# Mott VR Hopping Calculation Question

## Main Question or Discussion Point

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?

Related Atomic and Condensed Matter News on Phys.org
Physics Monkey
Homework Helper
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.

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.