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SOLID STATE THEORY
By Walter A. Harrison
Dover Publications, Inc.Copyright © 1979 The Harrison Family Trust
All rights reserved.
SOLID TYPES AND SYMMETRY
1. Crystal Structures
A familiar and useful classification of solids is by their electrical properties. We divide all materials into those which conduct and those which insulate. Metals are of course the classic conductors and ionic crystals are the classic insulators. Typically a metal will carry a large current when a voltage is applied across it. For a given voltage the current carried will decrease when the temperature is increased. An insulator, in contrast, will carry a negligible current for comparable voltages. Between these two extremes there are also semimetals and semiconductors. A semimetal, like a metal, will carry current and as in a metal that current will decrease as the temperature is increased. A semimetal is distinguished from a metal by the much smaller number of electronic carriers in a semimetal (smaller by a factor of perhaps 10-4). A semiconductor behaves as an insulator at low temperatures but will carry a current that increases as the temperature increases at fixed voltage.
We will see in detail how these differing electrical properties arise from the specific electronic structure. For the moment we may simply use the electrical properties as a criterion for distinguishing solids.
Each of these four solid types is characterized by a rather specific crystalline structure, or arrangement of constituent atoms. The exact details of the structure appear not to be the determining factor in the properties of solids, but it will be useful for our later considerations to discuss the typical structures and to consider those aspects of the properties which are determined by structure alone.
In metals the atoms tend to be closely packed; i.e., they are packed as we might put billiard balls in a box. The three most common metallic structures are face-centered cubic, body-centered cubic, and hexagonal close packed. These three simple structures are illustrated in Fig. 1.1. In the face-centered cubic and body-centered cubic structures there is a framework of metallic atoms in a cubic array. If no other atoms were present these two structures would be simple cubic structures. In the face-centered cubic lattice, however, there are also metallic atoms in the center of each of the cube faces formed by the cubic array; in the body-centered cubic structure there is an additional atom at the center of each cube but none on the cube faces.
We should pause here to specify some of the notation that is useful in describing directions in cubic crystals. We think of three orthogonal axes, each oriented parallel to a cube edge. We then can specify the direction of a vector by giving the magnitude of its three components along the three axes. A direction is customarily specified by writing these three numbers side by side enclosed in square brackets; a negative number is usually denoted with a minus above. Thus the direction  represents a direction parallel to one of the cube edges;  is in the opposite direction. The expression  represents a direction parallel to the diagonal of a cube face. The direction  represents the direction of a cube diagonal. The orientation of planes in crystals is similarly specified with parentheses; thus the expressions (100), (110), and (111) represent three planes perpendicular to the three directions given before. In common usage these expressions specify the orientation of the plane but not its position, although the position usually is taken through one of the atoms in the crystal. These crystallographic directions and planes are illustrated in Fig. 1.2. Similar notation has been generated for noncubic crystals, but none is universally used and the meaning of any such notation should be specified when it is used.
In the hexagonal close-packed structure there are planes of atoms closely packed, like billiard balls on a table. The next plane of atoms has the same arrangement, but is shifted so the atoms fit snugly between those in the first plane. The third plane has again the same arrangement, and each atom lies directly above one of the atoms in the first plane. The fourth plane is identical to the second and so on. Both the face-centered cubic and the hexagonal close-packed structures correspond to the densest possible packing of hard spheres, although this is not so easy to see for the face-centered cubic case because of the manner in which we have specified it. In the face-centered cubic lattice the close-packed planes are (111) planes.
A few metals, such as manganese and mercury, occur in more complicated structures, but those are the exception rather than the rule. In these cases also the metallic atoms are rather densely packed. It is interesting to note that only in this century has it been generally recognized that metals are in fact crystalline. They had long been thought to be amorphous, or without regular structure. This misconception remains in our language in the description of the embrittlement of metals resulting from extensive bending as "crystallization" of the metal. Although we now recognize that metals are crystalline in nature, we should perhaps emphasize that the essential feature of the metallic structure is its close packing rather than the details of its structure. If a metal is melted it loses all of its crystalline order but remains in a rather close-packed configuration. Even with this total loss of crystalline structure, however, the electrical properties remain very much the same.
A prototype insulator is the sodium chloride crystal, which is more appropriately thought of as a configuration of ions rather than of atoms. Sodium, of valence 1, gives up an electron to chlorine, of valence 7. The structure is illustrated in Fig. 1.3. Again it is based on a simple cubic array, but in this case alternate ions are sodium and chlorine. The essential feature of this insulating structure is the alternation between positive and negative ions. Sodium chloride provides perhaps the simplest structure for an insulator. Many ionic compounds form quite complex structures, but all correspond to alternate packing of positive and negative ions.
Most molecules and all rare gases form insulators when they crystallize. In cases such as ice the molecules themselves may be thought of as ionic, but in cases such as molecular hydrogen they are not. The important characteristic feature of molecular crystals is the close association of the atoms making up the molecules rather than the configuration in which the molecules are arranged.
Insulators occur also in the amorphous state, i.e., with no long-range repeating structure. Glass is the most familiar example. Thus we may think of the insulating behavior of ionic compounds as arising from the alternate arrangement of positive and negative ions rather than from the details of the crystalline structure. Similarly, when a molecular crystal is melted, the molecules generally remain intact and the insulating behavior is maintained.
Semimetallic and semiconducting behavior is more sensitive to the structure of the crystal. Bismuth is a prototype semimetal; its structure is noncubic and rather complicated. However, the packing density, viewed as the packing of spheres, is comparable to that in metals. When bismuth is melted it becomes disordered, with essentially the structure of a liquid metal, and its electrical properties become that of a metal.
Our prototype semiconductors are silicon and germanium. Both silicon and germanium have the diamond structure that is illustrated in Fig. 1.4. The diamond structure may be generated from the face-centered cubic structure by adding a second interlocking face-centered cubic structure displaced from the first one-quarter of the way along a cube diagonal . This leaves any given atom surrounded by four near neighbors arranged at the corners of a regular tetrahedron centered at the given atom. The diamond structure is quite open in comparison to the metallic and semimetallic structure. If hard spheres were arranged in this configuration there would be room in the interstices to insert a second sphere for each sphere included in the original structure. Thus we might say the packing density is half that of a close-packed metal. Such open structures are typically semiconductors. They conduct only if the temperature is raised or if defects are introduced into the crystal. The semiconducting properties may be thought to be associated with the open structures. When either silicon or germanium is melted the packing density increases and they become liquid metals.
Some compounds made up of equal numbers of atoms of valence 3 and valence 5, such as indium antimonide, are also semiconductors and have structures just like the diamond structure but with the first face-centered cubic lattice being made of, say, antimony, while the second is made of indium. This is called the zincblende structure and is a structure of zinc sulfide (valence 2 and valence 6). Such two-six compounds have properties lying somewhere between those of insulators and the more familiar semiconductors.
With these differences in the arrangement of the atoms for different solids there are also significant differences in the distribution of the electrons in the crystal. In a metal, with close packing, the electrons are quite uniformly distributed through the crystal except in the immediate neighborhood of the nucleus where the density due to the core electrons is very large (and that due to the valence electrons is small). In Fig. 1.5 is a plot of the valence-electron distribution in a (110) plane in aluminum.
In ionic insulators we generally think, as we have indicated, of the outer electron on the metallic atom as having been transferred to the nonmetallic ion; however, once the ions are packed in a crystal this transfer really corresponds to only a subtle change in distribution.
In semiconductors much of the charge density due to the valence electrons ends up near the lines joining nearest neighbors. A plot of the valence-electron density in silicon is given in Fig. 1.6. This localized density forms the so-called covalent bonds in these structures. In semimetals, as in metals, the electrons are rather uniformly distributed. The importance of these differences, as well as the importance of the differences in structure, is perhaps overestimated in much thought about solids. Though the charges seem to be distributed in fundamentally different ways, they can always be rather well approximated by the superimposed electronic-charge distribution of free atoms.
A great deal can be learned about the properties of crystalline solids simply from a knowledge of their structure. Our first task will be to develop some of this knowledge. Out of this analysis will arise much of the terminology that is used to describe solids, and we will find that much of this terminology will remain appropriate when we discuss systems that do not have the simple structures.
2. Symmetry of Crystals
The characteristic feature of all of the structures we have described is translational invariance. This means that there are a large number of translations,
T = n1τ1 + n2τ2 + n3τ3
which a perfect crystal may undergo and remain unchanged. This, of course, moves the boundaries, but we are interested in behavior in the interior of the crystal. The ni are integers and any set of ni leaves the crystal invariant.
To obtain the most complete description of the translational invariance we select the smallest τi, which are not coplanar, for which Eq. (1.1) is true. These then are called the primitive lattice translations. These are illustrated for a two-dimensional lattice in Fig. 1.7. We have let the lattice translations originate at a lattice site. The set of points shown in Fig. 1.7 and specified in Eq. (1.1) is called the Bravais lattice. The volume of the parallelopiped with edges, τ1, τ2, τ3, is called the primitive cell. Clearly the crystal is made up of identical primitive cells.
Note that in the face-centered cubic structure a set of lattice translations could be taken as the edges of a cubic cell. These would not, however, be primitive lattice translations since translations to the face center are smaller yet also take the lattice into itself. By constructing a primitive cell based upon translations from a cube corner to three face centers, we obtain a cell with a volume equal to the volume per ion; i.e., a primitive cell containing only one atom. This is illustrated in Fig. 1.8. Had we used a cubic cell, we would readily see that the volume would be four times that large. The complete translational symmetry of the lattice is specified by the primitive lattice translations. A translational lattice based upon a larger cell contains only part of the symmetry information.
The specification of this translational invariance has reduced the information required to specify the structure of the crystal to the information required to specify the structure of a primitive cell.
Note that there is an atom at every point of the Bravais lattice, but there also may be other atoms within the primitive cell. The points specified, for example, might be indium ions and there also might be an antimony ion in each cell, which will be the case in indium antimonide. The points might be silicon atoms and there might be another silicon atom in each cell, yet translation of the lattice by the vector distance between the two silicon atoms will not leave the crystal invariant. This is the case for the diamond structure.
There will be other symmetry operations that take the crystal into itself. An inversion of the lattice through a Bravais lattice point will take the Bravais lattice into itself and it may take the crystal into itself. Various rotations, reflections, or rotary reflections may also take the crystal into itself. Clearly the combination of any two operations, each of which takes the crystal into itself, will also take the crystal into itself. The collection of all such operations, rotations, reflections, translations and combinations of these, is called the space group of the crystal. The space group contains all symmetry operations which take the crystal into itself.
If we take all space group operations of the crystals and omit the translation portion, the collection of all distinct rotations, reflections, and rotary reflections which remains is called the point group. In some crystals, certain operations of the point group will leave no atom in the crystal in the same position. Such an operation is illustrated for a two-dimensional case in Fig. 1.9. An operation combining reflection in a plane with a "partial lattice translation" (smaller than a primitive lattice translation) in the reflection plane is called glide-plane symmetry and the corresponding plane is called a glide plane. A second common combined operation is called a screw axis and consists of a rotation combined with a partial lattice translation along the rotation axis.
The symmetry operations of a crystal form a group (in the mathematical sense) as we shall see. The possible different point groups which are allowed in a crystal are greatly reduced from those which might be allowed in a molecule because of the translational symmetry. We may illustrate this by showing that in a crystal only rotations of 60°, 90°, or multiples of these can occur.
We construct a plane perpendicular to the axis of the rotation in question, and plot from the axis the projections on that plane of the lattice translations that take the crystal into itself. Of these projections we select the shortest, which will form a "star" as shown in Fig. 1.10. Note that for each projection in the star there will be a projection in the opposite direction, since for every lattice translation there is an inverse lattice translation. Let us suppose now that the rotation (counterclockwise) of an angle 0 takes the crystal into itself. Then the projection a will be taken into the projection b which must also be the projection of a lattice translation since the crystal, and therefore the lattice translations, has been supposed to be invariant under the rotation. The vector difference between b and a, however, must also be the projection of a lattice translation, since the difference between any two lattice translations is also a lattice translation. Further, this difference must not be less than the length of b (or a) since a and b have been taken to be the shortest projections. Thus, θ must be greater than or equal to 60°.
Excerpted from SOLID STATE THEORY by Walter A. Harrison. Copyright © 1979 The Harrison Family Trust. Excerpted by permission of Dover Publications, Inc..
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