# Fermi level and chemical potential in doped and pure semiconductors

Recently I was studying about band theory of semiconductors and I have some questions.

I found two definitions of Fermi level

$$1)$$ The quantum state which has a probability of occupancy of $$0.5$$

(from my book)

$$2)$$ The Fermi level of a body is the thermodynamic work required to add one electron to the body

(from Wikipedia)

Everywhere the Fermi level of a semiconductor is talked about, its done on the basis of the $$1$$st definition and the Fermi-Dirac probability distribution.

My question is if the Fermi level of a semiconductor can be defined in terms of the $$2$$nd definition (also known as the chemical potential of the system) or does this definition fail when talking about semiconductors because in semiconductors (both doped and pure) the Fermi level lies in the band-gap and that confuses me because how can an electron in the highest occupied state in valence band (please assume at $$0$$K) have energy corresponding to somewhere in the band gap.

My second doubt is that in an extrinsic semiconductor, (both p-type and n-type), the extrinsic Fermi level is also defined in terms of the 1st definition everywhere.

(Please note I am assuming at $$0$$K)

But if we consider the $$2$$nd definition using the chemical potential, shouldn't the Fermi level (in a p-type doped) rise above the intrinsic Fermi level instead of being somewhere in the middle of the acceptor and the valence band? Because in the case of doped semiconductors, no matter p-type or n-type, we're adding dopant atoms into the system which now has more atoms, and hence more electrons in addition to that of the electrons in the valence band of the intrinsic semiconductor so we are adding more energy to the system in the process of adding those additional electrons of the dopant and it should shift the Fermi level above the middle (intrinsic level).

But it doesn't happen since Fermi level of a p-type doped semiconductor lies far below the intrinsic level near the valence band(somewhere in the middle of the acceptor and the valence band, to be precise) so it's sure that I'm missing something.

Why does nobody want to talk in terms of chemical potential?

I found some other posts some of which are this and this. I also went through them but I think no one actually addresses this.

Please try to answer in terms of chemical potential rather than redefining Fermi level in terms of the Fermi-Dirac probability distribution. That will be very helpful to me.

Thank you very much :)

You're right that this can be a tricky point if there is a gap, and I think that a number of common definitions fail in this case. It gets even trickier at zero temperature.

Let me give an alternative definition: the chemical potential (Fermi level) $$\mu$$ is a normalization constant that makes the following equation true:

$$N = \int \mathrm{d}E \, g\left(E\right) \frac{1}{e^{\frac{E-\mu}{k_B T}}+1},$$

where $$N$$ is the number of particles in your system and $$g\left(E\right)$$ is the density of states.

If the system has a gap, at zero temperature, $$\mu$$ need not be uniquely defined. At zero temperature

$$N = \int \mathrm{d}E \, g\left(E\right) H\left(\mu-E\right),$$

where $$H$$ is the step function.

Since $$g\left(E\right) = 0$$ for $$E$$ in the gap, it doesn't matter what value $$H$$ has in the gap. So, if $$\mu$$ falls somewhere in the gap, it could just as well fall anywhere else in the gap.

Note that this peculiarity only exists at zero temperature! At any other temperature, the definition will give you a well defined $$\mu$$. So, I wouldn't get hung up on the zero temperature case. Note that, in simple cases, the above definition will agree with the ones you listed.

• +1 Thanks for the answer very much. Could you please expand a bit on if it's possible to define chemical potential(Fermi level) in terms of the energy of the electron in the highest occupied quantum state(at any other temperature)?I mean is the Fermi level really related to the energy of the electron in any way other than probability(at any temperature you may like)? How does an electron in the valence band (HOMO) have an energy which corresponds to a point in the band gap? Oct 13 '19 at 4:42
• It's a little hard to say what the highest occupied state is at non-zero temperature; if the temperature is non-zero, then any state should have some probability of being occupied, with probability $\left(e^{\frac{E-\mu}{k_B T}}+1\right)^{-1}$. Since the levels are effectively continuous in many semiconductor systems (unlike in atoms or molecules), it's hard to draw a cut-off to say which levels are unoccupied. So, I'm not sure how to interpret your first question. Oct 13 '19 at 13:00
• For your second question, yes, I believe that the chemical potential is still the energy needed to add another electron to the system ($\mu = \frac{dE}{dN}$). That's fairly easy to show mathematically with Maxwell-Boltzmann statistics (I think that I've posted it to stack exchange before). It should also be doable for Fermi-Dirac statistics, but I haven't done the math before. Oct 13 '19 at 13:06
• For your third question, are you asking how it can be that the chemical potential is in the gap if you'd be adding an electron to a not-fully-occupied state below the gap? Oct 13 '19 at 13:09
• Ok, I think that I understand the question. Let me start with a question: if the chemical potential is in the band gap, why do you say that the new electron has to go in the valence band? If the system is at non-zero temperature, the added electron kind of ends up “everywhere”, in an average sense: The probability of any state being occupied increases; including states in the valence and conduction band. The chemical potential is kind of a weighted average over all the possible states the electron could be added to, and that average can fall in the gap, even if the electron can’t. Oct 13 '19 at 17:10

For ordinary semiconductors there is not much of a problem. The real problem occurs with compounds where correlation is important.

For example in free atoms there is a difference between adding an electron (the electron affinity) and removing an electron (the ionization energy). Same with molecules: the HOMO (highest occupied molecular orbital) and the LUMO (lowest unoccupied molecular orbital). It plays a role in correlated oxides like NiO.

In semiconductors like silicon it does not really matter. There will always be surface states that are partially occupied. States that are in the gap of the band structure of the bulk (dangling bonds etc) that will pin the Fermi level.

And although Fermi level is only defined at 0 K, in semiconductor terminology it is also used at higher temperatures where the correct term would be chemical potential. So while doping will put the low-temperature Fermi level at a band edge, raising the temperature high enough will put it near the middle of the gap, when the concentration of thermally generated holes and electrons is comparable.

• +1 Thanks a lot for the answer. So what does Fermi level mean here? Can we make sense of it thinking it as chemical potential in a semiconductor (both pure and doped) instead of the notion of probability occupation? Oct 12 '19 at 19:37
• @user8718165 I expanded a bit. I am not sure if it answers the question.
– user137289
Oct 12 '19 at 22:23
• thank you very much for answering my question. Both the answers have helped me a lot. Oct 15 '19 at 4:27