1.1 Introduction to semiconductors  (Page 2/2)

It turns out that each band has exactly $2N$ allowed states in it, where $N$ is the total number of atoms in the particular crystal sample we are talking about. (Since there are 10 cups in eachband in the figure, it must represent a crystal with just 5 atoms in it. Not a very big crystal at all!) Into these bandswe must now distribute all of the valence electrons associated with the atoms, with the restriction that we can only put one electron into each allowed state . (This is the result of something called the Pauli exclusion principle .) Since in the case of silicon there are 4 valence electrons per atom, we would just fill up the first two bands, and the next would be empty. (If we make the logical assumption that the electrons will fill inthe levels with the lowest energy first, and only go into higher lying levels if the ones below are already filled.) Thissituation is shown in .

Here, we have represented electrons as small black balls with a "-" sign on them. Indeed, the first two bands are completelyfull, and the next is empty. What will happen if we apply an electric field to the sample of silicon? Remember the diagramwe have at hand right now is an energy based one, we are showing how the electrons are distributed in energy, not how they are arranged spatially. On this diagram wecan not show how they will move about, but only how they will change their energy as a result of the applied field. Theelectric field will exert a force on the electrons and attempt to accelerate them. If the electrons are accelerated, then theymust increase their kinetic energy. Unfortunately, there are no empty allowed states in either of the filled bands. An electronwould have to jump all the way up into the next (empty) band in order to take on more energy. In silicon, the gap between thetop of the highest most occupied band and the lowest unoccupied band is 1.1 eV.(One eV is the potential energy gained by an electron moving across an electrical potential of one volt.)The mean free path or distance over which an electron would normally move before it suffers acollision is only a few hundred angstroms ( $\approx (30010^{-8})$ cm) and so you would need a very large electric field (several hundred thousand $\frac{\mathrm{volts}}{\mathrm{cm}}$ ) in order for the electron to pick up enough energy to "jump the gap". This makes it appear that silicon would be avery bad conductor of electricity, and in fact, very pure silicon is very poor electrical conductor.