The Peltier effect

An electric current of magnitude $I$ across the junction of two different conductors $A$ and $B$ with Peltier coefficients $\Pi_A$ and $\Pi_B$ produces heat at the rate

$$\dot W = \left( \Pi_A - \Pi_B \right) \cdot I$$ (2)

(Fig. 4). The sign of $\dot W$ can be positive as well as negative. A negative sign means cooling of the junction. Contrary to Joule heating, the Peltier effect is reversible and depends on the direction of the current.

Figure 4: Setup for observing the Peltier effect.

The Peltier effect is caused by the fact that an electric current is accompanied by a heat current in a homogeneous conductor even at constant temperature. The magnitude of this heat current is given by $\Pi \cdot I$. The Peltier heat eqn. (2) is the balance of the heat flows towards and away from the interface. The heat current accompanying an electric current is explained by the different flow velocities of the electrons carrying the electric current. The flow velocities depend on the energies of the conduction electrons. E.g., if the flow velocity of electrons with an energy above the chemical potential (Fermi energy) is higher than for electrons with a lower energy, the electric current is accompanied by a heat current in the opposite direction (since the electronic charge is negative!). In this case the Peltier coefficient is negative. The same situation occurs for a $n$-doped semiconductor, in which the electric current is carried by electrons in conduction-band states.

The Seebeck and Peltier coefficient $Q$ and $\Pi$ obey the relation

$$\Pi = T \cdot Q\, ,$$ (3)

found already by Lord Kelvin, but for which a valid derivation could be given only later using the kinetic theory of conduction electrons or irreversible thermodynamics. The Kelvin relation (3) connects the material constants for two very different physical effects, of which the Peltier effect has the simple explanation sketched above. Nevertheless, an independent explanation of the Seebeck effect is desirable.