# Diffusion current and possible electric field within interior of conductor in no-net-current equilibrium

Assume we have a stationary conductor with an externally applied electric field.

(image from courses.lumenlearning.com)

Further assume that everywhere in our system, the electric field, the magnetic field, and the charge densities do not vary with time. That is, everywhere,

$$\frac{\partial \vec{E}}{\partial t} = \frac{\partial \vec{B}}{\partial t} = \frac{\partial \rho}{\partial t}=0$$

Also, assume further that the current density $$\vec{J}$$ everywhere within the conductor is $$0$$.

By the following simple argument from electrostatics, the electric field within the interior of the conductor must be exactly $$0$$. If it were not, so the argument goes, electrons would move under the influence of that electric field, and charge densities would change, violating our assumptions that the charge densities are time invariant, and that current is $$0$$.

However, in semiconductor physics, we are taught that charges not only drift due to electric fields, but also diffuse due to the combination of thermal motion and gradients in charge density.

One rarely sees any discussion of diffusion currents in conductors. I assume that they are frequently negligible. However, I am going to assume (in accordance with the answers to this question) that diffusion currents do occur in conductors as happens in semiconductors. Is that a bad assumption?

If it is true that diffusion currents occur in conductors as well as semiconductors, in the equilibrium situation of our thought experiment, if the charge densities are not uniform, there must be a diffusion "current" in the conductor. (I put "current" in quotes, because it is not a net current, but only a component of a net current).

If the net current density $$\vec{J}$$ is everywhere $$0$$, but there is a diffusion "current", there must also be an equal and opposite drift "current" to cancel the diffusion "current".

Finally, if there is a counter drift "current" within the interior of the conductor, there must be an electric field there as well to cause that drift "current" (even though it may be small enough to be generally ignored). Note that this result contradicts the presentation often given in accounts of electrostatics that the electric field within a conductor in equilibrium, and with no current, is exactly $$0$$.

My question is this? Is the result correct? Is there a (very small) electric field within the interior a conductor situated within an electric field, where the system is in electrical equilibrium and with zero current? If not, then which assumption or step(s) in my derivation are mistaken?

• There is no net diffusion of carriers in a uniform material. There is no net diffusion in a bulk semiconductor chunk. Diffusion becomes important when there are varying types and densities of carriers. Feb 3, 2022 at 18:32
• @JonCuster Then you disagree with [physics.stackexchange.com/a/318191/290970]? Do you have arguments for your position? References? Feb 4, 2022 at 4:30
• In equilibrium (or steady state) in a uniform material there can be no net diffusion of carriers - that would lead to a charge imbalance which contradicts being in equilibrium (or steady state). Note the answer you point to talks about effects at the junction of two different materials. Feb 4, 2022 at 18:48
• @JonCuster The external field creates an imbalance in the charge density. The imbalance in charge density causes diffusion. The diffusion current is balanced by an equal and opposite drift current, leaving no net current. No net current is OK for equilibriuim. Feb 4, 2022 at 19:31