What is the Fermi level in semiconductors?

[aaaa202]

Can someone give me a clear, nonambiguous definition of the fermi level in a semiconductor? Is it the energy of the highest occupied state, the chemical potential at T=0 or what?
I don't think it is the first since the fermi level is typically put midway between the conduction and valence band. On the other hand I don't see how this follows from the second definition. And if the second definition is correct, why is that you then typically use the fermi energy in the the fermi distribution function at T>0? Shouldn't you use the chemical potential?
I am a bit confused

 

[Henryk]

The electrons in solids (and everywhere) obey Fermi-Dirac statistic given by

The quantity denoted by μ is the chemical potential for electrons. It denotes the energy where the probability of the state being occupied is exactly 1/2. And it does not matter if there is an allowable energy level or not, the statistics is still valid and the chemical potential is well defined.
Strictly speaking, Fermi energy is the chemical potential at T = 0 but in practice, it is often used as the synonym for electron chemical potential.

 

[DrDu]

Yes, the Fermi level is the chemical potential at T=0. It is even in intrinsic semiconductors not half way between the conduction and valence band (this holds only true in one-dimensional semiconductors, but not in 2 or 3 dimensions). The reason is that the density of states increases in 2 or 3 dimensions with energy. This is also the reason why mu is temperature dependent.

 

[aaaa202]

Okay thanks. I have a new question. Say we have to different semiconductiors GaAs and InAs. For the separate know the electron affinities for the separate systems and from that and the respective band gaps we can find the fermi levels. Now we sandwich the two semiconductors as show on the figure. I am told that I should align the fermi levels. But how do I calculate this new common fermi level? Is it the average of the two old ones?

 

[Henryk]

When you bring two materials together, the electrons will flow from the material with higher chemical potential to the one with lower chemical potential adding electrostatic energy to the later and subtracting some from the former until the chemical potentials are equal. This will create a potential barrier at the interface up to a depth that depends on concentration of electrons and holes. Away from the junction, the chemical potential relative to the band edges will stay the same. Near the interface the potential varies and that adds up to the band energy. The overall energy diagram would look like this.

 

[aaaa202]

Yes. But what I still don't understand is. How do I calculate the fermi level for the new heterostructure?
Before I could calculate it for the separate systems simply assuming it was halfway between the valence and conduction bands. How do I calculate the new, aligned fermi level?

 

[DrDu]

No, it ist half way between the conduction and valence Band.

 

[aaaa202]

But what is halfway between the conduction and valence band for a heterostructure with varying conduction and valence band?

 

[DrDu]

I wanted to say it isn't half way. But anyhow. Its Position relative to the conduction and valence Band remains fixed. The potential shift is a matter oft pure classical electrostatics.

 

[aaaa202]

I still don't get it. For the two separate systems you can calculate the fermi level by assuming it is halfway between the conduction and valence band. Now you merge the two systems to get one with a varying conduction and valence bond as shown on my drawing. The fermi level for the new system is a number, which I want to calculate. How do I do that? Before for the separate system the fermi level was a number I could calculate. How do I calculate it for the heterostructure? Is my question not clear? :( :(

 

[Henryk]

aaaa202,
When you say you can calculate the Fermi level for separate system, what do you mean? Calculate absolute value or relative to the band edge?
In practice, the Fermi level is usually calculated with respect to conduction band bottom or top of the valence band and for a homogenous semiconductor. It usually depends on the material and the concentration of impurities.
Calculating Fermi level relative to, say, vacuum outside the material is a totally different story. The value of the Fermi level (relative to the vacuum outside the material) is equal to - work function. In practice, work function is very difficult quantity to measure accurately. For one, the structure of any material near the surface is not exactly the same as in the bulk. Atoms at the surface experience on average half the interaction compared to what is inside the material. On top of that, there is usually a surface layer chemisorbed or physisorbed to the surface with unpredictable structure, dipole moment, etc. Plus semiconductors have surface states that can accumulate charges.
Coming to the your question. My answer is that away from the interface, the Fermi level is what you would expect in the bulk material for both materials of the heterostructure. Near the junction, bands are bend due to an extra electrostatic energy added to carriers. The total amount of band shift is equal to difference in work functions of the material plus any effects of permanent charges at the interface. The shape depends on carrier density in both materials.

 

[Useful nucleus]

Before for the separate system the fermi level was a number I could calculate. How do I calculate it for the heterostructure?

This requires solving Poisson equation for the electrostatic potential coupled with defect diffusion equation. Approximate solutions can be found in many texts on semiconductor devices.

Also notice that assuming that Fermi level lies in the middle of the band gap is a textbook idealization. In fact if you simply consider the difference in dispersion of the valence band and conduction band you will realize that mid-gap Fermi level is an idealization.
For real materials such as GaAs and InAs , the Fermi level will vary depending on temperature, species chemical potentials (Ga, As, In, ...), and of course extrinsic intentional or unintentional doping.

 

[DrDu]

If the charge double layers are thin in comparison with the thickness of the sheets of the two semiconductors, then inside the bulk of both semiconductors, the Fermi level will be at the same distance from the conduction or valence band as it has in the isolated semiconductor. The absolute value of the Fermi level is usually of little relevance and depends on the detailed probe geometry, even for a single semiconductor.