Principles of Semiconductor Devices-L33

8/8/2019 Principles of Semiconductor Devices-L33

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Alam ECE-606 S09 1

www.nanohub.org

NCN

ECE606: Solid State DevicesLecture 33: MOSCAP Electrostatics (II)

Muhammad Ashraful [email protected]


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Outline

Alam ECE-606 S09 2

1. Review

2. Induced charges in depletion and inversion

3. Exact solution of electrostatic problem

4. Conclusion

REF: Chapters 15-18 from SDF


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Topic Map

Alam ECE-606 S09 3

Equilibrium DC Small

signal

Large

Signal

Circuits

Diode

Schottky

BJT/HBT

MOSCAP


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Threshold for Inversion

4

s(V)

l o g1 0QS S( )C / c m2

~ S

~ eqS / 2kBT

~ eqS / 2kBT

0 0

0

2 A oxG s s

ox A

qN xV qN

= +

-Q

+QExposed

Acceptors

VG

VG >VT

2F

0 0

0

22 2 A ox

th

ox

F

A

F

qN xV

qN

= +

Electrons


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What happens when surface potential is 2F ?

5

-Q

+Q

Electrons

Exposed

Acceptors

Electron concentration equals

background acceptor concentration

0 0

0

2 2 2 A oxthox

F

A

FqN xVqN

= +

( )F isE E

in e=

1sn

) ( )(( ) ( )F i bulk i bulk isE E

i

E En e e

=

( )( )i bulk isFi

E En e e

=

2F Fi

n e e =

1sn

+qs EC

EV

Ei

Ei,s

EF

F

i An e N

= =


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A little bit about scaling .

Alam ECE-606 S09

0 0

0

22 2 A oxth

ox

F

A

F

qN xV

qN

= +

-Q

+Q

Electrons

Exposed

Acceptors

Reduce Vth by

Reducing oxide thickness

(from 1000 A in 1970s

to 10 A now)

Increase dielectric constant

(SiO2 historically, HfO2 now in

Intel Penryn)

+qs EC

EV

Ei

Ei,s

EF


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Outline

Alam ECE-606 S09 7

1. Review



4. Conclusion


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Induced charges below Threshold

Alam ECE-606 S09 8

( )( )i bulk isF

s

E E

i

q

n e e

Be

=

1sn

0 0

0

2 A ox

G s s

ox A

qN xV

qN

= +

-Q

+Q

Electrons

Exposed

Acceptors

s(V)

l o g1 0QS S( )

C / c m2

~ S

/

~ S Bq k T

e

~ eqS / 2kBT

+qs

EC

EV

Ei

Ei,sEF


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Integrated charges below Threshold

9

2( )

0

2( )

0

1

1

( )

q xi

B

q xi

B

ne d

dN

dxn

e dN x

=

=

E

2( )

0 0

( )q xi

B

i nn xQ

dx e dxq N

= =

2( )

0

1

( )

q xi

B

ne d

x N

E Winv

Qs

QiQB

+qs

EC

EV

Ei

Ei,s

EF

2

( )

sqis

B

B

inv

ne

k T

qW

xn

N

=

E


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Charges above Threshold

Alam ECE-606 S09 10

-Q

+Q

Electrons

Exposed

Acceptors

VG

VG >VT

0

( )G s o s o

ox

i s Fox

Q

x

Q

V x

+

= + = E

0

(2 )2 2

th F o F o

ox

i Fox

FV x xQQ

+= + =

E

( )0

2() 2 )(

G th s F i s F

o

x

i

o

V VQQ

x

= +

( )Gi ox thC VQ V=

+Q

VG

-Q


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Linear Charge Build-up Above Threshold?

11

~ 2 F

s(V)

( )10log S SQ

~ S

-Q

+

Q

Electrons

Exposed

Acceptors

VG

VG >VT

Small changes s in changes Qia lot .. Change in Qichanges Eox, because Eox=Qi/0e0 Vox is large because Vox=Eox x0, i.e. most of the

drop above 2F goes to Vox.

Acts like a parallel plate capacitor, hence the inversion equation.


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Tunneling Current

Alam ECE-606 S09 12

( ) ( )GqV

T i G m th J Q V qn e T E

=

EC

EV

EF

EG

T( )

( ) ox C

C oxC

T s g g s

m th

qV E

i G m t

i G

h

qVE E

J J J

e qn e e

Q V q

Q

e T

V

n T e

= =

=


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Outline

Alam ECE-606 S09 13

1. Review



4. Conclusion


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A step back: Exact Solution ofQS(

S)

Alam ECE-606 S09 14

EC

EV

EFEG

qS

S> 0

D=

Jn q( )= GR( )

Jp q

( )= GR( )

0 0

0

2 A ox

s sG

ox A

qN xV

qN

= +Approximate

2

0 02

0

( ) ( )D A

si

d qp x n x N N

dx

+ = +


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Normalized Variable (to save some writing)

Alam ECE-606 S09 15

q x( )qS x( )= 0

EC(x) = constant q(x)

qVG > 0

x

, ( )( )

C bulk C E E x

xq

=

( ) ( )( )/

i bulk i x

B B

E Exu

k T q k T

= =

( ) ( )

/

i bulk i surfaceS

B B

SuE E

k T q k T

= =

( )

/

i bu

F

lk FF

B B

E E

k T q k T u

= =


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Normalized Variable (to save some writing!)

Alam ECE-606 S09 16

q x( )qS x( )= 0

qVG > 0

x

[ ( ) ] ( )( ) F Fi

E x E U

i

U

i p x n e n e + = =

[ ( ) ] ( )( ) Fi F

E x E U

i

U

in x n e n e = =

,[ ] ( )F i bulk FE E

D i

U

i N n e n e + = =

,[ ] ( )F i bulk FE EA i

Ui N n e n e

= =


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Poisson-Boltzmann Equation

Alam ECE-606 S09 17

2

20

( ) ( )D A

s

d q

p x n x N N dx

+ = +

( ) ( )2

2

0

( , )F F F FU U U U U U

F

i

i i

B s

qnq d Ue e n e n e

k dg U

xU

T

+ = +

2

2

0

(2 , ) 2i B

s

F

n k Td U

d

dU dU

dx dxxg U U

=

2

2 2( , )12 D

Fd d g U UU dUdx dxdx dx L dx =

2( ) ( )

2

0 0

( , )1

q x kT U

F

x

D

dUd dU

LxUg

dU

=

E

Can be evaluated

at any U


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Exact Solution (continued)

Alam ECE-606 S09 18

2 ( ) 2

2 2

0

1 ( , )( ,

( ))

U x

D

F

D

FF U Ug U Uq x

dU

L LkT

=

E

2( ) ( )

2

0 0

( , )1

q x kT U

D

F

x

d dUL

dUdx

g U U

=

E

( , )B

D

Ss FF U

k T

qLU=E

0 0( , )s s B

G s s s

ox ox

s

D

F

k TV

LU x

qUx F

= + = +

E

0 0

0

2 A ox

s sG

ox A

qN xV

qN

= +

Compare

Vox


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How does the calculation go

19

2 ( ) 2

2 2

0

1 ( , )

( ,

( )

)

U x

D

F

D

FF U U

g U U

xq

dUkT L L

=

E

0 0( , )s s B

G s s s

ox ox D

s F

k TV x x

qLF U U

= + = +E

Begin with a surface potential

Calculate Us and then divide Us by N points.

Calculate g(U, UF) at those points

and integrate to find F(Us,UF)

Find VG.


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Exact Solution

Alam ECE-606 S09 20

Accumulation Depletion Inversion (weak)

Inversion (strong)


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Exact solution is not really exact

Alam ECE-606 S09 21

EC

EV

EF

EG

qS

S> 0

2

2

2( ) ( ) ( )

D A

xd q

p x n x N N dx

+ = +

Wave function should be accounted for

Bandgap widening near the interface

must also should be accounted for.

Assumption of nondegeneracy may not

always be valid

wavefunction, not potential !


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Conclusion

Alam ECE-606 S09 22

Our discussion today was focused on calculating the

induced charge in the depletion and inversion region as a

function of gate bias.

We found that we could calculate the tunneling currentfrom the inversion changes by using the thermionic

emission theory.

We also discussed the exact solution of the MOS-capacitor electrostatics. The exact solution is

mathematically exact, but not necessarily physically exact

solution of the electrostatic problem.

Principles of Semiconductor Devices-L33

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