# Why are electron and hole currents zero at thermal equilibrium?

• Tony
In summary, it is claimed that at thermal equilibrium, we can take J_e and J_h to be identically zero. As I understand it, the requirement of T.E. allows us to state that the derivatives w.r.t. time are zero. However, I'm not certain as to why we need to mandate that the currents are zero.
Tony
Hi, I'm currently studying in an introductory semiconductor course where we use the following equations (numbers 1-5 on the first page):

http://web.mit.edu/kimt/www/6.012/TheFiveEquations.pdf

as a model of the underlying physics.

Now, it is claimed that at thermal equilibrium, we can take J_e and J_h to be identically zero. As I understand it, the requirement of T.E. allows us to state that the derivatives w.r.t. time are zero. However, I'm not certain as to why we need to mandate that the currents are zero.

Of course there's the intuitive notion that if there are currents, then things are moving and thus are not "at equilibrium"; but what I'm looking for is an algebraic derivation from the physical equations that restrict the current to be zero. However, I have not had too much success so far, and most resources just seem to assume that J_i = 0 at thermal equilibrium as a trivial result.

Can anyone give me some suggestions on how to begin?

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Here's a justification approach: Any flux $J$ (energy, matter, momentum, etc.) can be related to the gradient of a generalized potential:

$$J\propto-\nabla\Phi$$

For charge carriers, we want to use the electrochemical potential:

$$\Phi=\mu+q\phi$$

where $\mu$ is the chemical potential (related to concentration), $q$ is the carrier charge, and $\phi$ is the electric field. Now, if we assume that the electric field is zero and the carrier concentration is uniform, then the flux must be zero. Does this sound reasonable?

Actually, I'm not sure I'm convinced with that argument (if I understand you correctly).

We use the claim that the flux is zero in order to obtain a relation between carrier concentration and the electric field. (Equations 1 and 2 in the pdf.) In particular, we use this to analyze the case of an abrupt p-n junction where at equilibrium neither the concentration nor the electric potential are uniform, but the current densities are zero.

This is identical (I think) to your formalism, where the conclusion is that $$\Phi$$ may be uniform while each of its components are not. Given this example I'm not so sure about the generality of your argument.

Well, another way of looking at it is, if the electron flux isn't zero at thermal equilibrium, why aren't all the electrons piled up on one side of the sample? That just isn't experimentally observed.

One could argue that electrons and holes are forming on the left side of a sample, diffusing, and recombining at the right side. But you could just as well argue that they're diffusing similarly towards the left side. Thermal generation and recombination do occur, but the diffusion process is undirected, and thus the electron and hole fluxes are zero.

I realize these are "softer" arguments than you're looking for. But the fact is that all diffusion relations are phenomenological. Fick's Law, whose general form I wrote above, is phenomenological.

The only other approach I can think is thermodynamical: the coordinated motion of carriers (a non-zero flux) in the absence of a driving force would decrease the entropy of the system, which is prohibited by the Second Law.

Tony said:
Now, it is claimed that at thermal equilibrium, we can take J_e and J_h to be identically zero. As I understand it, the requirement of T.E. allows us to state that the derivatives w.r.t. time are zero. However, I'm not certain as to why we need to mandate that the currents are zero.

Look at the pn-junction : at thermal equilibrium, both of the Fermi levels are aligned. So the Fermi level of the p type semiconductor is aligned with that one of the n type semiconductor. This implies that it's going to cost energy to :

1)get electrons from the n side to the p side
2)get holes from the p side to the n side.

So, without providing that energy, ie by a forward bias, there will be no current because of the potential difference over the junction.

More here :
http://hyperphysics.phy-astr.gsu.edu/hbase/solids/pnjun2.html#c1

marlon

## 1. Why are electron and hole currents zero at thermal equilibrium?

At thermal equilibrium, the number of electrons and holes in a semiconductor material are in balance. This means that for every electron that moves from the conduction band to the valence band, creating a hole, there is also an electron moving from the valence band to the conduction band, filling the hole. As a result, the net flow of electrons and holes is zero, leading to zero current.

## 2. What is the role of the energy band gap in determining zero current at thermal equilibrium?

The energy band gap is the minimum amount of energy required for an electron to jump from the valence band to the conduction band in a semiconductor material. At thermal equilibrium, the energy levels of the electrons in the conduction band and holes in the valence band are in balance, making it difficult for any electrons to jump across the energy band gap. This results in zero current at thermal equilibrium.

## 3. How does temperature affect electron and hole currents in a semiconductor material?

As temperature increases, the number of electrons and holes in a semiconductor material also increases due to thermal excitation. This results in a higher probability for electrons to jump across the energy band gap, leading to a small but non-zero current. However, at thermal equilibrium, this increase in current is balanced by an equal number of electrons moving in the opposite direction, resulting in zero net current.

## 4. Can electron and hole currents be zero in non-thermal equilibrium conditions?

Yes, electron and hole currents can be zero in non-thermal equilibrium conditions if there is a balance between the number of electrons and holes in a semiconductor material. This can occur in certain situations, such as when an external electric field is applied, creating an equal number of electrons and holes moving in opposite directions and canceling out any net current.

## 5. How does the presence of impurities affect electron and hole currents in a semiconductor material?

The presence of impurities in a semiconductor material can create an imbalance between the number of electrons and holes, resulting in a non-zero current at thermal equilibrium. This is because impurities can introduce additional energy levels in the energy band gap, making it easier for electrons to jump across and creating a net flow of electrons and holes. This is the basis for the functioning of semiconductor devices such as diodes and transistors.

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