ece606: solid state devices lecture 22 moscap frequency...

Klimeck – ECE606 Fall 2012 – notes adopted from Alam

ECE606: Solid State DevicesLecture 22

MOScap Frequency ResponseMOSFET I-V Characteristics

Gerhard [email protected]


Outline

2

1. Background2. Small signal capacitances3. Large signal capacitance4. Intermediate Summary

5. Sub-threshold (depletion) current6. Super-threshold, inversion current7. Conclusion

Ref: Sec. 16.4 of SDF


Topic Map

3

Equilibrium DC Small signal

Large Signal

Circuits

Diode

Schottky

BJT/HBT

MOSCAP


Small Signal Equivalent Circuit

4

p

G0

CD

CJ

CJ/Co

1

VG

For insulated devices, consider only majority carrier junction capacitance CJ

High frequency

Low frequency

Gate semiconductorG0 is small (only tunelling current)

kHz


Small Signal Equivalent Circuit

5

p

G0

CD

CJ

CJ/Co

1

VG

For insulated devices, consider only majority carrier junction capacitance CJ

High frequency

Low frequency

Gate semiconductorG0 is small (only tunelling current)

kHz


Junction Capacitance

6

p-Si

+-~

sinG SV tυ ω+ CG ≡dQG

dVG

=d −QS( )

dVG

SGS

OCV

Qψ= −

( ) ( )1G

S

S

OS C

dV d

d Q d Q

ψ= +− −

11 1

S OG CC C= +

Grey: gateRed: oxideBlue: inversionGreen: depletion


Junction Capacitance

7

1 1 1

SG OCC C= +

( )SS

S

Q

dC

d

ψ−

≡

VG

OC

SC

( )SSQ ψCs is not fixed

Remember the Qs vs phi_sfigure we mentioned in the previous lecture

which we already understand!


Definition of m for later use

8

( )1 S Om C C= + VG

OC

CS

ψ S

O GS G

O S

C VV

C C mψ ≡=

+

‘body effect coefficient’

( )01 S O Tm x Wκ κ= +

1.1 ≤ m ≤ 1.4

in practice:

WT depends on the voltage


Outline

9





J u n c t i o n C a p a c i t a n c e in accumulation

10

0, 0

0

oxj accC C

x

κ ε≈ ≡ ,

,

0, ,

s acc

s

sj acc s ac

ac a

o

o c

c

cc

C CWC

C

C

C κ ε= ≡+

Wacc

-Q

+Q

Low frequencyC/Co

1

+Q

Arrows is the charge induced by small signalTwo blue arrow� C0Two red arrow � CsThese two capacitors are in series


Junction Capacitance in depletion

1

+Q

VG

-Q

0 0

0,

01j ds

es

sp

C

C

C C

CC

C C= =

+ +

0

1 1o G

s

W

x

V

Vδ

κκ

= + −

2

0 00 2

A AG

o s

qN qNWV x

W

κ ε κ ε

= +

1

o

G

C

V

Vδ

=+

Low frequencyC/Co

VG

1

0

0

0

01 so

W

C

x

ε κκ ε=+

First term is the V drops on oxideSecond term is band bending


Junction capacitance in inversion

12

-Q

+Q

Electrons

ExposedAcceptors

VG

VG’>VT’

0, 0

0

sj invC C

x

κ ε≈ ≡

0,

o inv sj inv inv

o inv inv

C CC C

C C W

κ ε= ≡+

Low frequencyC/Co

VG

1

Time to generate inversion charge. ms to µs


Equivalent Oxide Thickness

13

Qi = −CG VG − VT( )

,ino

G j inv ov

v oin

C

C

CC C C

C= = <

+

s onv

inviC

W

κ ε≡0

o ooC

x

κ ε=

ox O

e cG

leEOTC

κ ε= ox OO Oinelec

sv

O

x xE WOTκ εκ ε

= + >

‘Equivalent oxide thickness - electrical ’

Low frequency

C/Cox

1

VG


High frequency curve at inversion

14

-∆Q

∆Q

WT

What about high frequency part of the curve?

0, 0

0

sj invC C

x

κ ε≈ ≡

High frequency

C/Co

1

VGVTH

The red region contribute to the C, as if it is still in depletion

-∆Q

∆Q

WT

High FrequencyLow frequency


Response Time

15

VG

VG’ >VT’

VGVG

Dielectric Relaxation

0s

στ =κ ε

SRH Recombination-Generation

2

1 1( ) ( )i i

n p n p

np n nR

p p n n

− −= →τ + + τ + τ + τ

Ref. Lecture no. 15


High frequency response in MOS-C

16

High frequency

Low frequency

C/Co

1

VG

-∆Q

∆Q

WT

Low Frequency

-∆Q

∆Q

WT

High Frequency


Ideal vs. Real C-V Characteristics

17

Blue dot: Flat band voltage …

Red dot: Threshold voltage …

C/CoC/Co


Outline

18





Topic Map

19


Large Signal

Circuits

Diode

Schottky

BJT/HBT

MOS


Large Signal Deep Depletion

20

0 0,

0

0

1

sj dep

oxs

s

C C CC

WC Cx

κκ

= =+ +

1

o

G

C

V

Vδ

=+

(even beyond threshold)High frequency

Low frequency

C/Co

VG

-∆Q

∆QWdm

NA

x0

ρ(x)

-∆Q

∆Q Wdm

NA

For large signal, the green do not have time to response;� continue to deplete.Small signal there is green because of the DC bias builds it.


Relaxation from Deep Depletion

21

High frequency

Low frequency

C/Cox

1

Deep depletion

-∆Q

∆Q

Wdm

NA

x0

ρ(x)

-∆Q

∆Q

Wdm

NA

x0

ρ(x)

-∆Q

∆Q

Wdm

ρ(x)

Depending on the measurement frequency, it will either merge with low-freq. or high-freq. curve.

High frequencylow frequency


Ideal vs. Real C-V Characteristics

22

Flat band voltage …

Threshold voltage …

C/CoC/Co

VG VGideal

Real case

If the signal rise slower, it will be closer to ideal case


Low or High frequency?

23

typically observe hi-frequency CV

G =ni

2τp-Si

typically observe low-frequency CVNo deep-depletion as well

p-Si

n+-Si n+-Si

What happens if I shine light on a MOS capacitor?


Intermediate Summary

24

1) Since current flow through the oxide is small, we are

primarily interested in the junction capacitance of the

MOS-capacitor.

2) High frequency of MOS-C is very different than low-

frequency C-V.

3) In MOSFET, we only see low frequency response.

4) Deep depletion is an important consideration for

MOS-capacitor that does not happen in MOSFETs.


Topic Map

25


Large Signal

Circuits

Diode

Schottky

BJT/HBT

MOSCAP

MOSFET


Outline

26





Subthreshold Region (VG < Vth)

27

n+ n+p

GS D

∆n

y

EC

EV

No voltages

Look

s lik

e 2

pnju

nctio

nsIn

2D

–no

t 1D

What happens with a gate bias?Remember this is a 2D device!

Back-gate grounded => fixed potential

MO

Sca

pas

d

iscu

ssed

bef

ore

wit

h s

urf

ace

psi

_s



28

∆n

y

EC

EV

1 2D n

ch

QI q

QD

L

−=

log ID

Ψs

SubthresholdSlope =>60mV/dec

( )2

/ 1GqViinv

c A

m

h

D nqWW e

L Nβ≈ −

Q1

Q2

invW

W

( )2

1sq

ch

iv

Ain

ne

NW

Dq W

Lψ β

= × ×

−

or VG/m

High injection

m = body coefficient typically 1.1~1.4



29

∆n

EC

EV

Q1

Q2


Recall the definition of body coefficient (m)

30

( )1 S Om C C= +VG

OC

CS

ψ S

O GS G

O S

C VV

C C mψ = ≡

+

‘Body Effect Coefficient’

( )01 S O Tm x Wκ κ= +

1.1 ≤ m ≤ 1.4

in practice:


Outline

31





Post-Threshold MOS Current (VG>Vth)

32

( )0

= − ∫DSV

D effch

iQI VVW

dL

µ

1) Square Law

2) Bulk Charge

3) Simplified Bulk Charge

4) “Exact” (Pao-Sah or Pierret-Shields)

[ ]) )( = − − −i G G TC V VQ mVV

[ ]) )( = − − −G Ti GC V VV VQ

( )2 22( )

+ = − − − − −

Si A BG Gi FB B

O

Qq N V

C V V VVC

ε φψ

Formula overview –derivation to follow


2ψ B

Effect of Gate Bias

33

P

S G D

B

VGS

n+ n+

Gated doped or p-MOS with adjacent n+ regiona) gate biased at flat-bandb) gate biased in inversion

A. Grove, Physics of Semiconductor Devices, 1967.

WDM

VGS > VT

WD

VBI

y

No source-drain bias


The Effect of Drain Bias

Alam ECE-606 S09 34

2D band diagram for an n-MOSFET

a) device

b) equilibrium (flat band)

c) equilibrium (ψS > 0)

d) non-equilibrium with VG and VD >0 applied

SM. Sze, Physics of Semiconductor Devices, 1981 and Pao and Sah.

FNDepletion very different in source and drain side Gate voltage must ensure channel formation=> LARGE


Effect of a Reverse Bias at Drain

35

ψ S = 2ψ B + VRVBI + VR

WDM

VGS > VT VR( )

WD

VRVR

Gated doped or p-MOS with adjacent, reverse-biased n+ regiona) gate biased at flat-bandb) gate biased in depletionc) gate biased in inversion

A. Grove, Physics of Semiconductor Devices, 1967.

VR


Inversion Charge in the Channel

36

P

S G

B

n+ n+ ( )

( ( 0)( ) )i ox G th

TA T

Q C V V V

q VVN W W

= − − −+ − =

VG=0VD=0

VVB

VG=0VD>0

VD

VB

VG>0VD>0

D


Inversion Charge at one point in Channel

37

VGVD

VBVG

( )* 2( )A

th Fox

Tq W VNV V

Cφ= + −2

( 0)A Tth F

ox

Wq

C

VNV φ= =−

( )* ( ( 0))T TAth th

ox

qNV V

W WVV

V

C

=−= + −

*( )i ox G thQ C V V= − −

V


Approximations for Inversion Charge

38

( )( ) ( ) ( 0)i O G th A T TQ C V V V qN W V W V= − − − + − =

( ) ( )2 2( ) 22 = − − − + −

+O G th S o A S oB A BC V V V q N q NVφ φκ ε κ ε

( )i ox G thQ C V V V≈ − − −

( )i ox G thQ C V VmV≈ − − −

Approximations:

Square law approximation …

Simplified bulk charge approximation …


The MOSFET

39

P

S G D

B

n+ n+

VS = 0 VD > 0

Fn = Fp = EF

Fn = Fp − qVD

Fn increasingly negative from source to drain(reverse bias increases from source to drain)


Elements of Square-law Theory

40

VG VD>00

y

GCA : Ey << Ex

[ ]( ) ( )i ox G thQ y C V V mV y= − − −

VG V VG


Another view of Channel Potential

41

FP

FNFN

FPFP

FN

EC

EFEV

N+ N+P-doped

Source Drain

x

x


Square Law Theory

42

VG VD>00

VD

Q

V1, 1,

1, 0

( )D

ii

i N i N

V

Dox G th

i N

J dyQ dV

Jdy C V V mV dV

µ

µ

= =

=

=

= − −

∑ ∑

∑ ∫

1 1 1 1

1

2 2 2 2

2

3 3 3 3

3

4 4 4 4

4

dVJ Q Q

dy

dVJ Q Q

dy

dVJ Q Q

dy

dVJ Q Q

dy

µ µ

µ µ

µ µ

µ µ

= =

= =

= =

= =

E

E

E

E

( )2

2ox D

D G th Dch

C VJ V V V m

L

µ = − −

Q1Q2 Q3 Q4


Square Law or Simplified Bulk Charge Theory

43

( )2

2o

D G Tch

W CI V V

mL

µ= −ID

VDS

VGS

( )D o G T D

WI C V V V

Lµ= −

VDSAT = VGS − VT( )/ m

( )2

2ox D

D G th Dch

C VI W V V V m

L

µ = − −

( )2

2ox D

D G th Dch

C VJ V V V m

L

µ = − −

( ) ( )*,0= = − − ⇒ = −D

G th D D sat G th

dIV V mV V V V m

dV


Why does the curve roll over?

44

( )2

2o

D G Tch

W CI V V

mL

µ= −

ID

VDS

VGS

VDSAT = VGS − VT( )/ m

( )i ox G thQ C V V mV≈ − − −

VG VD>00

Q

loss of inversion

Expression is only valid for voltages up to pinch-off


Linear Region (Low VDS)

45

ID

VDS small

VGS

( )D o G T Dch

DS

CH

WI C V V V

L

V

R

µ= −

=

Actual

SubthresholdConduction

VT

Mobility degradation at high VGS

Slope gives mobility

Intercept gives VT

Can get VT also from C-V


Summary

46

1) MOSFET differs from MOSCAP in that the field from

the S/D contacts now causes a current to flow.

2) Two regimes, diffusion-dominated Subthreshold and

drift-dominated super-threshold characteristics,

define the ID-VD-VG characteristics of a MOSFET.

3) The simple bulk charge theory allows calculation of

drain currents and provide many insights, but there

are important limitations of the theory as well.

ece606: solid state devices lecture 22 moscap frequency...

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