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Doping
In the previous section we derived expressions for electron and
hole densities in the undoped semiconductor. In this case each excited
electron leaves behind a hole, thus, n and p are equal. We
can calculate the position of the intrinsic Fermi level of the undoped
semiconductor.
Not very surprising, the intrinsic Fermi level is approximately at midgap
position with a small shift due to the different effective masses. A second
important quantitiy is the product of n and p which is independent
of the actual Fermi level position.
The expression ni is called intrinsic carrier density,
it is an important paramter because its value is independent of doping.
Donors
The semiconductor we considered so far was an ideal material of highest
crystal quality and purity. In reality we will deal with crystal defects
(vacancies, antisite atoms, interstitials, etc.) and impurities (atoms
other than the constituents of the semiconcuctor material). Each deviation
from the ideal crystal gives rise to distortions of the ideal band structure.
Let us start with a very special class of impurities, the donors.
Imagine we take out one atom and replace it with an atom from an adjacent
group in the periodic system. If the replacing atom is from a group to
the right, it will have an extra electron which is not needed in the original
bonding configuration. The most prominent example is phosphorus in silicon.
In the tetrahedral bonding configuration of silicon four electrons are
needed for the bonding to the four neighbours. Obviously, the fifth electron
of phosphorus is not needed for the bonding. We may assume that the single,
weakly bound extra electron feels a potential similar to the electron in
the hydrogen atom, but extending downwards from the conduction band.
For an estimation of the "donor depth" ED we consider
the ionisation energy of hydrogen:
We will use the dielectric properties of the semiconductor by using
ee0
instead of e0 and the effective electron
mass me instead of the free electron mass. The dielectric
constant e of silicon is 12, the effective mass
for conduction electrons is 0.26 me. Thus, we obtain
an ionisation energy of 13.8 meV. Compared to the thermal energy kT
of 25.8 meV at 300 K we can usally assume that all donor atoms are ionised
at room temperature.
Earlier we derived an equation for the density of electrons as function
of the Fermi level. This equation is still valid after introducing a given
number of electrons n by doping. Then, we can solve it for the Fermi
level position.
Doping with donors raises the Fermi level from the midgap position closer
to the conduction band.
As the product of n and p must still be equal to ni2,
the number of holes is drastically reduced in the n-doped semiconductor.
Acceptors
We can draw the analogy to the hydrogen atom a little further. The opposite
of the hydrogen ionisation process is the attraction of a second electron
in oder to fill the shell to the preferred configuration of helium. The
energy gain is eqivalent to the electron affinity and in case of hydrogen
amounts to 1.93 eV. Similarly in the semiconductor, impurity atoms with
one electron less than the original atom on the host site also have an
affinity to attract electrons in order to obtain the correct number of
bonding electrons. This is the reason why they are called acceptors.
In the electron picture the atom gains energy by the capture process. However,
we don't want to describe electrons bound to an acceptor atom, rather,
we prefer the hole description. Then, the energy scale counts downwards
and a hole "bound" to the acceptor requires some energy to be "excited"
into the valence band.
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