Mathematical Modeling of CVD Equipment in Semiconductor Manufacturing

1. Overview of CVD in Semiconductor Fabrication

Chemical Vapor Deposition (CVD) is a fundamental process in semiconductor manufacturing that deposits thin films onto wafer substrates through gas-phase and surface chemical reactions.

1.1 Types of Deposited Films

• Dielectrics: $\text{SiO}_2$, $\text{Si}_3\text{N}_4$, low-$\kappa$ materials
• Conductors: W (tungsten), TiN, Cu seed layers
• Barrier Layers: TaN, TiN diffusion barriers
• Semiconductors: Epitaxial Si, polysilicon, SiGe

1.2 CVD Process Variants

2. Governing Equations: Transport Phenomena

CVD modeling requires solving coupled partial differential equations for mass, momentum, and energy transport.

2.1 Mass Transport (Species Conservation)

The species conservation equation describes the transport and reaction of chemical species:

$$ \frac{\partial C_i}{\partial t} + abla \cdot (C_i \mathbf{v}) = abla \cdot (D_i abla C_i) + R_i $$

Where:

• $C_i$ — Molar concentration of species $i$ $[\text{mol/m}^3]$
• $\mathbf{v}$ — Velocity vector field $[\text{m/s}]$
• $D_i$ — Diffusion coefficient of species $i$ $[\text{m}^2/\text{s}]$
• $R_i$ — Net volumetric production rate $[\text{mol/m}^3 \cdot \text{s}]$

Stefan-Maxwell Equations for Multicomponent Diffusion

For multicomponent gas mixtures, the Stefan-Maxwell equations apply:

$$

abla x_i = \sum_{j eq i} \frac{x_i x_j}{D_{ij}} (\mathbf{v}_j - \mathbf{v}_i) $$

Where:

• $x_i$ — Mole fraction of species $i$
• $D_{ij}$ — Binary diffusion coefficient $[\text{m}^2/\text{s}]$

Chapman-Enskog Diffusion Coefficient

Binary diffusion coefficients can be estimated using Chapman-Enskog theory:

$$ D_{ij} = \frac{3}{16} \sqrt{\frac{2\pi k_B^3 T^3}{m_{ij}}} \cdot \frac{1}{P \pi \sigma_{ij}^2 \Omega_D} $$

Where:

• $m_{ij} = \frac{m_i m_j}{m_i + m_j}$ — Reduced mass
• $\sigma_{ij}$ — Collision diameter $[\text{m}]$
• $\Omega_D$ — Collision integral (dimensionless)

2.2 Momentum Transport (Navier-Stokes Equations)

The Navier-Stokes equations govern fluid flow in the reactor:

$$ \rho \left( \frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot abla \mathbf{v} \right) = - abla p + abla \cdot \boldsymbol{\tau} + \rho \mathbf{g} $$

Where:

• $\rho$ — Gas density $[\text{kg/m}^3]$
• $p$ — Pressure $[\text{Pa}]$
• $\boldsymbol{\tau}$ — Viscous stress tensor $[\text{Pa}]$
• $\mathbf{g}$ — Gravitational acceleration $[\text{m/s}^2]$

Newtonian Stress Tensor

For Newtonian fluids:

$$ \boldsymbol{\tau} = \mu \left( abla \mathbf{v} + ( abla \mathbf{v})^T \right) - \frac{2}{3} \mu ( abla \cdot \mathbf{v}) \mathbf{I} $$

Slip Boundary Conditions

At low pressures where Knudsen number $Kn > 0.01$, slip boundary conditions are required:

$$ v_{slip} = \frac{2 - \sigma_v}{\sigma_v} \lambda \left( \frac{\partial v}{\partial n} \right)_{wall} $$

Where:

• $\sigma_v$ — Tangential momentum accommodation coefficient
• $\lambda$ — Mean free path $[\text{m}]$
• $n$ — Wall-normal direction

Mean Free Path

$$ \lambda = \frac{k_B T}{\sqrt{2} \pi d^2 P} $$

2.3 Energy Transport

The energy equation accounts for convection, conduction, and heat generation:

$$ \rho c_p \left( \frac{\partial T}{\partial t} + \mathbf{v} \cdot abla T \right) = abla \cdot (k abla T) + Q_{rxn} + Q_{rad} $$

Where:

• $c_p$ — Specific heat capacity $[\text{J/kg} \cdot \text{K}]$
• $k$ — Thermal conductivity $[\text{W/m} \cdot \text{K}]$
• $Q_{rxn}$ — Heat from chemical reactions $[\text{W/m}^3]$
• $Q_{rad}$ — Radiative heat transfer $[\text{W/m}^3]$

Radiative Heat Transfer (Rosseland Approximation)

For optically thick media:

$$ Q_{rad} = abla \cdot \left( \frac{4\sigma_{SB}}{3\kappa_R} abla T^4 \right) $$

Where:

• $\sigma_{SB} = 5.67 \times 10^{-8}$ W/m²·K⁴ — Stefan-Boltzmann constant
• $\kappa_R$ — Rosseland mean absorption coefficient $[\text{m}^{-1}]$

3. Chemical Kinetics

3.1 Gas-Phase Reactions

Gas-phase reactions decompose precursor molecules and generate reactive intermediates.

Example: Silane Decomposition for Silicon Deposition

Primary decomposition:

$$ \text{SiH}_4 \xrightarrow{k_1} \text{SiH}_2 + \text{H}_2 $$

Secondary reactions:

$$ \text{SiH}_2 + \text{SiH}_4 \xrightarrow{k_2} \text{Si}_2\text{H}_6 $$

$$ \text{SiH}_2 + \text{SiH}_2 \xrightarrow{k_3} \text{Si}_2\text{H}_4 $$

Arrhenius Rate Expression

Rate constants follow the modified Arrhenius form:

$$ k(T) = A \cdot T^n \exp\left( -\frac{E_a}{RT} \right) $$

Where:

• $A$ — Pre-exponential factor $[\text{varies}]$
• $n$ — Temperature exponent (dimensionless)
• $E_a$ — Activation energy $[\text{J/mol}]$
• $R = 8.314$ J/(mol·K) — Universal gas constant

Species Source Term

The net production rate for species $i$:

$$ R_i = \sum_{r=1}^{N_r} u_{i,r} \cdot k_r \prod_{j=1}^{N_s} C_j^{\alpha_{j,r}} $$

Where:

• $

u_{i,r}$ — Stoichiometric coefficient of species $i$ in reaction $r$

• $\alpha_{j,r}$ — Reaction order of species $j$ in reaction $r$
• $N_r$ — Total number of reactions
• $N_s$ — Total number of species

3.2 Surface Reaction Kinetics

Surface reactions determine the actual film deposition.

Langmuir-Hinshelwood Mechanism

For bimolecular surface reactions:

$$ R_s = \frac{k_s K_A K_B C_A C_B}{(1 + K_A C_A + K_B C_B)^2} $$

Where:

• $k_s$ — Surface reaction rate constant $[\text{m}^2/\text{mol} \cdot \text{s}]$
• $K_A, K_B$ — Adsorption equilibrium constants $[\text{m}^3/\text{mol}]$
• $C_A, C_B$ — Gas-phase concentrations at surface $[\text{mol/m}^3]$

Eley-Rideal Mechanism

For reactions between adsorbed and gas-phase species:

$$ R_s = k_s \theta_A C_B $$

Sticking Coefficient Model (Kinetic Theory)

The adsorption flux based on kinetic theory:

$$ J_{ads} = \frac{s \cdot p}{\sqrt{2\pi m k_B T}} $$

Where:

• $s$ — Sticking probability (dimensionless, $0 < s \leq 1$)
• $p$ — Partial pressure of adsorbing species $[\text{Pa}]$
• $m$ — Molecular mass $[\text{kg}]$
• $k_B = 1.38 \times 10^{-23}$ J/K — Boltzmann constant

Surface Site Balance

Dynamic surface coverage evolution:

$$ \frac{d\theta_i}{dt} = k_{ads,i} C_i (1 - \theta_{total}) - k_{des,i} \theta_i - k_{rxn} \theta_i \theta_j $$

Where:

• $\theta_i$ — Surface coverage fraction of species $i$
• $\theta_{total} = \sum_i \theta_i$ — Total surface coverage
• $k_{ads,i}$ — Adsorption rate constant
• $k_{des,i}$ — Desorption rate constant
• $k_{rxn}$ — Surface reaction rate constant

4. Film Growth and Deposition Rate

4.1 Local Deposition Rate

The film thickness growth rate:

$$ \frac{dh}{dt} = \frac{M_w}{\rho_{film}} \cdot R_s $$

Where:

• $h$ — Film thickness $[\text{m}]$
• $M_w$ — Molecular weight of deposited material $[\text{kg/mol}]$
• $\rho_{film}$ — Film density $[\text{kg/m}^3]$
• $R_s$ — Surface reaction rate $[\text{mol/m}^2 \cdot \text{s}]$

4.2 Boundary Layer Analysis

Rotating Disk Reactor (Classical Solution)

Boundary layer thickness:

$$ \delta = \sqrt{\frac{ u}{\Omega}} $$

Where:

• $

u$ — Kinematic viscosity $[\text{m}^2/\text{s}]$

• $\Omega$ — Angular rotation speed $[\text{rad/s}]$

Sherwood Number Correlation

For mass transfer in laminar flow:

$$ Sh = 0.62 \cdot Re^{1/2} \cdot Sc^{1/3} $$

Where:

• $Sh = \frac{k_m L}{D}$ — Sherwood number
• $Re = \frac{\rho v L}{\mu}$ — Reynolds number
• $Sc = \frac{\mu}{\rho D}$ — Schmidt number

Mass Transfer Coefficient

$$ k_m = \frac{Sh \cdot D}{L} $$

4.3 Deposition Rate Regimes

The overall deposition process can be limited by different mechanisms:

Regime 1: Surface Reaction Limited ($Da \ll 1$)

$$ R_{dep} \approx k_s C_{bulk} $$

Regime 2: Mass Transfer Limited ($Da \gg 1$)

$$ R_{dep} \approx k_m C_{bulk} $$

General Case:

$$ \frac{1}{R_{dep}} = \frac{1}{k_s C_{bulk}} + \frac{1}{k_m C_{bulk}} $$

5. Step Coverage and Feature-Scale Modeling

5.1 Thiele Modulus Analysis

The Thiele modulus determines whether deposition is reaction or diffusion limited within features:

$$ \phi = L \sqrt{\frac{k_s}{D_{Kn}}} $$

Where:

• $L$ — Feature depth $[\text{m}]$
• $k_s$ — Surface reaction rate constant $[\text{m/s}]$
• $D_{Kn}$ — Knudsen diffusion coefficient $[\text{m}^2/\text{s}]$

Interpretation:

Knudsen Diffusion in Features

For high aspect ratio features where $Kn > 1$:

$$ D_{Kn} = \frac{d}{3} \sqrt{\frac{8RT}{\pi M}} $$

Where:

• $d$ — Feature diameter/width $[\text{m}]$
• $M$ — Molecular weight $[\text{kg/mol}]$

5.2 Level-Set Method for Surface Evolution

The level-set equation tracks the evolving surface:

$$ \frac{\partial \phi}{\partial t} + V_n | abla \phi| = 0 $$

Where:

• $\phi(\mathbf{x}, t)$ — Level-set function (surface at $\phi = 0$)
• $V_n$ — Local normal velocity $[\text{m/s}]$

Reinitialization Equation

To maintain $| abla \phi| = 1$:

$$ \frac{\partial \phi}{\partial \tau} = \text{sign}(\phi_0)(1 - | abla \phi|) $$

5.3 Ballistic Transport (Monte Carlo)

For molecular flow in high-aspect-ratio features, the flux at a surface point:

$$ \Gamma(\mathbf{r}) = \frac{1}{\pi} \int_{\Omega_{visible}} \Gamma_0 \cos\theta \, d\Omega $$

Where:

• $\Gamma_0$ — Incident flux at feature opening $[\text{mol/m}^2 \cdot \text{s}]$
• $\theta$ — Angle from surface normal
• $\Omega_{visible}$ — Visible solid angle from point $\mathbf{r}$

View Factor Calculation

The view factor from surface element $i$ to $j$:

$$ F_{i \rightarrow j} = \frac{1}{\pi A_i} \int_{A_i} \int_{A_j} \frac{\cos\theta_i \cos\theta_j}{r^2} \, dA_j \, dA_i $$

6. Reactor-Scale Modeling

6.1 Showerhead Gas Distribution

Pressure Drop Through Holes

$$ \Delta P = \frac{1}{2} \rho v^2 \left( \frac{1}{C_d^2} \right) $$

Where:

• $C_d$ — Discharge coefficient (typically 0.6–0.8)
• $v$ — Gas velocity through hole $[\text{m/s}]$

Flow Rate Through Individual Holes

$$ Q_i = C_d A_i \sqrt{\frac{2\Delta P}{\rho}} $$

Uniformity Index

$$ UI = 1 - \frac{\sigma_Q}{\bar{Q}} $$

6.2 Wafer Temperature Uniformity

Combined convection-radiation heat transfer to wafer:

$$ q = h_{conv}(T_{susceptor} - T_{wafer}) + \epsilon \sigma_{SB} (T_{susceptor}^4 - T_{wafer}^4) $$

Where:

• $h_{conv}$ — Convective heat transfer coefficient $[\text{W/m}^2 \cdot \text{K}]$
• $\epsilon$ — Emissivity (dimensionless)

Edge Effect Modeling

Radiative view factor at wafer edge:

$$ F_{edge} = \frac{1}{2}\left(1 - \frac{1}{\sqrt{1 + (R/H)^2}}\right) $$

6.3 Precursor Depletion

Along the flow direction:

$$ \frac{dC}{dx} = -\frac{k_s W}{Q} C $$

Solution:

$$ C(x) = C_0 \exp\left(-\frac{k_s W x}{Q}\right) $$

Where:

• $W$ — Wafer width $[\text{m}]$
• $Q$ — Volumetric flow rate $[\text{m}^3/\text{s}]$

7. PECVD: Plasma Modeling

7.1 Electron Kinetics

Boltzmann Equation

The electron energy distribution function (EEDF):

$$ \frac{\partial f}{\partial t} + \mathbf{v} \cdot abla_r f + \frac{e\mathbf{E}}{m_e} \cdot abla_v f = \left( \frac{\partial f}{\partial t} \right)_{coll} $$

Where:

• $f(\mathbf{r}, \mathbf{v}, t)$ — Electron distribution function
• $\mathbf{E}$ — Electric field $[\text{V/m}]$
• $m_e = 9.109 \times 10^{-31}$ kg — Electron mass

Two-Term Spherical Harmonic Expansion

$$ f(\varepsilon, \mathbf{r}, t) = f_0(\varepsilon) + f_1(\varepsilon) \cos\theta $$

7.2 Plasma Chemistry

Electron Impact Dissociation

$$ e + \text{SiH}_4 \xrightarrow{k_e} \text{SiH}_3 + \text{H} + e $$

Electron Impact Ionization

$$ e + \text{SiH}_4 \xrightarrow{k_i} \text{SiH}_3^+ + \text{H} + 2e $$

Rate Coefficient Calculation

$$ k_e = \int_0^\infty \sigma(\varepsilon) \sqrt{\frac{2\varepsilon}{m_e}} f(\varepsilon) \, d\varepsilon $$

Where:

• $\sigma(\varepsilon)$ — Energy-dependent cross-section $[\text{m}^2]$
• $\varepsilon$ — Electron energy $[\text{eV}]$

7.3 Sheath Physics

Floating Potential

$$ V_f = -\frac{T_e}{2e} \ln\left( \frac{m_i}{2\pi m_e} \right) $$

Bohm Velocity

$$ v_B = \sqrt{\frac{k_B T_e}{m_i}} $$

Ion Flux to Surface

$$ \Gamma_i = n_s v_B = n_s \sqrt{\frac{k_B T_e}{m_i}} $$

Child-Langmuir Law (Collisionless Sheath)

Ion current density:

$$ J_i = \frac{4\epsilon_0}{9} \sqrt{\frac{2e}{m_i}} \frac{V_s^{3/2}}{d_s^2} $$

Where:

• $V_s$ — Sheath voltage $[\text{V}]$
• $d_s$ — Sheath thickness $[\text{m}]$

7.4 Power Deposition

Ohmic heating in the bulk plasma:

$$ P_{ohm} = \frac{J^2}{\sigma} = \frac{n_e e^2 u_m}{m_e} E^2 $$

Where:

• $\sigma$ — Plasma conductivity $[\text{S/m}]$
• $

u_m$ — Electron-neutral collision frequency $[\text{s}^{-1}]$

8. Dimensionless Analysis

8.1 Key Dimensionless Numbers

u^2}$ | Buoyancy vs. viscous forces |

8.2 Temperature Sensitivity Analysis

The sensitivity of deposition rate to temperature:

$$ \frac{\delta R}{R} = \frac{E_a}{RT^2} \delta T $$

Example Calculation:

For $E_a = 1.5$ eV = $144.7$ kJ/mol at $T = 973$ K (700°C):

$$ \frac{\delta R}{R} = \frac{144700}{8.314 \times 973^2} \cdot 1 \text{ K} \approx 0.018 = 1.8\% $$

Implication: A 1°C temperature variation causes ~1.8% deposition rate change.

8.3 Flow Regime Classification

Based on Knudsen number:

9. Multiscale Modeling Framework

9.1 Modeling Hierarchy

┌─────────────────────────────────────────────────────────────────┐
│  QUANTUM SCALE (DFT)                                            │
│  • Reaction mechanisms and transition states                    │
│  • Activation energies and rate constants                       │
│  • Length: ~1 nm, Time: ~fs                                     │
├─────────────────────────────────────────────────────────────────┤
│  MOLECULAR DYNAMICS                                             │
│  • Surface diffusion coefficients                               │
│  • Nucleation and island formation                              │
│  • Length: ~10 nm, Time: ~ns                                    │
├─────────────────────────────────────────────────────────────────┤
│  KINETIC MONTE CARLO                                            │
│  • Film microstructure evolution                                │
│  • Surface roughness development                                │
│  • Length: ~100 nm, Time: ~μs–ms                                │
├─────────────────────────────────────────────────────────────────┤
│  FEATURE-SCALE (Continuum)                                      │
│  • Topography evolution in trenches/vias                        │
│  • Step coverage prediction                                     │
│  • Length: ~1 μm, Time: ~s                                      │
├─────────────────────────────────────────────────────────────────┤
│  REACTOR-SCALE (CFD)                                            │
│  • Gas flow and temperature fields                              │
│  • Species concentration distributions                          │
│  • Length: ~0.1 m, Time: ~min                                   │
├─────────────────────────────────────────────────────────────────┤
│  EQUIPMENT/FAB SCALE                                            │
│  • Wafer-to-wafer variation                                     │
│  • Throughput and scheduling                                    │
│  • Length: ~1 m, Time: ~hours                                   │
└─────────────────────────────────────────────────────────────────┘

9.2 Scale Bridging Approaches

Bottom-Up Parameterization:

• DFT → Rate constants for higher scales
• MD → Diffusion coefficients, sticking probabilities
• kMC → Effective growth rates, roughness correlations

Top-Down Validation:

• Reactor experiments → Validate CFD predictions
• SEM/TEM → Validate feature-scale models
• Surface analysis → Validate kinetic models

10. ALD-Specific Modeling

10.1 Self-Limiting Surface Reactions

ALD relies on self-limiting half-reactions:

Half-Reaction A (e.g., TMA pulse for Al₂O₃):

$$ \theta_A(t) = \theta_{sat} \left( 1 - e^{-k_{ads} p_A t} \right) $$

Half-Reaction B (e.g., H₂O pulse):

$$ \theta_B(t) = (1 - \theta_A) \left( 1 - e^{-k_B p_B t} \right) $$

10.2 Growth Per Cycle (GPC)

$$ GPC = \theta_{sat} \cdot \Gamma_{sites} \cdot \frac{M_w}{\rho N_A} $$

Where:

• $\theta_{sat}$ — Saturation coverage (dimensionless)
• $\Gamma_{sites}$ — Surface site density $[\text{sites/m}^2]$
• $N_A = 6.022 \times 10^{23}$ mol⁻¹ — Avogadro's number

Typical values for Al₂O₃ ALD:

• $GPC \approx 0.1$ nm/cycle
• $\Gamma_{sites} \approx 10^{19}$ sites/m²

10.3 Saturation Dose

The dose required for saturation:

$$ D_{sat} \propto \frac{1}{s} \sqrt{\frac{m k_B T}{2\pi}} $$

Where:

• $s$ — Reactive sticking coefficient
• Lower sticking coefficient → Higher saturation dose required

10.4 Nucleation Delay Modeling

For non-ideal ALD on different substrates:

$$ h(n) = GPC \cdot (n - n_0) \quad \text{for } n > n_0 $$

Where:

• $n$ — Cycle number
• $n_0$ — Nucleation delay (cycles)

11. Computational Tools and Methods

11.1 Reactor-Scale CFD

Typical mesh requirements:

• $10^5 - 10^7$ cells for 3D reactor
• Boundary layer refinement near wafer
• Adaptive meshing for reacting flows

11.2 Chemical Kinetics

11.3 Feature-Scale Simulation

11.4 Process/TCAD Integration

12. Machine Learning Integration

12.1 Physics-Informed Neural Networks (PINNs)

Loss function combining data and physics:

$$ \mathcal{L} = \mathcal{L}_{data} + \lambda \mathcal{L}_{physics} $$

Where:

$$ \mathcal{L}_{physics} = \frac{1}{N_f} \sum_{i=1}^{N_f} \left| \mathcal{F}[\hat{u}(\mathbf{x}_i)] \right|^2 $$

• $\mathcal{F}$ — Differential operator (governing PDE)
• $\hat{u}$ — Neural network approximation
• $\lambda$ — Weighting parameter

12.2 Surrogate Modeling

Gaussian Process Regression:

$$ f(\mathbf{x}) \sim \mathcal{GP}(m(\mathbf{x}), k(\mathbf{x}, \mathbf{x}')) $$

Where:

• $m(\mathbf{x})$ — Mean function
• $k(\mathbf{x}, \mathbf{x}')$ — Covariance kernel (e.g., RBF)

Applications:

• Real-time process control
• Recipe optimization
• Virtual metrology

12.3 Deep Learning Applications

13. Key Modeling Challenges

13.1 Stiff Chemistry

• Reaction timescales vary by orders of magnitude ($10^{-12}$ to $10^0$ s)
• Requires implicit time integration or operator splitting
• Chemical mechanism reduction techniques

13.2 Surface Reaction Parameters

• Limited experimental data for many chemistries
• Temperature and surface-dependent sticking coefficients
• Complex multi-step mechanisms

13.3 Multiscale Coupling

• Feature-scale depletion affects reactor-scale concentrations
• Reactor non-uniformity impacts feature-scale profiles
• Requires iterative or concurrent coupling schemes

13.4 Plasma Complexity

• Non-Maxwellian electron distributions
• Transient sheath dynamics in RF plasmas
• Ion energy and angular distributions

13.5 Advanced Device Architectures

• 3D NAND with extreme aspect ratios (AR > 100:1)
• Gate-All-Around (GAA) transistors
• Complex multi-material stacks

Summary

CVD equipment modeling requires solving coupled nonlinear PDEs for momentum, heat, and mass transport with complex gas-phase and surface chemistry. The mathematical framework encompasses:

• Continuum mechanics: Navier-Stokes, convection-diffusion
• Chemical kinetics: Arrhenius, Langmuir-Hinshelwood, Eley-Rideal
• Surface science: Sticking coefficients, site balances, nucleation
• Plasma physics: Boltzmann equation, sheath dynamics
• Numerical methods: FEM, FVM, Monte Carlo, level-set

The ultimate goal is predictive capability for film thickness, uniformity, composition, and microstructure—enabling virtual process development and optimization for advanced semiconductor manufacturing.