Lundstrom ECE 305 F15
ECE-305: Spring 2015
Gate Voltage and
Capacitance vs. Voltage
Professor Mark Lundstrom Electrical and Computer Engineering
Purdue University, West Lafayette, IN USA [email protected]
4/3/15
Pierret, Semiconductor Device Fundamentals (SDF) pp. 567-584
1
Gate voltage and CV
2
1) Quick review of band bending 2) Surface potential gate voltage 3) Example 4) Capacitance vs. voltage
Lundstrom ECE 305 F15
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.
band banding in an MOS device
3 Fig. 16.6, Semiconductor Device Fundamentals, R.F. Pierret Flat band Accumulation Depletion Inversion
!VG = 0 !VG < 0 0 < !VG <VT !VG > !VT!S = 0 !S < 0 0 < !S < 2!F !S > 2!F
depletion, inversion, and accumulation
4
EC
EV
Ei
EF
Si
q! x( ) ! = 0
W
x
q!F
!F =kBTqln NA
ni
"#$
%&'
!S
!S < 0
0 < !S < 2!F
!S > 2!F
accumulation:
depletion:
inversion:
Lundstrom ECE 305 F15
depletion approximation
5
E x( )
x
What gate voltage produced this surface potential?
P
E S =
qNA
! S"0W
12E SW = !S
E S
W
W = 2! S"0#SqNA
cm
E S =
2qNA!S" s#0
V/cm
QB !S( ) = " 2q# s$0NA!S C/cm2
QB = !qNAW "S( )C/cm2
0 < !S < 2!F
surface electric field and charge
6
E x( )
x
P E S
dEdx
= ! qNA
KS"0
dE = ! qNA
KS"0dx
dE
E W( )
E 0( )
! = " qNA
KS#0dx
W
0
!
E 0( ) =E S =
qNAWKS!0
Lundstrom ECE 305 F15
E ox
KS!0E S = DS = qNAW( ) = "QB
DS = !QB
Dox = !QB
DS = Dox(no charge at interface)
surface electric field and charge
7 Lundstrom ECE 305 F15
QS > 0 C cm2
D =QS
x
surface electric field and charge
8
E x( )
x
P E S
Lundstrom ECE 305 F15
E ox
QB = !qNAW C/cm2
Dox = !QB DS = !QB
W
Gate voltage and CV
9
1) Quick review of band bending 2) Surface potential gate voltage 3) Example 4) Capacitance vs. voltage
Lundstrom ECE 305 F15
gate voltage and surface potential
10 10
EC
EV
Ei
EF
Si
metal
!VS
!VOX
EFM
!VG = "VOX +#S
0 < !S < 2!F
!VG = ?Given the surface potential, what is the gate voltage?
relation to gate voltage
11
x
E S
W!xo
metal
!VG = "Vox +#S
!VG
E ox
!Vox = xoE ox
E x( )
Dox = KO!0E ox = "QB
!Vox = xoE ox = xo
"QB
KO#0
Volt drop across a capacitor
12
!Vox
C ! QV
= KO"0Axo
= Fxo
+ + + + + +
- - - - - -
CA= KO!0
xo= F/cm2
Cox =KO!0xo
= F/cm2
relation to gate voltage
13
E x( ) E S
W!xo
metal
!VG
E ox
x
!VG = "QB #S( )Cox
+#S
QB !S( ) = "qNAW !S( )
Cox = KO!0 xo
!Vox = xoE ox = xo
"QB
KO#0
14
E x( )
x
P
E S =
qNA
! S"0W
12E SW = !S
E S
W
W = 2! S"0#SqNA
cm
E S =
2qNA!S" s#0
V/cm
QB = ! 2q" s#0NA$S C/cm2
QB = !qNAW "S( )C/cm2
0 < !S < 2!F
!VG = "QB #S( )Cox
+#S
recap (depletion)
MOS electrostatics: inversion
15
EC
EV
Ei
EF
Si
! x( ) ! 0( )
x
!F!S " 2!F !F
WT
WT =2KS!0qNA
2"F#
$%
&
'(
1/2 Maximum depletion region depth
delta-depletion approximation
16
!
x
metal
!xo
WT
! = "qNA
QB = !qNAWT
Qn
WT =2! S"0 2#FqNA
delta-depletion approximation
17
E x( )
x
P
WT
E S
E 0+( ) = ! QB
KS"0
E 0( ) = ! QS
KS"0
Lundstrom ECE 305 F15
MOS electrostatics: inversion
18
EC
EV
Ei
EF
Si
! x( ) ! 0( )
x
!F
!S " 2!F
!F
WT
WT =2KS!0qNA
2"F#
$%
&
'(
1/2
!VG = "QB 2#F( ) +Qn
Cox
+ 2#F
!VT = "QB 2#F( )Cox
+ 2#F
Qn = !Cox "VG ! "VT( )
!VG = " QS
Cox
+ 2#F
Lundstrom ECE 305 F15
Gate voltage and CV
19
1) Quick review of band bending 2) Surface potential gate voltage 3) Example 4) Capacitance vs. voltage
Lundstrom ECE 305 F15
20
example
source drain
SiO
2
silicon
S G DAssume n+ poly Si gate with Φms = -1.0 V
NA = 1018 cm-3 channel doping
xo = 1.5 nm
What is VT?
Electric -field in oxide at VG = 1V?
21
example (cont.)
!VG = "QS #S( )Cox
+ #S
!VT = "QB 2#F( )Cox
+ 2#F
VT = !ms "QB 2!F( )Cox
+ 2!F
!F =kBTqln NA
ni
"#$
%&'
Cox = KO!0 xo
QB = ! 2q" s#0NA2$F
QB = !qNAW 2"F( )
!ms = "1.00 V
!F = 0.48 V
Cox = 2.36 !10"6 F/cm2
QB = !5.71"10!7 C/cm2
!ms = "1.00 VVT = 0.20 V
22
example (cont): electric field in the oxide
Qn = !Cox VG !VT( )
E ox = ! QS
Ko"0= !
Qn +QB 2#F( )Ko"0
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
23
example (cont): volt drop in the oxide at VG = 1 V
!Vox =E oxxo E ox = 7.1!106 V/cm !Vox = 1.07V
!VS = "s !s = 2!F !F = 0.48 V !s = 0.96
VG = !Vox + !VS = 1.07 + 0.96 = 2.03
VG = 1V
Gate voltage and CV
24
1) Quick review of band bending 2) Surface potential gate voltage 3) Example 4) Capacitance vs. voltage
Lundstrom ECE 305 F15
MOS capacitor
25
p-Si
vS sin!t
+ VG
+
-
-
VG + vS sin!t
~
MOS capacitor in depletion
26
VG
p-Si
W !S( ) W VG( )
27
MOS capacitor in depletion
xo
W !S( )
KO
KS
C = ?
Gate
Undepleted P-type semiconductor
28
a simpler problem
xo
W !S( )
KO
KS CS =KS!0W "S( )
Cox =KO!0xo
1C
=1Cox
+1CS
C =CSCox
CS + Cox
C =Cox
1+ Cox CS
C = Cox
1+KOW !S( )KSxo
result
xo
W !S( )
! ox
! Si
C = Cox
1+KOW !S( )KSxo
VG
Cox
CS
!S
30
s.s. gate capacitance vs. d.c. gate bias
C
VG!
C = Cox
1+KOW !S( )KSxo
accumulation depletion
inversion
VT!
flat band
Cox
31
s.s. gate capacitance vs. d.c. gate bias
C
VG!C = Cox
1+KOW !S( )KSxo
accumulation
depletion
inversion
VT!
flat band
Cox
32
capacitance vs. gate voltage
C
VG!C = Cox
1+KOW !S( )KSxo
accumulation depletion
inversion
VT!
flat band
Cox
33
high frequency vs. low frequency
C
VG!C = Cox
1+KOW !S( )KSxo
accumulation depletion
inversion
VT!
flat band
Cox
high frequency
low frequency
34
high frequency vs. low frequency
p-Si
n+-Si n+-Si
MOS capacitor
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