Lundstrom ECE-606 S13
Notes for ECE-606: Spring 2013
MOS Caps and FETs
Professor Mark Lundstrom
Electrical and Computer Engineering Purdue University, West Lafayette, IN USA
1 4/11/13
2
outline
1) MOS Electrostatics 2) MOS C-V 3) MOSFETS
Lundstrom ECE-606 S13
MOS capacitor
3
VG
p-Si or n-Si
metal or
heavily doped “polysilicon”
SiO2
tox ≈ 1− 2 nm
Qn = −Cox VG −VT( ) C/cm2
Cox =κ oxε0 tox F/cm2
VG >VT :
Lundstrom ECE-606 S13
EC
EV
Ei
SiO2
EG ≈ 8.9eV
χi
e-band diagram
4
EC
EV
Ei
EF
EG = 1.12eV
Si
aluminum
EFM
ΦM = 4.08 eV
E0
χS = 4.05eVΦS
Lundstrom ECE-606 S13
equilibrium e-band diagram
5
EC
EV
Ei
EF
metal
Δφ
Δφox
Vbi = −φms
φ x( ) = 0 in the bulk
φ x = 0( ) = φS surface potential
Vbi = Δφox +φSLundstrom ECE-606 S13
equilibrium e-band diagram (“ideal”)
6
EC
EV
Ei
SiO2
EG ≈ 8eV
EC
EV
Ei
EF
EG = 1.12eV
Sihypothetical
metal
EFM
ΦM
E0
χS
χi
Lundstrom ECE-606 S13
“ideal” MOS structure
7
EC
EV
SiO2
EC
EV
Ei
EF
EG = 1.12eV
Sihypothetical
metal
EFVbi = 0
“flat band at VG’ = 0
Lundstrom ECE-606 S13
band banding in an MOS device
8 Fig. 16.6, Semiconductor Device Fundamentals, R.F. Pierret
accumulation, depletion, inversion
9
EC
EV
Ei
EF
Si
φ x( ) φ 0( ) E S
WD
x
qφF
φF =kBTqln NA
ni
⎛⎝⎜
⎞⎠⎟
φS
φS < 0
0 < φS < 2φF
φS > 2φF
accumulation:
depletion:
inversion:
Lundstrom ECE-606 S13
MOS electrostatics: depletion
10
EC
EV
Ei
EF
Si
φ x( ) φ 0( )
WD
x
φF
0 < φS < 2φF
Assume the depletion approximation for the charge in the semiconductor.
E S φS( )
φS
WD φS( )QS φS( ) = −qNAWD φS( )
Given the surface potential:
Lundstrom ECE-606 S13
final answers (semiconductor)
11
E x( )
x
What gate voltage produced this surface potential?
P
E S =
qNA
κ Sε0WD
12E SW = φS
E S
WD
WD = 2κ Sε0φSqNA
E S =
2qNAφSκ sε0
QB = − 2qκ sε0NAφS
QB = −qNAWD φS( )
0 <φS < 2φF
Lundstrom ECE-606 S13
relation to gate voltage
12
E x( )
x
E S
WD−tOX
metal
φSΔφOX ′VG =E oxtox + φS
′VG = φS −QB φS( )Cox
QB φS( ) = −qNAWD φS( )Cox =κ oxε0 tox
′VG
Lundstrom ECE-606 S13
MOS electrostatics: inversion
13
EC
EV
Ei
EF
Si
φ x( ) φ 0( )
x
φF
φS ≈ 2φF
φF
WT
WT =2KSε0qNA
2φF⎡
⎣⎢
⎤
⎦⎥
1/2
′VG = 2φF −QB 2φF( ) +Qn
Cox
′VT = 2φF −QB 2φF( )Cox
Qn = −Cox VG −VT( )
′VG = 2φF −QS
Cox
Lundstrom ECE-606 S13
delta-depletion approximation
14
ρ
x
metal P
−tox
WT
ρ = −qNA
QB = −qNAWT
Qn = −Cox VG −VT( )
WT = 2κ Sε0 2φFqNA
Lundstrom ECE-606 S13
delta-depletion approximation
15
15
E x( )
x
P
WD
E S
E 0+( ) = 2qNA2φF
κ sε0
E 0( ) = − QS
κ Sε0
Lundstrom ECE-606 S13
16
“exact” MOS electrostatics
QS φS( )C/cm2
φS
~ φS
~ eqφS /2kBT ~ e−qφS /2kBT
φS = 0
′VG = 0
EC = 0
EV = 0
EF
Lundstrom ECE-606 S13
17
outline
1) MOS Electrostatics (Exact) 2) MOS C-V 3) MOSFETS
Lundstrom ECE-606 S13
18
MOS electrostatics
φS
QS ψ S( )C/cm2
~ φS
~ eqφS /2kBT ~ e−qφS /2kBT
φS = 0
′VG = 0
EC = 0
EV = 0
EF
Lundstrom ECE-606 S13
19
space charge and sheet charge density
ρr( ) = q p r( ) − n r( ) + ND
+ r( ) − NA− r( )⎡⎣ ⎤⎦ C/cm3
∇ •D = ρ r( )
QT = ρ r( )∫∫∫ dxdydz C
QT = ρ x( )∫ dx A C
QS =QT
A= ρ x( )∫ dx C cm2
ρ r( ) C/cm3
QS C cm2QT = ρ x( )∫∫∫ dxdydz C (uniform in y-z plane)
Lundstrom ECE-606 S13
20
‘exact’ solution of QS(φS)
�
∇ • D = ρ
∇ • J n −q( ) = G − R( )
∇ • J p q( ) = G − R( )
d 2φdx2
= −qεSi
p0 φ(x){ }− n0 φ(x){ }+ ND+ − NA
−⎡⎣ ⎤⎦
equilibrium
Lundstrom ECE-606 S13
21
Poisson-Boltzmann equation
qφ(x)
φS > 0
n0 (x) = NCeEF −EC (x )[ ]/kBT
p0 (x) = NVeEV (x )−EF[ ]/kBT
EC (x) = EC (∞)− qφ(x)
n0 (x) = nBeqφ (x )/kBT
p0 (x) = pBe−qφ (x )/kBT
EC
EV
EF
′VG > 0
Lundstrom ECE-606 S13
22
Poisson equation
d 2φdx2
= −ρε
n0 (x) = nBeqφ (x )/kBTp0 (x) = pBe
−qφ (x )/kBT
d 2φdx2
= −qε
p0 (x)− n0 (x)+ ND+ − NA
−⎡⎣ ⎤⎦
d 2φdx2
= −qε
pBe−qφ (x )/kBT − nBe
qφ (x )/kBT + ND+ − NA
−⎡⎣ ⎤⎦
Lundstrom ECE-606 S13
23
Poisson-Boltzmann equation d 2φdx2 x→∞
= 0 = −qε
pB − nB + ND+ − NA
−⎡⎣ ⎤⎦
ND+ − NA
−( ) = − pB + nB
d 2φdx2
= −qε
pBe−qφ (x )/kBT − nBe
qφ (x )/kBT + ND+ − NA
−⎡⎣ ⎤⎦
d 2φdx2
= −qε
pBe−qφ (x )/kBT − nBe
qφ (x )/kBT − pB + nB⎡⎣ ⎤⎦
pB ≈ NA − ND nB = ni2 pB
Lundstrom ECE-606 S13
24
depletion
QS φS( ) = −qNAWD = 2qNAεSiφS
QS φS( ) =QB φS( )
Lundstrom ECE-606 S13
25
inversion
QS φS( ) =QB φS( ) +Qn φS( )
VG
φS > 0
inversion
φS > 2φFQB φS( ) = 2qNAεSiφS
Qn φS( ) = −qn(0) kBT / q
E S
= −qn(0)Winv
n(0) = nBeqφS /kBT
Winv =
kBT / qE S
EC
EV
EF
Lundstrom ECE-606 S13
26
strong inversion (above threshold)
QS φS( ) ≈Qn φS( )
E S φS( ) =Qn εSi
Qn φS( ) = −qn(0) kBT / q
E S
Qn φS( ) = − εSi kBT nBeqφS /2kBT
Lundstrom ECE-606 S13
27
strong inversion criterion
n 0( ) ≈ pB
nBeqφS /kBT = ni
2
NA
eqφS /kBT ≈ pB = NA
φS = 2φF = 2kBTqln NA
ni
⎛⎝⎜
⎞⎠⎟
Lundstrom ECE-606 S13
28
weak inversion (sub-threshold)
QS φS( ) ≈QB φS( )
VG
εS > 0
weak inversion
φS < 2ψ B
E S φS( ) = 2qNAεSiφS εSi
Qn φS( ) = −qn(0) kBT / q
E S
Qi φS( ) = −qnBeqφS /kBT kBT / q
2qεSiNAφS εSi
EC
EV
EF
Lundstrom ECE-606 S13
29
summary
QS φS( )C/cm2
�
ψS
QS ≈Qacc ~ e−qφS /2kBT
QS ≈QB ~ φS − kBT q( )
QS ≈Qn ~ eqφS /2kBT
Wacc/inv ≈ (kBT / q) E S Lundstrom ECE-606 S13
Lundstrom ECE-606 S13 30
assumptions
1) Boltzmann statistics (not valid above threshold)
2) Fully ionized dopants, uniform doping (not valid in practice)
3) No quantum confinement (not valid above threshold)
EC
EV
EF VG
quantum well
31
outline
1) MOS Electrostatics 2) MOS C-V 3) MOSFETS
Lundstrom ECE-606 S13
band banding in an MOS device
32 Fig. 16.6, Semiconductor Device Fundamentals, R.F. Pierret
33
capacitance
tox
W φS( )
κ ox
κ Si
CS =κ Sε0WD φS( )
Cox =κ oxε0tox
1C
=1Cox
+1CS
C =CSCox
CS + Cox
C =Cox
1+ Cox CS
C = Cox
1+κ oxWD φS( )
κ Stox
Lundstrom ECE-606 S13
34
s.s. gate capacitance vs. d.c. gate bias
C
VG′C = Cox
1+κ oxWD φS( )
κ Stox
accumulation
depletion
inversion
VT′
flat band
Cox
Lundstrom ECE-606 S13
35
capacitance vs. gate voltage
C
VG′C = Cox
1+κ oxWD φS( )
κ Stox
accumulation depletion
inversion
VT′
flat band
Cox
Lundstrom ECE-606 S13
36
high frequency vs. low frequency
C
VG′C = Cox
1+κ oxWD φS( )
κ Stox
accumulation depletion
inversion
VT′
flat band
Cox
high frequency
low frequency
Lundstrom ECE-606 S13
37
VT equation
′VG = φS −QS φS( )Cox
′VT = 2φF −QB 2φF( )Cox
VT =VFB + 2φF −QB 2φF( )Cox
VFB = φms −QF
Cox
′VG =VG −VFB
Lundstrom ECE-606 S13
38
outline
1) MOS Electrostatics 2) MOS C-V 3) MOSFETS
Lundstrom ECE-606 S13
39
nanoscale MOSFETs
source drain
SiO
2
silicon
channel ~ 32 nm
gate oxide EOT ~ 1.1 nm
gate electrode
S G D
Lundstrom ECE-606 S13
40
MOSFET IV characteristic
VGS
(Courtesy, Shuji Ikeda, ATDF, Dec. 2007)
S
D
G
circuit
symbol
VDS
VGS
IDS
gate-voltage controlled resistor
gate-voltage controlled
current source
Lundstrom ECE-606 S13
41
MOSFET IV: low VDS
VG>VT VD 0
ID = −WQn x( )υx (x)
�
ID =W Cox VGS −VT( )µeffE x
Qn x( ) ≈ −Cox VGS −VT( )
�
ID = WL
µeffCox VGS −VT( )VDS
VGS
gate-voltage controlled resistor
E x = −
VDSL
L
Lundstrom ECE-606 S13
42
MOSFET IV: “pinch-off” at high VDS
VG VD 0
Qn x( ) = −Cox VGS −VT −V (x)( )
�
V x( ) = VGS −VT( )
Qn x( ) ≈ 0
Lundstrom ECE-606 S13
43
MOSFET IV: high VDS
VG VD 0
ID = −W Cox VGS −VT( )µeffE x (0)
�
ID = WL
µeffCox VGS −VT( )22
E x (0) ≈ −
VGS −VTL
�
Qi x( ) = −Cox VGS −VT −V (x)( )�
V x( ) = VGS −VT( )
VGS
gate-voltage controlled
current source
ID = −WQn x( )υx (x) = −WQn 0( )υx (0)
Lundstrom ECE-606 S13
44
velocity saturation
electric field V/cm --->
velo
city
cm
/s --
->
107
104
�
υ = µE�
υ = υ sat
VDSL
≈ 1.0V30nm
≈ 3×105 V/cm
105
Lundstrom ECE-606 S13
45
MOSFET IV: velocity saturation
VG VD 0
ID = −WQn x( )υx (x)
�
ID =W Cox VGS −VT( )υ satID =WCoxυsat VGS −VT( )
E x >> 104
(Courtesy, Shuji Ikeda, ATDF, Dec. 2007)
Lundstrom ECE-606 S13
µm( )
Velo
city
(cm
/s) à
46
carrier transport nanoscale MOSFETs
υSAT
D. Frank, S. Laux, and M. Fischetti, Int. Electron Dev. Mtg., Dec., 1992.
µm( )
EC
quasi-ballistic
Lundstrom ECE-606 S13
47
outline
1) MOS Electrostatics 2) MOS C-V 3) MOSFETS
Lundstrom ECE-606 S13
MOS summary (p-Si)
48
WD = 2κ Sε0φSqNA
cm
E S =
2qNAφSκ sε0
V/cm
QB = − 2qκ sε0NAφS C/cm2
′VG = −QB φS( )Cox
+φS
COX =κ oxε0 tox F/cm2
0 < φS < 2φF φS > 2φF
WT =2κ Sε0 2φFqNA
cm
E S = ? V/cm
QB = − 2qκ sε0NA2φF C/cm2
′VT = −QB 2φF( )Cox
+ 2φF
Qn = −Cox VG −VT( )
φF = kBTqln NA
ni
⎛⎝⎜
⎞⎠⎟
Lundstrom ECE-606 S13