548
21 Diodes
Fig. 21.32 Band diagram
close to the interface of a
p-type semiconductor MIS
diode. Accumulation
occurs for s < 0,
depletion for s > 0 and
inversion (as shown here)
for s > > B > 0
E V
E F
e S
E i
E C
e B
semiconductor
insulator
The corresponding phenomena occur for n-type semiconductors for the opposite signs of the voltage with electron accumulation and depletion. In the inversion case, p > n close to the insulator
(Fig. 21.35a–c).
21.3.2 Space-Charge Region
Now we calculate the charge and electric field distribution in an ideal MIS diode, following the treatment
in [1757]. We introduce the potential that measures the separation of the intrinsic bulk Fermi level
and the actual intrinsic level E i , i.e. −e = E i (x) − E i (x → ∞) (see Fig. 21.32). Its value at the
surface is termed s , the surface potential. The value is positive, i.e. s > 0, if the intrinsic Fermi
level at the surface is below the bulk Fermi level.
The electron and hole concentrations are given (for a p-type semiconductor) as
n p = n p 0 exp(ββ)
(21.79a)
p p = p p 0 exp(−ββ) ,
(21.79b)
where n p 0 ( p p 0 ) are the bulk electron (hole) concentrations, respectively, and β = e/kT > 0.
Therefore, the net free charge is given by
n p − p p = n p 0 exp(ββ) − p p 0 exp(−ββ) .
(21.80)
The electron and hole concentrations at the surface are denoted with an index ‘s’ and are given by
16
n s = n p 0 exp(ββ s )
(21.81a)
p s = p p 0 exp(−ββ s ) .
(21.81b)
We use the Poisson equation
d
2
dx 2 = −
ρ
s
with the charge given by
ρ(x) = e
p p (x) − n p (x) + N
+
D (x) − N
−
A (x)
.
(21.82)
As boundary condition we employ that far away from the surface (for x → ∞) there is charge neutrality
(cf. (7.40)), i.e.
n p 0 − p p 0 = N
+
D − N
−
A ,
(21.83)
16 s represents the voltage drop across the semiconductor that will be discussed in more detail in Sect. 21.3.3. In this
sense, s for the MIS diode is related to V bi − V for the Schottky contact.
21 Diodes
Fig. 21.32 Band diagram
close to the interface of a
p-type semiconductor MIS
diode. Accumulation
occurs for s < 0,
depletion for s > 0 and
inversion (as shown here)
for s > > B > 0
E V
E F
e S
E i
E C
e B
semiconductor
insulator
The corresponding phenomena occur for n-type semiconductors for the opposite signs of the voltage with electron accumulation and depletion. In the inversion case, p > n close to the insulator
(Fig. 21.35a–c).
21.3.2 Space-Charge Region
Now we calculate the charge and electric field distribution in an ideal MIS diode, following the treatment
in [1757]. We introduce the potential that measures the separation of the intrinsic bulk Fermi level
and the actual intrinsic level E i , i.e. −e = E i (x) − E i (x → ∞) (see Fig. 21.32). Its value at the
surface is termed s , the surface potential. The value is positive, i.e. s > 0, if the intrinsic Fermi
level at the surface is below the bulk Fermi level.
The electron and hole concentrations are given (for a p-type semiconductor) as
n p = n p 0 exp(ββ)
(21.79a)
p p = p p 0 exp(−ββ) ,
(21.79b)
where n p 0 ( p p 0 ) are the bulk electron (hole) concentrations, respectively, and β = e/kT > 0.
Therefore, the net free charge is given by
n p − p p = n p 0 exp(ββ) − p p 0 exp(−ββ) .
(21.80)
The electron and hole concentrations at the surface are denoted with an index ‘s’ and are given by
16
n s = n p 0 exp(ββ s )
(21.81a)
p s = p p 0 exp(−ββ s ) .
(21.81b)
We use the Poisson equation
d
2
dx 2 = −
ρ
s
with the charge given by
ρ(x) = e
p p (x) − n p (x) + N
+
D (x) − N
−
A (x)
.
(21.82)
As boundary condition we employ that far away from the surface (for x → ∞) there is charge neutrality
(cf. (7.40)), i.e.
n p 0 − p p 0 = N
+
D − N
−
A ,
(21.83)
16 s represents the voltage drop across the semiconductor that will be discussed in more detail in Sect. 21.3.3. In this
sense, s for the MIS diode is related to V bi − V for the Schottky contact.