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

First-Order Planar Growth Kinetics - The Linear Parabolic Model

Oxide grows by indiffusion

Chemical Reaction

$Si+O_{2}\rightarrow SiO_{2}$

$Si+2H_{2}O\rightarrow SiO_{2}+2H_{2}$

Three Fluxes

Transport of the oxidant to the oxide surface

 $F_{1}=h_{G}(C_{G}-C_{S})\;$

• $F_{1}$: flux in molecules
• $(C_{G}-C_{S})$: concentration difference between gas flow and surface
• $h_{G}$: mass transfer coefficient

Equilibrium concentration of a gas species

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

$C_{O}=HP_{S}$

$C*=HP_{G}$

• $C*$:oxidant concentration in oxide that would be in equilibrium with $P_{G}$
• $P_{G}$: bulk gas pressure

From the ideal gas law:

$C_{G}={\frac {P_{G}}{kT}}$

$C_{S}={\frac {P_{S}}{kT}}$

$F_{1}=h(C*-C_{O})$

Diffusion of oxidant through oxide to interface

In steady state,

 $F_{2}=-D{\frac {\partial C}{\partial x}}$

 $F_{2}=D\left({\frac {C_{O}-C_{t}}{x_{O}}}\right)$

• $C_{O}$ and $C_{I}$: concetration at two interfaces
• $x_{O}$: 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

$F_{3}=k_{S}C_{I}$

• $k_{S}$: interface reaction rate constant

Equating three fluxes

With $F_{1}=F_{2}=F_{3}$

 $C_{I}={\frac {C*}{1+{\frac {k_{S}}{h}}+{\frac {k_{S}x_{O}}{D}}}}$
 $C_{I}\approx {\frac {C^{*}}{1+{\frac {k_{S}x_{O}}{D}}}}$

 $C_{O}={\frac {C^{*}\left(1+{\frac {k_{S}x_{O}}{D}}\right)}{1+{\frac {k_{S}}{h}}+{\frac {k_{S}x_{O}}{D}}}}$
 $C_{O}\approx C^{*}$

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

Limiting cases

Reaction rate controlled - thin oxides

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

$C_{I}\approx C^{*}$

$k_{S}x_{O}/D<<1$

Diffusion controlled - thick oxides

$k_{S}x_{O}/D>>1$

${\frac {dx_{O}}{dt}}={\frac {F}{N_{1}}}$

${\frac {dx_{O}}{dt}}={\frac {k_{S}C^{*}}{N_{1}\left(1+{\frac {k_{S}}{h}}+{\frac {k_{S}x_{O}}{D}}\right)}}$

• $N_{1}$: number of oxidant molecules incorporated

Integrate from initial oxide thickness $x_{i}$ to final thickness $x_{o}$:

$N_{1}\int _{x_{i}}^{x_{o}}1+{\frac {k_{S}}{h}}+{\frac {k_{S}x_{O}}{D}}dx_{0}=k_{S}C^{*}\int _{0}^{t}dt$

${\frac {x_{O}^{2}-x_{i}^{2}}{B}}+{\frac {x_{O}-x_{i}}{B/A}}=t$

• $B={\frac {2DC^{*}}{N_{1}}}$
• ${\frac {B}{A}}={\frac {C^{*}}{N_{1}\left({\frac {1}{k_{S}}}+{\frac {1}{h}}\right)}}{\frac {C^{*}k_{S}}{N_{1}}}$

${\frac {x_{O}^{2}}{B}}+{\frac {x_{O}}{B/A}}=t+\tau$

$\tau ={\frac {x_{i}^{2}+Ax_{i}}{B}}$

$x_{O}={\frac {A}{2}}\left({\sqrt {1+{\frac {t+\tau }{A^{2}/4B}}}}-1\right)$

Limiting forms of the linear parabolic growth law

 $x_{O}\approx {\frac {B}{A}}(t+\tau )$

 $x_{O}^{2}\approx B(t+\tau )$