21.3 Metal-Insulator-Semiconductor Diodes
549
Fig. 21.33 Dependence of
the space charge on the
surface potential s for
p-type silicon with
N A = 10 15 cm −3 at
T = 300 K. The flat-band
condition is present for
s = 0, strong inversion
for s > 2 B . The dashed
blue (red) line is the
dependence of the
depletion (inversion)
charge
and that = 0. The ionization of the dopants only depends on temperature which is everywhere the
same
17 ; thus N
+
D − N
−
A is constant throughout the homogeneous semiconductor. Therefore (21.83)
(but not charge neutrality) holds everywhere in the semiconductor. Using (21.80) the Poisson equation
reads
∂
2
∂ x 2 = −
e
s
{ p p 0
exp(−ββ) − 1
− n p 0
exp(ββ) − 1
} .
(21.84)
The Poisson equation is integrated and with the notations
F (() =
exp(−ββ) + ββ − 1
+
n p 0
p p 0
exp(ββ) − ββ − 1
(21.85a)
L D =
s kT
e 2 p p 0
=
s
e β p p 0
,
(21.85b)
with L D being the Debye length (screening effect) for holes, the electric field can be written as
E = −
∂∂
∂ x
= ±
√
2 kT
e L D
F (() .
(21.86)
The positive (negative) sign is for > 0 ( < 0), respectively. At the surface, s will be taken as the
value for . The total charge Q s per unit area creating the surface field
E s = −
∂∂
∂ x
x=0
= ±
√
2 kT
e L D
F (( s )
(21.87)
is given by the one-dimensional Gauss law as Q s = − s E s .
The dependence of the space-charge density from the surface potential
18 is depicted in Fig. 21.33.
When s is negative, F is dominated by the first term in (21.85a) and the space charge is positive
(accumulation) and proportional to Q s ∝ exp(β| For s = 0 the (ideal) MIS diode is under flatband condition and the space charge is zero. For s > 0 the space charge is negative. For 0 < < s ≤ B
17 The donor and acceptor ionization energies do not change upon
18 We note that we discuss the space-charge region now only with regard to s , the voltage drop across the semiconductor,
and the dependence of s on the bias of the diode will be discussed in the next section.