Investigation of the thermal stability of Cu nanowires using atomisticsimulations

F. Granberg, S. Parviainen,a) F. Djurabekova, and K. NordlundDepartment of Physics and Helsinki Institute of Physics, P.O. Box 43, FIN-00014 University of Helsinki,Finland

(Dated: 3 January 2012)

We present a method for determining the melting point of copper nanowires based on classical moleculardynamics simulations and use it to investigate the dependence of the melting point on wire diameter. Themelting point is determined as the temperature at which there is a significant change in the fraction of liquidatoms in the wire, according to atomic bond angle analysis. The results for the wires with diameters inthe range 1.5 nm to 20 nm show that the melting point is inversely proportional to the diameter while thecross-sectional shape of the wire does not have a significant impact. Comparison of results obtained usingdifferent potentials show that while the absolute values of the melting points may differ substantially, themelting point depression is similar for all potentials. The obtained results are consistent with predictionsbased on the semi-empirical liquid drop model.

Keywords: finite size effects, nanowires, melting, molecular dynamics

I. INTRODUCTION

Nanowires (NW’s) are currently a hot research topicbecause of the many applications in e.g. electronics, com-posite materials and optics, where many of the uniqueproperties of these 1D structures have proven highly use-ful1. However, spontaneously growing nanostructureshave also been linked to short circuits in electronics2 anddestructive vacuum discharges in particle accelerators3

and fusion reactors4. To control both the advantageousand disadvantageous effects associated with the presenceof NW’s, it is necessary to gain additional insight intotheir properties.

It is qualitatively well known that finite size effects(FSE), such as high surface-to-volume ratio, lower co-hesive energy5 and phonon confinement6, give nanoscaleobjects properties that can differ substantially from bulkmaterials. However, as new synthesis techniques en-able the production of ever smaller nanostructures1,7, theneed for a clear quantitative understanding is becomingever more important.

Copper wires are often used electrical interconnects inelectronics due to their high electrical conductivity andlow costs8. Since it is advantageous to minimize the sizeof the interconnects, the electrical properties of Cu NWhave been extensively studied9–11 previously. These stud-ies show reduced electrical and thermal conductivities,which may lead to over-heating of the components wherethey are used12. To increase the reliability of electronicsit is, therefore, critical to understand the thermal stabil-ity of nanowires. Despite this, few systematic studies ofthe melting temperatures of copper nanowires have beenconducted.

It has been known for more than a century that FSE

a)Electronic mail: stefan.parviainen@helsinki.fi

can lower the melting point of nanosized objects, suchas nanoparticles and nanowires. During this time severalmodels explaining the FSE have been proposed, based on,among others, thermodynamics13, phonon instability14

and the liquid drop model15 (LDM). Most of the pro-posed models predict a melting point which is inverselyproportional to the wire diameter. Confirming the the-oretical models experimentally is difficult because thinnanowires break easily even at low temperatures, due toe.g. Rayleigh instability16, strain17 or shock18, makingthem both difficult to manufacture and perform exper-iments on. In spite of this, experimental results of thesize dependence of the melting point have been obtainedfor e.g. gold12, zinc19 and mercury20 nanowires.

Because of the difficulty of working with nanowires,most studies concerning their properties have been per-formed using computer simulations, in particular basedon classical molecular dynamics (MD)21–23. It is, how-ever, important to realize the limitations of this simu-lation technique as many factors, such as the choice ofinteratomic potential, and temperature control methodcan affect the results significantly.

This papers aims to rectify the situation by 1) describ-ing a new method for determining the melting point ofnanowires and its size dependence, 2) providing simula-tion results for a wide range of copper wires of differ-ing shapes, 3) comparing results obtained using severalwidely used potentials, 4) comparing new results with atheoretical estimate based on the liquid drop model and5) providing an analysis of the possible sources of errorsin the results.

II. SIMULATION AND ANALYSIS METHODS

The classical MD code PARCAS24 was used to simu-late Cu nanowires with diameters in the range 1.5 nmto 20 nm with at least 2 different lengths, except for the

2

(a) (b) (c)

FIG. 1: Examples of wires with initially a) circular, b)square and c) hexagonal cross-sections used in the

simulations. The wire axis is [001] for a) and b) and[111] for c).

15 nm and the 20 nm NWs, over a wide temperaturerange.

In order to obtain potential independent results weperformed the simulations using three different poten-tials: the Sabochick-Lam25(SL) and Foiles26 EmbeddedAtom Method potentials and the Corrected EffectiveMedium (MC/MD-CEM) 27 potential. The SL and Foilespotentials are known to give a fairly accurate bulk melt-ing point28, while the MC/MD-CEM potential gives arealistic surface energy29.

For every temperature we performed a separate set ofthe simulations using all three potentials. Firstly, in eachcase we constructed a {100} FCC copper block of the ap-propriate size using the lattice constant with the accountfor lattice expansion, calculated with the applied poten-tial for the given temperature. Such blocks were usedto construct NWs with square or circular cross section,by merely cutting out the required shape from the block.Hexagonal wires were created in a similar manner, butstarting from a {111} FCC block (Fig. 1). The side facetswere in this case {110} planes . The size of the block waschosen to ensure the length of the NWs to be at least1.8 times the diameter; via the replications over periodicboundaries in the lateral direction, the NW could, how-ever, be considered as infinitely long.

Each NW was allowed to evolve during 2 ns, with thelength of each MD timestep being 4.06 fs, at the sametemperature for which the initial block was constructedto allow the wire to reach a steady state, or break in casethe temperature was sufficiently high to cause the wireto melt. After this the liquid atoms were detected byanalysis of the structure factors Pst of all atoms, whichdepend on how far from their equilibrium angles in a per-fect crystal the angles between nearest-neighbour bondsof the atoms are30. We have accepted an atom to be“liquid” if during the simulation its structure factor isbetween 0.35 and 0.675, and in addition it has 5/6 of itsnearest neighbors with the structure factor in the samelimits. These values were obtained previously in Ref. 30and allow for a clear distinction between liquid and de-fect atoms. This is important for the present case as wehave a significant number of surface atoms compared to

the bulk.Since we are dealing with structures that have signif-

icant surface-to-volume ratio, special attention must bepaid to the surface atoms. In order to handle the surfaceatoms in the same analysis, we assume that the anglesbetween surface atoms and their missing neighbours tobe as in a perfect crystal. This artifact, as well as thefact that the surface atoms form an extended defect (withthe structure factor 1), means that the fraction of liq-uid atoms in the NW will never reach unity, even whenthe wire is fully molten. Hence we defined the meltingtemperature as the point at which the fraction of liquidatoms in the wire increases significantly compared withlower temperatures, indicating that a phase transitionhas occurred.

The temperature in the system was controlled by usingBerendsen temperature control31 with the time constant20 fs. The Berendsen-type pressure control set to 0 Pawith the time constant 1000 fs did not show a differentresult compared to no pressure control.

The Berendsen temperature control algorithm worksby scaling atomic velocities, so that the average totalkinetic energy corresponds to the desired temperature.However, in addition to the component due to thermalfluctuation, the kinetic energy also includes rotationaland translational energy. If the latter two componentsare significant, the temperature scaling may be exagger-ated. Since the thinnest NWs have the highest proba-bility of spontaneous rotation, we chose to analyze thecontribution of the rotational energy in the total kineticenergies only for the NW of 1.5 nm in diameter. Wecalculated the mean angular velocity for all the atomsin the system to minimize the effect of thermal fluctu-ations. While the rotational frequency was up to theorder of GHz in some cases, the rotational energy cor-responded to a temperature of merely a few K due tothe small diameter of the wires. Thus, the error in thereported temperature due to rotation is negligible.

III. RESULTS AND DISCUSSION

Fig. 2 shows distributions of atomic structure factors ina wire with a diameter of 3.0 nm, simulated using the SLpotential at different temperatures. At higher tempera-tures a shift towards higher values is clearly visible, indi-cating that a phase transition has occurred. The peak atPst = 1.1 corresponds to surface atoms and does, there-fore, not change significantly with temperature. Even forwires below the melting point there is a small peak in theliquid region of structure factors. In this case the atomswith structure factors in this range all lie close to thewire surface (Fig. 3).

The fraction of liquid atoms rises markedly at a specifictemperature corresponding to the melting point as shownby the example in Fig. 4. Below the melting point thereis a slight temperature dependence due to surface melt-ing, while the fraction of liquid atoms is approximately

3

0.0

0.02

0.04

0.06

0.08

0.1

0.12

Frac

tion

ofat

oms

0.0 0.2 0.4 0.6 0.8 1.0 1.2

Structure factor Pst

1000 K1100 K1200 K1300 K

FIG. 2: Distribution of structure factors in a copperwire with diameter of 3 nm heated at various

temperatures and using the SL potential. The shift inthe distribution at 1200 K indicates that a phase

transition has occurred.

(a) (b)

FIG. 3: Structure factor of individual atoms in across-section of a wire with diameter 3 nm at

temperature a) 1000 K and b) 1200 K

temperature independent above the melting point. Be-cause of the sharp and well-defined rise in liquid atomsthe melting point can be determined with good accuracy.

Even at temperatures below the melting point thereare atoms close to the surface with structure factors inthe liquid region (Fig. 5). This indicates that meltingbegins at the surface even below the conventional meltingpoint, and then spreads towards the centre of the wire.

In order to analyze the obtained result in the system-atic manner we have approximated the square and hexag-onal cross sections of NWs by the circular equivalent,whose diameter was used for the comparison with theanalytical model. The diameter is also calculated for theNW equivalent held at 0 K, not at the melting point. Themelting point is found to be inversely proportional to thewire diameter using all three potentials (Fig. 6), with thesize-dependent melting points given approximately by

TSLm (d) = 1334.35 K− 615.65 nKm/d, (1)

TFoilesm (d) = 1316.30 K− 638.68 nKm/d (2)

and

TMC/MD−CEMm (d) = 1656.72 K− 958.52 nKm/d (3)

FIG. 4: Fraction of liquid atoms as a function oftemperature in a cylindrical wire with a diameter of

3.5 nm simulated using the SL potential. The fractionof liquid atoms does not approach the value 1.0 due to

the way surface atoms are handled in the analysis.

0.2

0.350.4

0.60.675

0.8

1.0

Mea

nst

uctu

refa

ctor

2 4 6 8 10 12 14 16

Distance from centre [A]

300 K

800 K1150 K

1200 K

FIG. 5: Mean structure factor as a function of distancefrom the centre of a cylindrical wire with a diameter of

3 nm simulated using the SL potential at varioustemperatures.

for the SL, Foiles and MC/MD-CEM potentials, respec-tively.

The 1/d proportionality found is consistent with theliquid drop model which gives the melting point as

TLDMm (d) = T bulk

m − 6Vmγ

cd= 1357 K−1247 nKm/d (4)

where T bulkm = 1357 K is the bulk melting point, Vm =

0.012 nm3/mol the molar volume and γ = 9.94 eV/nm2

the surface energy per area for Cu, measured experimen-tally. The constant c = 5.736 × 10−4eV/K is the em-pirically determined ratio between cohesive energy andbulk melting point for different materials15. The sameproportionality has also been found in previous studiesof other metals32,33.

The MC/MD-CEM potential significantly overesti-mates the melting temperatures while the SL andFoiles potentials somewhat underestimate them, com-pared with melting temperatures given by Eq. 4. How-ever, when the experimental values of the bulk melting

4

CircularSquareHexagonalLDM/potLDM/exp

Tbulk

/expm

(a) Sabochick-Lam potential

CircularSquareHexagonalLDM/potLDM/exp

Tbulk

/expm

(b) Foiles potential

CircularSquareHexagonalLDM/potLDM/exp

Tbulk

/expm

(c) MC/MD-CEM potential

FIG. 6: Size-dependent melting points obtained usingthe different potentials and wire shapes. Values as given

by the LDM using both experimentally (LDM/exp)measured values for molar volume and surface energy

for Cu, and values determined from the usedinteratomic potential (LDM/pot) are also shown.

point and surface energy are replaced by values appropri-ate for each potential, the difference between potentialsbecomes much less pronounced. Thus we can concludethat the simulation method is applicable satisfactorily,while the deviation of the result from the experiment canbe attributed to the limitations of the used potentials inthe description of Cu properties on the nanoscale.

While the SL and Foiles potentials give more realistic

0.0

0.2

0.4

0.6

0.8

1.0

Nor

mal

ized

mel

ting

poin

t

0 2 4 6 8 10 12 14 16 18 20

Wire diameter [nm]

Sabochick-LamFoilesMC/MD-CEMLDM/exp

FIG. 7: The ratio of the NW melting point to the bulkmelting point as a function of the wire diameter as

obtained using the different simulation potentials, andas predicted by the liquid drop model.

values for the absolute melting points (compared with theLDM) than the MC/MD-CEM potential, the situation isreversed for the ratio of the melting point of a NW tothe bulk melting point (Fig. 7) which reflects FSE. Thus,even though the MC/MD-CEM potential is ill-suited fordetermining absolute melting points, it is appropriate touse when studying the impact of FSE.

While the surface energy clearly affects the meltingpoint of Cu NWs, the difference in melting temperaturesbetween different wire shapes is small. Although the dif-ference in surface energies between the initial shapes wereapproximately 10% to 20% for the used potentials, wedid not observe a visible effect of it on the melting point.This is explained by the fact, that as the temperatureincreases, the wire changes shape to minimize the freesurface.

Simulations of ultra-thin nanowires with a diameter of1 nm show them to be very volatile, even at low temper-atures. Because of the low number of non-surface atomsthat can be used in the structure factor analysis, themethod described above can not be used to determinethe melting point of such wires. In some cases we ob-served a pentagonal reconstruction of circular NW whenusing the SL potential (Fig. 8). Due to the stochasticnature of the reconstruction, it is, thus, not possible toaccurately calculate the melting point in the case of verythin wires. However, the lower limit for unstable wireslies at 400 ± 25 K when using the SL potential. Thesame reconstruction has been observed previously34,35,but only when an external strain has been applied to thewire. However, in our simulations this was not the case.

IV. CONCLUSIONS

In this paper we have described a method for determin-ing the melting point of nanowires in molecular dynam-ics based on the angular structure factor analysis of thewires. The melting points for Cu wires with diameters in

5

(a) (b)

FIG. 8: Section of wire with initial diameter of 1 nm a)before and b) after reconstruction.

the range 1.5 nm to 20 nm showed an inverse dependenceon wire diameter and are close to the values predicted us-ing the semi-empirical liquid drop model. This suggeststhat the simulation and analysis method presented hereis well suited for the task. However, some other methodmust be used for thinner wires, because of random recon-structions and the low number of non-surface atoms.

The choice of potential used in simulations is impor-tant: on one hand the SL and Foiles potentials resultin melting temperatures whose absolute values are closeto those predicted by the liquid drop model, while theMC/MD-CEM potential gives severely overestimated re-sults. On the other hand, the MC/MD-CEM potentialdescribes the size-dependence better due to the morerealistic surface energies given by the potential. Whenchoosing a potential one must, therefore, carefully con-sider which property is more important: the absolutevalue or the size-dependence of the melting point.

V. ACKNOWLEDGMENTS

This research was supported by the European Com-mission under the FP7 Research Infrastructures projectEuCARD (grant no. 227579) and the Finnish Centre ofExcellence in Computational Molecular Science (CMS),financed by The Academy of Finland and the Universityof Helsinki. Grants of computer time from the Centerfor Scientific Computing in Espoo, Finland, are grate-fully acknowledged.

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