility of having chiral current along the helical strand and consequently they may have self-inductance and nanocoil effect [18,37]. To demonstrate whether the electrical current passing through a chiral CuNT has chirality, together with the structure of the (6, 4) chiral CuNT, we show the band-decomposed charge densities onto specific k-points p1, p2, p3, p4 and p5 at the Fermi level in reproduced band structure of the (6, 4) CuNT in c1, c2, c3, c4 and c5 plots of Figure 6. The local favorable (6, 4) chiral tube in tip-suspended condition is taken as an example because it has more quantum conductance than the other local favorable (4, 3) chiral tube. In order to describe the flow of the current, three reference points A, B and C on the tube structure are considered and shown by dark spots on each charge density plot. When the current flows from the top to the bottom, the geometry of the atomic position in the tube provides three distinct circumferential directions for the flow of the current, which are defined as AB, AC and BC directions. When the flow of the current along the AB and AC directions is defined as the left-
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handed helices, the current with the opposite chiral direction motion along the BC direction is right-handed. The period and direction of chirality for different channels are quite different. From Figure 6 we can see c2 and c4 have right-handed chiral current, while c1, c3 and c5 have the opposite chiral current. Thus, the chirality of the net current depends on the combined “nanocoil” effect of the conduction channels and the resultant chirality effect may be weaker than one expects. This simple method was also used by Senger et al. to predict the chiral currents of the chiral (5, 3) gold single-wall nanotube [18].
3 Conclusions In summary, the structural, energetic and electronic properties of chiral (n, m) (3n6, n/2mn) single-wall CuNTs have been investigated by using the first-principles PAW potential within the DFT framework under GGA. The following conclusions are obtained: (1) Due to the curvature effect, the optimized cell length
Figure 6 (Color online) Left: Band structure of the local favorable (6, 4) chiral CuNT. The Fermi level is shown with a dashed line. Five bands crossing the Fermi level and the corresponding specific k-points p1, p2, p3, p4 and p5 at the Fermi level are denoted by the solid dots. Right: A schematic description of the tube structure, and isosurfaces for the band-decomposed charge densities c1, c2, c3, c4 and c5, respectively corresponding to the specific k-points p1, p2, p3, p4 and p5 at the Fermi level in band structure. The value for the isosurface is 0.0007 e/Å3.
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LZ along the tube axis is slightly shorter than the initial cell length LZ0, while the optimized diameter D is slightly larger than the initial diameter D0 of the (n, m) tube rolled up from 2D triangular Cu (111) lattice sheet. (2) Binding energy and string tension analysis reveal that the (4, 3) CuNT is energetically stable and should be observed experimentally in both free-standing and tipsuspended conditions, whereas the (5, 5) and (6, 4) CuNTs should be observed in free-standing and tip-suspended conditions, respectively. (3) The number of conductance channels in the CuNTs does not always correspond to the number of atomic strands comprising the nanotube. (4) Charge density contours show that there is an enhanced interatomic interaction in CuNTs compared with Cu bulk. (5) Current transporting states display different periods and chirality, the combined effects of which lead to weaker chiral currents on CuNTs.
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23 This work was supported by the State Key Development for Basic Research of China (Grant No. 2010CB631002) and the National Natural Science Foundation of China (Grant Nos. 51071098, 11104175 and 11214216).
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