I saw in a paper that the current in a metal is sigma (conductivity=e*mu*n) multiplied by dk/dx where k is the electrochemical energy (or fermi energy).
It seems to me that it is some kind of a generalized ohm's law (j=sigma*E) where the electric field is 1/e*dk/dx.
My questions are these:
# Is it truely a generalized ohm's law or is it comes from other more fundamental law?
# Is this law valid for semiconductors and/or outside equilibrium (steady state, external applied voltage)?
# I couldn't find anything on this equation and I'll be grateful if someone could direct me to some books referring this equation.
Hi, haven't come across the equation in this form but you are right:
i) energy W is the force F=eE integrated along a line, so dW/dx=F=eE. You will just have to be careful with the direction of x.
ii) I think this can only work for the special case of heavily n-doped semiconductors, where the hole conductivity p can be neglected (so your equation for the conductivity holds). Bias should not be a problem as this creates another field that superimposes with the above and effectively shifts the Fermi energy.
This is actually a very interesting question and has analogs in kinetic theory and statistical physics. I would look at the wikipedia page on the Einstein relation: https://en.wikipedia.org/wiki/Einstein_relation_(kinetic_theory)
The proof does a good job of explaining what is going on.
Essentially, to summarize: there is a detailed balance between drift and diffusion currents under equilibrium. Under non-equilibrium steady state, this can be used to derive how the current is related to potential gradients.
The more fundamental version of this is called the fluctuation dissipation theorem.
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