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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<j} \beta_{ij} x_i x_j + \varepsilon

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

Scale	Method	Outputs
Atomic (Å)	MD, DFT	Yield functions, surface chemistry
Feature (nm–μm)	Monte Carlo, Level Set	Profile evolution
Reactor (cm)	Fluid/hybrid plasma models	Plasma uniformity
Wafer (mm–cm)	Empirical/FEM	Loading, CD uniformity

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

Physics	Equation
Etch rate	$R = Y(E) \cdot \Gamma_{ion} \cdot \Theta$
Level set evolution	$\frac{\partial \phi}{\partial t} + V

abla\phi| = 0$ |

Selectivity	$S_{A:B} = R_A / R_B$
ARDE	$R(AR) = R_0 / (1 + \alpha \cdot AR)$
Bohm flux	$\Gamma_i = n_s \sqrt{k_B T_e / M_i}$
ALE EPC	$EPC = \theta_{sat} \cdot \delta_{ML}$
Knudsen diffusion	$D_K = \frac{d}{3}\sqrt{8k_BT/\pi m}$

Source: ChipFoundryServices — Search this topic — Ask CFSGPT

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