CVD/PVD Theory Guide - Comprehensive Deposition Physics

Gas-Phase Kinetics

1.1 Molecular Collision Theory

In CVD processes, gas-phase reactions occur through molecular collisions governed by kinetic theory. The collision frequency determines the rate of homogeneous reactions and precursor decomposition.

Mean Free Path

The average distance traveled between collisions is given by:

$$\lambda = \frac{k_B T}{\sqrt{2} \pi d^2 P}$$

$\lambda$: Mean free path (m)

$k_B$: Boltzmann constant (1.381×10⁻²³ J/K)

$T$: Temperature (K)

$d$: Molecular diameter (m)

$P$: Pressure (Pa)

Derivation from Kinetic Theory:

Step 1: Start with the collision cross-section $\sigma = \pi d^2$

Step 2: Number density from ideal gas law: $n = P/(k_B T)$

Step 3: Mean free path: $\lambda = 1/(\sqrt{2} n \sigma)$

Step 4: Combining yields the final expression

Collision Frequency

The rate at which molecules collide per unit volume:

$$Z = \frac{1}{2} n^2 \sigma \bar{v} = \frac{1}{2} n^2 \pi d^2 \sqrt{\frac{8k_B T}{\pi m}}$$

$Z$: Collision frequency (collisions/m³·s)

$n$: Number density (molecules/m³)

$\bar{v}$: Mean molecular velocity (m/s)

$m$: Molecular mass (kg)

1.2 Arrhenius Kinetics

Chemical reaction rates exhibit exponential temperature dependence through the Arrhenius equation, fundamental to understanding CVD process windows.

Arrhenius Rate Equation

$$k(T) = A \exp\left(-\frac{E_a}{R T}\right)$$

$k$: Reaction rate constant (units vary)

$A$: Pre-exponential factor (frequency factor)

$E_a$: Activation energy (J/mol)

$R$: Gas constant (8.314 J/mol·K)

$T$: Temperature (K)

Example: Silane Decomposition

For thermal decomposition of SiH₄:

$$\text{SiH}_4 \rightarrow \text{Si} + 2\text{H}_2$$

Typical parameters:

$A = 1.5 \times 10^{13}$ s⁻¹
$E_a = 230$ kJ/mol
At 650°C: $k \approx 0.1$ s⁻¹
At 750°C: $k \approx 2.5$ s⁻¹ (25× increase!)

Modified Arrhenius Equation

For more accurate modeling over wide temperature ranges:

$$k(T) = A T^n \exp\left(-\frac{E_a}{R T}\right)$$

$n$: Temperature exponent (typically 0.5 to 2)

1.3 Homogeneous vs. Heterogeneous Reactions

CVD involves both gas-phase (homogeneous) and surface (heterogeneous) reactions. Understanding their relative rates determines film quality and uniformity.

Damköhler Number

Ratio of reaction rate to transport rate:

$$\text{Da} = \frac{k_s \tau}{1} = \frac{\text{Reaction rate}}{\text{Transport rate}}$$

$k_s$: Surface reaction rate constant

$\tau$: Residence time

Da << 1: Reaction-limited (good uniformity)

Da >> 1: Transport-limited (poor uniformity)

Key Insights

Low pressure (LPCVD) favors heterogeneous reactions due to longer mean free path
High temperature increases homogeneous reactions, causing gas-phase nucleation
Optimal CVD occurs in surface-reaction limited regime for best uniformity
Precursor selection aims to minimize homogeneous decomposition

Surface Reactions

2.1 Langmuir Adsorption Isotherm

Surface coverage by adsorbed species follows the Langmuir model, which assumes monolayer adsorption with no interaction between adsorbed molecules.

Langmuir Isotherm

$$\theta = \frac{K P}{1 + K P}$$

Where the equilibrium constant is:

$$K = \frac{k_{\text{ads}}}{k_{\text{des}}} = K_0 \exp\left(\frac{E_{\text{ads}} - E_{\text{des}}}{R T}\right)$$

$\theta$: Fractional surface coverage (0 to 1)

$K$: Adsorption equilibrium constant (Pa⁻¹)

$P$: Partial pressure of adsorbate (Pa)

$k_{\text{ads}}$: Adsorption rate constant

$k_{\text{des}}$: Desorption rate constant

Derivation of Langmuir Isotherm:

Step 1: Rate of adsorption: $r_{\text{ads}} = k_{\text{ads}} P (1-\theta)$

Step 2: Rate of desorption: $r_{\text{des}} = k_{\text{des}} \theta$

Step 3: At equilibrium: $r_{\text{ads}} = r_{\text{des}}$

Step 4: $k_{\text{ads}} P (1-\theta) = k_{\text{des}} \theta$

Step 5: Solving for $\theta$ yields the Langmuir equation

2.2 Surface Reaction Mechanisms

Film growth involves multiple surface steps: adsorption, surface diffusion, reaction, and desorption of byproducts.

Overall Deposition Rate

For a simple Langmuir-Hinshelwood mechanism:

$$R_{\text{dep}} = k_s \theta = \frac{k_s K P}{1 + K P}$$

At low pressure (KP << 1):

$$R_{\text{dep}} \approx k_s K P \quad \text{(first-order, reaction-limited)}$$

At high pressure (KP >> 1):

$$R_{\text{dep}} \approx k_s \quad \text{(zero-order, transport-limited)}$$

Multi-Step Surface Kinetics

For complex reactions with multiple intermediates:

$$\frac{d\theta_i}{dt} = \sum_j r_j - \sum_k r_k$$

Example: TEOS decomposition on Si surface

$$\begin{align} \text{TEOS}_{\text{gas}} &\leftrightarrow \text{TEOS}_{\text{ads}} \\ \text{TEOS}_{\text{ads}} &\rightarrow \text{SiO}_2 + \text{byproducts}_{\text{ads}} \\ \text{byproducts}_{\text{ads}} &\rightarrow \text{byproducts}_{\text{gas}} \end{align}$$

2.3 Sticking Coefficient

The sticking coefficient represents the probability that an incident molecule will be incorporated into the film.

Sticking Coefficient Definition

$$s = \frac{\text{Number of molecules incorporated}}{\text{Number of molecules incident}}$$

Temperature dependence:

$$s(T) = s_0 \exp\left(-\frac{E_s}{k_B T}\right)$$

$s$: Sticking coefficient (0 to 1)

$s_0$: Pre-exponential factor

$E_s$: Activation energy for sticking

Typical Sticking Coefficients

Physical deposition (PVD): s ≈ 0.3-0.8 (weakly temperature dependent)
Chemical deposition (CVD): s ≈ 0.01-0.5 (strongly temperature dependent)
ALD: s → 1.0 in self-limiting regime
Reactive sputtering: s ≈ 0.5-0.9 (high for reactive species)

Transport Phenomena

3.1 Mass Transport Regimes

Transport of reactants to the surface depends on pressure regime, characterized by the Knudsen number.

Knudsen Number

$$\text{Kn} = \frac{\lambda}{L}$$

$\lambda$: Mean free path

$L$: Characteristic dimension (reactor/feature size)

Kn < 0.01: Continuum flow (viscous)

0.01 < Kn < 1: Transition regime

Kn > 1: Molecular flow (ballistic)

Regime Characteristics

Continuum (APCVD): Fick's law diffusion, boundary layers, convection important
Molecular (LPCVD): Direct impingement, excellent step coverage, long pump-down
Transition: Mixed transport, requires complex modeling

3.2 Diffusion in Boundary Layers

In atmospheric pressure CVD, a stagnant boundary layer forms above the substrate, creating a diffusion barrier.

Fick's First Law

$$J = -D \frac{dC}{dx}$$

$J$: Diffusive flux (molecules/m²·s)

$D$: Diffusion coefficient (m²/s)

$C$: Concentration (molecules/m³)

Binary Diffusion Coefficient

Chapman-Enskog theory for gas mixtures:

$$D_{AB} = \frac{3}{16} \frac{(4\pi k_B T)^{1/2}}{n \pi d_{AB}^2} \left(\frac{1}{m_A} + \frac{1}{m_B}\right)^{1/2}$$

Simplified temperature/pressure scaling:

$$D \propto \frac{T^{3/2}}{P}$$

Mass Transfer Coefficient

For flow over a flat plate substrate:

$$h_m = \frac{D}{\delta}$$

Boundary layer thickness:

$$\delta \approx \sqrt{\frac{D x}{v}}$$

$h_m$: Mass transfer coefficient (m/s)

$\delta$: Boundary layer thickness (m)

$x$: Distance from leading edge

$v$: Velocity (m/s)

3.3 Overall Deposition Rate Model

Combining mass transport and surface kinetics using the resistance model.

Series Resistance Model

$$\frac{1}{R_{\text{overall}}} = \frac{1}{R_{\text{transport}}} + \frac{1}{R_{\text{reaction}}}$$

Deposition rate:

$$R_{\text{dep}} = \frac{h_m k_s C_{\text{bulk}}}{h_m + k_s}$$

Surface concentration:

$$C_{\text{surf}} = \frac{k_s}{h_m + k_s} C_{\text{bulk}}$$

Limiting Cases:

Transport-limited (ks >> hm):

$$R_{\text{dep}} \approx h_m C_{\text{bulk}}$$

Surface concentration ≈ 0, weak temperature dependence

Reaction-limited (ks << hm):

$$R_{\text{dep}} \approx k_s C_{\text{bulk}}$$

Surface concentration ≈ bulk, strong temperature dependence

Nucleation and Growth

4.1 Classical Nucleation Theory

Film formation begins with nucleation of atomic clusters on the substrate surface. The process is governed by thermodynamic competition between surface energy and bulk energy.

Gibbs Free Energy of Nucleation

For a spherical nucleus of radius r:

$$\Delta G(r) = -\frac{4}{3}\pi r^3 \Delta G_v + 4\pi r^2 \gamma$$

$\Delta G_v$: Bulk free energy change per unit volume (J/m³)

$\gamma$: Surface energy (J/m²)

Critical Nucleus Size

Maximum in free energy curve occurs at:

$$r^* = \frac{2\gamma}{\Delta G_v}$$

Activation barrier for nucleation:

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

Nucleation Rate

Number of nuclei formed per unit area per unit time:

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

$N_s$: Surface site density (sites/m²)

$\nu$: Attempt frequency (s⁻¹)

4.2 Growth Modes

Thin film growth follows three primary modes depending on interfacial energies.

Wetting Criterion

Young's equation determines growth mode:

$$\gamma_{\text{substrate}} = \gamma_{\text{film}} + \gamma_{\text{interface}} + \gamma_{\text{film}} \cos\theta$$

Frank-van der Merwe (layer-by-layer): $\gamma_s > \gamma_f + \gamma_i$

Volmer-Weber (island): $\gamma_s < \gamma_f + \gamma_i$

Stranski-Krastanov (layer + island): Intermediate case

Growth Mode Examples

Layer-by-layer: Si on Si, GaAs on GaAs, most homoepitaxy
Island: Metals on oxides (Au/SiO₂, Ag/Al₂O₃), poor wetting
Layer + island: Ge on Si (lattice mismatch), InAs on GaAs

4.3 Film Thickness Evolution

After nucleation, film growth proceeds through island coalescence and continuous film formation.

Avrami Equation

Fraction of surface covered as a function of time:

$$\theta(t) = 1 - \exp\left(-kt^n\right)$$

$n$: Avrami exponent (dimensionality indicator)

$k$: Growth rate constant

n = 1: 1D growth (needles)

n = 2: 2D growth (disks)

n = 3: 3D growth (spheres)

Grain Growth Kinetics

Average grain size evolution during coalescence:

$$\frac{dD}{dt} = K_0 \exp\left(-\frac{E_g}{k_B T}\right) \frac{1}{D}$$

Parabolic growth law:

$$D^2 = D_0^2 + K t$$

Plasma Physics for PECVD

5.1 Plasma Generation and Properties

PECVD uses RF or microwave power to create weakly ionized plasma, enabling low-temperature deposition through enhanced chemical reactivity.

Degree of Ionization

$$\alpha = \frac{n_e}{n_e + n_n}$$

$n_e$: Electron density (m⁻³)

$n_n$: Neutral density (m⁻³)

Typical PECVD: α ≈ 10⁻⁶ to 10⁻⁴ (weakly ionized)

Electron Energy Distribution

Maxwellian distribution for electron energies:

$$f(E) = 2\pi n_e \left(\frac{1}{\pi k_B T_e}\right)^{3/2} \sqrt{E} \exp\left(-\frac{E}{k_B T_e}\right)$$

$T_e$: Electron temperature (eV), typically 1-10 eV

Ion temperature: ≈ gas temperature (0.03 eV at 300K)

Debye Shielding Length

Distance over which electric fields are screened:

$$\lambda_D = \sqrt{\frac{\epsilon_0 k_B T_e}{n_e e^2}}$$

For typical PECVD (Te = 3 eV, ne = 10¹⁶ m⁻³):

$$\lambda_D \approx 1.3 \times 10^{-4} \text{ m} = 130 \text{ μm}$$

Plasma Frequency

$$\omega_p = \sqrt{\frac{n_e e^2}{\epsilon_0 m_e}}$$

RF coupling requires $\omega_{RF} > \omega_p$ for efficient power transfer

5.2 RF Sheath Dynamics

Charged particle sheaths form at plasma-surface boundaries, creating DC self-bias and ion bombardment.

Child-Langmuir Law

Sheath voltage-current relationship:

$$J_i = \frac{4\epsilon_0}{9} \sqrt{\frac{2e}{M}} \frac{V_s^{3/2}}{d^2}$$

$J_i$: Ion current density (A/m²)

$V_s$: Sheath voltage (V)

$d$: Sheath thickness (m)

$M$: Ion mass (kg)

DC Self-Bias

For capacitively coupled RF plasma:

$$V_{DC} = -V_{RF} \left(\frac{A_{\text{powered}}}{A_{\text{ground}}}\right)^n$$

where n ≈ 2-4 depending on pressure and frequency

Ion Bombardment Effects

Positive: Enhanced surface mobility, densification, contamination removal
Negative: Radiation damage, preferential sputtering, stress buildup
Control: Adjust RF power, bias voltage, pressure for optimal balance

5.3 Electron Impact Processes

Plasma chemistry driven by energetic electron collisions with precursor molecules.

Electron Impact Cross-Section

Rate coefficient for electron impact process i:

$$k_i = \int_0^{\infty} \sigma_i(E) v(E) f(E) \, dE$$

$\sigma_i(E)$: Energy-dependent cross-section (m²)

$v(E)$: Electron velocity (m/s)

$f(E)$: Electron energy distribution function

Example: SiH₄ Dissociation in Plasma

Electron impact reactions (with threshold energies):

e⁻ + SiH₄ → SiH₃ + H + e⁻ (≈ 8.7 eV)
e⁻ + SiH₄ → SiH₂ + 2H + e⁻ (≈ 10.5 eV)
e⁻ + SiH₄ → SiH₄⁺ + 2e⁻ (≈ 11.6 eV ionization)
e⁻ + SiH₄ → SiH₂⁻ + H₂ (≈ 3 eV dissociative attachment)

SiH₃ is primary growth precursor in PECVD a-Si:H

Sputtering Physics

6.1 Sputter Yield Theory

Sputter yield quantifies the efficiency of material removal by ion bombardment.

Sputter Yield Definition

$$Y = \frac{\text{Number of sputtered atoms}}{\text{Number of incident ions}}$$

Sigmund Theory

For low-energy sputtering (E < 10 keV):

$$Y = \frac{3}{4\pi^2} \frac{\alpha S_n(E)}{U_0 N}$$

$\alpha$: Energy transfer efficiency ≈ 0.15-0.3

$S_n(E)$: Nuclear stopping power (eV·m²)

$U_0$: Surface binding energy (eV)

$N$: Atomic density (atoms/m³)

Energy Dependence

Simplified power-law approximation:

$$Y(E) = Y_0 \left(\frac{E - E_{th}}{E_0}\right)^n$$

$E_{th}$: Threshold energy ≈ 20-50 eV

$n$: Exponent ≈ 0.5-1

Angular Dependence

Sputter yield increases with incident angle:

$$Y(\theta) = Y(0) \cdot f(\theta)$$

Common forms:

$$f(\theta) = \cos^{-n}(\theta) \quad \text{or} \quad f(\theta) = \exp\left(\frac{1}{\cos\theta} - 1\right)$$

Maximum typically occurs at θ ≈ 60-70° from normal

Typical Sputter Yields (Ar⁺, 500 eV)

Cu: Y ≈ 2.4 atoms/ion
Al: Y ≈ 1.1 atoms/ion
Ti: Y ≈ 0.5 atoms/ion
W: Y ≈ 0.6 atoms/ion
Au: Y ≈ 3.0 atoms/ion
SiO₂: Y ≈ 0.3 molecules/ion (RF sputtering)

6.2 Deposition Rate

Translating sputter yield to film deposition rate on substrate.

Deposition Rate from Sputter Yield

$$R_{\text{dep}} = \frac{Y \cdot J_i \cdot M}{\rho \cdot e \cdot N_A} \cdot \frac{\Omega}{4\pi}$$

$J_i$: Ion current density at target (A/m²)

$M$: Atomic mass (g/mol)

$\rho$: Film density (g/cm³)

$\Omega/4\pi$: Geometric factor (substrate-target)

Cosine Law for Flux Distribution

Sputtered atom flux from target element dA:

$$d\Phi = \frac{Y J_i}{e} \frac{\cos\theta_1 \cos\theta_2}{\pi r^2} dA$$

where θ₁ is emission angle, θ₂ is incidence angle at substrate

Factors Affecting Deposition Rate

Power: Higher power → higher Ji → faster deposition
Pressure: Lower pressure → less scattering → better directionality
Target-substrate distance: Closer → higher flux, worse uniformity
Magnetron effect: 5-10× higher deposition rate vs. diode sputtering

6.3 Reactive Sputtering

Combining sputtered metal atoms with reactive gas (O₂, N₂) to form compound films.

Berg Model

Target coverage by reactive compound:

$$\frac{d\theta_t}{dt} = \alpha q_{react} (1-\theta_t) - Y_c J_i \theta_t$$

$\theta_t$: Fraction of target covered by compound

$\alpha$: Sticking coefficient of reactive gas

$q_{react}$: Reactive gas flux

$Y_c$: Sputter yield of compound

Hysteresis curve: Deposition rate vs. reactive gas flow showing metallic mode → transition → compound mode

Evaporation Thermodynamics

7.1 Vapor Pressure

Evaporation rate is determined by equilibrium vapor pressure, which depends strongly on temperature.

Clausius-Clapeyron Equation

$$\ln P = -\frac{\Delta H_{vap}}{R T} + C$$

More accurate Antoine equation:

$$\log_{10} P = A - \frac{B}{T + C}$$

$P$: Vapor pressure (Torr or Pa)

$\Delta H_{vap}$: Enthalpy of vaporization (J/mol)

$A, B, C$: Material-specific constants

Vapor Pressures at Various Temperatures

Material	T for 10⁻² Torr	Melting Point
Al	1050°C	660°C
Cu	1320°C	1085°C
Au	1520°C	1064°C
Ag	1150°C	962°C
Ti	1710°C	1668°C

7.2 Evaporation Rate

Maximum evaporation rate from kinetic theory of gases.

Langmuir Evaporation Equation

$$\Phi = \alpha \frac{P}{\sqrt{2\pi m k_B T}}$$

In practical units (g/cm²·s):

$$\Gamma = 5.83 \times 10^{-2} \alpha P \sqrt{\frac{M}{T}}$$

$\Phi$: Evaporation flux (molecules/m²·s)

$\Gamma$: Mass flux (g/cm²·s)

$\alpha$: Evaporation coefficient (≈0.01 to 1)

$P$: Vapor pressure (Torr)

$M$: Molecular weight (g/mol)

Derivation from Kinetic Theory:

Step 1: From Maxwell-Boltzmann distribution, mean velocity: $\bar{v} = \sqrt{8k_BT/\pi m}$

Step 2: Molecular flux: $\Phi = \frac{1}{4} n \bar{v}$

Step 3: Number density: $n = P/(k_B T)$

Step 4: Combining: $\Phi = P/\sqrt{2\pi m k_B T}$

7.3 Flux Distribution and Film Thickness

Evaporated atoms follow ballistic trajectories in high vacuum, creating directional deposition.

Point Source Cosine Distribution

Flux at substrate from small evaporation source:

$$\Phi(r, \theta) = \frac{\Phi_0 \cos\theta}{r^2}$$

Film thickness for normal incidence:

$$t(r) = \frac{t_0}{1 + (r/h)^2}$$

$r$: Radial distance from axis

$h$: Source-substrate distance

$\theta$: Emission angle from source normal

Evaporation Characteristics

Highly directional: Poor step coverage, shadowing effects
Line-of-sight: Useful for liftoff patterning
Planetary rotation: Improves uniformity over large areas
Contamination: Requires UHV for high purity films

Atomic Layer Deposition Chemistry

8.1 Self-Limiting Surface Chemistry

ALD achieves atomic-level precision through sequential, self-limiting surface reactions that ensure exactly one monolayer per cycle.

Saturation Curve

Growth per cycle as a function of precursor exposure:

$$\text{GPC} = \text{GPC}_{\infty} \left(1 - \exp\left(-\frac{t}{\tau}\right)\right)$$

GPC: Growth per cycle (Å/cycle)

GPC∞: Saturation value

$t$: Pulse time (s)

$\tau$: Time constant (s)

Example: Al₂O₃ ALD with TMA/H₂O

Half-reaction 1 (TMA pulse):

$$\text{-OH*} + \text{Al(CH}_3\text{)}_3 \rightarrow \text{-O-Al(CH}_3\text{)}_2\text{*} + \text{CH}_4$$

Half-reaction 2 (H₂O pulse):

$$\text{-Al(CH}_3\text{)}_2\text{*} + 2\text{H}_2\text{O} \rightarrow \text{-Al(OH)}_2\text{*} + 2\text{CH}_4$$

Net reaction per cycle:

$$2\text{Al(CH}_3\text{)}_3 + 3\text{H}_2\text{O} \rightarrow \text{Al}_2\text{O}_3 + 6\text{CH}_4$$

Typical GPC: 1.1-1.3 Å/cycle at 200-300°C

8.2 ALD Temperature Window

ALD operates within a temperature window where reactions are complete but decomposition/desorption are minimal.

Temperature Regime Boundaries

Lower bound: Incomplete reaction or condensation

$$T_{\text{min}} \approx \frac{\Delta H_{rxn}}{R \ln(P_0/P_{\text{vap}})}$$

Upper bound: Precursor decomposition or desorption

$$T_{\text{max}} \approx \frac{E_{\text{decomp}}}{R \ln(\nu t / \ln 2)}$$

ALD window plot: GPC vs. Temperature showing condensation regime, ALD window, decomposition regime

8.3 Conformality and Aspect Ratio

ALD provides excellent step coverage due to self-limiting chemistry, but high aspect ratio features require sufficient precursor exposure.

Aspect Ratio Dependent Exposure

Required exposure for complete penetration:

$$E = P \cdot t \propto \text{AR}^2$$

E: Precursor exposure (Langmuir, Pa·s)

AR: Aspect ratio (depth/width)

Knudsen Diffusion in Trenches

Effective diffusivity in high-AR features:

$$D_{eff} = \frac{d}{3} \bar{v} = \frac{d}{3} \sqrt{\frac{8RT}{\pi M}}$$

Penetration depth:

$$L_p = \sqrt{D_{eff} \tau_p}$$

$d$: Feature width (m)

$\tau_p$: Pulse time (s)

ALD Advantages and Challenges

Pros: Atomic precision, perfect conformality, digital thickness control
Cons: Slow (0.1-1 nm/min), requires reactive surface, limited materials
Applications: High-k dielectrics, diffusion barriers, 3D structures

Film Properties

9.1 Intrinsic Stress

Thin films develop stress during growth from lattice mismatch, thermal expansion differences, and microstructural evolution.

Stoney Equation

Stress from substrate curvature measurement:

$$\sigma_f = \frac{E_s t_s^2}{6(1-\nu_s) t_f} \left(\frac{1}{R} - \frac{1}{R_0}\right)$$

$\sigma_f$: Film stress (Pa, positive = tensile)

$E_s$: Substrate Young's modulus (Pa)

$\nu_s$: Substrate Poisson ratio

$t_s, t_f$: Substrate, film thickness (m)

$R, R_0$: Curved, initial radius of curvature (m)

Thermal Stress Component

$$\sigma_{thermal} = \frac{E_f}{1-\nu_f} (\alpha_s - \alpha_f) \Delta T$$

$\alpha_s, \alpha_f$: Substrate, film thermal expansion coefficients (K⁻¹)

$\Delta T$: Temperature change from deposition (K)

Typical Stress Values

LPCVD Si₃N₄: +1000 MPa (highly tensile)
PECVD Si₃N₄: -200 to +200 MPa (tunable)
Sputtered Al: -50 to +50 MPa (low stress)
E-beam evaporated Cr: +500 MPa (tensile)
PECVD SiO₂: -300 to -100 MPa (compressive)

9.2 Grain Structure and Texture

Polycrystalline film microstructure affects electrical, mechanical, and optical properties.

Hall-Petch Relation

Grain size effect on mechanical strength:

$$\sigma_y = \sigma_0 + \frac{k_y}{\sqrt{D}}$$

$\sigma_y$: Yield strength (Pa)

$D$: Grain diameter (m)

$k_y$: Hall-Petch constant (Pa·m^(1/2))

Texture Coefficient

Quantifies preferential crystallographic orientation:

$$TC(hkl) = \frac{I(hkl)/I_0(hkl)}{\frac{1}{N}\sum_N I(hkl)/I_0(hkl)}$$

I(hkl): Measured XRD intensity

I₀(hkl): Random powder reference intensity

TC = 1: Random orientation

TC > 1: Preferred orientation

Scherrer Equation for Grain Size

From XRD peak broadening:

$$D = \frac{K\lambda}{\beta \cos\theta}$$

$K$: Shape factor ≈ 0.9

$\lambda$: X-ray wavelength (m)

$\beta$: FWHM of diffraction peak (radians)

$\theta$: Bragg angle

9.3 Resistivity and Electrical Properties

Thin metal films exhibit higher resistivity than bulk due to surface scattering and grain boundary scattering.

Matthiessen's Rule

$$\rho_{total} = \rho_{bulk} + \rho_{surface} + \rho_{GB} + \rho_{impurity}$$

Fuchs-Sondheimer Model (Surface Scattering)

$$\frac{\rho_{film}}{\rho_{bulk}} = 1 + \frac{3\lambda}{8t}(1-p)$$

$\lambda$: Electron mean free path (m)

$t$: Film thickness (m)

$p$: Specularity parameter (0 = diffuse, 1 = specular)

Mayadas-Shatzkes Model (Grain Boundary Scattering)

$$\frac{\rho_{film}}{\rho_{bulk}} = 1 + \frac{3}{2}\alpha - 3\alpha^2 + 3\alpha^3 \ln\left(1 + \frac{1}{\alpha}\right)$$

where:

$$\alpha = \frac{\lambda}{D} \frac{R}{1-R}$$

$D$: Grain diameter (m)

$R$: Grain boundary reflection coefficient

Copper Interconnect Resistivity

Bulk Cu: ρ = 1.7 μΩ·cm, λ = 40 nm
100 nm Cu film: ρ ≈ 2.0 μΩ·cm (+18% increase)
20 nm Cu line: ρ ≈ 4.5 μΩ·cm (+165% increase)
Severe resistivity increase at advanced nodes (<10 nm)

Interface Physics

10.1 Adhesion Energy

Film adhesion depends on interfacial bonding energy, which can be estimated from thermodynamic considerations.

Work of Adhesion

$$W_{ad} = \gamma_f + \gamma_s - \gamma_{fs}$$

$W_{ad}$: Work of adhesion (J/m²)

$\gamma_f, \gamma_s$: Film, substrate surface energies

$\gamma_{fs}$: Film-substrate interfacial energy

Dupré Equation with Contact Angle

$$W_{ad} = \gamma_f (1 + \cos\theta)$$

where θ is the contact angle of liquid film material on substrate

Improving Adhesion

Surface cleaning: Remove oxides, organics before deposition
Adhesion layers: Ti, Cr, Ta for metal on oxide
Surface roughening: Mechanical interlocking
Interface reaction: Form chemical bonds (e.g., silicide)

10.2 Interfacial Reactions and Diffusion

At elevated temperatures, atomic interdiffusion across interfaces can degrade film properties or create useful phases.

Fick's Second Law

$$\frac{\partial C}{\partial t} = D \frac{\partial^2 C}{\partial x^2}$$

Solution for diffusion couple (semi-infinite boundaries):

$$C(x,t) = \frac{C_0}{2} \left[1 - \text{erf}\left(\frac{x}{2\sqrt{Dt}}\right)\right]$$

Diffusion Coefficient Temperature Dependence

$$D = D_0 \exp\left(-\frac{E_a}{RT}\right)$$

$D_0$: Pre-exponential factor (m²/s)

$E_a$: Activation energy for diffusion (J/mol)

Example: Al/Si Contact

At 450°C, Al and Si interdiffuse:

Si diffusion into Al: D ≈ 10⁻¹² cm²/s
After 30 min: diffusion length ≈ 0.2 μm
Can cause junction spiking in shallow junctions!
Solution: Add Si to Al (1-2% Si-Al alloy)

10.3 Contact Resistance

Metal-semiconductor contacts exhibit resistance due to Schottky barriers or tunneling through interfacial layers.

Specific Contact Resistivity

For moderately doped semiconductors:

$$\rho_c = \frac{k}{\sqrt{N_d}} \exp\left(\frac{2\phi_B}{\hbar} \sqrt{\frac{\epsilon_s m^*}{N_d}}\right)$$

$\rho_c$: Specific contact resistance (Ω·cm²)

$N_d$: Doping concentration (cm⁻³)

$\phi_B$: Barrier height (eV)

Transfer Length Method (TLM)

Total resistance vs. contact spacing:

$$R_{total} = 2R_c + \frac{\rho_s L}{W}$$

where the contact resistance is:

$$R_c = \frac{1}{W} \sqrt{\rho_c \rho_s} \coth\left(\frac{L_c}{L_T}\right)$$

$L_T$: Transfer length = $\sqrt{\rho_c / \rho_s}$

$\rho_s$: Sheet resistance of semiconductor

Advanced Transport Models

A.1 Rarefied Gas Dynamics

In the transition and molecular flow regimes, continuum fluid mechanics breaks down and kinetic theory is required.

Boltzmann Transport Equation

$$\frac{\partial f}{\partial t} + \vec{v} \cdot \nabla_r f + \vec{a} \cdot \nabla_v f = \left(\frac{\partial f}{\partial t}\right)_{\text{coll}}$$

$f(\vec{r}, \vec{v}, t)$: Distribution function

$\vec{a}$: Acceleration (external forces)

Collision term: Describes particle interactions

BGK Approximation

Simplified collision term:

$$\left(\frac{\partial f}{\partial t}\right)_{\text{coll}} = -\frac{f - f_{eq}}{\tau_c}$$

$f_{eq}$: Maxwellian equilibrium distribution

$\tau_c$: Collision time

Direct Simulation Monte Carlo (DSMC)

Numerical method for rarefied flows - particle tracking with statistical collisions

$$\Delta t < \frac{\lambda}{\bar{v}}, \quad \Delta x < \lambda$$

Number of particles per cell:

$$N_{cell} > 20 \text{ for statistical accuracy}$$

A.2 Multicomponent Diffusion

In CVD with multiple precursors and carrier gases, Stefan-Maxwell equations describe coupled diffusion.

Stefan-Maxwell Equations

$$\nabla x_i = \sum_{j \neq i} \frac{x_i x_j}{D_{ij}} (\vec{v}_j - \vec{v}_i)$$

$x_i$: Mole fraction of species i

$D_{ij}$: Binary diffusion coefficient

$\vec{v}_i$: Diffusion velocity of species i

Effective Diffusivity Matrix

For N-component system:

$$\vec{J}_i = -\rho \sum_j D_{ij}^{eff} \nabla Y_j$$

where mass fractions sum to unity: $\sum_i Y_i = 1$

A.3 Thermophoresis and Electrophoresis

Temperature and electric field gradients can induce particle drift, affecting film uniformity and contamination.

Thermophoretic Velocity

$$\vec{v}_{th} = -K_{th} \frac{\nabla T}{T}$$

$K_{th}$: Thermophoretic coefficient (m²/s)

Typical value: K_th ≈ 10⁻⁷ m²/s for particles in gases

Electrophoretic Mobility

$$\vec{v}_{ep} = \mu_{ep} \vec{E}$$ $$\mu_{ep} = \frac{q}{6\pi \eta r}$$

$q$: Particle charge (C)

$\eta$: Dynamic viscosity (Pa·s)

$r$: Particle radius (m)

Advanced Reaction Kinetics

B.1 Transition State Theory

Provides molecular-level understanding of activation energies and pre-exponential factors.

Eyring Equation

$$k = \frac{k_B T}{h} e^{\Delta S^‡/R} e^{-\Delta H^‡/RT}$$

$\Delta S^‡$: Activation entropy (J/mol·K)

$\Delta H^‡$: Activation enthalpy (J/mol)

$h$: Planck's constant (6.626×10⁻³⁴ J·s)

Transmission Coefficient

Accounts for quantum tunneling and recrossing:

$$k = \kappa \frac{k_B T}{h} e^{-\Delta G^‡/RT}$$

$\kappa$: Transmission coefficient (typically 0.5-1.5)

$\Delta G^‡$: Gibbs free energy of activation

B.2 Microkinetic Modeling

Detailed reaction mechanism with elementary steps for CVD surface chemistry.

Surface Coverage Dynamics

System of ODEs for each surface species:

$$\frac{d\theta_i}{dt} = \sum_r \nu_{ir} k_r \prod_j \theta_j^{\alpha_{jr}} P_j^{\beta_{jr}}$$

$\nu_{ir}$: Stoichiometric coefficient of species i in reaction r

$\alpha, \beta$: Reaction orders for surface and gas species

Example: TEOS CVD Microkinetic Model

Elementary steps:

TEOS(g) + * → TEOS*
TEOS* → Si(OC₂H₅)₃* + C₂H₅*
Si(OC₂H₅)₃* → Si(OC₂H₅)₂* + C₂H₅O*
Si(OC₂H₅)₂* + O* → SiO₂* + products
C₂H₅* → C₂H₄(g) + H*
2H* → H₂(g) + 2*

12-step model with rate constants from DFT calculations

B.3 Sensitivity Analysis

Identifies rate-limiting steps and important parameters in complex mechanisms.

Normalized Sensitivity Coefficient

$$S_{ij} = \frac{\partial \ln y_i}{\partial \ln k_j} = \frac{k_j}{y_i} \frac{\partial y_i}{\partial k_j}$$

$y_i$: Observable (e.g., deposition rate)

$k_j$: Rate constant for reaction j

|S| > 1: Highly sensitive parameter

Advanced Plasma Modeling

C.1 Plasma Fluid Equations

Self-consistent model coupling particle densities, fluxes, and electric fields.

Continuity Equations

For each charged species:

$$\frac{\partial n_s}{\partial t} + \nabla \cdot \vec{\Gamma}_s = S_s$$

Where flux includes drift and diffusion:

$$\vec{\Gamma}_s = \pm n_s \mu_s \vec{E} - D_s \nabla n_s$$

Poisson's Equation

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

$\phi$: Electric potential (V)

$n_i, n_e$: Ion, electron densities

Electron Energy Equation

$$\frac{3}{2}\frac{\partial (n_e T_e)}{\partial t} + \nabla \cdot \vec{q}_e = -e\vec{\Gamma}_e \cdot \vec{E} - \sum_k n_e n_k k_k \Delta E_k$$

$\vec{q}_e$: Electron energy flux

$\Delta E_k$: Energy loss in collision process k

C.2 Particle-in-Cell (PIC) Method

Kinetic simulation tracking individual macro-particles through self-consistent fields.

PIC Algorithm Cycle

Solve Poisson equation on grid: $\nabla^2 \phi = -\rho/\epsilon_0$
Calculate electric field: $\vec{E} = -\nabla \phi$
Interpolate field to particle positions
Move particles: $\frac{d\vec{v}}{dt} = \frac{q}{m}\vec{E}$, $\frac{d\vec{x}}{dt} = \vec{v}$
Weight particles to grid: $\rho(\vec{x}_i) = \sum_p q_p W(\vec{x}_i - \vec{x}_p)$
Apply collisions via Monte Carlo

C.3 Global Plasma Models

Volume-averaged model for rapid parameter space exploration.

Power Balance

$$P_{abs} = P_{ion} + P_{excit} + P_{diss} + P_{wall}$$

Absorbed RF power:

$$P_{abs} = \eta \cdot P_{RF}$$

$\eta$: Power transfer efficiency (0.6-0.9)

$P_{ion}$: Power to ionization

$P_{excit}$: Power to excitation

$P_{diss}$: Power to dissociation

Particle Balance

$$\frac{dn_e}{dt} = k_{iz} n_e n_g - \frac{n_e}{\tau_{loss}}$$

Steady-state electron density:

$$n_e = \sqrt{\frac{k_{iz} n_g}{\nu_{loss}/\Lambda}}$$

$\tau_{loss}$: Particle confinement time

$\Lambda$: Characteristic diffusion length

Defects and Stress Mechanisms

D.1 Point Defect Thermodynamics

Vacancies, interstitials, and impurities affect film properties and stress evolution.

Equilibrium Defect Concentration

$$c_v = \exp\left(-\frac{G_f}{k_B T}\right) = \exp\left(\frac{S_f}{k_B}\right) \exp\left(-\frac{H_f}{k_B T}\right)$$

$G_f$: Gibbs free energy of formation

$H_f$: Formation enthalpy (typically 1-3 eV)

$S_f$: Formation entropy

Defect Diffusion

$$D_v = a^2 \nu_0 \exp\left(-\frac{E_m}{k_B T}\right)$$

$E_m$: Migration energy barrier

$a$: Lattice parameter

$\nu_0$: Attempt frequency ≈ 10¹³ s⁻¹

D.2 Grain Boundary Structure

Grain boundaries are planar defects that dominate polycrystalline film properties.

Grain Boundary Energy

$$\gamma_{GB}(\theta) = \gamma_{max} \sin\theta \left[1 - \ln(\sin\theta)\right]$$

$\theta$: Misorientation angle

$\gamma_{max}$: High-angle GB energy (0.5-1 J/m²)

Read-Shockley Equation

For low-angle grain boundaries ($\theta < 15°$):

$$\gamma_{GB} = \gamma_0 \theta \left(1 - \ln\frac{\theta}{\theta_m}\right)$$

D.3 Stress Evolution During Growth

Film stress changes dynamically during deposition through multiple mechanisms.

Chason-Sheldon Model

Stress evolution rate:

$$\frac{d\sigma}{dt} = \frac{d\sigma_{dep}}{dt} + \frac{d\sigma_{relax}}{dt}$$

Deposition term (tensile during island coalescence):

$$\frac{d\sigma_{dep}}{dt} = \frac{2\gamma_{surf}}{h} \frac{dh}{dt}$$

Relaxation term (stress relief):

$$\frac{d\sigma_{relax}}{dt} = -\frac{\sigma}{\tau_{relax}}$$

Atomic Peening Model

Ion bombardment-induced densification in PECVD/PVD:

$$\sigma = K \cdot \left(\frac{E_{ion}}{E_d}\right)^{1/2} \cdot \frac{J_i}{J_a}$$

$E_{ion}$: Ion energy (eV)

$E_d$: Displacement energy ≈ 25 eV

$J_i/J_a$: Ion-to-atom flux ratio

D.4 Hillock and Void Formation

Stress-driven surface morphology evolution can cause film failure.

Critical Stress for Hillock Formation

$$\sigma_{crit} = \frac{\gamma_{surf}}{h} + \frac{2\gamma_{GB}}{d}$$

$h$: Film thickness

$d$: Grain size

Compressive stress > σ_crit causes hillocks

Void Nucleation Criterion

Tensile stress can nucleate voids at grain boundaries:

$$\sigma > \sigma_{void} = \frac{3\gamma_{GB}}{r_{void}}$$

Optical Properties of Thin Films

E.1 Fresnel Equations and Reflectance

Light interaction with thin films described by complex refractive index n + iκ.

Fresnel Reflection Coefficients

For s-polarization:

$$r_s = \frac{n_1 \cos\theta_i - n_2 \cos\theta_t}{n_1 \cos\theta_i + n_2 \cos\theta_t}$$

For p-polarization:

$$r_p = \frac{n_2 \cos\theta_i - n_1 \cos\theta_t}{n_2 \cos\theta_i + n_1 \cos\theta_t}$$

Thin Film Interference

Reflectance with interference from film of thickness d:

$$R = \left|\frac{r_{01} + r_{12} e^{i2\beta}}{1 + r_{01} r_{12} e^{i2\beta}}\right|^2$$

where phase thickness:

$$\beta = \frac{2\pi}{ \lambda} n_f d \cos\theta_t$$

Anti-Reflection Coating Design

Single-layer AR coating on substrate (n_s):

Optimal refractive index: $n_f = \sqrt{n_s}$
Optimal thickness: $d = \lambda/(4n_f)$ (quarter-wave)
Example for Si (n=3.88) at 550nm: n_f = 1.97, d = 70 nm
SiO₂/TiO₂ or Si₃N₄ commonly used

E.2 Ellipsometry and Film Characterization

Ellipsometry measures change in polarization state to determine n, κ, and thickness.

Ellipsometric Angles

$$\rho = \frac{r_p}{r_s} = \tan\Psi \cdot e^{i\Delta}$$

$\Psi$: Amplitude ratio angle

$\Delta$: Phase difference

Measured (Ψ, Δ) fitted to optical model → n, κ, d

Cauchy Dispersion Model

For transparent dielectrics:

$$n(\lambda) = A + \frac{B}{\lambda^2} + \frac{C}{\lambda^4}$$

A, B, C: Cauchy coefficients

Example for SiO₂: A=1.450, B=0.0036 μm²

Tauc-Lorentz Model

For amorphous semiconductors combining bandgap and oscillator:

$$\epsilon_2(E) = \frac{AE_0C(E-E_g)^2}{(E^2-E_0^2)^2 + C^2E^2} \cdot \frac{1}{E} \quad (E > E_g)$$

$E_g$: Optical bandgap

$E_0$: Oscillator energy

$A, C$: Amplitude and broadening

E.3 Absorption and Bandgap Engineering

Optical absorption determines bandgap and electronic structure of deposited films.

Beer-Lambert Law

$$I = I_0 e^{-\alpha d}$$

where absorption coefficient:

$$\alpha = \frac{4\pi \kappa}{\lambda}$$

Tauc Plot for Bandgap

For direct bandgap semiconductors:

$$(\alpha h\nu)^2 = A(h\nu - E_g)$$

For indirect bandgap:

$$(\alpha h\nu)^{1/2} = A(h\nu - E_g)$$

Plot (αhν)² vs. hν, extrapolate linear region to α=0

PECVD a-Si:H typically has E_g = 1.6-1.8 eV

Film Metrology and Characterization

F.1 Thickness Measurement Techniques

Multiple complementary techniques for film thickness measurement from Å to μm range.

Stylus Profilometry

Mechanical step height measurement:

$$\Delta z = \int_0^L \frac{dz}{dx} dx$$

Resolution: 1-10 Å vertical, 0.1 μm lateral

Requires step edge (masked or scratched)

Limitations: Destructive, no optical properties

X-Ray Reflectometry (XRR)

Oscillations in reflectivity vs. angle:

$$\Delta \theta = \frac{\lambda}{2d}$$

Thickness range: 10 Å to 500 nm

Also measures: Density, roughness, composition

Non-destructive, no patterning needed

Cross-Sectional TEM

Direct imaging with sub-nm resolution:

Resolution: 0.1-0.2 nm achievable

Reveals: Interfaces, grain structure, defects

Destructive: Requires thin sample prep (FIB milling)

F.2 Composition Analysis

Elemental and chemical state analysis of deposited films.

Rutherford Backscattering (RBS)

Energy of backscattered ion:

$$E_1 = K E_0$$

where kinematic factor:

$$K = \left[\frac{\cos\theta + \sqrt{(M_2/M_1)^2 - \sin^2\theta}}{1 + M_2/M_1}\right]^2$$

$M_1, M_2$: Projectile, target masses

Quantitative: Absolute atomic % without standards

Depth profiling: Energy loss ∝ depth

X-Ray Photoelectron Spectroscopy (XPS)

Photoelectron kinetic energy:

$$KE = h\nu - BE - \phi$$

BE: Binding energy (element-specific, ~0.1 eV resolution)

$\phi$: Work function

Chemical shifts: Oxidation state identification

Depth: 5-10 nm sampling depth

Secondary Ion Mass Spectrometry (SIMS)

Ultimate sensitivity for trace impurities:

Detection limit: ppt to ppb (10¹² - 10¹⁵ atoms/cm³)

Depth resolution: 2-5 nm

Destructive: Sputters material during analysis

Applications: Dopant profiles, contamination

F.3 Structural Characterization

Crystal structure, orientation, and phase identification.

X-Ray Diffraction (XRD)

Bragg's law:

$$n\lambda = 2d_{hkl}\sin\theta$$

$d_{hkl}$: Interplanar spacing

Peak position: Lattice parameter, strain

Peak width: Grain size (Scherrer), microstrain

Peak intensity: Texture, preferred orientation

Williamson-Hall Analysis

Separating size and strain broadening:

$$\beta \cos\theta = \frac{K\lambda}{D} + 4\epsilon\sin\theta$$

$D$: Crystallite size

$\epsilon$: Microstrain

Plot β cosθ vs. 4sinθ → intercept/slope give D and ε

Pole Figure Analysis

3D texture measurement:

Vary sample tilt (χ) and rotation (φ) at fixed 2θ

Maps crystallographic orientation distribution

Quantifies fiber texture vs. random polycrystalline

Process Control and Optimization

G.1 Run-to-Run Control

Adaptive control to compensate for equipment drift and maintain target specifications.

Exponentially Weighted Moving Average (EWMA)

$$u_n = u_{n-1} + K(T - y_n)$$

$u_n$: Control parameter (e.g., gas flow) at run n

$y_n$: Measured output (e.g., thickness)

$T$: Target value

$K$: Controller gain (0 < K < 2)

PID Control for In-Situ Monitoring

$$u(t) = K_p e(t) + K_i \int_0^t e(\tau) d\tau + K_d \frac{de}{dt}$$

$K_p$: Proportional gain

$K_i$: Integral gain (eliminates steady-state error)

$K_d$: Derivative gain (damps oscillations)

G.2 Design of Experiments (DOE)

Systematic approach to optimize multiple process parameters efficiently.

Response Surface Methodology

Second-order polynomial model:

$$y = \beta_0 + \sum_i \beta_i x_i + \sum_i \beta_{ii} x_i^2 + \sum_i \sum_{j>i} \beta_{ij} x_i x_j + \epsilon$$

$y$: Response (film property)

$x_i$: Coded process parameters

$\beta$: Fitted coefficients

Example: 3-Factor Central Composite Design

Optimize PECVD SiN with factors:

Temperature: 250-350°C
RF Power: 20-100 W
NH₃/SiH₄ ratio: 5-20

20-run design yields response surface for:

Deposition rate
Refractive index
Film stress
Hydrogen content

G.3 Virtual Metrology and Soft Sensors

Predict film properties from equipment parameters without physical measurement.

Multivariate Regression Model

$$\hat{y} = \mathbf{X}\boldsymbol{\beta}$$

where $\mathbf{X}$ contains process parameters, $\boldsymbol{\beta}$ trained from historical data

Principal Component Regression

PCA on process data: $\mathbf{X} = \mathbf{T}\mathbf{P}^T$
Regress on scores: $y = \mathbf{T}\boldsymbol{\beta}_{PCR}$
Reduces dimensionality and multicollinearity

Key References and Further Reading

Foundational Textbooks

Ohring, M. (2001). Materials Science of Thin Films, Academic Press
Glaser, H. J. (2000). Chemical Vapor Deposition, InTech
Mattox, D. M. (2010). Handbook of Physical Vapor Deposition (PVD) Processing, Elsevier
George, S. M. (2010). Atomic Layer Deposition: An Overview, Chem. Rev. 110, 111-131
Bunshah, R. F. (2001). Handbook of Deposition Technologies for Films and Coatings, Noyes
Lieberman, M. A. & Lichtenberg, A. J. (2005). Principles of Plasma Discharges and Materials Processing, Wiley

Advanced Topics

Venables, J. A. (2000). Introduction to Surface and Thin Film Processes, Cambridge
Freund, L. B. & Suresh, S. (2004). Thin Film Materials: Stress, Defect Formation and Surface Evolution, Cambridge
Heavens, O. S. (1991). Optical Properties of Thin Solid Films, Dover
Chapman, B. (1980). Glow Discharge Processes, Wiley

Research Journals

Journal of Vacuum Science & Technology A/B
Thin Solid Films
Surface and Coatings Technology
Journal of Applied Physics
Applied Physics Letters
Plasma Sources Science and Technology

Theoretical Foundations

Table of Contents

Gas-Phase Kinetics

1.1 Molecular Collision Theory

Derivation from Kinetic Theory:

1.2 Arrhenius Kinetics

Example: Silane Decomposition

1.3 Homogeneous vs. Heterogeneous Reactions

Key Insights

Surface Reactions

2.1 Langmuir Adsorption Isotherm

Derivation of Langmuir Isotherm:

2.2 Surface Reaction Mechanisms

2.3 Sticking Coefficient

Typical Sticking Coefficients

Transport Phenomena

3.1 Mass Transport Regimes

Regime Characteristics

3.2 Diffusion in Boundary Layers

3.3 Overall Deposition Rate Model

Limiting Cases:

Nucleation and Growth

4.1 Classical Nucleation Theory

4.2 Growth Modes

Growth Mode Examples

4.3 Film Thickness Evolution

Plasma Physics for PECVD

5.1 Plasma Generation and Properties

5.2 RF Sheath Dynamics

Ion Bombardment Effects

5.3 Electron Impact Processes

Example: SiH₄ Dissociation in Plasma

Sputtering Physics

6.1 Sputter Yield Theory

Typical Sputter Yields (Ar⁺, 500 eV)

6.2 Deposition Rate

Factors Affecting Deposition Rate

6.3 Reactive Sputtering

Evaporation Thermodynamics

7.1 Vapor Pressure

Vapor Pressures at Various Temperatures

7.2 Evaporation Rate

Derivation from Kinetic Theory:

7.3 Flux Distribution and Film Thickness

Evaporation Characteristics

Atomic Layer Deposition Chemistry

8.1 Self-Limiting Surface Chemistry

Example: Al₂O₃ ALD with TMA/H₂O

8.2 ALD Temperature Window

8.3 Conformality and Aspect Ratio

ALD Advantages and Challenges

Film Properties

9.1 Intrinsic Stress

Typical Stress Values

9.2 Grain Structure and Texture

9.3 Resistivity and Electrical Properties

Copper Interconnect Resistivity

Interface Physics

10.1 Adhesion Energy

Improving Adhesion

10.2 Interfacial Reactions and Diffusion

Example: Al/Si Contact

10.3 Contact Resistance

Advanced Transport Models

A.1 Rarefied Gas Dynamics

A.2 Multicomponent Diffusion

A.3 Thermophoresis and Electrophoresis

Advanced Reaction Kinetics

B.1 Transition State Theory

B.2 Microkinetic Modeling

Example: TEOS CVD Microkinetic Model

B.3 Sensitivity Analysis

Advanced Plasma Modeling

C.1 Plasma Fluid Equations

C.2 Particle-in-Cell (PIC) Method

C.3 Global Plasma Models

Defects and Stress Mechanisms

D.1 Point Defect Thermodynamics

D.2 Grain Boundary Structure

D.3 Stress Evolution During Growth