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:
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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.
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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?
Best Answer
Hopping and tunneling are often used as synonyms, but they are really very different terms with a fundamentally different basis.
Tunneling is an inherently quantum-mechanical feature which means that a particle wave-function tends to overlap into it's energetically disallowed area which leads to a non-zero probability of finding it "where it should not be". This isn't limited to materials or lattices and it happens more or less always.
Hopping, on the other hand, is a process reserved for statistical physics and it corresponds to a correction due to the approximation of using single-site wavefunctions as your basis. In a periodic lattice, your energy eigenstates are always periodic (see Bloch's theorem). So when you choose a non-periodic basis of single-site wavefunctions, your Hamiltonian will not be diagonal. The non-diagonal terms are the "hopping" terms which just express the fact that the stationary state is a particle spread throughout the lattice rather than staying at one point.
An instructive example
Consider a periodic lattice which at every site looks kinda like a harmonic potential. You thus choose the ground state of the harmonic oscillator as your one-site wave function. Recall that this looks like $$\psi_{x_0} \sim \exp \left(-\frac{(x-x_0)^2}{2a} \right)$$ I.e. this wave-function is everywhere non-zero and thus there is a non-zero probability of finding the particle anywhere in space. So quantum tunneling is happening either way. Now let us label the sites $x_0$ by an index $i$. You then express the Hamiltonian in the basis of $\psi_i$ as (higher excitations than ground state are often neglected, but it's easy include them also) $$H_{ij} = \langle\psi_i|H|\psi_j \rangle$$ The terms of the type $\langle\psi_i|H|\psi_{i+1} \rangle$ are exactly the hopping terms and they express the fact that there might be a non-zero energy contribution in being spread between the two sites (which you can intuitively understand as "being in the middle of a hop"). It all traces back to $\psi_i$ not being an eigenenergy state. Since different temperatures give different distributions over energy states, you will see a different occurrence of "being in the middle of a hop".