Image force in normal metal electrode tunneling

[Louis Fry]

Hi there,

I have a question that I think has an answer but I cannot find it in the literature in any convincing way:

I am using two models, WKB based and TMM based to calculate the transmission probability of electrons through a (initially rectangular) thin potential of the order of 1-2nm, but very high between 2 to 4eV (small affinity/wide gap insulators + high work function metals).

Further to this I am adding more insulator layers within this structure with different permittivities and potential heights, such that there are steps in the potential along the transmission axis.

Now when it comes to considering the image force it seems that people in the literature choose to include it mostly with WKB based models such as Simmons' and Brinkman's for fitting IV characteristics of experimental results, then use various values for permittivities ranging from optical to static values. Then with most theoretical papers I have come across that I can just about understand aspects of generally do not mention the image force, that is, in the context of direct tunneling between normal metal electrodes.

So here is my reasoning, and then followed by the problem:

If I consider it, for a thin and tall barrier in the range described above, the effect will be quite strong depending on the value of permittivity, and for any reasonable values the use of the WKB approximation is questionable.

Now choosing the value of permittivity one could take a sort of particle view similar to the way Schottky (thermionic) emission is understood (note I am not using the image potential commonly used in Schottky emission), and that is that since the transition of the electron across the material is very fast, an optical value should be used since the non-electronic components of the permittivity won't have time to respond. In this case the effect is huge, using a value like 3.0 completely rounds off the corners of such barriers. Now to justify the use of the static value, it's certainly unreasonable to consider transition times of a particle, rather it seems nice to say that since the evanescent wave in the dielectric (for single particle wavefunction) is static in time at least as far as the time-independent description is concerned, then a static value should be used.

In either case this effect appears to be so strong that there must be a physical justification for using it or not. Say I introduce a material layer (edit: of the order of a few atoms) at the interface that has a low permittivity and high barrier, the effect is so strong that layer is completely distorted almost as if it wasn't there. It doesn't seem physically reasonable, and it doesn't seem clear in the literature how exactly to treat this in quantum tunneling. There are old papers describing how to use it but it isn't convincing as all that was available at the time was the WKB approximation (pre-1970) until the TMM method of Tsu and Esaki came along, after which it seems to be quite unclear when to use it or not.

EDIT: Ok maybe I should be a little more specific, I guess it boils down to what kind of image force should I consider, writing this post actually made me chill for a sec, typical brainfart right there. I have been using Simmons' image force from his paper:
"Generalized Formula for the Electric Tunnel Effect between Similar Electrodes Separated by a Thin Insulating Film"

So sorry for the little essay here, hopefully it will trigger some interesting discussion or shed some light on this topic?
Thanks!

 

[Louis Fry]

If anyone comes across this, I have come to the following conclusion:

In my structures the tunnel probabilities are small, but the thickness is very small such that an electron tunnel event is expected to occur quite quickly. An accurate description of transition time of a tunnel electron is to my knowledge an unsolved problem, so it is hard to make any conclusions, however it is postulated that it is on the order of a few femptoseconds for thin barriers[1].

The first reasonable thing to consider before the dielectric properties of the insulators, are actually the dielectric (plasmonic) properties of the metal! Simply put, if the transition time of the electron is faster than the inverse of the plasma frequency of the metal (>10^12 Hz typically), then the "exchange correlation hole" (the image) hasn't enough time to spread, thus no image potential is expected to develop in the insulator. The physical way of seeing this is that the electron exists 'unscreened' in the insulator for such a short amount of time that both electrodes won't have time to polarise, increase that total time regardless of the actual transmission probability then the image potential may have a chance to develop.

This is still a handwave explanation in itself but there are of course many more details such as the question of bandstructure continuity and spatial functions of the relevant material parameters like polarizability/density which occur in real systems due to growth/deposition methods of thin films. It's clearly a very difficult problem to consider in detail in a real system!

Having followed up as much literature as I could about this subject when I posted, it looks as though it's still a controversial issue to this day, but no one really uses the classical and semi-classical (Simmons' expression) image potential anymore unless there are good physical reasons to do so, for instance in experiments where electrons are basically localized in front of a metallic plane[2].

Therefore I ignored it in my system. Has anyone here had to work with this before?

[1] "Modeling and simulation of tunneling through ultra-thin gate dielectrics"
Schenk, Andreas and Heiser, Gernot, Journal of Applied Physics, 81, 7900-7908 (1997)
[2] "Long-distance entanglement of soliton spin qubits in gated nanowires"
Paweł Szumniak, Jarosław Pawłowski, Stanisław Bednarek, and Daniel Loss
Phys. Rev. B 92, 035403

I could have included more references here but this was a little while ago now, these were the most interesting!