# Band Structure in Solid State

The electrons in molecules are described in terms of molecular orbital theory. In this the electrons occupy a set of molecular orbitals which belong to a corresponding set of molecular energy levels. These molecular orbitals are made up of linear combinations of atomic orbitals, adapted for the symmetry of the molecule. Often, a given molecular orbital will have the character, ie. be made up from, only a small number of atomic orbitals.

In extended solids, molecular orbital theory may also be applied to describe the energies of the electrons in the solid. However, as the number of atoms increases, the number of atomic orbitals increases, and so the number of molecular orbitals increases and the separation in energy of the molecular orbitals decreases.

In the limit of a very large number of atoms, the molecular orbitals merge to form a series of bands corresponding to the extended overlap of atomic orbitals of the same symmetry.

 The Band Structure of a system with N atoms

The diagram above shows how the number of levels increases with N, and how the separation between the energy levels decreases with N.

For very large N, the separation between the levels becomes very small, and the individual levels can no longer be considered as discreet levels, and instead they overlap and form a band.

The molecular symmetry of the band is the same as the symmetry of the molecular orbitals which make up the band.

When two atomic orbitals overlap to give two molecular orbitals, one of the molecular orbitals is bonding and the other is antibonding. In the extended solid, the same principal applies: the lowest energy level of the band is fully bonding in character, and the highest energy is fully antibonding in character. The degree of bonding character steadily decreases as one goes up through the band.

The occupation of the bands by the electrons within the extended solid is similar to the aufbau principle for atoms, ie. the bands lowest in energy are occupied first, and so on until all the electrons are accounted for.

 Occupation of levels in a band Consider a system of N atoms, each with one valence orbital and one valence electron. The number of levels in a band made up from N orbitals is N, and each orbital can accommodate 2 electrons. There are N electrons, so they occupy the N/2 levels which are lowest in energy. The other, highest energy, N/2 levels remain unoccupied. The band is therefore only half full.

When atomic and molecular orbitals of more than one symmetry type exist, different bands are formed from overlap of the different types of orbitals. As the interaction between the atoms increases, the width of the bands formed by the overlap of the atomic orbitals increases, and so the bands of different symmetry types may also overlap in the extended solid.

 Degree of overlap of different bands Here, s and p bands formed from overlap of the s- and p-orbitals on the atoms are separated by a gap, the band gap. Here, the s- and p- bands are wider, due to a stronger interaction between the atomic orbitals, so the s- and p- bands overlap and there is no band gap.

At T = 0 K, the energy of the highest occupied band is called the Fermi level. At temperatures greater than 0 K, some excitation of electrons from levels just below the Fermi level to levels just above the Fermi level occurs, but this number is small compared to the total number of electrons, so we can generally assume that electrons occupy all the levels in the band from the bottom up to the Fermi level.

When it is important to allow for the thermal excitation which leads to population of levels higher than the Fermi level, the distribution of the occupied levels is given by a Boltzmann distribution adapted to allow for the fact that each level can hold two electrons. This distribution is known at the Fermi-Dirac distribution.

 The Fermi-Dirac distribution EF is the Fermi energy, the energy of the level for which P = 1/2 (as T rises, the  Fermi energy rises above the Fermi level as more higher energy levels become occupied).

#### Metallic and Non-Metallic Solids

The degree of overlap of the bands and the occupancy of those bands determines the conductivity of the compound.

Metals have delocalized electrons in partially filled bands, whilst non-metals have localized electrons in filled bands, with large band gaps between the fully occupied and fully unoccupied bands.

#### Simple Metals

Good examples of the simple metals are the Group 1 and Group 2 elements, and Aluminium.

The atoms have high coordination numbers, in the fcc- or hcp-arrays, and so there is a large degree of orbital overlap.

The ns- and np-bands are very wide (due to the large overlap), and they merge because the energies of the atomic ns- and np-orbitals are close in energy.

The band therefore contains 4N levels (1 for the s- and 3 from the p-orbitals), and so can hold up to 8N electrons. The simple metals have only N (Group1), 2N (Group 2), or 3N (Al) electrons in the valence shell, and so the band is only partially filled and conduction occurs. This is because the energy required to promote an electron from an occupied level to an unoccupied level in a partially filled band is small, and so electrons can easily become delocalized and transport energy throughout the solid.

Semimetals are those compounds which have a fully occupied band which touches a fully unoccupied band, ie. the band-gap is zero. Although there is no partially filled band, the promotion energy is again small, and so conduction may occur. Graphite is a semimetal, in the direction of the planes of carbon atoms.

#### Non-Metals

An extended solid which does not display metallic conductivity is also known as an insulator. A solid with enough electrons to completely fill a band and with a considerable band gap before the next unoccupied band will be an insulator.

Simple ionic solids, made from closed-shell species will be insulators. In NaCl, the N Cl ions form a band from the overlapping 3s- and 3p-orbitals, containing 4N levels. The N Na+ ions also have orbitals which overlap to produce a band. Each Cl atom has 7 electrons, and each Na atom has 1 electron, and these 8N electrons all occupy the 4N-level band from the Cl ions, and so it will be completely filled. The band from the N Na+ ions will be empty, and also have a much higher energy that the Cl band (the band gap is 7 eV), and so no promotion of electrons is possible and NaCl is an insulator.

In covalent solids, the situation is largely the same as in ionic solids, except that the bands have the character of both species. In a simple molecule, the bonding orbital is made up from one combination of shared atomic orbitals, and the antibonding band is made up from another combination of shared atomic orbitals. Similarly we get bands in the extended solid made up of combinations of the orbitals.

#### Evidence for band formation

The evidence is spectroscopic:

The UV/Visible absorption spectrum shows no absorption until a particular frequency is reached, the absorption threshold, and then there is continuous absorption over a larger energy range. This is the band.

The X-ray absorption spectra are consistent with transitions from localized core levels to broad, empty levels.