# 1. Classification of Solids

Throughout the field of condensed matter physics, scientists classify different materials in a variety of different ways which we shall examine here. Fundamentally we can talk of the three broad states of matter: solid, liquid and gas....

## Solids, Liquids and Gases

In a solid, the atoms which make up the material are fixed in some kind of rigid structure. As you heat the solid up the atoms will jiggle about a bit, but they will still be essentially jiggling about a fixed position. This is why a solid feels rigid when you push against it. If you heat the solid up enough, however, the atoms will eventually break loose from their fixed positions and wander around randomly. If the atoms remain fairly closely packed together then the material will have become a liquid. Heat them up still more and it is possible to make the atoms fly apart to form a very loosely defined group in which the atoms can travel long distances without hitting each other. If this happens the material has become a gas. Condensed matter physics deals with the first two of these states of matter.

We can also, however, further subdivide the solids in a number of different ways. I shall outline two of these types of classification: classification due to thermal/electrical properties, and classification due to structural properties.

## The Structure of Solids

Broadly speaking, solids can fall into one of two categories: those which possess long-range-order in the disposition of their atoms, and those which do not. The first type of material is known as a crystal, while the second is termed an amorphous material.

That is, in a crystal the sites of atoms are determined simply by repeating some sub-unit of the crystal at regular intervals to fill all space. Mathematically we describe a crystal in terms of a regularly arranged set of points whose distribution throughout space looks identical from any point in the set (the lattice), and a prescription telling us how many atoms of each type to associate with each point and where they should go in relation to that point (the basis).

For example, the sodium chloride crystal structure is based upon the face centred cubic lattice in which the lattice points are arranged as if at the corners of an array of adjoining cubes, but with an additional lattice point at the centre of each face. The basis then dictates that each lattice site be given two atoms (one sodium and one chlorine) separated from each other by a distance equal to half the cube side length.

Different materials have different underlying lattices and different kinds of basis. There are an infinite number of possible atomic bases, but symmetry dictates that there are only 14 possible different types of lattice (in 3D), and that these can be further categorised into just 7 different types of symmetry.

The 14 different lattices are known as Bravais lattices, and the 7 different symmetry groups are known as the crystal systems.

Crystals are the most widely studied solids from the theoretical point of view, because we can learn about the behaviour of an entire crystal just by studying a very small portion (remember, the structure simply repeats itself at regular intervals). Furthermore, crystals are extremely important in everyday life, in industry, in science and technology: metals are crystalline, for example.

Amorphous materials are less well-studied, which is a shame since they are also very important in the real world. A good example of an amorphous material is glass. However, the lack of a repeating structure means that these materials are much more difficult to deal with from the theoretical point of view. Consequently they won't feature at all in the following material.

## Electronic and Thermal Properties of Solids

This section starts easy and then gets just a little bit hard....

Everybody knows that metals generally conduct electricity and heat very well. This is why they are used for electrical wiring and for saucepans. At the other end of the scale everybody knows that plastics conduct both electricity and heat relatively poorly. This is why they are used for insulating electrical wiring from the outside world. Carrying this a step further we can suggest that all solids can be classified into those which conduct electricity and heat well at room temperature (conductors-clever or what?) and those which do not (insulators-what a surprise!).

You can probably guess that if one actually tries to classify materials along these lines, one is likely to find a rather awkward bunch of materials whose properties lie midway between the two extremes. With characteristic imagination these are known as semiconductors. This rather belies the true nature of these materials because not all of their properties are simply related to those of conductors and insulators: semiconductors are rather special.

To understand just why they are special, we will need to look into the thermal and electronic properties of solids in a little more detail....

The atoms which make up a solid are made up of a compact heavy core (the nucleus) surrounded by several much-lighter electrons. The nucleus has a positive electric charge, while the electrons have a negative charge. This causes the electrons to be attracted to the nucleus. A neutral atom has just enough electrons to exactly balance the charge of the nucleus.

However, some of the electrons are so tightly bound to the nucleus by this attraction that we may as well think of the nucleus plus its core electrons as a single entity (the ion core whose positive charge is equal to that of the nucleus less the negative charge of the core electrons. This means that we only have to worry about the behaviour of the remaining loosely bound electrons (known as the valence electrons).

When atoms combine to form a solid it is the attraction between the positive ion cores and the valence electrons which holds the material together. While the ion cores occupy fixed positions (either in an amorphous or a crystalline structure) the valence electrons whiz around between them, forming a kind of electrostatic glue. In some materials this "glue" is piled up into distinct bonds between particular ion cores (so-called covalent bonding), but in others the electrons are more evenly distributed in the space between the ion cores (known as metallic bonding). A third form of bonding occurs when some of the valence electrons from one atomic species are donated wholesale to another atomic species. Atoms of the species which donates electrons become positive ions and atoms of the species which accepts the electrons become negative ions. This leads to a direct electrostatic attraction between the ions and is known as ionic bonding.

These different forms of bonding are largely responsible for the different thermal and electrical properties of conductors and insulators. Both electrical current and heat are transmitted through these solids by the motion of electrons (to be strictly accurate, heat is also transmitted through vibrations of the atomic structure, but we'll ignore this for the moment). In a metallically bonded material the electrons can drift easily between the ion cores, but in a covalently bonded material they have to "hop" from one bond to the next in order to move. In an ionically bonded material the valence electrons are tightly bound to ion-cores which are themselves "tied" to fixed ionic sites in the crystal structure. Thus ionic solids are generally poor conductors, covalent solids may be slightly better, and metals are the best of all.

The difference between conductors and insulators can also be seen by using the techniques of Quantum Mechanics to calculate the energies of possible states that the electrons can occupy. In both conductors and insulators the valence electrons settle into the lowest energy states available (the so-called valence band states). The electrons in these states are whizzing around very rapidly, but in totally random directions so that there is no overall motion in any one direction. Thus there is no electrical current and no flow of heat.

If we want to make electricity or heat flow through the material we must either apply an electric field or heat one part of the sample. When we do this we raise the energy of some of the electrons and they will flow in such a way to carry the electrical or thermal energy away from its source. However, there must be unoccupied quantum mechanically allowed states available for the electrons to occupy when we raise their energies (conduction band states ).

In conductors there are unoccupied conduction band states with energies ranging from the highest energy of the valence band upward, so it is easy to move electrons from the valence band (where they do not contribute to the current) into the conduction band (where they do). In insulators, however, the lowest energy unoccupied conduction band state is significantly higher in energy than the highest energy valence band state (i.e. there is a band gap between the valence and conduction bands). This means that electrons cannot enter the conduction band until they have been given extra energy at least equal to the band gap energy. Consequently it takes much more energy to make an insulator conduct electricity than it does to make a conductor do the same.

In terms of its valence and conduction bands (its band structure) a semiconductor looks just like an insulator with a very small band gap. Because of the small band gap the semiconductor can be adjusted to behave in a variety of novel ways by including small numbers of impurity atoms (doping the semiconductor). For example, if the impurity atoms have more valence electrons than each of the semiconductor atoms then an occupied band with energy just below the conduction band is created. These electrons can easily be excited into the conduction band and so the conduction of the material is improved. On the other hand, if impurity atoms are included which have fewer valence electrons than the semiconductor atoms then an unoccupied band with energy just above the valence band is created. Electrons from the valence band can easily be excited up to this band. Once there they are fairly tightly bound to the impurity atoms and so cannot move any great distance, but the positively charged holes that they left behind in the valence band can move, and it is these which can carry current.

Thus there are two types of doped semiconductors: n-type materials in which the current is carried by negatively charged electrons in the conduction band, and p-type materials in which the current is carried by positively charged holes in the valence band.

Most common semiconductors are either elements from group IV of the periodic table (Si, Ge, Sn), or are compounds formed from elements on either side of group IV. Thus there are so-called III-V semiconductors (GaAs, GaP, InP, AlAs, GaN, etc....) and also II-VI semiconductors (ZnSe, CdTe, etc....).

By combining these different materials in their undoped, n-type and p-type forms, semiconductor devices can be manufactured with very special electronic properties. It is these properties which allow modern computers and other electronic equipment to function, and this in turn provides the driving force behind much of the current effort to understand the physics of semiconductors.

Stephen Jenkins
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