# Fermi energy (in semiconducors) vs. chemical potential

• itaischles
The Fermi energy is a concept introduced in a different course, and it is not the same as the chemical potential. The Fermi energy is the state of the highest energy electrons in a system. It is not a temperature dependent concept.

#### itaischles

Hi all,

In the solid state physics course I took a year ago we used the chemical potential μ which appeared inside the fermi-dirac distribution function to describe the energy that above it no electrons resides and below it they all reside as the temperature reaches 0 kelvin.
Now, when I started the semi-conductors course this year, we learned about the "Fermi energy" which was kind of defined the same way as the chemical potential (it even had the same place in the equation of the fermi-dirac function). In this course they told us that when you have donor atoms, and the temperature reached 0 kelvin the fermi energy sits in the middle between the bottom of the conduction band and the donors' energy level.
Now, I don't get it: why isn't the fermi energy sits at the donor's level? when the temperature reaches 0 kelvin, the semiconductor is in "freezeout" and all the donors' electrons are sitting in the donors' energy levels and are the highest energy electrons so I would expect that this will be the boundary of the fermi-dirac distribution discontinuity.
Also, is there any difference between the chemical potential and the fermi energy? And can the chemical potential/fermi energy depend on the temperature or are they defined only at 0 kelvin?
I know it's a lot to ask but please try and help me understand this :)

Blue Painting
I just read up on a wiki, because I liked your question and I wanted to test if I could understand it. How does this sound? The wave properties of an electron manifest stationary states when they are bound, like the covalent bonds, but higher up in the conduction band electrons effectively overlap their wave functions and become free, so these are not quantised in the same way. For the Fermi layer, the Pauli exclusion principle means that adding electrons to these kinds of bound states forces them to occupy higher energy levels. Mixing two systems together would cause a redistribution of electrons, so you can see how that is a chemical/thermodynamic type of potential. The Fermi energy is the highest electron state added so these are the current potential state of those electrons, and apparently they have some dynamics even approaching zero so they are effectively thermally characterised.

The question has been asked frequently. Consult previous posts, or may be Chapter 28 in solid state physics by Ashcroft and Mermin.
In summary: referring to the chemical potential in semiconductors as "the Fermi level" is an unfortunate practice.