Lecture 17
OUTLINE
• The MOS Capacitor (cont’d)
– Small-signal capacitance
(C-V characteristics)
Reading: Pierret 16.4; Hu 5.6
fS and W vs. VG
(p-type Si)
2fF
fS:
2
qN A si 
2Cox (VG  VFB ) 
 1
fs 
 1
2
qN A si
2Cox 

0
accumulation V depletion V inversion
FB
T
0
accumulation
EE130/230M Spring 2013
(for VFB  VG  VT )
VG
WT 
W:
2
2ε Si (2fF )
qN A
2
2 SifS  Si 
2Cox (VG  VFB ) 
 1
W

 1 (for VFB  VG  VT )
qN A
Cox 
qN A si


VFB depletion VT inversion
Lecture 17, Slide 2
VG
Total Charge Density in Si, Qs
(p-type Si)
Qacc  Cox (VG  VFB )
depletion
0
accumulation
VFB
accumulation
inversion
VT
depletion
inversion
0
VFB
VG accumulation
depletion
inversion
VT
VG
Qdep  qN AW
accumulation
Qs  Qacc  Qdep  Qinv
VG
depletion
inversion
0
VFB
0
VT
VG
VT
Qinv
slope = -Cox
Qinv  Cox (VG  VT )
EE130/230M Spring 2013
VFB
Lecture 17, Slide 3
MOS Capacitance Measurement
C-V Meter
iac
vac
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MOS Capacitor
GATE
Semiconductor
Lecture 17, Slide 4
• VG is scanned slowly
• Capacitive current due
to vac is measured
dvac
iac  C
dt
dQGATE
dQs
C

dVG
dVG
MOS C-V Characteristics
(p-type Si)
accumulation
depletion
inversion
VG
VFB
VT
dQs
C
dVG
Qinv
C
slope = -Cox
Cox
Ideal C-V curve:
VG
VFB
accumulation
EE130/230M Spring 2013
Lecture 17, Slide 5
VT
depletion
inversion
Capacitance in Accumulation
(p-type Si)
• As the gate voltage is varied, incremental charge is added (or
subtracted) to (or from) the gate and substrate.
 The incremental charges are separated by the gate oxide.
M
O
S
DQ
Q
dQacc
C
 Cox
dVG
-Q
DQ
Cox
EE130/230M Spring 2013
Lecture 17, Slide 6
Flat-Band Capacitance
(p-type Si)
• At the flat-band condition, variations in VG give rise to the
addition/subtraction of incremental charge in the substrate,
at a depth LD
– LD is the “extrinsic Debye Length,” a characteristic screening distance,
or the distance where the electric field emanating from a perturbing
charge falls off by a factor of 1/e
LD 
Cox
CDebye
EE130/230M Spring 2013
 Si kT
q2 N A
1
1
LD


CFB Cox  Si
Lecture 17, Slide 7
Capacitance in Depletion
(p-type Si)
• As the gate voltage is varied, the depletion width varies.
 Incremental charge is effectively added/subtracted at a
depth W in the substrate.
M
O
DQ
S
C
Q
W
dQdep
dVG
2(VG  VFB )
1


2
qN A Si
Cox
DQ
-Q
1
1
1
1 W




C Cox Cdep Cox  Si
Cox Cdep
EE130/230M Spring 2013
Lecture 17, Slide 8
Capacitance in Inversion
(p-type Si)
CASE 1: Inversion-layer charge can be supplied/removed
quickly enough to respond to changes in gate voltage.
 Incremental charge is effectively added/subtracted at the
surface of the substrate.
DQ
M
O
S
WT
Time required to build inversion-layer
charge = 2NAto/ni , where
to = minority-carrier lifetime at surface
DQ
dQinv
C
 Cox
dVG
Cox
EE130/230M Spring 2013
Lecture 17, Slide 9
Capacitance in Inversion
(p-type Si)
CASE 2: Inversion-layer charge cannot be supplied/removed
quickly enough to respond to changes in gate voltage.
 Incremental charge is effectively added/subtracted at a
depth WT in the substrate.
1
1
1


C Cox Cdep
DQ
M
O
S
WT
DQ
Cox Cdep
EE130/230M Spring 2013
1 WT


Cox  Si
1
2(2fF )
1



Cox
qN A Si C min
Lecture 17, Slide 10
Supply of Substrate Charge
(p-type Si)
Accumulation:
Inversion:
EE130/230M Spring 2013
Depletion:
Case 1
Lecture 17, Slide 11
Case 2
MOS Capacitor vs. MOS Transistor C-V
(p-type Si)
C
MOS transistor at any f,
MOS capacitor at low f, or
quasi-static C-V
Cmax=Cox
CFB
MOS capacitor at high f
Cmin
accumulation
EE130/230M Spring 2013
VFB
depletion
Lecture 17, Slide 12
VT
inversion
VG
Quasi-Static C-V Measurement
(p-type Si)
C
Cmax=Cox
CFB
Cmin
accumulation
VFB
depletion
VT
inversion
VG
The quasi-static C-V characteristic is obtained by slowly ramping the
gate voltage (< 0.1V/s), while measuring the gate current IG with a
very sensitive DC ammeter. C is calculated from IG = C·(dVG/dt)
EE130/230M Spring 2013
Lecture 17, Slide 13
Deep Depletion
(p-type Si)
• If VG is scanned quickly, Qinv cannot respond to the change in
VG. Then the increase in substrate charge density Qs must
come from an increase in depletion charge density Qdep
 depletion depth W increases as VG increases
 C decreases as VG increases
C
Cox
Cmin
VFB
EE130/230M Spring 2013
VT
Lecture 17, Slide 14
VG
MOS C-V Characteristic for n-type Si
MOS transistor at any f,
MOS capacitor at low f, or
quasi-static C-V
C
Cmax=Cox
CFB
MOS capacitor at high f
accumulation
EE130/230M Spring 2013
Cmin
VT
depletion
Lecture 17, Slide 15
VFB
inversion
VG
Examples: C-V Characteristics
C
QS
Cox
HF-Capacitor
VT
VFB
VG
Does the QS or the HF-capacitor C-V characteristic apply?
(1) MOS capacitor, f=10kHz
(2) MOS transistor, f=1MHz
(3) MOS capacitor, slow VG ramp
(4) MOS transistor, slow VG ramp
EE130/230M Spring 2013
Lecture 17, Slide 16
Example: Effect of Doping
• How would the normalized C-V characteristic below change if
the substrate doping NA were increased?
– VFB
– VT
– Cmin
C/Cox
1
VFB
EE130/230M Spring 2013
VT
Lecture 17, Slide 17
Example: Effect of Oxide Thickness
• How would the normalized C-V characteristic below change if
the oxide thickness xo were decreased?
– VFB
– VT
– Cmin
C/Cox
1
VFB
EE130/230M Spring 2013
VT
Lecture 17, Slide 18
Derivation of Time to Build Inversion-Layer Charge
(for an NMOS device, i.e. p-type Si)
2
n

D
n

D
p
i  pn
The net rate of carrier generation is:


t
t
t p (n  n1 )  t n ( p  p1 )
(ref. Lecture 5, Slide 24)
( ET  Ei ) / kT
( Ei  ET ) / kT
n

n
e

n
and
p

n
e
 ni
where 1
i
i
1
i
since trap states that contribute most significantly to G-R have an associated
energy level near the middle of the band gap.
Within the depletion region, n and p are negligible, so
where tn  tp  to
n
Dn Dp

 i
t
t
2t o
Therefore, the rate at which the inversion-layer charge density Qinv (units: C/cm2)
increases due to thermal generation within the depletion region (of width W) is
dQinv
ni
 q
W
dt
2t o
EE130/230M Spring 2013
Lecture 17, Slide 19
For a fixed value of gate voltage, the total charge in the semiconductor is fixed:
QS  Qinv  Qdep  constant
Therefore
dQdep
dQinv
d  qN AW 


dt
dt
dt
 

qni
dW
W  qN A
2t o
dt
ni
dW

W 0
dt 2 N At o
The solution to this differential equation is
where
EE130/230M Spring 2013
W  e  ni / 2 N At o t  e  t /t
2 N At o
t
ni
Lecture 17, Slide 20