# Fermi level and band gap

hello to everyone. It is normally said that in semiconductors the Fermi level is half way between the valence and the conduction band. I have the following question on that: what does it mean that the Fermi level is in the band gap? It should mean that the maximum energy of an electron in the solid is in the forbidden zone?
And moreover: How can Fermi-Dirac distribution tell us the occupation number of a state when it does not even take into account the underlying crystal potential?
I have a lot of confusion on this point.
Thanks to all

Physics Monkey
Homework Helper
The notion of a Fermi level doesn't really make sense when the Fermi level would lie inside a gap. Better yet, it just depends on how you define the Fermi level. If your definition is the energy of the highest occupied state then the Fermi level would sit at the bottom of the gap by definition. This is clearly not the definition being used when one says the Fermi level lies in the middle of the gap.

In fact, what does lie in the middle of the gap is the chemical potential, and because the chemical potential of a free gas of electrons is equal to its Fermi level as traditionally defined, one says that the Fermi level of the semiconductor considered is at the center of the gap. However, let me emphasize again that it really just depends on how you define the Fermi level.

If one redefines the Fermi level from "energy of highest occupied state" to "chemical potential" then the Fermi level will lie in the gap in the semiconductor case considered and the Fermi level will sit at the energy of the highest occupied state in the free gas in agreement with the more familiar definition.

As an aside, the Fermi level as defined by the energy of the highest occupied state doesn't quite work at finite temperature because all states have at least some tiny probability of being occupied. This is related to the familiar statement that the Fermi surface of a metal gets blurred on the scale of T (temp) at finite T. Of course, the difference between a mathematically sharp zero temperature Fermi surface and one blurred by temperature effects can be a minor one from a physical point of view.

This may be going to far, but in an interacting Fermi liquid the Fermi energy is also not equal to the energy of the highest occupied state even at zero temperature. Instead, the Fermi energy is defined in terms of a discontinuity in momentum space occupation.

Physics Monkey
Homework Helper
The Fermi-Dirac function gives you the distribution for any fermionic energy level. You should think of it just as a consequence of thermal equilibrium (which is why its so universal). But you're quite right, the crystal structure must enter somehow. The crystal structure enters by telling you what energies to put into the Fermi function. In other words, the crystal structure determines the energy levels and the density of states, and the Fermi function tells you how these levels are occupied.

I do want to emphasize that you're right to be suspicious, the fact that the thermal distribution (Fermi function) depends only on the energy (and not on the momentum, for example) is a non-trivial statement. It comes from assumptions about the nature of thermal equilibrium and need not always be true. Nevertheless, for most applications you are safe using the standard rule.

The notion of a Fermi level doesn't really make sense when the Fermi level would lie inside a gap. Better yet, it just depends on how you define the Fermi level. If your definition is the energy of the highest occupied state then the Fermi level would sit at the bottom of the gap by definition. This is clearly not the definition being used when one says the Fermi level lies in the middle of the gap.

In fact, what does lie in the middle of the gap is the chemical potential, and because the chemical potential of a free gas of electrons is equal to its Fermi level as traditionally defined, one says that the Fermi level of the semiconductor considered is at the center of the gap. However, let me emphasize again that it really just depends on how you define the Fermi level.

If one redefines the Fermi level from "energy of highest occupied state" to "chemical potential" then the Fermi level will lie in the gap in the semiconductor case considered and the Fermi level will sit at the energy of the highest occupied state in the free gas in agreement with the more familiar definition.

As an aside, the Fermi level as defined by the energy of the highest occupied state doesn't quite work at finite temperature because all states have at least some tiny probability of being occupied. This is related to the familiar statement that the Fermi surface of a metal gets blurred on the scale of T (temp) at finite T. Of course, the difference between a mathematically sharp zero temperature Fermi surface and one blurred by temperature effects can be a minor one from a physical point of view.

This may be going to far, but in an interacting Fermi liquid the Fermi energy is also not equal to the energy of the highest occupied state even at zero temperature. Instead, the Fermi energy is defined in terms of a discontinuity in momentum space occupation.

Ok this is MUCH more clear to me on general grounds, thank you a lot, really. But, if I think about the chemical potential I have in mind that it is the energy required to add or remove a particle from the system. Now I have a further question: what does it mean to add an electron to a semiconductor? Does it mean to add a delocalized charge to the system? And how does that matches to the fact that the energy cost should be equal to our precise value of the chemical potential? Thanks a lot for your patience

Ok this is MUCH more clear to me on general grounds, thank you a lot, really. But, if I think about the chemical potential I have in mind that it is the energy required to add or remove a particle from the system. Now I have a further question: what does it mean to add an electron to a semiconductor? Does it mean to add a delocalized charge to the system? And how does that matches to the fact that the energy cost should be equal to our precise value of the chemical potential? Thanks a lot for your patience

Sorry, I quote myself but it is just to say that I have found an article that explains many of my doubts related to this subject. To whoever is also interested this article is American Journal of Physics -- May 2004 -- Volume 72, Issue 5, pp. 676-678

the fermi level in the midst of valancy band and conduction band indicates equal concentrations of electrons in conduction band and holes in valancy band.And at "0" kelvin the electrons reach upto that level only and they can't exceed it.