Band Theory: For the Intermediate Level
The electronic strucutre of solids are usually described as a series of molecular orbitals that are found ovber all of the structure, rather than just in one area. For example, in metals, it is said all of the electrons are delocalised over the entire strucutre and not situated with the nuclei from which they com form. Stated below are some rules that are derived from Molecular Orbital theory that will help in deriving the bands that are used to describe the electronic structure of solids.
The rules are:
- Choose the atomic orbitals that you wish to make your molecular orbitals out of. This is the Basis Set
- N molecular orbitals can be constructed from a basis set of N atomic orbitals.
- The Pauli exclusion principle implies that each molecular orbital may be occupied by the electrons that are spin paired.
- By solving the Schrodinger equation, it is possible to arrive at solutions that model the molecular orbitals formed by the atomic orbitals.
- Areas of constructive interference lead to overlap of the atomic orbitals due to accumulated electron density. This results in a bonding molecular orbital that is low in energy and contributes to keeping the molecules together.
- Areas of destructive interference lead to non-overlap of the atomic orbitals due to reduction of electron density. This results in an anti-bonding molecular orbital that is high in energy and tends to tear apart the molecule.
- The are of reduced electron density resulting in a node where there is considered to be no electron density.
- In a bonding molecular orbital, the probability of finding the electrons on one, both or between the atoms is quite high.
- In an anti-bonding orbital, the probability of finding an electron in between the atoms is 0, and 1 for on one or the other atom.
By following the above rules it is possible to determine the electronic structure of various atoms. For example oxygen, a diatomic is hsown below. In the beginning we start with 8 atomic orbitals, and by solving the Schrodinger equation we get out 8 molecular orbitals: 4 bonding and 4 anti-bonding which describe the electronic structure. It is also observed that there are more than one molecular orbital at the same energy. When this happens, we call them degenerate orbitals. The molecular orbitals for oxygen can be seen below.
If a liner chain of finite lenght of p orbitals are now considered, we can observe that as the number of atomic orbitals are increased, the number of molecular orbitals will increase as well. This results in a decrease in the energy difference between each individual molecular orbitals, while the energy difference between the lowest bonding molecular orbital and the highest anti-bonding molecular orbital increases. This range of energies for the molecular orbitals only increases to a maximum 4B, where B is the overlap integral, which describes the extent to which the atomic orbitals are able to overlap to form the molecular orbitals. Once 4B is achieved, and as the number of atomic orbitals further increase, the resulting molecular orbitals become close together and start to blur due to an increase in the amount of overlap of the molecular orbitals.
Now consider a solid where the atomic orbitals are considered to be infinite and we will stich again to the idea of using a chain of p-orbitals. Therefore, upon solving the Schrodinger equation, we would expect an infinite number of molecular orbitals as the result. These molecular orbitals all fit into an energy separation of 4B. Therefore, the energy difference between each individual molecular orbital is such that it cannot be resolved, so they blur into bands. The final result is that we now have the electronic structure of the solid described by a series of bands made up of overlapping molecular orbitals, separated by band gaps, in which there are no molecular orbitals.
The final image of molecular orbitals as bands is shown below
When filling the molecular orbitals contained within the band, it is important to start filling the bands up form the lowest bonding molecular orbital and must floow the Pauli exclusion principle: No more than two electrons may occupy a single orbital and if two do occupy a single orbital, then their spins must be spin paried.As well, Hunds rule must not be broken: When more than one orbital has the same energy, electrons occupy separate orbitals ans do so with parallel spins.Now, if each atomic orbital contributing to the molecular orbitals contains only one electron, it is observed the band is only half full, since there is only N electrons for N molecular orbitals. This leaves empty levels in the bands. If on the other hand each atomic orbital contains 2 electrons, then there is 2N electrons for N molecular orbitals. This means that the band is full. Therefore, as long as there is less than 2N electrons, then there will be empty levels within the band.
In the case where the band is only half full, the occupation of the energy levels is dictated by what is know as the Fermi-Dirac Distribution. Under this distribution, it is found that only at T=0 K does it hold true that the band is exactly half filled. The uppermost filled molecular orbital becomes known as the Fermi energy. At temperatures above 0 K, the energy rises above the Fermi energy level because the electrons start occupying higher states due to thermal excitation. The shape of the distribution can be seen below.
In the above image, it is clearly seen that at temperatures greater than 0 K, the occupation of the molecular orbitals tails into the empty bands above the Fermi energy. This indicates that the electrons close to the Fermi level are very mobile and can move relatively freely through the solid. At much higher energies, it is found that the occupation of the empty levels follows a Boltzmann distribution.
Now that the Band Theory of Solids has been discussed, there is perhaps one more important property of bands to discuss: The density of states. The density of states describes the energy levels per unit energy increment. The density of states in a band is found to be non-uniform across the band. The reason for this is that the levels are packed more closely together at some energies than others as a result of overlap of the molecular orbitals.
Well I hope you enjoyed this little tour of Band Theory. Question, comments, get back to me by email.