The Seebeck effect

The experimental setup is shown schematically in Fig. 1. The junctions 1 and 2 of the metallic wires made of different materials $A$ and $B$ are kept at different temperatures $T_1$ and $T_2$. The potential $U$ measured by the voltmeter $V$ inserted into wire $A$ is given by

$$U = (Q_A - Q_B)(T_1 - T_2)\, .$$ (1)

Here $Q_A$ and $Q_B$ denote the Seebeck coefficient (thermoelectric power) of metal $A$ and $B$. The dimension of $Q$ is energy/(charge $\cdot$ temperature). The natural unit of thermopower is $k_B/e \approx 10^{-4}V/K$. Typical $Q$ values for metals are lower than that by a factor of 10 to 100, for semiconductors they are higher by similar factors (see Figs. 2 and 3). If the temperature dependence of $Q$ is taken into account, eqn. (1) for the thermoelectric potental needs to be replaced by a path integral, taken along the pieces of metal between the exits $a$ and $b$ of the voltmeter in the circuit shown in Fig. 1 (see Sec. 5, eqn. (12), below). However, this modification is only of quantitative importance.

Figure 1: Thermoelectric circuit made of conductors $A$ and $B$ with junction temperatures $T_1$ and $T_2$. $z$ is the coordinate along the conductors joining the two exits $a$ and $b$ of the voltmeter.

If the circuit of Fig. 1 is short-circuited by removing the voltmeter, a stationary electric current flows. Approximately its magnitude is given by the ratio of the thermoelectric potential measured by the voltmeter and the total Ohmic resistance of the circuit without voltmeter. While the thermoelectric potential is low (in metals of order mV), this thermocurrent can be rather large if the resistance is small.

Figure 2: The temperature dependence of the thermoelectric power of some chosen metals (from [2])

Figure 3: The temperature dependence of the thermoelectric power of silicon containing different kinds of impurity (from [2]).