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Notes for ECE-606: Spring 2013 MOS Caps and FETs Professor Mark Lundstrom Electrical and Computer Engineering Purdue University, West Lafayette, IN USA [email protected] 1 4/11/13 2 outline 1) MOS Electrostatics 2) MOS C-V 3) MOSFETS Lundstrom ECE-606 S13
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Page 1: MOS Caps and FETsnanohub.org/groups/ece606lundstrom/File:MOS Caps and FETs.pdf · Notes for ECE-606: Spring 2013 MOS Caps and FETs Professor Mark Lundstrom Electrical and Computer

Lundstrom ECE-606 S13

Notes for ECE-606: Spring 2013

MOS Caps and FETs

Professor Mark Lundstrom

Electrical and Computer Engineering Purdue University, West Lafayette, IN USA

[email protected]

1 4/11/13

2

outline

1) MOS Electrostatics 2) MOS C-V 3) MOSFETS

Lundstrom ECE-606 S13

Page 2: MOS Caps and FETsnanohub.org/groups/ece606lundstrom/File:MOS Caps and FETs.pdf · Notes for ECE-606: Spring 2013 MOS Caps and FETs Professor Mark Lundstrom Electrical and Computer

MOS capacitor

3

VG

p-Si or n-Si

metal or

heavily doped “polysilicon”

SiO2

tox ≈ 1− 2 nm

Qn = −Cox VG −VT( ) C/cm2

Cox =κ oxε0 tox F/cm2

VG >VT :

Lundstrom ECE-606 S13

EC

EV

Ei

SiO2

EG ≈ 8.9eV

χi

e-band diagram

4

EC

EV

Ei

EF

EG = 1.12eV

Si

aluminum

EFM

ΦM = 4.08 eV

E0

χS = 4.05eVΦS

Lundstrom ECE-606 S13

Page 3: MOS Caps and FETsnanohub.org/groups/ece606lundstrom/File:MOS Caps and FETs.pdf · Notes for ECE-606: Spring 2013 MOS Caps and FETs Professor Mark Lundstrom Electrical and Computer

equilibrium e-band diagram

5

EC

EV

Ei

EF

metal

Δφ

Δφox

Vbi = −φms

φ x( ) = 0 in the bulk

φ x = 0( ) = φS surface potential

Vbi = Δφox +φSLundstrom ECE-606 S13

equilibrium e-band diagram (“ideal”)

6

EC

EV

Ei

SiO2

EG ≈ 8eV

EC

EV

Ei

EF

EG = 1.12eV

Sihypothetical

metal

EFM

ΦM

E0

χS

χi

Lundstrom ECE-606 S13

Page 4: MOS Caps and FETsnanohub.org/groups/ece606lundstrom/File:MOS Caps and FETs.pdf · Notes for ECE-606: Spring 2013 MOS Caps and FETs Professor Mark Lundstrom Electrical and Computer

“ideal” MOS structure

7

EC

EV

SiO2

EC

EV

Ei

EF

EG = 1.12eV

Sihypothetical

metal

EFVbi = 0

“flat band at VG’ = 0

Lundstrom ECE-606 S13

band banding in an MOS device

8 Fig. 16.6, Semiconductor Device Fundamentals, R.F. Pierret

Page 5: MOS Caps and FETsnanohub.org/groups/ece606lundstrom/File:MOS Caps and FETs.pdf · Notes for ECE-606: Spring 2013 MOS Caps and FETs Professor Mark Lundstrom Electrical and Computer

accumulation, depletion, inversion

9

EC

EV

Ei

EF

Si

φ x( ) φ 0( ) E S

WD

x

qφF

φF =kBTqln NA

ni

⎛⎝⎜

⎞⎠⎟

φS

φS < 0

0 < φS < 2φF

φS > 2φF

accumulation:

depletion:

inversion:

Lundstrom ECE-606 S13

MOS electrostatics: depletion

10

EC

EV

Ei

EF

Si

φ x( ) φ 0( )

WD

x

φF

0 < φS < 2φF

Assume the depletion approximation for the charge in the semiconductor.

E S φS( )

φS

WD φS( )QS φS( ) = −qNAWD φS( )

Given the surface potential:

Lundstrom ECE-606 S13

Page 6: MOS Caps and FETsnanohub.org/groups/ece606lundstrom/File:MOS Caps and FETs.pdf · Notes for ECE-606: Spring 2013 MOS Caps and FETs Professor Mark Lundstrom Electrical and Computer

final answers (semiconductor)

11

E x( )

x

What gate voltage produced this surface potential?

P

E S =

qNA

κ Sε0WD

12E SW = φS

E S

WD

WD = 2κ Sε0φSqNA

E S =

2qNAφSκ sε0

QB = − 2qκ sε0NAφS

QB = −qNAWD φS( )

0 <φS < 2φF

Lundstrom ECE-606 S13

relation to gate voltage

12

E x( )

x

E S

WD−tOX

metal

φSΔφOX ′VG =E oxtox + φS

′VG = φS −QB φS( )Cox

QB φS( ) = −qNAWD φS( )Cox =κ oxε0 tox

′VG

Lundstrom ECE-606 S13

Page 7: MOS Caps and FETsnanohub.org/groups/ece606lundstrom/File:MOS Caps and FETs.pdf · Notes for ECE-606: Spring 2013 MOS Caps and FETs Professor Mark Lundstrom Electrical and Computer

MOS electrostatics: inversion

13

EC

EV

Ei

EF

Si

φ x( ) φ 0( )

x

φF

φS ≈ 2φF

φF

WT

WT =2KSε0qNA

2φF⎡

⎣⎢

⎦⎥

1/2

′VG = 2φF −QB 2φF( ) +Qn

Cox

′VT = 2φF −QB 2φF( )Cox

Qn = −Cox VG −VT( )

′VG = 2φF −QS

Cox

Lundstrom ECE-606 S13

delta-depletion approximation

14

ρ

x

metal P

−tox

WT

ρ = −qNA

QB = −qNAWT

Qn = −Cox VG −VT( )

WT = 2κ Sε0 2φFqNA

Lundstrom ECE-606 S13

Page 8: MOS Caps and FETsnanohub.org/groups/ece606lundstrom/File:MOS Caps and FETs.pdf · Notes for ECE-606: Spring 2013 MOS Caps and FETs Professor Mark Lundstrom Electrical and Computer

delta-depletion approximation

15

15

E x( )

x

P

WD

E S

E 0+( ) = 2qNA2φF

κ sε0

E 0( ) = − QS

κ Sε0

Lundstrom ECE-606 S13

16

“exact” MOS electrostatics

QS φS( )C/cm2

φS

~ φS

~ eqφS /2kBT ~ e−qφS /2kBT

φS = 0

′VG = 0

EC = 0

EV = 0

EF

Lundstrom ECE-606 S13

Page 9: MOS Caps and FETsnanohub.org/groups/ece606lundstrom/File:MOS Caps and FETs.pdf · Notes for ECE-606: Spring 2013 MOS Caps and FETs Professor Mark Lundstrom Electrical and Computer

17

outline

1) MOS Electrostatics (Exact) 2) MOS C-V 3) MOSFETS

Lundstrom ECE-606 S13

18

MOS electrostatics

φS

QS ψ S( )C/cm2

~ φS

~ eqφS /2kBT ~ e−qφS /2kBT

φS = 0

′VG = 0

EC = 0

EV = 0

EF

Lundstrom ECE-606 S13

Page 10: MOS Caps and FETsnanohub.org/groups/ece606lundstrom/File:MOS Caps and FETs.pdf · Notes for ECE-606: Spring 2013 MOS Caps and FETs Professor Mark Lundstrom Electrical and Computer

19

space charge and sheet charge density

ρr( ) = q p r( ) − n r( ) + ND

+ r( ) − NA− r( )⎡⎣ ⎤⎦ C/cm3

∇ •D = ρ r( )

QT = ρ r( )∫∫∫ dxdydz C

QT = ρ x( )∫ dx A C

QS =QT

A= ρ x( )∫ dx C cm2

ρ r( ) C/cm3

QS C cm2QT = ρ x( )∫∫∫ dxdydz C (uniform in y-z plane)

Lundstrom ECE-606 S13

20

‘exact’ solution of QS(φS)

∇ • D = ρ

∇ • J n −q( ) = G − R( )

∇ • J p q( ) = G − R( )

d 2φdx2

= −qεSi

p0 φ(x){ }− n0 φ(x){ }+ ND+ − NA

−⎡⎣ ⎤⎦

equilibrium

Lundstrom ECE-606 S13

Page 11: MOS Caps and FETsnanohub.org/groups/ece606lundstrom/File:MOS Caps and FETs.pdf · Notes for ECE-606: Spring 2013 MOS Caps and FETs Professor Mark Lundstrom Electrical and Computer

21

Poisson-Boltzmann equation

qφ(x)

φS > 0

n0 (x) = NCeEF −EC (x )[ ]/kBT

p0 (x) = NVeEV (x )−EF[ ]/kBT

EC (x) = EC (∞)− qφ(x)

n0 (x) = nBeqφ (x )/kBT

p0 (x) = pBe−qφ (x )/kBT

EC

EV

EF

′VG > 0

Lundstrom ECE-606 S13

22

Poisson equation

d 2φdx2

= −ρε

n0 (x) = nBeqφ (x )/kBTp0 (x) = pBe

−qφ (x )/kBT

d 2φdx2

= −qε

p0 (x)− n0 (x)+ ND+ − NA

−⎡⎣ ⎤⎦

d 2φdx2

= −qε

pBe−qφ (x )/kBT − nBe

qφ (x )/kBT + ND+ − NA

−⎡⎣ ⎤⎦

Lundstrom ECE-606 S13

Page 12: MOS Caps and FETsnanohub.org/groups/ece606lundstrom/File:MOS Caps and FETs.pdf · Notes for ECE-606: Spring 2013 MOS Caps and FETs Professor Mark Lundstrom Electrical and Computer

23

Poisson-Boltzmann equation d 2φdx2 x→∞

= 0 = −qε

pB − nB + ND+ − NA

−⎡⎣ ⎤⎦

ND+ − NA

−( ) = − pB + nB

d 2φdx2

= −qε

pBe−qφ (x )/kBT − nBe

qφ (x )/kBT + ND+ − NA

−⎡⎣ ⎤⎦

d 2φdx2

= −qε

pBe−qφ (x )/kBT − nBe

qφ (x )/kBT − pB + nB⎡⎣ ⎤⎦

pB ≈ NA − ND nB = ni2 pB

Lundstrom ECE-606 S13

24

depletion

QS φS( ) = −qNAWD = 2qNAεSiφS

QS φS( ) =QB φS( )

Lundstrom ECE-606 S13

Page 13: MOS Caps and FETsnanohub.org/groups/ece606lundstrom/File:MOS Caps and FETs.pdf · Notes for ECE-606: Spring 2013 MOS Caps and FETs Professor Mark Lundstrom Electrical and Computer

25

inversion

QS φS( ) =QB φS( ) +Qn φS( )

VG

φS > 0

inversion

φS > 2φFQB φS( ) = 2qNAεSiφS

Qn φS( ) = −qn(0) kBT / q

E S

= −qn(0)Winv

n(0) = nBeqφS /kBT

Winv =

kBT / qE S

EC

EV

EF

Lundstrom ECE-606 S13

26

strong inversion (above threshold)

QS φS( ) ≈Qn φS( )

E S φS( ) =Qn εSi

Qn φS( ) = −qn(0) kBT / q

E S

Qn φS( ) = − εSi kBT nBeqφS /2kBT

Lundstrom ECE-606 S13

Page 14: MOS Caps and FETsnanohub.org/groups/ece606lundstrom/File:MOS Caps and FETs.pdf · Notes for ECE-606: Spring 2013 MOS Caps and FETs Professor Mark Lundstrom Electrical and Computer

27

strong inversion criterion

n 0( ) ≈ pB

nBeqφS /kBT = ni

2

NA

eqφS /kBT ≈ pB = NA

φS = 2φF = 2kBTqln NA

ni

⎛⎝⎜

⎞⎠⎟

Lundstrom ECE-606 S13

28

weak inversion (sub-threshold)

QS φS( ) ≈QB φS( )

VG

εS > 0

weak inversion

φS < 2ψ B

E S φS( ) = 2qNAεSiφS εSi

Qn φS( ) = −qn(0) kBT / q

E S

Qi φS( ) = −qnBeqφS /kBT kBT / q

2qεSiNAφS εSi

EC

EV

EF

Lundstrom ECE-606 S13

Page 15: MOS Caps and FETsnanohub.org/groups/ece606lundstrom/File:MOS Caps and FETs.pdf · Notes for ECE-606: Spring 2013 MOS Caps and FETs Professor Mark Lundstrom Electrical and Computer

29

summary

QS φS( )C/cm2

ψS

QS ≈Qacc ~ e−qφS /2kBT

QS ≈QB ~ φS − kBT q( )

QS ≈Qn ~ eqφS /2kBT

Wacc/inv ≈ (kBT / q) E S Lundstrom ECE-606 S13

Lundstrom ECE-606 S13 30

assumptions

1)  Boltzmann statistics (not valid above threshold)

2)  Fully ionized dopants, uniform doping (not valid in practice)

3)  No quantum confinement (not valid above threshold)

EC

EV

EF VG

quantum well

Page 16: MOS Caps and FETsnanohub.org/groups/ece606lundstrom/File:MOS Caps and FETs.pdf · Notes for ECE-606: Spring 2013 MOS Caps and FETs Professor Mark Lundstrom Electrical and Computer

31

outline

1) MOS Electrostatics 2) MOS C-V 3) MOSFETS

Lundstrom ECE-606 S13

band banding in an MOS device

32 Fig. 16.6, Semiconductor Device Fundamentals, R.F. Pierret

Page 17: MOS Caps and FETsnanohub.org/groups/ece606lundstrom/File:MOS Caps and FETs.pdf · Notes for ECE-606: Spring 2013 MOS Caps and FETs Professor Mark Lundstrom Electrical and Computer

33

capacitance

tox

W φS( )

κ ox

κ Si

CS =κ Sε0WD φS( )

Cox =κ oxε0tox

1C

=1Cox

+1CS

C =CSCox

CS + Cox

C =Cox

1+ Cox CS

C = Cox

1+κ oxWD φS( )

κ Stox

Lundstrom ECE-606 S13

34

s.s. gate capacitance vs. d.c. gate bias

C

VG′C = Cox

1+κ oxWD φS( )

κ Stox

accumulation

depletion

inversion

VT′

flat band

Cox

Lundstrom ECE-606 S13

Page 18: MOS Caps and FETsnanohub.org/groups/ece606lundstrom/File:MOS Caps and FETs.pdf · Notes for ECE-606: Spring 2013 MOS Caps and FETs Professor Mark Lundstrom Electrical and Computer

35

capacitance vs. gate voltage

C

VG′C = Cox

1+κ oxWD φS( )

κ Stox

accumulation depletion

inversion

VT′

flat band

Cox

Lundstrom ECE-606 S13

36

high frequency vs. low frequency

C

VG′C = Cox

1+κ oxWD φS( )

κ Stox

accumulation depletion

inversion

VT′

flat band

Cox

high frequency

low frequency

Lundstrom ECE-606 S13

Page 19: MOS Caps and FETsnanohub.org/groups/ece606lundstrom/File:MOS Caps and FETs.pdf · Notes for ECE-606: Spring 2013 MOS Caps and FETs Professor Mark Lundstrom Electrical and Computer

37

VT equation

′VG = φS −QS φS( )Cox

′VT = 2φF −QB 2φF( )Cox

VT =VFB + 2φF −QB 2φF( )Cox

VFB = φms −QF

Cox

′VG =VG −VFB

Lundstrom ECE-606 S13

38

outline

1) MOS Electrostatics 2) MOS C-V 3) MOSFETS

Lundstrom ECE-606 S13

Page 20: MOS Caps and FETsnanohub.org/groups/ece606lundstrom/File:MOS Caps and FETs.pdf · Notes for ECE-606: Spring 2013 MOS Caps and FETs Professor Mark Lundstrom Electrical and Computer

39

nanoscale MOSFETs

source drain

SiO

2

silicon

channel ~ 32 nm

gate oxide EOT ~ 1.1 nm

gate electrode

S G D

Lundstrom ECE-606 S13

40

MOSFET IV characteristic

VGS

(Courtesy, Shuji Ikeda, ATDF, Dec. 2007)

S

D

G

circuit

symbol

VDS

VGS

IDS

gate-voltage controlled resistor

gate-voltage controlled

current source

Lundstrom ECE-606 S13

Page 21: MOS Caps and FETsnanohub.org/groups/ece606lundstrom/File:MOS Caps and FETs.pdf · Notes for ECE-606: Spring 2013 MOS Caps and FETs Professor Mark Lundstrom Electrical and Computer

41

MOSFET IV: low VDS

VG>VT VD 0

ID = −WQn x( )υx (x)

ID =W Cox VGS −VT( )µeffE x

Qn x( ) ≈ −Cox VGS −VT( )

ID = WL

µeffCox VGS −VT( )VDS

VGS

gate-voltage controlled resistor

E x = −

VDSL

L

Lundstrom ECE-606 S13

42

MOSFET IV: “pinch-off” at high VDS

VG VD 0

Qn x( ) = −Cox VGS −VT −V (x)( )

V x( ) = VGS −VT( )

Qn x( ) ≈ 0

Lundstrom ECE-606 S13

Page 22: MOS Caps and FETsnanohub.org/groups/ece606lundstrom/File:MOS Caps and FETs.pdf · Notes for ECE-606: Spring 2013 MOS Caps and FETs Professor Mark Lundstrom Electrical and Computer

43

MOSFET IV: high VDS

VG VD 0

ID = −W Cox VGS −VT( )µeffE x (0)

ID = WL

µeffCox VGS −VT( )22

E x (0) ≈ −

VGS −VTL

Qi x( ) = −Cox VGS −VT −V (x)( )�

V x( ) = VGS −VT( )

VGS

gate-voltage controlled

current source

ID = −WQn x( )υx (x) = −WQn 0( )υx (0)

Lundstrom ECE-606 S13

44

velocity saturation

electric field V/cm --->

velo

city

cm

/s --

->

107

104

υ = µE�

υ = υ sat

VDSL

≈ 1.0V30nm

≈ 3×105 V/cm

105

Lundstrom ECE-606 S13

Page 23: MOS Caps and FETsnanohub.org/groups/ece606lundstrom/File:MOS Caps and FETs.pdf · Notes for ECE-606: Spring 2013 MOS Caps and FETs Professor Mark Lundstrom Electrical and Computer

45

MOSFET IV: velocity saturation

VG VD 0

ID = −WQn x( )υx (x)

ID =W Cox VGS −VT( )υ satID =WCoxυsat VGS −VT( )

E x >> 104

(Courtesy, Shuji Ikeda, ATDF, Dec. 2007)

Lundstrom ECE-606 S13

µm( )

Velo

city

(cm

/s) à

46

carrier transport nanoscale MOSFETs

υSAT

D. Frank, S. Laux, and M. Fischetti, Int. Electron Dev. Mtg., Dec., 1992.

µm( )

EC

quasi-ballistic

Lundstrom ECE-606 S13

Page 24: MOS Caps and FETsnanohub.org/groups/ece606lundstrom/File:MOS Caps and FETs.pdf · Notes for ECE-606: Spring 2013 MOS Caps and FETs Professor Mark Lundstrom Electrical and Computer

47

outline

1) MOS Electrostatics 2) MOS C-V 3) MOSFETS

Lundstrom ECE-606 S13

MOS summary (p-Si)

48

WD = 2κ Sε0φSqNA

cm

E S =

2qNAφSκ sε0

V/cm

QB = − 2qκ sε0NAφS C/cm2

′VG = −QB φS( )Cox

+φS

COX =κ oxε0 tox F/cm2

0 < φS < 2φF φS > 2φF

WT =2κ Sε0 2φFqNA

cm

E S = ? V/cm

QB = − 2qκ sε0NA2φF C/cm2

′VT = −QB 2φF( )Cox

+ 2φF

Qn = −Cox VG −VT( )

φF = kBTqln NA

ni

⎛⎝⎜

⎞⎠⎟

Lundstrom ECE-606 S13


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