Mathematics Modeling

1. Crystal Growth (Czochralski Process)

Growing single-crystal silicon ingots requires coupled models for heat transfer, fluid flow, and mass transport.

1.1 Heat Transfer Equation

Variables:

• $\rho$ — density ($\text{kg/m}^3$)
• $c_p$ — specific heat capacity ($\text{J/(kg·K)}$)
• $T$ — temperature ($\text{K}$)
• $\mathbf{v}$ — velocity vector ($\text{m/s}$)
• $k$ — thermal conductivity ($\text{W/(m·K)}$)
• $Q$ — heat source term ($\text{W/m}^3$)

1.2 Melt Convection Drivers

• Buoyancy forces — thermal and solutal gradients
• Marangoni flow — surface tension gradients
• Forced convection — crystal and crucible rotation

1.3 Dopant Segregation

Equilibrium segregation coefficient:

Effective segregation coefficient (Burton-Prim-Slichter model):

Variables:

• $C_s$ — dopant concentration in solid
• $C_l$ — dopant concentration in liquid
• $v$ — crystal growth velocity
• $\delta$ — boundary layer thickness
• $D$ — diffusion coefficient in melt

2. Thermal Oxidation (Deal-Grove Model)

The foundational model for growing $\text{SiO}_2$ on silicon.

2.1 General Equation

Variables:

• $x_o$ — oxide thickness ($\mu\text{m}$ or $\text{nm}$)
• $A$ — linear rate constant parameter
• $B$ — parabolic rate constant
• $t$ — oxidation time
• $\tau$ — time offset for initial oxide

2.2 Growth Regimes

• Linear regime (thin oxide, surface-reaction limited):

• Parabolic regime (thick oxide, diffusion limited):

2.3 Extended Model Considerations

• Stress-dependent oxidation rates
• Point defect injection into silicon
• 2D/3D geometries (LOCOS bird's beak)
• High-pressure oxidation kinetics
• Thin oxide regime anomalies (<20 nm)

3. Diffusion and Dopant Transport

3.1 Fick's Laws

First Law (flux equation):

Second Law (continuity equation):

For constant $D$:

3.2 Concentration-Dependent Diffusivity

Variables:

• $D_i$ — intrinsic diffusivity
• $D^{-}, D^{2-}$ — diffusivity via negatively charged defects
• $D^{+}, D^{2+}$ — diffusivity via positively charged defects
• $n, p$ — electron and hole concentrations
• $n_i$ — intrinsic carrier concentration

3.3 Point-Defect Mediated Diffusion

Effective diffusivity:

Point defect continuity equations:

Recombination rate:

Variables:

• $C_I, C_V$ — interstitial and vacancy concentrations
• $C_I^, C_V^$ — equilibrium concentrations
• $G_I, G_V$ — generation rates
• $R_{IV}$ — interstitial-vacancy recombination rate

3.4 Transient Enhanced Diffusion (TED)

Ion implantation creates excess interstitials causing:

• "+1" model: each implanted ion creates one net interstitial
• Enhanced diffusion persists until excess defects anneal out
• Critical for ultra-shallow junction formation

4. Ion Implantation

4.1 Gaussian Profile Model

Variables:

• $N(x)$ — dopant concentration at depth $x$ ($\text{cm}^{-3}$)
• $\phi$ — implant dose ($\text{ions/cm}^2$)
• $R_p$ — projected range (mean depth)
• $\Delta R_p$ — straggle (standard deviation)

4.2 Pearson IV Distribution

For asymmetric profiles using four moments:

• First moment: $R_p$ (projected range)
• Second moment: $\Delta R_p$ (straggle)
• Third moment: $\gamma$ (skewness)
• Fourth moment: $\beta$ (kurtosis)

4.3 Monte Carlo Methods (TRIM/SRIM)

Stopping power:

• $S_n(E)$ — nuclear stopping power
• $S_e(E)$ — electronic stopping power

Key outputs:

• Ion trajectories via binary collision approximation (BCA)
• Damage cascade distribution
• Sputtering yield
• Vacancy and interstitial generation profiles

4.4 Channeling Effects

For crystalline targets, ions aligned with crystal axes experience:

• Reduced stopping power
• Deeper penetration
• Modified range distributions
• Requires dual-Pearson or Monte Carlo models

5. Plasma Etching

5.1 Surface Kinetics Model

Variables:

• $\theta$ — fractional surface coverage of reactive species
• $J_i$ — incident ion/radical flux
• $s_i$ — sticking coefficient
• $k_r$ — surface reaction rate constant

5.2 Etching Yield

Dependence factors:

• Ion energy ($E_{ion}$)
• Ion incidence angle ($\theta$)
• Ion-to-neutral flux ratio
• Surface chemistry and temperature

5.3 Profile Evolution (Level Set Method)

Variables:

• $\phi(\mathbf{x}, t)$ — level set function (surface defined by $\phi = 0$)
• $V$ — local etch rate (normal velocity)

5.4 Knudsen Transport in High Aspect Ratio Features

For molecular flow regime ($Kn > 1$):

Key effects:

• Aspect ratio dependent etching (ARDE)
• Reactive ion angular distribution (RIAD)
• Neutral shadowing

6. Chemical Vapor Deposition (CVD)

6.1 Transport-Reaction Equation

Variables:

• $C$ — reactant concentration
• $\mathbf{v}$ — gas velocity
• $D$ — gas-phase diffusivity
• $k$ — reaction rate constant
• $n$ — reaction order

6.2 Thiele Modulus

Regimes:

• $\phi \ll 1$ — reaction-limited (uniform deposition)
• $\phi \gg 1$ — transport-limited (poor step coverage)

6.3 Step Coverage

Conformality factor:

Models:

• Ballistic transport (line-of-sight)
• Knudsen diffusion
• Surface reaction probability

6.4 Atomic Layer Deposition (ALD)

Self-limiting surface coverage:

Variables:

• $\theta(t)$ — fractional surface coverage
• $p$ — precursor partial pressure
• $\tau$ — characteristic adsorption time

Growth per cycle (GPC):

where $\Gamma_{ML}$ is the monolayer thickness.

7. Chemical Mechanical Polishing (CMP)

7.1 Preston Equation

Variables:

• $dz/dt$ — material removal rate (MRR)
• $K_p$ — Preston coefficient ($\text{m}^2/\text{N}$)
• $P$ — applied pressure
• $V$ — relative velocity

7.2 Pattern-Dependent Effects

Effective pressure:

where $\rho_{pattern}$ is local pattern density.

Key phenomena:

• Dishing: over-polishing of soft materials (e.g., Cu)
• Erosion: oxide loss in high-density regions
• Within-die non-uniformity (WIDNU)

7.3 Contact Mechanics

Hertzian contact pressure:

Pad asperity models:

• Greenwood-Williamson for rough surfaces
• Viscoelastic pad behavior

8. Lithography

8.1 Aerial Image Formation

Hopkins formulation (partially coherent):

Variables:

• $I(\mathbf{x})$ — intensity at image plane position $\mathbf{x}$
• $TCC$ — transmission cross-coefficient
• $M(\mathbf{f})$ — mask spectrum at spatial frequency $\mathbf{f}$

8.2 Resolution and Depth of Focus

Rayleigh resolution criterion:

Depth of focus:

Variables:

• $\lambda$ — exposure wavelength (e.g., 193 nm for DUV, 13.5 nm for EUV)
• $NA$ — numerical aperture
• $k_1, k_2$ — process-dependent factors

8.3 Photoresist Exposure (Dill Model)

Photoactive compound (PAC) decomposition:

Intensity attenuation:

Dill parameters:

• $A$ — bleachable absorption coefficient
• $B$ — non-bleachable absorption coefficient
• $C$ — exposure rate constant
• $m$ — normalized PAC concentration

8.4 Development Rate (Mack Model)

Variables:

• $r$ — development rate
• $r_{max}$ — maximum development rate
• $m$ — normalized PAC concentration
• $a, n$ — resist contrast parameters

8.5 Computational Lithography

• Optical Proximity Correction (OPC): inverse problem to find mask patterns
• Source-Mask Optimization (SMO): co-optimize illumination and mask
• Inverse Lithography Technology (ILT): pixel-based mask optimization

9. Device Simulation (TCAD)

9.1 Poisson's Equation

Variables:

• $\psi$ — electrostatic potential
• $\epsilon$ — permittivity
• $q$ — elementary charge
• $n, p$ — electron and hole concentrations
• $N_D^+, N_A^-$ — ionized donor and acceptor concentrations

9.2 Carrier Continuity Equations

Electrons:

Holes:

Variables:

• $\mathbf{J}_n, \mathbf{J}_p$ — electron and hole current densities
• $G$ — carrier generation rate
• $R$ — carrier recombination rate

9.3 Drift-Diffusion Current Equations

Electron current:

Hole current:

Einstein relation:

9.4 Advanced Transport Models

• Hydrodynamic model: includes carrier temperature
• Monte Carlo: tracks individual carrier scattering events
• Quantum corrections: density gradient, NEGF for tunneling

10. Yield Modeling

10.1 Poisson Yield Model

Variables:

• $Y$ — chip yield
• $A$ — chip area
• $D_0$ — defect density ($\text{defects/cm}^2$)

10.2 Negative Binomial Model (Clustered Defects)

Variables:

• $\alpha$ — clustering parameter
• As $\alpha \to \infty$, reduces to Poisson model

10.3 Critical Area Analysis

Variables:

• $D_i$ — defect density for defect type $i$
• $A_{c,i}$ — critical area sensitive to defect type $i$

Critical area depends on:

• Defect size distribution
• Layout geometry
• Defect type (shorts, opens, particles)

11. Statistical and Machine Learning Methods

11.1 Response Surface Methodology (RSM)

Second-order model:

11.2 Design of Experiments (DOE)

11.3 Statistical Process Control (SPC)

Process capability indices:

11.4 Machine Learning Applications

12. Multi-Scale Modeling

12.1 Modeling Hierarchy

12.2 Bridging Methods

• Coarse-graining: atomistic → mesoscale
• Parameter extraction: quantum → continuum
• Concurrent multiscale: couple different scales simultaneously

13. Key Mathematical Toolkit

13.1 Partial Differential Equations

• Diffusion equation: $\frac{\partial u}{\partial t} = D \nabla^2 u$
• Heat equation: $\rho c_p \frac{\partial T}{\partial t} = \nabla \cdot (k \nabla T)$
• Navier-Stokes: $\rho \frac{D\mathbf{v}}{Dt} = -\nabla p + \mu \nabla^2 \mathbf{v} + \mathbf{f}$
• Poisson: $\nabla^2 \phi = -\rho/\epsilon$
• Level set: $\frac{\partial \phi}{\partial t} + \mathbf{v} \cdot \nabla \phi = 0$

13.2 Numerical Methods

• Finite Difference Method (FDM): simple geometries
• Finite Element Method (FEM): complex geometries
• Finite Volume Method (FVM): conservation laws
• Monte Carlo: stochastic processes, particle transport
• Level Set / Volume of Fluid: interface tracking

13.3 Optimization Techniques

• Gradient descent and conjugate gradient
• Newton-Raphson method
• Genetic algorithms
• Simulated annealing
• Bayesian optimization

13.4 Stochastic Processes

• Random walk (diffusion)
• Poisson processes (defect generation)
• Markov chains (KMC)
• Birth-death processes (nucleation)

14. Modern Challenges

14.1 Random Dopant Fluctuation (RDF)

Threshold voltage variation:

14.2 Line Edge Roughness (LER)

Power spectral density:

Variables:

• $\sigma$ — RMS roughness amplitude
• $\xi$ — correlation length
• $H$ — Hurst exponent

14.3 Stochastic Effects in EUV Lithography

• Photon shot noise: $\sigma_N = \sqrt{N}$ where $N$ = absorbed photons
• Secondary electron blur
• Resist stochastics: acid generation, diffusion, deprotection

14.4 3D Device Architectures

Modern modeling must handle:

• FinFET: 3D fin geometry
• Gate-All-Around (GAA): nanowire/nanosheet
• CFET: stacked complementary FETs
• 3D NAND: vertical channel, charge trap

14.5 Emerging Modeling Approaches

• Physics-Informed Neural Networks (PINNs)
• Digital twins for real-time process control
• Reduced-order models for fast simulation
• Uncertainty quantification for variability prediction