To understand semiconductors one first has to understand the physics behind them.
0.0.1. Postulates of Quantum mechanicsUncertainty principle we can not now the position and the momentum at the same time:
0.0.2. How do we do quantum mechanicsWith the Schrödinger equations which is actually a wave equation.
or
or when we write in in complex form
0.0.3. The Electron Wavenumber kIn equation 2 and 3 we used k, because the differential equation is easier to solve with the constant k instead of the term mentioned in equation 4.
does directly relate to the momentum of the particle
. From standard physics we know equation 5
When we insert this in equation 7 we get after some rearinging equation 10 which is the same as equation 4.
0.0.4. Infinite Potential WellNow we want to find a solution to the differential equation. Let’s assume we have a cube of silicon. The electrons are for sure inside this cube/lattice. So the probability that they are outside of the latice is zero. This behaviour can also be described with the graphic below:
C
therefore
. So therefore k can only take certain values, which means the energy is restricted to certain values. The last thing that remains now is to solve for
. We further know equation 8, which says integral of the probability of the particle must be one.
. To do that we have to calculate the total number of states divided by the total volume of the semiconductor.
. The density apple per volume is in this case
which is one apple per
. The total number of apples is know the volume divided by the apple density
. From before we know that the states of an electron are defined like the following:
therefore the states are in a distance to each other of
. Futher more we can say that we have a state volume, therefore
in this volume is just
which is
and therefore
.
The density of states in a certain volume is then given by the following formula:
with
. We already know:
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