Metal Nanowires: Quantum Transport, Cohesion, and Stability

Metal nanowires exhibit a number of interesting properties: their electrical conductance is quantized, their shot-noise is suppressed by the Pauli principle, and they are remarkably strong and stable. We show that many of these properties can be understood quantitatively using a nanoscale generalization of the free-electron model. Possible technological applications of nanowires are also discussed. Introduction Metal nanowires represent nature’s ultimate limit of conductors down to a single atom in thickness. In the past eight years, experimental research on metal nanowires has burgeoned [1-13]. The simplest model of a metal is the free-electron model [14], which already describes many bulk properties of simple monovalent metals semiquantitatively. In this article, we discuss our generalization of the free-electron model to describe nanoscale conductors [15-22]. A remarkable feature of metal nanowires is the fact that they are stable at all. Fig. 1 shows electron micrographs by Kondo and Takayanagi [5] illustrating the formation of a gold nanowire. Under electron beam irradiation, the wire becomes ever thinner, until it is but four atoms in diameter. Almost all of the atoms are at the surface, with small coordination numbers. The surface energy of such a structure is enormous, yet it is observed to form spontaneously, and to persist almost indefinitely. Even wires one atom thick are found to be remarkably stable [8, 9, 13]. Naively, such structures might be expected to break apart into clusters due to surface tension [23], but we find that electron-shell effects can stabilize arbitrarily long nanowires [22]. A crucial clue to understanding the physics of metal nanowires is the observed correlation between their electrical and mechanical properties. In a seminal experiment [3] carried out in 1995, Rubio, Agräıt and Vieira simultaneously measured the electrical conductance and cohesive force of an atomic-scale gold wire as it formed and ruptured (see Fig. 2, left panel). They observed steps of order G0 = 2e /h in the conductance, which were synchronized with a sawtooth structure with an amplitude of order 1nN in the force. Similar results were obtained independently by Stalder and Dürig [4]. Note that the tensile strength of the nanowire in the final stages before rupture exceeds that of macroscopic gold by a factor of 20, and is of the same order of magnitude as the theoretical value in the absence of dislocations [3]. This is consistent with the recent finding of Rodrigues, Fuhrer, and Ugarte that such nanowires are, in fact, typically free of defects in their central region [13]. The standard description of nanoscale cohesion, pioneered by Landman and coworkers [24], is via molecular dynamics simulations [24, 25, 26], which utilize short-ranged interatomic potentials suitable to describe the bulk properties of metals. However, such an approach appears problematic when applied to metal nanowires, in which electron-shell effects [11] due to the transverse confinement are likely to be important. On the other hand, atomistic 1e-mail: [email protected]; fax: (520) 621-4721 phys. stat. sol. – 2 – Figure 1: Transmission electron micrographs showing the formation of a gold nanowire [5] (image courtesy of Y. Kondo): (a) an image of Au(001) film with closely spaced nanoholes, the initial stage of the nanowire; (b) a nanowire four atoms in diameter, resulting from further electron-beam irradiation. quantum calculations [27] using, e.g., the local-density approximation, are restricted to such small systems that their results can not really be disentangled from finite-size effects [20]. An alternative approach, developed by our group, is to replace the discrete ionic coordinates by a coarse-grained jellium background, in order to be able to treat the electronic degrees of freedom correctly. We have argued [15] that an atomic-scale contact between two pieces of metal can be thought of as a waveguide for conduction electrons (which are responsible for both electrical conduction and cohesion in simple metals): Each quantized mode transmitted through the contact contributes 2e/h to its conductance and a force of order εF/λF (roughly 1nN) to its cohesion, where λF is the de Broglie wavelength of an electron at the Fermi energy εF (see Fig. 2, right panel). To my knowledge, our approach is the only one in which the observed correlations between the cohesive and conducting properties of metal nanowires have been explained within a single theoretical model. The paper is organized as follows: The free-electron model of nanoscale conductors is introduced in the next section, followed by a discussion of quantum transport, including the effect of realistic contacts to the nanowire. Nanoscale cohesion is then analyzed within our model, followed by a discussion of the remarkable stability of nanowires. The paper concludes with some comments about the technological promise of metal nanowires. Free-electron model We investigate the simplest possible model [15, 16] for a metal nanowire: a free (conduction) electron gas confined within the wire by Dirichlet boundary conditions. A nanowire is an open quantum system, and so is treated most naturally in terms of the electronic scattering matrix S. The Landauer formula [28] expressing the electrical conductance in terms of the submatrix S12 describing transmission through the wire is