Universality in Metallic Nanocohesion: A Quantum Chaos Approach

Abstract

Convergent semiclassical trace formulas for the density of states and the cohesive force of a narrow constriction in an electron gas, whose classical motion is either chaotic or integrable, are derived. It is shown that mode quantization in a metallic point contact or nanowire leads to universal oscillations in its cohesive force: the amplitude of the oscillations depends only on a dimensionless quantum parameter describing the crossover from chaotic to integrable motion, and is of order 1 nN, in agreement with recent experiments. PACS numbers: 73.40.Jn, 03.65.Sq, 05.45.Mt, An intriguing question posed by Kac [1] is the following: "Can one hear the shape of a drum?" That is, given the spectrum of the wave equation [1] or Schrödinger's equation for free particles [2] on a domain, can one infer the domain's shape? This question was answered in the negative [1,2]; nevertheless there is an intimate relation between the two. In the context of metallic nanocohesion [3-10], a related question has recently emerged: "Can one feel the shape of a metallic nanocontact?" It was shown experimentally [3] that the cohesive force of Au nanocontacts exhibits mesoscopic oscillations on the nano-Newton scale, which are synchronized with steps of order 2e 2 ͞h in the contact conductance. In a previous article [4], it was argued that these mesoscopic force oscillations, like the corresponding conductance steps [11], can be understood by considering the nanocontact as a waveguide for the conduction electrons (which are responsible for both conduction and cohesion in simple metals). Each quantized mode transmitted through the contact contributes 2e 2 ͞h to the conductance [11] and a force of order´F͞l F to the cohesion, where l F is the de Broglie wavelength at the Fermi energy´F. It was shown by comparing various geometries [4] that the force oscillations were determined by the area and symmetry of the narrowest cross section of the contact, and depended only weakly on other aspects of the geometry. Subsequent studies confirmed this observation, both for generic geometries [5,7,8,10], whose classical dynamics is chaotic, and for special geometries [6,9], whose classical dynamics is integrable. The insensitivity of the force oscillations to the details of the geometry, along with the approximate independence of their rms size on the contact area, was termed universality in Ref. [4]. A fundamental explanation of the universality observed in both the model calculations [4-10] and the experiments [3] has so far been lacking.In this Letter, we derive semiclassical trace formulas for the force and charge oscillations of a metallic nanocontact, modeled as a constriction in an electron gas with hardwall boundary conditions (see Fig. 1, inset), by adapting methods from quantum chaos [12][13][14][15] to describe the quantum mechanics of such an open system. It is found that Gutzwiller-type trace formulas [12][13][14][15], which typically do not converge for closed systems, not only converge but give quantitatively accurate results for open quant...

Force, charge, and conductance of an ideal metallic nanowire

“…A more general expression for Ω, valid within a self-consistent mean-field treatment of interactions, will be presented elsewhere. 30 Here it is sufficient to note that it will be argued below that the corrections to Eq. (8) due to charging effects are negligible.…”

“…(62) will be given elsewhere. 30 Equation (62) indicates that the mesoscopic charge fluctuations of order e lead to a negligible correction to the free energy of the system, even in the limit C → 0 of perfect screening, justifying the independent-electron model of nanocohesion.…”

“…It can be shown that relaxation of this constraint, to include e.g., a small elastic deformation, does not modify the mesoscopic effects in an essential way. 30 The cohesive force of the nanowire is given by the derivative of the grand canonical potential with respect to the elongation:…”

Quantum suppression of the Rayleigh instability in nanowires

“…A free electron jellium model, treating the electrons as a noninteracting Fermi gas confined within the wire by hard-wall boundary conditions, provides an intuitive and even quantitative explanation of observed quantities like conductance [12][13][14], force [13][14][15][16][17], and shot noise [18]. To examine energetic stability, we consider a weakly deformed cylinder, and find its thermodynamic potential to quadratic order in the amplitude of the deformation.…”

Quantized conductance through an asymmetric narrow constriction in a three-dimensional electron gas

“…Utilizing semiclassical ideas, this model has been used, e.g., by Stafford et al [14][15][16] to develop a theory of nanocohesion in metallic wires. The constriction ͑1͒ has a "bottleneck" in the x-y plane which is bounded by the ellipse x 2 / c 2 + y 2 / b 2 = 1; see Fig.…”