CMOS Analog Design Using
All-Region MOSFET Modeling
Chapter 2
Advanced MOS transistor modeling
CMOS Analog Design Using All-Region MOSFET
Modeling
1
Semiconductors
 Four types of charge are present inside a
semiconductor: the fixed positive charge of ionized
donors, the fixed negative charge of ionized
acceptors, the positive mobile charge of holes, and
the negative mobile charge of electrons.
 We consider all donors and acceptors ionized
ND  N

D
and N A  N

A
 On this basis, the net positive charge density ρ is
  q( ND  N A  p  n)
CMOS Analog Design Using All-Region MOSFET
Modeling
2
Boltzmann’s Law – (1)
 In equilibrium electrons and holes follow Boltzmann’s
law and their concentrations (number per unit
volume) are proportional to
-( Energy / kT )
e

 k=1.38x10-23 J/K - Boltzmann constant
 T - absolute temperature (K).
 electron and hole densities in equilibrium are related
to electrostatic potential  by

p( 1)
e
p( 2)
q ( 1 2 )
kT
 q=1.6x10-19 C
CMOS Analog Design Using All-Region MOSFET
Modeling
3
Boltzmann’s Law – (2)
 n0 and p0 - equilibrium electron and hole concentrations
in the neutral bulk (=0 )
p  p0e
u   / t

q
kT
 p0e
u
n  n0e
q
kT
 n0eu
- normalized electrostatic potential
t  kT / q - thermal voltage
 the mass-action law is
np  ni2
ni - concentration of electrons (and holes) in the intrinsic
semiconductor
CMOS Analog Design Using All-Region MOSFET
Modeling
4
Example: Calculate the built-in potential for a Si p-n
junction with NA = 1017 atoms/cm3 and ND = 1018
atoms/cm3 ,T=300K
In equilibrium, if we choose the potential origin  = 0 where the
semiconductor is intrinsic (i.e., where p0=n0=ni), then
p0  ni e
 / t
 / t
n0  ni e
nregion / t
Far from the junction in the n-side
n0  ND  ni e
Far from the junction in the p-side
p0  N A  ni e
pregion / t
The built-in potential is given by
bi  n region   p region
 ND
 t ln 
 ni
bi  26  ln 1015   900 mV
 
 NA 
 ND N A 
   t ln 
   t ln 

2
n
n
 
 i 
 i 
CMOS Analog Design Using All-Region MOSFET
Modeling
5
The two-terminal MOS structure
CMOS Analog Design Using All-Region MOSFET
Modeling
6
The ideal two-terminal MOS structure
(VFB=0)
A
VG
Cox 
ox
tox
A - capacitor area,
Q
VG  s  G
Cox
+
s
_
QG  QC  0
QG
QC
M
tox - oxide thickness
O
ox - permittivity of oxide
S
QG
Cox  ox
 
QG 
; Cox

A
A
tox
QC
VG  s 
Cox
CMOS Analog Design Using All-Region MOSFET
Modeling
7
Example: oxide capacitance
(a) Calculate the oxide capacitance per unit area for tox=
5 and 20 nm assuming ox = 3.90, where 0= 8.85·10-14
F/cm is the permittivity of free space. (b) Determine the
area of a 1pF metal-oxide-metal capacitor for the two
oxide thicknesses given in (a).
Answer: (a) =690 nF/cm2 = 6.9 fF/m2 for tox=5 nm and =
172 nF/cm2= 1.7 fF/m2 for tox= 20 nm. The capacitor
areas are 145 and 580 m2 for oxide thicknesses of 5
and 20 nm, respectively.
CMOS Analog Design Using All-Region MOSFET
Modeling
8
The flat-band voltage
In equilibrium (with the two terminals shortened/open), the
contact potential between the gate and the semiconductor
substrate of the MOS induces charges in the gate and the
semiconductor for VGB=0.
Charges inside the insulator and at the semiconductor-insulator
interface also induce a semiconductor charge at zero bias.
The effect of the contact potential and oxide charges can be
counterbalanced by applying a gate-bulk voltage called the
flat-band voltage VFB.
VG  VFB
QC
 s 

Cox
CMOS Analog Design Using All-Region MOSFET
Modeling
9
Example: flat-band voltage
(a) Determine the expression for the flat-band voltage of n+
polysilicon-gate on p-type silicon (b) Calculate the flat-band voltage
for an n+ polysilicon-gate on p-type silicon structure with NA = 1017
atoms/cm3.
Answer: (a) In equilibrium, by analogy with an n+ p junction, the
potential of the n+-region is positive with respect to that of the pregion. The flat-band condition is obtained by applying a negative
potential to the n+ gate with respect to the p-type semiconductor of
value
VFB _ n  p  bi _ n  p
(b)
 NA 
 0.56 V  t ln 

 ni 
VFB  0.56 V  t ln 107   980 mV
CMOS Analog Design Using All-Region MOSFET
Modeling
10
Regions of operation of the MOSFET:
Accumulation (p-substrate)
G
QG
- - - - - - - - - - VGB
VGB  VFB
QC  0
s  0
Qo
+
+
+
+
++++++++++++++
Holes + accumulate in
QC
the p-type semiconductor
surface
B
CMOS Analog Design Using All-Region MOSFET
Modeling
11
Regions of operation of the MOSFET:
Depletion (p-substrate)
VGB  VFB
QC  0
G
QG
+ + + + + + + + +
VGB
Qo
+
- -- - - -Q - -- - - +
+
C
B
+
0  s   F
Holes evacuate from the P
semiconductor surface and
acceptor ion charges
become uncovered
-
F = Fermi potential ( to be defined)
CMOS Analog Design Using All-Region MOSFET
Modeling
12
Regions of operation of the MOSFET:
Inversion (p-substrate)
G
QG
+ + + + + + + + +
VGB
Qo
+
- -- - ---Q - -- -- -- - - - - - +
+
C
VGB  VFB
QC  0
s  F
+
electrons
surface!
approach the
B
CMOS Analog Design Using All-Region MOSFET
Modeling
13
Inversion for p-type substrate
Volume charge density inside the semiconductor:
  q( p0e  n0e  n0  p0 )
u
u
Depletion of holes prevails over electron charge when
p0e
t
u
 n0e or, equivalently
p0
  ln( )
2
n0
u
mass-action
law
t
p02
p0
 ln( 2 )  t ln( )  F
2
ni
ni
For  >F the concentration of minority carriers (n)
becomes higher than that of majority carriers (p); the
semiconductor operates in the inversion region
CMOS Analog Design Using All-Region MOSFET
Modeling
14
Small-signal equivalent circuit of the
MOS capacitor

dQG
dQC
 
Cgb

dVG
dVG
 
C gb
dQC
1
 
Cgb


dQ
d
1
d s  C  s 


Cox
dQC Cox
1
1
1

Cc Cox
Cc   dQC ds
Cc  
d  QB  QI 
ds
 Cb  Ci
CMOS Analog Design Using All-Region MOSFET
Modeling
15
Main approximation for compact MOS
modeling: the charge-sheet model
Minority carriers occupy a zero-thickness layer at the Si-SiO2
interface, where   s
dQI
QI
Ci  

ds
t
s /t
QI  e
Charge-sheet + depletion approximation for the bulk charge gives
QB  qN A xd   2q s N A s  t 
2q s N A
 Cox
Cb 

2 s  t 2 s  t
  2q s N A / Cox
is the body-effect coefficient
CMOS Analog Design Using All-Region MOSFET
Modeling
16
The three-terminal MOS structure
VC
VG
p
n+
Carrier concentrations in Si
substrate follow Boltzmann’s
law:
n, p  exp(-Energy/kT)
The origin of potential  is taken deep in the bulk
p  p0e

q
kT
 p0eu ;
n  n0e
q ( VC )
kT
 n0eu uC
electrons are no longer in equilibrium with holes due to the bias of
the source-bulk junction VC
pn  ni2euC  ni2eVC /t
CMOS Analog Design Using All-Region MOSFET
Modeling
17
Small-signal equivalent circuit of the 3terminal MOS device
dQI  Cb  Cox  Ci


dVC Ci  Cb  Cox

1
1
dQI 
   dVC
 Cox  Cb Ci 
Approximations:
1) depletion capacitance per unit area is constant along the
channel and is calculated neglecting inversion charge
2) Charge sheet model
Ci  QI / t
CMOS Analog Design Using All-Region MOSFET
Modeling
18
The linearization surface potential sa
Determination of sa  s Q 0
VG
I
Potential balance
VG  VFB
+
QB
 sa 
 sa   sa  t

Cox
sa  t  VG  VFB  t 
2
4

Cox
QI  0

2
_
Ci  0
Cb
s  sa
_
QB
+
dsa
Cox
1


dVG Cox  Cb n
CMOS Analog Design Using All-Region MOSFET
Modeling
19
Example: slope factor
For tox = 5 nm and 20 nm determine the minimum doping NA for
which the slope factor n < 1.25 at sa = 2F.
Answer: For sa=2F
2q s N A
2q s N A
Cb
n=1+
 1
 1

Cox
 sa
 2F
2Cox
2Cox
Thus, for n=1.25
 0.25 2F 4Cox2
2
NA=
2q s
where F is a weak (logarithmic) function of NA. Using 2F = 0.8 V
for the first calculation, we obtain after two iterations that NA>
4.9x1015 atoms/cm3 for tox=5nm, and NA > 2.3·1014 atoms/cm3 for
tox=20 nm.
CMOS Analog Design Using All-Region MOSFET
Modeling
20
The Unified Charge Control Model
(UCCM) - 1

1
1

dQI 
   dVC


 Cox  Cb Ci 
Approximations:
1) depletion capacitance per unit area is constant along the
channel and is calculated neglecting inversion charge
2) Charge sheet model
Ci  QI / t
CMOS Analog Design Using All-Region MOSFET
Modeling
21
The Unified Charge Control Model
(UCCM) - 2

1
1
dQI 
   dVC
 Cox  Cb Ci 
Ci  QI / t
 1
t 
dQI 
   dVC
 nCox QI 
Cb
 n VGB 
where n  1 

Cox
Integrating from an arbitrary channel potential VC to a reference
potential VP yields the unified charge control model (UCCM)
 QIP
 QI  
  QI
VP  VC  t 
 ln 

 t
 
 QIP
 nCox
CMOS Analog Design Using All-Region MOSFET
Modeling
  QI V
QIP
C
VP
22
The “regional” strong and weak
inversion approximations
 QIP
 QI
  QI
VP  VC  t 
 ln 
 t

 QIP
 nCox
VP  VC
t



t
VP  VC
weak inversion
strong inversion
QI  nCox VP VC 
  QI
VP  VC  t ln 

  QIP
 
  1
 
or, equivalently
VP VC t
e
QI  QIP
CMOS Analog Design Using All-Region MOSFET
Modeling
t
23
Example : approximate UCCM
(a) Calculate the value of the inversion charge density, normalized
 t , for which the value of the voltage VP-VS
to nCox
calculated using the SI approximation differs from that
calculated using UCCM by 10 %; (b) Same using the WI
approximation ; (c) comment on ‘moderate’ inversion (MI)
qI  (VP  VC ) / t
Answer: a)
SI approximation error of less than 10 % for q’I > 20
b) WI approximation
qI  e
VP VC t
t
WI approximation error of less than 10 % for q’I < 0.22.
(c) MI region : SI and WI approximations give errors greater than
10 % for the control voltage VP-VS. The inversion charge
density variation from the lower to the upper limit of the MI
region is approximately two orders of magnitude (20/0.22).
CMOS Analog Design Using All-Region MOSFET
Modeling
24
The pinch-off charge density
The channel charge density corresponding to the effective
channel capacitance times the thermal voltage, or thermal
charge, defines pinch-off
  (Cox
  Cb )t  nCox
 t
QIP
The name pinch-off is retained herein for historical reasons
and means the channel potential corresponding to a small
(but well-defined) amount of carriers in the channel.
CMOS Analog Design Using All-Region MOSFET
Modeling
25
The pinch-off voltage VP
The channel-to-substrate voltage (VC) for which the channel
charge density equals is called the pinch-off voltage VP.
in weak
inversion
 2 V /
 2 V /
QI  Cbt e sa F C  t  Cox (n  1)t e sa F C  t
UCCM is
asymptotically correct
in weak inversion if

 n 
VP  sa  2F  t 1  ln 

n

1



VP  sa  2F
CMOS Analog Design Using All-Region MOSFET
Modeling
26
Threshold voltage
Equilibrium threshold voltage VT0, for VC=0:
  nCox
 t
Gate voltage for which QI  QIP
or
Gate voltage for which VP=0
VP  sa  2F
Recalling that
it follows that
VG  VFB  sa   Cox sa  t
VT 0  VFB  2F   2F
CMOS Analog Design Using All-Region MOSFET
Modeling
27
Example: threshold voltage
+
Estimate VT0 for an n-channel transistor with n polysilicon gate,
NA=1017 atoms/cm3 and tox=5 nm.
Answer: The flat-band voltage (slide 10) is -0.98 V; F=0.419; C’ox=
690 nF/cm2. The body-effect factor is   2q s N A / Cox  0.264 V
VT 0  VFB  2F   2F =  0.98  0.838  0.264 0.838= 0.1V
For this low value of the threshold voltage, the off-current (for VGS=0)
is too high for digital circuits.
Solution to control the magnitude of the threshold voltage without an
exaggerated increase in the slope factor
a non-uniform
high-low channel doping.
CMOS Analog Design Using All-Region MOSFET
Modeling
28
Pinch-off voltage vs. gate voltage
4.0
4.00E+00
2
3.0
2.0
1.5
1.5
2.0
2.00E+00
1
1.0
1.0
1.00E+00
VP
0
0.00E+00

Cox
dVP dsa
1




dVG dVG Cb  Cox
n
-1.0
-1.00E+00
0.00E+00
0
1.00E+00
2.00E+00
3.00E+00
4.00E+00
5.00E+00
slope factor
pinch-off voltage
3.00E+00
0.5
0.5
0
0
6.00E+00
1.0
2.0
3.0
4.0
5.0 VG (V)
VT0 (equilibrium threshold voltage)
Useful approximation: VP 
VGB  VT0
n
CMOS Analog Design Using All-Region MOSFET
Modeling
29
The MOS transistor
W xi
xi
0 0
0
I D     J n dxdz  W  J n dx
CMOS Analog Design Using All-Region MOSFET
Modeling
30
‘Exact’ I-V model of the MOSFET (1)
 d 
dn
J n  qn n  
  qDn
dy
 dy 
drift
n  n0e
q ( VC )
kT
u uC
 n0e
Using the Einstein relationship
VS  VC  VD
diffusion
dn n  d dVC 
 


dy t  dy dy 
Dn  nt
 d dVC 
dVC
d
J n  qn n
 qn n 

  qn n
dy
dy
 dy dy 
CMOS Analog Design Using All-Region MOSFET
Modeling
31
‘Exact’ I-V model of the MOSFET (2)
W xi
xi
I D     J n dxdz  W  J n dx
0 0
0
dVC
J n  qn  n
dy
dVC
I D  qW  nn
dx
dy
0
xi
xi
QI  q  ndx
0
I D  W  nQI
dVC
dy
Since the current is constant along the channel
ID  
nW VD
L
 QI dVC
L is the channel length
VS
CMOS Analog Design Using All-Region MOSFET
Modeling
32
Charge-sheet formula for the current
Ci  
dVC
+
QI
VG
t
dQI
_
d s
dQI  Ci  dVC  ds 
dQI
dVC  ds  t
QI
I D  I drift  I diff
  nWQI
CMOS Analog Design Using All-Region MOSFET
Modeling
J n  qn  n
dVC
dy
ds
dQI
 nW t
dy
dy
33
Charge control compact model (1)
VG
dQI
dVC
+
_
Ci
ds
dQI
I D   nWQI
 nW t
dy
dy
Cox
d s
_
Cb
  Cb )ds  nCox
 ds
dQI  (Cox
dQB
+
nW
dQI
 )
ID  
(QI  t nCox

nCox
dy
Integrating along the channel yields
ID 
 2  QID
 2
nW  QIS

L 

2nCox

  QID
 
 t  QIS

CMOS Analog Design Using All-Region MOSFET
Modeling
34
Charge control compact model (2)
drift +
ID 
 2  QID
 2
 nW  QIS

L 

2nCox
diffusion


 
 Q  QID
  Q  QID
  QID
     nW  IS
 t   IS
 t  QIS
 nCox


2
nC
L


ox



“virtual” charge
 ds
dQI  nCox
  QID

 QIS
  s 0  sL 

I D   nW 
 nCoxt  

2
L



average
charge density
CMOS Analog Design Using All-Region MOSFET
Modeling
average
electric field
35
Charge control compact model (3)
To emphasize the symmetry of the rectangular geometry
MOSFET
ID  IF  IR
I F ( R)
2



Q
W
IS ( D )
 ( D) 
 n 
 t QIS

L
 2nCox

(compare with Ebers-Moll model of the BJT)
CMOS Analog Design Using All-Region MOSFET
Modeling
36
Drain current vs. gate-to-bulk voltage
CMOS Analog Design Using All-Region MOSFET
Modeling
37
Comparing UCCM and the surface potential
model with exact numerical solution of
Poisson equation
CMOS Analog Design Using All-Region MOSFET
Modeling
38
Modeling the bulk charge from
accumulation to inversion
  s  t (es / t  1)
Charge-sheet approximation QB   sgn(s )Cox
VG  VFB  s    QI  QB  / Cox
Potential balance
sa  s Q 0
I
VG  VFB  sa 2   2 sa  t e 
sa
1 dQB
n  1
Cox ds
s sa
t

1




sgn sa   1  esa / t
Cb
 1
 1
Cox
2 sa  t e sa t  1
CMOS Analog Design Using All-Region MOSFET
Modeling
39
Modeling from accumulation to inversion:
Surface potential and pinch-off voltage (VP)
CMOS Analog Design Using All-Region MOSFET
Modeling
40
Transistor symmetry
1.
I D  I D VG ,VS ,VD 
VG
Voltages referenced to local substrate: VS
VG VGB
VS VSB
VD
VD VDB
B
2. Symmetry
I D VG ,V1,V2   I D VG ,V2 ,V1 
ID
B
V1
ID V2
VG
CMOS Analog Design Using All-Region MOSFET
Modeling
41
Normalization
3. For a long-channel MOSFET
W
I D  I F  I R  I S i f  ir   I SQ  f VG ,VS   f VG ,VD  
L
I F  R   I S  qIS  D  2  2qIS  D  
qIS  D   QIS  D  /  nCox t 
D
ID
i f r   I F R / I S
t2 W
W
I S   Cox n
 I SQ
2 L
L
IF  IR
G
IS and ISQ are the normalization (specific)
current and the “sheet” normalization
current, slightly dependent on bias.
CMOS Analog Design Using All-Region MOSFET
Modeling
B
S
42
Forward and reverse currents
Long-channel MOSFET
I D  I F  I R  I (VG ,VS )  I (VG ,VD )
IF: forward current
IR: reverse current
IR=
IF=
CMOS Analog Design Using All-Region MOSFET
Modeling
43
Specific current
The specific (normalization) current
I S  Cox n
t2 W
W
 I SQ
2 L
L
ISQ : process parameter slightly dependent on VG and T
ISQ 25 nA (p-channel)
ISQ 75 nA (n-channel)
in 0.35 m CMOS
CMOS Analog Design Using All-Region MOSFET
Modeling
44
Pinch-off voltage and slope factor (1)
UCCM
 ( D )  1  ln qIS
 ( D ) 
VP  VS ( D )  t  qIS
qIS ( D )  1  i f ( r )  1
&
2
2





VP   VG  VT 0   2F      2F
2
2




Linearization:
VP
Slope =1
Slope =1/n
VP  VP 0 
VG  VG 0
n VG 0 
n VG 0   1 

2 VP 0  2F
In particular:
VP0
VP 
VT0
VG0
VG
VG  VT 0
n VT 0 
n VT 0   1 
CMOS Analog Design Using All-Region MOSFET
Modeling

2 2F
45
Pinch-off voltage and slope factor (2)
Determination of the pinch-off voltage and the slope factor as
functions of VG. NMOS transistor W=20 m, L=2 m, 0.18 m
CMOS technology.
CMOS Analog Design Using All-Region MOSFET
Modeling
46
The I-V relationship (1)
VP  VS  t  1  i f  2  ln



1  i f 1 

1,00E-03
10-3
VD = VG
ID (A)
VD
1,00E-04
ID
VS = 0 V
1,00E-05
0.5
1.0
10-6
1,00E-06
1.5
VG
2.0
VS
1,00E-07
2.5
3.0
1,00E-08
-9
1,00E-09
10
0,00E+00
0
5,00E-01
1,00E+00
1
1,50E+00
2,00E+00
2
2,50E+00
3,00E+00
3
3,50E+00
4,00E+00
4,50E+00
4 VG (V)
Common-source characteristics
CMOS Analog Design Using All-Region MOSFET
Modeling
47
The I-V relationship (2)
VP  VS  t  1  i f  2  ln

10-3
ID (A)


1  i f (r ) 1 

VD = VG
VG = 4.8 V
VD
ID
10-6
VG
0.8 V
10-9
VS
0
1
2
3
VS (V)
Common-gate characteristics VG=0.8, 1.2, 1.6,
2.0, 2.4, 3.0, 3.6, 4.2, and 4.8 V
CMOS Analog Design Using All-Region MOSFET
Modeling
48
Weak inversion model
Weak inversion
if(r)<1
VG  VT 0
 VS ( D )  t  1  i f ( r )  2  ln

n
-1
 VG VT 0


V
S  / t

n


I D  I 0e
1  eVDS / t 


I0   n


1  i f (r )  1 

if(r)/2
W
  t2e1  2 I S e1
nCox
L
CMOS Analog Design Using All-Region MOSFET
Modeling
49
Strong inversion model (1)
VG  VT 0
 VS ( D )  t  1  i f ( r )  2  ln

n
Strong inversion
if(r)>>1


1  i f (r )  1 

VG  VT 0
 VS ( D )  t i f ( r )  t I F ( R ) I S
n

I D  I F  I R   nCox
Moderate inversion
1<if(r) <100
W 
2
2
V

V

nV

V

V

nV
 G T0
 G T0
S
D

2nL 
Both sqrt(.) and ln(.) terms are important
CMOS Analog Design Using All-Region MOSFET
Modeling
50
Strong inversion model (2)
ID
VDS
VG
ID/IF
1
VDSsat=VP=(VG-VT0)/n
VDS
Transistor output characteristic
CMOS Analog Design Using All-Region MOSFET
Modeling
51
Strong inversion model (3)
ID 
ID
Cox W
2n L
VG  VT 0 
ID 
ID
Cox W
2n L
VG  VT 0  nVS 
SCE, , n,
“model”
VT0
VDD
VG  VT 0 
VG
n
VS
VDD
ID
ID
VG
VG
VS
CMOS Analog Design Using All-Region MOSFET
Modeling
52
Universal output characteristics
 1  i f 1 
 qIS 

  ln 
 qIS  qID
  1  i f  1  ir  ln 

 
t
 qID
 1  ir  1 
VDS
(o): measured
(—): model
(a) if= 4.5x 10-2 (VG=0.7 V); (b) if= 65(VG= 1.2 V); (c) if= 9.5x102 (VG= 2.0 V); (d) if=
3.1x 103 (VG= 2.8 V); (e) if= 6.8x 103 (VG= 3.6 V); (f) if= 1.2x 104 (VG= 4.4 V).
CMOS Analog Design Using All-Region MOSFET
Modeling
53
Saturation voltage
 / qIS  
Saturation voltage (VDSsat) – VDS such that qID
VDSsat  t ln 1    1   



1  i f 1 

1  
CMOS Analog Design Using All-Region MOSFET
Modeling
is the saturation level
54
Transconductances - 1
Transconductances
I D  g mg VG  g ms VS  g md VD  g mb VB
gmg  gms  gmd  gmb  0
g mg 
Calculation of gms
  IF  IR 
I
W
gms  
  F   QIS
 VS
 VS
L
g md
g mg  I S
UCCM
(i f  ir )
VG
I D
I
I
I
, g ms   D , g md  D , g mb  D
VG
VS
VD
VB
i f
VG
W

   QID
L

i f
nVS
ir
i
 r
VG
nVD
Pao-Sah ID (UCCM)
g mg
g mg
g ms  g md

n
g
in saturation
 ms
n
CMOS Analog Design Using All-Region MOSFET
Modeling
55
Transconductances - 2
VDD
ID
VG
VS
Source transconductance VG= 0.8, 1.2, 1.6, 2.0, 2.4, 3.0, 3.6,
4.2, and 4.8 V
(W=L=25 m, tox=280 Å)
CMOS Analog Design Using All-Region MOSFET
Modeling
56
Transconductances - 3
VDD
ID
VG
VS
Gate transconductance VS= 0, 0.5, 1.0,1.5, 2.0, 2.5, and 3.0 V .
W=L=25 m, tox=280 Å
CMOS Analog Design Using All-Region MOSFET
Modeling
57
The transconductance-to-current ratio - 1
Transconductance
-to-current ratio
g ms ( d )t
I F ( R)

1
2
2

i f (r )
1  i f (r )  1
WI (if <1)
SI (if >>1)
102
gms/IF
W=25 m
101
tox = 28 nm (IS = 26 nA)
L=25 m, tox= 280 Å
tox = 5.5 nm (IS = 111 nA)
L=20 m, tox= 55 Å
Seqüência1
Seqüência2
model
Seqüência3
100
10-4
1,00E-03
10-2
1,00E-01
100
102
if
104
CMOS Analog Design Using All-Region MOSFET
Modeling
58
The transconductance-to-current ratio - 2
Transconductance
-to-current ratio
g ms ( d )t

I F ( R)
1
2
1  i f (r )  1

WI (if <1)
2
i f (r )
SI (if >>1)
1,00E+02
2
10
gms/IF
V
W=L=25 m, tox= 280 Å
= 1.0 V (IS = 33 nA)
Seqüência1
GB
1,00E+01
1
10
VGB = 2.0 V (IS = 26 nA)
Seqüência2
VGB = 3.0 V (IS = 24 nA)
Seqüência3
model
Seqüência4
1,00E+00
0
1,00E-04
-4
10
10
1,00E-03
1,00E-02
-2
10
1,00E-01
1,00E+00
0
10
1,00E+01
1,00E+02
2
10
if
1,00E+03
1,00E+04
4
10
CMOS Analog Design Using All-Region MOSFET
Modeling
59
The transconductance-to-current ratio - 3
g ms ( d )t
Transconductance
-to-current ratio
101002
I F ( R)

1
2
1  i f (r )  1

WI (if <1)
2
i f (r )
SI (if >>1)
gms/IF
10101
L = 25 m (IS = 26 nA)
W=25 m, tox= 280 Å
Seqüência1
L = 2.5 m (IS = 260 nA)
Seqüência2
model
Seqüência3
0
1
101,00E-04
10-4
1,00E-03
1,00E-02
10-2
1,00E-01
1,00E+00
100
1,00E+01
1,00E+02
102
1,00E+03
if
1,00E+04
104
1,00E+05
CMOS Analog Design Using All-Region MOSFET
Modeling
60
The low-frequency small-signal model
G
g md v d
id
S
g mb vb
D
g ms v s
g mg vg
B
CMOS Analog Design Using All-Region MOSFET
Modeling
61
Quasi-static charge-conserving model
 The current entering each terminal of the transistor is split into
a transport component (IT) and a capacitive charging term.
dQD
I D (t )  IT (t ) 
dt
W VD t 
I T (t )  n
 QI (VC )dVC
L VS t 
 Quasi-static approximation: To calculate the stored charges
we suppose that the charge stored in the transistor depends
only on the instantaneous terminal voltages
Neglecting leakage currents
L
QG  W  QG dy
0
L
QB  W  QB dy
0
dQG
I G (t ) 
dt
dQB
I B (t ) 
dt
CMOS Analog Design Using All-Region MOSFET
Modeling
62
Ward-Dutton partition of the channel
charge
L
y
QS  W  (1  )QI dy
L
0
y
QD  W  QI dy
0 L
dQS
I S (t )  IT (t ) 
dt
dQD
I D (t )  IT (t ) 
dt
L
dQD dQS dQI
I D (t )  I S (t ) 


dt
dt
dt
As expected
L
QI  W  QI dy is the total inversion charge stored
0
in the channel
CMOS Analog Design Using All-Region MOSFET
Modeling
63
Calculation of stored charge - 1
I D  I drift  I diff 
ds
dQI
  nWQI
  nW t
dy
dy
 ds
dQI  nCox
nW
 t  dQI
dy   '  QI  nCox
nCox I D
It is convenient to define
dy  
nW
'
nCox
ID
 t
QIt  QI  nCox
QIt dQIt
CMOS Analog Design Using All-Region MOSFET
Modeling
64
Calculation of stored charge - 2
L
QI  W  QI dy
0
dy  
nW
'
nCox
ID
QI  
QIt dQIt
 ( D)  nCox
 t
QF ( R)  QIS
Using
or
ID 





Q

nC

Q
dQ
 
ox t  It
It 
  QR It
I D nCox

QI  
nt (W / L)
 2nCox t 
nW 2 QF
2
nW 2  QR3  QF3

 ID 
nCox
QF2  QR2 


3
QR2  QF2 
 t
 nCox

2

we find that
 2 QF2  QF QR  QR2


 nCoxt 
Q I  WL 


3
Q

Q
F
R


2  QIS
 QID
  QID
2 )  nCox
 t (QIS
  QID
 )
2 3(QIS
QI  WL
  QID
  2nCox
 t
QIS
CMOS Analog Design Using All-Region MOSFET
Modeling
In weak inversion
QI  WL
  QID
 )
(QIS
2
In strong inversion &
saturation

QI   2 3WLQIS
65
Total inversion, source and drain charges
Channel linearity
coefficient
  nCox
 t
QR QID


  nCox
 t
QF QIS
=1 in WI
0 in SI sat
=1 in SI for VDS=0
 2 1    2

  nCox
 t )  nCox
 t 
QI  WL 
(QIS
 3 1 

 6  12  8 2  4 3

n



QS  WL 
(QIS  nCoxt )  Coxt 
2
2


15 1   
 4  8  12 2  6 3

n
  nCox
 t )  Cox
 t 
QD  WL 
(QIS
2
2


15 1   
CMOS Analog Design Using All-Region MOSFET
Modeling
66
Capacitive coefficients - 1
Using the quasi-static approximation
Q j dVG Q j dVS Q j dVD Q j dVB




dt
VG dt
VS dt VD dt
VB dt
dQ j
Defining
 Qj
C jk  
 Vk
jk
0
 Qj
C jj 
Vj
0
 dQ G / dt   C gg  C gs C gd C gb   dVG / dt 


 


/
dt
dQ
dV
/
dt

C
C
C

C
sg
ss
sd
sb
 S
 S


 dQ / dt   C dg C ds
C dd C db   dVD / dt 
D


 
 dV / dt 
 dQ / dt   C


 B
  bg C bs C bd C bb   B
CMOS Analog Design Using All-Region MOSFET
Modeling
67
Capacitive coefficients - 2
 The 16 capacitive coefficients are not linearly
independent
Assume
VG t   VS t   VD t   VB t   V (t )
Under equal terminal voltage variations, the charging
currents are zero. For the gate charging current, e.g.,
we have
dQ G
dV
 (Cgg  Cgs  Cgd  Cgb )
0
dt
dt
Cgg  Cgs  Cgd  Cgb
Similarly, for S, D, and B nodes
Cdd  Cdg  Cds  Cdb
Css  Csg  Csd  Csb
Cbb  Cbg  Cbs  Cbd
CMOS Analog Design Using All-Region MOSFET
Modeling
68
Capacitive coefficients - 3
Assume that
dVS dVD dVB


0
dt
dt
dt
d QG
dVG d Q S
dVG
 C gg
,
 Csg
,
dt
dt
dt
dt
dQD
dVG d Q B
dV
 Cdg
,
 Cbg G
dt
dt
dt
dt
The sum of all the charging currents is
d QG d Q S d Q D d Q B
dVG



 (Cgg  Csg  Cdg  Cbg )
dt
dt
dt
dt
dt
Charge conservation,
d(QS+QD+QB+QG)/dt=0
Cgg  Csg  Cdg  Cbg
CMOS Analog Design Using All-Region MOSFET
Modeling
69
Capacitive coefficients - 4
 Linear relationships between capacitive
coefficients
Cgg  Cgs  Cgd  Cgb  Csg  Cdg  Cbg
Css  Csg  Csd  Csb  Cgs  Cds  Cbs
Cdd  Cdg  Cds  Cdb  Cgd  Csd  Cbd
Cbb  Cbg  Cbs  Cbd  Cgb  Csb  Cdb
 Only nine out of the sixteen capacitive
coefficients are linearly independent
CMOS Analog Design Using All-Region MOSFET
Modeling
70
A complete set of 9 capacitive coefficients
for the MOSFET
2
1  2 qIS
Cgs  Cox
2
3
1  1  qIS
C gd
2
2
 2 qID

 Cox
2
3
1  1  qID
Csd
Cbs(d )  (n 1)Cgs(d )
Cgb  C gb
n 1

(Cox  C gs  C gd )
n
4
  3 2   3 qID
  nCox
3
15
1  1  qID
4
1  3   2 qIS
Cds   nCox
3
15
1  1  qIS
Cdg  Cgd  Cm  (Csd  Cds ) / n
CMOS Analog Design Using All-Region MOSFET
Modeling
71
Simplified small-signal MOSFET model
G
g md vDB  Csd
Cgs
dvDB
dt
Cg
d
S
Cbs
dv
g mg vGB  Cm GB
dt
Cbd
g ms vSB  Cds
B
D
Cgb
dvSB
dt
CMOS Analog Design Using All-Region MOSFET
Modeling
72
The five capacitances of the simplified
model
Intrinsic capacitances simulated from (___) the charge-based and (o) from
the S- model (NMOS transistor, tox= 250Å, NA=2x1016 cm-3, and VT0=0.7V.
CMOS Analog Design Using All-Region MOSFET
Modeling
73
Capacitances of extrinsic transistor - 1
CMOS Analog Design Using All-Region MOSFET
Modeling
74
Capacitances of extrinsic transistor - 2
CMOS Analog Design Using All-Region MOSFET
Modeling
75
Non-quasi-static (NQS) small-signal
model
Channel segmentation: representation of the MOSFET
as a series combination of short transistors
CMOS Analog Design Using All-Region MOSFET
Modeling
76
Simplified high-frequency MOSFET model
G
Cgs
1  j 1   2 
g md vd
1  j 1
S
1  j 1   3 
D
g mg vg
Cbs
1  j 1   2 
B
Cgd
1  j 1
Cgb
1  j 1   4 
 Cgb
Cbd
g ms vs
1  j 1
CMOS Analog Design Using All-Region MOSFET
Modeling
1  j 1   3 
77
Time constants of the NQS MOSFET
model

4 1  3   2
1 
1  qIS 15 1   3


1 2  8  5
2 
1  qIS 15 (1   )2 (1  2 )
2
L2
t

1 5  8  2 2
3 
1  qIS 15 (1   ) 2(2   )
CMOS Analog Design Using All-Region MOSFET
Modeling
78
Quasi-static small-signal model
1<<1
2(3)<<1
non-quasi-static model
reduces to the five-capacitor model
G
Cgs
gmd vDB
C gd
gmg vGB
S
Cbs
Cgb
D
gms vSB
Cbd
B
CMOS Analog Design Using All-Region MOSFET
Modeling
79
Example: Small-signal parameters
Calculate ID, VDSsat and small-signal parameters of a saturated nchannel MOSFET in 0.35 m technology at if = 3 with
VSB= 0. W=10 m, L=1 m, tox=7 nm, n=1.2, n=400 cm2/V-s at 300 K.
Answer:
Cox =493 nF/cm2 and I SH  nCox
 nt2 / 2= 80 nA.
gmd = 0, Cgd = 0 and Cbd = 0 since the transistor is saturated.
I D  I F  (W / L)I SH i f  10  80  3  2.4 μA


VDSsat  t 3  1  i f  26(3  2)  130 mV

QIS
qIS  
 1  i f 1  1 3 1  1
 t
nCox
g mg  (2 I S / nt )


1 i f 1
g mg 
2  10  80 nA
 51 μA/V
1.2  0.026 V
CMOS Analog Design Using All-Region MOSFET
Modeling
80
Example: Small-signal parameters (continued)
2
1  2 qIS
Cgs  Cox
2
3
1  1  qIS
C gb
qID  1
1


 1
qIS  1 qIS
n 1

(Cox  C gs  C gd )
n
=1/(1+1)=0.5
  49 fF
Cox  WLCox
2
11
1
Cgs  49
 14.5 fF
2
3 10.5 1  1
C gb
0.2

(49  14.5  0)  5.75 fF
1.2
CMOS Analog Design Using All-Region MOSFET
Modeling
81
Intrinsic transition frequency
fT 

g mg
2 C gs  C gb


t
fT 
2
2
2 L

g ms
2 n C gs  C gb



1  i f 1
CMOS Analog Design Using All-Region MOSFET
Modeling
82
Example: Transition frequency
Determine the inversion level for which the transition
frequency of a minimum (nominal) length NMOS
transistor in the 0.35 m technology is 10 GHz at room
Answer:
Assuming that n=1.2 and n = 400 cm2/V-s at 300 K
it follows that

i f  1   L fT / nt
2

2
 1  21
Thus, operation in moderate inversion can be
considered for a design at 1 GHz, for example.
CMOS Analog Design Using All-Region MOSFET
Modeling
83
Main short-channel effects
 Mobility dependence on the electric field
 Channel length modulation
 Drain-induced barrier lowering
 Velocity saturation
CMOS Analog Design Using All-Region MOSFET
Modeling
84
Mobility dependence on the electric field
Inclusion of mobility variations in compact modeling: the constant
mobility is substituted with an effective mobility, which depends
on the applied voltages.
eff 
0
 QBS
  QIS
 QBD
  QID
 
1   


2

2

s
s


0, the low-field mobility and , the scattering constant, are fitting
parameters
Another simplification: the effective transversal field is assumed
constant along the channel and equal to its value at pinch-off.
eff 
CMOS Analog Design Using All-Region MOSFET
Modeling
0
1  

QBa
s
85
Channel length modulation
The dependence of the effective channel length on the drain-to-source
voltage is referred to as the channel length modulation (CLM).
D
S
ID
 VDS  VDSsat 
L  LC ln 1 

V


p
VDSsat
VDS
L
0
  2
qIS  qID

1
  qID
 
ID  IS
qIS

L 1    qIS  qID
 
1
L
CMOS Analog Design Using All-Region MOSFET
Modeling
Le
L
y
86
Drain-induced barrier lowering (DIBL)
 An increase in the drain voltage produces an increase in the
surface potential in the channel and, consequently, a reduction
in the potential barrier seen by the electrons at the source (
DIBL).
 The inclusion of the DIBL effect in MOSFET models is generally
through the threshold voltage.
VT  VT ,lc

 L 
6t ox
2bi  VBS   VDS exp  

d1
 4d 1  

VT  VT ,lc   bi  VSB   bi  VDB 
CMOS Analog Design Using All-Region MOSFET
Modeling
87
Velocity saturation effects - 1

s
1
F
FC

1
s
 s d S
vsat dy
dS
  F (longitudinal field)
dy
Allows analytical
integration for ID
v
vsat
vsat
s
s
 FC
FC
CMOS Analog Design Using All-Region MOSFET
Modeling
F
88
Velocity saturation effects - 2
 ds
dQI  nCox
I D   WQI
dVC
dy
s
s

1
nCox vsat
sWQI
dQI
dy
dQI
ID 
dQI dy
1
1
 FC dy
nCox
t  dVC
dQI  1
 

dy  nCox QI  dy
 1
t 





nC
Q
I 
 ox
  QIS
 
  QID


  QIS
 
ID  
 QIP   QID

 
Q  QIS
 L
nCox
2

1  ID

LFC nCox
sW
1
CMOS Analog Design Using All-Region MOSFET
Modeling
89
Velocity saturation effects - 3
Normalized current vs.
normalized charge densities
 t 
qI  QI /  nCox
IS 

t2
W

s nCox
L
2
iD  I D / I S
  2
qIS  qID

  qID
 
iD 
qIS

 
1    qIS  qID
st / L 


vsat
Normalization (specific) current
short-channel parameter : ratio of diffusion-related
velocity to saturation velocity
CMOS Analog Design Using All-Region MOSFET
Modeling
90
Velocity saturation effects - 4
Saturation: The minimum amount of electron charge flowing at
the saturation velocity, required to sustain the current is

QIDSAT
  I D / Wvsat
2
qIS  1  qIDsat  1  qIDsat

S
D
ID
QI
QIS

QID
VDS

QIDSAT
0
CMOS Analog Design Using All-Region MOSFET
Modeling
91
Velocity saturation effects - 5
2
qIS  1  qIDsat  1  qIDsat
QIDSAT QIS

1
 st L
Short channel
Long channel
strong
inversion
weak
inversion
vsat
10-2
100
102
104
 QIS nCox  t
CMOS Analog Design Using All-Region MOSFET
Modeling
92
Small dimension effects on charges
and capacitances
I D   WQI
dVC
dy

1
s
s
nCox vsat
dQI
dy
t  dVC
dQI  1
 

dy  nCox QI  dy
sW 
ID


dy  
 QI  nCoxt 
 ID 
nCox
Wvsat
CMOS Analog Design Using All-Region MOSFET
Modeling

 dQI

93
Virtual charge formalism - 1
Virtual inversion charge density
 t 
QV  QI  nCox
ID
Wvsat
inversion +pinch off -saturation charge
densities
Along the channel
sW 
dQV  dQI
ID


dy  
 QI  nCoxt 
 ID 
nCox
Wvsat

sW

QV dQV
 dQI  
 ID
nCox

CMOS Analog Design Using All-Region MOSFET
Modeling
94
Virtual charge formalism -2
sW
dQV
ds
ID  
QV
 sWQV

nCox
dy
dy
The drift of the virtual charge produces the
same current as the actual movement of
the real charge, which includes drift,
diffusion and velocity saturation
CMOS Analog Design Using All-Region
All Region MOSFET
Modeling
95
Channel linearity coefficient  with vsat
dQV
QV
The integration of I D  

nCox
dy
sW
from
source to drain results in
ID 
where
2  QVD
2
sW QVS
 L
Cox
2n

 I D0 1   2

  nCox
  t  I D Wvsat

QID
QVD



  nCox
  t  I D Wvsat
QVS
QIS
CMOS Analog Design Using All-Region MOSFET
Modeling
96
Stored charges including vsat
The stored charge

QI  W 0LL QI dy  W LQIDsat
is calculated changing the integration variable
from y to QV
dy  
sW
 ID
nCox
QV dQV
resulting in
 2 1    2
 LI D
  nCox
 t 
QVS
Q I  W ( L  L) 
 3 1 
 vsat
CMOS Analog Design Using All-Region MOSFET
Modeling
97
Source and drain charges including vsat
CMOS Analog Design Using All-Region MOSFET
Modeling
98
Capacitive coefficients including vsat - 1
Le g ms 1 
2
1  2 qIS

Cgs  WLeCox

3
(1   )2 1  qIS 3nvsat 1  2
2
Le gmd 1 
2
 2  2 qID

 WLeCox

3
(1   )2 1  qID 3nvsat 1  2
2
Cgd
2
Le gmg 1  
n 1 
 Cox  Cgso  Cgdo 

Cgb  Cbg 
2

n
3vsat 1 


2
2
2
2
3


7
1



g
q



4
1

3



1
L
L
e
IS
ms
W e
Cds   nCox

15
L 1 3 1  qIS 30 vsat L
1 3
2
3
2
qID
4
1 g md L e2  37 1 
L
e   3  
W
Csd   nCox

3
15
L 1 3 1  qID 30 vsat L
1


 
Cbs(d )   n 1 Cgs(d )
2
Cdg  Cgd  Csd  Cds  / n
CMOS Analog Design Using All-Region MOSFET
Modeling
99
Capacitive coefficients including vsat - 1
Normalized capacitances versus drain-source voltage
CMOS Analog Design Using All-Region MOSFET
Modeling
100
Gate-to-bulk capacitance with and
without the effect of velocity saturation
CMOS Analog Design Using All-Region MOSFET
Modeling
101