As previously described in Section 2.3.1, the DealGrove model can only describe oxidation growth for oxides with thicknesses above 30nm. At the time when the DealGrove model was introduced (1965), the semiconductor industry did not require such thin films. However, as the device sizes and geometries began to shrink, the limitations of the DealGrove model became evident. It has been observed experimentally that the oxidation rate is much faster than predicted by the DealGrove model at the initial stages of oxidation and for thin oxide growths [34]. Several researchers have suggested that the cause of the increased oxidation rate are electrochemical effects such as fieldenhanced oxidation, structural effects such as microchannels, stress effects modifying oxidant diffusivity, and changes in oxygen solubility in the oxide with little success. Those that had more success suggested the increased rate is due to parallel oxidation mechanisms such as silicon interstitials injected into the oxide, oxygen vacancies, diffusion of atomic oxygen, surface oxygen exchange, and the effects of a finite nonstoichiometric transition region between amorphous SiO and Si [34].
Massoud et al., in 1985 [143], [144] suggested an update to the DealGrove model for dry oxidation, which was to address the thin oxide growth regime. They provided an analytical model based on parallel oxidation mechanisms to fit experimental data with good success. The price for the improved model for thin oxides is an increased complexity of the model.
Massoud introduced additional terms to the oxidation rate equation (2.16) and changed the values of the linear and parabolic constants. The oxidation rate is given by
The values for the preexponential constants , and the activation energies , for different crystal orientations and temperatures are listed in Table 2.3. It should also be noted that the values for , , , and change with temperature, which was not the case with the original DealGrove model.

In (2.37), the second and third term on the right hand side are additional terms which represent the rate enhancement in the thin regime. They are defined by preexponential constants and and characteristic lengths and . The first characteristic length is in the order of 1nm and is meant to deal with the rate increase in the first 5nm of oxide growth, after which it vanishes. The second characteristic length , with a value in the order of 7nm, is meant to decay until approximately 25nm, after which it no longer influences the oxidation rate and the rate becomes linearparabolic once again.
Another way to express (2.37) in terms which are easier to manipulate and mathematically solve is presented in [142]
The preexponential constants and activation energies of the above expressions (2.41)(2.44) are given in Table 2.4 for dry oxidation in a temperature range from 8001000 C.

As already performed for the DealGrove expression in Section 2.3.1, inverting (2.40) gives a convenient expression for the oxide thickness as a function of oxidation time and viceversa. Therefore, (2.40) is rewritten as
Equation (2.46) can be solved in order to obtain an analytic expression for the oxide thickness as a function of oxidation time
The relationship (2.48) is meant to give a valid expression for the oxide thickness after an oxidation time in a dry ambient from the native oxide thickness conditions. Figure 2.13 shows the difference between the DealGrove model and the Massoud model for the initial stages of oxidation. It is evident that the Massoud model depicts a faster initial oxidation rate.
