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Mott VR Hopping Calculation Question

  1. Sep 6, 2012 #1
    I hope someone out there can help me with this fairly simple but specific question about the Mott 3-D variable range hopping conduction model.

    Mott theory has the energy difference between hopping sites as:

    ΔE ≈ 1 / g0rD

    Where g0 density of states near the Fermi surface, D is the dimension of hopping (1,2,3) and r is the distance between hopping sites.

    My question is about using this numerically. I don't know how to calculate or look up a specific number for g0 for a particular material. Also, what are the units of this quantity usually? States/volume? Energy/volume? The second makes sense because the units work out for 3D, but is that the usual unit of density of states?
  2. jcsd
  3. Sep 7, 2012 #2

    Physics Monkey

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    You can define the DOS as just [itex] G(E) = \sum_{i} \delta(E-E_i) [/itex] in which case the units are one over energy. However, in a large system, [itex] G(E) [/itex] diverges with system size, so it makes more sense to define the DOS per unit volume (sometimes also just called the DOS) [itex] g(E) = G(E)/V [/itex]. Obviously this has units of one over (volume times energy).

    Especially in a localized phase g(E) makes more physical sense since it as a local measure of the energy level spacing. If you're looking to mix states using an operator that acts within a range [itex] \xi [/itex], then the typical level spacing [itex] \Delta E(\xi) [/itex] of the states mixed by that operator would be roughly [itex] \frac{1}{\Delta E(\xi)} = g(E) \xi^d [/itex]. In other words, to find two states with a splitting much smaller than this you would typically need to look over a larger region.

    Unfortunately, I don't know of a place to look up values for this quantity. It's not clear that such a table can even exist since the value may depend on disorder. Nevertheless, you can relate this quantity to a number of other experimentally relevant quantities (not just the hopping conductivity), so that might be an avenue to investigate.

    Sorry I couldn't be more helpful.
  4. Sep 7, 2012 #3
    Not at all. That was very helpful. It makes me realize that there is nothing that straightforward about VR hopping, even though to me it seems like a relatively simple concept. I'm definitely not a solid state physicist, just a lowly chemist delving into solid state stuff.

    The thing is, all I'm trying to do is fit the conductivity over temperature of a certain material (semiconductor, of course) to a model. It certainly doesn't fit the Arrhenius model. It's fairly close to Mott 3D VR hopping, but not good enough. When I try to look up how others fit their curves to this model, there is a great deal of variation, but not much explanation as to how they arrived at their fit.

    Anyway, based on your explanation, it seems that ΔE ≈ 1 / g0rD refers to a very local density of states and only deals with two specific hopping points and shouldn't be used as a general statement over the bulk of the material. Generally speaking is there a large difference in ΔE between various different hopping sites over the bulk of the material? Or can one value of ΔE be a good approximation of all the other ones? I guess there would be differences depending on the material and even the sample, but maybe the range is small no matter what?

    I guess I'm just looking for a convenient general equation that I can plug into the plotting software, monkey around with some parameters, and fit the curve. It looks like it's not going to be that easy.
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