Video Transcript
In a semiconductor that contains
acceptor ions and is at thermal equilibrium, the number of vacancies can be modeled
as being equal to the number of acceptor 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 free
electrons in the semiconductor is represented by 𝑛, and the density of acceptor
ions is represented by N subscript 𝐴 superscript minus. (A) The electron density equals the
undoped density squared minus the acceptor density. (B) The electron density equals the
acceptor density plus the undoped density squared. (C) The electron density equals the
acceptor density divided by the undoped density squared. (D) The electron density equals the
undoped density squared divided by the acceptor density. (E) The electron density equals the
acceptor density minus the undoped density squared.
In this question, we’re thinking
about a semiconductor that’s been doped with acceptor ions. Recall that in a silicon lattice,
an acceptor ion has room for one more possible electron in its outermost electron
shell. Therefore, each acceptor ion
creates a new vacancy in the lattice without creating any new free electrons. We’ve been told that the amount of
vacancies, represented by 𝑝, can be modeled as equal to the amount of acceptor
ions. This basically tells us that all of
the vacancies in the sample are due to doping.
So we already know the amount of
vacancies in the sample. But in this question, we wanna
figure out a way to model the amount of free electrons in the sample as well. So let’s use our knowledge of
semiconductors to write an expression solved for 𝑛. Let’s begin by recalling that for a
pure undoped sample, free electrons and vacancies are created in pairs. So for a pure semiconductor, the
density of free electrons equals the density of vacancies.
Now we’ve been told that the
quantity 𝑛 sub 𝑖 represents the density of free electrons if the sample was
undoped. This means for a pure
semiconductor, the undoped density is the density of free electrons. And because the density of free
electrons equals the density of vacancies, the undoped density equals the density of
vacancies as well.
Next, recall that we can take the
product of or multiply the electron density and the vacancy density, which is the
same thing as the undoped density squared. Now, something really interesting
and useful is that this term, the undoped density squared, remains constant for a
sample at a given temperature, even if the density of electrons and vacancies change
due to doping. So this expression holds true
whether our sample is doped or totally pure. Therefore, we know this
relationship correctly represents our semiconductor, and it’s written in terms of
the electron density, which is what we want to solve for.
So we’re getting close, but notice
that none of the answer choices are written in terms of the vacancy density. Instead, we wanna model our
semiconductor in terms of the acceptor density. So since we’ve already established
that the vacancy density equals the acceptor density, let’s go ahead and make this
simple substitution. And now, our expression is written
in terms of the acceptor density. All that’s left to do is solve it
for the electron density. So let’s divide both sides of the
formula by the acceptor density and cancel that term over here.
And now we have our answer, which
corresponds to option (D). The density of free electrons
equals the undoped density squared divided by the acceptor density.
