Density of States and Temperature

[chill_factor]

The formula for density of states in a free electron gas is $g(E) = (3/2) (n/E_{F})\sqrt{E/E_F}$. However, this looks like it has no direct dependence on temperature. It seems that only the probability of electron occupation of a state changes with temperature, not the number of states itself.

Am I correct or am I missing something?

 

[Darwin123]

The formula for density of states in a free electron gas is $g(E) = (3/2) (n/E_{F})\sqrt{E/E_F}$. However, this looks like it has no direct dependence on temperature. It seems that only the probability of electron occupation of a state changes with temperature, not the number of states itself.

Am I correct or am I missing something?

1) This is an expression for the density of states near the Fermi-level, not a general expression for density of states.

-The electrons that are not near the Fermi-level are important in some phenomenon. The expression that you presented is a good approximation when calculating electrical conductivity in metals. However, you may want to use a more accurate expression for a high-resistivity semiconductor.

-The electrons with kinetic energies high above or way below the Fermi-energy will have a different density of states.

-The general equation for density of states does not depend on temperature.

2) If the electron gas is not degenerate, then the "Fermi-energy" can change with sample-temperature. According to your formula, this means the density of states will change, too.

-The expression for Fermi-energy that is usually given is specific to degenerate gases. A degenerate Fermi gas has a Fermi-energy that is determined by fermion density only. However, at high temperatures the Fermi gas won't be degenerate. Therefore, at high sample-temperatures the Fermi energy will vary with sample-temperature.

-You may want to use a more accurate expression for Ferm-energy in high temperature plasmas.

-The dark Fermi-level of an intrinsic semiconductor DOES NOT satisfy the metallic approximation for Fermi-level.

3) A nomograph for the temperature dependence of a Fermi level in a degenerate parabolic band is provided in the following reference.

"Optical Properties of Semiconductors" by Jaques I. Pankove (Dover, 1971) pages 414-415.

-The nomograph enables you to calculate the Fermi-energy even at high sample-temperatures.

-The usual approximation for the Fermi-energy is good enough for most purposes. However, the nomograph Fermi-level is more useful at either very high temperatures or very low carrier densities.

-This nomograph may be useful when you study semiconductors rather than metals.

 

[DrDu]

1)
2) If the electron gas is not degenerate, then the "Fermi-energy" can change with sample-temperature. According to your formula, this means the density of states will change, too.

You sure? I understood that the Fermi Energy always refers to zero temperature. What varies with temperature is the chemical potential (aka Fermi level) which tends to E_F for T to 0.
You also state that this isn't a general expression for the density of states. However the OP was referring to a gas of free electrons and for these I don't think there are deviations.

 

[Darwin123]

You sure? I understood that the Fermi Energy always refers to zero temperature. What varies with temperature is the chemical potential (aka Fermi level) which tends to E_F for T to 0.
You also state that this isn't a general expression for the density of states. However the OP was referring to a gas of free electrons and for these I don't think there are deviations.

The phrase "Fermi-energy" is often used ambiguously. Some authors use "Fermi-energy" interchangeably with "chemical-potential". I think this is the way the OP is using the term. After all, at zero degrees Kelvin the occupancy of electronic states is a simple step function. He is writing the density of states as a function of electron energy (E) which varies continuously. So I infer that what he is talking about is a condition that is slightly different from absolute zero. I may be wrong.

I know what you are talking about. Yet, I don't think what you are describing is a universal description.

I have read textbooks where the author uses the term "Fermi energy" where he really means "chemical-potential". I do not know whether the author's are formally correct, but these are often reputable science books. Two books which define a "Fermi-level" at a nonzero temperature are:

1) "Optical Processes in Semiconductors" by Jacques I. Pankove (Dover, 1971).

2) "Point Defects in Semiconductors II" by J. Bourgoin and M. Lannoo (Springer, 1983).

These are very good textbooks for scientists working in the field. Yet, they could be making a mistake. They don't really discriminate between chemical-potential and Fermi-level. I should also point out that they are not discussing true metals. Maybe semiconductor physicist have different conventions than metal physicists.

I admit to being a little confused by that point, myself. I would appreciate it if you could clarify the difference in meaning between "Fermi level" and "chemical potential".

 

[DrDu]

Yes, I am also often confused about that. But I think at least the symbol E_F is exclusively used for the Fermi Energy at zero absolute temperature.
I rather avoid to speak of Fermi level as, e.g. in a semi-conductor, the chemical potential may not coincide with an energy level when it falls into a gap.

 

[Useful nucleus]

I was always confused by this point until reading a very clear footnote in solid state physics by Ashcroft and Mermine (footnote #11 , chapter 28 Homogeneous Semiconductors).
The footnote addresses the question: Is Fermi Level a good terminology for semiconductors as it is for metals? The answer is no although Fermi Level is frequently used in practice. The reason is: Fermi Level is the level that separates the occupied from the unoccupied states. For a metal, this is a clear cut. For a semiconductor there is no unique level.
So essentially what people mean when they talk about Fermi Level in semiconductors is the chemical potential of electrons. An interesting thing to observe is that the chemical potential of electrons does not have to coincide with an "actual" electron level in the band gap. Basically it varies according to the thermodynamic conditions (T, chemical activities, doping, mechanical stress,...).

 

[Darwin123]

So essentially what people mean when they talk about Fermi Level in semiconductors is the chemical potential of electrons. An interesting thing to observe is that the chemical potential of electrons does not have to coincide with an "actual" electron level in the band gap. Basically it varies according to the thermodynamic conditions (T, chemical activities, doping, mechanical stress,...).

Based on what you just said, is it safe to say the following?

Chemical-potential and Fermi-level have the same value in a metal at absolute zero temperature.

If that is true, then I think that we can answer the OP's question as follows.

1) The equation that the OP gave is an approximation of the density of states that is accurate only in the limit of a metal near absolute zero for electrons close to the Fermi-level.

2) The density of states in the OP's formula formula varies with the kinetic energy of the electron, not on temperature.

3) The precise expression for density of states, which was not shown by the OP, does not explicitly vary with temperature.

4) I don't have the software or know how to insert equations in a post. However, a precise expression for density of state is given in the following reference.

"Optical Processes in Semiconductors" By J. Pankove (Dover, 1971) pages 6-7.



5) Dover textbooks are useful but inexpensive. I don't know if Dover still publishes textbooks. However, used copies of this book are still being sold.

6) The expression that the OP gave is an adequate approximation in a metal at room temperature for most electronic-engineering purposes.

7) The density-of-states expression that the OP gave is not a good approximation for an intrinsic or compensated semiconductor at any temperature.

 

[DrDu]

Based on what you just said, is it safe to say the following?

Chemical-potential and Fermi-level have the same value in a metal at absolute zero temperature.

Yes, that is correct.
I think all the points you gave are correct with the exception of 1.
The formula is correct for all energies as long as a gas of free electrons neglecting interaction is considered.

 

[Darwin123]

You sure? I understood that the Fermi Energy always refers to zero temperature. What varies with temperature is the chemical potential (aka Fermi level) which tends to E_F for T to 0.
You also state that this isn't a general expression for the density of states. However the OP was referring to a gas of free electrons and for these I don't think there are deviations.

I know, for sure, that Pankove uses the symbol E_F for chemical-potential. I also know that Pankove uses the phrase "Fermi-level" to refer to chemical-potential.


I am staring at the appropriate pages in Pankove right now.


This month, I am correcting an article that I submitted for publication. The referee has made statements indicating that he is confused by my use of the phrase "Fermi level." I was using the phrase the way Pankove uses it, to mean chemical-potential. However, the system that I am studying has two Fermi-levels, one for conduction-electrons and one for valence-holes. So there is a second issue as to which reference-energy to use when writing down the value of the Fermi-level/chemical-potential.

There is a third issue. "Valence-holes" are really quasiparticles made of "valence-electrons". The chemical-potential for the valence-electrons is opposite in sign. Further, the chemical potential of the conduction-electrons has an extra component in the band-gap.

With all three issues going on at once, I can see how the referee lost me. My ideas are perfectly clear to me with Pankove open in front of me. However, he probably doesn't have Pankove. I don't know what conventions he learned.

I think all these issues are making my article hard to read. So I will use the "internal definition" method.

I am going to try solving the problem by defining chemical-potential and Fermi-level separately within my article. The next manuscript will have internal-definitions of chemical-potential and Fermi-level.

I will write set down an article-limited convention as to what they are. In my definitions, chemical-potential and Fermi-level will have different values because they are measured from different references. However, these will be explicitly local definitions. I will tell the reader that I am choosing conventions specifically for this article.

I don't know if this will work but I don't know what else to do. It is all up to the referee.

 

[chill_factor]

Thank you guys greatly for clearing this up. I am used to using μ = E_F as a useful approximation, but do know that chemical potential is not, in general, the Fermi energy except at exactly 0 degrees.

What got me confused was someone asked me a question about the dependence of the density of states g(E) on the temperature for a free electron gas, which I found there was none and was baffled by the question; only the occupancy of states is determined by the temperature.