Compensation of the thermodiffusion current; origin of the thermoelectric power

Under conditions where electric currents cannot flow, the sum of drift and diffusion current given by eqn. (18) and the thermodiffusion current given by eqn. (24) must compensate each other. This condition yields the equation

$$\vec\nabla \varphi_{e-ch} = - \frac{l_{12}}{l_{11}} \frac{\vec \nabla T}{T}\, ,$$ (26)

according to which a temperature gradient inside a conductor is accompanied by a gradient of the electrochemical potential. We have thus derived the thermoelectric law (13) and obtained the expression

$$Q= l_{12}/(Tl_{11})$$ (27)

for the thermoelectric power.

It should be noted that in a more detailed treatment of thermodiffusion in metals the energy dependence of the electron's mean free path plays an important role [3]. For different scattering mechanisms of the conduction electrons this energy dependence can be very different. Therefore the thermopower may depend strongly on the prevailing type of scattering mechanism. This explains why the thermopower of a metal sometimes varies appreciably upon doping or generation of crystal defects, which can change the dominant scattering mechanism of the electrons.