lec16 mos es

ECE 663

MOSFETs

ECE 663

A little bit of history..

Substrate

Channel Drain

InsulatorGate

Operation of a transistorVSG > 0 n type operation

Positive gate bias attracts electrons into channelChannel now becomes more conductive

More electrons

Source

VSD

VSG

Substrate

Channel Drain

InsulatorGate

Operation of a transistor

Transistor turns on at high gate voltageTransistor current saturates at high drain bias

Source

VSD

VSG

Substrate

Channel Drain

InsulatorGate

Source

VSD

VSG

Start with a MOS capacitor

ECE 663

MIS Diode (MOS capacitor) – Ideal

W

QuestionsWhat is the MOS capacitance? QS(S)

What are the local conditions during inversion? S,cr

How does the potential vary with position? (x)

How much inversion charge is generated at the surface? Qinv(x,S)

Add in the oxide: how does the voltage divide? S(VG), ox(VG)

How much gate voltage do you need to invert the channel? VTH

How much inversion charge is generated by the gate? Qinv(VG)

What’s the overall C-V of the MOSFET? QS(VG)

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EC

EF

EV

Ei

Ideal MIS Diode n-type, Vappl=0

Assume Flat-band at equilibrium

qS

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02

B

gmms q

E

Ideal MIS Diode n-type, Vappl=0

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Ideal MIS Diode p-type, Vappl=0

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Ideal MIS Diode p-type, Vappl=0

02

B

gmms q

E

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Accumulation

Pulling in majority carriers at surface

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But this increases the barrier for current flow !!

n+ p n+

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Depletion

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Need CB to dip below EF. Once below by B, minority carrier density trumps the intrinsic density. Once below by 2B, it trumps the major carrier density (doping) !

Inversion

B

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Sometimes maths can help…

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P-type semiconductor Vappl0

Convention for p-type: positive if bands bend down

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Ideal MIS diode – p-type

enenenenn pkTq

pkTEqE

ikTEE

ipFiFi

0/

0/)(/)( '

epepp pkTq

pp 0/

0

kTq

CB moves towards EF if > 0 n increases

VB moves away from EF if > 0 p decreases

ECE 663

Ideal MIS diode – p-type

At the semiconductor surface, = s

senn ps 0

sepp ps 0

s < 0 - accumulation of holes

s =0 - flat band

B> s >0 – depletion of holes

s =B - intrinsic concentration ns=ps=ni

s > B – Inversion (more electrons than holes)

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Surface carrier concentrationsenn ps

0sepp ps

0

EC EF

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Want to find , E-field, Capacitance

• Solve Poisson’s equation to get E field, potential based on charge density distribution(one dimension)

s

s

dxd

dxd

Ddxdk

/

1//

2

2

0

)()( ppAD npNNqx

EE

E

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• Away from the surface, = 0

• and

00 ppAD pnNN

enepnp pppp 00

)1()1( 002

2

enepqdxd

pps

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Solve Poisson’s equation:

)1()1( 002

2

enepqdxd

pps

E = -d/dx

d2/dx2 = -dE/dx = (dE/d).(-d/dx) = EdE/d

)1()1( 002

2

enepqdxd

pps

EdE/d

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• Do the integral:• LHS:

• RHS:

• Get expression for E field (d/dx):

dxdxxxdx

x 0

2

2

x x

x dxdxe0 0

,

11

2 0

002

2 epn

eqp

qkTE

p

p

s

pfield

Solve Poisson’s equation:

ECE 663

2

1

0

0

0

0 11,

e

pn

epn

Fp

p

p

p

Define:

0

20 p

s

p

sD qpqp

kTL Debye Length

Then:

0

0,2

p

p

Dfield p

nF

qLkTE

+ for > 0 and – for < 0

> 0

E > 0

< 0E < 0

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0

0,2

p

ps

DSss p

nF

qLkTEQ

Use Gauss’ Law to find surface charge per unit area

2

1

0

0 112

sp

ps

Ds

sS epn

eqL

kTQ

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Accumulation to depletion to strong Inversion

• For negative , first term in F dominates – exponential• For small positive , second term in F dominates -

• As gets larger, second exponential gets big

10

0

p

p

pen

B = (kT/q)ln(NA/ni) = (1/)ln(pp0/√pp0np0)

(np0/pp0) = e-2B

S > 2B

Questions What is the MOS capacitance? QS(S)








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Charges, fields, and potentials

• Charge on metal = induced surface charge in semiconductor

• No charge/current in insulator (ideal)metal insulsemiconductor

depletioninversion

SAnM QWqNQQ

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Electric Field Electrostatic Potential

Charges, fields, and potentials

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Depletion Region

11

2 0

002

2 epn

eqp

qkTE

p

p

s

pfield

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= s(1-x/W)2

Wmax = 2s(2B)/qNA

B = (kT/q)ln(NA/ni)

Depletion Region

Couldn’t we just solve

this exactly?

U =

US = S

UB = B

Exact Solution

d/dx = -(2kT/qLD)F(B,np0/pp0)

dU/F(U) = x/LD

U

US

F(U) = [eUB(e-U-1+U)-e-UB (eU-1-U)]1/2

Exact Solution

dU’/F(U’,UB) = x/LDU

US

F(U,UB) = [eUB(e-U-1+U) + e-UB (eU-1-U)]1/2

= qni[eUB(e-U-1) – e-UB(eU-1)]

Exact Solution

NA = 1.67 x 1015

Qinv ~ 1/(x+x0)

x0 ~ LD . factor

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Threshold Voltage for Strong Inversion

• Total voltage across MOS structure= voltage across dielectric plus s

Bi

SSiT C

QVinversionstrongV 2)_(

)2(2)(2)( max BAsA

ssAAS qN

qNinvqNWqNSIQ

Bi

BAsT C

qNV

2

)2(2

oxVi/tox = ss/(W/2) Before Inversion

After inversion there is a discontinuity in D due to surface Qinv

Vox (at threshold) = s(2B)/(Wmax/2)Ci =

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Notice Boundary Condition !!

Bi

BAsT C

qNV

2

)2(2

Local Potential vs Gate voltage VG = Vfb + s + (stox/ox) √(2kTNA/0s)[s + es-2B)]1/2

Initially, all voltage drops across channel (blue curve). Above threshold, channel potential stays pinned to 2B, varying only logarithmically, so that most of the gate voltage drops across the oxide (red curve).

oxs

Look at Effective charge width

Initially, a fast increasing channel potential drops across increasing depletion widthEventually, a constant potential drops across a decreasing inversion layer width, so field keeps increasing and thus matches increasing field in oxide

~Wdm/2

~tinv

Charge vs Local Potential Qs ≈ √(20skTNA)[s + es-2B)]1/2

Beyond threshold, all charge goes to inversion layer

How do we get the curvatures?

EXACTAdd other terms and keepLeading term

NEW

Inversion Charge vs Gate voltage Q ~ es-2B), s

- 2B ~ log(VG-VT)Exponent of a logarithm gives a linear variation of Qinv with VG

Qinv = -Cox(VG-VT)

Why Cox?

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Capacitance

0

0

0

0

,

11

2

p

pS

p

p

D

SSD

pn

F

epne

LQC

ss

For s=0 (Flat Band):

D

SD L

bandflatC )_(

Expand exponentials….. ........!3!2

132

xxxe x

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Capacitance of whole structure

• Two capacitors in series:

Ci - insulator

CD - Depletion

Di CCC111 OR

Di

Di

CCCCC

dC i

i

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Capacitance vs Voltage

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Flat Band Capacitance

• Negative voltage = accumulation – C~Ci

• Zero voltage – Flat Band

i

Ds

i

si

Dis

D

siDiFB

LdLd

LdCCC

11111

FBCCV 00

D

iFB Ld

Cs

i

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CV• As voltage is increased, C goes through

minimum (weak inversion) where d/dQ is fairly flat

• C will increase with onset of strong inversion

• Capacitance is an AC measurement

• Only increases when AC period long wrt minority carrier lifetime

• At “high” frequency, carriers can’t keep up – don’t see increased capacitance with voltage

• For Si MOS, “high” frequency = 10-100 Hz

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CV Curves – Ideal MOS Capacitor

max

'min Wd

Cs

i

i

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But how can we operate gate attoday’s clock frequency (~ 2

GHz!)if we can’t generate minoritycarriers fast enough (> 100

Hz) ?

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MOScap vs MOSFET

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Substrate

DrainInsulator

Gate

Source Channel

Substrate

InsulatorGate

Channel

Minority carriers generated byRG, over minority carrier lifetime~ 100sSo Cinv can be << Cox if fast gateswitching (~ GHz)

Majority carriers pulled infrom contacts (fast !!)

Cinv = Cox

MOScap vs MOSFET

ECE 663

Example Metal-SiO2-Si

• NA = 1017/cm3

• At room temp kT/q = 0.026V• ni = 9.65x109/cm3

s = 11.9x1.85x10-14 F/cm

mcmW

XxxXxx

NqnNkT

WA

i

As

1.010

10106.11065.910ln026.01085.89.11ln4

5max

1719

91714

2max

ECE 663

Example Metal-SiO2-Si

• d=50 nm thick oxide=10-5 cm i=3.9x8.85x10-14 F/cm

Voltsinvx

xxxCWqNV

CC

cmFxx

xxWd

C

Voltsx

xxnN

qkTinv

cmFxxxd

C

sBi

ATH

i

i

i

ABs

ii

s

i

07.184.023.0)(109.6

1010106.12

13.0

/101.9109.119.3105

1085.89.3

84.01065.9

10ln026.02ln22)(

/109.610

1085.89.3

7

51719max

'min

2857

14

max

'min

9

17

275

14

ECE 663

Real MIS Diode: Metal(poly)-Si-SiO2 MOS

• Work functions of gate and semiconductor are NOT the same

• Oxides are not perfect– Trapped, interface, mobile charges– Tunneling

• All of these will effect the CV characteristic and threshold voltage

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Band bending due to work function difference

msFBV

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Work Function Difference

• qs=semiconductor work function = difference between vacuum and Fermi level

• qm=metal work function• qms=(qm- qs)• For Al, qm=4.1 eV• n+ polysilicon qs=4.05 eV• p+ polysilicon qs=5.05 eV• qms varies over a wide range depending

on doping

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SiO2-Si Interface Charges

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Standard nomenclature for Oxide charges:

QM=Mobile charges (Na+/K+) – can causeunstable threshold shifts – cleanlinesshas eliminated this issue

QOT=Oxide trapped charge – Can be anywherein the oxide layer. Caused by brokenSi-O bonds – caused by radiation damagee.g. alpha particles, plasma processes,hot carriers, EPROM

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QF= Fixed oxide charge – positive charge layernear (~2mm) Caused by incompleteoxidation of Si atoms(dangling bonds)Does not change with applied voltage

QIT=Interface trapped charge. Similar in originto QF but at interface. Can be pos, neg,or neutral. Traps e- and h during deviceoperation. Density of QIT and QF usuallycorrelated-similar mechanisms. Cureis H anneal at the end of the process.

Oxide charges measured with C-V methods

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Effect of Fixed Oxide Charges

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Surface Recombination

Lattice periodicity broken at surface/interface – mid-gap E levelsCarriers generated-recombined per unit area

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Interface Trapped Charge - QIT

• Surface states – R-G centers caused by disruption of lattice periodicity at surface

• Trap levels distributed in band gap, with Fermi-type distributed:

• Ionization and polarity will depend on applied voltage (above or below Fermi level

• Frequency dependent capacitance due to surface recombination lifetime compared with measurement frequency

• Effect is to distort CV curve depending on frequency• Can be passivated w/H anneal – 1010/cm2 in Si/SiO2 system

kTEEDD

DDFegN

N/)(1

1

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Effect of Interface trapped charge on C-V curve

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a – idealb – lateral shift – Q oxide, msc – distorted by QIT

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Non-Ideal MOS capacitor C-V curves

• Work function difference and oxide charges shift CV curve in voltage from ideal case

• CV shift changes threshold voltage

• Mobile ionic charges can change threshold voltage as a function of time – reliability problems

• Interface Trapped Charge distorts CV curve – frequency dependent capacitance

• Interface state density can be reduced by H annealing in Si-Si02

• Other gate insulator materials tend to have much higher interface state densities

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All of the above….

• For the three types of oxide charges the CV curve is shifted by the voltage on the capacitor Q/C

• When work function differences and oxide charges are present, the flat band voltage shift is:

d

iechoxideFB dxxx

dCV

0arg_ )(11

i

otmfmsFB C

QQQV

Some important equations in the inversion regime (Depth direction)

VT = ms + 2B + ox

Wdm = [2S(2B)/qNA]

Qinv = Cox(VG - VT)

ox = Qs/Cox

Qs = qNAWdm

VT = ms + 2B + ([4SBqNA] - Qf + Qm + Qot)/Cox

Substrate

Channel Drain

InsulatorGate

Source

x

lec16 mos es

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