CMP Modeling

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 - 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

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

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)})

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CMP Process Cross-Section — Polishing Head, Pad, and Slurry Interaction. CMP removes material by pressing a rotating wafer face-down against a polyurethane pad while a chemically active slurry of abrasive nanoparticles (30–200 nm silica or ceria) flows between them. The material removal rate follows the Preston equation: $MRR = k_p \times P \times V$ where $$P$$ is the applied pressure (1–7 psi) and $$V$$ is the relative velocity (0.5–2 m/s). The Preston coefficient $$k_p$$ encapsulates all the chemistry — slurry pH, oxidizer concentration, particle loading, and pad condition — and ranges from $10^{-14}$ to $10^{-12}$ m $$^2$$ /N depending on the film being polished.

CMP Dishing and Erosion — The Two Failure Modes That Limit Scaling. Dishing is the concave over-removal of soft material (copper) relative to the surrounding hard barrier (Ta/TaN), creating a recess of 10–50 nm in wide metal lines that increases resistance and degrades reliability. Erosion is the thinning of the surrounding dielectric in dense arrays where pattern density exceeds 50%, removing 5–30 nm of oxide that shifts via landing depths and causes opens. Both scale with feature geometry: dishing increases with line width (100 nm line dishes 5 nm vs 10 $\mu$ m line dishes 40 nm), while erosion increases with pattern density. The Preston equation cannot predict these non-uniformities because it assumes a flat wafer — real planarization requires pattern-density-dependent models (Stine, Ouma) that integrate the local effective pressure distribution across topography.

CMP Process Steps in a Modern Interconnect Flow. A 2 nm node BEOL integration requires 30–50 CMP steps across 13–15 metal layers. Each copper damascene level involves: (1) bulk Cu removal (high rate, 500–800 nm/min, non-selective), (2) barrier clear (moderate rate, 100–200 nm/min, Cu:Ta selectivity 50:1), and (3) buff/touch-up (low rate, 20–50 nm/min, final surface quality). Total CMP time per metal level: 2–4 minutes across these three platens. Front-end CMP includes STI oxide planarization (removing 200–400 nm step height to $$<$$ 5 nm remaining topography), poly-Si or replacement-metal-gate CMP (stopping on oxide with $$<$$ 2 nm dishing), and interlayer dielectric CMP before via patterning. The slurry chemistry switches completely between steps: acidic H $$_2$$ O $$_2$$ -based for copper (pH 2–4), alkaline KOH-based for oxide (pH 10–11), and ceria-based for STI (high selectivity oxide:nitride $$>$$ 50:1).

CMP Equipment Market and Productivity. The CMP equipment market reached approximately 4 billion USD in 2023, dominated by Applied Materials Reflexion platform ( $\sim$ 65% share) and Ebara FREX ( $\sim$ 25%). The consumables market (pads + slurry + conditioners) adds another 3 billion USD annually — Entegris/CMC Materials for slurries, DuPont for pads, and Fujimi/Cabot for specialty abrasives. A modern CMP tool processes 40–60 wafers per hour on a multi-platen architecture (3 platens + 1 cleaner), with each platen running a different recipe step. Downtime is dominated by pad replacement (every 500–1,000 wafers) and conditioner diamond disk changes (every 2,000–5,000 wafers). The largest productivity improvement of the past decade was the move from batch-conditioning (offline) to in-situ conditioning (diamond sweeps the pad during polishing), which eliminated 15% of non-productive time and improved pad lifetime by 30%.

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