Lundstrom ECE 305 F16
ECE-305: Spring 2016
Gate Voltage
Professor Mark Lundstrom Electrical and Computer Engineering
Purdue University, West Lafayette, IN USA [email protected]
3/31/16
Pierret, Semiconductor Device Fundamentals (SDF) pp. 567-584
band banding in an MOS device
2 Fig. 16.6, Semiconductor Device Fundamentals, R.F. Pierret Flat band Accumulation Depletion Inversion
′VG = 0 ′VG < 0 0 < ′VG <VT ′VG > ′VTφS = 0 φS < 0 0 < φS < 2φF φS > 2φF
Lundstrom ECE 305 F16
depletion approximation
3
E x( )
x
What gate voltage produced this surface potential?
P
E S
W
W = 2κ Sε0φSqNA
cm
E S =
2qNAφSκ sε0
V/cm
QB φS( ) = − 2qκ sε0NAφS C/cm2
QB = −qNAW φS( )C/cm2
0 < φS < 2φF
Gate voltage and surface potential
4
1) Surface potential à gate voltage 2) Example
Lundstrom ECE 305 F16
Note: Next week (MOSFET IV), will follow for the most part Lundstrom “Lecture Notes on MOSFETs,” which is posted on the course home page under Week 12.
surface electric field and charge
5
E x( )
x
P E S
dEdx
= − qNA
KSε0
E 0( ) =E S =
qNAWKSε0
Lundstrom ECE 305 F16
E ox
KSε0E S = DS = qNAW( ) = −QB
DS = −QB
Dox = −QB
DS = Dox(no charge at interface)
surface electric field and charge
7
E x( )
x
P E S
Lundstrom ECE 305 F16
E ox
QB = −qNAW C/cm2
Dox = −QB DS = −QB
W
gate voltage and surface potential
8
EC
EV
Ei
EF
Si
metal
ΔVS
ΔVOX
EFM
′VG = ΔVOX +φS
0 < φS < 2φF
′VG = ?Given the surface potential, what is the gate voltage?
relation to gate voltage
9
x
E S
W−xo
metal
′VG = ΔVox +φS
′VG
E ox
ΔVox = xoE ox
E x( )
Dox = KOε0E ox = −QB
ΔVox = xoE ox = xo
−QB
KOε0
Volt drop across a capacitor
10
ΔVox
C ≡ QV
= KOε0Axo
= Fxo
+ + + + + +
- - - - - -
CA= KOε0
xo= F/cm2
Cox =KOε0xo
= F/cm2
Lundstrom ECE 305 F16
relation to gate voltage
11
E x( ) E S
W−xo
metal
′VG
E ox
x
′VG = −QB φS( )Cox
+φS
QB φS( ) = −qNAW φS( )
Cox = KOε0 xo
ΔVox = xoE ox = xo
−QB
KOε0
Lundstrom ECE 305 F16
12
E x( )
x
P
E S =
qNA
κ Sε0W
12E SW = φS
E S
W
W = 2κ Sε0φSqNA
cm
QB = − 2qκ sε0NAφS C/cm2
QB = −qNAW φS( )C/cm2
0 < φS < 2φF
′VG = −QB φS( )Cox
+φS
recap (depletion)
MOS electrostatics: inversion
13
EC
EV
Ei
EF
Si
φ x( ) φ 0( )
x
φFφS ≈ 2φF φF
WT
WT =2KSε0qNA
2φF⎡
⎣⎢
⎤
⎦⎥
1/2 Maximum depletion region depth
delta-depletion approximation
15
15
E x( )
x
P
WT
E S
E 0+( ) = − QB
KSε0
E 0( ) = − QS
KSε0
Lundstrom ECE 305 F16
MOS electrostatics: inversion
16
EC
EV
Ei
EF
Si
φ x( ) φ 0( )
x
φF
φS ≈ 2φF
φF
WT
WT =2KSε0qNA
2φF⎡
⎣⎢
⎤
⎦⎥
1/2
′VG = −QB 2φF( ) +Qn
Cox
+ 2φF
′VT = −QB 2φF( )Cox
+ 2φF
Qn = −Cox ′VG − ′VT( )
′VG = − QS
Cox
+ 2φF
Lundstrom ECE 305 F16
18
example
source drain
SiO
2
silicon
S G DAssume n+ poly Si gate with Φms = -1.0 V NA = 1018 cm-3 channel doping xo = 1.5 nm What is VT? Electric -field in oxide at VG = 1V?
Lundstrom ECE 305 F16
19
example (cont.)
′VG = −QS φS( )Cox
+ φS
′VT = −QB 2φF( )Cox
+ 2φF
VT = φms −QB 2φF( )Cox
+ 2φF
φF =kBTqln NA
ni
⎛⎝⎜
⎞⎠⎟
Cox = KOε0 xo
QB = − 2qκ sε0NA2φF
QB = −qNAW 2φF( )
φms = −1.00 V
φF = 0.48 V
Cox = 2.36 ×10−6 F/cm2
QB = −5.71×10−7 C/cm2
φms = −1.00 VVT = 0.20 V
20
example (cont): electric field in the oxide
Qn = −Cox VG −VT( )
E ox = − QS
Koε0= −
Qn +QB 2φF( )Koε0
Qn = −1.89 ×10−6 C/cm2
E ox = 7.1×106 V/cm
Qn
−q= 1.2 ×1013 /cm2
VG = 1V
VT = 0.2 V
Lundstrom ECE 305 F16
21
example (cont): volt drop in the oxide at VG = 1 V
ΔVox =E oxxo E ox = 7.1×106 V/cm ΔVox = 1.07V
ΔVS = φs φs = 2φF φF = 0.48 V φs = 0.96
VG = ΔVox + ΔVS = 1.07 + 0.96 = 2.03
VG = 1VLundstrom ECE 305 F16