Seebeck coefficient

Physically, the magnitude and sign of the Seebeck coefficient can be approximately understood as being given by the entropy per unit charge carried by electrical currents in the material.

The voltmeter voltage is always dependent on relative Seebeck coefficients among the various materials involved.

Most generally and technically, the Seebeck coefficient is defined in terms of the portion of electric current driven by temperature gradients, as in the vector differential equation where

The sign is made explicit in the following expression: Thus, if S is positive, the end with the higher temperature has the lower voltage, and vice versa.

In most conductors, however, the charge carriers exhibit both hole-like and electron-like behaviour and the sign of S usually depends on which of them predominates.

This is because 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.

Determination of the absolute Seebeck coefficient therefore requires more complicated techniques and is more difficult, but such measurements have been performed on standard materials.

, which expresses the strength of the Thomson effect, can be used to yield the absolute Seebeck coefficient through the relation:

A publication in 1958 used this technique to measure the absolute Seebeck coefficient of lead between 7.2 K and 18 K, thereby filling in an important gap in the previous 1932 experiment mentioned above.

[6] The difficulty of these measurements, and the rarity of reproducing experiments, lends some degree of uncertainty to the absolute thermoelectric scale thus obtained.

[8] The Seebeck coefficient of platinum itself is approximately −5 μV/K at room temperature,[9] and so the values listed below should be compensated accordingly.

A material's temperature, crystal structure, and impurities influence the value of thermoelectric coefficients.

The Seebeck effect can be attributed to two things:[10] charge-carrier diffusion and phonon drag.

On a fundamental level, an applied voltage difference refers to a difference in the thermodynamic chemical potential of charge carriers, and the direction of the current under a voltage difference is determined by the universal thermodynamic process in which (given equal temperatures) particles flow from high chemical potential to low chemical potential.

In other words, the direction of the current in Ohm's law is determined via the thermodynamic arrow of time (the difference in chemical potential could be exploited to produce work, but is instead dissipated as heat which increases entropy).

Charge carriers (such as thermally excited electrons) constantly diffuse around inside a conductive material.

As they move, however, they occasionally scatter dissipatively, which re-randomizes their energy according to the local temperature and chemical potential.

The combination of diffusion and dissipation favours an overall drift of the charge carriers towards the side of the material where they have a lower chemical potential.

This means that high energy levels have a higher carrier occupation per state on the hotter side, but also the hotter side has a lower occupation per state at lower energy levels.

However, there is a competing process: at the same time low-energy carriers are drawn back towards the hot end of the device.

In materials with strong interactions, none of the above equations can be used since it is not possible to consider each charge carrier as a separate entity.

[13] The formulae above can be simplified in a couple of important limiting cases: In semimetals and metals, where transport only occurs near the Fermi level and

Note that whereas the free electron model predicts a negative Seebeck coefficient, real metals actually have complicated band structures and may exhibit positive Seebeck coefficients (examples: Cu, Ag, Au).

[15] In extrinsic (doped) semiconductors either the conduction or valence band will dominate transport, and so one of the numbers above will give the measured values.

In general however the semiconductor may also be intrinsic in which case the bands conduct in parallel, and so the measured values will be This results in a crossover behaviour, as shown in the figure.

,[16] or the thermoelectric figure of merit, and the optimum generally occurs at high doping levels.

If the phonon-electron interaction is predominant, the phonons will tend to push the electrons to one end of the material, hence losing momentum and contributing to the thermoelectric field.

At lower temperatures, material boundaries also play an increasing role as the phonons can travel significant distances.

[18] Practically speaking, phonon drag is an important effect in semiconductors near room temperature (even though well above