planned downtime, manufacturing operations
Planned downtime includes scheduled maintenance changeovers and other intended stops.
758 technical terms and definitions
Planned downtime includes scheduled maintenance changeovers and other intended stops.
Planned maintenance schedules preventive activities based on time condition or predictive indicators.
Scheduled downtime for maintenance.
Generate action sequences to achieve goals.
Plasma cleaning replaces wet chemical processes using ionized gases reducing waste and chemical usage.
Plasma decapsulation removes organic package materials using oxygen plasma with minimal die damage.
Concentration of ions and radicals in plasma.
RF frequency for plasma generation (13.56 MHz 27 MHz etc).
# Mathematical Modeling of Plasma Etching in Semiconductor Manufacturing ## 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 $$ \frac{\partial (nm\mathbf{u})}{\partial t} + \nabla \cdot (nm\mathbf{u}\mathbf{u}) + \nabla p = nq(\mathbf{E} + \mathbf{u} \times \mathbf{B}) - nm\nu_m \mathbf{u} $$ **Where:** - $p = nk_BT$ — Pressure - $q$ — Particle charge - $\mathbf{E}$, $\mathbf{B}$ — Electric and magnetic fields - $\nu_m$ — 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: $$ \sigma = \frac{n_e e^2}{m_e(\nu_m + i\omega)} $$ **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 - $\nu_m$ — 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: $$ \frac{d}{d\varepsilon}\left[\varepsilon D \frac{df_0}{d\varepsilon} + \left(V + \frac{\varepsilon\nu_{\text{inel}}}{\nu_m}\right)f_0\right] = 0 $$ #### 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: $$ P_{\text{collision}} = 1 - \exp(-\nu_{\text{max}} \Delta t) $$ **Where** $\nu_{\text{max}}$ 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 |
# Semiconductor Manufacturing Process: Plasma Physics Mathematical Modeling ## 1. The Physical Context Semiconductor manufacturing relies on **low-temperature, non-equilibrium plasmas** for etching and deposition. ### Key Characteristics - **Electron temperature**: $T_e \approx 1\text{–}10 \text{ eV}$ (~10,000–100,000 K) - **Ion/neutral temperature**: $T_i \approx 0.03 \text{ eV}$ (near room temperature) - **Non-equilibrium condition**: $T_e \gg T_i$ This disparity is essential—hot electrons drive chemistry while cool heavy particles preserve delicate nanoscale structures. ### Common Reactor Types - **CCP (Capacitively Coupled Plasmas)**: Used for reactive ion etching (RIE) - **ICP (Inductively Coupled Plasmas)**: High-density plasma etching - **ECR (Electron Cyclotron Resonance)**: Microwave-driven high-density sources - **Remote plasma sources**: Gentle surface treatment and cleaning ## 2. Fundamental Governing Equations ### 2.1 The Boltzmann Equation (Master Kinetic Equation) The foundation of plasma kinetic theory: $$ \frac{\partial f_s}{\partial t} + \mathbf{v} \cdot \nabla_{\mathbf{r}} f_s + \frac{q_s}{m_s}(\mathbf{E} + \mathbf{v} \times \mathbf{B}) \cdot \nabla_{\mathbf{v}} f_s = \left(\frac{\partial f_s}{\partial t}\right)_{\text{coll}} $$ Where: - $f_s(\mathbf{r}, \mathbf{v}, t)$ — Distribution function for species $s$ in 6D phase space - $q_s$ — Particle charge - $m_s$ — Particle mass - $\mathbf{E}$, $\mathbf{B}$ — Electric and magnetic fields - Right-hand side — Collision operator encoding all scattering physics ### 2.2 Fluid Approximation (Moment Equations) Taking velocity moments of the Boltzmann equation yields the fluid hierarchy: #### Continuity Equation (Zeroth Moment) $$ \frac{\partial n_s}{\partial t} + \nabla \cdot (n_s \mathbf{u}_s) = S_s $$ Where: - $n_s$ — Number density of species $s$ - $\mathbf{u}_s$ — Mean velocity - $S_s$ — Source/sink terms from chemical reactions #### Momentum Equation (First Moment) $$ m_s n_s \frac{D\mathbf{u}_s}{Dt} = q_s n_s (\mathbf{E} + \mathbf{u}_s \times \mathbf{B}) - \nabla p_s - \nabla \cdot \boldsymbol{\Pi}_s + \mathbf{R}_s $$ Where: - $p_s = n_s k_B T_s$ — Scalar pressure - $\boldsymbol{\Pi}_s$ — Viscous stress tensor - $\mathbf{R}_s$ — Momentum transfer from collisions #### Energy Equation (Second Moment) $$ \frac{\partial}{\partial t}\left(\frac{3}{2}n_s k_B T_s\right) + \nabla \cdot \mathbf{q}_s + p_s \nabla \cdot \mathbf{u}_s = Q_s $$ Where: - $\mathbf{q}_s$ — Heat flux vector - $Q_s$ — Energy source terms (heating, cooling, reactions) ### 2.3 Maxwell's Equations #### Full Electromagnetic Set $$ \nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0} = \frac{e}{\varepsilon_0}\sum_s Z_s n_s $$ $$ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} $$ $$ \nabla \cdot \mathbf{B} = 0 $$ $$ \nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t} $$ #### Electrostatic Approximation (Poisson Equation) For most processing plasmas: $$ \nabla^2 \phi = -\frac{e}{\varepsilon_0}(n_i - n_e) $$ Where $\mathbf{E} = -\nabla \phi$. ## 3. Critical Plasma Parameters ### 3.1 Debye Length The characteristic shielding scale: $$ \lambda_D = \sqrt{\frac{\varepsilon_0 k_B T_e}{n_e e^2}} $$ Numerical form: $$ \lambda_D \approx 7.43 \times 10^{3} \sqrt{\frac{T_e[\text{eV}]}{n_e[\text{m}^{-3}]}} \text{ m} $$ **Typical values**: 10–100 $\mu$m in processing plasmas. ### 3.2 Plasma Frequency The characteristic electron oscillation frequency: $$ \omega_{pe} = \sqrt{\frac{n_e e^2}{m_e \varepsilon_0}} $$ Numerical form: $$ \omega_{pe} \approx 56.4 \sqrt{n_e[\text{m}^{-3}]} \text{ rad/s} $$ ### 3.3 Collision Frequency Electron-neutral collision frequency: $$ \nu_{en} = n_g \langle \sigma_{en} v_e \rangle \approx n_g \sigma_{en} \bar{v}_e $$ Where: - $n_g$ — Neutral gas density - $\sigma_{en}$ — Collision cross-section - $\bar{v}_e = \sqrt{8 k_B T_e / \pi m_e}$ — Mean electron speed ### 3.4 Knudsen Number Determines the validity of fluid vs kinetic models: $$ \text{Kn} = \frac{\lambda_{\text{mfp}}}{L} $$ Where: - $\lambda_{\text{mfp}}$ — Mean free path - $L$ — Characteristic system length **Regimes**: - $\text{Kn} \ll 1$: Fluid models valid (collisional regime) - $\text{Kn} \gg 1$: Kinetic treatment required (collisionless regime) - $\text{Kn} \sim 1$: Transitional regime (most challenging) ## 4. Sheath Physics: The Critical Interface The **sheath** is the thin, non-neutral region where ions accelerate toward surfaces. This controls ion bombardment energy—the key parameter for anisotropic etching. ### 4.1 Bohm Criterion Ions must enter the sheath at or above the Bohm velocity: $$ u_s \geq u_B = \sqrt{\frac{k_B T_e}{m_i}} $$ This arises from requiring monotonically decreasing potential solutions. ### 4.2 Child-Langmuir Law (Collisionless Sheath) Space-charge-limited current density: $$ J = \frac{4\varepsilon_0}{9}\sqrt{\frac{2e}{m_i}}\frac{V_0^{3/2}}{s^2} $$ Where: - $J$ — Ion current density - $V_0$ — Sheath voltage - $s$ — Sheath thickness ### 4.3 Matrix Sheath Thickness For high-voltage sheaths: $$ s = \lambda_D \left(\frac{2V_0}{T_e}\right)^{1/2} $$ ### 4.4 RF Sheath Dynamics In RF plasmas, the sheath oscillates with the applied voltage, creating: - **Self-bias**: Time-averaged DC potential due to asymmetric current flow $$ V_{dc} = -V_{rf} + \frac{T_e}{e}\ln\left(\frac{m_i}{2\pi m_e}\right)^{1/2} $$ - **Ion Energy Distribution Functions (IEDF)**: Bimodal structure depending on frequency - **Stochastic heating**: Electrons gain energy from oscillating sheath boundary #### Frequency Dependence of IEDF | Condition | IEDF Shape | |-----------|------------| | $\omega \ll \omega_{pi}$ (low frequency) | Broad bimodal distribution | | $\omega \gg \omega_{pi}$ (high frequency) | Narrow peak at average energy | ## 5. Electron Energy Distribution Functions (EEDF) ### 5.1 Non-Maxwellian Distributions The EEDF is generally **not Maxwellian** in low-pressure plasmas. The two-term Boltzmann equation: $$ -\frac{d}{d\varepsilon}\left[A(\varepsilon)\frac{df}{d\varepsilon} + B(\varepsilon)f\right] = C_{\text{inel}}(f) $$ Where: - $A(\varepsilon)$, $B(\varepsilon)$ — Coefficients depending on E-field and cross-sections - $C_{\text{inel}}$ — Inelastic collision operator ### 5.2 Common Distribution Types #### Maxwellian Distribution $$ f_M(\varepsilon) = \frac{2\sqrt{\varepsilon}}{\sqrt{\pi}(k_B T_e)^{3/2}} \exp\left(-\frac{\varepsilon}{k_B T_e}\right) $$ #### Druyvesteyn Distribution (Elastic-Dominated) $$ f_D(\varepsilon) \propto \exp\left(-c\varepsilon^2\right) $$ #### Bi-Maxwellian Distribution $$ f_{bi}(\varepsilon) = \alpha f_M(\varepsilon; T_{e1}) + (1-\alpha) f_M(\varepsilon; T_{e2}) $$ ### 5.3 Rate Coefficient Calculation Reaction rates depend on the EEDF: $$ k = \langle \sigma v \rangle = \int_0^\infty \sigma(\varepsilon) v(\varepsilon) f(\varepsilon) \, d\varepsilon $$ For electron-impact reactions: $$ k_e = \sqrt{\frac{2}{m_e}} \int_0^\infty \varepsilon \, \sigma(\varepsilon) f(\varepsilon) \, d\varepsilon $$ ## 6. Plasma Chemistry Modeling ### 6.1 Species Rate Equations General form: $$ \frac{dn_i}{dt} = \sum_j k_j \prod_l n_l^{\nu_{jl}} - n_i \nu_{\text{loss}} $$ Where: - $k_j$ — Rate coefficient for reaction $j$ - $\nu_{jl}$ — Stoichiometric coefficient - $\nu_{\text{loss}}$ — Total loss frequency ### 6.2 Arrhenius Rate Coefficients For thermal reactions: $$ k(T) = A T^n \exp\left(-\frac{E_a}{k_B T}\right) $$ Where: - $A$ — Pre-exponential factor - $n$ — Temperature exponent - $E_a$ — Activation energy ### 6.3 Example: Chlorine Plasma Chemistry Simplified Cl₂ plasma reaction set: | Reaction | Type | Threshold | |----------|------|-----------| | $e + \text{Cl}_2 \rightarrow 2\text{Cl} + e$ | Dissociation | ~2.5 eV | | $e + \text{Cl}_2 \rightarrow \text{Cl}_2^+ + 2e$ | Ionization | ~11.5 eV | | $e + \text{Cl} \rightarrow \text{Cl}^+ + 2e$ | Ionization | ~13 eV | | $e + \text{Cl}^- \rightarrow \text{Cl} + 2e$ | Detachment | — | | $\text{Cl}_2^+ + e \rightarrow 2\text{Cl}$ | Dissociative recombination | — | | $\text{Cl} + \text{wall} \rightarrow \frac{1}{2}\text{Cl}_2$ | Surface recombination | — | Full models include 50+ reactions with rate constants spanning 10+ orders of magnitude. ## 7. Transport Models ### 7.1 Drift-Diffusion Approximation Standard flux expression: $$ \boldsymbol{\Gamma}_s = \text{sgn}(q_s) \mu_s n_s \mathbf{E} - D_s \nabla n_s $$ Where: - $\mu_s$ — Mobility - $D_s$ — Diffusion coefficient **Einstein Relation**: $$ \frac{D_s}{\mu_s} = \frac{k_B T_s}{|q_s|} $$ ### 7.2 Ambipolar Diffusion In quasi-neutral bulk plasma, electrons and ions diffuse together: $$ D_a = \frac{\mu_i D_e + \mu_e D_i}{\mu_e + \mu_i} $$ Since $\mu_e \gg \mu_i$: $$ D_a \approx D_i \left(1 + \frac{T_e}{T_i}\right) $$ ### 7.3 Tensor Transport (Magnetized Plasmas) In magnetic fields, transport becomes anisotropic: $$ \boldsymbol{\Gamma} = -\mathbf{D} \cdot \nabla n + n \boldsymbol{\mu} \cdot \mathbf{E} $$ The diffusion tensor has components: - **Parallel**: $D_\parallel = D_0$ - **Perpendicular**: $D_\perp = \frac{D_0}{1 + \omega_c^2 \tau^2}$ - **Hall**: $D_H = \frac{\omega_c \tau D_0}{1 + \omega_c^2 \tau^2}$ Where $\omega_c = qB/m$ is the cyclotron frequency. ## 8. Computational Approaches ### 8.1 Hierarchy of Models | Model | Dimensions | Physics Captured | Typical Runtime | |-------|------------|------------------|-----------------| | Global (0D) | Volume-averaged | Detailed chemistry | Seconds | | Fluid (1D-3D) | Spatial resolution | Transport + chemistry | Minutes–Hours | | PIC-MCC | Full phase space | Kinetic ions/electrons | Days–Weeks | | Hybrid | Mixed | Fluid electrons + kinetic ions | Hours–Days | ### 8.2 Fluid Model Implementation Solve the coupled system: 1. **Species continuity equations** (one per species) 2. **Electron energy equation** 3. **Poisson equation** 4. **Momentum equations** (often drift-diffusion limit) #### Numerical Challenges - **Nonlinear coupling**: Exponential dependence of source terms on $T_e$ - **Disparate timescales**: - Electron dynamics: ~ns - Ion dynamics: ~$\mu$s - Chemistry: ~ms - **Spatial scales**: Sheath ($\lambda_D \sim 100$ $\mu$m) vs reactor (~0.1 m) #### Common Numerical Techniques - Semi-implicit time stepping - Scharfetter-Gummel discretization for drift-diffusion fluxes - Multigrid Poisson solvers - Adaptive mesh refinement near sheaths ### 8.3 Particle-in-Cell with Monte Carlo Collisions (PIC-MCC) #### Algorithm Steps 1. **Push particles** using equations of motion: $$ \frac{d\mathbf{x}}{dt} = \mathbf{v}, \quad m\frac{d\mathbf{v}}{dt} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B}) $$ 2. **Deposit charge** onto computational grid 3. **Solve Poisson** equation for electric field 4. **Interpolate field** back to particle positions 5. **Monte Carlo collisions** based on cross-sections #### Applications - Low-pressure kinetic regimes - IEDF predictions - Non-local electron kinetics - Detailed sheath physics #### Computational Cost Scales as $O(N_p \log N_p)$ per timestep, with $N_p \sim 10^6\text{–}10^8$ superparticles. ## 9. Multi-Scale Coupling: The Grand Challenge ### 9.1 Scale Hierarchy | Scale | Phenomenon | Typical Model | |-------|------------|---------------| | Å–nm | Surface reactions, damage | MD, DFT | | nm–$\mu$m | Feature evolution | Level-set, Monte Carlo | | $\mu$m–mm | Sheath, transport | Fluid/kinetic plasma | | mm–m | Reactor, gas flow | CFD + plasma | ### 9.2 Feature-Scale Modeling #### Level-Set Method Track the evolving surface $\phi = 0$: $$ \frac{\partial \phi}{\partial t} + V_n |\nabla \phi| = 0 $$ Where $V_n$ is the local etch/deposition rate depending on: - Ion flux $\Gamma_i$ and energy $\varepsilon_i$ from plasma model - Neutral radical flux $\Gamma_n$ - Surface composition and local geometry - Angle-dependent yields $Y(\theta, \varepsilon)$ #### Etch Rate Model $$ R = Y_0 \Gamma_i f(\varepsilon) + k_s \Gamma_n \theta_s $$ Where: - $Y_0$ — Base sputter yield - $f(\varepsilon)$ — Energy-dependent yield function - $k_s$ — Surface reaction rate - $\theta_s$ — Surface coverage ### 9.3 Aspect Ratio Dependent Etching (ARDE) $$ \frac{R_{\text{bottom}}}{R_{\text{top}}} = f(\text{AR}) $$ #### Physical Mechanisms - Ion angular distribution effects (Knudsen diffusion in feature) - Neutral transport limitations - Differential charging in high-aspect-ratio features - Sidewall passivation dynamics ## 10. Electromagnetic Effects in High-Density Sources ### 10.1 ICP Power Deposition The RF magnetic field induces an electric field: $$ \nabla \times \mathbf{E} = -i\omega \mathbf{B} $$ Power deposition density: $$ P = \frac{1}{2}\text{Re}(\mathbf{J}^* \cdot \mathbf{E}) = \frac{1}{2}\text{Re}(\sigma_p)|\mathbf{E}|^2 $$ ### 10.2 Plasma Conductivity $$ \sigma_p = \frac{n_e e^2}{m_e(\nu_m + i\omega)} $$ Where: - $\nu_m$ — Electron momentum transfer collision frequency - $\omega$ — RF angular frequency ### 10.3 Skin Depth Electromagnetic field penetration depth: $$ \delta = \sqrt{\frac{2}{\omega \mu_0 \text{Re}(\sigma_p)}} $$ **Typical values**: $\delta \approx 1\text{–}3$ cm, creating non-uniform power deposition. ### 10.4 E-to-H Mode Transition ICPs exhibit hysteresis behavior: - **E-mode** (low power): Capacitive coupling, low plasma density - **H-mode** (high power): Inductive coupling, high plasma density The transition involves bifurcation in the coupled power-density equations. ## 11. Surface Reaction Modeling ### 11.1 Surface Reaction Mechanisms #### Langmuir-Hinshelwood Mechanism Both reactants adsorbed: $$ R = k \theta_A \theta_B $$ #### Eley-Rideal Mechanism One reactant from gas phase: $$ R = k P_A \theta_B $$ #### Surface Coverage Dynamics $$ \frac{d\theta}{dt} = k_{\text{ads}}P(1-\theta) - k_{\text{des}}\theta - k_{\text{react}}\theta $$ ### 11.2 Kinetic Monte Carlo (KMC) For atomic-scale surface evolution: 1. Catalog all possible events with rates $\{k_i\}$ 2. Calculate total rate: $k_{\text{tot}} = \sum_i k_i$ 3. Time advance: $\Delta t = -\ln(r_1)/k_{\text{tot}}$ 4. Select event $j$ probabilistically 5. Execute event and update configuration ### 11.3 Molecular Dynamics for Ion-Surface Interactions Newton's equations with empirical potentials: $$ m_i \frac{d^2 \mathbf{r}_i}{dt^2} = -\nabla_i U(\{\mathbf{r}\}) $$ **Potentials used**: - Stillinger-Weber (Si) - Tersoff (C, Si, Ge) - ReaxFF (reactive systems) **Outputs**: - Sputter yields $Y(\varepsilon, \theta)$ - Damage depth profiles - Reaction probabilities ## 12. Emerging Mathematical Methods ### 12.1 Machine Learning in Plasma Modeling - **Surrogate models**: Neural networks for real-time prediction - **Reduced-order models**: POD/DMD for parametric studies - **Inverse problems**: Inferring plasma parameters from sensor data ### 12.2 Uncertainty Quantification Given uncertainties in input parameters: - Cross-section data (~20–50% uncertainty) - Surface reaction coefficients - Boundary conditions **Propagation methods**: - Polynomial chaos expansions - Monte Carlo sampling - Sensitivity analysis (Sobol indices) ### 12.3 Data-Driven Closures Learning moment closures from kinetic data: $$ \mathbf{q} = \mathcal{F}_\theta(n, \mathbf{u}, T, \nabla T, \ldots) $$ Where $\mathcal{F}_\theta$ is a neural network trained on PIC simulation data. ## 13. Key Dimensionless Groups | Parameter | Definition | Significance | |-----------|------------|--------------| | $\Lambda = L/\lambda_D$ | System size / Debye length | Plasma character ($\gg 1$ for quasi-neutrality) | | $\omega/\nu_m$ | Frequency / collision rate | Collisional vs collisionless | | $\omega/\omega_{pe}$ | Frequency / plasma frequency | Wave propagation regime | | $r_L/L$ | Larmor radius / system size | Degree of magnetization | | $\text{Kn} = \lambda/L$ | Mean free path / system size | Fluid vs kinetic regime | | $\text{Re}_m$ | Magnetic Reynolds number | Magnetic field diffusion | ## 14. Example: Complete CCP Model ### 14.1 Governing Equations (1D) #### Electron Continuity $$ \frac{\partial n_e}{\partial t} + \frac{\partial \Gamma_e}{\partial x} = k_{\text{iz}} n_e n_g - k_{\text{att}} n_e n_g $$ #### Electron Flux $$ \Gamma_e = -\mu_e n_e E - D_e \frac{\partial n_e}{\partial x} $$ #### Ion Continuity $$ \frac{\partial n_i}{\partial t} + \frac{\partial \Gamma_i}{\partial x} = k_{\text{iz}} n_e n_g $$ #### Electron Energy Density $$ \frac{\partial n_\varepsilon}{\partial t} + \frac{\partial \Gamma_\varepsilon}{\partial x} + e\Gamma_e E = -\sum_j n_e n_g k_j \varepsilon_j $$ #### Poisson Equation $$ \frac{\partial^2 \phi}{\partial x^2} = -\frac{e}{\varepsilon_0}(n_i - n_e) $$ ### 14.2 Boundary Conditions At electrodes ($x = 0, L$): - **Potential**: $\phi(0,t) = V_{\text{rf}}\sin(\omega t)$, $\phi(L,t) = 0$ - **Secondary emission**: $\Gamma_e = \gamma \Gamma_i$ (with $\gamma \approx 0.1$) - **Kinetic fluxes**: Derived from distribution function at boundary ### 14.3 Numerical Parameters | Parameter | Typical Value | |-----------|---------------| | Grid points | ~1000 | | Species | ~10 | | RF cycles to steady state | $10^5\text{–}10^6$ | | Time step | $\Delta t < 0.1/\omega_{pe}$ | ## Summary The mathematical modeling of plasmas in semiconductor manufacturing represents a magnificent multi-physics, multi-scale scientific endeavor requiring: 1. **Kinetic theory** for non-equilibrium particle distributions 2. **Fluid mechanics** for macroscopic transport 3. **Electromagnetism** for field and power coupling 4. **Chemical kinetics** for reactive processes 5. **Surface science** for etch/deposition mechanisms 6. **Numerical analysis** for efficient computation 7. **Uncertainty quantification** for predictive capability The field continues to advance with machine learning integration, exascale computing enabling full 3D kinetic simulations, and tighter coupling between atomic-scale and reactor-scale models—driven by the relentless progression toward smaller feature sizes and novel materials in semiconductor technology.
# Semiconductor Manufacturing Process: Plasma Physics Mathematical Modeling ## 1. The Physical Context Semiconductor manufacturing relies on **low-temperature, non-equilibrium plasmas** for etching and deposition. ### Key Characteristics - **Electron temperature**: $T_e \approx 1\text{–}10 \text{ eV}$ (~10,000–100,000 K) - **Ion/neutral temperature**: $T_i \approx 0.03 \text{ eV}$ (near room temperature) - **Non-equilibrium condition**: $T_e \gg T_i$ This disparity is essential—hot electrons drive chemistry while cool heavy particles preserve delicate nanoscale structures. ### Common Reactor Types - **CCP (Capacitively Coupled Plasmas)**: Used for reactive ion etching (RIE) - **ICP (Inductively Coupled Plasmas)**: High-density plasma etching - **ECR (Electron Cyclotron Resonance)**: Microwave-driven high-density sources - **Remote plasma sources**: Gentle surface treatment and cleaning ## 2. Fundamental Governing Equations ### 2.1 The Boltzmann Equation (Master Kinetic Equation) The foundation of plasma kinetic theory: $$ \frac{\partial f_s}{\partial t} + \mathbf{v} \cdot \nabla_{\mathbf{r}} f_s + \frac{q_s}{m_s}(\mathbf{E} + \mathbf{v} \times \mathbf{B}) \cdot \nabla_{\mathbf{v}} f_s = \left(\frac{\partial f_s}{\partial t}\right)_{\text{coll}} $$ Where: - $f_s(\mathbf{r}, \mathbf{v}, t)$ — Distribution function for species $s$ in 6D phase space - $q_s$ — Particle charge - $m_s$ — Particle mass - $\mathbf{E}$, $\mathbf{B}$ — Electric and magnetic fields - Right-hand side — Collision operator encoding all scattering physics ### 2.2 Fluid Approximation (Moment Equations) Taking velocity moments of the Boltzmann equation yields the fluid hierarchy: #### Continuity Equation (Zeroth Moment) $$ \frac{\partial n_s}{\partial t} + \nabla \cdot (n_s \mathbf{u}_s) = S_s $$ Where: - $n_s$ — Number density of species $s$ - $\mathbf{u}_s$ — Mean velocity - $S_s$ — Source/sink terms from chemical reactions #### Momentum Equation (First Moment) $$ m_s n_s \frac{D\mathbf{u}_s}{Dt} = q_s n_s (\mathbf{E} + \mathbf{u}_s \times \mathbf{B}) - \nabla p_s - \nabla \cdot \boldsymbol{\Pi}_s + \mathbf{R}_s $$ Where: - $p_s = n_s k_B T_s$ — Scalar pressure - $\boldsymbol{\Pi}_s$ — Viscous stress tensor - $\mathbf{R}_s$ — Momentum transfer from collisions #### Energy Equation (Second Moment) $$ \frac{\partial}{\partial t}\left(\frac{3}{2}n_s k_B T_s\right) + \nabla \cdot \mathbf{q}_s + p_s \nabla \cdot \mathbf{u}_s = Q_s $$ Where: - $\mathbf{q}_s$ — Heat flux vector - $Q_s$ — Energy source terms (heating, cooling, reactions) ### 2.3 Maxwell's Equations #### Full Electromagnetic Set $$ \nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0} = \frac{e}{\varepsilon_0}\sum_s Z_s n_s $$ $$ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} $$ $$ \nabla \cdot \mathbf{B} = 0 $$ $$ \nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t} $$ #### Electrostatic Approximation (Poisson Equation) For most processing plasmas: $$ \nabla^2 \phi = -\frac{e}{\varepsilon_0}(n_i - n_e) $$ Where $\mathbf{E} = -\nabla \phi$. ## 3. Critical Plasma Parameters ### 3.1 Debye Length The characteristic shielding scale: $$ \lambda_D = \sqrt{\frac{\varepsilon_0 k_B T_e}{n_e e^2}} $$ Numerical form: $$ \lambda_D \approx 7.43 \times 10^{3} \sqrt{\frac{T_e[\text{eV}]}{n_e[\text{m}^{-3}]}} \text{ m} $$ **Typical values**: 10–100 μm in processing plasmas. ### 3.2 Plasma Frequency The characteristic electron oscillation frequency: $$ \omega_{pe} = \sqrt{\frac{n_e e^2}{m_e \varepsilon_0}} $$ Numerical form: $$ \omega_{pe} \approx 56.4 \sqrt{n_e[\text{m}^{-3}]} \text{ rad/s} $$ ### 3.3 Collision Frequency Electron-neutral collision frequency: $$ \nu_{en} = n_g \langle \sigma_{en} v_e \rangle \approx n_g \sigma_{en} \bar{v}_e $$ Where: - $n_g$ — Neutral gas density - $\sigma_{en}$ — Collision cross-section - $\bar{v}_e = \sqrt{8 k_B T_e / \pi m_e}$ — Mean electron speed ### 3.4 Knudsen Number Determines the validity of fluid vs kinetic models: $$ \text{Kn} = \frac{\lambda_{\text{mfp}}}{L} $$ Where: - $\lambda_{\text{mfp}}$ — Mean free path - $L$ — Characteristic system length **Regimes**: - $\text{Kn} \ll 1$: Fluid models valid (collisional regime) - $\text{Kn} \gg 1$: Kinetic treatment required (collisionless regime) - $\text{Kn} \sim 1$: Transitional regime (most challenging) ## 4. Sheath Physics: The Critical Interface The **sheath** is the thin, non-neutral region where ions accelerate toward surfaces. This controls ion bombardment energy—the key parameter for anisotropic etching. ### 4.1 Bohm Criterion Ions must enter the sheath at or above the Bohm velocity: $$ u_s \geq u_B = \sqrt{\frac{k_B T_e}{m_i}} $$ This arises from requiring monotonically decreasing potential solutions. ### 4.2 Child-Langmuir Law (Collisionless Sheath) Space-charge-limited current density: $$ J = \frac{4\varepsilon_0}{9}\sqrt{\frac{2e}{m_i}}\frac{V_0^{3/2}}{s^2} $$ Where: - $J$ — Ion current density - $V_0$ — Sheath voltage - $s$ — Sheath thickness ### 4.3 Matrix Sheath Thickness For high-voltage sheaths: $$ s = \lambda_D \left(\frac{2V_0}{T_e}\right)^{1/2} $$ ### 4.4 RF Sheath Dynamics In RF plasmas, the sheath oscillates with the applied voltage, creating: - **Self-bias**: Time-averaged DC potential due to asymmetric current flow $$ V_{dc} = -V_{rf} + \frac{T_e}{e}\ln\left(\frac{m_i}{2\pi m_e}\right)^{1/2} $$ - **Ion Energy Distribution Functions (IEDF)**: Bimodal structure depending on frequency - **Stochastic heating**: Electrons gain energy from oscillating sheath boundary #### Frequency Dependence of IEDF | Condition | IEDF Shape | |-----------|------------| | $\omega \ll \omega_{pi}$ (low frequency) | Broad bimodal distribution | | $\omega \gg \omega_{pi}$ (high frequency) | Narrow peak at average energy | ## 5. Electron Energy Distribution Functions (EEDF) ### 5.1 Non-Maxwellian Distributions The EEDF is generally **not Maxwellian** in low-pressure plasmas. The two-term Boltzmann equation: $$ -\frac{d}{d\varepsilon}\left[A(\varepsilon)\frac{df}{d\varepsilon} + B(\varepsilon)f\right] = C_{\text{inel}}(f) $$ Where: - $A(\varepsilon)$, $B(\varepsilon)$ — Coefficients depending on E-field and cross-sections - $C_{\text{inel}}$ — Inelastic collision operator ### 5.2 Common Distribution Types #### Maxwellian Distribution $$ f_M(\varepsilon) = \frac{2\sqrt{\varepsilon}}{\sqrt{\pi}(k_B T_e)^{3/2}} \exp\left(-\frac{\varepsilon}{k_B T_e}\right) $$ #### Druyvesteyn Distribution (Elastic-Dominated) $$ f_D(\varepsilon) \propto \exp\left(-c\varepsilon^2\right) $$ #### Bi-Maxwellian Distribution $$ f_{bi}(\varepsilon) = \alpha f_M(\varepsilon; T_{e1}) + (1-\alpha) f_M(\varepsilon; T_{e2}) $$ ### 5.3 Rate Coefficient Calculation Reaction rates depend on the EEDF: $$ k = \langle \sigma v \rangle = \int_0^\infty \sigma(\varepsilon) v(\varepsilon) f(\varepsilon) \, d\varepsilon $$ For electron-impact reactions: $$ k_e = \sqrt{\frac{2}{m_e}} \int_0^\infty \varepsilon \, \sigma(\varepsilon) f(\varepsilon) \, d\varepsilon $$ ## 6. Plasma Chemistry Modeling ### 6.1 Species Rate Equations General form: $$ \frac{dn_i}{dt} = \sum_j k_j \prod_l n_l^{\nu_{jl}} - n_i \nu_{\text{loss}} $$ Where: - $k_j$ — Rate coefficient for reaction $j$ - $\nu_{jl}$ — Stoichiometric coefficient - $\nu_{\text{loss}}$ — Total loss frequency ### 6.2 Arrhenius Rate Coefficients For thermal reactions: $$ k(T) = A T^n \exp\left(-\frac{E_a}{k_B T}\right) $$ Where: - $A$ — Pre-exponential factor - $n$ — Temperature exponent - $E_a$ — Activation energy ### 6.3 Example: Chlorine Plasma Chemistry Simplified Cl₂ plasma reaction set: | Reaction | Type | Threshold | |----------|------|-----------| | $e + \text{Cl}_2 \rightarrow 2\text{Cl} + e$ | Dissociation | ~2.5 eV | | $e + \text{Cl}_2 \rightarrow \text{Cl}_2^+ + 2e$ | Ionization | ~11.5 eV | | $e + \text{Cl} \rightarrow \text{Cl}^+ + 2e$ | Ionization | ~13 eV | | $e + \text{Cl}^- \rightarrow \text{Cl} + 2e$ | Detachment | — | | $\text{Cl}_2^+ + e \rightarrow 2\text{Cl}$ | Dissociative recombination | — | | $\text{Cl} + \text{wall} \rightarrow \frac{1}{2}\text{Cl}_2$ | Surface recombination | — | Full models include 50+ reactions with rate constants spanning 10+ orders of magnitude. ## 7. Transport Models ### 7.1 Drift-Diffusion Approximation Standard flux expression: $$ \boldsymbol{\Gamma}_s = \text{sgn}(q_s) \mu_s n_s \mathbf{E} - D_s \nabla n_s $$ Where: - $\mu_s$ — Mobility - $D_s$ — Diffusion coefficient **Einstein Relation**: $$ \frac{D_s}{\mu_s} = \frac{k_B T_s}{|q_s|} $$ ### 7.2 Ambipolar Diffusion In quasi-neutral bulk plasma, electrons and ions diffuse together: $$ D_a = \frac{\mu_i D_e + \mu_e D_i}{\mu_e + \mu_i} $$ Since $\mu_e \gg \mu_i$: $$ D_a \approx D_i \left(1 + \frac{T_e}{T_i}\right) $$ ### 7.3 Tensor Transport (Magnetized Plasmas) In magnetic fields, transport becomes anisotropic: $$ \boldsymbol{\Gamma} = -\mathbf{D} \cdot \nabla n + n \boldsymbol{\mu} \cdot \mathbf{E} $$ The diffusion tensor has components: - **Parallel**: $D_\parallel = D_0$ - **Perpendicular**: $D_\perp = \frac{D_0}{1 + \omega_c^2 \tau^2}$ - **Hall**: $D_H = \frac{\omega_c \tau D_0}{1 + \omega_c^2 \tau^2}$ Where $\omega_c = qB/m$ is the cyclotron frequency. ## 8. Computational Approaches ### 8.1 Hierarchy of Models | Model | Dimensions | Physics Captured | Typical Runtime | |-------|------------|------------------|-----------------| | Global (0D) | Volume-averaged | Detailed chemistry | Seconds | | Fluid (1D-3D) | Spatial resolution | Transport + chemistry | Minutes–Hours | | PIC-MCC | Full phase space | Kinetic ions/electrons | Days–Weeks | | Hybrid | Mixed | Fluid electrons + kinetic ions | Hours–Days | ### 8.2 Fluid Model Implementation Solve the coupled system: 1. **Species continuity equations** (one per species) 2. **Electron energy equation** 3. **Poisson equation** 4. **Momentum equations** (often drift-diffusion limit) #### Numerical Challenges - **Nonlinear coupling**: Exponential dependence of source terms on $T_e$ - **Disparate timescales**: - Electron dynamics: ~ns - Ion dynamics: ~μs - Chemistry: ~ms - **Spatial scales**: Sheath ($\lambda_D \sim 100$ μm) vs reactor (~0.1 m) #### Common Numerical Techniques - Semi-implicit time stepping - Scharfetter-Gummel discretization for drift-diffusion fluxes - Multigrid Poisson solvers - Adaptive mesh refinement near sheaths ### 8.3 Particle-in-Cell with Monte Carlo Collisions (PIC-MCC) #### Algorithm Steps 1. **Push particles** using equations of motion: $$ \frac{d\mathbf{x}}{dt} = \mathbf{v}, \quad m\frac{d\mathbf{v}}{dt} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B}) $$ 2. **Deposit charge** onto computational grid 3. **Solve Poisson** equation for electric field 4. **Interpolate field** back to particle positions 5. **Monte Carlo collisions** based on cross-sections #### Applications - Low-pressure kinetic regimes - IEDF predictions - Non-local electron kinetics - Detailed sheath physics #### Computational Cost Scales as $O(N_p \log N_p)$ per timestep, with $N_p \sim 10^6\text{–}10^8$ superparticles. ## 9. Multi-Scale Coupling: The Grand Challenge ### 9.1 Scale Hierarchy | Scale | Phenomenon | Typical Model | |-------|------------|---------------| | Å–nm | Surface reactions, damage | MD, DFT | | nm–μm | Feature evolution | Level-set, Monte Carlo | | μm–mm | Sheath, transport | Fluid/kinetic plasma | | mm–m | Reactor, gas flow | CFD + plasma | ### 9.2 Feature-Scale Modeling #### Level-Set Method Track the evolving surface $\phi = 0$: $$ \frac{\partial \phi}{\partial t} + V_n |\nabla \phi| = 0 $$ Where $V_n$ is the local etch/deposition rate depending on: - Ion flux $\Gamma_i$ and energy $\varepsilon_i$ from plasma model - Neutral radical flux $\Gamma_n$ - Surface composition and local geometry - Angle-dependent yields $Y(\theta, \varepsilon)$ #### Etch Rate Model $$ R = Y_0 \Gamma_i f(\varepsilon) + k_s \Gamma_n \theta_s $$ Where: - $Y_0$ — Base sputter yield - $f(\varepsilon)$ — Energy-dependent yield function - $k_s$ — Surface reaction rate - $\theta_s$ — Surface coverage ### 9.3 Aspect Ratio Dependent Etching (ARDE) $$ \frac{R_{\text{bottom}}}{R_{\text{top}}} = f(\text{AR}) $$ #### Physical Mechanisms - Ion angular distribution effects (Knudsen diffusion in feature) - Neutral transport limitations - Differential charging in high-aspect-ratio features - Sidewall passivation dynamics ## 10. Electromagnetic Effects in High-Density Sources ### 10.1 ICP Power Deposition The RF magnetic field induces an electric field: $$ \nabla \times \mathbf{E} = -i\omega \mathbf{B} $$ Power deposition density: $$ P = \frac{1}{2}\text{Re}(\mathbf{J}^* \cdot \mathbf{E}) = \frac{1}{2}\text{Re}(\sigma_p)|\mathbf{E}|^2 $$ ### 10.2 Plasma Conductivity $$ \sigma_p = \frac{n_e e^2}{m_e(\nu_m + i\omega)} $$ Where: - $\nu_m$ — Electron momentum transfer collision frequency - $\omega$ — RF angular frequency ### 10.3 Skin Depth Electromagnetic field penetration depth: $$ \delta = \sqrt{\frac{2}{\omega \mu_0 \text{Re}(\sigma_p)}} $$ **Typical values**: $\delta \approx 1\text{–}3$ cm, creating non-uniform power deposition. ### 10.4 E-to-H Mode Transition ICPs exhibit hysteresis behavior: - **E-mode** (low power): Capacitive coupling, low plasma density - **H-mode** (high power): Inductive coupling, high plasma density The transition involves bifurcation in the coupled power-density equations. ## 11. Surface Reaction Modeling ### 11.1 Surface Reaction Mechanisms #### Langmuir-Hinshelwood Mechanism Both reactants adsorbed: $$ R = k \theta_A \theta_B $$ #### Eley-Rideal Mechanism One reactant from gas phase: $$ R = k P_A \theta_B $$ #### Surface Coverage Dynamics $$ \frac{d\theta}{dt} = k_{\text{ads}}P(1-\theta) - k_{\text{des}}\theta - k_{\text{react}}\theta $$ ### 11.2 Kinetic Monte Carlo (KMC) For atomic-scale surface evolution: 1. Catalog all possible events with rates $\{k_i\}$ 2. Calculate total rate: $k_{\text{tot}} = \sum_i k_i$ 3. Time advance: $\Delta t = -\ln(r_1)/k_{\text{tot}}$ 4. Select event $j$ probabilistically 5. Execute event and update configuration ### 11.3 Molecular Dynamics for Ion-Surface Interactions Newton's equations with empirical potentials: $$ m_i \frac{d^2 \mathbf{r}_i}{dt^2} = -\nabla_i U(\{\mathbf{r}\}) $$ **Potentials used**: - Stillinger-Weber (Si) - Tersoff (C, Si, Ge) - ReaxFF (reactive systems) **Outputs**: - Sputter yields $Y(\varepsilon, \theta)$ - Damage depth profiles - Reaction probabilities ## 12. Emerging Mathematical Methods ### 12.1 Machine Learning in Plasma Modeling - **Surrogate models**: Neural networks for real-time prediction - **Reduced-order models**: POD/DMD for parametric studies - **Inverse problems**: Inferring plasma parameters from sensor data ### 12.2 Uncertainty Quantification Given uncertainties in input parameters: - Cross-section data (~20–50% uncertainty) - Surface reaction coefficients - Boundary conditions **Propagation methods**: - Polynomial chaos expansions - Monte Carlo sampling - Sensitivity analysis (Sobol indices) ### 12.3 Data-Driven Closures Learning moment closures from kinetic data: $$ \mathbf{q} = \mathcal{F}_\theta(n, \mathbf{u}, T, \nabla T, \ldots) $$ Where $\mathcal{F}_\theta$ is a neural network trained on PIC simulation data. ## 13. Key Dimensionless Groups | Parameter | Definition | Significance | |-----------|------------|--------------| | $\Lambda = L/\lambda_D$ | System size / Debye length | Plasma character ($\gg 1$ for quasi-neutrality) | | $\omega/\nu_m$ | Frequency / collision rate | Collisional vs collisionless | | $\omega/\omega_{pe}$ | Frequency / plasma frequency | Wave propagation regime | | $r_L/L$ | Larmor radius / system size | Degree of magnetization | | $\text{Kn} = \lambda/L$ | Mean free path / system size | Fluid vs kinetic regime | | $\text{Re}_m$ | Magnetic Reynolds number | Magnetic field diffusion | ## 14. Example: Complete CCP Model ### 14.1 Governing Equations (1D) #### Electron Continuity $$ \frac{\partial n_e}{\partial t} + \frac{\partial \Gamma_e}{\partial x} = k_{\text{iz}} n_e n_g - k_{\text{att}} n_e n_g $$ #### Electron Flux $$ \Gamma_e = -\mu_e n_e E - D_e \frac{\partial n_e}{\partial x} $$ #### Ion Continuity $$ \frac{\partial n_i}{\partial t} + \frac{\partial \Gamma_i}{\partial x} = k_{\text{iz}} n_e n_g $$ #### Electron Energy Density $$ \frac{\partial n_\varepsilon}{\partial t} + \frac{\partial \Gamma_\varepsilon}{\partial x} + e\Gamma_e E = -\sum_j n_e n_g k_j \varepsilon_j $$ #### Poisson Equation $$ \frac{\partial^2 \phi}{\partial x^2} = -\frac{e}{\varepsilon_0}(n_i - n_e) $$ ### 14.2 Boundary Conditions At electrodes ($x = 0, L$): - **Potential**: $\phi(0,t) = V_{\text{rf}}\sin(\omega t)$, $\phi(L,t) = 0$ - **Secondary emission**: $\Gamma_e = \gamma \Gamma_i$ (with $\gamma \approx 0.1$) - **Kinetic fluxes**: Derived from distribution function at boundary ### 14.3 Numerical Parameters | Parameter | Typical Value | |-----------|---------------| | Grid points | ~1000 | | Species | ~10 | | RF cycles to steady state | $10^5\text{–}10^6$ | | Time step | $\Delta t < 0.1/\omega_{pe}$ | ## Summary The mathematical modeling of plasmas in semiconductor manufacturing represents a magnificent multi-physics, multi-scale scientific endeavor requiring: 1. **Kinetic theory** for non-equilibrium particle distributions 2. **Fluid mechanics** for macroscopic transport 3. **Electromagnetism** for field and power coupling 4. **Chemical kinetics** for reactive processes 5. **Surface science** for etch/deposition mechanisms 6. **Numerical analysis** for efficient computation 7. **Uncertainty quantification** for predictive capability The field continues to advance with machine learning integration, exascale computing enabling full 3D kinetic simulations, and tighter coupling between atomic-scale and reactor-scale models—driven by the relentless progression toward smaller feature sizes and novel materials in semiconductor technology.
# Semiconductor Manufacturing Plasma Science ## Overview This document covers the physics, chemistry, and engineering of plasma processes in semiconductor manufacturing—the foundation of modern chip fabrication. ## 1. Fundamentals of Plasma Physics ### 1.1 What is Plasma? Plasma is the **fourth state of matter**—an ionized gas containing: - Free electrons ($e^-$) - Positive ions ($\text{Ar}^+$, $\text{Cl}^+$, $\text{F}^+$, etc.) - Neutral species (atoms, molecules, radicals) In semiconductor processing, we use **non-equilibrium** or **cold** plasmas where: $$ T_e \gg T_i \approx T_n \approx T_{\text{room}} $$ Where: - $T_e$ = electron temperature (~1–10 eV, equivalent to $10^4$–$10^5$ K) - $T_i$ = ion temperature (~0.025–0.1 eV) - $T_n$ = neutral temperature (~300 K) This asymmetry allows chemically reactive species to be generated without thermally damaging the substrate. ### 1.2 Key Plasma Parameters | Parameter | Symbol | Typical Value | Description | |-----------|--------|---------------|-------------| | Electron density | $n_e$ | $10^9$–$10^{12}$ cm$^{-3}$ | Number of electrons per unit volume | | Electron temperature | $T_e$ | 1–10 eV | Mean kinetic energy of electrons | | Ion temperature | $T_i$ | 0.025–0.1 eV | Mean kinetic energy of ions | | Debye length | $\lambda_D$ | 10–100 μm | Characteristic shielding distance | | Plasma frequency | $\omega_{pe}$ | ~GHz | Characteristic oscillation frequency | ### 1.3 Debye Length The **Debye length** characterizes the distance over which charge separation can occur: $$ \lambda_D = \sqrt{\frac{\varepsilon_0 k_B T_e}{n_e e^2}} $$ Where: - $\varepsilon_0$ = permittivity of free space ($8.85 \times 10^{-12}$ F/m) - $k_B$ = Boltzmann constant ($1.38 \times 10^{-23}$ J/K) - $T_e$ = electron temperature (K) - $n_e$ = electron density (m$^{-3}$) - $e$ = electron charge ($1.6 \times 10^{-19}$ C) ### 1.4 Plasma Frequency The **plasma frequency** is the natural oscillation frequency of electrons: $$ \omega_{pe} = \sqrt{\frac{n_e e^2}{\varepsilon_0 m_e}} $$ Or in practical units: $$ f_{pe} \approx 9 \sqrt{n_e} \text{ Hz} \quad \text{(with } n_e \text{ in m}^{-3}\text{)} $$ ## 2. The Plasma Sheath ### 2.1 Sheath Formation The **plasma sheath** is the most critical region for semiconductor processing. At any surface in contact with plasma: 1. Electrons (lighter, faster) escape more readily than ions 2. A positive space charge region forms adjacent to the surface 3. This creates a potential drop that accelerates ions toward the substrate ### 2.2 Sheath Potential The **Bohm criterion** requires ions entering the sheath to have a minimum velocity: $$ v_{\text{Bohm}} = \sqrt{\frac{k_B T_e}{M_i}} $$ Where $M_i$ is the ion mass. The **floating potential** (potential of an isolated surface) is approximately: $$ V_f \approx -\frac{k_B T_e}{2e} \ln\left(\frac{M_i}{2\pi m_e}\right) $$ For argon plasma with $T_e = 3$ eV: $$ V_f \approx -15 \text{ V} $$ ### 2.3 Child-Langmuir Law The **ion current density** through a collisionless sheath is given by: $$ J_i = \frac{4\varepsilon_0}{9} \sqrt{\frac{2e}{M_i}} \frac{V^{3/2}}{d^2} $$ Where: - $V$ = sheath voltage - $d$ = sheath thickness ### 2.4 Sheath Thickness The sheath thickness scales approximately as: $$ s \approx \lambda_D \left(\frac{2eV_s}{k_B T_e}\right)^{3/4} $$ Where $V_s$ is the sheath voltage. ## 3. Plasma Etching ### 3.1 Etching Mechanisms Three primary mechanisms contribute to plasma etching: 1. **Chemical etching** (isotropic): $$ \text{Rate}_{\text{chem}} \propto \Gamma_n \cdot S \cdot \exp\left(-\frac{E_a}{k_B T_s}\right) $$ Where $\Gamma_n$ is neutral flux, $S$ is sticking coefficient, $E_a$ is activation energy 2. **Physical sputtering** (anisotropic): $$ Y(E) = \frac{0.042 \cdot Q \cdot \alpha^* \cdot S_n(E)}{U_s} $$ Where $Y$ is sputter yield, $E$ is ion energy, $U_s$ is surface binding energy 3. **Ion-enhanced etching** (synergistic): $$ \text{Rate}_{\text{total}} > \text{Rate}_{\text{chem}} + \text{Rate}_{\text{phys}} $$ ### 3.2 Etch Rate Equation A general expression for ion-enhanced etch rate: $$ \text{ER} = \frac{1}{n} \left[ k_s \Gamma_n \theta + Y_{\text{phys}} \Gamma_i + Y_{\text{ion}} \Gamma_i (1-\theta) + Y_{\text{chem}} \Gamma_i \theta \right] $$ Where: - $n$ = atomic density of material - $\Gamma_n$ = neutral flux - $\Gamma_i$ = ion flux - $\theta$ = surface coverage of reactive species - $Y$ = yield coefficients ### 3.3 Ion Energy Distribution Function (IEDF) For sinusoidal RF bias, the IEDF is bimodal with peaks at: $$ E_{\pm} = eV_{dc} \pm eV_{rf} \cdot \frac{\omega_{pi}}{\omega_{rf}} $$ Where: - $V_{dc}$ = DC self-bias voltage - $V_{rf}$ = RF amplitude - $\omega_{pi}$ = ion plasma frequency - $\omega_{rf}$ = RF frequency The peak separation: $$ \Delta E = 2eV_{rf} \cdot \frac{\omega_{pi}}{\omega_{rf}} $$ ### 3.4 Common Etch Chemistries | Material | Chemistry | Key Radicals | Byproducts | |----------|-----------|--------------|------------| | Silicon | SF$_6$, Cl$_2$, HBr | F*, Cl*, Br* | SiF$_4$, SiCl$_4$ | | SiO$_2$ | CF$_4$, CHF$_3$, C$_4$F$_8$ | CF$_x$*, F* | SiF$_4$, CO, CO$_2$ | | Si$_3$N$_4$ | CF$_4$/O$_2$ | F*, O* | SiF$_4$, N$_2$ | | Al | Cl$_2$/BCl$_3$ | Cl* | AlCl$_3$ | | Photoresist | O$_2$ | O* | CO, CO$_2$, H$_2$O | ### 3.5 Selectivity **Selectivity** is the ratio of etch rates between target and mask (or underlayer): $$ S = \frac{\text{ER}_{\text{target}}}{\text{ER}_{\text{mask}}} $$ For oxide-to-nitride selectivity in fluorocarbon plasmas: $$ S_{\text{ox/nit}} = \frac{\text{ER}_{\text{SiO}_2}}{\text{ER}_{\text{Si}_3\text{N}_4}} \propto \frac{[\text{F}]}{[\text{CF}_x]} $$ ## 4. Plasma Sources ### 4.1 Capacitively Coupled Plasma (CCP) **Configuration**: Parallel plate electrodes with RF power **Power absorption**: Primarily through stochastic (collisionless) heating: $$ P_{\text{stoch}} \propto \frac{m_e v_e^2 \omega_{rf}^2 s_0^2}{v_{th,e}} $$ Where $s_0$ is the sheath oscillation amplitude. **Dual-frequency operation**: - High frequency (27–100 MHz): Controls plasma density - Low frequency (100 kHz–13 MHz): Controls ion energy Ion energy scaling: $$ \langle E_i \rangle \propto \frac{V_{rf}^2}{n_e^{0.5}} $$ ### 4.2 Inductively Coupled Plasma (ICP) **Power transfer**: Through induced electric field from RF current in coil: $$ E_\theta = -\frac{\partial A_\theta}{\partial t} = j\omega A_\theta $$ **Skin depth** (characteristic penetration depth of fields): $$ \delta = \sqrt{\frac{2}{\omega \mu_0 \sigma_p}} $$ Where $\sigma_p$ is plasma conductivity: $$ \sigma_p = \frac{n_e e^2}{m_e \nu_m} $$ **Power density**: $$ P = \frac{1}{2} \text{Re}(\sigma_p) |E|^2 $$ **Advantages**: - Higher plasma density: $10^{11}$–$10^{12}$ cm$^{-3}$ - Lower operating pressure: 1–50 mTorr - Independent control of ion flux and energy ### 4.3 Plasma Density Comparison | Source Type | Density (cm$^{-3}$) | Pressure Range | Ion Energy Control | |-------------|---------------------|----------------|-------------------| | CCP | $10^9$–$10^{10}$ | 10–1000 mTorr | Coupled | | ICP | $10^{11}$–$10^{12}$ | 1–50 mTorr | Independent | | ECR | $10^{11}$–$10^{12}$ | 0.1–10 mTorr | Independent | | Helicon | $10^{12}$–$10^{13}$ | 0.1–10 mTorr | Independent | ## 5. Plasma-Enhanced Deposition ### 5.1 PECVD Fundamentals **Reaction rate** in PECVD: $$ R = k_0 \exp\left(-\frac{E_a}{k_B T_{eff}}\right) [A]^a [B]^b $$ Where $T_{eff}$ is an effective temperature combining gas and electron contributions. The plasma reduces the effective activation energy by providing: - Electron-impact dissociation - Ion bombardment energy - Radical species ### 5.2 Common PECVD Reactions **Silicon dioxide** from silane and nitrous oxide: $$ \text{SiH}_4 + 2\text{N}_2\text{O} \xrightarrow{\text{plasma}} \text{SiO}_2 + 2\text{N}_2 + 2\text{H}_2 $$ **Silicon nitride** from silane and ammonia: $$ 3\text{SiH}_4 + 4\text{NH}_3 \xrightarrow{\text{plasma}} \text{Si}_3\text{N}_4 + 12\text{H}_2 $$ **Amorphous silicon**: $$ \text{SiH}_4 \xrightarrow{\text{plasma}} a\text{-Si:H} + 2\text{H}_2 $$ ### 5.3 Film Quality Parameters Film stress in PECVD films: $$ \sigma = \frac{E_f}{1-\nu_f} \left( \alpha_s - \alpha_f \right) \Delta T + \sigma_{\text{intrinsic}} $$ Where: - $E_f$ = film Young's modulus - $\nu_f$ = film Poisson's ratio - $\alpha_s, \alpha_f$ = thermal expansion coefficients (substrate, film) - $\sigma_{\text{intrinsic}}$ = intrinsic stress from deposition process ### 5.4 Plasma-Enhanced ALD (PEALD) **Growth per cycle (GPC)**: $$ \text{GPC} = \frac{\theta_{\text{sat}} \cdot \Omega}{A_{\text{site}}} $$ Where: - $\theta_{\text{sat}}$ = saturation coverage - $\Omega$ = molecular volume - $A_{\text{site}}$ = area per reactive site **Self-limiting behavior** requires: $$ \Gamma_{\text{precursor}} \cdot t_{\text{pulse}} > \frac{N_{\text{sites}}}{S_0} $$ Where $S_0$ is the initial sticking coefficient. ## 6. Advanced Topics ### 6.1 Aspect Ratio Dependent Etching (ARDE) Etch rate decreases with increasing aspect ratio due to: 1. **Ion shadowing**: Reduced ion flux at feature bottom 2. **Neutral transport**: Knudsen diffusion limitation 3. **Product redeposition**: Reduced volatile product escape **Knudsen number** for feature transport: $$ Kn = \frac{\lambda}{w} $$ Where $\lambda$ is mean free path, $w$ is feature width. For $Kn > 1$ (molecular flow regime): $$ \Gamma_{\text{bottom}} = \Gamma_{\text{top}} \cdot K(\text{AR}) $$ Where $K(\text{AR})$ is the Clausing factor, approximately: $$ K(\text{AR}) \approx \frac{1}{1 + \frac{3}{8}\text{AR}} $$ For high aspect ratio features. ### 6.2 Atomic Layer Etching (ALE) **Self-limiting surface modification**: $$ \theta(t) = \theta_{\text{sat}} \left[1 - \exp\left(-\frac{t}{\tau}\right)\right] $$ **Etch per cycle (EPC)**: $$ \text{EPC} = \frac{N_{\text{modified}} \cdot a}{n_{\text{film}}} $$ Where: - $N_{\text{modified}}$ = surface density of modified atoms - $a$ = atoms removed per modified site - $n_{\text{film}}$ = atomic density of film ### 6.3 Plasma-Induced Damage **Charging damage** occurs when: $$ V_{\text{antenna}} = \frac{J_e - J_i}{C_{\text{gate}}/A_{\text{antenna}}} \cdot t > V_{\text{breakdown}} $$ **Antenna ratio** limit: $$ \text{AR}_{\text{antenna}} = \frac{A_{\text{antenna}}}{A_{\text{gate}}} < \text{AR}_{\text{critical}} $$ **UV damage** from vacuum UV photons ($\lambda < 200$ nm): $$ N_{\text{defects}} \propto \int I(\lambda) \cdot \sigma(\lambda) \cdot d\lambda $$ ## 7. Plasma Diagnostics ### 7.1 Langmuir Probe Analysis **Electron density** from ion saturation current: $$ n_e = \frac{I_{i,sat}}{0.61 \cdot e \cdot A_p \cdot \sqrt{\frac{k_B T_e}{M_i}}} $$ **Electron temperature** from the exponential region: $$ T_e = \frac{e}{k_B} \left( \frac{d(\ln I_e)}{dV} \right)^{-1} $$ **EEDF** from second derivative of I-V curve: $$ f(\varepsilon) = \frac{2m_e}{e^2 A_p} \sqrt{\frac{2\varepsilon}{m_e}} \frac{d^2 I}{dV^2} $$ ### 7.2 Optical Emission Spectroscopy (OES) **Actinometry** for radical density measurement: $$ \frac{n_X}{n_{\text{Ar}}} = \frac{I_X}{I_{\text{Ar}}} \cdot \frac{\sigma_{\text{Ar}} \cdot Q_{\text{Ar}}}{\sigma_X \cdot Q_X} $$ Where: - $I$ = emission intensity - $\sigma$ = electron-impact excitation cross-section - $Q$ = quantum efficiency ## 8. Process Control Equations ### 8.1 Residence Time $$ \tau_{\text{res}} = \frac{p \cdot V}{Q \cdot k_B T} $$ Where: - $p$ = pressure - $V$ = chamber volume - $Q$ = gas flow rate (sccm converted to molecules/s) ### 8.2 Mean Free Path $$ \lambda = \frac{k_B T}{\sqrt{2} \pi d^2 p} $$ For argon at 10 mTorr and 300 K: $$ \lambda \approx 0.5 \text{ cm} $$ ### 8.3 Power Density **Effective power density** at wafer: $$ P_{\text{eff}} = \frac{\eta \cdot P_{\text{source}}}{A_{\text{wafer}}} $$ Where $\eta$ is power transfer efficiency (typically 0.3–0.7). ## 9. Critical Equations | Application | Equation | Key Parameters | |-------------|----------|----------------| | Debye length | $\lambda_D = \sqrt{\frac{\varepsilon_0 k_B T_e}{n_e e^2}}$ | $T_e$, $n_e$ | | Bohm velocity | $v_B = \sqrt{\frac{k_B T_e}{M_i}}$ | $T_e$, $M_i$ | | Skin depth | $\delta = \sqrt{\frac{2}{\omega \mu_0 \sigma_p}}$ | $\omega$, $n_e$ | | Selectivity | $S = \frac{\text{ER}_1}{\text{ER}_2}$ | Chemistry, energy | | ARDE factor | $K \approx (1 + 0.375 \cdot \text{AR})^{-1}$ | Aspect ratio | | Residence time | $\tau = \frac{pV}{Qk_B T}$ | $p$, $Q$, $V$ |
Use plasma to enhance bonding.
# Semiconductor Manufacturing Plasma Processes Plasma processes are foundational to modern semiconductor fabrication—nearly 40-50% of all processing steps in advanced chip manufacturing involve plasma in some form. ## 1. What is Plasma in Semiconductor Manufacturing? In semiconductor manufacturing, plasma refers to a **partially ionized gas** containing: - Free electrons ($e^-$) - Positive ions ($\text{Ar}^+$, $\text{Cl}^+$, etc.) - Neutral atoms and molecules - Highly reactive radicals ($\text{F}^{\bullet}$, $\text{Cl}^{\bullet}$, $\text{O}^{\bullet}$) ### Plasma Characteristics These are typically **"cold" or non-equilibrium plasmas**: | Parameter | Symbol | Typical Value | |-----------|--------|---------------| | Electron Temperature | $T_e$ | $1-10 \text{ eV}$ $(10^4 - 10^5 \text{ K})$ | | Ion/Gas Temperature | $T_i$ | $\sim 300-500 \text{ K}$ | | Electron Density | $n_e$ | $10^9 - 10^{12} \text{ cm}^{-3}$ | | Pressure | $P$ | $1-100 \text{ mTorr}$ | The electron temperature is related to thermal energy by: $$T_e [\text{eV}] = \frac{k_B T}{e} \approx \frac{T[\text{K}]}{11600}$$ ### Debye Length The characteristic shielding distance in plasma: $$\lambda_D = \sqrt{\frac{\varepsilon_0 k_B T_e}{n_e e^2}} = 743 \sqrt{\frac{T_e [\text{eV}]}{n_e [\text{cm}^{-3}]}} \text{ cm}$$ For typical process plasmas: $\lambda_D \approx 10-100 \text{ μm}$ ### Plasma Frequency The characteristic oscillation frequency of electrons: $$\omega_{pe} = \sqrt{\frac{n_e e^2}{m_e \varepsilon_0}} \approx 9000 \sqrt{n_e [\text{cm}^{-3}]} \text{ rad/s}$$ ## 2. Major Plasma Processes ### 2.1 Plasma Etching The most critical plasma application—removes material in precisely defined patterns. #### 2.1.1 Reactive Ion Etching (RIE) Combines **chemical attack** from radicals with **directional ion bombardment**. **Key Mechanism - Ion-Enhanced Etching:** $$\text{Etch Rate}_{total} >> \text{Etch Rate}_{chemical} + \text{Etch Rate}_{physical}$$ The synergistic enhancement factor: $$\eta = \frac{R_{ion+neutral}}{R_{ion} + R_{neutral}}$$ Typically $\eta = 5-20$ for common etch processes. **Common Chemistries:** - **Silicon etching:** - $\text{SF}_6 \rightarrow \text{SF}_x + \text{F}^{\bullet}$ (isotropic) - $\text{Cl}_2 \rightarrow 2\text{Cl}^{\bullet}$ (anisotropic with sidewall passivation) - $\text{HBr} \rightarrow \text{H}^{\bullet} + \text{Br}^{\bullet}$ (high selectivity) - **Silicon dioxide etching:** - $\text{CF}_4 + \text{O}_2 \rightarrow \text{CF}_x + \text{F}^{\bullet} + \text{CO}_2$ - $\text{C}_4\text{F}_8 \rightarrow \text{CF}_2 + \text{C}_2\text{F}_4$ (polymerizing) - $\text{CHF}_3$ (selective to Si) - **Metal etching:** - $\text{Cl}_2/\text{BCl}_3$ for Al, W - $\text{Cl}_2/\text{O}_2$ for Ti, TiN **Silicon Etch Reaction:** $$\text{Si}_{(s)} + 4\text{F}^{\bullet} \xrightarrow{\text{ion assist}} \text{SiF}_{4(g)} \uparrow$$ **Oxide Etch Reaction:** $$\text{SiO}_2 + \text{CF}_x \xrightarrow{\text{ion bombardment}} \text{SiF}_4 \uparrow + \text{CO}_2 \uparrow$$ #### 2.1.2 Deep Reactive Ion Etching (DRIE) Creates **high-aspect-ratio structures** using the Bosch process. **Bosch Process Cycle:** 1. **Etch step** (typically 5-15 seconds): $$\text{SF}_6 \rightarrow \text{SF}_5^+ + \text{F}^{\bullet} + e^-$$ $$\text{Si} + 4\text{F}^{\bullet} \rightarrow \text{SiF}_4 \uparrow$$ 2. **Passivation step** (typically 2-5 seconds): $$\text{C}_4\text{F}_8 \rightarrow n\text{CF}_2 \rightarrow (\text{CF}_2)_n \text{ polymer}$$ **Achievable Parameters:** - Aspect ratio: $> 50:1$ - Etch depth: $> 500 \text{ μm}$ - Sidewall angle: $90° \pm 0.5°$ - Scallop size: $< 50 \text{ nm}$ (optimized) #### 2.1.3 Atomic Layer Etching (ALE) Provides **angstrom-level precision** through self-limiting reactions. **Two-Step ALE Cycle:** 1. **Surface modification** (self-limiting): $$\text{Surface} + \text{Reactant} \rightarrow \text{Modified Layer}$$ 2. **Modified layer removal** (self-limiting): $$\text{Modified Layer} \xrightarrow{\text{ion/thermal}} \text{Volatile Products} \uparrow$$ **Example - Silicon ALE with Cl₂/Ar:** - Step 1: $\text{Si} + \text{Cl}_2 \rightarrow \text{SiCl}_x$ (surface chlorination) - Step 2: $\text{SiCl}_x + \text{Ar}^+ \rightarrow \text{SiCl}_y \uparrow$ (ion-assisted removal) **Etch per Cycle (EPC):** $$\text{EPC} \approx 0.5 - 2 \text{ Å/cycle}$$ **Total Etch Depth:** $$d = N \times \text{EPC}$$ where $N$ = number of cycles. ### 2.2 Plasma-Enhanced Chemical Vapor Deposition (PECVD) Deposits thin films at **lower temperatures** than thermal CVD. **Temperature Advantage:** $$T_{PECVD} \approx 200-400°\text{C} \quad \text{vs} \quad T_{thermal CVD} \approx 700-900°\text{C}$$ **Deposition Rate Model (simplified):** $$R_{dep} = k_0 \exp\left(-\frac{E_a}{k_B T}\right) \cdot f(n_e, P, \text{flow})$$ Where plasma activation effectively reduces $E_a$. #### Common PECVD Films **Silicon Dioxide:** $$\text{SiH}_4 + \text{N}_2\text{O} \xrightarrow{\text{plasma}} \text{SiO}_2 + \text{H}_2 + \text{N}_2$$ or using TEOS: $$\text{Si(OC}_2\text{H}_5)_4 + \text{O}_2 \xrightarrow{\text{plasma}} \text{SiO}_2 + \text{CO}_2 + \text{H}_2\text{O}$$ **Silicon Nitride:** $$3\text{SiH}_4 + 4\text{NH}_3 \xrightarrow{\text{plasma}} \text{Si}_3\text{N}_4 + 12\text{H}_2$$ Film composition varies: $\text{SiN}_x\text{H}_y$ where $x \approx 0.8-1.3$ **Film Properties (Typical):** | Film | Refractive Index | Stress (MPa) | Density (g/cm³) | |------|------------------|--------------|-----------------| | $\text{SiO}_2$ | $1.46-1.47$ | $-100$ to $+200$ | $2.1-2.3$ | | $\text{SiN}_x$ | $1.8-2.1$ | $-200$ to $+500$ | $2.4-2.8$ | #### High-Density Plasma CVD (HDP-CVD) Simultaneous deposition and sputtering for **gap fill**. **Deposition-to-Sputter Ratio:** $$D/S = \frac{R_{deposition}}{R_{sputter}}$$ Optimal gap fill: $D/S \approx 3-5$ **Gap Fill Mechanism:** - Deposition occurs everywhere - Sputtering preferentially removes material from corners/top - Net result: bottom-up fill ### 2.3 Physical Vapor Deposition (Sputtering) Argon ions bombard a solid target, ejecting atoms. #### Sputter Yield Number of target atoms ejected per incident ion: $$Y = \frac{3\alpha}{4\pi^2} \cdot \frac{4M_1 M_2}{(M_1 + M_2)^2} \cdot \frac{E}{U_s}$$ Where: - $M_1$ = ion mass - $M_2$ = target atom mass - $E$ = ion energy - $U_s$ = surface binding energy - $\alpha$ = dimensionless function of mass ratio **Typical Sputter Yields** (500 eV Ar⁺): | Target | Yield (atoms/ion) | |--------|-------------------| | Al | 1.2 | | Cu | 2.3 | | W | 0.6 | | Ti | 0.6 | | Ta | 0.6 | #### Ionized PVD (iPVD) Ionizes sputtered metal atoms for **directional deposition**. **Ionization Fraction:** $$f_{ion} = \frac{n_{M^+}}{n_{M^+} + n_M}$$ Modern iPVD: $f_{ion} > 70\%$ **Bottom Coverage Improvement:** $$\text{BC} = \frac{t_{bottom}}{t_{field}}$$ iPVD achieves BC > 50% in features with AR > 5:1 ### 2.4 Plasma-Enhanced Atomic Layer Deposition (PEALD) Uses plasma as one of the reactants in the ALD cycle. **Standard ALD Cycle:** 1. Precursor A exposure (self-limiting) 2. Purge 3. Precursor B exposure (self-limiting) 4. Purge **PEALD Advantage:** Plasma provides reactive species at lower temperatures: $$\text{O}_2 \xrightarrow{\text{plasma}} 2\text{O}^{\bullet}$$ vs thermal: $$\text{H}_2\text{O} \xrightarrow{T > 300°C} \text{OH}^{\bullet} + \text{H}^{\bullet}$$ **Example - HfO₂ PEALD:** - Step 1: $\text{Hf(NMe}_2)_4 + \text{Surface-OH} \rightarrow \text{Surface-O-Hf(NMe}_2)_3 + \text{HNMe}_2$ - Step 2: $\text{Surface-O-Hf(NMe}_2)_3 + \text{O}^{\bullet} \rightarrow \text{Surface-HfO}_2\text{-OH}$ **Growth per Cycle (GPC):** $$\text{GPC} \approx 0.5-1.5 \text{ Å/cycle}$$ **Film Thickness:** $$t = N \times \text{GPC}$$ ## 3. Plasma Sources ### 3.1 Capacitively Coupled Plasma (CCP) Two parallel plate electrodes with RF power (typically 13.56 MHz). **Sheath Voltage:** $$V_{sh} \approx \frac{V_{RF}}{2}$$ **Ion Bombardment Energy:** $$E_{ion} \approx eV_{sh} = \frac{eV_{RF}}{2}$$ For $V_{RF} = 500\text{ V}$: $E_{ion} \approx 250\text{ eV}$ **Plasma Density:** $$n_e \propto P_{RF}^{0.5-1.0}$$ Typical: $n_e \approx 10^9 - 10^{10} \text{ cm}^{-3}$ **Limitations:** - Ion flux and energy are coupled - Lower density than ICP ### 3.2 Inductively Coupled Plasma (ICP) RF coil induces plasma currents. **Power Transfer:** $$P_{plasma} = \frac{V_{ind}^2}{R_{plasma}}$$ Where induced voltage: $$V_{ind} = -\frac{d\Phi}{dt} = \omega \cdot N \cdot B \cdot A$$ **Key Advantage - Independent Control:** - **Source power** ($P_{source}$) → Ion flux ($\Gamma_i$) $$\Gamma_i \propto n_e \propto P_{source}^{0.5-1.0}$$ - **Bias power** ($P_{bias}$) → Ion energy ($E_i$) $$E_i \propto V_{bias} \propto \sqrt{P_{bias}}$$ **Typical Parameters:** | Parameter | CCP | ICP | |-----------|-----|-----| | $n_e$ (cm⁻³) | $10^9-10^{10}$ | $10^{11}-10^{12}$ | | Pressure (mTorr) | $50-500$ | $1-50$ | | Ion energy control | Limited | Independent | ### 3.3 Electron Cyclotron Resonance (ECR) Microwave power (2.45 GHz) + magnetic field. **Resonance Condition:** $$\omega = \omega_{ce} = \frac{eB}{m_e}$$ At 2.45 GHz: $B_{res} = 875 \text{ G}$ **Advantages:** - Very high density: $n_e > 10^{12} \text{ cm}^{-3}$ - Low pressure operation: $< 1 \text{ mTorr}$ - Efficient power coupling ### 3.4 Remote Plasma Plasma generated away from substrate—only **radicals** reach wafer. **Radical Flux at Wafer:** $$\Gamma_r = \Gamma_0 \exp\left(-\frac{L}{\lambda_{mfp}}\right) \cdot \exp\left(-\frac{t}{\tau_{recomb}}\right)$$ Where: - $L$ = distance from plasma - $\lambda_{mfp}$ = mean free path - $\tau_{recomb}$ = recombination lifetime **Benefits:** - No ion bombardment damage - Gentle surface treatment - Ideal for cleaning and selective processes ## 4. Plasma Sheath Physics The sheath is the region between bulk plasma and surfaces. ### 4.1 Sheath Formation Electrons are faster than ions: $$v_e = \sqrt{\frac{8k_BT_e}{\pi m_e}} >> v_i = \sqrt{\frac{8k_BT_i}{\pi m_i}}$$ Result: Surfaces charge **negatively**, forming a positive space-charge sheath. ### 4.2 Bohm Criterion Ions must reach sheath edge with minimum velocity: $$v_{Bohm} = \sqrt{\frac{k_B T_e}{m_i}}$$ **Ion flux to surface:** $$\Gamma_i = n_s \cdot v_{Bohm} = n_s \sqrt{\frac{k_B T_e}{m_i}}$$ Where $n_s \approx 0.61 n_e$ at sheath edge. ### 4.3 Child-Langmuir Law Ion current density through collisionless sheath: $$J_i = \frac{4\varepsilon_0}{9} \sqrt{\frac{2e}{m_i}} \cdot \frac{V^{3/2}}{d^2}$$ ### 4.4 Sheath Thickness $$s = \frac{\sqrt{2}}{3} \lambda_D \left(\frac{2V_s}{T_e}\right)^{3/4}$$ For $V_s = 100\text{ V}$, $T_e = 3\text{ eV}$: $s \approx 10-100 \text{ μm}$ ### 4.5 Ion Angular Distribution **Without collisions** (low pressure): $$\theta_{max} \approx \arctan\sqrt{\frac{T_i}{eV_s}}$$ Typically $\theta_{max} < 5°$ — highly directional! **With collisions** (high pressure): $$\theta \propto \frac{s}{\lambda_{mfp}}$$ Collisions broaden the angular distribution, reducing anisotropy. ## 5. Etch Process Metrics ### 5.1 Etch Rate $$R = \frac{\Delta d}{\Delta t} \quad [\text{nm/min}]$$ Typical values: - Si in $\text{SF}_6$: $200-1000$ nm/min - $\text{SiO}_2$ in $\text{CF}_4$: $50-200$ nm/min - Poly-Si in $\text{Cl}_2$: $100-500$ nm/min ### 5.2 Selectivity Ratio of etch rates between two materials: $$S_{A:B} = \frac{R_A}{R_B}$$ **Critical Selectivities:** | Process | Target/Stop | Required Selectivity | |---------|-------------|---------------------| | Gate etch | Poly-Si / $\text{SiO}_2$ | $> 50:1$ | | Contact etch | $\text{SiO}_2$ / Si | $> 20:1$ | | Spacer etch | $\text{SiN}$ / Si | $> 100:1$ | ### 5.3 Anisotropy $$A = 1 - \frac{R_{lateral}}{R_{vertical}}$$ - $A = 1$: Perfectly anisotropic (vertical sidewalls) - $A = 0$: Perfectly isotropic (hemispherical profile) ### 5.4 Uniformity $$U = \frac{R_{max} - R_{min}}{2 \cdot R_{avg}} \times 100\%$$ Target: $U < 3\%$ across 300mm wafer. ### 5.5 Aspect Ratio Dependent Etching (ARDE) Etch rate decreases with aspect ratio: $$R(AR) = R_0 \cdot f(AR)$$ **Knudsen Transport Model:** $$\frac{R(AR)}{R_0} = \frac{1}{1 + \frac{AR}{K}}$$ Where $K$ is a chemistry-dependent constant (typically 5-20). ## 6. Process Control Parameters ### 6.1 RF Power **Source Power** (ICP coil or CCP top electrode): - Controls plasma density: $n_e \propto P^{0.5-1.0}$ - Controls radical production - Typical: $100-3000$ W **Bias Power** (substrate electrode): - Controls ion energy: $E_i \propto \sqrt{P_{bias}}$ - Controls anisotropy - Typical: $0-500$ W ### 6.2 Pressure **Effects:** | Pressure | Mean Free Path | Ion Directionality | Radical Density | |----------|----------------|-------------------|-----------------| | Low ($< 10$ mTorr) | Long | High | Lower | | High ($> 100$ mTorr) | Short | Low | Higher | **Mean Free Path:** $$\lambda = \frac{k_B T}{P \cdot \sigma}$$ At 10 mTorr, 300K: $\lambda \approx 5 \text{ mm}$ ### 6.3 Gas Flow and Chemistry **Residence Time:** $$\tau_{res} = \frac{P \cdot V}{Q}$$ Where $Q$ = flow rate (sccm), $V$ = chamber volume. **Dissociation Fraction:** $$\alpha = \frac{n_{dissociated}}{n_{total}}$$ Higher power → higher $\alpha$ ### 6.4 Temperature **Wafer Temperature Effects:** - Reaction rates: $k \propto \exp(-E_a/k_BT)$ - Desorption rates - Selectivity - Film stress (PECVD) Typical range: $-20°C$ to $400°C$ ## 7. Advanced Topics ### 7.1 Pulsed Plasmas Modulate RF power on/off with period $T_{pulse}$. **Duty Cycle:** $$D = \frac{t_{on}}{t_{on} + t_{off}} = \frac{t_{on}}{T_{pulse}}$$ **Benefits:** - Narrower ion energy distribution - Reduced charging damage - Better selectivity control **Ion Energy Distribution (IED):** - CW plasma: Bimodal distribution - Pulsed plasma: Controllable, narrower distribution ### 7.2 Plasma-Induced Damage **Charging Damage:** $$V_{gate} = \frac{Q_{accumulated}}{C_{gate}} = \frac{(J_e - J_i) \cdot t \cdot A}{C_{gate}}$$ When $V_{gate} > V_{BD}$ → oxide breakdown! **Mitigation:** - Pulsed plasmas - Neutral beam sources - Process optimization **UV Damage:** VUV photons ($E > 9$ eV) can break Si-O bonds. $$\text{Si-O} + h\nu \rightarrow \text{defects}$$ ### 7.3 Loading Effects **Macro-loading:** $$R = R_0 \cdot \frac{1}{1 + \frac{A_{etch}}{A_0}}$$ More exposed area → lower etch rate (radical consumption). **Micro-loading:** Local pattern density affects local etch rate. $$\Delta R = R_{isolated} - R_{dense}$$ ### 7.4 Profile Control **Sidewall Passivation Model:** $$\theta = \arctan\left(\frac{R_{lateral}}{R_{vertical}}\right) = \arctan\left(\frac{R_V - R_P}{R_V}\right)$$ Where: - $R_V$ = vertical etch rate - $R_P$ = passivation deposition rate **Ideal Vertical Profile:** $R_P = R_{lateral}$ on sidewalls ## 8. Equipment and Monitoring ### 8.1 Chamber Components - **Chuck/Pedestal:** Temperature-controlled substrate holder - Electrostatic chuck (ESC) for wafer clamping - He backside cooling for thermal contact - **Gas Distribution:** - Showerhead or side injection - Mass flow controllers (MFCs): $\pm 1\%$ accuracy - **Pumping System:** - Turbo-molecular pump: base pressure $< 10^{-6}$ Torr - Throttle valve for pressure control - **RF System:** - Generator: 13.56 MHz, 2 MHz, 60 MHz common - Matching network: L-type or $\pi$-type ### 8.2 In-Situ Monitoring **Optical Emission Spectroscopy (OES):** Monitor plasma species by emission lines: | Species | Wavelength (nm) | |---------|-----------------| | F | 703.7 | | Cl | 837.6 | | O | 777.4 | | CO | 483.5 | | Si | 288.2 | | SiF | 440.0 | **Endpoint Detection:** $$\text{EPD Signal} = \frac{I_{product}}{I_{reference}}$$ Endpoint when signal changes (product species decrease). **Interferometry:** Film thickness from interference: $$2nd\cos\theta = m\lambda$$ Real-time thickness monitoring during etch/deposition. ## 9. Challenges at Advanced Nodes ### 9.1 Feature Dimensions At 3nm node: - Gate length: $\sim 12$ nm ($\sim 50$ atoms) - Fin width: $\sim 5-7$ nm - Metal pitch: $\sim 20-24$ nm **Precision Required:** $$\sigma_{CD} < 0.5 \text{ nm}$$ ### 9.2 New Architectures **Gate-All-Around (GAA) FETs:** - Requires isotropic etching for channel release - Selective removal of SiGe vs Si - Inner spacer formation **3D NAND:** - $> 200$ stacked layers - High aspect ratio etching ($> 60:1$) - Memory hole etch: $> 10$ μm deep ### 9.3 New Materials | Material | Application | Etch Chemistry Challenge | |----------|-------------|-------------------------| | $\text{HfO}_2$ | High-k gate | Low volatility of Hf halides | | $\text{Ru}$ | Contacts | RuO₄ volatility issues | | $\text{Co}$ | Interconnects | Selectivity to Cu | | $\text{SiGe}$ | Channel | Selectivity to Si | ## 10. Key Equations ### Plasma Parameters $$\lambda_D = \sqrt{\frac{\varepsilon_0 k_B T_e}{n_e e^2}}$$ $$v_{Bohm} = \sqrt{\frac{k_B T_e}{m_i}}$$ $$\Gamma_i = 0.61 \cdot n_e \cdot v_{Bohm}$$ ### Etch Metrics $$S_{A:B} = \frac{R_A}{R_B}$$ $$A = 1 - \frac{R_{lateral}}{R_{vertical}}$$ $$U = \frac{R_{max} - R_{min}}{2R_{avg}} \times 100\%$$ ### Process Dependencies $$n_e \propto P_{source}^{0.5-1.0}$$ $$E_i \propto \sqrt{P_{bias}}$$ $$R \propto \Gamma_i \cdot f(E_i) \cdot [X^{\bullet}]$$
DIP with plastic body.
PGA with organic substrate.
Plate heat exchangers transfer heat between air streams through conductive metal plates.
Rotating table holding CMP pad.
Platform designs create base architectures enabling derivative products efficiently.
Used for specific applications.
Calibration method using logistic regression.
NeRF without neural networks.
Plenoxels represent scenes as voxel grids enabling fast optimization without neural networks.
Multi-step diffusion sampler.
Design story plots.
Plotly creates interactive visualizations. Dash for dashboards.
Steer generation using attribute classifiers.
Pushes compound into cavity.
Scheduled maintenance to prevent equipment failure.
How well PM prevents failures.
PM overdue indicates missed maintenance requiring attention before production.
Preventive maintenance schedules plan routine service to prevent failures.
Probabilistic Matrix Factorization models user-item ratings using Gaussian distributions over latent factors with regularization through priors.
Principal Neighborhood Aggregation uses multiple aggregators and scalers enhancing GNN expressiveness on graph tasks.
Pneumatic valves use compressed air for actuation in chemical systems.
Pocket implants create localized high-doping regions near source-drain edges controlling short-channel effects.
Similar to halo selective doping.
Distance between components in tape.
Create poems in various forms.
Python dependency management.
Poetry manages Python dependencies. Lockfile, build, publish.
Generate poetry. Various styles, rhyme schemes.
Fill missing regions in point clouds.
Generate point clouds.
Start from SfM points.
Work with 3D point cloud data.
Classify points in 3D space.
Process 3D point cloud sequences.
Point clouds are 3D points from LiDAR or depth sensors. Process with PointNet. Autonomous driving.
Vacancies and interstitials.