Modern Physics of the Thermoelectric Phenomena: Achievements and Problems

Abstract

This chapter discusses internal discrepancies of contemporary conceptions of physics of thermoelectric phenomena (Seebeck, Peltier, and Thomson effects). These conceptions contradict also with experimental data obtained in a wide range of temperature for various materials (pure metals, alloys, Si, Ge, intermetallic and oxide compounds, borides, and silicides). One of these contradictions arises from the energy conservation law and definition of the Seebeck coefficient—the last cannot exceed 86.25 μV/K in any material. This limitation is met in metals and alloys, while in nonmetallic materials it exceeded hundreds and thousands of times. Experimental temperature dependence of the Seebeck coefficient demonstrates the polarity reversal and sharp extrema (increases up to 100–1000 times) for various materials, which are not followed from theory. Constancy of the Seebeck and Peltier coefficients (underlying the definitions of thermoEMF and Peltier heat) contradicts with Thomson formulae requiring temperature dependence of these coefficients (otherwise the Thomson effect is absent in any materials). The role of structural (spatial) inhomogeneity of the thermoelectric material and the wave nature of thermal radiation are discussed for potential physical mechanism of thermoEMF generation. Extension of expressions for charge and thermal energy flow to take into account nonlinear properties leads to huge mathematical complications.

1. Introduction

The thermoelectric materials are the new energy efficient materials for the green energy harvest. Owing to (i) threat to exhaustion of fossil fuels (oil, gas, and coal), so there is a need to create efficient and cheap methods of using renewable energy sources and (ii) the expansion of technical and technological capabilities of mankind, which leads to an increase in the amount of waste heat and pollution of the environment. This worsens the conditions of human existence and forces to look for methods of using waste heat; (iii) creation of the theory of “phonon glass, electronic crystal” [1] and the discovery of a new class of thermoelectric materials, the oxide compound NaCo₂O₄ having high value of thermopower [2], the research interest on the thermoelectric phenomena in various materials and structures has increased dramatically in the last two decades.

Scientific aspect of this interest is also important because thermoelectricity:

1. allows us to have information on the state of the electronic subsystem in various materials and structures;

2. has served an experimental basis for verification of concepts of equilibrium and nonequilibrium thermodynamics;

3. can serve as a scientific basis for creating new thermoelectric materials used in devices to convert waste heat and solar radiation into electricity.

Over two centuries of research on thermoelectric phenomena, many famous scientists have contributed to their study and practical applications. As a result, a physical concept of these phenomena was developed, which made it possible to explain the temperature dependence of the coefficient of thermoEMF (Seebeck coefficient) S in metals and semiconductors at high temperatures T, where the dependence S(T) is weak. These concepts are based on the laws of linear nonequilibrium thermodynamics and the solution of the Boltzmann kinetic equation (mainly with respect to the relaxation time approximation τ). However, such nonlinear properties of S(T) such as changes in sign and extreme value at low temperatures are often considered as anomalies not worthy of attention and therefore not reflected in these concepts.

A close examination of theoretical conceptions, including phonon drag, and a comparison of their results with experiment reveals a number of internal disagreements and inconsistencies. For example, researchers have contrary definitions of “absolute thermopower” and “differential thermopower” [3, 4]. The present chapter will discuss (1) the main theoretical and experimental results of thermoelectric phenomena, (2) the internal inconsistencies of the theoretical concepts and their discrepancies with the experiment, (3) the possible causes of these inconsistencies and discrepancies, and (4) the possible ways to overcome them. The effects of magnetic field as well as anisotropy of material will not be considered.

2. Theoretical basics

There are three thermoelectric phenomena (Figure 1):

1. Seebeck effect − generation of thermoEMF Δφ in material due to the temperature difference ΔT = T_H−T_C (discovered in 1821).

2. Peltier phenomenon − cooling (Q_P < 0) or heating (Q_P > 0) of contact of different conductors, connected in series, according to the direction of electrical current I (discovered in 1834).

3. Thomson phenomenon (1854) − evolution (Q_T > 0) or absorption (Q_T < 0) of additional heat in homogenous conductor, which is simultaneously affected by a temperature difference ΔT and an electric current I. Here, T_H and T_C are temperatures of the hot and cold ends of the sample, respectively. The most studied and frequently used phenomena are the Seebeck phenomenon (generators of electrical energy and thermocouples to measure temperature in wide range as well as thermopiles to measure thermal radiation intensity) and the Peltier phenomenon (refrigerators, heaters, and heat pumps), which are considered mutually reversible, while the Thomson phenomenon is of scientific interest mainly because of its small value.

Figure 1.

Setup for observing Seebeck (a), Peltier (b), and Thomson (c) effects.

The thermoelectric properties of a material characterized by Seebeck S (thermoEMF), Peltier π, and Thomson τ coefficients:

$$\mathrm{\Delta }\phi =S\mathrm{\Delta }T,$$E1

$$Q_{\mathrm{P}}=\mathit{\pi I},$$E2

$$Q_{\mathrm{T}}=\mathit{\tau I}\mathrm{\Delta }T.$$E3

The additional heat quantities Q_P and Q_T in expressions (2) and (3) are linear functions of current I, as opposed to Joule heat $Q_{\mathrm{J}}=I^{2}R$ of the sample of resistance R. Therefore, the emanation or absorption of Q_P and Q_T depends on the direction of the current. Thomson effect is considered to be positive (Q_T > 0) and causes heating of the conductor if the direction of the current (I) coincides with the gradient of the temperature (T). Samoylovich and Korenblit [3, 5] reported that the emanation of the heat is proportional to a current, which is caused by the inhomogeneity of the system through which the current flows. This inhomogeneity cannot be of chemical nature only but also due to difference in their structure. For example, thermoEMF is generated at the interface of mechanically deformed and non-deformed (hardened and non-hardened, quenched and unquenched) sections of the homogeneous conductor [6] as well.

The definitions of the Seebeck, Peltier, and Thomson effects, written in the form (1)–(3), imply by default that coefficients S and τ do not depend on T (or ΔT) while π and τ do not depend on I.

Thomson’s thermodynamic theory [7] has established an intercorrelation of coefficients S, π, and τ (the first and the second thermoelectric relations, respectively):

$$\tau =T\left(\mathit{dS}/\mathit{dT}\right),$$E4

$$\pi =\mathit{TS}.$$E5

Further development of thermodynamics of the thermoelectricity has led to reciprocity relations [8, 9] of kinetic coefficients $L_{12}=L_{21}$ in the generalized linear equations for charge flow j(T, E) and heat flow q(T, E):

$$j=-L_{11}\nabla \frac{\mu }{T}+L_{12}\nabla \frac{1}{T},$$E6

$$q=-L_{21}\nabla \frac{\mu }{T}+L_{22}\nabla \frac{1}{T}-\mathit{\mu j},$$E7

where μ = μ₀ + eφ is the electrochemical potential consisting of chemical μ₀ and electric φ parts.

The linear Eqs. (6) and (7) for the charge and heat flows, bringing to Onsager’s reciprocity relations [8, 9], deny the possibility of thermoEMF generation in the homogeneous isotropic conductor (Magnus’s law, 1851; see also [3, 5]). Thus, Samoylovich and Korenblit [3, 5] have emphasized that the assumption on reversibility of thermoelectric heat (Peltye and Thomson) does not follow from the second principle of thermodynamics and is the additional assumption. On the other hand, for developing Eqs. (4) and (5), W. Thomson considered the thermoelectric phenomena as reversible that is not rather reasonable. This idea of Thomson’s theory was criticized by Boltzmann as it has fail to confirm the results of various experiments [3, 5]. We will see below that the experimental data used for confirmation, which almost do not contain any nonlinearity. Herein, the nonlinearities (see, for example, [6]) were excluded intentionally as anomalies.

The situation is that the researchers ignore irreversibility of thermoelectric effects [3]: “Thomson applied the first and second principles of thermodynamics to the analysis of thermoelectric phenomena, considering thermoelectric processes to be reversible. The proportionality of the Peltier and Thomson heats to the strength of the electric current and, consequently, the fact that when the direction of the current changes, instead of releasing heat, there will be absorption, or, conversely, makes natural the assumption that in the case of the Thomson and Peltier effects, we are dealing with processes that are reversible in the thermodynamic sense of the word.” It can be seen here that the reversibility of thermoelectric phenomena is provided by excluding from consideration the fact that when an electric current passes through a homogeneous conductor placed in a medium with a uniform temperature, Joule heat Q_J = I²R is emanated and dissipated in the surrounding medium. Joule heat is the main part of the energy released in the conductor. One can say that the emanation of Joule heat when an electric current flows through a homogeneous conductor placed in a medium homogeneous in temperature is common, while no one has seen the reverse process - the generation of an electric current in a homogeneous conductor due to the uniform temperature along the conductor.

It has to be noted that in definition of Seebeck coefficient (1) $e\mathrm{\Delta }\phi$ represents (Figure 1) excess Coulomb energy of electrons on the cold end of a sample relative to the hot end, arising from their diffusion due to surplus of thermal energy $k_{\mathrm{B}}\mathrm{\Delta }T$ on the hot end relative to the cold one (e is the electron’s charge). Therefore, it is reasonable to write $e\mathrm{\Delta }\phi \le k_{\mathrm{B}}\mathrm{\Delta }T$, which in a steady state without a current in an external circuit (an open-circuit mode) leads to the maximum value of Seebeck coefficient: $\left|S\right|\le k_{\mathrm{B}}/e=86.25\mathrm{\mu V}/\mathrm{K}$. Here $k_{\mathrm{B}}$ is Boltzmann’s constant. It is important to note that this limit does not depend on both the studied substance and on the temperature range.

Apparent expressions for S(T) of metals and semiconductors have derived from Boltzmann kinetic equation [10]:

For metals, (degenerate electron gas)

$$S\left(T\right)=-\frac{{\pi }^2}{3}\frac{k_{\mathrm{B}}}{e}\frac{k_{\mathrm{B}}T}{\mu }\left(r+\frac{3}{2}\right),$$E8

For semiconductors (nondegenerate electron gas)

$$S\left(T\right)=-\frac{k_{\mathrm{B}}}{e}\left[\frac{\mu }{k_{\mathrm{B}}T}-\left(r+\frac{5}{2}\right)\right].$$E9

Here, r is the exponent of momentum relaxation. These expressions are applicable for higher temperatures (region I in Figure 2), while at lower temperatures, there are polarity reversal and extremum(s) of S(T). Increasing of |S(T)| at low temperatures (region II in Figure 2) is usually explained by phonon drag [10, 11, 12, 13] (Herring model, see also Seeger [4]):

Figure 2.

Typical temperature dependence of the thermopower S(T) (schematic).

$$S_{\mathrm{ph}}=\frac{k_{\mathrm{B}}}{e}\left(\frac{m^{\ast }{s}^2}{k_{\mathrm{B}}T}\right)\frac{⟨{\tau }_{\mathrm{e}}{\tau }_{\mathrm{ph}}/{\tau }_{\mathrm{eph}}⟩}{⟨{\tau }_{\mathrm{e}}⟩}\propto T^{-7/2}.$$E10

Here, s is the sound velocity, τ_e – full relaxation time of electrons momentum, τ_eph – relaxation time of electrons momentum due to scattering on phonons, and τ_ph – relaxation time of phonons averaged over momentum. For moderate temperatures, when phonon-electron scattering obeys Herring model, S_ph ∼ T^−7/2 [14]. Gurevich and Mashkevich [15] showed that phonon drag effects not only S but also the electrical conductivity of semiconductors.

This mechanism implies that phonons (thermal vibrations of atoms) move from the hot end of the sample to the opposite end simultaneously with electrons. Phonons average velocity (≈ 5∙10³ m/s) in solids is essentially higher than electrons average one (≈ 40 m/s for diffusion in Si). Therefore, phonons, colliding with electrons, pass them part of impulse, creating an additional charge flow and corresponding thermoEMF. The estimation of this additional thermoEMF gives 400 mV/K up to 10 V/K [4].

Based on thermodynamic considerations, it is assumed that near 0 K (region III in Figure 2), the Seebeck coefficient tends to zero. This assumption is experimentally confirmed in many pure metals and alloys and nonmetallic compounds (Figure 3a,b,d-g), but a deviation from it is observed in some doped alloys and nonmetallic materials (Figure 3c,h,i, and 4).

The opinions of researchers are differing about where the thermoEMF occurs. In many reports [4, 27], it is believe that thermoEMF occurs in the volume of a homogeneous material due to a temperature drop (absolute thermopower). Others [3, 5] associate the appearance of thermoEMF with the inhomogeneity of the material. Ioffe [28] related part of S to the contacts. However, this question is still open due to the lack of direct experimental evidence of one or the other. From an experimental point of view, absolute thermopower is irrelevant since it cannot be probed directly from a voltmeter.

The relationship of S with the electron energy structure (energy levels) of a metal is given by the Mott formula [29]:

$$S=\frac{{\pi }^2}{3}\frac{k_{\mathrm{B}}^{2}T}{e}{\left[\frac{dln\sigma \left(E\right)}{\mathit{dE}}\right]}_{E\mathit{‡}\mu },$$E11

$$\sigma \left(E\right)=-\frac{2e^2}{3m^{\ast }}\underset{0}{\overset{\infty }{\int }}g\left(E\right){\tau }_e\frac{{\mathit{df}}_0\left(E\right)}{\mathit{dE}}\mathit{dE},$$E12

$g\left(E\right)\mathit{dE}=\frac{4\pi {\left(2m^{\ast }\right)}^{3/2}}{h^3}\mathit{dE}$ is the density of electronic states; m* is an effective mass of charge carriers; h – Plank constant, f₀(E) is the Fermi distribution function

$$f_0\left(E\right)=\frac{1}{1+e^{\left(E-\mu \right)/k_{B}T}}.$$E13

Comparing distribution of energy flow in the thermopile branches (Figure 1) in the framework of Thomson’s theory and the computational results, Spry [30] concludes that Thomson’s theory is based on the erroneous precondition. Hence, for the description of the thermoelectric phenomena, four parameters (σ, κ, S, and π) of a material suffice, and Thomson’s phenomenon can be neglected.

3. Experimental results

For almost two-centuries history of a thermoelectricity, the thermoelectric properties of the various materials have been investigated and agreement of experimental data with expressions (1), (8) and (9) at high temperature, assuming weak variation in the values of S, π and τ while significant deviations from these expressions have been observed at a low temperatures. Temperature dependence S(T) of various materials are shown in Figures 3–8.

Figure 3.

Temperature dependence of thermopower in: Metals at low (a) [16] and high temperatures (b) [17]; (c) germanium [18]; (d) silicon [19]; (e) borides AlB₁₂ (1), B₁₄Si (2), FeB29:5 (3) [20]; (f) FeSb₂ [21]; (g) HfTe₅ and Hf_0.95Ti_0.05Te₅ monocrystals [22]; (h) CeB₆ thermopower S(T) (1), specific resistance ρ(T) (2), effective mass m*(T) (3) [11]; (i) (TMTSF)₂PF₆ [23]; (j) TlInSe₂ [24]; (k) the doped silicate glass [25].

Figure 4.

S/T as a function of temperature for four different Ce-based compounds [26] using previously published data by three different groups. In each panel, the horizontal vector points to the value corresponding to γ/(NAve). Note the semilogarithmic scale in the case of CeCu₆.

Figure 5.

Thermopower Q (as is in original, instead of S – G. A.) of a filament of p-germanium (Geballe, unpublished), compared with the Q of Eq. (9) [14]. The room temperature carrier concentration is 5.0∙10¹³ cm⁻³. The dashed curve is computed from (9), using the cyclotron masses; the bend over at the right takes account of intrinsic conduction near room temperature by placing the EMF for the holes in parallel with one of similar form for the electrons.

Figure 6.

Temperature dependence of thermopower of diamond with hole density p = 10¹⁵ cm⁻³, calculated from first principles [31]. Solid red curve is for natural diamond (1.1% ¹³C). Dashed blue curve is for isotopically pure diamond (100% ¹²C). Dotted black curve is the calculated magnitude of the thermopower for silicon at electron density 2.8∙10¹⁴ cm⁻³ [32]. Blue arrow shows the maximum measured thermopower for FeSb₂ at 10 K [21], while purple arrow shows the maximum measured thermopower for isotopically enriched Ge at 15 K [33].

Figure 7.

(a) Variation of thermopower (S) with temperature of compacted Ni-NPs samples for various TOP concentration [34]. Inset shows variation of S in the temperature range 100–300 K; (b) Thermopower of various Ni samples, as indicated [35]. Inset: Low temperature regime.

Figure 8.

Temperature dependence of thermopower S, electrical conductivity ρ, thermal conductivity κ, and thermoelectric figure of merit ZT for Cu₂Se around the structure transition [36].

Let us note some features of the experimental data presented in Figures 3–8: (1) sharp changes of S(T) at low temperatures; (2) polarity reversal of S in many cases; (3) the maximum value of S in many materials essentially exceeds the limit of 86 μV/K. For example, in the compound TlInSe₂S ≈ 10 V/K at T ≤ 100 K (Figure 3j) and in the organic semiconductor (TMTSF)₂PF₆, S(T) increases at T → 0 K (Figure 3i) instead of reduction according to the theory. Properties (1) and (2) have been found in metals more than 100 years ago [3, 5]. Consequence of these features signifies a strong temperature dependence of Peltier π and Thomson τ coefficients, defined by relations (4) and (5).

Seeger [4] gives dependence of S(T) in silicon (Figure 3d) where S(T) ≈ 100 mV/K is reached at T = 30 K, as confirmation of phonon drag. However, there is a question: what is a source of additional energy of electrons to carry out the energy conservation law? Such high magnitude of S has observed in Ge by Geballa at about 20 K (Figure 5) [14].

There are many results on exotic materials as well as on nanomaterials. Li et al. [31] investigated lightly doped diamond (Figure 6) and high S value (about 100 mV/K at 100 K) as well as its temperature dependence explained in terms of phonon drag.

The nanoparticle size (d) dependence of S was first observed by J. Singh et al. [34] and Soni and Okram (Figure 7) [35]. It was found that the magnitude of S increases as d decreases and achieves the large value about 100 mV/K at 20 K for d ≤ 7 nm, while for bulk nickel, this is about few μV/K. Authors have tried to explain that this huge value of S is a result of the enhanced grain-boundary scattering combined with quantum confinement. It is also noteworthy that the nanoparticles did not show sign reversal, in contrast to bulk nickel.

Nanotechnology approach has raised ZT value up to ∼3.6 at 1000 K for well-known Si-Ge solid solution [37], a high-temperature thermoelectric in use. Supersaturated solid solutions of Si-Ge containing ∼1 at.% Fe and 10 at.% P are prepared by high-energy ball milling. The bulk samples consisting of ultrafine nano-crystallites (9.7 nm) are obtained by the sophisticated “low-temperature and high-pressure sintering process.” Despite that the electrical resistivity is slightly high due to the localization of electrons associated with the highly disordered structure and low electrical density of states near the Fermi energy, a very low thermal conductivity κ (< 1 W m⁻¹ K⁻¹) and a very large value of Seebeck coefficient |S| exceeding 470 μV K⁻¹ have been achieved in association with the nanostructuring and the Fe 3d impurity states, respectively.

Hinterleitner et al. [38] have estimated ZT = 5–6 at 370 K for 1 μm thick films of the metastable Heusler alloy Fe₂V_0.8W_0.2Al. Rising of ZT is mainly due to very large slope of density of states near Fermi level. For such materials, ZT = 20 was predicted theoretically [39].

An interesting result has been achieved in Cu₂Se [36] – in the region of structure transition, thermoelectric figure of merit ZT has risen up to 470 – highest value in the history of thermoelectricity. On this basis, one can say that (1) an absence of any fundamental limitations for the value of ZT has been experimentally conformed; (2) the ideal heat engine has been created having efficiency of Carnot’s cycle; (3) structure of the thermoelectric is crucial for thermoelectric phenomena. Unfortunately, this heat engine is no practical applicable because of the very narrow (order of few K) working interval of temperature, and low total efficiency.

One possible way to improve thermoelectric properties is using two-dimensional materials with negative correlation between electrical and thermal conductivity [40], which leads to increasing of σ while κ decreases. For the 16-nm-thick, two-dimensional SnS₂ nanosheets obtained at 300 K thermopower 34.7 mV/K and ZT ≈ 0.13, last value is ∼1000 times greater than previously reported bulk single-crystal SnS₂.

The silicate glass doped by ruthenium dioxide [25, 41, 42] stands by itself among the materials of thermoelectric properties investigated. This glass has metallic properties (S ≈ 10−20 μV/K, $\rho \propto T^{\alpha },\alpha =1-2$) near room temperature, and the maximum of S(T) ≈ 1.7 mV/K is observed at T ≈ 970−1000 K instead of at low temperatures (Figure 3k, more detailed in Figure 9). Such behavior, being accompanied by a similar change of resistance R(T), is assumed to be a consequence of structural transformations in silicate nanocrystals leading to change of energy gap between the impurity band and the top of the valence band of glass [25, 43]. This energy gap is the pseudogap in reality, and nanocrystals act as localization centers for charge carriers [44, 45].

Figure 9.

Temperature dependence of resistivity and Seebeck coefficient for silicate glass doped with RuO₂ (thick film resistor). Insert is same about room temperature [25].

It has been reported in the nineteenth century that S depends not only on the composition of the materials but also on its mechanical deformation (change in structure) [6]. Mechanism of this effect remains unclear. Effect of isotopic substitution of impurities in metals on S remains unexplained for many years [27]. Progress in thermoelectric materials in last two decades (Figure 10) based on nanotechnology – nanosize in homogeneities generated in the material to enhance phonon scattering and decreasing the thermal conductivity κ. But this method has a fundamental lower limit of about 0.1 W m⁻¹ K⁻¹, inherent in amorphous materials. Lower value of κ in experiments is about 0.2–0.5 W m⁻¹ K⁻¹ for thermoelectric materials, so one can say that this method almost achieved the limit.

Figure 10.

Achievements in thermoelectric materials. Fe₂V_0.8W_0.2Al from [37], Si:Ge(P,Fe) from [38].

Effect of carrier concentration on S was directly conformed in experiments on thermoelectric profiling of p-n-junctions, made of GaAs (Figure 11) [46] and Si (Figure 12) [47]. In these experiments, the heated tip of the scanning tunnel microscope was in contact with the sample along a line perpendicular to p-n-junction (inset in Figure 11b) and the thermoEMF between the tip and the sample was measured at each point. Despite the nanometer resolution, these results do not solve the problem of location of the thermoEMF because of measuring the thermoEMF between the material of heated tip and the cool sample at the point of contact.

Figure 11.

(a) Schematic of the SThEM setup showing a nonuniform temperature zone in the sample; (b) measured thermoelectric voltage (circles) and calculated S multiplied by 22 K (line) across a GaAsp-n junction as a function of distance (x) [46].

4. Discussion of theoretical and experimental results

It was established in two center of investigations that thermoelectric phenomena are strong nonlinear for all materials (metals, alloys, and semiconductors), and these nonlinearities manifest mainly at low temperatures (Figures 3–9), so structure investigations of samples simultaneously with measuring of S(T) is complicated. Moreover, there are only few experimental data on simultaneously measured S(T) and R(T) for the same samples, especially for metals and alloys. As consequence, the features of S(T) in Figures 3–9 are interpreted mainly in itself. This situation is some corrected in last years [16, 17, 19].

The Mott formulas (11) and (12), which relate S(T) and σ(T) through the density of states N(E), are of little use in the interpretation of experimental data due to the fact that there are often no data on N(E).

Let us now to consider the problem of Seebek coefficient measurement. Direct measurement of S(T) in a homogeneous material is essentially impossible (Magnus law, 1851, [4]), and it is determined from measured τ(T) via the expression (4) [10, 48]:

$$S\left(T\right)={\int }_0^T\frac{\tau \left(T\right)}{T}\mathit{dT}=S\left(T\right)-S\left(0\right).$$E14

This situation gives rise to the following problem. In (14), τ(T) and temperature should be measured from T = 0 K, but the third law of thermodynamics forbids reaching 0 K. To get around this obstacle, the assumption is made that $\underset{x\to \mathrm{\infty }}{lim}S\left(T\right)=0$. Unfortunately, this assumption is not justified physically in any way. The third law of thermodynamics does prescribe that Δ(T = 0) = 0 and $\underset{T\to 0}{lim}\mathrm{\Delta }s=0$ (the Nernst theorem, s is an entropy). However, (i) the Nernst theorem is applicable only to equilibrium states, while thermoelectric phenomena are essentially nonequilibrium (there are mass, charge, and energy transport, simultaneously); (ii) the definition of the Seebeck coefficient S(T) = Δφ(T)/ΔT mathematically leads to an uncertainty of 0/0 at T → 0 K, and the value of S(0) may be quite different depending on the behavior of Δ(T) near 0 K. In metals and alloys, indeed S(0) = 0, but recently, materials have been found [11, 22, 23, 26] in which there is no indication that the assumption S(0) = 0 is valid (Figure 3h,i,4 and 8).

This conclusion is additionally confirmed by limit of S(T) from expressions (7) and (9):

$$\underset{T\to 0}{lim}S\left(T\right)=\left\{\begin{array}{ll}0& \text{for\hspace{0.17em}metals},\\ \infty & \text{for\hspace{0.17em}non}‐\text{degenerate\hspace{0.17em}semiconductors}.\end{array}\right.$$E15

As is noted above (page 2), definition (1) implies by default that S does not depend on temperature (or weakly dependence), so ΔT < < T is required in experiment to provide applicability of (1); that is, ΔT should be about few K. But then, average excess of phonon’s energy $k_{\mathrm{B}}\mathrm{\Delta }T\approx 100\mathrm{\mu eV}$ is essentially less than the excess Coulomb energy $e\mathrm{\Delta }\mathit{\phi }\approx 0.4-10\mathrm{eV}$ of electrons, and it is required to find out a source of their additional energy.

To avoid the difficulties associated with the Seebeck coefficient for the second branch of the thermocouple while measuring the absolute Seebeck coefficient of a conductor, it is advisable to use a superconductor in the superconducting state as the second (reference) branch [27]. The Seebeck coefficient of a superconductor is, by definition, zero. However, this raises the question of the contact of the superconductor with the conductor under study, which is not yet considered.

There is a contradiction between the definition of the Seebeck coefficient (1) and the first Thomson formula (4) as well. Namely, definition (1) requires that S does not depend on T, that is, dS/dT = 0. So, the Thomson coefficient τ = TdS/dT = 0. This means absence of the Thomson phenomenon in any materials.

A similar conclusion follows from the fact due to the assumption of small ΔT; definition (1) can be considered as an expansion of the unknown function Δφ(ΔT) in a series in powers of T near T₀ (compare with definitions of temperature coefficients of linear expansion, electric resistance, capacities, inductance): $\mathrm{\Delta }\phi \left(\mathrm{\Delta }T\right)=\left(T-T_0\right){\left(d\mathrm{\Delta }\phi /\mathit{dT}\right)}_{T_0}$, and $S={\left(d\mathrm{\Delta }\phi /\mathit{dT}\right)}_{T_0}$ must also be constant, so τ = TdS/dT = 0 is again in accordance with (4).

As seen from above, understanding the physical nature of thermoelectric phenomena is not far from the situation at the end of nineteenth century [6, 49] and in the 1950s [3, 5]:

“The subject, I have chosen is one intimately connected with the names of at least two well-known members of this University—the late Prof. Cumming and Sir William Thomson. It possesses at present peculiar interest for the physicist; for, though a great many general facts and laws connected with it are already experimentally, or otherwise, secured to science—the pioneers have done little more than map the rough outlines of some of the more prominent features of a comparatively new and almost unexplored region. Some of its experimental problems are extremely simple, others seem at present to present all but insuperable difficulties. And it does not appear that any further application of mathematical analysis can be safely, or at least usefully, made until some doubtful points are cleared up experimentally.” [49].

“The best currently used Bi₂Te₃-Sb₂Te₃ system was optimized in 1949 (Z increased from ≈ 10⁻³ K⁻¹ to ≈ 2.6∙10⁻³-K⁻¹). Over the following 50 years this system was reoptimized only insignificantly (by no more than 15%)… By now the method of solid solutions has exhausted itself. Numerous attempts to introduce third and subsequent systems gave only a 2–3% improvement.

Currently, the efficiency of thermoelectric materials (low-temperature) is Z ≈ 3∙10⁻³ K⁻¹. The next qualitative step is needed. Various options and new methods are tried and studied. However, they give a significant addition to the Z value only for materials with a low initial efficiency value. Therefore, again and again we can only approach the magic number Z = 3∙10⁻³ K⁻¹. Is it a law of nature? Rather, no. But it is an experimental fact. For bad materials the new methods give significant improvements, for good ones - small ones, for very good ones - practically none so far.

Unfortunately, this explains the barrier that thermoelectric physics and technology cannot overcome. New fundamental models and approaches are needed.” [50]

5. Possible ways to solve the problems in thermoelectricity

At present, two directions for finding solutions to existing problems of physics of thermoelectricity can be assumed.

The first one consists of the accounting of nonlinear correlations of thermodynamic forces and flows in expressions (6) and (7) by including the highest order terms (at least - the second order):

$$j=-L_{11}\nabla \frac{\mu }{T}+L_{12}\nabla \frac{1}{T}+\frac{1}{2}{L}_{13}\nabla \frac{\mu }{T}\nabla \frac{1}{T}+\frac{1}{2}{L}_{14}{\left(\nabla \frac{\mu }{T}\right)}^2+\frac{1}{2}{L}_{15}{\left(\nabla \frac{1}{T}\right)}^2,$$E16

$$q=-L_{21}\nabla \frac{\mu }{T}+L_{22}\nabla \frac{1}{T}+\frac{1}{2}{L}_{23}\nabla \frac{\mu }{T}\nabla \frac{1}{T}+\frac{1}{2}{L}_{24}{\left(\nabla \frac{\mu }{T}\right)}^2+\frac{1}{2}{L}_{25}{\left(\nabla \frac{1}{T}\right)}^2-\mathit{\mu j}.$$E17

Bakhareva [51] confirms that for coefficients of linear parts of expressions (16) and (17), the Onsager’s reciprocity relations are again carried out: $L_{12}=L_{21}$. The nonlinear terms cause bifurcation, turbulence, and a hysteresis, complex chemical processes (for example, Belousov-Zhabotinsky reactions), self-organizing of various systems, including biological (up to emergence of life). These phenomena are considered in Prigogine’s fundamental works [52] and Bakhareva’s [51] and Maksimov’s [53] monographs, mainly discussed with reference to the chemical and the biological processes. Stratanovich [54, 55] has considered the fluctuation-dissipation theorems for thermodynamics and their applications to some chemical and the physical systems, for example, to the Benar’s cells formation.

Nonlinear properties in thermoelectricity are considered in a number of works [56, 57, 58, 59], mainly in the case of the high gradient of temperature arising in the nanostructures. Cimmelli et al. [56] noted violation of Onsager’s reciprocity relations because of nonlinear processes. But more detailed consideration of the problem compels one to admit the impossibility of same gradients. Moreover, the conception of temperature comes some undefined at nanoscale.

There are mathematical obstacles for investigation of charge and heat flows (16) and (17) as well.

The second possible direction to eliminate the collected divergences of experiment and the theory of the thermoelectric phenomena, most likely, as supplementing previous one, is based on the wave nature of thermal radiation. As a matter of fact, the existing theories of the thermoelectric phenomena provide information regarding the interaction of the thermal radiation with the thermoelectric material as process of energy exchange only. This idea considered a process of energy transmission from a heat to an electronic gas (the heat engine) of density n and pressure of electronic gas $p={\mathit{nk}}_{\mathrm{B}}T$ changes. In the absence of the temperature gradient, the difference of pressure of electron gas in contacting conductors is counterbalanced by the difference of contact potential. Difference of temperatures in diverse conductors redistributes mobile charges and generates thermoEMF [6].

It is interesting to note that modern researchers of thermoelectric phenomena argue on the generation of thermoEMF in the volume of a homogeneous conductor by the temperature gradient (neglecting the presence of a contact separating two different conductors). However, in the absence of a temperature difference, the pressure difference of the electron gas in the conductors (due to different n and E_F in them) is compensated precisely by the contact potential difference; that is, the state of contact of conductors is important for the occurrence of thermoEMF. In this regard, we recall once again that the Seebeck and Peltier phenomena are mutually inverse, and the experiment of E. Lenz in 1838 unambiguously proves that the second takes place precisely in the contact of different conductors (see page 655 in [6]).

It is well-known that a thermoelectric system has spatial inhomogeneity of various scale (fluctuations of composition, contacts, polycrystallinity, etc.), and as a result, the concentration of free charge carriers n(x) is spatial inhomogeneous. The last leads to an uneven distribution of the electric field of incident thermal radiation at nanoscale. Interaction of inhomogeneous high-frequency electric field with free charge carriers generates unidirectional forces (ponderomotive forces) and corresponding flow of the carriers [60].

From this point of view, free charge carriers in conductors are considered as oscillating system (plasma or Langmuir oscillations) with own frequency [61]

$${\omega }_{\mathrm{p}}=\sqrt{\frac{{\mathit{ne}}^2}{m^{\ast }{\mathit{\epsilon \epsilon }}_0}},$$E18

This oscillating system interacts with an incident thermal radiation (which is a wideband electromagnetic waves, Planck formula). This concept leads to the origin of dispersion relation, resonance, and the nonlinearity. Here, m^* – effective mass of electrons, ε₀ - electric constant, and ε - dielectric permeability of a material. The spatial inhomogeneity of n(x) leads to variations of ω_p in the thermoelectric system, in the orientation and magnitude of the ponderomotive force from point to point (i.e. polarity reversal of the generated thermoEMF). If ω < ω_p, the ponderomotive force draws carriers into the higher field region, while for ω > ω_p, this force throw out carriers same regions; that is, at the frequency $\omega \simeq {\omega }_{\mathrm{p}}$, generated direct EMF changes the polarity [58].

Studies of semiconductor diodes of various designs [62, 63] showed that even diodes aimed for operation at microwave (frequency higher than 1 GHz) lose detector properties at about 100–200 MHz but become capable of detecting at 10 GHz (Figure 12a).

Figure 12b shows the dependence of the detected open-circuit voltage V_OC on microwave power P at 10 GHz for diode GA402B where one can see polarity reversal of V_OC and hysteresis. An arrow pointing down corresponds to an increase in power during measurements; an arrow pointing up corresponds to its decrease. Origin of the hysteresis is not clear yet.

The sign of detected voltage of serviceable diode (curve 1 in Figure 12b) is opposite to the sign of thermoEMF (curve 2), inherent for deliberate breakdown diode. Comparison of the Figure 12a and b makes us assume that the detected voltage (curve 1 in Figure 12b) has an unknown physical nature. Furthest experiments on point-contact, p-n-junction, and Schottky diodes [64] demonstrated the correlation of detector properties (Figure 12b) with the microwave electric field inhomogeneity, generated by various methods.

Figure 12.

(a) Principle of scanning Seebeck microscopy; (b) quantitative profiling of S across a silicon p − n junction with STM [47]. The theoretical S profile based on the doping density and the S profile reconstructed from V_S and ΔT.

Probably, observable features of S(T) in various materials (see Figures 3–9) are the results of interaction of infrared electromagnetic wave (heat radiation) with the inhomogeneous electron (hole) gas in the thermoelectric and corresponding local variations of m*, ε or n, especially in thin transitional contact region, where strong spatial variations of properties of materials and of the ponderomotive force take place (Figure 13).

Figure 13.

(a) Frequency dependence of the detected open-circuit voltage V_OC for the diode GA402B at fixed power P = 1 mW and T = 300 K [62]; (b) the detected open-circuit voltage V_OC versus power P at 10 GHz for same diode [63]. The germanium diffusion p-n-junction mesa-diode GA402B (from Russia) aims to operate as parametric amplifier at 10 GHz [64].

Such approach to the thermoelectric phenomena enters into consideration, unlike linear nonequilibrium thermodynamics, temporal characteristics of thermoEMF generation process.

6. Conclusion

Experiments have confirmed that there are no fundamental limitations for thermoelectric figure of merit ZT. Achieved maximum of ZT ≈ 470 allows us to say that the ideal heat engine has been realized. This engine has efficiency about Carnot’s cycle but does not have practical applications because of the narrow working interval of temperature (order of few K).

Origin of the high value of Seebeck coefficient (1000–100,000 μV/K) and nature of their polarity reversal remain unclear yet.

Comparison of the theory of the thermoelectric phenomena and experimental data in various materials (metals and their alloys, oxides, borides, silicides, and intermetallic compounds) shows that observed nonlinear properties (nonlinearity, an extremum, and sign of change) on the temperature dependence of Seebeck coefficient (and, respectively, Peltier and Thomson coefficients) are common for all materials and essentially differ from theoretical representations. It shows that the possible way to eliminate these divergences is the accounting of temperature dependence of parameters of a material (an electrical conductivity, thermal conductivity, and Seebeck, Peltier, and Thomson coefficients) with simultaneous consideration of wave properties of thermal radiation. Also, the comparative experimental study of S(T) and R(T) as well as structural changes on the same sample is necessary.

The correlations between S and ρ, on the one hand, and the impurity band position in the energy gap of the silicate glass need thorough investigation in understanding the physical mechanism of electrical conduction and the thermoEMF generation in doped silicate glass, which is considered as prospect thermoelectric material.

From the experimental point of view, it is required to locate the position of the contact where the thermoEMF arises in a homogeneous part of the sample. As far as Seebeck and Peltier phenomena are considered mutually reversible and appearance of the last on contact of two differing materials have unequivocally established, the Seebeck effect needs such experimental demonstration as well.

Conflict of interests

The authors declare no conflict of interests regarding the publication of this chapter.

G. Abdurakhmanov and G. S. Vokhidova acknowledge the Ministry of Higher Education, Science and Innovation of Uzbekistan for supporting the Uzbek-Indo joint project Uzb-Ind-2021-78 and Uzbek-Belarus joint project IL-4821091667.

D. P. Rai acknowledges the Government of India, Ministry of Science and Technology, Department of Science & Technology (International bilateral Cooperation Division) for supporting the Indo-Uzbek jont project via Sanction No. INT/UZBEK/P-02.