Conductance and Mechanical Properties of Atomic-Size Metallic Contacts: A Simple Model.

We present a simple model for the evolution of atomic-scale metallic contacts together with exact free-electron calculations of the conductance during an elongation-contraction process. The critical pressure that the neck can sustain before yielding is constant. Shortly before the neck breaks the pressure gradually increases up to a maximum for a single-atom contact. In force microscopy experiments this maximum value also depends on the cantilever constant. Conductance histograms show clear peaks at integer multiples of 2e 2 yh slightly shifted to lower values. [S0031-9007(96)01134-9] In the last few years, several techniques have been extensively used to study the conductance of atomic-scale metallic contacts with variable cross sections [1 ‐ 7]. The physics of these small contacts has been theoretically analyzed by molecular dynamics (MD) simulations [7 ‐ 10] and conductance calculations [11‐ 16]. Because of the interplay between conductance quantization effects and nanomechanical properties, the interpretation of the experimental results is still controversial. Recently, the combination of force and tunneling microscopy measurements has made it possible to study the electronic and mechanical properties of these contacts [17,18], even down to a single-atom contact [19]. In this Letter, we present a simple model for the evolution of atomic-scale metallic contacts in these experiments together with exact free-electron conductance calculations. Our approach is inspired by the results of MD simulations [7 ‐ 10] as well as by the recent experiments on contact nanomechanics [17‐ 19]. We model the initial relaxed contact (just before retraction) as a constriction with cylindrical symmetry of height L0. The contact is made of NL slices with a length, Ln, equal to the interlayer spacing d0 (d0 1.63 times the atomic radius r0 typical for a close packed arrangement). The cross section of each slice (pR 2 ) is an integer multiple of an effective atomic section a0 pr 2 0 . Both extremes of the neck always maintain the initial radius R0 and are connected to electron reservoirs through semi-infinite cylindrical leads. The total number of atoms forming the neck remains constant through the simulation. The conductance of the constriction is exactly calculated by solving the 3D free-electron stationary Schrodinger equation with hard-wall boundary conditions [20]. This is done by appropriate mode matching of the wave function in each slice together with a generalized scattering-matrix technique [21]. The elastic properties of the contact are determined by the macroscopic Young’s modulus Y and Poisson’s ratio s [22]. Each slice is characterized by an effective spring constant kn › YpR 2 yLn. The effect of the macroscopic part of the tip is included in a spring constant kt Y pR0. The spring constant of the total system ktot is obtained by adding in series all the constants including that of the cantilever kL [see inset in Fig. 1(a)]. The elongation of the contact proceeds in alternating elastic and yielding stages [8,17]. MD simulations performed by Landman et al. [8] show that the elastic stage is followed by a shorter atomic rearrangement period culminating in the formation of an added layer, with a relief of the accumulated stress. This suggests [18] an interpretation of the yielding process in FIG. 1. Representative simultaneous calculations of conductance and force during the elongation of atomic-scale Au contacts. (a) Conductance G versus elongation for different initial necks. Dotted lines correspond to the semiclassical conductance GS (see text). Continuous lines are the results of the exact free-electron conductance calculation. A sketch of the elastic model is also included. ( b) Force versus elongation curves for the same contacts. The histogram represents the forces to break a single-atom Au contact obtained from 500 different curves.