# EE-612 Lecture 9 MOSFET IV Part 3 Mark Lundst

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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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