Semiconductor Manufacturing: Dielectric Deposition Process (DDP) Modeling

Overview

DDP (Dielectric Deposition Process) refers to the set of techniques used to deposit insulating films in semiconductor fabrication. Dielectric materials serve critical functions:

• Gate dielectrics — $\text{SiO}_2$, high-$\kappa$ materials like $\text{HfO}_2$
• Interlayer dielectrics (ILD) — isolating metal interconnect layers
• Spacer dielectrics — defining transistor gate dimensions
• Passivation layers — protecting finished devices
• Hard masks — etch selectivity during patterning

Dielectric Deposition Methods

Primary Techniques

Selection Criteria

• Conformality requirements — ALD > LPCVD > PECVD
• Thermal budget — PECVD/ALD for low-$T$, thermal oxidation for high-quality
• Throughput — CVD methods faster than ALD
• Film quality — Thermal > LPCVD > PECVD generally

Physics of Dielectric Deposition Modeling

Fundamental Transport Equations

Modeling dielectric deposition requires solving coupled partial differential equations for mass, momentum, and energy transport.

Mass Transport (Species Concentration)

$$ \frac{\partial C}{\partial t} + abla \cdot (\mathbf{v}C) = D abla^2 C + R $$

Where:

• $C$ — species concentration $[\text{mol/m}^3]$
• $\mathbf{v}$ — velocity field $[\text{m/s}]$
• $D$ — diffusion coefficient $[\text{m}^2/\text{s}]$
• $R$ — reaction rate $[\text{mol/m}^3 \cdot \text{s}]$

Energy Balance

$$ \rho C_p \left(\frac{\partial T}{\partial t} + \mathbf{v} \cdot abla T\right) = k abla^2 T + Q $$

Where:

• $\rho$ — density $[\text{kg/m}^3]$
• $C_p$ — specific heat capacity $[\text{J/kg} \cdot \text{K}]$
• $k$ — thermal conductivity $[\text{W/m} \cdot \text{K}]$
• $Q$ — heat generation rate $[\text{W/m}^3]$

Momentum Balance (Navier-Stokes)

$$ \rho\left(\frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot abla \mathbf{v}\right) = - abla p + \mu abla^2 \mathbf{v} + \rho \mathbf{g} $$

Where:

• $p$ — pressure $[\text{Pa}]$
• $\mu$ — dynamic viscosity $[\text{Pa} \cdot \text{s}]$
• $\mathbf{g}$ — gravitational acceleration $[\text{m/s}^2]$

Surface Reaction Kinetics

Arrhenius Rate Expression

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

Where:

• $k$ — rate constant
• $A$ — pre-exponential factor
• $E_a$ — activation energy $[\text{J/mol}]$
• $R$ — gas constant $= 8.314 \, \text{J/mol} \cdot \text{K}$
• $T$ — temperature $[\text{K}]$

Langmuir Adsorption Isotherm (for ALD)

$$ \theta = \frac{K \cdot p}{1 + K \cdot p} $$

Where:

• $\theta$ — fractional surface coverage $(0 \leq \theta \leq 1)$
• $K$ — equilibrium adsorption constant
• $p$ — partial pressure of adsorbate

Sticking Coefficient

$$ S = S_0 \cdot (1 - \theta)^n \cdot \exp\left(-\frac{E_a}{RT}\right) $$

Where:

• $S$ — sticking coefficient (probability of adsorption)
• $S_0$ — initial sticking coefficient
• $n$ — reaction order

Plasma Modeling (PECVD/HDPCVD)

Electron Energy Distribution Function (EEDF)

For non-Maxwellian plasmas, the Druyvesteyn distribution:

$$ f(\varepsilon) = C \cdot \varepsilon^{1/2} \exp\left(-\left(\frac{\varepsilon}{\bar{\varepsilon}}\right)^2\right) $$

Where:

• $\varepsilon$ — electron energy $[\text{eV}]$
• $\bar{\varepsilon}$ — mean electron energy
• $C$ — normalization constant

Ion Bombardment Energy

$$ E_{ion} = e \cdot V_{sheath} + \frac{1}{2}m_{ion}v_{Bohm}^2 $$

Where:

• $V_{sheath}$ — plasma sheath voltage
• $v_{Bohm} = \sqrt{\frac{k_B T_e}{m_{ion}}}$ — Bohm velocity

Radical Generation Rate

$$ R_{radical} = n_e \cdot n_{gas} \cdot \langle \sigma v \rangle $$

Where:

• $n_e$ — electron density $[\text{m}^{-3}]$
• $n_{gas}$ — neutral gas density
• $\langle \sigma v \rangle$ — rate coefficient (energy-averaged cross-section × velocity)

Feature-Scale Modeling

Critical Phenomena in High Aspect Ratio Structures

Modern semiconductor devices require filling trenches and vias with aspect ratios (AR) exceeding 50:1.

Knudsen Number

$$ Kn = \frac{\lambda}{d} $$

Where:

• $\lambda$ — mean free path of gas molecules
• $d$ — characteristic feature dimension

Mean Free Path Calculation

$$ \lambda = \frac{k_B T}{\sqrt{2} \pi d_m^2 p} $$

Where:

• $d_m$ — molecular diameter $[\text{m}]$
• $p$ — pressure $[\text{Pa}]$

Step Coverage Model

$$ SC = \frac{t_{sidewall}}{t_{top}} \times 100\% $$

For diffusion-limited deposition:

$$ SC \approx \frac{1}{\sqrt{1 + AR^2}} $$

For reaction-limited deposition:

$$ SC \approx 1 - \frac{S \cdot AR}{2} $$

Where:

• $S$ — sticking coefficient
• $AR$ — aspect ratio = depth/width

Void Formation Criterion

Void formation occurs when:

$$ \frac{d(thickness_{sidewall})}{dz} > \frac{w(z)}{2 \cdot t_{total}} $$

Where:

• $w(z)$ — feature width at depth $z$
• $t_{total}$ — total deposition time

Film Properties to Model

Structural Properties

• Thickness uniformity:

$$ U = \frac{t_{max} - t_{min}}{t_{max} + t_{min}} \times 100\% $$

• Film stress (Stoney equation):

$$ \sigma_f = \frac{E_s t_s^2}{6(1- u_s)t_f} \cdot \frac{1}{R} $$ Where:

• $E_s$, $

u_s$ — substrate Young's modulus and Poisson ratio

• $t_s$, $t_f$ — substrate and film thickness
• $R$ — radius of curvature

• Density from refractive index (Lorentz-Lorenz):

$$ \frac{n^2 - 1}{n^2 + 2} = \frac{4\pi}{3} N \alpha $$ Where $N$ is molecular density and $\alpha$ is polarizability

Electrical Properties

• Dielectric constant (capacitance method):

$$ \kappa = \frac{C \cdot t}{\varepsilon_0 \cdot A} $$

• Breakdown field:

$$ E_{BD} = \frac{V_{BD}}{t} $$

• Leakage current density (Fowler-Nordheim tunneling):

$$ J = \frac{q^3 E^2}{8\pi h \phi_B} \exp\left(-\frac{8\pi\sqrt{2m^*}\phi_B^{3/2}}{3qhE}\right) $$

Where:

• $E$ — electric field
• $\phi_B$ — barrier height
• $m^*$ — effective electron mass

Multiscale Modeling Hierarchy

Scale Linking Framework

┌─────────────────────────────────────────────────────────────────────┐
│  ATOMISTIC (Å-nm)              MESOSCALE (nm-μm)        CONTINUUM   │
│  ─────────────────             ──────────────────       (μm-mm)     │
│                                                         ──────────  │
│  • DFT calculations            • Kinetic Monte Carlo    • CFD       │
│  • Molecular Dynamics          • Level-set methods      • FEM       │
│  • Ab initio MD                • Cellular automata      • TCAD      │
│                                                                     │
│  Outputs:                      Outputs:                 Outputs:    │
│  • Binding energies            • Film morphology        • Flow      │
│  • Reaction barriers           • Growth rate            • T, C      │
│  • Diffusion coefficients      • Surface roughness      • Profiles  │
└─────────────────────────────────────────────────────────────────────┘

DFT Calculations

Solve the Kohn-Sham equations:

$$ \left[-\frac{\hbar^2}{2m} abla^2 + V_{eff}(\mathbf{r})\right]\psi_i(\mathbf{r}) = \varepsilon_i \psi_i(\mathbf{r}) $$

Where:

$$ V_{eff} = V_{ext} + V_H + V_{xc} $$

• $V_{ext}$ — external potential (nuclei)
• $V_H$ — Hartree potential (electron-electron)
• $V_{xc}$ — exchange-correlation potential

Kinetic Monte Carlo (kMC)

Event selection probability:

$$ P_i = \frac{k_i}{\sum_j k_j} $$

Time advancement:

$$ \Delta t = -\frac{\ln(r)}{\sum_j k_j} $$

Where $r$ is a random number $\in (0,1]$

Specific Process Examples

PECVD $\text{SiO}_2$ from TEOS

Overall Reaction

$$ \text{Si(OC}_2\text{H}_5\text{)}_4 + 12\text{O}^* \xrightarrow{\text{plasma}} \text{SiO}_2 + 8\text{CO}_2 + 10\text{H}_2\text{O} $$

Key Process Parameters

Deposition Rate Model

$$ R_{dep} = k_0 \cdot p_{TEOS}^a \cdot p_{O_2}^b \cdot \exp\left(-\frac{E_a}{RT}\right) $$

Typical values: $a \approx 0.5$, $b \approx 0.3$, $E_a \approx 0.3 \, \text{eV}$

ALD High-$\kappa$ Dielectrics ($\text{HfO}_2$)

Half-Reactions

Cycle A (Metal precursor):

$$ \text{Hf(N(CH}_3\text{)}_2\text{)}_4\text{(g)} + \text{-OH} \rightarrow \text{-O-Hf(N(CH}_3\text{)}_2\text{)}_3 + \text{HN(CH}_3\text{)}_2 $$

Cycle B (Oxidizer):

$$ \text{-O-Hf(N(CH}_3\text{)}_2\text{)}_3 + 2\text{H}_2\text{O} \rightarrow \text{-O-Hf(OH)}_3 + 3\text{HN(CH}_3\text{)}_2 $$

Growth Per Cycle (GPC)

$$ \text{GPC} = \frac{\theta_{sat} \cdot \rho_{site} \cdot M_{HfO_2}}{\rho_{HfO_2} \cdot N_A} $$

Typical GPC for $\text{HfO}_2$: $0.8-1.2 \, \text{Å/cycle}$

ALD Window

           ┌────────────────────────────┐
     GPC   │     ┌──────────────┐       │
    (Å/    │    /│              │\      │
   cycle)  │   / │   ALD        │ \     │
           │  /  │   WINDOW     │  \    │
           │ /   │              │   \   │
           │/    │              │    \  │
           └─────┴──────────────┴─────┴─┘
                 T_min        T_max
                 Temperature (°C)

Below $T_{min}$: Condensation, incomplete reactions Above $T_{max}$: Precursor decomposition, CVD-like behavior

HDPCVD Gap Fill

Deposition-Etch Competition

Net deposition rate:

$$ R_{net}(z) = R_{dep}(\theta) - R_{etch}(E_{ion}, \theta) $$

Where:

• $R_{dep}(\theta)$ — angular-dependent deposition rate
• $R_{etch}$ — ion-enhanced etch rate
• $\theta$ — angle from surface normal

Sputter Yield (Yamamura Formula)

$$ Y(E, \theta) = Y_0(E) \cdot f(\theta) $$

Where:

$$ f(\theta) = \cos^{-f}\theta \cdot \exp\left[-\Sigma(\cos^{-1}\theta - 1)\right] $$

Machine Learning Applications

Virtual Metrology

Objective: Predict film properties from in-situ sensor data without destructive measurement.

$$ \hat{y} = f_{ML}(\mathbf{x}_{sensors}, \mathbf{x}_{recipe}) $$

Where:

• $\hat{y}$ — predicted property (thickness, stress, etc.)
• $\mathbf{x}_{sensors}$ — OES, pressure, RF power signals
• $\mathbf{x}_{recipe}$ — setpoints and timing

Gaussian Process Regression

$$ y(\mathbf{x}) \sim \mathcal{GP}\left(m(\mathbf{x}), k(\mathbf{x}, \mathbf{x}')\right) $$

Posterior mean prediction:

$$ \mu(\mathbf{x}^*) = \mathbf{k}^T(\mathbf{K} + \sigma_n^2\mathbf{I})^{-1}\mathbf{y} $$

Uncertainty quantification:

$$ \sigma^2(\mathbf{x}^) = k(\mathbf{x}^, \mathbf{x}^*) - \mathbf{k}^T(\mathbf{K} + \sigma_n^2\mathbf{I})^{-1}\mathbf{k} $$

Bayesian Optimization for Recipe Development

Acquisition function (Expected Improvement):

$$ \text{EI}(\mathbf{x}) = \mathbb{E}\left[\max(f(\mathbf{x}) - f^+, 0)\right] $$

Where $f^+$ is the best observed value.

Advanced Node Challenges (Sub-5nm)

Critical Challenges

Equivalent Oxide Thickness (EOT)

For high-$\kappa$ gate stacks:

$$ \text{EOT} = t_{IL} + \frac{\kappa_{SiO_2}}{\kappa_{high-k}} \cdot t_{high-k} $$

Where:

• $t_{IL}$ — interfacial layer thickness
• $\kappa_{SiO_2} = 3.9$
• Typical high-$\kappa$: $\kappa_{HfO_2} \approx 20-25$

Low-$\kappa$ Dielectric Design

Effective dielectric constant:

$$ \kappa_{eff} = \kappa_{matrix} \cdot (1 - p) + \kappa_{air} \cdot p $$

Where $p$ is porosity fraction.

Target for advanced nodes: $\kappa_{eff} < 2.0$

Tools and Software

Commercial TCAD

• Synopsys Sentaurus Process — full process simulation
• Silvaco Victory Process — alternative TCAD suite
• Lam Research SEMulator3D — 3D topography simulation

Multiphysics Platforms

• COMSOL Multiphysics — coupled PDE solving
• Ansys Fluent — CFD for reactor design
• Ansys CFX — alternative CFD solver

Specialized Tools

• CHEMKIN (Ansys) — gas-phase reaction kinetics
• Reaction Design — combustion and plasma chemistry
• Custom Monte Carlo codes — feature-scale simulation

Open Source Options

• OpenFOAM — CFD framework
• LAMMPS — molecular dynamics
• Quantum ESPRESSO — DFT calculations
• SPARTA — DSMC for rarefied gas dynamics

Summary

Dielectric deposition modeling in semiconductor manufacturing integrates:

1. Transport phenomena — mass, momentum, energy conservation 2. Reaction kinetics — surface and gas-phase chemistry 3. Plasma physics — for PECVD/HDPCVD processes 4. Feature-scale physics — conformality, void formation 5. Multiscale approaches — atomistic to continuum 6. Machine learning — for optimization and virtual metrology

The goal is predicting and optimizing film properties based on process parameters while accounting for the extreme topography of modern semiconductor devices.