Silicon Oxidation Model
Deal and Grove’s model describes
the kinematics of silicon oxidation. The model is generally valid for
temperatures between 700 and 1300°C,
partial pressures between 0.2 and 1 atmosphere, and for oxide thickness between
300 and 20,000°A for oxygen and water
Flux is the number of atoms or
molecules crossing a unit area in unit time.
Fig. 2 Basic model for thermal oxidation of
Basic model for thermal
oxidation of silicon 
The oxidizing species:
- Are transported from the bulk of the gas phase
to the gas-oxide interface with flux F1.
- Are transported across the existing oxide
toward the silicon with flux F2.
- React at Si-SiO2 interface with the silicon with flux F3.
Under steady state, all the
fluxes are equal, F1=F2=F3
- Assuming that the flux of the oxidant from the
bulk of the gas phase to the gas-oxide interface is proportional to the
difference between the oxidant concentration in the bulk of gas CG
and the oxidant concentration adjacent to the oxide surface CS,
hG is the gas phase mass-transfer coefficient.  
- From Henry’s law, which states that the
concentration of an adsorbed species at the surface of a solid is
proportional to the partial pressure of that species in the gas just above
the solid, we get
is the equilibrium concentration in the oxide at the outer surface.
C* is the equilibrium bulk concentration in the
the partial pressure in the gas adjacent to the oxide surface.
is partial pressure in the bulk of gas
Henry’s law constant.
- Using Henry’s law along with the ideal gas law
C*=HpG=HkTCG where pG=kTCG
C0 and C*, into
equation (1), we get the following:
F1=h(C*-C0) where h=hG/HkT
- Oxidation is a non-equilibrium process with
the driving force being the deviation of concentration from equilibrium.
Henry’s law is valid only in the absence of dissociation effects at the
gas-oxide interface. This implies that the species moving through the
oxide are molecular.
- Flux of the oxidizing species across the oxide
follows Fick’s law at any point d in the oxide layer.
D is the
is the oxidizing species concentration in the oxide adjacent to the
is the oxide thickness.
- Flux corresponding to the Si-SiO2
interface reaction is proportional to the concentration of oxidizing
species in the oxide adjacent to the oxide-silicon interface, Ci.
is the rate constant of chemical surface reaction for silicon oxidation.
- Under steady state conditions we have
and solving simultaneous
equations we can find Ci and C0.
- When diffusivity is very small Ci®0 and C0®C*.
This is called diffusion-controlled case. It results from the flux of
oxidant through the oxide (F2) being small compared to the flux
corresponding to the silicon-silicon dioxide interface reaction (F3).
Hence oxidation rate depends on the supply of oxidant to the interface and
not on the reaction at the interface.
- When diffusivity is large, Ci=C0.
This is called reaction-controlled case, because an abundant supply of
oxidant is provided at the silicon-silicon oxide interface and the
oxidation rate is controlled by the reaction rate constant kS
- Combining various equations and assuming that
an oxide may be present initially from a previous processing step, that
is, d0=di at t=0 the following equation can be
+ A d0 = B(t+t)
Where t represents a shift in the time axis to account for the
presence of the initial oxide layer di.
the number of oxidant molecules incorporated into a unit volume of the oxide