SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

2.8. DOPING OF SEMICONDUCTORS 63

and requires the solution of multiband effective mass theory. However, the acceptor level can be
reasonably predicted by using the heavy hole mass. Due to the larger hole mass, acceptor levels
are usually deeper in the bandgap than donor levels.

Population of dopant levels
The presence of a dopant impurity creates a bound levelEd(orEa) near the conduction (or va-
lence) bandedge. If the extra electron associated with the donor occupies the donor level, it does
not contribute to the mobile carrier density. The purpose of doping is to create a mobile electron
or hole. When the electron associated with a donor (or a hole associated with an acceptor) is in
the conduction (or valence) band, the dopant is said to be ionized. To calculate densities of elec-
trons and holes at finite temperatures in doped semiconductors we note that carrier densities the
electrons will be redistributed, but their numbers will be conserved and will satisfy the following
equality resulting from charge neutrality

(n−ni)+nd = Nd (2.8.7)

(p−pi)+pa = Na (2.8.8)

or

n+nd = Nd−Na+p+pa (2.8.9)

where

n = total free electrons in the conduction band

nd = electrons bound to the donors

p = total free holes in the valence band

pa = holes bound to the acceptors

The number density of electrons attached to the donors has been derived in equation 2.4.4 and
is given by
nd
Nd

=

1

1

2 exp

(

Ed−EF

kBT

)

+1

(2.8.10)

The factor^12 essentially arises from the fact that there are two states an electron can occupy at a
donor site corresponding to the two spin-states.
The probability of a hole being trapped to an acceptor level is given by

pa

Na

=

1

1

4 exp

(

EF−Ea

kBT

)

+1

(2.8.11)

The factor of^14 comes about because of the presence of the two bands, light hole, heavy hole,
and the two spin-states.
To find the fraction of donors or acceptors that are ionized, we have to use a computer pro-
gram in which the position of the Fermi level is adjusted so that the charge neutrality condition
given Eq. 2.8.9 is satisfied. OnceEFis known, we can calculate the electron or hole densities
in the conduction and valence bands. For doped systems, it is useful to use the Joyce–Dixon