3. Semiconductor Band Structure
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| Figure 3: GaAs band structure |
Semiconductors have an interesting electronic structure, called band structure. The electrons in a semiconductor at absolute zero exactly fill up all the states in what is known as the valence band. Lets define the highest energy level filled at absolute zero as zero energy. There is then a break in the available energy states for electrons. The next available states appear again after an energy gap known as the band gap, Eg. These states are known as the conduction band. Since all the valence states are filled, there are no states available for electrons to propogate through the semiconductor unless they are promoted to the conduction band. There is a technique known as doping (not discussed here) which allows one to control the number of electrons in the conduction band. Also if you are operating at a temperature above absolute zero, say T, there will be a thermal excitation proportional to exp(-Eg / kT) in the conduction band which the engineer has no control over.
Electrons which are promoted to the conduction band will fill up the lowest states, so an electron in the conduction band with no momentum will have an energy of just Eg. How do we treat an electron with wave vector k (the momentum divided by hbar)? Notice that the band structure at wave vector close to zero, i.e. at small applied voltages is approximately parabolic. Lets expand E(k) in a Taylor Series:
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Note that (5) is exactly the expression for a free particle. So our approach is to consider an electron traveling in the terribly complicated conduction band as a simple free particle with an energy zero at Eg and a mass of m*. m* is known as the effective mass since m* need not equal the mass of the electron. Now we have the necessary theoretical tools to build barriers to scatter our electrons as we discussed earlier.