Fermi Level of Electrons and Holes in Equilibrium and Non-Equilibrium

[dielectrics]

Hi all,
Why do the fermi level for electrons and holes coincide in equilibrium condition and why they separate as quasi fermi levels in non equilibrium situation?

 

[Darwin123]

Hi all,
Why do the fermi level for electrons and holes coincide in equilibrium condition and why they separate as quasi fermi levels in non equilibrium situation?

First, I need to clarify the nomenclature a bit. Nomenclature was discussed in a separate thread some time ago. However, allow me to repeat the general conclusions. This will help make my answer clearer (?).

What you are referring to as the "Fermi-level" probably is the "chemical potential". Some textbooks use the phrases "Fermi-level" and "chemical potential" interchangeably. However, some textbooks make an important distinction between the two.

For some authors, the two are not the same. "Fermi-level" is sometimes defined as the highest occupied energy-level of fermions in thermal equilibrium. The "chemical-potential" is defined as that threshold parameter in the Fermi-Dirac function. Defined this way, the Fermi-level is not always the same as a chemical potential. In many situations, they happen to be the same. However, I think it will be useful in answering your question to distinguish between the two.

Furthermore, the "electrons" in your question are not true electrons. Furthermore, the holes in your question are not positrons.

Conduction-electrons and valence-holes are not fundamental particles. Conduction-electrons and valence-holes are actually quasiparticles. Both quasiparticles are mixed states of electrons in the crystal.

The usual schematics showing the conduction band and the valence band are showing energy levels of electrons. The important electrons in a semiconductor crystal can be divided into two groups which I will refer to as the conduction-electrons and the valence-electrons.

Please understand that there is only one electron gas in the crystal. The conduction-electrons consist of electrons in a conduction band of energy states. The valence-electrons are electrons in the valence band of energy states. Holes are defined by replacing the momenta and energies of the valence-electrons by their negative values.

Replacing the phrase "conduction-electrons" with electrons is a short-cut which can be misleading. Defining holes by the negative of "valence-electrons" is also a short-cut which can be misleading.

The following conventions will be used in this post. I will use the full phrases "conduction-electron" and "valence-electrons" in my answer. I will use the phrase "chemical-potential" instead of Fermi-level. If your textbook uses another convention, then I hope you will adjust my answer to that convention.

The electrons in a semiconductor crystal form a Fermi-gas. If all the electrons are in thermal equilibrium, then the occupancy number of the electrons has to be equal to a Fermi-Dirac function. The Fermi-Dirac function has only one temperature and one chemical-potential. Obviously, the electrons in thermal equilibrium can not have two temperatures or two chemical-potentials.


The electrons in a semiconductor may not be in complete thermal equilibrium. The electrons may be in a steady-state conditions where energy is entering and leaving the crystal at the same rate. However, the some electrons may be incapable of interacting with the other electrons.

Suppose one has a steady-state condition where the conduction-electrons and the valence-electrons are not in thermal equilibrium. Suppose that all the conduction-electrons are in thermal equilibrium with each other. Suppose the valence-electrons are in thermal equilibrium with each other. However, the interaction between conduction-electrons and valence-electrons is so weak that the two bands are not in thermal equilibrium.

One could in that situation have a one temperature and one chemical-potential for the conduction-electrons. One could in that situation have another temperature and another chemical-potential for the valence-electrons.

The temperature of the conduction-electrons and the valence-electrons are usually the same because they scatter off of each other. Therefore, in most steady-state conditions that aren't a complete thermal equilibrium only the chemical-potentials are different. The chemical-potential for conduction-electrons is different from the chemical-potential of valence-electrons. Their temperatures are the same.

There may be no electrons located at either chemical-potential. Therefore, there may be a little confusion using the words "Fermi-energy" instead of chemical-potential. The phrase "Ferm-level" may also be ambiguous in a high-resistivity semiconductor in thermal equilibrium. However, some textbooks still use the two phrases interchangeably.

Just to make the situation more confusing, the electron temperature is often higher than the lattice (i.e., nucleii) temperature. You may ask a question about this later.

 

[dielectrics]

Thank you very much for the help!