MOSFETs -...
Transcript of MOSFETs -...
ECE 663
MOSFETs
ECE 663
A little bit of history..
Substrate
Channel Drain
Insulator
Gate
Operation of a transistor
VSG > 0n type operation
Positive gate bias attracts electrons into channelChannel now becomes more conductive
Moreelectrons
Source
VSD
VSG
Substrate
Channel Drain
Insulator
Gate
Operation of a transistor
Transistor turns on at high gate voltageTransistor current saturates at high drain bias
Source
VSD
VSG
Substrate
Channel Drain
Insulator
Gate
Source
VSD
VSG
Start with a MOS capacitor
ECE 663
MIS Diode (MOS capacitor) – Ideal
W
Questions
What is the MOS capacitance? QS(yS)
What are the local conditions during inversion? yS,cr
How does the potential vary with position? y(x)
How much inversion charge is generated at the surface? Qinv(x,yS)
Add in the oxide: how does the voltage divide? yS(VG), yox(VG)
How much gate voltage do you need to invert the channel? VTH
How much inversion charge is generated by the gate? Qinv(VG)
What’s the overall C-V of the MOSFET? QS(VG)
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EC
EF
EV
Ei
Ideal MIS Diode n-type, Vappl=0
Assume Flat-band at equilibrium
qfS
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02
yff B
g
mmsq
E
Ideal MIS Diode n-type, Vappl=0
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Ideal MIS Diode p-type, Vappl=0
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Ideal MIS Diode p-type, Vappl=0
02
yff B
g
mmsq
E
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Accumulation
Pulling in majority carriers at surface
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But this increases the barrier for current flow !!
n+ p n+
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Depletion
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Need CB to dip below EF. Once below by yB, minority carrier density trumps the intrinsic density. Once below by 2yB, it trumps the major carrier density (doping) !
Inversion
yB
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Sometimes maths can help…
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P-type semiconductor Vappl0
Convention for p-type: y positive if bands bend down
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Ideal MIS diode – p-type
yyy enenenenn p
kTq
p
kTEqE
i
kTEE
ipFiFi
0
/
0
/)(/)( '
yy epepp p
kTq
pp 0
/
0
kT
q
CB moves towards EF if y > 0 n increases
VB moves away from EF if y > 0 p decreases
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Ideal MIS diode – p-type
At the semiconductor surface, y = ys
senn ps
y 0
sepp ps
y 0
• ys < 0 - accumulation of holes
• ys =0 - flat band
• yB>ys >0 – depletion of holes
• ys =yB - intrinsic concentration ns=ps=ni
• ys > yB – Inversion (more electrons than holes)
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Surface carrier concentration
senn ps
y 0
sepp ps
y 0
ECEF
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Want to find y, E-field, Capacitance
• Solve Poisson’s equation to get E field, potential based on charge density distribution(one dimension)
s
s
dx
d
dx
d
Ddx
dk
y
y
/
1//
2
2
0
)()( ppAD npNNqx
EE
E
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• Away from the surface, = 0
• and
00 ppAD pnNN
yy enepnp pppp 00
)1()1( 002
2
y
yy enepq
dx
dpp
s
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Solve Poisson’s equation:
)1()1( 002
2
y
yy enepq
dx
dpp
s
E = -dy/dx
d2y/dx2 = -dE/dx
= (dE/dy).(-dy/dx)
= EdE/dy
)1()1( 002
2
y
yy enepq
dx
dpp
s
EdE/dy
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• Do the integral:
• LHS:
• RHS:
• Get expression for E field (dy/dx):
dx
dx
xxdx
x y
0
2
2
x xx dxdxe
0 0
,
yy
yy 11
2 0
00
2
2 ep
ne
qp
q
kTE
p
p
s
p
field
Solve Poisson’s equation:
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2
1
0
0
0
011,
yy
y yy e
p
ne
p
nF
p
p
p
p
Define:
0
2
0 p
s
p
sD
qpqp
kTL Debye Length
Then:
y
0
0,
2
p
p
D
fieldp
nF
qL
kTE
+ for y > 0 and – for y < 0
y > 0
E > 0
y < 0
E < 0
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y
0
0,
2
p
p
s
D
Sssp
nF
qL
kTEQ
Use Gauss’ Law to find surface charge per unit area
2
1
0
011
2
yy
yy
s
p
p
s
D
ssS e
p
ne
qL
kTQ
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Accumulation to depletion to strong Inversion
• For negative y, first term in F dominates – exponential
• For small positive y, second term in F dominates - y
• As y gets larger, second exponential gets big10
0
y
p
p
p
en
yB = (kT/q)ln(NA/ni) = (1/)ln(pp0/√pp0np0)
(np0/pp0) = e-2yB
yS > 2yB
Questions
What is the MOS capacitance? QS(yS)
What are the local conditions during inversion? yS,cr
How does the potential vary with position? y(x)
How much inversion charge is generated at the surface? Qinv(x,yS)
Add in the oxide: how does the voltage divide? yS(VG), yox(VG)
How much gate voltage do you need to invert the channel? VTH
How much inversion charge is generated by the gate? Qinv(VG)
What’s the overall C-V of the MOSFET? QS(VG)
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Charges, fields, and potentials
• Charge on metal = induced surface charge in semiconductor
• No charge/current in insulator (ideal)
metal insul semiconductor
depletion
inversion
SAnM QWqNQQ
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Electric Field Electrostatic Potential
Charges, fields, and potentials
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Electric Field Electrostatic Potential
Depletion Region
yy
yy 11
2 0
00
2
2 ep
ne
qp
q
kTE
p
p
s
p
field
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Electric Field Electrostatic Potential
y = ys(1-x/W)2
Wmax = 2s(2yB)/qNA
yB = (kT/q)ln(NA/ni)
Depletion Region
Questions
What is the MOS capacitance? QS(yS)
What are the local conditions during inversion? yS,cr
How does the potential vary with position? y(x)
How much inversion charge is generated at the surface? Qinv(x,yS)
Add in the oxide: how does the voltage divide? yS(VG), yox(VG)
How much gate voltage do you need to invert the channel? VTH
How much inversion charge is generated by the gate? Qinv(VG)
What’s the overall C-V of the MOSFET? QS(VG)
Couldn’t we just solvethis exactly?
U = y
US = yS
UB = yB
Exact Solution
dy/dx = -(2kT/qLD)F(yB,np0/pp0)
dU/F(U) = x/LD
U
US
F(U) = [eUB(e-U-1+U)-e-UB (eU-1-U)]1/2
Exact Solution
dU’/F(U’,UB) = x/LD
U
US
F(U,UB) = [eUB(e-U-1+U) + e-UB (eU-1-U)]1/2
= qni[eUB(e-U-1) – e-UB(eU-1)]
Exact Solution
NA = 1.67 x 1015
Qinv ~ 1/(x+x0)a
x0 ~ LD . factor
Questions
What is the MOS capacitance? QS(yS)
What are the local conditions during inversion? yS,cr
How does the potential vary with position? y(x)
How much inversion charge is generated at the surface? Qinv(x,yS)
Add in the oxide: how does the voltage divide? yS(VG), yox(VG)
How much gate voltage do you need to invert the channel? VTH
How much inversion charge is generated by the gate? Qinv(VG)
What’s the overall C-V of the MOSFET? QS(VG)
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Threshold Voltage for Strong Inversion
• Total voltage across MOS structure= voltage across dielectric plus ys
B
i
SSiT
C
QVinversionstrongV yy 2)_(
)2(2)(2
)( max BAs
A
ssAAS qN
qN
invqNWqNSIQ y
y
B
i
BAsT
C
qNV y
y 2
)2(2
oxVi/tox = sys/(W/2) Before Inversion
After inversion there is a discontinuity in D due to surface Qinv
Vox (at threshold) = s(2yB)/(Wmax/2)Ci =
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Notice Boundary Condition !!
B
i
BAsT
C
qNV y
y 2
)2(2
Local Potential vs Gate voltage
VG = Vfb + ys + (kstox/kox)√(2kTNA/0ks)[ys + eys-2yB)]1/2
Initially, all voltage drops across channel (blue curve). Above threshold, channel potential stays pinned to 2yB, varying only logarithmically, so that most of the gate voltage drops across the oxide (red curve).
yoxys
Look at Effective charge width
Initially, a fast increasing channel potential drops across increasing depletion width
Eventually, a constant potential drops across a decreasing inversion layer width, so field keeps increasing and thus matches increasing field in oxide
~Wdm/2
~tinv
Questions
What is the MOS capacitance? QS(yS)
What are the local conditions during inversion? yS,cr
How does the potential vary with position? y(x)
How much inversion charge is generated at the surface? Qinv(x,yS)
Add in the oxide: how does the voltage divide? yS(VG), yox(VG)
How much gate voltage do you need to invert the channel? VTH
How much inversion charge is generated by the gate? Qinv(VG)
What’s the overall C-V of the MOSFET? QS(VG)
Charge vs Local Potential
Qs ≈ √(20kskTNA)[ys + eys-2yB)]1/2
Beyond threshold, all charge goes to inversion layer
How do we get the curvatures?
EXACTAdd other terms and keepLeading term
Inversion Charge vs Gate voltage Q ~ eys-2yB), ys
- 2yB ~ log(VG-VT)Exponent of a logarithm gives a linear variation of Qinv with VG
Qinv = -Cox(VG-VT)
Why Cox?
Questions
What is the MOS capacitance? QS(yS)
What are the local conditions during inversion? yS,cr
How does the potential vary with position? y(x)
How much inversion charge is generated at the surface? Qinv(x,yS)
Add in the oxide: how does the voltage divide? yS(VG), yox(VG)
How much gate voltage do you need to invert the channel? VTH
How much inversion charge is generated by the gate? Qinv(VG)
What’s the overall C-V of the MOSFET? QS(VG)
ECE 663
Capacitance
y
y
yy
0
0
0
0
,
11
2
p
p
S
p
p
D
SSD
p
nF
ep
ne
L
QC
ss
For ys=0 (Flat Band):
D
SD
LbandflatC
)_(
Expand exponentials….. ........!3!2
132
xx
xex
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Capacitance of whole structure
• Two capacitors in series:
Ci - insulator
CD - Depletion
Di CCC
111 OR
Di
Di
CC
CCC
dC i
i
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Capacitance vs Voltage
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Flat Band Capacitance
• Negative voltage = accumulation – C~Ci
• Zero voltage – Flat Band
i
D
s
i
si
Dis
D
siDiFB
LdLd
Ld
CCC
11111
FBCCV y 00
D
iFB
LdC
s
i
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CV
• As voltage is increased, C goes through minimum (weak inversion) where dy/dQ is fairly flat
• C will increase with onset of strong inversion
• Capacitance is an AC measurement
• Only increases when AC period long wrt minority carrier lifetime
• At “high” frequency, carriers can’t keep up – don’t see increased capacitance with voltage
• For Si MOS, “high” frequency = 10-100 Hz
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CV Curves – Ideal MOS Capacitor
max
'
minWd
C
s
i
i
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MOScap vs MOSFET
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Substrate
Drain
InsulatorGate
Source Channel
Substrate
InsulatorGate
Channel
Minority carriers generated byRG, over minority carrier lifetime~ 100ms
So Cinv can be << Cox if fast gateswitching (~ GHz)
Majority carriers pulled infrom contacts (fast !!)
Cinv = Cox
MOScap vs MOSFET
ECE 663
Example Metal-SiO2-Si
• NA = 1017/cm3
• At room temp kT/q = 0.026V
• ni = 9.65x109/cm3
• s = 11.9x1.85x10-14 F/cm
mcmW
Xx
xXxx
Nq
n
NkT
WA
i
As
m
1.010
10106.1
1065.910ln026.01085.89.11ln4
5
max
1719
91714
2max
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Example Metal-SiO2-Si
• d=50 nm thick oxide=10-5 cm
• i=3.9x8.85x10-14 F/cm
Voltsinvx
xxx
C
WqNV
C
C
cmFxx
xx
WdC
Voltsx
xxn
N
q
kTinv
cmFxxx
dC
sB
i
ATH
i
i
i
ABs
ii
s
i
07.184.023.0)(109.6
1010106.12
13.0
/101.9109.119.3105
1085.89.3
84.01065.9
10ln026.02ln
22)(
/109.610
1085.89.3
7
51719
max
'
min
28
57
14
max
'
min
9
17
27
5
14
yy
yy
ECE 663
Real MIS Diode: Metal(poly)-Si-SiO2 MOS
• Work functions of gate and semiconductor are NOT the same
• Oxides are not perfect– Trapped, interface, mobile charges
– Tunneling
• All of these will effect the CV characteristic and threshold voltage
ECE 663
Band bending due to work function difference
msFBV f
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Work Function Difference
• qfs=semiconductor work function = difference between vacuum and Fermi level
• qfm=metal work function
• qfms=(qfm- qfs)
• For Al, qfm=4.1 eV
• n+ polysilicon qfs=4.05 eV
• p+ polysilicon qfs=5.05 eV
• qfms varies over a wide range depending on doping
ECE 663
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SiO2-Si Interface Charges
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Standard nomenclature for Oxide charges:
QM=Mobile charges (Na+/K+) – can causeunstable threshold shifts – cleanlinesshas eliminated this issue
QOT=Oxide trapped charge – Can be anywherein the oxide layer. Caused by brokenSi-O bonds – caused by radiation damagee.g. alpha particles, plasma processes,hot carriers, EPROM
ECE 663
QF= Fixed oxide charge – positive charge layernear (~2mm) Caused by incompleteoxidation of Si atoms(dangling bonds)Does not change with applied voltage
QIT=Interface trapped charge. Similar in originto QF but at interface. Can be pos, neg,or neutral. Traps e- and h during deviceoperation. Density of QIT and QF usuallycorrelated-similar mechanisms. Cureis H anneal at the end of the process.
Oxide charges measured with C-V methods
ECE 663
Effect of Fixed Oxide Charges
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Surface Recombination
Lattice periodicity broken at surface/interface – mid-gap E levelsCarriers generated-recombined per unit area
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Interface Trapped Charge - QIT
• Surface states – R-G centers caused by disruption of lattice periodicity at surface
• Trap levels distributed in band gap, with Fermi-type distributed:
• Ionization and polarity will depend on applied voltage (above or below Fermi level
• Frequency dependent capacitance due to surface recombination lifetime compared with measurement frequency
• Effect is to distort CV curve depending on frequency
• Can be passivated w/H anneal – 1010/cm2 in Si/SiO2 system
kTEE
DD
D
DFegN
N/)(
1
1
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Effect of Interface trapped charge on C-V curve
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a – idealb – lateral shift – Q oxide, fms
c – distorted by QIT
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Non-Ideal MOS capacitor C-V curves
• Work function difference and oxide charges shift CV curve in voltage from ideal case
• CV shift changes threshold voltage
• Mobile ionic charges can change threshold voltage as a function of time – reliability problems
• Interface Trapped Charge distorts CV curve – frequency dependent capacitance
• Interface state density can be reduced by H annealing in Si-Si02
• Other gate insulator materials tend to have much higher interface state densities
ECE 663
All of the above….
• For the three types of oxide charges the CV curve is shifted by the voltage on the capacitor Q/C
• When work function differences and oxide charges are present, the flat band voltage shift is:
d
i
echoxideFB dxxxdC
V0
arg_ )(11
i
otmfmsFB
C
QQQV
f
Some important equations in the inversion regime (Depth direction)
VT = fms + 2yB + yox
Wdm = [2S(2yB)/qNA]
Qinv = Cox(VG - VT)
yox = Qs/Cox
Qs = qNAWdm
VT = fms + 2yB + ([4SyBqNA] - Qf + Qm + Qot)/Cox
Substrate
Channel Drain
Insulator
Gate
Source
x