How to understand chemical potential?

Question

I wonder how to understand the chemical potential in terms of the condensed matter physics. Let’s say without the thermodynamic terms.

Let’s consider some example system that can conduct and its electronic properties (e.g. conductivity) depends on a chemical potential. What interpretation has this chemical potential in such a case? Is it energy that electron has? And I also find sentence like “if the chemical potential is in the band gap”. What does it mean and how to intérprete this? And how we can change it? By adding additional electron in the system?

Answer 1

The chemical potential of any system can be thought of as the energy that can be absorbed or released due to a change of the number of electrons:

$dU = \mu dN$

In the case of the small systems where the discrete charge can be observed, this equation can be rewritten as a finite difference:

$\mu=\frac{dU}{dN}=\frac{U(N+1)-U(N)}{\Delta N =1}=U(N+1)-U(N)$.

So it is energy difference of the system with $N$ and $N+1$ electrons. Roughly speaking it is a measure of how "hard" or "easy" it is to add or remove a electron to/from the system.

This quantity is related to electronegativity as $E=-\mu$. It is also related to the ionization potential and electron affinity as their average: $\mu=-\frac{IP+EA}{2}$.

As for conductance, the chemical potential determines the center of the energy window where the conductance occurs. The width of the window is determined by temperature. The last ingredient that is needed to determine conductance is the density of states or the density of transmission modes in this energy window. This answer is inspired by the room-temperature Landauer–Büttiker formula. This is well applicable to conductors.

As for semiconductors, the location of chemical potential within the band gap, relative to the conduction and valance band edges, determines the concentration of thermally activated charge carriers (due to doping). The closer it gets to the band edges the smaller the energy of thermal activation is.

Answer 2

Let’s consider some example system that can conduct and its electronic properties (e.g. conductivity) depends on a chemical potential. What interpretation has this chemical potential in such a case?

Metals
Chemical potential is a parameter in the distribution function, which separates the filled states from the empty ones. In (isotropic) metals the filled states occupy a Fermi sphere, in which case the energy of the edge of the sphere (Fermi energy) is identical with the chemical potential (from where is the common confusion between $\mu$ and $E_F$.) The linear conductivity is determined by the electrons at the edge of the sphere, that is by the electrons at the chemical potential - since these can be excited from below the Fermi energy to above it by infinitesimally small electric field. Thus the density-of-states corresponding to the Fermi energy would determine the number of electrons excited and the size of the conductivity.

In mathematical terms the conductivity usually contains the derivative of the Fermi Function, which is delta-function at zero temperature, but is broadened at the finite temperature. Hence, electrons in the layer $\mu\pm k_B T$ are the ones contributing to the conductivity the most.

Semiconductors
The density-of-the-states corresponding to the chemical potential becomes crucial when dealing with semiconductors - since the DOS in the gap is zero, there is no conductivity when the chemical potential is in the gap. The material becomes conducting only when it is doped (electrons or holes are added by impurities) - i.e., made effectively metallic) and/or when the temperature broadening is big enough in comparison to the size of the gap.

Nanostructures
In nanostructures one often models non-equilibrium situation by considering a low-conductance region sandwiched between two effectively metallic (highly doped) regions in quasi-equilibrium, where the applied potential is modeled as the difference between the chemical potentials. Semiconductor diode is a classical example here, but for more modern developments it is worth checking Landauer-Büttiker formalism and Conductance quantization.