Mott VR Hopping Calculation Question

[mesogen]

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 / g₀r^D

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?

 

[Physics Monkey]

You can define the DOS as just $G(E) = \sum_{i} \delta(E-E_i)$ in which case the units are one over energy. However, in a large system, $G(E)$ diverges with system size, so it makes more sense to define the DOS per unit volume (sometimes also just called the DOS) $g(E) = G(E)/V$. 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 $\xi$, then the typical level spacing $\Delta E(\xi)$ of the states mixed by that operator would be roughly $\frac{1}{\Delta E(\xi)} = g(E) \xi^d$. 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.

 

[mesogen]

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 / g₀r^D 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.