Tight-binding Green's-function calculation of electron tunneling. I. One-dimensional two-band model

Abstract

Using the Weaire and Thorpe two-band Hamiltonian, we calculate tight-binding Green's functions for a one-dimensional model of the metal-insulator-metal tunnel junction. These functions are then employed in the Caroli-Combescot-Nozi\`eres-Saint-James theory of electron tunneling in order to explore the relationship between the electronic structure of the tunnel junction and the tunneling current. These calculations demonstrate the dependence of the tunneling current on the width of the electrode and insulator bands, the gap between the insulator valence and conduction bands, and a parameter that describes the nature of the electrode-insulator interface. For small bias the Green's functions for the bounded insulator in the tunneling junction are expressed in terms of functions for infinite and semi-infinite lattices. We thereby obtain a closed-form expression for the tunneling current which is dependent on the properties of the bulk lattice and can be evaluated by known techniques. For larger bias, the insulator functions must be calculated directly. They can be expressed in terms of finite continued fractions which are convenient for computer calculation. The results of our model calculation are valid for any thickness of the insulator film. We find, in general, that the tunneling current is greater than that predicted by the independent electron model (IEM) which replaces the insulator structure by an external one-electron potential. Also, we find that the shape of the current vs bias curve is entirely different than predicted by the IEM. The logarithmic conductivity exhibits peaks whose position depends on all the parameters of the tunnel junction, and, in contrast to the IEM, there is no simple relation between bias and barrier height. Finally, we see that for certain values of the junction parameters we obtain zero-bias peaks in the conductance curves and, in some cases, negative values of conductance with corresponding dips in the tunneling current.