Deal-Grove Oxidation Kinetics & Stress-Induced Thin Film Growth Physics

01Mathematical Derivation of the Deal-Grove Equation

Consider an oxide film of thickness $x_{o}$ . The flux of oxidant molecules passing through the surface boundary layer ( $F_{1}$ ), diffusing through the oxide ( $F_{2}$ ), and reacting at the interface ( $F_{3}$ ) are expressed as:

F_{1} = h_{g} (C_{g} - C_{s})

F_{2} = D_{e ff} \frac{C _{o} - C _{i}}{x _{o}}

F_{3} = k_{s} C_{i}

Where:

$h_{g}$ is the gas-phase mass transfer coefficient.
$D_{e ff}$ is the effective diffusion coefficient of oxidant in $S i O_{2}$ .
$k_{s}$ is the chemical surface reaction rate constant at the $Si/SiO_{2}$ interface.
$C_{g}, C_{s}, C_{o}, C_{i}$ are oxidant concentrations in the bulk gas, at the surface, inside the top oxide surface, and at the interface, respectively.

Oxidant Concentration Profile Across Oxide LayerSchematic · source diagram

Applying Henry's Law ( $C_{o} = H P_{g} = N_{0}$ ) and setting $F_{1} = F_{2} = F_{3} = F$ at steady state yields the differential equation for oxide growth:

\frac{d x _{o}}{d t} = \frac{F}{N _{1}} = \frac{k _{s} C _{o} / N _{1}}{1 + \frac{k _{s}}{h} + \frac{k _{s} x _{o}}{D _{e ff}}}

Where $N_{1}$ is the number of oxidant molecules incorporated into a unit volume of $S i O_{2}$ ( $2.2 \times 1 0^{22} cm^{- 3}$ for $O_{2}$ ). Integrating with initial condition $x_{o} (t = 0) = x_{i}$ yields the quadratic Deal-Grove equation:

x_{o}^{2} + A x_{o} = B (t + τ)

Where:

A = 2 D_{e ff} (\frac{1}{k _{s}} + \frac{1}{h})

B = \frac{2 D _{e ff} C _{o}}{N _{1}}

τ = \frac{x _{i}^{2} + A x _{i}}{B}

Limiting Regimes:

Short Times / Thin Films ( $x_{o} ≪ A /2$ ): Linear Regime

x_{o} \approx \frac{B}{A} (t + τ) ⟹ Rate Constant k_{l in} = \frac{B}{A} \approx \frac{k _{s} C _{o}}{N _{1}}

Controlled entirely by interface reaction rate $k_{s}$ .

Long Times / Thick Films ( $x_{o} ≫ A /2$ ): Parabolic Regime

x_{o}^{2} \approx Bt ⟹ Rate Constant B = \frac{2 D _{e ff} C _{o}}{N _{1}}

Controlled entirely by oxidant diffusion through the bulk oxide.

02The Thin Oxide Anomaly & Extended Kinetic Models

For dry $O_{2}$ oxidation, experimental growth rates for $x_{o} < 20 nm$ are significantly faster than predicted by $x_{o} \approx (B / A) t$ . An extra initial thickness $x_{i} \approx 15 nm$ must be added artificially to fit Deal-Grove parameters.

Dry Oxidation Anomaly For Thin Films (<20Nm)Schematic · source diagram

Several mechanisms explain this phenomenon:

Space Charge & Oxygen Ion Solvation: Electric fields across ultra-thin oxides enhance ionic oxidant drift.
Han-Masaoud Dual-Mechanistic Model: Proposes parallel diffusion of atomic oxygen radicals ( $O^{*}$ ) alongside molecular $O_{2}$ .
Kao Stress-Dependent Model: Viscous flow of $S i O_{2}$ occurs above the transition temperature ( $T_{g} \approx 95 0^{\circ} C$ ). Below $95 0^{\circ} C$ , high compressive intrinsic stress builds at non-planar silicon corners, suppressing $k_{s}$ and $D_{e ff}$ .

→Prerequisites & continue reading

Prerequisites: ["p07-t7.1-t2-diffusion-furnace-dry-wet-oxidation-mechanics"]

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