A new formulation of elastic one-electron tunneling through three-dimensional (3D), nonseparable, spatially localized barriers is developed in terms of potential-scattering theory. To illustrate the principles of the method, a model metal-vacuum-metal junction is used, consisting of two parallel electrodes, one of which has a hemispherical protrusion. The electronic structure of each metal electrode is assumed to be free-electron-like, for simplicity. The bias and multiple-image tunneling barriers for this model are constructed on the basis of classical electrostatics and a simple quantum correction at the metal surfaces. Regarding the barrier as made of a planar, separable part plus a nonseparable, localized perturbation due to the spherical boss, the exact, unperturbed, one-electron Green’s function of the planar part is first obtained by numerical integration of the corresponding, effectively 1D Schrödinger equation. Then the localized boss potential is treated to all orders of perturbation by solving the Dyson equation for the full barrier Green’s function, using a real-space discretization of the integral equation on a finite grid.

New useful formulas are derived for correcting the discretization error associated with ignoring the singular diagonal matrix elements of the Green’s functions. The tunneling current density is then expressed in terms of the exact 3D wave functions which are obtained at the grid points by discretizing the Lippmann-Schwinger equation. The axial symmetry of the present barrier model leads to a reduction of the size of the Green’s matrices, since the wave functions of different axial angular momenta contribute independently to the tunneling. The m=0 wave functions are found to contribute 90% of the total tunnel current at the Fermi level. The new method is applied to a discussion of the lateral resolution of the scanning tunneling microscope. It is found that the current distribution peaks within a narrow angle around the boss axis, confirming earlier estimates based on the transfer-Hamiltonian formalism and in agreement with the observed atomic resolution of the microscope, when operating with atomic-size tips. The present Green’s-function method is applicable to several other problems of one-electron tunneling through localized barriers and may be extended to incorporate such effects as the corrugation and the band structure of the electrodes. Moreover, the method lends itself to a quantitative assessment of the accuracy of approximate tunneling theories such as the transfer-Hamiltonian formalism when applied to elastic one-electron tunneling problems.

• Received 4 November 1987

DOI:https://doi.org/10.1103/PhysRevB.37.10708