125
Metal-Oxide-Semiconductor System
iii. Energy levels (E c , E i , E f , E v , E g )
iv. Width of the depletion layer, X d
v. Width of the inversion layer, X inv
Clearly state any assumptions you make.
3.4 Use Equation 3.52 to calculate and plot the total charge (Q s ) in semiconductor as a function of surface potential f s of an MOS capacitor
system for –0.4 V ≤ f s ≤ 1.4 V. Label all the operating regions of the
MOS capacitor system. Assume that the MOS capacitor is fabricated
on a uniformly doped p-type substrate with doping concentration
N a = 1 × 10 17 cm –3 . Clearly state any assumptions you make.
3.5 Consider an MOS capacitor system with uniformly doped p-type
substrate with doping concentration, N a , to show the following:
a. n N
v
i
a
B
k T
2
2
2
/
exp(
/ )
=
− φ
, where n i , f B , and v kT are the intrinsic carrier concentration, bulk potential, and thermal voltage,
respectively.
b. Complete the mathematical steps to express Equations 3.52 to
3.53 in terms of Debye length L d given by Equation 3.54.
3.6 Consider a double gate MOS capacitor system shown in Figure E3.1
fabricated on a lightly doped, N a = 5 × 10 15 cm –3 , p-type substrate with
t si = 30 nm, L g = 45 nm, and T ox = 1.5 nm as shown in Figure E3.1, and
gates connected together. Clearly state any assumptions you make to
answer the following questions.
a. Calculate the total width of the depletion region in silicon at
strong inversion, if t si → ∝.
b. Calculate the total width of the inversion layer at strong inversion.
Assume that the inversion layer thickness by a single gate is given by
X
kT
q
K
qN
inv
s
a s
=
ε
φ
0
2
where:
k is the Boltzmann constant
T is the temperature
q is the electronic charge
K si is the dielectric constant in silicon
ε 0 is the permittivity of free space
N a is the channel doping concentration
f s is the surface potential
c. Use the values of the parameters in parts (a) and (b) to sketch
the band diagram into silicon from x = 0 at the top gate Si/SiO 2
Metal-Oxide-Semiconductor System
iii. Energy levels (E c , E i , E f , E v , E g )
iv. Width of the depletion layer, X d
v. Width of the inversion layer, X inv
Clearly state any assumptions you make.
3.4 Use Equation 3.52 to calculate and plot the total charge (Q s ) in semiconductor as a function of surface potential f s of an MOS capacitor
system for –0.4 V ≤ f s ≤ 1.4 V. Label all the operating regions of the
MOS capacitor system. Assume that the MOS capacitor is fabricated
on a uniformly doped p-type substrate with doping concentration
N a = 1 × 10 17 cm –3 . Clearly state any assumptions you make.
3.5 Consider an MOS capacitor system with uniformly doped p-type
substrate with doping concentration, N a , to show the following:
a. n N
v
i
a
B
k T
2
2
2
/
exp(
/ )
=
− φ
, where n i , f B , and v kT are the intrinsic carrier concentration, bulk potential, and thermal voltage,
respectively.
b. Complete the mathematical steps to express Equations 3.52 to
3.53 in terms of Debye length L d given by Equation 3.54.
3.6 Consider a double gate MOS capacitor system shown in Figure E3.1
fabricated on a lightly doped, N a = 5 × 10 15 cm –3 , p-type substrate with
t si = 30 nm, L g = 45 nm, and T ox = 1.5 nm as shown in Figure E3.1, and
gates connected together. Clearly state any assumptions you make to
answer the following questions.
a. Calculate the total width of the depletion region in silicon at
strong inversion, if t si → ∝.
b. Calculate the total width of the inversion layer at strong inversion.
Assume that the inversion layer thickness by a single gate is given by
X
kT
q
K
qN
inv
s
a s
=
ε
φ
0
2
where:
k is the Boltzmann constant
T is the temperature
q is the electronic charge
K si is the dielectric constant in silicon
ε 0 is the permittivity of free space
N a is the channel doping concentration
f s is the surface potential
c. Use the values of the parameters in parts (a) and (b) to sketch
the band diagram into silicon from x = 0 at the top gate Si/SiO 2