The Strength of Gold NanowiresKen Gall,* Jiankuai Diao, and Martin L. Dunn

Department of Mechanical Engineering, UniVersity of Colorado at Boulder,Boulder, Colorado

Received September 21, 2004; Revised Manuscript Received October 4, 2004

ABSTRACT

Atomistic simulations are used to investigate the yield strength of experimentally observed atomic and nanometer scale gold wires. Theatomistic predictions of strength are quantitatively consistent with discrete experimental measurements and they reveal the mechanisms forincreasing nanowire strength with decreasing dimensional scale. Distinct transitions in yield strength and yield mechanism are discovered.At nanometer scales (diamete r > 1 nm), the mechanism for strengthening involves the scarcity and low mobility of dislocations coupled withconstraint from tensile surface stresses. As the wires approach the atomic scale (diamete r < 1 nm), an increase in strength occurs concurrentwith a surface-stress-induced change in the stable structure of the nanowires and the absence of dislocation-mediated yield. The resultsconstitute a new fundamental understanding of strength in metallic nanowires spanning technologically relevant dimensional scales.

Strength is one of the most fundamental and significantmechanical properties of a material; it measures the basiccapacity to bear load. The yield, or fracture, of a materialresults in a decrease, or complete loss, of load bearingcapacity and, in most cases, loss of functionality. Throughouthistory, the implementation of stronger materials in technol-ogy has assisted the development of smaller, lighter, faster,and more robust devices. Recent advances in experimentaltools have facilitated strength measurements in metal nanow-ires and nanostructures down to single atomic chains.1-10

The fledging field of nanotechnology provides motivationfor a comprehensive understanding of the strength ofmaterials down to nanometer and atomic scales. The realiza-tion of devices with nanometer scale elements hinges on theability of low-dimension materials to sustain mechanicalloads during fabrication, assembly, and operation.

For quite some time, theoreticians have opined that smalleris stronger.1 That is, as the dimensional scale of a materialis reduced, yield strength should increase accordingly. Acollection of experimental yield strength measurements on99.99% pure gold (Au) is presented in Figure 1.2-10 In Figure1, stress, the basis for measuring strength, is defined as forceper unit cross-sectional area.11 Figure 1 reveals a substantialand systematic increase in measured strength with decreasingdimensional scale. Figure 1 includes classical and atomistictheoretical predictions of the “ideal” yield strength of bulksingle-crystal Au in the absence of crystalline defects.1,15Theideal theoretical strength is considered an upper bound onyield strength since it is based on the coordinated shearingof two adjacent atomic planes in the absence ofdefectnucleation from surfaces or structural inhomogeneities.15

In light of ideal strength predictions, the strength increaseobserved traversing the micrometer size scale is readilyexplained based on dislocation mechanics. Drawn microwiresand deposited nanofilms both exhibit yield under appliedstress2-4 and possess yield strengths below ideal theoreticalpredictions for bulk single crystal Au (Figure 1). In poly-crystalline drawn metal whiskers or deposited thin films, thetiny grain size and small lateral dimensions restrict disloca-tion activity, driving an increase in yield strength relative to

* Corresponding author. E-mail: kenneth.gall@colorado.edu. Tel. (303)735-2711.

Figure 1. Ideal predictions14 and experimental measurements2-10

regarding the yield strength of 99.99% pure Au as a function oflateral specimen dimension. The data show a systematic trendtoward stronger materials as the lateral specimen dimension isreduced toward the atomic scale. At discrete experimental lengthscales, the materials in the nanometer to micrometer range alldemonstrate yielding, while single atom chains undergo atomicseparation. The relative sizes of various natural substances areincluded for reference.

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normal bulk materials. However, nanofilms and microwiresultimately experience dislocation-governed plasticity, andthus their yield strength values are below the ideal theoreticalvalues for bulk materials. As the dimensional scale of theAu samples approaches the nanometer range, processingtechniques usually result in single-crystal structures thatdemonstrate quantized plasticity or discrete yielding events.5-8

The measured yield strength of the nanowires undergoingquantized plasticity is typically above theoretical predictionspredicated on a seemingly more difficult coordinated shearingof atomic planes.15

The transition from the nanometer to atomic scale pro-vokes another increase in strength and ultimately thedeparture of quantized yield behavior. As may be expected,single atom chains do not undergo yield per se, but ratherthey experience ideal tensile atomic separation. Previousexperimental12,16-21 and theoretical13,16,18,22-24 studies haveexamined the formation of a single atom chain through thenecking of an atomic point contact. However, prior studieshave not analyzed variations in stable nanowire strength withsystematically decreasing cross-sectional area approachingthe single chain wire. As a result, yielding and strength innanometer and atomic scale wires is not understood from auniversal perspective. In particular, it is critical to understandthe transition from experimentally observed quantized plasticflow in nanowires to perfect atomic separation in single atomchains.

We study the initial yield of atomic and nanometer scaleAu wires using computational atomistic simulations. Wefocus on initial yield behavior since nanowires experiencean instantaneous drop in load carrying capacity5-8 andfunctionality, e.g. conductance,8,17-21 upon yielding. Theaforementioned behavior contrasts that of bulk materials,which typically experience strain hardening after initial yieldup to their ultimate tensile strength. In this sense, the initialyield point of nanowires is analogous to the ultimate tensilestrength of bulk materials, both of which occur concurrentwith geometric instability (necking). We consider threeclasses of Au wires that are stable at lateral dimensionsspanning individual atoms to nanometers: (1) atomic wiresconsisting of single atom chains,9,10,12,13(2) multishell wireswith cylindrical or helical packing,25-28 and (3) face-centered-cubic (fcc) ⟨110⟩ wires with a rhombic cross section and{111} transverse surfaces.25,29 As indicated by respectivereferences, the structural stability of the various wires hasbeen confirmed by experimental observations. However, aseamless understanding of the mechanical behavior andyielding mechanisms for wires spanning these size scalesdoes not exist.

Materials and Methods.With the exception of the singleatom chain predictions, all simulations were performed usingthe embedded atom method (EAM).30 Simulation of a singleatomic chain with EAM is questionable unless potentials arefit to properties relevant to these extreme low-coordinationsystems. Results for single atomic chains were extracted froma prior study using ab initio simulations.10 In the EAMframework, the total energy of the atomic system consistsof a pair potential term and an embedding energy term. The

embedding energy is based on the energy required to placea host atom into the background electron density of sur-rounding atoms. The background electron density in EAMis composed of spherically averaged contributions fromneighboring atoms. The EAM framework facilitates com-putation of relatively larger wires while still capturingelectron density coordination effects critical to free surfacesand small wires. The EAM potentials for the presentsimulations were parameterized to fit the sublimation energy,equilibrium lattice constant, elastic constants, and vacancyformation energy in bulk Au.30

The ⟨110⟩ nanowires were created by initially placingatoms in positions representative of a bulk fcc lattice. Wethen anneal the nanowires in a molecular dynamics frame-work by gradually raising the temperature from 2 K toabout300 K, holding, and then gradually lowering the temperatureto 2 K. We then further relax the nanowire using an energyminimization method. Larger⟨110⟩ nanowires retain theirfcc structure during the entire process, while smaller nanow-ires form the experimentally observed multishell structure.The multishell structures generated this way are generallynot uniform throughout the length. We examined the shellstructures and then created wires with uniform shell struc-tures. Finally, we annealed the wires to determine theirequilibrium configurations.

Starting from the equilibrium configurations of the nanow-ires, we performed uniaxial tensile loading on the nanowiresuntil yield. Specifically, we displaced all the atoms inaccordance with a prescribed uniform strain in the lengthdirection, fixed the length, and then dynamically relaxed thenanowires at 2 K for 200 ps with a 0.004 ps per time step toobtain the equilibrium configurations of the nanowires at theprescribed strain. The equilibrium state of the nanowires wasdetermined by allowing the stress to saturate as a functionof time. The nanowires typically reach equilibrium after 100ps, and the stress averaged over the second 100 ps is usedas the stress of the nanowires. We use the virial stress for asystem of atoms, which is equivalent to the Cauchy stressin the average sense.31-34 Atoms in the fcc structure werevisualized according to the slip vector which locates atomsthat have sheared relative to one another, marking disloca-tions and stacking faults.35

Results and Discussion.Three-dimensional perspectiveimages of representative rhombic and multishell Au wiresare shown in Figures 2a and 2b, respectively. Consistent withexperimental observations, the wires in Figure 2 representstable structures for respective sizes determined by “anneal-ing” the wires, followed by energy minimization usingembedded atom method potentials for Au.30 The multishellwire in Figure 2b was predicted by annealing an originalstructure resembling a small⟨110⟩ fcc wire. Although therhombic wire in Figure 2a was created by initially placingatoms in their fcc positions with appropriate orientations,the atomistic model also predicts the formation of the low-energy rhombic wire in Figure 2a from a⟨100⟩ wire throughsurface-stress-driven reorientation.29 The rhombic and mul-

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tishell structures are both highly stable configurations fornanowires because they minimize the total wire energy,which consists of a significant surface energy contribution.

Tensile stress-strain curves for the representative wiresin Figures 2(a,b) are presented in Figure 2c. The rhombicand multishell wires show an elastic loading stage, character-ized by monotonically increasing stress with increasingapplied strain, followed by a yield point marked by a suddenstress drop. The elastic modulus (slope of the stress-straincurve) for the⟨110⟩ wire is lower than the modulus of themultishell wire. The calculated elastic modulus of the⟨110⟩wire is consistent with continuum mechanics predictions atsmall strains for this orientation aside from differences dueto the existence of the free surfaces and edges. At largerstrains, both materials demonstrate nonlinear elasticity aspredicted by the curvature in the atomic potential functionsat larger atomic separations.

The incipient yield points in the nanowire stress-strainresponses (Figure 2c) are defined as the yield strengths ofthe wires. Following yield, and the sudden stress drop, thewires undergo elastic deformation followed by additionalyield events or so-called quantized plastic flow.8 Based onthe results in Figure 2c, the multishell nanowire is consider-ably stronger than the fcc rhombic nanowire. Figure 2dpresents simulation predictions of nanowire yield strengthas a function of lateral wire dimension. Relevant experi-mental measurements and ideal strength predictions fromFigure 1 are included in Figure 2d, and the stable wirestructure is indicated along the top of the figure. Thesimulation results in Figure 2d represent predictions sincethe potentials were parameterized to bulk properties of Aurather than nanowire specific properties or dislocationproperties. The agreement between discrete experimentalmeasurements and modeling predictions is remarkable andsupports the utility of semiempirical atomistic calculations.Experimental strength measurements are currently not avail-able at intermediate wire sizes, particularly the size rangewhere the multishell wires are stable.

For the⟨110⟩ rhombic nanowires, the simulations predictstrength values commensurate with experimental observa-tions (Figure 2d). Detailed investigation of the simulationresults reveals that tensile surface stresses from the{111}side surfaces inhibit nanowire yield. The{111} surfacesnaturally contract relative to the nanowire core due to theexistence of tensile surface stress. Without any externallyapplied force, the tensile surface stresses generate an“intrinsic” compressive stress in the wire core.31 At equi-librium, the tensile forces on the surface balance thecompressive forces in the interior of the wire. Although thisintrinsic stress exists in all solid materials, its magnitudeincreases with decreasing dimensional scale and is significantonly for nanometer scale materials. For example, the intrinsiccompressive stress in a 7 nm⟨110⟩ nanowire with{111}side surfaces is on the order of 1 GPa based on a{111}surface stress of 1.8 J/m2. Intrinsic stresses on the order ofGPa can alter the yield strength of a nanowire relative toideal classical or atomistic theories. The surface stress

Figure 2. Equilibrium structures for two representative Aunanowires with different lateral dimensions. (a) Relaxed atomicpositions of a 2.2 nm rhombic wire with a⟨110⟩ axis orientationand{111} side surfaces. (b) Relaxed atomic positions of a 0.7 nmmultishell wire. The tensile stress-strain responses of the nanowiresin (a) and (b) are presented in (c). (d) Experimental measurementsand simulation predictions of tensile yield strength as a functionof size scale. All simulation predictions are from an embedded atommodel30 with exception of the single atom chain predictions, whichare from ab initio simulations.10 Experimental measurements andthe atomistic ideal strength prediction are the same as shown inFigure 1.6-10

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induced intrinsic stress in macroscopic solids, averaged overthe large core cross section, is on the order of Pa, a negligiblevalue.

During tensile loading, the stress in the interior of the wirechanges from a compressive to a tensile state, and ultimatelythe wire yields under tension. Thus, the applied tensile forcemust overcome the surface-stress-induced intrinsic compres-sive stress in the wire core. The effect of surface stressconstraint is not considered in ideal theoretical strengthpredictions for bulk materials (Figure 2d). For example,classical and atomistic ideal yield strengths for bulk Au are1.48 and 3.56 GPa, respectively (Figure 1). The classicaland atomistic tensile strength values increase by 67% and28%, respectively, with consideration of an opposing 1 GPaintrinsic compressive stress. Furthermore, the predictedincrease in the strength of the⟨110⟩ rhombic wires, with adecrease in size from 6 nm to nearly 1 nm (Figure 2d), iscaused by an increase in the intrinsic compressive stress,which scales with 1/d, whered is the lateral dimension ofthe wire. In fact, at very small wire sizes, a properly orientedwire can experience compressive yield under the exclusiveinfluence of surface stresses.29 Experimental measurementsof asymmetric strength in nanowires (lower strength incompression versus tension)8 partially support the existenceof intrinsic compressive stresses in nanowires, although thenucleation of unidirectional partial dislocation systems canalso contribute to strength asymmetry.36 The intrinsic stresseffect is only critical at very small nanowire sizes (less than10 nm in diameter), and other size dependent mechanisms,which alter resistance to plastic flow, are clearly operatingas size scale spans nanometers to microns (Figure 1).

A somewhat auspicious agreement is discovered betweenthe atomistic predictions ofnanowirestrength and the idealstrength calculated usingbulk atomistic simulations. Inparticular, Figure 2d shows quantitative agreement betweenthe ideal strength from atomistics and the predicted strengthof nanowires in the size range of 1-2 nm. In reality, thepredicted strength of the nanowires, relative to ideal theoreti-cal predictions (Figure 2d), is a counterbalance of strengthenhancement from intrinsic surface stress constraint andstrength reduction due to the availability of free surfaces andcorners for defect nucleation. The free surfaces and cornersact as favorable sites for heterogeneous defect nucleation,resulting in a lower predicted strength value relative tohomogeneous slip assumed in ideal strength predictions forbulk materials. On the other hand, as wire size increases,the surface-stress-induced intrinsic compressive stress di-minishes and the predicted nanowire strength gradually fallsbelow the ideal theoretical value. Additional surface features,such as ledges and steps, can further lower the barrier todislocation nucleation and decrease the strength of nanowires.Increased temperature may further lower atomistic predic-tions relative to experimental measurements, so the effectof temperature needs further study.

A distinct transition in yield strength is discoveredtraversing the 1 nm size scale. The strength values for wiressmaller than 1 nm are under-predicted by ideal strengththeories based on the fcc structure. The drastic increase in

yield strength occurs concurrently with a change in wirestructure from a fcc rhombic⟨110⟩ wire to a nonfcc multishellwire. The embedded atom model predicts this structuralchange in line with experimental observations.25-28 Thesystematic strength increases in the multishell nanowiresheads toward the strength of the single atom chain (Figure2d). The 1.0 nm multishell wire contains two shells sur-rounding a single atom chain, while the 0.7 nm multishellwire contains one shell surrounding a single atom chain(Figure 2b). Smallermultishell structures are not possiblebecause removing the final outer shell results in a single atomchain (Figure 2b). The strength level of single atom chainshas been previously examined using ab initio simulations,which account explicitly for electronic degrees of freedomin calulations.10 These first principal calculations havereasonable agreement with experimental measurements forsingle atom chains (Figure 2d). At present, ab initio simula-tions are impractical for the larger wires in the present studydue to computational limitations.

To explain the dependence of strength on wire size, weexamine the yield mechanisms in representative nanowires.Figure 3 provides a series of images during the initial yieldof the ⟨110⟩ rhombic wire in Figure 2a. Only a portion ofthe wire is shown and the wire is viewed edge-on (Figure3a). Blue atoms correspond to atoms that have not beensheared, while colored atoms correspond to slipped atoms.

Figure 3. Yield mechanism in the 2.2 nm⟨110⟩ rhombicnanowire: (a) indicates the viewing direction for images in (b-f).Atoms are colored according to their slip vector,35 which is equalto 1.67 for partial⟨112⟩ dislocations and 2.89 for perfect⟨110⟩dislocations. Yield occurs by the sudden nucleation and propagationof leading⟨112⟩ partial dislocations on{111} planes.

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Yielding in the rhombic⟨110⟩ nanowire is controlled by thenucleation and sudden propagation of partial dislocationsfrom near the wire edge. Figure 3b shows the wire at theonset of yield where a small group of edge atoms (coloredin green) begin to slip. Almost instantaneously, two leading{111}⟨112⟩ partial dislocations nucleate and propagate acrossthe wire cross section on{111} planes, leaving stackingfaults (Figure 3c). Following the onset of yield, a fewadjacent planes undergo slip (Figure 3d). During the yielding,the sample experiences a reduction in cross-sectional area(necking), evident in Figures 3(c-f). Ultimately, stackingfaults remain on only one of the initial slip planes (Figure3f). Rhombic⟨110⟩ nanowires with other sizes studied hereyield via the same mechanism presented in Figure 3. Theyield mechanisms in the⟨110⟩ rhombic nanowires arefundamentally the same as observed plastic flow mechanismsin bulk fcc Au. However, the nanowire yield is controlledby the nucleation and propagation of a few partial disloca-tions, and the wires yield under the combined influence ofapplied forces and surface-stress-induced intrinsic stresses.The nucleation of dislocations in a perfect⟨110⟩ nanowireoccurs near the edge of the wire, an unavoidable geometricalfeature attributed to the faceted nature of small single crystals.

The yield mechanisms in the multishell nanowires areillustrated in Figure 4 for the smaller multishell nanowire(one shell and single atom core). Yielding in the multishellnanowire is controlled by local tensile bond fracture at thewire surface, followed by relative shear of surrounding atoms.Figure 4a shows the atomic positions in the multishellnanowire just prior to yielding. Yield nucleates at an atomictriple junction on the wire periphery near the center of the

wire (Figure 4b). The local tensile fracture of the atomicbond between the three surface atoms results in stress reliefalong the fractured chain, a reduction in cross-sectional area,and the formation of a point-like defect at the wire surface(Figures 4b-e). Subsequent yield occurs through the shearingof adjacent atoms leading to further wire necking (Figures4f-j). Larger multishell nanowires (two shells and singleatom core) yield by a similar two-staged mechanism depictedin Figure 4. The yield mechanism in the multishell nanowirerepresents a noteworthy departure from dislocation plasticityobserved in bulk fcc solids and larger fcc nanowires. In fact,the yield mechanism in the multishell wires is an intriguingcompromise between perfect atomic separation forced insingle atom chains and dislocation plasticity in larger fccnanowires. The unique yield mechanism in the multishellnanowires explains the transitional strength values predictedin this nanowire relative to larger and smaller nanowires(Figure 2d).

In closing, we mention that our simulations have onlyconsidered perfect single crystal nanowires in a limited sizerange. It is still necessary to examine nanowires larger than10 nm as significant size scale effects are observed as wiresize traverses the 10 nm to 1000 nm range (Figure 1).Additionally, the effect of preexisting defects, temperature,and impurities on nanowire strength merits study. Owing totheir small overall volume, nanowires are extremely sensitiveto defects such as stacking faults,36 preexisting dislocations,surface ledges, and impurities. For example, Au nanowiresare often chemically grown in solution and may containimpurity atoms in their core or on their surface. Knowingthe significant influence impurity atoms have on bulkmaterials, it seems absolutely critical to understand the effectof impurities on the strength of nanowires. It is possible thatthe aforementioned structural features may change thenanowire yield response significantly from what is predictedherein.

Conclusions.We present a fundamental understanding ofthe strength of nanometer scale Au wires. Nanowire strengthis considerably higher than bulk materials, depends on wiresize, and cannot be predicted by ideal bulk strength theories.Atomistic simulations reveal that size-dependent nanowirestrengths for wires under 10 nm in diameter are influencedby wire structure, surface stress, and defect formationmechanism. Free surfaces are critical to the mechanicalbehavior of nanowires because they lead to intrinsic com-pressive stresses and dislocation resistant wire structures, bothof which drive increases in tensile yield strength consistentwith experimental measurements. Free surfaces also serveas favorable nucleation sites for defects in perfect nanowires,driving analogous decreases in tensile yield strength. Thequantitative and qualitative agreement between experimentalmeasurements and atomistic predictions highlights the utilityof semiempirical atomistic tools for examining the mechan-ical behavior of nanometer scale materials.

Acknowledgment. The work was supported by a Depart-ment of Energy (DOE) Presidential Early Career Award forScientists and Engineers (PECASE).

Figure 4. Yield mechanism in the 0.7 nm multishell nanowire.Yield occurs by the nucleation of local tensile fracture in an atomictriple junction on the wire periphery (see center of images a-e).Subsequent shearing of adjacent atoms propagates yield across thecross section of the nanowire (f-j).

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