NANO LETTERS
On the Reasons for Stepwise Changes in the Tunneling Current across Metallic Nanogaps
2003 Vol. 3, No. 12 1633-1637
Claudio F. Narambuena, Mario G. Del Po´polo, and Ezequiel P. M. Leiva* Unidad de Matema´ tica y Fı´sica, Facultad de Ciencias Quimicas, INFIQC, UniVersidad Nacional de Co´ rdoba, Cordoba 5000, Argentina Received July 3, 2003; Revised Manuscript Received August 15, 2003
ABSTRACT Tao and co-workers (Li, C. Z.; He, H. X.; Tao, N. J. Appl. Phys. Lett. 2000, 77, 3995−3997) have shown experimentally that the size of the nanogap between two electrodes changes with increments that are very often close to 0.5 Å. We employ a grand canonical Monte Carlo simulation using embedded atom potentials to analyze some aspects of this phenomenon. Although the simulations do not deliver a single distance but a log-normal-like distribution of distance increments that presents a maximum close to 0.5 Å, the present results provide insight for understanding why the changes in the distances in the nanogaps are considerably smaller that the distances between lattice planes.
Introduction. The study of how different structural, thermodynamic, and kinetic factors contribute to the processes and properties related to small metal clusters (nanostructures) appears to be a necessary condition in determining the limits set by nature on the control and stability of these systems. This leads to novel technological applications, among which the fabrication of electrodes separated by a few nanometers can be mentioned. One of the purposes of these achievements is to connect biomolecules between the electrodes to perform electronic studies with single molecules.1-3 To do that, it is necessary to control at will the separation between electrodes, according to the size of the molecule under study. One of the methods devised to adjust this separation is based on the mechanical rupture of a nanowire, with the subsequent formation of two electrodes.4 Among the nonmechanical methods are electromigration5 and electrochemical methods.6,7 Mechanical methods allow us to obtain a continuous range in the separation between two electrodes (nanogaps).4 However, electrochemical methods involving metal deposition and dissolution do not yield a continuous but a discrete range of nanogaps. This is the case in the experiments done by Tao and co-workers,7 who measured the tunneling current across a nanogap using tunneling microscopy and then translated the measured values of current into distances. In the electrochemical method developed by these authors, Cu atoms are deposited on or dissolved from one of the electrodes, the distance between the electrodes being controlled by a feedback setup. Tao and co-workers have shown that the size of the nanogap between the two electrodes is * Corresponding author. Present address: Facultad de Ciencias Quimicas, Haya de la Torre esq. Medina Allende, Ciudad Universitaria, 5000 Co´rdoba, Argentina. E-mail:
[email protected]. Tel/fax: ++54-351-4344972. 10.1021/nl034474i CCC: $25.00 Published on Web 11/01/2003
© 2003 American Chemical Society
restricted to take discrete values with changes that are often close to 0.5 Å.7 This remarkable observation must be deeply related to the discrete nature of matter. However, these changes in the nanogap are considerably smaller than the distances between the low-index lattice planes of the fcc structure of Cu (Table 1), which are on the order of 2.0 Å. For this reason, these authors have suggested that this phenomenon may originate from the existence of relaxation processes, proposing that when an atom is added to the surface an atomic reconfiguration occurs toward more stable atomic structures, leading to changes in the nanogap that are much lower than expected. Two types of experiments were performed.7 In one set of experiments, two Cu electrodes microfabricated on oxidized Si were employed to develop the nanogap. In the other set, Cu atoms were deposited on an STM tip, and the tunneling current between this and a Au substrate was measured as a function of the separation distance. To study the latter theoretically, we used a grand canonical simulation method and embedded atom potentials in the present work. The electrode (or tip) where the deposition and/or dissolution of Cu atoms occurs is approximated by an isolated cluster, and the surface of the substrate is replaced with a plane located at a distance Rp from the center of the cluster. The plane representing the substrate may either be smooth or present a crystalline structure. The cluster is of a crystalline nature and is characterized by a random orientation with respect to the surface. In this way, we assume that the apex of the real electrode is given by a well-ordered domain of unknown relative orientation. The present simulations show that the discrete distance changes in the nanogap can be described by a probability
Table 1. Some Relevant Distances for the fcc Structure of Cu (a ) lattice parameter; nm ) nearest neighbor) characteristic distances
Å
d(a) d(nn) d(111) d(100) d(110)
3.62 2.56 2.08 1.81 1.28
distribution function, originating from different lattice orientations, whose maximum is very close to 0.5 Å. Although this fact does not correspond exactly to the experiment, where very often 0.5-Å steps were reported, the simulations are enlightening to show why distances considerably smaller that the distance between low-index lattice planes are obtained. The introduction of a small corrugation to the plane representing the surface of the substrate (like that of a (111) lattice plane) changes the probability distribution function only slightly. Simulation Method. To simulate atomic clusters made of several hundreds of atoms, we use the embedded atom method (EAM),8 a many-body potential that provides a reliable description of static and kinetic properties of transition metals for bulk systems as well as for systems involving surfaces. The EAM represents an improvement on pair potentials such as the Lennard-Jones potential because the consideration of many-body properties by the former yields a better description of cohesion out from equilibrium. Within this formalism, the total energy of the system is given by Etotal ) Ei ) Fi(Fh,i) + Fh,i )
∑i Ei 1
∑ Φij(Rij)
2j(*i)
∑ Fj(Rij)
j(*i)
The total energy of the cluster Etotal is written as the sum of the contributions of each atom Ei. Each atom is assigned an energy term that is a function of the electronic density Fh,i at the point where the atom is located. This is calculated as the sum of the electronic densities of the isolated atoms Fj(Rij). The term ∑j(* i)Φij(Rij) is a sum of repulsive pseudoCoulombic potentials, and the parameters of the functional form Fi(Fh,i) are obtained from a fit of experimental properties (sublimation energies, elastic constants, etc.).8 In the experiment, the deposition or dissolution of atoms is driven by a feedback setup that controls the potential applied to the surface, which is equivalent to controlling the chemical potential of the atoms at the interphase. The natural computational counterpart is then a grand canonical simulation,9,10 where the chemical potential, the volume, and the temperature of the system are fixed and the number of particles is allowed to fluctuate. Because, according to Tao’s proposal, particle rearrangement at the interphase is also 1634
Figure 1. Initial configuration of the 1709 Cu atoms cluster employed in the present grand canonical Monte Carlo simulations.
important, we allowed the relaxation of the position of the particles. Thus, technically speaking, three types of movements are allowed in our model: (a) particle displacement, (b) particle removal, and (c) particle creation. Particle displacement is achieved through the Metro´polis algorithm as usually employed in simulations of condensed phases.11 That is, the displacement of a particle is accepted with a probability of min{1, exp(-∆E/kT)}, where ∆E is the change in potential energy between the initial and final configurations, k is the Boltzmann constant, and T is the absolute temperature. Similarly, particles are created at a random position or destroyed with a corresponding exponential probability that depends on the chemical potential applied to the system. The simulations involved 30 000 Monte Carlo steps. In each step, a motion and a removal/creation attempt was made for each of the particles of the system. The initial configuration, shown in Figure 1, was a 1709atom cluster. This structure was prepared from a piece of the bulk structure of Cu according to the following prescription: after the coordinate system centered on one of the atoms was chosen, a large sphere was drawn and cleaved in the directions given by the (111) and (100) lattice planes at distances from the center according to the Wulff construction for Cu.12 All atoms out of this surface were discarded; only the 1709 atoms mentioned above remained. Because we were attempting to emulate a deposit on a bulk metal structure, the chosen starting configuration was based on the fcc structure of the bulk metal. It is well known that in the case of small clusters the most stable structures do not correspond to the fcc structure and diverse configurations appear.13 However, the experiments were carried out by depositing Cu on a massive tip that was expected to provide the basis for growth with a fcc structure. Furthermore, because the simulations, like the experiments, are run at room temperature, there is practically no room for rearrangement of the cluster to the singular structures found for small-sized clusters. The reason for choosing a cluster for the simulations and not a tip-like geometry is due to the fact that with a single simulation dissolution in several directions can be analyzed, as described below. Nano Lett., Vol. 3, No. 12, 2003
Figure 3. Variation of the nanogap width as a function of the number of MC steps for various chemical potentials µ. The plane representing the surface of the substrate is oriented in the (100) direction of the cluster. The µ values are reported.
Figure 2. Scheme of the edge of a nanocluster close to plane P representing the substrate. Atoms 1, 2, and 3 are located closer to the plane. (a) The substrate is represented by a smooth plane. (b) The substrate is represented by a crystalline (111) lattice plane
Tao and co-workers’ experiments7 involve measurements of the tunneling current between an STM tip on which the metal is deposited and a substrate as a function of time. This current is transformed into a distance scale by means of a previous calibration. To mimic the experimental measurements, we shall calculate the closest distance between the cluster and an arbitrary plane representing the surface of the substrate. Figure 2a shows a diagram of the edge of a cluster and a plane P, whose orientation is characterized by the unit vector uˆ . The distances (d1, d2, and d3) between different atoms on the edge of the cluster and plane P are also shown there. The closest distance between the cluster and the plane, labeled D, is given in this case by d1. Because the tunneling current decreases exponentially with the distance, it is reasonable to assume that practically all of the current flows between the surface and the atom closest to it, located at distance D. The corrugation of the substrate was introduced by replacing geometrical plane P with Cu single-crystaline (111) lattice plane P′ (Figure 2b). In this case, distance D was measured between the closest atomic pair, one of the atoms of the pair belonging to the cluster and the other to plane P′. Because the cluster is constructed on the basis of an fcc structure, we shall define the corresponding planes (jkl) as, for example, (111), (100), (110) or higher indices. Rjkl denotes the vector normal to the jkl plane of the cluster. Nano Lett., Vol. 3, No. 12, 2003
From Figure 2a (2b), we find that if atom 1 is removed from the cluster then atom 2 is active in the tunneling process. In this way, D changes from d1 to d2. At this point, we begin to understand that distance D should, in principle, take discrete values and that its variation should be strongly dependent on the way in which the cluster dissolves. It is also evident that the changes in D will be sensitive to the orientation of plane P in relation to the surface of the cluster. If geometric plane P is substituted by a plane with a crystalline structure (Figure 2b), then the calculation of D is reduced to find the minimal separation between the atoms in the crystalline plane and the atoms of the cluster, D ) min(|Rj - Ri|), where Rj and Ri are the positions of the atoms in the plane and in the cluster, respectively. Discussion. In our simulations as well as in the experiments, the crystalline domain located on the apex of the tip may have different orientations in relation to the surface. For this reason, in the present studies the coordinate system of the cluster is kept fixed, and the plane representing the surface of the substrate is rotated to represent these different relative orientations. We shall define the orientation of this plane by the unit vector uˆ perpendicular to it. For example, when we say that the plane is oriented in the direction (100), we mean that this plane is located perpendicular to the x axis, intersecting it at a distance Rp on the positive axis. Figure 3 shows a typical variation of the distance in the nanogap as a function of the number of MC steps for various chemical potentials µ when the plane representing the surface of the substrate is oriented in the (100) direction. It can be noticed that the dissolution becomes faster as µ decreases. A moderate dissolution rate (µ ) -3.55 eV) was chosen to study the structural details of the dissolution. During this simulation, steps in the distance of the nanogap are clearly observed and correspond very closely to the distance between the (100) lattice planes of Cu (1.81 Å). The existence of noise in the distances caused by the thermal motion of the atoms is also apparent. The estimation of the rms deviation of this noise in the longer portions of the curve for µ ) -3.55 eV yielded a value close to 0.1 Å. This gives us an idea of 1635
Figure 4. Nanogap width as a function of the number of Monte Carlo steps for (a) specific orientations of plane P in the direction of a low index plane of the cluster. The orientations are given. (b) Three orientations of plane P chosen at random.
the accuracy with which steps can be estimated when an arbitrary orientation of plane P is chosen. Figure 4a shows the distance changes in the nanogap for orientations of plane P corresponding to the low-index planes. As expected, the steps in the curves again follow the distance between the corresponding lattice planes (Table 1). However, Figure 4b shows dissolutions for three different orientations of plane P chosen at random. In this case, steps that are no longer uniform but present a distribution can also be observed. To get information on the probability distribution for observing a given step size, independently of the relative orientation between the tip and the surface, we proceeded as follows. For a given orientation of plane P (or P′) chosen at random, a histogram of the nanogap width was constructed, employing a bin of 0.1 Å. Each of these histograms was smoothed, resulting in a set of peaks (Figure 5). The changes in the nanogap width ∆D were calculated from the distance between two neighboring peaks. The process was then repeated for a large number of planes (on the order of 5000) chosen at random, and the ∆D values were accumulated. In this way, a distribution of angles was calculated as an average over a representative sample of orientations. Figure 6a shows the histogram for the probability of obtaining a given ∆D in the case of a smooth substrate surface. A maximum close to 0.5 Å is clearly observed. When the corrugation of the lattice plane representing the surface of the substrate is considered, the histogram presents some slight changes at longer distances, as shown in Figure 6b. Corrugation effects are more remarkable at short nanogaps, where the atoms on the tip “see” the structure of the plane but become smaller at longer distances. Figure 6b shows that a log-normal distribution fits the data of our simulations reasonably well. The observation of the simulation frames for different orientations of the surface shows that, as a general rule, it can be stated that tunneling from or to the cluster should occur at one of its corners because alignment between a face of the cluster and the substrate is highly improbable. To 1636
Figure 5. Procedure employed to calculate the distance changes in the nanogap ∆D. (a) Typical variation of the nanogap width vs the number of MC steps. (b) Histogram obtained from part a using a 0.1-Å bin. A typical ∆D is shown there.
Figure 6. Probability of obtaining a certain change ∆D in the width of the nanogap. The histogram was constructed by accumulating ∆D values for 5000 planes oriented at random. (a) Histogram obtained with a smooth substrate surface. (b) Histogram obtained with a crystalline (111) lattice plane representing the surface of the substrate.
analyze which configurations contribute to the 0.5-Å steps, we selected some directions in which these steps are typically observed. Figure 7 shows the steps observed in such a direction, together with some configurations that yield such steps. The atoms between which the tunneling current should flow are also shown there. It is remarkable that in these Nano Lett., Vol. 3, No. 12, 2003
Figure 7. Changes in the width of the nanogap for a direction in which distance changes close to 0.5 Å are frequently observed. Configurations a-c show the corresponding geometries of the cluster in the regions marked by the arrows.
simulations the nanogap distance moves backward and forward, as observed in Figure 4 of ref 7. Conclusions. We have proposed a simple computational model based on a grand canonical Monte Carlo simulation and embedded atom potentials to study distance changes in nanogaps controlled by fixing the chemical potential of the atoms constituting the system. The predictions of the model indicate that, if the relative orientation of the nanoelectrode and the substrate is such that the surface of the substrate is parallel to a low-index plane (jkl) of the nanoelectrode, the distance changes will correspond to the interlattice distance djkl. However, if the dissolution is studied for a set of randomly oriented planes, a log-normal-like distribution of the distance changes is observed, with a maximum close to 0.5 Å. Because in experimental results in the literature stepwise changes of 0.5 Å with a small dispersion have been reported, it is possible that the actual distribution is considerably sharper than the one from our simulations. It must be emphasized that an important element of the electrochemical experiment is absent in our simulations, namely, the electric field, which may provide an important driving force to orient growth at the electrochemical interphase. This should be a further improvement to the current modeling. However, it is appealing that the feature of 0.5 Å appears in both the experimental and simulated results. Acknowledgment. Financial support from CONICET, Agencia Co´rdoba Ciencia, SecytUNC, Program BID1201/
Nano Lett., Vol. 3, No. 12, 2003
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