Lecture #22
OUTLINE
The MOS Capacitor
• Capacitance
• Effect of Oxide Charges
Reading: Course Reader (Part III, Chap. 2)
Spring 2003
EE130 Lecture 22, Slide 1
Review: Threshold Voltage
• For p-type Si (“NMOS”):
VT = VFB + 2ψ B +
2qN Aε Si (2ψ B )
Cox
• For n-type Si (“PMOS”):
VT = VFB + 2ψ B −
Spring 2003
2qN Dε Si 2ψ B
Cox
EE130 Lecture 22, Slide 2
1
ψs and Wd vs. VG (p-type Si)
2φ F
ψs:
ψs =
2
qN Aε si 
2C (V − VFB ) 
 1 + ox G
− 1
2
qN
2Cox 

Aε si
0
Wdm =
Wd =
0
2ε Si (2ψ F )
qN A
2
2ε Siψ S ε Si 
2C (V − VFB ) 
 1 + ox G
=
− 1 (for VFB < VG < VT )
qN A
Cox 
qN Aε si


VG
accumulation V depletion V inversion
FB
T
Spring 2003
(for VFB < VG < VT )
VG
accumulation V depletion V inversion
FB
T
Wd:
2
EE130 Lecture 22, Slide 3
Total Charge Density in Si, Qs
Qacc = −Cox (VG − VFB )
depletion
0
accumulation
0
depletion
inversion
VG
depletion
VFB
depletion
inversion
VG
Qdep = − qN AW
0
accumulation
VT
VFB
accumulation
Qs = Qacc + Qdep + Qinv
VG
VT
VFB
accumulation
inversion
0
inversion
VFB
VT
VG
VT
Qinv
slope = -Cox
Qinv = −Cox (VG − VT )
Spring 2003
EE130 Lecture 22, Slide 4
2
MOS Capacitance Measurement
• VG is scanned slowly
• Capacitive current due
to vac is measured
iac
iac = C
GATE
vac
C=
Si
C-V Meter
dvac
dt
dQGATE
dQs
=
dVG
dVG
MOS Capacitor
Spring 2003
EE130 Lecture 22, Slide 5
MOS C-V Characteristics (p-type Si)
accumulation
depletion
inversion
VG
VFB
VT
C=
Qinv
dQs
dVG
C
slope = -Cox
Cox
Ideal C-V curve:
VG
VFB
accumulation
Spring 2003
VT
depletion
inversion
EE130 Lecture 22, Slide 6
3
Capacitance in Accumulation (p-type Si)
• As the gate voltage is varied, incremental charge is
added/subtracted to/from the gate and substrate.
• The incremental charges are separated by the gate oxide.
M
O
S
∆Q
Q
C=
-Q
dQacc
= Cox
dVG
−∆Q
Spring 2003
Cox
EE130 Lecture 22, Slide 7
Flat-Band Capacitance
• 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”
– characteristic shielding distance, or the distance where the
electric field emanating from a perturbing charge falls off by
a factor of 1/e
LD =
ε Si kT
q2 N A
1
1
L
=
+ D
CFB Cox ε Si
Spring 2003
EE130 Lecture 22, Slide 8
4
Capacitance in Depletion (p-type Si)
• As the gate voltage is varied, the width of the depletion
region varies.
→ Incremental charge is effectively added/subtracted at a
depth Wd in the substrate.
M
O
∆Q
Q
S
C=
Wd
-Q
−∆Q
dQdep
dVG
=
2(VG − VFB )
1
+
2
qN Aε Si
Cox
1
1
1
1 Wd
=
+
=
+
C Cox Cdep Cox ε Si
Cox Cdep
Spring 2003
EE130 Lecture 22, Slide 9
Capacitance in Inversion (p-type Si)
CASE 1: Inversion-layer charge can be supplied/removed
quickly enough to respond to changes in the gate voltage.
→ Incremental charge is effectively added/subtracted at the
surface of the substrate.
∆Q
M
O
S
WT
−∆Q
Spring 2003
Cox
Time required to build inversion-layer
charge = 2NAτo/ni , where
τo = minority-carrier lifetime at surface
C=
dQinv
= Cox
dVG
EE130 Lecture 22, Slide 10
5
Capacitance in Inversion (p-type Si)
CASE 2: Inversion-layer charge cannot be supplied/removed
quickly enough to respond to changes in the gate voltage.
→ Incremental charge is effectively added/subtracted at a
depth Wd in the substrate.
1
1
1
=
+
C Cox Cdep
∆Q
M
O
S
Wdm
−∆Q
Cox Cdep
Spring 2003
=
1 Wdm
+
Cox ε Si
=
1
2(2ψ B )
1
+
≡
Cox
qN Aε Si C min
EE130 Lecture 22, Slide 11
Supply of Substrate Charge (p-type Si)
gate
gate
Accumulation:
Depletion:
Cox
Cox
+ + + + + +
C dep
p-type Si
p-type Si
Inversion:
Case 1
Case 2
gate
gate
Cox
N+
Wd
- - - - - -
Cox
DC
- - - - - Cdep,min
-
DC and AC Wdm
p-type Si
Spring 2003
AC
WT
p-type Si
EE130 Lecture 22, Slide 12
6
Capacitor vs. Transistor C-V
(or LF vs. HF 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
VFB
Spring 2003
depletion
VT
inversion
VG
EE130 Lecture 22, Slide 13
Quasi-Static C-V Measurement
C
p-type Si:
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.
Spring 2003
EE130 Lecture 22, Slide 14
7
Examples: C-V Characteristics
C
QS
Cox
HF-Capacitor
VG
VT
VFB
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.
Spring 2003
EE130 Lecture 22, Slide 15
Deep Depletion
• If VG is scanned quickly, Qinv cannot respond to the
change in VG. The increase in substrate charge
density Qs must then come from an increase in
depletion charge density Qdep
⇒ depletion depth Wd increases as VG increases
⇒ C decreases as VG increases
C
Cox
Cmin
VFB
Spring 2003
VG
VT
EE130 Lecture 22, Slide 16
8
Parameter Extraction from C-V
From a single C-V measurement, we can extract much
information about the MOS device.
• Suppose we know that the gate-electrode material is
heavily doped n-type poly-Si (ΦM=4.05eV), and that the
gate dielectric is SiO2 (εr=3.9):
– From Cmax = Cox we determine the oxide thickness tox
– From Cmin and Cox we determine substrate doping (by iteration)
– From substrate doping and Cox we calculate the flat-band
capacitance CFB
– From the C-V curve, we can find VFB = VG
C =C FB
– From ΦM, ΦS, Cox, and VFB we can determine Qf
Spring 2003
EE130 Lecture 22, Slide 17
Example: Effect of Doping
C/Cox
1
VFB
VT
VG
• How would C-V characteristic change if substrate
doping NA were increased?
– VFB
– VT
– Cmin
Spring 2003
EE130 Lecture 22, Slide 18
9
Example: Effect of Oxide Thickness
C/Cox
1
VFB
VG
VT
• How would C-V characteristic change if oxide
thickness tox were decreased?
– VFB
– VT
– Cmin
Spring 2003
EE130 Lecture 22, Slide 19
Oxide Charges
In real MOS devices, there is
always some charge in the oxide
and at the Si/oxide interface.
• In the oxide:
– Trapped charge Qot
• High-energy electrons
and/or holes injected into
oxide
– Mobile charge QM
• Alkali-metal ions, which
have sufficient mobility to
drift in oxide under an
applied electric field
• At the interface:
– Fixed charge QF
• Excess Si (?)
– Trapped charge Qit
• Dangling bonds
Spring 2003
EE130 Lecture 22, Slide 20
10
Effect of Oxide Charges
• In general, charges in the oxide cause a shift
in the gate voltage required to reach the
threshold condition:
∆VT = −
t ox
1
ε SiO
2
∫ xρ
ox
( x)dx
0
(x defined to be 0 at metal-oxide interface)
• In addition, they may alter the field-effect
mobility of mobile carriers (in a MOSFET)
due to Coulombic scattering
Spring 2003
EE130 Lecture 22, Slide 21
Fixed Oxide Charge QF
M
3.1 eV
O
S
qQF / Cox
Ec= EFM
|qVFB |
Ev
Ec
EFS
Ev
VFB = φ MS −
QF
Cox
4.8 eV
Spring 2003
EE130 Lecture 22, Slide 22
11
Determination of QF
Measure C-V characteristics of capacitors with different
oxide thicknesses. Plot VFB as a function of tox.
VFB
10nm
20nm
30nm
0
x ox
VFB = φ MS −
–0.15V
×
tox
ε SiO
QF
2
×
–0.3V
Spring 2003
×
EE130 Lecture 22, Slide 23
Mobile Ions
• Odd shifts in C-V characteristics were once a mystery:
∆VFB = −
QM
Cox
• Source of problem: Mobile charge moving to/away from
interface, changing charge centroid
Spring 2003
EE130 Lecture 22, Slide 24
12
Interface Traps
Traps cause “sloppy” C-V and also
greatly degrade mobility in channel
Q (ψ )
∆VG = − IT S
Cox
Spring 2003
EE130 Lecture 22, Slide 25
13