Models The first major model is that of Deal and Grove (1965) The first major model is that of Deal and Grove (1965) This lead to the linear/parabolic model This 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 rate the effect of oxidation of the diffusion rate the oxidation of shaped surfaces the oxidation of shaped surfaces the oxidation of very thin oxides in mixed ambients the oxidation of very thin oxides in mixed ambients The model is an excellent starting place for the other more complicated models The model is an excellent starting place for the other more complicated models

CHEMICAL REACTIONS Process for dry oxygen Process for dry oxygen Si + O 2  SiO 2 Si + O 2  SiO 2 Process for water vapor Process for water vapor Si + 2H 2 O  SiO 2 + 2H 2

OXIDE GROWTH Si is consumed as oxide grows and oxide expands. The Si surface moves into the wafer. Si is consumed as oxide grows and oxide expands. The Si surface moves into the wafer. 54% 46% SiO 2 Silicon wafer Original surface

MODEL OF OXIDATION Oxygen must reach silicon interface Oxygen must reach silicon interface Simple model assumes O 2 diffuses through SiO 2 Simple model assumes O 2 diffuses through SiO 2 Assumes no O 2 accumulation in SiO 2 Assumes no O 2 accumulation in SiO 2 Assumes the rate of arrival of H 2 O or O 2 at the oxide surface is so fast that it can be ignored Assumes the rate of arrival of H 2 O or O 2 at the oxide surface is so fast that it can be ignored Reaction rate limited, not diffusion rate limited Reaction rate limited, not diffusion rate limited

Deal-Grove Model of Oxidation Fick’s First Law of diffusion states that the particle flow per unit area, J (particle flux), is directly proportional to the concentration gradient of the particle. Fick’s First Law of diffusion states that the particle flow per unit area, J (particle flux), is directly proportional to the concentration gradient of the particle. We assume that oxygen flux passing through the oxide is constant everywhere. F 1 is the flux, C G is the concentration in the gas flow, C S is the concentration at the surface of the wafer, and h G is the mass transfer coefficient

J Distance from surface, x N NoNo NiNi Silicon dioxide Silicon SiO 2 Si XoXo

Deal-Grove Model of Oxidation Assume the oxidation rate at Si-SiO 2 interface is proportional to the O 2 concentration: Assume the oxidation rate at Si-SiO 2 interface is proportional to the O 2 concentration: Growth rate is given by the oxidizing flux divided by the number of molecules, M, of the oxidizing species that are incorporated into a unit volume of the resulting oxide: Growth rate is given by the oxidizing flux divided by the number of molecules, M, of the oxidizing species that are incorporated into a unit volume of the resulting oxide:

Deal-Grove Model of Oxidation The boundary condition is The boundary condition is The solution of differential equation is The solution of differential equation is

Deal-Grove Model of Oxidation x ox : final oxide thickness x i : initial oxide thickness B/A : linear rate constant B : parabolic rate constant x i : thickness of initial oxide layer  : equivalent time required to grow initial oxide layer

There are two limiting cases: There are two limiting cases: Very long oxidation times, t >>  Very long oxidation times, t >>  x ox 2 = B t x ox 2 = B t Oxide growth in this parabolic regime is diffusion controlled. Oxide growth in this parabolic regime is diffusion controlled. Very short oxidation times, (t +  ) << A 2 /4B Very short oxidation times, (t +  ) << A 2 /4B x ox = B/A ( t +  ) x ox = B/A ( t +  ) Oxide growth in this linear regime is reaction- rate limited. Oxide growth in this linear regime is reaction- rate limited.

(111) Si (100) Si Temperature ( 0 C) B/A (  m/hr) 1000/T (K -1 ) H 2 O (640 torr) E A = 2.05 eV Dry O 2 E A = 2.0 eV At short times, B/A is the linear rate constant Process is controlled by the reaction at the Si surface

B(  m 2 /hr) /T(K -1 ) Temperature ( 0 C) H 2 O (640 torr) E A =0.78eV Dry O 2 E A =1.23eV At long times, B is the parabolic rate constant (x O 2  B) Process is controlled by diffusion of O through oxide

Deal-Grove Model Predictions Once B and B/A are determined, we can predict the thickness of the oxide versus time Once B and B/A are determined, we can predict the thickness of the oxide versus time

Deal-Grove Model of Oxidation

Oxide as a Diffusion Barrier Diffusion of As, B, P, and Sb are orders of magnitude less in oxide than in silicon Diffusion of As, B, P, and Sb are orders of magnitude less in oxide than in silicon Oxide is excellent mask for high-temperature diffusion of impurities Oxide is excellent mask for high-temperature diffusion of impurities Boron Phosphorus Diffusion time (hr) Mask thickness (  m) 1200 C 1000 C 1200 C 1000 C 1100 C 900 C 1100 C 900 C B P

Other Models A variety of other models have been suggested, primarily to correct the deficiencies of the Deal-Grove model for thin oxides A variety of other models have been suggested, primarily to correct the deficiencies of the Deal-Grove model for thin oxides These include These include The Reisman power law model The Reisman power law model The Han and Helms model with parallel oxidation paths The Han and Helms model with parallel oxidation paths The Ghez and van Meulen model to account for the effect of oxygen pressure The Ghez and van Meulen model to account for the effect of oxygen pressure Some of these models do a much better job for thin oxides Some of these models do a much better job for thin oxides None are widely accepted None are widely accepted

Other Topics Several topics other than the simple planar growth of wet and dry oxide are important Several topics other than the simple planar growth of wet and dry oxide are important These include These include Thin oxide growth kinetics Thin oxide growth kinetics Dependence on oxygen pressure Dependence on oxygen pressure Dependence on crystal orientation Dependence on crystal orientation Mixed ambient growth kinetics Mixed ambient growth kinetics 2D growth kinetics 2D growth kinetics

Example: 2D Growth

There are several interesting observations There are several interesting observations There is significant retardation of the oxide growth in sharp corners There is significant retardation of the oxide growth in sharp corners The retardation is more pronounced for low temperature oxidation than for high temperature oxidation The retardation is more pronounced for low temperature oxidation than for high temperature oxidation Interior (concave) corners show a more pronounces retardation that exterior (convex) corners Interior (concave) corners show a more pronounces retardation that exterior (convex) corners

Example: 2D Growth

Several physical mechanisms are needed to understand these results Several physical mechanisms are needed to understand these results 1. Crystal orientation 2. Oxidant diffusion 3. Stress due to volume expansion Kao et al suggested changes to the linear-parabolic (Deal-Grove) model to correct for these effects Kao et al suggested changes to the linear-parabolic (Deal-Grove) model to correct for these effects Most of these effects are built into the modeling software such as SUPREM IV and ATHENA Most of these effects are built into the modeling software such as SUPREM IV and ATHENA

Measurement Methods The parameters of interest include The parameters of interest include Thickness Thickness Dielectric constant and strength Dielectric constant and strength Index of refraction Index of refraction Defect density Defect density There are three classes of measurement There are three classes of measurement Physical (usually destructive) Physical (usually destructive) Optical (usually nondestructive) Optical (usually nondestructive) Electrical (usually nondestructive) Electrical (usually nondestructive)

Physical Measurements Simple step height technique (DekTak) Simple step height technique (DekTak) Etch away oxide with HF Etch away oxide with HF Use a small stylus to measure the resulting step height Use a small stylus to measure the resulting step height The resolution is <10 nm The resolution is <10 nm More complex technique uses one or more of the SFM concepts (AFM, MFM, etc) More complex technique uses one or more of the SFM concepts (AFM, MFM, etc) Technique has atomic resolution Technique has atomic resolution SEM or TEM (electron microscopy) SEM or TEM (electron microscopy) All require sample preparation that makes the tests destructive and not easy to use in production All require sample preparation that makes the tests destructive and not easy to use in production

Optical Measurements Most optical techniques use the concept of measuring reflected monochromatic light Most optical techniques use the concept of measuring reflected monochromatic light If monochromatic light of wavelength shines on a transparent film of thickness x 0, some light is reflected directly and some is reflected from the wafer-film interface If monochromatic light of wavelength shines on a transparent film of thickness x 0, some light is reflected directly and some is reflected from the wafer-film interface For some wavelengths, the light will be in phase and for others it will be out of phase For some wavelengths, the light will be in phase and for others it will be out of phase constructive and destructive interference constructive and destructive interference Minima and maxima of intensity are observed as is varied Minima and maxima of intensity are observed as is varied

Optical Techniques

Color Chart r_chart_thermal_silicon_dioxide.htm r_chart_thermal_silicon_dioxide.htm

Optical Measurements Instrument from Filmetrics ( Instrument from Filmetrics (

Optical Measurements The positions of the minima and maxima are given by m=1,2,3… for maxima and ½,3/2,5/2,… for minima The positions of the minima and maxima are given by m=1,2,3… for maxima and ½,3/2,5/2,… for minima This is called reflectometry and works well for thicknesses over a few 10’s of nm This is called reflectometry and works well for thicknesses over a few 10’s of nm

Optical Measurements If one does not know n, or if the film is very thin, then ellipsometry is better If one does not know n, or if the film is very thin, then ellipsometry is better When multiple wavelengths of light are used, the instrument is known as a spectroscopic ellipsometer When multiple wavelengths of light are used, the instrument is known as a spectroscopic ellipsometer

Optical Measurements Here, one uses polarized light. Here, one uses polarized light. The measurement may be performed at multiple angles of incidence to obtain a higher degree of accuracy The measurement may be performed at multiple angles of incidence to obtain a higher degree of accuracy One can get the index of refraction as a function of wavelength as well as the extinction coefficient One can get the index of refraction as a function of wavelength as well as the extinction coefficient Can be used to measure thickness to <1 nm Can be used to measure thickness to <1 nm Fitting routines are necessary to take into account rough interfaces between Si and SiO 2 layers. Fitting routines are necessary to take into account rough interfaces between Si and SiO 2 layers.

Cauchy Equation Sellmeier Equation

Electrical Measurements These measure properties that correlate directly to the performance of the devices fabricated using the oxides These measure properties that correlate directly to the performance of the devices fabricated using the oxides The dominant techniques is the C—V measurement The dominant techniques is the C—V measurement The basic structure for the measurement is the MOS capacitor The basic structure for the measurement is the MOS capacitor The usual combination is Si-SiO 2 -(Al or pSi) The usual combination is Si-SiO 2 -(Al or pSi) Any conductor-dielectric-semiconductor can be used Any conductor-dielectric-semiconductor can be used

MOS Capacitor t ox Al Si wafer V + -

C-V Plot

Differences between high frequency and low frequency C-V data Differences between high frequency and low frequency C-V data Doping concentration in Si near Si-oxide interface Doping concentration in Si near Si-oxide interface Voltage shift proportional to charged defects within oxide Voltage shift proportional to charged defects within oxide