The electrochemical potential of electrons in a solid is called the Fermi level.
At this point, the Fermi levels for the two metals are equal, and there is no voltage difference between them.
In the present context, the electron is considered to come from the Fermi level of the Earth.
The density of states at the Fermi level does not play a special role.
In metals, it is typically only necessary to plot the Fermi level.
The original difference in Fermi levels, before contact, is referred to as the emf.
The conduction band, above the Fermi level, is normally nearly completely empty.
In this article the term "Fermi level" is used to mean the same thing as chemical potential.
Transitions to bound vacant states just above the Fermi level can be seen.
The distribution is characterized by the temperature of the electrons, and the Fermi level.