Effect of bending stress on structures and quantum conduction of CunanowiresC. He, W. X. Zhang, Z. Q. Shi, J. P. Wang, and H. Pan Citation: Appl. Phys. Lett. 100, 123107 (2012); doi: 10.1063/1.3696052 View online: http://dx.doi.org/10.1063/1.3696052 View Table of Contents: http://apl.aip.org/resource/1/APPLAB/v100/i12 Published by the American Institute of Physics. Related ArticlesFeasibility, accuracy, and performance of contact block reduction method for multi-band simulations of ballisticquantum transport J. Appl. Phys. 111, 063705 (2012) Length dependence of the resistance in graphite: Influence of ballistic transport J. Appl. Phys. 111, 033709 (2012) Spin-transfer torque in magnetic junctions with ferromagnetic insulators J. Appl. Phys. 111, 07C902 (2012) Subthreshold characteristics of ballistic electron emission spectra J. Appl. Phys. 111, 013701 (2012) Theory of quantum energy transfer in spin chains: Superexchange and ballistic motion J. Chem. Phys. 135, 234508 (2011) Additional information on Appl. Phys. Lett.Journal Homepage: http://apl.aip.org/ Journal Information: http://apl.aip.org/about/about_the_journal Top downloads: http://apl.aip.org/features/most_downloaded Information for Authors: http://apl.aip.org/authors

Effect of bending stress on structures and quantum conductionof Cu nanowires

C. He,1 W. X. Zhang,2,a) Z. Q. Shi,1 J. P. Wang,1 and H. Pan3

1State Key Laboratory for Mechanical Behavior of Materials, School of Materials Science and Engineering,Xi’an Jiaotong University, Xi’an 710049, China2School of Materials Science and Engineering, Chang’an University, Xi’an 710064, China3Department of Environmental Science and Engineering, Xi’an Jiaotong University, Xi’an 710049,People’s Republic of China

(Received 30 January 2012; accepted 3 March 2012; published online 23 March 2012)

The ballistic transport properties of Cu nanowires under different bending stresses are investigated

for future application in flexible displays and flexible solar cell using first-principles

density-function theory. The stability and quantum conduction of both nonhelical and helical

atomic strands are reduced by applying a bending stress f. With increasing of f, the helical wire

becomes disorder, suffering a phase transition to similar nonhelical one and collapsing eventually.

Our calculations show that the maximum bearable bending stress is fmax¼ 3 nN for the helical

atomic strands while is more stable than fmax¼ 2.5 nN for the nonhelical atomic strands. VC 2012American Institute of Physics. [http://dx.doi.org/10.1063/1.3696052]

Cu nanowires (NWs) have been intensively studied for

the fundamental interests in theory and possible applications

in flexible displays and flexible solar cell.1 After the realiza-

tion of the fabrication NWs, the physical properties of Cu

NWs are measured and calculated, especially their electron

transport properties.2 Moreover, the blending effect on elec-

tronic transport in NWs is one of the important characteris-

tics for its applications.3 When the length and width scales

of NWs are reduced to the mean free path of electrons, the

electron transport mechanism changes from diffusive to bal-

listic. It is now known that the electric conductance is inde-

pendent on the length of the NWs and the quantum

conduction G has been observed4 as predicted by the Lan-

dauer formula.5 G is quantized in units of G0¼ 2e2/h, where

e denotes the electronic amount and h the Planck constant.

Valence charge polarization by the locally entrapped core

electron could be possible mechanism for the ballistic trans-

portation in the NW.6–8 In the past decades, ultrathin metal

nanowires produced by the tip retracting from nanoindenta-

tion in scanning tunnelling microscopy (STM) or by a

mechanically controllable break junction (MCBJ) have been

subjects of numerous experimental and theoretical studies.8,9

Both atomic structures and size of NWs affect the transport

properties.10–12 It is found that G of the pentagonal Cu NWs

with [110] orientated structure is about 4.5G0 without bend-

ing stresses.4 When NW is sufficiently thin, beside, the con-

ventional structure will turn exotic.13,14 However, the effect

of bending stress on G, which is the precondition of the elec-

tronic transport in flexible displays and flexible solar cell,

has not been considered systemically up to now.

In this Letter, first-principles density functional theory

(DFT) calculations are performed to determine the changes

of atomic and electronic structures of Cu NWs in light of

density of states (DOS) under bending stresses. In the calcu-

lations, two optimized atomic structures of Cu NWs are used

as the starting points. It is found that atomic structures and

related structure changes decide the size of G(f) function of

Cu NWs, and the electronic distribution within NWs also

reveals the magnitude of G.

The first-principles DFT calculation is provided by

DMOL3 code.15,16 The generalized gradient approximation

with the Perdew–Burke–Ernzerhof correlation gradient cor-

rection15,16 is employed to optimize geometrical structures

and calculate properties of Cu NWs. The all-electron relativ-

istic Kohn–Sham wave functions are expanded in the local

atomic orbital basis set. The Cu NWs are modeled in a tet-

ragonal supercell with one-dimensional periodical boundary

conditions along the NWs. Our outlines of the used struc-

tures are directly referred to the results of Wang et al.17 The

length of Cu NWs (L), which is determined by the distance

of the projection of mean locations of atom centers in the 1st

and 20th layers on the axis, are chosen to be 3.28 nm. It is

because the distance between two neighbor layers with a

core atom in the NW is aboutffiffiffi

2p

/2 time than the distance

without a core atom in NW. 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 Ha¼ 27.2114 eV), maximum

force of 0.004 Ha/A, and maximum displacement of

0.005 A. G(f) values are determined by the Landauer for-

mula.18 The layer electronic distributions are carried out by

the Mulliken charge analysis.19

Two representative structures of Cu NWs with nonheli-

cal and helical structures are shown in Fig. 1. The structures

are referred to as 6-1a and 6-1b, where 6 shows the atomic

number of one atomic layer in the surface cell and 1 repre-

sents that in the core.17 The average stress strength f is

obtained by averaging stress strength F of each layer accord-

ing to our previous work.20 It is apparent that length of the

top layer atoms (Ltop) of 6-1a increases sharply when fincreases from 0 to 2.5 nN, while that length of the down

layer atoms (Ldown) of 6-1a decreases sharply in the same

a)Author to whom correspondence should be addressed. Electronic mail:

wxzhang@chd.edu.cn.

0003-6951/2012/100(12)/123107/3/$30.00 VC 2012 American Institute of Physics100, 123107-1

APPLIED PHYSICS LETTERS 100, 123107 (2012)

range. However, when f> 2.5 nN, Ltop of 6-1a decreases,

while Ldown of 6-1a keep constant. The reasons are as fol-

lows: First, during the bending process, the top layer atoms

are under tensile stress, while the down layer atoms are

under the compressive stress. Second, when f> 2.5 nN, the

distances between certain atoms of the down layer are close

to the critical value (0.23 nm), and repulsive forces domi-

nates the lattice distance. Due to this, there is not enough

space for the movement of atoms. The trend of Ltop and

Ldown of 6-1b are the same as 6-1a. In detail, with increasing

strength of bending stresses, the helical wire becomes disor-

der, suffering a phase transition to similar nonhelical one and

collapsing eventually. The only difference between them are

the critical value of 6-1b is 3 nN while the value of 6-1a is

2.5 nN. Moreover, the trend for G(f) for both structures are

almost the same, which is opposite with the trend of Ltop(f).It is because that the atom accumulation becomes weaker

when the distance among some atoms increases.14 Moreover,

when f> 2.5 nN for 6-1a and f> 3 nN for 6-1b, the stability

of structure of Cu nanowire is broken, and positions of atoms

are out of order, and then G(f) goes down to zero while

Ldown(f) keeps constant.

According to the Landauer formula, the number of

bands crossing Fermi level Ef attributes to the number of

conductional channels or the magnitude of G. The calculated

G(f) values of 6-1a and 6-1b with all conductional channels

are shown in Figs. 2(a), 2(b) and 2(c), 2(d) and Table I.

G(f¼ 0)¼ 4 agrees with other results.17 As f further

increases to 1.5 nN, G(f¼ 1.5nN)/G(f¼ 0)¼ 1 for 6-1a,

which means the structure still stable under relatively week

blending stress. However, G(f¼ 2.5 nN)/G(f¼ 0)¼ 0.5,

which means the structure is unstable under relatively large

blending stress. On the other hand, the 6-1b has the similar

situation. Hence, G(f) functions of the both structures indeed

have opposite tendencies on Ltop(f) before their structures

collapse.

For 6-1a, Ltop(f¼ 2.5 nN) reaches the maximum where

the atomic structure varies and distance between layers (D)

in the middle of 6-1a increases. The increase of D10-11

decreases the electron jump between the two layers due to

the ballistic transport nature and thus decreases G. The

atomic movement of inner atom in the 12th layer inserts into

the 11th layer and pushes the surface atoms of the 12th layer

away form the axis, which induce the larger value of D12-13.

Similar to 6-1a, when 0< f< 3nN, Ldown(f) functions of

6-1b drop monotonously, but Ddown(f) of the middle layers

increase a little. The reason is that Ddown(f) of two tips

decrease more strongly. Ldown drops from 3.46 to 3.38 nm at

f¼ 1.5 nN because D1-8 and Ddown13-20 decrease evidently.

When f¼ 3 nN, Ddown1-2¼ 0.08 nm, Ddown3-4¼ 0.11 nm,

Ddown17-18¼ 0.12 nm, and Ddown19-20¼ 0.13 nm where the

atoms in the odd layers go to the interstice of the next layer

and the structure tends to continuously shrink. The atoms

also accumulate on both tips of NWs, while the atomic num-

ber in the middle part of NWs decreases. When f¼ 3.2 nN,

G decreases to 0 because the structure collapses. The critical

collapse pressure of 6-1b is higher than f¼ 2.5 nN for 6-1a.

6-1b could stand higher stress than 6-1a and would collapse

under higher force intensity.

DOS of the two structures obtained DFT calculations

are presented in Figs. 2(e) and 2(f). For 6-1a, the largest

peak of DOS below Ef is located between �0.91 and

�0.33 eV under f¼ 0, �1.18 and 0.13 eV under f¼ 2.5nN,

respectively. For 6-1b, the case is similar. The largest peak

obviously shifts right as f increases, which implies the energy

increases of both structures. The largest peak obviously

shifts left when structure of Cu nanowire is collapse, which

implies that the energy decreases of both structures are

FIG. 1. (Color online) Morphologies of Cu NWs as a function of f in nN.

FIG. 2. (Color online) G(f), DOS of Cu NWs with 6-1a of a, c, e and with

6-1b of b, d, f in Fig. 1. Ef¼ 0 (vertical dotted line) is taken.

TABLE I. G(V) function of Cu nanowires obtained from DFT calculations.

G in G0. f in nN.

f¼ 0 f¼ 1.5 f¼ 2.5 f¼ 2.8 f¼ 3 f¼ 3.2

G(6-1a) 4 4 2 0 0 0

G(6-1b) 4 4 4 4 2 0

123107-2 He et al. Appl. Phys. Lett. 100, 123107 (2012)

separated into two or three pieces. The magnitude of DOS at

Ef under f¼ 2.5 nN is 0.5 times of that under f¼ 0 for 6-1a.

While f¼ 2.8 nN with the collapse of the structure, the mag-

nitude of DOS at Ef decreases to 0 and the value is 0.5 times

for 6-1b. In the same way, the above results for DOS ratios

at Ef confirm the calculated results from Landauer formula

shown in Figs. 2(a)–2(d), where the corresponding G ratios

are 0.5 and 1.

Mulliken charge e(V) functions of all structures are

shown in Figure 3. e(f) functions of 6-1a and 6-1b are shown

in Figs. 3(a) and 3(b). The layer is indexed in Fig. 1. When

f¼ 0, e(f) functions of the both structures are similar and ho-

mogeneous along the axis of NWs. The atoms in the center

row of NWs are positive charge and others are in reverse. For

6-1a, when f¼ 1.5nN, although the charge distribution has

been a little changed, the positive and negative charges are

still in sequence. There is no influence on the electronic trans-

port. However, when f¼ 2.5nN, the charge distribution are

totally changed. The positive charge in 9th and 13th becomes

negative, which induce the accumulation of electron. This

accumulation is unfavorable for the electronic transport and

decrease of G. While f increases to 2.8 nN, the symmetry dis-

tribution of charge is broken and the charge is out-of-order.

The electron transport is broken off, and the electron accumu-

lation is thus observed in the 6th and 12th layers.

In the case of 6-1b, electrons are equably distributed in

all layers when f¼ 0. The positive and negative charges are

alternately distributed and have the same absolute value;

however, when 0< f< 3 nN, the positive and negative

charges are still alternately distributed but absolute value of

them are no longer consistent. Once f¼ 3 nN, some of the

odd and even layers combine and form a new layer consist-

ing of seven Cu atoms, which increases the channel numbers

of ballistic transport. Meanwhile, the channel numbers of the

neighbor layer at the same time have been decreases. When

f¼ 3.2 nN, the total energy decreases and the structure trends

to be more stable. The above so-called new layer appears

more and more, which it does not benefit electronic trans-

port. The distribution of charge is out-of-order. The electrons

transport is broken off, and the electron also accumulates in

the 12th layers, just like the results of 6-1a under f¼ 2.8 nN.

In summary, we studied the effect of bending stress on

atomic and electronic structures and transport properties of

Cu NWs with nonhelical and helical structures. When the

bending stress is considered, both structures are more unsta-

ble than those without bending stresses. G(f) function of 6-1a

and 6-1b decreases as f increases. G(f) function of 6-1a

decreases as f increases when 0< f< 2.5 nN, while G(f)¼ 0

when f> 2.5 nN, and 6-1b has a similar trend. This is

directly proportional to the changes of D values even if the

changes are inhomogeneous. The helical atomic strands are

more stable than the nonhelical ones since the former owns

higher collapse-resistant f.

The authors acknowledge supports by National Key Ba-

sic Research and Development Program (Grant Nos.

2010CB631001 and 2012CB619400), National Natural Sci-

ence Foundation of China (NSFC, Grant No. 51101117),

Ph.D. Programs Foundation of Ministry of Education of

China (Grant No. 20110201120002), the Fundamental

Research Funds for the Central Universities and State Key

Laboratory for Mechanical Behavior of Materials.

1A. R. Rathmell, S. M. Bergin, Y. L. Hua, Z. Y. Li, and B. J. Wiley, Adv.

Mater. 22, 3558 (2010).2Y. F. Zhu, X. Y. Lang, W. T. Zheng, and Q. Jiang, ACS Nano 4, 3781 (2010).3S. Y. Chang and D. S. Chen, Appl. Phys. Lett. 94, 231909 (2009).4L. Olesen, E. Laegsgaard, I. Stensgaard, F. Besenbacher, J. Schiotz, P.

Stoltze, K. W. Jacobsen, and J. K. Norskov, Phys. Rev. Lett. 72, 2251

(1994).5B. J. Van Wees, H. Van Houten, C. W. J. Beenakker, J. G. Williamson, L.

P. Kouwenhoven, D. Van der Marel, and C. T. Foxon, Phys. Rev. Lett. 60,

848 (1998).6X. Zhang, J. L. Kuo, M. Gu, X. Fan, P. Bai, Q. G. Song, and C. Q. Sun,

Nanoscale 2, 412 (2010).7C. Q. Sun, H. L. Bai, S. Li, B. K. Tay, C. Li, T. P. Chen, and E. Y. Jiang,

J. Phys. Chem. B 108, 2162 (2004).8C. He, P. Zhang, Y. F. Zhu, and Q. Jiang, J. Phys. Chem. C 112, 9045

(2008).9N. Agrait, A. L. Yeyati, and J. M. van Ruitenbeek, Phys. Rep. 377, 81 (2003).

10C. He, W. X. Zhang, and J. L. Deng, J. Phys. Chem. C 115, 3327 (2011).11G. E. Tommei, F. Baletto, R. Ferrando, R. Spadacini, and A. Danani,

Phys. Rev. B 69, 115426 (2004).12J. J. Zhao, C. Buia, J. Han, and J. P. Lu, Nanotechnology 14, 501 (2003).13E. Tosatti, S. Prestipino, S. Kostlmeier, A. Dal Corso, and F. D. Di Tolla,

Science 291, 288 (2001).14Y. F. Zhu, W. T. Zheng, and Q. Jiang, Appl. Phys. Lett. 95, 083110 (2010).15J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865

(1996).16Z. M. Ao and F. M. Peeters, Appl. Phys. Lett. 96, 253106 (2010).17B. L. Wang, J. J. Zhao, X. S. Chen, D. Shi, and G. H. Wang, Nanotechnol-

ogy 17, 3178 (2006).18D. Sanchez-Portal, E. Atacho, and J. M. Soler, Solid State Commun. 95,

685 (1995).19W. Liu, Y. H. Zhao, J. Nguyen, Y. Li, Q. Jiang, and E. J. Lavernia, Carbon

47, 3452 (2009).20C. He, L. Qi, W. X. Zhang, and H. Pan, Appl. Phys. Lett. 99, 073105

(2011).

FIG. 3. (Color online) The Mulliken charge population of 6-1a (a) and 6-1b

(b). The charges show the sum of each layer where the layer number is

defined in Fig. 1.

123107-3 He et al. Appl. Phys. Lett. 100, 123107 (2012)