Chapter 16-2. MOS electrostatic:
Quantitative analysis
In this class, we will
• Derive analytical expressions for the charge density, electric field and the electrostatic potential.
• Expression for the depletion layer width
• Describe delta depletion solution
• Derive gate voltage relationship
– Gate voltage required to obtain inversion (VT)
Electrostatic variables in the ideal MOS
Let’s establish analytical relationship for the charge density (ρ), the electric field (E), and the electrostatic potential in an ideal MOS capacitor.
Because the metal is an equipotential region, charge
appearing near the M-O interface resides only a few Angstrom into the metal and may be modeled as a δ-function.
Because there is no charge in the oxide, the magnitude of the charge in the metal is simply equal to the sum of the charges inside the semiconductor.
Due to the absence in the oxide, the E-field is constant in the oxide. linear function of position.
Solving for the electrostatic variables in the ideal MOS
Electrostatic potential, (x)
Define a new term, (x) taken to be the potential inside the semiconductor at a given position x.
*The symbol instead of V used in MOS work to avoid confusion with externally applied voltage (V).
)]
( (bulk)
[ )
( Ei Ei x
x q1
(surface)]
(bulk)
[ i i
S E E
q
1
] (bulk)
[ i F
F E E
q
1
Electrostatic potential at x
Surface potential (x = 0)
| F |, reference voltage related to doping concentration.
Electrostatic parameters
S is positive if the
band bends downward
(surface)]
(bulk)
[ i i
S E E
q
1
] (bulk)
[ i F
F E E
q
1
)]
( (bulk)
[ )
( Ei Ei x
x q1
qS
] (bulk)
[ i F
F E E
q
1
A D
D kT
E E
ie N N N
n
nbulk F i(bulk) / ... if
D A
A /kT
E (bulk) E
i
bulk n e N ... if N N
p i F
Combining the equations for each type,
type -
n ...
type -
p ...
F
D i A
n N q
kT
n N q
kT
ln ln
Electrostatic parameters
Therefore,
Under flat band conditions,
S = 0 and (x) = constant
Ei(surface) – EF = EF – Ei(bulk)
or Ei(surface) – Ei(bulk) = 2[EF – Ei(bulk)]
qS
At the depletion-inversion transition point (VG = VT),
2
Surface potential, S
(surface)]
(bulk)
[ i i
S E E
q
1
Electrostatic parameters
For a p-type semiconductor (F > 0),
For a n-type semiconductor (F < 0),
Exercise 16.2
Consider the following F and S parameters. Indicate
whether the semiconductor is p-type or n-type, specify the biasing condition, and draw the energy band diagram at the biasing condition. F = 12 kT/q; S = 12 kT/q
F = +12 kT/q means that Ei – EF in the semiconductor is 12 kT (a
positive value); So, p-type.
EC
Ei EF EV
12kT
S=12 kT/q means Ei (bulk) – Ei(surface) = 12 kT; i.e. the
band bends downward near the surface. depletion of holes
(surface)]
(bulk)
[ i i
S E E
q
1
] (bulk)
[ i F
F E E
q
1
F = 9 kT/q; S = 18 kT/q
EC
EF Ei
-9kT
Exercise 16.2
] (bulk)
[ i F
F E E
q
1
(surface)]
(bulk)
[ i i
S E E
q
1
here F = 9 kT/q means [Ei(bulk) – EF] = 9 kT; i.e., Ei is below EF. Thus the semiconductor is n-type.
S= 18 kT/q means that Ei (bulk) – Ei(surface) = –18 kT; So band bends upwards near the surface.
The surface is “inverted” since the surface has the same number of holes as the bulk has electrons.
Delta-depletion approximation
Consider p-type MOS in an accumulation condition.
Charge on metal = QM
Charge on semiconductor = (QM )
Accumulation of holes
x
M O S
VG < 0
p-Si
Assume that the free carrier
concentration at O-S interface is a
-function.
The accumulation charges are mobile holes, and appear close to the surface and fall-off rapidly as x increases.
E
E = 0 for all x > 0 x
s = 0 for all accumulation biases
Consider p-type MOS in a depletion condition (applying a positive VG such that s < 2F).
Using the depletion approximation, the charge density in the depletion region is
) 0
( )
(p n N N qN x W
q D A A
...
Charges in Si are immobile ions - results in depletion layer.
Then the Poisson’s equation reduces to
) 0
(
0 0
W K x
qN K
dx d
S A S
...
E
Integrating with the boundary condition (E = 0 at x = W) yields
xW SA
(x) dx
ε K d qN
0 0
E E
W x
ε K x qN
S
A
0
)
E( 0 x W
Delta-depletion approximation
From the definition of E-field,
dx d
E
W x
ε K
qN dx
d
S
A
0
E
W x
0
Separating variables and integrating from an arbitrary point x to W ( = 0 at x = W),
xW S A
(x) W x dx
ε K d qN
0 0
22 0
)
( W x
ε K x qN
S
A
0 x WQuadratic function
Delta-depletion approximation
Applying = s at x = 0,
22 0
)
( W x
ε K x qN
S
A
0 x W
2
2 0 W
ε K qN
S A s
and therefore s
A S
qN ε W 2K 0
Note that for an n-type MOS, N in the preceding equations The depletion width at the deletion-inversion transition point (s = 2F) is the maximum attainable equilibrium depletion width (WT)
) 2 2 0 (
F A
S
T qN
ε
W K
Delta-depletion approximation
type -
n ...
type -
p ...
F
) / ln(
) / ln(
i D
i A
n q N
kT
n q N
kT
) 2 2 0 (
F A
S
T qN
ε
W K
Maximum equilibrium depletion width
Charges in Si are immobile ions - results in depletion layer.
Depletion of holes
QM w
|q NA A W| = |QM| () (+)
When surface potential is s ,
E
x
M O S
VG > 0
p-Si For a p-type MOS in a depletion
condition
W x
ε K
x) qNA E(
s A S
qN ε W 2K 0
ε W K
qN
S A S
0
E
0
) (
ε K
qN dx
x d
S
A
E
Delta-depletion approximation
M O S
VG >> 0
p-Si For a p-type MOS in an
inversion condition
Depletion of holes
wT QM
Inversion electrons:
-function-like
Once inversion charges
appear, they remain close to the surface since they are mobile.
The -function-like charge added in inversion precisely balances the charge added to the metal.
The charge in the depletion region and E-field (x > 0) remain
E
T S
A
S W
ε K
qN
0
E
)
(x qN
dE A
Delta-depletion approximation
Gate voltage relationship (V
Gversus
S)
The applied VG is dropped partly across the oxide and partly across the semiconductor.
VG= ox +semi
Consider a p-type MOS.
Because = 0 in the bulk,
semi = (x = 0) (bulk) = S
In the ideal insulator without any carriers and charge centers,
0 dx dEox
and E constant
dx d ox
ox
Integrating across the oxide of which thickness is x0,
ox(x 0) 0 0
VG > 0
M O S
p-Si
Semi
ox
-x0 0 x
Gate voltage relationship (V
Gversus
S)
ox ox
x
0E
According to the well-known boundary condition on the E-field normal to an interface between two dissimilar materials,
S - interface O
S -
O Q
D
Dsemi ox) (
where D = εE is the dielectric displacement and QO-S is the charge density (charge/unit area) located at the interface.
Since QO-S = 0 in the ideal MOS,
0semi x
ox
D
D
S S ox
O
K
K E E
where KO and KS are dielectric constants of the oxide and semiconductor, respectively (KO = 3.9 for SiO2 and KS = 11.8 for Si ).
E
ox 3 E
S KGate voltage relationship (V
Gversus
S)
Combining all equations,
S O
S
ox x
K
K 0E
VG= ox +semi semi = S
S O
S S
G x
K
V K 0E
s A S
qN ε W 2K 0
W x
ε K x qN
S
A
0
) E(
Employing the delta-depletion solution,
A combination of the two equations gives at x = 0. s
S A
S K ε
qN
0
2 E
Therefore s
S A O
S S
G K ε
x qN K
V K
0 0
2
0 S 2F
Gate voltage relationship (V
Gversus
S)
The exact solution can be found in Appendix B.
s = 2F for all inversion biases
s = 0 for all
accumulation biases
s S
A O
S S
G K ε
x qN K
V K
0 0
2
F
S
2
0
The VG-s relationship obtain using the delta-depletion approximation can be used only for the depletion bias
) 2 2 (
2
0
0 F
S A O
S F
T K ε
x qN K
V K
V
G-
Srelationship: Alternative method
VG= Semi + ox
ox= QM/Cox = Qs/Cox where Cox is oxide capacitance and Qs is the depletion layer charge in semiconductor
Qs = q A NAW Cox = ox A / xox
Consider a p-type MOS
s Si
ox A ox
s Si Si
ox A ox
s Si G
Si ox A
ox Si ox
ox ox A
2 /
x qN qN W
x V
qN W x x
A
W N
A q
Announcements
• Next lecture: p. 584 ~ 598