EE-612

Lecture 4/5
MOS Capacitors
Joerg Appenzeller
Electrical and Computer Engineering
Purdue University
West Lafayette, IN USA
Fall 2024

Slides were developed by Prof. Mark Lundstrom

Appenzeller EE-612 F’24 1

 outline

1) Short review
2) Gate voltage / surface potential relation
3) The flatband voltage
4) MOS capacitance vs. voltage
5) Gate voltage and inversion layer charge

Appenzeller EE-612 F’24 2

1) short review (bulk semiconductor)

log10 QS  S 
C/cm 2
QS  Qi ~ eq S / 2 kBT

QS  Qacc ~ e  q S / 2 kBT

accumulation depletion inversion

QS  QD ~  S   kBT q 

dQ
C S
dV

Appenzeller EE-612 F’24 3

1) short review

qS EC
 x   0
EF
 qVG  0
EFM EV

 S VG ?

Appenzeller EE-612 F’24 4

 outline

1) Short review
2) Gate voltage / surface potential relation
3) The flatband voltage
4) MOS capacitance vs. voltage
5) Gate voltage and inversion layer charge

Appenzeller EE-612 F’24 5

 2) flat-band conditions
“flat band” Evac
• when the gate electrode Fermi level
lines up with the semiconductor
Fermi level, the bands are flat in the
semiconductor M S
EC

• this occurs at V’G = 0 when the gate VG EF

electrode workfunction equals the
EV
semiconductor workfunction
 M  S eV
 M  S V
S  0
Appenzeller EE-612 F’24 6
 gate voltage and S

VOX  0 S  0     0
VG   S  VOX
EC
DOX VG   S  EOX tOX
EF

EV QS
qVG  0 VG   S  tOX
 OX
QS
VG   S 
DOX  OX EOX  QS COX
 OX
COX  F/cm 2
tOX
Appenzeller EE-612 F’24 7
 gate voltage and S

 S  VG
QS  S 
S VG   S  VOX   S 
COX
2 B
QS   2nB Si k BT eq S /2 kB T

QS   2q Si N A S

VT VG

Appenzeller EE-612 F’24 8

 threshold voltage, VT

 S  2 B onset of inversion

QS (2 B )
VG  VT  2 B 
COX

QS (2 B )  QD (2 B )   2q Si N A (2 B )

VT  2 B  2q Si N A (2 B ) COX

Appenzeller EE-612 F’24 9

 outline

1) Short review
2) Gate voltage / surface potential relation
3) The flatband voltage
4) MOS capacitance vs. voltage
5) Gate voltage and inversion layer charge

Appenzeller EE-612 F’24 10

 flatband voltage

In an ideal MOS-C, S = 0 when VG’ = VFB = 0.

In a real MOS-C, charges at the oxide-silicon

interface, in the oxide, and gate-
semiconductor workfunction differences all
shift the flatband voltage.

Appenzeller EE-612 F’24 11

 interface charge

VOX  0 S  0     0
EC
DOX
EF
qVG  0 EV

VG   S  EOX tOX

DOX  QS  QF  QIT  S 
VG   S  QS COX  QF COX  QIT ( S ) COX

Appenzeller EE-612 F’24 12

 charge in the oxide

DL  DR  QM

QM EC DR  QS
DL DR ER  QS OX
EF
EL  ER  QM OX
EV
VOX  xM EL  (tOX  xM )ER
QS  x M  QM
0 x M t ox x VOX  
COX  tOX  COX
QS  xM  QM
VG   S  
COX  tOX  COX
Appenzeller EE-612 F’24 13
 distributed charge in the oxide

tOX
EC
QM   Q(x)dx
0
EF
tOX
EV
 xQ(x)dx
xM  0
tOX

0 t ox x  Q(x)dx
0
QS  xM  QM
VG   S  
COX  tOX  COX
Appenzeller EE-612 F’24 14
 additional information
For more information on the oxide-silicon interface and the origin of
the various charges, see:
1) R.F. Pierret, Semiconductor Device Fundamentals, pp. 650-671
Addison-Wesley, 1996

2) J.A. Del Alamo, EE 6720J/ 3.43J Integrated Microelectronic

Devices,Fall 2002
Lecture 22: “The Si Surface and MOS Structure”

Available from MIT OpenCourseWare:

http://ocw.mit.edu
‘Electrical Engineering and Computer Science’
‘Graduate’
Appenzeller EE-612 F’24 15
 gate oxides 2008
• Typically SiON with k ~ 4.6 for 15% N2 (not SiO2 with k = 3.9).

• Use of thicker oxides with higher k gives less gate leakage.

• SiON is more resistant to boron penetration.

• But, SiON degrades mobility and reliability (NBTI). Engineering the N2

profile may help.

• Typically grown in dry 02, followed by plasma nitridation and rapid

thermal anneal. Results in 10-15% N2.

• NIT ~ 5 x 1010 cm-2 (corresponds to 500 traps/mm2)

• Oxide scaling has stopped at ~1.1 nm due to gate leakage.
Introduction of high-k beginning at 45-32 nm node.
(Source: M.A. Alam, Purdue Univ, 8/06)
Appenzeller EE-612 F’24 16
 re-cap

QS
VG   S  VOX S 
COX
fast charge
QS  xM  QM
VG   S   QF COX  QIT ( S ) COX  
COX  tOX  COX
fixed charge charge in oxide

Under flatband conditions:  S  QS  0

 x M  QM
   QF COX  QIT ( S  0) COX
VFB 
 t  C
OX OX

Appenzeller EE-612 F’24 17

 gate-semicond. workfunction differences
EVAC
flat band
S
 M  qm S   M
EC EC

EF VG  0 EF

EV EV

S  0
Appenzeller EE-612 F’24 18
 M < S
EVAC q S  0

 M  qm S
EC
S
EC
EF
EF
EV
EV

Vbi  ms  0
 
Appenzeller EE-612 F’24 19
 flatband voltage
VG  0 VFB  ms  Vbi  0

EC
VG  0 EC
EF VG  VFB  0
EF
EV
EV

Vbi  ms  0 S  0
 
Appenzeller EE-612 F’24 20
 flatband voltage

QS
recall: VG   S 
COX

VG  VG  VFB

QS
VG  VFB   S 
COX

QF QIT  S   xM  QM
VFB   ms   
COX COX  tOX  COX

Appenzeller EE-612 F’24 21

 outline

1) Short review
2) Gate voltage / surface potential relation
3) The flatband voltage
4) MOS capacitance vs. voltage
5) Gate voltage and inversion layer charge

Appenzeller EE-612 F’24 22

 capacitance

dQG d QS 
VG  vS sin  t CG  
dVG dVG
QS
+ VG  VFB   S 
COX
- p-Si dVG d S 1
 
d QS  d QS  COX

1 1 1
 
CG CS COX

Appenzeller EE-612 F’24 23

 capacitance

VG 1 1 1
 
CG CS COX
COX
S d QS 
CS 
CS d S

QS  S 

already
understood!
Appenzeller EE-612 F’24 24
 a closer look

QS
VG  VFB   S 
COX

QF QIT  S   xM  QM QIT  S 
VFB   ms      VFB 
COX COX  tOX  COX COX

1 1 1
 
CG CS  C IT  COX

Appenzeller EE-612 F’24 25

 surface state capacitance

VG
1 1 1
 
CG CS  C IT  COX
COX
S
d QS 
CS 
CSi C IT d S

d QIT
C IT 
d S

Appenzeller EE-612 F’24 26

 capacitance vs. voltage
CG

Cacc COX Cinv

CFB low frequency

VG

Appenzeller EE-612 F’24 27

 (i) accumulation capacitance

d Qacc  Qacc
CS  VG   S 
d S COX

Qacc ~ eq S /2kBT Qacc  COX VG   S 

CS 
Qacc COX VG   S 
CS 
(2k BT / q) (2kBT / q)

CS  VG   S 
  1
COX (2kBT / q)
Appenzeller EE-612 F’24 28
 another way to look at it

 OX
COX 
tOX
CS  Si tOX
  1
 Si COX  OX t acc
CS 
t acc

Appenzeller EE-612 F’24 29

accumulation capacitance

CG
Cacc COX Cinv

CFB
low frequency

VG

1 1 1
  Cacc  COX
Cacc CS COX

Appenzeller EE-612 F’24 30

 flat band capacitance

flat band
QS  0

dQS  Si
CS (FB)   EC
d S LD
VG  0 EF

 Si kBT EV
LD 
q2 N A

S  0
Appenzeller EE-612 F’24 31
 flat band capacitance

CG
Cacc COX Cinv

CFB
low frequency

VG
VFB

1 LD 1
  CFB  COX
CFB  Si COX
Appenzeller EE-612 F’24 32
 depletion capacitance
q S  0
QS  QD   2q Si N A S

EC
dQS  Si
CS  C D  
d S WD EF

EV
2 Si S
WD 
qN A

Appenzeller EE-612 F’24 33

 depletion capacitance

CG
Cacc COX Cinv

CFB
low frequency

VG

1 WD 1
  Cdepl  COX
Cdepl  Si COX
Appenzeller EE-612 F’24 34
 inversion capacitance

d Qinv   S  2 B
CS 
d S
EC
Qinv ~ eq S /2kBT
Qinv EF
CS 
(2k BT / q) EV

COX VG  VT 
CS 
(2kBT / q)

CS

VG  VT 
 1
COX (2kBT / q)
Appenzeller EE-612 F’24 35
 inversion capacitance

CG
Cacc COX Cinv

CFB
low frequency

VG

1 1 1
  Cinv  COX
Cinv CS COX

Appenzeller EE-612 F’24 36

 EOT(electrical) in inversion

CG
Cacc COX Cinv

CFB
low frequency

VG

 OX
Cinv  EOTelectrical  tOX
EOTelectrical

Appenzeller EE-612 F’24 37

 role of frequency

CG
Cacc COX Cinv

CFB
low frequency

VG

Appenzeller EE-612 F’24 38

 inversion capacitance (high frequency)

QS  QD ( S )  Qi  S   S  2 B

dQS dQD ( S ) dQI  S 

CS  
d S

d S

d S
X EC

EF
dQD  Si
CS    EV
d S  S  2 B
Wdm

2 Si 2 B 
Wdm 2 B  
qN A
Appenzeller EE-612 F’24 39
inversion capacitance (high frequency)

CG
Cacc COX Cinv

CFB

high frequency

VG

1 Wdm 1
  Cinv  COX
Cinv  Si COX
Appenzeller EE-612 F’24 40
 low vs. high frequency

VG
QS  QD  Qinv
COX
CS  CS dep   CS inv 
?
X
CS dep  CS (inv)

may (or may not) be able

to follow the ac signal

Appenzeller EE-612 F’24 41

 low or high frequency?

n+-Si n+-Si

ni
G
2
p-Si
p-Si

typically observe hi- typically observe low-

frequency CV frequency CV

Appenzeller EE-612 F’24 42

 MOS CV recap

CG
Cacc COX Cinv
 Si
 Si CS inv  
CS acc   CFB tinv
t acc
low frequency
 Si
CS depl  
WD high frequency
VG

Appenzeller EE-612 F’24 43

 outline

1) Short review
2) Gate voltage / surface potential relation
3) The flatband voltage
4) MOS capacitance vs. voltage
5) Gate voltage and inversion layer charge

Appenzeller EE-612 F’24 44

 inversion charge - gate voltage relation

QS  S 
VG  VFB   S 
COX

At threshold:
QD (2 B )
VT  VFB  2 B 
COX

Beyond threshold:

VG  VT   S  2 B 
 QI ( S )  QD ( S )  QD (2 B )

COX COX
Appenzeller EE-612 F’24 45
 surface potential beyond threshold

V  VG  VT 

COX
COX
 S  S  V
CS (inv) CS (inv)  COX

Appenzeller EE-612 F’24 46

 inversion charge - gate voltage relation (ii)

Beyond threshold:
QI ( S )  QD ( S ) QD (2 B )
VG  VT   S  2 B  
COX COX

COX
 S  V 0
Cinv  COX

Qi
VG  VT   QI  COX VG  VT 
COX

Appenzeller EE-612 F’24 47

 inversion charge - gate voltage relation (iii)

Beyond threshold:
QI ( S )  QD ( S ) QD (2 B )
VG  VT   S  2 B  
COX COX
QI
VG  VT   S  (neglect depletion charge variation)
COX
COX
 S  V
CS (inv)  COX QI  CG VG  VT 

COX CS (inv)
CG 
CS (inv)  COX

Appenzeller EE-612 F’24 48

 summary

1) Short review
2) Gate voltage / surface potential relation
3) The flatband voltage
4) MOS capacitance vs. voltage
5) Gate voltage and inversion layer charge

Appenzeller EE-612 F’24 49