This writeup is about semiconductor physics. If you have no idea what it means, you could read my writeup in that node. The nodes conductivity, Fick's Law, diffusion, and p-n junction will also help. While it will be difficult to follow my derivation without a background in semiconductor physics, I think the result itself can be understood if one reads those nodes I mentioned above. If, upon reading those nodes, this writeup still makes absolutely no sense, let me know.
The Einstein relation is an important equation that relates the diffusion constants of electrons and holes to their mobilities. This writeup presents an extremely common derivation of the Einstein relation. Since mobility (the ratio of drift velocity to electric field) is fairly easy to characterize (see conductivity), the Einstein relation completely characterizes the diffusion constant.
Things to note:
- While the Einstein relation is true in both equilibrium and nonequilibrium conditions, the ubiquitous derivation I've seen assumes equilibrium. I don't think it would be terribly difficult to extend it to nonequilibrium situations, but I have never considered a proof. A simple argument is the following. For typical values of applied electric field, mobility can be shown, both experimentally and theoretically, to be constant (though this breaks down in the velocity saturation regime). The same is true for the diffusion constant. Since both constants are independent of bias, their ratio should be as well.
- This is Einstein's only contribution to solid state physics and semiconductor physics of which I am aware. I have no idea why he was interested in this area.
- The Einstein relation assumes that Boltzmann statistics hold. While technically electrons obey Fermi-Dirac statistics, in a semiconductor, by great fortune, the "Boltzmann approximation" is accurate unless it is doped extremely heavily.
Derivation
In equilibrium, both electron current density and hole current density must be zero, by definition. I will derive the Einstein relation for electrons--the derivation for holes is identical. I will work in one dimension--it is straightforward to extend to three.
In the following, J is electron current density, q is electron charge, μ is mobility, E is the electric field, D is the diffusion constant, n is electron concentration (in cm-3), Ec(x) is the energy at the edge of the conduction band, Ef is the Fermi level, k is Boltzmann's constant, T is temperature, and N is a constant called the conduction band effective density of states.
J = Jdrift + Jdiffusion = 0
qμnE + qD(dn/dx) = 0
D/μ = -nE/(dn/dx)
From Boltzmann statistics, n = Ne(Ec(x)-Ef)/kT. Simple differentiation yields n/(dn/dx) = kT/(dEc(x)/dx). Furthermore, the electric field E = -(dEc(x)/dx)/q, since it is the negative gradient of potential.
I used the well-known fact that in equilibrium, almost by definition, the Fermi level is constant. Of course, T is also constant under equilibrium conditions.
We arrive at the magic formula:
D/μ = kT/q
It's easy to remember because it rhymes, even if you invert both fractions! This kind of analysis is important when dealing with semiconductor junctions like the p-n junction. In the p-n junction, there is a quite large built-in electric field (and a large drift current). Furthermore, there is a large diffusion current due to holes diffusing from the p-side to the n-side and electrons diffusing from the n-side to the p-side. In such junctions, diffusion cannot be neglected. Diffusion, not drift, accounts for the exponential forward-bias current in a p-n junction.