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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.

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3 Answers 3

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 $\leftrightarrow$ 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).

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Ah, I see. In that case, I'm definitely confused as to how nearest neighbor and variable range "hopping" would be classified. I've seen both referred to as tunneling, although nearest neighbor hopping is generally described as a phonon-assisted process. –  SMPAT Mar 20 '13 at 17:01

I agree broadly with the gist of the answer provided by Mark. However, since most authors do not distinguish between "tunneling" and "hopping", it would be best to refer to the paper you mentioned initially. In particular, the following is from the beginning of Section 1.3 (pp. 98):

Electrons may move from one localized state to another with the help of energy transfer to or from the phonon system; this is hopping conductivity. Alternatively quantum mechanical tunneling from one site to another may spread the eigenstates over the whole system so that the theory of metallic conductivity is applicable.

The author also states (pp. 95):

[...] hopping can only occur if phonons can be emitted or absorbed to compensate for the change in energy of the electron, or if energy can be supplied from some other source.

When the author mentions "energy transfer to or from the phonon system," he is considering what might be referred to as "inelastic transport" in more recent work (e.g. take a look at inelastic electron tunneling spectroscopy). This is supported by the quote from pp. 95 where he states that "hopping" only occurs with the emission or absorption of a phonon. Note that this is not the same as incoherent transport in general, wherein the key assumption is that the electron does not retain memory of its phase between hopping events (so-called "sequential tunneling").

Conversely, when he uses the phrase "quantum mechanical tunneling," it is implicit that he means a coherent process wherein there is no energy transfer out of the electron system. The standard approach in this context is now to apply the Landauer-Büttiker formalism, which treats quantum transport as a scattering problem. This technique was not well developed at time the paper you refer to was written.

As you may have guessed, if your interest is quantum transport, it's probably not a good idea to pick up nomenclature from a paper written in 1974. A good reference for these issues is the book Electronic Transport in Mesoscopic Systems by S. Datta. I've linked to the 1st edition, which I personally prefer to the newer one.

More generally (outside the field of quantum transport), there is no consistent distinction between "hopping" and "tunneling". Most authors would probably regard them as synonymous except in the context of specific work. For example, the "hopping" term in a Hubbard or tight-binding Hamiltonian has nothing to do with phonons or a loss of phase coherence.

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Simple and shortly tunneling is some like penetrating of electrons or tunneling of electrons from the barrier b/w low energy and high energy band. And hopping is like jumping of electrons from low to high energy band and in tunneling electrons carry all the info. But in the hopping electrons losses its all info.

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