Mathematical Modeling of Metal Deposition in Semiconductor Manufacturing

Keywords: metal deposition, CVD, PVD, ALD, sputtering, electroplating, copper

Mathematical Modeling of Metal Deposition in Semiconductor Manufacturing

1. Overview: Metal Deposition Processes

Metal deposition is a critical step in semiconductor fabrication, creating interconnects, contacts, barrier layers, and various metallic structures. The primary deposition methods require distinct mathematical treatments:

2. Transport Phenomena Models

2.1 Gas-Phase Transport (CVD/PECVD)

The precursor concentration field follows the convection-diffusion-reaction equation:

Where:

• $C$ — precursor concentration (mol/m³)
• $\mathbf{v}$ — velocity field vector (m/s)
• $D$ — diffusion coefficient (m²/s)
• $R_{gas}$ — gas-phase reaction source term (mol/m³$\cdot$s)

2.2 Flow Field Equations

The incompressible Navier-Stokes equations govern the velocity field:

With continuity equation:

Where:

• $\rho$ — gas density (kg/m³)
• $p$ — pressure (Pa)
• $\mu$ — dynamic viscosity (Pa$\cdot$s)

2.3 Knudsen Number and Transport Regimes

At low pressures, the Knudsen number determines the transport regime:

Where:

• $\lambda$ — mean free path (m)
• $L$ — characteristic length (m)
• $k_B$ — Boltzmann constant ($1.38 \times 10^{-23}$ J/K)
• $T$ — temperature (K)
• $d$ — molecular diameter (m)
• $p$ — pressure (Pa)

Transport regime classification:

• $Kn < 0.01$ — Continuum regime → Navier-Stokes CFD
• $0.01 < Kn < 0.1$ — Slip flow regime → Modified NS with slip boundary conditions
• $0.1 < Kn < 10$ — Transitional regime → DSMC, Boltzmann equation
• $Kn > 10$ — Free molecular regime → Ballistic/Monte Carlo methods

3. Surface Reaction Kinetics

3.1 Langmuir-Hinshelwood Mechanism

For bimolecular surface reactions (common in CVD):

Where:

• $r$ — reaction rate (mol/m²$\cdot$s)
• $k$ — surface reaction rate constant (mol/m²$\cdot$s)
• $K_A, K_B$ — adsorption equilibrium constants (Pa⁻¹)
• $p_A, p_B$ — partial pressures of reactants A and B (Pa)

3.2 Sticking Coefficient Model

The probability that an impinging molecule adsorbs on the surface:

Where:

• $S$ — sticking coefficient (dimensionless)
• $S_0$ — pre-exponential sticking factor
• $E_a$ — activation energy (J)
• $f(\theta) = (1 - \theta)^n$ — site blocking function
• $\theta$ — surface coverage (dimensionless, 0 to 1)
• $n$ — order of site blocking

3.3 Arrhenius Temperature Dependence

Where:

• $A$ — pre-exponential factor (frequency factor)
• $E_a$ — activation energy (J/mol)
• $R$ — universal gas constant (8.314 J/mol$\cdot$K)
• $T$ — absolute temperature (K)

4. Film Growth Models

4.1 Continuum Surface Evolution

Edwards-Wilkinson Equation (Linear Growth)

Kardar-Parisi-Zhang (KPZ) Equation (Nonlinear Growth)

Where:

• $h(\mathbf{x}, t)$ — surface height at position $\mathbf{x}$ and time $t$
• $

u$ — surface diffusion coefficient (m²/s)

• $\lambda$ — nonlinear growth parameter
• $F$ — mean deposition flux (m/s)
• $\eta$ — stochastic noise term (Gaussian white noise)

4.2 Scaling Relations

Surface roughness evolves according to:

Where:

• $W$ — interface width (roughness)
• $L$ — system size
• $\alpha$ — roughness exponent
• $z$ — dynamic exponent
• $f$ — scaling function

5. Step Coverage and Conformality

5.1 Thiele Modulus

For high-aspect-ratio features, the Thiele modulus determines conformality:

Where:

• $\phi$ — Thiele modulus (dimensionless)
• $L$ — feature depth (m)
• $k_s$ — surface reaction rate constant (m/s)
• $D_{eff}$ — effective diffusivity (m²/s)

Step coverage regimes:

• $\phi \ll 1$ — Reaction-limited → Excellent conformality
• $\phi \gg 1$ — Transport-limited → Poor step coverage (bread-loafing)

5.2 Knudsen Diffusion in Trenches

Where:

• $D_K$ — Knudsen diffusion coefficient (m²/s)
• $w$ — trench width (m)
• $R$ — universal gas constant (J/mol$\cdot$K)
• $T$ — temperature (K)
• $M$ — molecular weight (kg/mol)

5.3 Feature-Scale Concentration Profile

Solving for concentration in a trench with reactive walls:

General solution:

6. Atomic Layer Deposition (ALD) Models

6.1 Self-Limiting Surface Kinetics

Surface site balance equation:

Where:

• $\theta$ — fractional surface coverage
• $k_a$ — adsorption rate constant (m³/mol$\cdot$s)
• $k_d$ — desorption rate constant (s⁻¹)
• $C$ — gas-phase precursor concentration (mol/m³)

At equilibrium saturation:

6.2 Growth Per Cycle (GPC)

Where:

• $\Gamma_0$ — surface site density (sites/m²)
• $\Omega$ — volume per deposited atom (m³)
• $\eta$ — reaction efficiency (dimensionless)

6.3 Saturation Dose-Time Relationship

Impingement flux from kinetic theory:

Where:

• $\Phi$ — molecular impingement flux (molecules/m²$\cdot$s)
• $p$ — precursor partial pressure (Pa)
• $m$ — molecular mass (kg)

7. Plasma Modeling (PVD/PECVD)

7.1 Plasma Sheath Physics

Child-Langmuir law for ion current density:

Where:

• $J_{ion}$ — ion current density (A/m²)
• $\varepsilon_0$ — vacuum permittivity ($8.85 \times 10^{-12}$ F/m)
• $e$ — elementary charge ($1.6 \times 10^{-19}$ C)
• $M_i$ — ion mass (kg)
• $V_s$ — sheath voltage (V)
• $d_s$ — sheath thickness (m)

7.2 Ion Energy at Substrate

Bohm velocity:

Where:

• $T_e$ — electron temperature (K or eV)

7.3 Sputtering Yield (Sigmund Formula)

Where:

• $Y$ — sputtering yield (atoms/ion)
• $\alpha$ — dimensionless factor (~0.2–0.4)
• $M_1$ — incident ion mass
• $M_2$ — target atom mass
• $E$ — incident ion energy (eV)
• $U_0$ — surface binding energy (eV)

7.4 Electron Energy Distribution Function (EEDF)

The Boltzmann equation in energy space:

Where:

• $f$ — electron energy distribution function
• $\mathbf{E}$ — electric field
• $m_e$ — electron mass
• $C[f]$ — collision integral

8. MDP: Markov Decision Process for Process Control

8.1 MDP Formulation

A Markov Decision Process is defined by the tuple:

Components in semiconductor context:

• State space $S$: Film thickness, resistivity, uniformity, equipment state, wafer position
• Action space $A$: Temperature, pressure, flow rates, RF power, deposition time
• Transition probability $P(s' | s, a)$: Stochastic process model
• Reward function $R(s, a)$: Yield, uniformity, throughput, quality metrics
• Discount factor $\gamma$: Time preference (typically 0.9–0.99)

8.2 Bellman Optimality Equation

Q-function formulation:

8.3 Run-to-Run (R2R) Control

Optimal recipe adjustment after each wafer:

Where:

• $\mathbf{u}_k$ — process recipe parameters at run $k$
• $\mathbf{y}_k$ — measured output at run $k$
• $\mathbf{K}$ — controller gain matrix (from MDP policy optimization)

8.4 Reinforcement Learning Approaches

9. Electrochemical Deposition (Copper Damascene)

9.1 Butler-Volmer Equation

Where:

• $i$ — current density (A/m²)
• $i_0$ — exchange current density (A/m²)
• $\alpha_a, \alpha_c$ — anodic and cathodic transfer coefficients
• $F$ — Faraday constant (96,485 C/mol)
• $\eta = E - E_{eq}$ — overpotential (V)
• $R$ — gas constant (J/mol$\cdot$K)
• $T$ — temperature (K)

9.2 Mass Transport Limited Current

Where:

• $i_L$ — limiting current density (A/m²)
• $n$ — number of electrons transferred
• $D$ — diffusion coefficient of Cu²⁺ (m²/s)
• $C_b$ — bulk concentration (mol/m³)
• $\delta$ — diffusion layer thickness (m)

9.3 Nernst-Planck Equation

Where:

• $\mathbf{J}_i$ — flux of species $i$
• $z_i$ — charge number
• $\phi$ — electric potential

9.4 Superfilling (Bottom-Up Fill)

The curvature-enhanced accelerator mechanism:

Where:

• $v_n$ — local growth velocity normal to surface
• $v_0$ — baseline growth velocity
• $\kappa$ — local surface curvature (1/m)
• $\Gamma_{acc}$ — accelerator surface concentration

10. Multiscale Modeling Framework

10.1 Hierarchical Scale Integration

-
┌──────────────────────────────────────────────────────────────┐
│                     REACTOR SCALE                            │
│            CFD: Flow, temperature, concentration             │
│                   Time: seconds | Length: cm                 │
└─────────────────────────┬────────────────────────────────────┘
                          │ Boundary fluxes
                          ▼
┌──────────────────────────────────────────────────────────────┐
│                     FEATURE SCALE                            │
│        Level-set / String method for surface evolution       │
│                   Time: seconds | Length: $\mu$m             │
└─────────────────────────┬────────────────────────────────────┘
                          │ Local rates
                          ▼
┌──────────────────────────────────────────────────────────────┐
│                    MESOSCALE (kMC)                           │
│          Kinetic Monte Carlo: nucleation, island growth      │
│                    Time: ms | Length: nm                     │
└─────────────────────────┬────────────────────────────────────┘
                          │ Rate parameters
                          ▼
┌──────────────────────────────────────────────────────────────┐
│                  ATOMISTIC (MD/DFT)                          │
│       Molecular dynamics, ab initio: binding energies,       │
│              diffusion barriers, reaction paths              │
│                    Time: ps | Length: Å                      │
└──────────────────────────────────────────────────────────────┘

10.2 Kinetic Monte Carlo (kMC)

Event rate from transition state theory:

Total rate and time step:

Where $r \in (0, 1]$ is a uniform random number.

10.3 Molecular Dynamics

Newton's equations of motion:

Lennard-Jones potential:

Embedded Atom Method (EAM) for metals:

Where $\rho_i = \sum_{j eq i} f_j(r_{ij})$ is the electron density at atom $i$.

11. Uniformity Modeling

11.1 Wafer-Scale Thickness Distribution (Sputtering)

For a circular magnetron target:

Where:

• $t(r)$ — thickness at radial position $r$
• $\theta_t$ — emission angle from target
• $\theta_w$ — incidence angle at wafer

11.2 Uniformity Metrics

Within-Wafer Uniformity (WIW):

Wafer-to-Wafer Uniformity (WTW):

Target specifications:

• $\sigma_{WIW} < 1\%$ for advanced nodes (≤7 nm)
• $\sigma_{WTW} < 0.5\%$ for high-volume manufacturing

12. Virtual Metrology and Statistical Models

12.1 Gaussian Process Regression (GPR)

Squared exponential (RBF) kernel:

Predictive distribution:

12.2 Partial Least Squares (PLS)

Where:

• $\mathbf{X}$ — process parameter matrix
• $\mathbf{Y}$ — quality outcome matrix
• $\mathbf{B}$ — regression coefficient matrix
• $\mathbf{E}$ — residual matrix

12.3 Principal Component Analysis (PCA)

Hotelling's $T^2$ statistic for fault detection:

13. Process Optimization

13.1 Response Surface Methodology (RSM)

Second-order polynomial model:

13.2 Constrained Optimization

Example constraints:

• $g_1$: Non-uniformity ≤ 3%
• $g_2$: Resistivity within spec
• $g_3$: Throughput ≥ target
• $h_1$: Total film thickness = target

13.3 Pareto Multi-Objective Optimization

Common trade-offs:

• Uniformity vs. throughput
• Film quality vs. cost
• Conformality vs. deposition rate

14. Mathematical Toolkit

15. Physical Constants

Source: ChipFoundryServices — Search this topic — Ask CFSGPT