Environmental Engineering Reference
In-Depth Information
case, the carrier density goes to zero at low temperature, making the material
effectively insulating.)
The essential data needed to
nd
N
e
and
N
h
, are
(a) the Fermi energy
E
F
, the energy at which the available states have 0.5 occupation
probability, that is,
f
FD
(
E
)
¼
0.5 (see Equation 3.20);
(b) the size of the bandgap energy
E
g
;
(c) the temperature
T
; effective masses, number of equivalent band minima, and
(d) the concentrations of donor and acceptor impurities,
N
D
and
N
A
, respectively.
Smaller effects on the carrier densities comes from the values of the effective
masses (Equation 3.47) for electrons and holes. For an accurate numerical work,
note that for silicon, with indirect bandgap, the electron energy surfaces are
ellipsoidal and the correct density of states mass is
m
DOS
¼
(
m
l
m
t
m
t
)
1/3
¼
0.187.
For Si, with 6 equivalent band minima, the electron concentration must further be
multiplied by 6.
In addition, it is important to recognize that the semiconductor as a whole must be
electrically neutral
, because each constituent atom, including the added donor and
acceptor atoms, is an electrically neutral system.
To
find the expected number of electrons per conduction band minimum requires
integrating the density of states, multiplied by the occupation probability, over the
energy range in the band. In a useful approximation, valid when the Fermi energy is
several multiples of
k
B
T
below the conduction band edge, a standard result is derived,
which simpli
es the task of
finding carrier densities
N
e
and
N
h
.
In formulas for carrier concentrations, the energy
E
is always measured from the
top of the valence band. The standard formulas assume that the energies of electrons
in the conduction band (and of holes in the valence band) vary with wavevector
k
as
E¼
h
2
k
2
/2
m
, where
h
¼h
/2
p
, with
h
Plancks constant and
m
the effective mass, as
given by (3.47). In the formula for the energy
E
,
m
must be given in units of kg, while
the common shorthand notation, as in Table 3.2, is to quote
m
as a number, in which
case the readermust multiply that number by the electronmass
m
e
, before entering it
into a calculation. (The same applies to the mass of a hole
m
h
, recalling that the
motion of a hole is really the motion of an adjacent electron falling into the vacant
bond position.) The bandgap,
E
g
, is measured from the top of the valence band
E
V
¼
0, so that
E
C
¼E
g
.
The density of electron states (per unit energy per unit volume) in a bulk
semiconductor is
1
=
2
C
e
¼
4
pð
2
m
e
Þ
3
=
2
h
3
gðEÞ¼C
e
ðEE
C
Þ
;
=
:
ð
3
:
50
Þ
This formula (and the similar one for the density of hole states) is closely related
to (3.17), whichwas derived for a three-dimensional metal. Note also that the effective
density of states mass
m
DOS
¼
(
m
l
m
t
m
t
)
1/3
may be appropriate.
Similarly, the density of hole states in a bulk semiconductor is given by
1
=
2
C
h
¼
4
pð
2
m
e
Þ
3
=
2
h
3
gðEÞ¼C
h
ðEÞ
;
=
:
ð
3
:
51
Þ
