Tamm Bound States of Electrons on a Crystal Surface and Surface Vibrations of Lattice Atoms

I. M. Lifshits and S. I. Pekar

The first part of the article gives a review of the successive stages in the development of the theory of Tamm surface states of electrons. The second part considers phenomena in which surface states of electrons play an essential role, and presents experimental evidence for the actual existence of these states in real crystals. The third part of the article sets out the theory of surface vibrations of a crystal lattice, which is mathematically close to and genetically connected with the theory of surface states of electrons*).

1. General Theory of Surface States of Electrons

In 1932 I. E. Tamm, on the basis of quantum mechanics, showed for the first time that, alongside the electron “band” states in a crystal known at that time, states of electrons of a completely different type may also exist at the surface of a crystal1,2. These “surface” states of electrons possess a discrete energy spectrum and wave functions that decay exponentially with increasing distance from the surface both into the depth of the crystal and toward the vacuum.

For a quantitative treatment of the question it is necessary to solve the wave equation for an electron moving in a static potential field of the following type: inside the crystal (the region \(x > 0\)) the potential is assumed to be periodic, while outside the crystal (the region \(x < 0\)) the potential is constant. The plane \(x = 0\) represents the surface of the crystal. Since the periodic potential of an electron in a crystal is not known exactly, and also because solving the wave equation with a periodic potential in the general case is very difficult, it was necessary, as in the theory of “band”

*) The first two parts of the article were written by S. I. Pekar, the third part by I. M. Lifshits.

states of the electron, first to consider a number of simplified particular cases (the one-dimensional case, the simplest types of periodic potential), and then, generalizing the results obtained, to draw conclusions valid for all particular examples and, apparently, not dependent on the special choice of the periodic potential.

In the one-dimensional case the wave equation has the form:

\[ \frac{d^2\psi}{dx^2}+\frac{2m}{\hbar^2}[E-U(x)]\psi=0, \tag{1} \]

where \(U(x)\) is a function having the period of the lattice for \(x>0\), and \(U=U_0\) is a constant for \(x<0\). As is known from the general theory of differential equations with periodic coefficients, inside the crystal the solution of equation (1) has the form:

\[ \psi=A_1u_k(x)e^{ikx}+A_2u_{-k}(x)e^{-ikx}, \tag{2} \]

where \(u_k\) and \(u_{-k}\) are functions having the period of the lattice, \(A_1\) and \(A_2\) are arbitrary constants, and the coefficient \(k\) is a definite function of the electron energy: \(k=k(E)\).

In an unbounded lattice extending in the \(x\) direction from \(-\infty\) to \(+\infty\), in order that \(\psi\) be finite, it is necessary and sufficient that the coefficient \(k\) be real. Therefore only those ranges of values of \(E\) in which \(k(E)\) is real are allowed (allowed energy bands). The other ranges of values of \(E\), in which \(k(E)\) is complex, are forbidden. Thus allowed and forbidden energy bands are obtained in an infinite crystal.

The situation is different if the lattice extends only in the region \(x>0\), while in the region \(x<0\) there is vacuum with the constant potential \(U=U_0\). In this case, if \(E<U_0\), then the solution in the vacuum region, finite as \(x\to-\infty\), will be the function

\[ \psi=A\exp\frac{\sqrt{2m(U_0-E)}}{\hbar}\,x. \tag{3} \]

This function must be matched with the solution (2) in the region \(x>0\) so that in the plane where the solutions are joined (the plane \(x=0\)) \(\psi\) and \(d\psi/dx\) are continuous, which leads to the following relation between the coefficients \(A_1\), \(A_2\), and \(A\):

\[ A_1u_k(0)+A_2u_{-k}(0)=A, \tag{4} \]

\[ A_1[u'_k(0)+iku_k(0)]+A_2[u'_{-k}(0)-iku_{-k}(0)] = \frac{A\sqrt{2m(U_0-E)}}{\hbar}. \tag{5} \]

In addition to these relations, the choice of the coefficients \(A_1\), \(A_2\), and \(A\) is limited only by the condition of finiteness of the \(\psi\)-function (2) in the region \(x>0\).

TAMM BOUND STATES OF ELECTRONS

Two cases should be distinguished:

1) The value of the electron energy lies within one of the allowed zones of the unbounded crystal, i.e. \(k(E)\) is real. In this case the function (2) is finite for arbitrary \(A_1\) and \(A_2\). Thus, it is necessary to satisfy only conditions (4) and (5). The latter constitute two linear equations for three unknowns: \(A_1\), \(A_2\), and \(A\). As is known, they have solutions for arbitrary values of the coefficients, i.e. for any value of \(E\) within the allowed band. Consequently, all electron energy levels allowed in the unbounded crystal also turn out to be allowed in the bounded crystal.

2) Let us suppose now that the value of the electron energy lies in one of the forbidden zones of the unbounded crystal, i.e. \(k(E)\) is complex. In this case, in order that the function (2) be finite as \(x \to \infty\), one of the coefficients \(A_1\), \(A_2\) must be set equal to zero (depending on the sign of the imaginary part of \(k\)). As a result, relations (4) and (5) are transformed into two linear homogeneous equations with two unknowns. They have a solution not for every value of \(E\), but only for such a value for which the determinant of the system is equal to zero. Thus, in a bounded crystal, allowed discrete values of the electron energy are possible in that region of energies which in the unbounded crystal is a forbidden band. The electron wave functions corresponding to these discrete levels in the vacuum region decrease monotonically with distance from the crystal surface, as is evident from formula (3). In the crystal region, however, the wave functions decay while oscillating as one goes deeper into the crystal. The latter is easily seen from formula (2), if one takes into account that \(k\) is complex and that, for the term growing with \(x\), the coefficient is set equal to zero.

Fig. 1.

As the simplest example admitting an easy quantitative calculation, I. E. Tamm considered the one-dimensional Kronig and Penney model3, in which the periodic potential is specified as a series of identical rectangular potential barriers equally spaced from one another (Fig. 1); then the height of each barrier \(U_1\) tends to infinity, and the width of the barrier \(b\) tends to zero

so that the quantity \(\dfrac{mU_1}{\hbar^2}\,ab=p\) remains finite. Here \(a\) is the lattice constant. With such a choice of the periodic potential, the dependence of \(k\) on the electron energy is determined by the equation

\[ \cos ka=p\,\frac{\sin \varkappa a}{\varkappa a}+\cos \varkappa a,\qquad \varkappa=\frac{\sqrt{2mE}}{\hbar}. \tag{6} \]

If the electron energy \(E\) is such that the right-hand side of equation (6) does not exceed unity in absolute value, then \(k\) is real. Otherwise \(k\) is imaginary, which corresponds to a forbidden band in the case of an unbounded crystal.

For each value of the energy \(E\), equation (6) has two roots \(k\), differing in sign. These two roots correspond to the two particular solutions of the wave equation represented by the first and second terms of formula (2). For definiteness we shall agree that, in the case of real \(k\), \(k\) in formula (2) is to be regarded as positive. If, however, \(k\) is complex, then, putting \(k=i\mu+n\pi\), where \(n\) is the number of the forbidden band, we shall regard \(\mu\) as positive. In this latter case, in order that the function (2) remain finite as \(x\to\infty\), it is necessary to put \(A_2=0\).

As shown in Ref. \(^{3}\), the wave function of the electron in the first crystal cell, i.e. for \(0<x<a\), has the form:

\[ \psi=A_1(e^{i\varkappa x}+\beta e^{-i\varkappa x}), \tag{7} \]

where

\[ \beta=-\frac{1-e^{-i(k-\varkappa)a}}{1-e^{-i(k+\varkappa)a}}. \tag{8} \]

We now proceed to the calculation of the surface states of the electron for the simplified form of the potential chosen above (Fig. 1). Matching the solution (7) in the region \(x>0\) with the solution (3) in the region \(x<0\), so that \(\psi\) and \(\dfrac{d\psi}{dx}\) are continuous at the point \(x=0\), we obtain the following relations for \(A_1\) and \(A\):

\[ A_1(1+\beta)=A, \tag{9} \]

\[ A_1 i\varkappa(1-\beta)=A\frac{\sqrt{2m(U_0-E)}}{\hbar}. \tag{10} \]

These equations represent the relations (4) and (5) in the particular example under consideration. Equating the determinant of the system of equations (9), (10) to zero and substituting the value of \(\beta\) from formula (8), we obtain:

\[ e^{-\mu a+in\pi} =-\frac{a}{\hbar}\sqrt{2m(U_0-E)}\cdot\frac{\sin \varkappa a}{\varkappa a} +\cos \varkappa a, \tag{11} \]

\[ \varkappa=\sqrt{2mE}/\hbar;\qquad k=i\mu+n\pi;\qquad \mu>0. \]

The dependence of \(k\) (or \(\mu\)) on the energy is determined by equation (6), which in the case under consideration of complex \(k\) takes the form

\[ (-1)^n \operatorname{ch}\mu a = p\,\frac{\sin \varkappa a}{\varkappa a} + \cos \varkappa a . \tag{12} \]

Eliminating \(\mu\) from equations (11) and (12) leads to the equation for the energy \(E\):

\[ \xi \operatorname{ctg}\xi = \frac{q^2}{2p^2} - \sqrt{q^2-\xi^2}, \tag{13} \]

where

\[ \xi=\varkappa a=\frac{a}{\hbar}\sqrt{2mE}, \qquad q=\frac{a}{\hbar}\sqrt{2mU_0}. \tag{14} \]

The roots of equation (13) are Tamm surface levels of the electron if and only if the corresponding value of \(\mu\) proves to be positive.

In the case under consideration \(E<U_0\), i.e. \(\xi<q\), equation (13) has one and only one root \(\xi_n\) in each interval \(n\pi<\xi<(n+1)\pi\). According to work \(^{3}\), each such interval contains exactly one forbidden energy band (in the case of an infinite crystal). The only exception can be the first interval \(0<\xi<\pi\), where there is no root \(\xi_0\), if the inequality

\[ \frac{q^2}{2p}-q<1 \tag{15} \]

is violated.

Subtracting equation (12) from (11), one easily obtains the equation determining \(\mu\):

\[ (-1)^n \operatorname{sh}\mu_n a = \left(p-\sqrt{q^2-\xi_n^2}\right) \frac{\sin \xi_n}{\xi_n}. \tag{16} \]

Since for the Kronig–Penney potential \(^{3}\) in each forbidden band the sign of the fraction \(\dfrac{\sin \xi}{\xi}\) coincides with the sign of the factor \((-1)^n\), it is seen from (16) that \(\mu_n\) is positive if

\[ p>\sqrt{q^2-\xi_n^2}. \tag{17} \]

It follows from the above that surface levels exist for those roots \(\xi_n\) of equation (13) for which inequality (17) is satisfied. In particular, if

\[ p>q, \tag{18} \]

then inequalities (17) and (15) are always satisfied; in this case there exists one surface level in each forbidden band.

Further, in the same work \(^{1}\) I. E. Tamm gives a qualitative generalization of the results obtained in the one-dimensional case to the case of a three-dimensional lattice bounded by the plane \(x=0\). For approximate estimates the periodic potential \(U\) is assumed to depend—

depending only on the coordinate \(x\). Then in the wave equation the variables \(x, y\), and \(z\) are separated by the Fourier method, and along the coordinates \(y\) and \(z\) the motion of the electron has the same character as in the case of a completely free electron: the electron wave function has the form

\[ \psi = u_{k_1}(x)e^{i(k_1x+k_2y+k_3z)}, \tag{19} \]

where, for surface states, \(k_1\) is complex and has exactly the same allowed discrete values as in the one-dimensional case, whereas \(k_2\) and \(k_3\) may take any real values.

The energy spectrum of the surface states of the electron has the form

\[ E_{nk_2k_3}=E_n+\frac{\hbar^2}{2m}(k_2^2+k_3^2), \tag{20} \]

where \(E_n\) coincides with the surface level of the one-dimensional case. The second term in (20) has a continuous spectrum of values and characterizes the free motion of the electron along the coordinates \(y\) and \(z\).

If the crystal is bounded in the directions \(y\) and \(z\), or if a cyclicity condition in \(y\) and \(z\) is imposed on the wave function, then the spectrum of the quantum numbers \(k_2\) and \(k_3\) becomes quasi-discrete, which makes it possible to count the number of surface states of the electron in the energy interval \(\Delta E\). It turned out that, for example, in the energy interval in which the second term (20) increases from \(0\) to \(0.1\ \text{eV}\), the number of surface states per \(1\ \text{cm}^2\) of surface is of the order of \(10^{13}\). It is easy to see that if the spectrum of the surface states of the electrons were below the Fermi energy of the crystal by an amount of the order of \(1\ \text{eV}\), then the number of surface states filled by electrons would be so large that the surface electric charge produced by them would, by its field, be capable of tearing the crystal apart.

In reality this does not occur because, as the surface states are filled with electrons, the potential in the near-surface region of the crystal increases (for negative charges), and the spectrum of surface levels rises with respect to the Fermi energy until only a small number of surface levels remains below the Fermi energy. As a result, a stationary state is established in which the lowest level of the spectrum of surface states approximately coincides with the Fermi energy.

In the case of metals (in contrast to semiconductors and dielectrics) the field of the surface charge penetrates into the depth of the crystal only over a distance smaller than the lattice constant: it is very strongly screened by the positive charge that arises in the thin near-surface layer of the metal as a result of the partial departure of conduction electrons from this layer. Thus, the-

position of the lowest surface levels by electrons leads to the formation, at the surface of the metal, of an additional electric double layer. Owing to the very small thickness of this layer, it may be regarded as already included in the potential step \(0 \to U_0\) in the plane \(x=0\) (Fig. 1), and no other changes need be introduced into the course of the micropotential shown in Fig. 1. As a result, the entire calculation of the surface states given above remains valid, with only the additional proviso that the magnitude \(U_0\) depends on the degree of filling of the surface levels.

Let \(E_\nu\) be the lowest of the surface levels. In the stationary state it must approximately coincide with the Fermi energy \(\zeta\). The absence of surface levels in all forbidden bands below the \(\nu\)-th means that in them inequality (17) is violated. Consequently, in the \(\nu\)-th forbidden band inequality (17) is satisfied for the first time, and therefore this inequality is not satisfied very strongly. In order of magnitude one may put

\[ p \sim \sqrt{q^2-\xi_\nu^2} \tag{21} \]

or, putting \(\xi_\nu=\dfrac{a}{\hbar}\sqrt{2m\zeta}\), from this we obtain:

\[ U_0-\zeta=\frac{\hbar^2p^2}{2ma^2}. \tag{22} \]

Thus, \(U_0-\zeta\) is approximately expressed in terms of the “binding strength” of the electron at the lattice site \(p\). \(U_0-\zeta\) is that part of the thermoelectric work function of the metal which is realized in a layer of thickness of the order of an angstrom. The other part of the work function is associated with the comparatively slow growth of the electron potential in vacuum over many tens of angstroms, as the distance from the surface of the metal increases. This second part of the work function is connected with the forces of mirror image, with the field of the electron atmosphere existing in the vacuum near the metal surface, etc. I. E. Tamm\(^1\) also points out that, since the electrons bound to the surface can move freely along this surface, there must exist a corresponding surface conductivity of crystals, which should be investigated experimentally.

When the surface of a dielectric is electrically charged as a result of friction or by bombardment of the surface by electrons from outside, the surface charge that arises is interpreted as an additional filling of surface states by electrons, since the surface levels may be situated below the conduction band of the dielectric.

The further development of the theory of surface states of electrons proceeded along the line of considering other kinds of periodic po-

potential, consideration of deviations from periodicity in the volume of a crystal, and generalization of the theory to a three-dimensional crystal. In the work of A. A. Sokolov⁴, the one-dimensional model considered by Tamm is generalized by introducing violations of the periodicity of the potential inside the crystal: one of the rectangular potential barriers is chosen to have an anomalous height and width, which is intended to represent a defect in the crystal (for example, an impurity atom). It turned out that near such a violation of the periodicity of the potential a local state of the electron can also form, whose energy level lies in the forbidden energy band, while the wave function decays with increasing distance from the site of the defect.

E. M. Tsentzer⁵ considered a finite one-dimensional chain of atoms, applying Bloch’s tight-binding-electron method, i.e., assuming that the electron wave functions at neighboring atoms overlap only slightly, which makes it possible, in the zero approximation, to construct the electron wave function in the chain as a linear combination of atomic wave functions. In this method the shape of the atomic potential wells is not restricted in any way, except for the requirement of low transparency of the interatomic potential barriers. Thus, this treatment supplements the results obtained on the Kronig–Penney model, in which the transparency of the interatomic potential barriers may be arbitrary, but the atomic potential wells are chosen to be rectangular in shape. In work⁵ surface states of electrons were also obtained, whose energy levels lie in the forbidden bands, and whose wave functions decay with increasing distance from the edge of the chain in both directions. Just as in Tamm’s work, it was found here that the closer the surface level lies to the allowed band, the more slowly its wave function decays into the depth of the crystal. In⁵ the case was also considered in which foreign atoms are present at the boundaries of the atomic chain, corresponding in the three-dimensional case to a monatomic film on the surface. The perturbation of the periodic course of the density of the electronic space charge near the edge of the atomic chain was considered.

Next one should note a number of works in which the theory of surface states of electrons is generalized to the three-dimensional case. Ryzhanov⁶ considered a bounded three-dimensional crystal, proceeding from Bloch’s tight-binding-electron approximation (assuming that the atomic wave functions overlap weakly). As a result he arrived at the existence of surface electron bands, predicted earlier by Tamm. A. A. Sokolov, starting from a three-dimensionally periodic potential of the form⁷

\[ U(\mathbf r)=U_1(x)+U_2(y)+U_3(z), \tag{23} \]

where \(U_1, U_2, U_3\) are periodic functions of \(x, y, z\), respectively,

of the same type as that shown in Fig. 1. The choice of the potential in the form (23) makes it possible immediately to separate the variables \(x, y, z\) in the wave equation and to reduce the problem to three independent one-dimensional problems solved above. However, a potential of the form (23) is bad in that, if, moving in the direction \(x\), one crosses the surface of the crystal and emerges into the vacuum, then there the potential turns out to be of the form \(\mathrm{const}+U_2(y)+U_3(z)\), whereas it should be constant. Therefore, as Staats has shown by example\(^{23}\), the potential (23) can lead to incorrect results.

In the works of Mays\(^{8}\) and Goodwin\(^{9}\) a three-dimensional crystal bounded by a plane is considered. The periodic potential is specified in general form as a certain three-dimensional Fourier series with arbitrary coefficients. The nearly-free-electron approximation is used, i.e. the periodic potential is omitted in the zeroth approximation, and in the first approximation is introduced as a small perturbation. Thus this treatment complements all those mentioned above. Both authors arrived at agreeing results. Below we shall present Goodwin’s results, which seem to us more complete.

In the case of crystals with a cubic lattice, the periodic potential inside the crystal was specified in the form of the series

\[ U(\mathbf r)=\sum_n U_n e^{\frac{2\pi i(\mathbf n,\mathbf r)}{a}},\quad x>0, \tag{24} \]

where \(\mathbf n\) is a vector with integer components \(n_1, n_2\), and \(n_3\), and \(a\) is the lattice constant. In the vacuum, i.e. in the region \(x<0\), \(U(\mathbf r)=0\). In the nearly-free-electron approximation, as is well known, each harmonic of the series (24), independently of the others, causes Bragg scattering of the electron wave and leads to the appearance of the corresponding forbidden energy band. Therefore each forbidden band corresponds to one term in the sum (24) and will be numbered by the same index \(n\). Each forbidden band, in turn, corresponds to a Brillouin plane of discontinuity of the electron energy \(E(k)\) in \(k\)-space, and the equation of this plane of discontinuity has the form

\[ (\mathbf n,\mathbf k)=\frac{\pi}{a}n^2. \tag{25} \]

Here \(\mathbf k\) is the wave vector of the electron wave, and \(n\) is the number of the forbidden band. Further, if \(k\) is some point on the plane (25), then the boundaries of the forbidden energy band at this point are equal to

\[ U_{000}+E^0(\mathbf k)-|U_{n_1n_2n_3}|\quad \text{and}\quad U_{000}+E^0(\mathbf k)+|U_{n_1n_2n_3}|, \tag{26} \]

where

\[ E^0(\mathbf k)=\frac{\hbar^2 k^2}{2m}. \tag{27} \]

Goodwin’s results reduce to the following: surface states exist only in such forbidden zones for which \(n_2=n_3=0,\ n_1\ne0\). These are the forbidden zones corresponding to families of Bragg reflection planes parallel to the surface of the crystal (the plane \(x=0\)). The corresponding Brillouin planes of energy discontinuity in \(k\)-space are also parallel to the surface of the crystal. If, for these forbidden zones, we introduce the notation

\[ U_{l,0,0}=-U_l,\qquad U_{000}=-U_0,\qquad l=0,1,2,3,\ldots, \tag{28} \]

then the surface zones \(E_l\) are determined by the formula

\[ E_l=\frac{\hbar^2}{2m}\left(\frac{\pi l}{a}\right)^2 \left(\frac{U_0}{U_0+U_l}\right)^2 -\frac{U_0^2}{U_0+U_l} +\frac{\hbar^2(k_2^2+k_3^2)}{2m}. \tag{29} \]

Here \(k_2\) and \(k_3\) are the projections of the electron wave vector onto the directions \(y\) and \(z\). They may take any real values. For the existence of bound surface states of the electron it is necessary that, in the right-hand side of (29), the sum of the first two terms be negative, while \(E_l\) must lie within the limits (26). The wave functions corresponding to (29) have the form

\[ \psi_{l k_2 k_3}= \begin{cases} N e^{-qx+i(k_2y+k_3z)} \cos\left(\dfrac{\pi l}{a}x-\hat{\delta}\right), & x>0,\\[6pt] N e^{q\frac{U_0}{U_l}x+i(k_2y+k_3z)} \cos\hat{\delta}, & x<0, \end{cases} \tag{30,31} \]

where

\[ \left. \begin{aligned} q&=\frac{G_l}{U_0+U_l} \sqrt{\frac{2m}{\hbar^2}(U_0+U_l)-\left(\frac{\pi l}{a}\right)^2},\\[6pt] \hat{\delta}&=\arccos \frac{\dfrac{\pi l}{a}} {\sqrt{\dfrac{2m}{\hbar^2}(U_0+U_l)}}, \qquad 0<\hat{\delta}<\frac{\pi}{2}. \end{aligned} \right\} \tag{32} \]

If the wave functions of the surface states are normalized according to the condition

\[ \int_{-\infty}^{\infty}\int\int \psi^{*}_{l'k'_2k'_3}\psi_{lk_2k_3}\,dx\,dy\,dz = \hat{\delta}_{l'l}\hat{\delta}(k'_2-k_2)\hat{\delta}(k'_3-k_3), \tag{33} \]

then for the normalization coefficient \(N\) one obtains the value

\[ |N|^2= \frac{ q\left[q^2+\left(\dfrac{\pi l}{a}\right)^2\right] }{ \pi^2\left(\dfrac{\pi l}{a}\right)^2 \left(1+\dfrac{2U_l}{U_0}\right) }. \tag{34} \]

In Goodwin’s work,^9 a crystal of noncubic symmetry is also considered, but for the sake of brevity we shall not present the results here.

In the case of a one-dimensional chain consisting of a finite number of atoms, i.e., bounded on both sides, surface states arise simultaneously at both ends of the chain. If the length of the chain is large in comparison with the effective decay depth of the wave function of the surface state, then an electron in a surface state almost “does not feel” the presence of the second end of the chain; therefore its energy and wave function are the same as in the case of a semi-infinite chain. Thus, the surface states at the two ends of the chain have the same energy.

The wave function of such a doubly degenerate level is an arbitrary linear combination of the wave functions of the two surface states. If the influence of the end of the chain on an electron localized at the opposite end is now taken into account as a small perturbation, then the above-mentioned doubly degenerate level splits into two simple levels. The wave functions of the latter are expressed as the sum and the difference of the two surface local functions. This result is obvious if one takes into account that the potential of the finite chain possesses mirror symmetry and, consequently, the wave functions must be either mirror-symmetric (the sum of the surface local functions) or mirror-antisymmetric (the difference of the surface local functions).

Fig. 2.

The next substantial step in the development of the theory of surface states of electrons was made by W. Shockley,^10 who considered a one-dimensional finite chain of equidistant symmetric atoms with potential wells of arbitrary form and investigated the evolution of the electron wave functions and energy levels as the lattice constant decreases from infinity to small finite values. It is important to emphasize that the electron potential within the chain was strictly periodic, up to and including the outermost cells (see Fig. 2, solid curve).

The results obtained are presented in Fig. 3, which corresponds to a chain of eight atoms. From this figure one can trace how the energy spectrum of the chain arises from the spectrum of an isolated atom as the lattice constant \(a\) decreases. For large \(a\), the interaction between atoms may be neglected, and the spectrum of the chain repeats the spectrum of the isolated atom, with the only difference that all levels are eightfold degenerate. As \(a\) decreases to the value \(a_1\), the degenerate levels split, forming allowed bands. The corresponding wave functions are of a nondecaying type (within the limits of the chain) and, consequently, do not represent surface states. With a further decrease of \(a\), the allowed bands intersect, and only after this do two close levels \(p\) and \(p'\) appear in the forbidden band (see the position \(a_2\) in Fig. 3), corresponding to local surface states of the electron with a wave function that decays with distance from the boundaries of the chain. One of these surface levels, as is seen from Fig. 3, originates from the upper level of the lower allowed band, the other from the lower level of the upper allowed band. The remaining levels of the spectrum still correspond to wave functions of the nondecaying type.

Fig. 3.

Usually in a dielectric the number of electrons is such that, at the position \(a_1\), a certain number of bands (for example, the lowest one in Fig. 3) is completely filled with electrons, while the following ones are entirely empty. The chain of eight atoms under consideration, for example, will represent a dielectric if it contains 16 electrons (two electrons with oppositely oriented spins on each level). If we now pass to the position \(a_2\), then only 14 electrons can be accommodated in the lower band; two electrons will be on the surface level \(p\). Thus, in the forbidden band separating the empty conduction band from the highest of the filled bands, the surface levels must be half-filled with electrons. Shockley arrived at this conclusion as a consequence of ignoring the distortions of the periodic potential by the strong field of the surface charge created by the electrons in the surface states.

Shockley’s calculation showed that surface states exist only for small \(a\), when there is an intersection of the allowed energy bands. At first glance this result contradicted the conclusions of a number of other authors (Tamm, Center, Goodwin\(^{11}\)), who arrived at the existence of surface states in those cases when there was no intersection of bands. In reality there is no contradiction here: the authors mentioned considered chain potentials of a somewhat different type, in which the end cells had distorted potential wells (see Fig. 2, dashed line). These distortions of the potential wells can by themselves cause the appearance of local electron states even in the case of an infinite chain, as occurs for impurity local states in the bulk of a crystal. Thus, in passing from the potential shown in Fig. 2 by the solid curve to the potential shown by the dashed line, two additional surface states may appear. As a result, in the one-dimensional case with nonintersecting bands there will be two surface states, and with intersecting bands—four.

Let us consider, for example, the potential used in Tamm’s work, shown in Fig. 4, \(a\), where the heavy vertical lines depict very narrow and high potential barriers. This potential can be represented in the form of the curve shown in Fig. 4, \(b\), where the heavy vertical lines depict barriers of half the height. This figure shows that the potential well of the end cell is distorted in comparison with the wells of the other cells owing to the presence in it of a negative peak of the potential \(v\). If this peak is removed, then, as the calculation shows, the surface states also disappear. It follows from this that in the case of nonoverlapping bands the existence of surface states is connected with the distortion of the end potential well.

Fig. 4.

We do not agree with Shockley\(^{10}\) only in that part of his paper where he compares his results with the results of Mautz\(^{8}\) and Goodwin\(^{9}\), who used the nearly-free-electron approximation. Shockley believes that, since in these authors’ work the potential is given by expression (24), i.e., is strictly periodic, distortions of the end potential wells are not taken into account. Therefore the surface states obtained in \(^{8,9}\) Shockley considers to be connected with the intersection of the allowed bands. However, it should be noted that in the one-dimensional case, in the nearly-free-electron approximation (just as for a cosine-

soidal potential of Mathieu with arbitrary amplitude), intersection of the allowed bands is in general impossible, except for the trivial case when one of the coefficients \(U_n\) in (24) is equal to zero and when the corresponding forbidden band is altogether absent. On the other hand, in \(^{8,9}\), as indicated above, for the appearance of surface states only the terms of the sum (24) containing \(n_1 \ne 0\), \(n_2=n_3=0\) are essential, i.e., those forming precisely a one-dimensional potential. Therefore it seems to us that the surface states obtained in \(^{8,9}\) are not connected with the intersection of bands, but are connected with a distortion of the potential wells of the boundary cells, which is fully compatible with the strict periodicity of the potential within the chain and with formula (24). The latter is easily seen from Fig. 5, \(a\).

Fig. 5.

The investigations of surface states set forth above are substantially supplemented by the work of Statz \(^{12}\), who considered the one-dimensional wave equation with a potential periodic in a bounded region:

\[ \frac{d^2\psi}{dx^2}+\frac{2m}{\hbar^2}[E-\alpha U(x)]\psi=0. \tag{35} \]

The dependence of the electron energy spectrum on the values of the parameter \(\alpha\) was investigated. The origins for \(E\) and \(U(x)\) were chosen so that the mean value \(\overline{U(x)}=0\). In this case equation (35) is Hill’s equation, for which the dependence of the boundaries of the allowed bands on \(\alpha\) is already known \(^{13}\) for known forms of the periodic potential \(U(x)\).

If in equation (35) one sets \(E'=-\dfrac{E}{a}\) and \(x'=\sqrt{a}\,x\), then it becomes

\[ \frac{d^2\psi}{dx'^2}+\frac{2m}{\hbar^2}\left[E'-U\left(\frac{x'}{\sqrt{a}}\right)\right]\psi=0. \tag{36} \]

It follows from this that the introduction of the parameter \(\alpha\) (its variation from 1 to \(a\)) is equivalent to introducing new units for the energy and to increasing the period of the potential by a factor of \(\sqrt{a}\). The latter means that the dependence of the energy spectrum on \(\sqrt{a}\) is analogous to the dependence of the spectrum on \(a\), which was considered above (Fig. 3). The graphs of the dependences of \(E\) on \(\alpha\) and \(E\) on \(a\) are topologically equivalent.

Fig. 6.

Figures 5,a and 6,a present the dependence of the electron energy spectrum on the parameter \(\alpha\), respectively for a potential \(U(x)\) of cosinusoidal and rectangular form. The figures correspond to a chain of four or five atoms. Curves \(a\) and \(b\) depict the course of the potential, respectively for negative and positive \(\alpha\).

The dashed lines depict the boundaries of the bands. As the absolute value of \(\alpha\) increases, the allowed bands narrow, which corresponds to the increase of \(a\) in Fig. 3.

Fig. 5 shows that in the case of a cosinusoidal potential there is no intersection of the allowed bands, apart from the trivial tangency at \(\alpha=0\), i.e., in the case of a completely free electron.

For positive \(\alpha\), i.e. in the case of potential \(\delta\), there are no surface states. For negative \(\alpha\) (potential \(d\)), in the first, third, fifth, etc. forbidden bands there exist pairs of surface levels, which are associated with the distortion of the end atomic potential wells in curve \(a\).

Fig. 6 shows that, in the case of a rectangular potential, intersection of allowed bands is possible (the rectangular potential considered consists of barriers of finite width, which distinguishes it from the potential used by Tamm\(^4\), for which band intersection is impossible). Therefore, in contrast to the preceding case with a sinusoidal potential, here surface states also exist for positive \(\alpha\), when the end potential wells are not distorted. As is seen from Fig. 6, \(в\), the corresponding surface levels appear beginning with the third forbidden band and higher.

In Fig. 6, \(в\), in the third forbidden band for negative \(\alpha\) we see that in the region of large \(|\alpha|\) there exists a pair of surface levels associated with the distortion of the end potential wells. Further, as \(|\alpha|\) decreases, band intersection occurs; however, contrary to Shockley’s assumption\(^ {10}\), four surface levels are not formed, and the surface levels disappear altogether. Apparently, in the region where the distortion of the end potential wells is combined with band intersection, there may exist 0, 2, or 4 surface levels, depending on the specific form of the one-dimensional potential.

According to Figs. 5 and 6, pairs of surface levels, both in the case of distortion of the end potential wells and in the case of band intersection, are formed, as \(\alpha\) changes, from the lowest level of the higher-lying allowed band and the highest level of the lower-lying band. However, this cannot be regarded as a general law, since for another form of the one-dimensional potential there are cases in which both surface levels split off from one and the same allowed band (the higher-lying or the lower-lying one).

In the adsorption of a foreign atom on the surface of a crystal, formation or disappearance of surface states is also possible. This case may be considered as a particular case of the examples set out above. In this case the foreign atom should be treated as a distorted end potential well of the chain. For a closer approximation to reality, two adjacent end atomic potential wells should be regarded as distorted. In this case, too, surface states are possible, with the only difference that their number is proportional to the degree of surface coverage, and surface bands are formed only at a high degree of coverage (a monatomic layer).

The development of the theory of Tamm surface states gave rise to the theory of local states of electrons in the bulk of a crystal, near-

... of internal cracks, impurity atoms, and at the junctions of different crystals or microcrystals of a polycrystal. There also appeared the idea, expressed by L. D. Landau, of electron autolocalization in an ideal infinite crystal14, 15, 16, according to which the possibility was admitted of localization of an electron at a disturbance of the periodicity of the crystal caused by the field of the electron itself. An exposition of these questions lies beyond the scope of the present article.

2. THE INFLUENCE OF SURFACE STATES OF ELECTRONS ON THE PHYSICAL PROPERTIES OF CRYSTALS

Surface states of electrons should manifest themselves in the following phenomena:

a) A surface two-dimensional band, in the case where it is partially filled with electrons, must give rise to surface conductivity of metallic type. In the case of metals this surface conductivity must be masked by the large bulk conductivity. But in dielectrics and semiconductors the bulk electrical conductivity is considerably smaller than in metals, and, in the case of a sufficiently thin plate with current along the plate, the surface conductivity may compete with the bulk conductivity and be measured.

b) In fine polycrystalline semiconductors and dielectrics, surface conduction bands may exist on the surfaces of the microcrystals. For sufficiently small sizes of the crystallites, their surface conductivity may exceed the bulk conductivity. A distinguishing feature of such a case is a noticeable excess of the electrical conductivity of a polycrystal over the electrical conductivity of a single crystal; the dependence of the electrical conductivity of polycrystals on the sizes of the microcrystals; different values of the Hall constant, of the temperature coefficient of electrical conductivity, of the breakdown field strength, of the thermal e.m.f. temperature dependence, different lifetimes of current carriers, and different values of mobilities in single crystals and polycrystals.

The ratio between the currents of the surface and bulk conduction bands in case a) depends on the thickness of the plate, whereas in case b) it does not depend on the thickness.

c) If the linear conduction band formed along the edges of microcrystals is located energetically lower than the surface conduction band formed on the faces of the microcrystals, then, in the case of a fine polycrystal, it may turn out that the main part of the current flowing through the semiconductor passes along the linear bands along the edges of the crystallites. In this case all the features listed in b) should occur; however, in contrast to case b), in such a semiconductor there should in general be no Hall effect, and the dependence of the electrical conductivity on

of a magnetic field, since in the case of a linear conduction band the magnetic field does not deflect conduction electrons.

d) Surface levels at the surface of a semiconductor must perform the same functions as impurity levels perform in its bulk. Thus, for example, they may serve as donors or acceptors of conduction electrons, determine absorption bands of light whose intensity, unlike impurity bands, should depend on the thickness of the semiconductor; and, in the case of surface levels at the boundaries of microcrystals in a polycrystal, the intensity of the bands will be proportional to the thickness of the semiconductor. Surface levels may serve as sources of photoelectrons in the internal and external photoeffect and perform analogous functions when a crystal is irradiated with corpuscular radiation. They may serve as intermediate states in the recombination of current carriers.

e) If the surface band is partially filled at absolute zero, i.e., if surface conductivity of metallic type takes place, then the latter leads to enhanced reflection of light from the crystal, just as if the crystal were coated with a thin metallic foil. This additional reflection of light is especially large when the electric vector of the light wave is parallel to the crystal surface. This additional reflection of light, caused by the surface conduction band, as shown by G. E. Pikus[^17],[^18], is capable of increasing the reflection coefficient of a crystal severalfold.

f) In the case of a thin semiconductor or a fine polycrystal, the number of surface states may exceed the number of impurity local states in the bulk. In this case, when normalizing the electron distribution function over energies and determining the position of the chemical potential, it is necessary to take into account the surface states, which substantially affect the position of the chemical potential and the bulk electrical conductivity of the semiconductor[^18],[^19].

g) The role of the surface states of electrons is especially great in view of the fact that electrons that enter them are capable of creating a very large negative surface charge. At a surface concentration of electrons of the order of \(10^{14}\ \mathrm{cm}^{-2}\), a surface-charge field of the order of \(10^{7}\)—\(10^{8}\ \mathrm{V}\cdot\mathrm{cm}^{-1}\) arises. This field strongly bends the course of the energy bands near the crystal surface, affects the work function, and, in the case of semiconductors, creates near the surface layers depleted of conduction electrons (blocking layers), etc.

h) Surface states may be filled by electrons from atoms or molecules adsorbed on the crystal surface, or, conversely, give electrons to these molecules. Thus, surface states are important for sorption forces and for the processes of surface catalysis of chemical reactions.

Such is a far from complete list of phenomena in which surface states play an essential role.

As an example, let us consider the influence of surface states on the work function and on the formation of a blocking layer at the surface of a semiconductor. These phenomena have been rather well studied both theoretically and experimentally. The dependence of the electric field \(E(x,J)\) and of the concentration of conduction electrons \(n(x,J)\) in a semiconductor on the coordinate \(x\) and on the current density is determined by the equation of electrical conduction and diffusion

\[ J = eunE - eD\,\frac{dn}{dx} \tag{37} \]

and by Poisson’s equation

\[ \varepsilon\,\frac{dE}{dx}=4\pi\rho(n). \tag{38} \]

The density \(J\) of the current flowing perpendicular to the surface of the semiconductor, in the one-dimensional problem, does not depend on \(x\) and is assumed to be specified. Only the case of electronic conductivity is considered. \(u\) is the mobility, \(D\) is the diffusion coefficient of electrons in the semiconductor, with

\[ \frac{D}{u}=\frac{kT}{e}. \tag{39} \]

In order to express \(\rho\) explicitly through \(n\), we shall assume that even in the presence of a current one may speak of local thermal equilibrium of the electrons within a thin layer of thickness \(dx\), and that the normalization of the Fermi distribution function within each such layer is carried out not according to the condition of electrical neutrality of this layer, but so that the concentration of electrons in the conduction band is equal to \(n(x,J)\), determined from equations (37) and (38). This assumption follows strictly from the assumption that the probability of electron transitions between various local and band levels is independent of the current density.

Further, we shall assume that all bulk local levels (or, at least, those of them whose concentration is appreciable) are separated from the local value of the chemical potential \(\zeta(x)\) by more than \(kT\).

Under these assumptions an elementary calculation gives \(^{20}\):

\[ \rho=eP(T)\left[n-\frac{n_\infty^2}{n}\right], \tag{40} \]

where

\[ P(T)=1+\frac{h^3}{2(2\pi m_{\mathrm{eff}}kT)^{3/2}} \sum_{\varepsilon_i>\zeta} N_i e^{\frac{u-\varepsilon_i}{kT}}, \tag{41} \]

the summation being carried out over all levels situated above the chemical potential; \(N_i\) is the concentration of bulk local

levels with energy \(\varepsilon_i\), \(n_\infty\) is the concentration of conduction electrons in the depth of the semiconductor, where it is neutral, and \(u(x)\) is the energy of the bottom of the conduction band, coinciding with the macropotential in the semiconductor.

By eliminating \(dx\) from (37) and (38), one obtains an equation relating the field and the concentration:

\[ \frac{dE}{dn}=\frac{4\pi P kT}{\varepsilon}\cdot \frac{\dfrac{n_\infty}{n}-\dfrac{n}{n_\infty}} {E_\infty-\dfrac{n}{n_\infty}E}. \tag{42} \]

Here

\[ E_\infty=\frac{J}{eun_\infty} \tag{43} \]

is the field in the depth of the semiconductor, where the electron concentration is equal to \(n_\infty\) and Ohm’s law is valid. The reader will find the solution of equation (42) in works \(^{20,21}\). Here we shall restrict ourselves to the special case of the absence of current, when \(J=E_\infty=0^{22,23}\). In this case equation (42) is integrated elementarily by separation of variables, and one obtains

\[ E=\sqrt{\frac{8\pi P kT n_\infty}{\varepsilon}} \left( \sqrt{\frac{n}{n_\infty}}-\sqrt{\frac{n_\infty}{n}} \right), \tag{44} \]

where the boundary condition \(E=0\) at \(n=n_\infty\) has been used. In the absence of current the conduction electrons are in thermal equilibrium and obey the usual Boltzmann distribution

\[ \frac{n}{n_\infty}=e^{-\frac{u(x)}{kT}}, \tag{45} \]

with the potential \(u(x)\) calibrated so that in the depth of the semiconductor \(u(\infty)=0\). With the help of (45), equation (44) can be rewritten in the form

\[ E=-2\sqrt{\frac{8\pi P kT n_\infty}{\varepsilon}}\, \operatorname{sh}\frac{u}{2kT}. \tag{46} \]

This formula shows that at each point of the semiconductor the field \(E\) uniquely determines the rise of the conduction band \(u(x)\), shown in Fig. 7. In particular, in order to determine from formula (46) the rise of the potential \(u(0)\) in the plane of the semiconductor surface \(x=0\), it is necessary to substitute into (46) the value of the field near the surface \(E_h\).

If in the region \(x<0\) there is a vacuum and an electric field \(E_0\) has been created in it, then

\[ \varepsilon E_h=E_0+4\pi\delta, \tag{47} \]

where \(\delta\) is the surface charge density in the plane \(x=0\). The latter is most often interpreted as an accumulation of electrons on the surf-

...levels. If in the region \(x<l<0\) there is a metal or another semiconductor, while the region \(l<x<0\) is a “contact gap,” then \(E_0\) should be understood as the field in this gap. Equations (37)—(47) are valid irrespective of what is situated to the left of the semiconductor (Fig. 7) in the region \(x<0\), and of what causes the field \(E_0\), the surface charge \(\delta\), and the corresponding bending of the energy bands in the boundary region of the semiconductor.

Formulas (46) and (47) make it possible, for a given field \(E_0\) and surface charge \(\delta\), to determine the rise of the conduction band \(u(0)\). Depending on the sign of \(u(0)\), a layer with a reduced concentration of conduction electrons (a blocking layer) or a layer with an increased concentration of them must form in the boundary region of the semiconductor. An analogous phenomenon should also occur in a hole semiconductor.

Fig. 7.

The contact of a semiconductor with a metal is considered in detail in papers \(^{20,21,25}\), where the current-voltage characteristics (rectifying effect) of the near-electrode layer of the semiconductor are calculated. The particular case of extremely weak currents, for which the contact resistance still obeys Ohm’s law, was considered by Mott \(^{22}\) and Davydov \(^{23}\). Blocking layers at the contact of two semiconductors were considered by B. I. Davydov \(^{24}\) and A. I. Gubanov \(^{26}\).

If the semiconductor is a thermocathode, then, as is clear from Fig. 7, \(u(0)\) is the only term entering into the thermoelectronic work function \(e\varphi\), which depends on the field in vacuum \(E_0\) and on \(\delta\). From (46) and (47) one obtains

\[ u(0)=-2kT\,\operatorname{arcsh}\frac{E_0+4\pi\delta}{\sqrt[4]{2\pi PkT n_\infty e}},\qquad e\varphi=e\varphi_0+u(0). \tag{48} \]

This formula expresses the dependence of the work function on the field of the vacuum and on the surface charge, which was first noted by N. D. Morgulis \(^{27}\).

In the above-mentioned works it was assumed that all electron energy levels and, in particular, the surface bands are separated from the chemical potential by more than \(kT\). In this case the surface bands are either completely filled or empty. Therefore \(\delta\) is a constant.

Bardeen\(^{28}\) considered the case where the chemical potential falls inside the surface zone. It then turned out that the degree of filling of the surface zone depends on the limiting curvature of the bands, i.e., on \(u(0)\). Since the density of electron energy levels in the surface zone is large, changes in \(\delta\) are very considerable even for small changes in \(u(0)\). Formula (48) in this case remains valid, but in it \(\delta\) must be regarded as a function that varies rapidly with \(u(0)\). Equation (48) thereby becomes an equation that implicitly determines the dependence of \(u(0)\) on \(E_0\). If in this equation one specifies a small change of \(u(0)\), then it corresponds to a large change of \(\delta\) and, according to (48), a large change of \(E_0\). For example, for \(\Delta u(0)\sim 0.1\) ev one obtains \(\Delta E_0\sim 10^7 — 10^{-8}\ \mathrm{v\cdot cm^{-1}}\). This means that \(u(0)\) changes little with \(E_0\), in contrast to the case considered earlier, when the chemical potential did not fall inside the surface zone, i.e., when \(\delta\) was constant.

The weak dependence of \(u(0)\) on \(E_0\) can physically be interpreted as follows: as the field \(E_0\) entering the semiconductor from the left is increased, the slightest tendency toward an increase of \(u(0)\), owing to the large density of surface levels, is accompanied by a large change of the surface charge \(\delta\), which almost completely screens the added field \(E_0\), preventing it from penetrating into the semiconductor.

Since the resistance of the blocking layer, its temperature coefficient, the thermoelectronic work function of the semiconductor, etc., are determined by the quantity \(u(0)\), they too must depend little on \(E_0\), if the chemical potential falls into the middle of the surface zone.

In Bardeen’s work\(^{28}\) the dependence of \(\delta\) on \(u(0)\) is not calculated, but only estimated. More rigorously and fully this dependence was determined by Pekar\(^{19}\), who assumed that in the lower part of the surface zone the energy is proportional to \((k_2^2+k_3^2)\) (as in a two-dimensional gas). As a result, for the surface concentration of localized electrons the following value was obtained

\[ n'=\frac{m^* kT}{\pi\hbar^2}\ln\left(1+e^{\frac{W}{kT}}\right),\qquad \delta=-en'. \tag{49} \]

For the meaning of the quantity \(W\), see Fig. 7: \(W=\varepsilon' - u(0)+\zeta\) \((\zeta<0)\).

In addition, Pekar\(^{19}\) considers the case of a thin semiconductor whose thickness is comparable with the effective depth of penetration of the field into the semiconductor (with the Debye screening length). In this case the entire semiconductor plate is permeated by the influence of both boundaries, and electrical neutrality is violated throughout the whole volume of the semiconductor. The distribution function of the electrons over energies is normalized with allowance for the surface levels—

BOUND STATES OF ELECTRONS

... whose number may exceed the number of impurity levels in the volume of the semiconductor. The volume concentration of conduction electrons in this case may differ substantially from that in the case of a thick semiconductor, when the influence of the boundaries may be ignored.

Beginning in 1947, experimental works began to appear confirming the theoretical ideas set forth above concerning the surface band and its role in the formation of near-boundary curvature of the energy bands of a semiconductor and of the corresponding blocking layer. Let us first consider the resistance of the near-boundary layer of a semiconductor, caused by the decrease in the concentration of current carriers in this layer. This near-boundary resistance in the case of contact with a metal, at extremely weak currents, is equal (per \(1\ \text{cm}^2\) of surface) to\(^{23}\)

\[ R_k=\frac{1}{\sigma_\infty}\sqrt{\frac{\varepsilon kT}{2\pi P n_\infty e^2}}\left(\exp\frac{u(0)}{2kT}-1\right), \tag{50} \]

where \(\sigma_\infty\) is the bulk specific electrical conductivity of the semiconductor. B. I. Davydov\(^{23}\) takes as the boundary condition \(-\frac{1}{e}u(0)\), simply equal to the contact potential difference between the metal and the semiconductor. Therefore, proceeding from the theory, one would expect that a given semiconductor in contact with different metals may have different contact resistances (different not only in magnitude but also in sign).

We do not consider it correct to identify \(\frac{1}{e}u(0)\) with the contact potential difference \(\varphi_{\text{met}}-\varphi_{\text{semicond}}\), since this means neglecting the potential drop in the “contact gap” and assuming that the entire work function \(\varphi\) is localized in an interval of the order of one angstrom. In reality, a considerable part of the work function is localized in the region of action of the image forces, with a thickness of the order of hundreds of angstroms; this part of the work function is not realized in the direct contact of a semiconductor with a metal. The potential drop in the contact gap is also quite substantial, as will be seen below. S. I. Pekar\(^{20,21,29}\) takes as the boundary condition equality (47), after which \(u(0)\) is expressed through the field in the “contact gap” \(E_0\) by formula (48). However, in this case as well, since \(E_0\) depends substantially on the nature of the metal, one should expect a substantial dependence of the contact resistance (50) on the metal (if \(z\) is regarded as constant).

In a number of cases the dependence of the contact resistance on the metal was indeed observed experimentally, as, for example, in the work of A. V. Ioffe\(^{30}\) and in the work of V. I. Lyashenko and A. M. Pav...

lenko^31. However, this dependence proved to be much weaker than would have been expected from formula (50) for constant $\delta$. M. A. Krivoglaz and K. B. Tolpygo^32 generalized the theory of the contact to the case in which the chemical potential crosses impurity electron energy levels, and also took into account the possible tunneling effect for the transition of electrons from the metal into the conduction band of the semiconductor through the left upper edge of the forbidden band in Fig. 7. These authors showed that both additionally taken into account factors weaken the dependence of the contact resistance on the nature of the metal, and that an explanation of the results of Lyashenko and Pavlenko without invoking surface states of electrons is not excluded.

But in other cases a complete independence of the contact resistance from the metal was established experimentally. This was so in the experiments of Meyerhof^33, who studied the contact of silicon with various metals. Similar results were also obtained in the study of point contact of metals with “$n$-type” germanium^34.

It is especially necessary to note the experiments of Benière^34, which showed that the current-voltage characteristics of the contact of two germanium samples identical in composition are symmetrical with respect to the sign of the current and are similar to the characteristics of a metal—germanium contact in the blocking direction. In the germanium—germanium contact, in the absence of current the entire system is completely mirror-symmetric with respect to the contact plane. Therefore the field $E_0$ in the “contact gap” must be equal to zero. Then, according to formula (48), the entire rise of the conduction band $u(0)$ is due exclusively to the negative surface charge $\delta$. It follows from what has been said that also in the germanium—metal contact the blocking layer is mainly due to the same negative surface charge.

If two $p$-germanium samples identical in composition are brought into contact, then a contact blocking layer is not formed^36,37. This is explained by the fact that the same negative surface charge $\delta$ in the case of a hole semiconductor causes, at the contact, not a decrease but an increase in the concentration of current carriers. Such a thin layer with reduced resistance is usually not detected experimentally.

The above-mentioned experiments are supplemented by the study of the bending of the energy bands near the free (bordering on vacuum) surface of silicon and germanium. Thus, Brattain and Shockley^38 measured the work function from silicon of $n$- and $p$-types (the contact potential difference of silicon with respect to platinum was measured). The concentrations, respectively, of donors and acceptors were determined from the measurement of the specific electrical conductivity and the Hall constant. If there were no bending of the energy bands at the surface, then the position of the chemical potential with respect to the energy bands at the surface would be the same as

in the interior of the semiconductor; namely, in $n$-type silicon the chemical potential should pass near the bottom of the electron-conduction band, and in $p$-type silicon near the top of the filled band. Since the width of the forbidden band is equal to $1.2$ eV, one would expect a difference of the same order of magnitude in the thermoelectronic work functions of $n$- and $p$-type silicon. Experiment, however, has shown a considerably smaller difference in the work functions of samples of $n$- and $p$-type silicon, which also agrees with the results of Meyerhof[^33]. This shows, first, that near the free surface of silicon there is a bending of the bands $\bigl(u(0) \ne 0\bigr)$ and, second, that the position of the chemical potential relative to the bands at the surface of the crystal changes comparatively little, even when in the bulk of the semiconductor it changes substantially. This is explained by the fact that even a small displacement of the chemical potential relative to the energy spectrum near the surface of the crystal, owing to the large density of levels in the surface band, is accompanied by a considerable change in the surface charge and by the appearance of an additional electric field, which impedes further displacement of the chemical potential. An experiment carried out by Brattain and Shockley[^38] made it possible to estimate the density of levels in the surface band, which proved to be of the order of $10^{14}\ \mathrm{cm}^{-2}$. This experiment also showed that the work function depends substantially on the treatment of the surface and on the atmosphere in which the crystal is found.

In another experiment the free surface of silicon and germanium was illuminated with light, and the change in the contact potential difference caused by the illumination was measured. The latter was found to be $+0.12$ V for $n$-silicon, $-0.08$ V for $p$-silicon, and $+0.02$ V for $n$-germanium. The signs of the effect agree with the picture of bands bent at the surface, which permits the entire system to be treated as a valve-type photoelement.

Next comes a series of experiments in which a thin layer of semiconductor is placed inside a plane-parallel capacitor, or itself serves as a plate of the capacitor. The electric field created inside the capacitor penetrates into the semiconductor through its free surface and produces in it a bending of the energy bands. The latter causes the appearance at the surface of a layer of increased or decreased electrical conductivity. The dependence of the longitudinal conductivity of the semiconductor and, in particular, of the conductivity of its boundary layer on the capacitor field $E_0$ is measured. Shockley and Pearson[^40] investigated in this way a number of films of $p$-germanium, cuprous oxide, and $n$-silicon. In all cases the experimental dependence of the longitudinal electrical conductivity of the film on $E_0$ agreed qualitatively with the theory in which the charge of the semiconductor plate of the capacitor was regarded as partly localized on surface levels and partly distributed in the bulk, in the region of ...

of the boundary curvature of the zones. Comparison of theory with experiment made it possible to determine the only unknown parameter of the theory—the density of levels in the surface zone. It turned out to be, in order of magnitude, close to \(10^{14}\), in agreement with \(^{38}\). In this phenomenon the role of the surface states of the electrons consists in screening the field \(E_0\), as a result of which the dependence of \(u(0)\) on \(E_0\), i.e., the dependence of the conductivity of the surface layer on \(E_0\), is weakened. Under other conditions, when the chemical potential does not fall within the surface zone, i.e., when \(\delta\) is a constant, there is no such screening and a stronger dependence of the conductivity of the surface layer on \(E_0\) is observed \(^{41}\).

An analogous effect of almost complete screening of the field \(E_0\) by the surface charge was observed by V. E. Lashkarev and V. I. Lyashenko in cuprous oxide \(^{42}\). In this fundamental work, in addition to the dependence of the conductivity of the surface layer of a semiconductor on the field \(E_0\) penetrating into it, a number of other questions were also investigated: the influence of surface states on the work function of a semiconductor, on contact blocking layers, and on the photo-emf of an isolated semiconductor. In addition to the influence of an external electric field, the influence of the adsorption of dipole molecules on the semiconductor surface was also investigated (the case in which the molecules do not react chemically with the semiconductor and the adsorption is reversible). Adsorption is also capable of changing the filling of surface levels by electrons and in this sense acts analogously to an external field. A comprehensive study of all these phenomena led the authors to the conclusion that on the surface of cuprous oxide there exist numerous surface levels of acceptor type, creating a negative surface charge and leading to the formation, at the free surface of the semiconductor, of a layer with an increased concentration of conduction holes. For the concentration of surface charges, various methods of investigation give values close in order of magnitude \((10^{11}—10^{12}\ \text{cm}^{-2})\).

The influence of the surface charge on the work function and electrical conductivity of a semiconductor was also investigated theoretically by V. E. Lashkarev \(^{43}\). Next there follows a series of experimental works devoted to the influence of adsorption of molecules on the surface of a semiconductor on the conductivity of its surface layer (V. I. Lyashenko and I. I. Stepko \(^{44}\)), on the work function (V. I. Lyashenko \(^{45}\)), and on the photoconductivity of semiconductors (V. I. Lyashenko and O. V. Snitko \(^{46}\); V. I. Lyashenko, O. V. Snitko, and I. M. Semenenko \(^{47}\)). In all these works the effects are interpreted as a change in the degree of filling of surface states as a result of adsorption.

The temperature dependence of the work function in silicon and germanium was experimentally investigated by Smith \(^{48}\). When the temperature was raised from 250 to \(750^\circ\)C, the work function of silicon increased slightly (from 4.5 eV to 4.6 eV), while that of germanium remained un-

variable and equal to 4.5 eV. Were it not for the influence of the surface states of the electrons and the boundary curvature of the bands, the work function would have to change by the same amount by which the chemical potential shifts with respect to the energy spectrum in the bulk of the semiconductor, i.e., by several tenths of an electron-volt. The small change in the work function is explained by the fact that the quantity \(W\) cannot change appreciably. This question is considered theoretically in work \(^{49}\).

Interesting experiments were carried out by S. G. Kalashnikov and Ya. E. Pokrovskii \(^{50}\). The authors took a massive piece of semiconductor exhibiting electronic conductivity. The sign of the thermopower relative to a copper thermoprobe was determined at many points of the specimen. The temperature of the thermoprobe was monitored by a thermocouple and in all experiments was maintained one hundred degrees above room temperature. The semiconductor was then ground in a quartz mortar; the resulting powder was pressed, and on the pressed specimen, again at many points, the thermopower was measured once more. It turned out that as the grain size of the powder decreased, the thermopower diminished and changed sign. The latter means that the fine-grained semiconductor becomes hole-conducting.

The change in the type of conductivity of the semiconductor is explained by the influence of the surface states of the electrons on the position of the chemical potential with respect to the energy bands. This influence of the surface states is the greater, the more developed the surface of the semiconductor is, i.e., the smaller the grain size.

The authors of this experiment assert that the existence of surface states of electrons in this case is not connected with adsorption of gas by the small grains of the semiconductor, but is due to its own clean surface. This follows from the fact that the thermopower, measured after thorough degassing of the microcrystalline specimens by heating in vacuum, did not differ from the thermopower measured before this and after this in air. Oxidation of the surface of the grains likewise could not play an essential role; experiments carried out with thin films of this semiconductor showed that it oxidizes very slowly even at a temperature of \(500^\circ\)C.

These experiments shed light on the results of work \(^{51}\), in which the authors observed a change in the sign of the thermopower after polishing the surface of lead sulfide and germanium. By means of electron diffraction the authors established that as a result of polishing, a layer of amorphous or microcrystalline substance is formed on the surface. In exactly the same way, the hole conductivity of thin germanium films obtained in experiment \(^{52}\) by evaporating an electronic specimen in vacuum can be explained by their microcrystalline structure.

Ya. E. Pokrovskii also investigated fine-grained films of germanium, silicon, and tellurium \(^{53}\). Their specific resistance and po-

the Hall constant turned out to depend on the grain size, which the author explains by the influence of surface levels. The author came to the conclusion that the surface levels of germanium are acceptors, filled at room temperature with a density of about \(3.5\cdot 10^{11}\ \text{cm}^{-2}\). From the temperature dependence of the Hall constant, the activation energies of the surface levels were determined; they were found to be \(0.004\), \(0.02\), and \(0.055\ \text{eV}\). To explain the properties of fine-grained silicon films, Ya. E. Pokrovskii assumes the existence in silicon of empty and filled surface bands, located at a small distance from one another near the middle of the forbidden band. Tellurium possesses acceptor surface levels lying directly above the filled band.

The works considered above, of course, do not exhaust all the extensive material on the experimental investigation of surface Tamm states of electrons and their influence on the properties of semiconductors. However, these works already clearly show the importance of surface states for electronic processes in a solid. In them the existence of surface states has been experimentally proved both in the presence on the crystal surface of a film of adsorbed foreign molecules and on its clean surface.

Both experimental and theoretical investigations of surface states continue to develop. Recently a paper by G. E. Pikus\(^{54}\), devoted to the theory of the photoelectric effect from surface levels, was published. A fairly complete exposition of the theory of the influence of surface states of electrons on various phenomena in semiconductors (on the position of the chemical potential, on the work function, on boundary band bending and blocking layers, on the electrical conductivity of thin semiconductor plates and fine-grained polycrystals, on the reflection and absorption of light by the surface of a semiconductor) is given in G. E. Pikus’s dissertation\(^{18}\). The reader will find a brief review of the theory of surface barriers and surface conductivity in the recently published article by Bardeen and Morrison\(^{55}\). New experimental investigations of the influence of an external field and gas adsorption on the work function of a semiconductor, the boundary barrier, surface conductivity, and photoconductivity are collected in V. I. Lyashenko’s dissertation\(^{56}\).

Unfortunately, up to now it has not been possible to prove reliably by experiment the existence of surface conductivity along the surface two-dimensional energy band, separating this conductivity from the conductivity of the boundary layer of the semiconductor, enriched with current carriers owing to the bending of the energy bands.

There is an extensive literature in which surface states of electrons are invoked for constructing the theory of adsorption and surface catalysis of chemical reactions. However, these questions lie beyond the scope of the present article.

3. SURFACE VIBRATIONS OF THE LATTICE

The theory of electron surface states expounded in the preceding sections has a profound analogy with the theory of surface vibrations of the atoms of a lattice. These vibrations are rapidly damped in the direction normal to the surface; in the one-dimensional case they have a local character. The energy spectrum of such “surface phonons” has a distinctive character \(^{60,61}\); one of its branches, in the limiting case of long waves, passes into the well-known surface Rayleigh waves. Other vibrations form special surface optical branches.

Despite the far-reaching mathematical analogy between the theory of Tamm surface electronic levels and the theory of the corresponding phonon levels, there is an important difference between the two theories: electronic wave functions depend on continuously varying coordinates and satisfy a certain differential equation; elastic waves in a crystal (i.e., the wave functions of phonons in the quantum theory of the lattice) depend on a discrete argument and are described by the corresponding matrix equation, which only in the limiting case of long waves—for the acoustic branch of vibrations—passes into the differential equation of sound waves.

Despite a number of apparent complications to which the passage from a continuous argument to a discrete one, and from differential equations to matrix equations, leads, many questions become clearer precisely because of the possibility of operating with discrete quantities. In particular, the special method \(^{60}\), which makes it possible to obtain the spectrum of surface vibrations, is essentially based on the discreteness of the coordinate variables.

In order to set forth the idea of this method, as well as the basic results, we shall first recall some facts about vibrations in an unbounded crystal.

Consider a \(p\)-atomic unbounded crystal lattice. The vertices of its cells are located at the points \(\mathbf{R} = n_i \mathbf{a}_i\), \((i = 1, 2, 3)\), where \(n_i\) are integers and \(\mathbf{a}_i\) are the fundamental periods.

Let \(\psi^{sx}(\mathbf{R})\) be the \(x\)-th component of the displacement of the \(s\)-th atom in the \(\mathbf{R}\)-th cell from its equilibrium position \((s = 1,\ldots,p)\). Then the equations of lattice vibrations, after separation of the time factor \(e^{-i\omega t}\), have the form

\[ \sum_{\mathbf{R}'} A_{\mathbf{R}-\mathbf{R}'}^{\sigma\sigma'} \psi^{\sigma'}(\mathbf{R}') = m_s \omega^2 \psi^\sigma(\mathbf{R}), \tag{51} \]

where the symbol \(\sigma\) denotes the set of variables \(x\) and \(s\), and the coefficients \(A_{\mathbf{R}-\mathbf{R}'}^{\sigma\sigma'}\) characterize the interaction between atoms at the distance \(\mathbf{R} - \mathbf{R}'\).

The solutions of this equation are plane waves

\[ \psi^{a}(\mathbf R)=q^{a}e^{i\vec{\varkappa}\mathbf R}, \tag{52} \]

where the polarization vector \(q^{a}=q_{x}^{s}\) is determined from the system of \(3p\) equations

\[ \alpha^{\sigma\sigma'}q^{\sigma'}-\omega^{2}q^{\sigma}=0;\qquad \alpha^{\sigma\sigma'}=\sum_{\mathbf R}A_{\mathbf R}^{\sigma\sigma'}e^{i\vec{\varkappa}\mathbf R}. \tag{53} \]

The frequencies \(\omega=\omega(\vec{\varkappa})\) corresponding to the phonon wave vector \(\vec{\varkappa}\) are determined from the condition for the solvability of system (53)

\[ D\left(\alpha^{\sigma\sigma'}-\omega^{2}\delta_{\sigma\sigma'}\right)=0 \tag{54} \]

(the symbol \(D\) denotes the determinant). The solution of equation (54) gives \(3p\) roots

\[ \omega_{r}^{2}=\omega_{r}^{2}(\vec{\varkappa})\qquad (r=1,\ldots,3p), \]

each of which corresponds to a definite branch of vibrations. To each branch there correspond its own polarization vectors \(q_{r,\vec{\varkappa}}^{\sigma}\). If one takes into account the natural condition

\[ \sum_{\mathbf R,s'} A_{\mathbf R}^{\sigma\sigma'}=0, \]

which expresses the fact that there are no forces when the entire crystal is displaced as a whole, then one may verify that three of the branches \(\omega_{r}^{2}(\vec{\varkappa})\) start from zero and, for long waves (small \(\varkappa\)), have a dispersion law of the form

\[ \omega_{r}(\vec{\varkappa})=c_{r}(\mathbf n)\,\varkappa;\qquad \vec{\varkappa}=\varkappa\mathbf n;\qquad r=1,2,3. \tag{55} \]

These branches are called acoustic. The vibrations corresponding to them are characterized by the fact that, as \(\varkappa\to0\), each cell is displaced as a whole, i.e. the polarization vectors \(q^{s}\) do not depend on the number of the atom in the cell \((q^{s}=q)\). The remaining \((3p-3)\) branches do not possess this property; the corresponding frequencies, for small \(\varkappa\), have a dispersion law of the form

\[ \omega^{2}=\omega_{0}^{2}+c_{ik}\varkappa_{i}\varkappa_{k}. \tag{56} \]

These vibrations may, as \(\varkappa\to0\), have a nonzero dipole moment and are called optical.

In order to emphasize the analogy of the solutions (52) obtained with the wave functions of “zone” electrons, we write these solutions in the form

\[ \psi_{r}(\mathbf R,s)=q_{r}(s)\cdot e^{\pm i\vec{\varkappa}\mathbf R} \tag{57} \]

\[ \left(\psi^{s,x}(\mathbf R)\equiv\psi^{x}(\mathbf R,s);\quad q^{s,x}\equiv q^{x}(s)\right). \]

The expressions written have the form of modulated plane waves, if one takes into account that the variable \(s\) characterizes the “coordinate”

inside the cell, and \(R\) the coordinate of the cell. They completely correspond to equation (2) of § 1 (more precisely, to its three-dimensional analogue). The number of the branch of vibrations \(r\) corresponds to the band number for the electron functions. The dependence of the square of the frequency on the wave vector

\[ \omega^{2}=\omega^{2}(\vec{\chi}) \]

corresponds to the dispersion law \(E=E(\mathbf{k})\) for band electrons. It should be noted that the construction of the dispersion law \(\omega(\vec{\chi})\) for phonons from given force constants \(A_R\) reduces to the solution of an algebraic equation of degree \(3p\), whereas the analogous problem for the differential equation (1) has a transcendental character.

In particular, for a one-dimensional, monatomic chain the initial equation has the form

\[ \sum A_{n-n'}\psi(n')=m\omega^2\psi(n), \tag{58} \]

whence

\[ \omega^2(\chi)=\frac{1}{m}\sum A_n e^{i\chi n a}. \tag{59} \]

The interaction coefficients \(A_n\) rapidly decrease with the distance \(na\). If, in particular, we restrict ourselves to interaction only between nearest neighbors, then we shall have

\[ m\omega^2(\chi)=A_0+2A_1\cos a\chi. \tag{60} \]

The condition \(\sum A_n=0\) gives

\[ m\omega^2(\chi)=A_0(1-\cos a\chi), \tag{61} \]

and, in accordance with what was said earlier, for small \(\chi\)

\[ \omega(\chi)=c\chi;\qquad c=\sqrt{\frac{A_0a^2}{2m}}. \tag{62} \]

It should be noted that the more general case (60), where \(A_0+2A_1\ne0\), also has physical meaning if one considers not a monatomic chain but a diatomic one\({}^{61}\). Thus, for example, if in a diatomic chain the mass of the heavy atom is many times greater than the mass of the light one, then, for normal vibrations belonging to the optical branch, the amplitude of the vibrations of the heavy atoms will be vanishingly small. Therefore, in the equations of vibration the displacements of the heavy atoms may be neglected, and their role will reduce to the appearance of an additional elastic force binding the light atoms to their equilibrium position (i.e., to a violation of the condition \(A_0+2A_1=0\)). In this case, obviously,

\[ m\omega^2(0)=A_0+2A_1=\beta\ne0, \tag{63} \]

which is in agreement with expansion (56) for optical branches.

Bearing in mind the aim of clarifying the essence of the general method for studying surface vibrations on the simplest example of a one-dimensional chain with nearest-neighbor interaction, we shall adopt the more general relation (63) between the coefficients \(A_n\). Let us write the equation of vibrations of the infinite chain (58) in matrix form

\[ (\hat A-\omega^2)\psi=0; \qquad \psi=\psi(n);\quad A_{nn'}=A_{n-n'} \tag{64} \]

(for brevity, units have been chosen in which \(m=1,\ a=1\)).

In the case of nearest-neighbor interaction we introduce additional notation:

\[ A_0=2\alpha+\beta;\qquad A_1=-\alpha;\qquad \alpha,\beta>0 \tag{65} \]

and, consequently,

\[ \omega^2(\chi)=\beta+4\alpha\sin^2\frac{\chi}{2}. \tag{66} \]

Let us first consider an auxiliary problem: suppose that at the origin there is some inhomogeneity, say, a changed value of the coupling coefficient \(A'_{00}=A_0+\varepsilon\); it is necessary to determine how the presence of this inhomogeneity affects the spectrum of vibrations of the chain.

Mathematically, such a problem means replacing the original unperturbed equation (64) by the equation

\[ (\hat A-\omega^2)\psi=\hat \Lambda\psi, \tag{67} \]

where the perturbation matrix is

\[ \Lambda_{nn'}=\varepsilon\delta_{n0}\delta_{n'0}. \]

Perturbations of this type, “local” perturbations of general form, were studied in detail in works \(^{57-59}\)*).

In these works it was shown that the addition of a perturbation matrix of “local” character leads to the splitting off of discrete eigenvalues (discrete frequencies) from the boundaries of each of the intervals of the continuous spectrum of the unperturbed system. The maximum number of discrete frequencies split off from each of the intervals of the continuous spectrum does not exceed the rank of the perturbation matrix; concrete equations were also obtained which determine the positions of these frequencies. In the simplest case studied by us, regarding \(\hat \Lambda\psi\) as a given right-hand side and solving

*) Recently a number of papers \(^{62-65}\) and others have appeared in which only part of the results contained in the works of I. M. Lifshitz, published as early as 1947–1948, was obtained. These papers contain no references to the above-mentioned works, which apparently remained unknown to the authors of the cited papers.

equation (67) with respect to \(\psi\), we have, for frequencies \(\omega\) not lying in the region of the continuous spectrum \(\omega(\chi)\),

\[ \psi(n)=\frac{\varepsilon \psi(0)}{2\pi}\int\limits_{0}^{2\pi} \frac{e^{inx}\,dx}{\omega^2(\chi)-\omega^2}. \tag{68} \]

Putting \(n=0\) in (68) and cancelling \(\psi(0)\), we obtain the equation for determining the discrete eigenvalue \(\omega^2=z\) split off from the interval of the continuous spectrum \(\omega^2(\chi)\):

\[ 1=\frac{\varepsilon}{2\pi}\int\limits_{0}^{2\pi} \frac{dx}{\omega^2(\chi)-z}. \tag{69} \]

Let us note that the equation obtained is valid for an arbitrary difference matrix \(A_{n-n'}\) (with \(\omega^2(\chi)\) being determined by formula (59)). Investigation of the written equation (69) shows that for any \(\varepsilon\) there exists one root \(z\), and in the case \(\varepsilon<0\) (weakened bond) the discrete level \(z\) is situated to the left of the interval of the continuous spectrum, while in the case \(\varepsilon>0\) (strengthened bond) it is situated to the right of this interval.

After these remarks, let us return to the problem of determining surface levels; we shall begin again with the one-dimensional case.

The essence of the method being presented\({}^{60,61}\) is that the boundary of the crystal (in the present case, the boundary of a one-dimensional chain) is regarded as a certain local perturbation. The possibility of such a treatment follows from the following considerations: let us cut the chain at the origin of coordinates (between atoms \(n=0\), \(n=1\)); the new “perturbed” matrix of interaction coefficients \(A'_{nn'}\) differs from the original \(A_{n-n'}\) in that the interaction between atoms on the two sides of the cut disappears, while all the remaining elements of the matrix remain unchanged. Putting \(\hat A'=\hat A+\hat\Lambda\), we obtain for the perturbation matrix \(\hat\Lambda=\hat A'-\hat A\) the following expression, evident without additional explanation (in the notation (65)):

\[ \hat A'= \left\| \begin{array}{cccc|cccc} \cdots & 2\alpha+\beta & -\alpha & & & & & \\ \cdots & -\alpha & \alpha+\beta-\beta' & & & 0 & & \\ \hline & & & & \alpha+\beta-\beta & -\alpha & \cdots & \\ & 0 & & & -\alpha & 2\alpha+\beta & \cdots & \\ \cdots & & & & & & & \end{array} \right\|, \tag{70} \]

\[ \hat A = \left\| \begin{array}{ccccc} \cdots & \cdots & \cdots & \cdots & \cdots\\ \cdots\; 2\alpha+\beta & -\alpha & 0 & 0 & \cdots\\ \cdots\; -\alpha & 2\alpha+\beta & -\alpha & 0 & \cdots\\ \cdots\; 0 & -\alpha & 2\alpha+\beta & -\alpha & \cdots\\ \cdots\; 0 & 0 & -\alpha & 2\alpha+\beta & \cdots\\ \cdots & \cdots & \cdots & \cdots & \cdots \end{array} \right\| \tag{70a} \]

\[ \hat\Lambda = \left\| \begin{array}{ccccc} \cdots & \cdots & \cdots & \cdots & \cdots\\ \cdots\; 0 & 0 & 0 & 0 & \cdots\\ \cdots\; 0 & -\alpha-\beta_1 & \alpha & 0 & \cdots\\ \cdots\; 0 & \alpha & -\alpha-\beta_1 & 0 & \cdots\\ \cdots\; 0 & 0 & 0 & 0 & \cdots\\ \cdots & \cdots & \cdots & \cdots & \cdots \end{array} \right\| . \tag{70b} \]

In particular, in the case \(\beta=0\), as is easy to see, \(\beta_1=0\), and the matrix \(\hat\Lambda\) has the form*)

\[ \hat\Lambda = \left\| \begin{array}{ccccc} \cdots & \cdots & \cdots & \cdots & \cdots\\ \cdots\; 0 & 0 & 0 & 0 & \cdots\\ \cdots\; 0 & -\alpha & \alpha & 0 & \cdots\\ \cdots\; 0 & \alpha & -\alpha & 0 & \cdots\\ \cdots\; 0 & 0 & 0 & 0 & \cdots\\ \cdots & \cdots & \cdots & \cdots & \cdots \end{array} \right\| . \]

The rank of the perturbation matrix is evidently connected with the number of neighbors whose interaction we must take into account. A calculation similar to that carried out above gives, in this case, the discrete value

\[ \omega^2=-z=\beta-\frac{\beta_1^2}{\alpha+\beta_1}, \tag{71} \]

lying to the left of the left-hand boundary \(\beta\) of the interval of the continuous spectrum.

The amplitudes of the vibration corresponding to this discrete eigenvalue decay with increasing distance from the boundary of the chain:

\[ \psi(n)=C\left(\frac{\alpha}{\alpha+\beta}\right)^n;\quad n>0. \tag{72} \]

For \(\beta=0,\ \beta_1=0\), the continuous spectrum begins at zero and, naturally, no discrete frequency is obtained.

Passing to the real case of a three-dimensional crystal bounded by a plane interface (a crystalline half-space), let us choose the lattice period vectors \(\mathbf a_1,\ \mathbf a_2\) and \(\mathbf a_3\) in such a way that \(\mathbf a_1\) and \(\mathbf a_2\) lie in the plane of the boundary. Then, considering the presence of the boundary as a perturbation, just as this

*) In this case, at the free end \(\alpha[\psi(0)-\psi(1)]=m\omega^2\psi(0)\).

was done for the one-dimensional problem, we easily verify that the elements of the perturbation matrix \(\Lambda_{\mathbf R,\mathbf R'}\) in equations of type (67) have a difference character with respect to the components \(\mathbf R,\mathbf R'\) in the boundary plane and a “local” character with respect to the third components:

\[ \Lambda_{\mathbf R\mathbf R'}^{\sigma\sigma'} = \Lambda_{\,n_1-n_1',\, n_2-n_2',\, n_3 n_3'}^{\sigma\sigma'}, \qquad \mathbf R = n_1\mathbf a_1 + n_2\mathbf a_2 + n_3\mathbf a_3 . \tag{73} \]

We shall seek the phonon wave function (the solution of equation (67)) in the form

\[ \psi^\sigma(\mathbf R) = \chi^\sigma(n_3)\, e^{\,i(n_1 x_1+n_2 x_2)}, \qquad \mathbf x_1=\mathbf x\mathbf a_1;\quad \mathbf x_2=\mathbf x\mathbf a_2 . \tag{74} \]

This makes it possible to separate the variables and to write the equation for \(\chi^\sigma(n_3)\) in the form corresponding to the one-dimensional problem already discussed:

\[ \sum A_{n-n'}^{\sigma\sigma'}(x_1,x_2)\chi^{\sigma'}(n') -\omega^2\chi^\sigma(n) = \sum \Lambda_{nn'}^{\sigma\sigma'}(x_1,x_2)\chi^{\sigma'}(n'). \]

Here

\[ A_n^{\sigma\sigma'}(x_1,x_2) = \sum A_{n_1,n_2,n}^{\sigma\sigma'} e^{\,i(n_1x_1+n_2x_2)}, \]

\[ \Lambda_{nn'}^{\sigma\sigma'}(x_1,x_2) = \sum \Lambda_{n_1,n_2,nn'}^{\sigma\sigma'} e^{\,i(n_1x_1+n_2x_2)} . \]

Thus, by specifying the value of the wave vector in the boundary plane \(\vec x=(x_1,x_2)\), we obtain a one-dimensional equation in which the elements of the matrices \(A_{n-n'}(\vec x)\) and \(\Lambda_{nn'}(\vec x)\), in essence, describe the interaction between waves propagating in the layers of the lattice parallel to the boundary plane \(\mathbf a_1,\mathbf a_2\). For fixed values \(\vec x=(x_1,x_2)\), the frequency spectrum of the unperturbed (infinite) lattice, as was already shown, forms several continuous intervals (branches). The limiting values of these branches, corresponding to infinitely long waves in the direction normal to the boundary plane, are

\[ \omega_0=\omega(x_1,x_2,0). \]

Thus, even those branches which correspond to acoustic vibrations do not begin with zero frequencies (which, incidentally, is physically obvious, since the presence of nonzero values of \(x_1\) and \(x_2\) means a finite wavelength in the plane of the layer).

According to what was said earlier, the presence of a local perturbation \(\Lambda_{nn'}(\vec x)\), describing the boundary of the crystal, leads to the splitting off from the boundaries of the spectral intervals of discrete frequencies, the position of which depends on the value of the planar vector \(\vec x=(x_1,x_2)\). In particular, the eigenvalues split off from the acoustic

branches, are located for every finite $\vec{x}$ to the left of the left boundary of these branches, $\omega_s=\omega(x_1,x_2,0)^{60}$.

The frequencies separated from the branches of the continuous spectrum remain discrete only so long as the value of the wave vector $\vec{x}$ of the “surface wave” is fixed. When the vector $\vec{x}$ runs through all possible values, these frequencies form continuous “surface branches,” corresponding in the frequency spectrum of the crystal to the usual “bulk branches.” As was shown in Ref. 61 for the example of a simple model, “acoustic” surface waves, as a result of a limiting transition to the continuum, give the well-known Rayleigh waves. Alongside them, in complex lattices there exist, as follows from all that has been said, “optical” surface waves, which have no analogue in the theory of the continuum$^{60,61}$.

As is known, owing to translational symmetry only the limiting frequencies of optical branches are optically active; i.e., the normal vibrations of the lattice can possess a nonzero electric moment only in the limiting case of infinitely long waves. Thus, the number of frequencies of the infrared spectrum of an ideal crystal does not exceed $3p-3$, since restrictions associated with the symmetry of the crystal class can only reduce this number, leading to coincidence of limiting frequencies and to the loss of optical activity for some of them. Analogous considerations of translational symmetry in directions parallel to a free surface lead to the conclusion that, in the case of surface optical branches as well, only the limiting frequencies corresponding to infinitely long surface waves $(x_1,\ x_2=0)$ can be optically active. The presence of such frequencies may lead to the appearance, in the infrared absorption spectra and in the combination-scattering spectra of crystals, of additional spectral lines$^{68}$.

Since the amplitudes of surface waves decrease very rapidly into the depth of the crystal, the intensity of these lines may turn out to be at all noticeable only in the case of a very well-developed surface. On the other hand, it is easy to show that the appearance of discrete frequencies may be associated not only with a free surface, but also with any arbitrarily small homogeneous disturbance of the periodic structure of the lattice, localized near some crystallographic plane. Thus, for example, the boundary between blocks in a mosaic crystal or in a lamellar crystal of variable structure constitutes precisely a disturbance of this type. The thickness of the blocks is sufficiently large for the proposed treatment to be applicable, at least qualitatively. At the same time, the interfaces between blocks are repeated rather often and their number is very large, so that the intensity of the additional spectral lines associated with them may turn out to be noticeable. Unfortunately,

At present there are no direct experimental data on this question.

The change in the spectrum of vibrations near a free surface also affects the magnitudes of the surface values of the thermodynamic potentials. A method applicable to the calculation of such potentials is set forth in work \(^{67}\). Let us note, finally, that inhomogeneous problems of the statics and dynamics of a crystalline half-space can be solved in a similar way; in particular, a problem analogous to the Boussinesq problem for a discontinuum has been solved. Such problems are considered in work \(^{66}\).

In conclusion we note that the electron surface levels of I. E. Tamm, considered in detail in the first two sections, as is clear from all that has been set forth above, may be regarded as discrete levels split off from the corresponding energy bands as a result of the perturbation introduced by the free surface. Such an explanation of the origin of these levels seems to us quite natural and makes it possible, for their qualitative investigation, to use a number of considerations borrowed from the described method of local perturbations.

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