1.1 Doped semiconductors  (Page 2/3)

Again, if this were the end of the story, we still would not have any calculators, cell phones, or stereos, orat least they would be very big and cumbersome and unreliable, because they would have to work using vacuum tubes. We nowhave to focus on the few "empty" spots in the lower, almost full band (called the valence band .) We will take another view of this band, from a somewhat differentperspective. I must confess at this point that what I am giving you is even further from the way things really work, thenthe "cups at different energies" picture we have been using sofar. The problem is, that in order to do things right, we have to get involved in momentum phase-space, a lot more quantummechanics, and generally a bunch of math and concepts we don't really need in order to have some idea of how semiconductordevices work. What follow below is really intended as a motivation, so that you will have some feeling that what westate as results, is actually reasonable.

Consider [link] . Here we show all of the electrons in the valence, or almost fullband, and for simplicity show one missing electron. Let's apply an electric field, as shown by the arrow in the figure.The field will try to move the (negatively charged) electrons to the left, but since the band is almost completely full, the onlyone that can move is the one right next to the empty spot, or hole as it is called.

One thing you may be worrying about is what happens to the electrons at the ends of the sample. This is one of theplaces where we are getting a somewhat distorted view of things, because we should really be looking in momentum, or wave-vectorspace rather than "real" space. In that picture, they magically drop off one side and "reappear" on the other. This doesn'thappen in real space of course, so there is no easy way we can deal with it.

A short time after we apply the electric field we have the situation shown in [link] , and a little while after that we have [link] . We can interpret this motion in two ways. One is that we have anet flow of negative charge to the left, or if we consider the effect of the aggregate of all the electrons in the band we could picture what is going onas a single positive charge, moving to the right. This is shown in [link] . Note that in either view we have the same net effect in the way the totalnet charge is transported through the sample. In the mostly negative charge picture, we have a netflow of negative charge to the left. In the single positive charge picture, we have a net flow of positive charge to theright. Both give the same sign for the current!

Thus, it turns out, we can consider the consequences of the empty spaces moving through the co-ordinated motion of electronsin an almost full band as being the motion of positive charges, moving wherever these empty spaces happen to be. We call thesecharge carriers "holes" and they too can add to the total conduction of electricity in a semiconductor. Usingρ to represent the density (in cm ^-3 of spaces in the valence band and μ _e and μ _h to represent the mobility of electrons and holes respectively (they are usually not the same)we can modify to give the conductivity σ , when both electrons' holes are present, [link] .