Analysis and Design of Analog Integrated Circuits
Lecture 16
Subthreshold Operation and gm/Id Design
Michael H. Perrott
April 1, 2012
Copyright © 2012 by Michael H. Perrott
All rights reserved.
M.H. Perrott
A Closer Look at Transconductance
g
d
Id
Id
Vds > ΔV
M1
Vgs
NMOS

ΔId
gm =
ΔVgs
Vgs_op
Vgs
Id_op
s
VTH Vgs_op
ΔV
Assuming device is in strong inversion and in saturation:
μnCox W
(Vgs − VT H )2(1 + λVds)
ID =
2 L
s
W
W
δId
⇒ gm =
≈ μnCox (Vgs −VT H ) ≈ 2μnCox Id
δVgs
L
L
q
Id 2μnCox W/L
2Id
⇒ gm ≈
≈
√
Id
(Vgs − VT H )
M.H. Perrott
2
Unity Gain Frequency for Current Gain, ft
g
d
|Id/Iin|
Id
M1
Iin
s
Vds
f
0dB
NMOS

ft
Under fairly general conditions, we calculate:

ft is a key parameter for characterizing the achievable
gain·bandwidth product with circuits that use the device
M.H. Perrott
3
Current Density as a Key Parameter
W
Id
Id
M1
W
L
Id
W

Current density is defined as the ratio Id/W:
- We’ll assume that current density is altered by keeping
Id fixed such that only W varies
 Maintains constant power
 ro (i.e., 1/gds = 1/(Id)) will remain somewhat constant
M.H. Perrott
4
Investigating Impact of Current Density

For simplicity, let us assume that the CMOS device
follows the square law relationship
μnCox W
ID ≈
(Vgs − VT H )2
2 L
- This will lead to the formulations:
Vgs − VT H ≈
s
µ
2L
Id
μnCox W
¶
2Id
gm ≈
Vgs − VT H
- These formulations are only accurate over a narrow
region of strong inversion (with the device in saturation)
- However, the general trends observed from the above
expressions as a function of current density will provide
useful insight
M.H. Perrott
5
Investigate the Impact of Increasing Current Density
Vgs−Vth (Volts)
1.5
x 10
−6
Gate Overdrive versus Current Density
1
Gate overdrive increases
0.5
Vgs − VT H ≈
0
−7
10
gm (1/Ohms)
8
x 10
10
−3
−6
−5
10
Transconductance versus Current Density
10
−4
−3
µ
2L
Id
μnCox W
¶
6
gm decreases
4
2
0
−7
10
4
x 10
10
−6
11
−5
10
ft versus Current Density
10
−4
3
ft (Hz)
10
s
2
ft increases
1
0
−7
10
10
−6
−5
10
Current Density Id/W (Amps/micron)
10
−4
2Id
gm ≈
Vgs − VT H
10
−3
√
W
1 gm
ft =
∝
2π Cgs
W
10
−3
W decreased with fixed Id
M.H. Perrott
6
Transconductance Efficiency Versus ft
Vgs−Vth (Volts)
1.5
x 10
−6
Gate Overdrive versus Current Density
1
Gate overdrive increases
0.5
Vgs − VT H ≈
0
−7
10
gm (1/Ohms)
8
x 10
10
−3
−6
−5
10
Transconductance versus Current Density
10
−4
−3
µ
2L
Id
μnCox W
¶
6
gm decreases
4
2
0
−7
10
4
x 10
10
−6
11
−5
10
ft versus Current Density
10
−4
3
ft (Hz)
10
s
2
ft increases
1
0
−7
10
10
−6
−5
10
Current Density Id/W (Amps/micron)
Higher gm (more gain)
M.H. Perrott
10
−4
2Id
gm ≈
Vgs − VT H
10
−3
√
W
1 gm
ft =
∝
2π Cgs
W
10
−3
Higher ft (faster speed)
7
Transistor “Inversion” Operating Regions
Vgs−Vth (Volts)
1.5
x 10
−6
Gate Overdrive versus Current Density
1
Gate overdrive increases
0.5
Vgs − VT H ≈
0
−7
10
gm (1/Ohms)
8
x 10
10
−3
−6
−5
10
Transconductance versus Current Density
10
−4
−3
µ
2L
Id
μnCox W
¶
6
gm decreases
4
2
0
−7
10
4
x 10
10
−6
−5
10
ft versus Current Density
11
10
−4
3
ft (Hz)
10
s
2
ft increases
1
0
−7
10
10
−6
Weak
M.H. Perrott
−5
10
Current Density Id/W (Amps/micron)
Moderate
10
−4
Strong
2Id
gm ≈
Vgs − VT H
10
−3
√
W
1 gm
ft =
∝
2π Cgs
W
10
−3
8
Key Insights Related to Current Density

Current density sets the device operating mode
- Weak inversion (subthreshold): highest g efficiency
 Achieves highest g for a given amount of current, I
- Strong inversion: highest f
 Achieves highest speed for a given amount of current, I
- Moderate inversion: compromise between the two
m
m
d
t
d
 Often the best choice for circuits that do not demand the
highest speed but cannot afford the low speed of weak
inversion (subthreshold operation)

Key issue: validity of square law current assumption
μnCox W
(Vgs − VT H )2(1 + λVds)
ID =
2 L
- The above is only accurate over a narrow range of strong
inversion (i.e., the previous plots are inaccurate)
 General observations above are still true, though
M.H. Perrott
9
A Proper Model for Subthreshold Operation
ID
VGS<VTH
G
S

VDS
D
Drain current:
- Where:
´
W Vgs/(nVt) ³
−Vds/Vt
ID = ID0 e
1−e
L
kT
≈ 26mV at T = 300K
Vt =
q
Cox + Cdepl
n=
≈ 1.5
Cox
ID0 = μnCox (n − 1)Vt2e−VT H /(nVt)
- Note: channel length modulation, i.e., , is ignored here
M.H. Perrott
10
Saturation Region for Subthreshold Operation
Drain current Versus Vds and Vgs
140
Vgs = 0.44V
120
100
Vgs = 0.42V
d
I (nA)
80
60
Vgs = 0.4V
40
20
0

0
0.1
0.2
0.3
0.4
0.5
Vds (Volts)
Saturation occurs at roughly Vds > 100 mV
´
W Vgs/(nVt ) ³
W Vgs/(nVt)
−Vds/Vt
⇒ ID = ID0 e
1−e
≈ ID0 e
L
L
M.H. Perrott
11
Transconductance in Subthreshold Region
g
d
Id
Id
M1
Vgs
s
NMOS

Id_op
Vds > 100mV
ΔId
gm =
ΔVgs
Vgs_op
Vgs
Vgs_op
Assuming device is in subthreshold and in saturation:
W Vgs/(nVt)
gm purely a
ID ≈ ID0 e
function of Id!
L
δId
Id
W Vgs/(nVt) 1
⇒ gm =
≈ ID0 e
=
δVgs
L
nVt
nVt
2Id
Recall for strong inversion : gm ≈
(Vgs − VT H )
M.H. Perrott
12
Comparison of Strong and Weak Inversion for gm


Assumption: Id is constant with only W varying
Strong inversion formulation predicts ever increasing
gm with reduced overdrive voltage
2Id
gm ≈
(Vgs − VT H )
- Reduced current density leads to reduced overdrive

voltage and therefore higher gm
Weak inversion formulation predicts that gm will hit a
maximum value as current density is reduced
Id
gm =≈
nVt
- Note that the area of the device no longer influences g
m
when operating in weak inversion (i.e., subthreshold)
M.H. Perrott
13
Hybrid- Model in Subthreshold Region (In Saturation)
d
g
d
g
vgs
Cgs
gmvgs
gmbvs
ro
s
s

Looks the same in form as for strong inversion, but
different expressions for the various parameters
µ ¶
1 Id
gm ≈
n Vt
µ
¶
n − 1 Id
gmb ≈
n
Vt
1
ro ≈
λId
- We can use the very same Thevenin modeling
approach as in strong inversion
 We just need to calculate gm and gmb differently
M.H. Perrott
14
Noise for Subthreshold Operation (In Saturation)
d
g
d
g
vgs
Cgs
gmvgs
gmbvs
ro
2
ind
s
s

Recall transistor drain noise in strong inversion:
2
K
g
f
m
i2
=
4kT
γg
∆f
+
∆f
dso
nd
2
f W LCox
Thermal noise

1/f noise
In weak inversion (i.e., subthreshold):
2
Kf gm
2
ind = 2kT ngm∆f +
∆f
2
f W LCox
Thermal noise
M.H. Perrott
1/f noise
15
Strong Inversion Versus Weak Inversion

Strong inversion (Vgs > VTH)
- Poor g efficiency (i.e., g /I is low) but fast speed
- Need V > (V – V ) = V to be in saturation
- Key device parameters are calculated as:
m
ds

m d
gs
TH
2Id
gm ≈
(Vgs − VT H )
gmb ≈
Weak inversion (Vgs < VTH)
q
γgm
2 2|ΦF | + VSB
1
ro ≈
λId
- Good g efficiency (i.e., g /I is high) but slow speed
- Need V > 100mV to be in saturation
- Key device
parameters are calculated as:
µ
¶
µ ¶
m
m d
ds

1 Id
gm ≈
n Vt
n − 1 Id
gmb ≈
n
Vt
1
ro ≈
λId
Moderate inversion: compromise between the two
Thevenin Modeling Techniques Can Be Applied to All Cases
M.H. Perrott
16
gm/Id Design


gm/Id design is completely SPICE based
- Hand calculations of g , r , etc. are not performed
m
o
Various transistor parameters are plotted in terms of
gm/Id
- Low g /I corresponds to strong inversion
- High g /I corresponds to weak inversion
m d

m d
Once a given value of gm/Id is chosen, it constrains
the relationship between W, L, ft, etc. such that the
sizing of devices becomes a straightforward exercise
M.H. Perrott
17
Useful References Related to gm/Id Design

Prof. Bernhard Boser’s Lecture:
- B. E. Boser, "Analog Circuit Design with Submicron
Transistors," IEEE SSCS Meeting, Santa Clara Valley, May
19, 2005,
http://www.ewh.ieee.org/r6/scv/ssc/May1905.htm

Prof. Boris Murmann’s Course Notes:
- https://ccnet.stanford.edu/cgi-
bin/course.cgi?cc=ee214&action=handout_view&V_sect
ion=general
 See Slides 45 to 67 in particular

Prof. Reid Harrison’s paper on a low noise instrument
amplifier:
- http://www.ece.utah.edu/~harrison/JSSC_Jun_03.pdf
M.H. Perrott
18
Summary

CMOS devices in saturation can be utilized in weak,
moderate, or strong inversion
- Each region of operation involves different expressions
for drain current as a function of V and V
- It is best to use SPICE to calculate parameters such as
gs

ds
gm, gmb, ro due to the complexity of the device model in
encompassing these three operating regions
 gm/Id methodology is one such approach
Weak inversion offers large gm/Id but slow speed, and
strong inversion offers fast speed but lower gm/Id
Moderate inversion offers the best compromise between
achieving reasonable gm/Id and reasonable speed
Thevenin modeling approach is valid for all operating
regions once gm, gmb, and ro are known
M.H. Perrott
19