Seebeck coefficient

The 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, as induced by the Seebeck effect. The SI unit of the Seebeck coefficient is volts per kelvin, although it is more often given in microvolts per kelvin.
The use of materials with a high Seebeck coefficient is one of many important factors for the efficient behaviour of thermoelectric generators and thermoelectric coolers. More information about high-performance thermoelectric materials can be found in the Thermoelectric materials article. In thermocouples the Seebeck effect is used to measure temperatures, and for accuracy it is desirable to use materials with a Seebeck coefficient that is stable over time.
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. It may be positive or negative. In conductors that can be understood in terms of independently moving, nearly-free charge carriers, the Seebeck coefficient is negative for negatively charged carriers, and positive for positively charged carriers.

Definition

One way to define the Seebeck coefficient is the voltage built up when a small temperature gradient is applied to a material, and when the material has come to a steady state where the current density is zero everywhere. If the temperature difference ΔT between the two ends of a material is small, then the Seebeck coefficient of a material is defined as:
where ΔV is the thermoelectric voltage seen at the terminals.
Note that the voltage shift expressed by the Seebeck effect cannot be measured directly, since the measured voltage contains an additional voltage contribution, due to the temperature gradient and Seebeck effect in the measurement leads. 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 is the current density, is the electrical conductivity, is the voltage gradient, and is the temperature gradient. The zero-current, steady state special case described above has, which implies that the two electrical conductivity terms have cancelled out and so

Sign convention

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. The voltage gradient in the material will point against the temperature gradient.
The Seebeck effect is generally dominated by the contribution from charge carrier diffusion which tends to push charge carriers towards the cold side of the material until a compensating voltage has built up. As a result, in p-type semiconductors, S is positive. Likewise, in n-type semiconductors, S is negative.
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.

Relationship to other thermoelectric coefficients

According to the second Thomson relation, the Seebeck coefficient is related to the Peltier coefficient by the exact relation
where is the thermodynamic temperature.
According to the first Thomson relation and under the same assumptions about magnetism, the Seebeck coefficient is related to the Thomson coefficient by
The constant of integration is such that at absolute zero, as required by Nernst's theorem.

Measurement

Relative Seebeck coefficient

In practice the absolute Seebeck coefficient is difficult to measure directly, since the voltage output of a thermoelectric circuit, as measured by a voltmeter, only depends on differences of Seebeck coefficients. 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. Therefore, the measured Seebeck coefficient is a contribution from the Seebeck coefficient of the material of interest and the material of the measurement electrodes. This arrangement of two materials is usually called a thermocouple.
The measured Seebeck coefficient is then a contribution from both and can be written as:

Absolute Seebeck coefficient

Although only relative Seebeck coefficients are important for externally measured voltages, the absolute Seebeck coefficient can be important for other effects where voltage is measured indirectly. Determination of the absolute Seebeck coefficient therefore requires more complicated techniques and is more difficult, but such measurements have been performed on standard materials. In principle these measurements only have to be performed once for all time, and for all materials; for any other material, the absolute Seebeck coefficient can be obtained by performing a relative Seebeck coefficient measurement against another material with known absolute value.
A measurement of the Thomson coefficient, which expresses the strength of the Thomson effect, can be used to yield the absolute Seebeck coefficient through the relation:, provided that is measured down to absolute zero. The reason this works is that is expected to decrease to zero as the temperature is brought to zero—a consequence of Nernst's theorem. Such a measurement based on the integration of was published in 1932, though it relied on the interpolation of the Thomson coefficient in certain regions of temperature.
Superconductors have zero Seebeck coefficient, as mentioned below. By making one of the wires in a thermocouple superconducting, it is possible to get a direct measurement of the absolute Seebeck coefficient of the other wire, since it alone determines the measured voltage from the entire thermocouple. 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.
The combination of the superconductor-thermocouple technique up to 18 K, with the Thomson-coefficient-integration technique above 18 K, allowed determination of the absolute Seebeck coefficient of lead up to room temperature. By proxy, these measurements led to the determination of absolute Seebeck coefficients for all materials, even up to higher temperatures, by a combination of Thomson coefficient integrations and thermocouple circuits.
The difficulty of these measurements, and the rarity of reproducing experiments, lends some degree of uncertainty to the absolute thermoelectric scale thus obtained. In particular, the 1932 measurements may have incorrectly measured the Thomson coefficient over the range 20 K to 50 K. Since nearly all subsequent publications relied on those measurements, this would mean that all of the commonly used values of absolute Seebeck coefficient are too low by about 0.3 μV/K, for all temperatures above 50 K.

Seebeck coefficients for some common materials

In the table below are Seebeck coefficients at room temperature for some common, nonexotic materials, measured relative to platinum.
The Seebeck coefficient of platinum itself is approximately −5 μV/K at room temperature, and so the values listed below should be compensated accordingly. For example, the Seebeck coefficients of Cu, Ag, Au are 1.5 μV/K, and of Al −1.5 μV/K. The Seebeck coefficient of semiconductors very much depends on doping, with generally positive values for p doped materials and negative values for n doping.

Material	Seebeck coefficient relative to platinum
Selenium	900
Tellurium	500
Silicon	440
Germanium	330
Antimony	47
Nichrome	25
Iron	19
Molybdenum	10
Cadmium, tungsten	7.5
Gold, silver, copper	6.5
Rhodium	6.0
Tantalum	4.5
Lead	4.0
Aluminium	3.5
Carbon	3.0
Mercury	0.6
Platinum	0
Sodium	-2.0
Potassium	-9.0
Nickel	-15
Constantan	-35
Bismuth	-72
Bismuth telluride	-287

Physical factors that determine the Seebeck coefficient

A material's temperature, crystal structure, and impurities influence the value of thermoelectric coefficients. The Seebeck effect can be attributed to two things: charge-carrier diffusion and phonon drag.

Charge carrier diffusion

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 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. On the other hand, for the Seebeck effect not even the sign of the current can be predicted from thermodynamics, and so to understand the origin of the Seebeck coefficient it is necessary to understand the microscopic physics.
Charge carriers constantly diffuse around inside a conductive material. Due to thermal fluctuations, some of these charge carriers travel with a higher energy than average, and some with a lower energy. When no voltage differences or temperature differences are applied, the carrier diffusion perfectly balances out and so on average one sees no current:. A net current can be generated by applying a voltage difference, or by applying a temperature difference. To understand the microscopic origin of the thermoelectric effect, it is useful to first describe the microscopic mechanism of the normal Ohm's law electrical conductance—to describe what determines the in. Microscopically, what is happening in Ohm's law is that higher energy levels have a higher concentration of carriers per state, on the side with higher chemical potential. For each interval of energy, the carriers tend to diffuse and spread into the area of device where there are fewer carriers per state of that energy. As they move, however, they occasionally scatter dissipatively, which re-randomizes their energy according to the local temperature and chemical potential. This dissipation empties out the carriers from these higher energy states, allowing more to diffuse in. 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.
For the thermoelectric effect, now, consider the case of uniform voltage with a temperature gradient. In this case, at the hotter side of the material there is more variation in the energies of the charge carriers, compared to the colder side. 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. As before, the high-energy carriers diffuse away from the hot end, and produce entropy by drifting towards the cold end of the device. However, there is a competing process: at the same time low-energy carriers are drawn back towards the hot end of the device. Though these processes both generate entropy, they work against each other in terms of charge current, and so a net current only occurs if one of these drifts is stronger than the other. The net current is given by, where the thermoelectric coefficient depends literally on how conductive high-energy carriers are, compared to low-energy carriers. The distinction may be due to a difference in rate of scattering, a difference in speeds, a difference in density of states, or a combination of these effects.