Video Transcript
In a pure semiconductor at a
temperature of 320 kelvin, the number of free electrons in the semiconductor is 𝑛
minus and the number of holes in the semiconductor is 𝑛 plus. The temperature of the
semiconductor is reduced to 280 kelvin. What is the ratio of 𝑛 minus to 𝑛
plus? The semiconductor is in thermal
equilibrium at both temperatures.
In this example, we’re working with
a pure semiconductor. And let’s say we have a lattice of
such semiconductor atoms like this. Each one of these dots then
represents the nucleus of such an atom, where we haven’t shown the valence
electrons. These electrons are what become
free electrons when enough energy is transferred to them so that they’re liberated
from the bonds associated with their respective atoms.
At a temperature of 320 kelvin,
there will be some free electrons in our sample. Let’s say those free electrons look
like this, moving about all through the lattice. And we’re told that that number of
free electrons in our sample at this temperature is 𝑛 minus. An important thing to note is that
each one of these free electrons used to be a valence electron of an atom in this
lattice. When a valence electron is
liberated to become a free electron, it leaves behind a hole or a vacancy.
We’re told that the total number of
holes in this 320-kelvin semiconductor is 𝑛 plus. In a pure semiconductor like we
have here, the number of holes is always equal to the number of free electrons. Indeed, they can’t exist without
one another. If this is our semiconductor at 320
kelvin, if we then decrease its temperature down to 280 kelvin, the result would be
that there is less thermal energy available to transfer to valence electrons. And therefore, fewer valence
electrons become free electrons.
However, for any electrons that do
become free electrons, they also leave behind a hole or a vacancy. So, even though with the
temperature decreased the number of free electrons is less, the number of holes is
less too by the same amount. Our question asks us about the
ratio of 𝑛 minus to 𝑛 plus. Regardless of the temperature of a
pure semiconductor sample, that ratio is always one. For every free electron in a pure
semiconductor, there is one hole. Likewise, for every hole, there is
a corresponding free electron. The ratio 𝑛 minus to 𝑛 plus is
equal to one.
