128
Compact Models for Integrated Circuit Design
represent the QM and classical (CL) values of bandgap (E g ), respectively. Assume Q o = 0, V sub = 0, and n+ poly gate.
a. Show that the intrinsic carrier concentration due to QM effect is
given by
n
n
E
kT
i
i
g
QM
CL
=
−
exp
∆
2
b. Calculate the value of n i
QM
at the ambient temperature
T = 300 o K.
c. If the shift in the centroid of inversion charge, Δz ≅ 1 nm due to
the QM effect in the p-substrate, estimate the value of T ox (eff) at
strong inversion due to QM effect of the metal gate capacitor.
d. Estimate the ratio of C/C ox at strong inversion for metal gate
capacitor in part (c). Here, C = bias-dependent capacitance of
MOS system at strong inversion and C ox = gate oxide capacitance.
What is your conclusion on (C/C ox ) at strong inversion?
e. If the capacitor is fabricated using n+ polysilicon gate with doping concentration 1 × 10 19 cm –3 , calculate the value of T ox (eff) at
strong inversion. Assume that the depletion width of silicon and
polysilicon is given by the same expression.
f. Estimate the ratio of C/C ox at strong inversion for polysilicon
gate capacitor in part (e). Here, C = bias-dependent capacitance of MOS system at strong inversion and C ox = gate oxide
capacitance. What is your conclusion on (C/C ox ) at strong
inversion?
3.9 An MOS capacitor is built with the structure shown in Figure E3.3.
Both the n-type and p-type silicon regions are uniformly doped with
concentration 1 × 10 17 cm –3 and the areas of both the capacitors over
the n-type and p-type regions are the same and T ox = 200 nm. The
threshold voltages for the n- and p-substrates are –1 V and +1 V,
respectively. Sketch the shape of the HF (1 MHz) C–V curve that you
would expect to measure for this structure. Calculate the maximum
and minimum values of the composite capacitor per unit area. Label
as many points as possible and explain.
3.10 Consider an MOS capacitor system on a uniformly doped p-type
silicon substrate with doping concentration N a = ×
−
1 10
16
3
cm at room
temperature. Assume V fb = 0 and wherever necessary T ox = 20 nm.
a. Calculate and plot the inversion, bulk, and total charge Q i , Q b ,
and Q s , respectively, in the substrate of the capacitor on the same
plot. Clearly label all relevant parameters and explain.
Compact Models for Integrated Circuit Design
represent the QM and classical (CL) values of bandgap (E g ), respectively. Assume Q o = 0, V sub = 0, and n+ poly gate.
a. Show that the intrinsic carrier concentration due to QM effect is
given by
n
n
E
kT
i
i
g
QM
CL
=
−
exp
∆
2
b. Calculate the value of n i
QM
at the ambient temperature
T = 300 o K.
c. If the shift in the centroid of inversion charge, Δz ≅ 1 nm due to
the QM effect in the p-substrate, estimate the value of T ox (eff) at
strong inversion due to QM effect of the metal gate capacitor.
d. Estimate the ratio of C/C ox at strong inversion for metal gate
capacitor in part (c). Here, C = bias-dependent capacitance of
MOS system at strong inversion and C ox = gate oxide capacitance.
What is your conclusion on (C/C ox ) at strong inversion?
e. If the capacitor is fabricated using n+ polysilicon gate with doping concentration 1 × 10 19 cm –3 , calculate the value of T ox (eff) at
strong inversion. Assume that the depletion width of silicon and
polysilicon is given by the same expression.
f. Estimate the ratio of C/C ox at strong inversion for polysilicon
gate capacitor in part (e). Here, C = bias-dependent capacitance of MOS system at strong inversion and C ox = gate oxide
capacitance. What is your conclusion on (C/C ox ) at strong
inversion?
3.9 An MOS capacitor is built with the structure shown in Figure E3.3.
Both the n-type and p-type silicon regions are uniformly doped with
concentration 1 × 10 17 cm –3 and the areas of both the capacitors over
the n-type and p-type regions are the same and T ox = 200 nm. The
threshold voltages for the n- and p-substrates are –1 V and +1 V,
respectively. Sketch the shape of the HF (1 MHz) C–V curve that you
would expect to measure for this structure. Calculate the maximum
and minimum values of the composite capacitor per unit area. Label
as many points as possible and explain.
3.10 Consider an MOS capacitor system on a uniformly doped p-type
silicon substrate with doping concentration N a = ×
−
1 10
16
3
cm at room
temperature. Assume V fb = 0 and wherever necessary T ox = 20 nm.
a. Calculate and plot the inversion, bulk, and total charge Q i , Q b ,
and Q s , respectively, in the substrate of the capacitor on the same
plot. Clearly label all relevant parameters and explain.