Download - Lec17 Mosfet IV
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ECE 663
MOSFET I-Vs
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Substrate
Channel Drain
InsulatorGate
Operation of a transistorVSG > 0 n type operation
Positive gate bias attracts electrons into channelChannel now becomes more conductive
More electrons
Source
VSD
VSG
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Some important equations in the inversion regime (Depth direction)
VT = ms + 2B + ox
Wdm = [2S(2B)/qNA]
Qinv = -Cox(VG - VT)
ox = Qs/Cox
Qs = qNAWdm
VT = ms + 2B + [4SBqNA]/Cox
Substrate
Channel Drain
InsulatorGate
Source
x
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ECE 663
MOSFET Geometry
x
y
z
L
Z
S D
VG
VD
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ECE 663
How to include y-dependent potential without doing the whole problem over?
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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 inx and y
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ECE 663
How to include y-dependent potentials?
S = 2B + V(y)
VG = S + [2SSqNA]/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 todepletion first (Pinch off)
SATURATION [VDSAT = VG – VT]
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j = qninvv = (Qinv/tinv)v
I = jA = jZtinv = ZQinvv
ECE 663
So current:
Qinv = -Cox[VG – VT - V(y)]
v = -effdV(y)/dy
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ECE 663
So current:
I = eff ZCox[VG – VT - V(y)]dV(y)/dy
I = eff ZCox[(VG – VT )VD- VD2/2]/L
Continuity implies ∫Idy = IL
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ECE 663
But this current behaves like a parabola !!
ID
VD
IDsat
VDsat
I = eff ZCox[(VG – VT )VD- VD2/2]/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 = (CoxeffZ/2L)(VG-VT)2
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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
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Square law theory of MOSFETs
I = eff ZCox[(VG – VT )VD- VD2/2]/L, VD < VG - VT
I = eff ZCox(VG – VT )2/2L, VD > VG - VT
J = qnvn ~ Cox(VG – VT )v ~ effVD /L
NEW
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ECE 663
Ideal Characteristics of n-channel enhancement mode MOSFET
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ECE 663
Drain current for REALLY small VD
TGD
DTGinD
DDTGinD
VVV
VVVCLZI
VVVVCLZI
2
21
Linear operation
Channel Conductance:
)( TGinVD
DD VVC
LZ
VIg
G
Transconductance:
DinVG
Dm VC
LZ
VIg
D
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ECE 663
In Saturation
• Channel Conductance:
• Transconductance:
2
2 TGinD VVCLZsatI
0
GVD
DD V
Ig
TGinVG
Dm VVC
LZ
VIg
D
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ECE 663
Equivalent Circuit – Low Frequency AC
• Gate looks like open circuit• S-D output stage looks like current source with channel
conductance
gmdD
GVG
DD
VD
DD
vgvgi
VVIV
VII
DG
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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
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ECE 663
• Input circuit:
• Input capacitance is mainly gate capacitance
• Output circuit:
ggateggdgsin vfCjvCCji 2
gmout vgi
gate
m
in
out
fCg
ii
2
DinVG
Dm VC
LZ
VIg
D
Equivalent Circuit – Higher Frequency AC
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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
LV
ZLC
CVLZ
f
Cgf
Dn
i
iDn
gate
m
ZLCC igate
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ECE 663
Maximum Frequency (not in saturation)
2max 2 L
Vf Dn
LVv
vL
D /
/1
max
(Inverse transit time)
NEW
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ECE 663
Switching Speed, Power Dissipation
ton = CoxZLVD/ION
Trade-off: If Cox too small, Cs and Cd take over and you losecontrol of the channel potential (e.g. saturation)
(DRAIN-INDUCED BARRIER LOWERING/DIBL)
If Cox increases, you want to make sure you don’t controlimmobile charges (parasitics) which do not contribute tocurrent.
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ECE 663
Switching Speed, Power Dissipation
Pdyn = ½ CoxZLVD2f
Pst = IoffVD
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ECE 663
CMOS
NOT gate (inverter)
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ECE 663
CMOS
NOT gate (inverter)
Positive gate turns nMOS on
Vin = 1 Vout = 0
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ECE 663
CMOS
NOT gate (inverter)
Negative gate turns pMOS on
Vin = 0 Vout = 1
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ECE 663
So what?
• If we can create a NOT gate we can create other gates (e.g. NAND, EXOR)
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ECE 663
So what?
Ring Oscillator
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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
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ECE 663
Getting the inverter output
Gain
ON
OFF
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ECE 663
0
GVD
DD V
Ig
TGinVG
Dm VVC
LZ
VIg
D
What’s the gain here?
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ECE 663
Signal Restoration
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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 = (qni2Dn/WBND)exp(qVBE/kT) ID = CoxW(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
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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
BAsB
i
fms
i
BAsBFBT
CqN
CQ
CqN
VV
)2(22
)2(22
i
BSBAsBFBT C
VqNVV
)2(22
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ECE 663
Threshold Voltage Control
• Substrate Bias:
i
BSBAsBFBT C
VqNVV
)2(22
BBSBi
AsT
BSTBSTT
VCqN
V
VVVVV
222
)0()(
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ECE 663
Threshold Voltage Control-substrate bias
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ECE 663
It also affects the I-VVG
The threshold voltage is increased due to the depletion regionthat grows at the drain end because the inversion layer shrinksthere and can’t screen it any more. (Wd > Wdm)
Qinv = -Cox[VG-VT(y)], I = -effZQinvdV(y)/dy
VT(y) = + √2sqNA/Cox = 2B + V(y)
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ECE 663
It also affects the I-V
IL = ∫effZCox[VG – (2B+V) - √2sqNA(2B+V)/Cox]dV
I = (ZeffCox/L)[(VG–2B)VD –VD2/2
-2√2sqNA{(2B+VD)3/2-(2B)3/2}/3Cox]
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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 = s/Wdm
Q = -Cox[VG –VT - MV(y)], M = 1 + Cd/Cox
ID = (ZeffCox/L)[(VG-VT - MVD/2)VD]
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ECE 663
Comparison between different models
Square Law Theory
Body Coefficient
Bulk Charge Theory
Still not good below threshold or above saturation
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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
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ECE 663
Mobility dependence on gate voltage
)(10
TG VV
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ECE 663
Sub-Threshold Behavior
• For gate voltage less than the threshold – weak inversion
• Diffusion is dominant current mechanism (not drift)
LLnonqAD
ynqADAJI nnDD
)()(
kTVqi
kTqi
DBs
Bs
enLn
enn
/)(
/)(
)(
)0(
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ECE 663
Sub-threshold
kTqkTqVkT
inD
sD
B
eeLenqADI //
/
1
We can approximate s with VG-VT below threshold since all voltage drops across depletion region
kTVVqkTqVkT
inD
TGD
B
eeLenqADI //
/
1
•Sub-threshold current is exponential function of applied gate voltage•Sub-threshold current gets larger for smaller gates (L)
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ECE 663
Subthreshold Characteristic
GD VIS
log1
Subthreshold Swing
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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 !
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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 switchingLi, Ghosh, Stan
• Impact IonizationPlummer
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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
D s
B
V
p
p
V
D
nsD
pn
VF
eLL
ZI0
0
0,,
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ECE 663
Exact Charge Model (Pao-Sah)– Long Channel MOSFET
http://www.nsti.org/Nanotech2006/WCM2006/WCM2006-BJie.pdf
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ECE 663