ARTICLE pubs.acs.org/JPCC
Electric Field and Size Effects on Atomic Structures and Conduction Properties of Ultrathin Cu Nanowires Cheng He,†,* Wenxue Zhang,‡ and Juanli Deng‡,§ †
State Key Laboratory for Mechanical Behavior of Materials, School of Materials Science and Engineering, Xi’an Jiaotong University, Xi’an 710049, China ‡ School of Materials Science and Engineering, Chang’an University, Xi’an 710064, China § School of Natural and Applied Sciences, Northwestern Polytechnical University, Xi’an 710072, China ABSTRACT: The ballistic transport properties of Cu nanowires (NWs) with diameter of 0.2-1.0 nm under electric field (V = 1 V/Å) are reported for future applications as interconnections in microelectronics. Our density-functional calculations show that, under V = 1 V/Å, with the wire diameter increasing, the number of conduction channels of a helical atomic strand increases, whereas the number of a nonhelical atomic strand is constant within the considered size range. The structure, electronic, and charge properties of these two types of Cu NWs exhibit distinctly different behaviors.
1. INTRODUCTION There is increasing interest in developing metal nanowires (NWs) for possible applications for molecular electronic devices.1 Cu as an interconnection in microelectronics is the most useful metal and the electronic transport of nanosized interconnections is one of the important characteristics for future microelectronic applications.2,3 After the realization of the fabrication metallic NWs, physical properties of metallic NWs are measured and calculated, especially their electron transport properties. As the length and width scales of NWs are reduced to the mean free path of electrons, the electron transport mechanism changes from diffusive to ballistic. Now the electric conductance is independent of the length of the NWs and the quantum conduction G has been observed4 as expected from the Landauer formula.5 G is quantized in units of G0 = 2e2/h where e denotes the electronic amount and h the Planck constant. Valence charge polarization by the locally entrapped core electron could be a possible mechanism for the ballistic transportation in the NW.6-8 The quantum conductance of metallic nanowires is an exciting emerging field of both fundamental and applied relevance9-11 because metallic nanowires are building blocks for nanoelectronics12,13 and nanoelectromechanical systems (NEMS).14 In the past decades, ultrathin metal nanowires produced by the tip retracting from nanoindentation in scanning tunnelling microscopy (STM), or by a mechanically controllable break junction (MCBJ), have been subjects of numerous experimental and theoretical studies.9,12,15 Both atomic structures and size of NWs affect the transport properties.5,16-18 It is found that G of the pentagonal Cu NWs with a [110] orientated structure is about 4.5G0 without electric fields.4 When NW is sufficiently thin, beside the conventional structure,19 it can turn exotic.20 However, the effect of electric field strength V on G with different r 2011 American Chemical Society
sizes, which is the precondition of the electronic transport, has not been considered systemically up to now. In this contribution, the structures and conduction properties of ultrathin Cu NWs are determined by first-principles DFT calculations. The atomic structures with diameter from 0.21.0 nm have been optimized. The density of states (DOS), G(V) function and the electronic distribution are also performed to determine changes of atomic and electronic structures of Cu NWs under electric fields.
2. COMPUTATIONAL DETAILS The simulation is calculated by first-principles DFT, which is provided by DMOL3 code.21,22 The generalized gradient approximation is employed to optimize geometrical structures and calculate properties of Cu NWs with the Perdew-BurkeErnzerhof correlation gradient correction.23,24 The all-electron relativistic Kohn-Sham wave functions are expanded in the local atomic orbital basis set. The atomic orbitals are represented by the double numerical basis including a d-polarization function basis set. The Cu NWs are modeled in a tetragonal supercell with 1D periodical boundary conditions along the NWs. Our outlines of the used structures are directly referred to the results of Wang et al.25 The length of Cu NWs (L), which is determined by the distance of the projection of mean locations of atom centers in the first and 10th layers on the axis of the NWs, are chosen to be 1.64 and 2.04 nm. It is because the distance between two neighbor layers with a core atom in the NW is about (2)1/2/2 times the distance without a core atom in NW. If L is shorter than Received: December 17, 2010 Revised: January 12, 2011 Published: February 09, 2011 3327
dx.doi.org/10.1021/jp1119876 | J. Phys. Chem. C 2011, 115, 3327–3331
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Figure 1. Morphologies of Cu NWs as a function of V. V in V/Å. The arrow shows the direction of V. 3a, 3b, 4a, 4b are shown in (A) and 6-1a, 6-1b, 12-6-1 are shown in (B).
6 layers, the helicity is hard to represent. If L is too long, it is too costly in computational quantity and cannot improve the results for the periodical boundary conditions. Hence, we choose 10 layers. The k-point is set to 5 5 1 for all slabs, which brings out the convergence tolerance of energy of 2.0 10-5 Ha (1 Ha = 27.2114 eV), maximum force of 0.004 Ha/Å, and maximum displacement of 0.005 Å. V is directly applied along the axis of Cu NW in the DMOL3 program with values of 0 and 1 V/Å. G(V) values are determined by the Landauer formula.26 The layer electronic distributions are carried out by the Mulliken charge analysis,27,28 which is performed using a projection of a Linear Combination of Atomic Orbitals (LCAO) basis and to specify quantities such as atomic charge, bond population, charge transfer, and so forth. LCAO supplies better information regarding the localization of the electrons in different atomic layers than a plane wave basis set does. The obtained charge e, but not the absolute magnitude, displays a high degree of sensitivity to the atomic basis set and a relative distribution of charge with which e was determined.29,30
3. RESULTS AND DISCUSSION Figure 1 shows seven representative structures of Cu NWs obtained in the calculations. To characterize the structures of Cu nanowire, one can introduce the notation n1-n2-n3-n431
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(n1 > n2 > n3 > n4, from outer to inner) to describe a multishell nanowire consisting of coaxial tubes with n1, n2, n3, n4 atomic rows on each shell where n shows the atomic number of one atomic layer in the surface cell. Division as category, there are two different structures of Cu NWs where a represents a nonhelical strand and b is a helical one. Thus, the structures in part A of Figure 1 is defined as 3a, 3b, 4a, 4b and part B of Figure 1 is 6-1a, 6-1b, and 12-6-1, where 12 shows the atomic number of one atomic layer in the surface cell, 6 is in second-surface cell, and 1 represents that in the core.25 Note that there could be four possible Cu nanowires (including noncompact nonhelical, compact nonhelical, left helical, and right helical) for 6-1 according to the results of a genetic algorithm (GA) global search.25 There is no difference in G(V) functions found in our simulation. Thus, the helical direction in 6-1b is neglected in our simulation. Among these four structures, the noncompact nonhelical nanowire has higher energy than the compact nonhelical wire, and the helical nanowire has the lowest energy, so we choose the noncompact nonhelical as a and right helical as b and neglect the middle energy structure. At the same time, the noncompact structures of metal nanowires have been studied by Makita et al. and Gulseren et al.20 In general, the stable structures of Cu NWs are multishell packing composed of coaxial cylindrical shells or tubes. Each shell is formed by atom rows winding up helically side-by-side with different pitches of the helices, which have been theoretically predicted for Al and Pb NWs32-34 and experimentally observed in Au NWs.35 L(V) functions of seven structures are determined from Figure 1 and shown in Table 1. V, being similar to a stress field, deforms the atomic structures with atomic movements. Because the structures of 6-1a, 6-1b, 12-6-1 have the central axis atom and others do not have the central axis atom, the value of L(V = 0) is different. When V = 0, the distance between two neighboring layers at central axis (D) 0.24 nm for 3a, 3b, 4b, whereas D 0.18 nm for 6-1a, 6-1b, 12-6-1 in every layer. For nonhelical NWs, L(V = 1 V/Å) is a little longer than L(V = 0) for 3a, 4a, and L(V = 1 V/Å) is equal to L(V = 0) for 61a, where the atomic structure varies and D in the middle of nonhelical NWs increases. The corresponding results of D are shown in Table 2. For example, for 3a, D4-5 = 0.22 nm, D5-6 = 0.38 nm, and thus D4-6 = 0.60 nm. For 4a, D4-5 = 0.21 nm, D5-6 = 0.37 nm, and thus D4-6 = 0.58 nm, where the subscript numbers denote the corresponding layer numbers. As D4-6 increases, because of the nature of ballistic transport, the probability of an electron jump between the two layers decreases and thus G decreases. The atoms in the odd layer move toward the directions of both tips of the structure. Thus, D4-6(V = 1 V/Å) is about 1.25 times of that for 3a, 4a, and 1.6 times for 6-1a at V = 0. Moreover, we find that the value of D4-6(V = 1 V/Å) is about 0.59 nm for these three nonhelical NWs, which is only dependent on V but independent of the nanowire diameter (d). The larger value of D5-6 is induced by the atomic movement of the atom in the seventh layer putting into the sixth layer. Unlike nonhelical NWs, L(V = 1 V/Å) of helical NWs is shorter than L(V = 0). The reason is that D(V = 1 V/Å) values of two tips decrease more strongly, which is shown in Table 2. For instance, compared V = 1 V/Å with V = 0, L drops from 2.16 to 2.04 nm since D1-2 = 0.13 nm and D9-10 =0.14 nm for 3b, where the atoms in the odd layers go to the interstice of the next layer and the structure tends to continuously converge. Similar to nonhelical NWs, atoms also accumulate on the both tips of NWs, whereas the atomic number in the middle part of NWs decreases. 3328
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Table 1. Diameter (d), Length (L), and Quantum Conduction (G) Function of Cu NWs Obtained by DFT Calculationsa d1
wire
d2
d3
L1
L3
G1
G2
G3
3a
0.242
0.238
0.321
2.16
2.18
3
3
2
3b 4a
0.212 0.277
0.214 0.274
0.232 0.404
2.16 2.16
2.04 2.20
2 4
2 4
3 2
4b
0.288
0.286
0.312
2.16
2.08
3
3
5
6-1a
0.490
0.486
0.625
1.64
1.64
4
4
2
6-1b
0.418
0.424
0.433
1.64
1.54
4
4
6
12-6-1
0.942
0.936
0.956
1.64
1.61
11
10
13
a
Units: d in nm, L in nm, G in G0, V in V/Å; the subscript 1 and 3 denote diameters obtained from our simulation without electrical field, and under V = 1 V/Å, respectively. The subscript 2 denotes diameters obtained from ref 12 without electrical field.
Table 2. Distance between Two Neighboring Layers at Central Axis (D) in the Calculations under V = 1 V/Å. a wire
D1-2 D2-3 D3-4 D4-5 D5-6 D6-7 D7-8 D8-9 D9-10
3a
0.17
0.20
0.24
0.22
0.38
0.22
0.26
0.26
0.23
3b
0.13
0.28
0.26
0.24
0.26
0.24
0.24
0.25
0.14
4a
0.22
0.25
0.22
0.21
0.37
0.24
0.25
0.22
0.22
4b 6-1a
0.14 0.08
0.27 0.22
0.26 0.17
0.24 0.20
0.26 0.38
0.22 0.09
0.22 0.15
0.28 0.25
0.19 0.10
6-1b
0.10
0.13
0.13
0.16
0.22
0.24
0.21
0.20
0.15
12-6-1 0.11
0.15
0.16
0.17
0.24
0.26
0.20
0.23
0.12
a
When V = 0, the distance between two neighboring layers at central axis (D) 0.24 nm for 3a, 3b, 4b, while D 0.18 nm for 6-1a, 6-1b, 12-6-1 in every layers.
As L decreases, the rates of ballistic transport of even layers should be increased. According to the Landauer formula, the number of bands crossing Fermi level Ef attributes to the number of conductional channels or the size of quantum conductance G.9 The so determined G(V = 0) and G(V = 1 V/Å) values of seven structures with all conductional channels are shown in parts a-d of Figures 2 and Table 1. For NWs with the same size, the atomic configuration of different isomers plays a significant role in determining the details of energy bands at the vicinity of Fermi level and thus the number of conduction bands crossing the Fermi level. Namely, G(V) sensitively depends on the atomic structures of the wires. For instance, 3b has two ballistic conduction channels, whereas 3a has three channels. When d is constant, G(V) of helical NWs increases while G(V) of nonhelical NWs decreases. Moreover, when V = 1 V/Å, G(d) of helical NWs increases and G(d) of nonhelical is equal to 2 constantly. Table 1 shows the quantum conductance versus nanowire diameter at V = 0 and V = 1 V/Å. One can see that G (in unit of G0) approximately fit into a quadratic function of d (in unit of Å) at V = 0 as:25 GðV ¼ 0Þ ¼ 2:0 þ 0:12 d2
ð1Þ
2
This is understandable because d gives the area of the cross section of the NWs, and, at V = 1 V/Å, G could be also proximately fitted into the following quadratic function of wire d as: GðV ¼ 1V=ÅÞ ¼ 2:0 þ 11 d for helical NWs
ð2Þ
GðV ¼ 1V=ÅÞ ¼ 2:0 for nonhelical NWs
ð3Þ
Figure 2. G(V), DOS of nonhelical NWs of 3a, 4a, 6-1a and helical NWs of 3b, 4b, 6-1b, 12-6-1. Ef = 0 (vertical dotted line) is taken.
This means that the electric field effect of helical NWs on G is no longer decided by the area of the cross section but decided by the diameter. This is reasonable because the electric field is only in the axis direction, and the structure has been changed along the axis. Meanwhile, according to the results above, G(V = 1 V/Å) = 2.0 for 3a, 4a, 6-1a, which is independent of the size for nonhelical NWs. Because nonhelical NWs cannot remain stable when d > 9 Å in the simulation at present,12 we consider G(V = 1 V/Å) 2.0 for nonhelical NWs. It also proves that the smallest value of G at V = 1 V/Å should be existed. In other words, V = 1 V/Å is unable to make the nonhelical NWs collapse. DOS of the seven structures observed by DFT are present in Figure 2. For instance, the largest peak of DOS below Ef is located between -1.79 and -0.41 eV under V = 0, whereas -1.76 and -0.23 eV under V = 1 for 4a. The case is similar for other structures. The position of largest peak obviously shifts right as V increases, which implies the energy of all structures increase. The size of DOS at Ef under V = 1 is 0.45 times of that under V = 0 for 6-1a, whereas the comparison is 1.25 times that for 6-1b. These results for DOS ratios at Ef confirm the calculated results from Landauer formula shown in parts c and d of Figure 2. According to the figures, the size of DOS at Ef under V = 1 is 0.5 times of that V = 1 is 0.5 times of that under V = 0 for 6-1a, whereas the comparison is 1.5 times for 6-1b. Although the present experimentation and simulation cannot realize V > 1 V/Å, we predict that if V further increases without ruptures, the main peak of DOS can cross Ef, and G could be largely increased 3329
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Figure 3. Mulliken charge population of seven structures. The charges show the sum of each layer and the layer number is defined in Figure 1. 3330
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The Journal of Physical Chemistry C for helical NWs. However, for nonhelical NWs, this peak drops and would disappear as V further increases, where the rupture occurs and G = 0. Mulliken charge e(V) functions of all structures are shown in Figure 3. The layer number is indexed in Figure 1. Under V = 0, e(V) functions of the both nonhelical and helical NWs are similar and homogeneous along the axes of the NWs. In nonhelical NWs, the atoms in the center row are positively charged and others are in reverse. Namely, the charge is enriched on the surface due to the surface effect. When V = 1 V/Å, more electrons localize at 5-8th layers of the polarized nonhelical NWs. Thus, the charges are negative in the middle but positive in the both tips due to the movement of electrons. The minimum of e = -0.6 for 3a, e = -0.7 for 4a, e = -1.4 for 6-1a is located at the sixth layer. The largest D value is also emerged at the sixth layer of these structures, which resists the electronic course. So, the electron accumulation occurs only in the fifth and sixth layers, whereas electrons in other layers are positive. This accumulation is unfavorable for the electronic transport. However, in the case of helical NWs, electrons are homogeneous distributed in all layers under V. The odd and even layers combine and form a new layer, which increases the channel numbers of ballistic transport. In addition, the absolute value of e under V = 1 is larger than that under V = 0 due to smaller D values and the combination of layers.
4. CONCLUSIONS In summary, we have studied the effect of electric fields on atomic and electronic structures and transport properties of Cu NWs with diameter from 0.2-1.0 nm. The conduction of Cu NWs generally increases with d at V = 0. When V = 1 V/Å, G(V) of helical NWs increases as size increases and a linear relationship between G(V) and d is proposed, whereas that of a nonhelical NWs is constant within the considered size range. A homogeneous distribution of electrons along the axes of NWs benefits the ballistic electronic transport, whereas an inhomogeneous one deteriorates the transport. ’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected]. Tel: þ86 29 82668614.
’ ACKNOWLEDGMENT We acknowledge support from the National Key Basic Research and Development Program (Grant No. 2010CB631001). The authors also like to thank Professor Qing Jiang for fruitful discussions. The authors acknowledge the computer resources provided by Department of Materials Science and Engineering, Jilin University.
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