Bermel ECE 305 F16
ECE-305: Fall 2016
MOS Capacitors: Band Bending and Capacitance in the
Depletion Approximation
Professor Peter BermelElectrical and Computer Engineering
Purdue University, West Lafayette, IN [email protected]
Pierret, Semiconductor Device Fundamentals (SDF)Chapter 16 (pp. 567-584)
111/2/2016
band banding in p-type MOS
2Fig. 16.6, Semiconductor Device Fundamentals, R.F. Pierret
Flat band Accumulation Depletion Inversion
¢VG = 0 ¢VG < 0 0 < ¢VG <VT ¢VG > ¢VTfS = 0 fS < 0 0 <fS < 2fF fS > 2fF
MOS electrostatics: depletion
3
EC
EV
Ei
EF
Si
f x( ) f 0( )
W
x
fF
0 < fS < 2fF
Depletion approximation for the charge in the semiconductor.
E S fS( )
fS
W fS( )
QS fS( ) = -qNAW fS( )
Given the surface potential:
What gate voltage produced this surface potential?
space charge density vs. position
4
x
0
-xox
-qNA
r x( )
W
depletion charge
E = 0
dE
dx=
r
KSe0= -
qNAKSe0
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electric field (semiconductor)
5
E x( )
x
P
1)
1
2E SW = fS
E S
W
E S = ?
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surface electric field (semiconductor)
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E x( )
x
P
E S
W
dE
dx= -
qNAKSe0
dE = -qNAKSe0
dx
dEE W( )
E 0( )
ò = -qNA
KSe0dx
W
0
ò
1)1
2E SW = fS
2) E S =qNAW
KSe0Bermel ECE 305 F16
1)1
2E SW = fS
2) E S =qNAW
KSe0
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E x( )
x
P
E S =qNA
k Se0W
1
2E SW = fS
E S
W
W =2k Se0fSqNA
cm
E S =2qNAfSk se0
V/cm
QB = - 2qk se0NAfS C/cm2
QB = -qNAW fS( )C/cm2
0 <fS < 2fF
¢VG = -QB fS( )Cox
+fS
final answers (semiconductor)
What gate voltage produced this surface
potential?
gate voltage and surface potential
88
EC
EV
Ei
EF
Si
metal
DVS
DVOX
EFM
¢VG = DVOX +fS
0 < fS < 2fF
¢VG = ?
Given the surface potential, what is the gate voltage?
11/2/2016 Bermel ECE 305 F16
DVox = xoE ox
voltage drop across a capacitor
9
metal
V
C ºQ
V=KOe0A
xo= F
metalxo
C
AºQ A
V=KOe0xo
= Cox F/cm2
+V
++++++
------
Q A C/cm2
Cox = -Q A
V
V = -Q A
Cox
Bermel ECE 305 F1611/2/2016
relation to gate voltage
10
E x( )
E S
W-xo
metal
¢VG = DVox +fS
¢VG
E ox
DVox =-QB fS( )COX
x
¢VG = -QB fS( )Cox
+fS
QB fS( ) = -qNAW fS( )
Cox = KOe0 xo
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What is the max value of W?
MOS electrostatics: inversion
11
EC
EV
Ei
EF
Si
f x( ) f 0( )
x
fF
fS » 2fF
fF
WT
WT =2KSe0qNA
2fFé
ëê
ù
ûú
1/2
¢VG = -QB 2fF( ) +Qn
Cox+ 2fF
¢VT = -QB 2fF( )Cox
+ 2fF
Qn = -Cox ¢VG - ¢VT( )
¢VG = -QSCox
+ 2fF
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Maximum depletion region depth
delta-depletion approximation
12
r
x
metal
-xo
WT
r = -qNA
QB = -qNAWT
Qn
WT =2k Se0 2fFqNA
11/2/2016
delta-depletion approximation
13
E x( )
x
P
W
E S
E 0+( ) = - QBKSe0
E 0( ) = -QSKSe0
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14
example
source drain
SiO
2
silicon
S G D
Assume n+ poly Si gate1018 channel dopingtox = 1.5 nm
What is VT?e-field in oxide at VG = 1V?
Bermel ECE 305 F1611/2/2016
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example (cont.)
¢VG = -QS fS( )Cox
+fS
¢VT = -QB 2fF( )Cox
+ 2fF
VT = fms -QB 2fF( )Cox
+ 2fF
fF =kBT
qlnNAni
æ
èçö
ø÷
Cox = KOe0 xo
QB = - 2qk se0NA2fF
QB = -qNAW 2fF( )
fms = -kBT
qlnNANDni2
æ
èçö
ø÷
fF = 0.48 V
Cox = 2.36 ´10-6 F/cm2
QB = -5.71´10-7 C/cm2
fms = -1.06 VVT = 0.14 V
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example (cont)
Qn = -Cox VG -VT( )
E OX = -QSk oxe0
= -Qn +QB 2fB( )
k oxe0
Qn = -2.06 ´10-6 C/cm2
E OX = 7.3´106 V/cm
Qnq= -1.3´1013 C/cm2
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MOS capacitor
17
p-Si
vS sinwt
+VG
+
-
-
VG + vS sinwt
~
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MOS capacitor in depletion
18
VG
p-Si
W fS( ) W VG( )
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MOS capacitor in depletion
xo
W fS( )
KO
KSC = ?
Gate
Undepleted P-type semiconductor
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a simpler problem
xo
W fS( )
KO
KS CS =KSe0W fS( )
Cox =KOe0xo
1
C=1
Cox+1
CSC =
CSCoxCS + Cox
C =Cox
1+ Cox CS
C =Cox
1+KOW fS( )KSxo
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result
xo
W fS( )
k ox
k Si
C =Cox
1+KOW fS( )KSxo
VG
Cox
CS
fS
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s.s. gate capacitance vs. d.c. gate bias
C
VG¢
C =Cox
1+KOW fS( )KSxo
accumulationdepletion
inversion
VT¢
flat band
Cox
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s.s. gate capacitance vs. d.c. gate bias
C
VG¢
C =Cox
1+KOW fS( )KSxo
accumulation
depletion
inversion
VT¢
flat band
Cox
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capacitance vs. gate voltage
C
VG¢
accumulationdepletion
inversion
VT¢
flat band
Cox
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C =Cox
1+KOW fS( )KSxo
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high frequency vs. low frequency
C
VG¢
accumulationdepletion
inversion
VT¢
flat band
Cox
high frequency
low frequency
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C =Cox
1+KOW fS( )KSxo
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high frequency vs. low frequency
C
VG¢VT¢
Cox
high frequency
low frequency
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high frequency vs. low frequency
p-Si
n+-Si n+-Si
MOS capacitor
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conclusions Used the depletion approximation to calculate the
charge distribution and surface potentials of each MOS regime
This can then be translated into expressions for the gate voltage thresholds
Can then also calculate capacitance as a function of frequency and gate voltage
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