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1.2 Doped semiconductors  (Page 2/3)

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Again, if this were the end of the story, we still would not have any calculators, stereos or "Agent of Doom" video games (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 fullband (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 so far. The problem is, that in order to do things right, we haveto get involved in momentum phase-space, a lot more quantum mechanics, and generally a bunch of math and concepts we don'treally need in order to have some idea of how semiconductor devices work. What follow below is really intended as amotivation, so that you will have some feeling that what we state as results, is actually reasonable.

Consider . 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.

Band full of electrons, with one missing
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 , and a little while after that we have . 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 (whichwe have to do because of quantum mechanical considerations beyond the scope of this book) we could picture what is going onas a single positive charge, moving to the right. This is shown in . Note that in either view we have the same net effect in the way the total net 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!

Motion of the "missing" electron with an electric field
Further motion of the "missing electron" spot
Motion of a "hole" due to an applied electric field
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. Using ρ 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 this equation to give the conductivity σ , when both electrons' holes are present.
σ n q μ e ρ q μ h
How can we get a sample of semiconductor with a lot of holes in it? What if, instead of phosphorus, we dope our silicon sample with a group III element,say boron? This is shown in . Now we have some missing orbitals, or places where electrons could go if they were around. This modifies ourenergy picture as follows in . Now we see a set of new levels introduced by the boron atoms. They arelocated within the band gap, just a little way above the top of the almost full, or valence band. Electrons in the valence bandcan be thermally excited up into these new allowed levels, creating empty states, or holes, in the valence band. Theexcited electrons are stuck at the boron atom sites called acceptors , since they "accept" an electron from the valence band, and hence act as fixed negative charges, localized there. A semiconductor which is doped predominantly with acceptors is called p-type , and most of the electrical conduction takes place through the motion of holes. A semiconductor which isdoped with donors is called n-type , and conduction takes place mainly through the motion of electrons.
Silicon doped with Boron
P-type silicon, due to boron acceptors
In n-type material, we can assume that all of the phosphorous atoms, or donors , are fully ionized when they are present in the silicon structure. Since the number of donors isusually much greater than the native, or intrinsic electron concentration, ( ≈ 10 10 cm -3 ), if N d is the density of donors in the material, then n , the electron concentration, ≈ N d .
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Source:  OpenStax, Introduction to physical electronics. OpenStax CNX. Sep 17, 2007 Download for free at http://cnx.org/content/col10114/1.4
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