### Video Transcript

A pure semiconductor crystal has π
sub π equal to 1.6 times 10 to the eight per cubic centimeter. Upon adding impurities, the
concentration of vacancies increases to 3.2 times 10 to the 12 per cubic
centimeter. Calculate the concentration of free
electrons.

Here, weβve been asked about a
semiconductor to which impurities have been added. So this is a doped
semiconductor. And since we were told that doping
increased the concentration of vacancies, we know that this is a p-type
semiconductor.

Now, this question is asking us to
find the concentration of free electrons, π, in this doped sample. Thus, itβll be helpful to recall
that the concentration of free electrons π for a p-type semiconductor is given by
π equals π sub π squared divided by π sub Aβ, where π sub Aβ is the
concentration of negative acceptor ions. And π sub π is the free electron
and hole concentration for an undoped sample.

So long as we know values for both
the terms on the right-hand side of the formula, we can answer this question. We were told that before doping,
the material had an intrinsic charge carrier density, π sub π, of 1.6 times 10 to
the eight per cubic centimeter. We also need to recall that for a
p-type semiconductor, the concentration of vacancies is given by π equals π plus
π sub Aβ and is approximately equal to π sub Aβ, since a majority of electron
holes in our semiconductor are due to doping. And therefore π sub Aβ is much
greater than π.

Now, we were told that after
doping, the concentration of vacancies π is 3.2 times 10 to the 12 per cubic
centimeter. So we can take this as our value
for π sub Aβ. Letβs go ahead and substitute these
values into our formula for π. This gives us a result of 8000 free
electrons per cubic centimeter.

Choosing to write this in
scientific notation, our final answer is that this p-type semiconductor has a
concentration of eight times 10 to the three free electrons per cubic
centimeter.