Tunneling through arbitrary potential barriers and the apparent barrier height

A simple matrix formalism is presented that allows the tunneling current between two planar electrodes with an arbitrary barrier potential to be calculated conveniently. This formalism is a straightforward extension of the usual textbook example for tunneling through a rectangular barrier and can be easily implemented on a personal computer. A first principles derivation of the tunneling current based on free-electron electrodes is given that is useful for the teaching of solid-state physics. Examples showing the utility of the method are the transmission through a tunnel barrier modified by the image potential and tunneling via intermediate states (resonances). These issues are interesting topics in scanning tunneling microscopy where both phenomena are believed to contribute to the image formation.