UNIVERSITY OF CALIFORNIA
College of Engineering
Department of Electrical Engineering and Computer Sciences
EE 130/230M
Spring 2013
Prof. Liu & Dr. Xu
Homework Assignment #11 Solutions
Problem 1: MOSFET Subthreshold Leakage Current
For an n-MOSFET with Toxe = 2 nm and WT = 35 nm,
(a) The subthreshold swing is
 Cdep  kT
  W  kT
 3T 
kT
 
ln( 10)1 
ln( 10)1  Si T  
ln( 10)1  oxe 
q
Cox  q
WT 


  ox Toxe  q
 3 2 
S  0.06  1 
  70mV / dec
35 

S
(b) If the normalized current at VGS = 0V is 10 pA, then the threshold current level (100 nA) is 4 decades
higher. Therefore the minimum threshold voltage VT,min = 4 decades × 70 mV/decade = 0.28 V
(c) If the leakage current specification were to be relaxed to be 100 nA, the minimum threshold voltage
would be reduced to 0 V. For a fixed power supply voltage (maximum gate voltage) VDD, lower VT
would result in higher transistor on-state current, i.e. higher effective transistor drive current and
hence faster circuit operation.
(d) Small S is desirable for lower VT hence higher transistor on-state current, for a given off-state leakage
current specification.
Problem 2: MOSFET Small Signal Model
(a) The transconductance of a transistor is the change in output (drain) current that results from a small
change in input (gate) voltage. The equation for MOSFET transconductance is derived from the
equation for MOSFET on-state current (in the saturation region of operation) to be
W meff Coxe
gm =
(VGS -VT )
mL
Although VGS-VT increases with increasing gate bias voltage, the field-effect mobility decreases with
increasing gate bias voltage (due to increasing transverse electric field) and eventually causes gm to
decrease with increasing gate bias voltage. Therefore, there exists a maximum value of
transconductance as a function of the gate bias voltage.
(b) The intrinsic gain of a MOSFET is
gm W meff Coxe (VGS -VT )
=
gd
mL
l I Dsat 0
From Slide 18 of Lecture 20, we can see that the channel length modulation parameter  is inversely
proportional to the channel length L. That is, the length of the pinch-off region becomes more
significant relative to L, so that the channel length modulation effect of increasing drain current with
increasing drain voltage becomes more significant (i.e. gd increases), as L is reduced. Thus, intrinsic
gain degrades with decreasing L for deep-submicron MOSFETs.
Problem 3: Velocity Saturation
(a) First, we need to find the effective electron mobility, eff . In order to do so, we need to find the
effective transverse electric field in the inversion layer:
VGS +VT + 0.2V 1.0V + 0.3V + 0.2V
1.5V
=
=
= 0.83MV / cm
6Toxe
6 ´ 3nm
18 ´10-7 cm
From the plot on Slide 4 of Lecture 20,
meff » 260cm2 / V × s
The
which
electric
Esat  2
vsat
eff
field

strength
at
the
electron
velocity
reaches
saturation
is
2(8  10 cm / s )
 6.15  104V / cm  6.15  104V / cm
2
260cm / V  s
6
The bulk charge parameter,
m =1+
3Toxe
3´ 3
=1+
=1.18
WT
50
(i) For L = 1 m:
-1
VDsat
-1
æ m
æ 1.18
ö
1 ö
1
-1
=ç
+
+
= (1.69 + 0.16) = 0.54V
÷ =ç
5
-4 ÷
è 1- 0.3 0.615 ´10 ´10 ø
è VGS -VT Esat L ø
æ V -V
ö
( Esat L = 6.15) >> çè GS T = 0.59 ÷ø
m
\This MOSFET is a long-channel device.
(ii) For L = 0.1m:
-1
VDsat
æ 1.18
ö
1
-1
=ç
+
= (1.68 +1.63) = 0.3V
5
-5 ÷
è 0.7 0.615´10 ´10 ø
æ VGS -VT
ö
= 0.59 ÷
ø
m
( Esat L = 0.615) @ çè
 This MOSFET is a short-channel device.
(b) VDsat

m
1 

 

V

V
E
L
T
sat
 GS

1
 VDsat will decrease with decreasing (VGS-VT)/m or decreasing EsatL.
(i)
(ii)
When Toxe decreases, VT decreases (because Cox increases so that the voltage dropped across
the gate oxide, Qdep/Cox, decreases) and thus (VGS- VT) increases. Consequently, VDsat
increases.
When (VGSVT) gets smaller, VDsat decreases because it becomes easier to pinch off the
inversion-layer channel at the drain end.
Problem 4: MOSFET Short-Channel Effect
(a) For a n-channel MOSFET with channel dopant concentration NA = 41017/cm3, effective oxide
thickness Toxe = 3 nm, source/drain junction depth rj = 0.05 m:
The threshold voltage shift, DVT =
-qN AWT rj
2W
( 1+ T -1)
Coxe L
rj
Here, F F = (kT / q)ln(N A / ni ) = 0.026 * ln(4 ´10 7 ) = 0.455V
WT =
2e si | 2F F |
2 ´10-12 ´ | 2 ´ 0.455 |
=
= 53nm
qN A
1.6 ´10 -19 ´ 4 ´1017
Now,
i) at L=1m m,
DVT =
-qN AWT rj
2W
-1.6 ´10 -19 ´ 4 ´1017 ´ 53´10 -7 ´ 5 ´10-6 1
2 ´ 53
( 1+ T -1) =
( 1+
-1)
-7
-4
eox
3´10
L
rj
10
50
-14
Toxe
3.45´ 8.854 ´10
-1.696 ´10-12 1
(0.766) = -1.274 ´10-2 V = -12.74mV
1.02 ´10-6 10-4
ii) at L=0.05 m m,
DVT =
DVT =
-1.696 ´10-12
1
(0.766) = -0.2548V = -254.8mV
-6
1.02 ´10
0.05 ´10-4
From these calculations of VT it can be seen that the short channel effect (“VT roll-off”) becomes
more severe at shorter gate lengths.
(b) For a higher value of drain voltage, the drain junction depletion width is larger; so the drain
effectively supports a larger fraction of the depletion region charge underneath the gate electrode.
Thus, the short channel effect is enhanced and therefore |VT| decreases more rapidly with
decreasing L.