RESEARCH PAPER

Statistical analysis of stretched aluminum nanowires

Enrique Abad • Cesar Gonzalez •

Jose I. Martınez • Fernando Flores • Jose Ortega

Received: 17 June 2013 / Accepted: 8 January 2014

� Springer Science+Business Media Dordrecht 2014

Abstract We present a statistical analysis of the

mechanical and transport properties of stretched Al

nanowires. A molecular dynamics density functional

theory is used in combination with annealing tech-

niques to analyze a large amount of stretching

processes and new realistic geometries. From these

calculations, we generate a conductance histogram

that is compared with the experimental evidence. New

particular geometries appear frequently, and a corre-

lation between these new structures and the peaks in

the conductance histogram can be fairly established. In

particular, at the first stages of the nanowire elonga-

tion, we find a configuration with Al–Al bonds

oriented along the stretching direction that is related

to the peak appearing at 3 G0 in the conductance

histogram. Besides, an Al–Al dimer is found in most

of the cases at the nanowire neck in the last stage of the

nanowire stretching, just before the breaking point;

this configuration is reflected in the peak found in the

conductance histogram at 1 G0.

Keywords Density functional theory � Metal

nanowires � Aluminum � Annealing �Conductance histogram � Modeling and

simulation

Introduction

In the recent years, the great impulse on research in the

field of nanotechnology in its theoretical, experimen-

tal, and industrial branches has led to impressive

advances in the control and design of effective

electronic devices at the nanoscale (Datta 1997; Agraıt

et al. 2003; Xu and Tao 2003; Yeyati et al. 1999;

Gomez-Navarro et al. 2005; Rubio-Bollinger et al.

2001; Wang 2009; Tao 2006). It is well known that

how the conventional macroscopic physics dramati-

cally fails to predict their behavior at that scale, since

quantum phenomena govern these systems. This

implies that, before these devices can be developed

and constructed at large scale, having an accurate

description of their mechanical, chemical, and elec-

trical properties is of paramount importance, and

arises as a challenging task. In the last two decades,

with the development of the scanning probe micro-

scope techniques (Binnig et al. 1982, 1986), an

important step forward was made permitting a

E. Abad

Institute of Theoretical Chemistry, Universitat Stuttgart,

70569 Stuttgart, Germany

C. Gonzalez � J. I. Martınez � F. Flores � J. Ortega

Departamento de Fısica Terica de la Materia Condensada,

Condensed Matter Physics Center (IFIMAC), Universidad

Autonoma de Madrid, 28049 Madrid, Spain

e-mail: fernando.flores@uam.es

J. I. Martınez (&)

Departamento de Superficies y Recubrimientos, Instituto

de Ciencia de Materiales de Madrid (CSIC),

28049 Madrid, Spain

e-mail: joseignacio.martinez@icmm.csic.es

123

J Nanopart Res (2014) 16:2262

DOI 10.1007/s11051-014-2262-0

substantial advance in this field in both its theoretical

and experimental research (Datta 1997; Agraıt et al.

2003; Brandbyge et al. 1995; Gimzewski and Moller

1987; Sugimoto et al. 2007; Martınez et al. 2012).

In order to obtain nanosized electronic circuits, we

need a full comprehension and control of conducting

nanowires. In this line, a lot of research has been

devoted to carbon nanotubes, since they are highly

effective candidates for this kind of applications

(Gomez-Navarro et al. 2005; Tans et al. 1997; Frank

et al. 1998; Gonzalez et al. 2009). Nevertheless, other

class of nanowires formed by metallic atoms has also

been widely studied (Agraıt et al. 2003; Brandbyge

et al. 1995; Rodrigues and Ugarte 2002; Jelınek et al.

2003, 2004; Bhushan 2004; Hafner 2009; Talele et al.

2011; Sonawane et al. 2013), and found to be

extremely useful not only because of their improved

conducting properties, but also for their enhanced

catalytic behavior (Jelınek et al. 2006, 2008; Pieczyrak

et al. 2008), even for chemically inert metals such as

gold (Yoon et al. 2005; Chen and Goodman 2004).

The mechanical and transport properties of some

metallic nanowires (Al, Au, Ni...) have been carefully

measured by means of different techniques, like

Scanning Tunneling Microscopy (STM) or Mechan-

ically Controllable Break Junction (MCBJ) (Rodri-

gues and Ugarte 2002); a deep insight in the properties

of those systems was obtained by measuring statisti-

cally the properties of stretched metallic wires until

their breaking. In particular, the evolution of the

nanowire conductance as a function of its elongation

was measured for thousands of samples and, using

these data, a conductance histogram (Yanson and van

Ruitenbeek 1997) was obtained with some particular

conductance values—around n(2e2/h)—showing a

peaked frequency. Analyzing theoretically those sta-

tistical properties is not an easy task since the atomic

geometry of these stretched nanowires is very com-

plex and is not known. In particular, stretched

nanowires can be studied by classical molecular

dynamics methods or by first-principles calculations

based on density functional theory (DFT). Although

first-principles calculations provide a more accurate

description of the mechanical properties and the

electronic structure needed for the calculation of the

conductance, their large computational demands

restrict most of the applications to an analysis of

rather simple geometries for the contact. This explains

that the few attempts (Garcıa-Mochales et al. 2008,

2012) addressed to understanding theoretically the

statistical properties of stretched nanowires have been

performed using classical molecular dynamics

methods.

In this communication, we focus our interest in

performing a statistical study of stretched Al nano-

wires, using a local orbital DFT method, especially

devised to analyze efficiently complex systems (Lewis

et al. 2001, 2011; Jelınek et al. 2005b). This formu-

lation has an added value, as the transport properties of

the system can be easily calculated using non-

equilibrium Green’s function techniques (Jelınek

et al. 2003, 2004). We should say, however, that using

a DFT-approach is still computationally very demand-

ing, and this has forced us to restrict our statistical

analysis to around 20 stretching realizations for a

system with only 12 Al atoms in the nanowire neck

and a well-defined initial orientation (the 111-face). In

order to improve the statistical properties of this

ensemble, we have introduced some thermal annealing

in the calculation which helps to explore the config-

urational phase-space associated with the nanowire

structure along its stretching path. This theoretical

annealing (Ortega et al. 2000) has been simulated

giving random velocities to the mobile atoms; and,

after some steps of free DFT molecular dynamics

(MD), cooling the atoms slowly in order to quench the

system. We should also say that the case of a particular

Al-nanowire has already been analyzed in previous

works, showing that the last stages of the stretching

process is characterized by the appearance of an

aluminum dimer in the middle of the breaking-zone

(Jelınek et al. 2003, 2004, 2005a); in these studies, the

atomic geometries for the nanowire were obtained

from DFT calculations in which the elongation of the

nanowire was increased (0.2 A) step by step, relaxing

the atoms after each step. Then, part of the aim of this

work is to analyze up to which level that conclusion is

dependent on the initial preparation of the sample and

on the method used to obtain the nanowire configu-

rations as the nanowire is stretched.

In ‘‘Method’’ section, we present of our theoretical

approach, explaining the general calculation details,

the theoretical annealing technique, and the transport

calculation methodology. In ‘‘Results’’ section, we

will present our results, whose analysis and interpre-

tation will be given separately for different annealing

temperatures; the nanowire conductance is analyzed,

and the theoretical conductance histogram is presented

2262 Page 2 of 11 J Nanopart Res (2014) 16:2262

123

and compared to the experimental histogram. Finally,

the main conclusions of this study are presented in

‘‘Summary and conclusions’’ section.

Method

Computational details

The structural, mechanical, and electrical properties of

the Al nanowires have been calculated using the local-

orbital DFT code FIREBALL, whose main features have

been extensively explained elsewhere (Lewis et al.

2001, 2011; Jelınek et al. 2005b). The initial config-

uration of the wire corresponds to the FCC stacking of

4 (111) planes with three atoms per layer; the surface

layers simulating the two electrodes are also oriented

along the (111) direction. A supercell approach has

been used, with a 4 9 4 periodicity along the direction

parallel to the surface. We also impose periodicity

along the z-direction (perpendicular to the surface)

joining artificially the last two layers (top and bottom)

of the system, which implies that the electrodes appear

as having four effective layers. The atomic positions

for the edging electrode layers (upper and lower ones)

have been fixed during the MD relaxations. We use a

basis set of sp3 short-range atomic-like orbitals

(Sankey and Niklewski 1989; Basanta et al. 2007)

with cutoff radii (in atomic units) 5.4 and 6.2, for the

Al s and p orbitals, respectively. This yields a lattice

parameter of 4.09 A for bulk Al, to be compared with

the experimental value of 4.05 A (Witt 1997).

Annealing method

We look for new atomic geometries for the nanowire

along the stretching process by means of a DFT

annealing procedure (Ortega et al. 2000). The basic

idea of this approach is to generate new atomic

geometries, which are local minima of the total

energy, for the nanowire along the elongation process

by means of first-principles molecular dynamics

simulations; in the present calculations, we have

applied this approach, in each step of the elongation

process, using the following protocol: (i) first, each

non-fixed atom a is given a random initial velocity va

according to a Maxwell–Boltzmann distribution for a

given temperature T, such that hv2ai ¼ 3kBT=2m,

where m is the mass of the Al atom; (ii) the atoms

are then free to move for a few MD steps (tipically

50–100 fs, that correspond to 50–100 MD steps) using

these initial velocities and the atomic forces from the

DFT calculation; (iii) after the free dynamics step, we

gradually cool down the system to zero temperature

(va = 0) in the same amount of time as in the previous

step using DFT MD with velocity rescaling; (iv)

finally, the resulting configuration is relaxed using

standard techniques.1 Since for practical reasons, we

have to use very short annealing times, high values of

T are used to efficiently explore the configurational

phase-space. In order to simulate the mechanical

stretching process, the topmost layer has been moved

upwards in steps of 0.2 A. After each step, a new

atomic structure for the nanowire is calculated using

the theoretical annealing processes mentioned above.

Transport

For the electron transport calculations, the conductance

of the system has been obtained using a Keldysh-Green

function approach together with the first-principles

Hamiltonian obtained from FIREBALL. This formalism

has been explained in detail elsewhere (Jelınek et al.

2003; Blanco et al. 2004, 2006; Garcıa-Vidal et al.

2003), and here we summarize the main points. In our

approach, we rewrite the total electronic Hamiltonian

describing the system as bH ¼ bH1þ bH2 þ bT12, where

the total system is split into two subsystems, 1 and 2,

bT12 defining the coupling (a matrix) between both

parts. The zero bias differential conductance,

G ¼ dI

dV

� �

V!0, can be calculated as (Caroli et al.

1972; Martın-Rodero et al. 1988; Mingo et al. 1996):

G ¼ dI

dV

� �

V!0

¼ 4pe2

�hTr½q22ðEFÞTeff

21 q11ðEFÞTeff12 �;

ð1Þ

where q11 and q22 are the density of states matrices

associated with the subsystem sides 1 and 2, respec-

tively. T12eff is defined by the following matrix:

Teff11 ¼ ð1� T12G

022T21G

011Þ�1T21, G

0ii being the

1 We let the system evolve following free DFT MD until the

kinetic energy reaches a maximum. At this point, we set all the

atom velocities to zero, and repeat the processuntil the minimum

is obtained.

J Nanopart Res (2014) 16:2262 Page 3 of 11 2262

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Green-function of the uncoupled system i (with

qii ¼ �1=p=Gii). Now, we can rewrite Eq. (1) by

defining the matrix t = 2 p q22(EF)T21eff q11(EF)T12

eff to

obtain (Jelınek et al. 2003):

G ¼ dI

dV

� �

V!0

¼ 2e2

hTr½tty�; ð2Þ

which is the Landauer formula for the conductance

(Landauer et al. 1988). Thus, we can consider the

eigenvectors of the matrix t as the different conduc-

tance channels, and their associated values as the

transmission parameters of each corresponding

channel.

In our transport calculations, we have included

periodicity only in the XY plane, and we have included

two more undeformed layers both in the upper and in

the lower surfaces in order to obtain converged results.

For convenience, the division of the system has been

taken as symmetrical as possible, although the results

for the conductance do not depend on the particular

choice of that division.

Results

Nanowire geometries: T = 0 K

First of all, we have stretched the wire with T = 0 K,

i.e., the nanowire geometries along the stretching

process are obtained using a standard technique (see

Footnote 1) to minimize the energy at each step of the

stretching process, in similarity with previous calcu-

lations (Jelınek et al. 2003, 2004, 2005a). Energy and

conductance versus deformation are shown in Fig. 1.

Also, geometries in the most relevant points (labeled

in Fig. 1) are shown in Fig. 2. Note that in the right

panel of Fig. 1 the contribution of each individual

channel (see ‘‘Method’’ section) is also shown. The

energy versus distance curve shows that the wire

stretches elastically at the first stages of the process,

with a slightly parabolic behavior between the energy

and the deformation (see Fig. 6); this suggests that the

initial wire geometry is already submitted to some

stress, otherwise the energy curve would display a

parabolic behavior with a minimum at zero elonga-

tion. At some point (point A in Fig. 1), the geometrical

structure of the wire rearranges (see Fig. 2) and the

energy suddenly decreases (A ! B) due to this

configurational change. After this inelastic jump the

wire continues to deform elastically until a new

rearrangement occurs. In total, Fig. 1 presents 3

inelastic jumps before the nanowire finally breaks.

At the very last stages of the stretching process a dimer

appears at the nanowire neck (Jelınek et al. 2003), see

Fig. 2f. Notice that there is a direct link between these

structural rearrangements (or inelastic jumps) and

changes in the wire conductance, see Fig. 1(right), in

agreement with previous results (Jelınek et al. 2003,

2004, 2005a; Todorov and Sutton 1993). Each

Fig. 1 (Color online) Energy (in eV)—referred to the starting

geometry—(left panel) and conductance (right panel) as a

function of the elongation (in A) of the wire during the

stretching process at T = 0 K. Some representative points (A–

F) have been highlighted, which correspond to geometric

rearrangements of the wire along the stretching process (see

Fig. 2). In the right panel the total conductance (red line) and

each individual channel contribution (black lines) are shown

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geometrical rearrangement can be related with a

decrease of the conductance due to the closing of

some channels (black solid lines in right panel of

Fig. 1). When the dimer appears, there is a conduc-

tance plateau that slowly increases from around 0.5 G0

to 1 G0 (Jelınek et al. 2003, 2004). If we take a look at

the individual eigenchannel contributions of Fig. 1,

the most important one just before the wire breaks is

related to the spz bond of the Al dimer, while at the

beginning of the final plateau (point F in Fig. 1)

contributions from px py channels are also important

(Scheer et al. 1998).

Nanowire geometries: T = 500 K

In a next step, we have introduced the annealing effect

as explained in ‘‘Annealing method’’ section. In this

case, the Al atoms are given initially random velocities

corresponding to T = 500 K (see ‘‘Annealing

method’’ section), and the Al atoms in the nanowire

are allowed to move freely during 80 fs following

DFT MD. After that, the system is cooled down for

other 80 fs using DFT MD with gradual velocity

rescaling. The final structure is then obtained relaxing

this geometry (see Footnote 1). Figures 3 and 4 show

the results obtained along the process. The energy

versus elongation curve is quite complex, with more

inelastic changes than in the T = 0 K case previously

analyzed. In this case, due to the annealing procedure

introduced in each step in the elongation process, the

system can more easily escape from its local minimum

and explore other geometries. For example, Fig. 5

shows a comparison of the nanowire configurations

obtained in the first steps of the stretching process for

T = 0 K and T = 500 K. The geometry found using

the annealing method (T = 500 K) for short elonga-

tions of the wire, is more stable than the one found for

T = 0 K due to the bonds oriented along the

stretching direction. After this initial wire reconstruc-

tion, the first inelastic jump in the total energy in Fig. 3

occurs at 2.4 A (and the energy loss is small), while at

Fig. 2 (Color online) Geometries corresponding to highlighted

points in Fig. 1 along the stretching process

Fig. 3 (Color online) Energy (in eV)—referred to the starting

geometry—(left panel) and conductance (right panel) as a

function of the elongation (in A) of the wire during the

stretching process (at T = 500 K). Some representative points

(A–K) have been highlighted, which correspond to points where

the conductance changes along the stretching process—

although sometimes there is not a significant change in

energy—. In the left panel the total conductance (red line) and

each individual channel contribution (black lines) are shown

J Nanopart Res (2014) 16:2262 Page 5 of 11 2262

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zero temperature it occurred at 1.8 A, with a larger

energy loss; thus, for elongations of the nanowire

between 2.2 and 3.0 A, the energy of the nanowire

shown in Fig. 1 (T = 0 K) is slightly lower than the

energy obtained in Fig. 3 (T = 500 K). In other

words, the theoretical annealing method does not

always lead to a wire configuration with lower energy

for every step of the stretching process. This is due to

the presence of many configurations which are local

minima of the total energy.

Nanowire geometries: other temperatures

We have repeated the annealing process with different

values of T: 1,000, 1,500, and 2,000 K. For annealing

temperatures higher than 2,000 K, the wire breaks and

spontaneously rearranges as a new surface layer at the

very first stages of the process. We have carried out

two different calculations at every value of T, to check

the influence on the results of the random initial

velocities. The results are shown in Fig. 6.

We can extract some valuable information from

Fig. 6. First, calculations performed using the same

value of T do not lead to the exact same results, i.e., the

initial value of the kinetic energy of the nanowire

atoms in our annealing method does not completely

determine the mechanical stretching evolution of the

wire since the nanowire presents many different

structural local minima for a given elongation. Also,

we find that the geometry shown in Fig. 5 (right)

seems to be very attractive from the phase-space point

of view, since five other different calculations lead to

that geometry, for all different temperatures analyzed.

It is also worth mentioning that there is another similar

geometry that also appears in our relaxations in the

first stages of the elongation process, which also

presents Al–Al bonds oriented along the stretching

direction, but in a different configuration, see Fig. 7.

As mentioned below, these structures are related to the

peak found in the conductance histogram at 3 G0. After

these initial geometries in the first stages, each

nanowire follows its own stretching pattern, but most

of the wires reach a geometry that present a dimer in

the nanowire neck in the final stages, before the

breaking point. Only at the highest annealing temper-

atures (1,500 and 2,000 K) different structures are

found at the neck of the nanowire in these last stages,

such as a trimer and a four atoms chain, see Fig. 8. In

any case, in the last stages of the stretching process, we

always have a conductance plateau with a value

Fig. 4 (Color online) Geometries corresponding to highlighted points in Fig. 3 along the stretching process

Fig. 5 (Color online) Sketch of the geometry obtained during

the first stages of the stretching process for T = 0 K (left) and T =

500 K (right)

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starting below 1 G0 that slowly rises up to 1 G0 just

before reaching the nanowire breaking point. As in the

T = 0 K case, the main contribution at this moment is a

spz channel, while in the conductance plateaus the pxpy

channels also present important contributions (this is

also true for the trimer or long chains cases).

Nanowire geometries: mixed annealing/T = 0 K

calculations

In order to further analyze the nanowire configura-

tional phase-space, we have used a different strategy

to generate nanowire geometries. In this case, we take

as the initial nanowire structure some of the more

characteristic geometries that we have obtained using

the theoretical annealing method; from this starting

point, the nanowire geometries along the stretching

process are obtained relaxing the nanowire to mini-

mize the energy after each elongation step (i.e., T = 0

K relaxations) (see Footnote 1), see Figs. 9 and 10. In

this way, the wire is stuck around that initial local

minimum in the configurational-space, and new

geometric rearrangements only appear when this

minimum becomes unstable, yielding a simpler energy

versus elongation curve. Some of the initial geome-

tries that we have used in these calculations are those

shown in Figs. 5(right), 7 or 8c.

We have obtained some interesting results. First of

all, once again, we always see the characteristic

conductance plateau feature in the last stages of the

wire breaking for every case. The evolution of the

ladder geometry (Fig. 8c) leads to a huge Al chain, but

energetically unstable. Apart from this exception (and

a few more trimer cases), the dimer final configuration

is also the most common geometry in the last stages of

these simulations.

Conductance histogram

The significant number of theoretical stretching pro-

cesses carried out in this study has permitted us the

construction of a conductance histogram, which is

shown in Fig. 11 in comparison with the experimental

evidence (Yanson and van Ruitenbeek 1997). For the

construction of the theoretical histogram more than

700 conductance data-points have been broadened by

Lorentzian profiles, each of them with a width of 0.05

(in 2e2/h units) in order to mimic the experimental

histogram resolution (Yanson and van Ruitenbeek

1997); also, the theoretical histogram has been renor-

malized so that the normalized number of counts at

G = 1 G0 is the same in both histograms. A compar-

ison between the experimental and theoretical histo-

grams manifests a reasonable agreement of all the

representative conductance features.

Fig. 6 (Color online) Comparison of energy (in eV), referred to

the starting geometry, as a function of the elongation (in A) of

the wire during the stretching process for T = 0, 500, 1,000,

1,500, and 2,000 K. Two different calculations are shown for

each annealing temperature (explained in the text). All-cases-

average splines and polynomial fitting are shown superimposed

(solid lines). A visual parabolic guide (dashed blue line) has

been included for the low-elongation regime trend

Fig. 7 (Color online) ‘‘Alternative’’ geometry for the Al-

nanowires found in the initial stages of the elongation process

presenting also Al–Al bonds oriented along the stretching

direction

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The most pronounced peaks in the theoretical

histogram appear for conductances around 1 G0, 2 G0,

and 3 G0, and show slight shifts with respect to the

experimental case (0.9 G0, 1.8 G0, and 2 .9 G0 in the

theoretical case, and 0.8 G0, 1.9 G0, and 3.1 G0 in the

experimental case). The relative intensity between

them is quite accurately reflected in the theoretical

result for the first and third peaks. Nevertheless, we

find a noticeable mismatch between theory and

experiment for the second peak.

These results can be understood in terms of the most

stable structures found in our stretched wires. The peak

around 1 G0 is mainly associated with the dimer found

before the wire breaks; this is in agreement with the

results in previous literature (Jelınek et al. 2003). On the

other hand, the peak around 3 G0 can be associated with

structures such as the one shown in Fig. 5(right); notice

that this structure appears frequently in our calculations

and that its conductance value is 3 G0 (see Figs. 3, 4,

and the particular geometry named A in those figures).

The case of 2 G0 is different: in our calculations, we

have not found a well-defined structure associated with

that peak; on the contrary, many different structures

tend to give a very broad peak as the one found in the

Fig. 8 (Color online)

Geometries showing the

formation of a a trimer, b a

four atoms chain, and cshows the previous ‘‘ladder’’

configuration that leads to

these geometries

Fig. 9 (Color online) Representative mixed annealing/T = 0 K

geometries and energies. The dashed lines separate the initial

stretching part, obtained using the annealing method (T = 2,000

K), and the subsequent stretching geometries obtained simply

relaxing the nanowire to minimize the energy after each

elongation step (i.e., T = 0 K relaxations)

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calculations of Fig. 11. We should say at this point that

our results have been obtained starting in all the cases

with a (111)-surface orientation and it is very likely that

other orientations should be included to improve the

agreement with the experimental evidence. There are

also theoretical peaks around 4 G0 and 5 G0, just like in

the experimental histogram and with comparable

relative intensities. In conclusion, our theoretical

conductance histogram reproduces fairly well the

low-temperature experimental one, apart from the

small peak found around 2 G0. The theoretical statistics

for the construction of histogram seem to be good

enough; nevertheless, calculations with other surface

orientations will be necessary to improve the agreement

with experiments.

Summary and conclusions

In this communication, we have performed a statistical

analysis of the mechanical and transport properties of

stretched Al-nanowires, using first-principles local-

orbital DFT calculations combined with a theoretical

annealing to increase the quality of the ensemble used in

this analysis. Apart from the T = 0 K case, we have

obtained nanowire geometries along the stretching

process using a theoretical annealing method with

different values of the initial temperature, T, and also

using a mixed annealing/T = 0 K stretching. For

practical reasons, the annealing times are quite short,

and thus high values of T are required to efficiently

explore the nanowire configurational phase-space. In

particular, we have used the following values: T= 500,

1,000, 1,500, 2,000 K, in the present study. The

annealing method allows the system to switch frequently

Fig. 10 (Color online) Representative mixed annealing/T = 0

K geometries and energies. The dashed lines separate the initial

stretching part, obtained using the annealing method—(left

panel) T = 1,500 K, and (right panel) T = 500 K—, and the

subsequent stretching geometries obtained simply relaxing the

nanowire to minimize the energy after each elongation step (i.e.,

T = 0 K relaxations)

Fig. 11 (Color online) Comparison of experimental and theo-

retical conductance histograms. The experimental histogram

was constructed from 30,000 individual conductance curves for

two different samples of aluminum, using the mechanically

controlled break junction technique at 4.2 K, at a sample bias

voltage of 10 mV (reprinted from Yanson and van Ruitenbeek

1997 Copyright (1997) by the American Physical Society). The

theoretical conductance histogram is calculated from all the

realizations (more than 700 individual theoretical conductance

data-points) of the wire stretching calculated in this work

J Nanopart Res (2014) 16:2262 Page 9 of 11 2262

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between different energy minima, making the energy

versus distance curve more complex, and narrowing the

elastic deformation regions between atomic rearrange-

ments. The three nanowire rearrangements found at

T = 0 K turn into five at T = 500 K, and even more at

higher annealing temperatures T. Our results show that

the nanowire presents, for a given elongation, many

different meta-stable configurations; some of these

configurations are quite stable, in the sense that it is

not easy for the system to escape from them.

Since, in this study, we have performed more than

20 stretching realizations of the Al-nanowires, it is

possible to obtain some general properties. First, we

have found some particular configurations that appear

in almost all of the calculated structures. At the very

first stages, geometries of Figs. 5(right) and 7 are the

most favorable, this fact being reflected in the peak

found in our conductance histogram at G = 3 G0. At

more advanced stages, a full variety of structures

appear for each stretching process; but, at the last

stages, a dimer is found in most cases, just before the

wire breaks, this fact being reflected in the peak found

in the conductance histogram at 1 G0. In the last stages

of the stretching process, we always have a conduc-

tance plateau slightly below G = 1 G0, that slowly

rises up to G = 1 G0 just before reaching the nanowire

breaking point. In all these plateaus, there are three

channels contributing to the conductance, but two of

them vanish at the very last stages of the wire breaking,

and only one active channel remains. The eigenchan-

nels-channels are formed by one spz channel that is the

one that has the highest conductance contribution just

before wire breaking, and two pxpy ones. Similar results

were obtained in previous literature (Scheer et al. 1998;

Jelınek et al. 2005a).

Acknowledgements Work supported by Spanish MICIIN

(Grant FIS2010-16046) and CAM (Grant S2009/MAT-1467).

EA acknowledges financial support by the CAM and FSE. JIM

acknowledges funding from Spanish MICIIN and CSIC through

‘‘Juan de la Cierva’’ and ‘‘JaeDoc’’ fellowship Programs,

respectively.

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