Macroscopic description of quantum-mechanical tunneling

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Abstract
Recently a macroscopic description of electron transport in semiconductors was developed [M. G. Ancona and H. F. Tiersten, Phys. Rev. B 35, 7959 (1987)] that incorporates lowest-order quantum effects by endowing the electron gas with a density-gradient-dependent equation of state. Calculations made using this new description have been found to agree well with corresponding results obtained with use of one-electron quantum mechanics for various equilibrium (no current flow) situations. In the present paper, the density-gradient theory is applied to a quantum transport problem. The equations of nonequilibrium density-gradient theory are discussed first in general terms and then as applied to the specific example of steady-state tunneling through a metal-insulator-metal barrier with thermionic and space-charge effects (to be examined in a future paper) neglected. Two different tunneling regimes, which may be described as inertia dominated and bulk-scattering dominated, are analyzed, and approximate expressions for the current-voltage characteristics in each regime are given. For inertia-dominated tunneling, we devise ‘‘virtual-anode’’ boundary conditions to account for dissipation in the downstream contact and obtain results which compare favorably with those of standard elastic-tunneling theory.

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