Cross Sectional View of FET
ECE 584, Summer 2002
Brad Noble
Chapter 3 Slides
1
FET I-V Characteristic
ECE 584, Summer 2002
Brad Noble
Chapter 3 Slides
2
Saturation Voltage
• Vpinchoff = VDS,sat = VGS – VTH
– Separates resistive from saturation region
• The drain current is given by
I DS 
1
2
 N V GS  V TN  , for NFET
2
V TN  V TH and  N 
W
• Solving for VDS,sat:
L
 n C ox  SK PN
V DS , sat  V GS  V TN 
ECE 584, Summer 2002
Brad Noble
Chapter 3 Slides
2 I DS
N
3
Early Voltage Function of Length
ECE 584, Summer 2002
Brad Noble
Chapter 3 Slides
4
Early Voltage in MOSFETs
• Due to channel length modulation:
• Good to solve for quiescent voltage-current.
I DS 

1
 V TH
L
1
K PN S V GS  V TH

K PN S V GS  V TH

1
2
ECE 584, Summer 2002
V GS
2
2
I DS 
 n C ox
W
Brad Noble
Chapter 3 Slides
2
2
 1   V DS 
2
L 

1 

L 

V DS 

1 

VE 

5
Ex: Find VDS,sat for an NFET
ECE 584, Summer 2002
Brad Noble
Chapter 3 Slides
6
Body Effect
ECE 584, Summer 2002
Brad Noble
Chapter 3 Slides
7
Variations in VTH Across Channel
• We assume VTH is constant across channel
THIS IS NOT TRUE!
• Depletion region is thick at S and thin at D.
V TH   ms 
Q dep
 2 F
C ox
I DS 
N
V GS
2n
n  subthresho
 V TN 
2
Cox
Gate oxide
capacitance
inversion
layer
Cdep
ld slope factor  1 
Depletion cap,
function of x
C dep
 1 .5
C ox
ECE 584, Summer 2002
Brad Noble
Chapter 3 Slides
8
Small Signal Equivalent Ckt
ECE 584, Summer 2002
Brad Noble
Chapter 3 Slides
9
Parasitic Capacitance
ECE 584, Summer 2002
Brad Noble
Chapter 3 Slides
10
Capacitance Equivalent Circuit
ECE 584, Summer 2002
Brad Noble
Chapter 3 Slides
11
Variation in Capacitance
ECE 584, Summer 2002
Brad Noble
Chapter 3 Slides
12
Notes on PFETs
• PFETs typically have a shape factor 3 or 4
times larger than NFETs
• Body effect can be eliminated in PFETs by
tying the n-well to VDD
– Need 6m spacing between n-wells to isolate.
– Dr. Engel always does this on input devices, not
always elsewhere.
ECE 584, Summer 2002
Brad Noble
Chapter 3 Slides
13
Subthreshold Conduction
ECE 584, Summer 2002
Brad Noble
Chapter 3 Slides
14
Weak Inversion
• What really happens if VGS < VTN?
V GS  0 .95 V
V TH  1V
• In digital design, IDS = 0.
• We call it “weak inversion” or W.I.
• IDS is primarily due to Idrift in strong
inversion and Idiffusion in weak inversion.
ECE 584, Summer 2002
Brad Noble
Chapter 3 Slides
15
Modes of Inversion
• IDS = Idrift + Idiffusion
• If VGS > VTN the channel has been inverted.
• To be more precise, we can say the channel
has been “strongly inverted” (S.I.) due to an
abundance of carriers in the channel.
• Inversion is independent of whether the
FET is in the linear or saturation region.
ECE 584, Summer 2002
Brad Noble
Chapter 3 Slides
16
Weak Inversion Idiffusion
• Drain is more reverse biased than source:
N S  N 0 exp
  O  q V G  V S 
kT
N D  N 0 exp
  O  q V G  V S 
kT
• To find Idiff, compute gradient dN dx
• Because no carriers are lost as they travel
from S to D, current is the same for all x
and gradient is not a function of x.
• Note: This is not really true due to
recombination, but its close!
ECE 584, Summer 2002
Brad Noble
Chapter 3 Slides
17
W.I. Surface Potential
1
VG
C ox

S
s 
surface
potential
Cd
C ox
C d  C ox
n  1
Cd
C ox
I DS

vG
 SI D 0  exp
nU T

Exponentia
ECE 584, Summer 2002

-vS
  exp

UT


 1 .5,
j C d
1
j C d

1
j  C ox
subthresho ld
slope factor

- v DS
  1  exp

UT





l law device
Brad Noble
Chapter 3 Slides
18
Deriving Weak Inversion IDS
dN

ND  NS
dx
L
vG 
-vS 
-vD
 N0  N S 
 
 exp
 exp
  exp


L
UT 
UT
UT 


The current per unit width
I DS
W
  qD n
dN
dx
:
, where D n is diffusion
 kT 
 , a.k.a. the Einstein
D n   n 
 q 
I DS
coeff.
relationsh
ip

vG 
-vS
-vD 
W 

 exp
 I D 0  exp
  exp

nU T  
UT
UT 
 L 

ECE 584, Summer 2002
Brad Noble
Chapter 3 Slides
19
W.I. FET As Exp. Law Dev.
• S must be big for device to be useful.
- v DS
• If VDS = 100mV, exp
can be neglected.
UT
I DS

vG
 SI D 0  exp
nU T


-vS
  exp

UT

• For W.I. vDS,Sat  100mV
i
• Looks like a BJT 

 , for V DS  100 mV


C
v BE

ECE 584, Summer 2002
Brad Noble
Chapter 3 Slides
IC

v BE
 I S  exp
UT





20
Inversion Coefficient
• Let inversion
coefficien
t  
i DS
o
2nU
2
T
Weakly Inverted (W.I.)
 < 0.1
Strongly Inverted (S.I.)
 > 10
0.1 <  < 10 Moderately Inverted (M.I.)
• Shape factor as a function of  :
S 
i DS
o
2 n  K PN U T
2
Lets you chose shape to match inversion mode.
ECE 584, Summer 2002
Brad Noble
Chapter 3 Slides
21
Ex. Using Inversion Coeff.
i DS  1uA, S  100 ,  
1uA
O
2(1.5)(100 )(26mV)
2
(100)
  0 . 049  W.I.
i DS  100 uA, S  100 ,  
100uA
O
2
2(1.5)(100 )(26mV)
(100)
  4 . 9  M.I.
i DS  1mA, S  100 ,  
1mA
O
2(1.5)(100 )(26mV)
2
(100)
  49  S.I.
ECE 584, Summer 2002
Brad Noble
Chapter 3 Slides
22
Small Signal Analysis
Total Voltage

Small Change

DC Bias
v GS

v gs

V GS
Quiescent
Voltage
 v GS  V GS
O


v GS
O
ECE 584, Summer 2002




v BS
Brad Noble
Chapter 3 Slides
O
v DS
O
i DS
O
23
Ex: Quiescent Point
V T 0  0 . 75 V
v GS  3 V
V E  81 V
K PN  50  A
v DS  5 V
  0 . 59
S 
i DS 
O
i DS
O
O


O
V
2
v BS  0 V
v DS
O
O
100 μ m
v GS
O




v BS
O
i DS
O
10 μ m
 O
K PN S v GS  v T0   


2
1
1  100  m
 
2  10  m
2
2  F  v SB 
O
v DS 
 

2 F  1 


 
VE 
O

A 
5 
2
  50 2  3  0 . 75   1 
  1 . 34375 mA
V 
81 


Question: How many digits are significant?
ECE 584, Summer 2002
Brad Noble
Chapter 3 Slides
24
Small Signal Model Limits
• Suppose the previous circuit is the input
device of an amplifier.
• Small-signal model holds as long as the
deviations are small   kT q 
ECE 584, Summer 2002
Brad Noble
Chapter 3 Slides
25
Taylor Series Expansion
• Taking a Taylor expansion of one variable:
f ( x )  f ( x 0 )  f ( x )( x  x 0 ) 
1
2
2
f ( x )( x  x 0 )  
Linear Approx.
0
i DS  i
0
DS
0
0
  i DS 
  i DS 
  i SB 

  v GS  
  v DS  
  v SB  
  v GS 
  v DS 
  v SB 
gm
 v GS  v GS - v GS  v gs
O
g ds
 v DS  v ds
g mb
 v SB  v sb
i DS  iDS   iDS
O
ECE 584, Summer 2002
Brad Noble
Chapter 3 Slides
26
Small Signal Model Params
0
  i DS 
gm  
 
  v GS 
0
g ds
v DS  O

2  1 
 i DS , for large V E , g m 
VE 

O
2  i DS
O
i DS
1
  i DS 


 
VE
r ds
  v DS 
O
0
g mb

  i SB 

, typically
   g m where  
O
2 2  F  v SB
  v SB 


Gain  g m r ds
ECE 584, Summer 2002
Brad Noble
Chapter 3 Slides
27
1
Example: Small Signal Analysis
i DS  1 . 34375 mA, V E  81 V,   250
O
μA
V
2
Let v GS  3 . 026 V (up by kT q ) , v BS  0 V, v DS  5 . 026 V
Using full equation
(lots of work! ) :
iDS  1 . 3754 mA   iDS  31 . 65 μA
Using small signal analysis
iDS  g m v gs  g ds v ds 
 (1 . 1592
ECE 584, Summer 2002
mS

:
2  i DS
O

 i DS 
v gs  
v ds
 VE 
)( 26 mV )  ( 0 . 0166
Brad Noble
Chapter 3 Slides
O
mS
)( 26 mV )  30 . 57 μA
28
Small Signal Low-Freq Model
i ds
S.I.
signals
Sat
gm 

v gs

small
2  i DS
g m v gs
g mb v ds
rds
O
, where n  subthresho
ld slope factor  1.5
n
r ds 
VE
O

V E  L
i ds
ECE 584, Summer 2002
O
i ds
Brad Noble
Chapter 3 Slides
29
Ex: Find gm and rO
i DS  10 μA
O
W

L
Is it S.I. and saturated?
10 μm
If so,

2  4 10  100 μA/V
gm 
4 μm
2
10 uA   23 μS
1 .5
r ds 
V EN  L
i
ECE 584, Summer 2002
O
DS

10 V
μm 10 μm 
10 μA
Brad Noble
Chapter 3 Slides
 10 M 
30
Transconductance: W.I. & M.I.
• What is gm for a weakly inverted FET?
0
i DS
  i DS 
gm  
 
nU T
  v GS 
O
Not in textbooks!
• What is gm for a moderately inverted FET?
0

 i DS
1  exp  
  i DS 
gm  
, where  
 
nU T

  v GS 
ECE 584, Summer 2002
g ds 
O
i DS

O
for all modes
VE
Brad Noble
Chapter 3 Slides
31