Materials Science and Engineering/Derivations/Models of Micro and Nanoscale Processing

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First-Order Planar Growth Kinetics - The Linear Parabolic Model[edit | edit source]

Oxide grows by indiffusion

Chemical Reaction[edit | edit source]

Three Fluxes[edit | edit source]

Transport of the oxidant to the oxide surface[edit | edit source]

 
  • : flux in molecules
  • : concentration difference between gas flow and surface
  • : mass transfer coefficient

Equilibrium concentration of a gas species[edit | edit source]

The equilibrium concentration of a gas species dissolved in a solid is proportional to partial pressure of species at the surface.

  • :oxidant concentration in oxide that would be in equilibrium with
  • : bulk gas pressure

From the ideal gas law:

Diffusion of oxidant through oxide to interface[edit | edit source]

In steady state,

 
 
  • and : concetration at two interfaces
  • : oxide thickness

Oxygen and water seem to diffuse in different manners, though the effective diffusivities are of the same order.

Reaction at the Si/SiO2 interface[edit | edit source]

  • : interface reaction rate constant

Equating three fluxes[edit | edit source]

With

 
 
 
 

The approximations are based on the observation that is very large. Gas absorption occurs rapidly compared with chemistry at interface.

Limiting cases[edit | edit source]

Reaction rate controlled - thin oxides[edit | edit source]

Oxidant supplied to interface fast compared to that required to sustain the interface reaction

Diffusion controlled - thick oxides[edit | edit source]

  • : number of oxidant molecules incorporated

Integrate from initial oxide thickness to final thickness :

Limiting forms of the linear parabolic growth law[edit | edit source]