Tiwari 12 01 Technology 1

Technology

Sandip TiwariSandip [email protected]

A vast area (modern fabrication facilities are B$ operations with 100’s of major tools and 1000’s of processing steps each critically important … high yield) where

h l i h idl ( h l fi h b h I i dtechnologies change rapidly (the last five years e.g. have brought Immersion and double exposure for nanoscale lithography, or atomic layer deposition – ALD – for atomic scale thickness, …)

I will discuss only some of these important technologies because of the limited time and spend time on important physical concepts with a suitable description so that you can appreciate the power of these technologies

11Tiwari_12_2009_iWSG_Technology.pptx

(Some of the extra material here should be useful to you as background for later)

A Chip Cross-Section of These Days

Insulators(low k, Organic, SiO2, …)

Interconnects (Cu, Al, … with barrier layers)layers)

Vias (W, barrier layers)

Transistors


Transistors and MemoriesGate 3 different insulators

DRAMTransistorsTransistor Flash Memory

Gate (PolySi/Metal)

Capacitors (Electrodes &

Floating Gate (PolySi)

Source Drain

high κ insulators)(PolySi)

Control Gate(PolySi)Gate Insulators

SiO2, SiON, HfO2, …InsulatorsSiO2, SiON, …

1.2 nm


A Simple nMOS Lab Implementationphotoresist Arsenic implantphotoresist

Si3N4

SiO2Si 100

Arsenic implant

source drain+2p-Si <100>

B field implant

source drainn+

(a) (e)

active device area

n+

(b)

(f)

B channel implant

n+

p-Si <100>(g)

poly-silicon gate

(c)

GateDrain

W


(d) Source Lg

W(h)

PatterningLithography

O ti l ( d th ) d lf li t t h iOptical (and others) and self-alignment techniques

Etching

Dry (Ionized, Active, Gas phase)

W tWet

Material incorporationGate Insulator, Other insulators, Semiconductors growth

CVD, LPCVD, ALD, …

Interconnect metals

PVD, Electro and Electro-less Plating, …

Changing material propertiesDiffusion

Implantation and annealing

5

Silicidation, …

Tiwari_12_2009_iWSG_Technology.pptx

Mask (Qz)

AbsorberAbsorber (Chrome)

Resist

Substrate


Illumination


Exposure

Latent ImageImage


Develop


Pattern Transfer -- Addition

Deposit

Lift-Off


Pattern Transfer – Subtraction

Etch

(wet, release, etc.) Isotropic Anisotropic (RIE)

Sidewall

Profile


Critical

Pattern Transfer – Modification IonsIons

Electrons

Ph t

+ + + - - + + + - -

Photons

Chemicals

Electrical

Altered Properties

Mechanical

Optical


Optical

Chemical

Hot Processes: Deposition, Oxidation and Diffusion

13

Chemical Vapor Deposition (CVD)

Atmospheric Pressure CVD (APCVD)

Low Pressure CVD (LPCVD)

Plasma Enhanced CVD (PECVD)

Some Forms of EpitaxyVapor Phase Epitaxy (VPE)

Organometallic Vapor Phase Epitaxy (MOCVD, OMVPE)

Introduce ALL atoms/compounds for producing a film in the gas phasep

Ideally no consumption of substrate materialsSome level of surface reaction needed for adhesion

Provide energy (heat RF photons) to induce chemical reaction


Provide energy (heat, RF, photons) to induce chemical reaction

Deposited Dielectrics

15Tiwari_12_2009_iWSG_Technology.pptx 15

Pyrogenic CVD

Single Crystal Silicon

A h l iliAmorphous, poly-silicon(by cracking SiH4)

SiO d iti


SiO2 deposition

Laminar FlowViscous flow is characterized by Reynold’s number (a dimensionless parameter)

Diameter of pipe

G l it ( t f i )Gas velocity (center of pipe)

Fluid density (g/cm3)

Absolute viscosity (g/cm s)N d L i Fl ( h Absolute viscosity (g/cm.s)Need Laminar Flow (smooth, no turbulence)Most CVD Conducted in Viscous Flow Regime (Fluid Dynamics)

Laminar flow:

Regime (Fluid Dynamics)


Key Steps in Chemical Vapor Deposition

1. Transport of reactants to the deposition region

2. Transport of reactants from the main gas stream through the boundary layer to the wafer surface

3. Adsorption of reactants on the wafer surface

4. Surface reactions, including: chemical decomposition or reaction, surface migration to attachment sites (kinks and ledges); site incorporation; and other surface reactions (emission and re-deposition for example).

5. Desorption of byproducts

6. Transport of by-products through boundary layer


p y p g y y

7. Transport of by-products away from the deposition region

First Order Model of Deposition

Stagnant layer approximation:Diffusion coefficient (cm2/s)

Gas phase and surface concentration (molecules/cm3)

Flux Stagnant layer Mass transfer

Surface reaction rate approximation:

Flux(molecules/cm2.s)

Stagnant layerthickness (cm)

Mass transfer coefficient (cm/s)

Surface reaction rate approximation:

Steady state:

Surface reaction rate (cm/s)

Steady-state:


Number of atoms/unit volume

Limit Regions

Mass transfer limit

Surface reaction limit

Higher Temperature

Lower Temperature


Reduced Pressure (LPCVD)

Dependent on flows, geometry, …

Strongly dependent on T

Higher pressure higherHigher pressure, higher collisions, smaller diffusion

C t t i P if N t tConstant in P if NR constantUsually weaker function of P than gD

Therefore,

Higher Temperature

Lower Temperature

increases with

constant in


Reducing P allows surface reaction limited growth rates at higher T

CVD Poly-Silicon Deposition

SiH4 = Si + 2H2

Activated RegionR = R exp[ E /kT] E = 1 7 eV

reaction rate limited (with H2, N2, AsH3, PH3, B2H6, ..)

R = Ro exp[-Ea/kT], Ea = 1.7 eV

Surface reaction rate limit set by desorption of hydrogen

the activation energy is the similar for SiH4, SiH2Cl2, and SiCl4 with H2 as a carrier gas

Transport Regionmass transfer limited

p gLimit set by gas flow rate, flow geometry, …


Sze, Figs. 3, 6 (1988)

Rate vs 1/T Pascal: 1 N/m2

Atm = 101,325 Pa = 760 mm Hg = 760 torr = 14.7 psi

Some Common LPCVD Films

LPCVD Silicon Nitride850 C & SiH2Cl2 + NH3

LPCVD Silicon Oxynitride850 C & SiH2Cl2 + NH3 + N2O or O2

LPCVD Polysilicon620-650 C & SiH4 + optional PH3 or B2H6

LPCVD SiOLPCVD SiO2

420 C & SiH4 + O2

~500 C TEOS (Less dense)

LPCVD HTO SiO2

850 C & SiH2Cl2 + N2O


Atomic Layer Deposition: An Example

Pulse MeCln

Growth layer by layer

Purge with N2

Purge with H2O

MeCln + xH2O => MeOx + nHCl

Purge with N2


Repeat

Plasma Enhanced Chemical Vapor Deposition

Non-thermal energy to enhance processes at lower temperatures

Plasma consists of electronsPlasma consists of electrons, ionized molecules, neutral molecules, neutral and ionized fragments of broken-up molecules, excited molecules and free radicals

Free radicals are electrically neutral species that have incomplete bonding and areincomplete bonding and are extremely reactive (e.g. SiO, SiH3, F, …)

Net result from fragmentation, free g ,radicals, and ion bombardment is that the surface processes and deposition occur at much lower temperatures than in non plasma

Plasma Nitride: SiH4 & NH3 or N2

Plasma Oxide: SiH4 & N2O or O2

Organosilane (TEOS e.g.) & Oxidizer


temperatures than in non-plasma systems

g ( g )

Oxidation: Linear-Parabolic Model(also known as Deal-Grove Model)

ary

Laye

r Si + O2 -> SiO2

Si + 2H2O -> SiO2 + 2H2

ant/B

ound

a

Oxi

de

Mass transfer coefficient (cm/s)

The gas phase diffusion

CGC

x

Sta

gn is much faster than the other processes (diffusion in oxide and interface reaction Cs

Co

CiO

kinetics)

Oxidation is an example of self-assembly at high temperature (as is singleCi

Gas SiO2 Si

Oxygen or Water Vapor temperature (as is single crystal epitaxy that uses crystal template); both employ template


F1 F2 F3

p y pprovided by substrate

Deal-Grove Model #1SiO2 Si

C

CoCS

CG

in steady state

Ci

X

F2

F1

thickness of oxideF3

F2

reaction rate at interface with silicon

in steady state

So


Deal-Grove Model #2SiO2 Si

Ci

CoCS

CG

Ci

X

FF2

F1

F32

where and

Solution (X>0):

With


With

Deal-Grove Model # 3SiO2 Si

Ci

CoCS

CG

Ci

X

FF2

F1

and

F32

If X << A Linear regime: Growth is linear

If X >> A Parabolic regime


Process Condition EffectsAs remarked earlier, F1 is the fastest step of the oxidation (except at the very initial condition when X=0): C0 ~ CG scales linearly with pressure of oxidant

S l li l ith f id t ( t ti )Scales linearly with pressure of oxidant (concentration)ks :depends on Si surface (orientation/# of Si bonds)depends on doping (electrochemical conditions)

Scales linearly with pressure of oxidant (concentration)Independent of Si surface (orientation/# of Si bonds)D weakly dependent on oxide density

Quite independent of pressureDepends on oxidant speciesDepends on Si orientationD d kl id d it


Depends weakly on oxide density

Beyond Deal-Grove: Rapid Initial Oxidation Regime

Deal-Grove Empiricalp

(100) Si oxidation in dry O2 at 1 atmosphere

C3 = 7.48x106 μm/min

EA3 = 2.38 eVA3

L3 = 69 nm


Dry O2 (100) Si


Diffusion

Dopant atoms move through Si at significant rates at high temperatures

diffusiondiffusion

Useful for moving dopants from surface to desired depth

Diffusion is a limitation in design of shallow junction processes

We will discuss this through basic diffusion theory, explore concentration dependent diffusion and oxidation enhanced/retarded diffusion effects


Basic Mathematics of Diffusion: Fick’s 1st Law

Random thermal motion:

O di iOne dimension:


Fick’s 2nd Law

Fick’s 2nd law

Diffusion equation

If diffusion coefficient constant in positionp

In 3D


If D constant in (x,y,z)

Classic Examples: Constant D in Space and Time

Limited Source/Gaussian Diffusion

Infinite Source/Constant Surface Concentration Diffusion

Fixed Number/Area of Dopants at Surface

Obj ti fi d ( )C 1022

Gaussian Diffusion Example

Gaussian Diffusion

Objective, find ( , )

Boundary conditions

C x t

18

1020

1022

t=1 st=10 st=100 st=1000 st=10,000 s

0

(0,0) ( )

( , )

C Q x

C x t dx Q t

δ∞

=

= ∀∫1016

1018

onc

ent

ratio

n (

cm-3

)

Q=1x1015cm-2

D=2.59x10-14cm2/s

( )2

0

/ 2

Solution:

x DtQ

∫1012

1014Co


( )/ 2( , )

x DtQC x t e

Dtπ−

= 1010

0 0.5 1 1.5

x (um)

Gaussian Diffusion

1022

Gaussian Diffusion ExampleGaussian Diffusion Example

1018

1020

1022

t=1 st=10 st=100 st=1000 st=10,000 s

)

1020

1022

Gaussian Diffusion Example

1014

1016

10

Con

cent

ratio

n (

cm-3

)

Q=1x1015cm-2

D=2.59x10-14cm2/s

14

1016

1018

on

cen

tra

tion

(cm

-3)

Q=1x1015cm-2

D=2.59x10-14cm2/s

1010

1012

10

1010

1012

1014

t=1 st=10 st=100 st=1000 st=10,000 s

C

0 0.5 1 1.5

x (um)

100 0.1 0.2 0.3 0.4 0.5

x (um)


Infinite Source/Constant Surface Concentration Diffusion

( , )C x t

0

( , )

Boundary Conditions

(0, )

C x t

C t C t= ∀0(0, )

( ,0) 0 0

Solution

C t C t

C x x

∀= ∀ >

0

Solution

( , )2

xC x t C erfc

Dt

⎛ ⎞= ⎜ ⎟⎝ ⎠

( ) ( )

( ) 2

2

1z

Dt

erfc z erf z

f dη

⎝ ⎠≡ −

∫38Tiwari_12_2009_iWSG_Technology.pptx 38

( )0

erf z e dη η−≡ ∫

LithographyLithography

39

Photolithography (Production)Coat

PrimeCoat

Pre-Bake

Expose

Post-Bake

Develop


Hard Bake

Positive Resist Photolithography

Light

Areas exposed to Light

Island

plight are dissolved.

Shadow on photoresist

Chrome island on

Wi d

photoresist

Islandp

Exposed f

island on glass mask Window

PhotoresistPhotoresist

photoresist

silicon substrate

oxide oxide

silicon substrate

area of photoresist

Silicon substrateSilicon substrate

PhotoresistPhotoresistOxideOxide OxideOxide


Resulting pattern after the resist is developed.

Development of latent image


Negative Resist Photolithography

Light

Areas exposed to light are cross-linked and resist the developer chemical.Light

Island

p

Window

Exposed area of photoresist

Chrome island on

glass maskWindow

Shadow on photoresist

PhotoresistPhotoresist


PhotoresistPhotoresistOxideOxide OxideOxide


Resulting pattern after the resist is developed.

Development of latent image


Optical

Improving within Diffraction Limits:Att t d h hift (130 )

104

ngth

(nm

)

KrF248 nmG line I line

~3 μm - Attenuated phase shift (130nm)- Model-Based OPC (130nm)- Alternating phase shift (90nm)- Sub-resolution assist feat.( 65nm)

R t i t d d i l (45 )

103

& W

avel

e 248 nm

ArF193 nm

Immersion Doubling

436 nm 365 nm - Restricted design rules (45nm)- Immersion lithography (45nm)

100

atur

e S

ize

Immersion, Doubling, …

~50 nm

1980 1990 2000 2010

10Fea

50 nm


Photolithography Instruments

Contact Proximity Projection(Steppers)λ

resist thickness = z

s

2Lmin

s

R: resolution, k a constant that is a function of the design/set-up parameters

Fast, simple & inexpensive

Diffraction minimized by small (~0) mask-resist gap

Less mask wear/contaminationFast, simple & inexpensive

But, Greater diffraction & less

No mask contact/contaminationMask demagnified4x and 5x usuallyMask pattern at chip size with

g p p


But, mask-wear, defect generation & wafer-sized mask and light scattering in resist limits resolution

resolutionWafer sized mask

Mask pattern at chip size with wafer stepped for exposure

Expensive instrumentation

Lithography ToolsContact Projection


Wave Nature

For n=0 there is no diffraction (direct beam)

For n= 1 we have the first order diffracted beamsFor n= 1 we have the first order diffracted beams

For n= 2 we have the second order diffracted beam


Diffraction pattern is the Fourier Transform of the object

Wave Nature

Diffraction grating:

P

P is the repeat distance nλ

For constructive

P is the repeat distance (periodicity) that processing engineers call “pitch” φn

nλ

For constructive interference, the path difference must be equal to an qinteger number times λ

As P decreases φ increases


As P decreases, φn increasesn is the order of diffraction

Projection: Key Parameters

Resolution:

λ: wavelength of exposurek1: parameter characterizing system and

process dependence (typically between 0.25 and 1)NA i l tNA: numerical aperture

n: index of refraction (light transmission med 1 for air)f

θn: index of refraction (light transmission med., 1 for air)θ: half-angle of cone of light

Wafer image planeFocus plane

DOF

Depth of field:


lm

Need smaller lm and higher DOFCompromises of the optical design

Diffraction

The zero order beam does not contain any information on the spacing; is independent of the value P describing the repeat distance in the object

Intensity at maskdistance in the object

The first order beam contains information on the spacing. When made to interfere with the zero order beam, a sinusoidal beat

Optics

pattern forms

p

Intensity at wafer Imax

Imin


Modulation Transfer Function (MTF) quantitatively describes the relationship between source and image: (Imax-Imin)/(Imax+Imin)

Imaging

φn

P

nλ


“Ideal System”

Ideal Exposure System: 100% modulation of light over 0 distance

Ideal Positive Photoresist: 100% retention if exposed below Dcrit, 100% l if d b D100% removal if exposed above Dcrit

Relative Intensity Resist Retention

1

1.2

Perfect Exposure System

1

1.2

Perfect Resist

0 4

0.6

0.8

ativ

e In

tens

ity

0 4

0.6

0.8

sist

Re

ten

tion

0

0.2

0.4

Re

la

0

0.2

0.4

Re


-0.2-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

Position

-0.20 0.5 1 1.5 2

Dose/Dcrit

“Real” Exposure System

( )

I I

ModulationTransfer Function MTF

⎡ ⎤−I I

I Imax min

max min

MTF

This Example

⎡ ⎤−≡ ⎢ ⎥+⎣ ⎦

5 1 4 2

5 1 6 3

:

MTF

Also Note

−⎡ ⎤≈ = =⎢ ⎥+⎣ ⎦

1I I

1

1

min max

MTF

MTF

MTF

−⎡ ⎤= ⎢ ⎥+⎣ ⎦+⎡ ⎤1

I I1max min

MTF

MTF

+⎡ ⎤= ⎢ ⎥−⎣ ⎦


Photoresist PropertiesSensitivity: threshold energy (ET)

Contrast ratio

Sensitivity: threshold energy (ET)

ET

Sensitivity identifies energy needs of exposure

Contrast quantifies PR’s ability to distinguish light and dark; iti t d l t ft d t b ksensitive to development process, soft and post-exposure bake

and wavelength

Positive Resist exposure system should deliver D<D0 in unexposed regions D>D in exposed regions


unexposed regions D>D100 in exposed regions

Resist Critical Modulation Transfer Function (CMTF); Optics’ MTF should be better than resist’s

Resist Sensitivity

Resist Kodak 809 UV Positive ResistSensitivity S = 150 mJ/cm2

Exposure G-Line (436 nm)Thickness t = 1 μmThickness t = 1 μm

Exposure Dose 150 mJ/cm2

Photon Energy E = hf = hc/λ = 4.54x10-19 J (h = 6.62x10-34 Js, c = 2.99x1010 cm/s, λ = 436 nm)

Number of Photons: S/E = 3.3x1017 cm-2

Volume/Photon 3.3x10-22 cm3

Mean Photon Separation [3.3x10-22 cm3](1/3) = 1 μm

Resist Layer

Mean Photon Separation [3.3x10 cm ]6.72x10-8 cm = 0.67 nm

1 cm

1 cm

1 μm


Simulations Based on Huygen’s Principle

Wave-front of a propagating wave of light at any instant conforms to the envelope of spherical wavelets emanating from every point on the wave-front at the prior instant (with the understanding that the wavelets have the same speed as the overall wave)

– Christian Huygens (1629-1695)yg ( )

Mask

r

x1

dS(x1) = 0 in opaque regions

Exposure Surface (wafer)

r

x

d( 1) p q g

S(x1) = 1 in clear regions


Contact/Proximity 1 μm Gap/ 100 μm Exposure

1.4Exposure λ= 0.465μm Grating Period= 200μm Line Width= 100μm Separation= 1μm

1

1.2

0.8

zed

Inte

nsity

0.4

0.6

Nor

mal

i

0.2


-100 -80 -60 -40 -20 0 20 40 60 80 1000

Position (μm)



1

1.2

0.8

lized

Int

ensi

ty

0.4

0.6

Nor

mal

-5 -4 -3 -2 -1 0 1 2 3 4 50

0.2


5 4 3 2 1 0 1 2 3 4 5Position (μm)



1

lized

Int

ensi

ty

0.5

Nor

mal

0


-3 -2 -1 0 1 2 3Position (μm)

Contact/Proximity 0.5 μm Gap/ 3 μm Exposure

1.2

1.4Exposure λ= 0.465μm Grating Period= 6μm Line Width= 3μm Separation= 0.5μm

0.8

1

nten

sity

0.4

0.6

Nor

mal

ized

In

0

0.2


-3 -2 -1 0 1 2 30

Position (μm)

Contact/Proximity 0.1 μm Gap/ 3 μm Exposure

1.2

1.4Exposure λ= 0.465μm Grating Period= 6μm Line Width= 3μm Separation= 0.1μm

0.8

1

nten

sity

0.4

0.6

Nor

mal

ized

In

0

0.2


-3 -2 -1 0 1 2 30

Position (μm)

Photoresist Contrast

As an example, for A=1.55 and n=13


Functional fit adapted fromC. A. Mack, et al. SPIE Proc. 3677, pp. 415-434 (1999)

and

Use of MTF Curves


Standing Waves In Photoresist


Standing Waves: Example

~0 35 μm Lines/Spaces in~0.35 μm Lines/Spaces in Photoresist on ~31.7 nm SiO2 on Si


Phase Shifting


Introducing phase shift allows higher resolution by reducing intensity in overlap regionsSuitable only when two windows are placed close together

Pattern TransferPattern Transfer

66

Etchingmask materialmask material

thin film being etched

substrate

We are interested in removal of materials

Anisotropic/Directional Etch Isotropic Etch

Etch Raterate of material removal; working with thin films – so, 1-1000 nm/mindependence on concentration, agitation, temperature, density etc. of the thin film or substratethin film or substrate, …

Etch Selectivityrelative etch rates (ratio) of the thin film to the mask, substrate, or other films

Etch Geometry:Etch Geometry:sidewall slope (degree of anisotropy)

Reproducibility

Two principal approaches


Wet etching (reactants from liquid sources)Dry etching (reactants from gas/vapor phase – neutral or ionized)

Wet EtchingH O

Gnd

F-

H2O

HF

H+

F-

CF4 CF+2

O2

FSiO2 + 4HF = SiF4 + 2H2O

SiO + CF SiF + 2CO

Principal Sum ReactionsF-

HF

RF + DC Bias

SiO2 + CF4 = SiF4 + 2CO2

In detail, multiple charged species, movement of species multiple reactions

3*

4 CFFCF +⇔

At its simplest, the key steps in etching would be

movement of species, multiple reactions

4*

3*

4

4

2

SiFFSi

eCFFeCF

⇔+

++⇔+ +−

1: Etch species generation2: Movement to surface

Diffusion and Field-Aided

3: Adsorption

1

23 5

63: Adsorption4: Reaction5: Desorption6: Diffusion to bulk

3

4

5


6: Diffusion to bulkThere may be other competing and inhibiting simultaneous reactionsIn limits, slowest dominates

Wet Etching

Transport of reactants to surface

Surface reaction

Transport of products from surface

Key ingredientsOxidizer: H2O2, HNO3, …

Acid or base that dissolves the oxidized surface: H2SO4, NH4OH, HCl, …

Diluting agent for transporting reactants and products

H2O, CH3COOH, …

An electrochemical processOxidation: electron loss / increase in oxidation number

Reduction: electron gain/ decrease in oxidation number


HNA: HydroFluoric – Nitric Acid Etching


R. B. Darling

1

High HF concentrationsReaction limited by HNO3, follow constant HNO3% lines

Rate limited by oxidation

Etched wafer will have small oxideEtched wafer will have small oxide

2

High HNO3 concentrationsReaction limited by HF, follow y ,constant HF % lines

Rate limited by reduction

Etched wafer will have more oxide

33

Very little H2OFast etch rates followed by rapid drop with depletion


drop with depletion

Anisotropic Wet Etching

Wet etch processes can also be “anisotropic,” i.e., the etch rates are different in different directionsare different in different directions

<111> usually a stop plane for anisotropic etching


Hydroxide Etching of Si

Examples: KOH, NaOH, NH4OH, (CH3)4NOH (TMAH)

−+ +→ OHXXOH+− ++→ hHOHOH 4244 22

+ +→ OHXXOH

Oxidation of Si

Silicate forms water soluble complex

+++− →++ 2)(42 OHSihOHSi

Silicate forms water soluble complex

OHOHSiOOHOHSi 2222 2)(4)( +→+ −−−++

KOH example:250 g KOH: 200 g propanol, 800 g H2O at 80 C1000 nm/min of [100]


[ ]Etch stops at p++ layersAnisotropy: {111}:{110}:{100}::1:600:400

Anisotropic Wet Etching


Gaseous Chemical Etching

Example: XeF2 etching of Silicon

422 SiFXeSiXeF +⇔+

XeF2 absorptionSiF4 formationReaction product removal


pOccurs at few torrs at RT

Purely Physical Etching: Sputtering/Ion Milling


Ion Enhanced Etching


Barrel Reactor

Chemical etching dominant

Useful in non-critical stepsPhotoresist removal (ashing using O2 plasma and Ozone)


Parallel Plate or Capacitively Coupled Plasma (CCP)

Plasma modeElectrodes equal in area (or wafer electrode grounded with

Plasma Modeg

chamber and hence larger)Moderate sheath voltage (10-100 eV), so only moderate ionic componentcomponent

Strong chemical component

Etching fairly isotropic and selective

Reactive Ion Etching ModeWafers on RF powered electrodeGenerally higher inducedGenerally higher induced bias/stronger ion bombardment (~100-700V)Lower pressures (10-100mTorr)


Etching fairly anisotropic

79

Electron Cyclotron Resonance (ECR) and Inductively Coupled Plasma (ICP)

Remote, non-capacitively coupled plasma source (electron cyclotron resonance – ECR, or inductively coupled plasma source – ICP)

Separate RF source as wafer bias. Separation of plasma power/density from wafer bias/ion accelerating field

Very high density of plasma (1011-1012 ions/cm3) – faster etching

Lower pressures (1-10 mT) due to higher ionization efficiency (longerhigher ionization efficiency (longer mean free path/more anisotropy)

Currently an optimum compromise in high etch rates, good selectivity, good directionality while low iongood directionality, while low ion energy and damage


Inductively Coupled Plasma (ICP) or Transformer Coupled Plasma (TCP)


Principles of Glow Discharges

Sheath formation

RF PlasmasRF field provides energy to accelerate free electronsRF field provides energy to accelerate free electrons

Electron collisions with neutral molecules produce

Reactive neutrals

Ions

Excited (metastable neutrals) → Glow

Conditions of the glow discharge (charged species, separation of charge, …) produce an induced bias on the wafer surface

A quasineutral plasma is a mix of ~ equal densities of e-’s and ions (A+)e-’s are >2000x lighter than ions g

respond much more quickly to E-fields

e-’s have a much higher thermal velocity than ionsRandom thermal flow of e-’s and ions into electrodes (plane) is not equalB l t d b E fi ld b ild


Balance restored by E-field build-upLeads to Vp, the plasma potential (typically ~few kBT)

Glow Discharge

Fl drift

Fluxethermal

Assumes ions are fixed

VP

Fluxedrift

~few kBT

~Debye Screening LengthλD


Wall or grounded Electrode

λD

Parallel Plate Reactor: RF

Drive with AC Signal

Typical f=13.56 MHz (FCC) but others are also usedothers are also used

RF couples through insulating layers on wafer

In almost all cases, the RF powerIn almost all cases, the RF power supply must be impedance matched to the load (reactor)

VRF

Low PressureReactor

WaferLack of DC Current Means that the integral over 1 RF cycle of the current into the electrode must be zero

Wafer

must be zero

( )1

0full cycle

Idt

d

= ∫

∫Ion current


( )1

i e

full cycle

I I dt= +∫ electron current

84

Induced Bias Picture

1 5

VRF

Low PressureReactor

0 5

1

1.5

mal

ize

d)

Wafer

-0.5

0

0.5

olta

ge

(nor

m

-1.5

-1

0.5

Cat

hod

e V

o

Vbias

Matching Network(T, one common style)

-2-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

C

Time (normalized)


CF4

The 4 F can either etch Si or recombine with carbon:4F + C => CF4

This “reverse” reaction slows etching downThis reverse reaction slows etching down.So, we remove the Carbon by reacting it with an alternate species:

O2. C reacts strongly with O2 to make CO2.

C + O = COC + O2 = CO2

“Sticky” reaction products can cover the wafer with a film.The worst of these is teflon-like compound that is C-F based polymer

( CF CF )(-CF2CF2-)

Continuous ion bombardment (physical sputtering) with directional ions removes these films

Since directions are usually designed to be orthogonal, horizontalSince directions are usually designed to be orthogonal, horizontal surfaces are etched while vertical ones are not


Physical Vapor Deposition: MetallizationEvaporation and Sputtering

87

Physical Vapor Deposition

Thermal Evaporation E-Beam Evaporation Sputtering

target

plasma

substrate


Deposition Chamber

Th iThe vapor moves in line of sight

In a large systemIn a large system, the substrates are on a spherical carrier called planetary drive to ensure uniform metaluniform metal thickness over the wafer


Evaporation

Evaporation of Al:

For a source 1x1 cm2 in area

Langmuir: GE Labs, Nobel Prize with contributions

At a temperature 1200 C

Vapor Pressure: ~ 1 Pascal

Langmuir Equation for mass transport:contributions including LB film, use of Ar in W light bulb, research in X-

R(g/cm2s) = 4.43x10-4(amu/T)1/2 p (Pascal)

For Al , the square root has a value of about (30/1500)1/2 = (1/50)1/2 = 1/7 = 0.13

Ray, plasma, adsorption, ….

(30/1500) (1/50) 1/7 0.13

So, R will be ~4x10-4 g/cm2s

Source evaporates ~ 4x10-4 g/s

Now we need some knowledge of the source geometry If it is aNow we need some knowledge of the source geometry. If it is a hot ball, it will evaporate in all direction. If it is flat, over a half space. If the source is 20 cm away from the wafer, we need to now work on the geometrical aspects


now work on the geometrical aspects

Evaporation

With the wafer normal pointed towards the source, the mass per cm2, is be given by

M 4 10 4/ 2 (20)2Mdeposited = 4x10-4/ 2 π (20)2

or about 2x10-7 g/cm2s

Knowing density (2.7 g/cm2), we can determine deposition rate= 0.8x10-8 cm/s, i.e. ~0.1 nm/s

Most evaporations are between 0.1 and 1.0 nm/s

Smaller rates require higher vacuum levels to reduce contamination effects from competing reactions taking over the time scales

Melting point of Al is 660 C. At 1200 C, we exceed melting point significantly

High melting point (Tm) materials such as W therefore become difficult to evaporate because of its high melting point


This is not a complete picture!

Electron Beam

E-beam

5-15 KV, 1 A at filament

Molten Region

Al

Liquid


Water cooled jacket

Liquid puddle!

Sputtering Principle

Substrate Cathode (+)

Diffusion

+Ion

Ionization

NeutralAtom

DepositionDiffusion

EjectedIon

E(B)

Sputter DC or RF

Vacuum

p, T

Ion

+

Anode (-)Target

Sputter DC or RFFields

Pressure Range: 0.1 -1 Pa

Sputtering Modes: RF, DC. Magnetron


Applications: Metals, Metal Alloys, Semiconductors, Insulators

DC & RF Sputtering

DC sputtering requires DC current flow, so works for conducting substrates and targets

If one is not - usually the target such as SiO2 - there is a problem!

The solution is to quickly reverse the polarity before the positive ions hitting the insulating target generate a positive repulsive charge!

Upon polarity reversal, electrons will hit the target and neutralize any previous charge !

RF works best for insulating targetg g

Sputtering self-adjusts for alloy depositions


Sputtering self adjusts for alloy depositions

RF SputteringPlasma Bias

The plasma develops a DC biasdevelops a DC biasthat compensates for the differentfor the different mobilites of electrons and ions in an electric field.

It’s the DC bias that does the sputtering of insulators


Magnetron RF Sputtering

Magnetic fields force electrons to spiral, increasing collisions with neutral ions => denser plasmas => Magnetron sputtering.

Planar MagnetronReactor with ParallelPermanent MagnetsPermanent Magnetsp = 0.5 Pa, P = 5 kW/m2,B = 0.01 T Typical


Roosmalen, Fig. 5.17, p. 95

Reactive Sputtering

What if we want to make an oxide or nitride?

Sputtering a compound target may not give you what you want!

We can sputter in reactive gase.g.

Ti + (Ar, N2 ) ----------------> TiNx

Si + (Ar, O2 ) ----------------> SiOx

SiO2 + (Ar, O2 ) ----------------> SiOx

Problem: Compound control

Question: Is the compound synthesized at the target or at the substrate?

Answer: Both or either! (Depends primarily on the gas partial ( p p y g ppressures used. This determines sputter rate vs. reaction rate.)


Implantation and AnnealingImplantation and Annealing

98

Ion Implantation

Introducing dopants by diffusion has three major drawbacks:Surface concentration and depth are coupled.

E ti ll l G i d f fil ld b t dEssentially, only Gaussian and erfc profiles could be generated.

The process was not precisely reproducible because of various concentration of point defects (vacancies, interstitials etc).

Ion implantation provides precise control of dose and depth at theIon implantation provides precise control of dose and depth at the expense of implant damage

Variable profiles, precise control over amount of impurities

f SWide selection of masking material: Pr, oxide, polySi, metal, …

As+ As+ As+ As+

Gate


Ion Implanter

A typical implanter has an

ion sourceion source

accelerator

Filtering magnet

deflection/scanning coils or plates

incident current meter

Typical voltages areTypical voltages are 50 to 200 KeV, the trend is to lower voltages.


Ion Implantation Basics

How do the ions lose energy in the solid (Ion stopping)?

The final distribution of ions in the solid (Range, straggle)

The damage the ions do to the crystal (EOR damage)The damage the ions do to the crystal (EOR damage)

Annealing out the damage (Dopant activation, transient enhanced diffusion during that anneal)

If an ion is moving along specific crystallographic directions, e.g. [011], ions travel much deeper into the crystal because of “channeling”

Key ideas

Target atom recoilsAbsorption of Energyp gy

Generation of Vacancies

Deposited Energy Distribution

Point Defect and Ion Distributions


Critical Dose for Amorphization

Simulation of Implantation (Monte Carlo)SRIM: download from www.srim.org

Assumes amorphous target

Verify the parameters –energies, cross-sections


Si Crystal Faces

(100) Si Face (100) Si Face30


7o Tilt, 30o Twist

After Runyan & Bean, Fig. 9-15

Channeling

IonRecoilIon Scattered RecoilIon Scattered

into Channel

TargetAt

ChanneledIonAtom

Row

Ch l

Ion

Si


Channel

Implant Annealing/Activation

Ion Individual CollisionCascades for < 1012 cm-2

X-talSurface

D d

Cascades for < 1012 cm-2

Amorphous Layerfor > 1015 cm-2

DamagedX-tal orAmorphous

Annealing

TargetAtom

Annealing- SSER > 450 C- Point Defects- Extended Defects

X-talBulk

Extended Defects

Activation- Substitutional Sites


Channeling in Stop Profile

∫= dxxNDose )(

-3)

∫Dose is in /cm2

Concentration is in /cm3

atio

n (c

m

Implant300 keV As

Con

cent

ra 300 keV As Dose 1.5x1012 cm-2

(111) SiTilted (111) Si

C

( )

ElectricallyActive As


Depth (μm) Sze Fig. 15, p. 346

Damage to Crystal lattice

The incident ions have very high energy. Typically few to 100’s keV

It l t k b t 20 V t k k ili t ff it’ l tti itIt only takes about 20 eV to knock a silicon atom off it’s lattice sit and shoot as an interstitial somewhere into the lattice

Thus, a 100 Kev atom can dislodge, very roughly, on the order of 5000 Si t5000 Si atoms

If each dopant atom eventually lands on a lattice site, than there must be as many Si self interstitials as there are dopant atoms ( 1 1020 3) Th i t titi l l t i t {311} d f t d(e.g. 1x1020 cm-3). These interstitials cluster into {311} defects and end of range EOR loops (see Plummer)

Energy loss occurs through nuclear and electronic stopping which t h tcreates heat

Most crystalline damage at the end of range


Deposited Energy Distribution

n) I l t

(eV

/A/Io

n Implant300 keV Si -> Si

Ene

rgy

(


Depth (μm)Sze, Fig. 11, p. 342

Annealing/Activation

End-of-Range(EOR) DislocationLoops After Solid

entr

atio

n

pPhase EpitaxialRegrowth

Con

ce

Depth


Depth

PDG 8-25 (2000)

B -> Si Stop Profiles

B ImplantB Implanta-Si TargetNo AnnealingSIMS DataStop Profiles- Gaussian- Pearson IV

Sze Fig 6 p 335


Sze, Fig. 6, p. 335You can see that the Pearson 4 fits the experiment very well

Critical Dose for Amorphization

m-2

)

I l t > Si

Do

se (

cm

11B

Implant -> SiContinuousAmorphousLayer

Cri

tica

l D

122Sb

31PLayer

122Sb


1000/T (K-1)Sze, Fig. 12, p. 343

100 keV B Stop Profile Data

Target Density Range Straggleρ Rp ΔRp

(g/cm3) (nm) (nm)

silicon (Si) 2.33 296.8 73.5silicon dioxide (SiO2) 2.23 306.8 66.6silicon nitride (Si3N4) 3 45 188 3 40 8silicon nitride (Si3N4) 3.45 188.3 40.8resist (C8H12O) 1.37 1056.9 120.2titanium (Ti) 4.52 254.6 85.1titanium silicide (TiSi2) 4.04 215.4 56.3( 2)tungsten (W) 19.3 82.3 61.8tungsten silicide (WSi2) 9.86 144.0 55.5


Adapted from Sze, Table 1, p. 336

Back-Up InformationBack Up Information

113

Optical Microscopes

Camera

Reflected LightIlluminatorTrinocular Head

(tilting)

Eyepieces

Objective Lenses

X-Y Stage

Filters and Apertures

TransmittedLight

Condenser(Transmitted Light)


Illuminator

Magnification

Total Magnification Product of:Objective (5x - 250 x)Body (1x - 2x)E i (10 1 )Eyepiece (10 x- 15 x)

Camera

Magnification vs. ResolutionUseful magnificationUseful magnification

Reveals new informationEmpty magnification

Image is bigger but nothing new is resolved

Good Microscope1000X

Great Microscope2500X – 5000X


Reflected and Transmitted Imaging Modes

Si wafer with 100 mm diameter through wafer etched holes

Reflected image Transmitted image


Ray trace for reflected mode Ray trace for transmitted mode

Light Scattering at Surface

Planar surface

Angle in = Angle out

Edges reflect differently

For glancing illumination, only d ill fl t

Normal incidence illumination

edges will reflect up, perpendicular to the surface


Glancing incidence illumination

Bright Field and Dark Field Imaging

Bright Field

Light incidence ~perpendicular

Flat planar surfaces reflect well

Dark Field

Angular illumination

Plane surface reflect away from lensFlat planar surfaces reflect wellAppear bright

Edges and slopes reflect to the sideAppear dark

M t d f l

Plane surface reflect away from lensMost of sample is DARK

Sharp edges will scatter some light into lens

Edges and dirt sparkleMost common mode for general use Edges and dirt sparkle


Glancing incidence illuminationNormal incidence illumination

Bright Field vs. Dark Field

Bright Field Dark Field

You can see sub-100 nm features using dark field optical microscopy!

100 nm dots in PMMA exposed with e beam lithography

You can see sub 100 nm features using dark field optical microscopy!

This is an extremely useful technique for getting quick answers about exposures


Differential Interference ContrastBright Field

Polarized lightOptical index contrast

Differential interference contrast (DIC)

A false “depth” contrast achieved by interference

Also called Nomarski

Bright Fieldwith DIC


Confocal Microscopy

Confocal microscopy uses optical “tricks” to create a very shallow depth of focusdepth of focus

Only features at a specific depth are imaged clearly

Example:Requires high intensity source

Uses a spinning disk with a slit in it to block defocused light fromit to block defocused light from the sample

Excellent for pulling out one layer in a thin film stack

Gi l i f hi hGives cleaner images of high aspect ratio structures


Top: reflected light imageBottom: real time confocal image

Common Etchants


Common Gases in Dry Etching

Material Etchant CommentsSiO CF + O O Increases Etch RateSiO2 CF4 + O2 O2 Increases Etch Rate

CHF3 Better Selectivity over SiC2F6

C3F8 Increases Etch Rate over CF4

Si3N4 CF4 + O2

CHF3

C2F6

SF6 + HeSF6 + HeSi, Poly CClF3 + Cl2 Anisotropy and Selectivity to SiO2 Variable

CHCl3 + Cl2 Initiation and Selectivity to SiO2 VariableSF6 Medium SiO2 SelectivityNF I i Hi h E h RNF3 Isotropic, High Etch RateCCl4 Somewhat AnisotropicCF4 + H2 Poor SiO2 Selectivity, H Increases AnisotropyC2ClF5 Poor SiO2 Selectivity


C2ClF5 Poor SiO2 Selectivity

Adapted from Runyan & Bean, Table 6.7

Metal/Metal Alloy Properties

Material ρ T α Reaction StableMaterial ρ Tm α Reaction StableμΩcm C ppm/C with Si on Si to

C CAl 2.7-3.0 660 - ~250 250Mo 6-15 2620 5 400-700 ~400W 6-15 3410 4.5 600-700 ~600Cu 1.7 1083 17 ? NoTiSi2 13-16 1540 12 5 - 950TiSi2 13 16 1540 12.5 950MoSi2 40-100 1980 8.3 - >1000TaSi2 38-50 ~2200 8.8-10.7 - >1000WSi2 30-70 2165 6.3, 7.9 - >1000PtSi 28 35 1398 750PtSi 28-35 1398 - - <750CoSi2 10-18 1326 10.1 - <950TiN 40-150 2950 - 450-500 450


After Sze, Table 3, p. 383

Pressure

FAt FAtomMolecule

A

D fi iti F/A

Wall

Definition p = F/A

SI Unit [p] = [F]/[A] = Pascal (Pa)


The Manometer Gauge measures vacuum by pressure on a membrane.

Kinetic Gas Theory of Vacuum

Volume Density- 1 m3, 105 Pa, 22 C 2.5x1025 Molecules - 1 m3, 10-7 Pa, 22 C 2.5x1013 Molecules

vkTm

=8π

λ 1 66.

Average velocity

M F P th λπ

= ≈2 2don p Pa

mm( )

Γ =nv4

Mean Free Path

Particle Flux

Total Energy E mv kT= =12

2 32

Pressure p mnv=13


Pressure of Mixture p pi minivi nikT= ∑ =∑ = ∑13

126

Photoresists

Resists have many components:Resin - a base material that is a binder for obtaining the chemo-mechanical properties: chemical resistance for pattern transfer, …Sensitizer - Photo-active compound with radiation sensitivitySensitizer Photo active compound with radiation sensitivitySolvent - Control of properties for deposition - viscosity, providing liquid form Adhesion Promoter

SU8: a photosensitive thick photosensitive resist is an “epoxy”PMMA: Polymethyl methacrylate and solvent (usually chlorobenzene)PMMA: Polymethyl methacrylate and solvent (usually chlorobenzene)

DQ: Diazonquinone (20-50% of N: Novolac; Polymer containing

DQN: A positive resist for G- and I-line exposure

q (weight) Photosensitive part

; y garomatic ring with methyl and OH groups; dissolves in aqueous solutionsDQ

Solvent: adjusts viscosity but evaporates

UV Lightevaporates before exposure; little role in photo-Corboxylic Acid


photochemistry

Corboxylic Acid (…-C(=O)-OH)a dissolution enhancer

Positive Resist Example: DQNKetene anKetene, an intermediate short-lived molecule

Utilizes weak

DQ is insoluble in base solutions

bonding of Nitrogen and Carboxylic acid solubility in the developerDQ is insoluble in base solutions

Carboxylic acid reacts and dissolves in base solutionsResin/carboxylic mixture consumes water, assisted by release of N2

Dissolution occurs with breakdown of carboxylic acid into water-soluble

p

amines such as aniline (using developers containing KOH, NaOH)

Unexposed areas quite unchanged; pattern shapes retained

Novolac is long-chain aromatic ring polymer that is quite chemically i t t ki th h t i t d t d d t hi k


resistant, making these photoresists good wet and dry etching masks

Vapor Pressure

r)ss

ure

(to

rr Vap

or P

re

Vap

or

Pre

essure (P

aa)


Temperature (K) O’Hanlon, C.7, p. 371, (1980)

Annealing: Solid State Regrowth

Technical importance:

The source drain implants are so heavy that they usually hi th ili A li t hi h t t t thamorphize the silicon. Annealing at high temperature restores the

crystal structure, using deeper, undamaged part of the silicon wafer as a seed for regrowth

Velocity as function of temperatureVelocity as function of temperature

Velocity as function of crystal orientation

Velocity as a function of doping

I fl f I itiInfluence of Impurities


Crystallization following Implantation Amorphization

The growth velocity is thermally activated with EA = 2.76 eV.

The same process operates overThe same process operates over 10 orders of magnitude in velocity !

At 600 C, the speed is about 10 pAngstrom/sec


Summary: Solid State Epitaxial Growth

A single activation energies characterize regrowth over 10 orders of velocity.

Th ti ti 2 76 V i i d t li k d t thThe activation energy, ~ 2.76 eV, is unique and not linked to other activation energies in Silicon (point defect generation, migration etc)

M i th l it i /Maximum regrowth velocity is ~ m/sec

The regrowth velocity depends on doping and appear to be a Fermi level effect

It also depends strongly on impurities and orientation.


Comments

The key concept to rapid thermal annealing is that dopant diffusion (EA ~ 4 eV) varies less with temperature than lattice repair (EA ~ 5 eV)repair (EA ~ 5 eV)

The difference at high temperaturetemperature does not look like much but this is a log plot !


p

Annealing/Activation

1ImplantDoseBoron

ctio

n*

Implant150 keV B-> Si8x1012 or2.5x1014 or

Dose

vate

d F

ra

2.5x10 or2x1015 cm-2

Isochronal

Act

iv 30 min FurnaceAnneal

* Hole Charge/ PD ED ED Out

T t (C)

0.01

Hole Charge/Dose


Temperature (C)

After Campbell, Fig. 5-16, p. 115

Activated Fraction

SIMS

ImplantB-> Si70 keV

m-3

) Hall

800 C 1015 cm-2

Anneal800 or 900 C35 iat

ion

(cm

SIMS

800 C

35 min

Co

nce

ntr

a

SIMS

Hall

C

900 C

135Tiwari_12_2009_iWSG_Technology.pptxDepth (μm)

After Sze, Fig. 24, p. 357

Implant/Anneal Examples

1021

m-3

)

ImplantB-> Si35 keV

ratio

n (c

m

FAnnealsRTA 1100 C/10 sRTA 1100 C/30 s

Con

cent

r

I

RTA RTA 1100 C/30 sF 1000 C/30 m

C

1015


Depth (μm)0 1

Sze, Fig. 29, p. 362

Implant/Anneal Examples

m-3

)TransientEnhanced Diffusion

trat

ion

(cm

(TED)

AnomalousDiff i Aft

Con

cent Diffusion After

Ion Implantation

Depth (μm)


PDG 8-31 (2000)

Evaporation Variables

Base Pressure p [Pa, torr]

Mean Free PathkT

22λ =

Scattered Fraction

p22πσ

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−=

λd

nn

exp1

Geometric Factors: deposited Mass/Area (Cosine Law)

⎠⎝o

θφπ

coscos2r

eMDR =

Me = Mass of Evaporated Metalr, φ, θ = Geometry Parameters


Crucibles

Refractory Metals Melting Point Vapor Pressure at Temperature

W 3380 C 10-2 torr at 3230 C

Ta 3000 C 10-2 torr at 3060 CTa 3000 C 10-2 torr at 3060 C

Mo 2620 C 10-2 torr at 2530 C

Refractory Ceramics

Graphitic C 3700 C 10-2 at 2600 C

Al2O3 2030 C 10-2 at 1900 C

BN 2500 C 10-2 at 1600 C

Considerations: Thermal conductivity, Thermal expansion, Electrical conductivity, Wetting and Reactivity

BN 2500 C 10 2 at 1600 C

Aluminum: Tungsten dissolves in Aluminum. So, not quite compatible.graphite “boat,” but avoid cracking the “boat” due to stress/temperature gradients

N t ti i ll f t i l ith hi h lti i t i th


Next option, specially for materials with even higher melting points is the use of an electron gun!

Principles of

Chemical Vapor Deposition and OxidationInsulating dielectrics

LithographyOptical techniques for patterning photoresists

Dry and Wet EtchingPattern transfer techniques

Physical Vapor Depositiony p pVacuum techniques for evaporation and sputtering of metals and other materials

Diffusion, Implantation and Annealingp gMaterial modification techniques

CharacterizationOptical, other, and electrical measurements during and following


Optical, other, and electrical measurements during and following processing

2 Step Diffusion: Pre-Dep/Drive

Infinite Source Predep followedD tIf this condition is not met,

general result:1 1

2 2

Infinite Source Predep , followed

by Drive-in ,

From Infinite Source Pre-Dep (erfc):

D t

D t ( )21

01 2 2

0

1/ 2

general result:

2( , , )

1

zUC eC x t t dz

z

β

π

− +⎛ ⎞= ⎜ ⎟ +⎝ ⎠ ∫

1/ 2

1 102

If this is confined close enough to the surface

D tQ C

π⎛ ⎞= ⎜ ⎟⎝ ⎠

1/ 2

1 1

2 2

2

D tU

D t

xβ

⎛ ⎞= ⎜ ⎟⎝ ⎠

⎛ ⎞⎜ ⎟If this is confined close enough to the surface,

it will look like an impulse and we can use

Guassian diffusion solutions with this .Q( )

1 1 2 2

10

2

Final Surface Concentration will be:

2

D t D t

CC U

β = ⎜ ⎟⎜ ⎟+⎝ ⎠

⎛ ⎞⎜ ⎟

1/ 2

2 2

What is close enough?

3 Diffusion length of Drive vs. Pre-depD t

D t

⎛ ⎞≥⎜ ⎟

⎝ ⎠

( )1002

2tan

CC U

π−⎛ ⎞= ⎜ ⎟

⎝ ⎠


1 1D t⎝ ⎠

Tiwari 12 01 Technology 1

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