MOS Capacitors I - MIT OpenCourseWare · To analyze the MOS capacitor we will use the same...

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!


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.


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*


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



φ(x) Depletion: VFB < vGB < 0

-tox xd

φm x

φp

vGB < 0 ρ(x)

qNAxd xd x-tox

-qNAxd−qNA



φ(x) Flat band : vGB = VFB

-tox

x vGB = VFB

VFB = φp – φm φp ρ(x)

x-tox

φm

VFB = φp – φm



φ(x) Accumulation : vGB < VFB

-tox

x

vGB < VFB

ρ(x) φp

φm

- C *(vGB - VFB)ox -tox

C *(vGB - VFB)ox


x



-tox

φm x

vGB = VFB

VFB = φp – φm φp ρ(x)

x-tox



φ(x) Depletion: VFB < vGB < 0

-tox xd

φm x

φp

VFB < vGB < 0 ρ(x)

qNAxd xd x-tox

-qNAxd−qNA



φ(x) Depletion: vGB = 0

-tox xd

φmx

φp

ρ(x)

qNAxd xd x-tox

qD* = -qNAxd−qNA



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


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


VT = B + |2φp| + (2εSi|2φp|qNA)1/2/Cox

*


-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


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)


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 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





φ(0) = φp -φp

-tox

φm x0

vGB = VFB VFB φp= φ – φVFB p m

ρ(x)

x-tox

VFB = φp – φm





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



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


x

x




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





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


vGB

XDT

qN * = - C *(vGB - VT)ox

x


φm

XDT

-φp

-tox qD * = -qNAXDT

x


φ(x) Inversion: VT < vGB VT < vGB -φp < φ(0)

-tox |2φp |

0

φpVFB

ρ(x)qNAXDT + C *(vGB - VT)ox

−qNA


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.


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


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


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)


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)


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.

http://ocw.mit.edu

http://ocw.mit.edu/terms

MOS Capacitors I - MIT OpenCourseWare · To analyze the MOS capacitor we will use the same...

Documents

Transcript of MOS Capacitors I - MIT OpenCourseWare · To analyze the MOS capacitor we will use the same...