Can someone explain the difference between hopping and tunneling? The context I\
ID: 1392323 • Letter: C
Question
Can someone explain the difference between hopping and tunneling? The context I'm considering is conduction in semiconductors, specifically between impurity states within the bandgap. It's always been my understanding that hopping is tunneling. Variable range hopping and nearest neighbor hopping, as I understand it, are both forms of tunneling between overlapping states (for example, see http://igitur-archive.library.uu.nl/dissertations/2002-0806-101243/c4.pdf). However, in papers such as DJ Thouless 1974 Electrons in Disordered Systems and the Theory of Localization, hopping and tunneling are described as two different processes. I guess I don't exactly understand what "hopping" is supposed to mean, with respect to charge transport.
Explanation / Answer
It's difficult to know for sure without having access to the Thouless paper that you mentioned. However, in my own research field we sometimes talk about two processes called tunnelling and hopping, that are distinguished as follows. Tunnelling is a coherent process in which electrons move from one lattice site to another, maintaining a definite phase relationship between the amplitudes corresponding to finding an electron at different lattice sites. The probability of a tunnelling event is an exponential in the area under the energy barrier between sites. In this case one has ballistic wave-like propagation, and is it possible to define a meaningful wave function describing how the probability amplitude is distributed across the lattice.
Hopping is an incoherent, thermally activated process in which an electron moves from one site to another but loses all information about its phase in the process. In other words, there is no coherence between the amplitudes for finding an electron at different lattice sites. The hopping probability is an exponential in the height only of the (free) energy barrier between sites. In this case one has diffusive transport, and there is no need for a "wave function" as such, just a probability distribution describing where the electrons are likely to be found.
In the interesting intermediate case where both processes are present, one must describe the electron distribution by a density matrix that includes both the quantum position uncertainty due to tunnelling, and the classical position uncertainty due to the stochastic hopping.
edit in response to comment from OP: The term "hopping" is sufficiently imprecise that it will have many meanings in the literature. In the paper you link to, they appear to be referring to hopping as an incoherent process. I don't have time to read the paper properly, but I think that the reference to tunnelling comes from a semiclassical calculation of the hopping rates. The electron transfer occurs due to overlap between localised wavefunctions, and thus depends on the ratio of inter-site distance to localisation length. The phase information is however lost due to decoherence at a rate much larger than the coherent transfer time. The phonon-assisted nature of the hopping is (I think) a slightly separate issue that only becomes relevant when the binding energy of two sites is different: then energy conservation (and time reversal invariance ? detailed balance) requires the emission or absorption of a phonon. This explains the appearance of the Bose-Einstein distribution and Boltzmann factors in the rates. However, even if the process is not "phonon-assisted", the treatment of hopping is still classical/incoherent, because they assume that the total probability of a sequence of hops is the product of the probabilities, not the amplitudes (see Eq. 4.14).