My professor made a distinction between electron hopping (the closest wikipedia had an article on) and tunneling, saying that one (he didn't say which, but I assume hopping) was temperature dependent and the other wasn't.
However, I always thought they were the same mechanism. In fact, if you look at the Hubbard Model article above, it says "tunneling ('hopping')", implying that they are the same. I know WP isn't an authoritative source, but it at least shows that the two are commonly confused.
About the only two mentions of this question I could find online are from physics.SE and this physicsforums thread. They both definitely attempt to answer the question but neither are really satisfying to me.
To quickly cover a few of my confusions about the physics.SE top answer:
He says that while tunneling probability is exponential in the area under the energy barrier between sites, hopping probability is exponential only to the height. How is that possible? If you have two sites with some energy barrier between them, I can't imagine how it would have no effect on the hopping probability if you made the sites twice as far instead. When I have studied the Tight Binding Model in classes, we've always included the overlap of wave functions from neighboring atoms -- which seems like it must depend on distance between them.
He also says, about tunneling, that "is it possible to define a meaningful wave function describing how the probability amplitude is distributed across the lattice", but about hopping, "there is no need for a "wave function" as such, just a probability distribution describing where the electrons are likely to be found". Those two sound like the same exact thing to me -- the square of the wave function is the probability distribution, right?
In the physicsforums thread, one of the answers is:
In hopping, the hopping particle has to have energy greater than or equal to that of the height of the barrier in order to cross the barrier, in tunnling it can cross the barrier even with energy less than the height of the barrier.
That sounds nice, but in the TB Model, it physically represents atoms that are close to each other and their electrons. Their electrons have a much higher probability of being close to their "parent" atom, but some probability of being on neighboring ones. It seems like the "barrier" in between neighboring atoms is obviously of higher energy than the atomic states of the atoms, right?
Can anyone elucidate this for me?