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CURRENT STATE OF THE THEORY OF THERMOELECTRIC AND THERMOMAGNETIC PHENOMENA IN SEMICONDUCTORS
A. G. Samoilovich and L. L. Korenblit
CONTENTS
Part I. Thermodynamic Theory
§ 1. Basic thermoelectric phenomena . . . . . . . . . . . . . . . . . . . . . 244
§ 2. Thermodynamic treatment of thermoelectric phenomena . . . . . . . . . 246
§ 3. Basic results of classical thermodynamics . . . . . . . . . . . . . . . 247
§ 4. The concept of local equilibrium and slow processes . . . . . . . . . . 248
§ 5. Basic ideas of the thermodynamic theory of irreversible processes and an explanation using thermal conductivity as an example . . . . . . . . . . . . . . . . . . 249
§ 6. Generalized laws of electrical conductivity and thermal conductivity . . 251
§ 7. The first law of thermodynamics in differential form and the energy flux (Umov equation) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
§ 8. The first thermoelectric relation . . . . . . . . . . . . . . . . . . . . 254
§ 9. The second thermoelectric relation and the symmetry principle for kinetic coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
§ 10. The second law of thermodynamics in differential form and the entropy flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
§ 11. Critique of the derivation of the second thermoelectric relation from the second law of thermodynamics . . . . . . . . . . . . . . . . . . . . . 258
§ 12. Thermoelectric phenomena in anisotropic bodies . . . . . . . . . . . . 259
§ 13. Thermomagnetic and galvanomagnetic phenomena . . . . . . . . . . . . 262
§ 14. Efficiency of thermoelements . . . . . . . . . . . . . . . . . . . . . 268
Part II. Kinetic Theory
§ 15. Distribution function and kinetic equations. § 16. Formal solution of the kinetic equations. § 17. Generalized laws of electrical conductivity and thermal conductivity in kinetic theory. § 18. Kinetic equations in the case of high temperatures. Long-wave electron run. § 19. Thermoelectric phenomena in monovalent metals at high temperatures. § 20. Equilibrium of electrons in semiconductors. § 21. Thermo-
electric phenomena in semiconductors with an atomic lattice. § 22. Thermoelectric phenomena in ionic semiconductors. § 23. Distribution functions in the presence of weak magnetic fields. § 24. Thermomagnetic and galvanomagnetic phenomena in monovalent metals at high temperatures. § 25. Thermomagnetic and galvanomagnetic phenomena in semiconductors. § 26. Comparison of the theory of electrical phenomena in semiconductors with experiment. § 27. Conclusion.
The study of thermoelectric and thermomagnetic phenomena is of great importance for elucidating the mechanism of a whole series of processes occurring in semiconductors and metals. In the present article a survey is given of the basic theoretical views relating to these questions, and a comparison of theory with experimental data.
Our main task was to investigate the current state of the theory of thermoelectric and thermomagnetic phenomena in semiconductors. However, it is impossible to set forth the theory of thermoelectric and thermomagnetic phenomena in semiconductors in complete isolation from the theory of these same phenomena in metals. Therefore, in this article some questions relating to metals are also touched upon, though by no means all of them, but only those that are characteristic also of semiconductors.
The article sets forth the thermodynamic theory of thermoelectric and thermomagnetic phenomena, based on the thermodynamics of irreversible processes. This theory gives a general method for clarifying the connections between the coefficients characterizing the various thermoelectric and thermomagnetic effects. In the second part of the article the kinetic theory is presented, making it possible to find the numerical values of these coefficients. Kinetic theory is inevitably connected with definite model assumptions. Here the widely used “one-electron” theory of conductivity is taken as the basis.
It should be noted that in recent years a new approach to the construction of the theory of ionic semiconductors has been developed by Soviet scientists (Pekar, Bogoliubov, Tyablikov)\(^{1-3}\). We hope to return to these questions later.
Part I
THERMODYNAMIC THEORY
§ 1. Basic thermoelectric phenomena
Three different thermoelectric effects are observed experimentally in isotropic bodies, namely: thermoe.m.f., the Peltier effect, and the Thomson effect.
Thermo.e.m.f. arises in an open circuit of electrical conductors when there is a temperature gradient in it. In this case it is important that the temperatures of the junctions of different conductors be different. The occurrence of thermo.e.m.f. is connected with a redistribution of carri-
bodies of current carriers owing to the presence of a temperature gradient. The thermoelectric power of any pair of conductors depends both on the physical nature of the conductors themselves and on the temperature values at the junctions. The total thermoelectric power of a given pair of conductors is called integral \((A)\). To determine the dependence of the thermoelectric power only on the physical properties of the given pair of conductors, the concept of differential thermoelectric power \((\alpha)\) is introduced. The differential thermoelectric power is a characteristic of a given conductor and is defined as
\[ \alpha=\frac{\partial A}{\partial T}, \tag{1.1} \]
where the variable is taken to be the temperature of the “hot” junction.
The Peltier effect consists in the fact that, when an electric current passes through the junction of two different conductors, heat is released (or absorbed) at the boundary of their contact (Peltier heat), proportional to the current. For the amount of Peltier heat \((q_{\Pi a/b})\) released per unit time, the following relation holds:
\[ q_{\Pi a/b}=\Pi_{a/b} i, \tag{1.2} \]
where \(i\) is the current, and \(\Pi_{a/b}\) is the Peltier coefficient for the boundary of conductors \(a\) and \(b\). It should be noted that the release of heat proportional to the current is connected exclusively with the inhomogeneity of the system through which the current passes, and it is by no means necessary that this inhomogeneity be due to a chemical difference. For example, in a wire that is chemically homogeneous but has nonuniform density, Peltier heat will also be released. Therefore, in order to cover the most general case, it is more convenient for us to formulate Peltier’s law in differential form, i.e., in the form of a law referring to a given point of the substance.
Namely,
\[ \mathbf{Q}_{\Pi}=\Pi e\mathbf{j}, \tag{1.3} \]
where \(\mathbf{Q}_{\Pi}\) is the density of the Peltier heat flux through the given point, \(\Pi\) is the differential Peltier coefficient, \(e\) is the charge of a current carrier, and \(\mathbf{j}\) is the particle-flux density, so that \(e\mathbf{j}\) is the electric-current density. The Peltier heat released per unit volume per unit time is equal to
\[ q_{\Pi}=-\operatorname{div}\mathbf{Q}_{\Pi}. \tag{1.4} \]
The release of Peltier heat is connected with the fact that, in different parts of an inhomogeneous system, there is a different distribution of current carriers over energies. The transfer by the current of electric charges to different places of an inhomogeneous system is accompanied by a redistribution of the current carriers over energies, which gives rise to the release of Peltier heat.
It is necessary to note that the Peltier effect occurs in thermally homogeneous systems, i.e., in the absence of a temperature gradient in them.
The Thomson effect consists in the fact that, when an electric current passes through a thermally nonuniform system (even in the case of chemical homogeneity), i.e., in the presence of a temperature gradient, additional heat will be evolved, which was called Thomson heat. The Thomson effect in differential form may be written as
\[ q_T=-\tau e(\mathbf{j}\nabla T), \tag{1.5} \]
where \(q_T\) is the Thomson heat evolved per unit time, calculated per unit volume \((q_T>0\) if heat is evolved), and \(\tau\) is the Thomson coefficient.
The evolution of Thomson heat is connected with two causes: first, the temperature gradient creates a nonuniformity in the distribution of current carriers, which leads to a peculiar Peltier effect; second, when passing, the electric current performs additional work against the thermoelectric field.
It should be noted that the three effects listed are not the only thermoelectric effects. In anisotropic bodies (crystals) other effects are also observed (see § 12).
§ 2. Thermodynamic treatment of thermoelectric phenomena
As is known, in 1854 W. Thomson developed the thermodynamic theory of thermoelectric phenomena,^4 at the same time discovering a new effect, subsequently named after him in the literature.
Thomson applied the first and second laws of thermodynamics to the analysis of thermoelectric phenomena, considering thermoelectric processes to be reversible. The proportionality of the Peltier and Thomson heats to the electric current strength and, consequently, the circumstance that, when the direction of the current is changed, heat evolution is replaced by heat absorption, or conversely, makes natural the assumption that in the case of the Thomson and Peltier effects we are dealing with processes reversible in the thermodynamic sense of the word.
The principal results of the thermodynamic theory of thermoelectric phenomena reduce to establishing a mutual relation between the various thermoelectric phenomena. A consequence of the first law of thermodynamics is the relation
\[ \tau=\frac{\partial \Pi}{\partial T}-\alpha, \tag{2.1} \]
which bears the name of the first thermoelectric relation. From the second law of thermodynamics Thomson derived that
\[ \Pi=T\alpha. \tag{2.2} \]
The equality (2.2) is called the second thermoelectric relation. Combining (2.1) and (2.2), it is easy to obtain that
\[ \tau = T \frac{\partial \alpha}{\partial T}. \tag{2.3} \]
This is very important practically, since it is clear from this that, if the thermoelectric relations are valid, then for the study of all thermoelectric phenomena it is sufficient to measure only one quantity, for example, the differential thermoelectric power, as the one most easily amenable to measurement.
It should be emphasized, however, that Thomson himself\(^4\) expressed doubt as to the applicability of the thermodynamics of reversible processes to the analysis of thermoelectric phenomena, since the passage of an electric current is, in essence, an irreversible process associated with the irreversible liberation of Joule heat, which depends quadratically on the current.
Boltzmann\(^5\) subjected Thomson’s theory to still sharper criticism, pointing out that, if the irreversibility associated with the liberation of Joule heat can be neglected for small currents, then the irreversibility due to thermal conduction must be of substantial importance, since the heat flux caused by thermal conduction, being, like the Thomson effect, proportional to the temperature gradient, can no longer be considered negligibly small.
However, in view of the fact that the thermoelectric relations were repeatedly tested experimentally and, at least within the limits of experimental error, were justified, the discussion lost its urgency.
In recent years\({}^{6-8}\) a more detailed thermodynamics of irreversible processes has been developed, in which, instead of inequalities valid in classical thermodynamics and difficult to apply for obtaining any quantitative conclusions, there enter the corresponding equations allowing quantitative analysis of various irreversible processes.
The application of this theory to thermoelectric phenomena made it possible to establish that the first thermoelectric relation, being based on the first principle of thermodynamics, always remains valid. As for the second thermoelectric relation, under certain conditions it must be replaced by another analogous relation (see § 12).
§ 3. Basic results of classical thermodynamics
Classical thermodynamics is based mainly on the following propositions:
I. The quantity of heat \(Q(d)\) absorbed by a system, by virtue of the first principle of thermodynamics, can be expressed by the relation
\[ Q(d) = dE + A(d) - \mu_0\, dN. \tag{3.1} \]
where \(dE\) is the change in the internal energy of the system, \(A(d)\) is the work done by it against external forces, \(\mu_0\) is the chemical potential, and \(dN\) is the change in the number of particles in the system.
II. For reversible processes, the second law of thermodynamics leads to the relation
\[ T dS = Q(d). \tag{3.2} \]
III. For irreversible processes, equality (3.2) is replaced by the inequality
\[ T dS > Q(d). \tag{3.3} \]
For what follows, it is more convenient to give inequality (3.3) another form, known as the Clausius inequality.
Let us imagine a certain system which passes from one state into another and, in doing so, exchanges heat with a system of thermostats having temperatures \(T_1, T_2, T_3, \ldots\), and so on. Let \(\Delta S\) be the change in entropy of the system during such a transition, and let \(\Delta S_i\) be the change in entropy of the \(i\)-th thermostat. Since the system under consideration together with the thermostat as a whole forms an adiabatically isolated system, according to the second law of thermodynamics the total entropy must increase if irreversibility has occurred at least at some part of the process. Thus
\[ \Delta S + \sum_i \Delta S_i > 0. \tag{3.4} \]
But in the case of thermostats, regardless of whether reversible or irreversible heat exchange takes place,
\[ \Delta S_i = -\left(Q_i/T_i\right) \]
(here the minus sign is taken because we regard as positive that heat which is absorbed by the system under consideration). Substituting this into inequality (3.4), we obtain
\[ \Delta S > \sum_i \frac{Q_i}{T_i}. \tag{3.5} \]
§ 4. The Concept of Local Equilibrium
and Slow Processes
The thermodynamic theory of irreversible processes presented below is based on the concepts of local equilibrium and slow processes. Every macroscopic body may be divided into macroscopically small parts, i.e., parts which, being small from the macroscopic point of view, still contain a great many particles. These parts, interacting with the surrounding particles only at their surface, are almost isolated.
Thermodynamic equilibrium is established first of all in such a small part of a body, and therefore definite temperatures, chemical potentials, and other thermodynamic quantities may be assigned to these small parts of the body. Thus one may speak
THERMOELECTRIC AND THERMOMAGNETIC PHENOMENA
...speak of local equilibrium in small parts of a body, when the system as a whole is not yet in equilibrium. Mechanical processes throughout the whole body—transmission of pressure from one place to another—proceed considerably more slowly. Still more slowly is complete thermodynamic equilibrium established in a large volume, namely diffusion, heat conduction, etc. These sharp differences in rates allow us to speak of temperatures and chemical potentials of separate parts of a body, i.e., to describe slow processes by means of the concepts of equilibrium theory.
It is precisely on such assumptions that the classical theory of heat conduction is constructed. The thermodynamic theory of irreversible processes set forth below is a direct generalization of the classical theory of heat conduction\(^{9—10}\).
§ 5. Basic ideas of the thermodynamic theory of irreversible processes and an explanation by the example of heat conduction
The thermodynamic theory of irreversible processes proceeds from a certain generalization of the fundamental equalities (3.1) and (3.2) of classical thermodynamics. First, it is assumed that these equalities remain valid in the differential sense, i.e., with respect to small volumes within which local equilibrium takes place. In this connection, the quantities \(Q(d)\), \(E\), \(S\) should be understood as the corresponding specific quantities (for example, referred to one particle or to a unit volume). This is quite natural, since, for example, equality (3.2), in the case when the temperature is not the same in different parts of the system, has meaning only in such an interpretation. Second, it is assumed that the entropy depends only on those thermodynamic parameters of which it is a function in equilibrium. In an explicit way the entropy does not depend on coordinates and time. The dependence of entropy on coordinates and time enters only through the dependence of the energy, temperature, and chemical potential on these variables. This assumption is also quite natural under the assumption of local equilibrium. Finally, it is assumed that the total changes of energy and entropy within a given system are formed additively from the corresponding changes in the separate volumes.
Thus, the basic equations of the thermodynamics of irreversible processes may be written in the following form:
\[ \frac{Q(d)}{dt}=\frac{dE}{dt}+\frac{A(d)}{dt}-\int u_0\,\frac{\partial \rho}{\partial t}\,d\tau, \tag{5.1} \]
\[ \frac{dS}{dt}=\int \frac{1}{T}\,\frac{Q'(d)}{dt}\,d\tau, \tag{5.2} \]
where
\[ E=\int \varepsilon\,d\tau, \tag{5.3} \]
\[ S=\int s\,d\tau, \tag{5.4} \]
where \(\varepsilon\) and \(s\) are the volume densities of energy and entropy, \(\dfrac{Q'(d)}{dt}\,d\tau\) is the amount of heat supplied to the volume element \(d\tau\) per unit time, \(\mu_0\) is the chemical potential at the given point, and \(\dfrac{\partial \rho}{\partial t}\) is the change in the number of particles at the given point. The integration is carried out over the volume occupied by the system.
At first glance it seems that these assumptions are in sharp contradiction with classical thermodynamics, according to which, instead of equality (5.2), the corresponding inequality (3.3) holds. In fact, however, there is no contradiction here, since in passing from individual volume elements to the system as a whole one takes into account the change in entropy caused by irreversible processes. The assumption of local equilibrium makes it possible to calculate this additional change.
In order to make this clear, let us illustrate the above with the example of heat conduction. Let us single out, in a nonuniformly heated medium, some fixed volume and consider in it the balance of energy and entropy. Since the volume is fixed and there is no flux of particles, \(A(d)=0\), \(\dfrac{\partial \rho}{\partial t}=0\), and relation (5.1) takes the form
\[ \frac{dE}{dt}=\frac{Q(d)}{dt}. \tag{5.5} \]
Let \(\mathbf{Q}\) denote the heat-flux density. Then, obviously, the energy in the given volume can change only due to the influx of heat from outside. Therefore we may write that
\[ \frac{dE}{dt}=\frac{Q(d)}{dt} =-\oint(\mathbf{Q}\mathbf{n})\,d\sigma =-\int \operatorname{div}\mathbf{Q}\,d\tau, \tag{5.6} \]
where \(d\sigma\) is an element of the surface bounding the selected volume, and \(d\tau\) is an element of volume.
In an analogous way, the second law of thermodynamics may be written in the following form:
\[ \frac{dS}{dt} =-\oint \frac{1}{T}(\mathbf{Q}\mathbf{n})\,d\sigma +\int\left(\mathbf{Q}\nabla\frac{1}{T}\right)d\tau. \tag{5.7} \]
Let us now note that \(-(\mathbf{Q}\mathbf{n})\,d\sigma\) is the amount of heat which enters the system through the surface element \(d\sigma\). Let us denote it
through \(q(d)\). Next we shall use the law of heat conduction, according to which
\[ \mathbf{Q}=-\chi \nabla T, \tag{5.8} \]
where \(\chi\) is the coefficient of thermal conductivity, and always \(\chi>0\). Thus, finally, we have
\[ \frac{dS}{dt}=\oint \frac{q(d)}{T}+\int \chi\left(\frac{\nabla T}{T}\right)^2 d\tau . \tag{5.9} \]
Here the first integral represents the change in the entropy of the system due to the heat entering from outside; namely, this is the quantity that appears on the right-hand side of the Clausius inequality (3.5). The second term of equation (5.9) represents the change in entropy caused by irreversible heat conduction within the volume under consideration. Since this second term is always positive, expression (5.9), and also, of course, the general expression (5.2), do not contradict the Clausius inequality.
§ 6. Generalized laws of electrical conductivity and thermal conductivity
For what follows it is necessary to use the so-called generalized laws of electrical conductivity and thermal conductivity. Let us first consider a thermally homogeneous system, i.e. one in which the temperature is the same everywhere. In such a system, if it is in equilibrium, the chemical potential has the same value at all points. If, in addition, there is an electric field, then the condition of equilibrium requires that the so-called electrochemical potential remain constant:
\[ \mu=\mu_0+e\varphi . \tag{6.1} \]
Since a current is caused by a disturbance of equilibrium, i.e. by the fact that \(\mu\) becomes unequal at different points of the system, it is natural to suppose that the current density is proportional to the gradient of \(\mu\):
\[ \mathbf{j}\sim \nabla \mu . \]
Then, in the particular case of a physically homogeneous system \((\mu_0=\mathrm{const},\ \nabla \mu_0=0)\), we obtain
\[ e\mathbf{j}\sim \mathbf{E}, \]
i.e. Ohm’s law. In addition, diffusion of particles (and consequently an additional current) may be caused by a temperature gradient. Thus, the total particle-flux density may be written in the form
\[ \mathbf{j}=a\nabla\mu+b\nabla T, \tag{6.2} \]
which is the generalized law of electrical conductivity.
Similarly, the heat flux may be caused not only by the temperature gradient, but also by the transfer of energy by particles. This additional heat flux, just like the particle flux, will, naturally, be proportional to \(\nabla \psi\). Thus the generalized law of heat conduction may be written in the form
\[ \mathbf{q}=c\nabla\psi+d\nabla T, \tag{6.3} \]
where \(\mathbf{q}\) is the total heat flux with convection taken into account.
Let us note that, by postulating the proportionality of \(\mathbf{j}\) and \(\mathbf{q}\) to the gradients of the chemical potential and temperature [(6.2) and (6.3)], we thereby deny the possibility of the appearance of a thermoe.m.f. in homogeneous isotropic circuits (the so-called Benedicks effect). The reality of this effect, despite the numerous (and contradictory) investigations devoted to this question, cannot be regarded as established. The only conclusion that can be drawn with complete certainty on the basis of these investigations is that, even if the Benedicks effect exists, it must be negligibly small in comparison with the ordinary thermoe.m.f.
The coefficients of equations (6.2) and (6.3) are unambiguously related to the material constants of the conducting medium. Thus, it is easy to see that
\[ a=-\frac{1}{e^2}\sigma, \tag{6.4} \]
where \(\sigma\) is the specific electrical conductivity.
In an analogous manner, the thermal conductivity is
\[ \varkappa=\frac{bc}{a}-d. \tag{6.5} \]
If we eliminate \(\nabla\psi\) from (6.3) by means of (6.2), we obtain
\[ \mathbf{q}=-\varkappa\nabla T+\frac{c}{a}\mathbf{j} =-\varkappa\nabla T-\frac{e^2c}{\sigma}\mathbf{j}. \tag{6.6} \]
The second term of this expression represents the Peltier heat flux. Comparing the expression obtained with (1.3), we obtain for the differential Peltier coefficient
\[ \Pi=-\frac{ec}{\sigma}. \tag{6.7} \]
Thus, the coefficient \(c\) is related to the differential Peltier coefficient.
The coefficient \(b\) is related to the differential thermoe.m.f., which can be shown as follows. In the absence of current in a nonuniformly heated system there is a thermoelectric field, which, on the basis of (6.1), (6.2), and (6.4), may be written in the form
\[ \mathbf{E}=\frac{1}{e}\nabla\psi_0-\frac{eb}{\sigma}\nabla T. \tag{6.8} \]
The thermopower is defined as the contour integral along the entire circuit
\[ A=\int E_l\,dl. \]
Substituting \(E_l\) from equation (6.8) and taking into account the fact that
\(\int \nabla_l \mu_0\,dl=0\), owing to the equality of \(\mu_0\) at the ends of the circuit, we obtain
\[ A=-\int_{T_1}^{T_2} e\,\frac{b}{\sigma}\,dT. \tag{6.9} \]
According to formula (1.1),
\[ \alpha=-e\,\frac{b}{\sigma}, \tag{6.10} \]
where \(\alpha\) is the absolute thermopower characteristic of the given substance.
On the basis of the above, the generalized laws of electrical conductivity and thermal conductivity may be written in the form
\[ \mathbf{e}\mathbf{j}=-\frac{1}{e}\sigma\nabla\mu-\sigma\alpha\nabla T, \tag{6.11} \]
\[ \mathbf{q}=-\chi\nabla T+\Pi\,e\mathbf{j}. \tag{6.12} \]
In conclusion, let us note that the concepts of electrochemical potential and temperature as functions of a “point,” used, for example, in equations (6.11) and (6.12), are abstractions whose validity rests on the principle of local equilibrium. The limit of applicability of this principle, and consequently also of the concept of a thermodynamic function at a “point,” can be clarified only on the basis of kinetic theory. Therefore a detailed discussion of this question will be given below, in § 27.
§ 7. The first law of thermodynamics in differential form and the energy flux (Umov equation)
For the theory of thermoelectric phenomena it is convenient to introduce the first law of thermodynamics in differential form. In doing so we shall obtain an expression for the density of the energy flux. Let us again consider some fixed volume inside the system. The change of energy within this volume may be due to the following causes:
1) influx of heat,
2) influx of particles,
3) change in the potential energy of electric charges, and
4) as a result of the work of the electric current.
Using the results of §§ 5 and 6, one may write
\[ \frac{dE}{dt} = -\int \operatorname{div}\mathbf{q}\,d\tau + \int \mu_0\,\frac{\partial \rho}{\partial t}\,d\tau + e\int \varphi\,\frac{\partial \rho}{\partial t}\,d\tau - \int (\mathbf{i}\nabla\psi)\,d\tau. \tag{7.1} \]
Here the first term represents the change in energy due to the influx of heat, the second term—due to the influx of particles, the third—due to the redistribution of electric charges, and the fourth—due to the work of the electric current.
In differential form, taking into account the law of conservation of particles
\[ \frac{\partial \rho}{\partial t}+\operatorname{div}\mathbf{j}=0, \tag{7.2} \]
the first law of thermodynamics can be written in the following form:
\[ \frac{\partial \varepsilon}{\partial t}+\operatorname{div}\mathbf{w}=0, \tag{7.3} \]
where
\[ \mathbf{w}=\mathbf{q}+\mu\mathbf{j} \tag{7.4} \]
is the density of the energy flux. The equation of conservation of energy in this form was first obtained by N. A. Umov[^11].
Thus, the energy flux consists of the heat flux and the convection of the chemical potential. Since the chemical potential is calculated per particle, and
\[ \mathbf{j}=\rho\mathbf{v}, \tag{7.5} \]
where \(\mathbf{v}\) is the velocity of the particles, then
\[ \mu\mathbf{j}=\mu\rho\mathbf{v}=(\varepsilon-Ts+p+e\varphi)\mathbf{v}. \tag{7.6} \]
Thus, the convection of the chemical potential consists of the following parts:
1) transport of energy by particles \((\varepsilon\mathbf{v})\),
2) transport of heat due to convection of entropy \((Ts\mathbf{v})\),
3) transport of energy due to the work of pressure forces \((p\mathbf{v})\),
4) transport of potential energy.
§ 8. The First Thermoelectric Relation
It is now easy to obtain the first thermoelectric relation from the law of conservation of energy. In the case of stationary currents
\[ \operatorname{div}\mathbf{j}=0. \tag{8.1} \]
Using (7.4) and (6.12), we have
\[ \mathbf{w}=-\chi\nabla T+(\mu+e\Pi)\mathbf{j}. \tag{8.2} \]
Hence we obtain
\[ \frac{\partial \varepsilon}{\partial t} =\operatorname{div}(\chi\nabla T)-(\mathbf{j}\nabla[\mu+e\Pi]). \tag{8.3} \]
Expanding the last term, let us note that \(\Pi\) may depend on the coordinates directly (because of the inhomogeneity of the system) and through the temperature.
Thus we obtain
\[ \frac{\partial \varepsilon}{\partial t} = \operatorname{div}(\chi \nabla T) - e(\nabla \Pi_T \mathbf{j}) - e\frac{\partial \Pi}{\partial T}(\mathbf{j}\nabla T) - (\mathbf{j}\nabla\mu). \tag{8.4} \]
Eliminating from this expression \(\nabla \mu\) with the aid of (6.11), we obtain
\[ \frac{\partial \varepsilon}{\partial t} = \operatorname{div}(\chi \nabla T) + \frac{e^2 j^2}{\sigma} - e(\nabla \Pi_T \mathbf{j}) - e\left(\frac{\partial \Pi}{\partial T}-\alpha\right)(\mathbf{j}\nabla T). \tag{8.5} \]
Thus the change in energy is made up of the following parts:
1) liberation of energy as a result of heat conduction,
2) liberation of Joule heat,
3) liberation of Peltier heat,
4) the last term is the Thomson heat.
Comparing the expression obtained for it with formula (1.5), we have
\[ \tau=\frac{\partial \Pi}{\partial T}-\alpha, \tag{8.6} \]
i.e., the first thermoelectric relation.
Thus, as is clear from the derivation given, the first thermoelectric relation follows from the first law of thermodynamics if the generalized laws of electrical conductivity and heat conduction are adopted.
From formula (8.6) it is seen that the Thomson effect is determined by the two factors discussed in § 1.
§ 9. The second thermoelectric relation
and the principle of symmetry of kinetic coefficients
The assertion often encountered in the literature that the second thermoelectric relation follows from the second law of thermodynamics is incorrect. As will now be shown, the second thermoelectric relation in fact follows from the so-called principle of symmetry of kinetic coefficients (formulated in general form by Onsager \(^{12}\)), which is not equivalent to the second law of thermodynamics and contains more far-reaching assumptions. A critique of the derivation of the thermoelectric relation from the second law of thermodynamics will be given in § 11.
Let us write the generalized laws of electrical conductivity and heat conduction in the form
\[ \left. \begin{aligned} \mathbf{j} &= -\gamma_{11}\nabla\frac{\mu}{T} + \gamma_{12}\nabla\frac{1}{T},\\ \mathbf{q} &= -\gamma_{21}\nabla\frac{\mu}{T} + \gamma_{22}\nabla\frac{1}{T} -\mu\mathbf{j}, \end{aligned} \right\} \tag{9.1} \]
which is equivalent to (6.2) and (6.3). The coefficients of both
the representations are related to one another by the following relations:
\[ \begin{aligned} a&=-\frac{1}{T}\gamma_{11},\\ b&=\frac{1}{T^{2}}(\mu\gamma_{11}-\gamma_{21}),\\ c&=\frac{1}{T}(\mu\gamma_{11}-\gamma_{21}),\\ d&=-\frac{1}{T^{2}}\left[\mu^{2}\gamma_{11}-\mu(\gamma_{12}+\gamma_{21})+\gamma_{22}\right]. \end{aligned} \tag{9.2} \]
The principle of symmetry of the kinetic coefficients asserts that
\[ \gamma_{12}=\gamma_{21}. \tag{9.3} \]
Referring the reader for the derivation to the appropriate literature\(^{12—14}\), we note here only that the proof of the symmetry of the kinetic coefficients is based on the following assumptions:
1) reversibility of the microprocesses;
2) validity of the hypothesis of local equilibrium;
3) validity of the assumption that the dissipation of fluctuations obeys the same laws as the course of macroscopic irreversible processes (if the fluctuations are sufficiently large).
Comparing (9.3) and (9.2), we obtain
\[ c=bT, \tag{9.4} \]
whence, on the basis of (6.7) and (6.10),
\[ \Pi=T\alpha, \tag{9.5} \]
and this is the second thermoelectric relation. Consequently, the latter is valid only when those conditions are fulfilled which ensure the symmetry of the kinetic coefficients.
Thus, for example, in the presence of a magnetic field the microprocesses will no longer be reversible in the usual sense of the word. In this case the equations of motion remain invariant only if, simultaneously with changing the sign of the time, the direction of the magnetic field is reversed. In this case\(^{13}\)
\[ \gamma_{12}(\mathbf H)=\gamma_{21}(-\mathbf H). \tag{9.6} \]
In the general case the second thermoelectric relation must be replaced by the following\(^{6}\):
\[ \Pi-T\alpha=\frac{1}{e\eta}\left(\gamma_{21}-\gamma_{12}\right). \tag{9.7} \]
§ 10. The Second Law of Thermodynamics in Differential Form and the Entropy Flux
Let us consider some fixed volume inside the system. In this case, in accordance with formula (5.2), the entropy inside this volume can change owing to the redistribution of particles and energy within this volume \((A(d)=0)\). Thus, we may write
\[ \frac{dS}{dt} = \int \frac{\partial s}{\partial t}\,d\tau = \int \frac{1}{T}\frac{\partial \varepsilon}{\partial t}\,d\tau - \int \frac{\mu}{T}\frac{\partial \rho}{\partial t}\,d\tau, \]
where here \(\mu\) is understood as the electrochemical potential. Since this is valid for any volume, we obtain the expression of the second law of thermodynamics in differential form:
\[ \frac{\partial s}{\partial t} = \frac{1}{T}\frac{\partial \varepsilon}{\partial t} - \frac{\mu}{T}\frac{\partial \rho}{\partial t}. \tag{10.1} \]
Using the conservation laws (7.2) and (7.3), as well as the relation between the energy-flux density and the heat-flux density (7.4), we obtain
\[ \frac{\partial s}{\partial t} = -\operatorname{div}\frac{\mathbf q}{T} + \left(\mathbf w \nabla \frac{1}{T}\right) - \left(\mathbf j \nabla \frac{\mu}{T}\right). \tag{10.2} \]
The vector
\[ \mathbf S=\frac{\mathbf q}{T} \tag{10.3} \]
is called the entropy-flux density. Equation (10.2) expresses the fact that a change of entropy at a given point may occur both owing to an influx of entropy from outside and owing to irreversible processes taking place inside the given volume. It can be shown that the definition of the entropy-flux vector \(\mathbf S\) (10.3) and of the energy flux \(\mathbf w\) (7.4) by means of the differential equations (7.3) and (10.2) becomes completely unambiguous if one requires that these vectors be identically equal to zero everywhere outside the conductors of electric and heat currents[^14].
Substituting into (10.3) the expression (6.12) for \(\mathbf q\), we obtain
\[ \mathbf S = -\frac{\varkappa}{T}\nabla T + \frac{\Pi}{T}\,e\mathbf j . \tag{10.4} \]
The first term therefore represents the entropy flux due to thermal conductivity, and the second the transport of entropy by the particle flux. Denoting this part of the entropy flux by \(S_j\), we obtain
\[ \Pi=\frac{T}{e}S_j . \tag{10.5} \]
Similarly, we have
\[ \alpha=\frac{1}{e}S_j, \tag{10.6} \]
\[ \tau=\frac{T}{e}\frac{\partial S_j}{\partial T}. \tag{10.7} \]
Thus, thermoelectric phenomena are associated with the transfer of entropy by the current, as was first pointed out by Ehrenfest^15. It is precisely this circumstance that is connected with the peculiar “reversibility” of thermoelectric phenomena.
§ 11. Critique of the derivation of the second thermoelectric relation from the second law of thermodynamics
Thomson derived the relation \(\Pi = T\alpha\), assuming that thermoelectric heats are liberated or absorbed reversibly. As we have already indicated, he neglected the fact that thermoelectric processes are accompanied by irreversible phenomena. In later works on the thermodynamic theory of thermoelectric phenomena (see, for example,^7,15) the presence of irreversible processes was taken into account; however, the former assumption of the reversibility of the liberation (absorption) of thermoelectric heats was retained. We shall now show that, indeed, by making such an assumption, one can obtain the second thermoelectric relation without applying the principle of symmetry of kinetic coefficients. It is necessary, however, to emphasize that the assumption of the reversibility of thermoelectric heats does not follow from the second law of thermodynamics and is an additional assumption.
Let us consider a fixed volume in which, at a given instant of time, there is some distribution of heat and electric currents. According to (10.2), for the change of entropy inside this volume we may write
\[ \frac{dS}{dt} = \frac{\partial}{\partial t}\int s\,d\tau = \int\left\{\left(\mathbf{w}\nabla \frac{1}{T}\right) - \left(\mathbf{j}\nabla \frac{\mu}{T}\right)\right\}d\tau + \oint \frac{1}{T}(\mathbf{q}\mathbf{n})\,do. \tag{11.1} \]
After elementary transformations we obtain
\[ \frac{\partial}{\partial t}\int s\,d\tau = \int\left\{ \varkappa\left(\frac{\nabla T}{T}\right)^2 + \frac{(e\mathbf{j})^2}{T\sigma} \right\}d\tau + \int \frac{1}{T}\left(\alpha-\frac{\Pi}{T}\right)(e\mathbf{j}\nabla T)\,d\tau + \oint \frac{1}{T}(\mathbf{q}\mathbf{n})\,do. \tag{11.2} \]
The first term on the right-hand side of (11.2) represents the increase of entropy due to irreversible processes of heat conduction and the liberation of Joule heat. The second term represents the increase of entropy due to thermoelectric processes. The third term is the change of entropy due to the influx of heat through the surface. However, since, by assumption, thermoelectric phenomena ...
are reversible, then
\[ \int \frac{1}{T}\left(\alpha-\frac{\Pi}{T}\right)(\mathbf{e}\mathbf{j}\nabla T)\,d\tau=0. \]
In view of the fact that this must be valid for any volume, such an equality is possible only under the condition
\[ \Pi=T\alpha. \]
What was said in the preceding paragraph shows, however, that the assumption of reversibility of thermoelectric phenomena is an arbitrary assumption which cannot always be justified. Therefore the second thermoelectric relation is not a consequence of thermodynamics, but is an expression of regularities pertaining to the kinetics of the occurrence of thermoelectric phenomena.
§ 12. Thermoelectric phenomena in anisotropic bodies
The theory set forth in the preceding paragraphs can easily be generalized to the case of anisotropic bodies[^15]. In doing so, the theory predicts the existence of a new effect, occurring only in anisotropic bodies. The presence of this effect has been repeatedly confirmed experimentally. To construct the theory of thermoelectric phenomena in anisotropic bodies, it is sufficient to generalize in a natural way the basic relations (6.11) and (6.12). Namely, the coefficients entering into these equations must now be regarded as tensor quantities. In tensor notation these relations may be written as follows:
\[ \left. \begin{aligned} e_j j_i&=-\frac{1}{e}\,\sigma_{ik}\frac{\partial \mu}{\partial x_k} -\sigma_{il}\alpha_{lk}\frac{\partial T}{\partial x_k},\\ q_i&=-\varkappa_{ik}\frac{\partial T}{\partial x_k}+\Pi_{ik}e j_k, \end{aligned} \qquad i,k=1,2,3. \right\} \tag{12.1} \]
Here the usual convention of summation over repeated indices has been used.
Let us note that from the principle of symmetry of the kinetic coefficients it follows that the tensors of electrical conductivity \((\sigma_{ik})\) and thermal conductivity \((\varkappa_{ik})\) are symmetric[^12]:
\[ \sigma_{ik}=\sigma_{ki},\qquad \varkappa_{ik}=\varkappa_{ki}. \tag{12.2} \]
The tensors \(\alpha_{ik}\) and \(\Pi_{ik}\) are generalizations of the coefficients \(\alpha\) and \(\Pi\) and turn out to be connected with the thermoe.m.f. and the Peltier effect.
According to the first law of thermodynamics and on the basis of expression (12.1), we have
\[ \frac{\partial e}{\partial t} = \frac{\partial}{\partial x_i}\left(\chi_{ik}\frac{\partial T}{\partial x_k}\right) +\rho_{ik} e_{ji} e_{jk} - \left(\frac{\partial \Pi_{ik}}{\partial x_i}\right)_T e_{jk} + \left(\alpha_{ik}-\frac{\partial \Pi_{ik}}{\partial T}\right)e_{ji}\frac{\partial T}{\partial x_k} - \Pi_{ik}\frac{\partial}{\partial x_i}e_{jk}. \tag{12.3} \]
where \(\rho_{ik}\) is the tensor of electrical resistivity, inverse to the conductivity tensor \(\sigma_{ik}\). The introduction of the tensor Thomson coefficient \(\tau_{ik}\) (the analogue of the isotropic coefficient \(\tau\)) by the formula
\[ \tau_{ik}=\frac{\partial \Pi_{ik}}{\partial T}-\alpha_{ik} \tag{12.4} \]
makes it possible to rewrite equality (12.3) in the following form:
\[ \frac{\partial e}{\partial t} = \frac{\partial}{\partial x_i}\left(\chi_{ik}\frac{\partial T}{\partial x_k}\right) +\rho_{ik} e_{ji}e_{jk} - \frac{\partial \Pi_{ik}}{\partial x_i}e_{jk} - \tau_{ik} e_{ji}\frac{\partial T}{\partial x_k} - \Pi_{ik}\frac{\partial}{\partial x_i}e_{jk}. \tag{12.5} \]
The first two terms on the right-hand side of this equality are the heats released as a result of thermal conduction and the ohmic work of the current (Joule heat). The third and fourth terms are generalizations, to the case of anisotropic media, of the expressions already known to us for the Peltier and Thomson heats. The last term is connected exclusively with the anisotropy of the medium (it vanishes in an isotropic medium) and represents a new thermal effect peculiar only to crystals.
Let us rewrite the generalized laws of electrical conduction and thermal conduction (12.1) in the form:
\[ \left. \begin{aligned} j_i&=-\gamma_{11}^{ik}\frac{\partial}{\partial x_k}\frac{\mu}{T} +\gamma_{12}^{ik}\frac{\partial}{\partial x_k}\frac{1}{T},\\ q_i&=-\gamma_{21}^{ik}\frac{\partial}{\partial x_k}\frac{\mu}{T} +\gamma_{22}^{ik}\frac{\partial}{\partial x_k}\frac{1}{T}-\psi j_i, \end{aligned} \qquad i,k=1,2,3. \right\} \tag{12.6} \]
The system of equations (12.6), being the tensor analogue of system (9.1), obviously coincides with (12.1) under quite definite relations between the coefficients of these two systems. Namely, the following equalities must hold:
\[ \left. \begin{aligned} \gamma_{11}^{ik}&=\frac{1}{e}\,T\sigma_{ik},\\ \frac{1}{T^2}\left[\mu^2\gamma_{11}^{ik} -\mu\left(\gamma_{12}^{ik}+\gamma_{21}^{ik}\right) +\gamma_{22}^{ik}\right]&=\chi_{ik}. \end{aligned} \right\} \tag{12.7} \]
\[ \left. \begin{aligned} \frac{1}{T}\left(\mu\gamma_{11}^{ik}-\gamma_{21}^{ik}\right) &=-\frac{1}{e}\Pi_{il}\sigma_{lk},\\ \frac{1}{T^2}\left(\mu\gamma_{11}^{ik}-\gamma_{12}^{ik}\right) &=-\frac{1}{e}\sigma_{il}\alpha_{lk}. \end{aligned} \right\} \tag{12.8} \]
The symmetry principle for kinetic coefficients in the case of anisotropic bodies asserts that
\[ \gamma_{12}^{ik}=\gamma_{21}^{ki}. \tag{12.9} \]
The first equation (12.7) testifies to the symmetry of the coefficients \(\gamma_{11}^{ik}\):
\[ \gamma_{11}^{ik}=\gamma_{11}^{ki}, \tag{12.10} \]
whence it follows that
\[ \Pi_{ik}=T\alpha_{ki}, \tag{12.11} \]
i.e., the second thermoelectric relation.
With the aid of relation (12.11), equation (12.3) is rewritten in the form
\[ \frac{\partial \varepsilon}{\partial t} = \frac{\partial}{\partial x_i} \left( \chi_{ik}\frac{\partial T}{\partial x_k} \right) + \rho_{ik} e_{ji} e_{jk} - T\frac{\partial}{\partial x_i}(S_{ik}j_k), \tag{12.12} \]
where
\[ S_{ik}=\frac{\Pi_{ik}}{T} \tag{12.13} \]
is a component of the entropy flux, and the term
\[ q=-T\frac{\partial}{\partial x_i}(S_{ik}j_k) \tag{12.14} \]
represents the heat liberated as a result of thermoelectric processes.
Passing in (12.14) from the differential form to the integral one and using the Ostrogradsky–Gauss theorem, it is easy to obtain the following formula for the quantity of heat \(Q_{\Pi}\) liberated as a result of the Peltier effect per unit area of the interface between two media:
\[ Q_{\Pi}=T\left|S_{ik}j_k n_i\right|_{a}^{b}, \tag{12.15} \]
where \(n_i\) are the direction cosines of the normal to the surface.
It follows from (12.15) that in anisotropic bodies the Peltier effect may occur even in homogeneous regions, provided only that there are junctions in them of differently oriented crystals (Fig. 1). The liberation of Peltier heat may also occur, as is seen from formula (12.15), on the free surface of a crystalline conductor if this surface forms an acute angle with the direction of the principal axes of the crystal (Fig. 2).
Fig. 1.
Fig. 2.
Moreover, in crystals one can observe volume heat production in homogeneous isothermal parts of conductors, not connected with the presence of any separation surfaces. The production of this heat is due to the anisotropic term proper, \(-e\Pi_{ik}\dfrac{\partial}{\partial x_i}j_k\), in equation (12.14), and therefore takes place where the current lines bend (bend \(A\), Fig. 1).
In addition, a transverse thermoe.m.f. may occur, i.e., the appearance of a thermo-e.m.f. in a direction perpendicular to the direction of the temperature gradient, since now the thermoelectric coefficient has a tensor character.
§ 13. Thermomagnetic and galvanomagnetic phenomena
The theory set forth above is easily generalized to the case in which a magnetic field is present. The comparatively simple phenomena of heat conduction and electrical conduction in homogeneous media are complicated in the presence of a magnetic field: additional effects appear, the so-called thermomagnetic and galvanomagnetic effects. The former are due to the action of the magnetic field on the heat-conduction current, the latter to the action of the magnetic field on the electric current.
The influence of a homogeneous magnetic field on a medium may be interpreted as the appearance of a peculiar anisotropy in this medium, analogous to the anisotropy of rotation of the system about a fixed axis. Ya. I. Frenkel \(^{17}\) proposed calling such anisotropy gyrotropy. Any tensor \(L_{ij}\) characterizing a property of some system can be decomposed into symmetric and antisymmetric parts
\[ L_{ij}=L_{(ij)}+L_{[ij]}, \tag{13.1} \]
where
\[ L_{(ij)}=L_{(ji)},\quad L_{[ij]}=-L_{[ji]}. \tag{13.2} \]
Gyrotropic media are characterized by the fact that the symmetric part of the tensor \(L_{ij}\) is proportional to the unit tensor, i.e.
\[ L_{(ij)}=L\delta_{ij},\quad \delta_{ij}= \begin{cases} 0 & i\ne j,\\ 1 & i=j. \end{cases} \tag{13.3} \]
Thus, thermomagnetic and galvanomagnetic phenomena essentially reduce to the phenomena of heat and electrical conduction in a gyrotropic medium. From this point of view they will be considered below.
To construct a theory of galvanomagnetic and thermomagnetic phenomena, we shall proceed from the generalized laws of electrical conduction and heat conduction, written in tensor form (12.1),
where, for the sake of simplifying subsequent calculations, it is convenient to represent them in the following form*):
\[ \begin{gathered} E_i=\rho_{ik} e_{jk}-\alpha_{ik}\frac{\partial T}{\partial x_k},\\ q_i=\Pi_{ik} e_{jk}-\chi_{ik}\frac{\partial T}{\partial x_k}, \\ \text{where}\qquad i,k=1,2 . \end{gathered} \tag{13.4} \]
Taking into account the gyrotropy of the medium [cf. formulas (13.2), (13.3)] and taking the direction of the magnetic field as the \(z\)-axis, the system (13.4) can be rewritten in the following form:
\[ \begin{aligned} E_x&=\rho e_{jx}+\rho_{[xy]} e_{jy}+\alpha \frac{\partial T}{\partial x}+\alpha_{[xy]}\frac{\partial T}{\partial y},\\ E_y&=-\rho_{[xy]} e_{jx}+\rho e_{jy}-\alpha_{[xy]}\frac{\partial T}{\partial x}+\alpha \frac{\partial T}{\partial y},\\ q_x&=\Pi e_{jx}+\Pi_{[xy]} e_{jy}-\varkappa \frac{\partial T}{\partial x}-\varkappa_{[xy]}\frac{\partial T}{\partial y},\\ q_y&=-\Pi_{[xy]} e_{jx}+\Pi e_{jy}+\varkappa_{[xy]}\frac{\partial T}{\partial x}-\varkappa \frac{\partial T}{\partial y}. \end{aligned} \tag{13.5} \]
Galvanomagnetic and thermomagnetic phenomena are divided into “transverse” effects (electrical or thermal effects arise in a direction perpendicular to the direction of the primary electric or heat currents) and “longitudinal” effects (the effects are parallel to the primary currents); moreover, depending on the experimental conditions (the specimen under study is placed in a thermostat, or is adiabatically isolated from the external medium), a distinction is made between “isothermal” and “adiabatic” effects.
We shall give a brief list of the indicated phenomena.
I. Galvanomagnetic phenomena
A. Transverse effects
1. Hall effect. In a conductor through which an electric current flows, in the presence of a magnetic field perpendicular to the electric current \(e_{jx}\), there appears an electric-field strength \(E_y\),
*) We deliberately omit, for convenience in the subsequent exposition, terms depending on \(\partial \varphi_0/\partial x\) and \(\partial \varphi_0/\partial y\) and entering, generally speaking, into the system (12.1). The point is that this in no way affects the final result corresponding to the experimental situation, just as the term \(\nabla \varphi_0\) entering the definition of the thermoelectromotive field \(E\) [cf. (6.8)] drops out of the determination of the differential thermoelectric power \(\alpha\) (6.10) through the integral \(A\) (6.9).
proportional to the electric current density and the magnetic-field intensity at the point. The proportionality coefficient—the Hall constant—is determined by the condition
\[ R^{\perp}=\frac{E_y}{e j_x H_z}. \tag{13.6} \]
Under the action of this Hall intensity of the electric field \(E_y\), an electric current \(e j_y\) arises, which, because of the limited dimensions of the specimen, leads to the appearance on its lateral
Fig. 3. Fig. 4.
surfaces of a certain emf—the Hall emf—balancing the current in the direction of the \(y\)-axis (Fig. 3). It is possible, however, to create such conditions under which the Hall current will be closed, and then no Hall emf arises (Corbino effect, Fig. 4).
Depending on the thermal regime of the specimens under investigation, one may measure both the isothermal \((R^{\perp}_{\mathrm{is}})\) and the adiabatic \((R^{\perp}_{\mathrm{ad}})\) Hall coefficients. Usually \(R^{\perp}_{\mathrm{ad}}\) is most easily measured.
2. Ettingshausen effect. A transverse electric current in a magnetic field also creates a transverse temperature gradient (Ettingshausen effect). This effect is adiabatic and is characterized by the coefficient
\[ P^{\perp}=\frac{\dfrac{\partial T}{\partial y}}{e j_x H_z}. \tag{13.7} \]
B. Longitudinal effects
1. Isothermal and adiabatic change of electrical resistivity in a magnetic field. In the direction of the primary electric current, the magnetic field causes the appearance of an additional electric-field intensity, manifested as a change in electrical resistivity. It is characterized by the coefficient
\[ R^{\parallel}=\frac{E_x}{e j_x}. \tag{13.8} \]
- The isothermal and adiabatic Nernst effects are longitudinal analogues of the Ettingshausen effect. They are described by the coefficients
\[ P^{\parallel}=-\frac{\dfrac{\partial T}{\partial x}}{e j_x}. \tag{13.9} \]
II. Thermomagnetic phenomena
A. Transverse effects
- The isothermal and adiabatic Ettingshausen–Nernst effects are analogous to the Hall effect, with the sole difference that the primary current is not an electric current, but a heat current. The corresponding coefficient is
\[ Q^{\perp}=\frac{E_y}{\dfrac{\partial T}{\partial x} H_z}. \tag{13.10} \]
- The adiabatic Righi–Leduc effect is analogous to the Ettingshausen effect
\[ S^{\perp}=\frac{\dfrac{\partial T}{\partial y}}{\dfrac{\partial T}{\partial x} H_z}. \tag{13.11} \]
B. Longitudinal effects
- Isothermal and adiabatic effects of the change of thermal conductivity in a magnetic field:
\[ \varkappa^{\parallel}=-\frac{q_x}{\dfrac{\partial T}{\partial x}}. \tag{13.12} \]
- The longitudinal Ettingshausen–Nernst effect:
\[ Q^{\parallel}=\frac{E_x}{\dfrac{\partial T}{\partial x}}. \tag{13.13} \]
Let us proceed to the calculation of these coefficients on the basis of equations (13.5):
- The Hall effect. We start from the definition of the Hall constant (13.6), taking into account the conditions of isothermality or adiabaticity, which in the present case are written as follows:
\[ j_y=\frac{\partial T}{\partial x}=\frac{\partial T}{\partial y}=0 \tag{13.14} \]
(isothermal condition),
\[ j_y=\frac{\partial T}{\partial x}=q_y=0 \tag{13.15} \]
(adiabatic condition). Then from the second equation (13.5) it follows directly that
\[ R_{\mathrm{is}}^{\perp}=-\frac{1}{H_z}\rho_{[xy]}, \tag{13.16} \]
and from the second and fourth equations
\[ R_{\mathrm{ad}}^{\perp}=-\frac{1}{H_z}\left\{\rho_{[xy]}+\frac{\alpha}{\chi}\Pi_{[xy]}\right\}, \tag{13.17} \]
where the subscripts of the Hall coefficient indicate the isothermal or adiabatic character of the effect. From (13.16) and (13.17) it is seen that the adiabatic Hall effect takes into account the occurrence of a transverse temperature gradient [the Ettingshausen effect—compare with formula (13.18)], which gives rise to an additional emf superposed on the isothermal Hall emf.
2. The Ettingshausen effect. Using the definition (13.7) and the adiabatic condition (13.15), it is easy to obtain from the fourth equation (13.5):
\[ P^{\perp}=-\frac{1}{H_z}\frac{\Pi_{[xy]}}{\chi}. \tag{13.18} \]
3. Change of resistance in a magnetic field. From the first equation (13.5), in the isothermal case,
\[ R_{\mathrm{is}}^{\parallel}=\rho, \tag{13.19} \]
and from the first and second, in the adiabatic case,
\[ R_{\mathrm{ad}}^{\parallel}=\rho-\frac{\alpha_{[xy]}}{\chi}\Pi_{[xy]}. \tag{13.20} \]
In an analogous manner all the remaining galvanomagnetic and thermomagnetic constants are found:
\[ P_{\mathrm{is}}^{\parallel}=\frac{1}{\chi}\Pi, \tag{13.21} \]
\[ P_{\mathrm{ad}}^{\parallel}=\frac{\chi\Pi+\chi_{[xy]}\Pi_{[xy]}}{\chi^2+\chi_{[xy]}^2}, \tag{13.22} \]
\[ Q_{\mathrm{is}}^{\perp}=-\frac{1}{H_z}\alpha_{[xy]}, \tag{13.23} \]
\[ Q_{\mathrm{ad}}^{\perp}=-\frac{1}{H_z}\left\{\alpha_{[xy]}-\frac{\alpha}{\chi}\chi_{[xy]}\right\}, \tag{13.24} \]
$$ S^\perp=-\frac{1}{H_z}\frac{\varkappa_{[xy]}}{\varkappa}, \tag{13.25} $$
$$ Q_{\mathrm{is}}^{\parallel}=\alpha, \tag{13.26} $$
$$ Q_{\mathrm{ad}}^{\parallel}=\alpha+\frac{\alpha_{[xy]}}{\varkappa}\varkappa_{[xy]}, \tag{13.27} $$
$$ \varkappa_{\mathrm{is}}^{\parallel}=\varkappa, \tag{13.28} $$
$$ \varkappa_{\mathrm{ad}}^{\parallel}=\varkappa+\frac{\varkappa_{[xy]}^2}{\varkappa}. \tag{13.29} $$
From formulas (13.16)—(13.29) one can easily obtain certain relations between the galvanomagnetic and thermomagnetic effects that are not connected with the symmetry principle of the kinetic coefficients and are a consequence only of the isotropy of the medium.
Namely:
$$ R_{\mathrm{ad}}^\perp-R_{\mathrm{is}}^\perp=Q_{\mathrm{is}}^{\parallel}P^\perp, \tag{13.30} $$
$$ R_{\mathrm{ad}}^{\parallel}-R_{\mathrm{is}}^{\parallel}=-H_z^2 Q_{\mathrm{is}}^\perp P^\perp, \tag{13.31} $$
$$ Q_{\mathrm{ad}}^\perp-Q_{\mathrm{is}}^\perp=Q_{\mathrm{is}}^{\parallel}S^\perp, \tag{13.32} $$
$$ Q_{\mathrm{ad}}^{\parallel}-Q_{\mathrm{is}}^{\parallel}=-H_z^2 Q_{\mathrm{is}}^\perp S^\perp, \tag{13.33} $$
$$ \varkappa_{\mathrm{ad}}^{\parallel}-\varkappa_{\mathrm{is}}^{\parallel}=H_z^2\varkappa_{\mathrm{is}}^{\parallel}(S^\perp)^2. \tag{13.34} $$
To derive additional relations between the phenomena under consideration it is necessary to invoke the symmetry principle for the kinetic coefficients
$$ \gamma_{ij}^{xy}(\mathbf H)=\gamma_{ji}^{yx}(-\mathbf H),\quad i,\ j=1,\ 2. \tag{13.35} $$
It is not difficult to verify that, owing to the isotropy of the medium, it follows from (13.35) that the symmetric parts of the coefficients $\gamma_{ij}$ are even functions of the magnetic field, whereas the antisymmetric parts are odd functions, i.e.
$$ \gamma_{ij}^{(xy)}(\mathbf H)=\gamma_{ij}^{(xy)}(-\mathbf H),\quad \gamma_{ij}^{[xy]}(\mathbf H)=-\gamma_{ij}^{[xy]}(-\mathbf H). \tag{13.36} $$
Using this, it is easy to show that the coefficients $\sigma_{lk}$, $\varkappa_{lk}$, $\alpha_{lk}$, $\Pi_{lk}$ also possess the same property, namely
$$ \left. \begin{aligned} \sigma(\mathbf H)&=\sigma(-\mathbf H),& \sigma_{[xy]}(\mathbf H)&=-\sigma_{[xy]}(-\mathbf H),\\ \varkappa(\mathbf H)&=\varkappa(-\mathbf H),& \varkappa_{[xy]}(\mathbf H)&=-\varkappa_{[xy]}(-\mathbf H),\\ \alpha(\mathbf H)&=\alpha(-\mathbf H),& \alpha_{[xy]}(\mathbf H)&=-\alpha_{[xy]}(-\mathbf H),\\ \Pi(\mathbf H)&=\Pi(-\mathbf H),& \Pi_{[xy]}(\mathbf H)&=-\Pi_{[xy]}(-\mathbf H). \end{aligned} \right\} \tag{13.37} $$
It follows, furthermore, from (13.37) and (12.8) that the second thermoelectric relation in the presence of a magnetic field
has the form
\[ \left. \begin{aligned} \Pi &= T\alpha,\\ \Pi_{[yx]} &= -T\alpha_{[yx]} . \end{aligned} \right\} \tag{13.38} \]
Thus thermoelectric phenomena in a magnetic field, at least the longitudinal effects, should not differ in principle from the case of thermoelectric phenomena in ordinary anisotropic media.
With the aid of (13.38) one can derive three more relations:
\[ \chi_{\mathrm{iz}}^{\parallel} P_{\mathrm{iz}}^{\parallel} = T Q_{\mathrm{iz}}^{\parallel}, \tag{13.39} \]
\[ \chi_{\mathrm{iz}}^{\parallel} P_{\mathrm{iz}}^{\perp} = T Q_{\mathrm{iz}}^{\perp}, \tag{13.40} \]
\[ \chi_{\mathrm{ad}}^{\parallel} P_{\mathrm{ad}}^{\parallel} = T Q_{\mathrm{ad}}^{\parallel}. \tag{13.41} \]
These equalities are usually called the Bridgman relations and were derived by him on the basis of the assumption that thermoelectric phenomena in a magnetic field still remain reversible phenomena in the thermodynamic sense of the word.
§ 14. Efficiency of Thermoelements
The effectiveness of using one or another thermoelectric device as a generator of electric current ultimately depends on the efficiency of the thermoelectric process. This latter question will be considered in the present section on the basis of the thermodynamic theory of irreversible processes.
Let us consider a thermoelement, adiabatically insulated from the external medium except for the ends 1 and 2, which exchange heat with thermostats at temperatures \(T_1\) and \(T_2\) (Fig. 5). Here \(R_e\) is the resistance of the external load, \(R_i\) is the resistance of the thermoelement for the given temperature distribution in its parts, \(\rho_a\) and \(\rho_b\) are the specific resistivities of materials \(a\) and \(b\), respectively, and \(\chi_a\) and \(\chi_b\) are their thermal conductivities.
The energy balance in such a thermoelement, in the presence of stationarity, may be written in the form [cf. (5.1)]:
\[ Q_1 - Q_2 = I^2 R_e, \tag{14.1} \]
where \(Q_1\) is the heat taken from the heater per unit time; \(Q_2\) is the heat given to the refrigerator per unit time; \(I^2 R_e\) is the power of the total thermoelectric current in the external circuit.
The second law of thermodynamics in the stationary case considered by us asserts that
\[ \int \frac{\partial s}{\partial t}\, d\tau = 0, \]
Thermoelectric and Thermomagnetic Phenomena
where the integration is over the entire volume of the thermoelement. Using equalities (10.2), (6.11), (6.12), and (9.5), it is easy to obtain
\[ -\frac{Q_1}{T_1}+\frac{Q_2}{T_2} =\int k\left(\frac{1}{T}\frac{dT}{dl}\right)^2 dl +I^2\int \frac{1}{T}\frac{\rho}{s'}\,dl, \tag{14.2} \]
where \(k\) is the thermal conductivity of a unit length of the conductor, and \(s'\) is the cross-sectional area. The integrals are taken along the contour of the thermoelement.
Fig. 5.
The efficiency of a thermoelement, as, indeed, the efficiency of any heat engine, is determined by the ratio of the useful work produced by it to the amount of heat taken from the heater:
\[ \eta'=\frac{I^2R_e}{Q_1}=\frac{A^2}{4R_0Q_1}, \tag{14.3} \]
where \(R_0=R_e=R_i\), and \(A\) is the thermo-emf.
Comparing (14.3) and (14.2), we see that the decrease in the efficiency of the thermoelectric process is connected with the presence of irreversible phenomena of thermal conductivity and electrical conductivity (the terms \(\int k\left(\frac{1}{T}\frac{dT}{dl}\right)^2dl\) and \(I^2\int \frac{1}{T}\frac{\rho}{s'}\,dl\)), since, if they were absent, then \(\eta'\) would be exactly equal to the Carnot \(\eta\).
Combining (14.2) and (14.1), we obtain
\[ Q_1=\frac{T_1T_2}{\Delta T} \left\{ \frac{1}{T_1} \left[ \left(k\frac{dT}{dl}\right)_a + \left(k\frac{dT}{dl}\right)_b \right]_{T=T_1} - \frac{1}{T_2} \left[ \left(k\frac{dT}{dl}\right)_a + \left(k\frac{dT}{dl}\right)_b \right]_{T=T_2} + \frac{I^2R_0}{T_2} \right\}, \tag{14.4} \]
where \(\Delta T=T_1-T_2\) is the temperature difference in the thermoelement.
To find the values of the temperature gradient at the ends of the thermoelement, we use the law of conservation of energy:
\[ k\frac{d^2T}{dl^2}+I^2\frac{\rho}{s'}=0. \tag{14.5} \]
(We have neglected the weak dependence of \(\chi\) on temperature, as well as the Thomson effect.)
Assuming \(\rho\) to be a constant independent of temperature (such an approximation sometimes proves sufficient), it is easy to obtain
\[ k\frac{dT}{dl} = -k\frac{\Delta T}{L} -I^2Rl +\frac{1}{2}I^2RL, \tag{14.6} \]
where the coordinate \(l\) is measured from the ends \(1\) of the conductors,
\[ R=\rho \frac{L}{s'}, \quad L \]
is the length of the conductors. The efficiency is
\[ \eta'=\left\{\frac{4R_0(\chi_a\Phi_a+\chi_b\Phi_b)}{A^2}\Delta T+\frac{3T_1+T_2}{\Delta T}\right\}^{-1}, \tag{14.7} \]
where \(\Phi=\dfrac{s'}{L}\) is the shape factor.
The optimal geometric dimensions of the conductors correspond to the values \(\Phi_a\) and \(\Phi_b\) that minimize the product
\[ R_0(\chi_a\Phi_a+\chi_b\Phi_b) = \left(\frac{\rho_a}{\Phi_a}+\frac{\rho_b}{\Phi_b}\right) (\chi_a\Phi_a+\chi_b\Phi_b) \]
for given values of \(\rho_a,\rho_b,\chi_a,\chi_b\). By the usual rules we find:
\[ \left(\frac{\Phi_a}{\Phi_b}\right)_{\mathrm{opt}} = \left(\frac{\chi_a\rho_b}{\chi_b\rho_a}\right)^{1/2}. \]
Consequently, the most advantageous case is when the shape factors of conductors \(a\) and \(b\) are chosen in such a way that the heat conductance of the latter, as well as their electrical resistance, would be the same.
The efficiency of a thermoelement with optimally chosen values of the geometric parameters is
\[ \eta'=\Delta T \left\{ \frac{4\left[(\chi_a\rho_a)^{1/2}+(\chi_b\rho_b)^{1/2}\right]^2}{(\bar a)^2} +\frac{1}{2}(3T_1+T_2) \right\}^{-1}; \tag{14.8} \]
where \(\bar a\) is the mean differential thermoelectric emf in the temperature interval \(\Delta T\):
\[ \bar a=\frac{1}{\Delta T}\int_{T_1}^{T_2} a(T)\,dT. \tag{14.9} \]
Formula (14.8) makes it possible to calculate the efficiency of a thermoelement if the coefficients of thermal conductivity, electrical conductivity, thermoelectric emf, and the junction temperatures are known, for the case of a weak dependence of \(\chi\) and \(\rho\) on temperature. In the more general case one may formally preserve this form of the dependence of \(\eta'\) on the indicated quantities, but \(\chi_a,\rho_a\), etc., will already have the meaning of certain mean effective quantities depending, generally speaking, not only on the temperature but also on the temperature gradient.
For most metals and alloys the product \(\chi\rho \sim T\), and the proportionality coefficient, as is known, is equal to
\[ 2.45\cdot 10^{-8}\ \frac{\mathrm{W}\cdot\Omega}{\mathrm{deg}^2}. \]
Referring the values \(\chi_a\rho_a\) and \(\chi_b\rho_b\) in (15.12) to the mean temperature of the thermoelement,
\[ T_c=\frac{1}{2}(T_1+T_2), \]
we have
\[ \eta'=\Delta T\left\{\frac{19.6\cdot 10^{-8}}{\overline{\alpha}^{\,2}}-2T_c+\frac{1}{2}(3T_1+T_2)\right\}^{-1}. \]
Since the maximum mean differential thermopower of metals and metallic alloys is \(\sim 100\,\dfrac{\mu\mathrm{V}}{\mathrm{deg}}\), we obtain
\[ \eta'_{\text{metals}}\ll \Delta T(20.1T_1+21.1T_2)^{-1}. \tag{14.10} \]
If we take \(T_2 \simeq 300^\circ\mathrm{K}\) (room temperature), then from formula (14.10) it is clear that even with a temperature difference of \(500^\circ\mathrm{C}\), using the best metallic thermoelements we cannot attain an efficiency exceeding \(\sim 2\%\). Usually it is of the order of tenths of a percent.
Semiconductors differ from metals by a considerably greater electrical resistivity (this lowers the efficiency), but also by an incomparably greater thermopower. The latter fact is what determines the advantage of semiconductor materials over metallic ones for constructing thermoelements (since \(\overline{\alpha}\) enters formula (14.8) squared). Although, as was noted above, formula (14.8) is not directly applicable to this case (in semiconductors \(\rho \sim e^{\Delta E/2kT}\)), nevertheless, for estimating \(\eta'\) of semiconductor thermoelements one may proceed as follows. One may calculate its upper and lower limits by substituting into formula (14.8) the \(\rho\) corresponding to the “hot” (most electrically conducting) end, and then the \(\rho\) corresponding to the “cold” (least conducting) end, with one and the same \(\overline{\alpha}\), calculated by formula (14.9). In this way one can obtain, for a thermoelement consisting of PbS and ZnSb, at \(\Delta T=400^\circ\mathrm{C}\), \(\eta'\simeq 10\%\). Experimentally, under these conditions, \(\eta'\) proved to be \(\sim 7\%\)¹⁸.
Thus the problem of increasing the efficiency of thermoelements can be successfully solved on the basis of employing ever more perfect semiconductor materials. It is true that an increase in the thermopower of semiconductors is always accompanied by an increase in their resistance. However, these difficulties can be partly overcome by using intermetallic semiconductors of the ZnSb type. In the absence of disturbances in the stoichiometric composition, such compounds, possessing a large thermopower, are, in their electrically conducting properties, in the same category as insulators. However, the slightest disturbances of the stoichiometric composition (impurities) can reduce \(\rho\) by many orders of magnitude, while at the same time leaving the value of the thermopower still sufficiently large.
(Conclusion in the next issue.)
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