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

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

$$ Da = \frac{\text{reaction rate}}{\text{transport rate}} = \frac{k_s L}{D} $$

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

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

Dimensionless Numbers