Multi Scale Problems

Semiconductor Manufacturing: Multi-Scale Problems and Mathematical Modeling

1. The Multi-Scale Hierarchy

Semiconductor manufacturing spans roughly 12 orders of magnitude in length scale, each with distinct physics:

Scale	Range	Phenomena	Mathematical Approach
Quantum/Atomic	0.1–1 nm	Bond formation, electron tunneling, reaction barriers	DFT, quantum chemistry
Molecular	1–10 nm	Surface reactions, nucleation, atomic diffusion	Kinetic Monte Carlo, MD
Feature	10 nm – 1 μm	Line edge roughness, profile evolution, grain structure	Level set, phase field
Device	1–100 μm	Transistor variability, local stress	Continuum FEM
Die	1–10 mm	Pattern density effects, thermal gradients	PDE-based continuum
Wafer	300 mm	Global uniformity, edge effects	Equipment-scale models
Reactor	~1 m	Plasma distribution, gas flow	CFD, plasma fluid models

Fundamental Challenge

Physics at each scale influences adjacent scales, creating coupled nonlinear systems with vastly different characteristic times and lengths.

2. Key Processes and Mathematical Structure

2.1 Plasma Etching — The Most Complex Multi-Scale Problem

2.1.1 Reactor Scale (Continuum)

Electron density evolution:

\frac{\partial n_e}{\partial t} + \nabla \cdot \boldsymbol{\Gamma}_e = S_e - L_e

Ion density evolution:

\frac{\partial n_i}{\partial t} + \nabla \cdot \boldsymbol{\Gamma}_i = S_i - L_i

Poisson equation for electric potential:

\nabla^2 \phi = -\frac{e}{\epsilon_0}(n_i - n_e)

Where:

$$n_e$$ , $$n_i$$ = electron and ion densities
$\boldsymbol{\Gamma}_e$ , $\boldsymbol{\Gamma}_i$ = electron and ion fluxes
$$S_e$$ , $$S_i$$ = source terms (ionization)
$$L_e$$ , $$L_i$$ = loss terms (recombination)
$\phi$ = electric potential
$$e$$ = elementary charge
$\epsilon_0$ = permittivity of free space

2.1.2 Feature Scale — Profile Evolution via Level Set

Level set equation:

\frac{\partial \phi}{\partial t} + V_n |\nabla \phi| = 0

Where:

$\phi(x,t) = 0$ defines the evolving surface
$$V_n$$ = local etch rate (normal velocity)

The local etch rate $$V_n$$ depends on:

Ion flux and angle distribution (from sheath physics)
Neutral species flux (from transport)
Surface chemistry (from atomic-scale kinetics)

2.1.3 The Coupling Problem

The feature-scale etch rate $$V_n$$ requires:

Ion angular/energy distributions → from sheath models
Sheath models → depend on plasma conditions
Plasma conditions → affected by loading (total surface area being etched)

This creates a global-to-local-to-global feedback loop.

2.2 Chemical Vapor Deposition (CVD) / Atomic Layer Deposition (ALD)

2.2.1 Gas-Phase Transport (Continuum)

Navier-Stokes momentum equation:

\rho\left(\frac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u}\right) = -\nabla p + \mu \nabla^2 \mathbf{u}

Species transport equation:

\frac{\partial C_k}{\partial t} + \mathbf{u} \cdot \nabla C_k = D_k \nabla^2 C_k + R_k

Where:

$\rho$ = gas density
$\mathbf{u}$ = velocity field
$$p$$ = pressure
$\mu$ = dynamic viscosity
$$C_k$$ = concentration of species $$k$$
$$D_k$$ = diffusion coefficient
$$R_k$$ = reaction rate

2.2.2 Surface Kinetics (Stochastic/Molecular)

Adsorption rate:

r_{ads} = s_0 \cdot f(\theta) \cdot F

Where:

$$s_0$$ = sticking coefficient
$f(\theta)$ = coverage-dependent function
$$F$$ = incident flux

Surface diffusion hopping rate:

u = u_0 \exp\left(-\frac{E_a}{k_B T}\right)

Where:

- $
u_0$ = attempt frequency
- $$E_a$$ = activation energy
- $$k_B$$ = Boltzmann constant
- $$T$$ = temperature

2.2.3 Mathematical Tension

Gas-phase transport is deterministic continuum; surface evolution involves discrete stochastic events. The boundary condition for the continuum problem depends on atomistic surface dynamics.

2.3 Lithography

2.3.1 Aerial Image Formation (Wave Optics)

Hopkins formulation for partially coherent imaging:

I(\mathbf{r}) = \sum_j w_j \left| \iint M(f_x, f_y) H_j(f_x, f_y) e^{2\pi i(f_x x + f_y y)} \, df_x \, df_y \right|^2

Where:

$I(\mathbf{r})$ = image intensity at position $\mathbf{r}$
$$M(f_x, f_y)$$ = mask spectrum (Fourier transform of mask pattern)
$$H_j(f_x, f_y)$$ = pupil function for source point $$j$$
$$w_j$$ = weight for source point $$j$$

2.3.2 Photoresist Chemistry

Exposure (photoactive compound destruction):

\frac{\partial m}{\partial t} = -C \cdot I \cdot m

Post-exposure bake diffusion (acid diffusion):

\frac{\partial h}{\partial t} = D_h \nabla^2 h

Development rate (Mack model):

R = R_0 \frac{(1-m)^n + \epsilon}{(1-m)^n + 1}

Where:

$$m$$ = normalized photoactive compound concentration
$$C$$ = exposure rate constant
$$I$$ = intensity
$$h$$ = acid concentration
$$D_h$$ = acid diffusion coefficient
$$R_0$$ = maximum development rate
$$n$$ = dissolution selectivity parameter
$\epsilon$ = dissolution rate ratio

2.3.3 Stochastic Challenge at Advanced Nodes

At EUV wavelength (13.5 nm), photon shot noise becomes significant:

\text{Fluctuation} \sim \frac{1}{\sqrt{N}}

Where $$N$$ = number of photons per feature area.

This translates to line edge roughness (LER) of ~2-3 nm — comparable to feature dimensions.

2.4 Diffusion and Annealing

Classical Fick's law fails because:

Diffusion is mediated by point defects (vacancies, interstitials)
Defect concentrations depend on dopant concentration
Stress affects diffusion
Transient enhanced diffusion during implant damage annealing

Five-Stream Model

\frac{\partial C_s}{\partial t} = \nabla \cdot (D_s \nabla C_s) + \text{reactions with } C_I, C_V, C_{As}, C_{AV}, \ldots

Where:

$$C_s$$ = substitutional dopant concentration
$$C_I$$ = interstitial concentration
$$C_V$$ = vacancy concentration
$C_{As}$ = dopant-interstitial pair concentration
$C_{AV}$ = dopant-vacancy pair concentration

This creates a coupled nonlinear system of 5+ PDEs with concentration-dependent coefficients spanning time scales from picoseconds to hours.

3. Mathematical Frameworks for Multi-Scale Coupling

3.1 Homogenization Theory

For problems with periodic microstructure at scale $\epsilon$ :

-\nabla \cdot \left( A^\epsilon(x) \nabla u^\epsilon \right) = f

Where $A^\epsilon(x) = A(x/\epsilon)$ oscillates rapidly.

Two-Scale Expansion

u^\epsilon(x) = u_0\left(x, \frac{x}{\epsilon}\right) + \epsilon \, u_1\left(x, \frac{x}{\epsilon}\right) + \epsilon^2 \, u_2\left(x, \frac{x}{\epsilon}\right) + \ldots

This yields an effective coefficient $$A^*$$ that captures microscale physics in a macroscale equation.

Rigorous for linear elliptic problems; much harder for nonlinear, time-dependent cases in manufacturing.

3.2 Heterogeneous Multiscale Method (HMM)

Key Idea: Run microscale simulations only where/when needed to extract effective properties for the macroscale solver.

Mathematical Formulation

Macro equation:

\frac{\partial U}{\partial t} = F\left(U, D^*(U)\right)

Micro-to-macro coupling:

D^*(U) = \langle d(u) \rangle_{\text{micro}}

Where the micro simulation is constrained by the macroscopic state $$U$$ .

3.3 Kinetic-Continuum Transition

Boltzmann Equation

\frac{\partial f}{\partial t} + \mathbf{v} \cdot \nabla_x f + \frac{\mathbf{F}}{m} \cdot \nabla_v f = Q(f,f)

Where:

$f(\mathbf{x}, \mathbf{v}, t)$ = distribution function
$\mathbf{v}$ = velocity
$\mathbf{F}$ = external force
$$m$$ = particle mass
$$Q(f,f)$$ = collision operator

Chapman-Enskog Expansion

Derives Navier-Stokes equations in the limit:

Kn \to 0

Where the Knudsen number is defined as:

Kn = \frac{\lambda}{L}

$\lambda$ = mean free path
$$L$$ = characteristic length

Spatial Variation of Knudsen Number

Region	Knudsen Number	Valid Model
Bulk reactor	$Kn \ll 1$	Continuum (Navier-Stokes)
Feature trenches	$Kn \sim 1$	Transitional regime
Surfaces, small features	$Kn \gg 1$	Kinetic (Boltzmann)

3.4 Level Set and Phase Field Methods

3.4.1 Level Set Method

Interface definition: $\{\mathbf{x} : \phi(\mathbf{x},t) = 0\}$

Evolution equation:

\frac{\partial \phi}{\partial t} + V_n |\nabla \phi| = 0

Advantages:

Handles topology changes naturally (merging, splitting)
Implicit representation avoids mesh issues

Challenges:

Maintaining $|\nabla \phi| = 1$ (signed distance property)
Velocity extension from interface to entire domain

3.4.2 Phase Field Method

Diffuse interface evolution:

\frac{\partial \phi}{\partial t} = M\left[\epsilon^2 \nabla^2 \phi - f'(\phi) + \lambda g'(\phi)\right]

Where:

$$M$$ = mobility
$\epsilon$ = interface width parameter
$f(\phi)$ = double-well potential
$g(\phi)$ = driving force
$\lambda$ = coupling constant

Advantages:

No explicit interface tracking required
Natural handling of complex morphologies

Challenges:

Resolving thin interface requires fine mesh
Selecting appropriate interface width $\epsilon$

4. Fundamental Mathematical Challenges

4.1 Stiffness and Time-Scale Separation

Process	Characteristic Time
Electron dynamics	$10^{-12}$ s
Surface reactions	$10^{-9}$ – $10^{-6}$ s
Gas transport	$10^{-3}$ s
Feature evolution	$1$ – $10^{2}$ s
Wafer processing	$10^{2}$ – $10^{4}$ s

Time scale ratio: $\sim 10^{16}$ between fastest and slowest processes.

Direct simulation is impossible.

Solution Strategies

Implicit time integration with adaptive stepping
Quasi-steady state approximations for fast variables
Operator splitting: Treat different physics on different time scales
Averaging/homogenization to eliminate fast oscillations

4.2 High Dimensionality

The kinetic description $f(\mathbf{x}, \mathbf{v}, t)$ lives in 6D phase space.

Adding internal energy states and multiple species → intractable.

Reduction Strategies

Moment methods: Track $\langle 1, v, v^2, \ldots \rangle_v$ rather than full $$f$$
Monte Carlo: Sample from distribution rather than discretizing
Proper Orthogonal Decomposition (POD): Find low-dimensional subspace
Neural network surrogates: Learn mapping from inputs to outputs

4.3 Stochastic Effects at Nanoscale

At sub-10nm, continuum assumptions fail due to:

Discreteness of atoms: Can't average over enough atoms
Shot noise: Finite number of photons, ions, molecules
Line edge roughness: Atomic-scale randomness in edge positions

Mathematical Treatment

Stochastic PDEs (Langevin form):

du = \mathcal{L}u \, dt + \sigma \, dW

Where $$dW$$ is a Wiener process increment.

Master equation:

\frac{dP_n}{dt} = \sum_m \left( W_{nm} P_m - W_{mn} P_n \right)

Where:

$$P_n$$ = probability of state $$n$$
$W_{nm}$ = transition rate from state $$m$$ to state $$n$$

Kinetic Monte Carlo: Direct simulation of discrete events with proper time advancement.

4.4 Inverse Problems and Control

Forward problem: Given process parameters → predict outcome

Inverse problem: Given desired outcome → find parameters

Manufacturing Requirements

Recipe optimization
Run-to-run control
Fault detection/classification

Mathematical Challenges

Ill-posedness: Multiple solutions, sensitivity to noise
High dimensionality of parameter space
Real-time constraints for feedback control

Approaches

Regularization: Tikhonov, sparse methods
Bayesian inference: Uncertainty quantification
Optimal control theory: Adjoint methods
Surrogate-based optimization: Using ML models

5. Current Frontiers

5.1 Physics-Informed Machine Learning

Loss Function Structure

\mathcal{L} = \mathcal{L}_{\text{data}} + \lambda_{\text{physics}} \mathcal{L}_{\text{PDE}} + \lambda_{\text{BC}} \mathcal{L}_{\text{boundary}}

Where:

$\mathcal{L}_{\text{data}}$ = data fitting loss
$\mathcal{L}_{\text{PDE}}$ = physics constraint (PDE residual)
$\mathcal{L}_{\text{boundary}}$ = boundary condition constraint
$\lambda$ = weighting hyperparameters

Methods

Physics-Informed Neural Networks (PINNs): Embed governing equations as soft constraints
Neural operators (DeepONet, FNO): Learn mappings between function spaces
Hybrid models: Combine physics-based and data-driven components

Challenges Specific to Semiconductor Manufacturing

Sparse experimental data (wafers are expensive)
Extrapolation to new process conditions
Interpretability requirements for process understanding
Certification for high-reliability applications

5.2 Uncertainty Quantification at Scale

Manufacturing requires predicting distributions, not just means:

What is $P(\text{yield} > 0.95)$ ?
What is the 99th percentile of line width variation?

Polynomial Chaos Expansion

u(\mathbf{x}, \boldsymbol{\xi}) = \sum_{k} u_k(\mathbf{x}) \Psi_k(\boldsymbol{\xi})

Where:

$\boldsymbol{\xi}$ = random input parameters
$\Psi_k$ = orthogonal polynomial basis functions
$u_k(\mathbf{x})$ = deterministic coefficient functions

Challenge: Curse of Dimensionality

50+ random input parameters is common in semiconductor manufacturing.

Solutions

Sparse polynomial chaos
Active subspaces (dimension reduction)
Multi-fidelity methods (combine cheap/accurate models)

5.3 Quantum Effects at Sub-Nanometer Scale

As features approach ~1 nm:

Quantum tunneling through gate oxides
Quantum confinement affects electron states
Atomistic variability in dopant positions → device-to-device variation

Non-Equilibrium Green's Function (NEGF) Method

For quantum transport:

G^R(E) = \left[ (E + i\eta)I - H - \Sigma^R \right]^{-1}

Where:

$$G^R$$ = retarded Green's function
$$E$$ = energy
$$H$$ = Hamiltonian
$\Sigma^R$ = self-energy (contact + scattering)
$\eta$ = infinitesimal positive number

6. Conceptual Framework

Unified View of Multi-Scale Modeling

Key Mathematical Requirements

Consistency: Coarse-grained models recover fine-scale physics in appropriate limits
Conservation: Mass, momentum, energy preserved across scales
Efficiency: Computational cost scales with information content, not raw degrees of freedom
Adaptivity: Automatically refine where and when needed

7. Open Mathematical Problems

Problem	Current State	Mathematical Need
Stochastic feature-scale modeling	KMC possible but expensive	Fast stochastic PDE methods
Plasma-surface coupling	Often one-way coupling	Consistent two-way coupling with rigorous error bounds
Real-time model-predictive control	Simplified ROMs	Fast surrogates with guaranteed accuracy
Variability prediction	Expensive Monte Carlo	Efficient UQ for high-dimensional inputs
Atomic-to-device coupling	Sequential handoff	Concurrent adaptive methods
Inverse design	Local optimization	Global optimization in high dimensions

Key Equations Summary

Transport Equations

\text{Continuity:} \quad \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0

\text{Momentum:} \quad \rho \frac{D\mathbf{u}}{Dt} = -\nabla p + \mu \nabla^2 \mathbf{u} + \mathbf{f}

\text{Energy:} \quad \rho c_p \frac{DT}{Dt} = k \nabla^2 T + \dot{q}

\text{Species:} \quad \frac{\partial C_k}{\partial t} + \nabla \cdot (C_k \mathbf{u}) = D_k \nabla^2 C_k + R_k

Interface Evolution

\text{Level Set:} \quad \frac{\partial \phi}{\partial t} + V_n |\nabla \phi| = 0

\text{Phase Field:} \quad \tau \frac{\partial \phi}{\partial t} = \epsilon^2 \nabla^2 \phi - f'(\phi)

Kinetic Theory

\text{Boltzmann:} \quad \frac{\partial f}{\partial t} + \mathbf{v} \cdot \nabla_x f + \frac{\mathbf{F}}{m} \cdot \nabla_v f = Q(f,f)

\text{Knudsen Number:} \quad Kn = \frac{\lambda}{L}

Stochastic Modeling

\text{Langevin SDE:} \quad dX = a(X,t) \, dt + b(X,t) \, dW

\text{Fokker-Planck:} \quad \frac{\partial p}{\partial t} = -\nabla \cdot (a \, p) + \frac{1}{2} \nabla^2 (b^2 p)

Nomenclature

Symbol	Description	Units
$\rho$	Density	kg/m³
$\mathbf{u}$	Velocity vector	m/s
$$p$$	Pressure	Pa
$$T$$	Temperature	K
$$C_k$$	Concentration of species $$k$$	mol/m³
$$D_k$$	Diffusion coefficient	m²/s
$\phi$	Level set function or phase field	—
$$V_n$$	Normal interface velocity	m/s
$$f$$	Distribution function	—
$$Kn$$	Knudsen number	—
$\lambda$	Mean free path	m
$$E_a$$	Activation energy	J/mol
$$k_B$$	Boltzmann constant	J/K

Multi Scale Problems

Take multi scale problems further