3.2 Semiconductor

Crystal structure of silicon or germanium follows a tetrahedral pattern with each atom sharing one valence electron with each of four neighboring atoms. Let us examine this arrangement in two dimensions, for ease of representation. At low temperature most of the electrons are confined to the covalent bond region resulting in very low conductivity. At room temperature, a few electrons would have acquired enough energy to "jump": out of the valence band and enter into the conduction band. That is a few electrons that are delocalized are free to "float" around as in metallic materials. It is useful for you to think about the conduction band electrons as mobile while covalent bond electrons as essentially immobile and localized.

Now, the absence (or vacancy) of an electron in one of the covalent bond in the silicon lattice will create a local positive charge called "electron hole". On the other hand, presence of conduction band electrons, over and above that is needed to satisfy covalent bonds will create a local negative charge. You will note that in a pure silicon crystal, we will have exactly the same number of conduction electrons ( that is mobile electrons) as holes. If we apply an electric potential to this crystal, the electrons will migrate toward the positive electrode and the holes will migrate towards the negative terminal. The conductivity can be calculated from the relationship introduced in the last section.

= µ n q

 

Since there are two charge carriers, namely e and h, the above equation is modified to include both electrons and holes. That is,

 

In the above equation n and p refer to number density of negative and positive charge carriers respectively. The mobility of holes tend to be lower than that of electrons. Mobility of electrons and holes in pure silicon crystal at room temperature are given below.

 

Since the number of conduction electrons and holes are created simultaneously in a pure Si crystal, we can simplify the above equation by substituting ni for n and p. The conductivity relationship is therefore further simplified as

For pure silicon at room temperature, ni is about 1.5 x 1016 per m3. Let us consider a simple example.

Example: 1 Calculate the resistance of a 10 µm diameter cylinder of pure Si of 100 µm length at room temperature. Density of Si is 2.33 x 106 g/m3.

Solution: First calculate conductivity of Si using reported mobility values for e and h. That is:

 

Next, calculate resistance: