Video Transcript
In a semiconductor that contains donor ions and is at thermal equilibrium, the number of free electrons can be modeled as being equal to the number of donor ions. Using this model, which of the following formulas correctly represents the semiconductor? The density of free electrons in the semiconductor if it was undoped is represented by 𝑛 sub 𝑖. The density of vacancies in the semiconductor is represented by 𝑝. And the density of donor ions is represented by 𝑁 subscript 𝐷 superscript plus. (A) The vacancy density equals the undoped density squared minus the donor density. (B) The vacancy density equals the undoped density squared divided by the donor density. (C) The vacancy density equals the donor density divided by the undoped density squared. (D) The vacancy density equals the donor density minus the undoped density squared. (E) The vacancy density equals the donor density plus the undoped density squared.
In this question, we have a semiconductor that’s been doped with donor ions. Recall that a donor ion is introduced to a sample with one extra electron in its outermost electron shell. And that electron gets donated to the sample as a free electron. Therefore, donor ions bring new free electrons to the sample without creating any new vacancies. But still, the concentration of vacancies does change, since there are now more free electrons in the atomic lattice readily available and filling more vacancies. In this case, we want to model the concentration of vacancies. So we’re gonna figure out a formula solved for 𝑝.
Nevertheless, recall that for a pure sample, vacancies and free electrons are created in pairs, so the density of vacancies 𝑝 must be equal to the density of free electrons 𝑛. In this question, we’ve been told that the quantity 𝑛 sub 𝑖 represents the density of free electrons if the sample was undoped. So for a pure sample, 𝑛 sub 𝑖 equals 𝑛. And because 𝑛 equals 𝑝, we know that 𝑛 sub 𝑖 can represent the density of vacancies in an undoped sample as well. Now recall that multiplying 𝑝 and 𝑛 together can also be thought of as the undoped density times itself, so their product equals 𝑛 sub 𝑖 squared. The really cool thing is that this product has a constant value for any sample, doped or undoped, at a given temperature.
Therefore, if we dope a sample, the quantities 𝑝 and 𝑛 do change, but they change with respect to one another in a way that holds their product at a constant value. This is really useful since we know this relationship correctly represents our semiconductor, and it’s written in terms of the vacancy density 𝑝. But notice that none of the answer choices are written in terms of the electron density. Instead, we need to rewrite our formula in terms of the donor density.
Recall that we were told that the number of free electrons can be modeled as being equal to the number of donor ions. So we know that the electron density equals the donor density. So let’s make this substitution in our formula. And now all that’s left is to solve for the vacancy density. So let’s divide both sides of the formula by the donor density. So we can cancel that term over here and get the vacancy density by itself. Now we have our answer, which corresponds to option (B). So in a semiconductor that contains donor ions, the vacancy density equals the undoped density squared divided by the donor ion density.
