PRESENTED AT THE SLAFS7 MEETING HELD IN BARILOCHE, ARGENTINA, NOV. 14-21, 1992

THE INTERACTION OF SWIFT ELECTRONS WITH SURFACE EXCITATIONS

R. H. Ritchie Oak Ridge National Laboratory n,™*, ~ Oak Ridge, TN 37831-6123 USA CONP-9211193-1 and DE93 007245 Department of Physics University of Tennessee Knoxville, TN 37996 USA

For many decades swift electrons have comprised a powerful tool for the study of the dynamical properties of condensed matter. The development of this technique has involved much important physics. Here we sketch the historical background of the field and some important developments in theory and experiment. Possible directions for future research are indicated.

Research sponsored by the Office of Health and Environmental Research, U. S. Department of Energy under contract DE-AC05-84OR21400 with Martin Marietta Energy Systems, Inc., and the University of Tennessee, Knoxville, Tennessee. DISCLAIMER

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OF THSS DOCUMENT IS UNLIMITED THE INTERACTION OF SWIFT ELECTRONS WITH SURFACE EXCITATIONS

R. H. Ritchie Oak Ridge National Laboratory Oak Ridge, TN 37831-6123 USA and Department of Physics University of Tennessee Knoxville, TN 37996 USA

1. Introduction

Electron Energy Loss Spectroscopy (EELS) has developed into a notable tool for condensed matter diagnostics in the last several decades. It has contributed importantly to our knowledge about properties of matter in the excited state. Here we will survey some history as well as some recent developments in the field.

When a fast charged particle approaches and enters a piece of condensed matter, charges and currents are induced therein and act to modify the behavior of the incident particle. The eigenmodes of the struck system are very convenient to use in describing its response to such probes. These eigenmodes may consist of electron-hole pairs, bulk and surface plasmons, excitons, polaritons, magnons, etc. Below we describe the theory of the effect of the induced polarization on the probe self-energy, some current developments in the field, and possible directions for future research.

2. Background

Fundamental understanding of charged particle interactions with matter began in the dawning of modern physics with Niels Bohr (1913), contemporaneous with his famous theory of the hydrogenic spectrum. This pioneering theory of energy loss by alpha particles in matter was extended by Bethe (1930) in a fully quantal treatment that included an important sum rule which generalized the Kuhn- Reiche-Thomas sum rule (Kuhn 1925, Reiche and Thomas 1925) to arbitrary wavelengths. These classical and quantal treatments of energy loss were synthesized by Bloch (1933) in an analysis that made the first use of the quantal interaction picture. Bloch (1933a) also applied hydrodynamical theory to describe the dynamics of electrons in atoms, showing that the mean excitation potential in an atom should be proportional to its atomic number. Interestingly, Rayleigh (1906) anticipated the use of this model in the first decade of this century in a study of the eigenmodes of the Thompson "plum-pudding" model of the atom. The dielectric theory of matter has assumed great importance in recent times; Fermi (1939, 1940) was the first to use a local dielectric function to analyze the interaction of charged particles in condensed matter. Such active interest by these pioneers of modern physics is evidence for the intrinsic importance of charged particles as probes of the dynamic properties of matter.

3. Energy Losses in Condensed Matter

In the early decades of this century, electron energy loss spectroscopy was used by several workers (see, e.g., Birkhoff 1958) to study excitations in the solid state. The first extensive EELS study of several different solids was carried out by Rudberg (1936). A landmark quantal theory of plasmons was advanced by Fines and Bohm (1951, 1952, 1952a) to explain EELS discrete energy losses in metals, as well as other phenomena in metals that were not well understood previously. The Pines-Bohm description relied on quantization of longitudinal density fluctuations in the electron gas in terms of the harmonic oscillator using a Random Phase Approximation (RPA). Birkhoff and collaborators (Blackstock, et al 1955) confirmed this plasmon hypothesis by showing that their measured mean free paths for plasmon creation agreed with the results from the Bohm-Pines theory. They were also the first to show that the distribution of plasma losses obeys the Poisson law. A quantal theory of this distribution was later given (Ashley and Ritchie 1968); an elegant generalization by Sunjic and Lucas (1971) of the quantal theory is applicable to surface excitations as well.

Research following World War II saw a flood of papers on EELS showing increasing sophistication in both experimental and theoretical techniques (see, e.g., Raether 1965,1977, Egerton 1986). Dielectric theory was generalized (Gertsenshtein 1952) to include wave-vector dependence as well as frequency dependence. The fully quantal dielectric theory of the electron gas (Lindhard 1954, Hubbard 1957) led to a unified treatment of the plasmonic- and single-particle-behavior of bulk metallic systems, while quasi-particle concepts led to better understanding of condensed matter properties (see, e.g., Pines 1963, Nozieres 1964, Fetter and Walecka 1971).

Collective excitations occurring near the surface of condensed matter display different characteristics than the corresponding bulk excitations and in some cases may be more significant, e.g., in EELS measurements. Surface plasmons (Ritchie 1957), surface polarons (Evans and Mills 1972), surface excitons (Pekar 1957), surface polaritons (Fuchs and Kliewer 1965, Kliewer and Fuchs 1966), surface magnons and other elementary excitations existing near interfaces have played an important part in the development of modern surface physics and have led to significant increase in the understanding of condensed matter dynamics (Boardman 1982, Agranovich and Mills 1982, Pendry 1974, Garcia-Moliner and Flores 1979).

3.1 The Plasmon in Metals

The well-known plasmon in metals is simply modeled in the electron gas at long wavelengths

2 where the plasmon eigenfrequency is (o -JATZH e /nt * in terms of the effective electron mass m * and their density n • For a plane-bounded electron gas the corresponding surface plasmon (SP) eigenfrequency is <^ = or if covered by material with dielectric constant g becomes

(j =(OHA/1 +6 • As is now well-understood the SP consists of an oscillatory density fluctuation confined near the surface of condensed matter and may generate electric fields penetrating to large distances from the surface (Raether 1988). The wave-vector dependence of the SP and the splitting of its eigenmodes in thin films were predicted early on and confirmed experimentally (see, e.g., Ritchie 1957, Raether 1965). The quantal jellium model of the SP has received much attention from theorists (see, e.g., the review by Dasgupta and Beck 1982); the density functional approach, involving has been used to good effect in this connection and has been invoked recently in a study of the surface barrier at a jellium surface, including long-range correlations, by Eguiluz (1992). The surface response function has been codified in an important quantity (see, e.g., Persson and Apell 1983) d (to) > the centroid of density fluctuations near the surface. Sum rules based on the Kramers- Kronig relation have been given for this quantity. Multipole surface plasmons were found by Bennett (1970) in a hydrodynamical treatment of the SP dispersion relation and have been studied by other theorists (see, e.g., Dasgupta and Beck 1982). Tsuei et al (1990, 1991) have found experimental evidence for SP multipolarity in EELS.

3.2 The Surface Response Function

The response of a surface to external perturbation is of obvious importance in EELS work and has been represented in several approximate models, the simplest being obtained from local dielectric theory. Echenique (Milne and Echenique 1985, Echenique et al 1987, Echenique et al 1987a), and Ferrell and Echenique (1984) have found that such representations are very useful in analyzing experimental work in Scanning Transmission Electron Microscopy (STEM).

Hydrodynamical theory has received much attention in this connection. The limitations of this model have been discussed extensively (see, e.g., Doniach 1984, Boardman 1982, Barton 1979) but it is clear that it gives a good representation of the principal qualitative features of metallic systems. Echenique (1985) has evaluated the effect of plasmon dispersion on the excitation of interface modes in STEM, while Forstmann (1979, 1986) has used this model to treat a number of problems involving metal surfaces.

A number of different representations of the surface response function have been used by various workers. Ritchie and Marusak (1966) and Wagner (1966) independently used the specular reflection model (SRM), together with the RPA dielectric function of the electron gas, to study the dispersion relation of the SP; Newns (1970) defined a surface dielectric function using this model that has had extensive use (see, e.g., Garcia de Abajo, et a! 1992). Echenique et al (1981) gave an analytical representation of the surface dielectric function, using a plasmon-pole type approximation based on the SRM that contains collective as well as single-particle effects. Mahan (1973) presented a very convenient approximate Hamiltonian form to represent the electron-surface excitation as well as the electron-bulk excitation and is valid for both plasmons and optical phonons. Persson and Anderson (1984), working in the RPA, have derived a surface response function valid for small energy and momentum transfers. Evans and Mills (1973) showed how to account theoretically for both inelastic and elastic scattering of electrons from a surface.

33 The Boundary (or Begrenzung) Effect

In an early calculation of energy loss by a swift charged particle (Ritchie 1957) it was clearly shown that in a thin planar foil losses to surface plasmon modes is accompanied by decreased losses to bulk plasmons. Lucas and coworkers (Lucas et ai 1970, Lucas 1971; Sunjic and Lucas 1971) studied the structure of the quantal eigenmodes in thin dispersionless foils and showed that this effect involves a redistribution of oscillator strength from the bulk to the surface modes accompanying the introduction of surfaces. Feibelman, Duke and Bagchi (1972) reached similar conclusions for the bounded electron gas working in the RPA.

3.4 Sum Rules

Sum rules have played an important role in constraining theoretical approximations in many applications (Hamilton 1960, Bethe and Salpeter 1957). A useful sum rule for the inverse dielectric function of a homogeneous, isotropic system was first given by Nozieres and Pines (1957) based on 4

the approach of Bethe (1930). It reads

fda>u>Im (l)

and was generalized (Fano and Turner 1964, Inokuti 1971) to apply for higher powers of o> in the integrand following tri i work of Placzek (1952) in characterizing the response of matter to neutrons. Applications of these sum rules in approximating the full response functions of solids from their measured long-wavelength response functions have been made by Ritchie and Howie (1977), Ashley (1982), Penn (1987), Tougaard (1986) and others (see, e.g. Ritchie et al 1991).

A sum rule for the electronic response function of a surface was proved by Sham (1968) and used by others (see, e.g., Inglesfield and Wikborg 1974).

Apell (1992) has proved an interesting relation between the surface plasmon eigenfrequency of a given system and that of the 'complementary' system. The latter is defined as the original system with the condensed matter replaced by vacuum and vice versa. Thus M =6) +

inhomogeneous matter (Mukhopadhyay and Lundqvist 1979).

3.5 Relativistic Corrections to the Image Potential It is well-known that the image potential experienced by a classical point charge placed at fixed distance from a metal surface may be regarded as arising from the shift in the zero-point energy of the surface plasmon field due to the presence of the charge. The importance of surface modes to the image potential was emphasized independently by Lucas (1971), Feibelman (1971), Mahan (1972), and Ritchie (1972). This followed the earlier discovery that van der Waals forces also have their origin in surface modes (Van Kampen, et al 1968, Gerlach 1971). Attempts to generalize to the relativistic domain this approach to image potential theory were met with difficulties (Ritchie 1972, Tomas and Sunjic 1975). Ekardt (1981) and Babiker (1983) argue that it is necessary to include the bulk plasmons of the jellium model in a treatment in which the coupled electromagnetic field-electron gas eigenmodes are quantized in order to obtain a consistent answer for the image potential. This even though the bulk plasmon is thought to be unimportant at distances larger than a few Angstroms from the surface in the non-retarded domain. It is clear that this non-intuitive result should be studied further. 4. The Self-energy

The self-energy of a probe interacting with a many-particle system is due to the excitation, real or virtual, of all possible eigenmodes of that target and to the reaction of the probe to these excitations. A convenient representation of the self-energy was given by Manson and Ritchie (1981). Employing ordinary second-order perturbation theory in an unfolding procedure, they showed that the projected (or local) space-dependent self-energy can be written (2)

Here the projectile is described in terms of the basis set { |(j>J} with eigenenergies {eJ and a typical many-particle state vector of the target is taken to be I |jf ) with eigenenergy E • We write the

interaction energy between the target and the projectile as V(f)=\^ V(f-,f)> where f. is the i coordinate of the ith particle making up the target and f ls tne coordinate of the probe.

It is straightforward to show that this expression for the self-energy can be rewritten in a somewhat more familiar form as

idE< fdrtiir^rrtW)

tne orm for where WJff*) & time-ordered linear response function and EH=Ek -Ek- Thi* f the self-energy admits a nice interpretation in terms of semi-classical transition charge densities [Mott and Massey (1933)], used only a few years after the invention of quantum mechanics. The integral overr'

in Eq. 3 represents the effect at f °f 'he entire transition charge density t(f^(J>t (r') > while the

integral over E1 emphasizes the effect of this transition on the orbitals

It may be shown from the above that the nonloca! or two-sided self-energy EJfif*) can be written as

where GjXTif1) IS tne Green function for the probe and the energy shift in the interaction is Equation 5 expresses the non-local self-energy in the form of the "G-W" approximation of Hedin and Lundqvist (1969). Note that the local self-energy S (r) depends on the initial state

ty.(f)» but the non-local self-energy J^J^f^) depends only on the energy E- I* mav ako be noted that this exact form, written in slightly different notation, is given in the early work of Feynman (1949), who discussed in a typically lucid manner the self-energy of the electron due to its interaction with the vacuum electromagnetic field.

The self-energy formulation has been applied to many problems in EELS such as the theory of energy losses in STEM (Echenique et al 1987, Rivacoba and Echenique 1990, Zabala and Echenique 1990) and to questions of spatial localization of losses in STEM (Ritchie et al 1990). The early Echenique-Pendry (1975) formula for the excitation of surface modes by a swift electron traveling parallel with a condensed matter surface is nicely given in this formulation as well as the general theorem relating quantal losses in STEM to the classical representation for sufficiently fast electrons (Ritchie 1981, Ritchie and Howie 1988).

Higher-order contributions to the spatially-dependent self-energy of an electron interacting with surface plasmon modes may be found in a straightforward way (Zheng et al 1989). It has also been used to construct the quanta! wake potential of a swift charged particle in matter (Ritchie and Echenique 1982), in treating saturation effects in positronium-surface interactions (Manson and Ritchie 1984) and in many other applications (see, e.g., Echenique et al 1990, Echenique et al 1991).

5. Possible Future Directions

Many interesting aspects of EELS remain to be studied. With continuing improvements in the accuracy and comprehensiveness of experimental technique such as those that have taken place in the last few decades, we will surely learn much more about the excited states of condensed matter in the near future. The study of nonlinear interactions in systems such as the bounded electron gas that are fairly well-understood theoretically would seem to offer fruitful avenues for future research. For example, the Barkas effect in stopping power theory (Esbensen and Sigmund 1990, Pitarke et al, 1992) has received much attention from theorists and experimenters, but little notice has been taken of this effect in EELS. Similarly, the phenomenon of double-piasmon emission by swift electrons in the electron gas is not well understood even after much theoretical and experimental effort (Ashley and Ritchie, 1970 Schattschneider et al 1987).

Another quantity of importance in the interpretation of recent experiments is the image potential experienced by a highly charged ion located near a metal surface. Density functional theory could be used in evaluating the image potential appropriate to slowly-moving ions while hydrodynamic theory or time-dependent density functional theory could be used for swift ions. Nonlinear deviations from the usual image potential would be of interest in EELS, as well. A related question of interest is the following: how many plasmons can a metallic system support at the same time? Nonlinear effects in large systems are probably small (Ashley and Ritchie 1970), but multiplasmon generation in small metallic clusters may display interesting interference effects. Compton (1992) has measured multiphoton absorption in few-electron systems that may be due to the generation of several plasmons.

6. Acknowledgements

The author has had the great good fortune to work with many generous, outstanding people over the years; these include Pedro Echenique, Dick Manson, Archie Howie, Fernando Flores, Alberto Gras-Marti, Jim Ashley, Tom Ferrell and many others.

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