Current State of the Theory of Thermoelectric and Thermomagnetic Phenomena in Semiconductors

A. G. Samoilovich and L. L. Korenblit

Contents

Part II. Kinetic Theory*)

§ 15. Distribution function and kinetic equations . . . . . 338

§ 16. Formal solution of the kinetic equations . . . . . 340

§ 17. Generalized laws of electrical conductivity and thermal conductivity in kinetic theory . . . . . 341

§ 18. Kinetic equations in the case of high temperatures. Mean free path of electrons . . . . . 343

§ 19. Thermoelectric phenomena in monovalent metals at high temperatures . . . . . 346

§ 20. Equilibrium of electrons in semiconductors . . . . . 350

§ 21. Thermoelectric phenomena in semiconductors with an atomic lattice . . . . . 351

§ 22. Thermoelectric phenomena in ionic semiconductors . . . . . 357

§ 23. Distribution functions in the presence of weak magnetic fields . . . . . 360

§ 24. Thermomagnetic and galvanomagnetic phenomena in monovalent metals at high temperatures . . . . . 364

§ 25. Thermomagnetic and galvanomagnetic phenomena in semiconductors . . . . . 367

§ 26. Comparison of the theory of electrical phenomena in semiconductors with experiment . . . . . 372

§ 27. Conclusion . . . . . 379

Cited literature . . . . . 382

*) Part I, see UFN, Vol. XLIX, No. 2.

The kinetic theory of thermoelectric phenomena, in contrast to the thermodynamic theory, proceeds from definite model ideas about the structure of metals and other conductors of electricity in which thermoelectric effects may occur. Although, because of this, the generality of the conclusions is somewhat reduced, since it is difficult to establish which particular results of the theory are of a general character and which are the result of the adopted model and may subsequently be changed when another model is chosen, the exceptional importance of the kinetic theory consists in the fact that it makes it possible to calculate the kinetic coefficients entering into the generalized laws of electrical and thermal conductivity and thereby fills the conclusions of the thermodynamic theory with quite concrete content.

The calculation of kinetic coefficients is the principal task of the kinetic theory of thermoelectric phenomena.

§ 15. Distribution function and kinetic equations

We shall proceed from the assumption that the microstate of each particle of the system under consideration can be characterized by a certain set of parameters \(\alpha\) (the one-electron approximation of quantum mechanics).

We shall denote the distribution function by \(n_\alpha\) (the distribution of probabilities of microstates; \(n_\alpha\) is the probability that some one particle is in state \(\alpha\)).

The distribution function is found from the kinetic equation, which can be obtained from the following physical considerations. It is assumed that the distribution of particles over microstates may change under the influence of two causes:

1. Under the influence of external factors—applied fields, a temperature gradient.

2. Under the influence of interactions between particles (so-called “collisions”).

Let \((\partial n_\alpha/\partial t)_{\mathrm{ext}}\) denote the change of the distribution function under the influence of external factors, and \((\partial n_\alpha/\partial t)_{\mathrm{coll}}\) that due to “collisions.” Then, if the state of the system is stationary, the influences of these two factors mutually compensate each other:

\[ \left(\frac{\partial n_\alpha}{\partial t}\right)_{\mathrm{ext}} = - \left(\frac{\partial n_\alpha}{\partial t}\right)_{\mathrm{coll}}. \tag{15.1} \]

We shall denote by \(n_{\mathbf f}\) and \(N_\gamma\) the distribution functions of electrons and phonons, respectively, where \(\mathbf f\) is understood to mean the quasi-momentum of an electron, \(\gamma\) the quasi-momentum of a phonon, and let \(W_{\mathbf f,\mathbf f\pm\gamma}\) be the probability of transition of an electron from a state with quasi-momentum \(\mathbf f\) to a state ...

standing with quasi-momentum \(\mathbf f \pm \boldsymbol\gamma\). (\(W_{\mathbf f,\mathbf f\pm\boldsymbol\gamma}\) is symmetric with respect to \(\mathbf f\) and \(\mathbf f\pm\boldsymbol\gamma\).)

Assuming the distribution functions \(n_{\mathbf f}\) and \(N_{\boldsymbol\gamma}\) to differ little from their equilibrium values \(n_{\mathbf f}^0\) (Fermi distribution) and \(N_{\boldsymbol\gamma}^0\) (Bose–Einstein distribution), one may put

\[ n_{\mathbf f}=n_{\mathbf f}^0+\varphi_{\mathbf f}\,\frac{dn_{\mathbf f}^0}{dx_{\mathbf f}}, \tag{15.2} \]

\[ N_{\boldsymbol\gamma}=N_{\boldsymbol\gamma}^0+\Phi_{\boldsymbol\gamma}\,\frac{dN_{\boldsymbol\gamma}^0}{d\xi_{\boldsymbol\gamma}}, \tag{15.3} \]

where

\[ x_{\mathbf f}=\frac{1}{kT}\,[\varepsilon(\mathbf f)-\mu], \qquad \xi_{\boldsymbol\gamma}=\frac{1}{kT}\,\hbar\omega_{\boldsymbol\gamma}, \tag{15.4} \]

and \(\varphi_{\mathbf f}\) and \(\Phi_{\boldsymbol\gamma}\) are regarded as small corrections.

The kinetic equations for electrons and phonons, written to first order in \(\varphi_{\mathbf f}\) and \(\Phi_{\boldsymbol\gamma}\), have the form \({}^{10}\)

\[ T_e\,n_{\mathbf f}^0+\sum_{\boldsymbol\gamma} R_{\mathbf f,\boldsymbol\gamma}\,\Phi_{\boldsymbol\gamma} +\sum_{\mathbf f'} L_{\mathbf f,\mathbf f'}\,\varphi_{\mathbf f'}=0, \tag{15.5} \]

\[ T_\Phi\,N_{\boldsymbol\gamma}^0+S_{\boldsymbol\gamma}^{-1}\Phi_{\boldsymbol\gamma} +\sum_{\mathbf f} R_{\mathbf f,\boldsymbol\gamma}\,\varphi_{\mathbf f}=0, \tag{15.6} \]

where the coefficients of the equations have the following values:

\[ R_{\mathbf f,\boldsymbol\gamma} =(1-e^{-\xi_{\boldsymbol\gamma}})^{-1} \left[ \frac{ W_{\mathbf f,\mathbf f+\boldsymbol\gamma}\, \delta(\varepsilon_{\mathbf f+\boldsymbol\gamma}-\varepsilon_{\mathbf f}-\hbar\omega_{\boldsymbol\gamma}) }{ (e^{x_{\mathbf f+\boldsymbol\gamma}}+1)(e^{-x_{\mathbf f}}+1) } - \frac{ W_{\mathbf f,\mathbf f-\boldsymbol\gamma}\, \delta(\varepsilon_{\mathbf f}-\varepsilon_{\mathbf f-\boldsymbol\gamma}-\hbar\omega_{\boldsymbol\gamma}) }{ (e^{-x_{\mathbf f-\boldsymbol\gamma}}+1)(e^{x_{\mathbf f}}+1) } \right], \tag{15.7} \]

\[ \begin{aligned} L_{\mathbf f,\mathbf f'} ={}&\delta(\mathbf f'-\mathbf f)\sum_{\mathbf f''} \left[ \frac{ W_{\mathbf f',\mathbf f''}\, \delta(\varepsilon_{\mathbf f''}-\varepsilon_{\mathbf f'}-\hbar\omega_{\mathbf f''-\mathbf f'}) }{ (e^{x_{\mathbf f''}}+1)(e^{-x_{\mathbf f'}}+1)(1-e^{-\xi_{\mathbf f''-\mathbf f'}}) } \right.\\ &\left. \qquad\qquad + \frac{ W_{\mathbf f',\mathbf f''}\, \delta(\varepsilon_{\mathbf f'}-\varepsilon_{\mathbf f''}-\hbar\omega_{\mathbf f'-\mathbf f''}) }{ (e^{-x_{\mathbf f''}}+1)(e^{x_{\mathbf f'}}+1)(1-e^{-\xi_{\mathbf f'-\mathbf f''}}) } \right]\\ &- \left[ \frac{ W_{\mathbf f,\mathbf f'}\, \delta(\varepsilon_{\mathbf f'}-\varepsilon_{\mathbf f}-\hbar\omega_{\mathbf f'-\mathbf f}) }{ (e^{x_{\mathbf f'}}+1)(e^{-x_{\mathbf f}}+1)(1-e^{-\xi_{\mathbf f'-\mathbf f}}) } \right.\\ &\left. \qquad\qquad + \frac{ W_{\mathbf f,\mathbf f'}\, \delta(\varepsilon_{\mathbf f}-\varepsilon_{\mathbf f'}-\hbar\omega_{\mathbf f-\mathbf f'}) }{ (e^{-x_{\mathbf f'}}+1)(e^{x_{\mathbf f}}+1)(1-e^{-\xi_{\mathbf f-\mathbf f'}}) } \right], \end{aligned} \tag{15.8} \]

\[ S_{\gamma}= \left[ \sum_{\mathbf f} \frac{ W_{\mathbf f,\mathbf f+\gamma}\, \delta\!\left(\varepsilon_{\mathbf f+\gamma}-\varepsilon_{\mathbf f}-\hbar\omega_{\gamma}\right) }{ \left(e^{x_{\mathbf f+\gamma}}+1\right) \left(e^{-x_{\mathbf f}}+1\right) \left(1-e^{-\xi_{\gamma}}\right) } -\frac{1}{\tau_{\Phi}}\, \frac{dN_{\gamma}^{0}}{d\xi_{\gamma}} \right]^{-1}, \tag{15.9} \]

\[ \left. \begin{aligned} T_e n_{\mathbf f}^{0} &= \left[\nabla\mu+x_{\mathbf f}\nabla(kT)\right]\, \frac{\mathbf v_{\mathbf f}}{kT}\, \frac{dn_{\mathbf f}^{0}}{dx_{\mathbf f}}, \\[4pt] T_{\Phi}N_{\gamma}^{0} &= \xi_{\gamma}\nabla(kT)\, \frac{\mathbf c_{\gamma}}{kT}\, \frac{dN_{\gamma}^{0}}{d\xi_{\gamma}}, \\[4pt] \nabla\mu&=\nabla\mu_{0}-e\mathbf E . \end{aligned} \right\} \tag{15.10} \]

Here \(\mathbf v_{\mathbf f}\) and \(\mathbf c_{\gamma}\) are the mean (quantum-mechanical) velocities of the electron and phonon, respectively, \(\mathbf E\) is the electric field (the generalization to the case of a magnetic field will be given in § 23), and \(\tau_{\Phi}\) is the phonon relaxation time. It is easy to see that the coefficient \(L_{\mathbf f,\mathbf f'}\) is symmetric with respect to the indices \(\mathbf f\) and \(\mathbf f'\).

§ 16. Formal solution of the kinetic equations

Introduce the matrix \(K_{\mathbf f,\mathbf f'}\), inverse to \(L_{\mathbf f,\mathbf f'}\), so that

\[ \sum_{\mathbf f''} K_{\mathbf f,\mathbf f''}L_{\mathbf f'',\mathbf f'} = \delta(\mathbf f-\mathbf f'). \tag{16.1} \]

Then, with the aid of this matrix, one can solve equations (15.5), (15.6) with respect to \(\psi_{\mathbf f}\) and \(\Phi_{\gamma}\):

\[ \left. \begin{aligned} \psi_{\mathbf f} &= -\sum_{\mathbf f'} K_{\mathbf f,\mathbf f'}T_e n_{\mathbf f'}^{0} + \sum_{\mathbf f',\gamma} K_{\mathbf f,\mathbf f'}R_{\mathbf f',\gamma}S_{\gamma}T_{\Phi}N_{\gamma}^{0}, \\[4pt] \Phi_{\gamma} &= -S_{\gamma}T_{\Phi}N_{\gamma}^{0} + \sum_{\mathbf f,\mathbf f'} S_{\gamma}K_{\mathbf f,\mathbf f'}R_{\mathbf f,\gamma}T_e n_{\mathbf f'}^{0}, \end{aligned} \right\} \tag{16.2} \]

or, in more expanded form,

\[ \psi_{\mathbf f} = -\frac{\nabla\mu}{kT} \sum_{\mathbf f'} K_{\mathbf f,\mathbf f'}\mathbf v_{\mathbf f'} \frac{dn_{\mathbf f'}^{0}}{dx_{\mathbf f'}} - \frac{\nabla T}{T} \sum_{\mathbf f'} K_{\mathbf f,\mathbf f'}\mathbf v_{\mathbf f'}x_{\mathbf f'} \frac{dn_{\mathbf f'}^{0}}{dx_{\mathbf f'}} + \]

\[ + \frac{\nabla T}{T} \sum_{\mathbf f',\gamma} K_{\mathbf f,\mathbf f'}R_{\mathbf f',\gamma}S_{\gamma}\xi_{\gamma}\mathbf c_{\gamma} \frac{dN_{\gamma}^{0}}{d\xi_{\gamma}}, \tag{16.3} \]

\[ \Phi_{\gamma} = \frac{\nabla \mu}{kT}\sum_{f,f'} K_{f,f'} S_{\gamma} R_{f,\gamma} v_{f'}\, \frac{dn^0_{f'}}{dx_{f'}} + \]

\[ + \frac{\nabla T}{T}\sum_{f,f'} S_{\gamma} K_{f,f'} R_{f,\gamma} v_{f'} x_{f'}\, \frac{dn^0_{f'}}{dx_{f'}} - \frac{\nabla T}{T} S_{\gamma}\xi_{\gamma} C_{\gamma} \frac{dN^0_{\gamma}}{d\xi_{\gamma}} . \tag{16.4} \]

Usually in the theory of metals the nonequilibrium part of the phonon distribution is neglected. L. E. Gurevich\({}^{19}\) was the first to show that such an approximation is not always sufficient, since in many cases allowance for the nonequilibrium of the phonon distribution becomes necessary for a more complete description of transport phenomena in conductors, in particular, thermoelectric phenomena.

The additional perturbation of the electron distribution function under the influence of the nonequilibrium of the phonons was called by him the “drag” of electrons by phonons. Equation (16.3), as follows from its very derivation, takes account of the “drag” effect.

§ 17. Generalized laws of electrical conductivity and thermal conductivity in kinetic theory

Let us compute, with the aid of (16.3) and (16.4), the densities of the electric current and of the heat current. By definition,

\[ e j_i = e \sum_f \varphi_f \frac{dn^0_f}{dx_f} v_f^i, \tag{17.1} \]

\[ w_i = \sum_f \varphi_f \frac{dn^0_f}{dx_f}\,\varepsilon_f v_f^i + \hbar \sum_{\gamma} \Phi_{\gamma} \frac{dN^0_{\gamma}}{d\xi_{\gamma}}\, \omega_{\gamma} c_{\gamma}^i, \tag{17.2} \]

\[ i=1,\ 2,\ 3, \]

where \(e\mathbf{j}\) and \(\mathbf{w}\) are the densities of the electric current and of the energy flux, respectively. Replacing in (17.2) \(\varepsilon_f\) by \(kT x_f+\mu\) and recalling the definition of the heat current given in § 7, one may write the following formula for the components of the heat current:

\[ q_i = kT \sum_f \varphi_f \frac{dn^0_f}{dx_f}\, x_f v_f^i + kT \sum_{\gamma} \Phi_{\gamma} \frac{dN^0_{\gamma}}{d\xi_{\gamma}}\, \xi_{\gamma} c_{\gamma}^i . \tag{17.3} \]

If in (17.1) and (17.3) one substitutes the solutions (16.3) and (16.4) of the kinetic equations for electrons and phonons, then the equations for the electric and heat currents are brought to the form

\[ e j_i = e a^{il}\frac{\partial \mu}{\partial x_l} + e b^{il}\frac{\partial T}{\partial x_l}, \qquad q_i = c^{il}\frac{\partial \mu}{\partial x_l} + d^{il}\frac{\partial T}{\partial x_l}, \tag{17.4} \]

already familiar to us from the thermodynamic theory, where the coefficients have the following values:

\[ \left. \begin{aligned} a^{il}={}&-\frac{1}{kT}\sum_{f,f'} K_{f,f'} v_f^i v_{f'}^l \,\frac{dn_f^0}{dx_f}\,\frac{dn_{f'}^0}{dx_{f'}},\\[4pt] b^{il}={}&-\frac{1}{T}\sum_{f,f'} K_{f,f'} v_f^i v_{f'}^l x_{f'} \,\frac{dn_f^0}{dx_f}\,\frac{dn_{f'}^0}{dx_{f'}} \\ &+\frac{1}{T}\sum_{f,f',\gamma} K_{f,f'} R_{f',\gamma} S_\gamma v_f^i c_\gamma^l \xi_\gamma \,\frac{dn_f^0}{dx_f}\,\frac{dN_\gamma^0}{d\xi_\gamma},\\[4pt] c^{il}={}&-\sum_{f,f'} K_{f,f'} v_f^i v_{f'}^l x_f \,\frac{dn_f^0}{dx_f}\,\frac{dn_{f'}^0}{dx_{f'}} \\ &+\sum_{f,f',\gamma} K_{f,f'} R_{f,\gamma} S_\gamma v_{f'}^l c_\gamma^i \xi_\gamma \,\frac{dn_{f'}^0}{dx_{f'}}\,\frac{dN_\gamma^0}{d\xi_\gamma},\\[4pt] d^{il}={}&-k\sum_{f,f'} K_{f,f'} v_f^i v_{f'}^l x_f x_{f'} \,\frac{dn_f^0}{dx_f}\,\frac{dn_{f'}^0}{dx_{f'}} \\ &+k\sum_{f,f',\gamma} K_{f,f'} S_\gamma \left[ R_{f,\gamma} v_{f'}^l c_\gamma^i x_{f'}\xi_\gamma \,\frac{dn_{f'}^0}{dx_{f'}}\,\frac{dN_\gamma^0}{d\xi_\gamma} \right. \\ &\left. \qquad\qquad\qquad\qquad +R_{f',\gamma} v_f^i c_\gamma^l x_{f'}\xi_\gamma \,\frac{dn_f^0}{dx_f}\,\frac{dN_\gamma^0}{d\xi_\gamma} \right] \\ &-k\sum_\gamma S_\gamma c_\gamma^l c_\gamma^i \xi_\gamma^2 \left(\frac{dN_\gamma^0}{d\xi_\gamma}\right)^2 . \end{aligned} \right\} \tag{17.5} \]

In equations (17.4) summation over twice occurring indices is assumed. Of essential importance is the fact of the symmetry of the matrix \(L_{f,f'}\) and the symmetry, following from it, of the inverse matrix \(K_{f,f'}\).

Indeed, by virtue of this symmetry the following relations hold:

\[ a^{il}=a^{li},\qquad d^{il}=d^{li},\qquad c^{il}=Tb^{li}. \tag{17.6} \]

These relations express the principle of microscopic reversibility (the principle of symmetry of the kinetic coefficients) as applied to the case under consideration. Thus the “dragging” effect does not violate the validity of the second thermoelectric relation (which, as we have seen, is equivalent to the equality

\(c^i = T b^{il}\). This is connected with the circumstance that, although phonons directly participate only in the process of heat (energy) transfer, and not of electricity, nevertheless indirectly, by “dragging” electrons along with them, they produce an additional electric current—the “drag” electric current.

§ 18. Kinetic equations in the case of high temperatures. Mean free path of electrons

The explicit calculation of the inverse matrix \(K_{f,f'}\), by means of which we expressed the solution of the kinetic equations (15.5) and (15.6), is, generally speaking, associated with extraordinary mathematical difficulties, especially in the interval of intermediate temperatures. Bearing in mind that what interests us chiefly is the application of the theory to semiconductors, we shall confine ourselves to the comparatively simple case of high temperatures, which we shall analyze in the present paragraph in somewhat greater detail.

By the term “high temperatures” are understood temperatures much exceeding the Debye temperature \(\Theta\). In this case the mean energy of phonons is incomparably smaller than the mean energy of electrons:

\[ \hbar \bar{\omega}_\gamma \ll \bar{\varepsilon}_f . \]

Therefore at high temperatures it is possible, on the average, to regard “collisions” of electrons with phonons as taking place according to the laws of elastic impact, since as a result of these “collisions” the electrons, while changing the direction of their motion, almost do not change their energy. Moreover, at high temperatures electrons mainly create and absorb only the largest phonons, with energy \(\sim k\Theta\). Restricting ourselves to the consideration of isotropic media, the transition probabilities for electrons \(W_{f,f'}\) can always be written in the form \(W_{f,f'}(\cos \vartheta)\), where \(\vartheta\) is the angle between the directions of the vectors \(\mathbf f\) and \(\mathbf f'\). The same applies, of course, also to the coefficients \(R_{f,f'}\), \(S_\gamma\), and \(L_{f,f'}\), related to \(W_{f,f'}\) by formulas (15.7), (15.8), (15.9). In the case of high temperatures we may, in all these formulas, discard the terms \(\hbar \omega_\gamma\) in the arguments of the \(\delta\)-functions as negligible in comparison with \(\varepsilon_f\) and \(\varepsilon_{f'}\), and replace \((1-e^{-\xi_\gamma})^{-1}\) by \(T/\Theta\), which is also a consequence of the smallness of \(\hbar\omega_\gamma\). We shall use the orthogonality condition of the matrices \(K_{f,f'}\) and \(L_{f,f'}\):

\[ \frac{2}{(2\pi)^3}\int K_{f,f''}(\cos \vartheta'')\,L_{f''f'}(\cos \vartheta')\,f''^{\,2}\,df''\,\sin\vartheta'\,d\vartheta'\,d\varphi = -\frac{(2\pi)^3}{2}\, \frac{\delta(f-f')}{f'^{\,2}}\, \frac{\delta(\vartheta)\delta(\varphi)}{\sin\vartheta}, \qquad (18.1) \]

where \(\vartheta''\) is the angle between the directions \(\mathbf f\) and \(\mathbf f''\),

\(\vartheta'\) is the angle between the directions \(\mathbf f'\) and \(\mathbf f''\),

\(\vartheta\) is the angle between the directions \(\mathbf f\) and \(\mathbf f'\),

\(\dfrac{2}{(2\pi)^3}\) is the density of electron states in quasi-momentum space; moreover, at high temperatures

\[ \begin{aligned} L_{\mathbf f,\mathbf f'}(\cos\vartheta) &= -\,\frac{\delta(\mathbf f'-\mathbf f'')}{f'^2}\, \frac{\delta(\vartheta')\delta(\varphi')}{\sin\vartheta'}\, 4\pi\,\frac{T}{\Theta}\, \frac{dn^0_{\mathbf f'}}{dx_{\mathbf f'}}\, f'^2\,\frac{d f'}{d\varepsilon_{\mathbf f'}} \times \\ &\quad \times \int W_{\mathbf f,\mathbf f'}(\cos\chi)\sin\chi\,d\chi -\frac{T}{\Theta}\,\delta(\varepsilon_{\mathbf f''}-\varepsilon_{\mathbf f'}) W_{\mathbf f'',\mathbf f'}(\cos\vartheta') \times \\ &\quad \times \left[ \frac{1}{(e^{x_{\mathbf f''}}+1)(e^{-x_{\mathbf f'}}+1)} + \frac{1}{(e^{-x_{\mathbf f''}}+1)(e^{x_{\mathbf f'}}+1)} \right]. \end{aligned} \tag{18.2} \]

Expand \(K_{\mathbf f,\mathbf f''}(\cos\vartheta'')\) in a series in spherical functions:

\[ K_{\mathbf f,\mathbf f''}(\cos\vartheta'') = \sum_{l=0}^{\infty} A^l_{\mathbf f,\mathbf f''} P_l(\cos\vartheta''). \tag{18.3} \]

Using the addition theorem for spherical functions, as well as formulas (18.1)—(18.3), after simple transformations we obtain

\[ K_{\mathbf f,\mathbf f'}(\cos\vartheta) = \frac{1}{2\pi}\, \frac{(2\pi)^3}{2}\, \frac{\delta(f-f')}{f^2} \sum_{l=1}^{\infty} \frac{(2l+1)P_l(\cos\vartheta)}{2B^l_{\mathbf f'}}, \tag{18.4} \]

where

\[ B^l_{\mathbf f'} = -\frac{2}{(2\pi)^3}\,4\pi\, \frac{dn^0_{\mathbf f'}}{dx_{\mathbf f'}}\, f'^2\,\frac{d f'}{d\varepsilon_{\mathbf f'}}\times \]

\[ \times \frac{T}{\Theta} \int W_{\mathbf f,\mathbf f'}(\cos\chi) \{1-P_l(\cos\chi)\}\sin\chi\,d\chi . \]

For crystals with a center of symmetry, only terms of odd order in \(l\) remain in this expansion. Usually one restricts oneself to the first term, i.e., one sets

\[ K_{\mathbf f,\mathbf f'}(\cos\vartheta) = \frac{3}{4\pi}\, \frac{(2\pi)^3}{2}\, \frac{\delta(f-f')}{f'^2}\, \frac{\cos\vartheta}{B^1_{\mathbf f'}}, \tag{18.5} \]

where

\[ B^1_{\mathbf f'} = -\frac{2}{(2\pi)^3}\,4\pi\, \frac{dn^0_{\mathbf f'}}{dx_{\mathbf f'}}\, f'^2\,\frac{d f'}{d\varepsilon_{\mathbf f'}}\times \]

\[ \times \frac{T}{\Theta} \int W_{\mathbf f,\mathbf f'}(\cos\chi)(1-\cos\chi)\sin\chi\,d\chi . \tag{18.6} \]

Thus, in order to find the inverse matrix \(K_{f,f'}(\cos \vartheta)\), it is necessary to know the transition probabilities \(W_{f,f'}\). The latter are calculated with the aid of perturbation theory.

To find \(\varphi_f\) it is also necessary to calculate the sum

\[ I_{f'}=\sum_{\gamma} R_{f',\gamma} S_{\gamma}\xi_{\gamma}(c_{\gamma})_y \frac{dN^0_{\gamma}}{d\xi_{\gamma}} \]

(the temperature gradient and the field are directed along the \(y\)-axis).

At high temperatures the approximate equalities hold (see, for example, \(^{10,19}\)):

\[ S_{\gamma}\simeq -\frac{1}{\tau_{\Phi}}\frac{dN^0_{\gamma}}{d\xi_{\gamma}}, \]

\[ S_{\gamma}\xi_{\gamma}\frac{dN^0_{\gamma}}{d\xi_{\gamma}} \simeq -\tau_{\Phi}\xi_{\gamma}\simeq -\tau_{\Phi}\frac{\Theta}{T}, \]

\[ (c_{\gamma})_y \simeq \bar c\,\frac{\gamma_y}{\gamma} = \bar c\,\frac{\gamma_y}{\dfrac{\hbar\omega_{\gamma}}{\hbar c}} \simeq \hbar c^{-2}\frac{\gamma_y}{k\Theta}. \]

After simple calculations we obtain:

\[ I_{f'}=\frac{m^{*}\bar c^{\,2}}{kT}\,\tau_{\Phi} v_{f'}\cos \vartheta''\, B^1_f, \tag{18.7} \]

where \(\bar c\) is the velocity of sound, \(\vartheta''\) is the angle between the direction \(\mathbf f'\) and the \(y\)-axis, and \(m^*\) is the effective mass of the electron.

Thus,

\[ \varphi_f= \left[ \frac{l_f}{kT}\frac{\partial\mu}{\partial y} + x_f\frac{l_f}{T}\frac{\partial T}{\partial y} + \frac{m^*\bar c^{\,2}}{kT^2}\tau_{\Phi}v_f\frac{\partial T}{\partial y} \right]\cos\vartheta, \tag{18.8} \]

where \(\vartheta\) is the angle between the direction \(\mathbf f\) and the \(y\)-axis, and \(l_f\) is the so-called mean free path of the electrons:

\[ l_f=\frac{v_f}{B^1_f}\frac{dn^0_f}{dx_f} = \frac{\pi^3}{\hbar} \left[ f^2\left(\frac{\partial f}{\partial\varepsilon_f}\right)^2 T \int_0^{\pi} W_{f,f'}(\cos\chi)(1-\cos\chi)\sin\chi\,d\chi \right]^{-1}. \tag{18.9} \]

The last term in equality (18.8), which depends on the relaxation properties of the phonon system, takes into account the so-called effect of “dragging” of electrons by phonons.

§ 19. Thermoelectric phenomena in monovalent metals at high temperatures

The distinctive character of thermoelectric phenomena in metals is connected with the strong degeneracy of the electron gas. In this paragraph we shall restrict ourselves to considering the simplest—monovalent—metals, in whose case the one-electron scheme used in kinetic theory usually leads to satisfactory results. In this case it is convenient to use the effective-mass method, and the relation between the energy and the quasimomentum may be written in the form

\[ \varepsilon_f=\frac{\hbar^2 f^2}{2m^*}. \]

In connection with the strong degeneracy of the electron gas, the derivative \(\left(\partial n_f^0/\partial x_f\right)\) will be different from zero only in a small interval of energies near the chemical potential, which simplifies the calculations.

In doing so it is necessary to take into account that

\[ \int \frac{dn_f^0}{dx_f}\,dx_f=-1,\qquad \int x_f^2\frac{dn_f^0}{dx_f}\,dx_f=-\frac{\pi^2}{3}. \tag{19.1} \]

Then from (18.8) and (17.1), expanding the integrands in a series in powers of \(x_f\) and retaining terms with \(x_f\) to no higher than the first power, we obtain the following equation for the electric current (the index on \(j\) has been omitted):

\[ e\mathbf{j}=ea\nabla\mu+eb\nabla T, \tag{19.2} \]

where

\[ a=-\frac{1}{3\pi^2\hbar}\,f_0^2l_0, \]

\[ b=-\frac{k^2T}{9\hbar} \left(\frac{d}{d\varepsilon_f}f^2l_f\right)_{f=f_0} =-\frac{c^2}{3\pi^2T}\,f_0^3\tau_{\Phi}, \tag{19.3} \]

and the index “0” indicates that the corresponding quantities are taken at \(x_f=0\), i.e. at \(\varepsilon_f=\mu_0\).

According to the conclusions of the thermodynamic theory of thermoelectric phenomena, the coefficient of the reversible entropy current (see § 10) is

\[ S_j=-\frac{b}{a} =\frac{1}{3}\pi^2k^2T \left[ \frac{1}{\mu_0} + \left( \frac{1}{l_f}\frac{dl_f}{d\varepsilon_f} \right)_{f=f_0} \right] + \frac{c^2\hbar f_0}{Tl_0}\tau_{\Phi}. \tag{19.4} \]

(In (19.4) the linear dependence of \(f^2\) on \(\varepsilon_f\) has been taken into account.)

According to (18.9), the expression \(Tl_0\) does not depend on temperature. In the same approximation the term

\[ \frac{1}{l_f}\cdot\frac{dl_f}{d\varepsilon_f} \]

may also be regarded as independent of

temperature. As for the term \(\dfrac{1}{\mu_0}\), as is known, the dependence of \(\mu_0\) on temperature in the case of electrons in a metal is very weakly expressed.

Temperature enters the term with “drag” through the relaxation time \(\tau_\phi\), for which it was shown\({}^{19}\) that at high temperatures it is inversely proportional to temperature. Thus, the entropy flux is produced by two mechanisms, one of which leads to an effect \(\sim T\), and the other (“drag”) to an effect \(\sim T^{-1}\). This opens up the possibility of explaining the deviation of \(\alpha(T)\) from the linear law, which is sometimes observed experimentally in metals and which was not reflected within the framework of the old theories, which did not take the “drag” effect into account. Indeed, from (19.4) and on the basis of (10.5), (10.6), and (10.7) it follows at once that

\[ \Pi = \frac{1}{3e}\pi^2 k^2 T^2 \left[ -\frac{1}{\mu_0} + \left( \frac{1}{l_f}\frac{d l_f}{d\varepsilon_f} \right)_{f=f_0} \right] + \frac{c^2\hbar f_0}{e l_0}\,\tau_\phi, \tag{19.5} \]

\[ \alpha = \frac{1}{3e}\pi^2 k^2 T \left[ \frac{1}{\mu_0} + \left( \frac{1}{l_f}\frac{d l_f}{d\varepsilon_f} \right)_{f=f_0} \right] + \frac{c^2\hbar f_0}{e T l_0}\,\tau_\phi, \tag{19.6} \]

\[ \tau = \frac{1}{3e}\pi^2 k^2 T \left[ \frac{1}{\mu_0} + \left( \frac{1}{l_f}\frac{d l_f}{d\varepsilon_f} \right)_{f=f_0} \right] - \frac{c^2\hbar f_0}{e T l_0}\,\tau_\phi . \tag{19.7} \]

From the form of these equalities one may draw the following conclusions:

1. The term taking “drag” into account always gives a positive effect in the case of the Thomson phenomenon and a negative one in the case of the Peltier phenomenon.

2. The additional thermoelectric emf associated with “drag” is always of negative sign and is nonlinear in its temperature dependence.

3. It is possible, on the basis of measurements of the thermoelectric emfs, to estimate the magnitude of the thermoelectric emf caused by “drag.” Indeed, from (19.6), (19.7), and (2.3) it follows that

\[ \tau_\phi = \frac{e T l_0}{c^2\hbar f_0} \left( \alpha - T\frac{d\alpha}{dT} \right). \tag{19.8} \]

1. The sign of the thermoelectric emf depends on the sign and magnitude of the term

\[ \frac{1}{l_f}\frac{d l_f}{d\varepsilon_f}. \]

If this term is positive, then \(\alpha < 0\). If it is negative and, moreover, such that in absolute magnitude it is the determining term in equality (19.6), then the sign of \(\alpha\) may change to the opposite. Thus, the calculation of the dependence of the electron mean free path on energy acquires primary importance in the theory of thermoelectric phenomena. This calculation reduces to finding the probabilities of electron transitions under the action of the perturbing potential of the vibrating ions (atoms) by the methods of perturbation theory.

The simplest assumption that can be made concerning the perturbing potential of the vibrating atoms is the following: the ions vibrate together with their surrounding electron “clouds” as a single elastic continuum (the so-called model of “deformable” ions). For some monovalent metals such a scheme satisfactorily describes the main features of the interaction of an electron in a metal with the medium and gives the following result:

\[ l_f \sim \varepsilon_f^{2}. \tag{19.9} \]

Substituting (19.9) and (19.6) into (19.7) and neglecting the “drag” term, we obtain the relations known in the electron theory of metals

\[ \Pi=-\frac{1}{e}\pi^2 k^2 T^2 \frac{1}{\mu_0}, \]

\[ \alpha=\tau=-\frac{1}{e}\pi^2 k^2 T \frac{1}{\mu_0}, \tag{19.10} \]

from which it follows that at high temperatures \(\alpha\) and \(\tau\) must be negative and decrease proportionally with increasing temperature.

Table 1 gives a comparison of theoretical and experimental results

Table 1*)

Metal	\(t^\circ C\)	\(\alpha_{\mathrm{exp}}\cdot 10^6\) volt grad	\(\alpha_{\mathrm{theor}}\cdot 10^6\) volt grad	\(\left(\dfrac{\tau}{T}\right)_{\mathrm{exp}}\times 10^6\) volt grad	\(\left(\dfrac{\tau}{T}\right)_{\mathrm{theor}}\times 10^6\) volt grad
Na	400	\(-4\)	\(-8.8\)	\(-0.0282\)	\(-0.023\)
K	100	\(-7.25\)	\(-4.5\)	\(-0.0275\)	\(-0.036\)
Rb	200	\(-10\)	\(-6.9\)	\(-0.069\)	\(-0.041\)
Cs	400	\(-6\)	\(-8.8\)	\(-0.062\)	\(-0.048\)
Li	400	\(+41.2\)	\(-8.8\)	\(+0.040\)	\(0.016\)

for \(\alpha\) and \(\tau\) in the case of Na, K, Rb, Cs, Li at high temperatures (exceeding the melting points of these metals). The graphs in Fig. 6 also show that the temperature dependence of the thermoe.m.f. and of the Thomson coefficient (\(\alpha\) is expressed in \(\mu\mathrm{V}/\mathrm{degree}\)) agrees in order of magnitude with the predictions of the theory in the region of the liquid state of these metals.

As is seen from the table and the graphs, \(\tau_{\mathrm{theor}}\) predicts the experimental results rather well, with the exception of Li. The agreement is much worse in the case of the thermoe.m.f., although here too the theoretical

*) The data in Table 1 were calculated on the basis of the experimental curves given in work \(^{21}\).

and the experimental values agree in order of magnitude and in sign (with the exception of Li).

In some works of recent years\(^{21}\) there are indications that the Thomson coefficient is a linear function of temperature at \(T \gg \Theta\) not only for alkali metals, but also for certain metals of more complex structure (Au, Ag, Cu, Pd).

The model of “deformable” ions, on the basis of which the preceding results were obtained, does not take into account the circumstance that, in comparison with the heavy atomic cores, the electrons move almost without inertia owing to the large difference in masses. It follows from this that the problem of calculating the perturbing potential should more correctly have been posed as follows: the motion of the atoms is to be considered quasistatically, and for each configuration of the atoms the self-consistent potential of the electrons is to be found, since the latter, following the motion of the atoms without inertia, have time to redistribute themselves so that their potential energy in the field of the atoms is minimal at every instant (the adiabatic approximation). Such a physical scheme was made the basis of the calculation of the electrical conductivity of alkali metals by Bardeen\(^{23}\) and Serova\(^{23}\). Serova also calculated the Thomson effect for lithium\(^{24}\). In work\(^{22}\), plane unmodulated waves were taken as the wave functions for the electrons. This leads to the following result:

Fig. 6.

\[ \frac{1}{\mu_0}+\left(\frac{1}{l_f}\frac{dl_f}{d\varepsilon_f}\right)_{f=f_0} = \frac{3-0.06}{\mu_0}, \]

instead of \(3/\mu_0\), which is obtained in the theory of “deformable” ions. Thus, in this case the effect of the inertia-free redistribution of the electrons leads to a very small correction.

In Serova’s works on electrical conductivity and the Thomson effect in Li, refined wave functions of the electrons were used\(^{25}\).

As a result, the following was obtained:

\[ \frac{1}{\mu_0}+\left(\frac{1}{l_f}\frac{dl_f}{d\varepsilon_f}\right)_{f=f_0} = \frac{3-4.8}{\mu_0}. \]

Thus the effect of redistribution of the valence electrons of lithium proved to be predominant. Taking into account also the “entrainment” effect, Serova was able to explain satisfactorily the anomaly in the sign of the Thomson coefficient in the case of lithium. The use of more refined wave functions for the valence electrons also gives better agreement between the theoretical and experimental values for the electrical conductivity of Li[^23]. It is possible that the insufficient agreement between the theoretically calculated and experimental values of \(\alpha\) for the alkali metals is connected precisely with the circumstance that the behavior of the conduction electrons is described by ordinary Bloch functions, and not by more exact wave functions.

§ 20. Equilibrium of Electrons in Semiconductors

In metals the number of current carriers, and consequently also the chemical potential, are constant and depend only slightly on temperature. The fundamental difference between semiconductors and metals is that, in the case of semiconductors, electrons both of the valence band and of the conduction band participate in transport phenomena; moreover, various “impurity” (in the general sense of the word) energy levels may also enter the forbidden band.

When the temperature of a semiconductor changes, the electrons are redistributed among the valence band, the conduction band, and the impurity levels. Therefore the number of current carriers in semiconductors depends strongly on temperature.

As is known, the chemical potential of an equilibrium system of electrons can be found from the relation

\[ \frac{2}{(2\pi)^3}\int n_f^{0} f^2\,df\,d\Omega = n, \tag{20.1} \]

where \(n\) is the electron density, and the region of integration in the case of semiconductors is the valence band, the “impurity” levels, and the conduction band.

Depending on the particular form of the energy spectrum of the semiconductor, and also depending on the temperature, one can obtain from (20.1) various dependences between \(n\) and the chemical potential \(\mu_0\). The participation of electrons from one or another part of the energy spectrum in transport phenomena depends on the temperature of the semiconductor, or, more precisely, on the ratio of the mean thermal energy \(kT\) to the width of the various forbidden regions in the energy spectrum (according to Boltzmann’s principle).

The derivation of the dependence of \(\mu_0\) on \(n\) for various particular cases is given in the well-known review by Davydov and Shmushkevich[^26]. Here we give only a brief list of several formulas most important for us, valid at temperatures far from gas degeneracy, when Boltzmann statistics is applicable.

a) A pure semiconductor without impurities:

\[ \mu_0=-\frac{\Delta\varepsilon}{2}-\frac{1}{2}kT\ln\left(\frac{m_-^*}{m_+^*}\right)^{3/2}, \tag{20.2} \]

where \(m_-^*, m_+^*\) are the effective masses of electrons and holes, \(\Delta\varepsilon\) is the width of the forbidden band. If the densities of electrons and holes are denoted by \(n_{\pm}\), then

\[ n_- = n_+ = \frac{2}{(2\pi)^2}\frac{1}{\hbar} \left(\sqrt{m_-^*m_+^*}\,kT\right)^{3/2} e^{-\frac{\Delta\varepsilon}{2kT}} . \tag{20.3} \]

b) An impurity electronic semiconductor or an impurity hole semiconductor at temperatures far from saturation:

\[ \mu_0=-\frac{\Delta\varepsilon}{2} +kT\ln \frac{n_0^{1/2}h^{3/4}} {2^{1/2}\left(2\pi m_{\pm}^*kT\right)^{3/4}}, \tag{20.4} \]

\[ n= \frac{2^{1/2}n_0^{1/2}\left(2\pi m_{\pm}^*kT\right)^{3/2}} {h^2} e^{-\frac{\Delta\varepsilon}{2kT}}, \tag{20.5} \]

where \(n_0\) is the density of impurity electrons, \(\Delta\varepsilon\) is the energy gap between the impurity levels and the edge of the electron or hole bands.

Usually the temperature-dependent terms in formulas (20.2) and (20.4) are small, and the level of the chemical potential lies either midway between the normal band and the conduction band (an intrinsic semiconductor), or midway between the local levels and the band in which the charge carriers are found (the conduction band in the case of electronic conduction and the normal band in the case of hole conduction).

c) A semiconductor of the Neuber type:

\[ \mu_0=-\Delta\varepsilon +kT\ln\frac{n_0+\nu}{2(n_0-\nu)}, \tag{20.6} \]

\[ n= \frac{n_0+\nu}{n_0-\nu} \frac{\left(2\pi m_-^*kT\right)^{3/2}}{h^3} e^{-\frac{\Delta\varepsilon}{kT}}, \tag{20.7} \]

where \(n_0\) is the number of impurity levels, \(\nu/n_0\) is the degree of their filling.

§ 21. Thermoelectric phenomena in semiconductors with an atomic lattice

In the case of semiconductors with an atomic lattice, the interaction of conduction electrons with the medium has basically the same character as for metals, since it is due to “collisions” with phonons of the acoustic branch. The difference from the case of metals considered above consists, first, in the variety of specific mechanisms determining the electrical conductivity (electronic, hole, impurity conductivity), and, second, in the fact

absence of degeneracy of the electron gas in semiconductors (a small number of current carriers) at ordinary temperatures.

In the present section we shall consider semiconductors with mixed type of conductivity. Thermoelectric phenomena in electron or hole semiconductors are obtained from the general scheme as special cases. If the bottom of the conduction band is taken as the origin of energies and the chemical potential of the electrons is denoted by \(\mu_0\), then the chemical potential of the holes will be \(-\Delta \varepsilon-\mu_0\), where \(\Delta \varepsilon\) is the width of the energy gap between the electron and hole bands. In the absence of degeneracy, electrons and holes obey the Boltzmann distribution:

\[ n_f^{0}=e^{-x_f}, \qquad n_{f'}^{0}=e^{-x_{f'}}, \tag{21.1} \]

for electrons    for holes

where

\[ x_f=\frac{\varepsilon_f-\mu_0}{kT}, \qquad x_{f'}=\frac{\varepsilon_{f'}+\Delta \varepsilon+\mu_0}{kT}. \tag{21.2} \]

Since the condition of equilibrium between these two systems is taken into account in the chemical potential, we may regard electrons and holes as two independent systems and write for each of them the generalized laws of electrical conductivity quite analogously to the way we did this for the case of alkali metals at high temperatures, again using the concept of the mean free path of electrons.

Scattering of electron waves in semiconductors may be caused not only by lattice vibrations (scattering by acoustic or optical phonons), but also by various inclusions, disturbances of order in the lattice, etc. (impurities). This second source of scattering, in the case of metals, does not play a substantial role at ordinary temperatures, since the corresponding mean free path is very large (\(l\sim 0.1\ \mathrm{cm}\)) compared with the mean free path of electrons scattered by phonons, and therefore it begins to show itself in metallic alloys only at very low temperatures, when the two mean free paths become comparable. In some semiconductors, scattering of electrons by impurities, as scattering centers, may show itself already at room temperatures. In this case, just as in metallic alloys at low temperatures, it is necessary to distinguish ionized (more effective) from non-ionized impurities.

In contrast to metals, where redistribution of the electron system in the field of vibrating atoms plays an essential role in estimating the perturbation potential, in the case of semiconductors with an atomic lattice the excitation potential is determined only

with displacements of atoms. Calculations show that the transition probability for electrons is inversely proportional to the mean velocity of the electrons, and, consequently, the mean free path does not depend on the velocity and has the following form:

\[ \frac{1}{l}=\frac{4}{9\pi}\frac{kTm^{*2}}{\hbar^{4}c^{2}n_0M}\,C^2, \tag{21.3} \]

where \(n_0\) is the number of elementary cells per unit volume of the crystal,
\(M\) is the mass of the atoms in the elementary cell,
\(C^2\) is the so-called “interaction integral” in the theory of “deformable” ions.
Thus, \(l\sim T^{-1}\).

It is interesting to note that if the elementary cell of an atomic crystal contains more than one atom (2 atoms, for example, in the case of diamond), then in such a crystal there may occur not only acoustic, but also optical (although not polarization: the atoms of the elementary cell are identical!) vibrations.

In atomic crystals with high Debye temperatures (1800° K in the case of diamond), the optical phonons, naturally, may be regarded as practically not interacting with conduction electrons up to high temperatures. This can no longer be said of Ge and Si, whose Debye temperatures lie much closer to room temperatures.

As for the interaction of electrons with impurities, it has been established\(^{27}\) that \(l_{\mathrm{ip}}\) (the mean free path on ionized impurities) and \(l_{\mathrm{np}}\) (the mean free path on nonionized impurities) differ from \(l\) (the mean free path for interaction with phonons) in their dependence on the energy of the scattered electrons, and therefore the corresponding mean free paths differ in their dependence on temperature. (We note that in metallic alloys \(l_{\mathrm{ip}}\) and \(l_{\mathrm{np}}\) do not depend on temperature, owing to the strong degeneracy of the electron gas.)

Namely, \(l_{\mathrm{ip}}\sim \varepsilon_f^2\), \(l_{\mathrm{np}}\sim \varepsilon_f^{1/2}\). \(l_{\mathrm{ip}}\) and \(l_{\mathrm{np}}\) also depend on the concentrations of the corresponding impurities, as well as on the degree of their ordering. At small impurity concentrations the mean free path of the electrons can evidently be represented in the following form:

\[ \frac{1}{L}=\frac{1}{l}+\frac{1}{l_{\mathrm{ip}}}+\frac{1}{l_{\mathrm{np}}}. \]

In what follows, the effect of electron scattering by uncharged centers may be neglected in comparison with scattering by charged centers. Nor shall we take into account the “drag” of electrons by phonons, since, apparently, it does not play an essential role in the case of semiconductors\(^{28}\). (We shall estimate the “drag” term somewhat below.)

The expression for the total electric current obtained by superposing the currents of electrons and holes has the form:

\[ \mathbf{e j}=e^{2}a\mathbf{E}+eb\nabla T, \tag{21.4} \]

where

\[ a=-\frac{\hbar}{3\pi^{2}kT}\left[ \frac{1}{m_-^{*}}\int L_f f^{3}\frac{dn_f^{0}}{dx_f}\,df + \frac{1}{m_+^{*}}\int L_{f'} f'^{3}\frac{dn_{f'}^{0}}{dx_{f'}}\,df' \right], \tag{21.5} \]

\[ b=-\frac{\hbar}{3\pi^{2}T}\left[ \frac{1}{m_-^{*}}\int L_f x_f f^{3}\frac{dn_f^{0}}{dx_f}\,df - \frac{1}{m_+^{*}}\int L_{f'} x_{f'} f'^{3}\frac{dn_{f'}^{0}}{dx_{f'}}\,df' \right], \tag{21.6} \]

\[ e>0. \]

In (21.5) and (21.6) the first integrals are taken over all electron states, the second over all hole states.

As a result of integration we obtain:

\[ a=\frac{1}{3kT}\left[ L_- \overline{v_-}\,n_-+L_+ \overline{v_+}\,n_+ \right] =\frac{1}{e}\left(u_-n_-+u_+n_+\right), \tag{21.7} \]

\[ b=\frac{1}{3kT^{2}}\left\{ \left(\overline{\varepsilon_- L_- v_-}-\zeta_0\overline{L_- v_-}\right)n_- - \left[ \overline{\varepsilon_+ L_+ v_+}+(\Delta\varepsilon+\zeta_0)\overline{L_+ v_+} \right]n_+ \right\}, \tag{21.8} \]

where \(u_+\), \(u_-\) are the mobilities of holes and electrons, respectively. Horizontal bars denote averaging of the corresponding quantities over the distributions (21.1).

In the case of pure atomic semiconductors, or at such temperatures and impurity concentrations that the effect of scattering of electrons by impurities may be neglected, \(L_\pm=l_\pm\), and, as was noted above, \(l_\pm\) does not depend on the energy.

Therefore

\[ u_\pm=\frac{4}{3}\frac{el_\pm}{(2\pi m_\pm^{*}kT)^{1/2}}\sim T^{-1/2}, \tag{21.9} \]

\[ b=\frac{1}{eT}\left\{(2kT-\zeta_0)u_-n_- - \left[2kT+(\zeta_0+\Delta\varepsilon)\right]u_+n_+ \right\}. \tag{21.10} \]

In those cases where \(l_{\mathrm{imp}}\) is comparable in magnitude with \(l\), the mobility \(u\) acquires a more complicated dependence on temperature, namely

\[ u=a_1T^{-3/2}+a_2T^{3/2}. \]

In many substances (for example, Si, Ge, Mg\(_2\)Sn) precisely such a temperature dependence is observed \(a^{29,30}\).

THERMOELECTRIC AND THERMOMAGNETIC PHENOMENA

One can now estimate the role of “drag” by taking the ratio

\[ \frac{b_2}{b_1}\simeq \frac{ -\dfrac{m_-^*\bar c^{\,2}\tau_\Phi}{3kT^2}\,n_-v_-^{\,2} }{ -\dfrac{l_-}{3kT^2}\,n_-\mu_0\bar v_- } = 3\,\frac{kT}{|\mu_0|}\,\frac{\bar c}{\bar v_-}\,\frac{l_\Phi}{l_-}\ll 1, \]

since

\[ \mu_0\ll kT,\qquad \bar c\ll \bar v_- . \]

Here \(b_2\) is the coefficient at \(\nabla T\) in the equation for the electric current of “drag” [see (19.2)], and \(b_1\) is the corresponding coefficient in equation (21.4), which does not take “drag” into account;

\[ l_\Phi=\bar c\,\tau_\Phi . \]

Thus, when one type of current carrier predominates, the “drag” term is certainly very small.

From (21.4), taking into account (21.7) and (21.10), it follows that

\[ S_j=\frac{e}{\sigma} \left[ \frac{2kT-\mu_0}{T}\,u_-n_- - \frac{2kT+\mu_0+\Delta\varepsilon}{T}\,u_+n_+ \right], \tag{21.11} \]

where \(\sigma=e(u_-n_-+u_+n_+)\) is the specific electrical conductivity.

From (21.11) the thermoelectric coefficients are obtained directly:

\[ \alpha= -\frac{1}{T\sigma} \left[ (2kT-\mu_0)u_-n_- - (2kT+\mu_0+\Delta\varepsilon)u_+n_+ \right], \tag{21.12} \]

\[ \Pi= -\frac{1}{e} \left[ (2kT-\mu_0)u_-n_- - (2kT+\mu_0+\Delta\varepsilon)u_+n_+ \right], \tag{21.13} \]

\[ \tau=T\frac{\partial\alpha}{\partial T}. \tag{21.14} \]

The characteristic features of one or another type of semiconductor are taken into account in formulas (21.12)—(21.14) through \(\mu_0\) and \(m^*\). Thus, for example, in the case of an intrinsic semiconductor

\[ n_-=\frac{2}{h^3}(2\pi m_-^*kT)^{3/2}e^{\mu_0/kT}, \]

\[ n_+=\frac{2}{h^3}(2\pi m_+^*kT)^{3/2}e^{-(\Delta\varepsilon+\mu_0)/kT}, \]

and formulas (21.12) and (21.13) are transformed into the following ones:

take the form (first derived by N. L. Pisarenko \(^{26}\)):

\[ \alpha=\frac{k}{\sigma}\left\{ \left[2+\ln \frac{(2\pi m_+^* kT)^{3/2}}{4\pi^3\hbar^3 n_+}\right]u_+ n_+ - \left[2+\ln \frac{(2\pi m_-^* kT)^{3/2}}{4\pi^3\hbar^3 n_-}\right]u_- n_- \right\}, \tag{21.15} \]

\[ \Pi=\frac{kT}{\sigma}\left\{ \left[2+\ln \frac{(2\pi m_+^* kT)^{3/2}}{4\pi^3\hbar^3 n_+}\right]u_+ n_+ - \left[2+\ln \frac{(2\pi m_-^* kT)^{3/2}}{4\pi^3\hbar^3 n_-}\right]u_- n_- \right\}. \tag{21.16} \]

In the presence of carriers of one sign these formulas are simplified to

\[ \left. \begin{aligned} \alpha&=\pm \frac{k}{e}\left[2+\ln \frac{2(2\pi m^* kT)^{3/2}}{h^3 n}\right],\\ \Pi&=\pm \frac{kT}{e}\left[2+\ln \frac{2(2\pi m^* kT)^{3/2}}{h^3 n}\right], \end{aligned} \right\} \tag{21.17} \]

where the sign “\(+\)” refers to hole semiconductors, and “\(-\)” to electron semiconductors.

For impurity semiconductors with a predominance of carriers of one sign we have:

\[ \alpha=\pm \frac{1}{e}\left(\frac{\Delta\varepsilon}{2T}+k\ln\frac{n_{\pm}}{n_0}\right). \tag{21.18} \]

At temperatures far from the degeneracy temperature, \(\Delta\varepsilon\gg kT\). In this case, considering impurity semiconductors, the formulas (21.12) and (21.13) can be simplified directly. Setting \(\mu_0=-\frac{1}{2}\Delta\varepsilon\) and discarding terms containing \(kT\), we obtain:

\[ \left. \begin{aligned} \alpha&=-\frac{\Delta\varepsilon}{2T\sigma}(u_-n_- - u_+n_+),\\ \Pi&=-\frac{\Delta\varepsilon}{2\sigma}(u_-n_- - u_+n_+). \end{aligned} \right\} \tag{21.19} \]

In the case of a predominance of carriers of one sign we have:

\[ \alpha=\mp\frac{\Delta\varepsilon}{2eT},\qquad \Pi=\mp\frac{\Delta\varepsilon}{2e}, \tag{21.20} \]

whence

\[ \tau=\frac{\Pi}{T}=\mp\frac{\Delta\varepsilon}{2eT} \]

(the upper sign is for the case of the hole current mechanism, the lower for the electronic case).

From the formulas written above it is evident that the thermoelectric effects associated with different conduction mechanisms have different signs, which gives rise to partial (and sometimes complete) mutual compensation of the effects.

It also follows from (21.19) that for impurity semiconductors with a single conduction mechanism the dependence $\alpha = \alpha \left( \dfrac{1}{T} \right)$ must be essentially linear. Such a dependence is indeed observed experimentally in many semiconductor substances over certain temperature intervals$^{31}$. In the presence of both electron and hole conduction, such a simple dependence of $\alpha \left( \dfrac{1}{T} \right)$ should no longer be expected. The temperature dependence of $\alpha$ is complicated still further if impurity centers begin to play the role of electron scattering centers (see, for example, $^{32}$). In experiments, the most diverse dependences $\alpha(T)$ have indeed been found; however, it is not always possible to connect them with definite conclusions of the present theory. This will be discussed in more detail in § 26.

§ 22. Thermoelectric phenomena in ionic semiconductors

In ionic semiconductors, when the elementary cell of the crystal consists of ions of different polarity and with masses differing from one another, in addition to acoustic vibrations (with a linear dependence of frequency on quasimomentum) there also exist so-called optical, or polarization, vibrations.

In ionic crystals, unlike in valence crystals, electron scattering by phonons occurs mainly on optical phonons, since the interaction energy arising as a result of the appearance of an electric dipole moment is much greater than the interaction energy with acoustic vibrations, where such a moment is not excited.

It turns out that, just as in metals, electron scattering is produced only by the longest-wavelength longitudinal vibrations. Therefore, in order to calculate the perturbing potential one may proceed as follows: the vector of optical polarization associated with normal vibrations of unlike ions may be regarded as “smeared out” over space, and the additional potential produced by the polarization may be found under the assumption of a polarized continuum. The probabilities of electron transitions due to interaction with longitudinal optical phonons are then found from this perturbing potential according to the usual rules. In doing so, it is assumed that the spectrum of optical vibrations contains only one frequency $\omega_0$, coinciding with the maximum. Two cases should be distinguished.

a) \(\hbar\omega_0 \gg kT\). In this case electrons can emit phonons only after first being excited. But the lifetime of an electron in the excited state is

\[ \sim \frac{N_\gamma}{N_\gamma+1}=e^{-\frac{\hbar\omega_0}{kT}}\ll 1, \]

so that, considering the act of electron excitation (absorption of a quantum \(\hbar\omega_0\)) and the subsequent act of emission of a phonon only slightly different from \(\hbar\omega_0\) as a single “collision” process, we arrive at the same picture as in the case of metals at high temperatures, i.e., the “collision” processes between electrons and phonons may be regarded as “elastic,” with sharp changes in the direction of motion of the former.

b) \(\hbar\omega_0 \ll kT\). At first sight it seems that in this case there is no fundamental difference from the case of metals, apart from the fact that the place of acoustic phonons is now occupied by optical ones. However, this is not quite so. Calculations show that in ionic crystals

\[ \frac{1}{l_f} = \frac{\pi e^4 m^*}{\hbar M d^3 \omega_0}\, \frac{1}{\varepsilon_f} \left\{ \begin{array}{ll} 1+2\left(e^{\frac{\hbar\omega_0}{kT}}-1\right)^{-1}, & \varepsilon_f \gg \hbar\omega_0, \\[6pt] \left(e^{\frac{\hbar\omega_0}{kT}}-1\right)^{-1}, & \varepsilon_f \ll \hbar\omega_0, \end{array} \right. \tag{22.1} \]

where \(d\) is the distance between ions, and \(M\) is the reduced mass of the ions:

\[ \frac{1}{M}=\frac{1}{M^+}+\frac{1}{M^-}. \]

From (22.1) it is seen that, for \(\varepsilon_f=kT\gg \hbar\omega_0=k\Theta\),

\[ \frac{\varepsilon_f}{1+2\left(e^{\frac{\hbar\omega_0}{kT}}-1\right)^{-1}} \simeq \frac{k\Theta}{2}. \]

Conversely, for \(\varepsilon_f=kT\ll \hbar\omega_0=k\Theta\),

\[ \frac{\varepsilon_f}{1+2\left(e^{\frac{\hbar\omega_0}{kT}}-1\right)^{-1}} \simeq kT e^{\frac{\Theta}{T}}. \]

\(\Theta\) for many salts usually lies in the range from \(300^\circ\) to \(800^\circ\)K. A numerical calculation shows that only for \(kT\ll \hbar\omega_0\) is the mean free path sufficiently large in comparison with the lattice constant, whereas at \(T=\Theta\) and above the mean free path is, in order of magnitude, smaller than the lattice constant, and the concept of a mean free path, as well as the entire developed kinetic scheme, loses its physical meaning.

THERMOELECTRIC AND THERMOMAGNETIC PHENOMENA

We shall therefore restrict ourselves to considering only the case \(kT < \hbar\omega_0\), which is sufficient for many semiconductors, since this interval may include temperatures up to \(500\text{–}600^\circ\mathrm{K}\).

We give the final formula for the mean free path for the interaction of electrons with phonons at \(kT < \hbar\omega_0\),

\[ l_f= \frac{3\left(n^2-\varepsilon_0' +1\right)}{n^2-\varepsilon_0'} \left(\frac{\varepsilon_f}{\hbar\omega_0}\right)^{1/2} \frac{\hbar^2}{m_\pm^* e^2} \left(e^{\frac{\hbar\omega_0}{kT}}-1\right), \tag{22.2} \]

whence the mean free path of equilibrium electrons is

\[ l_\pm= \frac{6}{\sqrt{\pi}}\, \frac{n^2-\varepsilon_0' +1}{n^2-\varepsilon_0'}\, \frac{\hbar^2}{m_\pm^* e^2} \left(\frac{T}{\Theta}\right)^{1/2} \left(e^{\frac{\Theta}{T}}-1\right). \tag{22.3} \]

Here \(n\) is the refractive index, \(\varepsilon_0'\) is the dielectric constant of the medium.

In the range of applicability of this formula \(e^{\frac{\hbar\omega_0}{kT}} \gg 1\), and the unity in (22.3) may be neglected. Then

\[ l_f=l_0^\pm \left(\frac{\varepsilon_f}{\hbar\omega_0}\right)^{1/2} e^{\frac{\hbar\omega_0}{kT}}, \tag{22.4} \]

where

\[ l_0= \frac{6}{\sqrt{\pi}}\, \frac{n^2-\varepsilon_0' +1}{n^2-\varepsilon_0'}\, \frac{\hbar^2}{m_\pm^* e^2}. \tag{22.5} \]

We can now repeat all the calculations that were carried out for atomic semiconductors in finding the thermoelectric coefficients.

Finally we obtain:

\[ \left. \begin{aligned} \alpha&=\frac{1}{T^2} \left| \left(\frac{5}{3}kT-\mu_0\right)u_- n_- - \left(\frac{5}{3}kT+\mu_0+\Delta\varepsilon\right)u_+ n_+ \right|,\\[4pt] \Pi&=T\alpha,\qquad \tau=T\frac{d\alpha}{dT},\\[4pt] u_\pm&=\frac{e l_0 e^{\Theta/T}}{\sqrt{2m_\pm^* k\Theta}}, \end{aligned} \right\} \tag{22.6} \]

where, as in the preceding section, the specific structure of the energy spectrum of a given semiconductor substance is taken into account by means of the parameters \(\mu_0\), \(m_\pm^*\), \(\Theta\).

In the polaron model of ionic semiconductors, developed by S. I. Pekar\(^1\), somewhat different expressions are obtained for the mean free path and the mobility of the carriers (polarons)

for \(kT < \hbar\omega_0\), namely

\[ l_{\mathrm{pol}}=-\frac{c^{2}}{\hbar^{1/2}v_n}\sqrt{2M^{*}\omega_0}\, \frac{(1+1/\varkappa)^2}{(1+\varkappa)^8} \left(e^{\Theta/T}-1\right)^{-1}, \tag{22.7} \]

\[ u_{\mathrm{pol}}=u_0 e^{\Theta/T}, \tag{22.8} \]

where \(M^{*}\) is the mass of the polaron, \(v_n\) is its mean drift velocity,

\[ \varkappa=0.116\,\frac{\mu e^4 c^{2}}{\hbar^{3}\omega_0}, \qquad u_0=\left(\frac{\hbar}{M^{*}}\right)^{3/2} \frac{1}{c^{2}\sqrt{2\omega_0}}\, \frac{(1+\varkappa)^8}{(1+1/\varkappa)^2}. \tag{22.9} \]

§ 23. Distribution functions in the presence of weak magnetic fields

The kinetic equations for electrons and phonons in the presence of weak magnetic fields can easily be solved by the inverse-matrix method for the case of high temperatures. Here weak magnetic fields are understood to mean fields that distort the original energy spectrum of the electrons only insignificantly[^33]. Indeed, retaining the notation introduced in § 18, the term \((\partial n_{\mathbf f}/\partial t)_{\mathrm{ext}}\) in the present case has the following form:

\[ \left(\frac{\partial n_{\mathbf f}}{\partial t}\right)_{\mathrm{ext}} = T_e n_{\mathbf f}^{0} +\frac{e}{m^{*}c_1}\frac{dn_{\mathbf f}^{0}}{dx_{\mathbf f}} [\mathbf f\mathbf H]\,\nabla_{\mathbf f}\varphi_{\mathbf f}, \]

where \(c_1\) is the speed of light in vacuum. After elementary transformations it can be represented in the form

\[ \left(\frac{\partial n_{\mathbf f}}{\partial t}\right)_{\mathrm{ext}} = T_e n_{\mathbf f}^{0} - \frac{e}{m^{*}c_1}\int \frac{dn_{\mathbf f'}^{0}}{dx_{\mathbf f'}} [\mathbf f'\mathbf H]\, \varphi_{\mathbf f'}\nabla_{\mathbf f'}\delta(\mathbf f'-\mathbf f)\, (d\mathbf f'). \tag{23.1} \]

Let the magnetic field \(\mathbf H\) be directed along the \(z\)-axis \((H_x=H_y=0,\ H_z=H)\). Denote by \((\chi,\psi)\), \((\chi',\psi')\), and \((\chi'',\psi'')\) the coordinates of the vectors \(\mathbf f\), \(\mathbf f'\), and \(\mathbf f''\), respectively, and by \((\vartheta,\varphi)\), \((\vartheta',\varphi')\), \((\vartheta'',\varphi'')\), respectively, the angles between the directions of the following pairs of vectors: \((\mathbf f,\mathbf f')\), \((\mathbf f,\mathbf f'')\), \((\mathbf f',\mathbf f'')\).

Then the orthogonality condition for the direct and inverse matrices can be represented in the form

\[ \iint\left[ L_{\mathbf f,\mathbf f''}(\cos\vartheta')K_{\mathbf f'',\mathbf f'} \frac{2}{(2\pi)^3} f''^{2}\sin\vartheta'' + \right. \]

\[ \left. +\frac{eH}{m^{*}c_1}\frac{\partial K_{\mathbf f,\mathbf f'}}{\partial\psi''} \frac{dn_{\mathbf f''}^{0}}{dx_{\mathbf f''}} \delta(f''-f')\delta(\vartheta'')\delta(\varphi'') \right]\,df''d\vartheta''d\varphi'' = \frac{(2\pi)^3}{2}\, \frac{\delta(f'-f)}{f^2}\, \frac{\delta(\vartheta)\delta(\varphi)}{\sin\vartheta}. \tag{23.2} \]

If one expands \(K_{\mathbf f'',\mathbf f'}\) and \(L_{\mathbf f,\mathbf f''}\) in series in spherical functions

\[ K_{\mathbf f'',\mathbf f'}= \sum_{l=0}^{\infty}\sum_{m=-l}^{+l} A_{\mathbf f'',\mathbf f'}^{\,l,m} P_l^m(\cos\chi'')\,P_l^m(\cos\chi')\,e^{im(\psi''-\psi')}, \tag{23.3} \]

\[ L_{\mathbf f,\mathbf f''}=L_{\mathbf f,\mathbf f''}(\cos\vartheta') = \sum_{k=0}^{\infty}\sum_{m=-k}^{+k} B_{\mathbf f,\mathbf f''}^{\,k} P_k^n(\cos\chi)\,P_k^n(\cos\chi'') \times \]

\[ \times e^{in(\psi-\psi'')} \tag{23.4} \]

and substitute the expansions (23.2), then after uncomplicated, though cumbersome, calculations we find:

\[ K_{\mathbf f,\mathbf f'}= \frac{1}{2\pi}\cdot\frac{(2\pi)^3}{2}\, \frac{\delta(f'-f)}{f^2} \sum_{l=1}^{\infty} \left\{ \frac{2l+1}{2}\, \frac{P_l(\cos\chi)\,P_l(\cos\chi')}{B_{\mathbf f}^{\,l}} +\right. \]

\[ \left. +2\sum_{m=1}^{l} \frac{P_l^m(\cos\chi)\,P_l^m(\cos\chi')} {(B_{\mathbf f}^{\,l})^2+ m^2\left(\dfrac{eH}{m^*c_1}\dfrac{d\eta_{\mathbf f}^{0}}{dx_{\mathbf f}}\right)^2} \left[ B_{\mathbf f}^{\,l}\cos m(\psi-\psi') -\right.\right. \]

\[ \left.\left. -\frac{eH}{m^*c_1}\, \frac{d\eta_{\mathbf f}^{0}}{dx_{\mathbf f}}\, \sin m(\psi-\psi') \right] \right\}. \tag{23.5} \]

From (23.5) it is seen that \(K_{\mathbf f,\mathbf f'}\) is symmetric with respect to the interchange \(\mathbf f\rightleftarrows\mathbf f'\) only under the condition of a simultaneous change of sign of \(\mathbf H\):

\[ K_{\mathbf f,\mathbf f'}(\mathbf H)=K_{\mathbf f',\mathbf f}(-\mathbf H), \tag{23.6} \]

therefore the relations (17.6) following from the form (17.5) take, in the present case, the form\({}^{34}\):

\[ \left. \begin{aligned} a^{il}(\mathbf H)&=a^{li}(-\mathbf H),\\ Tb^{il}(\mathbf H)&=c^{li}(-\mathbf H),\\ d^{il}(\mathbf H)&=d^{li}(-\mathbf H). \end{aligned} \right\} \tag{23.7} \]

The equalities (23.7) constitute the expression of the symmetry principle for the kinetic coefficients in the case of magnetic fields.

In equation (23.5) (by analogy with § 18) it is sufficient to restrict ourselves to the first term of the series in \(l\).

Then we obtain:

\[ K_{\mathbf f,\mathbf f'}= -\frac{3}{4\pi}\cdot\frac{(2\pi)^3}{2}\, \frac{\delta(f'-f)}{f'^2} \left( \frac{d\eta_{\mathbf f'}^{0}}{dx_{\mathbf f'}} \right)^{-1} \frac{l_{\mathbf f}}{v_{\mathbf f}} \left\{ \cos\chi\cos\chi' +\right. \]

\[ \left. +\frac{\cos\psi-a_{\mathbf f}\sin\psi}{1+a_{\mathbf f}^{2}}\, \sin\chi\sin\chi'\cos\psi' +\right. \]

\[ \left. +\frac{\sin\psi+a_{\mathbf f}\cos\psi}{1+a_{\mathbf f}^{2}}\, \sin\chi\sin\chi'\sin\psi' \right\}, \tag{23.8} \]

where

\[ a_f=\frac{eH}{m^{*}c_1}\,\frac{l_f}{v_f} \]

and

\[ \varphi_i=\left\{\frac{l_f}{kT}\left[ \frac{\partial\mu}{\partial x}\, \frac{\cos\psi-a_f\sin\psi}{1+a_f^2} + \frac{\partial\lambda}{\partial y}\, \frac{\sin\psi+a_f\cos\psi}{1+a_f^2} \right]+\right. \]
\[ \left. +\left(x_f\frac{l_f}{T}+\frac{m^{*}\bar c^{\,2}}{kT^2}v_f\tau_\Phi\right) \left[ \frac{\partial T}{\partial x}\, \frac{\cos\psi-a_f\sin\psi}{1+a_f^2} + \frac{\partial T}{\partial y}\, \frac{\sin\psi+a_f\cos\psi}{1+a_f^2} \right]\right\}\sin\chi . \tag{23.9} \]

With the aid of (23.9) it is not difficult to calculate the coefficients of the system (17.4), namely:

\[ \begin{aligned} a_{xx}=a_{yy}=a &=\frac{4\pi}{3}\,\frac{1}{kT} \int \frac{l_f v_f}{1+a_f^2}\, \frac{dn_f^0}{dx_f}\,(df), \\[6pt] a_{xy}=-a_{yx} &=\frac{4\pi}{3}\,\frac{1}{kT} \int \frac{a_f}{1+a_f^2}\,l_f v_f\, \frac{dn_f^0}{dx_f}\,(df), \\[6pt] b_{xx}=b_{yy}=b=\frac{c}{T} &=\frac{4\pi}{3}\,\frac{1}{kT} \left\{ k\int \frac{l_f v_f x_f}{1+a_f^2}\, \frac{dn_f^0}{dx_f}\,(df) +\frac{m^{*}\bar c^{\,2}}{T}\tau_\Phi \int \frac{v_f^2\,\dfrac{dn_f^0}{dx_f}}{1+a_f^2}\,(df) \right\}, \\[6pt] b_{xy}=-b_{yx}=\frac{c_{xy}}{T} &=\frac{4\pi}{3}\,\frac{1}{kT} \left\{ k\int \frac{a_f}{1+a_f^2}\,l_f v_f x_f\, \frac{dn_f^0}{dx_f}\,(df) +\frac{m^{*}\bar c^{\,2}}{T}\tau_\Phi \int \frac{a_f}{1+a_f^2}\,v_f^2\, \frac{dn_f^0}{dx_f}\,(df) \right\}, \\[6pt] d_{xx}=d_{yy}=d &=\frac{4\pi k}{3} \left\{ \int \frac{l_f v_f x_f^2}{1+a_f^2}\, \frac{dn_f^0}{dx_f}\,(df) +\frac{2m^{*}\bar c^{\,2}}{kT}\tau_\Phi \int \frac{v_f^2 x_f}{1+a_f^2}\, \frac{dn_f^0}{dx_f}\,(df) -\bar c^{\,2}\tau_\Phi \right\}, \\[6pt] d_{xy}=-d_{yx} &=\frac{4\pi k}{3} \left\{ \int \frac{a_f}{1+a_f^2}\,l_f v_f x_f^2\, \frac{dn_f^0}{dx_f}\,(df) +\frac{2m^{*}\bar c^{\,2}}{kT}\tau_\Phi \int \frac{a_f x_f}{1+a_f^2}\,v_f^2\, \frac{dn_f^0}{dx_f}\,(df) \right\}; \qquad (df)=\frac{2}{(2\pi)^3}f^2\,df . \end{aligned} \tag{23.10} \]

Thus, in the presence of a magnetic field these coefficients have the form characteristic of gyrotropic media; moreover, as is easily seen, the symmetric parts are even, while the antisymmetric parts are odd functions of the field. The term

\[ \varkappa_p=\frac{4\pi k}{3}\,c\tau_\phi \]

on the right-hand side of the expression for (23.10) represents the contribution of the lattice to the phenomenon of thermal conductivity [see below formula (23.14)]. If the electronic thermal conductivity is small (high-resistance semiconductors), the coefficient of thermal conductivity \(\varkappa\) is practically reduced to \(\varkappa_p\). In metals, on the contrary, the main part of the thermal conductivity falls on the conduction electrons (\(\varkappa_p \ll \varkappa_{\mathrm{el}}\), where \(\varkappa_{\mathrm{el}}\) is the coefficient of electronic thermal conductivity).

For the study of galvanomagnetic and thermomagnetic phenomena it is convenient, as we have seen, to write the system (17.4) in the form (13.5). Simple calculations show that the following relation exists between the coefficients of the equations entering the systems (13.5) and (17.4):

\[ \rho=-\frac{a}{e^2\left(a^2+a_{xy}^2\right)},\qquad \rho_{[xy]}=\frac{a_{xy}}{e^2\left(a^2+a_{xy}^2\right)}, \tag{23.11} \]

\[ \alpha=-\frac{ab+a_{xy}b_{xy}}{e\left(a^2+a_{xy}^2\right)},\qquad \alpha_{[xy]}=-\frac{ab_{xy}-a_{xy}b}{e\left(a^2+a_{xy}^2\right)}, \tag{23.12} \]

\[ \Pi=-\frac{ac+a_{xy}c_{xy}}{e\left(a^2+a_{xy}^2\right)},\qquad \Pi_{[xy]}=-\frac{ac_{xy}-a_{xy}c}{e\left(a^2+a_{xy}^2\right)}, \tag{23.13} \]

\[ \left. \begin{aligned} \varkappa&=\frac{c\left(ab+a_{xy}b_{xy}\right)-c_{xy}\left(ab_{xy}-a_{xy}b\right)}{a^2+a_{xy}^2}-d,\\ \varkappa_{[xy]}&=\frac{c\left(ab_{xy}-a_{xy}b\right)+c_{xy}\left(ab+a_{xy}b_{xy}\right)}{a^2+a_{xy}^2}-d_{xy}, \end{aligned} \right\} \tag{23.14} \]

\[ e<0. \]

From formulas (23.11)—(23.14) it is clear that for the coefficients \(\rho_{ik}\), \(\alpha_{ik}\), \(\Pi_{ik}\), and \(\varkappa_{ik}\) as well, the fact noted above remains valid: the diagonal elements of these tensor coefficients are even, and the off-diagonal elements are odd functions of the field \(\mathbf H\).

Simultaneously with reducing the system of equations (17.4) to the form (13.5), and with the proof on the basis of the kinetic theory of the relations of microscopic reversibility (23.7) in the case of a magnetic field, the relations (13.30)—(13.34) and (13.39)—(13.41) between the coefficients characterizing the various longitudinal and transverse isothermal and adiabatic effects may, without further exposition, also be considered proved, since these relations follow, as we saw in § 13, precisely from the indicated properties of the generalized laws of electrical and thermal conductivity.

§ 24. Thermomagnetic and galvanomagnetic phenomena in monovalent metals at high temperatures

With the aid of definitions (13.16)—(13.29), using formulas (23.11)—(23.14) and (23.10), it is easy to calculate the thermomagnetic and galvanomagnetic coefficients. In the present paragraph we shall do this for the case of monovalent metals.

Analogous calculations were carried out for the case of thermoelectric phenomena in monovalent metals; therefore we shall not dwell on this here, and shall at once write down the values of the coefficients obtained as a result of integration:

\[ a=-\frac{1}{3\pi^{2}\hbar}\, \frac{l_{0}f_{0}^{2}}{1+a_{f_{0}}^{2}} -\frac{k^{2}T^{2}}{18\hbar} \left( \frac{d^{2}}{d\varepsilon_{f}^{2}} -\frac{l_{f}f^{2}}{1+a_{f}^{2}} \right)_{f=f_{0}}, \tag{24.1} \]

\[ a_{xy}=-\frac{eH}{3\pi^{2}\hbar^{2}c_{1}}\, \frac{l_{0}^{2}f_{0}}{1+a_{f_{0}}^{2}} -\frac{k^{2}T^{2}eH}{18\hbar^{3}c_{1}} \left( \frac{d^{2}}{d\varepsilon_{f}^{2}} -\frac{l_{f}^{2}f}{1+a_{f}^{2}} \right)_{f=f_{0}}, \tag{24.2} \]

\[ b=\frac{c}{T} =-\frac{k^{2}T}{9\hbar} \left( \frac{d}{d\varepsilon_{f}} -\frac{l_{f}f^{2}}{1+a_{f}^{2}} \right)_{f=f_{0}} -\frac{\bar c^{-2}\tau_{\Phi}}{3\pi^{2}T}\, \frac{f_{0}^{3}}{1+a_{f_{0}}^{2}}, \tag{24.3} \]

\[ \begin{aligned} b_{xy}=\frac{c_{xy}}{T} &=-\frac{k^{2}TeH}{9\hbar^{2}c_{1}} \left( \frac{d}{d\varepsilon_{f}} -\frac{l_{f}^{2}f}{1+a_{f}^{2}} \right)_{f=f_{0}} \\ &\quad -\frac{\bar c^{-2}\tau_{\Phi}eH}{3\pi^{2}\hbar c_{1}T}\, \frac{l_{0}f_{0}^{2}}{1+a_{f_{0}}^{2}}, \end{aligned} \tag{24.4} \]

\[ \begin{aligned} d &=-\frac{k^{2}T}{9\hbar}\, \frac{l_{0}f_{0}^{2}}{1+a_{f_{0}}^{2}} -\frac{2k^{2}T\bar c^{-2}\tau_{\Phi}}{9} \left( \frac{d^{2}}{d\varepsilon_{f}^{2}} -\frac{f^{3}}{1+a_{f}^{2}} \right)_{f=f_{0}} \\ &\quad -\frac{4\pi k}{3}\,\bar c^{-2}\tau_{\Phi}, \end{aligned} \tag{24.5} \]

\[ d_{xy}=-\frac{k^{2}TeH}{9\hbar^{2}c_{1}}\, \frac{l_{0}^{2}f_{0}}{1+a_{f_{0}}^{2}} -\frac{2kT\bar c^{-2}\tau_{\Phi}eH}{9\hbar c_{1}} \left( \frac{d^{2}}{d\varepsilon_{f}^{2}} -\frac{l_{f}f^{2}}{1+a_{f}^{2}} \right)_{f=f_{0}} . \tag{24.6} \]

Let us consider some galvanomagnetic and thermomagnetic effects.

a) Hall effect. According to (13.16) and (23.11),

\[ R_{\text{нз}}^{\perp} =-\frac{1}{e^{2}H}\, \frac{a_{xy}}{a^{2}+a_{xy}^{2}} . \tag{24.7} \]

In the case of the Hall effect it is sufficient to restrict ourselves to the first approximation (i.e., the system of electrons may be regarded as completely

degenerate), since in the present case we are dealing with a first-order effect.

Then

\[ R_{\mathrm{iz}}^{\perp}=\frac{1}{n_{-}ec},\qquad e<0, \tag{24.8} \]

where \(n_{-}\) is the electron density; \(f_0=(3\pi^2 n_{-})^{1/3}\).

The negative sign of \(R_{\mathrm{iz}}^{\perp}\) is connected with the electronic mechanism of the current (its sign depends on the sign of the carriers).

It also follows from (24.8) that the product \(R_{\mathrm{iz}}^{\perp}\sigma_{\mathrm{iz}}^{\parallel}\) (the mobility) must depend very weakly on temperature.

Qualitatively, the conclusions of the theory agree well with experiment in the case of monovalent metals (at least as far as the noble metals Au, Ag, Cu are concerned), i.e., they give the correct order of magnitude and the correct sign.

It must also be noted that in metals \(R_{\mathrm{iz}}^{\perp}\) differs little from \(R_{\mathrm{ad}}^{\perp}\) [see formula (13.20)], since \(\chi=\chi_{\mathrm{el}}+\chi_{\mathrm{p}}\) is very large (owing to the thermal conductivity of the electrons: \(\chi_{\mathrm{el}}\gg \chi_{\mathrm{p}}\)), while \(\alpha\) is usually small.

b) Change of resistance in a magnetic field. Usually this effect is described not by the coefficient \(\rho(\mathbf H)\), but by the quantity \((\Delta\rho/\rho_0)=[\rho(\mathbf H)-\rho(0)]/\rho(0)\), which characterizes the relative change in the resistance of a metal upon application of a magnetic field.

Let us note that in the present case we are dealing with a second-order effect.

Carrying out the necessary calculations, we obtain:

\[ \frac{\Delta \rho}{\rho_0} = -\frac{ \dfrac{1+a_{f_0}^{2}}{2a_0}\,\dfrac{k^2T^2}{9\hbar} \left( \dfrac{d^3}{d_i^2} - \dfrac{l_f f^2}{1+a_f^2} \right)_{f=f_1} }{ 1- \dfrac{1+a_{f_0}^{2}}{2a_0}\,\dfrac{k^2T^3}{9\hbar} \left( \dfrac{d^3}{d_i^2} - \dfrac{l_f f^2}{1+a_f^2} \right)_{f=f_0} } \simeq \]

\[ \simeq \frac{1+a_{f_0}^{2}}{2a_0}\, \frac{k^2T^2}{9\hbar} \left[ \frac{d^3}{d_i^2} - \frac{l_f f^3}{1+a_f^2} \right]_{f=f_0}, \qquad a_0=-\frac{l_0 f_0^2}{3\pi^2\hbar}. \]

When differentiating the expression in square brackets, one may take \(l_f\) to be independent of \(f\) and equal to \(l_0\). Then, retaining terms containing \(\mathbf H\) to powers no higher than the second, we obtain:

\[ \frac{\Delta\rho}{\rho_0} = \frac{B H^2}{1+CH^2}, \tag{24.9} \]

where

\[ B=-\frac{\pi^2}{3} \left( \frac{ekTl_0 m^*}{\hbar^3 f_0^2} \right)^3, \qquad C=\frac{a_{f_0}^{2}}{H^2}=(R_{\mathrm{iz}}^{\perp}\sigma_{\mathrm{iz}}^{\parallel})^2. \tag{24.10} \]

The dependence (24.9) gives a qualitatively good description of the course of the change in the resistance of metals in a magnetic field; however, there is a discrepancy, by a factor of \(10^4\), between the theoretically calculated value of \(B\) and its value found from the experimental curves. The reason for this discrepancy is that formula (24.9), with \(B\) taken from (24.10), takes into account not the principal, but the most secondary mechanism leading to an increase of the resistance in a magnetic field, namely, the presence of a Fermi “tail.” Indeed, owing to the assumed spherical symmetry of the distribution function of electrons in \(f\)-space in the case of complete degeneracy of the electron gas, there should be no change at all in the resistance in a magnetic field. Therefore, if one does not abandon the spherical symmetry of the distribution function, the only mechanism that leads to a change of \(\rho\) in a magnetic field is fluctuations of the electron energy in the Fermi “tail.” This negligible effect (owing to the high degree of degeneracy of the electron gas) is what is taken into account by formulas (24.9) and (24.10). The correct order of magnitude of \(B\), in agreement with experimental data, is obtained only on the basis of taking into account the actual anisotropy of the distribution, which must occur in any (even cubic) crystal. Indeed, this allows the magnetic field to produce a much more substantial redistribution of states. It is interesting to note that taking into account the anisotropy of the distribution over states has almost no effect on the magnitude of the Hall effect.

c) Change of the thermoelectric power in a magnetic field. Analogously to the resistance, the thermoelectric power (longitudinal) also changes in a magnetic field and determines the longitudinal Ettingshausen–Nernst effect. However, as a coefficient characterizing this effect, one may consider not \(Q_{\mathrm{in}}^{\parallel}=\alpha\), but \((\Delta\alpha,\alpha_0)=[\alpha(H)-\alpha(0)]/\alpha(0)\). Quite analogously to the change of resistance in a magnetic field, this is a second-order effect. Simple calculations show that

\[ \frac{\Delta\alpha}{\alpha_0}=\frac{CH^2}{1+CH^2}, \tag{24.11} \]

where \(C\) has been defined above [see formula (24.10)]. The thermoelectric power itself (or the coefficient of the longitudinal Ettingshausen–Nernst effect) then has the form

\[ \alpha=\alpha_0\left(1+\frac{CH^2}{1+CH^2}\right), \tag{24.12} \]

where \(\alpha_0=(b/ea)_{H=0}\) is the value of the thermoelectric power in the absence of a magnetic field.

d) Transverse Ettingshausen–Nernst effect. For this case we obtain:

\[ Q_{\mathrm{in}}^{\perp}\simeq \frac{\pi^2}{3}\frac{k^2Tl_0m^*}{\hbar^2 f_0^3 C_1}\,(1-CH^2). \tag{24.13} \]

In conclusion, let us note that so far we have considered only certain isothermal effects. The calculation of the remaining effects can be carried out in a completely analogous way.

§ 25. Thermomagnetic and galvanomagnetic phenomena in semiconductors

The specific features distinguishing transport phenomena in semiconductors from the case of metals were elucidated in §§ 20—22 in the investigation of the thermoelectric properties of semiconductors. Using the notation introduced there, it is easy to put the generalized laws of electrical and thermal conductivity in the form:

\[ \left. \begin{aligned} e j_x &=-e^2 a E_x-e^2 a_{xy}E_y+eb\,\frac{\partial T}{\partial x}+eb_{xy}\,\frac{\partial T}{\partial y},\\ e j_y &= e^2 a_{xy}E_x-e^2 aE_y-eb_{xy}\,\frac{\partial T}{\partial x}+eb\,\frac{\partial T}{\partial y},\\ q_x &=-ecE_x-ec_{xy}E_y+d\,\frac{\partial T}{\partial x}+d_{xy}\,\frac{\partial T}{\partial y},\\ q_y &= ec_{xy}E_x-ecE_y-d_{xy}\,\frac{\partial T}{\partial x}+d\,\frac{\partial T}{\partial y},\quad e>0, \end{aligned} \right\} \tag{25.1} \]

where

\[ a=\frac{\hbar}{3kT}\,4\pi\,\frac{2}{(2\pi)^3} \left( \frac{1}{m_+^*}\int \frac{l_{f'}f'^3}{1+a_{f'}^{\,2}}\, \frac{dn_{f'}^0}{dx_{f'}}\,df' + \frac{1}{m_-^*}\int \frac{l_f f^3}{1+a_f^{\,2}}\, \frac{dn_f^0}{dx_f}\,df \right), \]

\[ \left. \begin{aligned} a_{xy} &=\frac{\hbar}{3kT}\,4\pi\,\frac{2}{(2\pi)^3}\times \\ &\quad \times \left( \frac{1}{m_+^*}\int \frac{a_{f'}}{1+a_{f'}^{\,2}}\, l_{f'}f'^3\,\frac{dn_{f'}^0}{dx_{f'}}\,df' - \frac{1}{m_-^*}\int \frac{a_f}{1+a_f^{\,2}}\, l_f f^3\,\frac{dn_f^0}{dx_f}\,df \right), \\[1em] b=\frac{c}{T} &=\frac{\hbar}{3T}\,4\pi\,\frac{2}{(2\pi)^3}\times \\ &\quad \times \left( \frac{1}{m_+^*}\int \frac{l_{f'}f'^3x_{f'}}{1+a_{f'}^{\,2}}\, \frac{dn_{f'}^0}{dx_{f'}}\,df' - \frac{1}{m_-^*}\int \frac{l_f f^3x_f}{1+a_f^{\,2}}\, \frac{dn_f^0}{dx_f}\,df \right), \\[1em] b_{xy}=\frac{c_{xy}}{T} &=\frac{\hbar}{3T}\,4\pi\,\frac{2}{(2\pi)^3} \left( \frac{1}{m_+^*}\int \frac{a_{f'}}{1+a_{f'}^{\,2}}\, l_{f'}f'^3x_{f'}\,\frac{dn_{f'}^0}{dx_{f'}}\,df' \right.\\ &\quad \left. + \frac{1}{m_-^*}\int \frac{a_f}{1+a_f^{\,2}}\, l_f f^3x_f\,\frac{dn_f^0}{dx_f}\,df \right), \end{aligned} \right\} \tag{25.2} \]

\[ \left. \begin{aligned} d&=\frac{\hbar k}{3}\,4\pi\,\frac{2}{(2\pi)^3}\left( \frac{1}{m_+^*}\int \frac{l_{f'} f'^3 x_{f'}^2}{1+a_{f'}^2}\, \frac{dn_{f'}^0}{dx_{f'}}\,df' -\frac{1}{m_-^*}\int \frac{l_f f^3 x_f^2}{1+a_f^2}\, \frac{dn_f^0}{dx_f}\,df\right)-\frac{4\pi}{3}Rc^{-2}\tau_\varphi,\\ d_{xy}&=\frac{\hbar k}{3}\,4\pi\,\frac{2}{(2\pi)^3}\left( \frac{1}{m_+^*}\int \frac{a_{f'}}{1+a_{f'}^2}\, l_{f'} f'^3 x_{f'}^2\,\frac{dn_{f'}^0}{dx_{f'}}\,df' -\frac{1}{m_-^*}\int \frac{a_f}{1+a_f^2}\, l_f f^3 x_f^2\,\frac{dn_f^0}{dx_f}\,df\right). \end{aligned} \right\} \tag{25.2} \]

At the same time, if (25.1) is transformed to the form (13.5), then we again obtain formulas (23.11)—(23.14), where, however, \(e\) now has the meaning of the absolute, and not the algebraic, value of the electron charge.

The expression \(\left(1+a_f^2\right)^{-1}\), entering the integrands for the coefficients (25.2), may be expanded in powers of \(H\), and one may restrict oneself to terms in \(H\) not higher than quadratic. Integration with respect to \(f\) for the case of atomic lattices gives:

\[ \left. \begin{aligned} a&=-\frac{1}{e}\,(u_+n_+ + u_-n_-) +\frac{9\pi}{16c_1^2 e}\,(u_+^3 n_+ + u_-^3 n_-)\,H^2,\\ a_{xy}&=-\frac{3\pi}{8c_1 e}\,(u_+^2 n_+ - u_-^2 n_-)\,H,\\ b\equiv \frac{c}{T}&=-\frac{1}{eT}\big[(2kT+\mu_0+\Delta\varepsilon)\,u_+n_+ -(2kT-\mu_0)\,u_-n_-\big]\\ &\quad+\frac{9\pi}{16c_1^2 eT}\big[(kT+\mu_0+\Delta\varepsilon)\,u_+^3 n_+ -(kT-\mu_0)\,u_-^3 n_-\big]\,H^2,\\ b_{xy}\equiv \frac{c_{xy}}{T}&=-\frac{3\pi}{16c_1 eT}\left\{[3kT+2(\mu_0+\Delta\varepsilon)]\,u_+^2 n_+ +(3kT-2\mu_0)\,u_-^2 n_-\right\}H,\\ d&=-\frac{2k}{e}\left\{[3kT+2(\mu_0+\Delta\varepsilon)]\,u_+n_+ +(3kT-2\mu_0)\,u_-n_-\right\}\\ &\quad-\frac{1}{eT}\left[(\mu_0+\Delta\varepsilon)^3 u_+n_+ +\mu_0^2 u_-n_-\right]\\ &\quad+\frac{9k\pi}{8ec_1^2}\left[(kT+\mu_0+\Delta\varepsilon)\,u_+^3 n_+ +(kT-\mu_0)\,u_-^3 n_-\right]H^2\\ &\quad+\frac{9\pi}{16c_1^2 eT}\left[(\mu_0+\Delta\varepsilon)^3 u_+^3 n_+ +\mu_0^2 u_-^3 n_-\right]H^2 -\frac{4\pi k}{3}c^{-2}C\tau_\varphi, \end{aligned} \right\} \tag{25.3} \]

Thermoelectric and Thermomagnetic Phenomena

\[ \left. \begin{gathered} d_{xy}=-\frac{9\pi k}{8c_1e}\left\{[5kT+4(\mu_0+\Delta\varepsilon)]u_+n_+ -(5kT-4\mu_0)u_-n_-\right\}H \\ -\frac{3\pi}{8c_1eT}\left[(\mu_0+\Delta\varepsilon)^2u_+^2n_+ -\mu_0^2u_-^2n_-\right]H, \\ u_{\pm}=\frac{4el_{\pm}}{3(2\pi m_{\pm}^{*}kT)^{1/2}} . \end{gathered} \right\} \tag{25.3} \]

and for the case of ionic semiconductors

\[ \left. \begin{aligned} a&=-\frac{1}{e}(u_+n_+ + u_-n_-) +\frac{1}{ec_1^2}(u_+^3n_+ + u_-^3n_-)H^2,\\[4pt] a_{xy}&=-\frac{1}{ec_1}(u_+^2n_+ - u_-^2n_-)H,\\[4pt] b=\frac{c}{T}&=-\frac{k}{e}\left[\left(\frac{5}{2}kT+\mu_0+\Delta\varepsilon\right)u_+n_+\right.\\ &\qquad\left.-\left(\frac{5}{2}kT-\mu_0\right)u_-n_-\right]\\ &\quad+\frac{k}{c_1^2e}\left[\left(\frac{5}{2}kT+\mu_0+\Delta\varepsilon\right)u_+^3n_+\right.\\ &\qquad\left.-\left(\frac{5}{2}kT-\mu_0\right)u_-^3n_-\right]H^2,\\[4pt] b_{xy}=\frac{c_{xy}}{T}&=-\frac{k}{c_1e}\left[\left(\frac{5}{2}kT+\mu_0+\Delta\varepsilon\right)u_+^2n_+\right.\\ &\qquad\left.+\left(\frac{5}{2}kT-\mu_0\right)u_-^2n_-\right]H,\\[4pt] d&=-\frac{5k}{e}\left[\left(\frac{7}{4}kT+\mu_0+\Delta\varepsilon\right)u_+n_+\right.\\ &\qquad\left.+\left(\frac{7}{4}kT-\mu_0\right)u_-n_-\right]\\ &\quad-\frac{1}{eT}\left[(\mu_0+\Delta\varepsilon)^3u_+n_+ + \mu_0^2u_-n_-\right]\\ &\quad+\frac{5k}{ec_1^2}\left[\left(\frac{7}{4}kT+\mu_0+\Delta\varepsilon\right)u_+^3n_+\right.\\ &\qquad\left.+\left(\frac{7}{4}kT-\mu_0\right)u_-^3n_-\right]H^2\\ &\quad+\frac{1}{ec_1^2T}\left[(\mu_0+\Delta\varepsilon)^2u_+^3n_+ +\mu_0^2u_-^3n_-\right]H^2-\frac{4\pi}{3}kc^2\tau_{\phi},\\[4pt] d_{xy}&=-\frac{5k}{ec_1}\left[\left(\frac{7}{4}kT+\mu_0+\Delta\varepsilon\right)u_+^2n_+\right.\\ &\qquad\left.-\left(\frac{7}{4}kT-\mu_0\right)u_-^2n_-\right]H\\ &\quad-\frac{1}{ec_1T}\left[(\mu_0+\Delta\varepsilon)^3u_+^2n_+ -\mu_0^2u_-^2n_-\right]H,\\[4pt] u_{\pm}&=\frac{el_{\pm}^{0}}{(2m_{\pm}^{*}k\Theta)^{1/2}}e^{\Theta/T}. \end{aligned} \right\} \tag{25.4} \]

Various galvanomagnetic and thermomagnetic coefficients are calculated analogously to how this was done for the case of metals. Thus, for example, in the case of the Hall effect we obtain:

\[ R_{\text{iz}}^{\perp} = -\frac{1}{e^{2}H}\, \frac{a_{xy}}{a^{2}+a_{xy}^{2}} \simeq \frac{1}{e^{2}H}\,\frac{a_{xy}}{a} \simeq \frac{3\pi}{8c_{1}e}\, \frac{u_{+}^{2}n_{+}-u_{-}^{2}n_{-}} {(u_{+}n_{+}+u_{-}n_{-})^{2}} \tag{25.5} \]

for the case of atomic semiconductors, and

\[ R_{\text{iz}}^{\perp} \simeq \frac{1}{c_{1}e}\, \frac{u_{+}^{2}n_{+}-u_{-}^{2}n_{-}} {(u_{+}n_{+}+u_{-}n_{-})^{3}} \tag{25.6} \]

for the case of ionic semiconductors.

If there are current carriers of only one sign, these formulas take the following form:

\[ \left. \begin{aligned} R_{\text{iz}}^{\perp} &\simeq \pm \frac{3\pi}{8c_{1}en_{\pm}} &&\text{in the first case,}\\[4pt] R_{\text{iz}}^{\perp} &\simeq \pm \frac{1}{c_{1}en_{\pm}} &&\text{in the second.} \end{aligned} \right\} \tag{25.7} \]

In the case of semiconductors it cannot always be assumed that \(R_{\text{ad}}^{\perp}\simeq R_{\text{iz}}^{\perp}\), since the difference between them is aggravated.

Indeed, from the formula

\[ R_{\text{ad}}^{\perp}-R_{\text{iz}}^{\perp} = -\frac{1}{H}\frac{\alpha}{\chi}\Pi_{[xy]} = -\frac{T}{H}\frac{\alpha\chi_{[xy]}}{\chi_{\text{el}}+\chi_{\text{p}}} \]

it is clear that, in the case of high-resistance semiconductors, when the thermal conductivity is in fact reduced to the lattice thermal conductivity, small in magnitude,

\[ \left( \chi_{\text{p}}\sim 1\cdot 10^{-2}—1\cdot 10^{-3}\, \frac{\text{erg}}{\text{cm}\cdot\text{sec}\cdot\text{deg}} \right), \]

and when the thermo-e.m.f. reaches a considerable magnitude

\[ \left( \alpha\sim 5—10\cdot 10^{-4}\, \frac{\text{volt}}{\text{deg}} \right), \]

the additional e.m.f. associated with the occurrence of the Ettingshausen effect may sometimes (in the case when the sign of this e.m.f. is opposite to the sign of the Hall e.m.f.) not only distort the true value of the latter, but even completely compensate it and lead to an anomalous sign of \(R_{\text{iz}}^{\perp}\). This is apparently the case in tellurium\({}^{35}\), where, beginning at low temperatures \((t^\circ<-40^\circ\text{C})\) and up to \(t^\circ>230^\circ\text{C}\), \(R_{\text{iz}}^{\perp}\) changes its sign twice, from positive to negative and then again to positive. It has been possible to establish that, when the magnitude, sign, and temperature dependence of the e.m.f. are taken into account

Ettingshausen no sign changes of \(R_{\mathrm{iz}}^{\perp}\) should occur \(^{36}\).

To determine the change of resistance in a transverse magnetic field, we obtain, in the case of an atomic lattice:

\[ \frac{\Delta \rho}{\rho_0} = \frac{9\pi}{16c_1^2} \left[ \frac{u_+^3 n_+ + u_-^3 n_-}{u_+ n_+ + u_- n_-} \; [[\unclear: obscured sign or factor]] \; \frac{\pi}{4} \left( \frac{u_+^2 n_+ - u_-^2 n_-}{u_+ n_+ + u_- n_-} \right)^2 \right] H^2, \tag{25.8} \]

whereas in the case of an ionic lattice

\[ \frac{\Delta \rho}{\rho_0} = \frac{1}{c_1^2}\, n_+ u_+ n_- u_-\, \frac{(u_+ + u_-)^3}{(u_+ n_+ + u_- n_-)^3}\, H^2 . \tag{25.9} \]

It follows from (25.9) that, in the presence of carriers of one sign, the effect of the change in the resistance of ionic semiconductors in a magnetic field must vanish. This, generally speaking, contradicts experimental results and indicates that precisely in the case of ionic semiconductors the approximations of the present theory may prove to be completely inadequate to the real state of affairs.

In an analogous manner, the value obtained for the change of the thermo-emf in a magnetic field is too cumbersome. We shall give simplified formulas calculated for the case of carriers of one sign. Then

\[ \frac{\Delta x}{x} = \frac{9\pi^2}{64c_1^2} \begin{cases} \displaystyle u_+^2\, \frac{kT+\frac{3}{2}(\mu_0+\Delta\varepsilon)} {kT+\mu_0+\Delta\varepsilon}\, H^2 & \text{for holes,} \\[1.2em] \displaystyle u_-^2\, \frac{kT-\frac{3}{2}\mu_0} {kT-\mu_0}\, H^2 & \text{for electrons} \end{cases} \tag{25.10} \]

in the case of an atomic lattice, and

\[ \frac{\Delta \alpha}{\alpha_0} = -\frac{u_{\pm}^2}{c_1^2}H^2 \quad \text{for carriers of both signs} \tag{25.11} \]

in the case of an ionic lattice.

The Ettingshausen–Nernst coefficient has the following form:

\[ Q_{\mathrm{iz}}^{\perp} = -\frac{3\pi}{8c_1 e} \left\{ \frac{k}{2}\, \frac{u_+^3 n_+^2 + u_-^3 n_-^2} {(u_+ n_+ + u_- n_-)^3} - \frac{\frac{7}{2}kT+\Delta\varepsilon}{T}\, \frac{u_+ n_+ u_- n_- (u_+ + u_-)} {(u_+ n_+ + u_- n_-)^3} \right\}. \tag{25.12} \]

in the case of an atomic lattice, and

\[ Q_{\mathrm{нз}}=-\frac{1}{c_1 e}\, \frac{u_+ n_+ + u_- n_-}{(u_+ n_+ + u_- n_-)^2}\, \frac{(u_+ + u_-)\,\varepsilon kT+\Delta \varepsilon}{T}, \tag{25.13} \]

in the case of an ionic lattice.

As with the change in resistance, this effect should vanish in ionic semiconductors with carriers of only one sign.

§ 26. Comparison of the theory of electrical phenomena in semiconductors with experiment

The basis of the theory of the electrical properties of semiconductors (and of solids in general) is the undoubted fact of the quantization of the states of a system of electrons, the existence of allowed and forbidden energies for the motion of an individual electron (the one-electron band model), or of the entire aggregate of electrons (the many-electron, polar model).

Although the band character of the energy spectrum of solids (and, apparently, liquids) is undoubtedly one of the principal factors determining their electrical and other properties, nevertheless, since the theory set forth here was based on a highly simplified one-electron version of band theory, it is necessary to ascertain which facts are correctly conveyed by this theory.

The electrical (and other) properties of semiconductors depend essentially on the amount and character of the impurities they contain (example: Se[^37]), on their previous history (method of preparation, heat treatment), on the nature and pressure of the external atmosphere, and on a number of other factors (see, for example, [^38], [^39], [^41]).

Under such conditions, before proceeding to compare theory with experiment, it is necessary to establish a priori for what objects a (qualitative or quantitative) comparison of theory and experiment can be discussed, i.e., in what objects the premises and initial assumptions on which the theory is based are realized.

First of all, we note that the theory tacitly assumes complete homogeneity of the substance under study. However, this condition is not always satisfied in complex semiconductor substances of the type of metal oxides[^40] or in intermetallic semiconductors of the type \(\mathrm{Mg}_3\mathrm{Sb}_2\),[^38] if the latter have a polycrystalline structure, since very often the intercrystalline gaps may be filled with excess metal, which may substantially affect the magnitude and the temperature dependence of the electrical conductivity or thermoelectric force. Better agreement between theory and experiment should be expected for pure single-crystalline

of semiconductor specimens. Unfortunately, the overwhelming majority of experimental data concerning the energy properties of semiconductors pertain not to single-crystal specimens.

Usually the following are used:

1) massive polycrystalline specimens,
2) specimens made from powder of the given compound pressed (under one or another pressure),
3) film specimens of one thickness or another.

Massive polycrystalline specimens have the disadvantage that they are very often inhomogeneous, and their electrical properties in fact give only a certain statistical effect. Specimens made from pressed powders are still worse in this respect. Semiconductor specimens prepared in the form of homogeneous films can be a very valuable experimental object. However, here too there are a number of substantial drawbacks: film specimens, to a much greater degree than massive specimens, are subject to the action of external factors (pressure of the surrounding atmosphere, oxidation, occlusion, absorption and adsorption of gases, the influence of the cleanliness of the substrate surface and of its crystalline structure). Film specimens are very sensitive to the state of the surface, since in them surface energy levels begin to play a primary, dominant role. Undoubtedly, in thin films the small thickness of the specimens must also have a strong effect, apparently leading to a restructuring of the energy spectrum.

There are indications that, in the form of thin films, even metallic bodies can acquire characteristic semiconductor properties.

A number of semiconductors which do not exist in the massive form in the amorphous state can pass into such a state if they are made in the form of very thin films. All this indicates that between the electrical properties of some massive semiconductor and of a film specimen made from the same material as the massive specimen, there may sometimes exist a not entirely unambiguous relationship. The development of the theory of the film state has been begun only for the case of metals.

The existing theory of the electrical properties of semiconductors takes into account the influence of impurities in two ways: first, through a change in the energy spectrum introduced by impurities (impurity levels); second, impurities are taken into account simply as centers of additional scattering of electron waves with one or another “cross section” of scattering. The second effect is completely analogous to the additional scattering of electrons in metallic alloys, which determines the residual resistance at low temperatures. The difference from metals consists, apparently, in the fact that in the case of a nondegenerate electron gas, with which one deals in semiconductors, scattering by impurities may exert an influence not only

at low, but also at moderate temperatures, and this impurity-scattering effect depends substantially on temperature. In some semiconductors in which the mobility \(u \sim T^{-3/2}\), this effect may be disregarded as insignificant. In the case of other semiconductors (for example, the elements Ge, Si; the intermetallic compounds Mg\(_3\)Sb\(_2\), Mg\(_2\)Sn, etc.) the mobility has a more complicated temperature dependence, which indicates the need to take “impurity” scattering into account already at moderate temperatures.

This apparently occurs because different semiconductors, by virtue of their structure, have different sensitivity to impurities. Thus, for example, Ge and Si, at not very high temperatures, are in essence impurity semiconductors even at the most negligible impurity concentrations (\(\sim 10^{-3}\%\)), whereas other semiconductor substances (for example, a number of oxides of the ZnO type), all other conditions being equal, may in practice be regarded as “pure.” Secondly, this also depends on the degree of ordering of the impurities.

Two further circumstances should also be pointed out which must be borne in mind when comparing theory with experiment. The fact is that the usual formulae of the existing theory assume an electron gas far from the temperature of degeneracy, and they cannot be used when degeneracy begins to make itself felt. At the same time, in such substances as PbS, ZnSb, V\(_2\)O\(_3\), V\(_2\)O\(_5\), metallic degeneracy of the electrons is observed at not very high temperatures, so that it is always advisable to verify whether the electron gas in the particular case under consideration may be regarded as nondegenerate. In addition, it should be taken into account that formulae describing the electrical properties of semiconductors for the two limiting cases of purely atomic and purely ionic lattices may prove to be only approximately correct for substances with an intermediate type of bond (which is often encountered among intermetallic semiconductors).

From all that has been set forth it is clear that the question of comparing theory with experiment in the case of semiconductors must be approached by means of a comprehensive investigation of the influence of all possible factors on the electrical properties of semiconductors (impurity concentration, temperature, external atmosphere, degree of homogeneity, etc.).

At present, despite the fact that the electrical properties of very many semiconductor substances have been studied, the number of semiconductors that have been studied comprehensively and in depth is, unfortunately, not very large. These are primarily semiconductors of important technical interest (Cu\(_2\)O, PbS, Se, Te, Ge, Si, and a number of oxides). In the case of other semiconductor substances, the electrical properties have not been studied systematically enough to permit one to firmly establish the connection between their electrical properties

THERMOELECTRIC AND THERMOMAGNETIC PHENOMENA

with composition and structure, and this greatly complicates the comparison of theory with experiment.

It seems natural to classify the semiconductors studied up to now, from the standpoint of agreement between experiment and theory, in the following way:

1. Impurity-free semiconductors. To this group one may assign, in fact, all semiconductor substances, if they are sufficiently “pure” with respect to impurities; moreover, the maximum permissible concentration of impurities at which the substance still belongs to impurity-free semiconductors depends on the kind of substance and of impurities, and also on temperature.

Fig. 7.

Fig. 8.

In impurity-free semiconductors the basic conduction is intrinsic conduction. The electrical properties of impurity-free semiconductors are very well described by the ordinary theory (Figs. 7, 10 for the dependence \(\ln \sigma = f\!\left(\frac{1}{T}\right)\) in very pure specimens of various semiconductors, and Figs. 8, 9 for the dependence \(\alpha=\alpha\!\left(\frac{1}{T}\right)\)).

In the case of Ge and Si, whose properties have been studied in the greatest detail in the recent period, this is especially clearly seen. In a number of postwar works the electrical properties of both “pure” and \(p n\) specimens have been investigated. It is indicated that the electrical properties of very pure Ge (electrical conductivity, thermoelectric power, Hall effect) are very well described by the existing theory.

2. Impurity semiconductors at low impurity concentrations and high temperatures

From Figs. 7, 8, 10, 11 it is seen that, at sufficiently high temperatures, impurity semiconductors differ little in their electrical properties

Fig. 9.

Fig. 10.

from non-impurity samples of the same substances. The temperatures at which this transition takes place depend on the impurity concentrations and on the kind of substance, but, apparently, this situation is quite general.

Fig. 11.

Even if, at high temperatures, a simple coincidence of the properties of impurity and non-impurity samples of semiconductors is not observed, there is always a tendency toward a qualitative coincidence of the corresponding temperature dependences. Therefore, at high temperatures the electrical properties of semiconductors can be described quite well by a theory that does not take into account electron scattering at impurity centers.

This applies fully to semiconductors of the most diverse nature: the elements Ge and Si, the oxides V₂O₅, WO₃, MoO₃, ZnO, ionic semiconductors of the PbS type, intermetallic compounds of the Mg₃Sb₂, Mg₃Bi₂, Mg₂Sn type, and a number of others.

3. Semiconductors with pronounced impurity conductivity (high impurity concentration, low or moderate temperatures).

Very many semiconductors belonging to this group behave as the theory predicts. Especially illustrative is the example of Ge, which has been studied sufficiently comprehensively so that it has been possible to establish quite firmly the complete agreement of the theoretical predictions with the experimental data. It turned out that in this case scattering by impurity atoms plays an essential role. The theory of the electrical properties of Ge predicts in a very satisfactory manner all the studied properties of this semiconductor down to the very lowest temperatures.

Fig. 12.

Experiment showed, in full agreement with theory, that at very low temperatures (at these temperatures degeneration of the electron gas has time to set in) impurity \(p\)-\(n\) samples of Ge possess a temperature-independent residual resistance, quite analogous to metallic alloys (Matthiessen’s rule), in contrast to pure samples, where the resistance rises very sharply as the temperature is lowered. The thermoe.m.f., the Hall coefficient, and the Nernst and Ettingshausen coefficients depend on temperature as follows from theory (taking into account, of course, not only thermal but also impurity scattering of electrons).

Fig. 13.

However, along with this there are also semiconductors whose electrical properties do not quite fit within the framework of the theory. Thus, for example, in \(\mathrm{Cu_2O}\) and \(\mathrm{CuO}^{31}\) the thermoe.m.f. over a wide temperature interval does not depend on temperature (Fig. 12). A not entirely usual dependence \(\alpha(T)\) can be observed in the case of \(\mathrm{TaS_2}^{33}\) (Fig. 12); in some substances the absence (or smallness) of the Hall effect at first glance contradicts other electrical properties (\(\mathrm{TiO_2}^{41}\), etc.). The dependence

\[ \ln \sigma = f\!\left(\frac{1}{T}\right), \]

observed in the lower oxide of vanadium \(\mathrm{V_2O_3}^{42}\) (Fig. 13) or in pure polycrystalline \(\mathrm{Se}^{43}\), must seem very strange from the point of view of the theory.

Very often, with a more profound and comprehensive investigation not of a separately taken property, but of a whole series of electrical characteristics, these “anomalies” find a satisfactory explanation within the framework of band theory. Thus, for example, the absence of the Hall effect in \(\mathrm{TiO_2}\) can be explained\(^{41}\) if one assumes a large effective mass of the electron in \(\mathrm{TiO_2}\) (possibly, here there is an influence of inertial polarization of the crystal); the complicated temperature dependence of the thermoe.m.f. in \(\mathrm{TaS_2}\) is possibly connected with the presence of impurity scattering (with such scattering taken into account, \(\alpha(T)\) may possess a number of maxima and minima). Concerning the anomaly of the Hall effect in tellurium there are indications that it is connected either with structural changes, or with the neglect of the large difference between \(R_{\mathrm{add}}\) and \(R_{\mathrm{meas}}\).\(^{35,33}\) All these explanations, however, at the present time still remain to a greater degree working hypotheses than firmly established facts. Nevertheless, despite the existence of discrepancies between theory and experiment,

one still cannot as yet speak of the inapplicability of the theory to such substances.

4. Inhomogeneous semiconductors. Very often polycrystalline specimens of intermetallic semiconductors or metal oxides have microscopic inhomogeneities in the form of intercrystalline layers with an excess of metal (\(\mathrm{Mg_3Sb_2}\) \(^{38}\), \(\mathrm{ZnO}\) \(^{40}\), etc. \(^{44}\)). The electrical properties of such specimens constitute a certain statistical effect, and in such specimens the most varied electrical properties may be obtained. The theory should be compared with experimental data relating to such specimens with particular caution.

In general, it must be acknowledged that the existing theory of the electrical properties of semiconductors, based on the band model, proves successful in the simplest cases.

§ 27. Conclusion

In conclusion it is necessary to analyze in greater detail the basic premises of the theory, especially those assumptions which, from the modern point of view, give rise to the greatest doubts.

First of all it must be noted that in the theory there are two aspects of the question—one, if it may be so expressed, statistical, dealing with the averaging of microscopic currents and other processes in a nonequilibrium state, and the other—microscopic, which determines the energy structure of the system and permits one to calculate microcurrents and other characteristics. At the same time, as we shall see, such a division is nevertheless somewhat conventional, since macroscopic conditions in different parts of the object under consideration will influence the course of microprocesses in different ways. It is still convenient first to investigate the two aspects of the question separately.

As regards the statistical aspect of the problem, here it is necessary to note, as the principal assumption adopted both in the thermodynamic theory and in the kinetic theory, the hypothesis of local equilibrium. Without this hypothesis it is impossible to introduce either the concept of temperature for a nonuniformly heated body or other thermodynamic functions, in terms of which modern theory expresses its results. Therefore it is necessary first of all to establish the limits of applicability of the hypothesis of local equilibrium.

The present state of the theory makes it possible to give only rough estimates. Exact boundaries of the limits of applicability of the methods set forth can be established only on the basis of a more general theory of irreversible processes, one not founded on the concepts of the thermodynamics of equilibrium processes.

To obtain rough estimates of the limits of applicability of the theory of local equilibrium, it is necessary, first of all, to note that

the minimal dimensions of those regions whose state can still be characterized by thermodynamic parameters are determined first of all by the requirement that fluctuations of the density (and of other quantities) be small. If this requirement is not satisfied, then there can be no question at all of any thermodynamic description. The minimal dimensions of regions in which fluctuations are negligibly small can be estimated from Smoluchowski’s calculations of the mean return time of anomalous states and may be taken as \(d \geq 5 \cdot 10^{-5}\ \text{cm}^{45}\). In the case of systems in which there are free electric charges (metals, semiconductors), owing to the presence of slowly decreasing electrostatic forces these dimensions increase somewhat\(^{46}\).

In order that a small region may be assigned a definite temperature, it is necessary that the mean free path of the particles participating in thermal motion be significantly smaller than the dimensions of the region under consideration. For electrons in a metal \(l \sim 10^{-5}—10^{-6}\ \text{cm}\) at room temperature. For semiconductors with an atomic lattice the mean free path has the same order of magnitude, while for ionic lattices it is much smaller—\(10^{-8}\ \text{cm}\). Thus, regions of dimensions of the order of \(5 \cdot 10^{-5}—10^{-4}\ \text{cm}\) satisfy the requirements indicated above and may be regarded as those regions of local equilibrium which in the limit are treated as “points.”

Difficulties, however, may arise when quantum effects are taken into account. Evidently, the above considerations are valid when the mean de Broglie wavelength \(\lambda\) for a particle is significantly smaller than its mean free path \(l\). Indeed, since in calculating the relaxation time in kinetic theory transitions between stationary states are considered, the “smearing” of the energy due to the finite lifetime in the given state \(\delta \varepsilon \sim \dfrac{h}{\tau}\) must be much smaller than the energy of the particle in the same state. It follows directly from this that the condition

\[ \lambda \ll l . \]

must be satisfied.

However, in semiconductors this condition is often not fulfilled. Thus, for example, in the case of ionic semiconductors at high temperatures \(l \sim 10^{-8}\ \text{cm}\), whereas \(\lambda \sim 10^{-7}\ \text{cm}\). Thus the kinetic scheme proves to be unjustified here. Let us note that in some cases, for example in constructing the theory of the Hall effect and of the change of resistance in a magnetic field, this difficulty can be avoided and the theory formulated in such a way that the concept of the mean free path does not enter into it\(^{33}\).

There also arises the question of those limitations that are imposed on the possibility of characterizing the state thermodynamically

with functions in the presence of currents and temperature gradients. In this respect one can only say that the method set forth above is applicable if the distortion of the distribution of current carriers caused by external actions is small, i.e., if the distribution function differs only slightly from the equilibrium one (the condition of smallness of \(\varphi_f\) and \(\Phi_f\) relative to \(n_f^0\) and \(N_f^0\), respectively; see § 15). It is clear that the degree of deviation of the distribution function from the equilibrium function is determined by the processes of energy exchange of the current carriers with the lattice. This question was investigated in detail by Davydov and Shmushkevich \(^{34}\) with regard to the character of the processes in strong electric fields. They showed that in semiconductors, for fields \(\sim 10^5 \dfrac{\text{volts}}{\text{cm}}\), the distribution function already differs substantially from the Maxwellian one. In metals such fields are unattainable. Obviously, in practice there are not encountered such large temperature gradients as could produce similar fields, owing to the considerable thermal conductivity of the lattice. However, in some cases, at the contacts of two semiconductors, fields of great intensity may appear, and in these cases the usual treatment is no longer valid. One may suppose that in such cases the second thermoelectric relation will no longer be fulfilled.

Let us also note that, as Prigogine \(^{47}\) showed, the basic thermodynamic relation (5.2) in the case of irreversible processes is applicable only insofar as one may take into account only the first-order correction to the distribution function in kinetic theory. Taking into account corrections of the second and higher orders leads to other relations, in which the state can no longer be characterized by thermodynamic functions, but their gradients and higher derivatives must also be taken into account. Kohler \(^{16}\) likewise showed that the range of applicability of the method adopted in this review in fact coincides with the range of applicability of the generalized linear laws of electrical and thermal conductivity.

As for the second, so to speak, microscopic side of the question, it must be noted that the use of a one-electron scheme is a common defect of the modern theory of the solid state, and not specifically of the theory of kinetic processes. Recently Bonch-Bruevich, Vlasov, and Sokolov \(^{48}\) showed, using the example of the theory of electrical conductivity, how transport processes in conductors may be considered from the standpoint of the many-electron problem. There have as yet been no further applications of this method. Personally, it seems to us that the application of the many-electron treatment in connection with the method of second quantization (the quasiparticle method), in the spirit in which it is being developed by N. N. Bogolyubov and S. V. Tyablikov for semiconductors, is the most promising. One of the strongest features of this method is the possibility

considering the system as such an aggregate of quasiparticles in which the latter already prove to be comparatively weakly bound to one another, so that, with greater or lesser justification, the usual schemes for calculating transport phenomena can already be applied to these quasiparticles. The fruitfulness of such a method in the theory of metals was shown in the works of Vonsovskii \(^{48,49}\), and also in the works of Bohm, Gross, and Pines devoted to the theory of the electron plasma in metals \(^{50}\).

However, in connection with this there arises a whole series of new problems. Thus, for example, the problem of statistical averaging must here be posed in a new way, since some parameters characterizing these quasiparticles are themselves functions of temperature.

Still more important is the question of how one should consider the processes of energy exchange between current carriers and the lattice. In the usual kinetic theory these processes are considered as acts of emission and absorption of phonons, and these microprocesses are, in essence, reversible. Meanwhile, a closer consideration of this question from the new point of view gives grounds to suppose that a more adequate treatment of the process of energy exchange of current carriers with the lattice would be to interpret them as processes of damping. An important difference from the preceding would then be the circumstance that the microprocesses underlying transport phenomena would be irreversible. The problems we have listed show that the theory of transport phenomena in semiconductors in general, and of thermoelectric and thermomagnetic phenomena in particular, still has a broad and far-reaching path of development ahead of it.

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