6.012 - Microelectronic Devices and Circuits
Lecture 9 - MOS Capacitors I - Outline
• Announcements
Problem set 5 - Posted on Stellar. Due next Wednesday.
• Qualitative description - MOS in thermal equilibrium
Definition of structure: metal/silicon dioxide/p-type Si (Example: n-MOS)
Electrostatic potential of metal relative to silicon: φm
Zero bias condition: Si surface depleted if φm > φp-Si (typical situation)
Negative bias on metal: depletion to flat-band to accumulation
Positive bias on metal: depletion to threshold to inversion
• Quantitative modeling - MOS in thermal equilibrium, vBC = 0
Depletion approximation applied to the MOS capacitor:
1. Flat-band voltage, VFB
2. Accumulation layer sheet charge density, qA*
3. Maximum depletion region width, XDT
4. Threshold voltage, VT
5. Inversion layer sheet charge density, qN*
• Quantitative modeling - vBC ≠ 0; impact of vBC < 0
Voltage between n+ region and p-substrate: |2φp-Si | → |2φp-Si| - vBC
Clif Fonstad, 10/8/09
Lecture 9 - Slide 1
n-Channel MOSFET: Connecting with the npn MOSFET
A very similar behavior, and very similar uses.
iD
MOSET
D
+
Linear
iD or
Triode
iG
Saturation (FAR)
vDS
G+
iD ! K [vGS - V T(vBS)]2/2!
vGS
–
iC
BJT
C
–
S
+
Cutoff
vDS
iB
vCE
B+
Saturation
iC
i
vBE
–
B
–
E
FAR
iB ! IBSe qV BE /kT
vCE > 0.2 V
Cutoff
0.6 V
Input curve
Clif Fonstad, 10/8/09
vBE
Forward Active Region
iC ! !F iB
0.2 V
Cutoff
vCE
Output family
Lecture 9 - Slide 2
MOS
structures
S
–
vGS
G
+
iG
n+
vDS
D
iD
n+
p-Si
vBS
+
B
iB
An n-channel MOSFET
In an n-channel MOSFET, we have two n-regions (the source and
the drain), as in the npn BJT, with a p-region producing a potential
barrier for electrons between them. In this device, however, it is the
voltage on the gate, vGS, that modulates the potential barrier height.
The heart of this device is the MOS capacitor, which we will study
today. To analyze the MOS capacitor we will use the same depletion
approximation that we introduced in conjunction with p-n junctions.
Clif Fonstad, 10/8/09
Lecture 9 - Slide 3
S
C
The n-MOS
capacitor
–
vGS +
(= vGB)
G
SiO2
n+
p-Si
Right: Basic device
with vBC = 0
B
Below: One-dimensional structure for depletion approximation analysis*
vGB
+
–
SiO 2
G
-tox 0
Clif Fonstad, 10/8/09
p-Si
B
x
* Note: We can't forget the n+ region is there; we
will need electrons, and they will come from there.
Lecture 9 - Slide 4
Electrostatic potential and net charge profiles
φ(x)
-tox
Zero bias: vGB = 0
xd
x
φm
φp
ρ(x)
qNAxd
xd
x
-tox
−qNA
Clif Fonstad, 10/8/09
qD* = -qNAxd
Lecture 9 - Slide 5
Electrostatic potential and net charge profiles
φ(x)
-tox
Depletion: VFB < vGB < 0
xd
x
φm
φp
vGB < 0
ρ(x)
qNAxd
xd
x
-tox
−qNA
Clif Fonstad, 10/8/09
-qNAxd
Lecture 9 - Slide 6
Electrostatic potential and net charge profiles
φ(x)
Flat band : vGB = VFB
-tox
vGB = VFB
VFB = φp – φm
φp
ρ(x)
-tox
Clif Fonstad, 10/8/09
x
φm
VFB = φp – φm
x
Lecture 9 - Slide 7
Electrostatic potential and net charge profiles
φ(x)
Accumulation : vGB < VFB
-tox
φm
vGB < VFB
x
φp
ρ(x)
-tox
- Cox*(vGB - VFB)
x
Cox*(vGB - VFB)
Clif Fonstad, 10/8/09
Lecture 9 - Slide 8
Electrostatic potential and net charge profiles
φ(x)
Flat band : vGB = VFB
-tox
φm
vGB = VFB
VFB = φp – φm
x
φp
ρ(x)
-tox
Clif Fonstad, 10/8/09
x
Lecture 9 - Slide 9
Electrostatic potential and net charge profiles
φ(x)
-tox
Depletion: VFB < vGB < 0
xd
x
φm
φp
VFB < vGB < 0
ρ(x)
qNAxd
xd
x
-tox
−qNA
Clif Fonstad, 10/8/09
-qNAxd
Lecture 9 - Slide 10
Electrostatic potential and net charge profiles
φ(x)
-tox
Depletion: vGB = 0
xd
x
φm
φp
ρ(x)
qNAxd
xd
x
-tox
−qNA
Clif Fonstad, 10/8/09
qD* = -qNAxd
Lecture 9 - Slide 11
Electrostatic potential and net charge profiles
φ(x)
Depletion: 0 < vGB < VT
Weak inversion: φ(0) > 0
-tox
xd
x
φm
0 < vGB < VT
φp
J = 0 ⇒ n(x) = nie-qφ(x)/kT
and p(x) = nieqφ(x)/kT
ρ(x)
qNAxd
xd
-tox
−qNA
Clif Fonstad, 10/8/09
qD* =
-qNAxd
φ(0)↑ ⇒ n(0)↑
x
Weak inversion: φ(0) > 0 ⇒ n(0) > p(0)
Lecture 9 - Slide 12
Electrostatic potential and net charge profiles
φ(x)
vGB = VT
Threshold: vGB = VT
At threshold φ(0) = - φp
-φp
-tox
XDT
x
φm
φp
qNAXDT
ρ(x)
φ(0) = -φp ⇒ n(0) = NA
XDT
-tox
qD*
= -qNAXDT
x
−qNA
Clif Fonstad, 10/8/09
Lecture 9 - Slide 13
Electrostatic potential and net charge profiles
Threshold*: vGB = VT
φ(x)
vGB = VT
VT – VFB
-tox
-φp
qNAXDT/Cox*
XDT
XDT = (2εSi|2φp|/qNA)1/2
|2φp|
φm
x
φp
qNAXDT
ρ(x)
VT – VFB = |2φp| + qNAXDT/Cox*
1/2
**
VTT = VFFBB + |2φpp| + (2εSiSi|2φpp|qNAA)1/2
/Cox
ox
XDT
-tox
−qNA
Clif Fonstad, 10/8/09
qD*
= -qNAXDT
x
qD* = -qNAXDT = -(2εSi|2φp|qNA)1/2
* At threshold φ(0) = - φp
Lecture 9 - Slide 14
Electrostatic potential and net charge profiles
φ(x)
VT < vGB
-tox
Inversion: VT < vGB
-φp
XDT
|2φp |
x
φm
φp
qNAXDT +
Cox*(vGB - VT)
ρ(x)
qN* = Inversion layer charge
(sheet of mobile electrons in
Si near the Si-oxide interface)
XDT
-tox
qD*
= -qNAXDT
−qNA
qN* = - Cox*(vGB - VT)
Clif Fonstad, 10/8/09
x
qD*, depletion region
charge unchanged
Lecture 9 - Slide 15
Electrostatic potential and net charge profiles - regions and boundaries
φ(x)
φ(x)
φ(x)
-tox
vGB
φp
x
φm
vGB
-tox
φm
qNAxd
ρ(x)
-tox - Cox*(vGB - VFB)
x
xd
Acccumulation
vGB < VFB
qNAXDT +
Cox*(vGB - VT)
ρ(x)
-tox
-φp
-tox
φm
x
φp
xd
−qNA
vGB
Cox*(vGB - VFB)
vGB
XDT |2φp |
φp
ρ(x)
-tox
x
XDT
−qNA
qD* = -qNAxd
x
qN*
qD* = -qNAXDT
= - Cox*(vGB - VT)
x
Inversion
VT < vGB
Depletion
VFB < vGB < VT
vGB
Threshold Voltage
VT = VFB+|2φp|+(2εSi|2φp|qNA)1/2/Cox*
Flat Band Voltage
VFB = φp – φm
φ(x)
φ(x)
vGB
-tox
φm
φp
x
-φp
-tox
φm
qNAXDT
ρ(x)
-tox
vGB
x
-tox
−qNA
Clif Fonstad, 10/8/09
XDT
|2φp |
x
φp
ρ(x)
XDT
x
qD* = -qNAXDT
Lecture 9 - Slide 16
vGB
Electrostatic potential and net charge profiles
- the grand procession from accumulation to inversion -
VT
φ(x)
Accumulation : vGB < VFB
-φp
-tox
0
VFB
φm
vGB < VFB
x
φp
ρ(x)
-tox
- Cox*(vGB - VFB)
x
Cox*(vGB - VFB)
Clif Fonstad, 10/8/09
Lecture 9 - Slide 17
vGB
Electrostatic potential and net charge profiles
- the grand procession from accumulation to inversion -
VT
φ(x)
Flat band : vGB = VFB
φ(0) = φp
-φp
-tox
0
VFB
vGB = VFB
VFB = φp – φm φp
x
φm
ρ(x)
x
-tox
VFB = φp – φm
Clif Fonstad, 10/8/09
Lecture 9 - Slide 18
vGB
Electrostatic potential and net charge profiles
- the grand procession from accumulation to inversion -
VT
φ(x)
-φp
-tox
0
Depletion: VFB < vGB < 0
φp < φ(0)
xd
x
φm
φp
VFB
VFB < vGB < 0
ρ(x)
qNAxd
xd
x
-tox
−qNA
Clif Fonstad, 10/8/09
-qNAxd
Lecture 9 - Slide 19
vGB
Electrostatic potential and net charge profiles
- the grand procession from accumulation to inversion -
VT
φ(x)
Depletion: vGB = 0
φp < φ(0) < 0
-φp
-tox
0
xd
x
φm
φp
VFB
ρ(x)
qNAxd
xd
x
-tox
−qNA
Clif Fonstad, 10/8/09
qD* = -qNAxd
Lecture 9 - Slide 20
vGB
Electrostatic potential and net charge profiles
- the grand procession from accumulation to inversion -
VT
φ(x)
Depletion: 0 < vGB < VT
Weak Inversion: φ(0) > 0
-φp
-tox
0
VFB
xd
φm
x
0 < vGB < VT φ
p
ρ(x)
qNAxd
xd
-tox
qD* =
-qNAxd
x
−qNA
Clif Fonstad, 10/8/09
Lecture 9 - Slide 21
vGB
Electrostatic potential and net charge profiles
- the grand procession from accumulation to inversion -
VT
φ(x)
vGB = VT
Threshold: vGB = VT
φ(0) = -φp
-φp
-tox
0
XDT
x
φm
φp
VFB
qNAXDT
ρ(x)
XDT
-tox
qD*
= -qNAXDT
x
−qNA
Clif Fonstad, 10/8/09
VT = VFB + |2φp| + (2εSi|2φp|qNA)1/2/Cox*
Lecture 9 - Slide 22
vGB
Electrostatic potential and net charge profiles
- the grand procession from accumulation to inversion -
VT
φ(x)
VT < vGB
-tox
-φp
0
φm
VFB
φp
qNAXDT +
Cox*(vGB - VT)
Inversion: VT < vGB
-φp < φ(0)
XDT
|2φp |
x
ρ(x)
XDT
-tox
qD*
= -qNAXDT
x
−qNA
Clif Fonstad, 10/8/09
qN* = - Cox*(vGB - VT)
Lecture 9 - Slide 23
Bias between n+ region and substrate, cont.
Reverse bias applied to substrate, I.e. vBC < 0
C
vBC < 0
–
vBC
+
–
vGC
+
G
SiO2
n+
p-Si
B
Soon we will see how this will let us electronically adjust MOSFET
threshold voltages when it is convenient for us to do so.
Clif Fonstad, 10/8/09
Lecture 9 - Slide 24
vGB
With voltage between substrate
and channel, vBC < 0
VT(0)
φ(x)
Flat band: vGB = VFB
No difference from when vBC = 0
-tox
0
VFB
φm
vGB = VFB
φp
VFB = φp – φm
-tox
Clif Fonstad, 10/8/09
x
ρ(x)
x
Lecture 9 - Slide 25
vGB
With voltage between substrate
and channel, vBC < 0
φ(x)
Depletion: 0 < vGB < VT(vBC)
No difference from when vBC = 0
VT(0)
-φp – vBC
-φp
vGB
-tox
0
xd
φm
x
φp
VFB
ρ(x)
qNAxd
xd
-tox
qD* =
-qNAxd
x
−qNA
Clif Fonstad, 10/8/09
Lecture 9 - Slide 26
vGB
With voltage between substrate
and channel, vBC < 0
φ(x)
VT(0)
-φp – vBC
vGB
-φp
-tox
0
Depletion: 0 < vGB < VT(vBC)
No difference from when vBC = 0
XDT
x
φm
φp
VFB
qNAXDT
ρ(x)
XDT
-tox
qD*
= -qNAXDT
x
−qNA
Clif Fonstad, 10/8/09
Lecture 9 - Slide 27
VT(vBC)
VT(0)
vGB
With voltage between substrate
and channel, vBC < 0
φ(x)
vGB =
VT(vBC)
-φp – vBC
-tox
0
-φp
At threshold: vGB = VT(vBC)
Big difference from when vBC = 0
|2φp| – vBC
XDT(vCB = 0) X (v < 0)
DT BC
φm
x
φp
VFB
qNAXDT
ρ(x)
XDT(vBC < 0)
-tox
qN* = -qNAxDT
x
−qNA
Clif Fonstad, 10/8/09
Lecture 9 - Slide 28
With voltage between substrate
and channel, vBC < 0
φ(x)
Threshold: vGC = VT(vBC) with vBC < 0
vGB =
VT(vBC)
-φp – vBC
-tox
-φp
|2φp| – vBC
XDT(vCB = 0) X (v < 0)
DT BC
φm
x
φp
qNAXDT
1/2
*
ρ(x) VT(vBC) = VFB + |2φp| + [2εSi(|2φp|-vBC)qNA] /Cox
{This is vGC at threshold}
XDT(vBC < 0) = [2εSi(|2φp|-vBC)/qNA]1/2
XDT(vBC < 0)
-tox
qN* = -qNAxDT
x
−qNA
Clif Fonstad, 10/8/09
qN* = -[2εSi(|2φp|-vBC)qNA]1/2
Lecture 9 - Slide 29
6.012 - Microelectronic Devices and Circuits
Lecture 9 - MOS Capacitors I - Summary
• Qualitative description
Three surface conditions: accumulated, depleted, inverted
Two key voltages: flat-band voltage, VFB; threshold voltage, VT
The progression: accumulation through flat-band to depletion,
then depletion through threshold to inversion
• Quantitative modeling
Apply depletion approximation to the MOS capacitor, vBC = 0
Definitions: VFB ≡ vGB such that φ(0) = φp-Si
VT ≡ vGB such that φ(0) = – φp-Si
Cox* ≡ εox/tox
Results and expressions (For n-MOS example)
1. Flat-band voltage, VFB = φp-Si – φm
2. Accumulation layer sheet charge density, qA* = – Cox*(vGB – VFB)
3. Maximum depletion region width, XDT = [2εSi(|2φp-Si|-vBC)/qNA]1/2
4. Threshold voltage, VT = VFB – 2φp-Si + [2εSi qNA|(|2φp-Si|-vBC)]1/2/Cox*
5. Inversion layer sheet charge density, qN* = – Cox*(vGB – VT)
Clif Fonstad, 10/8/09
Lecture 9 - Slide 30
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6.012 Microelectronic Devices and Circuits
Fall 2009
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