T.E. Feuchtwang

A theory of vacuum tunneling microscopy

Abstract A theory of scanning electron microscope, recently developed by Binning et al., is presented. The theory follows Tersoff and Hamann in applying the transfer hamiltonian formalism to vacuum tunneling between a sample consisting of an infinitely extended single crystal plane and a field emission tip. A general expression for the tunneling current for any two electrodes of arbitrary shape is derived and discussed. Reasoning by analogy with field ion microscopy it is argued that the operation of the device depends on the strong enhancement of local electric fields due to microscopic imperfections of the topmost atomic layers of the central facet, such as small atomic clusters and kink sites. These local fields and the equally important multiple image potential lead to a strong dependence of the tunneling barrier on the polar angle. Accounting for this effect, which was not considered by Tersoff and Hamann, complicates the analysis significantly. The tunneling current is expressed as a convolution type integral involving the spectral densities of the samples and of the tip. It is argued that this integral can be interpreted in terms of an instrument function depending only on the tip or probe parameters, which is convoluted with a function depending only on characteristics of the sample. Simple limiting cases of the theory are considered. A discussion of the resolution predicted by the theory is presented.

Quantum size effects in the Fermi energy and electronic density of states in a finite square well thin film model

Abstract The dependence of the Fermi energy, E F , and the electronic density of states, ρ( E ), of thin metallic films ( L z ≲ 50 A) on film thickness, electron density, and potential well depth, is systematically investigated in a free-electron, finite square well model. Two size-dependent effects are observed: (1) oscillations in E F and ρ (E) due to the size-quantization of the energy levels, and (2) changes in the mean values of these quantities, averaged over several oscillation periods, relative to their bulk values. The mean value of E F is increased relative to its bulk value by as much as 5%–10% for physically reasonable well depths and typical metallic electron densities. For the special case in which the top energy level in the well is occupied, the mean value of E F is equal to its bulk value. The mean value ofρ( E F ) can be either greater than or less than its bulk value, depending on the well depth. In contrast to the small amplitude oscillations in E F , the oscillations in ρ( E F ) may have an amplitude as large as 25% of the mean value for sufficiently thin films. Accurate analytic expressions for the thickness dependence of the Fermi energy and density of states are derived.

Influence of the boundary conditions on the fermi energy and density of states in a free-electron solid of sub-micron dimensions

Abstract Asymptotic expressions for the distribution of the eigenvalues of the Helmholtz-Schrodinger equation are used to anlyze the dependence of the Fermi energy, E F , and the density of states, ρ( E ), on sample size, shape, and electron density, in a free-electron model with Dirichlet boundary conditions. It is found that for very small samples E F is increased relative to its asymptotic (i.e., bulk) value and ρ( E ) is decreased relative to its bulk value. These effects are more pronounced for samples with low electron density and with a large surface-to-volume ratio. In general E F and ρ( E F ) deviate significantly from their bulk values only for systems with fewer than 50,000 electrons and/or with linear dimensions of 100 A or less. The use of smoothing functions to represent the density of states obtained from the exact eigenvalue distribution is also discussed. It is shown that an oscillating density of states leads to small cusps in the plot of E F as a function of sample size. This is in qualitative agreement with the results of experiments on size-dependent oscillations in field emission from thin metallic films. Comparison is also made between photoemission experiments from thin films and other results obtained in this study.

Gauge Invariance and Gauge Independence of the S Matrix in Nonrelativistic Quantum Mechanics and Relativistic Quantum Field Theories

Abstract The gauge independence of transition rates as opposed to the gauge invariance of the equations of motion and gauge dependence of operators and state vectors is critically examined and explicitly demonstrated, both in nonrelativistic quantum mechanics and quantum field theory. Time independent as well as time dependent gauge transformations are explicitly analyzed using several techniques in order to clarify the physical content and significance of gauge independence and the conditions for its applicability.

Model studies of tunneling time

The precise definition and physical interpretation of a tunneling time is a fundamental problem in quantum mechanics. The lack of a well‐defined time operator in quantum theory precludes the calculation of time in terms of an expectation value. Consequently, model calculations and simulation studies have been proposed and used to determine the time for a particle to traverse a quantum‐mechanical barrier. The results offer both qualitatively and quantitatively, disparate predictions. One of the more commonly used approaches is the phase method, which we have applied to several one‐dimensional model potentials to show that the tunneling time is characterized by the ratio of the typical decay length in the barrier to the incident velocity, τ∼(1/κ)/v0. We also show that the phase and the spin precession methods are equivalent when the magnetic field is applied throughout the space containing the particle wave function and the tunneling barrier. In principle, the spin precession method can be regarded as an op...

Tunneling through localized barriers with application to scanning tunneling microscopy: New scattering theoretic approach and results

The general scattering theoretic technique of localized Green functions is applied to the calculation of one‐electron elastic tunneling current through nonseparable, microscopically localized barriers. The new technique is an exact theory of tunneling, not restricted, as the transfer‐Hamiltonian method of Bardeen, to weakly coupled electrodes. For illustration purpose and application to scanning tunneling microscopy (STM), we consider a model hemisphere–plane junction for which the multiple‐image barrier is taken into account for the first time. The full Green function is calculated at the points of a fine grid covering the tunnel region from the solution of Dyson’s equation which treats the localized tunnel barrier to all orders of perturbation of the planar barrier Green function. The tunnel current is obtained at the grid points from the Lippman–Schwinger equation, starting from the solution of the strictly planar, separable junction. The principal results of the calculation are presented as a current ...

An electrohydrodynamic formalism for ion and droplet formation in stressed conducting fluids

An electrohydrodynamic surface capillary wave theory is developed for ion and droplet formation in electrically stressed conducting viscous fluids. This theory can be used to describe properties of electrohydrodynamic sources such as the liquid metal ion sources. Using the mathematical models of Chung et al., and Grossman and Muller, we have obtained the first‐order corrections of the Navier–Stokes equation for a three‐dimensional axially symmetric fluid. Formal solutions are developed for the eigenmodes describing the deformation of the surface as a function of applied electric field with both radial and longitudinal components. It is argued that the spatial variation of the complex mode amplitudes yield both the size and directional distribution of the emitted particles. The real part of the eigenfrequency associated with the excited eigenmode is related to the temporal behavior of instabilities on the fluid surface. From this one obtains the growth and decay rates of the time‐dependent ion and droplet ...

An exact solution of Laplace's equation for cuspidal geometry

An exact solution of Laplaces equation is obtained for a system of conducting electrodes with cuspidal symmetry. The significance of this result in predicting and verifying the equilibrium configuration of a rotationally symmetric conducting fluid subject to electrostatic stress is discussed.

Computer simulation of a wave packet tunneling through a square barrier

It is shown that, for energies E<or=V/sub 0/, the barrier height, the typical tunneling time is on the order of 10/sup -15/ s. The traversal time as a function of kinetic energy is locally symmetric near E=V/sub 0/. The tunneling time depends on the shape of the wave packet. The tunneling time is linearly dependent on barrier thickness for a wave packet with finite width. These conclusions are compared with the results obtained from other methods for estimating tunneling times.

Reply to comment on “variational formulation for the equilibrium condition of a conducting fluid in an electric field”

In this comment we respond to the several criticisms of the paper by Sujatha et al. raised by Kingham and Bell. In particular, we demonstrate that, contrary to their assertion, Taylors solution for the electrostatic fields can never satisfy the boundary conditions for the actual experimental configurations involving field emission liquid metal ion sources and other experiments on electrostatically stressed conducting fluids. It is further argued that a careful analysis of Taylors experimental procedure and observations suggests that although the observed static structures have a macroscopic axial-symmetry they have not the idealized conical shapes of prescribed angle. Furthermore, the formation of the Taylor cone structure is shown to be inconsistent with the principle of energy minimization.It is concluded that none of the criticism raised by Kingham and Bell invalidate any of the analysis or conclusions presented in the paper by Sujatha et al.