Interface strength

1. Ab initio study of structural transition and pseudoelasticity in
Cu nanowires
Nguyen Tuan Hung a
, Do Van Truong a,b,
⁎
a
International Research Center for Computational Materials Science, Hanoi University of Science and Technology, Hanoi, Viet Nam
b
Department of Design of Machinery and Robot, Hanoi University of Science and Technology, Hanoi, Viet Nam
a b s t r a c ta r t i c l e i n f o
Article history:
Received 18 March 2015
Accepted 4 May 2015
Available online 12 May 2015
Keywords:
Metal nanowire
Size-dependent
Pseudoelasticity
Density function calculations
In this study, ab initio density functional theory calculations are used to investigate the intrinsic mechanical
responses of copper nanowires with an initial 〈100〉 axis and {110} side surfaces (〈100〉/{110} Cu NWs) under
a large tensile strain. For the small cross sectional diameters below 1.38 nm, surface stresses alone may cause
to transform from an initial face-centred-cubic (FCC) to a body-centred-tetragonal (BCT) structures. Under
loading and unloading conditions, the structural transition from a BCT to a face-centred-tetragonal (FCT)
structures reversibly occurs and is a key to explain the pseudoelastic behaviour of the 〈100〉/{110} Cu NWs.
The mechanical properties of the Cu NWs investigated depend not only on diameter size but also side surface.
© 2015 Elsevier B.V. All rights reserved.
1. Introduction
One-dimensional materials, such as metal nanowires (NWs), have
been attracting attention because of their fascinating physics and poten-
tial applications in next generation nanodevices. An experiment has
shown that the Au NWs with a diameter of 9 nm have good electron
conductivity with its resistivity at 260 Ω ⋅ nm and breakdown current
at 250 μA [1]. Cu NWs [2,3] and Ag NWs [4,5] can be coated from liquids
to create flexible, transparent conducting films that can potentially
replace the indium tin oxide (ITO) in bendable displays, solar cells,
and touch screens. Recently, the NWs have been fabricated with the
diameter of a few nanometers [1,6], and may be smaller in the near
future. The electromechanical properties of the NWs with nanometre
diameter are highly dependent on their strain states due to the changing
atomic structure after deformation. Thus, studying the mechanical
response of NWs under strain is a necessary task in order to improve
the future nanowire-based devices.
It is well known that the NW has a large ratio between surface area
and volume, and this ratio leads to the mechanical properties of NW dif-
ferent from those of bulk. For the NWs made by a top-down fabrication
process [7], both experimental [6] and theoretical studies [8,9,12] have
shown that the surface stress induced by intrinsic stress may be the
major cause of contract phenomenon. This contraction, which is
known as “spontaneously relaxing”, may be also the main cause of
structural transition and reorientation in the NWs. The structural
transition from a face-centred-cubic (FCC) to a helical multishell
(HMS) has been proposed for Au NWs (b1.5 nm in diameter) based
on high-resolution transmission electron microscopy observations [6].
The FCC–HMS transition has been predicted for Al NWs and Pb NWs
by molecular dynamics (MD) simulations [10]. MD simulations [9]
also have predicted that the structural transition from a body-centred-
cubic (BCC) to the FCC occurs in the Mo NWs. Besides, the density func-
tional theory (DFT) and the tight-binding (TB) methods also have been
used to study the structural transition [11]. The results obtained showed
that narrow Au and Pt NWs exhibit the spontaneous relaxation from the
initial FCC to a body-centred-tetragonal (BCT) structures. For Cu, Ni and
Ag NWs, in most cases, the spontaneous relaxation has caused the reori-
entation into a 〈110〉 axis and {111} side surfaces instead of the structural
transition [11–14]. Specifically, this reorientation can occur reversibly in
the Cu NWs under loading and unloading of tensile strain. The maximum
strain of this reversible transition can be estimated about 50%, leading to
a very large pseudoelastic or shape memory effects of the Cu NWs [12].
The large pseudoelastic effects have been also predicted in the interme-
tallic Ni–Al [15] and Cu–Zr NWs [16] by MD simulation.
In order to understand more the intrinsic mechanical responses of
Cu NWs, the present study aims to the Cu NWs with an initial 〈100〉
axis and {110} side surfaces. The focus is to find out the types of struc-
ture changing (structural transition or structural reorientation) and
the pseudoelastic behaviour of the Cu NWs under intrinsic surface stress
and large strain. The simulation is carried out by using ab initio based on
density function theory (DFT). The paper is organized as follows.
Section 2 describes the setup of the DFT calculations and the detailed
simulation procedure. Section 3 describes the relaxed structures,
FCC–BCT transition, pseudoelastic effects, and fracture mechanism of
Surface Science 641 (2015) 1–5
⁎ Corresponding author. Tel.: +84 43 868 0101.
E-mail address: truong.dovan@hust.edu.vn (D. Van Truong).
http://dx.doi.org/10.1016/j.susc.2015.05.004
0039-6028/© 2015 Elsevier B.V. All rights reserved.
Contents lists available at ScienceDirect
Surface Science
journal homepage: www.elsevier.com/locate/susc

2. the Cu NWs under large strain. We also investigate the effect of wire size
and phonon instability of bulk Cu along Bain path. Finally, Section 4
summarizes the results.
2. Computational details
2.1. Simulation method
First-principle (ab initio) simulations for tensile strain of Cu NWs are
performed. We use Quantum-ESPRESSO (QE) package [17] for first-
principle calculations, which is a full DFT [18,19] and DFPT [20,21] sim-
ulation package using a plane-wave basic set with pseudopotentials.
The Rabe–Rappe–Kaxiras–Joannopoulos (RRKJ) [22] type ultrasoft
pseudopotentials with an energy cutoff of 35 Ry for the wave functions.
Hermite–Gaussian smearing [23] with an energy width of 0.03 Ry is
adopted for the self-consistent calculations. The exchange–correlation
energy is evaluated by general-gradient approximation (GGA) using
the Perdew–Burke–Ernzerhof (PBE) [24] function. The dynamical
matrices are calculated based on DFPT within the linear response.
2.2. Models and simulation procedure
Fig. 1(a) shows the simulation models of the unit cell inside FCC bulk
Cu. Fig. 1(b), (c) shows the supercell of Cu NW with an initial 〈100〉 axis
and {110} side surfaces. The Cu NW has the cross section in which three
a0=
ffiffiffi
2
p
 a0=
ffiffiffi
2
p
 a0 unit cells are arranged in the [110] and 110
h i
directions, which is denoted as a 3 × 3 NW. Here, a0 is the theoretical lat-
tice constant of FCC bulk Cu (a0 = 3.673 Å). Since periodic boundary
condition is applied for three dimensions in all models, the thickness
of vacuum region is set to four times a0 perpendicular to the wire axis
to avoid undesirable interactions from neighbouring NWs. Thus, the
supercell dimensions in the cross section and axial direction are initially
set to 3=
ffiffiffi
2
p
þ 4
a0 and a0, respectively. We also investigate smaller
the Cu NWs with the cross sections of 2 × 2 and 1 × 1 cells in the
same manner to elucidate finite-size effect. The k-point grids in the
Brillouin-zone selected according to the Monkhorst–Pack method [25]
are a 14 × 14 × 14, 4 × 4 × 14, 3 × 3 × 14, and 2 × 2 × 14 for the bulk
Cu, 1 × 1 NW, 2 × 2 NW and 3 × 3 NW models, respectively. For the
DFPT phonon calculations, we used a primitive cell of the bulk Cu with
dynamical matrices are calculated on a 6 × 6 × 6 q-point.
To simulate the effect of tensile strain in the Cu NWs, first, the
models are fully relaxed by using the Broyden–Fretcher–Goldfarb–
Shanno (BFGS) minimization method for the atomic positions, and cell
dimensions in the z direction. These models are considered as the equi-
librium structures when all the Hellmann–Feynman forces and the
normal components of the stress σzz are less than 5.0 × 10−4
Ry/a.u.
and 1.0 × 10−2
GPa, respectively. Then the loading strain is applied to
the models by elongating the simulation cell along the z direction
with an increment of 2%. After each increment of the strain, the atomic
structure is fully relaxed under fixed cell dimensions. Similarly, the
unloading is applied to the models with a decrement of 2%.
3. Results and discussion
3.1. The FCC and BCT structures of bulk Cu
Fig. 2 shows the Bain path energies and stresses in the z direction,
σzz, for the bulk Cu as a function of tetragonality c/a. Where c and a
denote the lattice spacings in the z and x (or y) directions, respectively
[see the schematic illustration in Fig. 2]. In this calculation, c was fixed
and a was relaxed to satisfy the stress conditions σxx = σyy = 0 and
σzz ≠ 0 for each step of strain. The energy curve in the Bain path has a
deep minimum at c=a ¼
ffiffiffi
2
p
, corresponding to the FCC structure. A
shallow energy minimum is also found at c/a = 0.97 b 1, which is a
sign for the existence of the BCT structure. These results are also consis-
tent with the previous theoretical studies [11,26,27]. In addition, the re-
sult shows that both the FCC and BCT structures exist in the states with
zero-stress components. The BCT structure has a local energy minimum,
but it is not sufficient to confirm the structure in stability. A crystal
structure is considered as the stable or metastable structure only
when the soft phonon modes, elastic and other instabilities do not
occur. To examine the stability of the structures, the phonon frequencies
for both the FCC and BCT ones were calculated. The phonon dispersion
curves for the FCC structure [Fig. 3(a)] is in good agreement with the
experimental data [28]. The result shows that the FCC structure is stable
with all the positive phonon modes. In contract, the BCT structure is in-
stability related to the soft phonon mode: the transverse branch Γ → X
Fig. 1. (a) The simulation model of a unit cell inside FCC bulk Cu. (b) side views and
(c) cross sectional of supercell of Cu NW with a 3 × 3 cell cross section. The NW has a
〈100〉 axis and {110} side surfaces. The solid black box represents the supercell.
2 N.T. Hung, D. Van Truong / Surface Science 641 (2015) 1–5

3. ([110] direction in the Brillouin-zone) with a polarization along [110]
[Fig. 3(b)]. The slopes of this soft phonon curve are proportional to
square root of the shear elastic constant C66 [29]. This elastic constant
is reported as negative value (C66 = −78 GPa) [11] for BCT structure
using DFT–GGA. Since the instabilities are corresponding to the soft
phonon modes and negative elastic constants, the BCT structure is
unstable along [110] direction.
3.2. FCC–BCT transition of Cu nanowires
It is well known that the cross section diameter size of NWs is in-
versely proportional to surface stress. If the cross section diameter is
small enough, the large tensile surface stresses exist in the wire surfaces,
and cause the contract phenomenon of the NWs. A critical cross section
diameter has calculated about 2.44 nm by ab initio simulation for the
〈100〉/{100} Cu NWs [11]. In order to determine the critical cross section
diameter, Dc, of the 〈100〉/{110} Cu NWs, the surface stress of the Cu
with the {110} surfaces was considered, in which the Cu(110) surface
model with the slab geometries of 12 atomic layers was used. The
vacuum thickness was set to 12 Å between the periodic slabs and a
k-point sampling was 14 × 14 × 1. First, the surface stress, f, is defined as
f ¼
1
2
h Δ σ ð1Þ
where h is the height of the slab, the 1
/2 factor arises from the two
terminated surfaces of the slab, and Δσ is difference in the total stress
in the x and y directions between the slab and the reference bulk.
Second, Dc can be estimated as
Dc ¼
4f
σc
ð2Þ
where σc is the critical stress, which can be a main cause of the FCC–BCT
transition in the bulk Cu [Fig. 2]. The result obtained shows that the
critical diameter Dc is equal to 1.38 nm with the surface stress
f = 1.22 N/m and the critical stress σc = 3.45 GPa. The critical diameter
of the 〈100〉/{110} Cu NWs is lower than that of the 〈100〉/{100} Cu
NWs. This can be explained as the {110} surfaces have lower energy
and smaller surface area than those of the {100} surfaces [30]. Since
the cross sectional diameters D of the 1 × 1, 2 × 2, and 3 × 3 NWs are
smaller than the critical diameter (D b Dc = 1.38 nm), they can be spon-
taneously relaxed. The obtained results of the cross sectional diameters
of the models are recorded in Table 1.
Fig. 4 shows the unrelaxed and relaxed configurations of the Cu NW
with the cross section 3 × 3 cell. Only two adjacent layers of atoms are
showed as the layers repeating in an AB-stacking sequence with the dif-
ferent colours. After relaxation, the contraction in the axial direction is
up to 28% while the isotropic expansion in the lateral directions is
about 12% with a square cross sectional shape. This shows a transition
from the initial FCC to the BCT structures of 〈100〉/{110} Cu NWs,
which is similar to the FCC–BCT transition of Au NWs [11]. In contrast,
for the 〈100〉/{100} Cu NWs, MD simulation [12,14] has shown that a
reorientation occurs, the 〈110〉/{111} wire with the cross section of
rhombus is established. This may be explained by the change of the
atomic positions in the {100} and {110} surfaces under spontaneously
relaxing [Fig. 5]. The lattice structure of the {100} surface [Fig. 5a] can
be reconstructed into the hexagonal close-packed (HCP) lattice of
{111} surface while the {110} surface [Fig. 5b] cannot be reconstructed
under contraction. Thus, the Cu NW models designed with the initial
〈100〉 axis and {110} surfaces only exhibit the structural transition
instead of the reorientation mechanism.
3.3. Pseudoelasticity of Cu nanowires
Fig. 6 shows the tetragonality of the lattice c/a around the edge,
surface, and inner cells in the 3 × 3 NW as a function of the axial tensile
strain, εzz. Where the edge, surface, and inner cells are denoted by
different colours [see the schematic illustration in Fig. 6]. The
tetragonality c/a b 1 when no strain is applied, indicating that the 3 × 3
NW is in the BCT structure. As the axial strain is applied, the tetragonality
increases almost linear. The slope of the edge and surface cells are similar
and larger than that of inner. For the tensile strain over 36–38%, c=aN
ffiffiffi
2
p
,
indicating that the 3 × 3 NW is in the face-centred-tetragonal (FCT)
structure. This shows that the structure of the Cu NWs is transformed
from the BCT to the FCT under the large tensile strain.
Fig. 7 shows the changes of the atomistic and the electronic configu-
rations on the (110) plane of the 〈100〉/{110} Cu NW in the cross section
of 3 × 3 cell. At no strain (εzz = 0.00), the Cu NW shows the BCT struc-
ture with the strong Cu–Cu bonds, α, emphasized by the white solid
lines. The α-bond length is equal to the lattice constant c and longitudi-
nal. At the tensile strain εzz of 0.40, the Cu NW shows the FCT structure
Fig. 2. Bain path energies per atom and stresses σzz for the bulk Cu as a function of
tetragonality c/a, when c and a denote the lattice spacings in the lateral z and x (or y)
directions, respectively. The vertical dashed lines represent the position of the FCC
c=a ¼
ffiffiffi
2
p
and BCT (c/a = 0.97 b 1) structures corresponding to the local minimum
and at zero stress.
Fig. 3. DFPT calculated phonon frequencies for (a) FCC and (b) BCT structures for Cu bulk.
The negative vertical axis was used to plot the imaginary frequencies.
Table 1
Equilibrium configurations of the Cu NWs.
NW Diameter D (nm) Length L (nm)
Initial Equilibrium Initial Equilibrium
1 × 1 0.260 0.293 0.367 0.253
2 × 2 0.520 0.585 0.367 0.261
3 × 3 0.779 0.869 0.367 0.264
3N.T. Hung, D. Van Truong / Surface Science 641 (2015) 1–5

4. with the strong β-bonds formed. The β-bond length is equal to the
lattice constant a and transverse. The transition of the strong bonds
from the α-bonds into β-bonds is related the BCT–FCT transition of
the Cu NWs under the tensile strain.
Fig. 8 shows the stress–strain curve during the uniaxial loading and
unloading of the 1 × 1, 2 × 2, and 3 × 3 NWs. Three distinctive stages of
response during loading (Ai → Bi, Bi → Ci and Ci → Di, i = 1, 2, 3) are ob-
served. The first stage (Ai → Bi) corresponds to the elastic strain of the
BCT structure up to about 8–10%. A recent experiment [31] has shown
that the Cu NW with a diameter of ~ 5.8 nm has the maximum elastic
strain about 7.2%. The stress–strain curves of the 1 × 1, 2 × 2, and
3 × 3 NWs also show that “smaller is stronger and more elastic”. From
point Bi to Ci, the inelastic strain appears, which can be due to the
transition from the BCT into FCT structures. The non-smooth fluctua-
tions can be derived from the sensitivity on the reconstructed surface
structure during the BCT–FCT transition. The next stage (Ci → Di)
corresponds to the elastic strain of the FCT structure up to 88%, 68%,
and 66% for the 1 × 1, 2 × 2, and 3 × 3 NWs, respectively. From the
point Di, the fracture of Cu NWs is initiated. The fractured strain of the
nanowires about 50% has been observed by the experiments [32,33].
The unloading paths from the points Di to Ai show that the 1 × 1,
2 × 2, and 3 × 3 NWs at the high strains can recover their original shapes
when unloading. Therefore, the loading and unloading paths can form a
reversible switching between the BCT and FCT structures. The BCT–FCT
switch is a key to explain the pseudoelastic effects of the 〈100〉/{110} Cu
NWs.
4. Conclusion
Ab initio density-functional theory calculations with the general-
gradient approximation have been carried out to investigate the
structural transitions of Cu nanowires with the 〈100〉 axis and {110}
side surfaces under the intrinsic surface stress and large tensile strain.
The obtained results could be summarized below.
The bulk Cu can exist in both the FCC and BCT structures with the local
energy minimums and zero stress components. The FCC structure is
stable with all the positive phonon modes while the BCT structure is in-
stabilities related to the soft phonon mode carrying the negative energy.
Those are consistent with the previous theoretical studies [11,26,27].
Since the 1 × 1, 2 × 2, and 3 × 3 NWs have the cross sectional
diameters smaller than the critical diameter (Dc = 1.38 nm), they can
Fig. 4. (a) Cross-sections and (b) slide views of the unrelaxed and relaxed configurations
of the Cu NW with a cross section of 3 × 3 cells. Only two adjacent lattice planes of
atoms are shown, and atoms in different lattice planes are shown in different colours.
Fig. 5. The change atomic positions in the {100} and {110} surface under spontaneously
relaxing. The lattice structure of the {100} surface can be reconstructed into the {111}
surface (a) while the {110} surface cannot be reconstructed (b) under contraction.
Fig. 6. Tetragonality of lattice c = a around the edge, surface, and inner cells in the 3 × 3
NW as a function of axial tensile strain εzz. The edge, surface, and inner cells are denoted
by different colours.
Fig. 7. Change in atomic structures and charge density distributions on the (110) plane of
the 3 × 3 NW under tensile strain. The blue spheres indicate Cu atoms and the strong
Cu–Cu bonds are drawn with white solid lines. Units in a. u. −3
. (For interpretation of
the references to color in this figure legend, the reader is referred to the web version of
this article.)
4 N.T. Hung, D. Van Truong / Surface Science 641 (2015) 1–5

5. be spontaneously relaxed. This contraction leads to the structural
transition instead of the reorientation observed in the 〈100〉/{100} Cu
NWs [12,14]. The difference between the 〈100〉/{110} Cu NWs and
〈100〉/{100} Cu NWs under spontaneously relaxing may be due to
their different side surfaces. Therefore, the mechanical properties of
the Cu NWs depend not only on diameter size but also side surface.
The stress–strain curves of the 1 × 1, 2 × 2, and 3 × 3 NWs show that
“smaller is stronger and more elastic”. The Cu NWs are ultrahigh
strength and ultrahigh elastic strain because there is contribution of
three different behavioural states, as the elastic of the BCT structure,
the inelastic of the BCT–FCT structure and elastic of the FCT structure
in both the loading and unloading. The reversible switching between
the BCT and FCT structures is a key to explain the pseudoelastic effects
of the 〈100〉/{110} Cu NWs.
Acknowledgements
This work was supported by the Vietnam's National Foundation
for Science and Technology Development (NAFOSTED) with
No.107.02.2012.20.
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5N.T. Hung, D. Van Truong / Surface Science 641 (2015) 1–5