# Thermoelectric effect

(Redirected from Peltier-Seebeck effect)

The thermoelectric effect is the direct conversion of temperature differences to electric voltage and vice-versa. A thermoelectric device creates voltage when there is a different temperature on each side. Conversely, when a voltage is applied to it, it creates a temperature difference. At the atomic scale, an applied temperature gradient causes charge carriers in the material to diffuse from the hot side to the cold side.

This effect can be used to generate electricity, measure temperature or change the temperature of objects. Because the direction of heating and cooling is determined by the polarity of the applied voltage, thermoelectric devices can be used as temperature controllers.

The term "thermoelectric effect" encompasses three separately identified effects: the Seebeck effect, Peltier effect, and Thomson effect. Textbooks may refer to it as the Peltier–Seebeck effect. This separation derives from the independent discoveries of French physicist Jean Charles Athanase Peltier and Baltic German physicist Thomas Johann Seebeck. Joule heating, the heat that is generated whenever a voltage is applied across a resistive material, is related though it is not generally termed a thermoelectric effect. The Peltier–Seebeck and Thomson effects are thermodynamically reversible,[1] whereas Joule heating is not.

## Seebeck effect

A thermoelectric circuit composed of materials of different Seebeck coefficient (p-doped and n-doped semiconductors), configured as a thermoelectric generator. If the load is removed then the current stops, and the circuit functions as a temperature-sensing thermocouple.

The Seebeck effect is the conversion of temperature differences directly into electricity and is named after the Baltic German physicist Thomas Johann Seebeck, who, in 1821 discovered that a compass needle would be deflected by a closed loop formed by two metals joined in two places, with a temperature difference between the junctions. This was because the metals responded differently to the temperature difference, creating a current loop and a magnetic field. Seebeck did not recognize there was an electric current involved, so he called the phenomenon the thermomagnetic effect. Danish physicist Hans Christian Ørsted rectified the mistake and coined the term "thermoelectricity".

The Seebeck effect is a classic example of an electromotive force (emf) and leads to measurable currents or voltages in the same way as any other emf. Electromotive forces modify Ohm's law by generating currents even in the absence of voltage differences (or vice versa); the local current density is given by

$\mathbf J = \sigma (-\boldsymbol \nabla V + \mathbf E_{\rm emf}),$

where $V$ is the local voltage[2] and $\sigma$ is the local conductivity. In general the Seebeck effect is described locally by the creation of an electromotive field

$\mathbf E_{\rm emf} = - S \boldsymbol\nabla T,$

where $S$ is the Seebeck coefficient (also known as thermopower), a property of the local material, and $\boldsymbol \nabla T$ is the gradient in temperature $T$.

The Seebeck coefficients generally vary as function of temperature, and depend strongly on the composition of the conductor. For ordinary materials at room temperature, the Seebeck coefficient may range in value from −100 μV/K to +1000 μV/K (see Thermoelectric materials)

If the system reaches a steady state where $\mathbf J = 0$, then the voltage gradient is given simply by the emf: $-\boldsymbol \nabla V = S \boldsymbol\nabla T$. This simple relationship, which does not depend on conductivity, is used in the thermocouple to measure a temperature difference; an absolute temperature may be found by performing the voltage measurement at a known reference temperature. Conversely, a metal of unknown composition can be classified by its thermoelectric effect if a metallic probe of known composition, kept at a constant temperature, is held in contact with it (the unknown material is locally heated to the probe temperature). Industrial-quality control instruments use this as thermoelectric alloy sorting to identify metal alloys. Thermocouples in series form a thermopile, sometimes constructed in order to increase the output voltage, because the voltage induced over each individual couple is small. Thermoelectric generators are used for creating power from heat differentials and exploit this effect.

## Peltier effect

The Seebeck circuit configured as a thermoelectric cooler

The Peltier effect is the presence of heating or cooling at an electrified junction of two different conductors and is named for French physicist Jean Charles Athanase Peltier, who discovered it in 1834. When a current is made to flow through a junction between two conductors A and B, heat may be generated (or removed) at the junction. The Peltier heat generated at the junction per unit time, $\dot{Q}$, is equal to

$\dot{Q} = \left( \Pi_\mathrm{A} - \Pi_\mathrm{B} \right) I,$

where $\Pi_A$ ($\Pi_B$) is the Peltier coefficient of conductor A (B), and $I$ is the electric current (from A to B). Note that the total heat generated at the junction is not determined by the Peltier effect alone, as it may also be influenced by Joule heating and thermal gradient effects (see below).

The Peltier coefficients represent how much heat is carried per unit charge. Since charge current must be continuous across a junction, the associated heat flow will develop a discontinuity if $\Pi_A$ and $\Pi_B$ are different. The Peltier effect can be considered as the back-action counterpart to the Seebeck effect (analogous to the back-emf in magnetic induction): if a simple thermoelectric circuit is closed then the Seebeck effect will drive a current, which in turn (via the Peltier effect) will always transfer heat from the hot to the cold junction. The close relationship between Peltier and Seebeck effects can be seen in the direct connection between their coefficients: $\Pi = T S$ (see below).

A typical Peltier heat pump device involves multiple junctions in series, through which a current is driven. Some of the junctions lose heat due to the Peltier effect, while others gain heat. Thermoelectric heat pumps exploit this phenomenon, as do thermoelectric cooling devices found in refrigerators.

## Thomson effect

In many materials, the Seebeck coefficient is not constant in temperature, and so a spatial gradient in temperature can result in a gradient in the Seebeck coefficient. If a current is driven through this gradient then a continuous version of the Peltier effect will occur. This Thomson effect was predicted and subsequently observed by Lord Kelvin in 1851. It describes the heating or cooling of a current-carrying conductor with a temperature gradient.

If a current density $\mathbf J$ is passed through a homogeneous conductor, the Thomson effect predicts a heat production rate $\dot q$ per unit volume of:

$\dot q = - \mathcal K \mathbf J \cdot \boldsymbol \nabla T ,$

where $\boldsymbol \nabla T$ is the temperature gradient and $\mathcal K$ is the Thomson coefficient. The Thomson coefficient is related to the Seebeck coefficient as $\mathcal K = T\, dS/dT$ (see below). This equation however neglects Joule heating, and ordinary thermal conductivity (see full equations below).

## Full thermoelectric equations

Often, more than one of the above effects is involved in the operation of a real thermoelectric device. The Seebeck effect, Peltier effect, and Thomson effect can be gathered together in a consistent and rigorous way, described here; the effects of Joule heating and ordinary heat conduction are included as well. As stated above, the Seebeck effect generates an electromotive force, leading to the current equation[3]

$\mathbf J = \sigma (-\boldsymbol \nabla V - S \boldsymbol\nabla T).$

To describe the Peltier and Thomson effects we must consider the flow of energy. To start we can consider the dynamic case where both temperature and charge may be varying with time. The full thermoelectric equation for the energy accumulation, $\dot e$ is[3]

$\dot e = \boldsymbol \nabla \cdot (\kappa \boldsymbol \nabla T) - \boldsymbol \nabla \cdot (V\mathbf J + \Pi\mathbf J) + \dot q_{\rm ext},$

where $\kappa$ is the thermal conductivity. The first term is the Fourier's heat conduction law, and the second term shows the energy carried by currents. The third term $\dot q_{\rm ext}$ is the heat added from an external source (if applicable).

In the case where the material has reached a steady state, the charge and temperature distributions are stable so one must have both $\dot e = 0$ and $\boldsymbol \nabla \cdot \mathbf J = 0$. Using these facts and the second Thomson relation (see below), the heat equation then can be simplified to

$-\dot q_{\rm ext} = \boldsymbol \nabla \cdot (\kappa \boldsymbol \nabla T) + \mathbf J \cdot(\sigma^{-1} \mathbf J) - T \mathbf J \cdot\boldsymbol \nabla S,$

The middle term is the Joule heating, and the last term includes both Peltier ($\boldsymbol \nabla S$ at junction) and Thomson ($\boldsymbol \nabla S$ in thermal gradient) effects. Combined with the Seebeck equation for $\mathbf J$, this can be used to solve for the steady state voltage and temperature profiles in a complicated system.

If the material is not in a steady state, a complete description will also need to include dynamic effects such as relating to electrical capacitance, inductance, and heat capacity.

## Physical origin of the thermoelectric coefficients

A material's temperature, crystal structure, and impurities influence the value of the thermoelectric coefficients. The Seebeck effect can be attributed to two things[citation needed]: charge-carrier diffusion and phonon drag. Typically metals have small Seebeck coefficients because of partially filled bands, with a conductivity that is relatively insensitive to small changes in energy. In contrast, semiconductors can be doped with impurities that donate excess electrons or electron holes, allowing the value of S to be varied over a large range (both negative and positive). The sign of the Seebeck coefficients can be used to determine whether the electrons or the holes dominate electric transport in a semiconductor or semimetal.

### Thomson relations

In 1854, Lord Kelvin found relationships between the three coefficients, implying that the Thomson, Peltier, and Seebeck effects are different manifestations of one effect (uniquely characterized by the Seebeck coefficient).

The first Thomson relation is[3]

$\mathcal K \equiv {d\Pi \over dT} - S,$

where $T$ is the absolute temperature, $\mathcal K$ is the Thomson coefficient, $\Pi$ is the Peltier coefficient, and $S$ is the Seebeck coefficient. This relationship is is easily shown given that the Thomson effect is a continuous version of the Peltier effect.

The second Thomson relation is

$\Pi = TS .$

This relation expresses a fundamental connection between the Peltier and Seebeck effects. It was not satisfactorily proven until the advent of the Onsager relations, and it is worth nothing that this second Thomson relation is only valid in a time-reversal symmetric material (the material must be nonmagnetic, and the magnetic field must be zero). Using this relation, the first Thomson relation becomes $\mathcal K = T dS/dT$.

The Thomson coefficient is unique among the three main thermoelectric coefficients because it is the only one directly measurable for individual materials. The Peltier and Seebeck coefficients can only be easily determined for pairs of materials; hence, it is difficult to find values of absolute Seebeck or Peltier coefficients for an individual material.

If the Thomson coefficient of a material is measured over a wide temperature range, it can be integrated using the Thomson relations to determine the absolute values for the Peltier and Seebeck coefficients. This needs to be done only for one material, since the other values can be determined by measuring pairwise Seebeck coefficients in thermocouples containing the reference material and then adding back the absolute thermopower of the reference material.

### Charge-carrier diffusion

Charge carriers in the materials will diffuse when one end of a conductor is at a different temperature from the other. Hot carriers diffuse from the hot end to the cold end, since there is a lower density of hot carriers at the cold end of the conductor, and vice versa. If the conductor were left to reach thermodynamic equilibrium, this process would result in heat being distributed evenly throughout the conductor (see heat transfer). The movement of heat (in the form of hot charge carriers) from one end to the other is a heat current and an electric current as charge carriers are moving.

In a system where both ends are kept at a constant temperature difference, there is a constant diffusion of carriers. If the rate of diffusion of hot and cold carriers in opposite directions is equal, there is no net change in charge. The diffusing charges are scattered by impurities, imperfections, and lattice vibrations or phonons. If the scattering is energy dependent, the hot and cold carriers will diffuse at different rates, creating a higher density of carriers at one end of the material and an electrostatic voltage. This electronic contribution to the Seebeck coefficient is described by the Mott relation,[4]

$S = \frac{k_{\rm B}}{-e}\frac{1}{\sigma} \int \frac{E - \mu}{k_{\rm B}T} \sigma(E) \left( -\frac{df(E)}{dE} \right) \, dE$

where $\sigma(E)$ is the conductivity of electrons at an energy $E$, $\sigma$ is the whole conductivity given by $\textstyle\sigma = \int \sigma(E) ( -\frac{df(E)}{dE} ) \, dE$, and the function $f(E)$ is the energy occupation function. The Fermi level $\mu$ is defined by $f(\mu) = \tfrac{1}{2}$. The fact that the Seebeck coefficient depends on the structure of $\sigma(E)$ near $\mu$ means that the thermopower of a material depends greatly on impurities, imperfections, and structural changes, all of which can vary with temperature and electric field.

### Phonon drag

Phonons are not always in local thermal equilibrium; they move against the thermal gradient. They lose momentum by interacting with electrons (or other carriers) and imperfections in the crystal. 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. This contribution is most important in the temperature region where phonon-electron scattering is predominant. This happens for

$T \approx {1 \over 5} \theta_\mathrm{D} ,$

where $\theta_D$ is the Debye temperature. At lower temperatures there are fewer phonons available for drag, and at higher temperatures they tend to lose momentum in phonon-phonon scattering instead of phonon-electron scattering. This region of the thermopower-versus-temperature function is highly variable under a magnetic field.[citation needed]

### Relationship with entropy

The thermopower or Seebeck coefficient, represented by S, of a material measures the magnitude of an induced thermoelectric voltage in response to a temperature difference across that material, and the entropy per charge carrier in the material.[5] S has units of V/K, though μV/K is more common.

Superconductors have S = 0 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.

## Applications

### Thermoelectric generators

The Seebeck effect is used in thermoelectric generators, which function like heat engines, but are less bulky, have no moving parts, and are typically more expensive and less efficient. They have a use in power plants for converting waste heat into additional electrical power (a form of energy recycling), and in automobiles as automotive thermoelectric generators (ATGs) for increasing fuel efficiency. Space probes often use radioisotope thermoelectric generators with the same mechanism but using radioisotopes to generate the required heat difference. Commercially available examples can be found in self-powered fans and chargers designed for use on wood stoves.[6][7][8]

### Peltier effect

The Peltier effect can be used to create a refrigerator which is compact and has no circulating fluid or moving parts; such refrigerators are useful in applications where their advantages outweigh the disadvantage of their very low efficiency.

### Temperature measurement

Thermocouples and thermopiles are devices that use the Seebeck effect to measure the temperature difference between two objects, one connected to a voltmeter and the other to the probe. The temperature of the voltmeter, and hence that of the material being measured by the probe, can be measured separately using cold junction compensation techniques.

## References

1. ^ As the "figure of merit" approaches infinity, the Peltier–Seebeck effect can drive a heat engine or refrigerator at closer and closer to the Carnot efficiency. Disalvo, F. J. (1999). "Thermoelectric Cooling and Power Generation". Science 285 (5428): 703–6. doi:10.1126/science.285.5428.703. PMID 10426986. Any device that works at the Carnot efficiency is thermodynamically reversible, a consequence of classical thermodynamics.
2. ^ The voltage in this case does not refer to electric potential but rather the 'voltmeter' voltage $V = -\mu/e$, where $\mu$ is the Fermi level.
3. ^ a b c "A.11 Thermoelectric effects". Eng.fsu.edu. 2002-02-01. Retrieved 2013-04-22.
4. ^ Cutler, Melvin; Mott, N. (1969). "Observation of Anderson Localization in an Electron Gas". Physical Review 181 (3): 1336. doi:10.1103/PhysRev.181.1336.
5. ^ Rockwood, Alan L. (1984). "Relationship of thermoelectricity to electronic entropy". Phys. Rev. A 30 (5): 2843–4. Bibcode:1984PhRvA..30.2843R. doi:10.1103/PhysRevA.30.2843.
6. ^ Canadian company Caframo offers such products under the Ecofan label.
7. ^ The Biolite stove company http://biolitestove.com/campstove/camp-overview/how-it-works/#sub
8. ^ "Thermoelectric Generator For Sale". Thermoelectric-generator.com. Retrieved 2013-04-22.

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• Rowe, D. M., ed. (2006). Thermoelectrics Handbook:Macro to Nano. Taylor & Francis. ISBN 0-8493-2264-2.
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