Thermoelectric cooling

The Peltier effect can be exploited for cooling. The sample to be cooled (at temperature $T_1$) and a large heat bath (at temperature $T_0$) are connected by an electric circuit built of two conductors $A$ and $B$ (Fig. 5).The Peltier heat is absorbed from the sample at one junction, and transferred to the heat bath at the other. Let the wires have Peltier coefficients $\Pi_A$ and $\Pi_B$. Let's assume for simplicity that they have equal length and cross section as well as equal electrical ($\sigma$) and thermal conductivity ($\lambda$). The temperature difference achieved between heat bath and sample also depends on the magnitude of the electric current. For optimal choice of this current the maximum temperature difference is given by

$$\left( T_0-T_1 \right)_{\max} = \frac{1}{8} {\left( \Pi_A-\Pi_B \right)}^2 \sigma/\lambda .$$ (4)

According to this formula the cooling effect is strongest if, for given difference of Peltier coefficients, the electrical conductivity ($\sigma$) is as high as possible and the thermal conductivity ($\lambda$) as low as possible - two opposing requirements.

Figure 5: Thermoelectric cooling of a sample at temperature $T_1$. The electric current of magnitude $I$ is supplied by a battery.