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

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

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

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

$$ \frac{1}{E^*} = \frac{1 - u_1^2}{E_1} + \frac{1 - u_2^2}{E_2} $$

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

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

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

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

Symbol Glossary

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 $$ \hat{y} = \sigma(\mathbf{W}^{(n)} \cdot ... \cdot \sigma(\mathbf{W}^{(1)} \mathbf{x} + \mathbf{b}^{(1)}) + \mathbf{b}^{(n)}) $$