482 CHAPTER 9. FIELD EFFECT TRANSISTORS: MOSFET
Problem 9.4An Al-gate MOS capacitor has an oxide thickness of 100A and an oxide ̊
charge density of 3× 1011 cm−^2. The charge is positive. Calculate (a) the flat band
voltage, (b) the turn-on voltage. Also, draw the energy band diagram and electric field
profile of the structure at the onset of inversion.Na=5× 1015 cm−^3.
Problem 9.5An Al-gate transistor is fabricated on ap-type substrate with an oxide
thickness of 600A. The measured threshold voltage is ̊ VT=1.0V,andthep−type doping
is 5× 1016 cm−^3. Calculate the fixed charge density in the oxide.
- Section 9.4
Problem 9.6Ann-channel MOS capacitor has a doping ofNa=10^15 cm−^3 .Thegate
oxide thickness is 500A. Calculate the capacitances ̊ Cox,Cfb,andCminfor the capacitor.
Problem 9.7Show that ifρ(x)is the distribution of charge density in the SiO 2 2regionof
thicknessdox, the shift in the flat band voltage is given by
ΔVfb=−
1
Cox
∫dox
0
xρ(x)dx
dox
(Use Gauss’ law for electric field due to a thin sheet of charge density. Then use the
superposition principle.)
Problem 9.8Calculate the shift in the flat band voltage using the result of problem 9.7 for
the following oxide charge distributions: (a)Q
′
ss=10^11 cm−^2 is at the Si-SiO^2 interface;
(b) the same charge is uniformly distributed in the oxide; (c) the charge is at the gate-SiO 2
interface. The oxide thickness is 500A. Assume that the charge is positive. ̊
Problem 9.9The small signal capacitance of a (Metal−SiO 2 −Si−Metal)MOS
capacitor is equal to a series connection of two capacitors.
(a) One capacitor is formed by a plate in bulk Si and the other plate at theSiO 2 −Si
interface.
(b) The second capacitor, has its plates separated by the oxide layer.
Prove this using Gauss’ law.
- Section 9.5
Problem 9.10Consider ann-channel MOSFET with aZ/Lratio of 15, a threshold
voltage of 0.5 volt, mobility,μn= 500 cm^2 /V·s, anddox= 700A. Calculate the drain ̊
current and transconductance of the device (a) atVDS= 0.2 V; (b) in the saturation region.
The gate voltage is 1.5 V for both cases. Assume that thep-type doping is small.
Problem 9.11Consider an idealn-channel MOSFET with the following parameters:
Flat band voltage, Vfb = − 0 .9V
Channel width, Z =25μm
Channel mobility, μn = 450 cm^2 /V·s
Channel length, L =1. 0 μm
Oxide thickness, dox = 500A ̊
Channel doping, Na =5× 1014 cm−^3
