UNIVERSITY OF CALIFORNIA
College of Engineering
Department of Electrical Engineering and Computer Sciences
EE 130/230M
Spring 2013
Prof. Liu & Dr. Xu
Homework Assignment #10
Due at the beginning of class on Thursday, 4/11/13
Problem 1: The MOSFET as a Resistor
Given: A long-channel n-MOSFET with W/L = 10, effective gate-oxide thickness Toxe = 2 nm, substrate
(body) dopant concentration NA = 1018 cm-3, and gate work function M = 4.05 eV.
a)
The bulk potential, F 
kT N A
1018
ln
 0.026ln 10  0.48V
q
ni
10
The flatband voltage, VFB   M   S   M  (   (
The oxide capacitance, Coxe 
 ox
Toxe

EG
 F ))  4.05  (4.05  (0.56  0.48))  1.04V
2q
3.45  1013
 1.725  10 6 F / cm 2
7
2  10
The threshold voltage, VT  VFB  2F 
2qN D si | 2F |
Coxe
2  1.6  1019  1012  1018  0.96
1.725  106
 1.04  0.96  0.32  0.24V
V  VT  0.2
Now, the effective vertical field, Eeff  G
6Toxe
 1.04  0.96 
At low VDS , the channel resistance is
RDS 
L
WCoxe eff (VGS  VT )
Using these equations, and finding eff for the calculated Eeff, we can iteratively solve for RDS
VGS (V)
Eeff (MV/cm)
eff (cm2 V-1S-1)
RDS ()
0.5
0.78
300
732
0.7
0.95
250
496
0.3
0.61
350
2721
0.4
0.7
325
1098
0.418
0.71
~325
1002
The MOSFET presents a resistance of 1 k between the source and drain at the gate-to-source
voltage VGS0.4 V.
b) The inversion layer electron density,
Q
C (V  V ) 1.725  106  (0.418  0.24) 0.307  106
Ninv  inv  oxe GS T 

 1.919  1012 cm 2
19
19
q
q
1.6  10
1.6  10
c) From the graph on Slide 14 of Lecture 18, the inversion layer thickness Tinv≈1nm corresponding to
Eeff = 0.71 MV/cm. Hence the inversion layer electron density is
N inv 1.919  1012 cm 2

 1.92  1019 cm 3
7
Tinv
1  10
Problem 2: Long-Channel MOSFET I-V Characteristics
For the MOSFET of Problem 1:
a) The bulk charge factor is
C
3T
m = 1+ dep,min = 1+ oxe
Coxe
WT
Here, WT =
2e Si 2 | F F |
2 ´10 -12 ´ 2 ´ 0.48
=
= 3.46 ´10-6 cm
qN A
1.6 ´10-19 ´1018
3Toxe
3´ 2 ´10-7
= 1+
= 1.173
WT
3.46 ´10-6
b) For VGS = 0.5 V, based on the results of Problem 1, eff =325 cm2 V-1s-1. The saturation voltage,
V -V 0.5- 0.24
VDsat = GS T =
= 0.22V . And for VGS =1 V, following the same procedure as Problem 1, eff
m = 1+
m
1.173
=200 cm V S . The saturation voltage, VDsat = VGS -VT = 1- 0.24 = 0.648V . We use the following
m
1.173
equations for calculating the drain current in the linear and saturation regions.
W
m
Linear region current, I Dlin = Coxemeff (VGS -VT - VDS )VDS
L
2
W
Saturation region current, I Dsat =
Coxemeff (VGS -VT )2 [1+ l (VDS -VDsat )]
2mL
2
-1 -1
The IDS vs. VDS curves are plotted below.
c)
If the body dopant concentration NA were to be increased, VT would increase due to increased
semiconductor bulk potential (so that a larger voltage drop across the Si is needed to strongly invert
the surface, with a larger corresponding voltage drop across the oxide); eff would decrease due to
increased effective vertical electric field, so that the slope of the IDS vs. VDS characteristic would
decrease in the linear region of operation; the bulk charge factor m would increase due to increased
depletion capacitance Cdep (due to decreased depletion width WT); the channel-length modulation
factor  would decrease due to reduced depletion width in the pinch-off region (closer to the drain
side). Considering these effects, VDsat would decrease and IDsat would decrease. The qualitative
impact on the IDS vs. VDS characteristic is illustrated below, for VGS=1.0V:
Problem 3: Impact of Body Biasing on MOSFET VT
-12
-19
18
a) The body effect parameter, g = 2qN AeSi = 2 ´10 ´1.6 ´10 ´10 = 0.327
-6
Coxe
1.725´10
b)
The threshold voltage, VT = VT 0 + g ( (2fF +VSB ) - 2fF ) = 0.24 + 0.327( (0.96 +VSB ) - 2fF )V
The threshold voltage as a function of VSB is shown below.
If VSB is negative then the source-body n-p junction is forward biased resulting in significant
current flow between the source and body, which is undesirable because this current is not
controlled by the gate voltage. Hence, care should be taken to avoid strongly forward biasing the
body-source junction of a MOSFET, when body biasing is used to dynamically adjust VT. For an
n-channel MOSFET, we are only interested in positive values of VSB. In digital CMOS circuits, it
is common for the body terminals of the n-MOSFETs to be connected to the lowest potential (e.g.
ground).
c) The body effect can be minimized by decreasing the body dopant concentration and/or
increasing the oxide capacitance (i.e. by reducing the oxide thickness).