MOS Caps and FETsnanohub.org/groups/ece606lundstrom/File:MOS Caps and FETs.pdf · Notes for...

transcript

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

lundstro@purdue.edu

1 4/11/13

outline

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

MOS capacitor

p-Si or n-Si

metal or

heavily doped “polysilicon”

tox ≈ 1− 2 nm

Qn = −Cox VG −VT( ) C/cm2

Cox =κ oxε0 tox F/cm2

VG >VT :

EG ≈ 8.9eV

e-band diagram

EG = 1.12eV

aluminum

ΦM = 4.08 eV

χS = 4.05eVΦS

equilibrium e-band diagram

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”)

EG ≈ 8eV

EG = 1.12eV

Sihypothetical

“ideal” MOS structure

EG = 1.12eV

Sihypothetical

EFVbi = 0

“flat band at VG’ = 0

band banding in an MOS device

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

accumulation, depletion, inversion

φ x( ) φ 0( ) E S

φF =kBTqln NA

⎛⎝⎜

⎞⎠⎟

φS < 0

0 < φS < 2φF

φS > 2φF

accumulation:

depletion:

inversion:

MOS electrostatics: depletion

φ x( ) φ 0( )

0 < φS < 2φF

Assume the depletion approximation for the charge in the semiconductor.

E S φS( )

WD φS( )QS φS( ) = −qNAWD φS( )

Given the surface potential:

final answers (semiconductor)

E x( )

What gate voltage produced this surface potential?

κ Sε0WD

12E SW = φS

WD = 2κ Sε0φSqNA

2qNAφSκ sε0

QB = − 2qκ sε0NAφS

QB = −qNAWD φS( )

0 <φS < 2φF

relation to gate voltage

E x( )

WD−tOX

φSΔφOX ′VG =E oxtox + φS

′VG = φS −QB φS( )Cox

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

MOS electrostatics: inversion

φ x( ) φ 0( )

φS ≈ 2φF

WT =2KSε0qNA

2φF⎡

⎣⎢

⎦⎥

′VG = 2φF −QB 2φF( ) +Qn

′VT = 2φF −QB 2φF( )Cox

Qn = −Cox VG −VT( )

′VG = 2φF −QS

delta-depletion approximation

metal P

−tox

ρ = −qNA

QB = −qNAWT

WT = 2κ Sε0 2φFqNA

delta-depletion approximation

E x( )

E 0+( ) = 2qNA2φF

κ sε0

E 0( ) = − QS

κ Sε0

“exact” MOS electrostatics

QS φS( )C/cm2

~ eqφS /2kBT ~ e−qφS /2kBT

φS = 0

′VG = 0

EC = 0

EV = 0

outline

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

MOS electrostatics

QS ψ S( )C/cm2

~ eqφS /2kBT ~ e−qφS /2kBT

φS = 0

′VG = 0

EC = 0

EV = 0

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)

‘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

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

′VG > 0

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

−⎡⎣ ⎤⎦

Poisson-Boltzmann equation d 2φdx2 x→∞

= 0 = −qε

pB − nB + ND+ − NA

−⎡⎣ ⎤⎦

ND+ − NA

−( ) = − pB + nB

d 2φdx2

= −qε

qφ (x )/kBT + ND+ − NA

−⎡⎣ ⎤⎦

d 2φdx2

= −qε

qφ (x )/kBT − pB + nB⎡⎣ ⎤⎦

pB ≈ NA − ND nB = ni2 pB

depletion

QS φS( ) = −qNAWD = 2qNAεSiφS

QS φS( ) =QB φS( )

inversion

QS φS( ) =QB φS( ) +Qn φS( )

φS > 0

inversion

φS > 2φFQB φS( ) = 2qNAεSiφS

Qn φS( ) = −qn(0) kBT / q

= −qn(0)Winv

n(0) = nBeqφS /kBT

Winv =

kBT / qE S

strong inversion (above threshold)

QS φS( ) ≈Qn φS( )

E S φS( ) =Qn εSi

Qn φS( ) = − εSi kBT nBeqφS /2kBT

strong inversion criterion

n 0( ) ≈ pB

nBeqφS /kBT = ni

eqφS /kBT ≈ pB = NA

φS = 2φF = 2kBTqln NA

⎛⎝⎜

⎞⎠⎟

weak inversion (sub-threshold)

QS φS( ) ≈QB φS( )

εS > 0

weak inversion

φS < 2ψ B

E S φS( ) = 2qNAεSiφS εSi

Qi φS( ) = −qnBeqφS /kBT kBT / q

2qεSiNAφS εSi

summary

QS φS( )C/cm2

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)

quantum well

outline

band banding in an MOS device

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

capacitance

W φS( )

CS =κ Sε0WD φS( )

Cox =κ oxε0tox

C =CSCox

CS + Cox

C =Cox

1+ Cox CS

C = Cox

1+κ oxWD φS( )

κ Stox

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

VG′C = Cox

1+κ oxWD φS( )

κ Stox

accumulation

depletion

inversion

flat band

capacitance vs. gate voltage

VG′C = Cox

1+κ oxWD φS( )

κ Stox

accumulation depletion

inversion

flat band

high frequency vs. low frequency

VG′C = Cox

1+κ oxWD φS( )

κ Stox

accumulation depletion

inversion

flat band

high frequency

low frequency

VT equation

′VG = φS −QS φS( )Cox

′VT = 2φF −QB 2φF( )Cox

VT =VFB + 2φF −QB 2φF( )Cox

VFB = φms −QF

′VG =VG −VFB

outline

nanoscale MOSFETs

source drain

silicon

channel ~ 32 nm

gate oxide EOT ~ 1.1 nm

gate electrode

MOSFET IV characteristic

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

circuit

symbol

gate-voltage controlled resistor

gate-voltage controlled

current source

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

gate-voltage controlled resistor

E x = −

MOSFET IV: “pinch-off” at high VDS

VG VD 0

Qn x( ) = −Cox VGS −VT −V (x)( )

V x( ) = VGS −VT( )

Qn x( ) ≈ 0

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( )

gate-voltage controlled

current source

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

velocity saturation

electric field V/cm --->

υ = µE�

υ = υ sat

≈ 1.0V30nm

≈ 3×105 V/cm

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)

µm( )

/s) à

carrier transport nanoscale MOSFETs

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

µm( )

quasi-ballistic

outline

MOS summary (p-Si)

WD = 2κ Sε0φSqNA

2qNAφSκ sε0

QB = − 2qκ sε0NAφS C/cm2

′VG = −QB φS( )Cox

COX =κ oxε0 tox F/cm2

0 < φS < 2φF φS > 2φF

WT =2κ Sε0 2φFqNA

E S = ? V/cm

QB = − 2qκ sε0NA2φF C/cm2

′VT = −QB 2φF( )Cox

+ 2φF

φF = kBTqln NA

⎛⎝⎜

⎞⎠⎟

MOS Caps and FETsnanohub.org/groups/ece606lundstrom/File:MOS Caps and FETs.pdf · Notes for...

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