Thermopower

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The thermopower, or thermoelectric power, or Seebeck coefficient of a material is a measure of the magnitude of an induced thermoelectric voltage in response to a temperature difference across that material. The thermopower has units of $(V / K)$ . The term thermopower is a misnomer since it measures the voltage or electric field (not the electric power) induced in response to a temperature difference. An applied temperature difference causes charged carriers in the material, whether they be electrons or holes, to diffuse from the hot side to the cold side, similar to a classical gas that expands when heated. Mobile charged carriers migrating to the cold side leave behind their oppositely charged and immobile nuclei at the hot side thus giving rise to a thermoelectric voltage (thermoelectric refers to the fact that the voltage is created by a temperature difference). Since a separation of charges also creates an electric field, the buildup of charged carriers onto the cold side eventually ceases at some maximum value since there exists an equal amount of charged carriers drifting back to the hot side as a result of the electric field at equilibrium. Only an increase in the temperature difference can resume a buildup of more charge carriers on the cold side and thus lead to an increase in the thermoelectric voltage. Incidentally the thermopower also measures the entropy per charge carrier in the material.

The thermopower of a material, represented as $S$ , depends on the material's temperature, and crystal structure. Typically metals have small thermopowers because most have half-filled bands. Electrons (negative charges) and holes (positive charges) both contribute to the induced thermoelectric voltage thus canceling each other's contribution to that voltage and making it small. In contrast, semiconductors can be doped with an excess amount of electrons or holes and thus can have large positive or negative values of the thermopower depending on the charge of the excess carriers. The sign of the thermopower can determine which charged carriers dominate the electric transport in both metals and semiconductors.

If the temperature difference $Δ T$ between the two ends of a material is small, then the thermopower of a material is defined as:

$S = {\Delta V \over \Delta T}$

and a thermoelectric voltage ΔV is seen at the terminals.

This can also be written in relation to the electric field $E$ and the temperature gradient $\nabla T$ , by the equation:

$S = {E \over \nabla T}$

In practice one rarely measures the absolute thermopower of the material of interest. This is due to the fact that electrodes attached to a voltmeter must be placed onto the material in order to measure the thermoelectric voltage. The temperature gradient then also typically induces a thermoelectric voltage across one leg of the measurement electrodes. Therefore the measured thermopower is a contribution from the thermopower of the material of interest and the material of the measurement electrodes. This arrangement of two materials is usually called a thermocouple.

The measured thermopower is then a contribution from both and can be written as:

$S_{AB} = S_B-S_A = {\Delta V_B \over \Delta T} - {\Delta V_A \over \Delta T}$

Superconductors have zero thermopower since the charged carriers produce no entropy. This allows a direct measurement of the absolute thermopower of the material of interest, since it is the thermopower of the entire thermocouple as well. In addition, a measurement of the Thompson coefficient, $μ$ , of a material can also yield the thermopower through the relation: $S = \int {\mu \over T} dT$

The thermopower is an important material parameter that determines the efficiency of a thermoelectric material. A larger induced thermoelectric voltage for a given temperature gradient will lead to a larger efficiency. Ideally one would want very large thermopower values since only a small amount of heat is then necessary to create a large voltage. This voltage can then be used to provide power.