25
Review of Basic Device Physics
E E
E E
i
f
c
v
=
≅
+
2
(2.13)
Thus, the intrinsic Fermi level in a semiconductor material is very close to
the midpoint between the CB and the VB, and for all practical purposes, it
can be assumed that E i is in the middle of the energy gap. Thus, E i is commonly referred to as the mid-gap energy level.
In order to derive an expression for the intrinsic carrier concentration as a
function of T, we multiply Equations 2.9 and 2.10 to get
np n T
N N
E E
kT
N N
E T
kT
n
i
c v
c
v
c v
g
i
=
=
−
−
=
−
2 ( )
exp
e xp
( )
(
or
T T
CT
E T
kT
g
)
e xp
( )
/
=
−
3 2
2
(2.14)
where:
C is a constant
E g is the bandgap energy defined in Equation 2.1
k is the Boltzmann constant (8.62 × 10 –5 eV K –1 )
The term kT has the dimension of energy and is called thermal energy and
is equal to 25.86 meV at T = 300° K
Substituting the values for N c and N v [6], we can express Equation 2.14 as
n T
T
E T
kT
i
g
( )
.
e xp
( )
=
×
−
3 9 10
2
16 3 2
(2.15)
If E g (T NOM ) and n i (T NOM ) are the values of E g and n i at the nominal or the reference temperature T NOM , respectively, then we can show
n T
n T
T
T
E T
kT
E T
kT
i
i
NOM
NOM
g
g
NOM
NOM
( )
.
e xp
( )
= (
)
−
+
(
)
3 2
2
2
(2.16)
where E g (T) is given by Equation 2.2. The above expression is used in circuit
CAD for calculation of n i at any temperature T with n i = 1.45 × 10 10 cm –3 at
T = 300° K [6].
2.2.3.2 Effective Mass of Electrons and Holes
The electrons in the CB and holes in the VB move freely throughout the crystal
like free particles, suffering only occasional scattering by impurities and
defects present in the crystal. The free electrons experience Coulomb force
Review of Basic Device Physics
E E
E E
i
f
c
v
=
≅
+
2
(2.13)
Thus, the intrinsic Fermi level in a semiconductor material is very close to
the midpoint between the CB and the VB, and for all practical purposes, it
can be assumed that E i is in the middle of the energy gap. Thus, E i is commonly referred to as the mid-gap energy level.
In order to derive an expression for the intrinsic carrier concentration as a
function of T, we multiply Equations 2.9 and 2.10 to get
np n T
N N
E E
kT
N N
E T
kT
n
i
c v
c
v
c v
g
i
=
=
−
−
=
−
2 ( )
exp
e xp
( )
(
or
T T
CT
E T
kT
g
)
e xp
( )
/
=
−
3 2
2
(2.14)
where:
C is a constant
E g is the bandgap energy defined in Equation 2.1
k is the Boltzmann constant (8.62 × 10 –5 eV K –1 )
The term kT has the dimension of energy and is called thermal energy and
is equal to 25.86 meV at T = 300° K
Substituting the values for N c and N v [6], we can express Equation 2.14 as
n T
T
E T
kT
i
g
( )
.
e xp
( )
=
×
−
3 9 10
2
16 3 2
(2.15)
If E g (T NOM ) and n i (T NOM ) are the values of E g and n i at the nominal or the reference temperature T NOM , respectively, then we can show
n T
n T
T
T
E T
kT
E T
kT
i
i
NOM
NOM
g
g
NOM
NOM
( )
.
e xp
( )
= (
)
−
+
(
)
3 2
2
2
(2.16)
where E g (T) is given by Equation 2.2. The above expression is used in circuit
CAD for calculation of n i at any temperature T with n i = 1.45 × 10 10 cm –3 at
T = 300° K [6].
2.2.3.2 Effective Mass of Electrons and Holes
The electrons in the CB and holes in the VB move freely throughout the crystal
like free particles, suffering only occasional scattering by impurities and
defects present in the crystal. The free electrons experience Coulomb force