Deal-Grove model

The Deal–Grove model mathematically describes the growth of an oxide layer on the surface of a material. In particular, it is used to predict and interpret thermal oxidation of silicon in semiconductor device fabrication. The model was first published in 1965 by Bruce Deal and Andrew Grove, of Fairchild Semiconductor.

The model assumes that oxidation reaction occurs at the interface between the oxide and the substrate, rather than between the oxide and the ambient gas. Thus, it considers three phenomena that the oxidizing species undergoes, in this order:

The model assumes that each of these stages proceeds at a rate proportional to the oxidant's concentration. In the first case, this means Henry's law; in the second, Fick's law of diffusion; in the third, a first-order reaction with respect to the oxidant. It also assumes steady state conditions, i.e. that transient effects do not appear.

Given these assumptions, the flux of oxidant through each of the three phases can be expressed in terms of concentrations, material properties, and temperature.

By setting the three fluxes equal to each other, each may be found. In turn, the growth rate may be found readily from the oxidant reaction flux.

In practice, the ambient gas (stage 1) does not limit the reaction rate, so this part of the equation is often dropped. This simplification yields a simple quadratic equation for the oxide thickness. For oxide growing on an initially bare substrate, the thickness X_o at time t is given by the following equation:

where the constants A and B encapsulate the properties of the reaction and the oxide layer, respectively. These constants are given as:

where ${\displaystyle C_{s}=HP_{g}}$, with ${\displaystyle H}$ being the gas solubility parameter of the Henry's law and ${\displaystyle P_{g}}$ is the partial pressure of the diffusing gas. ${\displaystyle N_{i}}$ denotes the oxidant molecules/unit volume needed to produce a unit volume of the oxide.

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