reward modeling, training techniques
Reward modeling predicts human preferences using learned reward functions.
145 technical terms and definitions
Reward modeling predicts human preferences using learned reward functions.
Learn reward function from human preferences.
Model high-frequency behavior of devices.
Relational GCN with neighbor sampling reduces computational cost by sampling subsets of edges per relation type.
# 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 |
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