Environmental Engineering Reference
In-Depth Information
attracted to the donor impurity site by the Coulomb force, and the same physics that
was described for the Bohr model should apply! However, in the semiconductor
medium, the Coulomb force is reduced by the relative dielectric constant,
e
.
Referring to Table 3.2, values of
e
are large, 11.8 and 16, respectively, for Si and Ge.
The second important consideration for the motion of the electron around the
donor ion is the effective mass that it acquires because of the band curvature in the
semiconductor conduction band. These are again large corrections,
m
/
m
for elec-
trons is about 0.2 for Si and about 0.1 for Ge. So the Bohr model can usefully be scaled
by the change in dielectric constant and also by the change of electron mass to
m
.
The energy and Bohr radius are, from Equations 3.1
-
3.3,
E
n
¼k
C
Ze
2
/2
r
n
and
r
n
¼n
2
a
o
/
Z
, where
a
o
¼
h
2
/
mk
C
e
2
¼
0.053 nm. Consider the radius
first, and notice
that its equation contains both
k
C
, the Coulomb constant (proportional to 1/
e
) and the
mass. So the scaled Bohr radius will be
48
Þ
Similarly, considering the energy,
E
n
¼k
C
Ze
2
/2
r
n
¼E
o
Z
2
/
n
2
,
n¼
1,2,
...
, where
E
o
¼mk
C
2
e
4
/2
h
a
o
¼a
o
ðem
=
mÞ:
ð
3
:
2
, it is evident that
E
scales as
m
/(
me
2
):
E
n
¼ E
n
½m
=ðm e
2
Þ: ð
3
:
49
Þ
For donors in Si, we
nd
a
¼
59
a
o
¼
3.13 nm, and
E
o
¼
0.0014
E
o
¼
0.0195 eV.
The large scaled Bohr radius is an indication that the continuum approximation is
reasonable, and the small binding energy means that most of the electrons coming
fromdonors in Si at roomtemperature escape the impurity site andmove freely in the
conduction band. An entirely analogous situation occurs with the holes circling
acceptor sites.
An exciton is a bound state of an electron and hole created by light absorption. The
important point in calculating the behavior of the exciton is to consider the reduced
mass of the electron and hole as they circle about the center of mass. The exciton is an
analogue of positronium, a hydrogen-like atom formed by a positron and an electron.
Excitons play a role in energy transport in solar cells made of organic compounds.
3.6.2
Carrier Concentrations in Semiconductors
The objectives of this section are,
finding
N
e
and
N
h
, the densities of conduction electrons and holes, respectively, in a semicon-
ductor at temperature
T
. Second, an important special case occurs when the number
of donors or acceptors becomes large and the semiconductor becomes metallic.
At high doping, these systems behave like metals, with the Fermi energy actually
lying above the conduction band edge in the N
þ
case, or below the valence band edge,
in the P
þ
case. These separate rules apply to the so-calledN
þ
regions used inmaking
contacts, and also to the two-dimensional electron gas (2DEG) cases that are
important in making injection lasers and in forming certain charge qubit devices.
In these metallic or degenerate cases, the number of mobile carriers remains
large even at extremely low temperature, as in a metal. (In the usual semiconductor
first, to explain a standard method for
