Monthly Archives: April 2012

I’m With The Band Theory

We have already talked about the mechanical aspects of how a solid is assembled from individual atoms, when the atoms are in a repeating periodic structure called a crystal lattice. The lattice type can determine many of the material properties, but one of the most interesting and useful properties to think about is electrical conduction. One of the simplest ways to measure conduction is by creating a potential difference, so that one side of the material is more energetically favorable to charged particles than the other. This is called applying a voltage, named after Alessandro Volta who invented the battery, a device that uses chemical differences to create a voltage. When there is a voltage applied, electrons are drawn through the material, but how many electrons flow (which determines how large the measured electrical current is) varies widely by material. In some solids where electrons move freely, it’s easy to pass a lot of electrons through, creating an electrical current. This is characteristic of a metal, where electrical conduction is easy. But in other solids, it takes a lot of energy to move electrons through so electrical conduction is difficult, and we say these materials are insulators. And there is a third class of material, a semiconductor, which can be switched between conducting and non-conducting states.

To understand what causes the different electrical behaviors, we can think about how the atomic energy states available to electrons scale up to a bulk material. Each individual atom has a set of available electron states, that can be mathematically described using quantum mechanics. Some of these states are occupied by electrons, and some are not. A collection of atoms will have a collection of states, some occupied and some free, and an electron has to have available states that it can move between in order to traverse a solid. If there are no available states, it doesn’t matter how energetically favorable it is for the electron to go somewhere else, it has no way of getting there.

For a solid made up of only one kind of atom, the electronic states in each atom will be similar but may vary slightly due to varying conditions throughout a non-perfect crystal. This means that if we sum up all the states, instead of the precisely delineated states we found in a single atom, we’ll have smeared out bands of available states and forbidden regions which give a rough approximation of what energies electrons can have. As usual, the lowest energy states will be mostly occupied, and the highest energy states will be mostly empty. It’s the states right at the top of the electron occupancy which turn out to be the most useful for conduction, because of the minimal energy cost involved in moving electrons. (The line demarcating this is called the Fermi energy or Fermi level.) And how these available electron states look when we depict them as a function of energy can be very different, as shown below:

The various bands of allowed electronic states can overlap with each other, can have a small separation in energy, or can have a large separation in energy. Overlapping bands mean that in both bands, electrons have many available states to transition to, and that is why materials with overlapping bands have high electrical conductivity. These are metals, like gold or copper. For materials with a large separation between bands, the lower energy band is completely full and the higher energy band is completely empty. If an electron in the full band is tempted to move through the material, it must first scrounge up the energy to jump up to an available state, which is so considerable a task that most electrons can’t manage it. These are insulators, which may only pass one electron for every 1030 (a billion billion billion, or more electrons than stars in the universe) that pass through a metal at the same potential.

But the most interesting case, at least as far as modern electronics is concerned, is the material with a small separation between bands, the semiconductor. Only a small energy is needed to boost an electron from the full band to the empty band, and if the energy required is provided by thermal energy at room temperature, semiconductors can have significant electrical conduction at room temperature. But the most useful semiconductors are not quite conductive under normal conditions, but can easily be turned on by applying an electric field. That means they can operate as an electrical switch, acting as a metal or an insulator depending on what’s required. Silicon is the most widely used semiconductor in the present day,

Where the bands of available states fall exactly is determined by the crystal lattice type and the interatomic spacing, two factors which are themselves determined by the outer electrons of the atoms themselves. And for amorphous solids without a periodic structure, like glass, we still get energy bands. In fact, one way to think of the transparency of glass is that visible photons entering the material do not have enough energy to excite electrons from a filled band into an empty band, so they pass through the material without interacting with it. And that’s why you can see through glass!

All the justification for band theory involves a lot of math, of course. But just the basic idea, that bulk materials have bands of available states for electrons and the energy and grouping of these states determines electrical behavior, is pretty amazing because it puts a framework around the broad variety of electrical behaviors that we see in materials in nature. And, if you want to understand how electronics work, band theory is the first big piece of that puzzle.

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Crystal Lattices and Atomic Ordering

Back when we talked about the polymorphism of chocolate, we mentioned crystal phases, which are the differing configurations that atoms can take in order to make a solid. As we saw with chocolate, what crystal phase a material has can greatly affect its properties. If we want to know whether a material can conduct electrons, looks shiny, or is optically transparent, then it’s useful to know how its atoms are assembled.

Let’s start simple, with a solid composed of only a single element. How many ways are there to arrange identical atoms in three-dimensional space? We can do it randomly, so that there is no regular relationship between the positions of various atoms. This is called an amorphous, meaning not shaped, or glassy solid. Many materials (such as glass!) are amorphous, but often it’s more energetically favorable for there to be some kind of order in the arrangement. Remember that each atom has an electron cloud with a specific shape and orientation, and thus if we align the cloud shapes in some clever way, we may be able to fit more atoms into a given space. This is sometimes called a ‘close-packing’ problem, because of the similarity to packing M&Ms into a jar, dice into a container, and other practical mathematics problems.

Depending on the character of the electron cloud of the atom in question, there may be a specific distance between atoms that’s energetically preferred, or an angle between chemical bonds which yields the lowest energy configuration. These atomic traits largely dictate the crystal structure, but as it turns out if you want to create a repeating pattern in three dimensions there are only fourteen ways to do it. These possible lattices are often called the Bravais lattices, and some straightforward examples include the cubic lattice and the hexagonal lattice, pictured below. Most of the Bravais lattices are found in nature, but the denser lattices tend to be a lot more common.

                                http://en.wikipedia.org/wiki/File:Cubic.svg

Does the exact same lattice type continue throughout an entire solid, such as a metal table? Well, if it is the lowest energy configuration, yes! But what can happen is that one part of the table has the same lattice type, but slightly rotated from the part next to it. This could occur during solidification if crystal lattices begin growing at two points in a liquid, and gradually expand until they meet. The point where they meet is called a grain boundary, and because the crystal order is disrupted there, it’s usually a point of mechanical instability in the solid. For example, dropping something heavy onto a table is most likely to break the table if the heavy object lands on a grain boundary, and if the table is made of something which can corrode, that’s most likely to begin at a grain boundary. Grain boundaries are also really important in magnetism, which we’ll discuss in more detail another time.

But,  given a sample of some material, how can we find out what crystal structure it has? Well, these days there are some very powerful microscopes which can actually see the arrangement of individual atoms. But long before the development of those microscopes in the 1980s, the Braggs, a father and son physics team in the early 1900s,  thought of another way to verify crystal structure. With a lattice of atoms that repeats periodically, from some angles there will appear to be a series of planes. X-rays, very high energy photons, have a wavelength which is similar to the spacing between these planes. So when x-rays are sent into the sample, they will reflect off the planes of atoms, and along some angles these reflections will add up to give a strong scattered x-ray signal. This phenomena is known as Bragg reflection, and is the basis of x-ray diffraction, a family of very common techniques to determine the structure and composition of materials. As you can see from the example below, typical x-ray diffraction patterns have a lot of symmetry, but the locations of the bright spots can be used to mathematically calculate what the crystal structure of the sample is.

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Although the discussion above focused on materials made from a single element, it is also possible to have a periodic crystal whose building blocks have multiple elements, or even complicated organic structures like DNA. Now that we’ve seen how material response to external probing can depend on crystal structure, and how material strength can be affected by breaks in structure. But there are many other properties that are affected by which crystal lattice a material assumes, such as whether it is a metal, an insulator, or a solid. More next time!