I have an electron potential energy landscape for a metal surface to vacuum transition from a DFT calculation. The following pictures show a 2D slice of the potential, where potential energy is given in the z axis (also shown by color). The surface has an electric field applied to it.
The potential is periodic in x direction and can be considered constant in y direction.
I want to (numerically) find the tunnelling probability of an electron that comes from the metal and is incident on the barrier. The electrons in the metal can be assumed to be described by the free electron model (i.e. they are plane waves).
In 1D, there are multiple fairly simple numerical treatments for tunnelling probably (WKB approximation, transfer matrix method), is there something similar in 3D?
Or perhaps it is possible to define some "tunneling trajectories" at different points on the barrier and use 1D WKB method on them, and get the total 3D tunneling through some averaging?