Environmental Engineering Reference
In-Depth Information
Table 3.1 includes linear combinations of 3D wavefunctions that have character-
istic shapes, with lobes pointing in particular directions. These wavefunctions can
also participate in hybrid bonding.
3.6
Donor and Acceptor Impurities; Charge Concentrations
On the other hand, the theory predicts that a
filled band will lead to no conduction,
that is, an insulator, because no empty states are available to allow motion of the
electrons in response to an electric
field. Such a situation is present in the pure
semiconductors Si, GaAs, and Ge.
The four electrons per atom (eight electrons per cell) that fill the covalent bonds of
the diamond-like structure, completely ll the lowest two valence bands (because of
the spin degeneracy, mentioned above), leaving the 3D band empty. In concept, this
would correspond to the
first two bands in Figure 3.8 as being completely
lled, and
thus supporting no electrical conduction at zero temperature. Referring to Figure 3.8
one sees that at
k¼
0, just above the second band, there is a forbidden gap,
E
g
. There
are no states allowing conduction until the bottom of band 3, which is about 1 eV
higher in these materials. For this reason, at least at low temperatures, pure samples
of these materials do not conduct electricity.
Electrical conductivity at low temperature and room temperature in these materi-
als is accomplished by doping; substitution for the 4-valent Si or Ge atoms either
acceptor atoms of valence 3 or donor atoms of valence 5. In the case of 5-valent donor
atoms like P, As, or Sb, four electrons are incorporated into tetrahedral bonding and
the extra electron becomes a free electron at the bottom of the next empty band. In
useful cases, the number of free electrons,
n
, is close to the number of donor atoms,
N
D
. This is termed an N-type semiconductor. In the case of boron, aluminum, and
other 3-valent dopant atoms, one of the tetrahedral bonding states is un
lled, creating
a hole. Ahole acts like a positive charge carrier, whichmoves when an electron from
an adjacent bond jumps into the vacant position. Electrical conductivity by holes is
dominant in a P-type semiconductor.
The band structures for Si and GaAs are sketched in Figure 3.15. These results are
calculated from approximations to Schrodingers equation using more realistic 3D
forms for the potential energy
U
(
x
,
y
,
z
). The curves shown have been verified over a
period of years by various experiments.
In these semiconductors, the charge carriers of importance are either electrons at a
minimum in a nearly empty conduction band or holes at the top of a nearly
lled
valence band. In either case, themobility
m¼ et
/
m
, such that
, is an important
performance parameter. A high mobility is desirable as increasing the frequency
response of a device such as a transistor. A useful quantity, which can be accurately
predicted from the band theory, is the effective mass,
m
. This parameter is related to
the inverse of the curvature of the energy band. The curvature,
q
v ¼m E
2
E
/
q
k
2
, can be
2
k
2
/2
m
,is
calculated, and the formula, simply related to
E¼
h
