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,[1] building on Mohamed M. Atalla's work on silicon surface passivation by thermal oxidation at Bell Labs in the late 1950s.
[2] This served as a step in the development of CMOS devices and the fabrication of integrated circuits.
[3] 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.
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.
is the reaction rate coefficient for oxidation at the surface of the substrate.
the following relations can be derived: Assuming a diffusion controlled growth i.e. where
The solution to this equation gives the oxide thickness at any time t. where the constants
encapsulate the properties of the reaction and the oxide layer respectively, and
Solving the quadratic equation for x yields: Taking the short and long time limits of the above equation reveals two main modes of operation.
The quantities B and B/A are often called the quadratic and linear reaction rate constants.
The following table lists the values of the four parameters for single-crystal silicon under conditions typically used in industry (low doping, atmospheric pressure).
The linear rate constant depends on the orientation of the crystal (usually indicated by the Miller indices of the crystal plane facing the surface).
However, experimental data shows that very thin oxides (less than about 25 nanometres) grow much more quickly in
In silicon nanostructures (e.g., silicon nanowires) this rapid growth is generally followed by diminishing oxidation kinetics in a process known as self-limiting oxidation, necessitating a modification of the Deal–Grove model.
[3] If the oxide grown in a particular oxidation step greatly exceeds 25 nm, a simple adjustment accounts for the aberrant growth rate.
The Massoud model is analytical and based on parallel oxidation mechanisms.
The Deal-Grove model also fails for polycrystalline silicon ("poly-silicon").
First, the random orientation of the crystal grains makes it difficult to choose a value for the linear rate constant.
[citation needed] Dopant atoms strain the silicon lattice, and make it easier for silicon atoms to bond with incoming oxygen.
This effect may be neglected in many cases, but heavily doped silicon oxidizes significantly faster.
The pressure of the ambient gas also affects oxidation rate.
