Thermopower und thermodiffusion

In the presence of a temperature gradient in a conductor a second contribution to the electric current due to thermodiffusion occurs in addition to the current given by Ohm's law. Under conditions of zero electric current the two current contributions must compensate each other. Since the Ohmic current is proportional to a potential difference and the thermodiffusion current is proportional to a temperature difference, a proportionality between potential and temperature difference results, which corresponds to eqn. (1). This is explained in detail in the following.

We first note that the difference (10) of the electrochemical potential as measured in the voltmeter can be expressed as a path integral of its gradient along the three portions of metal between the exits $a$ and $b$ (see Fig. 1):

$$U = \int_a^b \vec{ds} \cdot \vec\nabla \varphi_{e-ch}$$ (11)

Since, according to (7), the electrochemical potential is continuous at the junctions of the two metals, there is no contribution to (11) from the junctions. On the other hand, the thermoelectric potential is given by the exact expression

$$U=-\int_a^b QdT = -\int_a^b Q \vec \nabla T \cdot \vec{ds}.$$ (12)

In case the Seebeck coefficients of both metals are independent of temperature eqn. (12) reduces to the eqn. (1). Comparing the path integrals (11) and (12) we find the equation

$$-\vec\nabla \varphi_{e-ch} = Q \cdot \vec\nabla T,$$ (13)

which is the local form of the law for the thermoelectric potential. It is now shown that the relation (13) results from the thermodiffusion of the conduction electrons. This holds both for metals and semiconductors.