6.012 - Microelectronic Devices and Circuits - Fall 2005
Lecture 11-1
Lecture 11 - MOSFET (III)
MOSFET Equivalent Circuit Models
October 18, 2005
Contents:
1. Low-frequency small-signal equivalent circuit model
2. High-frequency small-signal equivalent circuit model
Reading assignment:
Howe and Sodini, Ch. 4, §4.5-4.6
6.012 - Microelectronic Devices and Circuits - Fall 2005
Lecture 11-2
Key questions
• What is the topology of a small-signal equivalent circuit model of the MOSFET?
• What are the key dependencies of the leading model
elements in saturation?
6.012 - Microelectronic Devices and Circuits - Fall 2005
Lecture 11-3
1. Low-frequency small-signal equivalent circuit model
Regimes of operation of MOSFET:
VDSsat=VGS-VT
ID
saturation
linear
ID
VDS
VGS
VGS
VBS
VGS=VT
0
0
cutoff
VDS
• Cut-off:
ID = 0
• Linear:
W
VDS
ID = µnCox (VGS −
− VT )VDS
L
2
• Saturation:
W
µnCox (VGS −VT )2 [1+λ(VDS −VDSsat)]
ID = IDsat =
2L
Effect of back bias:
r
r
VT (VBS ) = VT o + γ( −2φp − VBS − −2φp )
6.012 - Microelectronic Devices and Circuits - Fall 2005
Lecture 11-4
Small-signal device modeling
In many applications, interested in response of device to
a small-signal applied on top of bias:
ID+id
vgs +
-
VGS
+
v
- ds
+
v
- bs
VBS
VDS
Key points:
• Small-signal is small
⇒ response of non-linear components becomes linear
• Can separate response of MOSFET to bias and small
signal.
• Since response is linear, superposition can be used
⇒ effects of different small signals are independent
from each other
6.012 - Microelectronic Devices and Circuits - Fall 2005
Lecture 11-5
MOSFET
small-signal
equivalent
circuit model
ID+id
+
v
- ds
+
v
- bs
VBS
vgs +
-
VGS
VDS
ID
id
=
VDS
VGS
+
VBS
vgs +
-
Mathematically:
iD (VGS + vgs, VDS + vds, VBS + vbs) '
∂ID
∂ID
∂ID
ID (VGS , VDS , VBS )+
| vgs +
| vds +
| vbs
∂VGS Q
∂VDS Q
∂VBS Q
where Q ≡ bias point (VGS , VDS , VBS )
Small-signal id:
id ' gm vgs + govds + gmb vbs
Define:
gm ≡ transconductance [S]
go ≡ output or drain conductance [S]
gmb ≡ backgate transconductance [S]
Then:
gm '
∂ID
|Q
∂VGS
go '
∂ID
|Q
∂VDS
gmb '
∂ID
|Q
∂VBS
+
v
- ds
+
v
- bs
6.012 - Microelectronic Devices and Circuits - Fall 2005
Lecture 11-6
2 Transconductance
In saturation regime:
W
µnCox (VGS − VT )2 [1 + λ(VDS − VDSsat)]
ID =
2L
Then (neglecting channel length modulation):
∂ID
W
| ' µnCox (VGS − VT )
gm =
∂VGS Q
L
Rewrite in terms of ID :
v
u
u
u
u
t
gm = 2
W
µnCox ID
L
gm
gm
saturation
saturation
cut-off
0
0
0
VT
VGS
0
ID
6.012 - Microelectronic Devices and Circuits - Fall 2005
Lecture 11-7
Transconductance of 3 µm nMOSFET (VDS = 2 V ):
Equivalent circuit model representation of gm :
id
G
vgs
S
B
D
+
-
gmvgs
6.012 - Microelectronic Devices and Circuits - Fall 2005
Lecture 11-8
2 Output conductance
In saturation regime:
ID =
W
µnCox (VGS − VT )2 [1 + λ(VDS − VDSsat)]
2L
Then:
go =
∂ID
W
ID
µnCox (VGS − VT )2 λ ' λID ∝
|Q =
∂VDS
2L
L
Output resistance is inverse of output conductance:
ro =
go
1
L
∝
go ID
go
saturation
saturation
cut-off
0
0
0
VT
VGS
0
ID
6.012 - Microelectronic Devices and Circuits - Fall 2005
Lecture 11-9
Output conductance of 3 µm nMOSFET:
Equivalent circuit model representation of go :
id
G
vgs
S
B
D
+
-
ro
6.012 - Microelectronic Devices and Circuits - Fall 2005
Lecture 11-10
2 Backgate transconductance
In saturation regime (neglect channel-length modulation):
W
µnCox (VGS − VT )2
ID '
2L
Then:
gmb
∂ID
W
∂VT
=
| = µnCox (VGS − VT )(−
| )
∂VBS Q
L
∂VBS Q
Since:
r
r
VT (VBS ) = VT o + γ( −2φp − VBS − −2φp )
Then:
∂VT
−γ
r
| =
∂VBS Q 2 −2φp − VBS
All together:
gmb =
γgm
2 −2φp − VBS
r
gmb inherits all dependencies of gm
6.012 - Microelectronic Devices and Circuits - Fall 2005
Lecture 11-11
Body of MOSFET is a true gate: output characteristics
for different values of VBS (VBS = 0 − (−3) V, ∆VBS =
−0.5 V , VGS = 2 V ):
Equivalent circuit model representation of gmb :
id
G
vgs
S
-
vbs
B
D
+
+
gmbvbs
6.012 - Microelectronic Devices and Circuits - Fall 2005
Lecture 11-12
Complete MOSFET small-signal equivalent circuit model
for low frequency:
id
G
vgs
S
-
vbs
B
D
+
+
gmvgs
gmbvbs
ro
6.012 - Microelectronic Devices and Circuits - Fall 2005
Lecture 11-13
2. High-frequency small-signal equivalent circuit model
Need to add capacitances. In saturation:
gate
Cfringe
source
Cgs,i
n+
n+
Cov
Cjsw
Cfringe
drain
Cov
n+
Csb,i
Cj
Cjsw
Cj
p
body
Cgs ≡ intrinsic gate capacitance
+ overlap capacitance, Cov (+fringe)
Cgd ≡ overlap capacitance, Cov
(+fringe)
Cgb ≡ (only parasitic capacitance)
Csb ≡ source junction depletion capacitance
+sidewall (+channel-substrate capacitance)
Cdb ≡ drain junction depletion capacitance
+sidewall
6.012 - Microelectronic Devices and Circuits - Fall 2005
Lecture 11-14
Complete MOSFET high-frequency small-signal equivalent circuit model:
id
Cgd
G
vgs
S
D
+
Cgs
gmbvbs
ro
-
vbs
B
gmvgs
Csb
+
Cdb
Plan for development of capacitance model:
• Start with Cgs,i
– compute gate charge QG = −(QN + QB )
– compute how QG changes with VGS
• Add pn junction capacitances
6.012 - Microelectronic Devices and Circuits - Fall 2005
Lecture 11-15
Inversion layer charge in saturation
QN (VGS ) = W
Z
L
0 Qn (y)dy
=W
Z
VGS −VT
0
Qn(Vc )
dy
dVc
dVc
But:
dVc
ID
=−
dy
W µnQn(Vc )
Then:
W 2Lµn Z VGS −VT 2
Qn(Vc )dVc
QN (VGS ) = −
0
ID
Remember:
Qn(Vc ) = −Cox (VGS − Vc − VT )
Then:
2 Z
W 2Lµn Cox
VGS −VT
2
QN (VGS ) = −
(V
−
V
−
V
)
dVc
GS
c
T
0
ID
6.012 - Microelectronic Devices and Circuits - Fall 2005
Lecture 11-16
Do integral, substitute ID in saturation and get:
2
QN (VGS ) = − W LCox (VGS − VT )
3
Gate charge:
QG(VGS ) = −QN (VGS ) − QB,max
Intrinsic gate-to-source capacitance:
Cgs,i
dQG
2
=
= W LCox
dVGS 3
Must add overlap capacitance:
2
Cgs = W LCox + W Cov
3
Gate-to-drain capacitance - only overlap capacitance:
Cgd = W Cov
6.012 - Microelectronic Devices and Circuits - Fall 2005
body
Lecture 11-17
polysilicon gate
source
drain
gate
n+
p+
n+
n+
p
inversion layer
channel
n
gate oxide
gate length
p+
p
n+
n+
n+
gate width
n
n+
STI edge
Body-to-source capacitance = source junction capacitance:
v
u
u
u
u
t
diff u
Csb = Cj +Cjsw = W L
qsNa
+(2Ldiff +W )CJ SW
2(φB − VBS )
Body-to-drain capacitance = drain junction capacitance:
v
u
u
u
u
t
diff u
Cdb = Cj +Cjsw = W L
qs Na
+(2Ldiff +W )CJ SW
2(φB − VBD )
6.012 - Microelectronic Devices and Circuits - Fall 2005
Lecture 11-18
Key conclusions
High-frequency small-signal equivalent circuit model of
MOSFET:
id
Cgd
G
vgs
S
Cgs
gmbvbs
gmvgs
-
vbs
B
D
+
Csb
+
Cdb
In saturation:
v
u
u
u
u
t
W
gm ∝
ID
L
go ∝
ID
L
Cgs ∝ W LCox
ro