ECE-305: Spring 2015
MOSFET IV
Professor Mark Lundstrom
Electrical and Computer Engineering
Purdue University, West Lafayette, IN USA
lundstro@purdue.edu
Lundstrom’s lecture notes: Lecture 4
Lundstrom ECE 305 F15
4/10/15
outline
1)
2)
3)
4)
5)
6)
7)
Introduction
Linear region
Saturation region
Full range (VDS = 0 à VDS)
DIBL
Series resistance
VS model level 0
Lundstrom ECE 305 F15
Long vs. short channel MOSFETs
Square Law
Velocity saturated
I DSAT ∝ (VGS − VT )
2
I DSAT ∝ (VGS − VT )
Fig. E17.2, Semiconductor Device
Fundamentals, R.F. Pierret
the MIT VS Model
ß 32 nm technology à
Lundstrom ECE 305 F15
outline
1)
2)
3)
4)
5)
6)
7)
Introduction
Linear region
Saturation region
Full range (VDS = 0 à VDS)
DIBL
Series resistance
VS model level 0
Lundstrom ECE 305 F15
MOSFET IV characteristic
gate-voltage
controlled
current source
circuit
symbol
D
ID
VGS
VDS
G
VGS
S
gate-voltage
controlled
resistor
(Courtesy, Shuji Ikeda, ATDF, Dec. 2007)
Lundstrom ECE 305 F15
MOSFET e-band (equilibrium)
L
0
VGS = 0
E
VGS > VT
EC ( y )
VDS
n-Si
VGS = VDD
n-Si
EF
p-Si
y=0
y=0
y
Qn ≈ −Cox (VGS − VT )
y
C cm 2
Lundstrom ECE 305 F15
MOSFET e-band (high VGS, low VDS)
L
0
E
VGS > VT
VDS
n-Si
n-Si
EC ( y )
Fn
p-Si
y=0
VGS = VDD
1 dEC
Ey=
q dx
y
y
Qn ≈ −Cox (VGS − VT )
Fn
C cm 2
Lundstrom ECE 305 F15
MOSFET IV
L
0
VGS > VT
current is charge per unit time
VDS
n-Si
n-Si
p-Si
I D = −W Qn ( y ) υ y (y)
C≡
y
y=0
MOS electrostatics
Q
V
F
Qn ≈ −Cox (VGS − VT )
(
C cm 2
)
Qn ≈ 0 VGS < VT
K ε
Cox = O 0 F cm 2
xo
Lundstrom ECE 305 F15
MOSFET IV: low VDS
L
0
VGS > VT
VDS
VGS
Qn ( y ) ≈ −Cox (VGS − VT )
I D = −W Qn ( y ) υ y (y)
Qn = − Cox (VGS − VT )
υ y ( y ) = − µnE y
E y = −VDS L
gate-voltage
controlled
resistor
ID =
Lundstrom ECE 305 F15
W
µ C (V − VT )VDS
L n ox GS
✓
outline
1)
2)
3)
4)
5)
6)
7)
Introduction
Linear region
Saturation region
Full range (VDS = 0 à VDS)
DIBL
Series resistance
VS model level 0
Lundstrom ECE 305 F15
MOSFET e-band (high VGS, low VDS)
L
0
“virtual source”
E
VGS > VT
VDS
n-Si
n-Si
VGS = VDD
EC ( y )
Fn
p-Si
Fn
y=0
y
Qn ≈ −Cox (VGS − VT )
Lundstrom ECE 305 F15
y=0
C cm 2
y
MOSFET IV: “pinch-off” at high VDS
0
VGS > VT
VD
(
Note: thickness of channel
illustrates the areal density of
electrons – not the actual
thickness.
)
V y pinch = (VGS − VT )
Qn ( y ) = −Cox (VGS − VT − V (y))
(
)
Qn y pinch ≈ 0
Electric field is very large in
the pinch-off region.
Lundstrom ECE 305 F15
“pinch off” in the channel
pinch-off
“point”
0
VGS > VT
VD
EC
VGS
y ( nm )
Lundstrom ECE 305 F15
MOSFET IV: high VDS
0
VGS > VT
VD
(
)
V y pinch = (VGS − VT )
VGS
Qn ( y ) = −Cox (VGS − VT − V ( y) )
I D = −W Qn ( y ) υ y (y) = W Qn ( 0 ) υ y (0)
Qn ( 0 ) = −Cox (VGS − VT )
υ ( 0 ) = − µnE y ( 0 )
(
E y (0) ≈ −V y pinch
)
L = − (VGS − VT ) L
2
W
I D = µ nCox (VGS − VT )
2L
short channel MOSFETs
I DSAT ∝ (VGS − VT )
Lundstrom ECE 305 F15
High VDS : velocity saturation
velocity cm/s --->
VDS 1.0V
≈
≈ 5 × 10 5 V/cm
L
20 nm
107
υ = υ sat
υ = µ nE
105
104
electric field V/cm --->
Lundstrom ECE 305 F15
MOSFET IV: velocity saturation
0
VGS > VT
VDS
E y >> 10 4
I D = −W Qn ( y ) υ y (y)
Qn = − Cox (VGS − VT )
υ y = υ sat
(Courtesy, Shuji Ikeda, ATDF, Dec. 2007)
I D = W Cox υ sat (VGS − VT )
Lundstrom ECE 305 F15
✓
outline
1)
2)
3)
4)
5)
6)
7)
Introduction
Linear region
Saturation region
Full range (VDS = 0 à VDS)
DIBL
Series resistance
VS model level 0
Lundstrom ECE 305 F15
MOSFET: IV (re-cap)
I DLIN =
W
µnCox (VGS − VT )VDS
L
ID
I DSAT = W Cox υ sat (VGS − VT )
VDSAT =
VDSAT
υ sat L
µn
VDS
We have developed a 2-piece approximation to the MOSFET IV
characteristic.
Lundstrom ECE 305 F15
piecewise model for ID(VGS, VDS)
I D W = −Qn (VGS ) υ (VDS )
VGS ≥ VT : Qn (VGS ) = −Cox (VGS − VT )
⎛ V ⎞
VDS ≤ VDSAT : υ (VDS ) = ⎜ µn DS ⎟
⎝
L ⎠
VGS < VT : Qn (VGS ) = 0
VDS > VDSAT : υ (VDS ) = υ sat
If we can make the average velocity go smoothly from the
low VDS to high VDS limits, then we will have a smooth model
for ID(VGS, VDS) – above threshold.
Lundstrom ECE 305 F15
From low VDS to high VDS
1
1
1
=
+
→
µnVDS L υ sat
υ (VDS )
υ (VDS ) = FSAT (VDS )υ sat
⎡ V V
υ (VDS ) = ⎢ DS DSAT
⎣ 1+ VDS VDSAT
FSAT (VDS ) =
VDS VDSAT
⎤
⎥υ sat
⎦
1/ β
⎡1+ (VDS VDSAT )β ⎤
⎣
⎦
The extra parameter, β, is empirically adjusted to fit the
IV characteristic. Typically, β ≈ 1.4 − 1.8 for both NMOSFETs and for P-MOSFETs.
Lundstrom ECE 305 F15
empirical saturation function
υ (VDS ) = FSAT (VDS )υ sat
FSAT (VDS ) ≡
VDS << VDSAT : FSAT (VDS ) →
υ (VDS ) →
VDS
υ sat
VDSAT
υ (VDS ) →
VDS
υ
υ sat L µn sat
υ (VDS ) → µn
VDS
L
VDS
VDSAT
VDS VDSAT
1/ β
⎡1+ (VDS VDSAT )β ⎤
⎣
⎦
VDS >> VDSAT : FSAT (VDS ) → 1
υ (VDS ) → υ sat
✓
✓
Lundstrom ECE 305 F15
saturation function: FSAT (VD)
υ (VDS ) = FSAT (VDS )υ sat
FSAT (VDS ) =
VDS VDSAT
1/ β
⎡1+ (VDS VDSAT )β ⎤
⎣
⎦
Although this is just an empirical method to produce smooth
curve that properly goes between the small and large VD limits,
it works very well in practice, which suggests that it captures
something important about MOSFETs.
Lundstrom ECE 305 F15
outline
1)
2)
3)
4)
5)
6)
7)
Introduction
Linear region
Saturation region
Full range (VDS = 0 à VDS)
DIBL
Series resistance
VS model level 0
Lundstrom ECE 305 F15
output resistance
need to treat the finite
output resistance
ID
I D = −W Qn υ sat
Qn = −Cox (VGS − VT )
VT = VT 0 − δ VDS
VDSAT
⎛ mV ⎞
δ = DIBL ⎜
⎝ V ⎟⎠
Lundstrom ECE 305 F15
VDS
outline
1)
2)
3)
4)
5)
6)
7)
Introduction
Linear region
Saturation region
Full range (VDS = 0 à VDS)
DIBL
Series resistance
VS model level 0
Lundstrom ECE 305 F15
intrinsic vs. extrinsic voltages
VD
ID
RD
VG′ = VG
D
VG = VG′
VDS
′
G
VGS
′
VGS
silicon
S
VDS
VD′ = VD − I D (VG′ , VS′, VD′ ) RD
VS′ = VS + I D (VG′ , VS′, VD′ ) RS
RS
VS
Lundstrom ECE 305 F15
effect of series resistances
I D = W Cox υ sat (VGS′ − VT )
ID
VGS′ = VGS − I D RS
I D = VDS
′ Rch
I D = VDS
(R
ch
+ RS + RD )
VDS
Lundstrom ECE 305 F15
outline
1)
2)
3)
4)
5)
6)
7)
Introduction
Linear region
Saturation region
Full range (VDS = 0 à VDS)
DIBL
Series resistance
Simple VS model
Lundstrom ECE 305 F15
Simple VS model
1) I D W = −Qn (VGS′ ) υ (VDS′ )
2) Qn (VGS′ ) = − Cox (VGS′ − VT )
(VGS′ > VT )
VT = VT 0 − δ VDS
′
3) υ (VDS′ ) = FSAT (VDS′ )υ sat
4) FSAT (VDS′ ) =
5) VDSAT =
VDS
′ VDSAT
Cox ,VT , δ ,υ sat , µn , L
RSD = RS + RD
1/ β
β
⎡1+ (VDS
′ VDSAT ) ⎤⎦
⎣
υ sat L
µn
Lundstrom ECE 305 F15
outline
1)
2)
3)
4)
5)
6)
7)
8)
There are only 8 devicespecific parameters in
this model:
Introduction
Linear region
Saturation region
Full range (VDS = 0 à VDS)
DIBL
Series resistance
Simple VS model
Summary
Lundstrom ECE 305 F15
+ β
MOSFET IV: low VDS
I D = W Cox υ sat (VGS − VT )
VGS
gate-voltage
controlled resistor
ID =
W
µ C (V − VT )VDS
L n ox GS
Lundstrom ECE 305 F15
sub-micron MOSFETs
VGS
square law theory
(pinch-off)
2
W
I D = µ nCox (VGS − VT )
2L
velocity saturation theory
I D = W Cox υ sat (VGS − VT )
the MIT VS Model
ß 32 nm technology à
Lundstrom ECE 305 F15
the MIT VS Model
1
1
→
µn
µapp
“apparent mobility”
υ sat → υinj
“injection velocity”
https://nanohub.org/resources/21703