Etch Film Stack Mathematical Modeling

Keywords: etch film stack modeling, etch film stack, etch modeling, etch film stack math, film stack etch modeling

Etch Film Stack Mathematical Modeling

1. Introduction and Problem Setup

A film stack in semiconductor manufacturing consists of multiple thin-film layers that must be precisely etched. Typical structures include:

• Photoresist (masking layer)
• Hard mask (SiN, SiO₂, or metal)
• Target film (material to be etched)
• Etch stop layer
• Substrate (Si wafer)

Objectives

• Remove target material at a controlled rate
• Stop precisely at interfaces (selectivity)
• Maintain profile fidelity (anisotropy, sidewall angle)
• Achieve uniformity across the wafer

2. Fundamental Etch Rate Models

2.1 Surface Reaction Kinetics

The Langmuir-Hinshelwood model captures competitive adsorption of reactive species:

$$ R = \frac{k \cdot \theta_A \cdot \theta_B}{\left(1 + K_A[A] + K_B[B]\right)^2} $$

Where:

• $R$ = etch rate
• $k$ = reaction rate constant
• $\theta_A, \theta_B$ = fractional surface coverage of species A and B
• $K_A, K_B$ = adsorption equilibrium constants
• $[A], [B]$ = gas-phase concentrations

2.2 Temperature Dependence (Arrhenius)

$$ R = R_0 \exp\left(-\frac{E_a}{k_B T}\right) $$

Where:

• $R_0$ = pre-exponential factor
• $E_a$ = activation energy
• $k_B$ = Boltzmann constant ($1.38 \times 10^{-23}$ J/K)
• $T$ = absolute temperature (K)

2.3 Ion-Enhanced Etching Model

Most plasma etching exhibits synergistic behavior—ions enhance chemical reactions:

$$ R_{total} = R_{chem} + R_{phys} + R_{synergy} $$

The ion-enhanced component dominates in RIE/ICP:

$$ R_{ie} = Y(E, \theta) \cdot \Gamma_{ion} \cdot \Theta_{react} $$

Where:

• $Y(E, \theta)$ = ion yield function (depends on energy $E$ and angle $\theta$)
• $\Gamma_{ion}$ = ion flux to surface (ions/cm²·s)
• $\Theta_{react}$ = fractional coverage of reactive species

3. Profile Evolution Mathematics

3.1 Level Set Method

The evolving surface is represented as the zero-contour of a level set function $\phi(\mathbf{x}, t)$:

$$ \frac{\partial \phi}{\partial t} + V(\mathbf{x}, t) \cdot | abla \phi| = 0 $$

Where:

• $\phi(\mathbf{x}, t)$ = level set function
• $V(\mathbf{x}, t)$ = local etch velocity (material and flux dependent)
• $

abla \phi$ = gradient of the level set function

• $|

abla \phi|$ = magnitude of the gradient

The surface normal is computed as:

$$ \hat{n} = \frac{ abla \phi}{| abla \phi|} $$

3.2 Visibility and Shadowing Integrals

For a point $\mathbf{p}$ inside a feature, the effective flux is:

$$ \Gamma(\mathbf{p}) = \int_{\Omega_{visible}} f(\hat{\Omega}) \cdot (\hat{\Omega} \cdot \hat{n}) \, d\Omega $$

Where:

• $\Omega_{visible}$ = solid angle visible from point $\mathbf{p}$
• $f(\hat{\Omega})$ = ion angular distribution function (IADF)
• $\hat{n}$ = local surface normal

3.3 Ion Angular Distribution Function (IADF)

Typically modeled as a Gaussian:

$$ f(\theta) = \frac{1}{\sqrt{2\pi}\sigma} \exp\left(-\frac{\theta^2}{2\sigma^2}\right) $$

Where:

• $\theta$ = angle from surface normal
• $\sigma$ = angular spread (related to $T_i / T_e$ ratio)

4. Multi-Layer Stack Modeling

4.1 Interface Tracking

For a stack with $n$ layers at depths $z_1, z_2, \ldots, z_n$:

$$ \frac{dz_{etch}}{dt} = -R_i(t) $$

Where $i$ indicates the current material being etched. Material transitions occur when $z_{etch}$ crosses an interface boundary.

4.2 Selectivity Definition

$$ S_{A:B} = \frac{R_A}{R_B} $$

Design requirements:

• Mask selectivity: $S_{target:mask} < 1$ (mask erodes slowly)
• Stop layer selectivity: $S_{target:stop} \gg 1$ (typically > 10:1)

4.3 Time-to-Clear Calculation

For layer thickness $d_i$ with etch rate $R_i$:

$$ t_{clear,i} = \frac{d_i}{R_i} $$

Total etch time through multiple layers:

$$ t_{total} = \sum_{i=1}^{n} \frac{d_i}{R_i} + t_{overetch} $$

5. Aspect Ratio Dependent Etching (ARDE)

5.1 General ARDE Model

Etch rate decreases with aspect ratio (AR = depth/width):

$$ R(AR) = R_0 \cdot f(AR) $$

5.2 Neutral Transport Limited (Knudsen Regime)

$$ R(AR) = \frac{R_0}{1 + \alpha \cdot AR} $$

The Knudsen diffusivity in a cylindrical feature:

$$ D_K = \frac{d}{3}\sqrt{\frac{8 k_B T}{\pi m}} $$

Where:

• $d$ = feature diameter
• $m$ = molecular mass of neutral species
• $T$ = gas temperature

5.3 Clausing Factor for Molecular Flow

For a tube of length $L$ and radius $r$:

$$ W = \frac{1}{1 + \frac{3L}{8r}} $$

5.4 Ion Angular Distribution Limited

$$ R(AR) = R_0 \cdot \int_0^{\theta_{max}(AR)} f(\theta) \cos\theta \, d\theta $$

Where $\theta_{max}$ is the maximum acceptance angle:

$$ \theta_{max} = \arctan\left(\frac{w}{2h}\right) $$

6. Plasma and Transport Modeling

6.1 Sheath Physics

Child-Langmuir Law (Collisionless Sheath)

$$ J = \frac{4\varepsilon_0}{9}\sqrt{\frac{2e}{M}}\frac{V_0^{3/2}}{d^2} $$

Where:

• $J$ = ion current density
• $\varepsilon_0$ = permittivity of free space
• $e$ = electron charge
• $M$ = ion mass
• $V_0$ = sheath voltage
• $d$ = sheath thickness

Sheath Thickness (Matrix Sheath)

$$ s = \lambda_D \sqrt{\frac{2eV_0}{k_B T_e}} $$

Where $\lambda_D$ is the Debye length:

$$ \lambda_D = \sqrt{\frac{\varepsilon_0 k_B T_e}{n_e e^2}} $$

6.2 Ion Flux to Surface

At the sheath edge, ions reach the Bohm velocity:

$$ u_B = \sqrt{\frac{k_B T_e}{M_i}} $$

Ion flux:

$$ \Gamma_i = n_s \cdot u_B = n_s \sqrt{\frac{k_B T_e}{M_i}} $$

Where $n_s \approx 0.61 \cdot n_0$ (sheath edge density).

6.3 Neutral Species Balance

Continuity equation for neutral species:

$$

abla \cdot (D abla n) + \sum_j k_j n_j n_e - k_{loss} n = 0 $$

Where:

• $D$ = diffusion coefficient
• $k_j$ = generation rate constants
• $k_{loss}$ = surface loss rate

7. Feature-Scale Monte Carlo Methods

7.1 Algorithm Overview

1. Sample particles from flux distributions at feature entrance 2. Track trajectories (ballistic for ions, random walk for neutrals) 3. Surface interactions: React, reflect, or stick with probabilities 4. Accumulate statistics for local etch rates 5. Advance surface using accumulated rates

7.2 Reflection Probability Models

Specular Reflection

$$ \theta_{out} = \theta_{in} $$

Diffuse (Cosine) Reflection

$$ P(\theta_{out}) \propto \cos(\theta_{out}) $$

Mixed Model

$$ P_{reflect} = (1 - s) \cdot P_{specular} + s \cdot P_{diffuse} $$

Where $s$ is the scattering coefficient.

7.3 Sticking Coefficient Model

$$ \gamma = \gamma_0 \cdot (1 - \Theta)^n $$

Where:

• $\gamma_0$ = bare surface sticking coefficient
• $\Theta$ = surface coverage
• $n$ = reaction order

8. Loading Effects

8.1 Macroloading (Wafer Scale)

$$ R = \frac{R_0}{1 + \beta \cdot A_{exposed}} $$

Where:

• $A_{exposed}$ = total exposed etchable area
• $\beta$ = loading coefficient

8.2 Microloading (Pattern Scale)

Local etch rate depends on pattern density $\rho$:

$$ R_{local} = R_0 \cdot \left(1 - \gamma \cdot \rho\right) $$

Dense patterns etch slower due to local reactant depletion.

8.3 Reactive Species Depletion Model

For a feature with area $A$ in a cell of area $A_{cell}$:

$$ R = R_0 \cdot \frac{1}{1 + \frac{k_{etch} \cdot A}{k_{supply} \cdot A_{cell}}} $$

9. Atomic Layer Etching (ALE) Models

9.1 Two-Step Process

Step 1 - Surface Modification:

$$ A_{(g)} + S_{(s)} \rightarrow A\text{-}S_{(s)} $$

Step 2 - Removal:

$$ A\text{-}S_{(s)} + B_{(g/ion)} \rightarrow \text{volatile products} $$

9.2 Self-Limiting Kinetics

Surface coverage during modification:

$$ \theta_{mod}(t) = 1 - \exp\left(-\Gamma_A \cdot s_A \cdot t\right) $$

Where:

• $\Gamma_A$ = flux of modifying species
• $s_A$ = sticking probability
• $t$ = exposure time

9.3 Etch Per Cycle (EPC)

$$ EPC = \theta_{sat} \cdot \delta_{ML} $$

Where:

• $\theta_{sat}$ = saturation coverage (ideally 1.0)
• $\delta_{ML}$ = monolayer thickness (typically 0.1–0.5 nm)

9.4 Synergy Factor

$$ S_f = \frac{EPC_{ALE}}{EPC_{step1} + EPC_{step2}} $$

Values $S_f > 1$ indicate synergistic enhancement.

10. Process Window Modeling

10.1 Response Surface Methodology

$$ CD = \beta_0 + \sum_{i=1}^{k} \beta_i x_i + \sum_{i=1}^{k} \beta_{ii} x_i^2 + \sum_{i

Where:

• $CD$ = critical dimension (response variable)
• $x_i$ = process parameters (pressure, power, gas flows, time, etc.)
• $\beta$ = regression coefficients
• $\varepsilon$ = random error

10.2 Sensitivity Analysis

$$ \frac{\partial CD}{\partial x_i} = \beta_i + 2\beta_{ii}x_i + \sum_{j eq i} \beta_{ij} x_j $$

10.3 Process Capability

$$ C_p = \frac{USL - LSL}{6\sigma} $$

$$ C_{pk} = \min\left(\frac{USL - \mu}{3\sigma}, \frac{\mu - LSL}{3\sigma}\right) $$

Where:

• $USL, LSL$ = upper and lower specification limits
• $\mu$ = process mean
• $\sigma$ = process standard deviation

11. Computational Implementation

11.1 Multi-Scale Hierarchy

11.2 Level Set Discretization

Using upwind finite differences:

$$ \phi_i^{n+1} = \phi_i^n - \Delta t \cdot V_i \cdot | abla \phi|_i $$

With the gradient approximated by:

$$

abla \phi| \approx \sqrt{\max(D^{-x}, 0)^2 + \min(D^{+x}, 0)^2 + \max(D^{-y}, 0)^2 + \min(D^{+y}, 0)^2} $$

Where $D^{\pm x}$ are forward/backward differences.

11.3 CFL Condition for Stability

$$ \Delta t < \frac{\Delta x}{V_{max}} $$

12. Advanced Considerations

12.1 High Aspect Ratio (HAR) Challenges

For 3D NAND (AR > 50:1):

$$ R_{HAR} = R_0 \cdot \exp\left(-\frac{AR}{AR_c}\right) $$

Where $AR_c$ is a characteristic decay constant.

12.2 Stochastic Effects at Atomic Scale

Line edge roughness (LER) from statistical fluctuations:

$$ \sigma_{LER} \propto \sqrt{\frac{1}{N_{atoms}}} \propto \frac{1}{\sqrt{CD}} $$

12.3 Pattern-Dependent Charging

Electron shading leads to differential charging:

$$ V_{bottom} = V_{plasma} - \frac{J_e - J_i}{C_{feature}} $$

This causes notching and profile distortion in HAR features.

12.4 Etch-Induced Damage

Ion damage depth follows:

$$ R_p = \frac{E}{S_n + S_e} $$

Where:

• $E$ = ion energy
• $S_n$ = nuclear stopping power
• $S_e$ = electronic stopping power

13. Equations

abla\phi| = 0$ |

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