cluster detection, yield enhancement
Cluster detection algorithms locate groups of failing die indicating localized defects.
1,005 technical terms and definitions
Cluster detection algorithms locate groups of failing die indicating localized defects.
Tool with central vacuum transfer chamber and multiple process modules.
Group similar clients.
Clustering indices quantify degree of spatial correlation in defect distributions for yield modeling.
Clustering groups similar items. K-means, DBSCAN, hierarchical.
Compositional generalization.
Methods to build nFET and pFET together.
Polish transferred layer.
Optimize CMP to reduce metal dishing.
# Chemical Mechanical Planarization (CMP) Modeling in Semiconductor Manufacturing ## 1. Fundamentals of CMP ### 1.1 Definition and Principle Chemical Mechanical Planarization (CMP) is a hybrid process combining: - **Chemical etching**: Reactive slurry chemistry modifies surface properties - **Mechanical abrasion**: Physical removal via abrasive particles and pad The fundamental material removal can be expressed as: $$ \text{Material Removal} = f(\text{Chemical Reaction}, \text{Mechanical Abrasion}) $$ ### 1.2 Process Components | Component | Function | Key Parameters | |-----------|----------|----------------| | **Wafer** | Substrate to be planarized | Material type, pattern density | | **Polishing Pad** | Provides mechanical action | Hardness, porosity, asperity distribution | | **Slurry** | Chemical + abrasive medium | pH, oxidizer, particle size/concentration | | **Carrier** | Holds and rotates wafer | Down force, rotation speed | | **Platen** | Rotates polishing pad | Rotation speed, temperature | ### 1.3 Key Process Parameters - **Down Force ($F$)**: Pressure applied to wafer, typically $1-7$ psi - **Platen Speed ($\omega_p$)**: Pad rotation, typically $20-100$ rpm - **Carrier Speed ($\omega_c$)**: Wafer rotation, typically $20-100$ rpm - **Slurry Flow Rate ($Q$)**: Typically $100-300$ mL/min - **Temperature ($T$)**: Typically $20-50°C$ ## 2. Classical Physical Models ### 2.1 Preston Equation (Foundational Model) The foundational model for CMP is the **Preston equation** (1927): $$ \boxed{MRR = k_p \cdot P \cdot v} $$ Where: - $MRR$ = Material Removal Rate $[\text{nm/min}]$ - $k_p$ = Preston's coefficient $[\text{m}^2/\text{N}]$ - $P$ = Applied pressure $[\text{Pa}]$ - $v$ = Relative velocity $[\text{m/s}]$ The relative velocity between wafer and pad: $$ v = \sqrt{(\omega_p r_p)^2 + (\omega_c r_c)^2 - 2\omega_p \omega_c r_p r_c \cos(\theta)} $$ Where: - $\omega_p, \omega_c$ = Angular velocities of platen and carrier - $r_p, r_c$ = Radial positions - $\theta$ = Phase angle ### 2.2 Modified Preston Models #### 2.2.1 Pressure-Velocity Product Modification $$ MRR = k_p \cdot P^a \cdot v^b $$ Where $a, b$ are empirical exponents (typically $0.5 < a, b < 1.5$) #### 2.2.2 Chemical Enhancement Factor $$ MRR = k_p \cdot P \cdot v \cdot f(C, T, pH) $$ Where $f(C, T, pH)$ represents chemical effects: - $C$ = Oxidizer concentration - $T$ = Temperature - $pH$ = Slurry pH #### 2.2.3 Arrhenius-Modified Preston Equation $$ MRR = k_0 \cdot \exp\left(-\frac{E_a}{RT}\right) \cdot P \cdot v $$ Where: - $k_0$ = Pre-exponential factor - $E_a$ = Activation energy $[\text{J/mol}]$ - $R$ = Gas constant $= 8.314$ J/(mol$\cdot$K) - $T$ = Temperature $[\text{K}]$ ### 2.3 Tribocorrosion Model For metal CMP (e.g., tungsten, copper): $$ MRR = \frac{M}{z F \rho} \cdot \left( i_{corr} + \frac{Q_{pass}}{A \cdot t_{pass}} \right) \cdot f_{mech} $$ Where: - $M$ = Molar mass of metal - $z$ = Number of electrons transferred - $F$ = Faraday constant $= 96485$ C/mol - $\rho$ = Density - $i_{corr}$ = Corrosion current density - $Q_{pass}$ = Passivation charge - $f_{mech}$ = Mechanical factor ### 2.4 Contact Mode Classification | Mode | Condition | Preston Constant | Friction Coefficient | |------|-----------|------------------|---------------------| | **Contact** | $\frac{\eta v_R}{p} < (\frac{\eta v_R}{p})_c$ | High, constant | High ($\mu > 0.3$) | | **Mixed** | $\frac{\eta v_R}{p} \approx (\frac{\eta v_R}{p})_c$ | Transitional | Medium | | **Hydroplaning** | $\frac{\eta v_R}{p} > (\frac{\eta v_R}{p})_c$ | Low, variable | Low ($\mu < 0.1$) | Where: - $\eta$ = Slurry viscosity - $v_R$ = Relative velocity - $p$ = Pressure ## 3. Pattern Density Models ### 3.1 Effective Pattern Density Model (Stine Model) The local material removal rate depends on effective pattern density: $$ \frac{dz}{dt} = -\frac{K}{\rho_{eff}(x, y)} $$ Where: - $z$ = Surface height - $K$ = Blanket removal rate $= k_p \cdot P \cdot v$ - $\rho_{eff}$ = Effective pattern density #### 3.1.1 Effective Density Calculation $$ \rho_{eff}(x, y) = \iint_{-\infty}^{\infty} \rho_0(x', y') \cdot W(x - x', y - y') \, dx' \, dy' $$ Where: - $\rho_0(x, y)$ = Local pattern density - $W(x, y)$ = Weighting function (planarization kernel) #### 3.1.2 Elliptical Weighting Function $$ W(x, y) = \frac{1}{\pi L_x L_y} \cdot \exp\left(-\frac{x^2}{L_x^2} - \frac{y^2}{L_y^2}\right) $$ Where $L_x, L_y$ are planarization lengths in x and y directions. ### 3.2 Step Height Evolution Model For oxide CMP with step height $h$: $$ \frac{dh}{dt} = -K \cdot \left(1 - \frac{h_{contact}}{h}\right) \quad \text{for } h > h_{contact} $$ $$ \frac{dh}{dt} = 0 \quad \text{for } h \leq h_{contact} $$ Where $h_{contact}$ is the pad contact threshold height. ### 3.3 Integrated Density-Step Height Model Combined model for oxide thickness evolution: $$ z(x, y, t) = z_0 - K \cdot t \cdot \frac{1}{\rho_{eff}(x, y)} \cdot g(h) $$ Where $g(h)$ is the step-height dependent function: $$ g(h) = \begin{cases} 1 & \text{if } h > h_c \\ \frac{h}{h_c} & \text{if } h \leq h_c \end{cases} $$ ## 4. Dishing and Erosion Models ### 4.1 Copper Dishing Model Dishing depth $D$ for copper lines: $$ D = K_{Cu} \cdot t_{over} \cdot f(w) $$ Where: - $K_{Cu}$ = Copper removal rate - $t_{over}$ = Overpolish time - $w$ = Line width - $f(w)$ = Width-dependent function Empirical relationship: $$ D = D_0 \cdot \left(1 - \exp\left(-\frac{w}{w_c}\right)\right) $$ Where: - $D_0$ = Maximum dishing depth - $w_c$ = Critical line width ### 4.2 Oxide Erosion Model Erosion $E$ in dense pattern regions: $$ E = K_{ox} \cdot t_{over} \cdot \rho_{metal} $$ Where: - $K_{ox}$ = Oxide removal rate - $\rho_{metal}$ = Local metal pattern density ### 4.3 Combined Dishing-Erosion Total copper thickness loss: $$ \Delta z_{Cu} = D + E \cdot \frac{\rho_{metal}}{1 - \rho_{metal}} $$ ### 4.4 Pattern Density Effects | Pattern Density | Dishing Behavior | Erosion Behavior | |-----------------|------------------|------------------| | Low ($< 20\%$) | Minimal | Minimal | | Medium ($20-50\%$) | Moderate | Increasing | | High ($> 50\%$) | Saturates | Severe | ## 5. Contact Mechanics Models ### 5.1 Pad Asperity Contact Model Assuming Gaussian asperity height distribution: $$ P(z) = \frac{1}{\sigma_s \sqrt{2\pi}} \exp\left(-\frac{(z - \bar{z})^2}{2\sigma_s^2}\right) $$ Where: - $\sigma_s$ = Standard deviation of asperity heights - $\bar{z}$ = Mean asperity height ### 5.2 Real Contact Area $$ A_r = \pi n \int_{d}^{\infty} R(z - d) \cdot P(z) \, dz $$ Where: - $n$ = Number of asperities per unit area - $R$ = Asperity tip radius - $d$ = Separation distance For Gaussian distribution: $$ A_r = \pi n R \sigma_s \cdot F_1\left(\frac{d}{\sigma_s}\right) $$ Where $F_1$ is a statistical function. ### 5.3 Hertzian Contact For elastic contact between abrasive particle and wafer: $$ a = \left(\frac{3FR}{4E^*}\right)^{1/3} $$ $$ \delta = \frac{a^2}{R} = \left(\frac{9F^2}{16RE^{*2}}\right)^{1/3} $$ Where: - $a$ = Contact radius - $F$ = Normal force - $R$ = Particle radius - $\delta$ = Indentation depth - $E^*$ = Effective elastic modulus $$ \frac{1}{E^*} = \frac{1 - \nu_1^2}{E_1} + \frac{1 - \nu_2^2}{E_2} $$ ### 5.4 Material Removal by Single Abrasive Volume removed per abrasive per pass: $$ V = K_{wear} \cdot \frac{F_n \cdot L}{H} $$ Where: - $K_{wear}$ = Wear coefficient - $F_n$ = Normal force on particle - $L$ = Sliding distance - $H$ = Hardness of wafer material ### 5.5 Multi-Scale Model Framework ``` - ┌─────────────────────────────────────────────────────────────┐ │ WAFER SCALE (mm-cm) │ │ Pressure distribution, global uniformity │ ├─────────────────────────────────────────────────────────────┤ │ DIE SCALE ($\mu$m-mm) │ │ Pattern density effects, planarization │ ├─────────────────────────────────────────────────────────────┤ │ FEATURE SCALE (nm-$\mu$m) │ │ Dishing, erosion, step height evolution │ ├─────────────────────────────────────────────────────────────┤ │ PARTICLE SCALE (nm) │ │ Abrasive-surface interactions │ ├─────────────────────────────────────────────────────────────┤ │ MOLECULAR SCALE (Å) │ │ Chemical reactions, atomic removal │ └─────────────────────────────────────────────────────────────┘ ``` ## 6. Machine Learning and Neural Network Models ### 6.1 Overview of ML Approaches Machine learning methods for CMP modeling: - **Supervised Learning** - Artificial Neural Networks (ANN) - Convolutional Neural Networks (CNN) - Support Vector Machines (SVM) - Random Forests / Gradient Boosting - **Deep Learning** - Deep Belief Networks (DBN) - Long Short-Term Memory (LSTM) - Generative Adversarial Networks (GAN) - **Transfer Learning** - Pre-trained models adapted to new process conditions ### 6.2 Neural Network Architecture for CMP #### 6.2.1 Input Features $$ \mathbf{x} = [P, v, t, \rho, w, s, pH, C_{ox}, T, ...]^T $$ Where: - $P$ = Pressure - $v$ = Velocity - $t$ = Polish time - $\rho$ = Pattern density - $w$ = Feature width - $s$ = Feature spacing - $pH$ = Slurry pH - $C_{ox}$ = Oxidizer concentration - $T$ = Temperature #### 6.2.2 Multi-Layer Perceptron (MLP) $$ \mathbf{h}^{(1)} = \sigma(\mathbf{W}^{(1)} \mathbf{x} + \mathbf{b}^{(1)}) $$ $$ \mathbf{h}^{(2)} = \sigma(\mathbf{W}^{(2)} \mathbf{h}^{(1)} + \mathbf{b}^{(2)}) $$ $$ \hat{y} = \mathbf{W}^{(out)} \mathbf{h}^{(2)} + \mathbf{b}^{(out)} $$ Where: - $\sigma$ = Activation function (ReLU, tanh, sigmoid) - $\mathbf{W}^{(i)}$ = Weight matrices - $\mathbf{b}^{(i)}$ = Bias vectors #### 6.2.3 Activation Functions | Function | Formula | Use Case | |----------|---------|----------| | **ReLU** | $\sigma(x) = \max(0, x)$ | Hidden layers | | **Sigmoid** | $\sigma(x) = \frac{1}{1 + e^{-x}}$ | Output (binary) | | **Tanh** | $\sigma(x) = \frac{e^x - e^{-x}}{e^x + e^{-x}}$ | Hidden layers | | **Softmax** | $\sigma(x_i) = \frac{e^{x_i}}{\sum_j e^{x_j}}$ | Classification | ### 6.3 CNN-Based CMP Modeling (CmpCNN) #### 6.3.1 Architecture ``` Input: Layout Image (Binary) + Density Map ↓ Conv2D Layer (3×3 kernel, 32 filters) ↓ MaxPooling2D (2×2) ↓ Conv2D Layer (3×3 kernel, 64 filters) ↓ MaxPooling2D (2×2) ↓ Flatten ↓ Dense Layer (256 units) ↓ Dense Layer (128 units) ↓ Output: Post-CMP Height Map ``` #### 6.3.2 Convolution Operation $$ (I * K)(i, j) = \sum_m \sum_n I(i+m, j+n) \cdot K(m, n) $$ Where: - $I$ = Input image (layout) - $K$ = Convolution kernel - $(i, j)$ = Output position ### 6.4 Loss Functions #### 6.4.1 Mean Squared Error (MSE) $$ \mathcal{L}_{MSE} = \frac{1}{N} \sum_{i=1}^{N} (y_i - \hat{y}_i)^2 $$ #### 6.4.2 Root Mean Square Error (RMSE) $$ RMSE = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (y_i - \hat{y}_i)^2} $$ #### 6.4.3 Mean Absolute Percentage Error (MAPE) $$ MAPE = \frac{100\%}{N} \sum_{i=1}^{N} \left| \frac{y_i - \hat{y}_i}{y_i} \right| $$ ### 6.5 Transfer Learning Framework For adapting models across process nodes: $$ \mathcal{L}_{transfer} = \mathcal{L}_{target} + \lambda \cdot \mathcal{L}_{domain} $$ Where: - $\mathcal{L}_{target}$ = Target domain loss - $\mathcal{L}_{domain}$ = Domain adaptation loss - $\lambda$ = Regularization parameter ### 6.6 Performance Metrics | Metric | Formula | Target | |--------|---------|--------| | $R^2$ | $1 - \frac{\sum(y_i - \hat{y}_i)^2}{\sum(y_i - \bar{y})^2}$ | $> 0.95$ | | RMSE | $\sqrt{\frac{1}{N}\sum(y_i - \hat{y}_i)^2}$ | $< 5$ Å | | MAE | $\frac{1}{N}\sum|y_i - \hat{y}_i|$ | $< 3$ Å | ## 7. Slurry Chemistry Modeling ### 7.1 Kaufman Mechanism Cyclic passivation-depassivation process: $$ \text{Metal} \xrightarrow{\text{Oxidizer}} \text{Metal Oxide} \xrightarrow{\text{Abrasion}} \text{Removal} $$ ### 7.2 Electrochemical Reactions #### 7.2.1 Copper CMP **Oxidation:** $$ \text{Cu} \rightarrow \text{Cu}^{2+} + 2e^- $$ **Passivation (with BTA):** $$ \text{Cu} + \text{BTA} \rightarrow \text{Cu-BTA}_{film} $$ **Complexation:** $$ \text{Cu}^{2+} + n\text{L} \rightarrow [\text{CuL}_n]^{2+} $$ Where L = chelating agent (e.g., glycine, citrate) #### 7.2.2 Tungsten CMP **Oxidation:** $$ \text{W} + 3\text{H}_2\text{O} \rightarrow \text{WO}_3 + 6\text{H}^+ + 6e^- $$ **With hydrogen peroxide:** $$ \text{W} + 3\text{H}_2\text{O}_2 \rightarrow \text{WO}_3 + 3\text{H}_2\text{O} $$ ### 7.3 Pourbaix Diagram Integration Stability regions defined by: $$ E = E^0 - \frac{RT}{nF} \ln Q - \frac{RT}{F} \cdot m \cdot pH $$ Where: - $E$ = Electrode potential - $E^0$ = Standard potential - $Q$ = Reaction quotient - $m$ = Number of H⁺ in reaction ### 7.4 Abrasive Particle Effects #### 7.4.1 Particle Size Distribution (PSD) Log-normal distribution: $$ f(d) = \frac{1}{d \sigma \sqrt{2\pi}} \exp\left(-\frac{(\ln d - \mu)^2}{2\sigma^2}\right) $$ Where: - $d$ = Particle diameter - $\mu$ = Mean of $\ln(d)$ - $\sigma$ = Standard deviation of $\ln(d)$ #### 7.4.2 Zeta Potential $$ \zeta = \frac{4\pi \eta \mu_e}{\varepsilon} $$ Where: - $\eta$ = Viscosity - $\mu_e$ = Electrophoretic mobility - $\varepsilon$ = Dielectric constant ### 7.5 Slurry Components Summary | Component | Function | Typical Materials | |-----------|----------|-------------------| | **Abrasive** | Mechanical removal | SiO₂, CeO₂, Al₂O₃ | | **Oxidizer** | Surface modification | H₂O₂, KIO₃, Fe(NO₃)₃ | | **Complexant** | Metal dissolution | Glycine, citric acid | | **Inhibitor** | Corrosion protection | BTA, BBI | | **Surfactant** | Particle dispersion | CTAB, SDS | | **Buffer** | pH control | Phosphate, citrate | ## 8. Chip-Scale and Full-Chip Models ### 8.1 Within-Wafer Non-Uniformity (WIWNU) $$ WIWNU = \frac{\sigma_{thickness}}{\bar{thickness}} \times 100\% $$ Where: - $\sigma_{thickness}$ = Standard deviation of thickness - $\bar{thickness}$ = Mean thickness ### 8.2 Pressure Distribution Model For a flexible carrier: $$ P(r) = P_0 + \sum_{i=1}^{n} P_i \cdot J_0\left(\frac{\alpha_i r}{R}\right) $$ Where: - $P_0$ = Base pressure - $J_0$ = Bessel function of first kind - $\alpha_i$ = Bessel zeros - $R$ = Wafer radius ### 8.3 Multi-Zone Pressure Control For zone $i$: $$ MRR_i = k_p \cdot P_i \cdot v_i $$ Target uniformity achieved when: $$ MRR_1 = MRR_2 = ... = MRR_n $$ ### 8.4 Full-Chip Simulation Flow ``` - ┌─────────────────────┐ │ Design Layout (GDS)│ └──────────┬──────────┘ ↓ ┌─────────────────────┐ │ Density Extraction │ │ ρ(x,y) for each │ │ metal/dielectric │ └──────────┬──────────┘ ↓ ┌─────────────────────┐ │ Effective Density │ │ ρ_eff = ρ * W │ └──────────┬──────────┘ ↓ ┌─────────────────────┐ │ CMP Simulation │ │ z(t) evolution │ └──────────┬──────────┘ ↓ ┌─────────────────────┐ │ Post-CMP Topography │ │ Dishing/Erosion Map │ └──────────┬──────────┘ ↓ ┌─────────────────────┐ │ Hotspot Detection │ │ Design Rule Check │ └─────────────────────┘ ``` ## 9. Process Control Applications ### 9.1 Run-to-Run (R2R) Control #### 9.1.1 EWMA Controller $$ \hat{y}_{k+1} = \lambda y_k + (1 - \lambda) \hat{y}_k $$ Where: - $\hat{y}_{k+1}$ = Predicted output for next run - $y_k$ = Current measured output - $\lambda$ = Smoothing factor $(0 < \lambda < 1)$ #### 9.1.2 Recipe Adjustment $$ u_{k+1} = u_k + G^{-1} (y_{target} - \hat{y}_{k+1}) $$ Where: - $u$ = Process recipe (time, pressure, etc.) - $G$ = Process gain matrix - $y_{target}$ = Target output ### 9.2 Virtual Metrology $$ \hat{y} = f_{VM}(\mathbf{x}_{FDC}) $$ Where: - $\hat{y}$ = Predicted wafer quality - $\mathbf{x}_{FDC}$ = Fault Detection and Classification sensor data ### 9.3 Endpoint Detection #### 9.3.1 Motor Current Monitoring $$ I(t) = I_0 + \Delta I \cdot H(t - t_{endpoint}) $$ Where $H$ is the Heaviside step function. #### 9.3.2 Optical Endpoint $$ R(\lambda, t) = R_{film}(\lambda, d(t)) $$ Where reflectance $R$ changes as film thickness $d$ decreases. ## 10. Current Challenges and Future Directions ### 10.1 Key Challenges - **Sub-5nm nodes**: Atomic-scale precision required - Thickness variation target: $< 5$ Å (3σ) - Defect density target: $< 0.01$ defects/cm² - **New materials integration**: - Low-κ dielectrics ($\kappa < 2.5$) - Cobalt interconnects - Ruthenium barrier layers - **3D integration**: - Through-Silicon Via (TSV) CMP - Hybrid bonding surface preparation - Wafer-level packaging ### 10.2 Future Model Development - **Physics-informed neural networks (PINNs)**: $$ \mathcal{L} = \mathcal{L}_{data} + \lambda_{physics} \cdot \mathcal{L}_{physics} $$ Where: $$ \mathcal{L}_{physics} = \left\| \frac{\partial z}{\partial t} + \frac{K}{\rho_{eff}} \right\|^2 $$ - **Digital twins** for real-time process optimization - **Federated learning** across multiple fabs ### 10.3 Industry Requirements | Node | Thickness Uniformity | Defect Density | Dishing Limit | |------|---------------------|----------------|---------------| | 7nm | $< 10$ Å | $< 0.05$/cm² | $< 200$ Å | | 5nm | $< 7$ Å | $< 0.03$/cm² | $< 150$ Å | | 3nm | $< 5$ Å | $< 0.01$/cm² | $< 100$ Å | | 2nm | $< 3$ Å | $< 0.005$/cm² | $< 50$ Å | ## Symbol Glossary | Symbol | Description | Units | |--------|-------------|-------| | $MRR$ | Material Removal Rate | nm/min | | $k_p$ | Preston coefficient | m²/N | | $P$ | Pressure | Pa, psi | | $v$ | Relative velocity | m/s | | $\rho$ | Pattern density | dimensionless | | $\rho_{eff}$ | Effective pattern density | dimensionless | | $L$ | Planarization length | $\mu$m | | $D$ | Dishing depth | Å, nm | | $E$ | Erosion depth | Å, nm | | $w$ | Feature width | nm, $\mu$m | | $h$ | Step height | nm | | $t$ | Polish time | s, min | | $T$ | Temperature | K, °C | | $\eta$ | Viscosity | Pa$\cdot$s | | $\mu$ | Friction coefficient | dimensionless | ## Key Equations ### Preston Equation $$ MRR = k_p \cdot P \cdot v $$ ### Effective Density $$ \rho_{eff}(x,y) = \iint \rho_0(x',y') \cdot W(x-x', y-y') \, dx' dy' $$ ### Material Removal (Density Model) $$ \frac{dz}{dt} = -\frac{K}{\rho_{eff}(x,y)} $$ ### Dishing Model $$ D = D_0 \cdot \left(1 - e^{-w/w_c}\right) $$ ### Erosion Model $$ E = K_{ox} \cdot t_{over} \cdot \rho_{metal} $$ ### Neural Network $$ \hat{y} = \sigma(\mathbf{W}^{(n)} \cdot ... \cdot \sigma(\mathbf{W}^{(1)} \mathbf{x} + \mathbf{b}^{(1)}) + \mathbf{b}^{(n)}) $$
Route wires considering CMP effects.
# Chemical Mechanical Planarization (CMP) Modeling in Semiconductor Manufacturing ## 1. Fundamentals of CMP ### 1.1 Definition and Principle Chemical Mechanical Planarization (CMP) is a hybrid process combining: - **Chemical etching**: Reactive slurry chemistry modifies surface properties - **Mechanical abrasion**: Physical removal via abrasive particles and pad The fundamental material removal can be expressed as: $$ \text{Material Removal} = f(\text{Chemical Reaction}, \text{Mechanical Abrasion}) $$ ### 1.2 Process Components | Component | Function | Key Parameters | |-----------|----------|----------------| | **Wafer** | Substrate to be planarized | Material type, pattern density | | **Polishing Pad** | Provides mechanical action | Hardness, porosity, asperity distribution | | **Slurry** | Chemical + abrasive medium | pH, oxidizer, particle size/concentration | | **Carrier** | Holds and rotates wafer | Down force, rotation speed | | **Platen** | Rotates polishing pad | Rotation speed, temperature | ### 1.3 Key Process Parameters - **Down Force ($F$)**: Pressure applied to wafer, typically $1-7$ psi - **Platen Speed ($\omega_p$)**: Pad rotation, typically $20-100$ rpm - **Carrier Speed ($\omega_c$)**: Wafer rotation, typically $20-100$ rpm - **Slurry Flow Rate ($Q$)**: Typically $100-300$ mL/min - **Temperature ($T$)**: Typically $20-50°C$ ## 2. Classical Physical Models ### 2.1 Preston Equation (Foundational Model) The foundational model for CMP is the **Preston equation** (1927): $$ \boxed{MRR = k_p \cdot P \cdot v} $$ Where: - $MRR$ = Material Removal Rate $[\text{nm/min}]$ - $k_p$ = Preston's coefficient $[\text{m}^2/\text{N}]$ - $P$ = Applied pressure $[\text{Pa}]$ - $v$ = Relative velocity $[\text{m/s}]$ The relative velocity between wafer and pad: $$ v = \sqrt{(\omega_p r_p)^2 + (\omega_c r_c)^2 - 2\omega_p \omega_c r_p r_c \cos(\theta)} $$ Where: - $\omega_p, \omega_c$ = Angular velocities of platen and carrier - $r_p, r_c$ = Radial positions - $\theta$ = Phase angle ### 2.2 Modified Preston Models #### 2.2.1 Pressure-Velocity Product Modification $$ MRR = k_p \cdot P^a \cdot v^b $$ Where $a, b$ are empirical exponents (typically $0.5 < a, b < 1.5$) #### 2.2.2 Chemical Enhancement Factor $$ MRR = k_p \cdot P \cdot v \cdot f(C, T, pH) $$ Where $f(C, T, pH)$ represents chemical effects: - $C$ = Oxidizer concentration - $T$ = Temperature - $pH$ = Slurry pH #### 2.2.3 Arrhenius-Modified Preston Equation $$ MRR = k_0 \cdot \exp\left(-\frac{E_a}{RT}\right) \cdot P \cdot v $$ Where: - $k_0$ = Pre-exponential factor - $E_a$ = Activation energy $[\text{J/mol}]$ - $R$ = Gas constant $= 8.314$ J/(mol·K) - $T$ = Temperature $[\text{K}]$ ### 2.3 Tribocorrosion Model For metal CMP (e.g., tungsten, copper): $$ MRR = \frac{M}{z F \rho} \cdot \left( i_{corr} + \frac{Q_{pass}}{A \cdot t_{pass}} \right) \cdot f_{mech} $$ Where: - $M$ = Molar mass of metal - $z$ = Number of electrons transferred - $F$ = Faraday constant $= 96485$ C/mol - $\rho$ = Density - $i_{corr}$ = Corrosion current density - $Q_{pass}$ = Passivation charge - $f_{mech}$ = Mechanical factor ### 2.4 Contact Mode Classification | Mode | Condition | Preston Constant | Friction Coefficient | |------|-----------|------------------|---------------------| | **Contact** | $\frac{\eta v_R}{p} < (\frac{\eta v_R}{p})_c$ | High, constant | High ($\mu > 0.3$) | | **Mixed** | $\frac{\eta v_R}{p} \approx (\frac{\eta v_R}{p})_c$ | Transitional | Medium | | **Hydroplaning** | $\frac{\eta v_R}{p} > (\frac{\eta v_R}{p})_c$ | Low, variable | Low ($\mu < 0.1$) | Where: - $\eta$ = Slurry viscosity - $v_R$ = Relative velocity - $p$ = Pressure ## 3. Pattern Density Models ### 3.1 Effective Pattern Density Model (Stine Model) The local material removal rate depends on effective pattern density: $$ \frac{dz}{dt} = -\frac{K}{\rho_{eff}(x, y)} $$ Where: - $z$ = Surface height - $K$ = Blanket removal rate $= k_p \cdot P \cdot v$ - $\rho_{eff}$ = Effective pattern density #### 3.1.1 Effective Density Calculation $$ \rho_{eff}(x, y) = \iint_{-\infty}^{\infty} \rho_0(x', y') \cdot W(x - x', y - y') \, dx' \, dy' $$ Where: - $\rho_0(x, y)$ = Local pattern density - $W(x, y)$ = Weighting function (planarization kernel) #### 3.1.2 Elliptical Weighting Function $$ W(x, y) = \frac{1}{\pi L_x L_y} \cdot \exp\left(-\frac{x^2}{L_x^2} - \frac{y^2}{L_y^2}\right) $$ Where $L_x, L_y$ are planarization lengths in x and y directions. ### 3.2 Step Height Evolution Model For oxide CMP with step height $h$: $$ \frac{dh}{dt} = -K \cdot \left(1 - \frac{h_{contact}}{h}\right) \quad \text{for } h > h_{contact} $$ $$ \frac{dh}{dt} = 0 \quad \text{for } h \leq h_{contact} $$ Where $h_{contact}$ is the pad contact threshold height. ### 3.3 Integrated Density-Step Height Model Combined model for oxide thickness evolution: $$ z(x, y, t) = z_0 - K \cdot t \cdot \frac{1}{\rho_{eff}(x, y)} \cdot g(h) $$ Where $g(h)$ is the step-height dependent function: $$ g(h) = \begin{cases} 1 & \text{if } h > h_c \\ \frac{h}{h_c} & \text{if } h \leq h_c \end{cases} $$ ## 4. Dishing and Erosion Models ### 4.1 Copper Dishing Model Dishing depth $D$ for copper lines: $$ D = K_{Cu} \cdot t_{over} \cdot f(w) $$ Where: - $K_{Cu}$ = Copper removal rate - $t_{over}$ = Overpolish time - $w$ = Line width - $f(w)$ = Width-dependent function Empirical relationship: $$ D = D_0 \cdot \left(1 - \exp\left(-\frac{w}{w_c}\right)\right) $$ Where: - $D_0$ = Maximum dishing depth - $w_c$ = Critical line width ### 4.2 Oxide Erosion Model Erosion $E$ in dense pattern regions: $$ E = K_{ox} \cdot t_{over} \cdot \rho_{metal} $$ Where: - $K_{ox}$ = Oxide removal rate - $\rho_{metal}$ = Local metal pattern density ### 4.3 Combined Dishing-Erosion Total copper thickness loss: $$ \Delta z_{Cu} = D + E \cdot \frac{\rho_{metal}}{1 - \rho_{metal}} $$ ### 4.4 Pattern Density Effects | Pattern Density | Dishing Behavior | Erosion Behavior | |-----------------|------------------|------------------| | Low ($< 20\%$) | Minimal | Minimal | | Medium ($20-50\%$) | Moderate | Increasing | | High ($> 50\%$) | Saturates | Severe | ## 5. Contact Mechanics Models ### 5.1 Pad Asperity Contact Model Assuming Gaussian asperity height distribution: $$ P(z) = \frac{1}{\sigma_s \sqrt{2\pi}} \exp\left(-\frac{(z - \bar{z})^2}{2\sigma_s^2}\right) $$ Where: - $\sigma_s$ = Standard deviation of asperity heights - $\bar{z}$ = Mean asperity height ### 5.2 Real Contact Area $$ A_r = \pi n \int_{d}^{\infty} R(z - d) \cdot P(z) \, dz $$ Where: - $n$ = Number of asperities per unit area - $R$ = Asperity tip radius - $d$ = Separation distance For Gaussian distribution: $$ A_r = \pi n R \sigma_s \cdot F_1\left(\frac{d}{\sigma_s}\right) $$ Where $F_1$ is a statistical function. ### 5.3 Hertzian Contact For elastic contact between abrasive particle and wafer: $$ a = \left(\frac{3FR}{4E^*}\right)^{1/3} $$ $$ \delta = \frac{a^2}{R} = \left(\frac{9F^2}{16RE^{*2}}\right)^{1/3} $$ Where: - $a$ = Contact radius - $F$ = Normal force - $R$ = Particle radius - $\delta$ = Indentation depth - $E^*$ = Effective elastic modulus $$ \frac{1}{E^*} = \frac{1 - \nu_1^2}{E_1} + \frac{1 - \nu_2^2}{E_2} $$ ### 5.4 Material Removal by Single Abrasive Volume removed per abrasive per pass: $$ V = K_{wear} \cdot \frac{F_n \cdot L}{H} $$ Where: - $K_{wear}$ = Wear coefficient - $F_n$ = Normal force on particle - $L$ = Sliding distance - $H$ = Hardness of wafer material ### 5.5 Multi-Scale Model Framework ``` ┌─────────────────────────────────────────────────────────────┐ │ WAFER SCALE (mm-cm) │ │ Pressure distribution, global uniformity │ ├─────────────────────────────────────────────────────────────┤ │ DIE SCALE (μm-mm) │ │ Pattern density effects, planarization │ ├─────────────────────────────────────────────────────────────┤ │ FEATURE SCALE (nm-μm) │ │ Dishing, erosion, step height evolution │ ├─────────────────────────────────────────────────────────────┤ │ PARTICLE SCALE (nm) │ │ Abrasive-surface interactions │ ├─────────────────────────────────────────────────────────────┤ │ MOLECULAR SCALE (Å) │ │ Chemical reactions, atomic removal │ └─────────────────────────────────────────────────────────────┘ ``` ## 6. Machine Learning and Neural Network Models ### 6.1 Overview of ML Approaches Machine learning methods for CMP modeling: - **Supervised Learning** - Artificial Neural Networks (ANN) - Convolutional Neural Networks (CNN) - Support Vector Machines (SVM) - Random Forests / Gradient Boosting - **Deep Learning** - Deep Belief Networks (DBN) - Long Short-Term Memory (LSTM) - Generative Adversarial Networks (GAN) - **Transfer Learning** - Pre-trained models adapted to new process conditions ### 6.2 Neural Network Architecture for CMP #### 6.2.1 Input Features $$ \mathbf{x} = [P, v, t, \rho, w, s, pH, C_{ox}, T, ...]^T $$ Where: - $P$ = Pressure - $v$ = Velocity - $t$ = Polish time - $\rho$ = Pattern density - $w$ = Feature width - $s$ = Feature spacing - $pH$ = Slurry pH - $C_{ox}$ = Oxidizer concentration - $T$ = Temperature #### 6.2.2 Multi-Layer Perceptron (MLP) $$ \mathbf{h}^{(1)} = \sigma(\mathbf{W}^{(1)} \mathbf{x} + \mathbf{b}^{(1)}) $$ $$ \mathbf{h}^{(2)} = \sigma(\mathbf{W}^{(2)} \mathbf{h}^{(1)} + \mathbf{b}^{(2)}) $$ $$ \hat{y} = \mathbf{W}^{(out)} \mathbf{h}^{(2)} + \mathbf{b}^{(out)} $$ Where: - $\sigma$ = Activation function (ReLU, tanh, sigmoid) - $\mathbf{W}^{(i)}$ = Weight matrices - $\mathbf{b}^{(i)}$ = Bias vectors #### 6.2.3 Activation Functions | Function | Formula | Use Case | |----------|---------|----------| | **ReLU** | $\sigma(x) = \max(0, x)$ | Hidden layers | | **Sigmoid** | $\sigma(x) = \frac{1}{1 + e^{-x}}$ | Output (binary) | | **Tanh** | $\sigma(x) = \frac{e^x - e^{-x}}{e^x + e^{-x}}$ | Hidden layers | | **Softmax** | $\sigma(x_i) = \frac{e^{x_i}}{\sum_j e^{x_j}}$ | Classification | ### 6.3 CNN-Based CMP Modeling (CmpCNN) #### 6.3.1 Architecture ``` Input: Layout Image (Binary) + Density Map ↓ Conv2D Layer (3×3 kernel, 32 filters) ↓ MaxPooling2D (2×2) ↓ Conv2D Layer (3×3 kernel, 64 filters) ↓ MaxPooling2D (2×2) ↓ Flatten ↓ Dense Layer (256 units) ↓ Dense Layer (128 units) ↓ Output: Post-CMP Height Map ``` #### 6.3.2 Convolution Operation $$ (I * K)(i, j) = \sum_m \sum_n I(i+m, j+n) \cdot K(m, n) $$ Where: - $I$ = Input image (layout) - $K$ = Convolution kernel - $(i, j)$ = Output position ### 6.4 Loss Functions #### 6.4.1 Mean Squared Error (MSE) $$ \mathcal{L}_{MSE} = \frac{1}{N} \sum_{i=1}^{N} (y_i - \hat{y}_i)^2 $$ #### 6.4.2 Root Mean Square Error (RMSE) $$ RMSE = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (y_i - \hat{y}_i)^2} $$ #### 6.4.3 Mean Absolute Percentage Error (MAPE) $$ MAPE = \frac{100\%}{N} \sum_{i=1}^{N} \left| \frac{y_i - \hat{y}_i}{y_i} \right| $$ ### 6.5 Transfer Learning Framework For adapting models across process nodes: $$ \mathcal{L}_{transfer} = \mathcal{L}_{target} + \lambda \cdot \mathcal{L}_{domain} $$ Where: - $\mathcal{L}_{target}$ = Target domain loss - $\mathcal{L}_{domain}$ = Domain adaptation loss - $\lambda$ = Regularization parameter ### 6.6 Performance Metrics | Metric | Formula | Target | |--------|---------|--------| | $R^2$ | $1 - \frac{\sum(y_i - \hat{y}_i)^2}{\sum(y_i - \bar{y})^2}$ | $> 0.95$ | | RMSE | $\sqrt{\frac{1}{N}\sum(y_i - \hat{y}_i)^2}$ | $< 5$ Å | | MAE | $\frac{1}{N}\sum|y_i - \hat{y}_i|$ | $< 3$ Å | ## 7. Slurry Chemistry Modeling ### 7.1 Kaufman Mechanism Cyclic passivation-depassivation process: $$ \text{Metal} \xrightarrow{\text{Oxidizer}} \text{Metal Oxide} \xrightarrow{\text{Abrasion}} \text{Removal} $$ ### 7.2 Electrochemical Reactions #### 7.2.1 Copper CMP **Oxidation:** $$ \text{Cu} \rightarrow \text{Cu}^{2+} + 2e^- $$ **Passivation (with BTA):** $$ \text{Cu} + \text{BTA} \rightarrow \text{Cu-BTA}_{film} $$ **Complexation:** $$ \text{Cu}^{2+} + n\text{L} \rightarrow [\text{CuL}_n]^{2+} $$ Where L = chelating agent (e.g., glycine, citrate) #### 7.2.2 Tungsten CMP **Oxidation:** $$ \text{W} + 3\text{H}_2\text{O} \rightarrow \text{WO}_3 + 6\text{H}^+ + 6e^- $$ **With hydrogen peroxide:** $$ \text{W} + 3\text{H}_2\text{O}_2 \rightarrow \text{WO}_3 + 3\text{H}_2\text{O} $$ ### 7.3 Pourbaix Diagram Integration Stability regions defined by: $$ E = E^0 - \frac{RT}{nF} \ln Q - \frac{RT}{F} \cdot m \cdot pH $$ Where: - $E$ = Electrode potential - $E^0$ = Standard potential - $Q$ = Reaction quotient - $m$ = Number of H⁺ in reaction ### 7.4 Abrasive Particle Effects #### 7.4.1 Particle Size Distribution (PSD) Log-normal distribution: $$ f(d) = \frac{1}{d \sigma \sqrt{2\pi}} \exp\left(-\frac{(\ln d - \mu)^2}{2\sigma^2}\right) $$ Where: - $d$ = Particle diameter - $\mu$ = Mean of $\ln(d)$ - $\sigma$ = Standard deviation of $\ln(d)$ #### 7.4.2 Zeta Potential $$ \zeta = \frac{4\pi \eta \mu_e}{\varepsilon} $$ Where: - $\eta$ = Viscosity - $\mu_e$ = Electrophoretic mobility - $\varepsilon$ = Dielectric constant ### 7.5 Slurry Components Summary | Component | Function | Typical Materials | |-----------|----------|-------------------| | **Abrasive** | Mechanical removal | SiO₂, CeO₂, Al₂O₃ | | **Oxidizer** | Surface modification | H₂O₂, KIO₃, Fe(NO₃)₃ | | **Complexant** | Metal dissolution | Glycine, citric acid | | **Inhibitor** | Corrosion protection | BTA, BBI | | **Surfactant** | Particle dispersion | CTAB, SDS | | **Buffer** | pH control | Phosphate, citrate | ## 8. Chip-Scale and Full-Chip Models ### 8.1 Within-Wafer Non-Uniformity (WIWNU) $$ WIWNU = \frac{\sigma_{thickness}}{\bar{thickness}} \times 100\% $$ Where: - $\sigma_{thickness}$ = Standard deviation of thickness - $\bar{thickness}$ = Mean thickness ### 8.2 Pressure Distribution Model For a flexible carrier: $$ P(r) = P_0 + \sum_{i=1}^{n} P_i \cdot J_0\left(\frac{\alpha_i r}{R}\right) $$ Where: - $P_0$ = Base pressure - $J_0$ = Bessel function of first kind - $\alpha_i$ = Bessel zeros - $R$ = Wafer radius ### 8.3 Multi-Zone Pressure Control For zone $i$: $$ MRR_i = k_p \cdot P_i \cdot v_i $$ Target uniformity achieved when: $$ MRR_1 = MRR_2 = ... = MRR_n $$ ### 8.4 Full-Chip Simulation Flow ``` ┌─────────────────────┐ │ Design Layout (GDS)│ └──────────┬──────────┘ ↓ ┌─────────────────────┐ │ Density Extraction │ │ ρ(x,y) for each │ │ metal/dielectric │ └──────────┬──────────┘ ↓ ┌─────────────────────┐ │ Effective Density │ │ ρ_eff = ρ * W │ └──────────┬──────────┘ ↓ ┌─────────────────────┐ │ CMP Simulation │ │ z(t) evolution │ └──────────┬──────────┘ ↓ ┌─────────────────────┐ │ Post-CMP Topography │ │ Dishing/Erosion Map │ └──────────┬──────────┘ ↓ ┌─────────────────────┐ │ Hotspot Detection │ │ Design Rule Check │ └─────────────────────┘ ``` ## 9. Process Control Applications ### 9.1 Run-to-Run (R2R) Control #### 9.1.1 EWMA Controller $$ \hat{y}_{k+1} = \lambda y_k + (1 - \lambda) \hat{y}_k $$ Where: - $\hat{y}_{k+1}$ = Predicted output for next run - $y_k$ = Current measured output - $\lambda$ = Smoothing factor $(0 < \lambda < 1)$ #### 9.1.2 Recipe Adjustment $$ u_{k+1} = u_k + G^{-1} (y_{target} - \hat{y}_{k+1}) $$ Where: - $u$ = Process recipe (time, pressure, etc.) - $G$ = Process gain matrix - $y_{target}$ = Target output ### 9.2 Virtual Metrology $$ \hat{y} = f_{VM}(\mathbf{x}_{FDC}) $$ Where: - $\hat{y}$ = Predicted wafer quality - $\mathbf{x}_{FDC}$ = Fault Detection and Classification sensor data ### 9.3 Endpoint Detection #### 9.3.1 Motor Current Monitoring $$ I(t) = I_0 + \Delta I \cdot H(t - t_{endpoint}) $$ Where $H$ is the Heaviside step function. #### 9.3.2 Optical Endpoint $$ R(\lambda, t) = R_{film}(\lambda, d(t)) $$ Where reflectance $R$ changes as film thickness $d$ decreases. ## 10. Current Challenges and Future Directions ### 10.1 Key Challenges - **Sub-5nm nodes**: Atomic-scale precision required - Thickness variation target: $< 5$ Å (3σ) - Defect density target: $< 0.01$ defects/cm² - **New materials integration**: - Low-κ dielectrics ($\kappa < 2.5$) - Cobalt interconnects - Ruthenium barrier layers - **3D integration**: - Through-Silicon Via (TSV) CMP - Hybrid bonding surface preparation - Wafer-level packaging ### 10.2 Future Model Development - **Physics-informed neural networks (PINNs)**: $$ \mathcal{L} = \mathcal{L}_{data} + \lambda_{physics} \cdot \mathcal{L}_{physics} $$ Where: $$ \mathcal{L}_{physics} = \left\| \frac{\partial z}{\partial t} + \frac{K}{\rho_{eff}} \right\|^2 $$ - **Digital twins** for real-time process optimization - **Federated learning** across multiple fabs ### 10.3 Industry Requirements | Node | Thickness Uniformity | Defect Density | Dishing Limit | |------|---------------------|----------------|---------------| | 7nm | $< 10$ Å | $< 0.05$/cm² | $< 200$ Å | | 5nm | $< 7$ Å | $< 0.03$/cm² | $< 150$ Å | | 3nm | $< 5$ Å | $< 0.01$/cm² | $< 100$ Å | | 2nm | $< 3$ Å | $< 0.005$/cm² | $< 50$ Å | ## Symbol Glossary | Symbol | Description | Units | |--------|-------------|-------| | $MRR$ | Material Removal Rate | nm/min | | $k_p$ | Preston coefficient | m²/N | | $P$ | Pressure | Pa, psi | | $v$ | Relative velocity | m/s | | $\rho$ | Pattern density | dimensionless | | $\rho_{eff}$ | Effective pattern density | dimensionless | | $L$ | Planarization length | μm | | $D$ | Dishing depth | Å, nm | | $E$ | Erosion depth | Å, nm | | $w$ | Feature width | nm, μm | | $h$ | Step height | nm | | $t$ | Polish time | s, min | | $T$ | Temperature | K, °C | | $\eta$ | Viscosity | Pa·s | | $\mu$ | Friction coefficient | dimensionless | ## Key Equations Summary ### Preston Equation $$ MRR = k_p \cdot P \cdot v $$ ### Effective Density $$ \rho_{eff}(x,y) = \iint \rho_0(x',y') \cdot W(x-x', y-y') \, dx' dy' $$ ### Material Removal (Density Model) $$ \frac{dz}{dt} = -\frac{K}{\rho_{eff}(x,y)} $$ ### Dishing Model $$ D = D_0 \cdot \left(1 - e^{-w/w_c}\right) $$ ### Erosion Model $$ E = K_{ox} \cdot t_{over} \cdot \rho_{metal} $$ ### Neural Network $$ \hat{y} = \sigma(\mathbf{W}^{(n)} \cdot ... \cdot \sigma(\mathbf{W}^{(1)} \mathbf{x} + \mathbf{b}^{(1)}) + \mathbf{b}^{(n)}) $$
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Co-training trains multiple models on different views of data where each model labels examples for others' training sets.
Train multiple models on different views of data and share predictions.
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Train from coarse to fine details.
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Cobalt contacts provide low resistivity at small dimensions through conformal selective deposition.
Use Co for small vias and local interconnects.
Common silicide for contacts.
Cocktail party problem refers to separating individual speakers from mixed audio in multi-talker scenarios.
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Similar to code-switching.
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Codebook learning discovers discrete symbols for efficient representation of continuous data.