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Mathematical Modeling of Epitaxy in Semiconductor Front-End Processing (FEP)

Keywords: fep modeling, front end processing, feol, ion implantation, diffusion modeling, oxidation modeling, dopant activation, junction formation, thermal processing, annealing

Mathematical Modeling of Epitaxy in Semiconductor Front-End Processing (FEP)

1. Overview

Epitaxy is a critical Front-End Process (FEP) step where crystalline films are grown on crystalline substrates with precise control of:

Thickness
Composition
Doping concentration
Defect density

Mathematical modeling enables:

Process optimization
Defect prediction
Virtual fabrication
Equipment design

1.1 Types of Epitaxy

Homoepitaxy: Same material as substrate (e.g., Si on Si)
Heteroepitaxy: Different material from substrate (e.g., GaAs on Si, SiGe on Si)

1.2 Epitaxy Methods

Vapor Phase Epitaxy (VPE) / Chemical Vapor Deposition (CVD)

Atmospheric Pressure CVD (APCVD)

Low Pressure CVD (LPCVD)

Metal-Organic CVD (MOCVD)

Molecular Beam Epitaxy (MBE)
Liquid Phase Epitaxy (LPE)
Solid Phase Epitaxy (SPE)

2. Fundamental Thermodynamic Framework

2.1 Driving Force for Growth

The supersaturation provides the thermodynamic driving force:

\Delta \mu = k_B T \ln\left(\frac{P}{P_{eq}}\right)

Where:

$\Delta \mu$ = chemical potential difference (driving force)
$k_B$ = Boltzmann's constant ($1.38 \times 10^{-23}$ J/K)
$T$ = absolute temperature (K)
$P$ = actual partial pressure of precursor
$P_{eq}$ = equilibrium vapor pressure

2.2 Free Energy of Mixing (Multi-component Systems)

For systems like SiGe alloys:

\Delta G_{mix} = RT\left(x \ln x + (1-x) \ln(1-x)\right) + \Omega x(1-x)

Where:

$R$ = universal gas constant (8.314 J/mol·K)
$x$ = mole fraction of component
$\Omega$ = interaction parameter (regular solution model)

2.3 Gibbs Free Energy of Formation

\Delta G = \Delta H - T\Delta S

For spontaneous growth: $\Delta G < 0$

3. Growth Rate Kinetics

3.1 The Two-Regime Model

Epitaxial growth rate is governed by two competing mechanisms:

Overall growth rate equation:

G = \frac{k_s \cdot h_g \cdot C_g}{k_s + h_g}

Where:

$G$ = growth rate (nm/min or μm/min)
$k_s$ = surface reaction rate constant
$h_g$ = gas-phase mass transfer coefficient
$C_g$ = gas-phase reactant concentration

3.2 Temperature Dependence

The surface reaction rate follows Arrhenius behavior:

k_s = A \exp\left(-\frac{E_a}{k_B T}\right)

Where:

$A$ = pre-exponential factor (frequency factor)
$E_a$ = activation energy (eV or J/mol)

3.3 Growth Rate Regimes

Temperature Regime	Limiting Factor	Growth Rate Expression	Temperature Dependence
Low T	Surface reaction	$G \approx k_s \cdot C_g$	Strong (exponential)
High T	Mass transport	$G \approx h_g \cdot C_g$	Weak (~$T^{1.5-2}$)

3.4 Boundary Layer Analysis

For horizontal CVD reactors, the boundary layer thickness evolves as:

\delta(x) = \sqrt{\frac{ u \cdot x}{v_{\infty}}}

Where:

$\delta(x)$ = boundary layer thickness at position $x$
$

u$ = kinematic viscosity (m²/s)

$x$ = distance from gas inlet (m)
$v_{\infty}$ = free stream gas velocity (m/s)

The mass transfer coefficient:

h_g = \frac{D_{gas}}{\delta}

Where $D_{gas}$ is the gas-phase diffusion coefficient.

4. Surface Kinetics: BCF Theory

The Burton-Cabrera-Frank (BCF) model describes atomic-scale growth mechanisms.

4.1 Surface Diffusion Equation

D_s abla^2 n_s - \frac{n_s - n_{eq}}{\tau_s} + J_{ads} = 0

Where:

$n_s$ = adatom surface density (atoms/cm²)
$D_s$ = surface diffusion coefficient (cm²/s)
$n_{eq}$ = equilibrium adatom density
$\tau_s$ = mean adatom lifetime before desorption (s)
$J_{ads}$ = adsorption flux (atoms/cm²·s)

4.2 Characteristic Diffusion Length

\lambda_s = \sqrt{D_s \tau_s}

This parameter determines the growth mode:

Step-flow growth: $\lambda_s > L$ (terrace width)
2D nucleation growth: $\lambda_s < L$

4.3 Surface Diffusion Coefficient

D_s = D_0 \exp\left(-\frac{E_m}{k_B T}\right)

Where:

$D_0$ = pre-exponential factor (~$10^{-3}$ cm²/s)
$E_m$ = migration energy barrier (eV)

4.4 Step Velocity

v_{step} = \frac{2 D_s (n_s - n_{eq})}{\lambda_s} \tanh\left(\frac{L}{2\lambda_s}\right)

Where $L$ is the inter-step spacing (terrace width).

4.5 Growth Rate from Step Flow

G = \frac{v_{step} \cdot h_{step}}{L}

Where $h_{step}$ is the step height (monolayer thickness).

5. Heteroepitaxy and Strain Modeling

5.1 Lattice Mismatch

f = \frac{a_{film} - a_{substrate}}{a_{substrate}}

Where:

$f$ = lattice mismatch (dimensionless, often expressed as %)
$a_{film}$ = lattice constant of film material
$a_{substrate}$ = lattice constant of substrate

Example values:

System	Lattice Mismatch
Si₀.₇Ge₀.₃ on Si	~1.2%
Ge on Si	~4.2%
GaAs on Si	~4.0%
InAs on GaAs	~7.2%
GaN on Sapphire	~16%

5.2 Strain Components

For biaxial strain in (001) films:

\varepsilon_{xx} = \varepsilon_{yy} = \varepsilon_{\parallel} = \frac{a_s - a_f}{a_f} \approx -f

\varepsilon_{zz} = \varepsilon_{\perp} = -\frac{2C_{12}}{C_{11}} \varepsilon_{\parallel}

Where $C_{11}$ and $C_{12}$ are elastic constants.

5.3 Elastic Energy

For a coherently strained film:

E_{elastic} = \frac{2G(1+ u)}{1- u} f^2 h = M f^2 h

Where:

$G$ = shear modulus (Pa)
$

u$ = Poisson's ratio

$h$ = film thickness
$M$ = biaxial modulus = $\frac{2G(1+

u)}{1- u}$

5.4 Critical Thickness (Matthews-Blakeslee)

h_c = \frac{b}{8\pi f(1+ u)} \left[\ln\left(\frac{h_c}{b}\right) + 1\right]

Where:

$h_c$ = critical thickness for dislocation formation
$b$ = Burgers vector magnitude
$f$ = lattice mismatch
$

u$ = Poisson's ratio

5.5 People-Bean Approximation (for SiGe)

Empirical formula:

h_c \approx \frac{0.55}{f^2} \text{ (nm, with } f \text{ as a decimal)}

Or equivalently:

h_c \approx \frac{5500}{x^2} \text{ (nm, for Si}_{1-x}\text{Ge}_x\text{)}

5.6 Threading Dislocation Density

Above critical thickness, dislocation density evolves:

\rho_{TD}(h) = \rho_0 \exp\left(-\frac{h}{h_0}\right) + \rho_{\infty}

Where:

$\rho_{TD}$ = threading dislocation density (cm⁻²)
$\rho_0$ = initial density
$h_0$ = characteristic decay length
$\rho_{\infty}$ = residual density

6. Reactor-Scale Modeling

6.1 Coupled Transport Equations

6.1.1 Momentum Conservation (Navier-Stokes)

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

Where:

$\rho$ = gas density (kg/m³)
$\mathbf{v}$ = velocity vector (m/s)
$p$ = pressure (Pa)
$\mu$ = dynamic viscosity (Pa·s)
$\mathbf{g}$ = gravitational acceleration

6.1.2 Continuity Equation

\frac{\partial \rho}{\partial t} + abla \cdot (\rho \mathbf{v}) = 0

6.1.3 Species Transport

\frac{\partial C_i}{\partial t} + \mathbf{v} \cdot abla C_i = D_i abla^2 C_i + R_i

Where:

$C_i$ = concentration of species $i$ (mol/m³)
$D_i$ = diffusion coefficient of species $i$ (m²/s)
$R_i$ = net reaction rate (mol/m³·s)

6.1.4 Energy Conservation

\rho c_p \left(\frac{\partial T}{\partial t} + \mathbf{v} \cdot abla T\right) = k abla^2 T + \sum_j \Delta H_j r_j

Where:

$c_p$ = specific heat capacity (J/kg·K)
$k$ = thermal conductivity (W/m·K)
$\Delta H_j$ = enthalpy of reaction $j$ (J/mol)
$r_j$ = rate of reaction $j$ (mol/m³·s)

6.2 Silicon CVD Chemistry

6.2.1 From Silane (SiH₄)

Gas phase decomposition:

\text{SiH}_4 \xrightarrow{k_1} \text{SiH}_2 + \text{H}_2

Surface reaction:

\text{SiH}_2(g) + * \xrightarrow{k_2} \text{Si}(s) + \text{H}_2(g)

Where $*$ denotes a surface site.

6.2.2 From Dichlorosilane (DCS)

\text{SiH}_2\text{Cl}_2 \rightarrow \text{SiCl}_2 + \text{H}_2

\text{SiCl}_2 + \text{H}_2 \rightarrow \text{Si}(s) + 2\text{HCl}

6.2.3 Rate Law

r_{dep} = k_2 P_{SiH_2} (1 - \theta)

Where:

$P_{SiH_2}$ = partial pressure of SiH₂
$\theta$ = surface site coverage

6.3 Dimensionless Numbers

Number	Definition	Physical Meaning
Reynolds	$Re = \frac{\rho v L}{\mu}$	Inertia vs. viscous forces
Prandtl	$Pr = \frac{\mu c_p}{k}$	Momentum vs. thermal diffusivity
Schmidt	$Sc = \frac{\mu}{\rho D}$	Momentum vs. mass diffusivity
Damköhler	$Da = \frac{k_s L}{D}$	Reaction rate vs. diffusion rate
Grashof

u^2}$ | Buoyancy vs. viscous forces |

7. Selective Epitaxial Growth (SEG) Modeling

7.1 Overview

In SEG, growth occurs on exposed Si but not on dielectric (SiO₂/Si₃N₄).

7.2 Loading Effect Model

G_{local} = G_0 \left(1 + \alpha \cdot \frac{A_{mask}}{A_{Si}}\right)

Where:

$G_{local}$ = local growth rate
$G_0$ = baseline growth rate
$\alpha$ = pattern sensitivity factor
$A_{mask}$ = dielectric (mask) area
$A_{Si}$ = exposed silicon area

7.3 Pattern-Dependent Growth

Sources of non-uniformity:

Local depletion of reactants over Si regions
Species reflected/desorbed from mask contribute to nearby Si
Gas-phase diffusion length effects

7.4 Selectivity Condition

For selective growth on Si vs. oxide:

r_{deposition,Si} > 0 \quad \text{and} \quad r_{deposition,oxide} < r_{etching,oxide}

Achieved by adding HCl:

\text{Si}(nuclei) + 2\text{HCl} \rightarrow \text{SiCl}_2 + \text{H}_2

Nuclei on oxide are etched before they can grow, maintaining selectivity.

7.5 Faceting Model

Growth rate depends on crystallographic orientation:

G_{(hkl)} = G_0 \cdot f(hkl) \cdot \exp\left(-\frac{E_{a,(hkl)}}{k_B T}\right)

Typical growth rate hierarchy:

G_{(100)} > G_{(110)} > G_{(111)}

8. Dopant Incorporation

8.1 Segregation Coefficient

Equilibrium segregation coefficient:

k_0 = \frac{C_{solid}}{C_{liquid/gas}}

Effective segregation coefficient:

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

Where:

$k_0$ = equilibrium segregation coefficient
$G$ = growth rate
$\delta$ = boundary layer thickness
$D_l$ = diffusivity in liquid/gas phase

8.2 Dopant Concentration in Film

C_{film} = k_{eff} \cdot C_{gas}

8.3 Dopant Profile Abruptness

The transition width is limited by:

Surface segregation length: $\lambda_{seg}$
Diffusion during growth: $L_D = \sqrt{D \cdot t}$
Autodoping from substrate

\Delta z_{transition} \approx \sqrt{\lambda_{seg}^2 + L_D^2}

8.4 Common Dopants for Si Epitaxy

Dopant	Type	Precursor	Segregation Behavior
B	p-type	B₂H₆, BCl₃	Low segregation
P	n-type	PH₃, PCl₃	Moderate segregation
As	n-type	AsH₃	Strong segregation
Sb	n-type	SbH₃	Very strong segregation

9. Atomistic Simulation Methods

9.1 Kinetic Monte Carlo (KMC)

9.1.1 Event Rates

Each atomic event has a rate following Arrhenius:

\Gamma_i = u_0 \exp\left(-\frac{E_i}{k_B T}\right)

Where:

$\Gamma_i$ = rate of event $i$ (s⁻¹)
$

u_0$ = attempt frequency (~10¹²-10¹³ s⁻¹)

$E_i$ = activation energy for event $i$

9.1.2 Events Modeled

Adsorption: $\Gamma_{ads} = \frac{P}{\sqrt{2\pi m k_B T}} \cdot s$
Desorption: $\Gamma_{des} =

u_0 \exp(-E_{des}/k_B T)$

Surface diffusion: $\Gamma_{diff} =

u_0 \exp(-E_m/k_B T)$

Step attachment: $\Gamma_{attach}$
Step detachment: $\Gamma_{detach}$

9.1.3 Time Advancement

\Delta t = -\frac{\ln(r)}{\Gamma_{total}} = -\frac{\ln(r)}{\sum_i \Gamma_i}

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

9.2 Density Functional Theory (DFT)

Provides input parameters for KMC:

Adsorption energies
Migration barriers
Surface reconstruction energetics
Reaction pathways

Kohn-Sham equation:

\left[-\frac{\hbar^2}{2m} abla^2 + V_{eff}(\mathbf{r})\right]\psi_i(\mathbf{r}) = \varepsilon_i \psi_i(\mathbf{r})

9.3 Molecular Dynamics (MD)

Newton's equations:

m_i \frac{d^2 \mathbf{r}_i}{dt^2} = - abla_i U(\mathbf{r}_1, \mathbf{r}_2, ..., \mathbf{r}_N)

Where $U$ is the interatomic potential (e.g., Stillinger-Weber, Tersoff for Si).

10. Nucleation Theory

10.1 Classical Nucleation Theory (CNT)

10.1.1 Gibbs Free Energy Change

\Delta G(r) = -\frac{4}{3}\pi r^3 \cdot \frac{\Delta \mu}{\Omega} + 4\pi r^2 \gamma

Where:

$r$ = nucleus radius
$\Delta \mu$ = supersaturation (driving force)
$\Omega$ = atomic volume
$\gamma$ = surface energy

10.1.2 Critical Nucleus Radius

Setting $\frac{d(\Delta G)}{dr} = 0$:

r^* = \frac{2\gamma \Omega}{\Delta \mu}

10.1.3 Free Energy Barrier

\Delta G^* = \frac{16 \pi \gamma^3 \Omega^2}{3 (\Delta \mu)^2}

10.1.4 Nucleation Rate

J = Z \beta^* N_s \exp\left(-\frac{\Delta G^*}{k_B T}\right)

Where:

$J$ = nucleation rate (nuclei/cm²·s)
$Z$ = Zeldovich factor (~0.01-0.1)
$\beta^*$ = attachment rate to critical nucleus
$N_s$ = surface site density

10.2 Growth Modes

Mode	Surface Energy Condition	Growth Behavior	Example
Frank-van der Merwe	$\gamma_s \geq \gamma_f + \gamma_{int}$	Layer-by-layer (2D)	Si on Si
Volmer-Weber	$\gamma_s < \gamma_f + \gamma_{int}$	Island (3D)	Metals on oxides
Stranski-Krastanov	Intermediate	2D then 3D islands	InAs/GaAs QDs

10.3 2D Nucleation

Critical island size (atoms):

i^* = \frac{\pi \gamma_{step}^2 \Omega}{(\Delta \mu)^2 k_B T}

11. TCAD Process Simulation

11.1 Overview

Tools: Synopsys Sentaurus Process, Silvaco Victory Process

11.2 Diffusion-Reaction System

\frac{\partial C_i}{\partial t} = abla \cdot (D_i abla C_i - \mu_i C_i abla \phi) + G_i - R_i

Where:

First term: Fickian diffusion
Second term: Drift in electric field (for charged species)
$G_i$ = generation rate
$R_i$ = recombination rate

11.3 Point Defect Dynamics

Vacancy concentration:

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

Interstitial concentration:

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

Where $k_{IV}$ is the recombination rate constant.

11.4 Stress Evolution

Equilibrium equation:

abla \cdot \boldsymbol{\sigma} = 0

Constitutive relation:

\boldsymbol{\sigma} = \mathbf{C} : (\boldsymbol{\varepsilon} - \boldsymbol{\varepsilon}^{thermal} - \boldsymbol{\varepsilon}^{intrinsic})

Where:

$\boldsymbol{\sigma}$ = stress tensor
$\mathbf{C}$ = elastic stiffness tensor
$\boldsymbol{\varepsilon}$ = total strain
$\boldsymbol{\varepsilon}^{thermal}$ = thermal strain = $\alpha \Delta T$
$\boldsymbol{\varepsilon}^{intrinsic}$ = intrinsic strain (lattice mismatch)

11.5 Level Set Method for Interface Tracking

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

Where:

$\phi$ = level set function (interface at $\phi = 0$)
$v_n$ = interface normal velocity

12. Advanced Topics

12.1 Atomic Layer Epitaxy (ALE) / Atomic Layer Deposition (ALD)

Self-limiting surface reactions modeled as Langmuir kinetics:

\theta = \frac{K \cdot P \cdot t}{1 + K \cdot P \cdot t} \rightarrow 1 \quad \text{as } t \rightarrow \infty

Growth per cycle (GPC):

GPC = \theta_{sat} \cdot d_{monolayer}

Typical GPC values: 0.5-1.5 Å/cycle

12.2 III-V on Silicon Integration

Challenges and models:

Anti-phase boundaries (APBs): Form at single-step terraces
Threading dislocations: $\rho_{TD} \propto f^2$ initially
Thermal mismatch stress: $\sigma_{thermal} = \frac{E \Delta \alpha \Delta T}{1-

u}$

12.3 Quantum Dot Formation (Stranski-Krastanov)

Critical thickness for islanding:

h_{SK} \approx \frac{\gamma}{M f^2}

Island density:

n_{island} \propto \exp\left(-\frac{E_{island}}{k_B T}\right) \cdot F^{1/3}

Where $F$ is the deposition flux.

12.4 Machine Learning in Epitaxy Modeling

Physics-Informed Neural Networks (PINNs):

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

Where:

$\mathcal{L}_{data}$ = data fitting loss
$\mathcal{L}_{physics}$ = PDE residual loss
$\mathcal{L}_{boundary}$ = boundary condition loss
$\lambda$ = weighting parameters

Applications:

Surrogate models for reactor optimization
Inverse problems (parameter extraction)
Process window optimization
Defect prediction

13. Key Equations

Phenomenon	Key Equation	Primary Parameters
Growth rate (dual regime)	$G = \frac{k_s h_g C_g}{k_s + h_g}$	Temperature, pressure, flow
Surface diffusion length	$\lambda_s = \sqrt{D_s \tau_s}$	Temperature
Lattice mismatch	$f = \frac{a_f - a_s}{a_s}$	Material system
Critical thickness

u)}\left[\ln\frac{h_c}{b}+1\right]$ | Mismatch, Burgers vector |

Elastic strain energy	$E = M f^2 h$	Mismatch, thickness, modulus
Nucleation rate	$J \propto \exp(-\Delta G^*/k_BT)$	Supersaturation, surface energy
Species transport

abla C = D abla^2 C + R$ | Diffusivity, velocity, reactions |

KMC event rate

u_0 \exp(-E_a/k_BT)$ | Activation energy, temperature |

Physical Constants

Constant	Symbol	Value
Boltzmann constant	$k_B$	$1.38 \times 10^{-23}$ J/K
Gas constant	$R$	8.314 J/mol·K
Planck constant	$h$	$6.63 \times 10^{-34}$ J·s
Electron charge	$e$	$1.60 \times 10^{-19}$ C
Si lattice constant	$a_{Si}$	5.431 Å
Ge lattice constant	$a_{Ge}$	5.658 Å
GaAs lattice constant	$a_{GaAs}$	5.653 Å

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