I am simulating quantum tunneling through a rectangular potential barrier, as a function of applied voltage across the barrier as well as barrier thickness. I am following the theory from the book "Exploring scanning probe microscopy with Mathematica". In it, the tunneling current takes the form:

enter image description here

$v$ is the applied voltage, $\Delta s$ is the "effective barrier thickness", and $\bar\phi$ is the "effective barrier height". The effective barrier height and thickness are caused by the electron seeing its image charge while in the barrier, and have to be figured out numerically (as outlined in the book, which I am doing).

I've plugged in the parameters I could find in the literature for the physical situation I'm trying to simulate (effective mass, permittivity, etc).

I have simulated the tunneling current as a function of voltage, for an array of barrier widths, then normalized the current for each width so that they have the same maximum, and can be plotted on the same graph, so their curvatures can be compared. Here is a plot, of normalized current vs applied voltage, for a range of barrier widths 0.2nm to 2.5nm, where the 0.2nm one is the (nearly) linear one and they get more curved for the wider barriers:

enter image description here

What's worrying me is that the 0.2nm barrier curve clearly has downward curvature for lower applied voltage, and that strikes me as wrong. My intuition tells me that as the barrier tends to 0 width, it should become an Ohmic contact, so linear. At higher voltages, it seems to behave as I'd expect, but not at lower voltages.

My only current hypothesis is either that I made a coding mistake (possible, but I want to be sure my simulation is sound), or, the book mentions using the WKB approximation to calculate the transmission probability at a given electron energy/etc. I read in a paper that the WKB approximation can often be a little inaccurate because in doesn't account for reflections off the back wall of the barrier, and subsequent reflections. However, it's very difficult for me to try and analyze it because of the numerical steps involved.

Would using the WKB approximation give rise to the distortion I'm seeing? Or is the distortion possibly expected?


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