# Probability of quantum tunneling?

As simply as possible, could someone try to explain how one would go about calculating the probability of a electron/(electric voltage) quantum tunneling through an insulator (preferably using an example please)?
And how small would the insulator, and how large would the current/voltage have to be for it to be a practical to observe that tunneling took place, and how much would you expect to observe?

Also, as a bit of a side not, how small is the insulator generally in a MIM diode? and what about in a MIIM diode? and how much electricity goes through each of them, and how quickly?

Anyone know?

Anyone know?
Please be patient. You can't expect a reply 25 mins after posting.

Please be patient. You can't expect a reply 25 mins after posting.
I meant it as is it common knowledge, estimates, incredibly variant or something else? (also I wanted to get my post to the top of the forum :P, which I now realize is not allowed, sorry for that)
Though I think (after looking through some other stuff) you can calculate the probability with the Schrodinger equation, it would still be nice if someone could present a brief example please.

Not sure exactly what you want to know, but for a free electron of energy E, hitting a rectangular potential barrier ## V_0 ## of width a, the tunneling amplitude is,
## \frac{1}{T} = 1+\frac{k^2+q^2}{2kq} \sinh^2(qa)##
where ## q=\sqrt{\frac{2m}{\hbar^2} (E-V_0)} ##
and ## k =\sqrt{\frac{2m}{\hbar^2} \ E} ##

You can just plug in numbers yourself to see various situations if you want.
As an side note: I was told by a professor that since all exposed wires quickly have an insulating oxide layer forming around them, that the only way electrons go between touching wires is via tunneling.

Not sure exactly what you want to know, but for a free electron of energy E, hitting a rectangular potential barrier ## V_0 ## of width a, the tunneling amplitude is,
## \frac{1}{T} = 1+\frac{k^2+q^2}{2kq} \sinh^2(qa)##
where ## q=\sqrt{\frac{2m}{\hbar^2} (E-V_0)} ##
and ## k =\sqrt{\frac{2m}{\hbar^2} \ E} ##

You can just plug in numbers yourself to see various situations if you want.
As an side note: I was told by a professor that since all exposed wires quickly have an insulating oxide layer forming around them, that the only way electrons go between touching wires is via tunneling.

Okay, thank you, I'll try it out.
Also, I may be completely wrong here but anyway, doesn't oxidation require rogue oxidizer molecule/atom, so wouldn't it be highly improbably every atom/molecule of the conductor immediately oxidized (though it would attract any rouge oxidizers)? And couldn't the oxidized state still be (at least somewhat) conductive (though obviously not as conductive, not necessarily an insulator right)?
Though due to the way things interact in general, referring to how things don't touch due to virtual photons, could you consider that to always cause things to need to quantum tunnel?

And do you have any idea about MIM (metal-insulator-metal) diodes?

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Okay, thank you, I'll try it out.
Also, I may be completely wrong here but anyway, doesn't oxidation require rogue oxidizer molecule/atom, so wouldn't it be highly improbably ever atom/molecule of the conductor immediately oxidized? And couldn't the oxidized state still be (at least somewhat) conductive?

Not every copper atom, just those exposed on the surface to oxygen ("the only way" is really just figurative if your being technical). And, most oxides are pretty poor conductors at normal temperatures. Though I do know that some oxides are being incorporated into alloys to make superconductors - but that's only for temperatures around the boiling point of liquid nitrogen.

Not every copper atom, just those exposed on the surface to oxygen ("the only way" is really just figurative if your being technical). And, most oxides are pretty poor conductors at normal temperatures. Though I do know that some oxides are being incorporated into alloys to make superconductors - but that's only for temperatures around the boiling point of liquid nitrogen.
By every atom, I was referring to every atom at the surface, I just worded it poorly. but don't you still need the rouge oxidizers? Oxygen won't always oxidize a substance, isn't it like 1/1000 or something like that?

By every atom, I was referring to every atom at the surface, I just worded it poorly. but don't you still need the rouge oxidizers? Oxygen won't always oxidize a substance, isn't it like 1/1000 or something like that?

I really don't know. But, I wouldn't think 1/1000 is really that improbable considering how many atoms were talking about. Moreover, only the point of contact needs to have a small layer of oxide for what we're discussing.

I really don't know. But, I wouldn't think 1/1000 is really that improbable considering how many atoms were talking about. Moreover, only the point of contact needs to have a small layer of oxide for what we're discussing.
That is completely correct, as the source that told my that number (I can't remember were it was, but it gave a number that was in the 1/1000 range) but I would still think that some of the atoms wouldn't immediately oxidize (its not as though they are touching 5000 more times more atoms, since oxygen makes up approximately 1/5 of the troposphere), and all you would need it one atom not to be oxidized (that touches another none oxidized atom) to get a current able to pass through without quantum tunneling.

Also, I would like to point out this is getting a weee bit off topic, and is starting to be more relevant to chemistry.