The contact potential between two metals is caused by their different work functions , which are the energies needed to remove an electron from the metal. The work function of a metal is the difference between the energy of a free electron (with no kinetic energy) outside the metal, which we choose as the zero of energy, and the chemical potential of the conduction electrons (Fig. 6) according to

The mean occupation number of a one-electron state with energy in a metal or semiconductor is given by the Fermi distribution as

(6) |

At room temperature, apart from a relatively narrow thermal energy shell of width , states with energies are occupied, states with are vacant (Fig. 7). As the temperature decreases, the transition between occupied and vacant states sharpens. In the limit the chemical potential is also known as Fermi energy. If two metals and with different work functions and are brought in contact, electrons pass over from the metal with the lower work function to that with the higher one, whereby an electric double layer is formed (Fig. 8). The electric double layer leads to a discontinuous jump of the electrostatic potential at the junction, which compensates the difference of the chemical potentials and :

Here denotes the electronic charge. The potential difference is the contact potential. In terms of the electrochemical potential of the conduction electrons defined by

eqn. (7) expresses the equality of this electrochemical potential in the two metals at the junction. This is the general condition for thermodynamic equilibrium between two metals in contact.

Since the conduction electrons on both sides of a junction are in thermodynamic equilibrium, the contact potentials - more precisely: the difference of contact potentials between two such junctions - cannot drive an electric current. In conclusion, the electric current which flows in a short-circuited thermoelectric circuit cannot be explained by the difference of contact potentials which results from the temperature difference between the junctions.

The erroneous explanation of the thermoelectric potential in terms of the difference between the contact potentials at two junctions of different temperature would read

However, the voltmeter in the thermoelectric circuit drawn in Fig. 1 measures the difference between the

In the second expression of eqn. (10), the difference vanishes, since the exits of the voltmeter are of the same metal at the same temperature. Therefore the thermoelectric potential measured in the thermoelectric circuit of Fig. 1 is given by the