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EE-612:
Lecture 9
MOSFET IV: Part 3
Mark Lundstrom
Electrical and Computer Engineering
Purdue University
West Lafayette, IN USA
Fall 2006
NCN
www.nanohub.org
Lundstrom EE-612 F06 1
outline
1) Quick review
2) Velocity saturation theory
3) Discussion
Lundstrom EE-612 F06 2
velocity saturation in bulk silicon
velocity cm/s --->
107 υ = υ sat
υ = μE
104
electric field V/cm --->
Lundstrom EE-612 F06 3
velocity saturation and MOSFETs
I D = W Qi (y )υ y (y)
υ y (y) = μeff E y (y) ?
VDD
Ey ~ << 10 4 V/cm OK for L > 1 micrometer
L
VDD
L >> 4
10
Lundstrom EE-612 F06 4
bulk charge theory of MOSFETs
W⎡ m 2 ⎤
I D = μ eff CG ⎢(VGS − VT )VDS − VDS ⎥
L ⎣ 2 ⎦ before
VGS > VT
channel pinch-off
VDS < (VGS − VT ) m
W (VGS − VT )
2
I D = μ eff CG
2 L′ m beyond
VGS > VT
channel pinch-off
VDS > (VGS − VT ) m
Lundstrom EE-612 F06 5
expected result
VDSAT = (VGS − VT ) / m
ID
VDSAT reduced IDSAT reduced
VDSAT VDS
Lundstrom EE-612 F06 6
outline
1) Brief review
2) Velocity saturation theory
3) Discussion
Lundstrom EE-612 F06 7
velocity vs. field characteristic (electrons)
− μE
υd =
2 1/ 2
⎡1 + ( E Ec ) ⎤
⎣ ⎦
velocity cm/s --->
107 υ = υ sat
− μE
υd =
1 + ( E Ec )
υ = μE
μ EC = υ sat
104
electric field V/cm --->
Lundstrom EE-612 F06 8
I-V derivation
0 VG ID VD I D = W Qi (y )υ y ( y )
− μ eff E
υ (y) =
y 1 + ( E Ec )
x
−WQi μ eff E y
ID =
1 + E y Ec
⎛ 1 dV ⎞ dV
ID ⎜1 +
⎝ ⎟ = −WQi μ eff dy
Ec dy ⎠
Lundstrom EE-612 F06 9
derivation (ii)
⎛ 1 ⎞
I D ⎜ dy + dV ⎟ = −WQi (V )μ eff dV
⎝ Ec ⎠
L VDS VDS
ID
∫I
0
D dy + ∫
0
Ec
dV = − ∫ WQ (V )μ
0
i eff dV
VDS
I D (L + VDS / Ec ) = − ∫ WQ (V )μi eff dV
0
exactly the same as for the bulk charge theory
Lundstrom EE-612 F06 10
derivation (iii)
W⎡ VDS ⎤
2
I D = Fv μ eff COX ⎢(VGS − VT )VDS − m
L ⎣ 2 ⎥⎦
(1)
1 1
Fv = =
(1 + VDS (
/ LEc ) 1 + μ eff VDS / υ sat L )
VDS / L = average electric field in the channel
when VDS / L > Ec then Fv < 1
(1) valid when:
VGS > VT VDS < ?
Lundstrom EE-612 F06 11
VDSAT
dI D
=0
dVDS
W⎡ VDS ⎤
2
I D = Fv μ eff COX ⎢(VGS − VT )VDS − m
L ⎣ 2 ⎥⎦
2 (VGS − VT ) / m (VGS − VT )
VDSAT = <
1 + 1 + 2 μ eff (VGS − VT ) mυ sat L m
eqn. (3.77) of Taur and Ning
Lundstrom EE-612 F06 12
IDSAT
1 + 2 μ eff (VGS − VT ) mυ sat L − 1
I DSAT = W CGυ sat (VGS − VT )
1 + 2 μ eff (VGS − VT ) mυ sat L + 1
eqn. (3.78) of Taur and Ning
Examine two limits:
i) L → ∞
ii) L → 0
Lundstrom EE-612 F06 13
L --> inf
2 (VGS − VT ) / m
VDSAT =
1 + 1 + 2 μ eff (VGS − VT ) mυ sat L
VDSAT →
(VGS − VT )
m
1 + 2 μ eff (VGS − VT ) mυ sat L − 1
I DSAT = W CGυ sat (VGS − VT )
1 + 2 μ eff (VGS − VT ) mυ sat L + 1
W (VGS − VT )
2
I DSAT → μ eff CG
2L m
Lundstrom EE-612 F06 14
L --> 0
2 (VGS − VT ) / m
VDSAT =
1 + 1 + 2 μ eff (VGS − VT ) mυ sat L
VDSAT → 2υ sat L (VGS − VT ) m μ eff
1 + 2 μ eff (VGS − VT ) mυ sat L − 1
I DSAT = W CGυ sat (VGS − VT )
1 + 2 μ eff (VGS − VT ) mυ sat L + 1
I DSAT = W CGυ sat (VGS − VT )
“complete velocity saturation”
current independent of L
Lundstrom EE-612 F06 15
near threshold
2 (VGS − VT ) / m 2 μ eff (VGS − VT )
VDSAT = << 1
1 + 1 + 2 μ eff (VGS − VT ) mυ sat L mυ sat L
VDSAT → (VGS − VT ) m
1 + 2 μ eff (VGS − VT ) mυ sat L − 1
I DSAT = W CGυ sat (VGS − VT )
1 + 2 μ eff (VGS − VT ) mυ sat L + 1
W (VGS − VT )
2
near threshold is
I DSAT → μ eff CG
2L m like long channel
Lundstrom EE-612 F06 16
near threshold
0 VG VD 2 μ eff (VGS − VT )
<< 1
mυ sat L
(VGS − VT ) m < Ec
V (x ) = (VGS − VT ) / m
L 2
y
x
Lundstrom EE-612 F06 17
‘signature’ of velocity saturation
ID ID
VGS VGS
VDS VDS
ID =
W
μeff Cox
(VGS − VT )
2
I D = W υ sat Cox (VGS − VT )
2L m
Lundstrom EE-612 F06 18
ID and (VGS - VT)
I D (VDS = VDD ) ~ (VGS − VT )
α
ID
1<α < 2
VGS
complete long channel
velocity
VDS saturation
Lundstrom EE-612 F06 19
outline
1) Brief review
2) Velocity saturation theory
3) Discussion
Lundstrom EE-612 F06 20
what happens near the drain?
Qi (y) = −CG [VG − VT − mV (y)]
Qi (y = L) = −CG [VG − VT − mVDSAT ]
2 (VGS − VT ) / m
VDSAT =
1 + 1 + 2 μ eff (VGS − VT ) mυ sat L
1 + 2 μ eff (VGS − VT ) mυ sat L − 1
Qi (y = L) = −CG (VGS − VT )
1 + 2 μ eff (VGS − VT ) mυ sat L + 1
Qi (y = L) > 0
Lundstrom EE-612 F06 21
what happens near the drain?
1 + 2 μ eff (VGS − VT ) mυ sat L − 1
Qi (y = L) = −CG (VGS − VT )
1 + 2 μ eff (VGS − VT ) mυ sat L + 1
1 + 2 μ eff (VGS − VT ) mυ sat L − 1
I DSAT = W CGυ sat (VGS − VT )
1 + 2 μ eff (VGS − VT ) mυ sat L + 1
I DSAT = W υ sat Qi (y = L)
Lundstrom EE-612 F06 22
what happens when L--> 0?
Qi ( y = L ) > 0
0 VG VD
L→0
Qi (y = L) → Qi (y = 0) = CG (VGS − VT )
V (x ) = (VGS − VT ) / m
y
x
Lundstrom EE-612 F06 23
velocity overshoot
103 V/cm 105 V/cm 103 V/cm
7
2.0 10 0.35
Kinetic energy per electron (eV)
Average velocity (cm/s)
0.3
7
1.5 10 0.25
υ sat 0.2
7
1.0 10
0.15
6 0.1
5.0 10
0.05
0
0.0 10 0
0 0.5 1 1.5
Position (μm)
υ ≠ μ n (E )E
Lundstrom EE-612 F06 24
velocity and transconductance
I DSAT = W CGυ sat (VGS − VT )
dI DSAT
gm ≡ = W CGυ sat
dVG
Lundstrom EE-612 F06 25
velocity overshoot in a MOSFET
Lundstrom EE-612 F06 26
outline
1) Brief review
2) Velocity saturation theory
3) Discussion
Lundstrom EE-612 F06 27
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