What Is Band Theory?

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Electron states

Band theory describes solids by using the quantum states of their electrons. In the graphs above, we see that as atoms get closer together, the number of possible energy levels increase. Each level represented by a straight line that branches off is an orbital. Starting from the bottom level orbital, we see the 1s, 2s, 3p, and 3s orbitals. The separation d is the distance between atoms that minimizes the potential energy. At low energies or large separations, the orbital energies may just be straight lines. These are no fun, so we go higher and see that these straight lines split off into multiple possible states for electrons to occupy. Each of the lines that are split off are so close together in energy that we can effectively fill the spaces. Then we could put little rectangles to occupy these areas to simplify things. These rectangles are the bands.

Bands

The bottom bands are the valence bands (VBs). These are the energy states occupied by the valence electrons in the outermost shell. These are the most occupied bands. The top bands are the conduction bands (CBs) and are normally empty. These are the lowest unfilled energy levels. For an electron to be used for conduction, it needs to gain enough energy to cross the band gap (BG) from the VB to the CB. Notice how the metal doesn’t have a BG. This is because it experiences an overlap of energies. Figure (b) simplifies things by not showing the overlap, but in reality, for the 2s, 2p, 3s orbitals and so on, numerous energy states overlap. All of those splitting lines actually cross over each other. This results in the lack of a BG for metals and allows for such effective conduction*. Insulators, on the other hand, have a large bandgap. In Figure (a), there is a pronounced gap between two orbitals. For conduction to take place, the electrons must gain enough energy to cross the BG. So what about semiconductors?

*In truth, metals do have bandgaps between orbitals, but they all overlap so there effectively is none. This is described more thoroughly with the theory of Brillouin zones.

Semiconductors

Semiconductors like to hybridize their s and p orbitals into sp3 orbitals. Semiconductor atoms on their own don’t do this because it increases their energy, but when multiple atoms get together, it actually reduces the overall potential energy. One result is a strange energy versus distance graph as shown above. As with insulators, a BG is formed, but semiconductors’ are typically not large. While insulators could have BG energies higher than 9 electron volts (eV), semiconductors get up to 4 eV. They are closer to around 0.5-2.3 eV. The internal motions of the atoms have enough energy to push some electrons across the BG, but very few. When heated, an electron loosens from its bond and leaves behind a positive hole. This hole will get filled by a neighboring electron, which will in turn leave behind another hole, and so on. Since the hole is effectively moving, we can actually calculate hole mobility and conduction. The same goes for insulators. So semiconductors and insulators have two charge carriers: holes and electrons.

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