Welcome to
the PV online course, in this chapter,
we cover the topic of how a solar cell works.
In this teaching unit, we deal with the doping of semiconductors.
Semiconductors generally
have a relatively low conductivity.
By doping semiconductors,
their conductivity can be changed or improved.
This is done by selective contamination
with impurity atoms of the otherwise very pure crystal.
Why is it necessary to dope silicon?
The covalent
bonds, here circled in red, represent a very stable connection,
thus the contained electrons are locally bound / fixed.
Therefore an intrinsic (pure)
Silicone crystal has a very poor
conductivity.
As previously
mentioned in a Silicon crystal lattice only a few electrons break free
from their bonds, leaving a hole behind.
This process is strongly dependent on temperature.
The higher the temperature, the more FREE moving
electrons and holes are in the intrinsic crystal.
As an example, the values for silicon are displayed.
n: Electron density,
p: Hole density, Ni is called the intrinsic charge carrier density
k: Boltzmann constant
and T the temperature in Kelvin.
At any temperature, the product of electron density and hole
density equals the intrinsic carrier density squared.
For a better understanding, let us first look at the distribution
of energy states in a metal.
On the y-axis the energy is plotted,
on the x-axis the probability F
with which an energy state is occupied is plotted
The red curve describes the occupation probability
as a function of the energy.
This function is called Fermi-Dirac-function.
At low energies
ALL states are occupied,
the Fermi-Dirac function has the value 1.
At high energies, none of the states is occupied
by an electron, the Fermi-Dirac-function has the value 0.
The Fermi energy is the ENERGY
at which the occupation probability is exactly 50%.
At a temperature of 0K,
at absolute zero, the Fermi-Dirac-function assumes a jump function.
All electrons are in the energetically
most favorable state.
In the conduction band, all energy states are occupied
with low energy state, are occupied
so the occupation probability is 1.
Above the Fermi level,
the occupation probability jumps to 0.
Thus, the Fermi energy corresponds to the highest occupied energy level
in a METAL at zero Kelvin.
With increasing temperature,
the electrons also assume states with higher energy.
Thus the Fermi-Dirac-function runs more continuously.
The electrons distribute themselves around the Fermi level.
As mentioned before: the Fermi energy is the ENERGY
where the occupation probability is exactly 50%.
With increasing temperature, the Fermi-Dirac-function changes
accordingly.
We now consider the band
model in an intrinsic semiconductor.
The Fermi level is in the middle
of the forbidden zone, the band gap.
Therefore, no states exist in the region of the Fermi level.
The occupation of the states in the conduction band
depends on the temperature.
This then leads to the fact that at 0 Kelvin, the valence band is fully occupied
and the conduction band is not occupied at all.
But also
at increasing temperature, there are only a few electrons
in the conduction band and correspondingly few holes in the valence band.
The missing electrons in the valence band,
the holes, were shown with purple circles.
For better representation, the other electrons in the valence band
have now been removed.
The red curve again describes the Fermi-Dirac-function
meaning the probability with which an electron is encountered
in the corresponding energy state.
The gray curve is 1 MINUS the Fermi-Dirac-function
meaning the probability with which a hole can be encountered in the valence band
depending on the energy.
Doping, as already
mentioned, can have a significant effect
on the conductivity of an semiconductor.
To do this, let's first look at the periodic table of elements.
Silicone is in the fourth main group
and has 4 valence electrons, meaning
4 electrons in the outer shell.
Next to it, in the 5th main group, is phosphorus
with 5 valence electrons.
So 1 electron more than silicone.
In the
3rd main group there is boron with 3 electrons in the outer shell.
So 1 electron less
in the outer shell than in the case of silicone.
If the base material, in this case a silicone crystal,
is mixed with elements
that each have one electron more in the outer shell,
this is called negative doping.
Phosphorus has 5
valence electrons, 1 electron more than the silicone
in the surrounding silicone crystal lattice.
This extra electron can move
freely in the Si crystal lattice.
The conductivity of the
semiconductor crystal increases.
After the spatial representation,
we now want to understand the energetic representation.
We consider the band diagram of a silicon crystal
which has been contaminated with donors (e.g. phosphorus).
These free electrons of the donors
are energetically at the donor level.
This energy level is much closer to the conduction band than to the valence band.
Very little thermal energy is required
to lift electrons from the donor level to the conduction band.
As a result, almost as many electrons enter
the conduction band of the silicone as dopant
atoms have been incorporated into the crystal lattice.
n-doping leads to an increase in the charge carrier
concentration in the conduction band.
This also increases the Fermi energy.
Accordingly, the Fermi level is very close
to the conduction band.
In this way, there are
then many more electrons in the conduction band than holes in the valence band.
The electrons are in the majority,
therefore they are also called majority charge carriers.
The free-moving holes in the valence
band, on the other hand, are outnumbered
in the n region and are called minority charge carriers.
Accordingly, p n
is the hole concentration in the n-doped region.
To calculate the conductivity of the semiconductor,
we need the electron concentration nn in the n-doped region.
This is approximately
equal to the concentration of ionized donor atoms,
and this in turn is approximately equal
to the concentration of donor atoms.
The semiconductor is n-conducting.
Thus, its conductivity is approximately
proportional to the donor density.
E is the elementary charge
and µ_n the mobility of the electrons.
If the basic material, here again our silicone crystal,
is mixed with elements, each of which has one electron
less in the outer shell, this is called positive doping.
Boron has 3 valence electrons, that is 1 electron
less than the silicone atoms in the silicone crystal lattice
This missing electrons or holes
can move freely in the silicone crystal lattice.
The conductivity of the semiconductor crystal increases.
However, the conductivity is lower
than in the n-doped semiconductor.
We also want to look
at the p-doping in the energetic representation.
You can see here the band diagram of a silicon crystal
which has been contaminated with acceptors (e.g. boron)
This results in an acceptor level.
This level is much closer to the valence band
than to the conduction band.
Very little thermal energy is required to lift electrons
from the fully occupied valence band to the acceptor level.
Remember the fully occupied parking deck
where parking spaces are created?
There are almost as many holes in the valence
band of the semiconductor as there are doping atoms in the crystal lattice.
p-Doping leads to an increase of the carrier
concentration in the valence band.
This also decreases the Fermi energy.
The Fermi level is very close to the valence band.
There are many more holes in
the valence band than electrons in the conduction band.
The holes outnumber the electrons,
so they are called majority charge carriers.
Pp is the hole concentration in the p-doped region.
In contrast, the free-moving electrons in the conduction band are outnumbered
in the p-doped region and are called
minority charge carriers.
Accordingly, n p
is the electron concentration in the p-doped region.
The concentration of the majority charge carriers
(holes) is approximately equal to the concentration
of the ionized acceptor atoms, and these in turn are approximately equal
to the concentration of the acceptor atoms.
The semiconductor is p-type.
So its conductivity is approximately proportional
to the acceptor density.
E is the elementary charge and µ_p
the mobility of the holes.
We now summarize this teaching unit.
In intrinsic
semiconductors, the total number of negative charge carriers
equals the total number of positive charge carriers.
In n-doped semiconductors a large number of free electrons is present.
Electrons are the majority charge carriers.
In n-doped semiconductors a large number of holes is present.
Holes are the majority charge carriers.
In both cases, the charge carrier density
can be estimated equal to the dopant concentration.