A silicon crystal is doped with aluminum atoms at a concentration of 10 to the 13th per cubic centimeter. Calculate the concentration of free electrons in the pure silicon crystal knowing that the concentration of free electrons in the doped crystal is 10 to the 11th per cubic centimeter.
What we have in this example then is a sample of pure silicon crystal, which is then doped with aluminum atoms. Through this doping process, a number of the silicon atoms in the lattice are replaced with aluminum. Not only do we know the silicon crystal was doped, but we also know the concentration of atoms it was doped with. The name we can give to that concentration is 𝑁 sub 𝐴 minus. This is the concentration of the negative impurity that we’ve doped the silicon with, that is, the aluminum atoms. In addition to knowing that concentration, we also know the concentration of free electrons in the doped crystal after the aluminum is added in. That quantity, 10 to the 11th per cubic centimeter, we’ll call 𝑛 to represent the free electron density in our doped crystal.
Knowing all this, we look back to our original undoped silicon crystal. And we want to know what is the concentration of free electrons in this sample. There’s an equation we can recall that connects this free electron concentration with the two bits of information we already know. This relationship says that 𝑛 sub 𝑖 squared, where 𝑛 sub 𝑖 is the charge carrier concentration, that is, the concentration of free electrons, is equal to 𝑁 sub 𝐴 minus multiplied by 𝑛.
Applying this relationship, as a quick review of terms, 𝑛 sub 𝑖 is the free electron concentration in our undoped silicon sample. 𝑁 sub 𝐴 minus is our negative impurity concentration, that is, the concentration of our aluminum atoms. And then, 𝑛 by itself is the free electron density in our doped crystal.
Since we know both 𝑁 sub 𝐴 minus and 𝑛, we can plug those values in to solve for 𝑛 sub 𝑖. 𝑛 sub 𝑖 squared is equal to 10 to the
third [13th] per cubic centimeter times 10 to the 11th per cubic centimeter. These results can be combined by adding their exponents. And we get a result of 10 to the 24th per centimeter to the sixth. Now remember, this is 𝑛 sub 𝑖 squared. And we want to solve for 𝑛 sub 𝑖. So to do that, we’ll take the square root of this result.
This gives us an answer of 10 to the 12th per cubic centimeter. This is the concentration of free electrons in the pure silicon crystal.