any of various phenomena due to an interrelation between thermal and electrical processes in metals or semiconductors. The three principal thermoelectric effects are the Seebeck effect, the Peltier effect, and the Thomson effect.

In the Seebeck effect an electromotive force is produced in a closed circuit containing dissimilar conductors if the junctions between the conductors are maintained at different temperatures; this electromotive force is called a thermoelectromotive force. In the simplest case the circuit consists of two different conductors and is known as a thermocouple. The magnitude of the thermoelectromotive force depends only on the temperature of the hot junction T₁, the temperature of the cold junction T₂ and the materials from which the conductors are made. When the temperature range T₁ – T₂ is small, the thermoelectromotive force E can be regarded as proportional to the difference in temperatures, that is, E = α(T₁ – T₂). The coefficient α is called the thermoelectric power, or Seebeck coefficient. It depends on the temperature range and on the conductor materials. In some cases α changes sign with a change in temperature.

Table 1 gives the value of α for a number of metals and alloys against lead as a reference metal for a temperature range of 0°–100°C. Here, α is regarded as positive when the current at the hot junction flows from the lead to the material in question.

The values in the table should not be regarded as absolutely reliable, since the thermoelectromotive force developed in a material is sensitive to the presence of microscopic quantities of impurities (in some cases, quantities beyond the sensitivity limits of chemical analysis); to crystal grain orientation; and to heat treatment or even cold working of the material. This property of the thermoelectromotive force is the basis of a method for sorting materials by composition. Because of the sensitivity of the Seebeck effect, a thermoelectromotive force can be produced in a circuit consisting of a single material, when a temperature drop is present, if different sections of the circuit have undergone different fabrication processes. On the other hand, the electromotive force of a thermocouple is not changed by connecting to the circuit any number of other materials in series if the additional junctions are kept at the same temperature.

The Peltier effect is the reverse of the Seebeck effect. Consider a current flowing in a circuit consisting of different conductors. In addition to the Joule heat produced, a certain amount of heat Q_Π is either evolved or absorbed at each junction, depending on the current direction. The heat Q_Π is proportional to the amount of electricity—that is, the product of the current I and time t—passing through the junction: Q_Π = ПIt. The coefficient Π is called the Peltier coefficient and is dependent on the temperature and on the nature of the materials forming the junction.

W. Thomson (Lord Kelvin) derived a thermodynamic relation between Peltier and Seebeck coefficients that is a special case of the symmetry of kinetic, or phenomenological, coefficients (seeONSAGER THEOREM): П = αT, where T is the absolute temperature. He also predicted the existence of a third thermoelectric effect, which is now called the Thomson effect. It consists in the following: if a current flows through a conductor along which a temperature gradient exists, heat is either evolved or absorbed, depending on the current direction, in addition to the Joule heat. This additional heat is called the Thomson heat Q_τ = τ(T₂ – T₁)It. The quantity τ is known as the Thomson coefficient and depends on the nature of the material. According to Thomson’s theory, the relation between the thermoelectric power of a thermocouple and the Thomson coefficients of the conductors forming the couple is given by the equation dα/dT = (τ₁ – τ₂)/T.

The Seebeck effect can be explained by reference to the conduction electrons in a conductor. The average energy of the conduction electrons depends on the nature of the conductor and increases in different ways with an increase in temperature. If a temperature gradient exists along a conductor, the electrons at the hot end have higher energies and velocities than the electrons at the cold end. Furthermore, in semiconductors the concentration of conduction electrons increases with temperature. As a result, electrons drift from the hot to the cold end. Negative charge accumulates at the cold end and an uncompensated positive charge remains at the hot end. The induced potential difference causes an electron current to flow from the cold end to the hot end. The accumulation of charge continues until the electron current becomes equal to the drift of electrons due to the temperature difference—that is, until equilibrium is reached. The algebraic sum of such potential differences in a circuit constitutes what is called the electron diffusion component of the thermoelectromotive force.

A second, contact component of the thermoelectromotive force is a consequence of the temperature dependence of the contact potential difference. If both junctions of a thermocouple are at the same temperature, the contact and electron diffusion components vanish.

The dragging along of electrons by phonons also makes a contribution to the thermoelectromotive force. If a temperature gradient exists in a solid, the number of phonons moving from the hot end to the cold end is greater than the number moving in the reverse direction. As a result of collisions with electrons, the phonons drag electrons along. Negative charge accumulates at the cold end of the specimen—and positive charge accumulates at the hot end—until the induced potential difference balances the drag effect. This potential difference is the third component of the thermoelectromotive force. At low temperatures the phonon drag component may be tens or hundreds of times greater than the previously discussed components.

In magnetic materials an additional component of the thermoelectromotive force is observed. This component is due to the dragging along of electrons by magnons.

In metals the concentration of conduction electrons is high and does not depend on temperature. Since the energy of the electrons is also practically independent of temperature, the thermoelectric power is very small for metals. The thermoelectric power reaches comparatively higher values in semimetals and their alloys, where the carrier concentrations are considerably smaller and dependent on temperature, and in certain transition metals and alloys thereof. For example, the thermoelectric power of palladium-silver alloys reaches 86 microvolts per °C (μV/°C). The concentration of electrons is high in the transition metal case. The thermoelectric power, however, is also high, because the average energy of the conduction electrons differs considerably from the Fermi energy. Sometimes fast electrons diffuse less readily than do slow electrons, and the thermoelectric power accordingly changes sign. The magnitude and sign of the thermoelectric power also depend on the shape of the Fermi surface. In metals and alloys with a complex Fermi surface, different regions of the Fermi surface may make contributions of opposite sign to the thermoelectric power, and the thermoelectric power may be zero or close to zero. In some metals the sign of the thermoelectric power changes at low temperatures as a result of the phonon-dragging of electrons.

In p-type semiconductors holes accumulate at the cold junction, and an uncompensated negative charge remains at the hot junction (if an anomalous scattering mechanism or phonon-drag does not cause a change in the sign of the thermoelectric power). The thermoelectric powers are additive in a thermocouple consisting of a p-type and an n-type semiconductor. In semiconductors with mixed conductivity, on the other hand, electrons and holes diffuse toward the cold junction, and their charges compensate each other. If the electrons and holes have equal concentrations and equal mobilities, the thermoelectric power is zero.

An explanation, to a first approximation, of the Thomson effect follows. Suppose a temperature gradient exists along a conductor through which a current is flowing, and suppose the direction of the current is such that the electrons move from the hot end to the cold end. As the electrons pass from a warmer region to a cooler region, they give up their excess energy to the surrounding atoms; in other words, heat is evolved. If the current flows in the reverse direction, the electrons, on passing from a cooler region to a warmer region, increase their energy at the expense of the surrounding atoms by absorbing heat. It should be noted that in the first case the field of the thermoelectromotive force slows down the electrons whereas in the second case it accelerates them. This circumstance influences the value of τ and may even change the sign of the effect.

The Peltier effect occurs because the average energy of the electrons participating in the transfer of current is different in different conductors. This average energy depends on the energy spectrum of the electrons (the band structure of the material), their concentration, and the mechanism of their scattering. As the electrons pass from one conductor to the other, they either transfer their excess energy to the surrounding atoms or increase their energy at the expense of the atoms, depending on the direction of the current. Peltier heat is evolved at the junction in the first case and absorbed in the second case.

Let us examine the case where the current direction is such that the electrons pass from a semiconductor to a metal. The flow of the current through the junction would not disturb thermal equilibrium (Q_Π = 0) if the electrons in the impurity levels of the semiconductor could move, under the action of an electric field, exactly like conduction electrons and if the average energy of the electrons were equal to the Fermi energy in the metal. In the semiconductor, however, the electrons in the impurity levels are localized, and the energy of the conduction electrons is much greater than the Fermi level in the metal; moreover, the energy of the conduction electrons depends on the scattering mechanism. When they pass into the metal, the conduction electrons give up their excess energy, and Peltier heat is thereby evolved.

If the current flows in the reverse direction, the only electrons that can pass from the metal to the semiconductor are those whose energy is above the bottom of the conduction band of the semiconductor. Here, the thermal equilibrium in the metal is disturbed; it is restored by means of the lattice vibrations. In this case Peltier heat is absorbed.

Peltier heat is also evolved or absorbed at a junction between two semiconductors or two metals because the average energy of the electrons participating in the current is different on the two sides of the junction.

Thus, the cause of all the thermoelectric effects is the disturbance of thermal equilibrium in a flux of charge carriers—that is, the difference of the average energy of the electrons in the flux from the Fermi energy. Since the absolute values of the thermoelectric coefficients increase with a decrease in carrier concentration, they are tens or hundreds of times greater in semiconductors than in metals and alloys.