MOSFET Cross-Section
A MOSFET Transistor
Source
Drain
Gate
Source
Substrate
Gate
Drain
MOSFET Schematic Symbols
Formation of the Channel for an
Enhancement MOS Transistor
Water Analogy of a
(subthreshold) MOSFET
Channel Current vs. Gate Voltage
Sub-Threshold
450
10-3
400
10-4
In linear scale, we
have a quadratic
dependence
350
300
250
200
VT
150
10-7
10-9
50
10-11
0.5
1
1.5
2
2.5
3
Gate voltage (V)
3.5
4
4.5
5
In log-scale, we
have an exponential
dependence
10-8
10-10
0
Ith
10-6
100
0
VT
10-5
Channel Current (A)
Channel Current (mA)
Above-Threshold
10-12
0.4
0.6
0.8
1
1.2
1.4
Gate voltage (V)
1.6
1.8
2
MOS Capacitor Picture
MOSFET Channel Picture
MOS Capacitor Picture
MOS Electrostatics
Vfb
Condition is called flatband --- the voltage when this occurs
is called flatband
This state is the baseline operating case --- a capacitive
divider has one free parameter
MOS Electrostatics
Depletion Condition --- gate charge is terminated by charged
ions in the depletion region
Part of this region is often referred to as weak-inversion
MOS Electrostatics
Inversion --- further gate charge is terminated by carriers
at the silicon--silicon-dioxide interface
MOS Structure Electrostatics
MOS Capacitor Picture
MOS-Capacitor Regions
Qs = e
(Y - Vs)/UT
Depletion (k(Vg - VT) - Vs < 0)
Qs = e
(k(Vg - VT) - Vs)/UT
Qs = ln( 1 + e
(k(Vg - VT) - Vs)/UT
)
Inversion (k(Vg - VT) - Vs > 0)
Qs = (k(Vg - VT) - Vs)/UT
MOS Capacitor Picture
MOSFET Channel Picture
Calculation of Drain Current
2
No recombination
0 0
0
dn
d Dn 2
= Dn
+G-R
2
dt
dx
d n
Dn 2 = 0
dx
Dn = Ax + B
 SC = VS -
 dC = Vd -
nsource  e - SC / uT
ndrain  e
0
l
 varies as kVG
nsource - ndrain
dn
= qDn
J = qDn
dx
l
(
(
I=I e
kVg -VS ) /UT
0
-e
- d C / uT
)
(kVg -Vd )/UT
MOSFET Current-Voltage Curves
(
I = I0 e
(kVg -VS )/ uT
-e
(kVg -Vd )/ uT
(
-V / u
- /
kV / u
I DS = I 0 e G T e S T - e VD uT
= I 0e
(kVg -VS )/ uT
(1 - e (
= I 0e
)
- Vd -VS )/ uT
(
= I 0 e ( kV G -VS ) / uT 1 - e -Vds / uT
( kV G -VS ) / uT
)
)
Vds > 4UT
Saturation
)
MOSFET Current-Voltage Curves
(
I = I0 e
(kVg -VS )/ uT
-e
(kVg -Vd )/ uT
(
-V / u
- /
KV / u
I DS = I 0 e G T e S T - e VD uT
= I 0e
(kVg -VS )/ uT
(1 - e (
= I 0e
)
- Vd -VS )/ uT
(
= I 0 e ( KVG -VS ) / uT 1 - e -VdS / uT
( KVG -VS ) / uT
)
)
VdS > 4uT
Saturation
)
Subthreshold MOSFETs
-6
10
nFET
pFET
-7
Drain current (A)
10
S
D
G
-8
10
In linear scale, we
have a quadratic
dependence
G
B
D
S
In log-scale, we
have an exponential
dependence
-9
10
k = 0.58680
Io = 1.2104fA
-10
10
-11
10
0.4
0.45
0.5
0.55
0.6
0.65
0.7
Gate voltage (V)
0.75
0.8
0.85
0.9
Determination of
Threshold Voltage
1.1
1
Drain current / subthreshold fit
0.9
0.8
0.7
0.6
0.5
VT = 0.86
0.4
0.3
0.2
0.1
0.4
0.5
0.6
0.7
Gate voltage (V)
0.8
0.9
1
Drain Current --- Source Voltage
-7
10
-8
10
Drain current (A)
-9
10
UT = 25.84mV
-10
10
-11
10
k = 0.545
-12
10
0.6
0.65
0.7
0.75
Source voltage (V)
0.8
0.85
0.9
Drain Characteristics
Origin of Drain Dependencies
Increasing Vd effects
the drain-to-channel
region:
• increases barrier
height
• increases depletion
width
Current versus Drain Voltage
Not flat due to Early effect
(channel length modulation)
Id = Id(sat) (1 + (Vd/VA) )
or
Id = Id(sat) e
Vd/VA
Iout
Id(sat)
GND
Rout
Early Voltage Length Dependence
Width of depletion region
depends on doping, not L
Might expect Vo to linearly
vary with L
MOSFET Operating Regions
Surface potential
moving from depletion
to inversion
Band-diagram
picture moving
from subthreshold to
above-threshold
Qualitative Above-Threshold
I = (K/2k)
(( k(Vg - VT) - Vs )2 - (( k(Vg - VT) - Vd )2 )
Above Threshold MOSFET Equations
I = (K/2k) ( (k(Vg - VT) - Vs)2
- (k(Vg - VT ) - Vd) 2 )
If k = 1 (ignoring back-gate effects):
I = (K/2) ( 2(Vgs - VT) Vds - Vds2 )
Saturation: Qd = 0
I = (K/2k) ( (k(Vg - VT) - Vs)2
Detailed MOSFET Derivation
Q(x) = CT ( k(Vg - VT) - V(x))
Qs = CT ( k(Vg - VT) - Vs),
CT = CD + Cox
Qd = CT ( k(Vg - VT) - Vd)
(k = Cox / CT)
I = m Q(x) E(x)
Current is constant through
the channel (no loss)
( E(x) = Electric Field )
E(x) = - d V(x)
dx
= (1 / CT ) d Q(x)
dx
I = (m / CT ) Q(x)
d Q(x)
dx
Detailed MOSFET Derivation
Integrate with respect to length: I = (m / CT ) Q(x)
I L = (m / 2 CT ) Q(x)2
d Q(x)
dx
Qs = CT ( k(Vg - VT) - Vs)
|Q
d
= CT ( k(Vg - VT) - Vd)
(
I = (m / 2 CT ) ( 1 / L) Qs2
(
I = (m CT / 2 ) ( 1 / L) ( k(Vg - VT) - Vs) 2
-Q )
d
2
- ( k(V
2
V
)
V
)
g
T
d
)
K = m Cox (W/L)
I = (K/2k)
(( k(Vg - VT) - Vs )2 - (( k(Vg - VT) - Vd )2 )
MOSFET Equations
When ignoring
back-gate effects
(we often do):
k=1
I = (K/2) ( (Vg - VT - Vs)2 - (Vg - VT - Vd) 2 )
Above-Threshold:
I = (K/2) ( 2(Vgs - VT) Vds - Vds2 )
Saturation: (Qd = 0) I = (K/2) (Vgs - VT )2
(Vd > Vg - VT )
I = Is e
Subthreshold:
Vgs/UT
(1 – e
-Vds/UT
)
Vgs/UT
Saturation: (Vds> 4 UT) I = Is e
Output Characteristics of the
Above-Threshold MOSFET
iD /ID0
vDS = vGS - VT
1.0
Active
Region
Interpretation of large-signal model
Saturation Region
0.75
iD
Channel modulation effects
vDS = vGS - VT
Active Region
Saturation Region
0.5
0.25
Cutoff Region
0
0
Increasing
values of vGS
vDS
0.5
1.0
1.5
2.0
vGS-VT = 1.0
VGS0 - VT
vGS-VT = 0.867
VGS0 - VT
vGS-VT
= 0.707
VGS0 - VT
vGS-VT = 0.5
VGS0 - VT
vGS-VT = 0
VGS0 - VT
vDS
V
GS0 - VT
2.5
MOSFETs
350
nFET
300
Current (mA)
S
D
250
200
pFET
G
G
B
D
S
150
100
K = 37.861 mA/V2
50
VT = 0.806
0
0
0.5
1
1.5
2
2.5
3
Gate voltage (V)
3.5
4
4.5
5
Drain Current - Gate Voltage
0.02
0.018
sqrt(Drain current (A))
0.016
0.014
0.012
0.01
0.008
0.006
VT = 0.806
0.004
K k = 37.861 mA/V2
0.002
0
0.5
1
1.5
2
2.5
3
3.5
Gate voltage (V)
4
4.5
5
Drain Current --- Source Voltage
4
3.5
K/k = 74.585 mA/V2
sqrt(Drain current (mA))
3
(k = 0.7)
2.5
2
1.5
k (Vg - VT) = 0.595
1
(k = 0.54)
0.5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Gate voltage (V)
0.7
0.8
0.9
1
An Ohmic MOSFET
I = (K/2k) ( (k(Vg - VT) - Vs)2
- (k(Vg - VT ) - Vd) 2 )
If Vd ~ Vs, (small difference)
I = K (Vg - VT)(Vd - Vs)
(Vd - Vs = 50mV)
A MOSFET in Saturation
Saturation: Qd = 0
I = (K/2k) ( (k(Vg - VT) - Vs)2
Vs = 0
I = (Kk/2) (Vg - VT) 2
More Ohmic Region Data
I = (K/2k) ( (k(Vg - VT) - Vs)2
- (k(Vg - VT ) - Vd) 2 )
Take the derivative of I with
respect to Vd (Vs = 0 )
dI / d Vd = (K/2k)( 0 (-2) (k(Vg - VT ) - Vd) )
= (K/2k)(k(Vg - VT ) - Vd)
Influence of VDS on the Output
Characteristics
Current versus Drain Voltage
Not flat due to Early effect
(channel length modulation)
Id = Id(sat) (1 + (Vd/VA) )
or
Id = Id(sat) e
Vd/VA
Iout
Id(sat)
GND
Rout
Early Voltage Length Dependence
Width of depletion region
depends on doping, not L
Might expect Vo to linearly
vary with L
Small-Signal Modeling
V3
V1
V1
V2 I
I
V
V1
V2
V3
Above VT
MOSFET
Sub VT
MOSFET
2I /(V1-V2 -VT)
I / UT
gmV
ro
-
V2
gm
V3
+
ro
VA / I
VA / I
V2
Av
2VA/(V1-V2 -VT)
VA / UT
Small-Signal Modeling
V3
V3
I
V1
I
V1
rp
V1
V2
V2
rp
V3
+
V gmV
ro
-
V2
gm
V2
ro
Av
BJT
(UT b) / I
I / UT
VA / I
VA / UT
Above VT
MOSFET

2I /(V1-V2 -VT)
VA / I
2VA/(V1-V2 -VT)
Sub VT
MOSFET

I / UT
VA / I
VA / UT
Capacitances in a MOSFET
MOSFET Depletion Capacitors
Overlap Capacitances
Capacitance Modeling
Capacitance Modeling
Velocity Saturation
Si Crystal Velocity Limit
Ideal Drift (Ohm’s Law)
Effect of Velocity Saturation
Square-law
region
VT
L = 76 nm MOSFET
Small-Signal Modeling (with kappa)
V3
V1
V1
V2 I
I
V
V1
V2
V3
gmV
ro
-
V2
gm
V3
+
ro
V2
Av
Above VT
MOSFET
2I /(V1-V2 -VT)
VA / I
2VA/(V1-V2 -VT)
Sub VT
MOSFET
kI / UT
VA / I
kVA / UT
Small-Signal Modeling (with kappa)
V3
V3
I
V1
I
V1
rp
V1
V2
V2
rp
V3
+
V gmV
ro
-
V2
gm
V2
ro
Av
BJT
(UT b) / I
I / UT
VA / I
VA / UT
Above VT
MOSFET

2I /(V1-V2 -VT)
VA / I
2VA/(V1-V2 -VT)
Sub VT
MOSFET

kI / UT
VA / I
kVA / UT