Computational Materials Science 67 (2013) 182–187
Contents lists available at SciVerse ScienceDirect
Computational Materials Science
journal homepage: www.elsevier.com/locate/commatsci
Investigation on the most probable breaking behaviors of copper nanowires
with the dependence of temperature
Fenying Wang a,b,c, Wei Sun c, Yajun Gao c, Yunhong Liu c, Jianwei Zhao c,⇑, Changqing Sun d
a
Education Center for Basic Chemistry Experiments, School of Science, Nanchang University, Nanchang 330031, PR China
School of Material Science and Engineering, Nanchang University, Nanchang 330031, PR China
c
State Key Laboratory of Analytical Chemistry for Life Science, School of Chemistry and Chemical Engineering, Nanjing University, Nanjing 210008, PR China
d
School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore
b
a r t i c l e
i n f o
Article history:
Received 25 May 2012
Received in revised form 4 July 2012
Accepted 8 July 2012
Available online 5 October 2012
Keywords:
Molecular dynamics simulations
Copper nanowires
The most probable breaking position
a b s t r a c t
The deformation and breaking behaviors of metallic nanowires have raised concerns owing to their
applied reliability in a nanoelectromechanical system. In this paper, molecular dynamics simulations
are used to study the deformation and breaking properties of the [1 0 0] oriented single-crystal copper
nanowires subjected to uniaxial tension at different temperatures. With a dependence of temperature,
statistical samples identify a most probable breaking position of the nanowire, and the ‘‘most probable’’
feature reveals that the breaking behavior is correlated with nanoscale compression wave propagation at
different temperatures. Macro-breaking position distributions confirm the influence of temperature on
micro-atomic fluctuation during the symmetric stretching of the nanowires.
Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction
In recent years, the deformation character of the metallic nanowire has attracted extensive attention because the character is related with its fascinating mechanical [1,2], thermal [3], electrical
[4,5] and magnetic [6] properties. To investigate metallic nanowires, the experimental approaches are mainly using scanning tunneling microscopy (STM) [7,8], atomic force microscopy (AFM)
[9,10], transmission electron microscope (TEM) [11,12] and
mechanically controllable break junctions (MCBJs) [13,14]. However, owing to their small scale [15], mechanical measurements
of individual nanowires are challenging because of some difficulties in performing standard tensile or bending tests on the
nanoscale systems. In addition, the deformation and breaking of
nanowires could be easily affected by many factors, such as temperature [16,17], orientation [18], loading rate [17–19], length
[20], surface conditions and boundary conditions [21,22], so the
metallic nanowires always behave their special and unpredictable
behaviors under experimental conditions. That is, there are some
difficulties to control and to manipulate the deformation and
breaking behaviors of nanowires in nanoelectromechanical systems. In contrast, molecular dynamics (MD) simulation [23,24],
which solves Newton’s equation of motion for a collection of interacting particle over a number of time steps, is an effective method
to evaluate deformation and breaking characters of the nanowire
⇑ Corresponding author. Tel./fax: +86 25 83596523.
E-mail address: [email protected] (J. Zhao).
0927-0256/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.commatsci.2012.07.048
subjected to stretching. It is also helpful for improving the applied
reliability of nanowires.
Using MD simulations, Sankaranarayanan et al. [17] studied
elastic properties and deformation behaviors of pure palladium
and platinum nanowires at different temperatures, and gave that
the crystal order and stability of the nanowire was highly preserved
at low temperature. Park and Ji [25] proposed the thermomechanical deformation of silver nanowires, and gave its strong dependence
of shape memory upon the temperature. Ju et al. [26] studied
mechanical properties of the ultrathin gold nanowire at different
temperatures and gave the influence of temperature on the
detected one atom chain. For the properties of metallic nanowires,
the deformation and breaking behavior of the [1 0 0] single-crystal
copper nanowire was studied under continuous strain [27]. It was
observed that one sample could not predetermine the breaking
position of the nanowire, and the breaking behavior is found to
present a statistic feature. In experiments, similar microscopic
phenomena were also found for the metallic and molecular junction
conductance, which follows a statistical distribution [28,29]. These
results indicate that analyzing of enormous samples might be the
only way to investigate the material failure in the nanoscale
devices. While the size of the device component reaches several
nanometers, the most probable breaking has been expected to be
more profound in dominating the device failure.
One question is raised, is any ‘‘most probable’’ quality in the
deformation and breaking of metallic nanowires at different
conditions? The ‘‘most probable’’, a concept in statistical mechanics, describes a massive physical system that is the result of the
F. Wang et al. / Computational Materials Science 67 (2013) 182–187
motions of molecules or atoms in the system. A well-known example is the most probable speed in Maxwell–Boltzmann distribution.
More evidences for the existence of the statistical distribution at
atomic/molecular levels with a ‘‘most probable’’ quality have been
reported with the atomic/molecular junction conductance, which
is motivated by the potential application of molecular electronic
devices [28–33]. However, there is no any sign of the existence
of the statistic distribution for the macroscopic objects, because
they follow Newton’s laws of motions that precisely determine
the displacement if the mass and force are known.
In this study, we design different initial equilibrium states to
investigate statistically the deformation and breaking behaviors of
[1 0 0] single-crystal copper nanowires at different temperatures.
A most probable breaking position (MPBP) is identified using
the breaking position distribution of the nanowires, and it should
be the evidence of the ‘‘most probable’’ quality in the deformation
behavior of the nanowire. According to these, we also note that
macro-breaking position distributions reflect micro-atomic fluctuation in the deformation of the nanowires, which are related with the
deformation mechanism of the nanowires at different temperatures.
183
Johnson. This methodology may also reduce the systematic error
to ensure the reliability of MD simulations. Therefore, Johnson’s
EAM potential might be the best choice for this particular purpose.
The total energy was given by:
E¼
X
1X
Vðr ij Þ þ
Fðqi Þ;
2 ij
i
qi ¼
X
uðrij Þ
ð1Þ
ð2Þ
i–j
where E is the total internal energy of the system, V is the pair potential between atoms i and j, and rij is the distance between them, F(qi)
is the energy to embed atom i in an electron density qi, u(rij) is the
electron density at atom i due to atom j as a function of the distance
rij. The stress (r) within the nanowire was calculated by the Virial
scheme [35]. All the presented MD simulations and visualization
process were performed with the software NanoMD [36], of which
reliability has been validated by a large amount of theoretical simulations [27,41–43] and the experimental measurements [48,49].
3. Results and discussion
2. Methodology
For all MD simulations of the [1 0 0] single-crystal copper nanowires, the size was 5a 5a 10a (a stands for copper’s lattice
constant of 0.362 nm), corresponding to 1000 atoms. Ten temperatures were set from 100 to 800 K. Before uniaxial stretching, the
single-crystal copper nanowire was relaxed firstly under zero
traction and zero stress in three-dimensional space to reach a
metastable equilibrium state. In general, equilibration states were
confirmed to be achieved when the recorded average potential energy per atom reached a stable state and the stress in the stretching direction was fluctuating slightly around 0.0 GPa with an
increase in the relaxed time. For each temperature, 300 initial equilibrium states were set for the system. In general, 300 states were
enough, reliable and suitable. It was because too little (less than
200) could not give an accurate distribution of the breaking positions, and too many samples (more than 300) would bring some
difficulty in calculating the deformation of nanowires. After relaxation, the nanowire was subjected to uniaxial stretching by uniformly moving two fixed layers in the tensile direction. In all, a
total samples of 3000 (10 300) were used to study the influence
of temperatures on the deformation and breaking behaviors of
[1 0 0] single-crystal copper nanowires. The strain rate was defined
as e = de/dt which was the rate of change in strain with respect to
time. The strain (e) was defined as e = (l l0)/l0, where l was the
current stretching length and l0 was the length just after relaxation. In this work, the strain rate of 1.3% ps1 was set in a quasistatic tensile region in order to avoid the influence of strain rate
as much as possible [19,34,35].
In MD calculations, Verlet leapfrog algorithm was used for the
integration of motion equations to obtain velocity and trajectories
of atoms with a simulation time step of 1.6 fs [36,37]. Nośe–Hoover
thermostat [38,39] was used as a rescaling method of velocity. Free
boundary condition was adopted [20,27,18,40–43]. For potential
functions, tight-binding molecular dynamics (TB-MD) [44,45] or
Car–Parrinello molecular dynamics (CP-MD) [46] simulations
might give the accurate results. However, these potential functions
have some difficulty in the calculations of the nanowires with
many samples. In contrast, the interaction between copper atoms
was described EAM potential function developed by Johnson
[47], which could provide an effective description of the transition
metals with the face centered cubic (FCC) structure. In addition, a
statistical analysis and a series of comparisons have been applied
in the present study using EAM potential function developed by
Using MD simulations, the deformation behaviors, deformation
mechanisms and stress–strain relationships of nanowires are studied to know the influence of temperatures on the most probable
breaking of nanowires subjected to uniaxial tension. Fig. 1a shows
the typical stress–strain relationships of [1 0 0] single-crystal copper nanowires and the representative breaking snapshots at different temperatures from 100 to 600 K. For all the temperatures,
stress increases linearly with the strain increasing before the first
yield point (the critical point between elastic and plastic deformation), which is consistent with elastic law. That is r1 = Ye1, Y is
Young’s modulus. r1 and e1 are strain and stress at the first yield
point, respectively, and they are defined as the first yield stress
and the first yield strain. Upon further stretching, a sudden drop
in stress–strain response exhibits the appearance of the plastic
deformation of the nanowire. Subsequently, the yield cycle repeats
continuously with a decreasing trend. When the yield cycle is completely over, it corresponds to the finally breaking of the nanowire.
Fig. 1a also shows the breaking strain increases at the final yield
cycle when the temperature increases. It indicates the temperature
is helpful for improving the ductility of the metallic nanowire. In
addition, the first yield point, that is the landmark between elastic
and plastic deformations, shows the strong dependence of temperature. Fig. 1b and c give the first yield stress and the first yield
strain against the corresponding temperatures, respectively. The
first yield stress decreases almost linearly, but the first yield strain
keeps almost constant with an increase in the temperature. From
the fitting line, we may obtain the temperature of 1060 K while
the first yield stress is zero GPa, which is very close to the melting
point of copper nanocluster. The first yield strain falls into a range
between 0.10 and 0.12, and it almost keeps constant at the temperature range from 100 to 600 K. Whereas, the higher temperature
(700 K) leads to a wide span of the yield strain. Young’s modulus
is derived from the slope of the linear part of the stress and strain
response before the first yield point. Fig. 1d shows Young’s modulus against the temperature, and it decreases when the temperature increases. The temperature dependence indicates increasing
temperature could lower the mechanical strength of metallic
nanowires.
The influences of the temperature on the mechanical properties
of nanowires are attributed to their different deformation mechanisms at different temperatures. Videos S1, S2, S3 and S4 show
the deformation behaviors of the nanowires at 100, 300, 600 and
700 K, respectively. At the lower temperature of 100 K, the
184
F. Wang et al. / Computational Materials Science 67 (2013) 182–187
Fig. 1. (a) Typical stress–strain responses with the strain rate 1.3% ps1 at the temperature range from 100 to 600 K. The snapshots of the nanowires corresponding to the
specimen size 5a 5a 10a (a stands for copper’s lattice constant 0.362 nm). (b) The first yield stress plotted against temperature. (c) The first yield strain plotted against
temperature. (d) Young’s modulus plotted against temperature.
stretching initially generates local amorphous that serves as
lubrication and the sliding along the (1 1 1) plane. When the temperature is up to 300 K, the slippage plane appears in the initial
stretching of the nanowire. However, amorphous structures dominate the whole deformation of the nanowire. A neck forms in
the middle region and the nanowire breaks around the neck. As
the temperature increases to 600 or 700 K, such amorphous structures become more and wider-spread over the nanowire. The
whole deformation exhibits a clear transformation from crystalline
to amorphous state. As also evidenced in the stress–strain curves,
the periodic fluctuation gradually vanishes. Instead, the random
stress fluctuation increases monotonously with an increase in the
temperature. It reflects the lowering of the structural order of the
nanowire.
The above studies show the higher temperature easily causes an
amorphous state of crystalline structures. It is attributed that acute
thermal movements of metallic atoms have higher potential energy when the nanowire is stretching at higher temperature. As
shown in Fig. 2, statistical results give the maximum average potential energy per atom in the tensile deformation processes of
nanowires at different temperatures. The average potential energy
increases with the temperature increasing, and the increasing
F. Wang et al. / Computational Materials Science 67 (2013) 182–187
Fig. 2. The maximum average potential energy per atom plotted against
temperature.
trend accords with a function of E (eV) = 1.655 + 0.018 exp(T/359.4), in which, E is the maximum average potential energy
per atom, T is the temperature. It illustrates the temperature
dependence is obvious especially for the higher temperatures,
and the relationship between the average potential energy and
the temperature reflects the influence of temperature on the degree of lattice order, which results from the breaking of metallic
bonds. It is a direct consequence of atomic movement overcoming
the interatomic cohesive energy in the tensile deformation process
of the nanowire. At the high temperature, a larger magnitude of
atomic oscillation around its equilibrium configuration makes the
atoms overcome the interatomic cohesive energy to form a disordered amorphous structure, which induces the increasing of the
average potential energy per atom. Whereas, the crystalline order
is relatively perfect at the low temperature, and the average potential energy per atom is also low in the deformation process of the
nanowire.
Knowing that the deformation behaviors and mechanical
properties of metallic nanowires are sensitive to the temperatures,
185
we perform a systematic study of 300 samples for each temperature, and we give histograms of the final breaking positions to
illustrate the relationships between macro-breaking position distribution and deformation mechanism induced by micro-atomic
fluctuation, which is related with thermal movements of metallic
atoms. As shown in Fig. 3a, it exhibits the analyzed breaking position distribution at different temperatures. Here, we considered
the breaking distribution has not a completely symmetric character. The relative breaking position is used to illustrate the breaking
of the nanowire at different temperatures. Instead of a smooth/uniform distribution, we find a pair of peaks existing on the both sides
symmetrically to the center of the nanowire though they are
merged at low temperatures, and two peaks are separated when
the temperature is 700 K. For the histogram of the breaking position distribution, Gaussian function is used to fit the histogram,
and the fitting peak corresponds to the most probable breaking position (MPBP) of the nanowire. With the temperature increasing,
the MPBP propagates from the middle to two ends of the nanowire.
The MPBP transformation is related with the microatomic fluctuation ways at different temperatures. As is well-recognized, the
stretching gives the nanowire a perturbation, leading to the lowering of the atom density. The nanoscale compression wave is originated from the stretching shocks at two ends of the nanowire.
Compared to the thermal motion of the large amount atoms, it is
weak for such fluctuation in atom density caused by stretching,
but the propagation of the density wave leads to a certain structural dislocation at a moment far from the apparent yielding point.
This microscopic structural change can statistically affect the MPBP
of the nanowire subjected to uniaxial tension. Only this point is the
true elastic limit. Getting inspired from the classical wave theory,
one may expect the wave speed is dependent on the Young’s
modulus and its density [50–56]. However, the temperature may
modulate Young’s modulus and the material density when the
nanowire is stretching at different temperatures, and it is different
from the simply changing of the nanowire with different sizes. It is
Fig. 3. (a) The breaking position distribution of the nanowire at the temperatures from 100 K to 800 K. (b) The corresponding peak width of the breaking position distribution.
(c) The comparison of the statistical and calculated most probable breaking positions.
186
F. Wang et al. / Computational Materials Science 67 (2013) 182–187
because different temperatures can generate different atomic flocculation and movements. Especially for the higher temperature,
the deformation of the nanowire belongs to a fast stage of the damage process, and the disordered crystalline structures induced by
temperatures dominate the damage progress of the nanowires.
With the structural observation and the stress–strain curves at different temperatures, we may understand how the temperature affects the compression wave as well as its propagation speed along
the tensile direction of the nanowire.
Fig. 3b gives the peak width of peaks in Fig. 3a at different temperatures, and the linear variation of the peak width is considered
as the concentration of the wave energy as a result of the increasing local amorphous and the decreasing compression ability. At
lower temperature, the nanowire is harder and more difficult to
be compressed. At the beginning of symmetrical loading, the waves
start to propagate from both ends and then meet at the center of
the nanowire. Due to superposition of two waves, the central
cross-section is loaded by twice larger stress, which causes significant local stretching of materials. It leads to the decrease of crosssection area and net tensile strength at the center of wire. This moment one may consider as quasi-static stretching of nanowire
when the applied stress is more or less uniformly distributed along
the nanowire having the weak central part. Therefore, the compression wave is wide-spread, leading to the wide breaking distribution. On the contrary, the deformation of the nanowire is in a
fast stage of the damage process at high temperature, and disordered crystalline structures induced by temperatures dominate
the damage progress of the nanowires. This moment one may consider as a non-equilibrium stretching of nanowire. In comparison
with the deformation at low temperature, tensile strength becomes lower than a stress in single wave (before superposition),
and the damage can be initiated at the vicinities of nanowire ends.
It leads to formation of a neck and damage of the nanowire at the
ends. Under these effects, the softer materials are easier to be compressed at high temperatures, so that the wave energy can be concentrated under the uniaxial stretching. Therefore, the peak width
of the breaking position distribution is narrow at higher temperatures. As shown in Fig. 3b, a good linear relationship can be observed in the wide temperature range from 100 to 700 K. When
the temperature increases to 800 K, the nanowire undergoes a
transformation from crystalline to amorphous at such high temperature. So the data from 800 K are not shown for the peak width
of the breaking distribution. An important feature is that by the
extrapolation of the linear relationship to the zero wave width,
which corresponds to the predetermined situation instead of probability distribution, we obtained a critical temperature of ca. 900 K.
This value is very close to the melting point of the FCC metallic
nanoparticles, such as Cu and Au [57,58].
One of the most intriguing questions is how to correlate the
MPBP with the wave propagation. From the macroscopic viewpoint, the stretching may lead to a pressure wave, though the continuous stretching generates a much complicated overlap of wave
packet. The traverse of the pressure wave has significant effect on
the MPBP, since the perturbation from the pressure wave can be
significantly compared to the intermolecular interaction in nanoscale systems. For macroscopic bulk materials, the fluctuation of
the atomic thermal motion is completely negligible. However, in
the microscopic scale, the pressure wave energy is comparable to
the energy fluctuation due to a small amount of atoms at nanoscale. Therefore, the effect of the pressure wave is prominent.
Although the particular position is unpredictable for a onesample breaking, the MPBP can be accurately determined while
the mechanical pressure wave plays the predominant role in the
process. In order to give the wave propagation, one requires giving
particular correlation between the MPBP with a macroscopic
quality. In a simply model [19–21], the longitudinal elastic wave
velocity
pffiffiffiffiffiffiffiffiffican be derived from the wave equation given by
t ¼ E=q, in which E is the Young’s modulus of the medium,
and q is the materials density. Although E is a function of the temperature and it can be calculated accurately in classical materials
physics, it varies for the nano-sized metallic wires due to the fluctuation of the atomic motion [17,25,59]. With the average Young’s
modulus as a function of temperature (see Fig. 1d), we calculate
the wave speed using the above formula and then give the comparison with the statistical and calculated MPBP (see Fig. 3c). It displays a linear relationship between MPBP and velocity with
going through the axis origin. In principle, the slope can give the
time in which the pressure wave traverses to the MPBP. The nanowire reaches a true elastic limit, corresponding to the weakest
stress at which dislocations move. Since dislocations move at very
weak stresses, and detecting such movement is difficult even with
tremendous amount of comparison. At the very beginning of
stretching, the nanowire remains a perfect crystal structure without dislocation. Further elongation makes the nanowire becoming
more and more unstable, though the original structure still remains. At the critical point that should correspond to the true elastic limit, the nanowire is located on an unstable state. Any
perturbation, in spite of how small it is, may lead to the structural
change of the nanowire. At this moment, the shock originated from
compression wave plays an important role in determining the final
breaking distribution [56]. When the nanoscale compression wave
helps the stretching overcome the first dislocation, further propagation continues as avalanche, facilitating the final breaking
around the concentration point of the shock wave. Therefore the
determination of the particular wave position, at which the nanowire just falls into the true elastic limit, is a key step in prediction
of the final MPBP.
4. Conclusions
In conclusion, we have performed the MD simulations of the
breaking of the nanowire at temperatures from 100 to 800 K. The
results prove that the nanoscale compression wave plays an
important role in the shift of the MPBP in the breaking position distribution of the nanowire. It also gives insight into how and at
what extent the macroscopic wave theory affects microscopic
behaviors of nanoscale materials.
Acknowledgments
This project was supported by the National Natural Science
Foundation of China (Grant Nos. 21273113 and 51071084), Natural
Science Foundation of Jiangsu Province (BK2010389), and Fundamental Research Funds for the Central Universities, and Special
Postdoctoral Science Foundation funded project of Jiangxi
Province.
Appendix A. Supplementary material
Supplementary data associated with this article can be found, in
the online version, at http://dx.doi.org/10.1016/j.commatsci.2012.
07.048.
References
[1]
[2]
[3]
[4]
J.K. Diao, K. Gall, M.L. Dunn, Nat. Mater. 2 (2003) 656–660.
B. Wu, A. Heidelberg, J.J. Boland, Nat. Mater. 4 (2005) 525–529.
L. Miao, V.R. Bhethanabotla, B. Joseph, Phys. Rev. B 72 (2005) 134109.
N.A. Melosh, A. Boukai, F. Diana, B. Gerardot, A. Badolato, P.M. Petroff, J.R.
Heath, Science 300 (2003) 112–115.
[5] M.A. Reed, C. Zhou, C.J. Muller, T.P. Burgin, J.M. Tour, Science 278 (1997) 252–
254.
[6] A.S. Alexandrov, V.V. Kabanov, Phys. Rev. Lett. 95 (2005) 076601.
F. Wang et al. / Computational Materials Science 67 (2013) 182–187
[7] N. Agrait, J.G. Rodrigo, C. Sirvent, S. Vieira, Phys. Rev. B 48 (1993) 8499.
[8] U. Landman, W.D. Luedtke, B.E. Salisbury, R.L. Whetten, Phys. Rev. Lett. 77
(1996) 1362.
[9] N. Agrait, G. Rubio, S. Vieira, Phys. Rev. Lett. 74 (1995) 3995.
[10] P.E. Marszalek, W.J. Greenleaf, H.B. Li, A.F. Oberhauser, J.M. Fernandez, PNAS 97
(2000) 6282–6286.
[11] S.B. Legoas, D.S. Galvao, V. Rodrigues, D. Ugarte, Phys. Rev. Lett. 88 (2002)
076105.
[12] V. Rodrigues, T. Fuhrer, D. Ugarte, Phys. Rev. Lett. 85 (2000) 4124.
[13] C.J. Muller, J.M. Vanruitenbeek, L.J. Dejongh, Phys. Rev. Lett. 69 (1992) 140.
[14] C.J. Muller, J.M. Vanruitenbeek, L.J. Dejongh, Physica C 191 (1992) 485–504.
[15] K.J. Hemker, Science 304 (2004) 22.
[16] F. Wang, T. Chen, T. Zhu, Y. Gao, J. Zhao, J. Appl. Phys. 110 (2011) 084307.
[17] S.K.R.S. Sankaranarayanan, V.R. Bhethanabotla, B. Joseph, Phys. Rev. B 76
(2007) 134117.
[18] F. Wang, Y. Gao, T. Zhu, J. Zhao, Nanoscale Res. Lett. 6 (2011) 291.
[19] A.S.J. Koh, H.P. Lee, Nano Lett. 6 (2006) 2260.
[20] Y. Liu, J. Zhao, F. Wang, Phys. Rev. B 80 (2009) 115417.
[21] A.S.J. Koh, H.P. Lee, Nanotechnology 17 (2006) 3451–3467.
[22] V.K. Sutrakar, D.R. Mahapatra, Nanotechnology 20 (2009) 045701.
[23] M.P. Allen, D.J. Tildesley, Computer Simulation of Liquids, Clarendon, New
York, 1997.
[24] D.C. Rapaport, The Art of Molecular Dynamics Simulation, second ed.,
Cambridge University Press, 2004.
[25] H.S. Park, C.J. Ji, Acta Mater. 54 (2006) 2645–2654.
[26] S.P. Ju, J.S. Lin, W.J. Lee, Nanotechnology 15 (2004) 1221–1225.
[27] D. Wang, J. Zhao, S. Hu, X. Yin, S. Liang, Y. Liu, S. Deng, Nano Lett. 7 (2007)
1208–1212.
[28] B.Q. Xu, N.J.J. Tao, Science 301 (2003) 1221–1223.
[29] P. Reddy, S.Y. Jang, R.A. Segalman, A. Majumdar, Science 315 (2007) 1568–
1571.
[30] Z.F. Huang, F. Chen, P.A. Bennett, N.J. Tao, J. Am. Chem. Soc. 129 (2007) 13225–
13231.
[31] M. Kiguchi, R. Stadler, I.S. Kristensen, D. Djukic, J.M. van Ruitenbeek, Phys. Rev.
Lett. 98 (2007) 146802.
[32] P. Garcia-Mochales, R. Paredes, S. Pelaez, P.A. Serena, Nanotechnology 19
(2008) 225704.
[33] A. Hasmy, A.J. Perez-Jimenez, J.J. Palacios, P. Garcia-Mochales, J.L. CostaKramer, M. Diaz, E. Medina, P.A. Serena, Phys. Rev. B 72 (2005) 245405.
187
[34] P. Heino, H. Hakkinen, K. Kaski, Phys. Rev. B 58 (1998) 641.
[35] H.A. Wu, Eur. J. Mech. A – Solids 25 (2006) 370–377.
[36] J. Zhao, X. Yin, S. Liang, Y. Liu, D. Wang, S. Deng, J. Hou, Chem. Res. Chin. Univ.
24 (2008) 367–370.
[37] G.M. Finbow, R.M. LyndenBell, I.R. McDonald, Mol. Phys. 92 (1997) 705–
714.
[38] S. Nośe, J. Chem. Phys. 81 (1984) 511–519.
[39] W.G. Hoover, Phys. Rev. A 31 (1985) 1695–1697.
[40] Y. Liu, F. Wang, J. Zhao, L. Jiang, M. Kiguchi, K. Murakoshi, Phys. Chem. Chem.
Phys. 11 (2009) 6514–6519.
[41] F. Wang, Y. Liu, X. Yin, N. Wang, D. Wang, Y. Gao, J. Zhao, J. Appl. Phys. 108
(2010) 074311.
[42] F. Wang, Y. Liu, T. Zhu, Y. Gao, J. Zhao, Nanoscale 2 (2010) 2818–2825.
[43] F. Wang, Y. Gao, T. Zhu, J. Zhao, Nanoscale 3 (2011) 1624–1631.
[44] E.Z. da Silva, F.D. Novaes, A.J.R. da Silva, A. Fazzio, Phys. Rev. B 69 (2004)
115411.
[45] M.J. Mehl, D.A. Papaconstantopoulos, Phys. Rev. B 54 (1996) 4519–4530.
[46] J. Hutter, A. Curioni, Chemphyschem 6 (2005) 1788–1793.
[47] R.A. Johnson, Phys. Rev. B 39 (1989) 12554–12559.
[48] J. Zhao, K. Murakoshi, X. Yin, M. Kiguchi, Y. Guo, N. Wang, S. Liang, H. Liu, J.
Phys. Chem. C 112 (2008) 20088–20094.
[49] L. Jiang, X. Yin, J. Zhao, H. Liu, Y. Liu, F. Wang, J. Zhu, J. Phys. Chem. C 113 (2009)
20193–20197.
[50] B.L. Holian, Phys. Rev. A 37 (1988) 2562–2568.
[51] B.L. Holian, G.K. Straub, Phys. Rev. Lett. 43 (1979) 1598–1600.
[52] G.K. Straub, B.L. Holian, R.G. Petschek, Phys. Rev. B 19 (1979) 4049–4055.
[53] K. Kadau, T.C. Germann, P.S. Lomdahl, B.L. Holian, Science 296 (2002) 1681–
1684.
[54] E.M. Bringa, J.U. Cazamias, P. Erhart, J. Stolken, N. Tanushev, B.D. Wirth, R.E.
Rudd, M.J. Caturla, J. Appl. Phys. 96 (2004) 3793–3799.
[55] M.F. Horstemeyer, M.I. Baskes, S.J. Plimpton, Acta Mater. 49 (2001) 4363–
4374.
[56] V.V. Zhakhovskii, S.V. Zybin, K. Nishihara, S.I. Anisimov, Phys. Rev. Lett. 83
(1999) 1175–1178.
[57] K. Koga, T. Ikeshoji, K. Sugawara, Phys. Rev. Lett. 92 (2004) 115507.
[58] Y. Shibuta, T. Suzuki, Chem. Phys. Lett. 498 (2010) 323–327.
[59] D. Wolf, V. Yamakov, S.R. Phillpot, A. Mukherjee, H. Gleiter, Acta Mater. 53
(2005) 1–40.