## Thermoelectricity

Understanding the physical mechanisms of thermoelectric effects

The purpose of this article is to explain the mechanism of the observed thermoelectic effects in electrical conductors, metals and semiconductors. Let's begin with the most commonl thermoelectric device, the thermocouple, used to measure temperature. A closed loop is made from two conductors, A and B, intimately joined at two points. The junctions are maintained at temperatures T1 and T2, with T1 > T2. T1 is the temperature of the hot juncti9n, and T2 the temperature of the cold junction. Often, T1 is the temperature to be measured, and T2 is held at a reference temperature, often 0°C by a bath of ice and water. The loop is broken at some point where the temperature is T0 and a potentiometer is connected that provides an emf opposite and equal to the sum of the emf's in the loop, and so reducing the current to zero. The value of T0 is not important. The observed emf around the loop is the Seebeck Effect, the firs thermoelectric effect to be observed, in 1824 and reported by T. J. Seebeck.

When the two conductors are joined, a difference in potential is created between them that aligns the Fermi levels, so that it is possible for charge carriers to cross the junction freely. The value of this contact potential is the difference in work functions of the conductors, and varies little with temperature. The Seebeck Effect is not dependent on any conditions at the junction, which merely serves to permit current to pass between the conductors.

Charge carriers diffuse from the hot junction towards the cold junction, driven by the temperature gradient. Since the net current is zeo, there must be a reverse conduction current of the same magnitude, driven by an electric field. If the charge carriers are electrons, the hot junction becomes positive and the cold junction negative.The potential difference is equal to the Seebeck coefficient S times dT, and the net potential difference between the hot and cold junctions in conductor A is the integral if SAdT from T1 to T2. The net potential around the loop is the integral of (SA - SB)dT from T1 to T2. If the charge carriers are positive--that is, holes--the sense of the potential difference is reversed, and the Seebeck coefficient is negative. These considerations allow one to determine the amount and sign of the Seebeck effect in any case. S depends sensitively on the band theory properties of the charge carriers, which can differ significantly from the behavior of free electrons. A higher value of S is also aided by significant resisitivity, so many thermocouple materials are also high-resistance alloys. Since thermocouples may be used at high temperatures, they are also often refractory alloys.

If E is the emf of a thermocouple, the quantity P = dE/dT is called the thermoelectric power, though, of course, it is not a power. It generally varies with temperature, so using this quantity for accurate work is difficult. For such applications, tables are available of the measured E as a function of T. If dE/dT is plotted against T, the result is an approximate straight line, dE/dT = a + bT., or E = aT +bT^2/2. In these expressions, T is T1-T2. P is the same as the value of SA - SB, and can be added or subtracted for different pairs of conductors. The absolute value, such as SA, cannot be measured for a single material. As an example, alumel has P = 17.48 &mu:V/K and chromel has P = -24.40 μV/K, in both cases with small b, both relative to lrad. Therefore, a chromel-alumel thermocouple has P = 41.88 μV/K. If the chromel wires are connected to the potentiometer, the positive terminal should be connected to the chromel wire to the hot junction. These Seebeck voltages are really rather small. The largest values for elemental conductors is 43.69 μV/K for Bismuth, and -35.58 μV/K for Antimony, in both cases with considerable values of b. However, their low melting points restrict their use. The largest values of P are for semiconductors, n-type giving positive values and p-type negative values.

We are concerned here with transport effects in which the charge carriers producw an electric current and a current of heat under the driving fores of an electric field and a temperature gradient. The theory is presented by Ziman (see References) and its result is that the electric current and the heat current both depend jointly on the electric field and the temperature gradient. All thermoelectric phenomena are described by these equations. In the case we have considered, the Seebeck effect, we have made the electric current zero. Setting it ti zero in the equation results in a relation between the electric field and the temperature gradient, whiich has been described above.

Now suppose we make the thermal gradients zero by making T1 = T2. Then we have transport equations for the electric current and the thermal current, both driven by the applied electric field. This implies that if we cause an electric current J to flow in the loop, it is accompanied by a thermal current U = ΠJ, where Π = ST. Here, S is the thermoelectric power of the material, and T is the absolute temperature. When the current passes from conductor A to conductor B, heat SA TJ arrives at the junction and SB TJ leaves it, so heat (SA - SB)T is liberated at this junction, while an equal and opposite amount is absorbed at the other junction. The net effect is the transfer of heat from one junction to the other, the direction depending on the direction of the current. This is the Peltier effect.

The Peltier effect is thermodynamically reversible, and results in no entropy change. We are neglecting the Joule heat that always accompanies current flow in a resistive conductor. It is not part of the transport processes and is not consisered a thermoelectric effect. Suppose now we replace the potentiometer of a thermocouple circuit with a resistor, and make T1 greater than T2. A current will flow, and electrical work will be done in the resistor. This looks very much like a heat engine, accepting heat at temperature T1, performing reversible work, and rejecting the balance of the energy at temperature T2. If this were the case, applying the same analysis as for a Carnot cycle, we should have efficiency = E/Π = ΔT/T1. Since E = SΔT, we find Π; = ST1, a relation we obtained from the transport equations above. Although there is some truth here, when we allow both a current in the loop and thermal gradients, we must take the transport equations into account. The result is some potential differences and heat transfer along the wires of the loop. These additional effects represent the Thomson Effect, described by William Thomson (Lord Kelvin), who did not have the advantage of the transport theory.

### References

Ziman, J. M., Principles of the Theory of Solids (Cambridge: Cambridge University Press, 1965). Chapter 7.

Reimann, A. L., Physics (New York: Barnes and Noble, 1971). Chapter 28.