Quantum tunneling probability for a 3d potential

Question

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?

Answer 1

You may be mixing paradigms here. The terminology "tunneling trajectories" may be specific domain related jargon. Also, are you asking whether or not code exists to do the calculation that is numerically stable? Or are you asking about the procedure. The same approach to tunneling in 1-D QM should work in 2 and 3-dim. You need to set up the wave equation and solve it. For bound states match BC and for free states (beam problems) start with an initial beam far removed from the potential structure (plane wave as you mention, or it could be a modeled Gaussian beam, something that represents your experimental setup) and numerically solve the PDE with BC imposed. If you have a fairly smooth representation for the surface there is no reason why you can't use it for U(x,y,z) in Schrodinger's equation and apply some numeric paradigm for solving (Finite Diff, Relaxation, RK ODE solver if separable). You may need to create the solver for your problem. Would WKB apply for your model? Is the energy high enough near the potential? You may create a bound state in the continuum for free low energy states.