103
Metal-Oxide-Semiconductor System
It is to be noted that the first term in Equation 3.45 represents the charge
density in the p-type substrate due to the majority carrier concentration and
the second term represents the charge density due to minority carrier concentration. Now, substituting Equation 3.45 in Equation 3.30, we get Poisson’s
equation in terms of band-bending potential f(x) at a depth x near the surface
of a p-type substrate as
d x
dx
q
K
N e
n
N
e
si
a
x v
i
a
x v
kT
kT
2
2
0
2
1
1
φ
ε
φ
φ
( )
( )
( )
= −
−
(
) −
−
(
)
−
(3.46)
We will solve Equation 3.46 for f(x) to obtain MOS capacitor behavior under
different operating conditions. Again, we notice from Equation 3.46 that
the first term inside the square bracket is due to the majority carrier charge
density whereas the second term is due to minority carriers in a p-type semiconductor substrate.
3.4.2 Electrostatic Potentials and Charge Distribution
In order to solve Equation 3.46 for potential distribution in silicon, we use the
mathematical identity
d
dx
d
dx
d
dx
d
dx
φ
φ φ
=
2
2
2
2
(3.47)
Then multiplying both sides of Equation 3.46 by 2 d x dx
φ( )
, we get
2
2
1
2
2
0
2
d x
dx
d x
dx
q
K
N e
n
N
e
si
a
x v
i
a
x v
kT
k
φ
φ
ε
φ
φ
( )
( )
( )
( )
= −
−
(
) −
−
T T
d x
dx
−
(
)
1
φ( ) (3.48)
Now, using Equation 3.47 in the left-hand side of Equation 3.48, we can show
that
d
dx
d x
dx
q
K
N e
n
N
e
si
a
x v
i
a
x v
kT
k
φ
ε
φ
φ
( )
( )
( )
= −
−
(
) −
−
2
0
2
2
1
T T
d x
dx
−
(
)
1
φ( ) (3.49)
We integrate Equation 3.49 from the bulk φ
φ
( )
, ( )/
x
d x dx
=
=
(
)
0
0 toward the
surface at any point x φ
φ
( ), ( )/
x d x dx
(
) near the surface shown in Figure 3.11
so that
d
d x
dx
q
K
N e
n
N
d x dx
si
a
x v
i
a
kT
φ
ε
φ
φ
( )
( )
( )
= −
−
(
) −
∫
−
2
0
0
2
2
1
e e
d x
x v
x
kT
φ
φ
φ
( )
( )
( )
−
(
)
∫
1
0
(3.50)
We know that the electric field at any point x near the silicon surface is
given by E x
d x dx
( )
( )
= −
φ
; therefore, after integration and simplification
of Equation 3.50, we can show
Metal-Oxide-Semiconductor System
It is to be noted that the first term in Equation 3.45 represents the charge
density in the p-type substrate due to the majority carrier concentration and
the second term represents the charge density due to minority carrier concentration. Now, substituting Equation 3.45 in Equation 3.30, we get Poisson’s
equation in terms of band-bending potential f(x) at a depth x near the surface
of a p-type substrate as
d x
dx
q
K
N e
n
N
e
si
a
x v
i
a
x v
kT
kT
2
2
0
2
1
1
φ
ε
φ
φ
( )
( )
( )
= −
−
(
) −
−
(
)
−
(3.46)
We will solve Equation 3.46 for f(x) to obtain MOS capacitor behavior under
different operating conditions. Again, we notice from Equation 3.46 that
the first term inside the square bracket is due to the majority carrier charge
density whereas the second term is due to minority carriers in a p-type semiconductor substrate.
3.4.2 Electrostatic Potentials and Charge Distribution
In order to solve Equation 3.46 for potential distribution in silicon, we use the
mathematical identity
d
dx
d
dx
d
dx
d
dx
φ
φ φ
=
2
2
2
2
(3.47)
Then multiplying both sides of Equation 3.46 by 2 d x dx
φ( )
, we get
2
2
1
2
2
0
2
d x
dx
d x
dx
q
K
N e
n
N
e
si
a
x v
i
a
x v
kT
k
φ
φ
ε
φ
φ
( )
( )
( )
( )
= −
−
(
) −
−
T T
d x
dx
−
(
)
1
φ( ) (3.48)
Now, using Equation 3.47 in the left-hand side of Equation 3.48, we can show
that
d
dx
d x
dx
q
K
N e
n
N
e
si
a
x v
i
a
x v
kT
k
φ
ε
φ
φ
( )
( )
( )
= −
−
(
) −
−
2
0
2
2
1
T T
d x
dx
−
(
)
1
φ( ) (3.49)
We integrate Equation 3.49 from the bulk φ
φ
( )
, ( )/
x
d x dx
=
=
(
)
0
0 toward the
surface at any point x φ
φ
( ), ( )/
x d x dx
(
) near the surface shown in Figure 3.11
so that
d
d x
dx
q
K
N e
n
N
d x dx
si
a
x v
i
a
kT
φ
ε
φ
φ
( )
( )
( )
= −
−
(
) −
∫
−
2
0
0
2
2
1
e e
d x
x v
x
kT
φ
φ
φ
( )
( )
( )
−
(
)
∫
1
0
(3.50)
We know that the electric field at any point x near the silicon surface is
given by E x
d x dx
( )
( )
= −
φ
; therefore, after integration and simplification
of Equation 3.50, we can show