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

enter image description here

(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?

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    $\begingroup$ 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. $\endgroup$
    – Jon Custer
    Feb 3, 2022 at 18:32
  • $\begingroup$ @JonCuster Then you disagree with [physics.stackexchange.com/a/318191/290970]? Do you have arguments for your position? References? $\endgroup$ Feb 4, 2022 at 4:30
  • $\begingroup$ 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. $\endgroup$
    – Jon Custer
    Feb 4, 2022 at 18:48
  • $\begingroup$ @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. $\endgroup$ Feb 4, 2022 at 19:31

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The net current in a semiconductor in equilibrium is zero, just as in a metal. What might be misleading here is that p-n junctions are often discussed in terms of two currents: the diffusion current which tends to level the carrier concentration and the drift current that tends to screen the potential of the junction. However, in equilibrium the two currents add up to zero.

The reason why the p-n junctions are discussed in this way is because we are interested in their response when they are driven out of equilibrium and a current flows - in this case the two currents do not balance each other anymore - this is, e.g., how Shockley diode equation is derived.

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  • $\begingroup$ This does not answer my question. I have taken as an assumption that the system has reached equilibrium and that the net current (density) is zero. However, it seems there might be, as there is in a semiconductor in equilibrium, a diffusion "current" with an equal and opposite drift "current" (giving a net 0 current). However, the drift current requires an electric field within the conductor. $\endgroup$ Feb 4, 2022 at 15:28
  • $\begingroup$ @MathKeepsMeBusy Perhaps I see better now what you mean: you are bothered by the fact that the field inside a semiconductor jucntion is not fully screened? Still, note that we are talking about a junction, not about a bulk semiconductor. $\endgroup$
    – Roger V.
    Feb 4, 2022 at 15:33
  • $\begingroup$ I am aware of how drift and diffusion "currents" behave in semiconductors. I am interested in conductors. I believe that if there is a charge gradient in a conductor, then in equilibrium, then a diffusion "current" will occur. And, because the system is in equilibrium, an equal and opposite drift "current" will occur. But the drift current can happen only if there is an electric field. $\endgroup$ Feb 4, 2022 at 15:38
  • $\begingroup$ @MathKeepsMeBusy you are comparing apples and oranges: on the one hand you have a junction with limited charge density, which is connected to infinite bulk p and n semiconductors; on the other you have a finite conductor with huge charge density, where surface charge screens everything - that charge is on the surface is an important part of that argument. $\endgroup$
    – Roger V.
    Feb 4, 2022 at 15:57
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    $\begingroup$ Thank you. The Wikipedia article only discusses Debye length in plasmas, electrolytes, semiconductors, but not conductors. However tf.uni-kiel.de/matwis/amat/elmat_en/kap_2/backbone/r2_4_2.html discusses the Debye length in conductors as well. This seems to be precisely what I had in mind. $\endgroup$ Feb 4, 2022 at 16:35

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