Surfaces and Contact Mechanics, Part One, Page 3

Surfaces and Contact Mechanics

5. Describing a Surface using the T-L-K Model

Imagine if you will, the cleaving (alternate words: fracturing, comminuting, splitting) of a perfect crystal along a crystal plane. Suppose that the crystal is simple cubic in which the atoms occupy the corners of a cube. These cubes are repeated ad infinitum, filling space. A perfect crystal with only one kind of atom occupying all positions equally with no vacancies on any of the sites. Of course, no such crystal really exists (at least I don't think so). Suppose now, that the crystal was cleaved along the (100) direction, a plane passing through the top surface of a plane of cubes. In this very special case, the atoms on the resulting surface have the same arrangement on the surface as those in the bulk. The situation can be represented as in Figure 8 below. Each cube in the figure represents a surface atom occupying its position in space.

Figure 8. A T-L-K Representation of a Singular, (100) Surface of a Simple Cubic Crystal [3]

The T-L-K model is used to describe the structure of equilibrium surfaces. The letters in the name stand for terrace, ledge, and kink. If an atom were to be removed or was missing from the surface in the figure above, that would be a defect called a surface vacancy. Here the word defect means a deviation from the ordered nature of the idealized surface. Figure 9 shows examples of various kinds of defects that can occur on a solid surface. In the figure, we see several things going on. This is an example of a vicinal, or high index, surface. Beginning with the top left, we see a terrace which is similar to the (100) surface discussed earlier, except that it ends in a ledge and we step down to another terrace. Some atoms are missing in the ledge, forming kinks in the chain of atoms. If an atom is moved into any position, it is called an adatom. If an atom is missing from a terrace, it may be referred to as a terrace vacancy.

Figure 9.

Let's consider now a surface in thermodynamic equilibrium above absolute zero [2]. The surface defects are similar, in terms of energy of formation considerations, to the kinds of defects observed in bulk crystalline solids, such as interstitial atoms and vacancies. To add or remove an atom from a site and to move it somewhere else requires energy. An atom in the crystalline bulk can be bonded to as many six nearest neighbors. Assume that all bonds are equal in this solid, to move an atom from a terrace site to a kink site requires breaking five bonds and making three bonds at the kink site. The difference in energy in this process is the energy of two bonds. The change in the Gibbs free energy function (final state minus initial state) for this situation is

As one might expect, the number of atoms undergoing some transition into or out of a particular energy state (kink, ledge, etc.) is dependent on the temperature and the number of such atoms available for that transition:

The above mentioned analysis is a nice approximation of what happens on a real surface, but there are some issues not addressed:
1. The model assumes that atoms rigidly hold there positions; and therefore, it doesn't account for relaxation around the defect location.
2. No attempt is made to account for interaction with second or third nearest neighbors.
3. When the temperature is high, the concentration of defects may become so large that they begin effecting the formation energies of the defects. This is an example of a cooperative effect. Since the defect formation energy can decrease with an increase in the concentration of defects, the situation may arise where there is a lack of order to the surface. This occurs above the so-called surface melting temperature.

6. Relaxation and Reconstruction [3]

As mentioned briefly earlier, the atomic configuration at the surface is very different than it is in the bulk because of the lack of interatomic forces outside the material. The reconfiguration of the surface is strongly dependent upon the type(s) of bonding present. In metals, the bonding is considered the result of the electrons of the solid being spread throughout a lattice of positive atomic cores and determining the stable configuration (or structure). The other types of bonds are the weak van der Waals, the ionic, and covalent. These are the classifications that have been made for chemical bonds and are well-established in chemistry and physics. Real materials, however, most exhibit mixtures of these types of bonds. In the case of many semiconductors which are tetrahedrally-bonded such as silicon, germanium, GaAs, and InP, the directionality of the bonds in the bulk will dramatically affect and may result in directed bonds at the surface.

Figure 10 below shows some possible arrangements of atoms that are typically observed on surfaces. The figure caption explains much of what is being shown in the figure. Relaxation occurs when there is a reduction or expansion in the atomic spacing in the direction perpendicular to the surface, the c direction. Reconstruction occurs when there is an in plane reconfiguration of the atoms.

Figure 10. A schematic view of the characteristic rearrangements of surface atoms of the simple cubic lattice is shown. The lattice constant of the bulk material isa. In part (a),relaxation results in differing lattice spacings in thec direction. In part (b), the atoms reconfigure in thea direction. A third configuration is shown in (c) in which every other atom at the surface is missing.

As one might expect, real surfaces can be quite a bit more complex than those in Figure 10. The reconstructed surface that was shown in Figure 2 is the result of the strong, directional nature of the covalent bonding in GaAs. Determining these atomic postions is difficult and involves a variety of experimental techniques, some of which we'll learn about in the next sections of the course.

Here are some final points before we move on. Surfaces do not exist in isolation. They are always found as an interface, whether it is solid to solid, solid to vapor, or even solid to vacuum. The solid/vacuum interfaces, lie those in Figure 10, are, of course, the simplest. Surface physics utilizes many techniques in the study of this type of interface in the hope of extending that information to the more complex interfaces, such as solid/solid. Figure 11 shows some interactions one might expect to occur at a solid/solid interface.

Figure 11. A thin film on a substrate may be used to illustrate various types of solid/solid interfaces. The film atoms are shown as filled circles; and the substrate surface atoms as blue open circles. (a) and (b) are abrupt interfaces since there is no mixing that occurs. The abrupt interface may either be crystalline, amorphous, or anything in between. The non-abrupt interfaces may be the result of mixing (or interdiffusion) or reactive (forming new chemical compounds, possibly multiple phases, the stability of which are dependent on thermodynamic parameters).

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