Semiconductor Manufacturing Process Chemical Kinetics: Mathematics
Introduction
Semiconductor manufacturing relies heavily on chemical kinetics to control thin film deposition, etching, oxidation, and dopant diffusion. This document provides the mathematical framework underlying these processes.
Fundamental Kinetic Concepts
Reaction Rate Expression
The general rate expression for a reaction $A + B \rightarrow C$ is:
$$
r = k[A]^m[B]^n
$$
Where:
- $r$ = reaction rate $\left(\frac{\text{mol}}{\text{m}^3 \cdot \text{s}}\right)$
- $k$ = rate constant
- $[A], [B]$ = concentrations $\left(\frac{\text{mol}}{\text{m}^3}\right)$
- $m, n$ = reaction orders (empirically determined)
Arrhenius Equation
The temperature dependence of rate constants follows the Arrhenius equation:
$$
k = A \exp\left(-\frac{E_a}{RT}\right)
$$
Where:
- $A$ = pre-exponential factor (frequency factor)
- $E_a$ = activation energy $\left(\frac{\text{J}}{\text{mol}}\right)$
- $R$ = universal gas constant $\left(8.314 \frac{\text{J}}{\text{mol} \cdot \text{K}}\right)$
- $T$ = absolute temperature (K)
Linearized Form (for Arrhenius plots):
$$
\ln(k) = \ln(A) - \frac{E_a}{R} \cdot \frac{1}{T}
$$
Chemical Vapor Deposition (CVD)
Overall Rate Model
CVD involves both gas-phase transport and surface reaction. The overall deposition rate is:
$$
R = \frac{C_g}{\frac{1}{h_g} + \frac{1}{k_s}}
$$
Where:
- $R$ = deposition rate $\left(\frac{\text{mol}}{\text{m}^2 \cdot \text{s}}\right)$
- $C_g$ = gas-phase reactant concentration
- $h_g$ = gas-phase mass transfer coefficient $\left(\frac{\text{m}}{\text{s}}\right)$
- $k_s$ = surface reaction rate constant $\left(\frac{\text{m}}{\text{s}}\right)$
Regime Analysis
Surface-Reaction Limited (low temperature, $k_s \ll h_g$):
$$
R \approx k_s \cdot C_g = A \exp\left(-\frac{E_a}{RT}\right) \cdot C_g
$$
Mass-Transport Limited (high temperature, $h_g \ll k_s$):
$$
R \approx h_g \cdot C_g
$$
Mass Transfer Coefficient
For laminar flow over a flat plate:
$$
h_g = \frac{D_{AB}}{L} \cdot 0.664 \cdot Re_L^{1/2} \cdot Sc^{1/3}
$$
Where:
- $D_{AB}$ = binary diffusion coefficient
- $L$ = characteristic length
- $Re_L = \frac{\rho v L}{\mu}$ = Reynolds number
- $Sc = \frac{\mu}{\rho D_{AB}}$ = Schmidt number
Thermal Oxidation: Deal-Grove Model
Governing Equation
The Deal-Grove model describes silicon oxidation ($\text{Si} + \text{O}_2 \rightarrow \text{SiO}_2$):
$$
x^2 + Ax = B(t + \tau)
$$
Where:
- $x$ = oxide thickness (m)
- $t$ = oxidation time (s)
- $\tau$ = initial time correction (accounts for native oxide)
Rate Constants
Linear Rate Constant:
$$
\frac{B}{A} = \frac{k_s C^*}{N_{ox}}
$$
Parabolic Rate Constant:
$$
B = \frac{2D_{eff} C^*}{N_{ox}}
$$
Where:
- $D_{eff}$ = effective diffusion coefficient of oxidant through oxide
- $C^*$ = equilibrium oxidant concentration in oxide
- $N_{ox}$ = number of oxidant molecules incorporated per unit volume of oxide
- $k_s$ = surface reaction rate constant
Limiting Cases
Thin Oxide Regime (short times, $x \ll A$):
$$
x \approx \frac{B}{A}(t + \tau)
$$
- Linear growth (surface-reaction controlled)
Thick Oxide Regime (long times, $x \gg A$):
$$
x \approx \sqrt{B \cdot t}
$$
- Parabolic growth (diffusion controlled)
Explicit Solution
Solving the quadratic equation:
$$
x = \frac{A}{2}\left[\sqrt{1 + \frac{4B(t+\tau)}{A^2}} - 1\right]
$$
Plasma Etching Kinetics
Ion-Enhanced Etching Model
The etch rate combines thermal and ion-assisted components:
$$
R = k_{thermal} \cdot P \cdot \exp\left(-\frac{E_a}{RT}\right) + k_{ion} \cdot \Gamma_{ion}^\alpha \cdot \theta
$$
Where:
- $k_{thermal}$ = thermal etching rate constant
- $P$ = reactive gas partial pressure
- $\Gamma_{ion}$ = ion flux $\left(\frac{\text{ions}}{\text{m}^2 \cdot \text{s}}\right)$
- $\alpha$ = ion flux exponent (typically 0.5–1.5)
- $\theta$ = surface coverage of reactive species
Sputter Yield Model
Physical sputtering rate:
$$
R_{sputter} = Y(\theta, E) \cdot \frac{\Gamma_{ion}}{n}
$$
Where:
- $Y$ = sputter yield (atoms removed per incident ion)
- $E$ = ion energy
- $\theta$ = ion incidence angle
- $n$ = atomic density of target material
Selectivity
Selectivity between materials A and B:
$$
S = \frac{R_A}{R_B}
$$
Surface Reaction Kinetics
Langmuir Adsorption Isotherm
For single-species adsorption at equilibrium:
$$
\theta = \frac{K \cdot P}{1 + K \cdot P}
$$
Where:
- $\theta$ = fractional surface coverage $(0 \leq \theta \leq 1)$
- $K$ = adsorption equilibrium constant
- $P$ = partial pressure
Temperature Dependence of K:
$$
K = K_0 \exp\left(\frac{-\Delta H_{ads}}{RT}\right)
$$
Multi-Species Competitive Adsorption
For species A and B competing for the same sites:
$$
\theta_A = \frac{K_A P_A}{1 + K_A P_A + K_B P_B}
$$
$$
\theta_B = \frac{K_B P_B}{1 + K_A P_A + K_B P_B}
$$
Surface Reaction Rate
Langmuir-Hinshelwood Mechanism (both reactants adsorbed):
$$
r = k_s \cdot \theta_A \cdot \theta_B = k_s \cdot \frac{K_A P_A \cdot K_B P_B}{(1 + K_A P_A + K_B P_B)^2}
$$
Eley-Rideal Mechanism (one reactant from gas phase):
$$
r = k_s \cdot \theta_A \cdot P_B = k_s \cdot \frac{K_A P_A \cdot P_B}{1 + K_A P_A}
$$
Limiting Behavior
| Condition | Rate Expression | Order |
|-----------|-----------------|-------|
| $K \cdot P \ll 1$ | $r \approx k_s K P$ | First-order |
| $K \cdot P \gg 1$ | $r \approx k_s$ | Zero-order |
Diffusion Processes
Fick's Laws
First Law (steady-state flux):
$$
J = -D \frac{\partial C}{\partial x}
$$
Second Law (transient diffusion):
$$
\frac{\partial C}{\partial t} = D \frac{\partial^2 C}{\partial x^2}
$$
For 3D:
$$
\frac{\partial C}{\partial t} = D
abla^2 C = D \left(\frac{\partial^2 C}{\partial x^2} + \frac{\partial^2 C}{\partial y^2} + \frac{\partial^2 C}{\partial z^2}\right)
$$
Concentration-Dependent Diffusion
For dopants where $D = D(C)$:
$$
\frac{\partial C}{\partial t} = \frac{\partial}{\partial x}\left[D(C) \frac{\partial C}{\partial x}\right]
$$
Analytical Solutions
Constant Surface Concentration (semi-infinite medium):
$$
C(x,t) = C_s \cdot \text{erfc}\left(\frac{x}{2\sqrt{Dt}}\right)
$$
Where $\text{erfc}$ is the complementary error function:
$$
\text{erfc}(z) = 1 - \text{erf}(z) = 1 - \frac{2}{\sqrt{\pi}}\int_0^z e^{-u^2} du
$$
Fixed Total Dose (Gaussian profile):
$$
C(x,t) = \frac{Q}{\sqrt{\pi D t}} \exp\left(-\frac{x^2}{4Dt}\right)
$$
Where $Q$ = total dose $\left(\frac{\text{atoms}}{\text{m}^2}\right)$
Diffusion Coefficient Temperature Dependence
$$
D = D_0 \exp\left(-\frac{E_a}{kT}\right)
$$
Where $k = 8.617 \times 10^{-5} \frac{\text{eV}}{\text{K}}$ (Boltzmann constant)
Reactor-Scale Modeling
Species Conservation Equation
The convection-diffusion-reaction equation:
$$
\frac{\partial C_i}{\partial t} +
abla \cdot (\mathbf{v} C_i) =
abla \cdot (D_i
abla C_i) + R_i
$$
Expanded form:
$$
\frac{\partial C_i}{\partial t} + \mathbf{v} \cdot
abla C_i = D_i
abla^2 C_i + R_i
$$
Coupled Equations
Navier-Stokes (momentum):
$$
\rho \left(\frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot
abla \mathbf{v}\right) = -
abla P + \mu
abla^2 \mathbf{v} + \rho \mathbf{g}
$$
Continuity (mass):
$$
\frac{\partial \rho}{\partial t} +
abla \cdot (\rho \mathbf{v}) = 0
$$
Energy:
$$
\rho c_p \left(\frac{\partial T}{\partial t} + \mathbf{v} \cdot
abla T\right) = k
abla^2 T + Q_{rxn}
$$
Where $Q_{rxn} = \sum_j (-\Delta H_j) r_j$ is the heat of reaction.
Boundary Conditions
Surface reaction flux:
$$
-D_i \frac{\partial C_i}{\partial n}\bigg|_{surface} = R_{s,i}
$$
Inlet conditions:
$$
C_i = C_{i,inlet}, \quad T = T_{inlet}, \quad \mathbf{v} = \mathbf{v}_{inlet}
$$
Dimensionless Analysis
Damköhler Number
$$
Da = \frac{\text{reaction rate}}{\text{transport rate}} = \frac{k_s L}{D}
$$
| Da Value | Regime | Characteristics |
|----------|--------|-----------------|
| $Da \gg 1$ | Reaction-limited | Uniform deposition, strong T dependence |
| $Da \ll 1$ | Transport-limited | Non-uniform, weak T dependence |
Thiele Modulus
For reactions in porous structures:
$$
\phi = L \sqrt{\frac{k}{D_{eff}}}
$$
Effectiveness Factor:
$$
\eta = \frac{\tanh(\phi)}{\phi}
$$
Peclet Number
$$
Pe = \frac{vL}{D} = \frac{\text{convective transport}}{\text{diffusive transport}}
$$
Stanton Number
$$
St = \frac{h}{\rho v c_p} = \frac{\text{heat transfer}}{\text{thermal capacity of flow}}
$$
Advanced Modeling Techniques
Microkinetic Modeling
System of coupled ODEs for surface species:
$$
\frac{d\theta_i}{dt} = \sum_j \left[
u_{ij}^+ r_j^+ -
u_{ij}^- r_j^-\right]
$$
Where:
- $\theta_i$ = coverage of species $i$
- $
u_{ij}$ = stoichiometric coefficient
- $r_j^+, r_j^-$ = forward and reverse rates of reaction $j$
Example: Adsorption-Desorption-Reaction:
$$
\frac{d\theta_A}{dt} = k_{ads} P_A (1-\theta_A-\theta_B) - k_{des} \theta_A - k_{rxn} \theta_A \theta_B
$$
Stochastic Methods
Kinetic Monte Carlo (KMC):
Transition rates:
$$
W_i =
u_i \exp\left(-\frac{E_i}{kT}\right)
$$
Time step:
$$
\Delta t = -\frac{\ln(r)}{\sum_i W_i}
$$
Where $r \in (0,1]$ is a random number.
Master Equation:
$$
\frac{dP_n}{dt} = \sum_m \left[W_{mn} P_m - W_{nm} P_n\right]
$$
Multi-Scale Coupling
| Scale | Size | Method | Output |
|-------|------|--------|--------|
| Quantum | ~Å | DFT | Reaction barriers, adsorption energies |
| Atomic | ~nm | MD, KMC | Surface morphology, growth modes |
| Feature | ~$\mu$m | Level-set, FEM | Profile evolution |
| Reactor | ~cm | CFD | Uniformity, gas dynamics |
Computational Methods
Numerical Discretization
Finite Difference (1D diffusion):
$$
\frac{C_i^{n+1} - C_i^n}{\Delta t} = D \frac{C_{i+1}^n - 2C_i^n + C_{i-1}^n}{(\Delta x)^2}
$$
Stability Criterion (explicit method):
$$
\frac{D \Delta t}{(\Delta x)^2} \leq \frac{1}{2}
$$
Operator Splitting
For stiff reaction-diffusion systems:
1. Diffusion step: Solve $\frac{\partial C}{\partial t} = D
abla^2 C$ for $\Delta t/2$
2. Reaction step: Solve $\frac{dC}{dt} = R(C)$ for $\Delta t$
3. Diffusion step: Solve $\frac{\partial C}{\partial t} = D
abla^2 C$ for $\Delta t/2$
Newton-Raphson for Nonlinear Systems
$$
\mathbf{x}^{(k+1)} = \mathbf{x}^{(k)} - \mathbf{J}^{-1}(\mathbf{x}^{(k)}) \cdot \mathbf{F}(\mathbf{x}^{(k)})
$$
Where $\mathbf{J}$ is the Jacobian matrix:
$$
J_{ij} = \frac{\partial F_i}{\partial x_j}
$$
Key Equations Summary
Rate Expressions
| Process | Equation |
|---------|----------|
| Arrhenius | $k = A \exp\left(-\frac{E_a}{RT}\right)$ |
| CVD Rate | $R = \frac{C_g}{1/h_g + 1/k_s}$ |
| Deal-Grove | $x^2 + Ax = B(t + \tau)$ |
| Langmuir | $\theta = \frac{KP}{1+KP}$ |
| Fick's 2nd Law | $\frac{\partial C}{\partial t} = D \frac{\partial^2 C}{\partial x^2}$ |
Dimensionless Numbers
| Number | Definition | Physical Meaning |
|--------|------------|------------------|
| Damköhler ($Da$) | $\frac{k_s L}{D}$ | Reaction vs. transport rate |
| Thiele ($\phi$) | $L\sqrt{k/D_{eff}}$ | Reaction-diffusion penetration |
| Peclet ($Pe$) | $\frac{vL}{D}$ | Convection vs. diffusion |
| Reynolds ($Re$) | $\frac{\rho vL}{\mu}$ | Inertial vs. viscous forces |