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.
MOSET
G
S
D
+
––
+
vGS
vDS
iG
iD
iB
vBE vCE
iC
0.6 V 0.2 V
Forward Active RegionFAR
CutoffCutoff
Saturation
iC ! !F iBvCE > 0.2 V
iB ! IBSe qVBE /kT
Input curve Output family
BJT
B
E
C
+
––
+
vBE
vCE
iB
iC
vDS
iD
Saturation (FAR)
Cutoff
Linear
or
Triode
iD ! K [vGS - VT(vBS)]2/2!
Clif Fonstad, 10/8/09 Lecture 9 - Slide 2
p-Si
B
G+vGS
n+
D
n+
S– vDS
vBS +
iG
iB
iD
MOS structures
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
The n-MOS capacitor
Right: Basic device with vBC = 0
p-Si
n+
B
SG
SiO2+–
vGS
(= vGB)
C
Below: One-dimensional structure for depletion approximation analysis*
Clif Fonstad, 10/8/09 Lecture 9 - Slide 4
BG
+ –
p-SiSiO 2
x-tox 0
vGB
* Note: We can't forget the n+ region is there; we will need electrons, and they will come from there.
Electrostatic potential and net charge profiles
φ(x) Zero bias: vGB = 0
-tox xd
φmx
φp
ρ(x)
qNAxd xd x-tox
qD* = -qNAxd−qNA
Clif Fonstad, 10/8/09 Lecture 9 - Slide 5
Electrostatic potential and net charge profiles
φ(x) Depletion: VFB < vGB < 0
-tox xd
φm x
φp
vGB < 0 ρ(x)
qNAxd xd x-tox
-qNAxd−qNA
Clif Fonstad, 10/8/09 Lecture 9 - Slide 6
Electrostatic potential and net charge profiles
φ(x) Flat band : vGB = VFB
-tox
x vGB = VFB
VFB = φp – φm φp ρ(x)
x-tox
φm
VFB = φp – φm
Clif Fonstad, 10/8/09 Lecture 9 - Slide 7
Electrostatic potential and net charge profiles
φ(x) Accumulation : vGB < VFB
-tox
x
vGB < VFB
ρ(x) φp
φm
- C *(vGB - VFB)ox -tox
C *(vGB - VFB)ox
Clif Fonstad, 10/8/09 Lecture 9 - Slide 8
x
Electrostatic potential and net charge profiles
φ(x) Flat band : vGB = VFB
-tox
φm x
vGB = VFB
VFB = φp – φm φp ρ(x)
x-tox
Clif Fonstad, 10/8/09 Lecture 9 - Slide 9
Electrostatic potential and net charge profiles
φ(x) Depletion: VFB < vGB < 0
-tox xd
φm x
φp
VFB < vGB < 0 ρ(x)
qNAxd xd x-tox
-qNAxd−qNA
Clif Fonstad, 10/8/09 Lecture 9 - Slide 10
Electrostatic potential and net charge profiles
φ(x) Depletion: vGB = 0
-tox xd
φmx
φp
ρ(x)
qNAxd xd x-tox
qD* = -qNAxd−qNA
Clif Fonstad, 10/8/09 Lecture 9 - Slide 11
Electrostatic potential and net charge profiles
φ(x) Depletion: 0 < vGB < VT Weak inversion: φ(0) > 0
xd x
0 < vGB
-tox
φm
< VT φp
J = 0 ⇒ n(x) = nie-qφ(x)/kT ρ(x)
and p(x) = nieqφ(x)/kT
qNAxd
φ(0)↑ ⇒ n(0)↑ xd
x qD
* = -qNAxd-tox
−qNA
Weak inversion: φ(0) > 0 ⇒ n(0) > p(0) Clif Fonstad, 10/8/09 Lecture 9 - Slide 12
Electrostatic potential and net charge profiles
φ(x) Threshold: vGB = VT vGB = VT
-φp
-tox
x
φp
φm
XDT
At threshold φ(0) = - φp
ρ(x) qNAXDT φ(0) = -φp ⇒ n(0) = NA
XDT
x qD
* = -qNAXDT -tox
−qNA
Clif Fonstad, 10/8/09 Lecture 9 - Slide 13
VT = B + |2φp| + (2εSi|2φp|qNA)1/2/Cox
*
Electrostatic potential and net charge profiles
-tox
x
φ(x)
φp
φm
-tox x
−qNA
XDT
XDT
ρ(x)qNAXDT
VT – VFB
Threshold*: vGB = VT
-φp
qD * = -qNAXDT
|2φp|
qNAXDT/Cox *
vGB = VT
VT – VFB = |2φp| + qNAXDT/Cox *
VF
XDT = (2εSi|2φp|/qNA)1/2
qD * = -qNAXDT = -(2εSi|2φp|qNA)1/2
VT = VFB + |2φp| + (2εSi|2φp|qNA)1/2/Cox *
Clif Fonstad, 10/8/09 * At threshold φ(0) = - φp Lecture 9 - Slide 14
Electrostatic potential and net charge profiles
Inversion: VT < vGB
-tox
x
φ(x)
φp
φm
XDT
VT < vGB
-φp
|2φp |
-tox x
−qNA
XDT
ρ(x) qNAXDT + Cox
*(vGB - VT)
qN * = - Cox
*(vGB - VT)
qD * = -qNAXDT
qD*, depletion regioncharge unchanged
qN* = Inversion layer charge(sheet of mobile electrons inSi near the Si-oxide interface)
Clif Fonstad, 10/8/09 Lecture 9 - Slide 15
Electrostatic potential and net charge profiles - regions and boundaries
φXDT
p
φ(x)
-tox φ
φ(x) φ(x)
φp φ
vGB -φ
|2φp |-tox vGB -tox xd xxx mm φp
m φpvGB qNAXDT +
xd ox ox
ρ(x) ρ(x) C *(vGB - VT)ρ(x) qNAxd ox
C
- C *(vGB - VFB) -t -tox XDT x-tox xx qD* = -qNAXDT −qNA * −qNAqD = -qNAxdox
*(vGB - VFB) vGB * qN = - Cox *(vGB - VT)
Acccumulation Depletion Inversion
Flat Band Voltage Threshold Voltage
vGB < VFB VFB < vGB < VT VT < vGB
vGB
|qNA)1/2/C– φ VT = VFB+|2φp|+(2εSi|2φp ox *VFB = φp m
φ(x)
φm φ
φ(x)
XDT φ
-φpvGB |2φ |p-tox -tox xx mvGB φp
p qNAXDT ρ(x) ρ(x)
-tox -tox XDT xx −qNA qD
* = -qNAXDT
Clif Fonstad, 10/8/09 Lecture 9 - Slide 16
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) Accumulation : vGB < VFB
-φp
-tox
x0
vGB < VFBVFB
ρ(x) φp
φm
- C *(vGB - VFB)ox-tox x
Cox *(vGB - VFB)
Clif Fonstad, 10/8/09 Lecture 9 -- Slide 17
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) Flat band : vGB = VFB
φ(0) = φp -φp
-tox
φm x0
vGB = VFB VFB φp= φ – φVFB p m
ρ(x)
x-tox
VFB = φp – φm
Clif Fonstad, 10/8/09 Lecture 9 -- Slide 18
Clif Fonstad, 10/8/09 Lecture 9 Slide 19
vGB Electrostatic potential and net charge profiles
- the grand procession from accumulation to inversion -VT
φ(x) Depletion: VFB < vGB < 0 φp < φ(0)
-φp
-tox xd x0 φm
φpVFB
VFB < vGB < 0 ρ(x)
qNAxd xd x-tox
-qNAxd−qNA
Clif Fonstad, 10/8/09 Lecture 9 -- Slide 19
Clif Fonstad, 10/8/09 Lecture 9 Slide 20
0
vGB
VT
-φp
-tox
φpVFB
qNAxd
−qNA
Electrostatic potential and net charge profiles- the grand procession from accumulation to inversion -
φ(x) Depletion: vGB = 0 φp < φ(0) < 0
xd
φm
-tox
ρ(x)
xd
qD* = -qNAxd
Clif Fonstad, 10/8/09 Lecture 9 -- Slide 20
x
x
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) Depletion: 0 < vGB < VT
Weak Inversion: φ(0) > 0 -φp
xd x0
0 <
-tox
φm
vGB < VT φpVFB
ρ(x) qNAxd
xd x
qD* = -qNAxd
-tox
−qNA
Clif Fonstad, 10/8/09 Lecture 9 -- Slide 21
Clif Fonstad, 10/8/09 Lecture 9 Slide 22
vGB Electrostatic potential and net charge profiles
- the grand procession from accumulation to inversion -VT
φm x
φ(x) Threshold: vGB = VT vGB = VT φ(0) = -φp
-φp
-tox XDT
0
φpVFB
ρ(x)qNAXDT
XDT
x qD
* = -qNAXDT -tox
−qNA
|qNA)1/2/Cox * Clif Fonstad, 10/8/09 Lecture 9 -- Slide 22VT = VFB + |2φp| + (2εSi|2φp
Clif Fonstad, 10/8/09 Lecture 9 Slide 23
vGB
XDT
qN * = - C *(vGB - VT)ox
x
Electrostatic potential and net charge profiles
φm
XDT
-φp
-tox qD * = -qNAXDT
x
- the grand procession from accumulation to inversion -VT
φ(x) Inversion: VT < vGB VT < vGB -φp < φ(0)
-tox |2φp |
0
φpVFB
ρ(x)qNAXDT + C *(vGB - VT)ox
−qNA
Clif Fonstad, 10/8/09 Lecture 9 -- Slide 23
Bias between n+ region and substrate, cont. Reverse bias applied to substrate, I.e. vBC < 0
vBC < 0
p-Si
n+
B
CG
SiO2+– vGC
vBC +
–
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
Flat band:VT(0) vGB = VFB φ(x) No difference from when vBC = 0
-tox
ρ(x)
φm x
0 vGB = VFB
VFB φp
= φ – φVFB p m
x-tox
Clif Fonstad, 10/8/09 Lecture 9 - Slide 25
vGB
φ(x) With voltage between substrate and channel, vBC < 0
VT(0) Depletion: 0 < vGB < VT(vBC) No difference from when vBC = 0
-φp – vBC
-φp
xdvGB -tox
φm x0
φpVFB
ρ(x) qNAxd
xd x
qD* = -qNAxd
-tox
−qNA Clif Fonstad, 10/8/09 Lecture 9 - Slide 26
0
VT(0)
VFB
vGB
vGB
φ(x)
-φp – vBC
With voltage between substrate and channel, vBC < 0 Depletion: 0 < vGB < VT(vBC)
No difference from when vBC = 0
XDT
x
φp
qNAXDT ρ(x)
XDT
x
−qNA
-tox
φm
-φp
-tox qD * = -qNAXDT
Clif Fonstad, 10/8/09 Lecture 9 - Slide 27
vGB
VT(vBC)
VT(0)
0
VFB
At threshold: vGB = VT(vBC) Big difference from when vBC = 0
-tox
x
φ(x)
φp
φm
XDT(vBC < 0)
-φp
vGB = VT(vBC)
XDT(vCB = 0)
-φp – vBC
|2φp| – vBC
With voltage between substrate and channel, vBC < 0
-tox x
−qNA
ρ(x) qNAXDT
qN * = -qNAxDT
XDT(vBC < 0)
Clif Fonstad, 10/8/09 Lecture 9 - Slide 28
φm
-φp XDT(vCB = 0)
-φp – vBC
x
φ(x) With voltage between substrate and channel, vBC < 0
Threshold: vGC = VT(vBC) with vBC < 0 vGB = VT(vBC)
|2φp| – vBC
-tox XDT(vBC < 0)
φp
|-vBC)qNA]1/2/Cox *
ρ(x) VT(vBC) = VFB + |2φp| + [2εSi(|2φpqNAXDT {This is vGC at threshold}
|-vBC)/qNA]1/2 XDT(vBC < 0) = [2εSi(|2φp
XDT(vBC < 0) x
qN* = -qNAxDT -tox
−qNA
|-vBC)qNA]1/2 Clif Fonstad, 10/8/09 Lecture 9 - Slide 29 qN* = -[2εSi(|2φp
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
MIT OpenCourseWarehttp://ocw.mit.edu
6.012 Microelectronic Devices and Circuits Fall 2009
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.