As is clear from the previous chapters, a considerable amount of information has become available about the diffusivity of metal atoms on metal surfaces. This knowledge will have an impact on understanding topics such as crystal growth and dissolution, annealing, sintering, as well as chemical surface reactions. We will not examine any of these important topics, but will instead focus on attempts to measure atomic surface interactions, as an example of effects which have become much clearer through gains in the knowledge of surface diffusion.

Interactions can be divided into two categories, direct, for example van der Waals, dipolar, as well as electronic interactions, and indirect, which are mediated by the substrate, such as coupling by electronic states, elastic effects, or vibrational coupling. In Chapters 8 and 9 we described the movement of clusters created mostly by direct interactions of adatoms. In this chapter we will concentrate on the second type of interactions, mediated by the lattice – indirect interactions. The theory of interactions between adsorbed atoms has been nicely reviewed by Einstein, so here we just want to point out that with two atoms in the gas phase, the wave function for a higher state will be confined to the vicinity of the core. Placed on a metal surface, however, quantum interference of wave functions enters and adatom waves combine with metal wave functions from the lattice. The contributions from two atoms may overlap, as shown in Fig. 10.1.

In this chapter, we continue the task started in the last one: we will list diffusion characteristics determined on a variety of two-dimensional surfaces. For better orientation, ball models of fcc(111) and (100) planes have already been shown in Fig. 3.3, together with the (110), (100), and (111) planes of the bcc lattice in Fig. 3.4. In experiments it has been observed that on fcc(111) planes there are two types of adsorption sites: fcc (sometimes called bulk or stacking) and hcp (also referred to as surface or fault sites); these sites, indicated in Fig. 3.3b, can have quite different energetic properties. Two types of adsorption sites also exist on bcc(111) and hcp(0001) structures. However, on bcc(110) as well as on bcc(100) and fcc(100) planes, only one type of adsorption site has so far been observed, which makes it easier to follow adsorption on these surfaces.

Aluminum: Al(100)

Experimental work on two-dimensional surfaces of aluminum was long in coming, and was preceded by considerable theoretical work, with which we therefore begin here. The first effort, by Feibelman in 1987, was devoted to the Al(100) surface, and relied on local-density-functional theory (LDA–DFT). The primary aim of the work was to examine the binding energy of atom pairs, but he also estimated a barrier of 0.80 eV for the diffusive hopping of Al adatoms. Two years later, Feibelman investigated surface diffusion which takes place on Al(100) by exchange of an adatom with one from the substrate.

In the previous chapter we presented possible mechanisms which can contribute to cluster diffusion; in this chapter, we will concentrate on the energetics of cluster movement, mostly the movement of the center of mass. Early studies of cluster diffusion were all done on tungsten surfaces using FIM, but as techniques other than field ion microscopy were applied to learning more about this subject, other surfaces came under scrutiny. The biggest change in the level of activity, however, was made by theoretical calculations. These now dominate the field and have usually covered several surfaces of different materials in one examination. The number of experimental studies of cluster behavior decreased markedly as computational efforts reached new intensities. Unfortunately, theoretical investigations still are quite uncertain and experiments are urgently needed for comparison and verification. Nevertheless we will try to arrange our comments chronologically in the description of each material, but with experiments and theoretical calculations separated.

Early investigations

Experiments

Work on the diffusion of single adatoms on a metal surface had been going on for just a few years when Bassett began to look at clusters formed by association of several atoms. In 1969 he noted that after depositing several atoms on the (211) and (321) planes of tungsten, clusters formed, with a mobility smaller than that of single atoms, provided that deposition took place with the atoms in the same channel. On these planes, clusters moved in only one dimension, along the channels of the planes.

With background information about the kinetic and experimental aspects of surface diffusion in hand we will now turn our attention to the atomistics of diffusion about which much has been learned through modern instrumentation. The usual picture of surface diffusion, which seems to agree at least qualitatively with experiments on diffusivities, is that atoms carry out random jumps between nearest-neighbor sites. Is this picture correct? Has it been tested in reasonable experiments? What can be said about the jumps which move an atom in surface diffusion? What is the nature of the sites at which atoms are bound? These are among the topics that will be considered at length now.

In these matters the geometry of the various surfaces studied plays an important role, and at the very start we therefore show hard-sphere models of planes that will emerge as significant. Presented first are channeled surfaces, bcc (211) and (321) in Fig. 3.1, as well as fcc(110), (311), and (331) in Fig. 3.2. These are followed in Fig. 3.3 by fcc(100) and (111) and bcc(100), (110), and (111) planes in Fig. 3.4. Their structures are sometimes quite similar, but their diffusion behavior may be quite different.

Adatom binding sites

Location

Where on a surface are metal adatoms bonded? That is an important question for understanding how atoms progress over a surface in diffusion. LEED has been very important in providing information about the geometry of adsorbed layers.

In Appendix A, Section A.2, we considered the oscillations of particles using Newton's laws of classical mechanics. The distinctive feature of Newton's mechanics, when considering interactions between particles (for example, the gravitational interaction), is the instantaneous transmission of the interaction between particles, i.e., transmission occurs with infinite speed. The interaction between charged particles is realized through an electromagnetic field, which possesses energy and momentum and is carried through space with finite speed. Electromagnetic waves can exist without any charges in a space in which there is no substance, i.e., in vacuum. This is substantiated by the fact that the equations of classical electrodynamics allow solutions in the form of electromagnetic waves for such conditions.

The main equations of classical electrodynamics are Maxwell's equations, which were formulated after analyzing numerous experimental data. In this sense they are analogous to Newton's equations of classical mechanics. Maxwell's equations are the basis for electrical and radio engineering, television and radiolocation, integrated and fiber optics, and numerous phenomena and processes that take place in materials placed in an electromagnetic field. Together with Newton's equations, Maxwell's equations are the fundamental equations of classical physics. Just like Newton's equations, Maxwell's equations have their limits of applicability. For example, they do not sufficiently well describe the state of an electromagnetic field in a medium at frequencies higher than 1014–1015 Hz.

The solution of most problems associated with electron quantum states in physical systems and structures (atoms, molecules, quantum nanostructure objects, and crystals) is hard to find because of the mathematical difficulties of getting exact solutions of the Schrödinger equation. Therefore, approximate methods of solving such problems are of special interest. We will consider some of these methods, such as the adiabatic approximation now and later the effective-mass method, using real physical systems as examples. In this chapter we will consider several widely used approximation methods for finding the wavefunctions and energies of quantum states as well as the probabilities of transitions between quantum states. First of all, we will consider stationary and non-stationary perturbation theories. What is common to these two theories is that it is assumed that the perturbation is weak and that it changes negligibly the state of the unperturbed system. Stationary perturbation theory is used for the approximate description of a system's behavior if the Hamiltonian of the quantum system being considered does not directly depend on time. In the opposite case, non-stationary theory is used. Then, we will briefly consider the quasiclassical approximation, which is used for the problems of quantum mechanics which are close to analogous problems of classical mechanics.

Stationary perturbation theory for a system with non-degenerate states

This theory is used for the approximate calculation of the energy levels and the wavefunctions of stationary states of systems that are subjected to the influence of small perturbations.

Three-dimensional quantum-dot superlattices can be considered as nanocrystals. Spherical nanoparticles consisting of a big enough number (from 10 to 1000) of atoms or ions, which are connected with each other and are ordered in a certain fashion, can be considered as the structural units of such nanocrystals. Examples of nanocrystals that are of natural origin are the crystalline modifications of boron and carbon which have as their structural units the molecules B12 and C60. The boron molecule B12 consists of 12 boron atoms, and the carbon molecule C60, which is called fullerene, consists of 60 carbon atoms. The fullerene molecule resembles a soccer ball, i.e., it consists of 12 pentagons and 20 hexagons, with carbon atoms at their corners. These nanoparticles form face-centered superlattices with a period of about 1−10 nm. At these distances between molecules of C60 weak molecular forces, which provide the crystalline state of fullerene, act.

In addition to nanocrystals of natural origin, numerous artificial three-dimensional superlattices consisting of various types of nanoparticles have been fabricated. The variety of nanocrystalline structures as well as of conventional crystals is defined by the differences in the distribution of electrons over the quantum states of atoms. The most significant role in the formation of individual nanoparticles as well as of crystals is played by the electrons in the outer shells of atoms.

In the previous chapter we have analyzed the peculiarities of quantized electron motion in layered structures with one-dimensional potential wells. From the mathematical point of view, the solution of the Schrödinger equation for one-dimensional potential profiles is much simpler. However, many quantum objects, such as atoms, molecules, and quantum dots, are three-dimensional objects. Thus, in order to analyze electron motion in such objects we need to find solutions of the Schrödinger equation for three-dimensional potential profiles. The electron motion in spaces with dimensionality higher than one, especially for rectangular potential profiles with infinite potential barriers, is not so difficult to analyze. At the same time we have to keep in mind that such potential profiles frequently represent some approximation of the more complex, real potential profiles. Depending on the type of structure and on the form of the potential profile, the electron motion may be limited in two directions (two-dimensional quantization) or in three directions (three-dimensional quantization). In this chapter we will show that the existence of discrete energy levels in the electron spectrum is an intrinsic feature of electron motion in potential wells of any form and dimensionality.

An electron in a rectangular potential well (quantum box)

In the previous chapter we studied the electron motion in one-dimensional potential wells. An electron's motion along one direction was confined by the potential profile and the momentum in this direction was quantized.