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
$$Si + O_2 \rightarrow SiO_2$$

$$Si + 2 H_2O \rightarrow SiO_2 + 2H_2$$

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:

$$C_G = \frac{P_G}{k T}$$

$$C_S = \frac{P_S}{kT}$$

$$F_1 = h (C* - C_O)$$

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 $$F_1 = F_2 = F_3$$

$$C_I = \frac{C*}{1 + \frac{k_S}{h} + \frac{k_S x_O}{D}}$$ $$C_I \approx \frac{C^*}{1+\frac{k_Sx_O}{D}}$$

$$C_O = \frac{C^* \left (1 + \frac{k_Sx_O}{D} \right )}{1 + \frac{k_S}{h} + \frac{k_S x_O}{D}}$$ $$C_O \approx C^*$$

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

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

$$x_O^2 \approx B (t + \tau)$$