Nanowire formation at metal–metal contacts

Nanowire formation at metal–metal contacts

Solid State Communications 135 (2005) 610–617 www.elsevier.com/locate/ssc Nanowire formation at metal–metal contacts E. Tosattia,b,* a International...

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Solid State Communications 135 (2005) 610–617 www.elsevier.com/locate/ssc

Nanowire formation at metal–metal contacts E. Tosattia,b,* a

International School for Advanced Studies (SISSA), INFM Democritos National Simulation Center, Via Beirut 2-4, I-34014 Trieste, Italy b International Centre for Theoretical Physics (ICTP), P.O. Box 586, I-34014 Trieste, Italy Received 10 March 2005; accepted 1 May 2005 by the guest editors Available online 4 June 2005

Abstract The thinnest nanocontacts between two metal bodies take sometimes the shape of regular ultra-thin suspended nanowires. I will qualitatively review here some of the theory work done in our group in connection with this phenomenon. I will discuss why nanowires arise, their stability, and their evolution with time. I will touch on the formation mechanism of ‘magic’ long lived nanowires endowed with especially stable structures, on their spontaneous thinning, and on the helical and the monatomic magic nanowires observed mostly in gold. q 2005 Published by Elsevier Ltd. PACS: 61.46.tw; 68.65.-k; 73.63.-b; 81.071.k Keywords: Burstein; ICTP; A. Nanowires; A. Nanocontacts; A. Magic nanowires; D. Ballistic conductance; A. Monatomic nanowires

1. Introduction When at night we turn the lights off to go to sleep or otherwise, with that simple flicking of a switch we carry out (most probably unawares) a very delicate physics experiment. We move in that moment two metal blocks apart, suddenly straining and mechanically rupturing a myriad of metal–metal nanocontacts, each carrying a fraction of the total current in the circuit. What are the nature, the geometry and shape, the electronic structure, the ultimate conductance of these atomic sized contacts? And what is the connection between these, the structure and the electronic and may be magnetic properties, and all of them and the electrical properties? Trying to answer some of these questions has been the theme of some research in our theory group, which I shall * Address: International School for Advanced Studies (SISSA), INFM Democritos National Simulation Center, Via Beirut 2-4, I-34014 Trieste, Italy. Tel.: C39 40 3787 443; fax: C39 40 3787 528. E-mail address: [email protected]

0038-1098/$ - see front matter q 2005 Published by Elsevier Ltd. doi:10.1016/j.ssc.2005.04.045

review in this short paper in honor of Eli Burstein. While this is not an area, where Eli himself contributed, the curiosity that inspired me to enter this area is very much in the spirit of Eli’s own ever curious approach to physics and to life, a spirit which he has been spreading contagiously throughout his long career. This paper covers in part material presented in a special Lundquist conference that was held honoring Eli Burstein at the International Centre for Theoretical Physics (ICTP) in Trieste in May 2004. The association of Eli with ICTP has been long and important, dating back to the 1970s. In close contact with Abdus Salam, who established ICTP, but especially with Franco Bassani and Stig Lundquist, each of them in different ways founding fathers of Condensed Matter theory at ICTP, and later with Bob Schrieffer, Praveen Chaudhari, Phil Anderson, and Yu Lu, Eli powerfully helped pulling together our international community and shaping our future along the lines that continue through different people today. I cannot resist showing at this point a nice picture of Eli among these and other main actors, taken at a meeting in ICTP in the early 90s, in Fig. 1. Now to business. What about nanocontacts, their geometry and structure, first of all? A generally acceptable notion seems that, uncontrolled as they are, metal–metal

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Fig. 1. Eli Burstein (center, between P. Chaudhari and E. Tosatti) at a meeting in ICTP, Trieste in the early 1990s. I recognize among others A. Salam, Yu Lu, F. G. Bassani, P. W. Anderson, K. S. Singwi, H. Singwi, A. Sjolander, F. Garcia Moliner, H. Rohrer, H. Stoermer, K. Schoenhammer, B. A. Huberman, L. P. Gorkov, R. Fieschi, G. Srinivasan, H. R. Ott, V. Sa-yakanit, C. W. Lung. X. G. Gong, S. M. Girvin, O. M. Gunnarsson, B. I. Lundquist, and S. O. Lundquist (whose anniversary was being celebrated) wearing two ties.

nanocontacts will as a rule differ from one another, all of them probably wildly irregular in shape and properties. As such, it would seem impossible for a theorist to give anything better than some kind of statistical discussion of their properties, along perhaps with simulations [1,2,23] of the way the rupture may occur from a plausible starting geometry of the nanocontact. The last few years nevertheless brought several direct pieces of experimental evidence offering a less discouraging view, and moving the focus to some specific microscopic geometries that are definitely non-random. The first kind of evidence comes from break junction conductance jump histograms [3], demonstrating contact a drop of conductance taking just before contact breaking the form of sharp steps, as shown for example in Fig. 2. The majority of steps are clearly—if approximately— quantized close to multiples of the ballistic conductance quantum G0Z2e2/h [3]. In fact metal nanocontacts are a notable instance, where quantized ballistic conductance can be observed even at room temperature. Besides that, this kind of conductance evidence also proves that the thinnest contact neck must really have atomically small dimensions, both lengthwise (otherwise conductance would not be ballistic) and across (otherwise the smallest steps would not be of order G0). From the conductance point of view the nanocontacts, far from being random, come in some kind of preferential forms, such as to produce well defined

abundance maxima and minima as reflected by the conductance histogram peaks. A second evidence came as a an additional and wonderful surprise when Takayanagi and his Tokyo group

Fig. 2. Break junction conductance histograms for Au, Ag and Cu. Sharp peaks, particularly the first, indicate quantization in units of G0Z2e2/h. After Ref. [3].

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Fig. 3. TEM image of a gold nanocontact displaying a perfect nanowire. This relatively thick nanowire has a crystalline inner structure, sheathed inside a denser, reconstructed outer layer. After Ref. [4].

produced transmission electron microscope (TEM) snapshots and movies showing that gold nanocontacts could under appropriate conditions take the shape of beautifully straight and regular nanowire segments, hanging as shown in Fig. 3 between two supporting electrodes, or tips [4–6]. While there still appears to be a great overall variety of behavior including less regular nanocontacts as well, both of crystalline and of amorphous type [7–9], the very existence of such regular nanowires is galvanizing. This paper is a short qualitative exposition of some of the present understanding developed in our theory group about the reasons for nanowire formation, their thinning, their structural and electronic properties, and their ballistic conductance.

2. Why do nanowires appear? First of all, talking of non-equilibrium structures, the first consideration should be one of time scales. When a metal– metal nanocontact is suddenly made or broken, that will encompass schematically three different time scales, say t1/t2/t3. During t1 there is a rapid, plastic flow of atoms. Within t2 some form of quasi equilibrium is reached and maintained so that the nanocontact is stationary in a metastable state. On a longer time scale t3 the nanocontact undergoes a successive thinning process, often reported in the form of jumps between quasi equilibrium structures, until final breaking—the only true equilibrium. Although several experiments and simulations really address only the shortest time scale t1, often as short as nanoseconds, we shall be concerned here mostly with quasi equilibrium at time scales t2 and t3, which in gold appears to reach milliseconds to seconds to minutes [4–6]. To set the scene, consider as a first unrealistic model two solid tips with some wetting liquid in between. Macroscopically, the liquid bridge between the two tips will take the well known, smooth catenoid shape that minimizes the surface area, and thus the total surface free energy, as one finds if for example one replaces the tips with two rings of radius R [10]. The catenoid shape actually becomes unstable

when the tips are moved far enough, above a distance LZ 1.33R. At that point the bridge midpoint thins down to zero, and the bridge breaks. Hence one would not expect stable liquid bridges with a length L much larger than the radius. The microscopic picture we wish to obtain for metal nanocontacts differs profoundly. The bridge and the tips in this case are made of the same metal and exchange atoms. Besides that, the atomic size of the contact makes it questionable to use purely macroscopic arguments. Moreover, even if at the nanocontact flow of atoms always occurs (as we shall discuss later) the metal–metal bridge is generally far from liquid. Because of that, except close to the melting point, equilibrium will be hard to reach. Even at the time scale of real TEM observations, say milliseconds, we must consider a situation of dynamical evolution. A description of that regime thus amount to be able to identify and to specifically earmark the all important long-lived quasi equilibrium geometries that the nanocontact will, even if temporarily, adopt during its evolution. Experiments [13] showed that even when the nanocontact could be thought of as locally liquid, for example close to the tip metal melting point, very long necks could be realized. Early atomistic simulations [11,12] of metal nanocontacts close to the melting point could unexpectedly evolve long thin regular nanowire geometries, such as that of Fig. 4. Theory easily explains why in the first place quasi equilibrium solid nanowires can arise. Consider a crystalline, solid nanowire of radius R and length L between two tips. The nanowire is well defined if the tip-wire neck size d is of order R and not of order L as in the catenoid. Provided the nanocontact is assumed to be a solid nanowire with flat surfaces, then it can be shown by a straightforward surface physics argument that dwR. This result provides the fundamental justification why solid nanowires exist [12]. A similar argument cannot be straightforwardly applied

Fig. 4. Simulation picture showing the formation of a nanowire at an Au–Pb nanocontact at a temperature close to the melting point of Pb, where the Pb surface behaves as locally liquid. Despite the liquid-like mobility, this nanocontact is very different from the catenoid neck form expected macroscopically (see text). After Refs. [11,12].

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to a liquid nanowire, because a macroscopic liquid surface will not be flat, technically speaking, due to capillary fluctuations. Here one would expect a catenoid neck and dwL, implying non-existence of proper liquid nanowires, which as was said is in some case denied by facts, particularly by simulations, that predicts nanowire-like contacts like that of Fig. 4 [11,12]. The theoretical question of the geometry of liquid nanocontacts remains open, but it appears that things may become really different at the nanoscale. In essence a sufficiently thin liquid nanowire (or other shape of nano-object), may be so strongly structured inside (layering), that its surface is probably not available to deform or to fluctuate at all wavelengths as is required for ideal liquid-like behavior. In presence of some layeringinduced surface rigidity, the outer layer might behave more like solid, reducing the neck extension to wR here too, and stabilizing a long liquid nanowire instead of a catenoid. This is an interesting subject that requires more research, which we will not further discuss here.

3. String tension, thinning, and magic nanowires Suppose that a solid metal nanowire forms between two tips at a given moment of time, say for example one as shown in Fig. 3. In order to consider theoretically the subsequent evolution of this nanocontact, we should identify the proper thermodynamical potential governing its evolution. For that scope, we can consider two options. The first option is to include in the description both the nanowire and the tips, the whole system. The evolution at temperature T will of course be governed by the canonical free energy F(T, N) at constant number N, because evaporation can generally be neglected in metals. This is the more fundamental approach; but its practical implementation will be a tour de force. A second simplified option is to consider the tips merely as bulk-like atom reservoirs, and to concentrate attention on the nanowire, now described through a grand canonical potential GZF(T, N)KmN. Now the free energy F refers to a stand-alone, arbitrarily long nanowire made up of N atoms, while the tips only enter through the bulk chemical potential m—the free energy required to remove an atom from the bulk metal. In this simplified approach, important complications such as the finite size of the tips, and the existence and nature of the tipwire junctions are completely neglected. In return one obtains an extreme simplicity of both calculation and understanding. Let us adopt, therefore, the second approach, where a perfect infinite wire whose grand canonical equilibrium with bulk reservoirs is described by the nanowire grand potential G [15]. For a nanowire of length L we should further normalize G per unit (projected) length, in the very same way one must normalize the surface free energy per unit projected area [14]. In this manner the thermodynamic evolution of a sufficiently long nanowire

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in full atom-exchange equilibrium between two tips will be governed by f Z LK1G Z ð1=LÞ½FðT; NÞ K mN

(1)

This quantity has the dimensions of a force, and is in fact nothing else than the thermodynamic string tension of the nanowire. Why does a string tension physically arise? In essence it is because the free energy F/N of an atom in the nanowire is strictly higher than that of m of the same atom in the bulk metal, where binding and cohesion is much better. Atoms in the suspended nanowire are, as it were, disputed between the two tips. The resulting string tension is a positive definite force pulling the two tips together, simply reflecting the suction exerted by the tips on the atoms of the nanowire [15]. Any externally added tension, caused for instance by a sudden prying apart of the tips, will cause additional evolution of the nanowire, in the attempt to reach again the equilibrium tension—in this case by pulling extra atoms out of the tips. In a very thick nanowire of radius R and surface energy g, f will be of order 2pgR, decreasing with decreasing radius, and only reaching zero at breaking. A decrease of string tension with decreasing radius directly explains the spontaneous thinning [12,15, 16] of nanowires which is observed to occur with time [4– 6], due to loss of atoms to the tips until final breaking, as sketched in Fig. 5. The thinning process in, e.g. gold, is reported to occur in steps, from one long lived, regularly structured ‘magic nanowire’ to the next one until final breaking [5]. We interpret the metastable magic nanowires as local minima of string tension [15]. The long lifetime of a magic nanowire is due, we propose, to the string tension barrier, which protects a local minimum. In order to abandon a magic structure for the next one, the tip-suspended

Fig. 5. Schematic plot of a nanowire thermodynamic string tension against radius. The necessity to decrease the string tension drives the spontaneous thinning of any nanowire with loss of atoms to the tips, until final breaking. The local minima signify metastable nanowire geometries, corresponding to the magic nanowires with longest lifetime. After Ref. [15].

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nanowire must wait for a fluctuation sufficiently large to overcome that barrier and nucleate a thinner magic structure with lower string tension [15]. The nucleation event causing the thinning instability seems in practice to take place at the tip-wire junctions [17]. Interesting thinning mechanisms have been proposed theoretically [16,12] and discussed experimentally [18].

4. Variety of mechanisms leading to magic nanowires If magic nanowires correspond, as we proposed, to local minima of the string tension G of Eq. (1), these minima will generally correspond by particularly happy nanowire structures with a low free energy F, (low energy E if entropy is irrelevant) and a large length L. A low energy structure can arise by a variety of different physical mechanisms. First of all, for tips that possess the same crystalline orientation, and for particular orientations, one can realize very perfect tip-suspended crystalline nanowires. This is certainly the case for sufficiently large radii [4], as is physically very clear [19]. Various groups have demonstrated experimentally that even very thin gold nanocontacts and nanowires can be crystalline [7–9]. I will not be further concerned here with this case. A second mechanism stabilizing a particular nanowire is electronic shell closing. In cluster physics it was shown two decades ago that the peculiar abundance peaks observed for example in Na clusters could be explained by considering them as super-atoms endowed with an approximate spherical symmetry. In this model free electrons fall into shells very much like they do in atoms. Magic clusters with spectacularly large stability and abundance correspond to the closing of electronic shells, the analog of rare gas configurations. Metal nanowires may be stabilized by a qualitatively similar mechanism, and in fact abundance peaks in Na have been shown to fit very well the corresponding shell closing criterion. In the cylindrical geometry appropriate for the nanowire, super-atom levels are replaced by 1D sub-bands, and shell closing occurs when the thickness is exactly such that the Fermi level reaches the brink of a new sub-band [3]. Shell closing has been recently shown to occur in gold nanocontacts too [18]. Another interesting possibility leading to very stable and thus magic nanowires is that of regular but non-crystalline structures that display a very good packing at the nanoscale. Clusters again provide here the basic example, with icosahedral structures—that are regular but non-crystalline in the sense that they cannot grow into a space-filling regular lattice—often dominate the structure of the smallest clusters. The equivalent possibility for nanowires is surprisingly rich, as was theoretically discovered in Ref. [19]. Besides wires with pentagonal cross-sections— the straight equivalent of icosahedral clusters—other unexpected helical structures appear with very low energy and thus with great stability. One such structure was found

for example to consist of a central straight atom chain (or strand), inside a metal nanotube, in turn inside a second larger nanotube. These metal nanotubes are helical and chiral, moreover with surprisingly different helical pitches. For this reason we dubbed these nanowires ‘weird’ [19,20]. The reason why such weird helical structures can be stabilized at sufficiently small thicknesses is that they possess a lower surface energy than the crystalline ones. Nanotubes have no facets and no edges, and the resulting surface energy saving can compensate for the worse inner packing.

5. Gold’s ‘weird’ helical nanowires The ideal testing metal for the study of nanowires is gold. The exceptionally large atomic mobility of gold appears to permit, in some cases already at room temperature, the grand canonical quasi equilibrium necessary for the realization of magic nanowires. Kondo and Takayanagi studied gold, and strikingly demonstrated that helical magic nanowires are a reality in this metal [6]. As Fig. 6 shows, the thinnest gold nanowires observed are (15-8-1), (14-7-1), (13-6), (11, 4), (7-1), where the figures in brackets stand for the number of atom strands in successive nanotubes. Thus for example (15-8-1) means a central monatomic strand inside an 8-strand tube, inside a 15-strand tube. The gold nanotubes can be thought as obtained by wrapping a triangular (1 1 1) lattice plane to form an (n, m) tube very much as in the carbon nanotubes. The ubiquity of the Fig. 7 around here is remarkable—note that 15K8Z7, 14K7Z7, 11K4Z7. We will come back to that later. While these structures beautifully verify the prediction of weird nanowires—in fact the (14-7-1) is essentially identical to the weird structure predicted for lead—the precise structures observed and their occurrence in gold represents an excellent occasion for a test of the string tension theory. Unlike the early empirical simulations [19], crucial for their ability to predict new phenomena, but very unreliable at the quantitative level, a convincing test can only be based on first principles, electronic structure based calculations. A theoretical test can be done, and was done, by the following protocol. First one builds a sufficient variety of trial nanowire structures, obtained for example by means of classical simulation, or otherwise. Any trial structure with N atoms per cell can be used as the starting point for a first principles density functional calculation of the ideal infinite model nanowire electronic structure and total energy (E(N)). Minimization of total energy, with full allowance for atoms to move, will lead from an initial structure to a final refined structure, where all parameters, including the length L, the free energy F(N, T) (that in a metal can be replaced with the zero temperature total energy E(N) even at room temperature) are known at zero stress. The bulk chemical potential m

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Fig. 7. Calculated first principles string tension f for the family of ideal gold nanowires consisting of a central atom chain, inside an n-strand nanotube with (n, m) folding. The plot against radius shows that the [(7, 3)K1] chiral nanowire is a local minimum, and should be a magic nanowire, in agreement with experimental TEM observations. After Ref. [15].

Fig. 6. TEM images and simulated model images of the nonmonatomic magic gold nanowires. After Ref. [6].

(again well approximated by the zero temperature cohesive energy) can easily be calculated separately. Thus we have all the required ingredients to substitute in Eq. (1) to get a first approximation for the string tension f. The nanowire unit cell can then be allowed to elongate, introducing a tensile stress equivalent to this approximate string tension. In the strained structure the string tension can be recalculated, etc. until final self-consistent geometry and string tension f are obtained. In this manner one can explore different geometries, to find which structure is best. In order to do that, it is

necessary to calculate and plot f against some geometrical parameter, such as the wire radius, and find if there is one structure that represents a local minimum of f for the family of structures considered. Fig. 7 is the kind of string tension plot one obtains for the family of [(n, m)K1] gold nanowires, including nZ5, 6, 7, 8. Remarkably, the ab initio theory confirms that the [(7, 3)K1] nanowire is a local minimum in its range of radii, and should be magic. The agreement of this prediction with TEM observations provides a strong support to the string tension theory. An additional confirmation is the agreement of the calculated number of conducting channels of the [(7, 3)K1] nanowire, equal to six [15], with recent conductance results showing precisely a conductance of 6 G0 [17] (and not of eight G0 as was guessed from the number of gold strands [18]) for that nanowire. It will remain for future work to show if the other, thicker, magic gold nanowires similarly correspond to local string tension minima. Straightforward in principle, the necessary calculations are just a bit too massive at the present stage. It is worth stressing here that a local string tension minimum, although triggered by an underlying well packed, low energy structure, does not coincide with the local energy minimum. For example for the strand plus single nanotube family the total energy minimum is attained by the [(8, 4)K1] nanowire and not by the [(7, 3)K1]. Nonetheless [(8, 4)K1] is not magic, because it does not minimize string tension. More generally, contrary to the string tension which increases with the wire thickness, the total energy per atom E(N)/N is a decreasing function of N, so that thicker and thicker wires are increasingly stable energetically. That would of course be correct in the absence of tips, when the drop of E(N)/N for increasing N simply anticipates the collapse of nanowires toward bulk-like clusters. In presence of suspending tips, the string tension replaces the total

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energy, and the trend is reversed, with string tension minimized by the thinnest and longest nanowires, for which L is largest. Remarkably, the geometrical necessity to maximize L in order to minimize string tension actually provides the basic explanation why the magic nanowires are helical, endowed with a large chirality. If a sheet of paper is rolled up to form a cylinder, we can increase its length L and reduce its radius R by going from a normal, perpendicular rollup to a skew, helical rollup (try it to convince yourself). Similarly, the magic nanotube-based nanowires end up being preferentially helical because the same number of strands can be stretched to a longer length by going from zero to a large chirality. This reasoning leads to expect that the outer tube—forming the largest portion of the nanowire, and thus supporting the largest part of the string tension—should tend to be the folded triangular lattice closest to a (n, n/2) folding. A second qualitative fact to be explained is the pervasiveness of the number 7 in the magic gold nanowires. This appears to be a geometrical consequence of wrapping tubes onto tubes. A hard disk of radius R can be uniquely and tightly surrounded in a plane by a ring of no more than six equal disks. A second ring surrounding the first has room for a bit more than six. Asymptotically, the circumference between a ring and the next will differ by 2pR. The tube surrounding a linear chain of hard spheres can in principle consist of more than six hard sphere strands, depending on the relative stacking and on the chirality of the tube folding. Similarly the tube surrounding another tube will be able to accommodate more that 2p strands in excess. Moreover in gold the outer tube in gold has the tendency to be laterally extra dense [4], with a larger than normal distance to the inner tube, both elements typical of gold surface reconstructions [21]. For this reason the strand number differs form a tube to the next by the closest integer larger than 2p, which is just 7.

6. Monatomic nanowires, in gold and elsewhere In 1998, the first TEM pictures of a monatomic gold nanowire, consisting of a tip-suspended 4-atom chain, stunned the community [5]. Other researchers confirmed them although with differences that depend on the experimental circumstances, and that are partly still to be clarified [8]. Monatomic chains were also very directly implied to exist by long Gw1 plateaus in break junction data of gold, platinum and iridium [22]. On the theoretical side, computer simulations, while nicely showing how single atoms uncoil out of tips to give rise to a monatomic wire of increasing length [2], do not provide a quantitative predictive criterion as to when monatomic nanowires should or should not be expected to form in a given material. I note here that long lifetimes imply the notion that the observed monatomic nanowires must be magic too. They should correspond to local minima of the string tension, although not as a function of radius, which is no longer

Fig. 8. Calculated string tension f of a gold monatomic wire. The local minimum shows that it should be stable. The barrier height, 1.5 nN, is the calculated force for breaking of an ideal infinite Au monowire.

negotiable. The only available parameter of a regular monatomic chain is the interatomic spacing a. Fig. 8 shows the calculated string tension Eq. (1) of a monatomic gold nanowire plotted against the 1/a, the atom linear density. This plot shows that the string tension indeed possesses a shallow but well-defined local minimum, which confirms that magic monatomic nanowires are to be expected to exist for gold. Similar minima are found for Pf and Ir, whereas the local minimum disappears for other metals [24], in agreement with experiments [22]. One extra bonus of the string tension plot is that it gives an upper limit for the breaking force of the monatomic nanowire, corresponding to the top of the string tension barrier protecting the shallow local minimum. For gold, we obtain 1.5 nN, which as it happens is in excellent agreement with the observed breaking force value of the monatomic nanowire [25]. Break junction experiments demonstrated additional phenomena connected with pulling long monatomic nanowires between gold tips. The nanowire pulling is signaled by a long extension of the tip-tip distance with several breaks in the force, against a sustained uninterrupted plateau of conductance GwG0 [25]. At a closer look, the conductance shows tiny fluctuations that take it slightly below G0. These conductance fluctuations most likely reflect scattering due to irregularities in the nanowire geometry. In an infinitely long nanowire the irregularities would cause Anderson localization of electrons, with exponential decay of their wavefunctions, leading to an insulating nanowire. The actual tip-suspended nanowire segment is too short for that; but the earliest onset of exponential decay is still visible and shows up as these percent sized fluctuating decreases of conductance below G0 [26]. The nanowire stretching by the tips has been simulated by classical molecular dynamics [2,25,27] as well as by ab initio methods [28]. It was found among other things that the tip-wire junctions are the nanocontact weakest point, which extends the most under stress, taking in practice as much as

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half the total strain [27]. One implication of this is that the nanowire longitudinal phonon vibrations, whose straininduced softening gauges quite accurately the actual interatomic spacing, soften only about half as much as they would if the strain were entirely taken by the wire [27]. These phonons and their softening are experimentally detected in the form of losses that reduce the conductance below its ballistic value as soon as the applied voltage surpasses the phonon energy [22], giving rise to inelastic losses [29]. Various instabilities, including zig-zag [30] and Peierls dimerizations are often invoked for monatomic nanowires. The former indicates a tendency of a nanowire to crumble that would actually show up in the absence of string tension. The presence of string tension appears to remove the zig-zag instabilities. A Peierls dimerization is also theoretically expected for a long gold nanowire, which has a half filled 1D band. Dimerization would drive the nanowire from metallic to insulating, but that is not generally observed. Again as in the case of Anderson localization by disorder, it would take a very long wire and very low temperature for a Peierls distortion to be observable. In a short wire at most 8 or so atoms long dimerization is very strongly hampered by size, by the presence of tips and by fluctuations, and is essentially washed away.

7. Magnetism of nanowires, and other work for the future There are many other points that would be worth addressing, but for space and time limitations. This review is limited essentially to gold as the nanocontact metal. There is now a quantity of other data relating not just to other noble metals, but also to transition metals. Theory indicates that there magnetism should be relevant, both in connection with ballistic conductance [31], and in connection with the possible emergence of local magnetism at nanocontacts of metals like Pt and Pd, that are non-magnetic in bulk [32]. While I will not develop this subject here, this is likely to represents another area of future development.

Acknowledgements My heartfelt thanks go to Eli Burstein for much support and guidance, and for the gift of his wonderful friendship and inspiration throughout my scientific life. I also wish to thank warmly all my collaborators on the nanowire projects, particularly A. Dal Corso, A. Delin, F. Di Tolla, F. Ercolessi, O. Gulseren, E. A. Jagla, S. Kostlmeyer, S. Prestipino, A. Smogunov, R. Weht, P. Gava, M. Wierzbowska. I also wish to thank D. Ceresoli, J. Tobik, and T. Zykova-Timan for their help. This work was sponsored by MIUR FIRB RBAU017S8 R004, FIRB RBAU01LX5H, and MIUR COFIN 2003 and PRIN/COFIN2004, as well as by INFM (Iniziativa trasversale calcolo parallelo).

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