PASSAGE OF ELECTRONS THROUGH THE SURFACE AND SURFACE LAYERS1

R. Fowler, Cambridge

I have entitled my lecture today “Passage of Electrons through the Surface and Surface Layers.” I shall therefore speak not about general principles, but about the application of modern theoretical physics to one special question. I shall try, however, to set out my subject in proper connection with the general theory. I shall try to show you that wave, or quantum, mechanics—call it what you will—in connection with the long-accepted view of the nature of the metal, leads us to the idea that the question of the emission of electrons by metals may, in simplified form, be treated as the problem indicated in the title of this lecture. I shall further set out how this problem is solved theoretically, show you that the theoretical calculations are in complete agreement with observations, and give explanations of a number of facts which until now have not found their proper place.

By way of apology for my choice of a particular, rather technical question, I shall say that this is one of the fields most closely familiar to me. The development of this question owes its successes chiefly to the work of Nordheim, and is the result of his work on this subject during the past year in Cambridge. I have little

I can add to what is set forth in his excellent review article published in Physikalische Zeitschrift, Vol. XXX, p. 177 (1929).1 The question of the emission of electrons by metals is in fact only a part of the general electron theory of metals, but in some respects it is simpler than any problem of internal structure, and therefore it is understandable that precisely this question has now already reached the first stage of its completion. We may consider that nonrelativistic quantum mechanics (which alone concerns us in the present question) is now already sufficiently developed, although to this day each of us encounters difficulties when he wants to set it forth clearly. In this theory in particular, Schrödinger has accustomed us to look at electrons, according to the circumstances, sometimes as wave packets, sometimes as trains of waves. Many of the most successful theorems explaining this question belong to Darwin [Proc. Roy. Soc. A. Vol. CXVII, 258 (1927)]. According to this theory, electrons must in every detail behave like waves of a definite length, which turns out to be equal to $\frac{h}{mV}$. This wavelength is often quite rightly called the de Broglie wavelength. The correspondence with radiation of the same wavelength turns out to be quite exact in all those questions that we have been able to work out in detail. Relying on this correspondence, we therefore have full grounds, making use of our optical experience and keeping in mind de Broglie waves, to introduce the corresponding approximations.

The length of the de Broglie wave of a “1-volt” electron, i.e. of an electron that has acquired acceleration in passing through a field with a potential drop of 1 V over 1 cm, is approximately $1.2 \times 10^{-7}$ cm.

For other field voltages,

\[ \text{wavelength}=\frac{1.2 \times 10^{-7}}{\sqrt{\text{voltage}}}. \]

From this it follows that slow electrons have a wavelength considerably greater than the distance between atoms in the crystalline lattice of the metal (3–4 times greater), and, by optical analogies, we are entitled to expect that such electrons will pass freely through the lattice, if it is perfectly regular.

I shall now say a few words about the picture of the structure of metals which I adopt. This picture is familiar to you: a conglomerate of positive ions more or less firmly fixed in space, immersed in an atmosphere of quite freely moving electrons, present in such a relative concentration that the mean volume charge is equal to zero. This conception, it seems to me, has been in force since the time of Drude’s theory of the metallic conduction of heat and electricity. In this conception the chief role in conduction is played by the so-called mean free path, which serves as an exact measure of the share of scattering in all possible directions and with all possible velocities of a beam of electrons moving in a definite direction and with a definite velocity.

This scattering is produced by the collision of electrons with the ions of the metal. In other words, the mean free path measures the spatial loss of directed momentum in a given beam. This mean free path must have a quite precisely defined value, whether we imagine electrons and atoms in the form of billiard balls or in the form of interacting sequences of waves, although the calculations, of course, will be different.

Using, as mentioned above, elementary optical analogies, we arrive at the conclusion that this free path in a pure metal must be fairly large—considerably longer than the mean distance between atoms. Studies on the influence of small traces of impurity in a metal on conductivity lead us to the conviction—independently of whatever theory we adopt—that electrons move freely through an undisturbed crystalline

lattice in a metal over distances hundreds of times greater than the distances between atoms. This was shown with sufficient clarity in Brillouin’s report at the Solvay Congress in 1924. Thus, theory and facts are in excellent agreement. I shall allow myself to emphasize this point more strongly. Long mean free paths were often considered difficult to explain. It may even be difficult for us to understand how they arise, given our ingrained conception of particles; but from the point of view of the theory of the structure of metals proposed by Sommerfeld and other scientists, they are easy to explain. Long mean free paths are, on the contrary, the fundamental requirement that the facts impose on any theory. We should regard it as a great success that wave mechanics, when applied to the electronic theory of metals, readily explains long mean free paths. Were this not so, one would have to declare either wave mechanics or the electronic theory of the metal to be incorrect.

Our main task is to consider the question of what will happen to an electron when it reaches the boundary of the metal. We shall think of the electron as a wave train, and imagine that the atomic structure of the metal has almost no effect on it, so that the group of waves reaches the boundary of the metal almost undistorted. In a first approximation we shall picture the boundary of the metal as a very narrow region (a plane layer—if the wavelength of the electron is taken as the scale), in which the principal change takes place, consisting, from the point of view of an individual electron, in its transition from the interior of the metal to the exterior. The initial stage of this change can evidently be represented figuratively as the ascent of the electron up a hill—corresponding to an increase in its potential energy at the expense of kinetic energy. We have abstracted from the structure of the electrical interactions in the metal, but in doing so we have obtained a certain average potential difference for electrons inside and outside the metal. This potential difference is only another expression of the forces of mutual attraction that hold the particles of the metal together and make the body precisely a metal and not a gas-

Thus we arrive logically and naturally at the starting point of those applications of the theory which I wish to discuss in this lecture. First of all we assert that the emission of electrons by metals can, in a first approximation, be treated as a phenomenon occurring in one dimension, as the passage of plane electron waves through (or across) a “ridge” of potential energy. In saying this, I am perhaps getting ahead of myself. The further explanations justifying the possibility of treating this problem in one dimension will show the possibility of reducing it to the question of the incidence of a group of waves in any direction upon a plane boundary field in which the potential energy depends on one Cartesian coordinate \(x\). But the wave equation given by Schrödinger has the property that for such a potential energy the motion along \(y\) and \(z\), at right angles to \(x\), is completely independent of the field, and the whole question is thus reduced to motion along the \(x\)-axis normal to the boundary, as I asserted above.

Typical cases of real and schematic boundary fields are shown in Figs. 1–5. Fig. 1 represents a boundary “hill” of energy which the electron must climb if one does not take into account the small forces of attraction acting on the electron when it has already escaped outside and is slowly moving away from the boundary. For simplified calculations with quantities of the first order this case may be represented by the step of Fig. 2. In both of these treatments the attractive forces, well known from elementary electrostatics, are disregarded; for a stationary, and hence also for a slowly moving, electron they give an attraction decreasing as \(\frac{e^{2}}{4x^{2}}\) and therefore may be represented by a potential energy \(\frac{e^{2}}{4x}\), where \(x\) is to be measured from the surface of the metal.

Such a rise is shown in Fig. 3. All these “hills” rise to the final height \(C\), which is thus the summit of the rise. There is, however, no reason why the summit of the rise should not be at a height \(B\) above the final level \(C\). Such a rise of the boundary curve—

physically depicted in Fig. 4 and, in a schematized form for a simplified calculation, in Fig. 5.

The problem of an electron incident on the boundary of a metal is the problem of a chain of electron waves incident on such a one-dimensional barrier.

In order to calculate the number of electrons escaping from the surface approximated by the above method, we must solve two further problems.

Fig. 1

Fig. 2

Fig. 3

1) We must determine what fraction of the electrons incident on the surface escape outward if the kinetic energy of their motion normal to the surface is \(W\). Let us call this fraction \(D(W)\). 2) We need to know how many such electrons arrive from within at unit area per unit time. Let us denote the number of electrons with energy varying within the limits \(W, W+dW\) by

\[ N(W)\cdot dW. \]

The emission of electrons will then give a current

\[ J=e\int_{0}^{\infty}N(W)\cdot D(W)\,dW \]

through unit area, where \(e\) denotes the charge of the electron. This, in general, is all that we need to know in order to compare our calculation with experiments, which for the most part con-

stand in the capture of electrons from unit area and in the determination of the current produced by them. We shall not dwell here on any detailed analysis of the velocity distribution of the emitted electrons.

The function \(N(W)\) can quite simply be obtained, apparently with an accuracy quite sufficient for our purpose, by using Sommerfeld’s theory, which gives the dependence

Fig. 4

Fig. 5

of \(N(W)\) on the temperature \(T\), the particular thermodynamic potential of the electrons \(\mu\), and, of course, \(W\).

We shall use Sommerfeld’s results without further comment. The formula for \(N(W)\) is, obviously, a simple consequence of the better-known distribution function \(f\). This distribution function, instead of the usual Maxwellian form, has the character graphically represented for the temperatures absolute zero (\(0^\circ\mathrm{K}\)) and \(1{,}500^\circ\mathrm{K}\) in Fig. 6. The corresponding values of the functions \(N(W)\) are shown in Fig. 7.

Fig. 6

There remains \(D(W)\), whose determination constitutes our special problem. It is well known that any surface emission of electrons is extremely sensitive to the quality of the surface layer. \(D(W)\) is a function of the surface and deserves more detailed study.

The value \(D(W)\) for the simple potential step shown in Fig. 2 can easily be computed exactly. If the electron were a classical particle, we would obtain:

\[ \begin{aligned} D(W)&=0 \qquad (W<C)\\ D(W)&=1 \qquad (W>C) \end{aligned} \]

In wave mechanics there is no such abrupt jump. In this case we obtain:

\[ D(W)=0 \qquad (W<C) \]

\[ D(W)=\frac{4\{(W-C)W\}^{1/2}}{\{W^{1/2}+(W-C)^{1/2}\}^{2}} \qquad (W>C) \]

The current produced by the emission of electrons at ordinary temperatures of thermionic emission corresponds to the emission of 20% of all electrons reaching the boundary with energy greater than \(C\). For gentler slopes of the curve the fraction of emitted electrons is larger, and as the slope decreases it rapidly approaches unity. For the force barrier due to the “image effect” and shown in Fig. 3, the fraction is about 95%. Wave mechanics does not introduce significant changes into the classical calculations, such as, for example, the Richardson and Dushman formulas.

If, however, the height of the barrier \(B\) is greater than \(C\), then wave mechanics gives an entirely different picture. According to classical mechanics we would obtain

\[ D(W)=0 \qquad (W<B) \]

But according to wave mechanics electrons can pass through a barrier of finite height, and we shall still obtain only

\[ D(W)=0 \qquad (W<C) \]

Here, with particles, we obtain an analogy to a well-known optical phenomenon. If light falls on the boundary between two media at an angle greater than the limiting angle, then total internal reflection occurs and the ray does not pass into the second medium at all. Nevertheless, in the second medium there arises an electromagnetic disturbance, decaying exponentially—

law and already at a distance of several wavelengths falls to zero. It varies in time with the ordinary frequency, but carries no energy. If, however, the second medium has a thickness of several wavelengths, the perturbation can pass through it and again emerge into the first medium in the form of a ray directed forward, only with a considerably reduced amplitude. Precisely the same thing also occurs with electrons.

A typical curve of the transparency coefficient \(D(W)\) is shown in Fig. 8.

Fig. 7.
Fig. 8.

Calculation of the current strength by means of the integral of \(D(W)\cdot N(W)\) leads to an expression of the following form

\[ J = 120\,D^{*}T^{2}e^{(C-\mu)/kT} \tag{1} \]

for the current through unit area, expressed in amperes. In this expression \(T\) denotes the absolute temperature, \(k\) the universal Boltzmann constant, and \(D^{*}\) the mean transparency coefficient for those electrons which can break out.

It is assumed here that no potential is applied outside the metal. In reality, however, a small potential difference is applied in order to collect the electrons, but its influence on the potential jump and on the distortion is so negligible that it may be disregarded. Under these conditions it was found experimentally that the current strength can be quite well expressed by the formula:

\[ J = AT^{2}e^{-\chi/kT}, \tag{2} \]

which precisely coincides with that found theoretically by the path-

therefore. For analysis, however, one must plot \(\log \dfrac{J}{T^2}\) as a function of \(\dfrac{1}{T}\). The resulting curve does not make it possible to determine the degree \(T\) with sufficient accuracy, or, if the degree \(T\) is known, to determine \(A\) with any precision.

For the bright surface of tungsten and metals similar to it, the observed value of \(A\) is approximately equal to 60. If the observed quantity \(\chi\) can be identified with the theoretical value \(C-\mu\), then \(D^*\) will correspondingly take the value \(1/2\). The most probable theoretical value is close to 1, but in such a rough calculation this may be regarded as a sufficiently good qualitative agreement, and no doubt arises as to the possibility of identifying \(C-\mu\) and \(\chi\). Here there is one attendant difficulty. There are metals, for example pure platinum, for which \(A\) apparently turns out to be approximately 10000. It is very probable that this is explained by the incorrectness of identifying the expressions \(C-\mu\) and \(\chi\), and that in reality the relation \(C-\mu=\chi_0-aT\) approximately holds. Then the observed value of \(A\) is expressed by the formula

\[ A=120D^*e^{\frac{a}{k}} \]

and the abnormal value of \(A\) will depend on \(a\). These points, however, are still not clear, and we still do not have sufficient theoretical knowledge concerning possible temperature oscillations.

Small contaminations of a surface that is pure until then can exert an astonishing effect both in the sense of increasing and in the sense of decreasing the emission of electrons. We shall consider only those cases in which the emission (at a given temperature) increases. It usually turns out that the new emission can also be expressed by formula (2), only with reduced coefficients \(A\) and \(\chi\). At the temperatures commonly used to determine the new emission, the effect of the decrease of \(\chi\) considerably exceeds the influence of the decrease of \(A\). The changes in \(A\) and \(\chi\) may be represented, in a rough approximation, by a formula of the type

\[ \log A=\xi-\eta(\Delta\chi), \tag{3} \]

in which \(\xi\) and \(\eta\) are constants, and \(\Delta \chi\) gives the decrease of \(\chi\) in comparison with a clean surface. An extremely pleasant feature of this theory is the possibility of obtaining from it, for \(A\) and \(\chi\), a relation of such a general form. A surface film, charged positively relative to the remaining mass of the metal, must form above the surface a double electric layer so arranged as to reduce the final work of emission of the electron \(C\). Thus \(C\) turns out to be reduced, but, on the other hand, the electrons must first pass through a barrier of the former height \(C\) (or close to it), and \(D^*\) correspondingly decreases. For the simplified picture of a rectangular barrier in Fig. 5, the relation between \(A\) and \(\chi\) has the form

\[ \log A = \xi - \eta l(\Delta \chi)^{1/2}. \tag{4} \]

where \(l\) denotes the thickness of the barrier. This formula is sufficiently similar to (3), because the thickness \(l\) may, with sufficient justification, be regarded as the thickness of a monomolecular layer.

In addition to the well-known emission of electrons at high temperatures and weak (insignificant) fields, electrons are also emitted at low temperatures and very strong fields; in this case the emission depends on the field strength and does not depend on the temperature (in the first approximation). In this second case, one may assume that there are in fact no electrons possessing sufficient energy to “jump over” the top of the barrier under the conditions of the problem described by us. All of them can only overcome the barrier, shown in simplified form in Fig. 9; if the “image effect,” which creates a force pulling the electrons back, is taken into account, then we obtain Fig. 10; if the existence of the surface layer is taken into account, a simplified representation of the process is given by Fig. 11. For a clean surface the theory gives, in this case, the following formula for the emission of electrons from unit area

\[ J = BF^{2} e^{-\frac{4}{3}\varkappa (C-\mu)^{3/2}/F} \qquad \left(\varkappa^{2}=\frac{8\pi^{2}m}{h^{2}}\right) \tag{5} \]

and the experimental data fit excellently into a formula of the type

\[ J = B F^{2} e^{-\frac{b}{F}} \tag{6} \]

The influences of surface layers, which simultaneously reduce \(b\) and \(B\), may also be taken into account quite well

Fig. 9

Fig. 10

with the aid of the rise in Fig. 11. An analysis of this film effect was carried out by me together with Gossling and Stern, on the basis of experimental work done several years ago by Gossling for the General Electric Co. A full account of our work was published in Proc. Royal Society, July 1929.

Fig. 11

The magnitudes of the field strength \(F\) required in order to obtain, from (5), an emission with the observed current–voltage characteristic usually, though not always, considerably exceed those values which can be calculated from the applied voltage and the dimensions of the apparatus. In some cases such an effective field exceeds the “geometrical” one by a factor of one hundred, but we have also found examples in which the ratio is no more than 2. This proves that, under the influence of a strong field, electrons are emitted not from the entire surface of the metal, but only from individual submicroscopic regions or irregularities of the surfaces.

Such an assumption agrees well with the well-known wandering character of the discharge in strong fields and with its intense localization. On the other hand, it may also be regarded as favorable that in some cases the number of emitting centers differs only slightly from the entire emitting surface.

Combined effects of temperature and strong field have also been discovered and satisfactorily reconciled with theory. One of the best-known effects of this kind is the Schottky effect, which is the action of a strong field on ordinary thermionic emission. This effect becomes immediately understandable if one takes into account the “image field”; and its presence serves precisely as convincing proof of the reality of the action of this image field on the emitted electrons. The effect is expressed (see Fig. 10) in a lowering of the height of the barrier and, consequently, in a decrease of the effective work function proportional to \(\sqrt{F}\). Therefore the emission is increased by the factor

\[ e^{\frac{a\sqrt{F}}{T}} \]

in comparison with the value for a weak field. The correctness of introducing this factor has been verified many times, and in one of the recent papers by de Bruyne the theoretically calculated value of \(a\) was confirmed by experimental data to within a few percent.

In a recently published paper, Millikan asserts that he has discovered a small temperature effect in the emission of electrons in strong fields, as a result of which formula (6) must be correspondingly modified, so that one obtains the formula:

\[ J=(B+B'T)F^2 e^{-\frac{b}{F}} \]

According to the calculations of Houston, who computed formula (5) to a higher approximation, such a change in formula (6) is, in order of magnitude, consistent with the theory.

For completeness, I shall mention in conclusion also the last type of electron emission—the photoelectric effect. The theory easily explains both the existence of a photoelectric threshold, rather sharp at ordinary temperatures, and its identity with the work function of thermionic emission. All this is quite simple; but when one turns to the question of the magnitude of the photoelectric effect and of the relative intensities, a considerably more detailed investigation is already required. The first and very successful work on this question was done by Wentzel and published in the volume in honor of Sommerfeld: Sommerfeld’s Festschrift, Probleme der modernen Physik.

The question of electron emission in the first stage may thus be regarded as completed, as I asserted at the beginning of my lecture, and I bring my address to an end. But the history of electron emission does not end here, and one must now proceed to the second stage of solving the question, no longer making such severe simplifications as those which allowed us to disregard the atomic structure of the metallic field.

1. I take this opportunity to express my gratitude to Dr. Nordheim for permission to reproduce in this article some of his drawings. ↩↩