Ddp Modeling

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

Method	Full Name	Temperature Range	Typical Applications
PECVD	Plasma-Enhanced CVD	$$200-400°C$$	$\text{SiO}_2$ , $\text{SiN}_x$ for ILD, passivation
LPCVD	Low-Pressure CVD	$$400-800°C$$	High-quality $\text{Si}_3\text{N}_4$ , poly-Si
HDPCVD	High-Density Plasma CVD	$$300-450°C$$	Gap-fill for trenches and vias
ALD	Atomic 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-on	SOG/SOD	$$100-400°C$$	Planarization layers

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} + \nabla \cdot (\mathbf{v}C) = D\nabla^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 \nabla T\right) = k\nabla^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 \nabla \mathbf{v}\right) = -\nabla p + \mu \nabla^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

Regime	Knudsen Number	Transport 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:

$$d_m$$ — 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

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}

$V_{ext}$ — 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

Parameter	Typical Range	Effect
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

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

Challenge	Technical Details	Modeling Complexity
Ultra-high AR	3D NAND: 100+ layers, AR > 50:1	Knudsen transport, ballistic modeling
Atomic precision	Gate dielectrics: 1-2 nm	Monolayer-level control, quantum effects
Low- $\kappa$ integration	$\kappa < 2.5$ porous films	Mechanical integrity, plasma damage
Selective deposition	Area-selective ALD	Nucleation control, surface chemistry
Thermal budget	BEOL: $$< 400°C$$	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:

$t_{IL}$ — 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.

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