Thermo electricity

Discovered then included/understood during the XIXe century thanks to work of Seebeck, Peltier or Lord Kelvin, the thermoelectric effect is a physical phenomenon present in certain materials: it binds to it the heat flow which crosses them to the electric current which traverses them. This effect is at the base of applications of refrigeration and generation of electricity: a thermoelectric material will make it possible to directly transform heat into electricity, or to move calories by the application of an electric current.
A great having number of materials of the interesting thermoelectric properties were discovered during the decades 1950 and 1960. It is in particular the case of the bismuth telluride (Bi2Te3) used in the commercial Peltier modules, or of the alloys silicon-germanium (SiGe) used for the power supply of the space probes in thermoelectric generators with radioisotope.
Until now, the outputs relatively low and the high costs of the thermoelectric conversion systems limited them to a niche market. Nevertheless, of recent progress as well as a new interest for these systems, due at the same time to the rise in the costs of energy and the environmental requirements, led to an important revival of the scientific research dedicated to this technology.

Historical aspects

The first thermoelectric effect was discovered by the German physicist Thomas Johann Seebeck in 1821. This one noticed that a metal needle is deviated when it is placed between two conductors of different nature bound by junctions at their ends and is subjected to a heat gradient. He interprets his observations by postulating a link between magnetic field and difference in temperature between the two junctions and establishes the direction of deviation of the needle for a great number of couples. He thus thinks of having found an explanation at the origin of the terrestrial magnetic field. Actually, the effect observed is of electric origin: a potential difference appears with the junction of two materials subjected to a difference in temperature. The most known use of the Seebeck effect is the temperature measurement using thermocouples.
A few years later, in 1834, the French physicist Jean Peltier discovered the second thermoelectric effect: a difference in temperature appears with the junctions of two materials of different nature subjected to an electric current. In 1838, Heinrich Lenz shows that heat is absorbed or released with a junction according to the direction of the current.
The English physicist William Thomson showed into 1851 that the Seebeck effects and Peltier are dependant: a material subjected to a heat gradient and traversed by an electric current heat transfer with the external medium. Reciprocally, an electric current is generated by a material subjected to a heat gradient and traversed by a heat flow. The basic difference between the Seebeck effects and Peltier considered separately and the Thomson effect is the existence of this last for only one material and the uselessness of a junction.

Potential applications

The current and potential applications of thermoelectric materials benefit from the two aspects of the Thomson effect:
On the one hand, the establishment of a heat flow, opposed to the thermal diffusion process, when a material subjected to a heat gradient is traversed by a current, makes it possible to consider applications of thermoelectric refrigeration. This alternative solution with the traditional refrigeration using of the cycles of compression-relaxation does not require any moving part, from where more a greater reliability, the absence of vibration and noise. These properties are fundamental in applications for which the temperature must be controlled in a very precise and reliable way, such as for example for the containers used for the transport of bodies to transplant, or for applications in which the vibrations constitute a considerable embarrassment, such as for example the guidance systems laser or integrated circuits. Moreover, the possibility of creating a heat flux starting from an electric current in a direct way makes useless the use of gases of the freon type, which contribute to degrade the layer of ozone.
In addition, the possibility of converting a heat flow into electric current makes it possible to consider applications of generation of electricity per thermoelectric effect, in particular starting from sources of heat lost like the mufflers of the cars, the chimneys of incinerators, the circuits coolant of the nuclear plants the thermoelectric systems would constitute clean auxiliary energy sources then, since, using unutilised existing sources of heat.
Moreover, very the greater reliability and durability of the systems led to their use for the electricity supply of the space probes. It is in particular the case Voyager probe, launched in 1977, in which the heat flow established between of fissile PuO2 PuO2 is radioactive and disintegrates, it is thus a source of heat and the external medium crosses a thermoelectric conversion system containing SiGe (alloy of silicon and germanium), allowing the power supply of the probe in electricity (indeed, the space probes moving away beyond Mars cannot be supplied with solar panels, solar flow being too weak).
The conversion systems using the thermoelectric effect have weak outputs however, whether it is in generation of electricity or refrigeration. Obtaining better output is expensive, which limits for the moment this technology to commercial niches in which reliability and durability are more important than the costs and the output.

Basic rules, in details

The energy transformation per thermoelectric effect (heat - electricity or electricity - heat) is based at the same time on the Seebeck effects, Peltier and Thomson.

Recall on the coefficients Seebeck, Peltier and Thomson

Seebeck coefficient
A difference in temperature dT between the junctions of two materials has and B implies a potential difference electric FD according to: Sab = dV ⁄ dT
The Seebeck coefficient, also called Thermoelectric Pouvoir is expressed in V.K-1 (or more generally in µV.K-1 within sight of the values of this coefficient in usual materials).
The Seebeck coefficients of two materials are connected to the Seebeck coefficient of the couple according to: Sab = Sa - Sb
Peltier coefficient
In the case of the Peltier effect, an electric current I is imposed on a circuit made up of two materials, which involves a release of heat Q to a junction and an absorption of heat to the other junction, according to : IIab = Q ⁄ I
Thomson coefficient
Contrary to the coefficients Seebeck and Peltier, the Thomson coefficient can be defined directly for only one material. When simultaneously a variation in temperature and an electric current are present, there are generation or absorption of heat in each material segment taken individually. The gradient of heat flux within material is then given by:dQ ⁄ dx = I * (dT ⁄ dx) * τ where X is the space coordinate and T is the Thomson coefficient of material.
Relations between the coefficients Seebeck, Peltier and Thomson
Kelvin showed that the three coefficients Seebeck, Peltier and Thomson are not independent from to each other. They are bound by the two relations :
IIab = SabT ↔ τa - τb = T * (dSab ⁄ dT)

Principles of the energy transformation per thermoelectric effect

Modulate series-connected electrically and in parallel thermically
For the refrigeration or the generation of electricity per thermoelectric effect, a module consists of electrically connected couples. Each couple makes up of a semiconductor material of type p (S>0) and of a semiconductor material of type N (S<0). These two materials are joined by a conducting material whose thermoelectric capacity is supposed to be null. Two branches (p and N) of the couple and all the other couples composing the module are connected in series electrically and parallel thermically (see diagram on the right). This provision makes it possible to optimize the heat flux which crosses the module and its electrical resistance. By preoccupation with a simplicity, we will reason in the continuation on only one couple, formed of two materials of constant sections.

Modulate thermoelectric refrigeration
The figure on the right presents the schematic diagram of a couple p-n used for the thermoelectric refrigeration. The electric current is imposed in such a way that the charge carriers (electrons and holes) move cold source with the hot source (with the thermodynamic direction) in the two branches of the couple. By doing this, they contribute to a transfer of entropy of the cold source to the hot source, and thus to a heat flux which will be opposed to that of thermal conductivity. If the selected materials have good thermoelectric properties (we will see thereafter which are the important parameters), this heat flux created by the movement of the charge carriers will be more important than that of thermal conductivity. The system will thus make it possible to evacuate heat since the cold source towards the hot source, and will act then like a refrigerator.
In the case of the generation of electricity, it is the heat flow which involves a displacement of the charge carriers and thus the appearance of an electric current.

Output of conversion and parameters important

Calculation of the output of conversion of a thermoelectric system
The calculation of the output of conversion of a thermoelectric system is carried out by determining the relation between the heat flow and the electric current in material. It requires the use of the relations of Seebeck, Peltier and Thomson (see higher), but also laws of propagation of heat and electric current.
The following example presents the calculation of the output of conversion in the case of the refrigeration (that of the generation of electricity can be carried out by similar reasoning).
Thus let us take again the preceding diagram. In each of the two branches of the couple, the heat flow generated by the Peltier effect is opposed to thermal conductivity. Total flows are thus in the branch p and branch N : Qp = SpIT - λpAp * dT ⁄ dx and Qn = - SnIT - λnAn * dT ⁄ dx
with X coordinates space, λp and λn thermal conductivities of materials, and Ap and An their sections.
Heat is thus extracted from the cold source with a Qf flow: Qƒ = (Qn + Qp) | x = 0
In same time, the current which traverses both branches is at the origin of a creation of heat per Joule effect I2ρ ⁄ A per unit of length of the branches. By using the equation of Domenicali and by supposing that the Thomson coefficient is null, the conservation of energy in the system is written in the two branches : - λpAp * d²T ⁄ dx² = I²ρp ⁄ Ap and - λnAn * d²T ⁄ dx² = I²ρp ⁄ Ap
By considering boundary conditions T = Tf in x=0 and T = Tc in x = Lp or x = Ln with LP and Ln the lengths of the branches p and N, Tf and Tc the temperatures of the sources cold and hot, Qf is written: Qƕ = (Sp - Sn) ITƒ - KΔT - 1 ⁄ 2 * I²R
with K and R thermal conductance and electrical resistance total of the branches of the couple: K = λpAp ⁄ Lp + λnAn ⁄ Ln and R = Lpρp ⁄ Ap + Lnρn ⁄ An
The electric output W provided to the couple corresponds to the Joule effect and with the Seebeck effect, that is to say: W = I[(Sp - Sn) ΔT + IR]
The output of the thermoelectric system of refrigeration corresponds to the report ⁄ 2 ratio of the heat extracted the cold source to the dissipated electric output, that is to say: η = Qƒ ⁄ W =[(Sp - Sn) ITƒ - KΔT - 1 ⁄ 2 2 * RI²)] ⁄ I[(Sp - Sn) ΔT + IR]
For a difference in temperature ΔT given, the output depends on the imposed electric current. Do two particular values of the current make it possible to maximize is the output of conversion that is to say the heat extracted the cold source Q_f.
By a similar reasoning, the output of a couple p-n used in generation of electricity will be given by the report ratio of the electric output useful delivered for a resistance of load R to the heat flux crossing material: η = Pu ⁄ Qc = I[(Sp - Sn) ΔT - IR] ⁄ (Sp - Sn) ITc + KΔT - 1 ⁄ 2 2 (R + r) I²
Here still, two particular values of I maximize either the output of conversion or the electric power output by the system.
Important parameters to obtain a good output
By maximizing these two outputs of conversion, one can show that they depend only on the Tf temperatures and Tc and an adimensional size ZpnTM called figure of merit (TM is the average temperature of the system, TM= (Tf+Tc) ⁄ 2 2) whose expression is: Zpn = (Sp - Sn)² ⁄ RK
It is noticed that Zpn for a couple is not an intrinsic quantity with material but depends on relative dimensions of the branches of the module through R and K electrical resistance and thermal conductance. The output of conversion of the system (in generation of electricity as in cooling) is maximum when Zpn is maximum, therefore when product RK is minimum, which is checked when : LnAp ⁄ LpAn = (ρpλn ⁄ ρpAn
The figure of Zpn merit then becomes function only of intrinsic parameters to materials: Zpn = (Sp - Sn)² ⁄ (√λpρp + √λnρn
To obtain a maximum output of conversion, it is thus advisable to choose materials constituting the couple so as to maximize Zpn. In general, that simply does not amount individually optimizing two materials to optimize their respective figures of merit Z = S2 ⁄ 2 #40;ρλ). With the majority of the temperatures used in practice, and in particular those used for the generation of electricity, the thermoelectric properties of best materials of the type p and type N are similar. In this case, the figure of merit of the couple is close to the average of the individual figures of merit, and it is reasonable to optimize two materials independently one of the other.
The material optimization for a use in the energy transformation per thermoelectric effect thus passes necessarily by the optimization of their electric and thermal properties of transport so as to maximize the figure of merit: ZT = S²T ⁄ ρλ
A good thermoelectric material will thus have simultaneously raised a Seebeck coefficient, a good electric conductivity, and a low thermal conductivity.

Evolution of the output of conversion according to the figure of merit.
The figure opposite watch evolution of the output of conversion of a thermoelectric system under ideal conditions according to the figure of merit ZT. For example, if ZT=1 and that the difference in temperature is of 300°C, the output of conversion will be of 8%, which means according to the case that 8% of heat crossing material will be converted into electricity, or although the heat extracted by cooling will correspond to 8% of the electric output employed.

Thermoelectric modules

Geometrical optimization
We saw that the properties of conversion of a thermoelectric material couple constituting a module are not only intrinsic: they also depend on the geometry of the system (length and section of the branches of the module) whose electrical resistance R and the thermal conductance depend K on the branches. It is necessary indeed that K is sufficiently low so that a heat gradient can be maintained, while being sufficiently high so that heat crosses the module: if K is null no heat does not cross the module and there is thus no conversion. In the same way, R must be selected so as to have the best possible compromise between the electric output and the potential difference electric. Once the selected materials constituting the module (thanks to the figure of merit ZT), it is thus necessary to optimize the geometry of the system to be able to obtain the output of conversion, the electric output or maximum heat extracted according to the application module.
Segmented modules
The materials used in the modules of thermoelectric conversion are generally effective only in one range of restricted temperature. Thus, the SiGe alloy used for the feeding Voyager probe is effective only with higher temperatures with 1000K approximately. It can thus be interesting, for applications where the variation in temperature is very large, to use several thermoelectric materials in each branch, each one in the range of temperature for which it is most effective. One speaks then about segmented thermoelectric module.

Segmented thermoelectric module.
The figure opposite illustrates the concept of segmented module thermoelectric. We have a very important variation in temperature here (700 K of difference between the hot zone and the cold zone), and no known material is not effective in all the range of temperature. Each of the two branches of the couple is thus formed of several materials. The length of each one of these materials is selected so that it is used in the range of temperature where it is most effective.
Such a module will thus make it possible to obtain an output of conversion, an electric output, or an extracted heat, definitely higher than if each branch were made up only of one material. Thus, the best outputs obtained in laboratory with this type of modules are at present close to 15%. The segmented modules are however price much higher than the simple modules, which restricts them with applications for which the cost is not the factor of decisive choice.
Thermoelectric materials
Materials used in the current devices
The thermoelectric material most usually used with the low-temperatures (150 K-200 K), is formed on the basis of Bi1-xSbx (alloy of bismuth and antimony) and unfortunately presents good thermoelectric properties only in type N (conduction the electrons), which restricts the output of conversion of the system since no material is effective in type p in this range of temperature (let us recall that a thermoelectric conversion system is made up at the same time of branches p and N).
Curiously, whereas its properties are relatively average (ZT 0,6), the application of a magnetic field makes it possible to double the figure of merit which exceeds the unit then. This material is thus generally used in partnership with a permanent magnet.
Vicinity of the ambient temperature
The material more studied at present is Bi2Te3 (tellurium and bismuth alloy). It is used in all the devices functioning in the vicinity of the ambient temperature, which includes the majority of the devices of thermoelectric refrigeration. The best performances are obtained when it is combined in Sb2Te3 (tellurium and antimony alloy) which has the same crystal structure.
Samples of the type p as of type N can be obtained by small variations of composition in the vicinity of the stochiometry. In both cases, values of the figure of merit ZT close to 1 are obtained in the vicinity of the ambient temperature. These good values of ZT are obtained partly thanks to very low thermal conductivity λ, near to 1 W.m-1.K-1 in best materials.
Intermediate temperatures
For a use with average temperature (550K-750 K approximately), the material more used is the telluride of PbTe lead and its alloys (PbSn) Te (lead-tin telluride). Both made up PbTe and SnTe can form a complete solution solid what makes it possible to optimize the gap (forbidden band of the semiconductor) with the desired value. The best materials obtained have figures of merit close to the unit around 700 K.
However, these values are obtained only in materials of the type N. PbTe cannot thus at present constitute with him only the two branches of a thermoelement. The branch p thus generally consists of a material of the type TAGS (for Tellurium-Antimony-Germanium-Money), which, as for him allows, to obtain figures of merit higher than the unit to 700 K only in p. type.
It thus appears crucial to develop a new material which can be used at the same time in type p and type N in this range of temperature. It is indeed easier industrially to use the same type of material for the two branches (and that would make it possible of more than eliminate strongly toxic tellurium).
High temperatures
The alloys containing silicon and germanium have good thermoelectric characteristics with the high temperatures (above 1000 K) and are in particular used for the generation of electricity in the space field. It is in particular of the alloys of this type which are used for the electricity supply Voyager probe.

Optimization of thermoelectric materials

The expression of the figure of merit ZT = (S2T) ⁄ 2 (ρλ) summarizes with it only the difficulty of optimizing the properties of transport of a thermoelectric material. Intuitively, it appears difficult for a material simultaneously to have a good electric conductivity and a bad thermal conductivity, characteristic of insulators. Ideally, a good thermoelectric material should thus have all at the same time the electric conductivity of a metal and the thermal conductivity of glass
The numerator of the figure of merit ZT, S2σ (σ is electric, opposite conductivity of the electrical resistance: σ=1 ⁄ 2 ρ) is named power-factor. In generation of electricity per thermoelectric effect, the useful output will be all the more large as the power-factor will be large. Unfortunately, the Seebeck coefficient and electric conductivity are not independent one of the other, and vary in an opposite way with the concentration out of charge carriers: the best thermoelectric capacities will be obtained in materials of weak concentration out of carriers while best electric conductivities will be it in materials with strong concentration of carriers. By compromise, the best thermoelectric materials will thus belong to the class of the semiconductors.
The second big factor in the expression of the figure of merit ZT is thermal conductivity: a material will have optimal thermoelectric properties for a low thermal conductivity. Indeed, in an intuitive way, a good thermal conductivity would tend to be opposed to the establishment of the heat gradient: heat would cross material without meeting resistance. The optimization of materials will thus require to seek to decrease thermal conductivity, without degrading electric conductivity. Only the contribution of the vibrations of the network will have to thus be decreased, and not the contribution due to the charge carriers.

Ways of research

We saw in the preceding paragraph that the best materials used at present in the devices of thermoelectric conversion have figures of merit ZT close to 1. This value does not make it possible to obtain outputs of conversion which economically make these systems profitable for general public applications. For example, one would need materials for which ZT=3 to be able to develop a competing domestic refrigerator. For the systems of generation of electricity, two means would make it possible to increase the profitability of the systems: a significant growth of their outputs, or a reduction in the costs. The goal of this paragraph is to present in a nonexhaustive way some currently followed ways of research, as well in industrial laboratories as public.

Basic structures dimensionnality

One names basic structure dimensionnality a working of a material for which one or more dimensions are very small compared to the others. It is for example the case of the thin layers into micro-electronic (structure 2D), of nanofils (structure 1D) or nanopoudres (structure 0D), in opposition to the massive material which has 3 dimensions.
These structures generally have properties rather different from massive material of the same composition. In the field of thermoelectricity, the goal of search is double: to seek to improve the output of conversion by using basic structures dimensionnality, while profiting from the systems of mass production used in micro-electronics.
The study of the basic structures dimensionnality became very important since notable improvements of the figure of merit ZT were predicted there theoretically then observed in experiments. The two principal effects observed are a strong diffusion of the phonons by the grain boundaries inducing a reduction in the thermal conductivity of network, and effects of containment of the charge carriers which strongly modify the properties of electric transport. Very high values of the figure of merit ZT, about 2,5 with the ambient temperature, were thus observed in laboratory in structures in thin layers.
At present, these structures are mainly considered for applications to low or average temperatures (<150-200°C). One of the main difficulties is indeed to obtain thermoelectric thin layers whose properties are not degraded with the temperature.

Identification and optimization of new materials

We saw previously that to obtain a good output of conversion, the materials must have the thermal conductivity lowest possible and the electric conductivity strongest possible. It must thus ideally conduct the electric current like a metal, and the heat like glass.
Various properties can make it possible the thermal conductivity of a crystal to approach that of glass. It is mainly:
A complex crystal structure. Indeed most of heat is transported by the acoustic phonon methods. However a material having NR atoms by will have 3 acoustic modes, and 3 (N-1) optical modes, from where the interest of complex structures for which NR is large and the majority of the phonon methods are optical modes which transport heat little.
Atoms slightly related to the remainder of the crystal lattice, or whose positions are not perfectly defined. These atoms induce an important disorder which contributes to the diffusion of the phonons and thus to the reduction in thermal conductivity. On the other hand, as they take part little in electric conductivity, the disorder does not cause too important degradation of this conductivity.
Particularly studied promising materials
Currently, three material classes are particularly studied according to these recommendations It are:
Compounds of the semi-Heusler type, general formula XYZ with X and Y of metals of transition and Z a metalloid or a metal, for example ZrNiSn. These compounds present very high power-factors S2σ, at the same time in type p and type N. One the their most interesting characteristics is the possibility of doping on each of the three sites, which tends moreover to modify the vibrations of the network. However their thermal conductivities are too high, and the best ZT obtained at present are about 0,7 to 700 K-800 K.
The second family of compounds, which presents a very great number of structural varieties, is that of the clathrates. These compounds have a relatively open structure made up, for the most studied compounds at present, of a network of If, GaGe or forming GaSn of large cages in which can be inserted heavy atoms. Their thermal conductivity is similar to that of glass whereas the electronic properties, which are mainly function of the network, are good. The best figures of merit obtained approach the unit around 800 K.
The third very studied family is that of skutterudites. These compounds have a formed cubic structure of a network of the type MX3, with in the center of this network a large cage in which can be inserted heavy atoms. These compounds have very high Seebeck coefficients as well as a good electric conductivity, but their thermal conductivities remain too high. The best figures of merit obtained are close to 1,4 around 800 K

Thermoelectric cooling

It is possible to cool objects by using thermoelectricity. One uses for that of the named components Peltier modules which transform an electric current into a difference in temperature.

General information

Modulate Peltier
The Peltier modules are named thus because they put in opens thermoelectricity and more precisely the Peltier effect. This module is fed by a current and presents two faces, one known as cold and the other heat. The object to be cooled must be put on the cold face, while it is necessary to have a mechanism of dissipation of the heat on other side.


Diagram of a cell with Peltier effect
A Peltier module consists of a series of couples made up of a selected semiconductor material so that the electrons can play the part of coolant.


In this section, it is used the following notations:
I is the current one crossing the Peltier module
Qf is the heat absorptive by the cold side of Peltier
Qc is the heat rejected by Peltier
Πab is the coefficient of Peltier effect of the module
Sm is the coefficient of Seebeck effect of the module
Km is the thermal resistance of the module
Rm is the electrical resistance of the module
Tf is the temperature on the cold side
Tc is the temperature on the hot side
ΔT = Tc - Tf
To model a thermoelectric module, the first idea is to give a transferred heat corresponding to the Peltier effect Qƒ = IIab * I or Qƒ = Sm * Tƒ * I. It is also necessary to consider heat due to the Joule effect which will apply to the 2 faces of the modules and which will increase with the feeding of this one. Absorptive heat is thus to decrease by 1 ⁄ 2 2 * Rm * I². It is also necessary to take account of the thermal conductivity which is opposed to the wanted effect, it is thus necessary to remove a heat of Km * ΔT
Finally there is an absorptive heat which is Qƒ = Sm * Tƒ * I - 1 ⁄ 2 2 * Rm * I² - Km * ΔT. This expression is not easily exploitable, more especially as coefficients Sm, km, Rm vary according to the temperature. To be able to use the Peltier modules correctly, the manufacturers provide curves giving the difference of temperature according to the current applied and transferred heat.
The terminal voltage of the module is V = Sm * ΔT + I * Rm
The heat rejected by the module is Qc = Qƒ + V * I

Advantages and disadvantages

Compared to a cooling system by compressor, thermoelectric cooling by Peltier effect has as main advantages:
its simplicity and thus its low costs of manufacture
the absence of use of gas
little of maintenance necessary
the low noise level (including with the addition of a ventilator)
the absence of vibrations
It has on the other hand like principal disadvantages
a high electricity consumption (output 0,6)
a less effectiveness (dependence with the ambient temperature)

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