mathematics
**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:
$$
y = \beta_0 + \sum_{i=1}^{k} \beta_i x_i + \sum_{i=1}^{k} \beta_{ii} x_i^2 + \sum_{i 1 μm | FEM, FDM | Process simulation |
| System | Wafer/die | Statistical | Yield modeling |
12.2 Bridging Methods
- Coarse-graining: atomistic → mesoscale
- Parameter extraction: quantum → continuum
- Concurrent multiscale: couple different scales simultaneously
13. Key Mathematical Toolkit
13.1 Partial Differential Equations
- Diffusion equation: $\frac{\partial u}{\partial t} = D
abla^2 u$
- Heat equation: $\rho c_p \frac{\partial T}{\partial t} =
abla \cdot (k
abla T)$
- Navier-Stokes: $\rho \frac{D\mathbf{v}}{Dt} = -
abla p + \mu
abla^2 \mathbf{v} + \mathbf{f}$
- Poisson: $
abla^2 \phi = -\rho/\epsilon$
- Level set: $\frac{\partial \phi}{\partial t} + \mathbf{v} \cdot
abla \phi = 0$
13.2 Numerical Methods
- Finite Difference Method (FDM): simple geometries
- Finite Element Method (FEM): complex geometries
- Finite Volume Method (FVM): conservation laws
- Monte Carlo: stochastic processes, particle transport
- Level Set / Volume of Fluid: interface tracking
13.3 Optimization Techniques
- Gradient descent and conjugate gradient
- Newton-Raphson method
- Genetic algorithms
- Simulated annealing
- Bayesian optimization
13.4 Stochastic Processes
- Random walk (diffusion)
- Poisson processes (defect generation)
- Markov chains (KMC)
- Birth-death processes (nucleation)
14. Modern Challenges
14.1 Random Dopant Fluctuation (RDF)
Threshold voltage variation:
$$
\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