Can someone simply describe the definition of the chemical potential?

Welcome to the forums sinayu71,

If you would permit me, could I ask what your level of education is? Why do you need to know it and why isn't it in your lecture notes?

 

The way I always interpret the chemical potential is by looking at temperature.

When two systems are equal in temperature, we say that they are in thermal equilibrium. This means there is no net energy flow from one system to the other (there can be an energy exchange, but the amount of energy of both systems is the same).

The chemical potential plays, in some sense, the same role as temperature does, but now for the exchange of particles. This means if the chemical potentials of two systems differs, then there will be a net flow of particles from one system to the other.

This means that for some big system the chemical potential must be the same everywhere (just as temperature), or else the system isn't in equilibrium and there would be some particle flow throughout the system.

There's another side to this story: In the Grand Canonical Ensemble you usually picture the system you're observing to be in contact with some bigger reservoir. The reservoir has an infinite amount of energy and particles. The reservoir also has a temperature T and a chemical potential u. If your system has a lower chemical potential, then particles would flow from the reservoir to the system. The system is therefore in equilibrium if it has the same chemical potential as the reservoir. (same goes for temperature and energy)

With this interpretation you can then also say that u is the minimal amount of energy a particle must have, when you add it to the system. Because if the energy of the particle would be less then u, then that "spot" in the system would already have been filled up by a particle of the reservoir.

I don't know if it's the best way to describe it, but for me it's the most intuitive.

(Note that most of the time we don't even have a reservoir of particles, because the number of particles is simply fixed, and therefore we can't even really assign a chemical potential to the system. We need to be able to vary the number of particles for that. But still we work in this Grand Canonical Ensemble, and "pretend" as if the system is in contact with some reservoir. The chemical potential is then fixed by the expectation value of the number of particles).

 

The previous post gives the classic meaning of chemical potential, as used directly in thermodynamics and chemistry. To apply it to semiconductors, you need to know about electron energy levels and bands. Electrons in an atom fill up the available energy levels starting from the lowest. When atoms are packed into a solid crystal, with a density of about 10^22 atoms per cc, atoms' discrete energy levels smear into bands of continuous levels. In a metal, electrons fill the available levels from the lowest, such that the topmost band is partially filled. The topmost occupied level (at low temperature, so there's no thermal agitation) is called the Fermi energy or Fermi level. Since there are plenty of empty levels nearby, an applied electric field can readily free the topmost electrons to provide electrical conduction.

In a semiconductor, the topmost band is completely filled and there is an energy gap (no occupation allowed) until the next higher band, called the conduction band, begins. At low temperature, then, the core bands are filled and the conduction band is empty, with a gap between. There's no conduction. At room temperature, however, thermal agitation can boost some electrons over the gap into the conduction band, leaving behind unfilled levels or "holes". The chemical potential is the energy at the center of the gap, at which the aggregate electron and hole energies are equal. This makes sense in terms of the previous discussion--it describes the energy at which the numbers of carriers (electrons and holes) on either side of the barrier or gap are in equilibrium. The chemical potential in semiconductors is also commonly called the Fermi energy, although technically that was defined for metals.

 

The enthalpy contribution (in the mixing and pure component terms) to Chemical Potential is easy to define. What is really hard to define for Chemical Potential is the Entropy contribution. Entropy is one of the hardest quantities to define in the physical sciences today. Most physics people will tell you that entropy is a measure of the "randomness" of the system, but that is a purely statistical approach which clouds our true understanding of this important quantity. Hope this helps.

Best Regards
Modey3

 

I'm confused by your post. Chemical potential is not the same as energy, so what are the enthalpy and entropy contributions to chemical potential? What role do they play in semiconductors, which is what the original poster asked about?

 

Well unless there are two definitions of chemical potential it is defined in only one way. Any book on thermodynamics defines chemical potential as the Gibbs Free Energy change per atom (or mole) at constant Temperature and Pressure. Since Gibbs Free Energy is a function of enthalpy and entropy chemical potential must also be a function enthalpy (H) and entropy (S). H and S can then be broken down into their pure material and mixing contributions. Chemical potential applies to all matter.

 

Actually the chemical potential can be defined in terms of the total internal energy, the Helmholtz free energy or the Gibbs free energy, depending on what quantities are held constant.

How does this help explain, in a simple way, the Fermi level of a bulk semiconductor?

 

"Actually the chemical potential can be defined in terms of the total internal energy, the Helmholtz free energy or the Gibbs free energy, depending on what quantities are held constant."

Yes, Yes, Yes. There are different ways of defining it for specific case (ie. for Helmholtz energy the system is isolated) , but the most general definition is the Gibbs Free Energy.

"How does this help explain, in a simple way, the Fermi level of a bulk semiconductor?"

I don't think that was the original question. sinayu71 wanted to know how chemical potential related to semiconductors. Since it was a general question I gave him a general answer.