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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:

Dielectric Deposition Methods

Primary Techniques

MethodFull NameTemperature RangeTypical Applications
PECVDPlasma-Enhanced CVD$200-400°C$$\text{SiO}_2$, $\text{SiN}_x$ for ILD, passivation
LPCVDLow-Pressure CVD$400-800°C$High-quality $\text{Si}_3\text{N}_4$, poly-Si
HDPCVDHigh-Density Plasma CVD$300-450°C$Gap-fill for trenches and vias
ALDAtomic Layer Deposition$150-350°C$Ultra-thin gate dielectrics ($\text{HfO}_2$, $\text{Al}_2\text{O}_3$)
Thermal Oxidation—$800-1200°C$Gate oxide ($\text{SiO}_2$)
Spin-onSOG/SOD$100-400°C$Planarization layers

Selection Criteria

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)

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

Where:

Energy Balance

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

Where:

Momentum Balance (Navier-Stokes)

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

Where:

Surface Reaction Kinetics

Arrhenius Rate Expression

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

Where:

Langmuir Adsorption Isotherm (for ALD)

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

Where:

Sticking Coefficient

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

Where:

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:

Ion Bombardment Energy

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

Where:

Radical Generation Rate

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

Where:

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:

RegimeKnudsen NumberTransport Type
Continuum$Kn < 0.01$Viscous flow
Slip$0.01 < Kn < 0.1$Transition
Transition$0.1 < Kn < 10$Mixed
Free molecular$Kn > 10$Ballistic/Knudsen

Mean Free Path Calculation

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

Where:

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:

Void Formation Criterion

Void formation occurs when:

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

Where:

Film Properties to Model

Structural Properties

$$ U = \frac{t_{max} - t_{min}}{t_{max} + t_{min}} \times 100\%$$
$$ \sigma_f = \frac{E_s t_s^2}{6(1- u_s)t_f} \cdot \frac{1}{R}$$

Where:

u_s$ — substrate Young's modulus and Poisson ratio

$$ \frac{n^2 - 1}{n^2 + 2} = \frac{4\pi}{3} N \alpha$$

Where $N$ is molecular density and $\alpha$ is polarizability

Electrical Properties

$$ \kappa = \frac{C \cdot t}{\varepsilon_0 \cdot A}$$
$$ E_{BD} = \frac{V_{BD}}{t}$$
$$ J = \frac{q^3 E^2}{8\pi h \phi_B} \exp\left(-\frac{8\pi\sqrt{2m^*}\phi_B^{3/2}}{3qhE}\right)$$

Where:

Multiscale Modeling Hierarchy

Scale Linking Framework

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  <text x="380" y="48" fill="#8b98a5" font-size="12" text-anchor="middle">Atomistic Surface Reaction Kinetics, Step Coverage, and Trench Profile Simulation</text>

  <!-- Left Panel: Gas Phase Transport -->
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    <text x="110" y="220" fill="#d2a8ff" font-size="11" font-weight="600" text-anchor="middle">Knudsen Number Kn</text>
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    <text x="110" y="310" fill="#3fb950" font-size="10" font-weight="600" text-anchor="middle">Ballistic Transport Regime</text>
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    <text x="110" y="185" fill="#e6edf3" font-size="11" text-anchor="middle">Level-Set Method Simulation</text>

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    <text x="110" y="220" fill="#58a6ff" font-size="11" font-weight="600" text-anchor="middle">Step Coverage %</text>
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  <text x="380" y="452" fill="#6b7684" font-size="11" text-anchor="middle">Predictive TCAD DDP Simulation for High-K Metal Gate (HKMG) &amp; 3D NAND Deep Trench Dielectrics</text>
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DFT Calculations

Solve the Kohn-Sham equations:

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

Where:

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

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

ParameterTypical RangeEffect
RF Power$100-1000 \, \text{W}$↑ Power → ↑ Density, ↓ Dep rate
Pressure$0.5-5 \, \text{Torr}$↑ Pressure → ↑ Dep rate, ↓ Conformality
Temperature$300-400°C$↑ Temp → ↑ Density, ↓ H content
TEOS:O₂ ratio$1:5$ to $1:20$Affects stoichiometry, quality

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

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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:

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:

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

ChallengeTechnical DetailsModeling Complexity
Ultra-high AR3D NAND: 100+ layers, AR > 50:1Knudsen transport, ballistic modeling
Atomic precisionGate dielectrics: 1-2 nmMonolayer-level control, quantum effects
Low-$\kappa$ integration$\kappa < 2.5$ porous filmsMechanical integrity, plasma damage
Selective depositionArea-selective ALDNucleation control, surface chemistry
Thermal budgetBEOL: $< 400°C$Kinetic limitations, precursor chemistry

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:

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

Multiphysics Platforms

Specialized Tools

Open Source Options

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.

ddp modelingdielectric depositionhigh-k dielectricsaldpecvdgap fillhdpcvdfeature-scale modeling

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