This document discusses MOSFET device physics and modeling. It begins with an overview of MOSFET operation and important equations. It then discusses modeling the current-voltage characteristics using gradual channel approximation. The document also covers subthreshold behavior, mobility effects, threshold voltage control, and more complete models that include both drift and diffusion currents.

1. ECE 663
MOSFET I-Vs

2. Substrate
Channel Drain
Insulator
Gate
Operation of a transistor
VSG > 0
n type operation
Positive gate bias attracts electrons into channel
Channel now becomes more conductive
More
electrons
Source
VSD
VSG

3. Some important equations in the
inversion regime (Depth direction)
VT = φms + 2ψB + ψox
Wdm = √[2εS(2ψB)/qNA]
Qinv = -Cox(VG - VT)
ψox = Qs/Cox
Qs = qNAWdm
VT = φms + 2ψB + √[4εSψBqNA]/Cox
Substrate
Channel Drain
Insulator
Gate
Source
x

4. ECE 663
MOSFET Geometry
x
y
z
L
Z
S D
VG
VD

5. ECE 663
How to include y-dependent potential
without doing the whole problem over?

6. ECE 663
Assume potential V(y) varies slowly along
channel, so the x-dependent and y-dependent
electrostats are independent
(GRADUAL CHANNEL APPROXIMATION)
i.e.,
Ignore ∂Ex/∂y
Potential is separable in
x and y

7. ECE 663
How to include y-dependent potentials?
ψS = 2ψB + V(y)
VG = ψS + √[2εSψSqNA]/Cox
Need VG – V(y) > VT to invert
channel at y (V increases
threshold)
Since V(y) largest at drain end, that
end reverts from inversion to
depletion first (Pinch off) 
SATURATION [VDSAT = VG – VT]

8. j = qninvv = (Qinv/tinv)v
I = jA = jZtinv = ZQinvv
ECE 663
So current:
Qinv = -Cox[VG – VT - V(y)]
v = -µeffdV(y)/dy

9. ECE 663
So current:
I = µeff ZCox[VG – VT - V(y)]dV(y)/dy
I = µeff ZCox[(VG – VT )VD- VD
2
/2]/L
Continuity implies ∫Idy = IL

10. ECE 663
But this current behaves like a parabola !!
ID
VD
IDsat
VDsat
I = µeff ZCox[(VG – VT )VD- VD
2
/2]/L
We have assumed inversion in our model (ie, always above pinch-off)
So we just extend the maximum current into saturation…
Easy to check that above current is maximum for VDsat = VG - VT
Substituting, IDsat = (CoxµeffZ/2L)(VG-VT)2

11. What’s Pinch off?
0
0 0
0
VG VG
Now add in the drain voltage to drive a current. Initially you get
an increasing current with increasing drain bias
0 VD
VG VG
When you reach VDsat = VG – VT, inversion is disabled at the drain
end (pinch-off), but the source end is still inverted
The charges still flow, just that you can’t draw more current
with higher drain bias, and the current saturates

12. Square law theory of MOSFETs
I = µeff ZCox[(VG – VT )VD- VD
2
/2]/L, VD < VG - VT
I = µeff ZCox(VG – VT )2
/2L, VD > VG - VT
J = qnv
n ~ Cox(VG – VT )
v ~ µeffVD /L
NEW

13. ECE 663
Ideal Characteristics of n-channel
enhancement mode MOSFET

14. ECE 663
Drain current for REALLY small VD
( )
( )[ ]
( )TGD
DTGinD
DDTGinD
VVV
VVVC
L
Z
I
VVVVC
L
Z
I
−<<
−≈




−−=
µ
µ
2
2
1
Linear operation
Channel Conductance:
)( TGin
VD
D
D VVC
L
Z
V
I
g
G
−µ=
∂
∂
≡
Transconductance:
Din
VG
D
m VC
L
Z
V
I
g
D
µ=
∂
∂
≡

15. ECE 663
In Saturation
• Channel Conductance:
• Transconductance:
( )2
2
TGinD VVC
L
Z
satI −µ=
0=
∂
∂
≡
GVD
D
D
V
I
g
( )TGin
VG
D
m VVC
L
Z
V
I
g
D
−µ=
∂
∂
≡

16. ECE 663
Equivalent Circuit – Low Frequency AC
• Gate looks like open circuit
• S-D output stage looks like current source with channel
conductance
gmdD
G
VG
D
D
VD
D
D
vgvgi
V
V
I
V
V
I
I
DG
+=
δ
∂
∂
+δ
∂
∂
=δ

17. ECE 663
• Input stage looks like capacitances gate-to-source(gate) and
gate-to-drain(overlap)
• Output capacitances ignored -drain-to-source capacitance
small
Equivalent Circuit – Higher Frequency AC

18. ECE 663
• Input circuit:
• Input capacitance is mainly gate capacitance
• Output circuit:
( ) ggateggdgsin vfCjvCCji π≈+ω= 2
gmout vgi ≈
gate
m
in
out
fC
g
i
i
π
=
2
Din
VG
D
m VC
L
Z
V
I
g
D
µ=
∂
∂
≡
Equivalent Circuit – Higher Frequency AC

19. ECE 663
Maximum Frequency (not in saturation)
• Ci is capacitance per unit area and Cgate is total capacitance of
the gate
• F=fmax when gain=1 (iout/iin=1)
2max
max
22
2
L
V
ZLC
CV
L
Z
f
C
g
f
Dn
i
iDn
gate
m
π
µ
=
π
µ
=
π
=
ZLCC igate =

20. ECE 663
Maximum Frequency (not in saturation)
2max 2 L
V
f Dn
π
µ
=
LVv
vL
D /
/
1
max
µ
ω
=
=
(Inverse transit time)
NEW

21. ECE 663
Switching Speed, Power Dissipation
ton = CoxZLVD/ION
Trade-off: If Cox too small, Cs and Cd take over and you lose
control of the channel potential (e.g. saturation)
(DRAIN-INDUCED BARRIER LOWERING/DIBL)
If Cox increases, you want to make sure you don’t control
immobile charges (parasitics) which do not contribute to
current.

22. ECE 663
Switching Speed, Power Dissipation
Pdyn = ½ CoxZLVD
2
f
Pst = IoffVD

23. ECE 663
CMOS
NOT gate
(inverter)

24. ECE 663
CMOS
NOT gate
(inverter)
Positive gate turns nMOS on
Vin = 1 Vout = 0

25. ECE 663
CMOS
NOT gate
(inverter)
Negative gate turns pMOS on
Vin = 0 Vout = 1

26. ECE 663
So what?
• If we can create a NOT gate
we can create other gates
(e.g. NAND, EXOR)

27. ECE 663
So what?
Ring Oscillator

28. ECE 663
So what?
• More importantly, since one is open and one is shut at steady
state, no current except during turn-on/turn-off
 Low power dissipation

29. ECE 663
Getting the inverter output
Gain
ON
OFF

30. ECE 663
0=
∂
∂
≡
GVD
D
D
V
I
g
( )TGin
VG
D
m VVC
L
Z
V
I
g
D
−µ=
∂
∂
≡
What’s the gain here?

31. ECE 663
Signal Restoration

32. ECE 663
BJT vs MOSFET
• RTL logic vs CMOS logic
• DC Input impedance of MOSFET (at gate end) is infinite
Thus, current output can drive many inputs  FANOUT
• CMOS static dissipation is low!! ~ IOFFVDD
• Normally BJTs have higher transconductance/current (faster!)
IC = (qni
2
Dn/WBND)exp(qVBE/kT) ID = µCoxW(VG-VT) 2
/L
gm = ∂IC/∂VBE = IC/(kT/q) gm = ∂ID/∂VG = ID/[(VG-VT)/2]
• Today’s MOSFET ID >> IC due to near ballistic operation
NEW

33. ECE 663
What if it isn’t ideal?
• If work function differences and oxide charges are present,
threshold voltage is shifted just like for MOS capacitor:
• If the substrate is biased wrt the Source (VBS) the threshold
voltage is also shifted
i
BAs
B
i
f
ms
i
BAs
BFBT
C
qN
C
Q
C
qN
VV
)2(2
2
)2(2
2
ψε
+ψ+





−φ=
ψε
+ψ+=
i
BSBAs
BFBT
C
VqN
VV
)2(2
2
+ψε
+ψ+=

34. ECE 663
Threshold Voltage Control
• Substrate Bias:
i
BSBAs
BFBT
C
VqN
VV
)2(2
2
+ψε
+ψ+=
( )BBSB
i
As
T
BSTBSTT
V
C
qN
V
VVVVV
ψ−+ψ
ε
=∆
=−=∆
22
2
)0()(

35. ECE 663
Threshold Voltage Control-substrate bias

36. ECE 663
It also affects the I-V
VG
The threshold voltage is increased due to the depletion region
that grows at the drain end because the inversion layer shrinks
there and can’t screen it any more. (Wd > Wdm)
Qinv = -Cox[VG-VT(y)], I = -µeffZQinvdV(y)/dy
VT(y) = ψ + √2εsqNAψ/Cox
ψ = 2ψB + V(y)

37. ECE 663
It also affects the I-V
IL = ∫µeffZCox[VG – (2ψB+V) - √2εsqNA(2ψB+V)/Cox]dV
I = (ZµeffCox/L)[(VG–2ψB)VD –VD
2
/2
-2√2εsqNA{(2ψB+VD)3/2
-(2ψB)3/2
}/3Cox]

38. ECE 663
We can approximately include this…
Include an additional charge term from the
depletion layer capacitance controlling V(y)
Q = -Cox[VG-VT]+(Cox + Cd)V(y)
where Cd = εs/Wdm
Q = -Cox[VG –VT - MV(y)], M = 1 + Cd/Cox
ID = (ZµeffCox/L)[(VG-VT - MVD/2)VD]

39. ECE 663
Comparison between different models
Square Law Theory
Body Coefficient
Bulk Charge Theory
Still not good below threshold or above saturation

40. ECE 663
Mobility
• Drain current model assumed constant mobility in channel
• Mobility of channel less than bulk – surface scattering
• Mobility depends on gate voltage – carriers in inversion
channel are attracted to gate – increased surface scattering
– reduced mobility

41. ECE 663
Mobility dependence on gate voltage
)(1
0
TG VV −θ+
µ
=µ

42. ECE 663
Sub-Threshold Behavior
• For gate voltage less than the threshold – weak inversion
• Diffusion is dominant current mechanism (not drift)
L
Lnon
qAD
y
n
qADAJI nnDD
)()( −
−≈
∂
∂
−==
kTVq
i
kTq
i
DBs
Bs
enLn
enn
/)(
/)(
)(
)0(
−ψ−ψ
ψ−ψ
=
=

43. ECE 663
Sub-threshold
( ) kTqkTqV
kT
in
D
sD
B
ee
L
enqAD
I //
/
1 ψ−
ψ−
−=
We can approximate ψs with VG-VT below threshold since all
voltage drops across depletion region
( ) ( ) kTVVqkTqV
kT
in
D
TGD
B
ee
L
enqAD
I //
/
1 −−
ψ−
−≈
•Sub-threshold current is exponential function of applied gate voltage
•Sub-threshold current gets larger for smaller gates (L)

44. ECE 663
Subthreshold Characteristic
( )( )GD VI
S
∂∂
≡
log
1
Subthreshold Swing

45. Tunneling transistor
– Band filter like operation
J Appenzeller et al, PRL ‘04
Ghosh, Rakshit, Datta
(Nanoletters, 2004)
(Sconf)min=2.3(kBT/e).(etox/m)
Hodgkin and Huxley, J. Physiol. 116, 449 (1952a)
Subthreshold slope = (60/Z) mV/decade
Much of new research depends on reducing S !

46. Much of new research depends on reducing S !
• Increase ‘q’ by collective motion (e.g. relay)
Ghosh, Rakshit, Datta, NL ‘03
• Effectively reduce N through interactions
Salahuddin, Datta
• Negative capacitance
Salahuddin, Datta
• Non-thermionic switching (T-independent)
Appenzeller et al, PRL
• Nonequilibrium switching
Li, Ghosh, Stan
• Impact Ionization
Plummer

47. ECE 663
More complete model – sub-threshold to
saturation
• Must include diffusion and drift currents
• Still use gradual channel approximation
• Yields sub-threshold and saturation behavior for long
channel MOSFETS
• Exact Charge Model – numerical integration
∫ ∫








βψ
µε
=
ψ
ψ
β−βψD s
B
V
p
p
V
D
ns
D
p
n
VF
e
LL
Z
I
0
0
0
,,

48. ECE 663
Exact Charge Model (Pao-Sah)
– Long Channel MOSFET
http://www.nsti.org/Nanotech2006/WCM2006/WCM2006-BJie.pdf

49. ECE 663