Photoelectric Effect in Metals

S. V. Vonsovskii, A. V. Sokolov, A. Z. Veksler

§ 1. Introduction

The study of phenomena involving the interaction of an electromagnetic field with matter is of very great scientific interest, since the elucidation of the details of the mechanism of this interaction makes it possible to penetrate into the subtleties of the structure of various substances and to obtain more complete and deeper conceptions of the nature of their properties. It is also of enormous practical interest in connection with the technical use of these phenomena in modern electrical engineering, radio engineering, and electronics.

Among the numerous specific types of interaction of electromagnetic radiation with matter, in the present review we shall be concerned with the case of the so-called external photoelectric effect in metallic bodies. From the phenomenological point of view, this effect is understood to mean the emission of electrons from the surface of a metal when it is illuminated, i.e., when it is acted upon by electromagnetic radiation in the infrared, visible, and ultraviolet frequency regions. The phenomenon of electron emission by means of x-ray and gamma radiation, strictly speaking, should also be called the photoelectric effect. Naturally, the photoelectric effect is observed not only in metals, but also in other solids (semiconductors, dielectrics, molecular crystals, etc.), as well as in liquids. Here, however, we shall in the main confine ourselves to consideration of the photoelectric effect only for the case of solid metallic bodies. Of other types of interaction of an electromagnetic field with matter that are to some extent related to the external photoelectric effect, mention should also be made of: 1) the phenomenon of photoionization, i.e., the emission of electrons when gases and vapors are illuminated; 2) the phenomenon of photoconductivity, or the internal photoelectric effect, i.e., the appearance of conduction electrons in bodies that do not conduct electric current (semiconductors or dielectrics) when they are illuminated; 3) the photogalvanic effect, i.e., the appearance of an emf.

in a closed circuit under the influence of illumination, but without an external source of current. A strict distinction between these phenomena and the external photoeffect cannot be drawn because of their common physical content, but here we shall confine ourselves to considering only the external photoeffect.

The discovery of the external photoelectric effect belongs to H. Hertz¹ (1887). It is noteworthy that the discovery of this phenomenon, which played an outstanding role in the development of quantum theory, was made by Hertz incidentally, in his famous experiments on the experimental substantiation of the electromagnetic theory of light. Hertz found that the length of the electric spark in the spark gap of an auxiliary oscillatory circuit depended on whether light from another spark, produced in the spark gap of the main circuit, fell upon it or not. It was found that if the spark gap of the auxiliary circuit was shielded from light, then the length of the spark in it was smaller than when it was illuminated. It also turned out that this effect is observed in the ultraviolet part of the spectrum of the incident radiation, and that it is observed best of all if the negative electrode of the spark gap is illuminated. These results of H. Hertz’s experiments served as the beginning of an intensive study of the photoelectric effect. Among the subsequent experimental investigations, mention should first of all be made of the works of A. G. Stoletov², in which the fundamental laws of the photoeffect were first discovered, and also the works of Hallwachs³ and other investigators. Lenard⁴ and Thomson⁵ proved that, when metals are illuminated, electrons are emitted—electrons that had earlier been discovered in experiments with cathode rays in discharge tubes.

In investigating the phenomenon of the emission of electrons by a substance subjected to illumination, it was necessary first of all to establish the connection between the characteristics of the incident radiation and the characteristics of the stream of emitted photoelectrons. The basic characteristics of the radiation should be chosen as the intensity, spectral composition (frequency), and polarization of the light; and, for the electrons, their number, distribution by velocity, and the direction of their emission. The task of theory is to explain the regular connections between these quantities and to compare them with experiment.

As a result of the experimental investigations mentioned above, as well as many others, three basic experimental laws of the external photoeffect were established:

1) The strength of the photoelectric current \(j\) is directly proportional to the intensity \(I\) of the light flux causing the photoeffect, provided the spectral composition of this flux remains unchanged (Stoletov’s law).

2) There exists a long-wavelength boundary (\(\lambda_0\) or \(\nu_0\)) in the radiation spectrum, starting from which (for \(\lambda < \lambda_0\), or \(\nu > \nu_0\)) the photoeffect can occur in the given substance (the law of the “red” boundary).

3) The energy of photoelectrons does not depend on the intensity of the light, and the maximum value of this energy \(\frac{1}{2}(mv^2)\) is linearly related to the frequency of the incident light, i.e.

\[ \frac{1}{2}(mv^2)_{\max}=a+b\nu, \tag{1.1} \]

where \(a\) and \(b\) are constants, \(m\) is the mass and \(v\) the velocity of the photoelectron.

It is known that the laws of the photoelectric effect (2) and (3) could not be explained on the basis of the classical wave theory of light and led to yet another “catastrophe” of classical physics, which was incapable of explaining the quantum character of optical phenomena. Einstein\(^6\) was the first to give a theoretical explanation of these laws, applying for this purpose Planck’s conceptions of light quanta. Einstein’s work on the photoelectric effect marked the beginning of the modern theory of the interaction of light with matter. Assuming that light of frequency \(\nu\) can be absorbed and emitted in quanta of energy \(h\nu\) (\(h\) is Planck’s constant), Einstein proposed that the energy of a light quantum incident on a metal is wholly absorbed by an electron, being spent partly on the work of extracting the electron from the metal, and for the rest being converted into the kinetic energy of the electron ejected from the metal. Thus, the law of conservation of energy in the present case has the form:

\[ h\nu=\chi+\frac{1}{2}mv_{\max}^{2}, \tag{1.2} \]

where \(\chi\) is the minimum energy required to remove an electron from the metal into the external free space, i.e. the so-called work function, and \(v_{\max}\) is the maximum velocity of the emitted electrons. From equation (1.2) the second and third laws of the photoelectric effect immediately follow. Namely, for the “red” limit we obtain

\[ h\nu_0=\chi,\qquad \nu_0=\frac{\chi}{h}. \tag{1.3} \]

Thus, the minimum frequency \(\nu_0\) is determined by the work function \(\chi\); from equation (1.2) it follows directly that the maximum energy of the photoelectrons is a linear function of the frequency of the light, i.e. the experimental law (3) is obtained. Stole­tov’s first law is also well explained by Einstein’s theory, since the number of photoelectrons increases directly in proportion to the number of photons. Einstein’s equations were verified in a number of experiments. Among these, first of all, one should mention the fundamental works of A. F. Ioffe\(^7\) on the elementary photoelectric effect, in which the elementary act of quantum absorption of light, predicted by Einstein in formula (1.1), was observed for the first time. An important role

...in the experimental study of the photoelectric effect was played by the very subtle—in terms of experimental skill—work of P. I. Lukirskii and S. S. Prilezhaev,^8 who developed a very precise method for measuring the photoelectric effect with the aid of a spherical condenser and carried out a careful investigation of photoelectron velocities for a whole series of metals, and also gave a precision method for the quantitative determination of Planck’s constant.

From formula (1.3) it follows that the photoelectric threshold is determined by the work function, while the latter is determined by the conditions on the surface of the metal. Thus, one may expect that the limiting frequency \(\nu_0\) depends strongly on the state of the metal surface; this is fully confirmed by experiment. Treatment of the surface, the presence of adsorbed gases, etc., can greatly change the magnitude of the work function and, together with it, the magnitude of the red threshold, and thereby make the problem of determining the red threshold of pure metals very difficult.

Fig. 1. Distribution curve of photoelectrons by energy for aluminum.^8

The temperature of the metal may also have a substantial influence on the value of \(\nu_0\). Experiments with clean surfaces in vacuum have shown that the photocurrent \(j\) changes only slightly with temperature for frequencies far from the red threshold \((\nu-\nu_0 \gg \nu_0)\), and increases sharply at frequencies close to it, especially for the smallest frequencies \((\nu \sim \nu_0)\), i.e., temperature effectively shifts the red threshold into the region of lower frequencies, and this threshold ceases to be sharp as the temperature rises. In a manner analogous to temperature, an accelerating electric field at the surface of the photocathode acts on the photocurrent. The influence of this field is slight for frequencies \(\nu\) far from the red threshold \((\nu \gg \nu_0)\), and very substantial for \(\nu\) close to \(\nu_0\).

In addition to the photoelectric threshold, of essential importance for the characterization of the photoelectric effect are the energy of the photoelectrons and the spectral distribution of the photoeffect, i.e., the dependence of the photocurrent on the frequency of the light. In this case it is important to determine the distribution function of electrons over velocities \(v\) or energies \(\varepsilon\), into which the frequency of the light \(\nu\) enters as a parameter: \(f(\varepsilon,\nu)\). In Fig. 1, as an example, the curve is given for the energy distribution of photoelectrons for aluminum,^8 illuminated by light of wavelength

\[ \lambda = 2536\ \text{\AA}. \]

Along the ordinate axis is plotted the number of photoelectrons \(n_{\phi}\) (in arbitrary units), and along the abscissa axis—the ratio of the electron energy to its maximum energy, \(E/E_{\max}\). These curves are little...

differ for different metals, but may be strongly deformed in the transition to thin films (for thicknesses \(d < 10^{-6}\) cm)\(^{8}\).

The curves of the dependence of photocurrent on light frequency have the form shown in Fig. 2. Here the ordinate gives the ratio of the photocurrent \(j\) to the intensity \(I\) of the light producing it (\(j/I\)—the sensitivity of the photocathode), and the abscissa gives the light frequency \(\nu\). Curve of type \(A\) in Fig. 2 corresponds to the “normal” photoelectric effect

Fig. 2. Schematic curves of the spectral distribution of photocurrent. \(A\)—“normal” photoelectric effect, \(B\)—photoelectric effect with spectral selectivity.

(a normal characteristic) and has a monotonically increasing character with increasing frequency. Curve \(B\) in Fig. 2 has a maximum in one or another frequency region and corresponds to the “selective” photoelectric effect\(^{9}\). In this case the selectivity may be simply spectral, when in some frequency region there is a maximum on the curve of Fig. 2 independent of the polarization of the light, and it may also be polarization selectivity, when this maximum is obtained only for a definite polarization of the light incident on the photocathode.

The sensitivity of the photocathode \(j/I\) can also be written in a somewhat different form, if one takes the quotient of the photocurrent divided by the electron charge, \(j/e = n_{\phi}\), i.e. the number of photoelectrons, and instead of the light intensity—its ratio to the quantum energy, \(I/h\nu = n_{\mathrm{kv}}\), i.e. the number of photons of the given frequency. Then we obtain a sensitivity expressed as the number of photoelectrons per one absorbed light quantum (the quantum yield of the photoelectric effect \(n_{\phi}/n_{\mathrm{kv}}\)). This quantity turns out to be very small. In the case of very pure metals the quantum yield is of the order of \(10^{-5}\)—\(10^{-3}\). This quantity is a function of the “distance” from the red limit, i.e. of the difference \(\nu - \nu_0\). According to the experimental data of Luk’yanov\(^{10}\), the quantum yield for the maximum of the spectral characteristic

of complex cathodes may reach a value of 0.3 electron per one absorbed light quantum.

We shall not dwell here on a review of the extensive experimental material, and refer the reader to monographs and surveys on the photoelectric effect[^11]. Below we shall set forth only the principal results of certain works on the quantum-mechanical theory of the photoelectric effect in metals.

§ 2. BASIC PREREQUISITES FOR CONSTRUCTING A QUANTUM THEORY OF THE PHOTOEFFECT AND THE SEMIPHENOMENOLOGICAL THEORY (ACCORDING TO FOWLER)

The theory of the photoeffect must explain the spectral characteristics of photocathodes, the law of distribution of photoelectrons over energies, the temperature dependence of the photocurrent and its dependence on the accelerating field, as well as the influence on the photocurrent of the state of the surface and of the nature of the metal, and of the form and dimensions of the photocathode. The elementary theory of Einstein[^6] could not solve all these problems. This proved to be within the power only of the modern quantum-mechanical theory of crystalline solids. In its most general form the problem is posed as follows: the system of conduction electrons in a crystal interacts with its ionic lattice and with one another; electromagnetic radiation of specified intensity, polarization, and direction falls from the vacuum onto the metal. It is required to calculate the electron current in the vacuum caused by the interaction of the electron system with the electromagnetic field of the light. For this purpose it is necessary to solve the wave equation of such a system, determine the wave function (which may also be nonstationary), and find the electron current by the general formulas of quantum mechanics. To take temperature effects into account, one must solve not a quantum-mechanical but a quantum-statistical problem, for which it is necessary to find not the wave function but the density matrix (statistical operator) and, with its aid, determine the photocurrent as a function of the temperature of the photocathode. However, a solution of both the first and, still more, the second of these general problems is practically impossible at the present time. Therefore one has to proceed by way of approximate solutions, using perturbation-theory methods and other approximations.

Already within the framework of the quantum model of free electrons one can obtain certain valuable indications for understanding the nature of the effect under study and its basic features in metals. From the point of view of the simplest model of a metal, the latter is a potential “box” (Fig. 3) with a potential-barrier height \(W_a = h\nu_a\). At \(0^\circ\mathrm{K}\) the electrons inside the “box,” by virtue of the Pauli principle, fill all the lower energy levels up to a certain maximum energy level \(\varepsilon_F = h\nu_\varphi\) (the limiting Fermi energy, or the chemical potential of the electron gas at \(0^\circ\mathrm{K}\)). From

it is seen from Fig. 3 that the difference between the height of the potential barrier at the metal boundary and the limiting Fermi energy is equal to the work function, \(W_a-\varepsilon_\phi=\chi\), i.e., to the minimum energy that must be expended in order to transfer an electron from the upper energy level in the metal into the vacuum. This quantity determines the red limit of the photoeffect: \(\chi=h\nu_0\) (see § 1). It is obvious that, strictly speaking, such a limit exists only at \(0^\circ\mathrm{K}\), when there is a definite value of the limiting energy \(\varepsilon_\phi\), above which there is not a single occupied electronic energy level. As the temperature is raised, in the Fermi distribution for the electrons there appears the so-called “Maxwellian tail” of thermally excited electrons. Therefore at \(T>0^\circ\mathrm{K}\) there arises a probability not only of the emission of photoelectrons, but also of spontaneous thermionic emission of electrons from the metal. However, owing to the strong degeneracy of the electron gas in typical metals, the “smearing” of the abrupt Fermi drop in the distribution of electrons over energies is very insignificant up to room temperatures and even higher \((100—1000^\circ\mathrm{K})\); therefore one may expect sufficient sharpness of the red limit also in the region of room temperatures. Its precise determination requires carrying out experiments at sufficiently low temperatures and applying a method of extrapolation of the spectral-distribution curves. This prediction of the theory is fully justified by experiment \(^{11}\).

Fig. 3. Potential barrier at the metal boundary, work function, and limiting Fermi energy.

It should be expected that, with increasing light frequency \(\nu\), the number of photoelectrons will at first increase at the expense of electrons occupying levels in the energy interval between \(W_a-h\nu\) and the limiting energy \(\varepsilon_\phi\) (Fig. 3). If the light frequency \(\nu\) becomes such that the quantum energy is greater than the barrier height, \(h\nu>W_a\), then the number of photoelectrons no longer has reason to increase (if one neglects the ionization of electrons not included among the conduction electrons by the incident light wave). Moreover, since with increasing frequency the probability of an elementary act of light absorption by electrons decreases, then, in passing through the critical frequency \(h\nu_a=W_a\), one may expect a maximum in the spectral curve of the photocurrent (selective effect). This is also observed experimentally, in particular especially for the alkali metals, which is evidently explained by the relatively small value of the potential \(W_a\) for these substances.

The large magnitude of the free path of electrons in a metal compels one to assume that the processes of collision of electrons

with lattice vibrations (phonons) should not have any appreciable influence on the magnitude of the photocurrent. The broad energy (velocity) interval of the photoelectrons is determined not by processes of electron collisions inside the crystal, but by the fact that, owing to the special properties of the Fermi distribution function, the conduction electrons in a metal, even at \(0^\circ\mathrm{K}\), are distributed over a broad energy interval (from 0 to \(\varepsilon_{\phi}\), which amounts to values \(\sim 10^{-12}\) erg \(\sim 0.1\) eV).

Within the framework of the free-electron model one can also carry out a more detailed quantitative calculation, which makes it possible, in particular, to give a fairly satisfactory explanation of a number of quantitative regularities of the photoeffect (Fowler \(^{12}\)). Let the surface of the metal coincide with the \(xy\) plane, and the normal to it with the \(z\) axis. Then the electron current through the surface is determined by the formula

\[ j=\int_{0}^{\infty} N(\varepsilon_{\perp})D(\varepsilon_{\perp})\,d\varepsilon_{\perp}, \tag{2.1} \]

where \(N(\varepsilon_{\perp})\) is the number of electrons whose normal component of the energy \(\varepsilon_{\perp}=mv_{\perp}^{2}/2\) lies in the interval \(d\varepsilon_{\perp}\) and which fall on the metal surface in 1 second, while \(D(\varepsilon_{\perp})\) is the probability that an electron with normal energy \(\varepsilon_{\perp}\) passes through the potential barrier. As Nordheim \(^{13}\) showed, the quantity \(N(\varepsilon_{\perp})\) is equal to*):

\[ N(\varepsilon_{\perp})=\frac{4\pi m}{h^{3}}kT\ln\left(1+e^{-\frac{(\varepsilon_{\perp}-\varepsilon_{\phi})}{kT}}\right). \tag{2.2} \]

*) Indeed \(^{13}\), from the expression for the Fermi distribution function it follows that the number of electrons whose component of momentum \(p_z\) lies in the interval \(dp_z\) is equal to

\[ n(p_z)\,dp_z=dp_z\,\frac{2}{h^{3}}\int_{-\infty}^{+\infty}\int \frac{dp_xdp_y}{ \exp\left(\frac{\frac{p_x^{2}+p_y^{2}+p_z^{2}}{2m}-\varepsilon_{\phi}}{kT}\right)+1 }. \]

Introduce new coordinates \(p_x=\rho\cos\varphi,\ p_y=\rho\sin\varphi\) and integrate over the angle \(\varphi\) from 0 to \(2\pi\), which gives

\[ n(p_z)=\frac{4\pi}{h^{3}}\int_{0}^{\infty} \frac{\rho\,d\rho}{ \exp\left(\frac{\frac{\rho^{2}+p_z^{2}}{2m}-\varepsilon_{\phi}}{kT}\right)+1 }. \]

Introduce the new integration variable \(\xi=\rho^{2}/2mkT\) and denote-

In particular, at absolute zero we obtain:

\[ \left. \begin{aligned} N(\varepsilon_\perp)&=\frac{4\pi m}{h^3}\,(\varepsilon_\phi-\varepsilon_\perp) &&\text{for } \varepsilon_\perp<\varepsilon_\phi,\\ N(\varepsilon_\perp)&=0 &&\text{for } \varepsilon_\perp>\varepsilon_\phi . \end{aligned} \right\} \tag{2.3} \]

The function (2.3) obviously has the form of a straight line.

To determine the probability \(D(\varepsilon_\perp)\), it is necessary to know the form of the potential barrier that confines the conduction electrons in the metal. For a given type of potential barrier, the probability can be determined by the usual quantum-mechanical methods.

When attempting to use formula (2.1) to calculate the photocurrent, we encounter a number of difficulties that arise from the need to determine the influence of the incident electromagnetic field on the distribution of conduction electrons over energies, and from the need to use the distribution function altered by this influence in determining the magnitude of the photocurrent. To solve this problem it is necessary: 1) to know the initial distribution of conduction electrons over energies in the absence of incident light; 2) to determine the probability of an elementary act of absorption of a photon of frequency \(\nu\) by an electron of energy \(\varepsilon\), as a result of which the electron acquires a definite normal component of the energy \(\varepsilon'_\perp\) (this probability, besides \(\nu\) and \(\varepsilon\), will depend on the intensity and polarization of the light and on the character of the potential field in the metal); 3) to find the distribution function of the electrons excited by the light with respect to the values of the normal energy at the metal surface, \(N'(\varepsilon'_\perp)\); 4) to determine the probability \(D'(\varepsilon'_\perp)\) that such electrons pass through the potential barrier. Multiplying the distribution function of the excited electrons by \(D'(\varepsilon'_\perp)\), we obtain an expression for the number of electrons, with a definite value of the normal energy, outside the metal, created by

denoting

\[ \beta=\frac{\varepsilon_\perp-\varepsilon_\phi}{kT}, \]

where \(p_z^2/2m=\varepsilon_\perp\), we then obtain

\[ n(p_z)\,dp_z=n(\varepsilon_\perp)\,d\varepsilon_\perp =\frac{4\pi m^{3/2}kT}{h^3\sqrt{2\varepsilon_\perp}} \int_0^\infty \frac{d\xi}{\exp(\xi+\beta)+1}\,d\varepsilon_\perp, \]

where

\[ \int_0^\infty \frac{d\xi}{\exp(\xi+\beta)+1} = \int_{e^\beta}^{\infty}\frac{dz}{z(z+1)} = \{\ln z-\ln(z+1)\}\Big|_{e^\beta}^{\infty} = \ln(1+e^{-\beta}), \]

whence formula (2.2) follows if the expression \(n(\varepsilon_\perp)\) is multiplied by the velocity

\[ v_z=\sqrt{\frac{2\varepsilon_\perp}{m}}, \]

which gives the flux of electrons per 1 sec.

for the given frequency of light, i.e., the distribution curve of the photoelectrons with respect to velocities \(N'(\varepsilon'_{\perp})D'(\varepsilon'_{\perp})\). Finally, integrating this expression over all possible values of the “normal” energies, we obtain the total photocurrent as a function of frequency and temperature \(j=f(\nu,T)\), i.e., the spectral characteristics of the effect at various temperatures.

In view of the great complexity of carrying out this exact calculation scheme, Fowler\(^{126}\) proposed a semiphenomenological solution for a special case of this problem. Namely, he confined himself to considering a comparatively narrow interval of frequencies near the “red threshold” (approximately from \(\nu_0\) to \(1.5\nu_0\)). The solution of this problem is undoubtedly also of practical interest, since experimenters and practitioners are most interested not so much in determining the complete curve of the spectral distribution as in the question of the exact behavior of this curve in a narrow frequency region near the red threshold (see, for example, the review\(^{11\text{г}}\), p. 82). If one restricts oneself to considering this narrow frequency region, it immediately follows that the photoelectrons have initial energies also in a narrow interval near the limiting Fermi energy \(\varepsilon_{\mathrm{f}}\). Since the distribution of these “thermally excited” electrons is strongly affected by temperature, it should be expected that the state of the photoelectrons in this frequency range will also depend strongly on \(T\). Therefore, in this case one can no longer assume that the surface of the photocathode is at \(0^\circ\mathrm{K}\). Taking advantage of the narrowness of the interval of initial energies, one may regard all quantities that do not depend very sharply on the electron energy as practically constant. Quantities depending on the frequency \(\nu\) to a small degree may also be regarded as constant, in comparison with quantities that depend on the frequency difference \((\nu-\nu_0)\).

If all these simplifications are taken into account, the calculation procedure is considerably facilitated. Namely, it is no longer necessary to determine the probability of the elementary act of absorption of a photon by an electron, since this probability may in the present case be taken as constant. Although the probability \(D(\varepsilon_{\perp})\), apparently, also depends on the difference \((\nu-\nu_0)\), Fowler assumes that \(D(\varepsilon_{\perp})=0\) for electrons with energy \(\varepsilon_{\perp}<W_a\) and is equal to unity for \(\varepsilon_{\perp}>W_a\). Thus one may use formula (2.2) for the number of electrons incident on \(1\ \mathrm{cm}^2\) of the metal boundary per 1 sec in the interval of “normal” energies from \(\varepsilon_{\mathrm{f}}\) to \(\varepsilon_{\mathrm{f}}+h\nu\). To determine the number of electrons that can leave the metal boundary, it is obviously necessary to integrate the quantity (2.2) over the limits from \(W_a-h\nu\) to \(\infty\). The choice of the lower limit is connected with the fact that light of frequency \(\nu\) as it were shifts the boundary energy by the amount \(h\nu\) and makes it equal to \(\varepsilon_{\mathrm{f}}+h\nu\), and therefore electrons appear with energy greater than the height of the potential barrier \(W_a\) (see Fig. 3). Instead of the probability of the elementary act of absorption of a photon by an electron, we introduce, following Fowler—

… a constant coefficient \(\alpha\), which shows how many times the value of the distribution function of electrons excited by light is smaller than this function for a normal conduction-electron gas. Thus, for the photocurrent we obtain:

\[ j=\alpha\,\frac{4\pi mekT}{h^3} \int_{W_a-h\nu}^{\infty} \ln\left(1+e^{\frac{\varepsilon_\phi-\varepsilon_\perp}{kT}}\right)d\varepsilon_\perp; \tag{2.4} \]

introducing the new variables

\[ \frac{\varepsilon_\phi-\varepsilon_\perp}{kT}=x', \qquad \frac{\varepsilon_\phi-(W_a-h\nu)}{kT}=x=\frac{h(\nu-\nu_0)}{kT} \]

and making the substitution \(d\varepsilon_\perp=-kT\,dx'\), we obtain:

\[ j=\alpha\,\frac{4\pi mek^2T}{h^3} \int_{-\infty}^{x}\ln(1+e^{x'})\,dx'. \tag{2.5} \]

The function

\[ f_i(x)=\int_{-\infty}^{x}\ln(1+e^{x'})\,dx' \]

can be found in tables or represented in the form of series:

\[ f_1(x)=e^x-\frac{e^{2x}}{2^2}+\frac{e^{3x}}{3^2}-\ldots \qquad \text{for } i=1 \quad x\leq 0, \tag{2.6} \]

\[ f_2(x)=\frac{x^2}{2}+\frac{\pi^2}{6}-e^{-x}+\frac{e^{-2x}}{2^2}-\frac{e^{-3x}}{3^2}+\ldots \]

\[ \text{for } i=2 \quad x\geq 0, \tag{2.7} \]

\[ f(0)=1-\frac{1}{2^2}+\frac{1}{3^2}-\frac{1}{4^2}+\ldots=\frac{\pi^2}{12} \qquad \text{for } x=0. \]

The equation of the spectral distribution can finally be written in the following form:

\[ j=\alpha A_0T^2 f_i(x) =\alpha A_0T^2 f_i\left[\frac{h}{kT}(\nu-\nu_0)\right]. \tag{2.8} \]

The universal constant

\[ A_0=\frac{4\pi mek^2}{h^3}=120\ \text{a}/\text{cm}^2\,\text{deg}^2. \]

It is easy to verify that for \(\nu=0\) and \(\alpha=1\) (in this case \(x=-\frac{h\nu_0}{kT}\ll 0\)), from formula (2.8) we immediately obtain Richardson’s equation for thermionic emission:

\[ j=A_0T^2e^{-\frac{h\nu_0}{kT}}. \]

Next, for the case \(T=0^\circ\mathrm{K}\), when \(x=\pm\infty\) for \(\nu>\nu_0\) or \(\nu<\nu_0\), respectively, we obtain:

\[ \begin{aligned} j&=0 &&(\nu<\nu_0),\\ j&=\frac{1}{2}\,\frac{\alpha A_0 h^2}{k^2}\,(\nu-\nu_0)^2 &&(\nu>\nu_0). \end{aligned} \tag{2.9} \]

The first of equations (2.9) gives the red boundary, while the second shows that, for frequencies close to it, the spectral characteristic has a quadratic character. At temperatures different from \(0^\circ\mathrm{K}\), it is no longer possible to speak of a red boundary, since for \(T>0\) and for \(\nu\leqslant\nu_0\), \(x\leqslant0\), the function \(f_i(x)\) is different from zero, to which it tends only asymptotically as \(x\to-\infty\). Therefore, for \(T>0\) a photocurrent exists not only for \(\nu>\nu_0\), but also for \(\nu\leqslant\nu_0\). However, one can preserve the notion of an effective red boundary also for \(T>0\), defining it as the frequency at \(x=0\) from Fowler’s equations. Fowler, and also DuBridge\(^{14}\), developed a very convenient graphical method that makes it possible to compare theory with experiment. We shall briefly discuss it in § 5. Readers wishing to become acquainted with this question in greater detail are referred to the appropriate literature (see, for example, \(^{11a-11г}\) and others). Let us also dwell on the question of the temperature dependence of the photocurrent. It follows from formula (2.8) that this dependence is determined to a large extent by the magnitude of the light frequency. For three special cases we obtain:

a) If \(\nu=\nu_0\) (i.e. \(x=0\)) and \(f(0)=1\), then

\[ j=\alpha A_0 T^2. \]

b) If \(\nu\gg\nu_0\), then \(x\gg0\), and by virtue of (2.7) we have:

\[ j=\alpha A_0T^2\left(\frac{x^2}{2}+\frac{\pi^2}{6}\right) =\frac{\alpha A_0}{2}\left[\frac{h^2(\nu-\nu_0)^2}{k^2}+\frac{\pi^2}{3}T^2\right]. \]

It should be noted that even for the frequency interval \((\nu-\nu_0)\) corresponding to a wavelength interval of \(\sim100\,\text{\AA}\), the first term in the right-hand side is much larger than the second, and therefore the photocurrent depends only very weakly on \(T\).

c) If \(\nu\ll\nu_0\), then in this case \(x\ll0\), and by virtue of (2.6) we have approximately:

\[ j=\alpha A_0T^2 e^{\frac{h(\nu-\nu_0)}{kT}}, \]

i.e. we obtain a very sharp dependence on temperature. If the constants \(\alpha\) and \(\nu_0\) in Fowler’s formulas are chosen by superposing the experimental and theoretical curves, then a sufficiently satisfactory agreement is obtained. This convinces us that the semiphenomenological theory presented, at least qualitatively, well reflects the real nature of the photoelectric effect in me-

metal. This theory can also be applied to the calculation of the energy distribution of photoelectrons, determined from volt-ampere curves for a retarding potential (see, for example, the review \(^{11\mathrm{r}}\), pp. 96–112). The further development of the theory will be discussed below, in §§ 3 and 4).

§ 3. FUNDAMENTALS OF THE QUANTUM-MECHANICAL THEORY OF THE PHOTOEFFECT IN METALS

Despite the great successes of the semiphenomenological theory of the photoeffect in metals, it cannot explain the magnitudes of a number of the most important parameters (for example, \(\nu_0\), \(\alpha\), etc.); it cannot be extended to the entire frequency interval and used to explain the dependence of the photocurrent on the intensity and polarization of light, to predict the character of the dependence of the photocurrent on the nature of the substance of the photocathode, on its structural state, and so forth. Modern nonrelativistic quantum mechanics, in principle, makes it possible to explain all the regularities of the photoeffect. However, the difficulties arising in the concrete solution of this problem, as already noted above, require the use of substantial simplifying assumptions and approximations.

In attempting to construct a consistent theory of the photoeffect, we must at once take into account one essential circumstance, namely that a free electron cannot completely absorb a light quantum, since in that case it is impossible simultaneously to satisfy the laws of conservation of energy and momentum. This is easily illustrated by the simplest example. Suppose that a free electron at rest has completely absorbed a photon with energy \(h\nu\) and momentum \(h\nu/c\). In this process the law of conservation of energy

\[ m_0 c^2\left[\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}-1\right]=h\nu \]

and the law of conservation of momentum

\[ \frac{m_0 v}{\sqrt{1-\frac{v^2}{c^2}}}=\frac{h\nu}{c}. \]

must be satisfied simultaneously.

Here \(c\) is the speed of light, \(m_0\) the rest mass of the electron, and \(v\) its velocity. It is easy to see that these equalities cannot be compatible for any velocity \(v\) less than the speed of light \(c\). Thus, for photoelectric absorption of light to be possible, the presence of a “third” body is necessary, which makes it possible to satisfy both conservation laws simultaneously. In other words, photoelectric absorption can take place only for

“bound” electron, i.e., either an electron in an atom or in some other system, for example, in a crystal. This conclusion can also be obtained if one considers the expression for the probability of an elementary act of absorption of a photon by a free electron. According to the laws of quantum mechanics this probability is determined, by the formulas of perturbation theory, by the square of the matrix element of the operator of the energy of interaction of light with the electron

\[ H_{\mathbf{k}\mathbf{k}'}=\int \psi_{\mathbf{k}'}^{*}(\mathbf{A}\nabla\psi_{\mathbf{k}})\,d\mathbf{r}, \tag{3.1} \]

where \(\mathbf{A}\) is the vector potential of the electromagnetic field, and \(\psi_{\mathbf{k}'}^{*}\) and \(\psi_{\mathbf{k}}\) are respectively the wave functions for the initial and final states of the electron, whose energies differ exactly by the energy of the light quantum \(h\nu\). In the case of a free electron the wave functions \(\psi_{\mathbf{k}'}^{*}\) and \(\psi_{\mathbf{k}}\) are given by the plane waves \(\exp[-2\pi i(\mathbf{k}'\cdot\mathbf{r})]\) and \(\exp[2\pi i(\mathbf{k}\cdot\mathbf{r})]\), and therefore the matrix element in (3.1), to within constant factors, is equal to the integral

\[ H_{\mathbf{k}\mathbf{k}'}\sim \int \exp 2\pi i\mathbf{r}(\mathbf{k}-\mathbf{k}')\,d\mathbf{r}, \tag{3.2} \]

which is exactly equal to zero for \(\mathbf{k}\ne\mathbf{k}'\) by virtue of the periodic character of the integrand. Thus, the probability of complete absorption of light by a free electron is indeed equal to zero. If, however, instead of the wave functions of a free electron, the functions of a bound electron are substituted in (3.1) and (3.2), a result different from zero is obtained, i.e., a finite probability of photoelectric absorption.

Wentzel attempted to overcome the difficulty with the violation of the conservation laws not by abandoning the model of free electrons, but by taking into account the damping of the light wave as a result of its absorption in the metal. In this case the vector potential of the electromagnetic field of the light wave will have an exponential factor \(e^{-\chi z}\), describing the attenuation of the wave in the direction of the normal \(z\) to the plane of the surface of the metal \(xy\). Therefore, instead of formula (3.2) we shall have another formula:

\[ H_{\mathbf{k}\mathbf{k}'}\sim \int \exp[2\pi i(k_z-k'_z)z-\chi z]\,dz, \]

which does not contain a purely periodic function under the integral sign, and therefore the matrix element, also for \(\mathbf{k}\ne\mathbf{k}'\), is in general different from zero. However, as I. E. Tamm and S. P. Shubin \(^{16}\) pointed out, Wentzel’s theory is not free from substantial objections, since in it, as in a microscopic theory, the probability of an elementary act of the photo-process is calculated with the aid of a macroscopically introduced absorption coefficient, which itself is essentially determined by elementary acts of photoabsorption. In addition, this

theory predicts that for thin layers of metal (transparent to light) the photoeffect should decrease sharply, which clearly contradicts experiments with thin metal films[^8]. This theory also gives incorrect predictions concerning the independence of the magnitude of the photocurrent from the polarization of light.

Freilich[^17], noting the difficulties with the treatment of the electromagnetic field inside the metal, proposed considering so thin a layer of metal that the absorption of light in it could be completely neglected. In doing so, Freilich assumes that the potential discontinuity at the boundary of the metal binds the electrons, and that taking this potential discontinuity into account should eliminate the difficulty with the laws of conservation of momentum.

Despite the importance of the works of Wentzel and Freilich, the beginning of a consistent quantum-mechanical theory of the external photoeffect in metals and in crystals should be recognized as the well-known work of I. E. Tamm and S. P. Shubin[^16]. These authors first of all pose the question: what cause makes possible, in general, the absorption of light quanta by the conduction electrons of a metal, and, in particular, the photoeffect? If we restrict ourselves to the case of visible and ultraviolet light, then Tamm and Shubin indicated two reasons for such absorption of light: 1) the presence of a potential discontinuity at the metal–vacuum boundary, owing to which the electron wave functions decay exponentially outside the metal (which Wentzel completely failed to take into account); 2) the coupling of the conduction electrons with the periodic field of the ionic lattice of the metal. Tamm and Shubin called the first effect surface, and the second volume. Both effects determine both the absorption of light and the emission of photoelectrons; however, for absorption the volume effect plays the decisive role, while for the emission of electrons—the surface effect. Indeed, in the surface layer of the metal (\(\sim 10^{-7}\) cm) only about one thousandth of the entire light energy is absorbed; at the same time, for every 10 quanta one photoelectron is emitted; thus the photoelectric yield is of the order of \(\sim 10^{-4}\). Therefore, although the surface absorption of light by a metal is comparatively very small in comparison with its total absorption in the volume of the metal and is insignificant when considering the optical properties of the metal, it is very important for the photoelectric effect. The remaining photons are absorbed inside the metal; they can eject electrons from the metal into vacuum only at high frequencies exceeding the “second red boundary” \(\nu'_0\), much larger than the first boundary \(\nu_0\), which is essential for the surface effect. As Tamm and Shubin showed, this follows from the selection rules for “optical” transitions in a crystal.

In calculating the surface part of the external photoeffect according to Tamm and Shubin, one may, in the first approximation, neglect

periodic potential of the metal lattice and to use the model of free electrons. The difficulty with satisfying the conservation laws is removed here by the fact that the jump of the potential at the boundary of the crystal and the exponential decay of the electron wave function in the vacuum make the electrons of the metal at the boundary “bound.” On the basis of these ideas, Tamm and Shubin used the simplest model for calculating the photocurrent of the surface part of the effect. The difficulty arising in this calculation is connected with the fact that we do not know the exact form of the potential barrier, nor the influence of image forces outside the metal. The shape of the barrier must have a substantial effect on the number of photoelectrons produced under the action of light in a thin surface layer. However, in a first approximation one may apparently assume that the shape of the barrier will not have a very substantial effect on the general character of the photoeffect. The barrier shape may have a substantial influence in the region of frequencies close to the red limit. The calculation of the surface effect by Tamm and Shubin, set out in detail in the work of D. I. Blokhintsev[^18], was then refined by Mitchell[^19]. Below we give an account of the latter work.

We shall regard the action of the electromagnetic field as a small perturbation in comparison with the height of the potential barrier and shall apply the method of perturbation theory. In Mitchell’s work it is shown that the latter can be applied both in stationary and in nonstationary form. Below a version of the calculation by the stationary method will be set out. The method of variation of constants was first used in the description of the photoeffect by Tamm and Shubin[^16] (in this review it will be used in § 6).

The wave equation of an electron moving in a potential field \(V\) and in the field of a light wave described by the vector potential \(\mathbf A\) has the form

\[ \frac{\hbar^{2}}{2m}\Delta u+\frac{\hbar}{i}\frac{\partial u}{\partial t}-Vu = -\frac{ie\hbar}{mc}(\mathbf A\nabla u). \tag{3.3} \]

Coinciding the plane surface of the metal, which fills the entire infinite half-space \(x\leq 0\), with the plane \(x=0\), we shall take for the dependence of the potential energy on \(x\) a form with a sharp jump at the boundary:

\[ \begin{aligned} V&=-\hbar\omega_a \qquad (x<0),\\ V&=0 \qquad\qquad (x>0). \end{aligned} \left\} \tag{3.4} \right. \]

The unperturbed stationary wave function of the electron in the absence of light \((\mathbf A=0)\) is given by the known expression

\[ u_0=u_k=\psi_k e^{\frac{iE_k t}{\hbar}}, \tag{3.5} \]

and equation (3.3) takes the form:

\[ \Delta \psi_k+\frac{2m}{\hbar^2}(E_k-V)\psi_k=0. \tag{3.6} \]

Since the potential-energy operator \(V\) depends only on the coordinate \(x\), the variables in (3.6) separate. Equation (3.6) should be solved for bound states with negative energy, i.e., for those for which

\[ E_k<0<E_k+\hbar\omega_a . \tag{3.7} \]

In addition, the usual condition of finiteness of the function \(\psi_k\) throughout all space must be satisfied, as well as the condition of continuity of this function and of its derivative \(\dfrac{\partial\psi_k}{\partial x}\) on the plane \(x=0\). Next, let us introduce three quantities \(k_x, k_y, k_z\) satisfying the equality

\[ \frac{2m}{\hbar^2}(E_k+\hbar\omega_a)=k_x^2+k_y^2+k_z^2, \tag{3.8} \]

and also introduce the notation

\[ p=(\mu\omega_a-k_x^2)^{1/2},\qquad \mu=\frac{2m}{\hbar}, \tag{3.9} \]

where \(p\) denotes the arithmetic (positive) value of the root. Under condition (3.7), the solutions of equation (3.6) will have the form:

\[ \begin{aligned} \psi_k&=a_k\left(e^{-ik_x x}+a_k e^{ik_x x}\right)e^{i(k_y y+k_z z)} \qquad (x<0),\\ \psi_k&=a_k b_k e^{-px}e^{i(k_y y+k_z z)} \qquad (x>0), \end{aligned} \tag{3.10} \]

where \(\alpha_k\) is an arbitrary constant, determined from the conditions of “matching” the solutions (3.10) on the plane \(x=0\):

\[ b_k=1+a_k;\qquad pb_k=ik_x(1-a_k). \tag{3.11} \]

It follows from (3.10) that the values \(k_x, k_y\), and \(k_z\) are real, and \(k_x\), moreover, is only positive.

Assuming further that the amplitude of the light wave is small, i.e., the condition

\[ \left|\frac{ie\hbar}{mc}(\mathbf A\nabla)u\right|\ll Vu \]

holds, one may apply perturbation theory to solve the full equation (3.3) and seek its solution in the form of the sum of the zero-order solution, determined by (3.5) and (3.10), and a small addition \(v\), i.e.,

\[ u=u_k+v. \tag{3.12} \]

Neglecting terms of second order of smallness and taking equation (3.6) into account, we find the equation for the first approximation of perturbation theory:

\[ \frac{\hbar^2}{2m}\Delta v+\frac{\hbar}{i}\frac{\partial v}{\partial t}-Vv =-\frac{ie\hbar}{mc}(\mathbf A\nabla u_k). \tag{3.13} \]

We assume that the vector potential has the form:

\[ \mathbf A=2\mathbf a\cos\omega\left\{t+\frac{x\cos\theta+y\sin\theta}{c}\right\}, \tag{3.14} \]

where \(\theta\) is the angle of incidence and \(\mathbf a\) is a constant vector. Replacing the cosine by exponential functions, it is easy to see that the right-hand side of (3.13), and consequently the function \(\vartheta\), split into two terms having time factors of the form \(\exp\left[i\frac{(E_k+\hbar\omega)t}{\hbar}\right]\). Omitting the radiation term with \(-\hbar\omega\), we find:

\[ \vartheta=\Phi(x,y,z)\exp\frac{i(E_k+\hbar\omega)t}{\hbar}, \tag{3.15} \]

where the function \(\Phi(x,y,z)\) satisfies the equation

\[ \Delta\Phi+\frac{2m}{\hbar^2}(E_k+\hbar\omega-V)\Phi= \]

\[ =-\frac{i2e}{c\hbar}(\mathbf a,\nabla\psi_k)\exp\left(i\omega\frac{x\cos\theta+y\sin\theta}{c}\right). \tag{3.16} \]

The right-hand side of expression (3.16) splits into the sum of three terms, containing respectively \(a_x, a_y, a_z\), which may be regarded as independent, since the angle of incidence \(\theta\) and the plane of polarization of the light wave can vary independently. Therefore the function \(\Phi\) may also be sought in the form of a superposition of three terms:

\[ \Phi=\lambda_x\Phi_x+\lambda_y\Phi_y+\lambda_z\Phi_z, \tag{3.17} \]

where \(\lambda_i\) are constants, and \(\Phi_i\) correspond to \(a_i\) \((i=x,y,z)\).

Let us first consider the determination of \(\Phi_x\). We introduce the abbreviated notation:

\[ \lambda_x=-\frac{2ie}{\hbar c}a_x\alpha_k,\quad q=(k_x^2+\mu\omega)^{1/2}; \]

\[ r=\{k_x^2+\mu(\omega-\omega_a)\}^{1/2}. \tag{3.18} \]

Here the roots are understood to mean their arithmetic values, and \(r\) is therefore assumed to be real. Thus, from (3.16) and (3.17) we obtain equations for \(\Phi_x\):

\[ \Delta\Phi_x+(q^2+k_y^2+k_z^2)\Phi_x =-ik_x\left[e^{-ik_xx}-a_ke^{ik_xx}\right]e^{i(k_yy+k_zz)}\times \]

\[ \times \exp\left(i\omega\frac{x\cos\theta+y\sin\theta}{c}\right)\quad (x<0), \tag{3.19} \]

\[ \Delta\Phi_x+(r^2+k_y^2+k_z^2)\Phi_x =-pb_ke^{-px}e^{i(k_yy+k_zz)}\times \]

\[ \times \exp\left(i\omega\frac{x\cos\theta+y\sin\theta}{c}\right)\quad (x>0). \tag{3.20} \]

In these expressions one may neglect the terms containing \(\omega/c\) in comparison with \(k_x\) and \(k_y\), since the order of magnitude of \(\omega/c\) for visible light is \(10^5\ \text{cm}^{-1}\), whereas the quantity \(k_x\) for electrons which lie near the Fermi surface and have the greatest value in the present discussion is equal to \(10^8\ \text{cm}^{-1}\). The particular integrals of equations (3.19) and (3.20) depend on \(y\) and \(z\) through the factor \(\exp[i(k_y y+k_z z)]\). In order that these integrals satisfy the “matching” condition at \(x=0\), it is necessary to represent them as a superposition of incident and reflected waves propagating along the \(x\)-axis. Namely:

\[ \Phi_x=\left\{c_x e^{iqx}-\frac{ik_x}{\mu\omega}\left(e^{-ik_xx}-a_k e^{ik_xx}\right)\right\}e^{i(k_y y+k_z z)} \quad (x<0), \tag{3.21} \]

\[ \Phi_x=\left\{b_x e^{-irx}-\frac{pb_k}{\mu\omega}e^{-px}\right\}e^{i(k_y y+k_z z)} \quad (x>0), \tag{3.22} \]

where \(b_x\) and \(c_x\) are constants determined from the conditions of continuity at \(x=0\):

\[ \left. \begin{aligned} c_x-\frac{ik_x}{\mu\omega}(1-a_k)&=b_x-\frac{pb_k}{\mu\omega},\\[4pt] iqc_x-\frac{k_x^2}{\mu\omega}(1+a_k)&=-irb_x+\frac{p^2b_k}{\mu\omega}. \end{aligned} \right\} \tag{3.23} \]

Eliminating \(c_x\) and using (3.11), we obtain:

\[ b_x=\frac{\omega a b_k}{i\omega(q+r)} =\frac{2k_x(p-ik_x)}{\mu\omega(q+r)}. \tag{3.24} \]

The functions \(\Phi_y\) and \(\Phi_z\) are determined in an analogous way. However, if the quantity \(\omega/c\) is again neglected, then in these cases the equations analogous to (3.19) and (3.20) will have in their right-hand sides the expressions \(\dfrac{\partial\psi_k}{\partial y}\) and \(\dfrac{\partial\psi_k}{\partial z}\) in both intervals \(x<0\) and \(x>0\). The particular integrals will then be the expressions \(\dfrac{1}{\mu\omega}\dfrac{\partial\psi_k}{\partial y}\) or \(\dfrac{1}{\mu\omega}\dfrac{\partial\psi_k}{\partial z}\); they and their first derivatives with respect to \(x\) will be continuous at \(x=0\). It is easy to verify that these solutions give nothing for the current. Taking into account the terms with \(\omega/c\) gives small corrections, in comparison with the current, caused by the function \(\Phi_x\).

Outside the metal \((x>0)\) the complete wave function, by virtue of (3.12), (3.10), (3.17), and (3.22), has the form:

\[ u= \left[ a_k b_k e^{-px} e^{\frac{iE_k t}{\hbar}} +\lambda_x\left(b_x e^{-irx}-\frac{pb_k}{\mu\omega}e^{-px}\right) e^{\frac{i(E_k+\hbar\omega)t}{\hbar}} \right]\times \]

\[ \times e^{i(k_y y+k_z z)}, \tag{3.25} \]

if the terms with \(\Phi_y\) and \(\Phi_z\), which contain the factor \(e^{-px}\), are omitted.

Substituting (3.25) into the usual quantum-mechanical expression for the current density

\[ j_x=\frac{e\hbar}{2mi}\left(u\frac{\partial u^*}{\partial x}-u^*\frac{\partial u}{\partial x}\right) \]

and noting that a nonzero result is obtained only from terms with \(e^{-irx}\), we find:

\[ j_x=\frac{e\hbar r}{m}|\lambda_x b_k|^2,\quad \text{if } k_x^2+\mu\omega>\mu\omega_a, \tag{3.26} \]

\[ j_x=0,\quad \text{if } k_x^2+\mu\omega\leq \mu\omega_a. \tag{3.27} \]

The total photocurrent according to (2.1) is found by summing expression (3.26) over all conduction electrons. For this purpose we multiply (3.26) by the quantity

\[ \frac{1}{8\pi^3} f(E)\,dk_x\,dk_y\,dk_z, \]

where \(f(E)\) is the Fermi distribution function, and integrate over all possible states for which \(r\) is a real quantity, i.e. \(k_x^2+\mu\omega>\mu\omega_a\). This gives

\[ J_x=\frac{e^3 a_x^2\omega_a}{\pi^3 m^2 c^2\omega^2} \iiint \frac{r k_x^2}{(q+r)^2} \frac{dk_x\,dk_y\,dk_z}{ 1+e^{\frac{\hbar^2(k^2-\mu\bar{\omega})}{2mkT}} }, \tag{3.28} \]

where \(\mu\bar{\omega}=(3\pi^2 n)^{2/3}\), and \(n\) is the density of conduction electrons. For frequencies not lying too close to the red boundary, it may be assumed that at ordinary temperatures the electron gas is completely degenerate and the electrons fill in \(k\)-space a sphere of radius \(k_0=(\mu\bar{\omega})^{1/2}\). The exponential function in the denominator will be equal to zero or infinity depending on whether the expression \(k_x^2+k_y^2+k_z^2-\mu\bar{\omega}\) is positive or negative. The variables \(k_x\) and \(k_y\) do not enter here into the integrand, and therefore integration over these coordinates is easily carried out by introducing polar coordinates \(dk_y\,dk_z=\rho\,d\rho\,d\varphi\) and integrating within the limits from \(0\) to \(2\pi\) in \(\varphi\) and from \(0\) to \((\mu\bar{\omega}-k_x^2)^{1/2}\) in \(\rho\). Using the second and third of equations (3.18), we find for the current (3.28):

\[ J_x= \frac{e^3 a_x\omega_a}{\pi^2 m^2 c^2\omega^2} \times \int_{0,[\mu(\omega_a-\omega)]^{1/2}}^{(\mu\bar{\omega})^{1/2}} -\frac{k_x^2(\mu\bar{\omega}-k_x^2)\{k_x^2+\mu(\omega-\omega_a)\}^{1/2}} {\left\{(k_x^2+\mu\omega)^{1/2}+[k_x^2+\mu(\omega-\omega_a)]^{1/2}\right\}^2} \,dk_x, \tag{3.29} \]

where the lower limit is equal to \(0\) for \(\omega>\omega_a\) and to \([\mu(\omega_a-\omega)]^{1/2}\) for \(\omega<\omega_a\). If the light is polarized in such a way that the vector of electric-field strength lies in the plane of incidence, then

\[ \mathbf{a}=|\mathbf{a}|(-\sin\theta,\cos\theta,0), \tag{3.30} \]

PHOTOEFFECT IN METALS

and, consequently, the flux of light energy incident on the surface of the metal will be equal to

\[ \frac{\omega^2}{2\pi c}\cos\theta\,|\mathbf a|^2 = \frac{\omega^2\cos\theta\,a_x^2}{2\pi c\sin^2\theta}. \tag{3.31} \]

Dividing expression (3.28) by (3.31), we obtain the sensitivity of the photocathode, expressed in electrons per \(1\ \mathrm{cm}^2\):

\[ P= \frac{2\pi e^3\omega_a\sin^2\theta}{m^2c\cos\theta}\times \]
\[ {}\times \iiint \frac{2}{8\pi^3}\, \frac{k_x^2}{\omega^4\left(k_x^2+\mu\omega\right)^{1/2}}\, \frac{ 4\left(k_x^2+\mu\omega\right)^{1/2} \left[k_x^2+\mu(\omega-\omega_a)\right]^{1/2} }{ \left\{ \left(k_x^2+\mu\omega\right)^{1/2} + \left[k_x^2+\mu(\omega-\omega_a)\right]^{1/2} \right\}^2 } \,f_0(E)\,d\mathbf k . \tag{3.32} \]

The first factor in the integrand is proportional to the probability of the elementary photoexcitation of a conduction electron, while the second is the transparency coefficient of the potential barrier. It should be recalled that this derivation has been obtained here under the assumption of a rectangular barrier. Makison\({}^{20}\) proved the validity of such a separation of factors in expression (3.32) also for a barrier of arbitrary shape. In the work of Hill\({}^{21}\) and others, the image-force field was also taken into account in the calculation.

Formula (3.32), like formulas (24.23) in Bethe and Sommerfeld\({}^{22}\), is not fully exact and, in particular, for grazing incidence of light \((\theta=\pi/2)\) leads to a divergent result. This shortcoming of the theory can be readily removed if one introduces a correction for reflection of light, assuming that the intensities of the transmitted and reflected light waves can be calculated from macroscopic electrodynamics, using optical constants, and that reflection takes place strictly at the metal surface.

Using the method of calculation set forth above and introducing the symbols \(a_i, a_r, a_t\) to denote the vector potentials, respectively, of the incident, reflected, and transmitted light (taking into account, of course, the exponential factor), instead of (3.23) we obtain:

\[ \left. \begin{aligned} c_x-\frac{ik_x}{\mu\omega}(1-a_k) &= b_x-\frac{pb_k}{\mu\omega}\, \frac{a_{ix}+a_{rx}}{a_{tx}}, \\ iqc_x-\frac{k_x^2}{\mu\omega}(1+a_k) &= -irb_x+\frac{p^2b_k}{\mu\omega}\, \frac{a_{ix}+a_{rx}}{a_{tx}} . \end{aligned} \right\} \tag{3.33} \]

Eliminating \(c_x\) from (3.33), we find:

\[ b_x= \frac{ \left(k_x^2-ipq\right)+p(p+iq)(a_{ix}+a_{rx})a_{tx} }{ i\mu\omega(q+r) }\,b_k, \tag{3.34} \]

where it has been taken into account that

\[ \lambda_x=-\,\frac{2ie}{\hbar c}\,a_{tx}a_k . \tag{3.35} \]

Using expression (3.34), we obtain for the photocurrent, instead of (3.29):

\[ J_x=\frac{e^3\omega_a}{\pi^3 m^2c^3\omega^2} \int_{0,\,[\mu(\omega_a-\omega)]^{1/2}}^{(\mu\omega)^{1/2}} \frac{k_x^2(\mu\omega-k_x^2)\,r}{(q+r)^2}\times \]

\[ \times \frac{\left|(k_x^2-ipq)+p(p+iq)(a_{ix}+a_{rx})\right|^2} {\mu^2\omega_a^2}\,dk_x . \tag{3.36} \]

This expression, for \(a_{ix}=a_{tx}=a_x,\ a_{rx}=0\), goes over into (3.29). If one takes into account the continuity conditions at the boundary of the metal \((x=0)\)

\[ \begin{aligned} a_{iy}+a_{ry}&=a_{ty},\\ a_{iz}+a_{rz}&=a_{tz}, \end{aligned} \qquad \tag{3.37} \]

then it is easy to show that the \(y\)- and \(z\)-components of the vector potential do not contribute to the current. Considering the case in which the electric-field vector lies in the plane of incidence, and using the continuity conditions for the tangential components of the electric and magnetic field intensities at the surface of the metal, we find the following expressions for the vector potentials:

\[ \left. \begin{aligned} \mathbf A_i&=(a_{ix},-a_{ix}\operatorname{ctg}\theta,0)\exp i\omega(x\cos\theta+y\sin\theta+ct)/c,\\ \mathbf A_r&=(a_{rx},a_{rx}\operatorname{ctg}\theta,0)\exp i\omega(-x\cos\theta+y\sin\theta+ct)/c,\\ \mathbf A_t&=\left(a_{tx},-a_{tx}\frac{n-i\chi}{\sin\theta},0\right) \exp i\omega\bigl((n-i\chi)x+y\sin\theta+ct\bigr)/c, \end{aligned} \right\} \tag{3.38} \]

where

\[ \left. \begin{aligned} \frac{a_{rx}}{a_{ix}} &= \frac{\sin^2\theta+(n-i\chi)^2-\dfrac{n-i\chi}{\cos\theta}} {\sin^2\theta+(n-i\chi)^2+\dfrac{n-i\chi}{\cos\theta}}, \\[6pt] \frac{a_{tx}}{a_{ix}} &= \frac{2} {\sin^2\theta+(n-i\chi)^2+\dfrac{n-i\chi}{\cos\theta}}, \end{aligned} \right\} \tag{3.39} \]

\(n\) is the refractive index and \(\chi\) the absorption coefficient. From equations (3.39) we find that the flux of light energy incident on the surface is equal to

\[ \frac{\omega^2|a_{tx}|^2}{8\pi c\cos\theta\,\sin^2\theta} \left\{(n^2+\chi^2+\sin^2\theta)\cos^2\theta+\right. \]

\[ \left. +\,2n\cos\theta\,(n^2+\chi^2+\sin^2\theta)+n^2+\chi^2\cos2\theta \right\}, \tag{3.40} \]

and the flux of absorbed energy

\[ \frac{\omega^{2}\,|a_{tx}|^{2}}{8\pi c\cos\theta\sin^{2}\theta}\, 4n\,(n^{2}+\chi^{2}+\sin^{2}\theta)\cos\theta . \tag{3.41} \]

Thus we find:

\[ -\frac{a_{ix}+a_{rx}}{a_{tx}}=\sin^{2}\theta+(n-i\chi)^{2}. \tag{3.42} \]

Dividing (3.36) by (3.40) and (3.41), we obtain an expression for the photoemission referred, respectively, to the unit incident or absorbed energy. The expressions obtained will be finite for all angles of incidence. The final formula is conveniently written if one introduces the abbreviated notation

\[ \omega_g=\omega_a-\overline{\omega},\qquad k_x^{2}=\mu\omega_g X^{2},\qquad \overline{\omega}=\varepsilon\omega_g,\qquad \omega=\eta\omega_g; \tag{3.43} \]

then finally, for emission per unit incident energy in coulombs per calorie (conversion factor \(1.395\cdot10^{-2}\)), we have the following expression:

\[ P=C\left[ \frac{ \sin^{2}\theta\cos\theta\left\{\left[(n^{2}+\chi^{2})^{2}-2(n^{2}-\chi^{2})\cos^{2}\theta+\cos^{4}\theta\right]\zeta_1(\eta)+ \right. }{ \omega_g\left\{(n^{2}+\chi^{2}+\sin^{2}\theta)^{2}\cos^{2}\theta+2n\cos\theta\times\right. } \right. \]
\[ \left. \left. \frac{ +\left[2(n^{2}-\chi^{2}-\cos^{2}\theta)+1\right]\zeta_2(\eta)+n\chi\zeta_3(\eta)+\zeta_4(\eta) }{ \times (n^{2}+\chi^{2}+\sin^{2}\theta)+n^{2}+\chi^{2}\cos^{2}2\theta \right\}} \right], \tag{3.44} \]

where

\[ C=\frac{32e^{3}}{\pi\hbar^{2}c}\,1.395\cdot10^{-2}=4.628\cdot10^{13} \tag{3.45} \]

and

\[ \zeta_1(\eta)= \frac{1+\varepsilon+\eta}{(1+\varepsilon)\eta^{4}}\times \]

\[ \times \int_{0,\,(1+\varepsilon-\eta)^{1/2}}^{\varepsilon^{1/2}} \frac{ X^{2}(\varepsilon-X^{2})(1+\varepsilon-X^{2})(X^{2}+\eta-1-\varepsilon)^{1/2} }{ \left\{(X^{2}+\eta)^{1/2}+(X^{2}+\eta-1-\varepsilon)^{1/2}\right\}^{2} }\,dX, \tag{3.46} \]

\[ \zeta_2(\eta)=\frac{1+\varepsilon}{1+\varepsilon+\eta}\,\zeta_1(\eta), \tag{3.47} \]

\[ \zeta_3(\eta)=\frac{4}{\eta^{4}} \int_{0,\,(1+\varepsilon-\eta)^{1/2}}^{\varepsilon^{1/2}} \times \]

\[ \times \frac{ X^{2}(\varepsilon-X^{2})(X^{2}+\eta-1-\varepsilon)^{1/2}(X^{2}+\eta)^{1/2}(1+\varepsilon-X^{2})^{1/2} }{ \left\{(X^{2}+\eta)^{1/2}+(X^{2}+\eta-1-\varepsilon)^{1/2}\right\}^{2} }\,dX, \tag{3.48} \]

\[ \zeta_4(\eta)=\frac{1}{\eta^{4}} \int_{0,\,(1+\varepsilon-\eta)^{1/2}}^{\varepsilon^{1/2}} \frac{ X^{4}(\varepsilon-X^{2})(X^{2}+\eta-1-\varepsilon)^{1/2} }{ \left\{(X^{2}+\eta)^{1/2}+(X^{2}+\eta-1-\varepsilon)^{1/2}\right\}^{2} }\,dX. \tag{3.49} \]

Formula (3.44) gives the dependence of photoemission on the optical constants of the metal, the angle of incidence of the light, the threshold frequency, and the dimensionless integrals (3.46)—(3.49).

§ 4. COMPARISON WITH EXPERIMENT AND FURTHER DEVELOPMENT OF THE THEORY

In proceeding to compare the theory of the photoelectric effect set out above with experiment, it is necessary to take into account the essential simplifications that were made in carrying out the calculations, and above all the simplification connected with the choice of the potential barrier. One may expect that the conditions at the surface of the metal are very sensitive to its structural state (roughness, the presence of adsorbed atoms, etc.). To date there are still no exhaustive experimental data obtained for a clean surface of a metallic crystal with minimal structural disturbances. Moreover, there is not a sufficient amount of data on the study of the dependence of the photocurrent on the state of polarization of the incident light.

Mitchell \(^{23}\) subsequently refined the formulas given above by taking into account the action of the periodic field of the lattice (by introducing the effective electron mass \(m^*\) into formula (3.28)) and replacing the quantities \(\overline{\omega}\) by \(\omega_0\) according to the formula \(\omega_0/\overline{\omega}=m/m^*\). The width of the energy band of the conduction electrons then became equal to \(\hbar\omega_0\) (instead of \(\hbar\overline{\omega}\)), and therefore the density of these electrons, for an unchanged total number of them, falling within a unit interval in quasimomentum space, changes by the factor

\[ \left(\frac{\overline{\omega}}{\omega_0}\right)^{3/2}. \]

Thus, the expression for the photocurrent must also be multiplied by this quantity.

In addition, Mitchell took account of the roughness of the surface of the crystal, assuming that the microscopic normal at each point of the surface is inclined to the mean normal and that the direction cosines of this “true” normal with respect to the axes \(x, y, z\) are \(l, m, n\). If the components of the vector potential are denoted by \(A_x, A_y\), and \(A_z\), then its component along the microscopic normal will be equal to \(lA_x+mA_y+nA_z\). Thus, the expression for the photocurrent emitted from a surface element of the metal \(dS\) will differ, for example, from (3.36) in that the quantity

\[ \left|a_{tx}(k^2-ipq)+(a_{ix}+a_{rx})p(p+iq)\right|^2 \]

must be replaced by the quantity

\[ \begin{aligned} \bigl|&(la_{tx}+ma_{ty}+na_{tz})(k^2-ipq)+p(p+iq)\times \\ &\times\{l(a_{ix}+a_{rx})+m(a_{iy}+a_{ry})+n(a_{iz}+a_{rz})\}\bigr|^2, \end{aligned} \tag{4.1} \]

\(\tilde{\omega}\) is replaced by the quantity \(\omega_0\), and the factor \(\left(\dfrac{\bar{\omega}}{\omega_0}\right)^{3/2}\) is added. After averaging over the directions \((l, m, n)\), the terms with the products \(lm\), \(mn\), \(nl\) drop out of (4.1) by virtue of symmetry; the tangential components of the current also drop out, and instead of (3.36) we obtain:

\[ J_x=-\frac{e^3\omega_a}{\pi m^2c^2\omega^2} \left(\frac{\bar{\omega}}{\omega_0}\right)^{3/2} \times \]

\[ \times \int_{0,\,[\mu(\omega_a-\omega)]^{1/2}}^{(\mu\omega_0)^{1/2}} dk_x\, \frac{ k_x^2(\mu\omega_0-k_x^2)\{k_x^2+\mu(\omega-\omega_a)\}^{1/2} }{ \left[(k_x^2+\mu\omega)^{1/2}+\{k_x^2+\mu(\omega-\omega_a)\}^{1/2}\right]^2 } \times \]

\[ \times \left[ \frac{ \cos^2\omega\,\left|(k_x^2-ipq)a_{tx}+p(p+iq)(a_{ix}+a_{rx})\right|^2 }{ \mu^2\omega_a^2 } + \frac{1}{2}\sin^2\tilde{\omega}\,\{|a_{ty}|^2+|a_{tz}|^2\} \right], \tag{4.2} \]

where \(\tilde{\omega}\) is a parameter characterizing the roughness of the surface.

We shall now obtain the photoemission \(P\) per unit incident energy for polarized light with the electric vector parallel \((E_{\parallel})\) or perpendicular \((E_{\perp})\) to the plane of incidence. We shall denote it respectively by \(P_{\parallel}\) and \(P_{\perp}\). The energy flux incident on the metal surface is given by formula (3.31). Using, in addition, formula (3.38), we find:

\[ P_{\parallel} = \frac{2e^3\omega_a\sin^2\theta}{m^2c\omega^4\cos\theta} \left(\frac{\bar{\omega}}{\omega_0}\right)^{3/2} \times \]

\[ \times \int_{0,\,[\mu(\omega_a-\omega)]^{1/2}}^{(\mu\omega_0)^{1/2}} dk_x\, \frac{ k_x^2(\mu\omega_0-k_x^2)\{k_x^2+\mu(\omega-\omega_a)\}^{1/2} }{ \left[(k_x^2+\mu\omega)^{1/2}+\{k_x^2+\mu(\omega-\omega_a)\}^{1/2}\right]^2 } \times \]

\[ \times \left[ \frac{\cos^2\tilde{\omega}}{\mu^2\omega_a^2} \left| (k_x^2-ipq)\frac{a_{tx}}{a_{ix}} +p(p+iq)\frac{a_{ix}+a_{rx}}{a_{ix}} \right|^2 + \frac{1}{2}\sin^2\tilde{\omega}\, \frac{n^2+x^2}{\sin^2\theta} \left| \frac{a_{tx}}{a_{ix}} \right|^2 \right], \tag{4.3} \]

\[ \omega_g=\omega_a-\omega_0,\qquad k_x^2=\mu\omega_gX^2,\qquad \omega_0=\varepsilon\omega_g,\qquad \omega=\eta\omega_g. \]

Using expressions (3.39), (3.42), and a somewhat modified for-

module (3.43) with the replacement of \(\bar{\omega}\) by \(\omega_0\), we obtain the generalization of formula (3.44):

\[ P_{\parallel} = \]

\[ = C\, \frac{ \varepsilon^{-3/2}\omega_g^{-1}\left(\dfrac{\bar{\omega}}{\omega_g}\right)^{3/2} }{ (n^2+\chi^2+\sin^2\theta)^3\cos^2\theta +2n(n^2+\chi^2+\sin^2\theta)\cos\theta +n^2+\chi^2\cos^2\theta } \times \]

\[ \times \left\{ \cos^2\tilde{\omega}\,\sin^2\theta\,\cos\theta \left[ (n^2+\chi^2)^2 -2(n^2-\chi^2)\cos^2\theta +\cos^4\theta \right]\zeta_1(\eta,\varepsilon) \right. \]

\[ \left. +\left[2(n^2-\chi^2-\cos^2\theta)+1\right]\zeta_2(\eta,\varepsilon) +n\chi\zeta_3(\eta,\varepsilon) +\zeta_4(\eta,\varepsilon) \right. \]

\[ \left. +\frac{1}{2}\sin^2\tilde{\omega}\,\cos\theta\,(n^2+\chi^2) \left[\zeta_2(\eta,\varepsilon)+\zeta_4(\eta,\varepsilon)\right] \right\}, \tag{4.4} \]

where the notation (3.45)—(3.49) has been used.

In the case of \(E_{\perp}\), only the \(z\)-components of the vector potential remain. The boundary conditions are then written in the form

\[ \left. \begin{aligned} \frac{a_{rz}}{a_{iz}}&= -\frac{(n-i\chi)-\cos\theta}{(n-i\chi)+\cos\theta},\\[4pt] \frac{a_{tz}}{a_{iz}}&= \frac{2\cos\theta}{(n-i\chi)+\cos\theta}, \end{aligned} \right\} \tag{4.5} \]

and for the photoemission \(P_{\perp}\) we obtain:

\[ P_{\perp} = C\,\frac{1}{\omega_g} \left(\frac{\bar{\omega}}{\omega_g}\right) \varepsilon^{-3/2} \{\zeta_2(\eta,\varepsilon)+\zeta_4(\eta,\varepsilon)\} \times \]

\[ \times \frac{\frac{1}{2}\sin^2\tilde{\omega}\cos\theta} {n^2+\chi^2+2n\cos\theta+\cos^2\theta}. \tag{4.6} \]

Let us first clarify the dependence of \(P_{\parallel}\) and \(P_{\perp}\) on the angle of incidence \(\theta\) at a constant frequency of light. For normal incidence of light \((\theta=0)\), \(P_{\perp}=P_{\parallel}\). As the angle increases, \(P_{\perp}\) decreases monotonically to zero. The emission \(P_{\parallel}\), on the contrary, first increases slowly, then passes through a maximum at \(\theta=70^\circ\), and then falls to zero at \(\theta=\pi/2\). These conclusions of the theory are in good qualitative agreement with experiment (see, for example, \(^{11a}\), pp. 131–132). Quantitative agreement depends to a considerable extent on the roughness of the surface (i.e., on the value of \(\tilde{\omega}\)), which is introduced into the theory in a purely phenomenological way. In order to obtain quantitatively acceptable results, it is necessary to assume a very large roughness \((\tilde{\omega}\sim45^\circ)\).

Mitchell \(^{23}\), as well as Tamm and Shubin \(^{16}\), compared the theoretical curve \(P(\nu)\) with experimental data. The theoretical curves (3.44) or (4.4) (but not 4.6!) have a maximum. However, it can be observed in the accessible part of the visible spectrum only for alkali metals, for which the work function is relatively small. Up to now there have been no exhaustive data from accurate and simultaneous measurements of the optical constants and threshold frequencies of these metals, carefully purified of gases and of lattice distortions at the surface. Therefore one can count only on qualitative agreement between theory and experiment. Taking for potassium the work function \(2.05\ \mathrm{ev}\) and the optical constants equal to \(n = 0.068,\ \varkappa = 1.5\), we obtain the theoretical curves for \(\theta = 60^\circ,\ \omega_g = \overline{\omega} = 5 \cdot 10^{14}\) and \(\omega = 45^\circ,\ \varepsilon = 0.4\) and \(\varepsilon = 0.6\). These curves are shown in Fig. 4, where the interval of curves obtained by Klauser \(^{24}\) for a clean sodium surface is also indicated. Although the theoretical and experimental curves are qualitatively similar and the order of magnitude of the effect is predicted correctly by the theory (at \(\varepsilon = 0.6 \cdot 10^{-4}\) coulomb/cal), the position of the maximum on the curve \(P(\nu)\) is given incorrectly by the theory. It should be noted that the shape of the curve may be changed by varying the optical constants. Consequently, the shape of the curve cannot be regarded as definitively determined until the exact spectral dependences of the optical constants are known. In this connection we draw attention to the works of Schiff and Thomas \(^{25}\) and of Makinson \(^{26}\), in which an attempt is made to solve jointly the quantum-mechanical problem of the optical properties and the photoelectric effect of metals. These authors emphasize that the assumption of an abrupt change in the magnitudes of the optical constants at the surface of a metal is a poor approximation even within the framework of the simplest model with a rectangular potential barrier, for the electron density falls at the boundary not immediately to zero, but decreases over distances of order \(3 \cdot 10^{-8}\ \mathrm{cm}\) from its maximum value inside the metal to a vanishingly small value outside it. Schiff and Thomas \(^{25}\) gave a quantum-mechanical calculation of metallic reflection. However, their calculations proved to be so complicated that it was impossible to carry out quantitative estimates and comparison with experiment. Therefore Makinson \(^{26}\) proposed a semiclassical theory of metallic reflection, which is very close in its results to the quantum theory, but at the same time much simpler than the latter. Makinson calculates the dependence of the electron density near a rectangular barrier on the distance along the normal to the metal surface. He then determines the field of the light wave in the region of the falloff of the electron density and the optical constants. In calculating the photocurrent Makinson draws attention to one inaccuracy in the calculations of Mitchell \(^{19,23}\), connected with the fact that the latter neglected in the right-hand

part of his perturbation-theory equation (3.13) by terms with \(\operatorname{div}\mathbf A\) and the scalar potential \(\Phi\). If one assumes that \(\Phi=0\) (as Mitchell and Schif and Thomas do), then \(\operatorname{div}\mathbf A \ne 0\) (for \(\operatorname{div}\mathbf A + \dfrac{1}{c}\dfrac{\partial \Phi}{\partial t}=0\)).

Fig. 4. Theoretical (solid) and experimental (dashed) curves of the spectral distribution of the photocurrent for sodium (unpolarized light). (According to the data of work \(^{33}\).)

However, Makinson’s corrections do not qualitatively change Mitchell’s results and do not improve the quantitative agreement of the theoretical and experimental curves of the spectral distribution of the photocurrent. Makinson points to the necessity of abandoning the simplifications associated with using a rectangular potential barrier.

Attempts to generalize the theoretical calculations of the photoeffect, connected with abandoning the simplified rectangular form of the potenti-

PHOTOEFFECT IN METALS

of the potential barrier, and also with allowance for the forces of electrical image interaction, have been made repeatedly\(^{27}\). Let us mention here the already cited works of Myers\(^{20a}\), Hill\(^{21}\), and Makinnon\(^{20b}\). However, in them, even in Makinnon’s most detailed work, no great progress was achieved in comparison with the simpler calculations in which a simple model of a rectangular barrier was used.

The question of the influence of the form of the potential barrier on the properties of the surface photoeffect in metals was analyzed in greatest detail in Buckingham’s work\(^{28}\). The latter took into account the dependence, indicated by Bardeen\(^{29}\), of the effective potential barrier at the metal boundary on the momentum of the electron incident on the boundary. This dependence is due to the exchange and correlation interaction of electrons inside the metal. Buckingham showed that allowance for this circumstance substantially lowers (approximately by a factor of 3) the absolute value of the photoemission yield for alkali metals predicted by the theory. However, the shape of the theoretical curves in this calculation poorly reflects the course of the experimental curves of the spectral distribution, especially near the red limit. An interesting point in Buckingham’s work is the author’s indication of new possibilities for the experimental study of the dependence of the transmission coefficient of the potential barrier for electrons participating in photo- or thermoemission on their energy, which can provide valuable information also on the form of the potential barrier (see \(^{28}\), § 5).

Mitchell\(^{23}\) also carried out a calculation of the energy distribution \(g(E)\) of photoelectrons knocked out of a metal. For this purpose, in formula (3.28) one must pass from the variables \(k_x, k_y, k_z\) to spherical coordinates:

\[ \left(\frac{2m}{\hbar^2}E\right)^{1/2}\cos\vartheta = \left[k_x^2+\mu(\omega-\omega_a)\right]^{1/2}, \]

\[ \left(\frac{2m}{\hbar^2}E\right)^{1/2}\sin\vartheta\cos\varphi = k_y,\qquad \left(\frac{2m}{\hbar^2}E\right)^{1/2}\sin\vartheta\sin\varphi = k_z, \tag{4.7} \]

where \(E\) is the total energy, equal to

\[ E=\frac{\hbar^2}{2m}\left(k_x^2+k_y^2+k_z^2\right)+\hbar(\omega-\omega_a). \tag{4.8} \]

Carrying out the integration over the angle \(\varphi\) from \(0\) to \(2\pi\) and passing from \(\vartheta\) to the new variable \(y=\cos\vartheta\), we find:

\[ g(E)\simeq \]

\[ \simeq \frac{E^{3/2}} {1+\exp\left[\dfrac{E-\hbar(\omega-\omega_g)}{kT}\right]} \int_{0}^{\left[\dfrac{\hbar(\omega-\omega_a)}{E}\right]^{1/2}} \frac{ y^2\left[Ey^2+\hbar(\omega_a-\omega)\right]^{1/2}\,dy }{ \left[E^{1/2}y+\left(Ey^2+\hbar\omega_a\right)^{1/2}\right]^2 }, \tag{4.9} \]

where the lower limit is equal to 0 if \(\hbar\omega<\hbar\omega_a\), or

\[ \left[\frac{\hbar(\omega-\omega_a)}{E}\right]^{1/2}, \]

if \(\hbar\omega>\hbar\omega_a\). Function (4.9) is shown in Fig. 5a for the case of sodium \((\omega_g=2\pi\cdot 5\cdot 10^{14},\ \omega_a=2\pi\cdot 10^{15},\ T=300^\circ\mathrm{K})\). In Fig. 5b are given the data of Brady’s experiments\(^{30}\), carried out with a well-degassed photocathode. Between theory and experiment there is also, in the present case, good qualitative agreement. The most probable energy of the emitted electrons on these curves lies very close to the maximum. Mitchell noted that one can compute the energy distribution of photoelectrons obtained in experiments,

Fig. 5a. Theoretical curve of the distribution of photoelectrons by total energy. (According to Mitchell’s calculation\(^{23}\).)

Fig. 5b. Experimental curve of the distribution of photoelectrons by total energy for potassium. (According to the data of work\(^{30}\).)

from the magnetic deflection of photoelectrons, and also from the values of the energy associated with the normal component of the velocity to the surface of the metal. In both cases one obtains not bad agreement with experiment (see\(^{23}\), §§ 4.3 and 4.4).

As already mentioned above, the quantum theory of the photoelectric effect was generalized also to the case of temperatures different from \(0^\circ\mathrm{K}\). The first such calculation was carried out by D. I. Blokhintsev\(^{18}\), who generalized the calculation of Tamm and Shubin\(^{15}\), and a more general calculation was given by Mitchell\(^{31}\). Mitchell additionally took into account the forces of the electric image (the smoothed form of the potential barrier), and also used the method of variation of constants. The final expression for the photocurrent near the threshold frequency \(\omega_0\) has the form:

\[ J_x \simeq \left(\frac{kT}{\hbar}\right)^2 \int_0^\infty dy \ln \left[1+\exp\left\{-y+\frac{\hbar(\omega-\omega_g)}{kT}\right\}\right]. \tag{4.10} \]

This result may be regarded as a theoretical substantiation of Fowler’s phenomenological formula (2.4)\(^{12}\). Later the scheme for calculations of temperature effects was developed by Houston\(^{32}\).

PHOTOEFFECT IN METALS

The generalization of the theory of Tamm and Shubin \(^{16}\) to the case of the photoeffect in metals in the presence of an external constant electric field was given by Savel’ev \(^{33}\). The presence of a “cutoff” of the potential barrier, caused by the action of the accelerating field, leads to the fact that even at \(0^\circ\text{K}\) there appears the fundamental possibility of exciting photoelectrons at an arbitrarily small frequency of light (absence of a red limit). For frequencies \(\omega < \omega_a\) Savel’ev obtained a formula similar to the formula for emission in the cold extraction of electrons by a field (Fowler and Nordheim \(^{34}\)). For frequencies \(\omega > \omega_a\) the formula for the photocurrent changes very little in comparison with the formula calculated without an accelerating field. Savel’ev also took into account the influence of electric image forces. The value predicted by him for the displacement of the effective red limit,

\[ \delta \omega = \frac{e\sqrt{F}}{\hbar} \]

toward the red side of the spectrum agrees quite well with the experimental data of Lawrence and Linford \(^{35}\). In the work of Guth and Mullin \(^{36}\) the derivation of Savel’ev’s formula was repeated (without reference to him), and a generalization was carried out for the case of temperatures above \(0^\circ\text{K}\).

Recently, Makinson and Buckingham \(^{37}\) calculated the probability of two-photon absorption and the photocurrent of second order. Although the intensity of this current is very small, this effect may in some cases be significant.

Quite special methods of calculation are used to compute the photoelectric effect with sensitized photoelectrically active cathodes. Here we shall confine ourselves only to indicating some of the principal works on the theory of this effect \(^{38–42}\).

A theoretical investigation of the volume photoeffect in an elementary semiclassical treatment was also given by Tamm and Shubin \(^{16}\). They made an approximate estimate of this effect and found that in the region of frequencies close to the usual red limit, and even considerably above it, the photocurrent due to absorption of light inside the metal is much smaller than the photocurrent from the surface effect. Only at frequencies above the “second red limit” \(\omega_{\nu\mathrm{ob}}\) does the volume effect become noticeable. And since in this frequency region the surface effect, on the contrary, decreases, the volume effect practically determines there the entire magnitude of the photocurrent. Tamm and Shubin compared their calculation with the data of the experiments of Shurman and Teissing \(^{43}\) and obtained satisfactory qualitative agreement for the alkali metals.

A more detailed calculation of the volume photoeffect was carried out much later by Fan \(^{44}\). Fan used the one-electron quantum theory of the metal, took into account the interference conditions for the quasi-momenta of the electrons, and obtained a quantitative expression for the spectral distribution of photoelectrons. He also took into account the effect of reflection of light from the surface of the photocathode. From the spectral reflection curve it proved possible to determine the effective second

red boundary, which in the case of sodium and potassium turned out to be respectively equal to \(\nu_{0 \text{ vol}} = 5.91 \cdot 10^{14}\) and \(\nu_{0 \text{ vol}} = 5.69 \cdot 10^{14}\). The red boundary of the surface effect for these metals is respectively equal to \(\nu_0 = 5.53 \cdot 10^{14}\ \mathrm{sec}^{-1}\) and \(\nu_0 = 4.84 \cdot 10^{14}\ \mathrm{sec}^{-1}\). An estimate of the magnitude of the photoemission yield due to the volume effect, for frequencies greater than the second red boundary, for these metals proves to be of the same order of magnitude as for the surface effect, and agrees with the experimental data. Cashman and Basso\(^{45}\) measured the photoeffect in barium and observed a volume effect. However, in this metal, which has a larger work function than the alkali metals, the maxima on the spectral-distribution curve of the photoelectrons overlap (the two red boundaries are comparatively close). Therefore for barium it is no longer possible to consider separately the surface and volume photoeffects; one must also abandon the free-electron model (see § 7).

§ 5. THE SURFACE PHOTOEFFECT IN ORDERING ALLOYS

It is known that all “anomalies” of various physical properties of ordering metallic alloys are due to the dependence of these properties on the degree of long-range order of atoms of the different components of the alloy over the various sites of the crystal lattice\(^{46}\). Therefore there is reason to expect that an anomaly of the photoelectric current should also be observed, associated with sharp changes in the degree of long-range order near the Curie point of ordering.

To generalize the semi-phenomenological theory of the photoeffect to the case of ordering alloys, Sokolov\(^{47}\) used the theory, proposed by A. A. Smirnov\(^{48}\), of the motion of an electron in the crystal lattice of a binary metallic alloy with arbitrary composition and degree of long-range order. In particular, for the energy of an electron in the body-centered cubic lattice of an alloy, Smirnov gave the following expression:

\[ E(\eta)=w+\varepsilon^{*}+\left(c-\frac{1}{2}\right)\Omega \pm \left[(q-c)^2\Omega^2\eta^2+64\gamma^2\cos^2\frac{k_x}{2}\cos^2\frac{k_y}{2}\cos^2\frac{k_z}{2}\right]^{1/2}, \tag{5.1} \]

where \(k_x, k_y, k_z\) are the components of the electron quasi-momentum, \(c\) is the relative concentration of one of the components of the alloy, \(\eta\) is the degree of long-range order; \(w, \varepsilon^{*}, \Omega, \gamma\) are constants independent of \(c\) and \(\eta\); the parameter \(q=2c\) for \(c \leq 1/2\); \(q=1\) for \(c \geq 1/2\).

Using the general expression for the effective mass

\[ m_{\mathrm{eff}}=\frac{\hbar^2}{\dfrac{\partial^2 E}{\partial k^2}}, \]

one can determine it for the upper, almost filled po-

the electron energy bands in the crystal:

\[ m_{\mathrm{eff}}=\frac{\hbar^{2}}{16\gamma^{2}a^{2}}\left[(q-c)^{2}\Omega^{2}\eta^{2}+64\gamma^{2}\right]^{1/2}, \tag{5.2} \]

where \(a\) is the lattice constant. The number of valence electrons per unit volume possessing a component of the quasi-momentum perpendicular to the surface of the alloy sample is determined by the well-known formula of type (2.2), which in the present case, in accordance with formulas (5.1) and (5.2), takes the form

\[ n(k_x)\,dk_x=\frac{dk_x}{8\pi^{2}a^{3}} \int_{0}^{\infty}\int_{0}^{2\pi} \frac{\rho\,d\rho\,d\vartheta} {\exp\left\{\dfrac{(\gamma-\alpha\eta^{2})(k_x^{2}+\rho^{2})-\varepsilon_{0}}{kT}\right\}+1}, \tag{5.3} \]

where

\[ \alpha=\frac{(q-c)^{2}\Omega^{2}}{128\gamma^{3}},\qquad \rho^{2}=k_y^{2}+k_z^{2}, \]

\(\vartheta\) is the angle with respect to the axis perpendicular to the surface of the alloy.

Assuming that the potential jump of the alloy at the boundary with the vacuum depends on the degree of long-range order, and expanding it in a series in this parameter, we obtain:

\[ W(\eta)=W(0)+\left(\frac{\partial W}{\partial \eta}\right)_0\eta +\frac{1}{2}\left(\frac{\partial^{2}W}{\partial \eta^{2}}\right)_0\eta^{2}+\cdots . \tag{5.4} \]

Below we shall be interested in the region of temperatures close to the Curie point. In this temperature region the degree of long-range order may be regarded as small \((\eta\ll 1)\), and in all calculations powers of \(\eta\) higher than the second may be neglected. From the equivalence of the sites of the crystal lattice, it is not difficult to show that the linear terms in expression (2.2) must be absent. Indeed, upon replacing \(\eta\) by \(-\eta\), expression (5.4) takes the form:

\[ W(-\eta)=W(0)-\left(\frac{\partial W}{\partial \eta}\right)_0\eta +\frac{1}{2}\left(\frac{\partial^{2}W}{\partial \eta^{2}}\right)_0\eta^{2}+\cdots . \tag{5.5} \]

Furthermore we have \(W(\eta)=W(-\eta)\), since by virtue of the equivalence of the sites of the crystal lattice the physical properties of the alloy must not change when \(\eta\) is replaced by \(-\eta\). Comparing (5.4) and (5.5), we obtain

\[ \left(\frac{\partial W}{\partial \eta}\right)_0 \eta=0, \]

and since \(\eta\ne 0\), then

\[ \left(\frac{\partial W}{\partial \eta}\right)_0=0. \]

Further, on the basis of the general theory of second-order phase transitions\(^{49}\), one may conclude that below the Curie point

\[ \left(\frac{\partial^{2}W}{\partial \eta^{2}}\right)_0<0. \]

Consequently, if we introduce the notation

\[ \frac{1}{2}\left(\frac{\partial^{2}W}{\partial \eta^{2}}\right)_0=-W_0 f, \]

then the magnitude of the potential jump of the ordering alloy takes the form:

\[ W(\eta)=W_0(1-f\eta^{2}). \tag{5.6} \]

It is clear that the effective work function \(\chi(\eta)\) is also a quadratic function of the long-range order parameter and is expressed as follows:

\[ \chi(\eta)=W(\eta)-\varepsilon(\eta)=\chi_0-\delta\eta^2, \tag{5.7} \]

where \(\delta=W_0 f-\alpha\varepsilon_0\). Using this formula, it is easy to determine the “ordering anomaly” of the alloy’s effective work function:

\[ \frac{\Delta\chi(\eta)}{\chi_0}=\frac{\chi_0-\chi}{\chi_0}=\frac{\delta}{\chi_0}\eta^2. \tag{5.8} \]

Let us determine the number of electrons torn from the surface of an ordering alloy at temperature \(T\) under the action of light of frequency \(\nu\), close to the threshold frequency \(\nu_0\). We shall assume that it is proportional to the number of electrons per unit volume of the metallic alloy that have a component of the quasimomentum perpendicular to the surface exceeding the critical value, i.e.

\[ (\gamma-\alpha\eta^2)k_x^2+h\nu=W(\eta). \]

This number of photoelectrons, which we shall denote by \(N_\phi(\eta)\), is determined by the expression

\[ N_\phi(\eta)= \int\limits_{(\gamma-\alpha\eta^2)k_x^2=W(\eta)-h\nu}^{\infty} n(k_x)\,dk_x. \tag{5.9} \]

To compute \(N_\phi(\eta)\), substitute (5.3) into (5.9) and introduce the variable \(z\) according to the following relation:

\[ (\gamma-\alpha\eta^2)k_x^2-W(\eta)+h\nu=kTz. \tag{5.10} \]

As a result we obtain:

\[ N_\phi(\eta)= \frac{(kT)^{3/2}}{16\pi^2 a^3(\gamma-\alpha\eta^2)^{1/2}} \int\limits_0^\infty \frac{\ln\left[1+\exp\left\{-z+\frac{h\nu-\chi(\eta)}{kT}\right\}\right]} {\left(z+\frac{W(\eta)-h\nu}{kT}\right)^{1/2}} \,dz. \tag{5.11} \]

Since we are considering the value of \(h\nu\) near the photoeffect threshold \([h\nu\sim\chi(\eta)]\), the quantity \(h\nu-\chi(\eta)\) is comparable with \(kT\), i.e. with the width of the “Maxwellian tail” at the Fermi drop, whereas the quantity \(|W(\eta)-h\nu|\) is always considerably greater than \(kT\), namely of the order of the limiting energy of the alloy electrons. Therefore, in a sufficiently good approximation one may neglect the quantity \(z\) in the expression

\[ \left(z+\frac{W(\eta)-h\nu}{kT}\right)^{1/2}. \]

In fact, this can be done because for large \(z\) the numerator of the integrand (5.11) tends to zero as \(e^{-z}\). Then we shall have:

\[ N_{\phi}(\eta)= \frac{k^{2}T^{2}}{16\pi^{2}a^{3}(\gamma-\alpha\eta^{2})^{3/2}(W(\eta)-h\nu)^{1/2}} \times \]

\[ \times \int_{0}^{\infty}\ln\left[1+\exp\left\{-z+\frac{h\nu-\chi(\eta)}{kT}\right\}\right]\,dz. \tag{5.12} \]

As a result of integrating (5.12), we obtain:

\[ N_{\phi}(\eta)= \frac{k^{2}T^{2}}{(\gamma-\alpha\eta^{3})^{1/2}(W-h\nu)} \Phi_i[x(\eta)], \tag{5.13} \]

where \(i=1\) for \(x(\eta)=\dfrac{h\nu-\chi(\eta)}{kT}<0\), and \(i=2\) for \(x(\eta)>0\); here

\[ \Phi_1[x(\eta)]=\frac{1}{16\pi^{2}a^{2}}\,f_1[x(\eta)], \]

and

\[ \Phi_2[x(\eta)]=\frac{1}{16\pi^{2}a^{2}}\,f_2[x(\eta)], \]

while \(f_1\) and \(f_2\) are defined by formulas (2.6) and (2.7).

Substituting (5.6) into (5.13) and separating out the term with \(\eta^2\), we obtain:

\[ N_{\phi}(\eta)= \frac{2k^{2}T^{2}}{\gamma^{3/2}(W_0-h\nu)^{1/2}} \Phi_i[x(\eta)](1-\Gamma\eta^{2}), \tag{5.14} \]

where

\[ \Gamma=\frac{W_0 f}{2(W_0-h\nu)}-\frac{3}{2}\alpha. \tag{5.15} \]

Since the photoemission current is proportional to \(N_{\phi}(\eta)\), one may write:

\[ \frac{j}{T^{2}(1-\Gamma\eta^{2})} = A(W_0-h\nu)^{-1/2}\Phi_i[x(\eta)], \tag{5.16} \]

where \(A\) is a constant independent of \(\nu\) and \(T\).

In formula (5.16) we are interested not in the absolute value of the photoelectric current \(j\), but in its “anomaly,” corresponding to the transition from the disordered state to the ordered state (the “ordering anomaly”), i.e.,

\[ \frac{\Delta j}{j_0}=\frac{j_0-j}{j_0}=D\eta^{2}, \tag{5.17} \]

where \(j_0\) is the value of \(j\) at \(\eta=0\), and

\[ D=\left(\Gamma-\frac{\Phi'_i[x(0)]\,W_0 f}{\Phi_i[x(0)]\,kT}\right). \tag{5.18} \]

The coefficient \(D\) is a function of frequency. Therefore, a change in the frequency of the light producing the photoeffect entails a change in the coefficient \(D\) itself (which should correspond in experiment to different photocurrent curves).

It would be highly desirable to carry out an experimental verification of the theoretical formula (5.17) as applied to ordering alloys. Here the curves of the dependence of the photoelectric current on temperature should have a characteristic break at the Curie point, below which there should be observed the decrease of the curve so characteristic of ordering alloys (compare, for example, with the temperature dependence of the electrical resistance for \(\mathrm{Cu}_3\mathrm{Au}^{46}\)).

In expression (5.17), \(\Delta j\) denotes the difference between the photoelectric current obtained by extrapolating the photocurrent curve above the Curie point, \(j_0\), and the photocurrent actually observed below the Curie point.

It is known that for frequencies close to the threshold frequency \(\nu_0\), the quantity \((W-h\nu)^{-1/2}\) is practically constant and can be combined with the coefficient \(A\). Taking the logarithm of (5.16), we obtain:

\[ \ln\left[\frac{j}{T^2}\left(1+\Gamma\eta^2\right)\right] = B+\ln \Phi_i[x(\eta)] = B+\ln \Phi_i\left[\frac{h(\nu-\nu_0)}{kT}\right], \tag{5.19} \]

where \(B\) is a new constant independent of temperature and \(\nu\).

Formula (5.19) differs from the corresponding expression of work\({}^{12}\) in that, on the left-hand side under the logarithm sign, there is the factor \((1+\Gamma\eta^2)\), while on the right-hand side \(\nu_0\) is a function of \(\eta\). To determine the true photoelectric threshold of an ordering alloy (which is a function of the degree of long-range order), one should plot the theoretical curve (the right-hand side of formula (5.19)) as a function of \(x\). If the experimental photocurrent values \(j\) are now plotted as a function of \(\ln\left[\dfrac{j}{T^2}(1+\Gamma\eta^2)\right]\) versus \(h\nu/kT\), and the experimental curve is shifted horizontally by \(\chi(\eta)/kT\) and vertically by the amount \(B\), then coincidence with the theoretical curve is obtained. The horizontal displacement gives the threshold frequency \(\nu_0\) or \(\chi/h\), since \(h\nu_0/kT\) is the point on the \(x\)-axis of the experimental curve corresponding to the point \(x=0\) on the abscissa axis of the theoretical curve.

The calculations presented above do not claim to provide a quantitative description of the photoeffect in ordering alloys, since they were carried out within the framework of a crudely simplified model of a metal. Nevertheless, there apparently is no reason to doubt that the principal qualitative conclusions concerning the nature of the influence of order on the photoelectric effect correspond to real metallic alloys.

§ 6. Photoelectric Effect in Ferromagnetic Metals

A. Elementary Theory

Experimental investigations of the photoelectric and thermionic properties of ferromagnetic nickel, carried out by Cardwell\({}^{50}\), showed that: 1) in nickel, near the Curie point, an anomalous course of the photoelectric current with temperature is observed (a break in the photocurrent–temperature curve); this anomaly cannot be explained by the existing theory\({}^{12}\); 2) the work of photoelectric emission increases with temperature.

It is known\({}^{51}\) that all “anomalies” of ferromagnetic metals are due to the existence of spontaneous magnetization. Therefore one should expect that the photocurrent anomaly in ferromagnets is also connected with the disappearance of spontaneous magnetization on passing through the Curie point. In work\({}^{52}\) an attempt was made to give a theoretical explanation of this anomaly.

In the case of ferromagnetic metals one may use the model of exchange interaction between the outer \(s\)- and inner \(d\)-electrons\({}^{53}\). According to this model, we regard the system of \(s\)-electrons in a ferromagnet as a mixture of two electron “gases,” corresponding to the two possible orientations of spin. The energy of an \(s\)-electron in the effective-mass approximation is equal to

\[ E(k_x, k_y, k_z)=\alpha-\alpha' y\sigma+(\beta+\beta' y\sigma)(k_x^2+k_y^2+k_z^2), \]

where \(\alpha,\alpha',\beta\), and \(\beta'\) are parameters depending on the exchange integrals of \(s\)- and \(d\)-electrons and on the transfer integrals of the \(s\)-electron, \(y\) is the mean relative atomic magnetic moment of the \(d\)-electron, and \(\sigma\) is the spin vector of the \(s\)-electron. The number of \(s\)-electrons with right spin orientation (\(s^{+}\)) per unit volume, having a component of quasimomentum perpendicular to the metal surface lying in the interval \(k_x^{+}, k_x^{+}+dk_x^{+}\) (for arbitrary values of the other two components \(k_y^{+}\) and \(k_z^{+}\)), is determined by the expression

\[ n(k_x^{+})\,dk_x^{+} = \frac{dk_x^{+}}{8\pi^{3}a^{3}} \times \]

\[ \times \int_{0}^{\infty}\int_{0}^{2\pi} \left\{ \exp\left[ \alpha-\alpha'y+(\beta+\beta'y)\bigl(k_x^{+2}+\rho^{2}\bigr)-\varepsilon_{0}^{+} \right]+1 \right\}^{-1} \rho\,d\rho\,d\vartheta = \]

\[ = \frac{kT}{8\pi^{3}a^{3}(\beta+\beta'y)} \times \]

\[ \times \ln\left[ 1+\exp\left( \frac{\varepsilon_{0}^{+}-\left[(\alpha-\alpha'y)+(\beta+\beta'y)k_x^{+}\right]} {kT} \right) \right]\,dk_x^{+}, \tag{6.1} \]

where \(\rho^{2}=k_x^{2}+k_y^{2}\), \(\vartheta\) is the azimuth about the axis perpendicular to the surface of the metal, and \(\varepsilon_{0}^{+}\) is the chemical potential of the \(s^{+}\)-electrons.

Let us denote the potential jump for \(s^{+}\)-electrons by \(W^{+}\); then their effective work function will be \(\chi^{+}=W^{+}-\varepsilon_{0}^{+}\). Similarly, for \(s\)-electrons with spin of left orientation, \(\chi^{-}=W^{-}-\varepsilon_{0}^{-}\). Thus, it is assumed that the magnitude of the potential jump \(W\) depends on the magnetization of the ferromagnet owing to the influence of the exchange interaction.

Let us determine the number of \(s^{+}\)-electrons emitted from the surface of a ferromagnet at temperature \(T\) under the action of light of frequency \(\nu\), close to the threshold frequency \(\nu_{0}\). We shall assume that it is proportional to the number of these electrons per unit volume of the ferromagnetic metal which have a component of the quasi-momentum perpendicular to the surface greater than its critical value, which we find from the equality

\[ \alpha-\alpha' y+(\beta+\beta' y)k_x^{+\,2}+h\nu=W^{+}. \]

This number of photoelectrons, which we shall denote by \(N_{\phi}^{+}\), is determined by the expression

\[ N_{\phi}^{+}= \int\limits_{(\alpha-\alpha'y)+(\beta+\beta'y)k_x^{+\,2}=W^{+}-h\nu} n(k_x^{+})\,dk_x^{+}. \tag{6.2} \]

Restricting ourselves to the case of frequencies near the photoelectric threshold \((h\nu\sim \chi^{+})\), it is not difficult to show that the number \(N_{\phi}^{+}\) is determined by the formula

\[ N_{\phi}^{+}= \frac{k^{2}T^{2}}{(\beta+\beta'y)^{1/2}(W^{+}-h\nu)^{1/2}}\, \varphi_i(x^{+}), \tag{6.3} \]

where \(i=1\) for \(x^{+}=\dfrac{h\nu-\chi^{+}}{kT}\leq 0\) and \(i=2\) for \(x^{+}>0\); here

\[ \varphi_1(x^{+})=-\frac{1}{16\pi^{2}a^{3}}\,f_1(x^{+}), \]

\[ \varphi_2(x^{+})=-\frac{1}{16\pi^{2}a^{3}}\,f_2(x^{+}), \]

where \(f_1\) and \(f_2\) are defined by formulas (2.6) and (2.7).

Analogous expressions, but with indices \((-)\) and with \((\beta+\beta'y)\) replaced by \((\beta-\beta'y)\), hold for \(s^{-}\)-electrons.

We further assume that the effective work function of the \(s^{+}\)-electrons is equal to the effective work function of the \(s^{-}\)-electrons, i.e.

\[ \nu_0=\frac{W^{+}-\varepsilon_{0}^{+}}{h} =\frac{W^{-}-\varepsilon_{0}^{-}}{h} =\frac{\chi^{+}}{h} =\frac{\chi^{-}}{h} =\frac{\chi}{h}. \tag{6.4} \]

The equality of the effective work functions \(\chi^{+}\) and \(\chi^{-}\) follows from thermodynamic considerations. Moreover, if the equality (6.4)

although it were not satisfied even approximately, then the magnitude of the photocurrent would change when the direction of magnetization was reversed, which would contradict symmetry considerations. It follows from (6.4) that \(\chi^+=\chi^-=\chi\).

Using the known expressions \(^{51,52}\) for the chemical potentials of \(s\)-electrons with spins of right and left orientation and relation (6.4), one can obtain the following expressions:

\[ W^+=W+\gamma y,\quad W^-=W-\gamma y, \tag{6.5} \]

where \(\gamma=4\pi^2\left(\dfrac{3}{8\pi}\right)^{1/3}\beta\delta\) and \(\delta=\dfrac{2}{3}k_1+\dfrac{\beta'}{3}\), while \(W\) is the potential jump in the absence of magnetization. The total number of photoelectrons \(N_{\phi}(y)\), obviously, will be equal to the sum of \(N_{\phi}^{+}\) and \(N_{\phi}^{-}\). After simple, but cumbersome transformations, we obtain for the total number of photoelectrons \(N_{\phi}(y)\) the following expression:

\[ N_{\phi}(y)= \frac{(kT)^2\psi\{|x(y)|\}} {8\pi^2 a^3\beta^{5/2}(W-h\nu)^{1/2}} (1-\Gamma y^2), \tag{6.6} \]

where

\[ \Gamma=-\left[ \frac{3}{4}\frac{\beta'\gamma\beta}{W-h\nu} + \frac{3}{8}\frac{\gamma^2}{(W-h\nu)^2} + \frac{15}{8}\left(\frac{\beta'}{\beta}\right)^2 \right]. \tag{6.7} \]

Generally speaking, \(\Gamma\) may have either a positive or a negative sign, depending on the signs of \(\beta\), \(\beta'\), \(\gamma\), and \((W-h\nu)\), and on their ratios. For \(y=0\), expression (6.6) coincides exactly with the corresponding Fowler formula \(^{12}\). Assuming that the photoemission current is proportional to \(N_{\phi}(y)\), one may write:

\[ j=AT^2(W-h\nu)^{-1/2}\varphi_i(x)(1-\Gamma y^2), \tag{6.8} \]

where \(A\) is a constant independent of \(\nu\) and \(T\).

In formula (6.8) we are interested not in the absolute value of the photoelectric current \(j\), but in its ferromagnetic “anomaly,” i.e.,

\[ \frac{\Delta j}{j_0}=Dy^2, \tag{6.9} \]

where \(j_0\) is the value of \(j\) at \(y=0\), and

\[ D=\Gamma-\frac{\psi'\{|x(0)|\}\varepsilon_{\beta}\delta_1} {\psi\{|x(0)|\}kT}. \tag{6.10} \]

The coefficient \(D\) (as also in the case of ordering alloys) is a function of frequency. Therefore, a change in the frequency of the light producing the photoeffect causes a change in the coefficient \(D\) itself (which corresponds to the different photocurrent curves in Cardwell). In expression (6.9), \(\Delta j\) denotes the difference between the photoelectric current obtained by extrapolating the curve

of the photocurrent above the Curie point \((j_j)\), and the actually observed photocurrent below the Curie point. Cardwell’s experimental data \({}^{30}\) qualitatively confirm the theoretical formula obtained (6.9).

It can be shown by direct calculations that the effective work function of a ferromagnetic metal is a function of the spontaneous magnetization. Indeed, using the relations for the chemical potentials of the \(s\)-electrons and formula (6.4), and restricting ourselves to terms quadratic in \(y\), it is easy to show the validity of the following relation:

\[ \chi(y)=W-\varepsilon_\beta\left(1+\hat{\delta}_1 y^2\right), \tag{6.11} \]

where

\[ \varepsilon_\beta=4\pi^2 a^2 \beta\left(\frac{3n}{8\pi}\right)^{2/3}, \qquad \hat{\delta}_1=\left(\frac{2}{3}k_1\frac{\beta'}{\beta}-\frac{1}{9}k_1^2\right). \]

For \(y=0\), formula (6.11) becomes the usual expression for the effective work function. If, in formula (6.11), the temperature dependence of \(W\) and \(\varepsilon_\beta\) is neglected, then the effective work function will depend on temperature only through the spontaneous magnetization. With a positive sign of \(\hat{\delta}_1\), it follows from formula (6.11) that, as the temperature rises, the value of \(\chi\) will increase, which is in agreement with Cardwell’s experimental data.

Using formula (6.11), it is easy to determine the ferromagnetic “anomaly” of the effective work function of a ferromagnet:

\[ \frac{\Delta\chi}{\chi_0}=\frac{\chi_0-\chi}{\chi_0} =\frac{\varepsilon_\beta \hat{\delta}_1}{W-\varepsilon_\beta}\,y^2, \tag{6.12} \]

where \(\chi_0\) is the value of \(\chi\) at \(y=0\). An experimental check of the theoretical conclusions represented by formula (6.12) would be highly desirable. Thus, the photoelectric current of ferromagnets should depend on the magnitude of their spontaneous magnetization. Near the temperature of the ferromagnetic transformation, this dependence has a simple quadratic character (see formulas (6.8) and (6.9)). In addition, the dependence of the effective work function of a ferromagnet on the magnitude of the spontaneous magnetization and its ferromagnetic “anomaly” has been obtained.

B. A More Rigorous Theory

a) A more consistent theory of the photoelectric effect in ferromagnetic metals was proposed by A. Z. Veksler. It is based, on the one hand, on the \((s-d)\)-exchange model \({}^{55}\), and on the other hand, on the rigorous theory of Tamm—Shubin \({}^{16}\)—Mitchell \({}^{23}\). All electron states in a ferromagnetic crystal are divided into two groups: bound and free. The probability

PHOTOEFFECT IN METALS

to find an electron in a bound state at a very large distance from the metal–vacuum interface in the region outside the crystal is very small, and therefore the photocurrent is equal to zero. The probability of finding an electron in a free state at a very large distance from the interface is not equal to zero, and therefore the photocurrent differs from zero. Electrons in free states correspond to a real value of the wave vector. Electrons filling bound states have an imaginary component of the wave vector along one of the coordinate axes. The photoeffect arises only in those cases when the electrons fill free states. In the photoeffect, the electrons receive the additional energy necessary for transition into a free state through absorption of a quantum of light.

The theory of the photoeffect set forth in Section A, in which the periodic potential of the crystal is not taken into account, is inconsistent because the peculiarities of the motion of electrons in ferromagnets are determined by their interaction, which is a “volume” factor and is only weakly connected with the properties of the metal–vacuum interface. Therefore the motion of a conduction electron in a ferromagnetic metal is more correctly described by a wave function not in the form of a sum of two plane waves (incident and reflected), as is usually done in the theory of the photoeffect \(^{16,23}\), but by a function describing the motion of an electron in the periodic potential field of the crystal lattice \(V(\mathbf r)=V(\mathbf r+\mathbf n)\). This wave function has the form:

\[ \psi(\mathbf r)=e^{i\mathbf k\cdot \mathbf r}u_{\mathbf k}(\mathbf r), \]

where

\[ u_{\mathbf k}(\mathbf r+\mathbf n)=u_{\mathbf k}(\mathbf r), \]

and \(\mathbf n\) is a lattice vector (in units of the parameter \(a\)).

Simultaneous allowance for the periodic potential of the crystal and for the boundary potential of the metal barrier makes it possible to obtain a general expression for the current in which the volume and surface effects are taken into account in a single scheme.

Let the metal fill the entire infinite half-space \(x \leq 0\), while the region \(x \geq x_0\) is vacuum. We shall consider photoemission of \(s\)-electrons from the metal, caused by the electromagnetic field of a light wave.

The eigenfunction of the unperturbed state of the electron is determined from the wave equations:

\[ \Delta\psi_1+\frac{2m}{\hbar^2}\,[E-V(\mathbf r)]\,\psi_1=0 \qquad x\leq x_0, \tag{6.13} \]

\[ \Delta\psi_2+\frac{2m}{\hbar^2}\left[E+\frac{e^2}{4x}\right]\psi_2=0 \qquad x\geq x_0, \tag{6.14} \]

S. V. Vonsovskii, A. V. Sokolov, A. Z. Veksler

where \(-\dfrac{e^{2}}{x}\) is the potential arising owing to the presence of image forces at the boundary outside the metal.

The solution of equation (6.13) can be represented in the form1:

\[ \psi_1=\sum_n \left\{ a_n \exp \left[i(k_x+q_{n_1})(x-x_0)\right]+\right. \]
\[ \left. +b_n \exp \left[-i(k_x+q_{n_1})(x-x_0)\right]+ \exp i\left[(k_y+q_{n_2})y+(k_z+q_{n_3})z\right]\right\}, \tag{6.15} \]

where \(q_{n_i}=\dfrac{2\pi n_i}{a}\), \(n_i\) are integers, \(a\) is the lattice constant, and \(k_x, k_y, k_z\) are components of the electron quasimomentum \((i=x,y,z)\).

Equation (6.14) can be solved by the method of separation of variables, putting

\[ \psi_2=f_1(x)f_2(y)f_3(z). \tag{6.16} \]

Substituting (6.16) into (6.14), we obtain:

\[ f_2(y)=e^{ig_y y},\quad f_3(z)=e^{ig_z z}, \tag{6.17} \]

and \(f_1(x)\) satisfies the equation

\[ \frac{d^2 f_1(x)}{dx^2} +\frac{2m}{\hbar^2}\left(\frac{\hbar^2}{2m}g_x^2+\frac{e^2}{4x}\right)f_1(x)=0, \tag{6.18} \]

with

\[ E=-\frac{\hbar^2}{2m}(g_x^2+g_y^2+g_z^2). \tag{6.19} \]

The general solution of equation (6.14) has the form

\[ \psi_2(x,y,z)=\sum_{g_y,g_z} C_g f_1(x)e^{i(g_y y+g_z z)}. \tag{6.20} \]

The coefficients \(C_g\) are found from the boundary conditions

\[ \psi_1\big|_{x=x_0}=\psi_2\big|_{x=x_0},\quad \frac{d\psi_1}{dx}\bigg|_{x=x_0}= \frac{d\psi_2}{dx}\bigg|_{x=x_0}. \tag{6.21} \]

The boundary conditions (6.21) are satisfied only when the following equalities hold:

\[ \begin{aligned} g_y&=k_y+q_{n_2},\\ g_z&=k_z+q_{n_3}. \end{aligned} \tag{6.22} \]

Taking (6.19) into account, we find:

\[ g_x^2=\frac{2m}{\hbar^2}E-(k_y+q_{n_2})^2-(k_z+q_{n_3})^2. \tag{6.23} \]

It follows from this that the summation in formula (6.20) is carried out over \(n_2\) and \(n_3\), and therefore it can be written in the following form:

\[ \psi_z(x,y,z)=\sum_{n_2,n_3} C_{n_2n_3} f_1(x)\exp\{i[(k_y+q_{n_2})+(k_z+q_{n_3})]\}. \tag{6.24} \]

At large distances from the interface, this function can be represented in two forms:

\[ \psi_z(x,y,z)\underset{x\to\infty}{\sim} \sum_{n_2,n_3} C'_{n_2n_3}\exp\{i[g_x x+ \]

\[ +(k_y+q_{n_2})y+(k_z+q_{n_3})z]\}, \tag{6.25} \]

\[ \psi_z(x,y,z)\underset{x\to\infty}{\sim} \sum_{n_2,n_3} C'_{n_2n_3}\exp\{-\chi_x x+ \]

\[ +i[(k_y+q_{n_2})y+(k_z+q_{n_3})z]\}, \tag{6.26} \]

where \(\chi_x=-ig_x\), if \(g_x\) is an imaginary number.

The state of the electrons described by function (6.25) will be called free, and the state described by function (6.26), bound.

A free state corresponds to a finite probability, different from zero, of finding an electron at an arbitrarily large distance from the metal—vacuum interface. For a bound state the probability of finding an electron at a large distance from the metal surface is small and, as \(x\) increases, tends to zero. The calculation of the current corresponding to the state described by function (6.25) leads to the following expression:

\[ j_x=\frac{e\hbar}{m}\sum_{n_2n_3}|C'_{n_2n_3}|^2 g_x . \tag{6.27} \]

Let us consider the photoeffect at a light frequency close to the threshold frequency. In this case the overwhelming majority of photoelectrons leaving the metal have energies of the order of the thermal energy. As shown in Ref. \(^{55}\), formula (6.27) is then simplified and takes the form

\[ j_x=\frac{e\hbar}{m}|C'_{0,0}|^2 g_x(0). \tag{6.28} \]

b) We start from the equation of the first approximation of perturbation theory (3.13). We seek the solution of this equation in the form of an expansion

in the Fourier integral16, 19, 23 with respect to the eigenfunctions of the unperturbed problem:

\[ v=\iint\limits_{-\infty}^{\infty} dk_y dk_z \left\{ \int\limits_0^\infty C_{\mathbf{k}}^+(t)U_{\mathbf{k}}^+(t)\,dk_x + \int\limits_0^\infty C_{\mathbf{k}}^-(t)U_{\mathbf{k}}^-(t)\,dk_x \right\}, \tag{6.29} \]

where

\[ U_{k_x,k_y,k_z}^{-}(x,y,z,t) = U_{-k_x,k_y,k_z}^{+}(x,y,z,t) = U_{-k_x,k_y,k_z}(x,y,z,t). \]

Substituting (6.29) into (3.13) and taking into account that \(U_{\mathbf{k}}(\mathbf{r},t)\) is a solution of an equation of the same form as (3.3), but with right-hand side equal to zero, we obtain:

\[ \iint\limits_{-\infty}^{\infty} dk_y dk_z \left\{ \int\limits_0^\infty dk_x\,\frac{dC_{\mathbf{k}}^+}{dt}U_{\mathbf{k}}^+ + \int\limits_0^\infty dk_x\,\frac{dC_{\mathbf{k}}^-}{dt}U_{\mathbf{k}}^- \right\} = \frac{e}{mc}(\mathbf{A}\nabla)U_{\mathbf{k}} . \tag{6.30} \]

Multiplying both sides of this equation by \(U_{\mathbf{k}'}^{\alpha *}\), where \(\alpha\) denotes \(+\) or \(-\), and integrating over the coordinates:

\[ \iint\limits_{-\infty}^{\infty} dk_y dk_z \left\{ \int\limits_0^\infty dk'_x\,\frac{dC_{\mathbf{k}}^+}{dt} \int U_{\mathbf{k}}^+U_{\mathbf{k}'}^{\alpha *}\,d\tau\,dk_x + \int\limits_0^\infty dk'_x\,\frac{dC_{\mathbf{k}}^-}{dt} \int U_{\mathbf{k}}^-U_{\mathbf{k}'}^{\alpha *}\,d\tau\,dk_x \right\} = \frac{e}{mc}\int U_{\mathbf{k}'}^{\alpha *}(\mathbf{A}\nabla)U_{\mathbf{k}}\,d\tau . \tag{6.31} \]

Thanks to this transformation, an equation has been obtained that does not depend explicitly on the potential. It is now not difficult to determine the amplitudes \(C_{\mathbf{k}}^\pm(t)\), bearing in mind that \(U_{\mathbf{k}}^\pm(\mathbf{r},t)\) are orthogonal:

\[ \left. \begin{aligned} \int U_{\mathbf{k}}^+U_{\mathbf{k}'}^{+*}\,d\tau &= \int U_{\mathbf{k}}^-U_{\mathbf{k}'}^{-*}\,d\tau = N_{\mathbf{k}}^{(1)}\delta(\mathbf{k}-\mathbf{k}'),\\ \int U_{\mathbf{k}}^+U_{\mathbf{k}'}^{-*}\,d\tau &= \int U_{\mathbf{k}}^-U_{\mathbf{k}'}^{+*}\,d\tau = N_{\mathbf{k}}^{(2)}\delta(\mathbf{k}-\mathbf{k}'), \end{aligned} \right\} \tag{6.32} \]

where \(N_{\mathbf{k}}^{(1)}\) and \(N_{\mathbf{k}}^{(2)}\) are normalization factors. Therefore equations (6.31) take the following form:

\[ \left. \begin{aligned} N_{\mathbf{k}}^{(1)}\frac{dC_{\mathbf{k}}^+}{dt} + N_{\mathbf{k}}^{(2)}\frac{dC_{\mathbf{k}}^-}{dt} &= \frac{e}{mc}\int U_{\mathbf{k}}^{+*}(\mathbf{A}\nabla)U_{\mathbf{k}}^+\,d\tau,\\ N_{\mathbf{k}}^{(2)}\frac{dC_{\mathbf{k}}^+}{dt} + N_{\mathbf{k}}^{(1)}\frac{dC_{\mathbf{k}}^-}{dt} &= \frac{e}{mc}\int U_{\mathbf{k}}^{-*}(\mathbf{A}\nabla)U_{\mathbf{k}}^+\,d\tau . \end{aligned} \right\} \tag{6.33} \]

PHOTOEFFECT IN METALS

Excluding \(\dfrac{dC_{\bar{k}}}{dt}\) and integrating with respect to time, under the assumption that for \(t<0\) the metal was not illuminated, we obtain:

\[ C_{\bar{k}}^{\pm}(t)= \frac{\exp\left[\frac{i}{\hbar}(E_k+\hbar\omega-E_{k'})\right]-1} {\frac{i}{\hbar}(E_k+\hbar\omega-E_{k'})}\, (\mathbf{k}|a|\mathbf{k}')^{\pm}, \tag{6.34} \]

where

\[ (\mathbf{k}|a|\mathbf{k}')^{\pm}= \frac{e}{mc\left\{[N_k^{(1)}]^2-[N_k^{(2)}]^2\right\}} \int (a\nabla\psi_k)\,d\tau\, \left[N_k^{(1)}\psi_{k'}^{\pm *}-N_k^{(2)}\psi_{k'}^{\mp}\right]. \tag{6.35} \]

In calculating \(C_{\bar{k}}^{\pm}(t)\), the term corresponding to radiation has been omitted, as well as the factor
\(\exp\left(i\omega \dfrac{x\cos\theta+y\sin\theta}{c}\right)\), since the case considered is that in which the light frequency is close to the threshold frequency, whence \(\dfrac{\omega}{c}\ll k\). Since the energy of an \(s^{+}\)-electron of a ferromagnetic metal can be represented in the form

\[ E=A+A'y+(B+B'y)(\cos k_xa+\cos k_ya+\cos k_za), \tag{6.36} \]

the normalization factors \(N_k^{(1)}\) and \(N_k^{(2)}\) will be equal to:

\[ N_k^{(1)}= \frac{4\pi^3\hbar^2}{m(B+B'y)\sin k_xa} \left\{ \sum_{n,n'} (2ak_x+2\pi n_1+2\pi n_1') \times \right. \]

\[ \left. \times (a_na_{n'}^{*}-b_nb_{n'}^{*}) + \sum_{n_2,n_3}|C_{n_2n_3}|^2\,ag_x \right\}, \tag{6.37} \]

\[ N_k^{(2)}= \frac{4\pi^3\hbar^2}{m(B+B'y)\sin k_xa} \times \]

\[ \times \left\{ \sum_{n,n'} (2ak_x+2\pi n_1+2\pi n_1') (a_nb_{n'}^{*}-a_{n'}^{*}b_n) \right\}. \tag{6.38} \]

The matrix element will contain only the \(x\)-component of the vector potential, since its \(y\)- and \(z\)-components give expressions equal to zero because of the orthogonality of the functions \(\exp(ik_yy)\) and \(\exp(ik_zz)\). This leads to the following expression for the matrix element:

\[ (\mathbf{k}|a|\mathbf{k}')^{\pm} = (2\pi)^3(\mathbf{k}|a_x|\mathbf{k}')^{\pm} \delta(k_y-k_y')\delta(k_z-k_z'), \tag{6.39} \]

where \((\mathbf{k}|a_x|\mathbf{k}')^{\pm}\) contains only the integral over \(x\). Consequently,

For the amplitudes \(C_{\mathbf k}^{\pm}(t)\) we have the formula

\[ C_{\mathbf k}^{\pm}(t)=(2\pi)^3 \frac{ \exp\left\{\frac{i}{\hbar}\left[(E_k+\hbar\omega-E_{k'})t\right]\right\}-1 }{ \frac{i}{\hbar}(E_k+\hbar\omega-E_{k'}) } \times (\mathbf k|a_x|\mathbf k')^{\pm}\delta(k_y-k_y')\delta(k_z-k_z'). \tag{6.40} \]

It is now not difficult to find the addition \(\dot v\) to the wave function, using the fact that the time \(t\) elapsed since the beginning of the illumination is very large:

\[ \dot v=(2\pi)^3 \left\{ \int_{0}^{\infty} U_{\mathbf k'}^{+}(\mathbf k|a_x|\mathbf k')^{+}\,dk_x' + \int_{0}^{\infty} U_{\mathbf k'}^{-}(\mathbf k|a_x|\mathbf k')^{-}\,dk_x' \right\} \times \frac{ \exp\left[\frac{i}{\hbar}(E_k+\hbar\omega-E_{k'})t\right]-1 }{ \frac{i}{\hbar}(E_k+\hbar\omega-E_{k'}) }. \tag{6.41} \]

Carrying out the integration, one can find the final expression for \(\dot v\):

\[ \dot v= \frac{(2\pi)^3(\mathbf k|a_x|\overline{\mathbf k}')_{0}^{+}} {(B+B'y)a\sin \overline{k}_x a} \,U_{\overline{\mathbf k}'}(x,y,z), \tag{6.42} \]

where \(\overline{\mathbf k}'\) is the value of \(\mathbf k'\) determined by the conditions

\[ \left. \begin{aligned} k_{y,z}'&=k_{y,z}+q_{n_{2},\,3}-q_{n_{2},\,3}',\\ E_{k'}&=E_k+\hbar\omega. \end{aligned} \right\} \tag{6.43} \]

We note that the second integral in (6.41) vanishes, as in paper \(^{19}\), and therefore the sign \(+\) may be omitted. The matrix element \((\mathbf k|a|\overline{\mathbf k})\) splits into two parts, one of which corresponds to the region inside the metal, and the other outside it. Carrying out the integration over the fundamental region, we find that the part of the matrix element associated with the wave function of the electron in the metal is equal to:

\[ \begin{aligned} (\mathbf k|a|\overline{\mathbf k})_{0,1} &= a_x \left\{ N_k^{(1)} \sum_{n_i,n_i'} (\overline{k}_x+q_{n_i}) \times \left[ \frac{a_n^{*}a_{n'}+b_n b_{n'}^{*}} {g_x+q_{n_i}-\overline{g}_x-q_{n_i'}} - \frac{a_n^{*}b_{n'}+b_n^{*}a_{n'}} {g_x+q_{n_i}+\overline{g}_x+q_{n_i'}} \right]\right.\\ &\qquad\left. + N_k^{(2)} \sum_{n_i,n_i'} (\overline{k}_x+q_{n_i}) \times \left[ \frac{a_n^{*}a_{n'}+b_n^{*}b_{n'}} {g_x+q_{n_i}+\overline{g}_x+q_{n_i}} - \frac{a_n^{*}b_{n'}+b_n^{*}a_n} {g_x+q_{n_i}-\overline{g}_x-q_{n_i'}} \right] \right\}. \end{aligned} \tag{6.44} \]

For the region outside the metal, the calculation of the matrix element presents considerable difficulties. However, one can establish the dependence of the matrix element on \(g_x\), and hence also the temperature dependence of the photocurrent. In calculating the matrix element, instead of the wave function of the bound state one may use its asymptotic expression \({}^{31}\)

\[ f_x(x)=b_x e^{-\varkappa x}(2\varkappa x)^{S/\varkappa}, \tag{6.45} \]

where \(S=-\dfrac{me^2}{4\hbar^2}\) and \(\varkappa=i g_x\).

This expression represents the function with sufficient accuracy down to distances from the interface of order \((3\div 4)\cdot 10^{-8}\) cm. For free states such a representation cannot be used, since for the region close to the interface \(2g_x x \ll 1\).

As shown in work \({}^{31}\), for light frequencies close to the threshold, i.e., for small \(g_x\),

\[ f_{g_x}(x)=C_1\lambda(x) \]

and, to first approximation, does not depend on \(g_x\). Thus, for the second part of the matrix element we have the expression

\[ (\mathbf{k}|a_x|\mathbf{k}')_{0,2} = a_x c_1\left[N_{k'}^{(1)}-N_{k'}^{(2)}\right] \int_{x_0}^{\infty}\lambda(x) \left[ -\varkappa e^{-\varkappa x}(2\varkappa x)^{S/\varkappa} + \right. \]

\[ \left. +\,2S e^{-\varkappa x}(2\varkappa x)^{\frac{S}{\varkappa}-1} \right]dx, \tag{6.46} \]

which does not depend on \(g_x\).

In deriving formula (6.42), the term corresponding to the conservation law of the quasimomentum along the \(x\)-direction was omitted:

\[ k_x' = k_x+q_{n_1}-q_{n_1}'. \tag{6.47} \]

This condition must be satisfied simultaneously with equalities (6.43), but this is impossible if, as a result of excitation, the electron does not pass into another band, since the energy in a band is a periodic function of the quasimomentum \({}^{35}\):

\[ E_l(\mathbf{k}+\mathbf{q}_n)=E_l(\mathbf{k}). \tag{6.48} \]

If the transition takes place into another band, then, for a definite relation between \(k_x, k_y, k_z\), equalities (6.43) and (6.48) are compatible. This case corresponds to the so-called optical transitions. Consequently, optical transitions, or, as this phenomenon was already called by Tamm and Shubin \({}^{16}\), the volume photoeffect, can contribute to photoemission only when the electron is transferred to the next energy band. In Tamm and Shubin, the volume photoeffect was considered without taking this circumstance into account,

that it is possible only for transitions into the next band. This turned out to be possible because the electron energy was taken in the form \(E=\dfrac{\hbar^2 k^2}{2m}\), i.e., it did not have the periodicity of the lattice and the band character of the electron energy spectrum was not taken into account. Naturally, therefore, the conclusion obtained in the cited work concerning the “red limit of the volume effect” is not quite accurate. The same remark also applies to the work \({}^{44}\).

Usually \({}^{16,19}\) the volume photoeffect was considered separately from the surface one. From Weksler’s calculation it is clear that there is no need to make such an artificial separation, since both parts of the photoeffect are obtained automatically, as component parts of the matrix element. On the basis of the foregoing it follows that the “second red boundary,” corresponding to the volume photoeffect, must be determined by the energy that must be imparted to an electron situated at the upper edge of a partially filled band in order for it to pass into the lower state of the next empty band. For metals this energy is much greater than the energy required for the transition of an electron into the lowest free state.

In calculating the photocurrent one should take into account only the current due to the addition \(\upsilon\) to the wave function, since the current arising from the unperturbed wave function (without the field) nevertheless vanishes. We determine the current density from the well-known formula

\[ j_x=\frac{e\hbar}{2mi}\left(\upsilon^*\frac{\partial \upsilon}{\partial x}-\upsilon\frac{\partial \upsilon^*}{\partial x}\right), \]

where \(\upsilon\) is given by formula (6.42). In order to find the current density of the electrons in vacuum at a large distance from the boundary surface, as the function \(U_k(x,y,z)\) entering into (6.42) we use the asymptotic expression for the hypergeometric function \({}^{57}\):

\[ U_k(x,y,z)=C\exp\left[ig_xx+\frac{iS}{g_x}\ln(2ig_xx)\right]\exp[i(g_yy+g_zz)]. \tag{6.49} \]

Then the current density is equal to:

\[ j_x=\frac{(2\pi)^6\left|(\mathbf{k}\mid a_x\mid \bar{\mathbf{k}})\right|^2 e\hbar}{ma^2(B+B')^2\sin^2 k_xa}\,g_x|C|^2. \tag{6.50} \]

In the work \({}^{55}\) it is shown that \(g_x|C|^2\) does not depend on \(g_x\). Therefore, if this quantity is denoted by \(A(\bar{k}_x)\), then the current can be represented in the form

\[ j_x=\frac{(2\pi)^6 e\hbar}{m}\, \frac{\left|(\mathbf{k}\mid a\mid \bar{\mathbf{k}})_0\right|^2 A(\bar{k}_x)} {(B+B')^2 a^2\sin^2 k_xa}. \tag{6.51} \]

Integrating over all initial states of the electrons in the metal, we obtain the total photocurrent:

\[ J_x=\frac{1}{(2\pi)^3}\int_{-\frac{\pi}{a}}^{\frac{\pi}{a}} dk_y \int_{-\frac{\pi}{a}}^{\frac{\pi}{a}} dk_z \int_{g_x=0}^{\infty} j_x \frac{1}{\exp\frac{E-\varepsilon_0}{kT}+1}\, dk_x . \tag{6.52} \]

Expansion of the Fermi function in a series in \(\exp \frac{E-\varepsilon_0}{kT}\) leads to the following expressions:

\[ \left[\exp\left(\frac{E-\varepsilon_0}{kT}\right)+1\right]^{-1} = \sum_n (-1)^n \exp\left(-\frac{(n+1)(E-\varepsilon_0)}{kT}\right), \quad E>\varepsilon_0, \tag{6.53} \]

\[ \left[\exp\left(\frac{E-\varepsilon_0}{kT}\right)+1\right]^{-1} = \sum_n (-1)^n \exp\left(\frac{n(E-\varepsilon_0)}{kT}\right), \quad E<\varepsilon_0. \tag{6.54} \]

In carrying out the integration (6.52) explicitly, we shall use the second relation (6.43), which can be represented in the form

\[ E_k=\frac{\hbar^2}{2m}\left(g_x^2+g_y^2+g_z^2\right)-\hbar\omega, \]

and we pass to the new integration variables \(g_x\), \(\rho^2=g_y^2+g_z^2\) and \(\vartheta=\operatorname{arctg}\frac{g_y}{g_z}\). Integration leads to the following result:

\[ J_x= \frac{(2\pi)^4 me(kT)^3 |(\mathbf{k}|\alpha_x|\overline{\mathbf{k}}')|^2 A(\bar{k}_x)} {\hbar^3 c^2 a^3 (B+B'y)^3 \sin^3 k'_{0x}a} \, \Phi_{1,2}\left(\frac{\varepsilon+\hbar\omega}{kT}\right), \tag{6.55} \]

where

\[ \Phi_1(x)=\sum_{n=0}^{\infty}\frac{(-1)^n}{(n+1)^3}\exp(n+1)x \qquad \text{for } x<0, \]

\[ \Phi_2(x)=\frac{\pi^2}{6}+\frac{x^2}{2} -\sum_{n=0}^{\infty}\frac{(-1)^n}{(n+1)^2}\exp[-(n+1)x] \qquad \text{for } x>0. \]

The total current of electrons with right and left spin orientations is equal (in the case of weak filling of the band):

\[ J=(A_0+B_0y^2)T^2\Phi_{1,2}\left[\frac{\varepsilon(\omega)+By^2}{kT}\right], \tag{6.56} \]

where

\[ A_0=-\frac{16\pi^2 m e k^2 D}{\hbar^2 a^4 a_1^{3/4}}\,M_0\sum_{n_1}|a_{n_1 00}|^2(\gamma_1+q_{n_1}), \]

\[ \begin{aligned} B_0={}&-\frac{16\pi^3 m e k^2 D}{\hbar^2 a^4 a_1^{3/2}} \sum_{n_1}\Biggl\{M_0\gamma_3+ \left(M_1-\frac{3}{2}\frac{a_2}{a_1}M_0\right)\gamma_2\\ &\quad+\left[\left(\frac{15}{8}\frac{a_2^2}{a_1^2} -\frac{3}{2}\frac{a_2}{a_1}\right)M_2 -\frac{3}{2}M_1\frac{a_2}{a_1}\right](\gamma_1+q_{n_1})\Biggr\} |a_{n_1 00}|^2 . \end{aligned} \]

\(M_0, M_1, M_2\) are the first, second, and third coefficients in the expansion of the square of the modulus of the matrix element in powers of the relative magnetization \(y\):

\[ \left|(\mathbf{k}|a_x|\mathbf{k})\right|^2 \simeq M_0+M_1y+M_2y^2;\quad a_1=A^2+3B^2+\varepsilon(\omega)(2B'+A'); \]

\[ a_3=2B'^2+A'^2+4A'B'+B'(A+2B);\quad \gamma_1=\arccos \xi_0; \]

\[ \gamma_2=\beta_2(1-\xi_0^2)^{-1/2};\quad \gamma_3=-(\xi_0\beta_2^2+\beta_3)(1-\xi_0^2)^{-1/2}; \]

\[ \xi_0=\frac{\varepsilon(\omega)-(A+2B)}{B};\quad \beta_2=\frac{2B'-A'}{B}+\xi_0\frac{B'}{B}; \]

\[ \beta_3=\xi_0\frac{B'^2}{B}+\frac{2B'-A'}{B^3}B' +\frac{B'}{B}; \]

\[ \beta_1=a^2(3\pi^2 n)^{1/2}k_1 \left(\frac{3}{8}k_1\beta+\frac{3}{2}\beta'\right); \]

\[ \varepsilon(\omega)=\alpha+a^2(3\pi^2 n)^{1/2}\beta+\hbar\omega; \]

\[ a_2=2BB'+2\varepsilon(\omega)(A'+2B'). \]

The theory of the photoelectric effect considered here can evidently also be used for nonferromagnetic metals, if one everywhere sets \(y=0\). Formula (6.56), which determines the dependence of the saturation current on temperature and frequency, has the same form as (6.8).

However, there is an essential difference between these two formulas: the parameters entering the formula for the current (6.56) have been obtained in the most general form, without using any additional assumptions whatever (for example, the assumption concerning the dependence of the jump of the potential on the relative magnetization, as was done in work \(^{52}\)). Moreover, the difference between a simplified treatment of the photoelectric effect and its more consistent treatment, taking into account the periodicity of the potential inside the metal, is especially noticeable when comparing the expressions determining the distribution of photoelectrons with respect to velocity (see the expression under the integral sign in (6.52)). This dependence proves to be more complicated than in the simplified theory, which, apparently, is what occurs in reality \(^{54}\).

In considering the photoelectric effect produced by light with a frequency much greater than the limiting frequency, one should also take into account the second

component of the matrix element relating to the volume effect. Comparison of the formula for the photocurrent (6.56) with the known relation determining its temperature dependence shows that the work function is equal to \(-\varepsilon_0^+ = -\varepsilon_0\). This result can be proved not only by a formal comparison of the relations, but also on the basis of a thermodynamic consideration analogous to that carried out in work \(^{55}\).

§ 7. CONCLUSION

As follows from all that has been set forth above, the one-electron model of the quantum-mechanical theory of the external photoelectric effect in metals already gives a sufficiently complete qualitative, and in many cases also quantitative, explanation of this important electronic phenomenon in crystals. However, the one-electron model has a very large and fundamental shortcoming—the neglect of interelectron interaction. Therefore there arises a very urgent problem of the quantum theory of metals: the development of a many-electron scheme for calculating the optical and photoelectric properties in metals and in solid bodies of other types.

In this connection one should mention the work of Shubin \(^{58}\), in which the method of the Dirac density matrix (the statistical operator in \(\mu\)-space) was applied to the theory of metals within the framework of the one-electron approximation. He obtained an expression for the density of the electric current caused by the action of an external alternating electric field \(F\cos\omega t\) of frequency \(\omega\):

\[ j=-\frac{e^3Fa}{\hbar m}\frac{\sin\omega t}{\omega}\sum_{\xi}(\xi|\hat p|\xi)\frac{\partial\rho_0}{\partial \xi} +\frac{2e^3F}{\hbar m}\sin\omega t\sum_{\xi\ne\xi'}[\rho_0(\xi)-\rho_0(\xi')] \frac{\omega}{\omega(\xi,\xi')}\cdot \frac{|(\xi|\hat p|\xi')|^2}{[\omega^2(\xi,\xi')-\omega^2]}, \tag{7.1} \]

where \(\xi\) is the quasimomentum vector of the stationary state of an electron in the metal, \(a\) is the lattice constant, \((\xi|\hat p|\xi)\) and \((\xi|\hat p|\xi')\) are respectively the diagonal and nondiagonal matrix elements of the electron momentum operator \(\hat p\), \(\rho_0(\xi)\) is the initial value of the density matrix in the state \(\xi\), and

\[ \omega(\xi,\xi')=\frac{1}{\hbar}\,[E(\xi)-E(\xi')], \]

where \(E(\xi)\) is the energy of the electron in the state \(\xi\).

The first term on the right-hand side of formula (7.1) describes the effects of the accelerating action of the field \(F\), while the second describes the quantum (opti-

cal) transitions. In principle, this formula can also be applied to the treatment of the photoelectric effect.

It proved possible to generalize the work \(^{58}\) also to the case of a many-electron model \(^{59}\), without making any limiting approximations as to the magnitude and character of the interelectron interaction*). Subsequently it was possible to obtain general dispersion formulas of the quantum optics of metals in such a many-electron scheme \(^{60}\) and to take into account in these calculations, in a phenomenological way, the effect of electron damping \(^{61}\).

The many-electron theory of crystals opens up natural possibilities for constructing a unified theory of the optical and photoelectric properties of metals and semiconductors.

As the first attempts at constructing a more detailed model many-electron theory of optical properties show, taking electron interaction into account may lead to a deeper and more complete explanation of the phenomena of the photoeffect in crystals. However, such calculations do not yet exist in finished form. One can only note that, in the particular case of the polar many-electron model of an atomic semiconductor, results are obtained that differ substantially from the conclusions of the one-electron band theory. Namely, whereas the latter predicted transitions of an electron from the ground state of the system in the crystal only to the lowest state of the energy conduction band, independently of the type of excitation (collisions with phonons, photons, etc.), within the many-electron model these transitions are different for different excitations. Under optical excitation the system passes into a state with an energy corresponding to some level in the middle of the conduction band, whereas under phonon excitation it passes, as in the one-electron band theory, into the state with the lowest energy in the conduction band. Calculation of the accelerating effect of the electric field of infrared light, with allowance for electron interaction, leads to a substantial increase in the effective mass of the carriers of electric current in the crystal (conduction electrons, excitons, etc.).

In conclusion, it should also be noted that a more intensive development of the theory of the photoeffect in semiconductors and semimetals is necessary. As A. F. Ioffe noted in his latest monograph \(^{63}\),

* ) One cannot agree with the remark about the quantum-mechanical theory of the optical properties of metals, made in the review article \(^{62}\) (see p. 493), which says that the derivation of formulas of type (7.1) by means of quantum calculations “adds little that is new” to the theory. Apart from the fact that these formulas show how the effective mass of conduction electrons (considered within this theory as a certain type of elementary excitation of the entire electronic system as a whole) should be correctly calculated, they make it possible, from a single point of view, to approach the treatment of all “optical” processes taking place in the electron ensemble of a crystalline lattice.

this question has been developed very little (see also \(^{116}\)). In constructing a theory of the photoelectric effect in semiconductors, as shown in work \(^{64}\), it proves necessary to take into account the influence of the surface Tamm energy levels of electrons in the crystal \(^{65}\).

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1. As indicated in the source text. ↩