Video Transcript
In a doped semiconductor that is at
thermal equilibrium, the density of free electrons in the semiconductor is
represented by 𝑛, and the density of vacancies in the semiconductor is represented
by 𝑝. The density of either free
electrons or vacancies in pure silicon is represented by 𝑛 sub 𝑖. Which of the following formulas
correctly models the semiconductor? (A) The vacancy density times the
electron density equals the undoped density squared. (B) The vacancy density times the
electron density equals the undoped density to the power of one-half. (C) The vacancy density times the
electron density equals the undoped density. (D) The vacancy density times the
electron density equals two times the undoped density. (E) The vacancy density times the
electron density equals the undoped density divided by two.
To begin, recall that in a pure
semiconductor such as pure silicon, free electrons and vacancies are created in
pairs, so the density of free electrons must be equal to the density of
vacancies. This is why the quantity 𝑛 sub 𝑖 is capable of modeling the concentration of either free electrons or vacancies in a
pure sample. If we know the undoped density
equals the electron density, then because of this relationship, we also know the
undoped density equals the vacancy density.
Now we wanna find out what happens
when we multiply the vacancy density and the electron density and state their
product in terms of the undoped density. Luckily, the math is pretty simple
at this point. On the right-hand side of our
formula, let’s make these two substitution, and we have 𝑛 sub i times 𝑛 sub i, or
𝑛 sub i squared. This might seem redundant in the
case of a pure semiconductor, but it’s an interesting result that this formula
actually holds true for any semiconductor at thermal equilibrium whether it’s pure
or doped.
Recall that as we dope a
semiconductor, the densities of electrons and vacancies change so that they’re no
longer equal. However, they change in such a way
that the quantity 𝑛 sub i squared remains constant, which makes this formula really
versatile and useful.
So answer choice (A) is
correct. A doped semiconductor at thermal
equilibrium is correctly modeled using the formula the vacancy density times the
electron density equals the undocked density squared.
