Introduction

Electrons entering a crystal undergo elastic and inelastic scattering. The main elastic process is Bragg scattering by the atoms. The inelastic scattering processes include plasmon excitations, thermal diffuse scattering and single-electron excitations. As a first step, we describe scattering by a single atom. In this case the inelastic processes are much simpler, being due only to excitations of atomic states. Plasmon scattering and thermal diffuse scattering are described in Chapter 16. In this chapter we describe elastic and inelastic scattering processes for atoms in the Born approximation and calculate differential and total cross sections for special cases.

The main results of this chapter will be expressions for the strength of the scattering and its angular dependence as well as an estimate of the relative elastic and inelastic mean free paths of electrons traveling in a crystal. From the point of view of the subsequent development, though, this chapter need only be examined briefly. Specific references will be made when required.

As described in later chapters, the angular dependences of elastic scattering amplitudes i.e. the electron scattering factors, are used to obtain crystal potentials for the dynamical theory of electron diffraction. The angular dependences of the inelastic scattering intensities contribute to the imaginary potentials in crystals used is the dynamical theory.

Kikuchi lines

Inelastically scattered electrons play an important role in the formation of the RHEED pattern. The primary inelastic process is a scattering event involving phonons and plasmons (Horio, 1996; Müller and Henzler, 1997). The intensity of these Kikuchi patterns depends strongly on the surface morphology, since scattering from small terraces and steps broadens them. Sharp Kikuchi lines are obtained from crystals with perfect surfaces and perfect bulk lattices. They seem to be stronger for heavier materials; for example, Si and SiC show strong Kikuchi patterns but GaAs, GaN and PbS show weaker patterns.

As will be discussed in Chapter 16, the inelastic scattering is peaked in the forward direction in a diffuse cone of about 0.1° for single-plasmon scattering and of more than 10° when multiple thermal diffuse scattering is important (Ichimiya, 1972). The inelastically scattered electrons in this diffuse cone can subsequently be diffracted by the crystal lattice planes, depending on their angle. These two scattering processes combine to give rise to the appearance of Kikuchi lines at specific exit angles. Figure 7.1 shows the energy distribution of the electrons scattered into a Kikuchi line which crosses the specular beam from Si(111) (Nakahara et al., 2003). The main contributions in the spectrum are surface and bulk plasmons, because the energy-loss spectra from thermal diffuse scattering (or phonon scattering) are not resolved owing to poor resolution of the spectrometer for RHEED.

Early experiments

The first RHEED experiment was conducted by Nishikawa and Kikuchi in 1928. Their interest at that time was whether the Kikuchi patterns that had been observed previously in transmission electron diffraction (Kikuchi, 1928a, b) were also observed in reflection. Later they were interested in effects due to the refraction of electrons by a mean inner potential (Kikuchi and Nakagawa, 1934).

At the outset, efforts were made to understand the angles at which the diffracted beams showed intensity maxima. For reflection diffraction, the angular positions of the diffraction maxima do not follow Bragg's law. These shifts were explained to some extent by considering the beam to be refracted by an inner potential (Thomson, 1928).

Owing to refraction, the lowest-order diffracted beams are totally internally reflected and so are not observed. Using this effect, efforts were made to determine the mean inner potentials, the values of which are related to paramagnetic susceptibilities (see Chapter 9). In order to determine the mean inner potentials, the RHEED intensity was measured as a function of incident angle, a measurement that has become known as a rocking curve. From the systematic deviation of the positions of diffraction maxima from Bragg's law, the values of the inner potentials for several materials were determined for the first time by Yamaguti (1930, 1931). The refraction effects of the inner potential are also observed in RHEED patterns as parabolic Kikuchi lines and envelopes (Shinohara, 1935).

Introduction

The goal of this chapter is to develop both a quantitative and qualitative understanding of the diffraction from disordered systems as measured by actual instruments. We start by examining the diffraction from well-ordered GaAs and determine the changes produced by small amounts of disorder. To some extent this has also been how our historical understanding has progressed, beginning with well-ordered molecular beam epitaxy (MBE) surfaces of GaAs and Si. At the start of efforts to use MBE for the growth of high-quality films, the cause of the diffraction streaks was not completely clear. It was known that the films were of very high quality, from their electrical properties and X-ray diffraction spectra; hence the idea of finite crystallite size as discussed by Raether (1932) was not felt to apply. Some discussions examined the role of thermal diffuse scattering, and this can certainly be important. But for systems of interest in MBE, for example GaAs(100), it quickly became apparent that steps on the surface, and in some cases anti-phase disorder, produced the streaks that were typically observed.

One of the clearest examples of the role of steps is the change in the diffraction pattern one observes when growth is initiated on GaAs(100). This is a surface that is easy to prepare, using current technologies, without significant extended defects. Figure 17.1 shows an example of the diffraction patterns before and after growth has begun.

Measured diffraction patterns exhibit a range of features, depending upon the degree of surface and subsurface perfection. Low-index Si surfaces and smooth GaAs surfaces prepared by MBE can show near-ideal behavior, while epitaxial films of GaN can be weak and diffuse or show features reminiscent of transmission electron diffraction. Even so, these patterns are often interpretable without analysis of the diffracted intensities, just with consideration of their geometrical aspects, as described in Chapter 6. One can not only determine the symmetries of atomic arrangements but also characterize the presence of some types of imperfections, the degree of surface roughness and the sizes of domains and terraces–in short, crucial information for many types of surface study. In this chapter we develop straightforward methods for analyzing patterns, giving a number of examples.

We will continue to use the Ewald construction described in Chapter 5, but now applied to samples with a range of domain sizes and domain orientations. This becomes complicated except for the simple case in which the incident beam is directed along a principal axis of a perfect surface. At off-symmetry conditions or for defected surfaces the patterns are often difficult to interpret. Off-symmetry incident azimuths are especially interesting, since for these the analysis of the diffracted intensities proves simpler and often they are the only azimuths available to a film grower. Misoriented surfaces, having been discussed in Section 6.4, will not be considered here.

RHEED intensity oscillations are now routinely used for measuring growth rates during molecular beam epitaxy. They are used to determine whether the growth mode is via island nucleation or step flow, whether growth occurs in layer or bilayer growth modes and to obtain estimates of surface diffusion. But little use has been made of quantitative measurements of the damping or of the strong angular dependence. RHEED oscillations are widely believed to be related to step density and to have strong path-length interference components. The main difficulty in separating these two mechanisms is that dynamical calculation from a disordered surface, with distributions of islands and adatoms randomly arranged at surface lattice sites, is exceedingly difficult. Our assessment and the topic for discussion in this chapter is that, depending upon the predominant island sizes, different mechanisms can dominate. We argue that step density, kinematic calculations and shadowing arguments are each inadequate to explain all the measured angular dependences and wave forms. The intensity oscillations measured with RHEED are fundamentally a dynamical effect, and each of the mechanisms contributes various aspects. Nonetheless, it appears that the coverages of the layers that comprise the growth front are the major factor producing the intensity oscillations. From a practical point of view, these measurements are at least as important as symmetry and structural determinations. In this chapter we examine their diverse characteristics from the various points of view in a variety of systems.

Plan of Chapters 12–14

There are now a number of different dynamical calculations of the intensity of high-energy electrons reflected from surfaces. They face similar difficulties and give results of similar accuracy, but there is no obviously best method. In Chapters 12–14 we present methods due to Ichimiya (1983, 1985), Zhao et al. (1988) and Maksym and Beeby (1981) in sufficient detail that the reader should be able to perform these calculations for real surfaces. The discussions of each are nearly self-contained and simple examples are provided to illustrate the methods. Our intent is to present these dynamical theories so that their important contributions to RHEED can be seen, as well as to allow surface structures to be determined. These are simple theories with some complicated algebra. We will try to present the work in such a way as to convey the overall method first, leaving detailed computations to later subsections. In order to calculate RHEED intensities, optimization is needed. The details of the optimization are described in Appendix F.

Introduction

The dynamical theory is based on a Bloch-wave solution of the Schrödinger equation for a system with a fast electron and a crystal potential that is periodic in two dimensions. Only elastic scattering is considered, with the electron momentum conserved up to a reciprocal lattice vector of the two-dimiensional crystal.

Introduction

In RHEED, an electron beam, at an energy usually between 8 and 20 keV for epitaxial growth systems, is incident on a crystal surface at a grazing angle of a few degrees. At the surface there is a scattering process in which there can be energy loss. Diffracted beams leave the crystal, also near grazing incidence, and strike a detector. It is a very open geometry, with the incident beam and detector as much as 20 cm from the sample. It is exceedingly surface sensitive. As a result, RHEED is an ideal measurement to combine with atom deposition, Auger electron spectroscopy, scanning tunneling microscopy, scanning electron microscopy and other surface probes. The appropriate experimental methods depend on the measurements desired and on the sample. In this chapter we describe several designs.

The optimal energy for electron diffraction depends somewhat on the purpose of the measurement. Electron optics become easier as the energy is increased but there does not seem to be any overriding issue. For dynamical analysis, as will be seen later, a planewave expansion is performed since at high energies the scattering is mainly in the forward direction and this is efficient. A spherical-wave expansion could also be used, but this is inefficient since at high energies many diffracted beams will be needed. So for dynamical analysis, energies greater than about 10 keV are essential. However, at lower energies it is possible to go to higher incident glancing angles and still maintain surface sensitivity.

Introduction

The embedded R-matrix method formulated by Tong's group (Tong et al., 1988a, b; Zhao et al., 1988), like Ichimiya's matrix method, seeks to manipulate the differential form of the Schrödinger equation. It is also a multi-slice method, in which the crystal is modelled as a slab divided into many thin slices parallel to the surface. The wave inside the crystalline slice is decomposed into a set of beams by making use of the perfect translational symmetry parallel to the surface, and the Schrödinger equation is then solved. To determine the RHEED intensity, the wave transmitted out of the bottom of the slab is sequentially matched to the wave functions found for each slice until it can be matched to the incident and reflected components of the wave at the surface, allowing the reflection coefficient to be calculated. The key feature of the method is in the procedure for connecting the ratio of the wave function and its first derivative at the interface between each slice. The method relies on the computation of this ratio using a simple recursion that only manipulates N × N matrices. Tong and coworkers argued that this ratio, a generalized logarithmic derivative, is fundamentally well behaved, lessening the numerical convergence problems associated with all these computational schemes.

Like the other methods that make use of the translational symmetry parallel to the surface, this method decomposes the wave function and the crystal potential into their Fourier components or “beams.”

Reflection high-energy electron diffraction (RHEED) is widely used for surface structural analysis in monitoring epitaxial growth. The purposes of this book are to serve as an introduction to RHEED for beginners and to describe detailed experimental and theoretical treatments for experts. This book consists of three parts. From Chapter 1 to Chapter 8 the principles of electron diffraction and many examples of RHEED patterns are described for beginners. Chapters 9-14 and Chapter 16 give detailed descriptions of RHEED theory. The third part consists of applications of RHEED. In Chapter 15, methods for the determination of atomic structures of surfaces using RHEED are explained with some examples. Chapters 17 and 18 give detailed descriptions of RHEED in the study of surface disordering and epitaxial growth. In Chapter 19 we describe RHEED intensity oscillations for various growth systems.

A. I. expresses many thanks to the late Professor R. Uyeda for his encouragement, to Drs T. Emoto and H. Nakahara for assistance in drawing many figures and to Ms M. Miwa, Ms Y. Mashita, Ms K. Hosono and Ms T. Arakawa for typing the text and checking references and indexes. P. I. C. is grateful to Ms A. D. Cohen for assistance with the references and especially to Drs J. M. Van Hove, C. S. Lent, P. R. Pukite and A. M. Dabiran for their help in understanding diffraction.

Introduction

In the dynamical theory of electron diffraction we intend to make use of the translational periodicity of a surface to write the scattered electron wave function as a sum of beams or Fourier components. The scattering properties of these beams will depend in turn upon the Fourier components of the crystalline potential. Because of the nature of the interaction between high-energy electrons and the crystal, this potential can be parametrized in a particularly simple and useful form. As a comparison, for LEED calculations this Fourier expansion method could be used in principle, but in that case one would need a large number of Fourier components for the calculation to converge (Pendry, 1974). For LEED, one also needs to include exchange scattering in the calculation, complicating the potential. For RHEED calculations, the cross section for exchange is negligibly small because of the large difference in energy between the atomic electrons and the incident and diffracted electrons.

The main goal of this chapter will be to determine an expression for the Fourier components of the crystalline potential that will be suitable for use in a dynamical calculation of surface structure. The only approximation that we will make will be to assume that the distribution of electrons around the atoms in a crystal is spherically symmetric. This means we are assuming that scattering from the atom cores dominates.

The Fermi liquid possesses an infinite number of electron–hole pair excitations in the vicinity of the Fermi surface and can be considered an infinitely degenerate system. Because of this nature the Fermi surface is largely transformed even by a small perturbation. The fragility of the Fermi surface gives rise to interesting physics, such as superconductivity and the various many-body problems related to the Fermi surface. A typical example is the Kondo effect. In this chapter, we consider the system of dilute impurity atoms in metals.

First, in Section 3.1, we consider the screening effect of an impurity charge in metals, following the Friedel theory. In Section 3.2, we explain Anderson's orthogonality theorem as a singularity originating from the infinite number of lowenergy excitations near the Fermi energy in metals. In Sections 3.3 and 3.4, we introduce the photoemission due to soft X-rays and the quantum diffusion of charged particles in metals, respectively, as typical examples in which the orthogonality theorem plays an important role.

Friedel sum rule

Let us assume that there exists an impurity atom in a normal metal such as Cu, Ag or Al. Owing to the local impurity potential V(r), the electron distribution around the impurity changes. The local change of electron number was calculated by Friedel in 1958. We introduce it here.