MODERN THEORY OF SOLIDS1

F. Seitz and R. P. Johnson, New York

Part III

This part of the article is devoted to the properties of the surfaces of solids and to the influence of impurities and cracks on certain of their volume characteristics. In this field there are still only very few theoretical works based on modern points of view. Therefore a considerable part of what we shall discuss here will be more or less conjectural. We set forth our ideas in the hope that they may perhaps suggest good ideas to the reader, but at the same time we caution against accepting them uncritically.

VI. CRYSTAL SURFACE

1. Surface barrier; thermionic emission

At the surface of a crystal the potential energy of an electron increases and tends asymptotically to the value zero (chosen arbitrarily), corresponding to removal of the electron to an infinitely large distance. The form of the potential barrier at large distances from the surface is determined by the usual law of the electric image force. According to Sommerfeld’s picture, at \(0^\circ\mathrm{K}\) the valence electrons in a metal fill a group of energy levels from the bottom of the “potential box” up to a height given by the Fermi–Dirac distribution function. The energy gap between this upper level and the top of the barrier is the work function \(W\). At high temperatures some electrons occupy energy states lying above the barrier, and a certain fraction of these electrons moves in a direction favorable for escape from the metal. An accelerating field at the surface effectively lowers the barrier and allows a larger number of electrons to escape. These simple considerations are illustrated in Fig. 28,a.

The theory predicts that the density of the electron current from a metal at \(T^\circ\mathrm{K}\), in the absence of an external electric field, should be:

\[ i = A \cdot T^2 \cdot e^{-\frac{W}{kT}} . \]

The coefficient \(A\) should be the universal constant \(\dfrac{4\pi emk^{2}}{h^{3}}\), numerically equal to \(120.4\ \mathrm{A}/\mathrm{cm}^{2}\ \mathrm{grad}^{2}\). Experiments1 on the most varied surfaces have confirmed the exponential dependence on temperature and have given values of the work function \(W\) that agree well with values obtained by other methods, for example by the photoelectric method. The constant \(A\) agrees with the theoretical value only in order of magnitude. The experimental values are different for different metals and even for different samples of one and the same metal. This discrepancy is usually ascribed to three factors: the slightest inhomogeneities of the surface, which increase the true emitting surface in comparison with the apparent one; a possible small dependence of the work function on temperature; and the “reflection” from the metal surface of electrons that are energetically capable of escaping. Recent experiments on the energy distribution of emitted electrons[^2] also apparently require a significant reflection of electrons having energy “sufficient” for escape. In the theoretical study of such reflection it is necessary to apply the wave picture of electronic motion, since a classical electron-particle moving toward the surface with energy in excess of the work function must necessarily escape. It turns out that the probability of reflection of an electron, and also the shape of the surface barrier, depends on the energy of the electron wave. For example, for a rectangular barrier with vertical walls it is greater than for a rounded barrier of the image-force type. Up to the present, attempts to devise a plausible shape of the barrier that would give the required reflection coefficient have not been sufficiently successful. A complete investigation is complicated by the circumstance that the experimental specimens are polycrystalline wires, whereas it has been experimentally proved that the work function for different crystal faces1 is different. Difficulties of the same kind arise, of course, in

Fig. 28. Thermionic emission according to Sommerfeld’s theory.

1 — electrons capable of creating a thermoelectron current; 2 — transparent barrier for electrons.

a. Right — the course of the potential energy at the surface in the absence of an applied field. Left — the Fermi–Dirac distribution of energy at temperature above \(0^\circ\mathrm{K}\).

b. An accelerating field lowers the height of the surface barrier and increases, in this way, the thermoelectron current (Schottky effect). If the field is very strong, the barrier becomes so narrow that many electrons (in the wave picture) can pass through it.

interpretation of photoelectric data. Apparently, these questions will be clarified only after a picture of the metal surface has been created that is more detailed than the one existing at present.

With very large accelerating fields, electron emission from cold metals is observed, increasing with the field strength according to an exponential law of the same type as the law governing the change of the thermoelectronic current with temperature. The wave picture makes it possible to explain this phenomenon of cold emission. When the applied electric field becomes very large, the surface potential barrier, besides being lowered, at the same time becomes very narrow near its summit. Therefore a considerable number of electrons, having energies somewhat smaller than that required to pass over the summit of this narrowed potential ridge, can pass through it and reach the external collector (Fig. 28, b). Of course, such penetration through the barrier would be impossible for a classical electron-particle. Calculations lead to an equation for the current which agrees rather well with observations, if the experimental difficulties are taken into account. It must be noted that this effect of passage through the barrier is additional to the above-mentioned effect of reflection. The potential barrier divides the electron-particles incident upon it into two sharply separated groups: electrons whose energy is greater than the top of the barrier pass completely through the barrier; those whose energy is less than this value are completely reflected. On the other hand, electron-waves in the presence of a barrier are not separated so sharply into groups, and the energy distribution of the electrons that have passed through the barrier depends on its shape, width, and height.

Fig. 29. Above—the band energy spectrum at the surface of a metal; below—the distribution of electronic charge near the surface.

If we take into account the wave nature of electrons inside a solid, and also at its surface, and take into consideration the periodic field of the lattice, we arrive at the band picture shown in Fig. 29; this figure also shows the distribution of the electronic charge, which decreases exponentially with distance from the surface. The levels in the energy spectrum are not distributed as uniformly as they were in Sommerfeld’s theory. We may expect that these nonuniformities, which are a consequence of the separation of states into bands, will somehow appear in the energy distribution. However, it can be shown that, in the first approximation, the change in the density of states is precisely compensated by the change of velocity with energy.

within the zone (see V, 7). Except for special cases, the zone theory and the Sommerfeld theory predict the same result for such an experiment.

What can be said about the magnitude of the work function? Wigner and Bardeen\(^4\) showed that it can be divided into two terms; one of them, arising as a result of the attraction of the departing electron by the ions of the lattice, does not depend on the nature of the surface and is approximately proportional to the binding energy; the second term depends on the nature of the surface, namely, on the dipole moment per unit area. The center of gravity of the electron distribution associated with the layer of ions on the surface of the body may be displaced to the left or to the right of this ionic layer (Fig. 30). An electron passing through such a double layer of charge changes its potential energy by \(4\pi eP\), where \(P\) is the moment per unit area. Bardeen’s calculations for the surface of sodium show that this moment is very small; this, apparently, is generally true for pure metals. Table 5, from which it is evident that the work function can be associated with the heat of sublimation, is experimental evidence that the volume term in the magnitude of the work function predominates. Direct experiments also show that differences in the work functions for different faces of the crystal of one and the same metal are of the order of several tenths of an electron-volt.

Fig. 30. Dependence of the sign of the surface dipole moment on the distribution of the electronic charge outside the outer layer of ions

Table 5

If a layer of foreign atoms is deposited on the surface of a metal, then its effect on the work function will mainly be expressed in a change of the dipole moment. Thus, if cesium atoms fall on

tungsten, they become ionized, losing their valence electrons; the surface acquires a large positive dipole moment and the work function decreases. Conversely, if the cesium surface is oxidized, the oxygen atoms become negatively charged and the work function increases. In what follows we shall consider such adsorption processes in greater detail.

2. Absorption of Light by a Surface

The strict selection rules for bulk absorption of light (see V, 5) are inapplicable to electrons at the surface of a crystal. Electrons in the exponential tail of the charge-distribution curve (Fig. 30) can undergo transitions in which the two components of the wave-vector $\mathbf{g}$ parallel to the surface remain unchanged, while only the component normal to the surface changes. Roughly speaking, the surface may be regarded as a third body, participating in the conservation of momentum, in the process of collision of a photon with an electron. Thus the range of possible transitions in metals and insulators is greatly expanded. Transitions to higher levels of the same band from which the transition begins are allowed, provided that these higher levels are unoccupied.

This difference between the selection rules for surface and bulk absorption is especially important in discussing the photoelectric properties of solids. For example, if all three components of $\mathbf{g}$ remained unchanged in surface absorption, the threshold of the photoelectric effect would not in general correspond to the thermionic work function. If the electrons excited in surface absorption have sufficient energy to escape from the metal, they can be detected by the appearance of an external photocurrent. In the curves of spectral photoelectric sensitivity this surface component of the photocurrent usually passes through a maximum corresponding to the maximum of surface absorption (see Fig. 26), and only at shorter wavelengths is there superposed on it a component caused by bulk absorption of light.

Atoms adsorbed on the surface not only change the magnitude of the work function, but also strongly affect the distribution of electronic charge near the surface and, consequently, the character of its optical absorption. A highly sensitive photoelectric surface combines a low work function with a high probability of surface optical absorption. On technical photoelectric surfaces—cesium on silver oxide or on silver—these conditions are apparently just fulfilled. A layer of cesium ions on the outer surface accounts for the presence of a large dipole moment and, as a result, a low work function, while the oxide layer between the surface and the metallic substrate is a region with a large number of electrons free from the restrictions imposed by bulk selection rules.

3. Contact Equilibrium

The properties of contacts can be studied with the aid of the simple principle that, in the steady state, the flux of electrons through the interface in one direction must be identical in magnitude and in distribution over velocities to the reverse flux. If two different metallic surfaces \(A\) and \(B\) are brought together, then before contact the electrons of greatest energy in \(A\), which has the smaller work function, lie energetically above the fastest electrons in \(B\) by precisely the amount of the difference between the work functions. When contact has been made, the electrons flow from \(A\) to \(B\), raising the internal potential of \(B\) until the energies of the highest levels in the two metals become equal. Even in the case of a poor contact, when there exists a potential barrier between the two metals (Fig. 31, \(b\)), thermally excited electrons ultimately establish the same equilibrium state, in which the resultant flux through the separating potential barrier is equal to zero. As Wilson\(^{5}\) noted, an equilibrium system requires special consideration if one of its constituent solids is a semiconductor. The value of the energy in the semiconductor corresponding to the upper edge of the filled band in the metal is not the upper edge of the highest filled band of the semiconductor, as one might suppose, but lies approximately in the middle of the forbidden region between this filled band and the lowest unfilled band (Fig. 32).

Fig. 31.

\(a\)—the contact potential difference between two different metallic surfaces is equal to the difference between their work functions; \(b\)—after contact has been established, at equilibrium, the tops of the filled bands in both metals are at the same level

Fig. 32. Relative position of the energy levels of a metal and a semiconductor at equilibrium.

\(1\)—vacant levels, \(2\)—forbidden region, \(3\)—filled regions. The upper edge of the filled levels of the metal lies midway between the filled and unfilled bands of the semiconductor

This occurs because only those electrons can cross the boundary separating the solids which, in one of the bodies,

occupy energy states that are allowed but not filled in the other body. The electrons in both solids obey Fermi–Dirac statistics. In both bodies there is some energy value about which, at \(T > 0^\circ\mathrm{K}\), thermally excited electrons and thermally created “holes” are distributed symmetrically. In a metal this energy value is the peak of the distribution at \(T = 0^\circ\mathrm{K}\); in a semiconductor it evidently lies in the middle of the forbidden region. At equilibrium these points must coincide, since the resultant current must be equal to zero.

Of course, the very same considerations apply to the case of contact between a metal and an insulator, since the difference between an insulator and a semiconductor is only quantitative. It may be noted that the time required to establish equilibrium after the contact is made is determined mainly by the rate at which electrons are thermally excited across the forbidden energy region of the nonconductor. If this energy gap is large and the temperature low, the transition to equilibrium may occur very slowly.

4. Contact Rectifiers

Contact rectifiers consist of a semiconductor clamped between two pieces of metal. In old radio detectors with galena crystals, these two metals were the crystal holder and the contact wire. In copper-oxide rectifiers the semiconductor is cuprous oxide, one metallic contact is the copper base (substrate), and the other contact is an arbitrarily chosen metal deposited on the surface of the oxide. This system has been investigated rather fully\(^{6,5}\), and in our discussion we shall dwell chiefly on it. Since such a rectifier operates satisfactorily even when the upper metallic contact is also made of copper, it may be assumed that this effect is connected mainly with the difference between the contacts of the oxide with the two metallic surfaces.

Fig. 33. Diagram of a copper-oxide rectifier with a blocking layer.

A thin film (\(\sim 10^{-7}\) cm) of pure \(\mathrm{Cu_2O}\) is a conductor with high resistance. In Schottky’s theory, rectification is explained by an asymmetric cold emission through this layer. Wilson’s theory uses only its ability to withstand potential differences between copper and contaminated \(\mathrm{Cu_2O}\).

The first interpretation of the rectifying action was proposed by Schottky and his co-workers. They assume that the greater part of the oxide layer contains an excess of oxygen and therefore has a very low resistance, but that at the boundary between the oxide and the copper substrate a thin “blocking layer” of pure \(\mathrm{Cu_2O}\) with relatively high resistance is formed (Fig. 33).

This picture is consistent with what is known to us about the mechanism of the oxidation process. It is assumed that the second contact between the oxide and the upper metal has a low resistance. If

to this system, then most of the potential drop occurs across the blocking layer, and in this layer a noticeable cold emission appears (VI, 1). Of course, in order to explain the observed rectifying action this cold emission must be asymmetric: the electrons must flow more freely from the copper to the copper oxide contaminated by copper impurity than in the opposite direction. Such an asymmetric current could occur if the form of the potential barrier were asymmetric, but, as Schottky pointed out, it is difficult to devise a plausible form of barrier that could explain the observed magnitude of the effect. de Boer and van Geel suggested that if one emitting surface is rougher than the other, the concentration of the field at the protrusions may give the required rectifying effect.

Fig. 34. Illustration of Wilson’s theory of contact rectification.

a—The metal is at a higher negative potential than in the equilibrium state (see Fig. 32), and thermally excited electrons can flow from energy region A into the semiconductor. b—The potential is shifted from energy region B, and electrons which, in the case of equilibrium, could flow into the semiconductor, can now no longer move in this way. Since \(OA < OB\), there are more electrons in A than in B, and rectifying action takes place. It is assumed that below \(O\) no current flow occurs.

Wilson proposed a theory based mainly on the idea of equilibrium between the metal and the semiconductor, which we briefly outlined above. At equilibrium the energy levels have the arrangement shown in Fig. 32, and the resultant electron flux through the contact is zero. In this theory of rectification we must assume that the fraction of the current carried in any direction by charged particles of all energy levels below the forbidden region—

Fig. 35. Characteristic of an oxide rectifier predicted by Wilson’s theory, assuming the presence of a blocking layer \(10^{-7}\ \mathrm{cm}\) thick. The scales are given only for clarity.

of the semiconductor, is practically equal to zero. This property is quite satisfactorily explained by the point of view of Gooden and Wilson, who indicate that the region of the semiconductor, marked in Fig. 33 as filled, contains bound electrons and holes incapable of motion and, as a result, cannot conduct current even in the presence of holes. We shall discuss this further in Section VII. Let us now suppose that the metal has a higher negative potential (Fig. 34, a). All the excited electrons in the energy region \(A\) can now flow into the semiconductor, so that in this direction a resultant flow of them appears. If the potential is reversed (Fig. 34, b), then the number of electrons capable of flowing from the semiconductor into the metal does not change, but the number of electrons capable of flowing from the metal into the semiconductor is reduced to the number in the energy region \(B\). In the region \(A\) there are many more excited electrons than in \(B\), since \(OA < OB\). As a result, an asymmetric form of the current-voltage characteristic is obtained, of the type shown in Fig. 35, taken from Wilson’s paper.

This theory assumes that the contact between solids is so good that an appreciable current can flow, but at the same time it is so poor that an appreciable potential difference can be maintained in the boundary layer. Schottky’s blocking layer is precisely a contact of this kind. The curves in Fig. 36 were calculated under the assumption of the presence of a barrier of thickness \(10^{-7}\) cm, having a rectangular form at equilibrium; the passage of electrons through such a barrier, distorted by the applied electric field, was also taken into account here. In this special case cold emission tends to neutralize rectification.

5. Adsorption on Crystal Surfaces

If an additional atom of the same material is deposited on the clean surface of a pure single crystal (of one element), then the energy states of this atom broaden into bands that become identical with the bands of the crystal; as a result, this additional atom enters the lattice as a constituent part. The energy levels of the initial crystal will, of course, also be perturbed, but the effect of this on each electron of the crystal is small, and in the resulting crystal one additional energy level will appear in each zone.

Now, instead of what was assumed earlier, let us suppose that an atom with a low ionization potential (for example Cs; 3.85 eV) is deposited on the surface of a substance with a large work function (such as W; 4.5 eV). We may expect that the energy states of the cesium atom broaden into bands of one kind or another, depending on the structure of the energy spectrum of W (Fig. 36). In the resulting energy spectrum of the complete system (solid body plus adsorbed atom), the states in which the valence electron is found as often near the atomic core of Cs as near one of the lattice ions of W lie appreciably higher than the normally filled states. This follows from the fact that the potential energy of an electron near a W ion

lower than near the Cs ion; therefore the amplitude of the wave function determining the charge distribution is greater near the W ions, and this difference becomes unimportant only if the total energy of the electron is comparatively large. The unequal distribution of charge around Li and F ions in the LiF lattice (Fig. 20, b) may be cited as an example of the very same effect. In the established equilibrium, the valence electron of a Cs atom occupies one of the lowest states, and therefore the valence charge (its distribution function) around the Cs ion is small. Consequently, Cs will be effectively ionized and, in the form of a positive ion, will increase the dipole moment of the surface. A small electronic charge around the ion will hold it at the surface even if it receives sufficient thermal energy for detachment. We note that, since this charge nevertheless exists, the atom is in fact not “completely ionized.”

Fig. 36. Above—schematic diagram of the broadening of Cs atomic levels as the Cs atom approaches the W surface.

Below—the upper charge distribution corresponds to an electron with energy \(B\) in the upper diagram; the lower distribution corresponds to an electron with energy \(A\) in the upper diagram. Since the low-lying, \(A\)-type states are free in the metal, the Cs valence electron can occupy one of them and the atom becomes effectively ionized.

Fig. 37. Illustration of how, when an atom is adsorbed on an insulating crystal, some atomic levels broaden, while others remain narrow.

\(1\)—forbidden region, \(2\)—filled band.

Gurney\(^7\) emphasized that one should expect the presence of some degree of ionization of adsorbed atoms even when their ionization potential exceeds the work function. The relative charge distribution is determined by the relative potential fields around the lattice ions and around the adsorbed atomic residue; these are best compared by comparing the ionization potentials of both types of atoms, rather than the work function of the solid with the ionization potential of the adsorbed atom. For example, Ca, Sr, and Ba all have ionization potentials higher than the work function of W, but lower than its ionization potential, and they are all partially ionized when adsorbed on it.

If the adsorbing crystal were a stable insulator with a filled lower band, then for the valence electron of the adsorbed cesium atom there would necessarily appear some higher state. This lowest possible level could easily form in the forbidden region of the insulator spectrum (Fig. 37). The distribution of the electronic charge would then be completely localized near the Cs ions, and its energy level would not broaden, since the corresponding periodic states do not exist in the lattice.

Fig. 38 depicts the hypothetical behavior of the energy levels when an atom with a high ionization potential (for example, O) is deposited on the surface of a crystal with a small work function (for example, Cs). Two of the atomic levels of oxygen are shown as leading into the forbidden region of the energy spectrum of Cs and below the band occupied by the valence electrons of Cs; the levels are not broadened, since there are no corresponding states in the lattice. At equilibrium, these discrete levels, lying below the filled region in the spectrum of Cs, will be occupied by valence electrons from the Cs lattice, and these electrons are completely localized near the O atoms. Such a redistribution of charge may be interpreted as the formation of a Cs$_2$O molecule; however, since the whole lattice as a whole gives up these two valence electrons, it is difficult to legitimize the notion that two particular Cs atoms are included in this “molecule.” In other words, the mobility of the oxygen atom on the surface of Cs is not directly related to the heat of dissociation of Cs$_2$O.

Fig. 38. Adsorption of an atom with a high ionization potential on a crystal with a low work function

In the case where the difference between the ionization potential and the work function is not so large (an example of this is possibly O on W), the atomic states could lead into the filled region of the solid’s spectrum. They could therefore broaden, and some valence electron of the crystal could be found near the adsorbed atomic core. If the field of this atomic core is strongly negative in comparison with the field of a lattice ion, then the wave functions will have a larger amplitude near the adsorbed ion than in the lattice; the atom then acquires, on average, a negative charge, the surface acquires a negative dipole moment, and its work function increases.

It is interesting to return to the case of Cs atoms deposited on W and to see how, in the end, the surface will be transformed into continuous Cs. As an ever greater number of Cs atoms is added and ionized, W acquires an ever increasing negative charge, and the energy of its valence electrons

becomes ever higher, closer to the energy of the valence electron in an isolated Cs atom. Finally, such a surface concentration of cesium is reached at which the added Cs atoms are not ionized to any appreciable extent, and the distribution of the electronic charge, extended outside the W lattice, becomes similar to the charge distribution around Cs ions in metallic Cs. There is experimental evidence that this state is reached when the surface is covered with cesium to approximately \(2/3\). In this case the lowest state of the Cs electron will be one in which it is found approximately as often near a Cs ion as near an ion of the W lattice. Moreover, the way in which the atomic levels broaden begins to resemble the way in which they split when added Cs atoms are deposited on a cesium lattice, and gradually becomes identical with the latter when the surface is covered with one or two atomic layers of Cs. With further accumulation, the Cs atoms form crystalline Cs, and finally we obtain two homogeneous metals in close contact.

6. Surface states of electrons

There exists the following very important effect associated with the presence of a surface in a crystal: additional allowed energy states appear in its energy spectrum. Electrons in these states cannot move freely in the lattice, but are bound at the surface. To see how these states arise, it is necessary to look somewhat more deeply at the reason for the occurrence of an energy band spectrum in an infinite lattice. When the Schrödinger equation is solved for a periodic potential field (see III, 1), solutions of two types are obtained. One of them, valid for certain values of the total energy, is a periodic function leading to a periodic probability distribution and to a finite probability of finding the electron at any point of the lattice. The energy levels associated with these solutions we called “allowed states,” and the energy intervals for which these solutions are valid are the “allowed regions” of the energy spectrum. Solutions of the second type are real exponential nonperiodic functions. They lead to probability distributions that decrease exponentially in one direction and increase exponentially in the other. Such solutions, if no restrictions are imposed on them, correspond to none of the physical properties of electrons; therefore they are excluded from consideration, and those values of the total energy of the electrons for which such solutions occur are the “forbidden regions” of the spectrum.

Tamm\(^8\) noted that if the crystal is not infinite but has a surface, then there is no need to discard all these solutions of the second type. If we choose from them those which decrease exponentially inward from the surface, then their exponential increase in the opposite direction, which would occur in the case of a crystal of infinite dimensions, is cut off at the surface. Outside the crystal,

where the potential energy of the electron is greater than the total energy, such a function rapidly dies out. As a result, a bounded probability distribution appears, admissible on physical grounds and corresponding to an electron localized at the surface in an energy state intermediate between the allowed bands of periodic states (Fig. 39). In both directions along the surface the probability distribution is periodic; electrons in these states can move freely along the surface. It turns out that the number of such localized levels is approximately equal to the number of atoms on the surface. Therefore the bulk properties of a crystal of ordinary dimensions are not greatly changed by the presence of these states, even if they are all occupied.

Fig. 39.
a—charge distribution for a state corresponding to an electron localized at the surface of a crystal; b—the localized state at the surface is depicted by a short line

In the case of metals with overlapping bands of periodic states, one cannot expect these surface levels to have a strong influence even on the surface properties of the metal, since an electron in one of these states can easily make a transition to the nearest periodic state, thereby obtaining the possibility of moving in the lattice. In an insulator, on the contrary, an electron can be “trapped” in such a localized state between two allowed regions of the bulk spectrum. It should be noted that localized states can form not only at the external surface of a crystal, but also near cracks and boundaries inside the specimen.

VII. INFLUENCE OF IMPURITIES ON BULK PROPERTIES

1. Impurity energy states

We have seen that in an ideally regular lattice, as a result of the interaction of all atoms, an electronic spectrum is created, consisting of allowed and forbidden regions of energy. An impurity atom,

regardless of whether it is a “substitutional” or “interstitial” impurity, disrupts the regularity of the lattice and causes certain changes in the energy spectrum, analogous to the changes introduced by a surface (VI, 6).

Fig. 40. Behavior of the atomic states of an impurity atom when it is placed inside a solid (shown schematically).

Some of the levels broaden, others become discrete levels in the “forbidden regions” of energy. 1, 2, 3 — bands of allowed levels.

We can obtain an idea of the character of the changes caused by an impurity atom if we imagine the process of introducing such an atom into the lattice. Some of the energy levels of this atom may be broadened into bands merging with the allowed bands in the solid; the others pass into discrete states in regions forbidden for the solid (Fig. 40). The probability distribution associated with a level passing into any of the allowed bands of the solid will be periodic at large distances from the impurity atom, as it would be if there were no inhomogeneities in this body (Fig. 41, a). In contrast to this, the probability of the charge distribution for discrete levels must decay exponentially in all directions away from the impurity atom (Fig. 41, b). An electron occupying one of these discrete levels proves to be localized near the impurity atom.

Fig. 41. Charge distribution around an impurity atom.

a — associated with one of the periodic states; b — associated with a state localized in one of the forbidden regions. B — impurity atom, A — atoms of the pure crystal.

If the impurity atom has a dense energy spectrum (for example, Fe or Mn), and the solid is a good insulator with broad forbidden energy regions between the allowed

bands, then several such localized levels may form around each impurity atom.

To clarify the influence of additional states (periodic and localized levels) on electrical conductivity, let us consider several hypothetical cases. First suppose that an atom of the insulator is replaced by an impurity atom with a set of energy levels, each of which passes into an allowed band of the basic energy spectrum. The number of levels in the new spectrum will then be the same as in the old one. Next suppose that the impurity atom has fewer valence electrons than the atom it replaces. Then a “positive hole” (i.e., vacant levels) is formed at the top of the formerly filled band of the insulator, and the presence of these holes creates electrical conductivity in the solid. At first this electrical conductivity increases linearly with the growth in the concentration of such impurity atoms and, like ordinary metallic conductivity, increases as the temperature decreases. When the number of impurity atoms becomes so large that they are in a “combinational relation” with the remaining atoms, a completely regular lattice may again be formed, with its own characteristic energy spectrum. Similarly, the presence of a discrete impurity level below the filled band of the insulator (Fig. 42, a) also imparts electrical conductivity to the crystal, if this level is normally not filled by an electron of the impurity atom. In the equilibrium state this lower level is occupied by an electron from the upper band, and the hole remaining in the upper band can create conductivity.

Fig. 42.
a — A localized state arising from the presence of an impurity atom, lying between two “filled” bands of a solid and not occupied by an electron of the impurity atom; it can be occupied by an electron from the upper filled band, and the presence of a vacant state (positive hole) in the upper band may lead to the appearance of electrical conductivity in this solid; b — A localized state lying between the filled and unfilled bands of a solid and normally occupied by an electron of the impurity atom. This electron is thermally excited into the upper band, where it acts as an electron of the lower filled band. Such states, arising from the presence of impurities, may lead to the high conductivity of many semiconductors.

If the localized impurity state lies between the filled and unfilled bands of the insulator and is normally occupied by an electron of the impurity atom (Fig. 42, b), then this electron can be thermally excited into the upper unfilled band, where it will then give rise to the appearance of conductivity. If the energy gap between the localized state and the upper band is small compared with the gap between the bands, thermal excitation from the localized state will be very effective compared with thermal excitation from the lower band. Such a substance will behave as a semiconductor with an energy gap smaller

…intrinsic pure state. It is usually considered that the observed small values of energy gaps for most semiconductors (see Table 2) are due to the presence of such states, caused by impurity. Gudden showed that the energy gap measured from the threshold of photoconductivity is usually much larger than the value obtained from data on the change of conductivity with temperature. Because of the small number of states caused by impurities, it is difficult to expect that they would noticeably absorb light; the photoconductivity threshold should therefore give the true value of the gap between filled and unfilled bands in the pure substance. For example, in cuprous oxide this quantity \(E\), calculated from the curve of the logarithm of the conductivity as a function of \(1/T\), is equal to 0.6 eV, whereas experiments with photoconductivity give a value close to 2 eV.

If the original pure substance is a metal, then the additional states associated with the impurity atom will evidently not have any noticeable effect on the conductivity. The impurity atom increases the resistance of the metal chiefly because it disturbs the regularity of the normal lattice, as a scattering system for electron waves. This influence of impurities in a metal is very noticeable at low temperatures, since it remains even after the irregularities caused by thermal vibrations of the lattice have largely disappeared (see V, 7).

2. Fluorescence and Phosphorescence in Crystals⁹

The phenomena belonging to this field may be roughly outlined as follows: a solid absorbs light of one wavelength and, sooner or later, emits light of another (longer) wavelength. If the time elapsed between absorption and radiation is immeasurably small, the phenomenon is called fluorescence; if there is a measurable delay, the phenomenon is called phosphorescence. Both of these phenomena may exist in one and the same material. The behavior of substances observed in different cases is extremely varied, and nothing resembling a general theory on any basis has yet been developed. The ideas proposed here are intended to indicate how this general phenomenon can be included in the band picture.

Let us suppose that the crystal is a pure insulator with a wide energy gap between the highest filled band and the lowest unfilled one. Upon absorption of a quantum, an electron may pass from some state of the filled band into the corresponding state of the unfilled band. This electron and the positive hole left by it can move in the lattice quite independently of one another, if the possibility of interaction between them is neglected (see V, 6). As a result of collisions with the lattice, the electron and the hole rapidly lose the kinetic energy acquired by them upon absorption of the quantum, converting its excess into thermal energy. As a result, the electron falls to a level close to the top of the lower band (Fig. 43, a).

Assuming that the mean free path is of the order of \(10^{-5}\) cm and that in each collision several percent of the excess kinetic energy is lost, one may conclude that the time required for the establishment of this thermal equilibrium is of the order of \(10^{-9}\) sec. At low temperatures the mean free path increases, and the time for reaching equilibrium becomes proportionally longer. How long can an electron remain in an excited state? Simple considerations show that if the electron and the hole cannot “recombine” until they are at a distance of \(10^{-8}\) cm from one another, and if in the crystal there is only one excited electron per \(1\ \text{cm}^3\), then their mean lifetime is about \(10^9\) sec. This mean lifetime, of course, decreases with increasing concentration of excited electrons and holes: if at the time \(t=0\) there were \(n_0\) excited electrons in \(1\ \text{cm}^3\), and the collision radius is taken to be about \(10^{-8}\) cm, then the number of electrons existing at time \(t\) can be found at once:

\[ n(t)=\frac{n_0}{1+\frac{n_0\cdot t}{10^9}}. \]

If the bands are narrow and the gap between them is wide, then the frequency of the quantum emitted upon recombination will be approximately the same as the frequency of the quantum initially absorbed—the fluorescence will be resonant.

Fig. 43.

a — schematic representation of how, after optical excitation, an electron separates from the lower free state and how this electron and the positive hole lose excess kinetic energy in the lattice; b — the possibility is shown of capture of an electron and a hole in localized states of impurity atoms with emission of fluorescence quanta of frequencies \(\nu_1\) and \(\nu_2\). The energy \(h\nu_3\) is still sufficient to obtain a fluorescence quantum of frequency \(\nu_3\).

If impurity atoms are present in a solid, forming localized levels between the allowed bands of the normal energy spectrum, then the electron or the hole (or both together), instead of direct recombination, may be captured into these states. Such a case is shown in Fig. 43, b, where two impurity levels are shown. The valence electron of an impurity atom, normally occupying the lower level, neutralizes the hole left in the lower band during optical excitation of an electron into the upper band. On the other hand, this excited electron falls into the upper normally free state of another impurity atom. Obviously, the fluorescence quantum emitted in either of these capture processes will have a greater wavelength than the initially absorbed quantum. The ratio between the probabilities of such capture and of direct recombination will evidently depend on the concentration of impurity atoms. For example, technical willemite contains about 1% Mn as an “activator” in a zinc-silicate base. This

corresponds to a density of Mn atoms of about \(10^{21}\) per \(\text{cm}^3\); for such a concentration, capture (taking the capture radius to be \(10^{-8}\ \text{cm}\)) proves much more probable than direct recombination.

After the electron and the hole have been trapped, the lattice still contains energy corresponding to the difference between the upper localized state, containing the electron, and the lower one, containing the hole (i.e., unoccupied). One may imagine two kinds of processes that lead to the release of this energy in the form of a quantum of phosphorescence light. The electron may be thermally excited from the bound state into the upper band, which will allow it to move freely through the lattice and have a finite probability of meeting the hole and neutralizing it; or the impurity atoms themselves may move through the crystal, through the lattice or along internal cracks. When one of these impurity atoms, containing an excess captured electron, meets another atom containing a captured hole, one may expect their neutralization and the emission of a quantum of phosphorescence. The rate at which either of these processes proceeds depends exponentially on the temperature. Indeed, the observed rate of decay of phosphorescence becomes very small at low temperatures.

We have already said that the phenomena belonging to this field are extremely diverse. Apparently, the modern theory has the possibilities necessary for explaining these phenomena, but before any quantitative comparison with experiment can be made, it is necessary to give a much more complete picture of the energy spectrum.

3. Latent photographic image

The formation of a latent image upon absorption of light by silver-halide crystals of a photographic plate is, apparently, a special type of the general process of fluorescence and phosphorescence of crystals. Some authors have tried, not without success, to explain qualitatively the basic photographic phenomenon on the basis of the modern picture of electron energy levels in halide crystals\(^{10}\). Gurney proposed that the formation of a latent image occurs when electrons, optically excited into the conduction band, are captured at localized states belonging to impurity atoms. These electrons were not necessarily initially in the filled band of the crystal, but could have been on levels localized above the filled band. It is very probable that the various methods of sensitizing plates to light of greater wavelength are effective because they create a large number of such levels, normally filled with electrons, situated not very far below the upper band.

To explain certain observations, such as, for example, the development of a plate grain by grain, it seems necessary to add a new complication: it must be assumed that the localized levels, after capturing electrons, somehow, in number

of the order of ten, coalesce into clouds and form centers of the developing image. Such aggregates are apparently more stable than isolated levels, although they can be broken up by light of a certain frequency (the Herschel effect). Collisions of the bound electron with a hole in the course of displacement during the formation of the first developing centers lead to the neutralization of this pair and to an effect opposite to the initial absorption of light. The probability of such a process is proportional to the square of the number of existing pairs. The formation of the latent image must therefore be less effective at high light intensities than at low ones—this is the explanation of the failure of the reciprocity law (blackening proportional to intensity and time) under strong illumination. A failure of the reciprocity law is also observed under weak illumination: the effectiveness of latent-image formation is small at very low light intensity. Webb assumed that this may be caused by dissociation of the developing center at the beginning of its formation as a result of thermal excitation of several electrons first captured. It is supposed that the stability of the center increases with the number of electrons captured in the center. At low light intensities the centers grow very slowly, and small centers at the beginning of formation have a relatively high probability of thermal destruction. The inefficiency of latent-image formation at low light intensities becomes more significant with increasing temperature, which is consistent with Webb’s hypothesis.

We note once again that the modern picture of the electronic structure of solids is only beginning to be applied to such phenomena as fluorescence and the latent image, and that the present theory is, for the most part of necessity, built on hypotheses and subject to considerable change. In this field it is extremely necessary to carry out a large number of experiments guided by the modern point of view. Highly efficient fluorescent powders, very important for technology, and, to an even higher degree, complex photographic emulsions, probably are not ideal materials for experiments given the present crude state of our knowledge. Experiments with single crystals of alkali-halide compounds, similar to those investigated by Pohl and his collaborators ^11 over a number of years, apparently open the most direct path to an understanding of these phenomena.

VIII. PLASTICITY AND FRACTURE OF CRYSTALS

In recent years many investigations have been carried out on the mechanical properties of single crystals, and significant progress has been achieved with regard to the atomic picture of the behavior of polycrystalline material. These questions, in the main, lie outside the bounds of band theory in its present state of development. We shall give a very brief survey of some of the simplest experiments and theoretical ideas ^12.

1. Shearing displacements (slip) in a single crystal

Single-crystal rods of all materials obey Hooke’s law under tension up to a certain critical force; beyond this point they undergo permanent inelastic deformations. The critical force depends on the history of the given specimen and on its crystallographic orientation. Examination of a stretched rod shows that such deformation consists of the slipping of portions of the rod relative to one another along certain lattice planes and in certain directions in these planes. Usually slip takes place only along one family of planes with small indices, although in some cases (for example, tetragonal $\beta$ Sn) two groups of slip planes appear. The direction of slip in the slip plane is likewise a definite simple lattice direction.

The critical shearing stress (i.e., the component of the tensile stress in the direction of slip at which plastic deformation just begins) is characteristic of the given material. It may vary from specimen to specimen by roughly a factor of five, depending on purity and on its previous history. Characteristic values are given in Table 6. The critical shearing stress decreases rather slowly with increasing temperature almost up to the melting point. It increases greatly if the material has previously been subjected to stress. An elongation of 100% may

Table 6

increase the critical stress by a factor of 20; this means that the range of stresses in which the material obeys Hooke’s law is greatly expanded. Annealing a stretched specimen returns the critical shearing stress to its initial low value.

Ionic crystals, if they are initially well annealed, exhibit sliding deformation of the same kind. For example, rock salt slips along the planes (110) in the direction [110] when the critical shearing stress exceeds approximately 0.075 kg/mm². This value is of the same order as for metallic crystals (Table 6). The breaking tensile stress is usually much smaller than for metals. Work hardening of the same type as in metals is observed, as is its removal by subsequent annealing. The critical shearing stress decreases with temperature, apparently more rapidly.

2. Stress that destroys a single crystal

A single crystal that has been stretched to a certain limit breaks along one of several simple crystallographic planes. Fracture occurs when the stress normal to the fracture plane exceeds a critical value which, like the critical shear stress, varies somewhat for different specimens depending on their purity and history. The degree to which the crystal must be stretched before it breaks depends on the ratio of the critical stresses for slip and fracture; the crystal is ductile if this ratio is small, and brittle if it is relatively large. The fracture and slip planes do not necessarily coincide. Characteristic values of the breaking stresses are given in Table 7. The breaking stress, like the critical slip stress, depends slightly on temperature, increasing at low temperatures. In NaCl the breaking stress, unlike the critical shear stress, increases at temperatures above room temperature; at 600° C it is of the order of 8 kg/mm².

Table 7

3. Theoretical values of critical stresses

One of the first attempts to calculate the breaking stress for an ideal lattice was made by Polanyi[^13]. From the surface tension of liquid NaCl he determined the energy \(\alpha\) required to form a unit surface of crystalline NaCl, and assumed that the work performed in the destruction of a crystal is spent mainly on the formation of two new surfaces. If the mean range of action of the forces between two surfaces is \(l\), and the normal fracture stress is \(S\), one should expect, at least roughly, the relation \(2\alpha = S \cdot l\). Using the values of \(\alpha\) obtained by Polanyi and the observed values of \(S\), we find: \(l = 1.4 \cdot 10^{-5}\) cm, which is improbably large. Taking \(l = 5 \cdot 10^{-8}\) cm as an acceptable value, we find \(S_{\text{id}} = 61\) kg/mm², which exceeds the observed value by about 300 times. Zwicky[^14] carried out more accurate calculations on the basis of the Madelung–Born theory and found \(S_{\text{id}} = 200\) kg/mm² for NaCl. Similar calculations were carried out for a large number of other substances; Table 8[^15] gives the most characteristic of these values.

Table 8

Similar determinations of the critical shearing stress were made by Frenkel, who used the Madelung–Born model. Just as for the breaking stress, the theoretical values turn out to be approximately 1,000 times greater than the observed ones.

Although the Madelung–Born model is unsuitable for exact calculations (see IV, 2), it should, of course, give a result closer to the true value than to a value differing from the true one by a factor of 1,000. To explain the observed low critical stresses we must take into account the fundamental difference between the real crystals on which the observations are made and the ideal crystal studied in theory.

Griffith¹⁶ showed that the tiniest cracks on the surface, at which stress accumulates, are very probably the cause of the low experimental values of the stresses. He calculated the stress distribution near a long cylindrical hole of elliptical cross-section in an isotropic elastic solid and found that the stress at each of the ends of the major axis exceeds the average stress by approximately 1,000 times if the length of the major axis of the ellipse exceeds the minor one by a factor of 500. Narrow and thin surface cracks of this kind are, in general, possible. It is easy to see that a rupture that begins near such a crack will develop like an “explosion.” As the cross-section decreases under a constant load, the average stress increases and the stress at the corners of the crack increases proportionally. Griffith found experimentally that thin rods of glass and quartz were very strong and flexible immediately after being made, but became quite brittle if the surface was scratched. Ioffe’s experiments on rock salt and Orowan’s on mica confirm this explanation of the small magnitude of the breaking stress.

On the other hand, the low value of the critical shearing stress observed in experiments is probably not connected with the existence of surface cracks. Taylor¹⁷ suggested that small, narrow cracks are distributed throughout the volume of bodies and that the stress increases at their corners. This idea, cast in quantitative form,

gives fairly good agreement with observations of shear stress. Work hardening under shear is thought to be caused by a decrease in the mean length of cracks during slip, which may reduce the growth of stress at their ends. It should be noted that narrow cracks, distributed according to Taylor’s theory throughout the volume of a well-annealed crystal, divide the crystal into a large number of blocks. There is much evidence that a mosaic structure of this type exists in most crystals, and several suggestions have been made as to the cause of its appearance. Zwicky^18 at one time proposed that such a mosaic is thermodynamically more stable than an ideal lattice, but this view has not found general support^19. The theory of the “structure of origin,” developed by Buerger^20, which ascribes lattice imperfection to inhomogeneities inevitably appearing in the crystal during its growth, seems especially probable and very general.

LITERATURE FOR PART III

1. Dushman, Rev. Mod. Phys., 2, 381, 1930; Reimann, Thermionic Emission, London, 1934.
2. Nottingham, Phys. Rev., 49, 78, 1936.
3. Brüche, Z. Physik, 98, 77, 1935; see also Becker, Rev. Mod. Phys., 7, 95, 1935.
4. Wigner and Bardeen, Phys. Rev., 48, 84, 1935; Bardeen, Phys. Rev., 49, 640, 653, 1936.
5. Wilson, Proc. Roy. Soc., A136, 487, 1932.
6. Grondahl, Rev. Mod. Phys., 5, 141, 1933; De Boer, Electron Emission and Adsorption Phenomena, Cambridge, 1935, ch. XV.
7. Gurney, Phys. Rev., 47, 479, 1935.
8. Tamm, Sow. Phys., 1, 722, 1932.
9. Pringsheim, Fluorescenz und Phosphorescenz, Berlin, 1928; Lenard, Schmid u. Tomaschek, Handb. d. Experiment. Physik, XXIII/1 u. 2.
10. An excellent review is given by Webb, JOSA, 26, 367, 1936.
11. Many of these studies were published in Nachr. Gött. Ges. Recent reviews were given by Hilsch, Ang. Chemie, 49, 69, 1936; Hughes, Rev. Mod. Phys., 8, 294, 1936.
12. The experiments described here are taken from the book Schmid u. Boas, Kristallplastizität, Berlin, 1936; Smekal, Handb. d. Physik, XXIV/2.
13. Polanyi, Z. Physik, 7, 323, 1921.
14. Zwicky, Z. Physik, 24, 131, 1923.
15. Taken from the article by Smekal, reference 10.
16. Griffith, Phil. Trans. Roy. Soc., 221, 163, 1921.
17. Taylor, Trans. Farad. Soc., 24, 121, 1928; Proc. Roy. Soc., A145, 362, 368, 1934.
18. Zwicky, Proc. Nat. Acad. Sci., 15, 816, 1930.
19. For the latest data on the general problem of real crystals, see vol. II, Report on the International Conference on Physics, London, 1934 (Cambridge, 1935).
20. Buerger, Z. Krist., 89, 195, 1934.

1. Brüche[^3] discusses some of the latest proofs and the conclusions drawn from them. ↩↩↩