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Mathematics Modeling

Keywords: mathematics,mathematical modeling,semiconductor math,crystal growth math,czochralski equations,dopant segregation,heat transfer equations,lithography math

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

\rho c_p \frac{\partial T}{\partial t} + \rho c_p \mathbf{v} \cdot abla T = abla \cdot (k abla T) + Q

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:

k_0 = \frac{C_s}{C_l}

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

k_{eff} = \frac{k_0}{k_0 + (1 - k_0) \exp\left(-\frac{v \delta}{D}\right)}

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

x_o^2 + A x_o = B(t + \tau)

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):

x_o \approx \frac{B}{A}(t + \tau)

Parabolic regime (thick oxide, diffusion limited):

x_o \approx \sqrt{B(t + \tau)}

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):

\mathbf{J} = -D abla C

Second Law (continuity equation):

\frac{\partial C}{\partial t} = abla \cdot (D abla C)

For constant $D$:

\frac{\partial C}{\partial t} = D abla^2 C

3.2 Concentration-Dependent Diffusivity

D(C) = D_i + D^{-} \frac{n}{n_i} + D^{2-} \left(\frac{n}{n_i}\right)^2 + D^{+} \frac{p}{n_i} + D^{2+} \left(\frac{p}{n_i}\right)^2

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:

D_{eff} = D_I \frac{C_I}{C_I^*} + D_V \frac{C_V}{C_V^*}

Point defect continuity equations:

\frac{\partial C_I}{\partial t} = D_I abla^2 C_I + G_I - R_{IV}

\frac{\partial C_V}{\partial t} = D_V abla^2 C_V + G_V - R_{IV}

Recombination rate:

R_{IV} = k_{IV} \left( C_I C_V - C_I^* C_V^* \right)

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

N(x) = \frac{\phi}{\sqrt{2\pi} \Delta R_p} \exp\left[ -\frac{(x - R_p)^2}{2 (\Delta R_p)^2} \right]

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:

\frac{dE}{dx} = S_n(E) + S_e(E)

$S_n(E)$ — 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

\frac{\partial \theta}{\partial t} = J_i s_i (1 - \theta) - k_r \theta

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

Y = \frac{\text{atoms removed}}{\text{incident ion}}

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)

\frac{\partial \phi}{\partial t} + V | abla \phi| = 0

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 ($Kn > 1$):

\frac{1}{\lambda} \frac{dI}{dx} = -I + \int K(x, x') I(x') dx'

Key effects:

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

6. Chemical Vapor Deposition (CVD)

6.1 Transport-Reaction Equation

\frac{\partial C}{\partial t} + \mathbf{v} \cdot abla C = D abla^2 C - k C^n

Variables:

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

6.2 Thiele Modulus

\phi = L \sqrt{\frac{k}{D}}

Regimes:

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

6.3 Step Coverage

Conformality factor:

S = \frac{\text{thickness at bottom}}{\text{thickness at top}}

Models:

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

6.4 Atomic Layer Deposition (ALD)

Self-limiting surface coverage:

\theta(t) = 1 - \exp\left( -\frac{p \cdot t}{\tau} \right)

Variables:

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

Growth per cycle (GPC):

\text{GPC} = \theta_{sat} \cdot \Gamma_{ML}

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

7. Chemical Mechanical Polishing (CMP)

7.1 Preston Equation

\frac{dz}{dt} = K_p \cdot P \cdot V

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:

P_{eff} = \frac{P_{applied}}{\rho_{pattern}}

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:

P(r) = P_0 \sqrt{1 - \left(\frac{r}{a}\right)^2}

Pad asperity models:

Greenwood-Williamson for rough surfaces
Viscoelastic pad behavior

8. Lithography

8.1 Aerial Image Formation

Hopkins formulation (partially coherent):

I(\mathbf{x}) = \iint TCC(\mathbf{f}, \mathbf{f}') \, M(\mathbf{f}) \, M^*(\mathbf{f}') \, e^{2\pi i (\mathbf{f} - \mathbf{f}') \cdot \mathbf{x}} \, d\mathbf{f} \, d\mathbf{f}'

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:

R = k_1 \frac{\lambda}{NA}

Depth of focus:

DOF = k_2 \frac{\lambda}{NA^2}

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:

\frac{\partial m}{\partial t} = -I(z, t) \cdot m \cdot C

Intensity attenuation:

I(z, t) = I_0 \exp\left( -\int_0^z [A \cdot m(z', t) + B] \, dz' \right)

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)

r = r_{max} \frac{(a + 1)(1 - m)^n}{a + (1 - m)^n}

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

abla \cdot (\epsilon abla \psi) = -q(p - n + N_D^+ - N_A^-)

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:

\frac{\partial n}{\partial t} = \frac{1}{q} abla \cdot \mathbf{J}_n + G - R

Holes:

\frac{\partial p}{\partial t} = -\frac{1}{q} abla \cdot \mathbf{J}_p + G - R

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:

\mathbf{J}_n = q n \mu_n \mathbf{E} + q D_n abla n

Hole current:

\mathbf{J}_p = q p \mu_p \mathbf{E} - q D_p abla p

Einstein relation:

D = \frac{k_B T}{q} \mu

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

Y = e^{-A D_0}

Variables:

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

10.2 Negative Binomial Model (Clustered Defects)

Y = \left(1 + \frac{A D_0}{\alpha}\right)^{-\alpha}

Variables:

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

10.3 Critical Area Analysis

Y = \exp\left( -\sum_i D_i \cdot A_{c,i} \right)

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:

y = \beta_0 + \sum_{i=1}^{k} \beta_i x_i + \sum_{i=1}^{k} \beta_{ii} x_i^2 + \sum_{i<j} \beta_{ij} x_i x_j + \epsilon

11.2 Design of Experiments (DOE)

Design Type	Application
Full factorial	Complete parameter space exploration
Fractional factorial	Screening many factors
Central composite	RSM fitting
Box-Behnken	Efficient quadratic modeling
Taguchi	Robust design optimization

11.3 Statistical Process Control (SPC)

Process capability indices:

C_p = \frac{USL - LSL}{6\sigma}

C_{pk} = \min\left( \frac{USL - \mu}{3\sigma}, \frac{\mu - LSL}{3\sigma} \right)

11.4 Machine Learning Applications

Method	Application
Neural Networks	Process-property prediction
Gaussian Process Regression	Surrogate modeling
Random Forest	Defect classification
Bayesian Optimization	Recipe tuning
Convolutional Neural Networks	Defect detection in images
Recurrent Neural Networks	Time-series process data

12. Multi-Scale Modeling

12.1 Modeling Hierarchy

Scale	Length	Method	Example
Quantum	< 1 nm	DFT, ab initio MD	Reaction barriers
Atomistic	1–100 nm	Classical MD	Surface diffusion
Mesoscale	100 nm – 1 μm	Kinetic Monte Carlo	Dopant clustering
Continuum	> 1 μm	FEM, FDM	Process simulation
System	Wafer/die	Statistical	Yield modeling

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

abla^2 u$

Heat equation: $\rho c_p \frac{\partial T}{\partial t} =

abla \cdot (k abla T)$

Navier-Stokes: $\rho \frac{D\mathbf{v}}{Dt} = -

abla p + \mu abla^2 \mathbf{v} + \mathbf{f}$

Poisson: $

abla^2 \phi = -\rho/\epsilon$

Level set: $\frac{\partial \phi}{\partial t} + \mathbf{v} \cdot

abla \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:

\sigma_{V_T} \propto \frac{1}{\sqrt{W \cdot L}} \cdot \frac{t_{ox}}{\sqrt{N_A}}

14.2 Line Edge Roughness (LER)

Power spectral density:

PSD(f) = \frac{2\sigma^2 \xi}{1 + (2\pi f \xi)^{2(1+H)}}

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

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