Mathematical Modeling of Plasma…

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Mathematical Modeling of Plasma Etching in Semiconductor Manufacturing

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  <text x="380" y="32" fill="#e6edf3" font-size="20" font-weight="700" text-anchor="middle">Reactive Ion Etching (RIE) — Plasma &amp; Anisotropic Profile Mechanics</text>
  <text x="380" y="52" fill="#8b98a5" font-size="12" text-anchor="middle">Synergy of Physical Ion Bombardment (Directionality) &amp; Chemical Radical Reaction (Selectivity)</text>

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      <text x="157.5" y="112" fill="#34d399" font-size="8.5" font-weight="700" text-anchor="middle">Anisotropy A ≈ 1 | Sub-3nm Nanometer Pattern Transfer</text>
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  <text x="380" y="430" fill="#fbbf24" font-size="9.5" font-weight="700" text-anchor="middle">Anisotropy Factor A = 1 - (V_lateral / V_vertical) | RF Sheath Acceleration drives A → 1 while Radical Chemistry delivers Selectivity (&gt;50:1)</text>

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  <text x="380" y="458" fill="#6b7684" font-size="11" text-anchor="middle">Critical dimension (CD) transfer standard for sub-3nm nanometer nodes · High ICP/CCP plasma density control</text>
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Introduction

Plasma etching is a critical process in semiconductor manufacturing where reactive gases are ionized to create a plasma, which selectively removes material from a wafer surface. The mathematical modeling of this process spans multiple physics domains:

Electromagnetic theory — RF power coupling and field distributions
Statistical mechanics — Particle distributions and kinetic theory
Reaction kinetics — Gas-phase and surface chemistry
Transport phenomena — Species diffusion and convection
Surface science — Etch mechanisms and selectivity

Foundational Plasma Physics

Boltzmann Transport Equation

The most fundamental description of plasma behavior is the Boltzmann transport equation, governing the evolution of the particle velocity distribution function $f(\mathbf{r}, \mathbf{v}, t)$:

\frac{\partial f}{\partial t} + \mathbf{v} \cdot \nabla f + \frac{\mathbf{F}}{m} \cdot \nabla_v f = \left(\frac{\partial f}{\partial t}\right)_{\text{collision}}

Where:

$f(\mathbf{r}, \mathbf{v}, t)$ — Velocity distribution function
$\mathbf{v}$ — Particle velocity
$\mathbf{F}$ — External force (electromagnetic)
$m$ — Particle mass
RHS — Collision integral

Fluid Moment Equations

For computational tractability, velocity moments of the Boltzmann equation yield fluid equations:

Continuity Equation (Mass Conservation)

\frac{\partial n}{\partial t} + \nabla \cdot (n\mathbf{u}) = S - L

Where:

$n$ — Species number density $[\text{m}^{-3}]$
$\mathbf{u}$ — Drift velocity $[\text{m/s}]$
$S$ — Source term (generation rate)
$L$ — Loss term (consumption rate)

Momentum Conservation

\frac{\partial (nm\mathbf{u})}{\partial t} + \nabla \cdot (nm\mathbf{u}\mathbf{u}) + \nabla p = nq(\mathbf{E} + \mathbf{u} \times \mathbf{B}) - nm u_m \mathbf{u}

Where:

$p = nk_BT$ — Pressure
$q$ — Particle charge
$\mathbf{E}$, $\mathbf{B}$ — Electric and magnetic fields
$

u_m$ — Momentum transfer collision frequency $[\text{s}^{-1}]$

Energy Conservation

\frac{\partial}{\partial t}\left(\frac{3}{2}nk_BT\right) + \nabla \cdot \mathbf{q} + p\nabla \cdot \mathbf{u} = Q_{\text{heating}} - Q_{\text{loss}}

Where:

$k_B = 1.38 \times 10^{-23}$ J/K — Boltzmann constant
$\mathbf{q}$ — Heat flux vector
$Q_{\text{heating}}$ — Power input (Joule heating, stochastic heating)
$Q_{\text{loss}}$ — Energy losses (collisions, radiation)

Electromagnetic Field Coupling

Maxwell's Equations

For capacitively coupled plasma (CCP) and inductively coupled plasma (ICP) reactors:

\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}

\nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}}{\partial t}

\nabla \cdot \mathbf{D} = \rho

\nabla \cdot \mathbf{B} = 0

Plasma Conductivity

The plasma current density couples through the complex conductivity:

\mathbf{J} = \sigma \mathbf{E}

For RF plasmas, the complex conductivity is:

\sigma = \frac{n_e e^2}{m_e( u_m + i\omega)}

Where:

$n_e$ — Electron density
$e = 1.6 \times 10^{-19}$ C — Elementary charge
$m_e = 9.1 \times 10^{-31}$ kg — Electron mass
$\omega$ — RF angular frequency
$

u_m$ — Electron-neutral collision frequency

Power Deposition

Time-averaged power density deposited into the plasma:

P = \frac{1}{2}\text{Re}(\mathbf{J} \cdot \mathbf{E}^*)

Typical values:

CCP: $0.1 - 1$ W/cm³
ICP: $0.5 - 5$ W/cm³

Plasma Sheath Physics

The sheath is a thin, non-neutral region at the plasma-wafer interface that accelerates ions toward the surface, enabling anisotropic etching.

Bohm Criterion

Minimum ion velocity entering the sheath:

u_i \geq u_B = \sqrt{\frac{k_B T_e}{M_i}}

Where:

$u_B$ — Bohm velocity
$T_e$ — Electron temperature (typically 2–5 eV)
$M_i$ — Ion mass

Example: For Ar⁺ ions with $T_e = 3$ eV: $$ u_B = \sqrt{\frac{3 \times 1.6 \times 10^{-19}}{40 \times 1.67 \times 10^{-27}}} \approx 2.7 \text{ km/s} $$

Child-Langmuir Law

For a collisionless sheath, the ion current density is:

J = \frac{4\varepsilon_0}{9}\sqrt{\frac{2e}{M_i}} \cdot \frac{V_s^{3/2}}{d^2}

Where:

$\varepsilon_0 = 8.85 \times 10^{-12}$ F/m — Vacuum permittivity
$V_s$ — Sheath voltage drop (typically 10–500 V)
$d$ — Sheath thickness

Sheath Thickness

The sheath thickness scales as:

d \approx \lambda_D \left(\frac{2eV_s}{k_BT_e}\right)^{3/4}

Where the Debye length is:

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

Ion Angular Distribution

Ions arrive at the wafer with an angular distribution:

f(\theta) \propto \exp\left(-\frac{\theta^2}{2\sigma^2}\right)

Where:

\sigma \approx \arctan\left(\sqrt{\frac{k_B T_i}{eV_s}}\right)

Typical values: $\sigma \approx 2°–5°$ for high-bias conditions.

Electron Energy Distribution Function

Non-Maxwellian Distributions

In low-pressure plasmas (1–100 mTorr), the EEDF deviates from Maxwellian.

Two-Term Approximation

The EEDF is expanded as:

f(\varepsilon, \theta) = f_0(\varepsilon) + f_1(\varepsilon)\cos\theta

The isotropic part $f_0$ satisfies:

\frac{d}{d\varepsilon}\left[\varepsilon D \frac{df_0}{d\varepsilon} + \left(V + \frac{\varepsilon u_{\text{inel}}}{ u_m}\right)f_0\right] = 0

Common Distribution Functions

Distribution	Functional Form	Applicability
Maxwellian	$f(\varepsilon) \propto \sqrt{\varepsilon} \exp\left(-\frac{\varepsilon}{k_BT_e}\right)$	High pressure, collisional
Druyvesteyn	$f(\varepsilon) \propto \sqrt{\varepsilon} \exp\left(-\left(\frac{\varepsilon}{k_BT_e}\right)^2\right)$	Elastic collisions dominant
Bi-Maxwellian	Sum of two Maxwellians	Hot tail population

Generalized Form

f(\varepsilon) \propto \sqrt{\varepsilon} \cdot \exp\left[-\left(\frac{\varepsilon}{k_BT_e}\right)^x\right]

$x = 1$ → Maxwellian
$x = 2$ → Druyvesteyn

Plasma Chemistry and Reaction Kinetics

Species Balance Equation

For species $i$:

\frac{\partial n_i}{\partial t} + \nabla \cdot \mathbf{\Gamma}_i = \sum_j R_j

Where:

$\mathbf{\Gamma}_i$ — Species flux
$R_j$ — Reaction rates

Electron-Impact Rate Coefficients

Rate coefficients are calculated by integration over the EEDF:

k = \int_0^\infty \sigma(\varepsilon) v(\varepsilon) f(\varepsilon) \, d\varepsilon = \langle \sigma v \rangle

Where:

$\sigma(\varepsilon)$ — Energy-dependent cross-section $[\text{m}^2]$
$v(\varepsilon) = \sqrt{2\varepsilon/m_e}$ — Electron velocity
$f(\varepsilon)$ — Normalized EEDF

Heavy-Particle Reactions

Arrhenius kinetics for neutral reactions:

k = A T^n \exp\left(-\frac{E_a}{k_BT}\right)

Where:

$A$ — Pre-exponential factor
$n$ — Temperature exponent
$E_a$ — Activation energy

Example: SF₆/O₂ Plasma Chemistry

Electron-Impact Reactions

Reaction	Type	Threshold
$e + \text{SF}_6 \rightarrow \text{SF}_5 + \text{F} + e$	Dissociation	~10 eV
$e + \text{SF}_6 \rightarrow \text{SF}_6^-$	Attachment	~0 eV
$e + \text{SF}_6 \rightarrow \text{SF}_5^+ + \text{F} + 2e$	Ionization	~16 eV
$e + \text{O}_2 \rightarrow \text{O} + \text{O} + e$	Dissociation	~6 eV

Gas-Phase Reactions

$\text{F} + \text{O} \rightarrow \text{FO}$ (reduces F atom density)
$\text{SF}_5 + \text{F} \rightarrow \text{SF}_6$ (recombination)
$\text{O} + \text{CF}_3 \rightarrow \text{COF}_2 + \text{F}$ (polymer removal)

Surface Reactions

$\text{F} + \text{Si}(s) \rightarrow \text{SiF}_{(\text{ads})}$
$\text{SiF}_{(\text{ads})} + 3\text{F} \rightarrow \text{SiF}_4(g)$ (volatile product)

Transport Phenomena

Drift-Diffusion Model

For charged species, the flux is:

\mathbf{\Gamma} = \pm \mu n \mathbf{E} - D \nabla n

Where:

Upper sign: positive ions
Lower sign: electrons
$\mu$ — Mobility $[\text{m}^2/(\text{V}\cdot\text{s})]$
$D$ — Diffusion coefficient $[\text{m}^2/\text{s}]$

Einstein Relation

Connects mobility and diffusion:

D = \frac{\mu k_B T}{e}

Ambipolar Diffusion

When quasi-neutrality holds ($n_e \approx n_i$):

D_a = \frac{\mu_i D_e + \mu_e D_i}{\mu_i + \mu_e} \approx D_i\left(1 + \frac{T_e}{T_i}\right)

Since $T_e \gg T_i$ typically: $D_a \approx D_i (1 + T_e/T_i) \approx 100 D_i$

Neutral Transport

For reactive neutrals (radicals), Fickian diffusion:

\frac{\partial n}{\partial t} = D\nabla^2 n + S - L

Surface Boundary Condition

-D\frac{\partial n}{\partial x}\bigg|_{\text{surface}} = \frac{1}{4}\gamma n v_{\text{th}}

Where:

$\gamma$ — Sticking/reaction coefficient (0 to 1)
$v_{\text{th}} = \sqrt{\frac{8k_BT}{\pi m}}$ — Thermal velocity

Knudsen Number

Determines the appropriate transport regime:

\text{Kn} = \frac{\lambda}{L}

Where:

$\lambda$ — Mean free path
$L$ — Characteristic length

Kn Range	Regime	Model
$< 0.01$	Continuum	Navier-Stokes
$0.01–0.1$	Slip flow	Modified N-S
$0.1–10$	Transition	DSMC/BGK
$> 10$	Free molecular	Ballistic

Surface Reaction Modeling

Langmuir Adsorption Kinetics

For surface coverage $\theta$:

\frac{d\theta}{dt} = k_{\text{ads}}(1-\theta)P - k_{\text{des}}\theta - k_{\text{react}}\theta

At steady state:

\theta = \frac{k_{\text{ads}}P}{k_{\text{ads}}P + k_{\text{des}} + k_{\text{react}}}

Ion-Enhanced Etching

The total etch rate combines multiple mechanisms:

\text{ER} = Y_{\text{chem}} \Gamma_n + Y_{\text{phys}} \Gamma_i + Y_{\text{syn}} \Gamma_i f(\theta)

Where:

$Y_{\text{chem}}$ — Chemical etch yield (isotropic)
$Y_{\text{phys}}$ — Physical sputtering yield
$Y_{\text{syn}}$ — Ion-enhanced (synergistic) yield
$\Gamma_n$, $\Gamma_i$ — Neutral and ion fluxes
$f(\theta)$ — Coverage-dependent function

Ion Sputtering Yield

Energy Dependence

Y(E) = A\left(\sqrt{E} - \sqrt{E_{\text{th}}}\right) \quad \text{for } E > E_{\text{th}}

Typical threshold energies:

Si: $E_{\text{th}} \approx 20$ eV
SiO₂: $E_{\text{th}} \approx 30$ eV
Si₃N₄: $E_{\text{th}} \approx 25$ eV

Angular Dependence

Y(\theta) = Y(0) \cos^{-f}(\theta) \exp\left[-b\left(\frac{1}{\cos\theta} - 1\right)\right]

Behavior:

Increases from normal incidence
Peaks at $\theta \approx 60°–70°$
Decreases at grazing angles (reflection dominates)

Feature-Scale Profile Evolution

Level Set Method

The surface is represented as the zero contour of $\phi(\mathbf{x}, t)$:

\frac{\partial \phi}{\partial t} + V_n |\nabla \phi| = 0

Where:

$\phi > 0$ — Material
$\phi < 0$ — Void/vacuum
$\phi = 0$ — Surface
$V_n$ — Local normal etch velocity

Local Etch Rate Calculation

The normal velocity $V_n$ depends on:

1. Ion flux and angular distribution $$\Gamma_i(\mathbf{x}) = \int f(\theta, E) \, d\Omega \, dE$$

2. Neutral flux (with shadowing) $$\Gamma_n(\mathbf{x}) = \Gamma_{n,0} \cdot \text{VF}(\mathbf{x})$$ where VF is the view factor

3. Surface chemistry state $$V_n = f(\Gamma_i, \Gamma_n, \theta_{\text{coverage}}, T)$$

Neutral Transport in High-Aspect-Ratio Features

Clausing Transmission Factor

For a tube of aspect ratio AR:

K \approx \frac{1}{1 + 0.5 \cdot \text{AR}}

View Factor Calculations

For surface element $dA_1$ seeing $dA_2$:

F_{1 \rightarrow 2} = \frac{1}{\pi} \int \frac{\cos\theta_1 \cos\theta_2}{r^2} \, dA_2

Monte Carlo Methods

Test-Particle Monte Carlo Algorithm

1. SAMPLE incident particle from flux distribution at feature opening
   - Ion: from IEDF and IADF
   - Neutral: from Maxwellian
   
2. TRACE trajectory through feature
   - Ion: ballistic, solve equation of motion
   - Neutral: random walk with wall collisions
   
3. DETERMINE reaction at surface impact
   - Sample from probability distribution
   - Update surface coverage if adsorption
   
4. UPDATE surface geometry
   - Remove material (etching)
   - Add material (deposition)
   
5. REPEAT for statistically significant sample

Ion Trajectory Integration

Through the sheath/feature:

m\frac{d^2\mathbf{r}}{dt^2} = q\mathbf{E}(\mathbf{r})

Numerical integration: Velocity-Verlet or Boris algorithm

Collision Sampling

Null-collision method for efficiency:

P_{\text{collision}} = 1 - \exp(- u_{\text{max}} \Delta t)

Where $ u_{\text{max}}$ is the maximum possible collision frequency.

Multi-Scale Modeling Framework

Scale Hierarchy

Scale	Length	Time	Physics	Method
Reactor	cm–m	ms–s	Plasma transport, EM fields	Fluid PDE
Sheath	µm–mm	µs–ms	Ion acceleration, EEDF	Kinetic/Fluid
Feature	nm–µm	ns–ms	Profile evolution	Level set/MC
Atomic	Å–nm	ps–ns	Reaction mechanisms	MD/DFT

Coupling Approaches

Hierarchical (One-Way)

Atomic scale → Surface parameters
     ↓
Feature scale ← Fluxes from reactor scale
     ↓
Reactor scale → Process outputs

Concurrent (Two-Way)

Feature-scale results feed back to reactor scale
Requires iterative solution
Computationally expensive

Numerical Methods and Challenges

Stiff ODE Systems

Plasma chemistry involves timescales spanning many orders of magnitude:

Process	Timescale
Electron attachment	$\sim 10^{-10}$ s
Ion-molecule reactions	$\sim 10^{-6}$ s
Metastable decay	$\sim 10^{-3}$ s
Surface diffusion	$\sim 10^{-1}$ s

Implicit Methods Required

Backward Differentiation Formula (BDF):

y_{n+1} = \sum_{j=0}^{k-1} \alpha_j y_{n-j} + h\beta f(t_{n+1}, y_{n+1})

Spatial Discretization

Finite Volume Method

Ensures mass conservation:

\int_V \frac{\partial n}{\partial t} dV + \oint_S \mathbf{\Gamma} \cdot d\mathbf{S} = \int_V S \, dV

Mesh Requirements

Sheath resolution: $\Delta x < \lambda_D$
RF skin depth: $\Delta x < \delta$
Adaptive mesh refinement (AMR) common

EM-Plasma Coupling

Iterative scheme:

1. Solve Maxwell's equations for $\mathbf{E}$, $\mathbf{B}$ 2. Update plasma transport (density, temperature) 3. Recalculate $\sigma$, $\varepsilon_{\text{plasma}}$ 4. Repeat until convergence

Advanced Topics

Atomic Layer Etching (ALE)

Self-limiting reactions for atomic precision:

\text{EPC} = \Theta \cdot d_{\text{ML}}

Where:

EPC — Etch per cycle
$\Theta$ — Modified layer coverage fraction
$d_{\text{ML}}$ — Monolayer thickness

ALE Cycle

1. Modification step: Reactive gas creates modified surface layer $$\frac{d\Theta}{dt} = k_{\text{mod}}(1-\Theta)P_{\text{gas}}$$

2. Removal step: Ion bombardment removes modified layer only $$\text{ER} = Y_{\text{mod}}\Gamma_i\Theta$$

Pulsed Plasma Dynamics

Time-modulated RF introduces:

Active glow: Plasma on, high ion/radical generation
Afterglow: Plasma off, selective chemistry

Ion Energy Modulation

By pulsing bias:

\langle E_i \rangle = \frac{1}{T}\left[\int_0^{t_{\text{on}}} E_{\text{high}}dt + \int_{t_{\text{on}}}^{T} E_{\text{low}}dt\right]

High-Aspect-Ratio Etching (HAR)

For AR > 50 (memory, 3D NAND):

Challenges:

Ion angular broadening → bowing
Neutral depletion at bottom
Feature charging → twisting
Mask erosion → tapering

Ion Angular Distribution Broadening:

\sigma_{\text{effective}} = \sqrt{\sigma_{\text{sheath}}^2 + \sigma_{\text{scattering}}^2}

Neutral Flux at Bottom:

\Gamma_{\text{bottom}} \approx \Gamma_{\text{top}} \cdot K(\text{AR})

Machine Learning Integration

Applications:

Surrogate models for fast prediction
Process optimization (Bayesian)
Virtual metrology
Anomaly detection

Physics-Informed Neural Networks (PINNs):

\mathcal{L} = \mathcal{L}_{\text{data}} + \lambda \mathcal{L}_{\text{physics}}

Where $\mathcal{L}_{\text{physics}}$ enforces governing equations.

Validation and Experimental Techniques

Plasma Diagnostics

Technique	Measurement	Typical Values
Langmuir probe	$n_e$, $T_e$, EEDF	$10^{9}–10^{12}$ cm⁻³, 1–5 eV
OES	Relative species densities	Qualitative/semi-quantitative
APMS	Ion mass, energy	1–500 amu, 0–500 eV
LIF	Absolute radical density	$10^{11}–10^{14}$ cm⁻³
Microwave interferometry	$n_e$ (line-averaged)	$10^{10}–10^{12}$ cm⁻³

Etch Characterization

Profilometry: Etch depth, uniformity
SEM/TEM: Feature profiles, sidewall angle
XPS: Surface composition
Ellipsometry: Film thickness, optical properties

Model Validation Workflow

1. Plasma validation: Match $n_e$, $T_e$, species densities 2. Flux validation: Compare ion/neutral fluxes to wafer 3. Etch rate validation: Blanket wafer etch rates 4. Profile validation: Patterned feature cross-sections

Key Dimensionless Numbers Summary

Number	Definition	Physical Meaning
Knudsen	$\text{Kn} = \lambda/L$	Continuum vs. kinetic
Damköhler	$\text{Da} = \tau_{\text{transport}}/\tau_{\text{reaction}}$	Transport vs. reaction limited
Sticking coefficient	$\gamma = \text{reactions}/\text{collisions}$	Surface reactivity
Aspect ratio	$\text{AR} = \text{depth}/\text{width}$	Feature geometry
Debye number	$N_D = n\lambda_D^3$	Plasma ideality

Physical Constants

Constant	Symbol	Value
Elementary charge	$e$	$1.602 \times 10^{-19}$ C
Electron mass	$m_e$	$9.109 \times 10^{-31}$ kg
Proton mass	$m_p$	$1.673 \times 10^{-27}$ kg
Boltzmann constant	$k_B$	$1.381 \times 10^{-23}$ J/K
Vacuum permittivity	$\varepsilon_0$	$8.854 \times 10^{-12}$ F/m
Vacuum permeability	$\mu_0$	$4\pi \times 10^{-7}$ H/m

riereactive ion etchreactive ion etchingdry etchplasma etchetch modelingplasma physics

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