Thermoelectric potential and contact potential

In introductory texts the thermoelectric potential is usually attributed to the temperature dependence of the contact potential between different metals. As one of the junctions in a thermoelectric circuit is heated the contact potential there changes compared with the other junction. According to this explanation, the difference of the contact potentials yields the thermoelectric potential of the circuit. In the following it is shown that this explanation is incorrect.

The contact potential between two metals is caused by their different work functions $W$, which are the energies needed to remove an electron from the metal. The work function of a metal is the difference between the energy of a free electron (with no kinetic energy) outside the metal, which we choose as the zero of energy, and the chemical potential $\mu$ of the conduction electrons (Fig. 6) according to

$$\mu = - W.$$ (5)

Figure 6: Illustration of the work function $W$ of metals.

The mean occupation number of a one-electron state $\vec k$ with energy $\varepsilon_{\vec k}$ in a metal or semiconductor is given by the Fermi distribution as

$$n_{0}(\vec k) = \frac{1}{e^{(\varepsilon_{\vec k}-\mu)/k_B T}+1}$$ (6)

At room temperature, apart from a relatively narrow thermal energy shell of width $k_BT$, states with energies $\varepsilon_{\vec k} < \mu$ are occupied, states with $\varepsilon_{\vec k} > \mu$ are vacant (Fig. 7). As the temperature decreases, the transition between occupied and vacant states sharpens. In the limit $T \to 0$ the chemical potential is also known as Fermi energy.

Figure 7: The Fermi distribution of a degenerate electron gas.

If two metals $A$ and $B$ with different work functions $W_A$ and $W_B$ are brought in contact, electrons pass over from the metal with the lower work function to that with the higher one, whereby an electric double layer is formed (Fig. 8).

Figure 8: Electric double layer at the junction of two metals with different work functions.

The electric double layer leads to a discontinuous jump of the electrostatic potential $\varphi$ at the junction, which compensates the difference of the chemical potentials $\mu_A$ and $\mu_B$:

$$-e (\varphi_A-\varphi_B) = -(\mu_A-\mu_B) = W_A-W_B.$$ (7)

Here $(-e)$ denotes the electronic charge. The potential difference $\varphi_A - \varphi_B$ is the contact potential. In terms of the electrochemical potential of the conduction electrons defined by

$$\varphi_{e-ch}= \varphi - \frac{\mu}{e}$$ (8)

eqn. (7) expresses the equality of this electrochemical potential in the two metals at the junction. This is the general condition for thermodynamic equilibrium between two metals in contact.

Since the conduction electrons on both sides of a junction are in thermodynamic equilibrium, the contact potentials - more precisely: the difference of contact potentials between two such junctions - cannot drive an electric current. In conclusion, the electric current which flows in a short-circuited thermoelectric circuit cannot be explained by the difference of contact potentials which results from the temperature difference between the junctions.

The erroneous explanation of the thermoelectric potential $U$ in terms of the difference between the contact potentials at two junctions of different temperature would read

$$'' U = \left( \varphi_A - \varphi_B\right)_{T_1} - \left( \varphi_A-\varphi_B \right)_{T_2} \, .''$$ (9)

However, the voltmeter in the thermoelectric circuit drawn in Fig. 1 measures the difference between the electrochemical potentials at the exits $a$ and $b$, viz.

$$U = \varphi_{e-ch} (b)-\varphi_{e-ch}(a)$$

$$= \varphi (b)- \varphi (a) - \frac{\mu(b)-\mu (a)}{e}\, .$$ (10)

In the second expression of eqn. (10), the difference $\mu(b) - \mu(a)$ vanishes, since the exits of the voltmeter are of the same metal at the same temperature. Therefore the thermoelectric potential $U$ measured in the thermoelectric circuit of Fig. 1 is given by the purely electrostatic potential difference $\varphi(b) - \varphi(a)$ between the two exits of the voltmeter!

Let us see where the difference between the expressions (9) and (10) comes from. The total difference (10) of electrostatic potential $\varphi$ can be decomposed into the contribution from the discontinuous jumps of potential at the junctions with temperatures $T_1$ and $T_2$, which is identical with expression (9), plus the sum of the continuous potential changes along the three portions of metal between $a$ and $b$. The latter sum is the difference between (9) and (10). Contrary to (9), the result (10) is not determined by the temperature dependence of work functions alone, but depends on transport properties of the charge carriers. As a consequence, the thermoelectric potential (10) - contrary to a contact potential - can be very sensitive to the doping of a metal by impurity atoms and to structural defects. The difference of contact potentials (9) usually agrees with the thermoelectric potential (10) only in its order of magnitude, but not necessarily in its sign.