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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:
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.
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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)