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Rydberg Redivivus in Surface Physics
RYDBERG REDIVIVUS IN SURFACE PHYSICS J. Rundgren and G.
Malmstr~m
Department of Theoretical Physics Royal Institute of Technology 100 44 Stockholm, Sweden Beam threshold effects have been observed in low-energy electron diffraction from a very early date. These effects were discovered &s fine structures in the intensity of the specularly reflected beam. They occur in a 1-2 eV range below the energy of grazing emergence of a new beam. In 1977 it became clear that at least some of the fine structures were induced by the image potential barrier in front of a metal surface. The coulombic potential on one side of the surface plane and the metal reflectivity on the other act as a two-dimensional waveguide which can hold metastable states arranged in a Rydberg series of spectral constant 1/16 Ryd. At present up to three levels of the series can be disentangled by a high-resolution LEED apparatus.
52
1. Introduction The explanation of the Rydberg series, the energy spectrum of a hydrogenic atom, can be taken as a symbol of the success of early quantum theory. Now, some seventy years later, the Rydberg series again appears as an interesting feature of a late development of quantum mechanics: surface physics. In classical electrostatics, the repertoire of the Coulomb force law does not only comprise the interaction between charged points, but also the interactio~ between a charged particle and the surface of a dielectric solid. Naturally, the latter kind of interaction is a manifestation of surface physics. If now, without bothering too much about the details, we assume that an electron could be bound just outside the surface of a dielectric solid, what would be the measurable consequenses? To specify, we thin~ of a geometry in which the half-space z > 0 is filled with material of dielectric constant £, and the part of space z < 0 is empty but for a single electron. This electron would then feel the image potential
V(z)
----
£+1
2z
,z
where Rydberg atomic units have been used, as they will be throughout this paper. This potential is equivalent to the radial s-state potential of the three-dimensional problem of an electron interacting with a point centre of charge 1/2 (£-1)/(£+1), and so we can immediately say that the electron bound at the surface, if it exists, will have the energy spectrum
53
(1)
En • - (
£-1 .. £+1
) 2 - -....... 2 ' (n+a)
n"'1,2, .••
(2)
which is commonly referred to as the Rydberg series. A quantity ~, the quantum defect or the Rydberg correction, which is more or less constant, has to be added to n as soon as the boundary conditions at z = 0 are different from those of the hydrogenic problem. The Rydberg series (2) has been proposed and measured in two widely different areas of surface physics, in lowtemperature physics in investigations of the surface of liquid helium, and in low-energy electron diffraction (LEED) experiments on the structure of metallic surfaces. The surface of liquid helium actually can hold electrons in bound states. The liquid helium application of the Rydberg series came firstf it was suggested in 1964 and definitely confirmed in 1974. For a rewiev of this field, see Cole (1974). The surface of a metal, on the other hand, can not hold extra electrons, but if the surface is exposed to an electron beam supplying electrons continously, the metallic surface is able to detain the electrons for short periods of time in metastable states. These states are called surface resonances, and they are accompanied by a fine structure in the observed LEED intensities, which proves to be arranged in a Rydberg series. The theoretical interpretation was proposed in 1977 (Rundgren and MalmstrOm 1977a) and confirmed by a measurement on the surface of tungsten in the same year (Adnot and Carette 1977a). But the Rydberg resonances had been trying to announce their existence for a long time. It is believed that already Davisson and Germer did see a low resolution picture of the Rydberg resonances in their LEEO experiments in the late 1920's. In the 1960's, when LEEO became a key method in the
54
rapidly expanding field of surface physics, low resolution pictures of Rydberg resonances were again observed, but because of the low resolution the experiments gave no clear hint as to the mechanism behind. In section 2 we will tell how we came across the resonances in our LEED calculations. Section 3 deals with the development of LEED after the electronic surface resonances were explained. Finally, in section 4, we discuss the type of information that the Rydberg surface resonances can give about the surface of a metal.
55
2. Computer discovery of a Rydberg series This section tells about how a computer program turned out to know more about surface physics than the scientists who wrote the code. First, let us consider some fundamental facts about LEED. A beam of electrons, with energies in the interval
o-
200 eV, is directed towards the plane surface of a
metal, giving rise to electron diffraction. As a result a bunch of beams are reflected back into the vacuum from the crystal surface. The energies of propagation parallel
to
the surface available to the beams are
(3)
where
k"
is the surface component of the wavevector of the
g is
primary beam and the vector
a two-dimensional reci-
procal lattice vector of the surface. In consequence, the energies available for propagation normal to the surface will be
in the vacuum:
E+ g.Lvac
=E
in the crystal: EgLcrys
- E+ gil
E - V
o - i6 -
(4) E g lI
where E is the primary electron energy in the vacuum and V
o
+ i6 is the inner potential of the crystal. The inner
potential is taken to be complex so as to be able to explain the decrease in intensity due to inelastic collisions (in LEED experiments only the elastically scattered electrons are detected).
56
An interesting situation occurs when the primary beam
has such an energy that some beam g is evanescent (dies off) in the vacuum, Eg~vac < 0, while at the same time it is free to propagate in the crystal,
Re(Eg~crys)
> O.
These inequalities are the necessary conditions for surface resonances to occur. A beam of
the kind considered
coming from within the crystal and travelling in the negative z direction is reflected by the potential barrier of height Vo in front of the crystal. Subsequently, it may again be reflected back towards the surface by the crystal, and so
forth. Under circumstances, these multiple scattered beams can interfere constructively, and a surface resonance will be maintained. In more physical terms, the slice of space between the outermost atomic layer and the image potential barrier acts as a two dimensional waveguide which is capable of trapping the electrons of the beam g in metastable states. For a moment one would perhaps think that a metastable state in a beam that does not come out into the vacuum would be impossible to observe. But the law of conservation of current associated with the diffraction ensures that the beams propagating in the vacuum are all influenced by a resonance in one of the beams. Observations of surface resonances in LEED are usually done in an energy range, 10 - 20 eV, where one single beam, the specular one, is reflected by the crystal. The primary energy is then increased until a new beam emerges, and the surface resonances are observed as a modulation of the intensity of the specular beam immediately before the new beam emerges. Ob~ervations
of LEED intensities indicate that electronic
surface resonances occur over roughly a 1 eV range below the emergence of a new beam. So much for the theoretical and experimental description of LEED resonances up to 1974 (McRae 1966, 1971, Jennings 1971, Jennings and Read 1974). Towards the end of 1974 our LEED group set out to
57
calculate the reflectivity of the potential barrier at a metal surface. A computer code was written which was based on a model potential which changed smoothly from a constant value V
o
inside the metal to the image potential
outside, and tended to zero far away from the metal surface. The inner potential was assumed to extend just through the outermost atomic layer. To this real potential we added an imaginary, absorptive part, which tailed off -2 like z , or in an exponential manner, in the vacuum, and approached the value i8 just inside the crystal surface. The motivation for this work was to be able to do very accurate
LEED studies of surface structures, i.e. to
find interatomic distances and interlayer separations. None of us thought of the Rydberg spectrum (2). Moreover, in LEED studies of crystal geometry, intensities are usually calculated with 1 or 2 eV increments in energy, so that a dense spectrum like (2) would be very likely to slip out of the energy grid and escape observation. In 1975 Henrich made recordings of the energy derivative of LEED intensity profiles, with the aim of studying resonance features associated with the emergence of new beams. He used the (100) surface of aluminium and various angles of incidence near the normal, and described the electronic surface resonance structures near 18 eV, where the first non-specular beams begin to emerge. Out of curiosity we simulated the experiment, doing a LEED calculation with
an energy grid of only 0.01 eV spacing.
The intensity structure revealed by the computer was amazing (figure 1). So, in this way we were told by the computer about the Rydberg spectrum (2), and because Rydberg resonances must be intrinsically connected. with LEED and metal surfaces we understood that Henrich's intensities, which only vaguely did ressemble our computed ones, were in fact deformed by the limited resolution of the LEED apparatus he had used.
58
It may seem astonishing that perfectly sharp resonance fringes accumulated as densely as the Rydberg series (2) are compatible with the use of an absorptive potential. Absorption usually implies line broadening. The explanation is that the image potential, which decreases as z-1 in the vacuum, has an infinitely long range, whereas the absorptive part of the potential, which decreases as z-2, has not. This means that the reflection against the barrier is effectively elastic. Shortly after our ideas about the electronic surface resonances of the Rydberg type were published (Rundgren and MalmstrOm 1977a), the existence of such resonances was confirmed by a high resolution LEED experiment, in which three different resonance fringes were disentangled. (Adnot and Carette 1977a). Afterwards we extracted some analytical, more or less model independent results for the Rydberg resonances (Rundgren and MalmstrOm 1977b). The most interesting property of the image potential barrier with absorption is its crysta1-barrier-crysta1 reflectivity r+-. In an energy interval IE+ I < 1 eV, it is natural to ~epresent + g.1.vac 1/2 r+= as a function of (Eg~vac), and it turns out that r has an essential singularity at Eg.1.vac = O. This is a typical feature of the Schr6dinger equation with a Coulomb potential. The way the reflectivity r+- depends on the energy is illustrated most clearly by a graph in the complex plane, see figure 2. It turns out that r+- tends to a limiting circle as E+.1.V approaches zero from negative +_ g ac energies, and that r makes one revolution on the limiting circle for each level in the Rydberg series
n = 1,2, •••
where a n varies slowly with n. When E+ goes to zero g~vac
59
(5)
through positive values, on the other hand, r+- tends to a constant. A remarkable fact is that the limiting circle of r+- is transferred through the LEED theory in such a way that the complex amplitudes of the diffracted beams revvlve on limiting circles synchronously with r+-.(The complex amplitude is generally a long radius vector, the point of which revolves on a ~ circle.) Hence, it is easily seen that the beam intensities, beeing absolute amplitudes squared, exhibit intensity fringes corresponding to the levels En of the Rydberg series (5), see figure 3. Recently we have presented a more complete mathematical description of the scattering of electrons by the image potential barrier (Malmstrom and Rundgren 1980a). We establish that the S-matrix for LEED at energies above the threshold .for beam emergence is closely related to the S-matrix for s-wave scattering by a Coulomb potential. Therefore the Rydberg resonances in LEED have a perfectly smooth continuation above the beam emergence to which they belong.
60
3. Current work on LEEO surface resonances The suggestion that a Rydberg series of metastable surface states is the mechanism behind the beam threshold effects in LEEO gave the impetus to new work on electronic surface resonances in several laboratories. Adnot and Carette (1977b) were the first to accept the interpretation given. Echenique and Pendry (1978) published a paper on Rydberg states at surfaces in which they studied the line width of the resonance fringes measured in LEEO. Jennings and coworkers at the University of Murdoch, Australia, have continued their LEEO calculations on resonance structures (Read and Jennings 1978, Jennings 1978, 1979, Jennings and Price 1980a,b, Price et al. 1979). They have started a series of case studies of LEEO resonances using a computing machinery similar to ours. Until today only Adnot and Carette have done measurements with a sUfficently high resolution to be able to confirm the existence of Rydberg surface states. In his computer simulation of this experiment, Jennings (1978) claimed to have found a fine structure ~ the energy of emergence of the new beam. However, according to our (1980a) paper the Rydberg resonance fringes should exist only below the emergence of a new beam, and the observed intensity above the emergence should be perfectly smooth, in contradiction to this part of Jennings's result. The disagreement is most likely a result of the difference in numerical methods used. Whereas Jennings integrates the Schrodinger equation from the metal surface out about 100 A and joins the wavefunction to a plane wave in the vacuum, we have integrated over only a distance of 2 - 5 A and joined the wavefunction to the correct solutions (Whittaker functions) in the vacuum. Our barrier transmission program is now published (Malmstrom and Rundgren 1980b).
61
McRae (1971) introduced the concept of a cwo-dimensional wave guide to explain the mechanism whereby a crystal surface is able to trap electrons in metastable states. But it seems that his work on beam threshold effects was delayed by the assumption (1971) that the inelastic electron processes in the crystal would prevent any resonance feature to have structure on a scale less than 2 - 8 eV (depending on Whether the energy of the incoming electrons lies below or above the threshold for plasmon creation). We have seen that this restriction does not apply, since the metastable, resonant states are localized outside the crystal, in a region of space Where there is very little damping of the electrons. McRae has recently concentrated his attention on the beam threshold effects in LEED, utilizing experimental data from Bell Laboratories (McRae, Landwehr and Caldwell 1977, McRae and Caldwell 1978) and from the CNRS surface physics laboratory in Grenoble (McRae, Aberdam, Baudoing and Gauthier 1978). He has also written two rewiev articles (McRae 1979a,b) on the subject. The LEED apparatuses used in these experiments are capable of resolving one fringe at the low end of the Rydberg series (2), i.e. one lying approximately - 1/16 Ryd or - 0.85 eV below the beam emergence. However, in the papers from 1978 just mentioned, the authors place the observed resonance features at or above the emergence in the first case, and 3.5 eV below in the second case. Both these results, which seemingly contradict each other, are at variance with our (1980a)1 prediction that the resonance structure caused by the image potential is to be found in an interval of length: 0.85 eV below the emergence of a new beam, with ~ structure above. It will be interesting to see if there is another type of electronic resonances at metal surfaces in addition to the Rydberg states, or if the experimentators were mistaken about the energy scale in their measurements.
62
The surface physics group at the National Bureau of Standards have designed a spin polarized electron source (Pierce et al. 1980) which they use in experiments on spin polarized LEEO. Rydberg resonances are observed in LEEO on tungsten, and it turns out that the fine structure looks different for the two spin directions (Pierce, private communication). The question arises whether the reflectivity of the image potential has a substantial spin dependence. A calculation shows that the dependence is minute (MalmstrOm and Rundgren 1980c). The spin up and spin down reflectivities are separated by a~ of the order of a 2, where a is the fine structure constant 1/137. This is in perfect analogy with the spin dependence of the hydrogenic spectrum where the spin up and spin down levels are of the order of a 2 apart. The spin correction to the surface barrier reflec~ivity can be calculated exactly by means of our computer program. The conclusion with respect to the Rydberg resonances of tungsten is therefore that the fine structures are different because the bulk reflectivity is very sensitive to spin.
63
4. Conclu8ion and prospects What can the surface resonances as observed in LEED tell us about the properties of the metal surface? To begin with, the resonances are likely to be useful for the experimental technique of LEED. They offer a possibility of a quick determination of the resolving power of the apparatus used by simply doing a count of the number of resonance fringes observed. At the same time, the n + m end of the series of fringes give accurate ·calibration marks· on the recorded intensity curves, and give a precise relationship between the energy and angle of incidence of the primary electron beam. In our opinion the Rydberg surface states in LEED can also give information about the electron density near a metal surface. Firstly, the radius of the limiting circle of the reflectivity r+- (shown in figure 2) depends strongly on the absorptive part of the potential at the surface, or rather on the integrated effect of the absorptive tail of the potential in vacuum. The absorption can be measured from the height of the fringes relative to the average LEED intensity of the background in the region where the resonance occurs. Secondly, by a very accurate LEED measurement, the Rydberg correction an of (5) could be determined for a number of different values of n. As indicated in the introduction, an depends of the reflectivity of the crystal bulk, and McRae (1979b) has made theoretical predictions of the sensitivity of an to various physical parameters of the surface. It is at present an open question to what extent an depends on that portion of the surface barrier potential which connects the Coulomb tail in the vacuum to the inner potential in the bulk. We plan an investigation of an by means of LEED calculations on the surface of aluminium.
64
The metal surface of classical electrostatics, i.e. the plane acting as a mirror for an external point charge, is defined to be the plane z = 0 in the expression (1). We believe that the physical location of the plane z = 0 with respect to the outermost atomic layer can be found from recordings of LEED intensities. We have begun such an investigation, but have found that only a very restricted selection of existing LEED data, if any, is accurate enough for this purpose. The location of the classical surface is an important parameter in the theory of a metal surface as developed by Lang and Kohn (1973) who also were able to predict a numerical value for this parameter. To obtain an experimental value for this parameter would be a most interesting check on the soundness of the theory.
65
References Adnot A and Carette J 0 1977a Phys. Rev. Lett. 38 1084-7 Adnot A and Carette J 0 1977b Phys. Rev. B ~ 4703-4 Cole M W 1974 Rev. Mod. Phys. i i 451-64 Echenique P M and Pendry J B 1978J. Phys. C 11 2065-75 Henrich V E 1975 Surface Sci. 12 675-80 Jennings P J 1971 Surface Sci. l1 513-25 Jennings P J 1978 Surface Sci. 75 773-6 Jennings P J 1979 Surface Sci. 88 L25-8 Jennings P J and Price G L 1980a Surface Sci. 2l L124-8 Jennings P J and Price G L 1980b Surface Sci. ~ L205-9 Jennings P J and Read M N 1974 Surface Sci. i l 113-24 Lang N 0 and Kohn W 1973 Phys. Rev. B 2 3541-50 McRae E G 1966 J. Chern. Phys. 45 3258-76 McRae E G 1971 Surface Sci. 25 491-512 McRae E G 1979a J. Vac. Sci: Technol. ~ 654-9 McRae E G 1979b Rev. Mod. Phys. 21 541-68 McRae E G, Aberdam 0, Baudoing R and Gauthier Y 1978 Surface Sci. ~ 518-30 McRae E G and Caldwell C W 1978 Surface Sci. 74 285-306 McRae E G, Landwehr J M and Caldwell C W 1977 Phys. Rev. Lett. 38 1422-5 Malmstram G and Rundgren J 1980a J. Phys. C 11 L61-5 Malmstram G and Rundgren J 1980b Comput. Phys. Commun.
12
263~70
MalmstrOm G and Rundgren J 1980c (to be published) Pierce 0 T, Celotta R J, Wang G C, Unertl W N, Galejs A, Kuyatt C E and Mielczarek 5 R 1980 Rev. Sci. Instrum. ~ 478-99. Price G L, Jennings P J, Best P E and Cornish J C L Surface Sci. ~ 151-8 Read M N and Jennings P J 1978 Surface Sci. 2! 54-68 Rundgren J and MalmstrOm G 1977a Phys. Rev. Lett. 38 836-9 Rundgren J and MalmstrOm G 1977b J. Phys. C 10 4671-87
66
1.'2.
1.1
'" I 0
::
...8
1.0
., .S 17- 5
18.0 Energy leV)
Figure 1 A Rydberg resonance in LEED from aluminium near 18 eV primary energy and normal incidence. The fine structure in the intensity 1 of the spec00 ular beam ('the beam reflected as by a mirror) indicates that new beams are about to emerge at 18.3 eVe
67
I
..
+
,';Ot-----'~_\"~-~_t_-__::__~-I_--_1
-.J
-·3
o Re r+-
Figure 2 The crystal-harrier-crystal reflection coefficient r+- illustrated by a graph in the complex plane. Along the curves are marks indicating the energy of propagation normal to the surface, EgLvac. The branch for negative energy, below beam emergence, terminates in a limiting circle when E+ + -0. The branch for positive energy, above gLvac beam emergence, is the short piece of curve, beginning at a finite value for terminating at zero for
Eg~vac
68
Eg~vac
+
~.
=
0 and
7,...-----....-----,r--------r---::o..----.
....I
...o
(,
o
c -e
s
-g
Figure 3 The LEED beams in the vacuum in front of the crystal are plane waves exp(ik+.~) having complex amplig tudes A The graph shows the amplitude AOO for the specular beam for the Rydberg resonance in figure 1. ADD is represented by a radius vector, the point of which moves, like r+-, on a limiting circle when the energy of beam emergence is approached.
g.
69
Malmstr~m
Department of Theoretical Physics Royal Institute of Technology 100 44 Stockholm, Sweden Beam threshold effects have been observed in low-energy electron diffraction from a very early date. These effects were discovered &s fine structures in the intensity of the specularly reflected beam. They occur in a 1-2 eV range below the energy of grazing emergence of a new beam. In 1977 it became clear that at least some of the fine structures were induced by the image potential barrier in front of a metal surface. The coulombic potential on one side of the surface plane and the metal reflectivity on the other act as a two-dimensional waveguide which can hold metastable states arranged in a Rydberg series of spectral constant 1/16 Ryd. At present up to three levels of the series can be disentangled by a high-resolution LEED apparatus.
52
1. Introduction The explanation of the Rydberg series, the energy spectrum of a hydrogenic atom, can be taken as a symbol of the success of early quantum theory. Now, some seventy years later, the Rydberg series again appears as an interesting feature of a late development of quantum mechanics: surface physics. In classical electrostatics, the repertoire of the Coulomb force law does not only comprise the interaction between charged points, but also the interactio~ between a charged particle and the surface of a dielectric solid. Naturally, the latter kind of interaction is a manifestation of surface physics. If now, without bothering too much about the details, we assume that an electron could be bound just outside the surface of a dielectric solid, what would be the measurable consequenses? To specify, we thin~ of a geometry in which the half-space z > 0 is filled with material of dielectric constant £, and the part of space z < 0 is empty but for a single electron. This electron would then feel the image potential
V(z)
----
£+1
2z
,z
where Rydberg atomic units have been used, as they will be throughout this paper. This potential is equivalent to the radial s-state potential of the three-dimensional problem of an electron interacting with a point centre of charge 1/2 (£-1)/(£+1), and so we can immediately say that the electron bound at the surface, if it exists, will have the energy spectrum
53
(1)
En • - (
£-1 .. £+1
) 2 - -....... 2 ' (n+a)
n"'1,2, .••
(2)
which is commonly referred to as the Rydberg series. A quantity ~, the quantum defect or the Rydberg correction, which is more or less constant, has to be added to n as soon as the boundary conditions at z = 0 are different from those of the hydrogenic problem. The Rydberg series (2) has been proposed and measured in two widely different areas of surface physics, in lowtemperature physics in investigations of the surface of liquid helium, and in low-energy electron diffraction (LEED) experiments on the structure of metallic surfaces. The surface of liquid helium actually can hold electrons in bound states. The liquid helium application of the Rydberg series came firstf it was suggested in 1964 and definitely confirmed in 1974. For a rewiev of this field, see Cole (1974). The surface of a metal, on the other hand, can not hold extra electrons, but if the surface is exposed to an electron beam supplying electrons continously, the metallic surface is able to detain the electrons for short periods of time in metastable states. These states are called surface resonances, and they are accompanied by a fine structure in the observed LEED intensities, which proves to be arranged in a Rydberg series. The theoretical interpretation was proposed in 1977 (Rundgren and MalmstrOm 1977a) and confirmed by a measurement on the surface of tungsten in the same year (Adnot and Carette 1977a). But the Rydberg resonances had been trying to announce their existence for a long time. It is believed that already Davisson and Germer did see a low resolution picture of the Rydberg resonances in their LEEO experiments in the late 1920's. In the 1960's, when LEEO became a key method in the
54
rapidly expanding field of surface physics, low resolution pictures of Rydberg resonances were again observed, but because of the low resolution the experiments gave no clear hint as to the mechanism behind. In section 2 we will tell how we came across the resonances in our LEED calculations. Section 3 deals with the development of LEED after the electronic surface resonances were explained. Finally, in section 4, we discuss the type of information that the Rydberg surface resonances can give about the surface of a metal.
55
2. Computer discovery of a Rydberg series This section tells about how a computer program turned out to know more about surface physics than the scientists who wrote the code. First, let us consider some fundamental facts about LEED. A beam of electrons, with energies in the interval
o-
200 eV, is directed towards the plane surface of a
metal, giving rise to electron diffraction. As a result a bunch of beams are reflected back into the vacuum from the crystal surface. The energies of propagation parallel
to
the surface available to the beams are
(3)
where
k"
is the surface component of the wavevector of the
g is
primary beam and the vector
a two-dimensional reci-
procal lattice vector of the surface. In consequence, the energies available for propagation normal to the surface will be
in the vacuum:
E+ g.Lvac
=E
in the crystal: EgLcrys
- E+ gil
E - V
o - i6 -
(4) E g lI
where E is the primary electron energy in the vacuum and V
o
+ i6 is the inner potential of the crystal. The inner
potential is taken to be complex so as to be able to explain the decrease in intensity due to inelastic collisions (in LEED experiments only the elastically scattered electrons are detected).
56
An interesting situation occurs when the primary beam
has such an energy that some beam g is evanescent (dies off) in the vacuum, Eg~vac < 0, while at the same time it is free to propagate in the crystal,
Re(Eg~crys)
> O.
These inequalities are the necessary conditions for surface resonances to occur. A beam of
the kind considered
coming from within the crystal and travelling in the negative z direction is reflected by the potential barrier of height Vo in front of the crystal. Subsequently, it may again be reflected back towards the surface by the crystal, and so
forth. Under circumstances, these multiple scattered beams can interfere constructively, and a surface resonance will be maintained. In more physical terms, the slice of space between the outermost atomic layer and the image potential barrier acts as a two dimensional waveguide which is capable of trapping the electrons of the beam g in metastable states. For a moment one would perhaps think that a metastable state in a beam that does not come out into the vacuum would be impossible to observe. But the law of conservation of current associated with the diffraction ensures that the beams propagating in the vacuum are all influenced by a resonance in one of the beams. Observations of surface resonances in LEED are usually done in an energy range, 10 - 20 eV, where one single beam, the specular one, is reflected by the crystal. The primary energy is then increased until a new beam emerges, and the surface resonances are observed as a modulation of the intensity of the specular beam immediately before the new beam emerges. Ob~ervations
of LEED intensities indicate that electronic
surface resonances occur over roughly a 1 eV range below the emergence of a new beam. So much for the theoretical and experimental description of LEED resonances up to 1974 (McRae 1966, 1971, Jennings 1971, Jennings and Read 1974). Towards the end of 1974 our LEED group set out to
57
calculate the reflectivity of the potential barrier at a metal surface. A computer code was written which was based on a model potential which changed smoothly from a constant value V
o
inside the metal to the image potential
outside, and tended to zero far away from the metal surface. The inner potential was assumed to extend just through the outermost atomic layer. To this real potential we added an imaginary, absorptive part, which tailed off -2 like z , or in an exponential manner, in the vacuum, and approached the value i8 just inside the crystal surface. The motivation for this work was to be able to do very accurate
LEED studies of surface structures, i.e. to
find interatomic distances and interlayer separations. None of us thought of the Rydberg spectrum (2). Moreover, in LEED studies of crystal geometry, intensities are usually calculated with 1 or 2 eV increments in energy, so that a dense spectrum like (2) would be very likely to slip out of the energy grid and escape observation. In 1975 Henrich made recordings of the energy derivative of LEED intensity profiles, with the aim of studying resonance features associated with the emergence of new beams. He used the (100) surface of aluminium and various angles of incidence near the normal, and described the electronic surface resonance structures near 18 eV, where the first non-specular beams begin to emerge. Out of curiosity we simulated the experiment, doing a LEED calculation with
an energy grid of only 0.01 eV spacing.
The intensity structure revealed by the computer was amazing (figure 1). So, in this way we were told by the computer about the Rydberg spectrum (2), and because Rydberg resonances must be intrinsically connected. with LEED and metal surfaces we understood that Henrich's intensities, which only vaguely did ressemble our computed ones, were in fact deformed by the limited resolution of the LEED apparatus he had used.
58
It may seem astonishing that perfectly sharp resonance fringes accumulated as densely as the Rydberg series (2) are compatible with the use of an absorptive potential. Absorption usually implies line broadening. The explanation is that the image potential, which decreases as z-1 in the vacuum, has an infinitely long range, whereas the absorptive part of the potential, which decreases as z-2, has not. This means that the reflection against the barrier is effectively elastic. Shortly after our ideas about the electronic surface resonances of the Rydberg type were published (Rundgren and MalmstrOm 1977a), the existence of such resonances was confirmed by a high resolution LEED experiment, in which three different resonance fringes were disentangled. (Adnot and Carette 1977a). Afterwards we extracted some analytical, more or less model independent results for the Rydberg resonances (Rundgren and MalmstrOm 1977b). The most interesting property of the image potential barrier with absorption is its crysta1-barrier-crysta1 reflectivity r+-. In an energy interval IE+ I < 1 eV, it is natural to ~epresent + g.1.vac 1/2 r+= as a function of (Eg~vac), and it turns out that r has an essential singularity at Eg.1.vac = O. This is a typical feature of the Schr6dinger equation with a Coulomb potential. The way the reflectivity r+- depends on the energy is illustrated most clearly by a graph in the complex plane, see figure 2. It turns out that r+- tends to a limiting circle as E+.1.V approaches zero from negative +_ g ac energies, and that r makes one revolution on the limiting circle for each level in the Rydberg series
n = 1,2, •••
where a n varies slowly with n. When E+ goes to zero g~vac
59
(5)
through positive values, on the other hand, r+- tends to a constant. A remarkable fact is that the limiting circle of r+- is transferred through the LEED theory in such a way that the complex amplitudes of the diffracted beams revvlve on limiting circles synchronously with r+-.(The complex amplitude is generally a long radius vector, the point of which revolves on a ~ circle.) Hence, it is easily seen that the beam intensities, beeing absolute amplitudes squared, exhibit intensity fringes corresponding to the levels En of the Rydberg series (5), see figure 3. Recently we have presented a more complete mathematical description of the scattering of electrons by the image potential barrier (Malmstrom and Rundgren 1980a). We establish that the S-matrix for LEED at energies above the threshold .for beam emergence is closely related to the S-matrix for s-wave scattering by a Coulomb potential. Therefore the Rydberg resonances in LEED have a perfectly smooth continuation above the beam emergence to which they belong.
60
3. Current work on LEEO surface resonances The suggestion that a Rydberg series of metastable surface states is the mechanism behind the beam threshold effects in LEEO gave the impetus to new work on electronic surface resonances in several laboratories. Adnot and Carette (1977b) were the first to accept the interpretation given. Echenique and Pendry (1978) published a paper on Rydberg states at surfaces in which they studied the line width of the resonance fringes measured in LEEO. Jennings and coworkers at the University of Murdoch, Australia, have continued their LEEO calculations on resonance structures (Read and Jennings 1978, Jennings 1978, 1979, Jennings and Price 1980a,b, Price et al. 1979). They have started a series of case studies of LEEO resonances using a computing machinery similar to ours. Until today only Adnot and Carette have done measurements with a sUfficently high resolution to be able to confirm the existence of Rydberg surface states. In his computer simulation of this experiment, Jennings (1978) claimed to have found a fine structure ~ the energy of emergence of the new beam. However, according to our (1980a) paper the Rydberg resonance fringes should exist only below the emergence of a new beam, and the observed intensity above the emergence should be perfectly smooth, in contradiction to this part of Jennings's result. The disagreement is most likely a result of the difference in numerical methods used. Whereas Jennings integrates the Schrodinger equation from the metal surface out about 100 A and joins the wavefunction to a plane wave in the vacuum, we have integrated over only a distance of 2 - 5 A and joined the wavefunction to the correct solutions (Whittaker functions) in the vacuum. Our barrier transmission program is now published (Malmstrom and Rundgren 1980b).
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McRae (1971) introduced the concept of a cwo-dimensional wave guide to explain the mechanism whereby a crystal surface is able to trap electrons in metastable states. But it seems that his work on beam threshold effects was delayed by the assumption (1971) that the inelastic electron processes in the crystal would prevent any resonance feature to have structure on a scale less than 2 - 8 eV (depending on Whether the energy of the incoming electrons lies below or above the threshold for plasmon creation). We have seen that this restriction does not apply, since the metastable, resonant states are localized outside the crystal, in a region of space Where there is very little damping of the electrons. McRae has recently concentrated his attention on the beam threshold effects in LEED, utilizing experimental data from Bell Laboratories (McRae, Landwehr and Caldwell 1977, McRae and Caldwell 1978) and from the CNRS surface physics laboratory in Grenoble (McRae, Aberdam, Baudoing and Gauthier 1978). He has also written two rewiev articles (McRae 1979a,b) on the subject. The LEED apparatuses used in these experiments are capable of resolving one fringe at the low end of the Rydberg series (2), i.e. one lying approximately - 1/16 Ryd or - 0.85 eV below the beam emergence. However, in the papers from 1978 just mentioned, the authors place the observed resonance features at or above the emergence in the first case, and 3.5 eV below in the second case. Both these results, which seemingly contradict each other, are at variance with our (1980a)1 prediction that the resonance structure caused by the image potential is to be found in an interval of length: 0.85 eV below the emergence of a new beam, with ~ structure above. It will be interesting to see if there is another type of electronic resonances at metal surfaces in addition to the Rydberg states, or if the experimentators were mistaken about the energy scale in their measurements.
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The surface physics group at the National Bureau of Standards have designed a spin polarized electron source (Pierce et al. 1980) which they use in experiments on spin polarized LEEO. Rydberg resonances are observed in LEEO on tungsten, and it turns out that the fine structure looks different for the two spin directions (Pierce, private communication). The question arises whether the reflectivity of the image potential has a substantial spin dependence. A calculation shows that the dependence is minute (MalmstrOm and Rundgren 1980c). The spin up and spin down reflectivities are separated by a~ of the order of a 2, where a is the fine structure constant 1/137. This is in perfect analogy with the spin dependence of the hydrogenic spectrum where the spin up and spin down levels are of the order of a 2 apart. The spin correction to the surface barrier reflec~ivity can be calculated exactly by means of our computer program. The conclusion with respect to the Rydberg resonances of tungsten is therefore that the fine structures are different because the bulk reflectivity is very sensitive to spin.
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4. Conclu8ion and prospects What can the surface resonances as observed in LEED tell us about the properties of the metal surface? To begin with, the resonances are likely to be useful for the experimental technique of LEED. They offer a possibility of a quick determination of the resolving power of the apparatus used by simply doing a count of the number of resonance fringes observed. At the same time, the n + m end of the series of fringes give accurate ·calibration marks· on the recorded intensity curves, and give a precise relationship between the energy and angle of incidence of the primary electron beam. In our opinion the Rydberg surface states in LEED can also give information about the electron density near a metal surface. Firstly, the radius of the limiting circle of the reflectivity r+- (shown in figure 2) depends strongly on the absorptive part of the potential at the surface, or rather on the integrated effect of the absorptive tail of the potential in vacuum. The absorption can be measured from the height of the fringes relative to the average LEED intensity of the background in the region where the resonance occurs. Secondly, by a very accurate LEED measurement, the Rydberg correction an of (5) could be determined for a number of different values of n. As indicated in the introduction, an depends of the reflectivity of the crystal bulk, and McRae (1979b) has made theoretical predictions of the sensitivity of an to various physical parameters of the surface. It is at present an open question to what extent an depends on that portion of the surface barrier potential which connects the Coulomb tail in the vacuum to the inner potential in the bulk. We plan an investigation of an by means of LEED calculations on the surface of aluminium.
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The metal surface of classical electrostatics, i.e. the plane acting as a mirror for an external point charge, is defined to be the plane z = 0 in the expression (1). We believe that the physical location of the plane z = 0 with respect to the outermost atomic layer can be found from recordings of LEED intensities. We have begun such an investigation, but have found that only a very restricted selection of existing LEED data, if any, is accurate enough for this purpose. The location of the classical surface is an important parameter in the theory of a metal surface as developed by Lang and Kohn (1973) who also were able to predict a numerical value for this parameter. To obtain an experimental value for this parameter would be a most interesting check on the soundness of the theory.
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References Adnot A and Carette J 0 1977a Phys. Rev. Lett. 38 1084-7 Adnot A and Carette J 0 1977b Phys. Rev. B ~ 4703-4 Cole M W 1974 Rev. Mod. Phys. i i 451-64 Echenique P M and Pendry J B 1978J. Phys. C 11 2065-75 Henrich V E 1975 Surface Sci. 12 675-80 Jennings P J 1971 Surface Sci. l1 513-25 Jennings P J 1978 Surface Sci. 75 773-6 Jennings P J 1979 Surface Sci. 88 L25-8 Jennings P J and Price G L 1980a Surface Sci. 2l L124-8 Jennings P J and Price G L 1980b Surface Sci. ~ L205-9 Jennings P J and Read M N 1974 Surface Sci. i l 113-24 Lang N 0 and Kohn W 1973 Phys. Rev. B 2 3541-50 McRae E G 1966 J. Chern. Phys. 45 3258-76 McRae E G 1971 Surface Sci. 25 491-512 McRae E G 1979a J. Vac. Sci: Technol. ~ 654-9 McRae E G 1979b Rev. Mod. Phys. 21 541-68 McRae E G, Aberdam 0, Baudoing R and Gauthier Y 1978 Surface Sci. ~ 518-30 McRae E G and Caldwell C W 1978 Surface Sci. 74 285-306 McRae E G, Landwehr J M and Caldwell C W 1977 Phys. Rev. Lett. 38 1422-5 Malmstram G and Rundgren J 1980a J. Phys. C 11 L61-5 Malmstram G and Rundgren J 1980b Comput. Phys. Commun.
12
263~70
MalmstrOm G and Rundgren J 1980c (to be published) Pierce 0 T, Celotta R J, Wang G C, Unertl W N, Galejs A, Kuyatt C E and Mielczarek 5 R 1980 Rev. Sci. Instrum. ~ 478-99. Price G L, Jennings P J, Best P E and Cornish J C L Surface Sci. ~ 151-8 Read M N and Jennings P J 1978 Surface Sci. 2! 54-68 Rundgren J and MalmstrOm G 1977a Phys. Rev. Lett. 38 836-9 Rundgren J and MalmstrOm G 1977b J. Phys. C 10 4671-87
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1.'2.
1.1
'" I 0
::
...8
1.0
., .S 17- 5
18.0 Energy leV)
Figure 1 A Rydberg resonance in LEED from aluminium near 18 eV primary energy and normal incidence. The fine structure in the intensity 1 of the spec00 ular beam ('the beam reflected as by a mirror) indicates that new beams are about to emerge at 18.3 eVe
67
I
..
+
,';Ot-----'~_\"~-~_t_-__::__~-I_--_1
-.J
-·3
o Re r+-
Figure 2 The crystal-harrier-crystal reflection coefficient r+- illustrated by a graph in the complex plane. Along the curves are marks indicating the energy of propagation normal to the surface, EgLvac. The branch for negative energy, below beam emergence, terminates in a limiting circle when E+ + -0. The branch for positive energy, above gLvac beam emergence, is the short piece of curve, beginning at a finite value for terminating at zero for
Eg~vac
68
Eg~vac
+
~.
=
0 and
7,...-----....-----,r--------r---::o..----.
....I
...o
(,
o
c -e
s
-g
Figure 3 The LEED beams in the vacuum in front of the crystal are plane waves exp(ik+.~) having complex amplig tudes A The graph shows the amplitude AOO for the specular beam for the Rydberg resonance in figure 1. ADD is represented by a radius vector, the point of which moves, like r+-, on a limiting circle when the energy of beam emergence is approached.
g.
69