models the first major model is that of deal and grove (1965) the first major model is that of deal...

ModelsModels

The first major model is that of Deal The first major model is that of Deal and Grove (1965)and Grove (1965)This lead to the linear/parabolic modelThis lead to the linear/parabolic model

Note that this model cannot explain Note that this model cannot explain the effect of oxidation of the diffusion ratethe effect of oxidation of the diffusion rate the oxidation of shaped surfacesthe oxidation of shaped surfaces the oxidation of very thin oxides in mixed the oxidation of very thin oxides in mixed

ambientsambients

The model is an excellent starting place The model is an excellent starting place for the other more complicated modelsfor the other more complicated models

CHEMICAL REACTIONSCHEMICAL REACTIONS

Process for dry oxygenProcess for dry oxygen

Si + OSi + O22 SiO SiO22

Process for water vaporProcess for water vapor

Si + 2HSi + 2H22O O SiO SiO22 + 2H + 2H22

OXIDE GROWTHOXIDE GROWTHSi is consumed as oxide grows and oxide expands. Si is consumed as oxide grows and oxide expands.

The Si surface moves into the wafer.The Si surface moves into the wafer.

54%

46%SiO2

Siliconwafer

Originalsurface

MODEL OF OXIDATIONMODEL OF OXIDATIONOxygen must reach silicon interfaceOxygen must reach silicon interface

Simple model assumes OSimple model assumes O22 diffuses diffuses through SiOthrough SiO22

Assumes no OAssumes no O22 accumulation in SiO accumulation in SiO22

Assumes the rate of arrival of HAssumes the rate of arrival of H22O or OO or O22 at the oxide surface is so fast that it can at the oxide surface is so fast that it can be ignoredbe ignoredReaction rate limited, not diffusion rate Reaction rate limited, not diffusion rate

limitedlimited

Deal-Grove Model of Deal-Grove Model of OxidationOxidation

Fick’s First Law of diffusion states that Fick’s First Law of diffusion states that the particle flow per unit area, J (particle the particle flow per unit area, J (particle flux), is directly proportional to the flux), is directly proportional to the concentration gradient of the particle.concentration gradient of the particle. We assume that oxygen flux passing through We assume that oxygen flux passing through

the oxide is constant everywhere.the oxide is constant everywhere.

FF11 is the flux, C is the flux, CGG is the concentration in the is the concentration in the gas flow, Cgas flow, CSS is the concentration at the is the concentration at the surface of the wafer, and hsurface of the wafer, and hGG is the mass is the mass transfer coefficienttransfer coefficient

)(1 SGG CChF

J

Distance from surface, x

N

No

Ni

Silicondioxide

Silicon

SiO2 Si

Xo

OiO

OiO

xCCDF

xNNDJ

/)(

/)(

2


Assume the oxidation rate at Si-SiOAssume the oxidation rate at Si-SiO22 interface is proportional to the Ointerface is proportional to the O22 concentration:concentration:

Growth rate is given by the oxidizing flux Growth rate is given by the oxidizing flux divided by the number of molecules, M, of divided by the number of molecules, M, of the oxidizing species that are incorporated the oxidizing species that are incorporated into a unit volume of the resulting oxide: into a unit volume of the resulting oxide:

is

is

CkF

NkJ

3

skDxM

DN

M

J

dt

dx

0

00


The boundary condition isThe boundary condition is

The solution of differential equation isThe solution of differential equation is

AD

kB

DN

M

x

B

x

B As

o i i 2 2 2

ixtx 00

AB

x

B

xt 0

20


xxox ox : final oxide thickness: final oxide thickness

xxi i : initial oxide thickness : initial oxide thickness

B/AB/A : linear rate constant: linear rate constant

B : parabolic rate constantB : parabolic rate constant

xxii: thickness of initial oxide layer: thickness of initial oxide layer

: equivalent time required to grow initial oxide : equivalent time required to grow initial oxide layerlayer

There are two limiting cases:There are two limiting cases:Very long oxidation times, Very long oxidation times, t >> t >>

xxoxox2 2 = B t= B t

Oxide growth in this parabolic regime is Oxide growth in this parabolic regime is diffusion controlled.diffusion controlled.

Very short oxidation times, (Very short oxidation times, (t + t + ) << ) << AA22/4B/4B xxoxox

= B/A ( t + = B/A ( t + ) )Oxide growth in this linear regime is Oxide growth in this linear regime is

reaction-rate limited.reaction-rate limited.

(111) Si(100) Si

10.0

1.0

0.1

0.01

0.001

0.0001

1200 1100 1000 900 800 700

Temperature (0C)

B/A (m/hr)

1000/T (K-1)

0.6 0.7 0.8 0.9 1.0 1.1

H2O (640 torr)EA = 2.05 eV

Dry O2

EA = 2.0 eV

At short times, B/A is the linear rate constantProcess is controlled by the reaction at the Si surface

11001200 1000 900 8001.0

0.1

0.01

0.001

B(

m2/h

r)

0.7 0.8 0.9 1.00.61000/T(K-1)

Temperature (0C)

H2O (640 torr)EA=0.78eV

Dry O2

EA=1.23eV

At long times, B is the parabolic rate constant (xO2

B)Process is controlled by diffusion of O through oxide

Deal-Grove Model PredictionsDeal-Grove Model Predictions Once B and B/A are determined, we can Once B and B/A are determined, we can

predict the thickness of the oxide versus predict the thickness of the oxide versus timetime

Oxide as a Diffusion BarrierOxide as a Diffusion Barrier Diffusion of As, B, P, and Sb are orders of magnitude Diffusion of As, B, P, and Sb are orders of magnitude

less in oxide than in siliconless in oxide than in silicon Oxide is excellent mask for high-temperature Oxide is excellent mask for high-temperature

diffusion of impuritiesdiffusion of impurities

BoronPhosphorus

10

1

0.1

0.010.1 1.0 10 100

Diffusion time (hr)

Mask t

hic

kn

ess

(m

)

1200 C

1000 C

1200 C

1000 C 1100 C

900 C

1100 C

900 C

BP

10

1

0.1

0.010.1 1.0 10 100

Other ModelsOther Models

A variety of other models have been suggested, A variety of other models have been suggested, primarily to correct the deficiencies of the Deal-primarily to correct the deficiencies of the Deal-Grove model for thin oxidesGrove model for thin oxides

These includeThese include The Reisman power law modelThe Reisman power law model The Han and Helms model with parallel The Han and Helms model with parallel

oxidation pathsoxidation paths The Ghez and van Meulen model to account for The Ghez and van Meulen model to account for

the effect of oxygen pressurethe effect of oxygen pressure Some of these models do a much better job for Some of these models do a much better job for

thin oxidesthin oxides None are widely acceptedNone are widely accepted

Other TopicsOther Topics

Several topics other than the simple Several topics other than the simple planar growth of wet and dry oxide planar growth of wet and dry oxide are importantare important

These includeThese includeThin oxide growth kineticsThin oxide growth kineticsDependence on oxygen pressureDependence on oxygen pressureDependence on crystal orientationDependence on crystal orientationMixed ambient growth kineticsMixed ambient growth kinetics2D growth kinetics2D growth kinetics

Example: 2D GrowthExample: 2D Growth


There are several interesting There are several interesting observationsobservationsThere is significant retardation of the There is significant retardation of the

oxide growth in sharp cornersoxide growth in sharp cornersThe retardation is more pronounced for The retardation is more pronounced for

low temperature oxidation than for high low temperature oxidation than for high temperature oxidationtemperature oxidation

Interior (concave) corners show a more Interior (concave) corners show a more pronounces retardation that exterior pronounces retardation that exterior (convex) corners(convex) corners


Several physical mechanisms are needed to Several physical mechanisms are needed to understand these resultsunderstand these results

1.1. Crystal orientationCrystal orientation

2.2. Oxidant diffusionOxidant diffusion

3.3. Stress due to volume expansionStress due to volume expansion Kao et al suggested changes to the linear-Kao et al suggested changes to the linear-

parabolic (Deal-Grove) model to correct for these parabolic (Deal-Grove) model to correct for these effectseffects

Most of these effects are built into the modeling Most of these effects are built into the modeling software such as SUPREM IV and ATHENA software such as SUPREM IV and ATHENA

Measurement MethodsMeasurement Methods

The parameters of interest includeThe parameters of interest includeThicknessThicknessDielectric constant and strengthDielectric constant and strengthIndex of refractionIndex of refractionDefect densityDefect density

There are three classes of measurementThere are three classes of measurementPhysical (usually destructive)Physical (usually destructive)Optical (usually nondestructive)Optical (usually nondestructive)Electrical (usually nondestructive)Electrical (usually nondestructive)

Physical MeasurementsPhysical Measurements

Simple step height technique (DekTak)Simple step height technique (DekTak) Etch away oxide with HFEtch away oxide with HF Use a small stylus to measure the resulting Use a small stylus to measure the resulting

step heightstep height The resolution is <10 nmThe resolution is <10 nm

More complex technique uses one or more of the More complex technique uses one or more of the SFM concepts (AFM, MFM, etc)SFM concepts (AFM, MFM, etc) Technique has atomic resolutionTechnique has atomic resolution

SEM or TEM (electron microscopy)SEM or TEM (electron microscopy) All require sample preparation that makes the All require sample preparation that makes the

tests destructive and not easy to use in tests destructive and not easy to use in productionproduction

Optical MeasurementsOptical Measurements

Most optical techniques use the concept of Most optical techniques use the concept of measuring reflected monochromatic lightmeasuring reflected monochromatic light If monochromatic light of wavelength If monochromatic light of wavelength shines shines

on a transparent film of thickness xon a transparent film of thickness x00, some , some light is reflected directly and some is reflected light is reflected directly and some is reflected from the wafer-film interfacefrom the wafer-film interface

For some wavelengths, the light will be in For some wavelengths, the light will be in phase and for others it will be out of phasephase and for others it will be out of phaseconstructive and destructive interferenceconstructive and destructive interference

Minima and maxima of intensity are observed as Minima and maxima of intensity are observed as is varied is varied

Optical TechniquesOptical Techniques

Color ChartColor Chart

http://www.htelabs.com/appnotes/sio2_color_chart_thermal_silicon_dioxide.htm


Instrument from Instrument from FilmetricsFilmetrics(http://www.filmetrics.com)(http://www.filmetrics.com)


The positions of the minima and maxima The positions of the minima and maxima are given byare given by

m=1,2,3… for maxima and ½,3/2,5/2,… for minimam=1,2,3… for maxima and ½,3/2,5/2,… for minima

This is called reflectometry and works well This is called reflectometry and works well for thicknesses over a few 10’s of nmfor thicknesses over a few 10’s of nm

1

01

01maxmin,

sinsin

cos2

n

n

m

xn


If one does not know n, or if the film is very thin, If one does not know n, or if the film is very thin, then ellipsometry is betterthen ellipsometry is better

When multiple wavelengths of light are used, the When multiple wavelengths of light are used, the instrument is known as a spectroscopic instrument is known as a spectroscopic ellipsometerellipsometer


Here, one uses polarized light.Here, one uses polarized light. The measurement may be performed at The measurement may be performed at

multiple angles of incidence to obtain a higher multiple angles of incidence to obtain a higher degree of accuracydegree of accuracy

One can get the index of refraction as a One can get the index of refraction as a function of wavelength as well as the function of wavelength as well as the extinction coefficientextinction coefficient Can be used to measure thickness to <1 nmCan be used to measure thickness to <1 nm

Fitting routines are necessary to take into Fitting routines are necessary to take into account rough interfaces between Si and account rough interfaces between Si and SiOSiO22 layers. layers.

Cauchy EquationCauchy Equation

...)(42

CB

An

Sellmeier EquationSellmeier Equation

...1)(3

2

23

22

22

12

21

C

B

C

B

C

Bn

http://en.wikipedia.org/wiki/Cauchy%27s_equation

Electrical MeasurementsElectrical Measurements

These measure properties that correlate directly These measure properties that correlate directly to the performance of the devices fabricated to the performance of the devices fabricated using the oxidesusing the oxides

The dominant techniques is the C—V The dominant techniques is the C—V measurementmeasurement The basic structure for the measurement is the MOS The basic structure for the measurement is the MOS

capacitorcapacitor The usual combination is Si-SiOThe usual combination is Si-SiO22-(Al or pSi)-(Al or pSi)

Any conductor-dielectric-semiconductorAny conductor-dielectric-semiconductor can be usedcan be used

MOS CapacitorMOS Capacitor

tox

Al

Al

Si wafer V

+

-

http://www.mtmi.vu.lt/pfk/funkc_dariniai/transistor/mos_capacitors.htm

http://ece-www.colorado.edu/~bart/book/book/chapter6/ch6_3.htm#fig6_3_5

C-V PlotC-V Plot

C-V PlotC-V Plot

Differences between high frequency Differences between high frequency and low frequency C-V dataand low frequency C-V dataDoping concentration in Si near Si-oxide Doping concentration in Si near Si-oxide

interfaceinterfaceVoltage shift proportional to charged Voltage shift proportional to charged

defects within oxidedefects within oxide

models the first major model is that of deal and grove (1965) the first major model is that of deal...

Documents

oxidation rate

oxide surface

dry oxide

helms model

major model

resulting oxide

growth rate

initial oxide layerthere