coefficient of thermal expansion of emc, cte, packaging
Thermal expansion matching.
923 technical terms and definitions
Thermal expansion matching.
Interdigitated pattern for leakage testing.
Total uncertainty from all sources.
Compare dimensions to standard.
Packaging for automated placement.
# Semiconductor Material Mathematical Modeling **Materials Covered:** Germanium (Ge), Silicon (Si), Gallium Arsenide (GaAs), Silicon Carbide (SiC) ## 1. Material Properties Overview | Property | Si | Ge | GaAs | 4H-SiC | |:---------|:--:|:--:|:----:|:------:| | **Bandgap (eV)** | 1.12 (indirect) | 0.66 (indirect) | 1.42 (direct) | 3.26 (indirect) | | **Lattice constant (Å)** | 5.431 | 5.658 | 5.653 | a=3.07, c=10.05 | | **Electron mobility (cm²/V·s)** | 1400 | 3900 | 8500 | 1000 | | **Hole mobility (cm²/V·s)** | 450 | 1900 | 400 | 120 | | **Thermal conductivity (W/cm\cdotK)** | 1.5 | 0.6 | 0.5 | 4.9 | | **Melting point (°C)** | 1414 | 937 | 1238 | 2730 (sublimes) | | **Intrinsic carrier conc. (cm⁻³)** | $1.5 \times 10^{10}$ | $2.4 \times 10^{13}$ | $1.8 \times 10^{6}$ | $\sim 10^{-9}$ | ### Key Characteristics - **Silicon (Si)** - Most widely used semiconductor - Excellent native oxide ($\text{SiO}_2$) - Mature processing technology - Diamond cubic crystal structure - **Germanium (Ge)** - Higher carrier mobility than Si - Unstable native oxide (water-soluble) - Lower thermal budget (lower melting point) - Used for high-speed devices - **Gallium Arsenide (GaAs)** - Direct bandgap → optoelectronics - Highest electron mobility - No stable native oxide - III-V compound semiconductor - **Silicon Carbide (SiC)** - Wide bandgap → high-power applications - Excellent thermal conductivity - High breakdown field - Multiple polytypes (3C, 4H, 6H) ## 2. Crystal Growth ### 2.1 Czochralski (CZ) Method — Si, Ge, GaAs #### Heat Transfer in Melt The temperature distribution in the melt is governed by the convection-diffusion equation: $$ \rho c_p \frac{\partial T}{\partial t} + \rho c_p (\mathbf{v} \cdot \nabla)T = \nabla \cdot (k \nabla T) $$ **Where:** - $\rho$ — density (kg/m³) - $c_p$ — specific heat capacity (J/kg·K) - $T$ — temperature (K) - $\mathbf{v}$ — velocity field (m/s) - $k$ — thermal conductivity (W/m·K) #### Melt Convection Navier-Stokes equation with Boussinesq approximation for buoyancy: $$ \rho \left( \frac{\partial \mathbf{v}}{\partial t} + (\mathbf{v} \cdot \nabla)\mathbf{v} \right) = -\nabla p + \mu \nabla^2 \mathbf{v} + \rho \mathbf{g} \beta (T - T_m) $$ **Where:** - $p$ — pressure (Pa) - $\mu$ — dynamic viscosity (Pa·s) - $\mathbf{g}$ — gravitational acceleration (m/s²) - $\beta$ — thermal expansion coefficient (K⁻¹) - $T_m$ — melting temperature (K) #### Stefan Condition at Crystal-Melt Interface The interface position is determined by the heat balance: $$ k_s \left( \frac{\partial T}{\partial n} \right)_s - k_l \left( \frac{\partial T}{\partial n} \right)_l = \rho_s L v_n $$ **Where:** - $k_s$, $k_l$ — thermal conductivity of solid and liquid - $L$ — latent heat of fusion (J/kg) - $v_n$ — interface velocity normal to surface (m/s) - $\rho_s$ — solid density (kg/m³) #### Dopant Segregation — Burton-Prim-Slichter (BPS) Model The effective segregation coefficient accounts for boundary layer effects: $$ k_{\text{eff}} = \frac{k_0}{k_0 + (1-k_0)\exp\left( -\frac{v_g \delta}{D} \right)} $$ **Where:** - $k_0$ — equilibrium segregation coefficient (dimensionless) - $v_g$ — crystal growth rate (m/s) - $\delta$ — boundary layer thickness (m) - $D$ — diffusion coefficient in melt (m²/s) **Limiting cases:** - Slow growth ($v_g \delta / D \ll 1$): $k_{\text{eff}} \rightarrow k_0$ - Fast growth ($v_g \delta / D \gg 1$): $k_{\text{eff}} \rightarrow 1$ ### 2.2 Physical Vapor Transport (PVT) — SiC SiC sublimes rather than melts. Growth occurs via vapor species transport. #### Sublimation Species $$ \text{SiC}_{(s)} \rightleftharpoons \text{Si}_{(g)} + \text{C}_{(s)} $$ $$ 2\text{SiC}_{(s)} \rightleftharpoons \text{Si}_2\text{C}_{(g)} + \text{C}_{(s)} $$ $$ \text{SiC}_{(s)} + \text{Si}_{(g)} \rightleftharpoons \text{SiC}_2{}_{(g)} $$ #### Mass Transport Equation $$ \frac{\partial C_i}{\partial t} + \nabla \cdot (C_i \mathbf{v}) = \nabla \cdot (D_i \nabla C_i) + R_i $$ **Where:** - $C_i$ — concentration of species $i$ (mol/m³) - $D_i$ — diffusion coefficient of species $i$ (m²/s) - $R_i$ — reaction rate for species $i$ (mol/m³·s) #### Supersaturation at Growth Interface $$ \sigma = \frac{P_{\text{source}} - P_{\text{eq}}(T_{\text{seed}})}{P_{\text{eq}}(T_{\text{seed}})} $$ **Growth rate approximation:** $$ G \propto \frac{\sigma \cdot D}{L} $$ **Where:** - $L$ — source-to-seed distance (m) - $P_{\text{eq}}$ — equilibrium vapor pressure at seed temperature ## 3. Epitaxial Growth ### 3.1 Chemical Vapor Deposition (CVD) — Si, SiC #### Grove Model for Growth Rate $$ R = \frac{k_s C_g}{1 + \dfrac{k_s}{h_g}} $$ **Where:** - $R$ — growth rate (m/s) - $k_s$ — surface reaction rate constant (m/s) - $C_g$ — gas-phase reactant concentration (mol/m³) - $h_g$ — gas-phase mass transfer coefficient (m/s) #### Temperature Dependence (Arrhenius) $$ k_s = k_0 \exp\left(-\frac{E_a}{kT}\right) $$ **Where:** - $k_0$ — pre-exponential factor (m/s) - $E_a$ — activation energy (eV or J) - $k$ — Boltzmann constant ($8.617 \times 10^{-5}$ eV/K) - $T$ — temperature (K) #### Two Limiting Regimes | Regime | Condition | Growth Rate | Temperature Dependence | |:-------|:----------|:------------|:-----------------------| | **Reaction-limited** | $k_s \ll h_g$ | $R \approx k_s C_g$ | Strong (exponential) | | **Mass-transport-limited** | $k_s \gg h_g$ | $R \approx h_g C_g$ | Weak ($\sim T^{1/2}$) | #### Boundary Layer Thickness $$ \delta \approx \sqrt{\frac{\mu L}{\rho v}} = \sqrt{\frac{\nu L}{v}} $$ **Where:** - $\nu$ — kinematic viscosity (m²/s) - $L$ — characteristic length (m) - $v$ — gas flow velocity (m/s) **Mass transfer coefficient:** $$ h_g \approx \frac{D}{\delta} $$ ### 3.2 Molecular Beam Epitaxy (MBE) — GaAs, Ge #### Knudsen Cell Flux (Effusion) $$ J = \frac{P \cdot A_e \cdot \cos\theta}{\sqrt{2\pi m k T}} \cdot \frac{1}{\pi r^2} $$ **Where:** - $J$ — flux at substrate (atoms/cm²·s) - $P$ — vapor pressure in cell (Pa) - $A_e$ — effusion orifice area (m²) - $m$ — atomic mass (kg) - $r$ — source-to-substrate distance (m) - $\theta$ — angle from normal #### Growth Rate $$ R = \frac{J_{\text{Ga}}}{n_0} $$ **Where:** - $J_{\text{Ga}}$ — Ga flux at substrate (atoms/cm²·s) - $n_0$ — surface atomic density ($\sim 6.3 \times 10^{14}$ cm⁻² for GaAs (100)) #### Surface Diffusion **Diffusion coefficient:** $$ D_s = D_0 \exp\left(-\frac{E_d}{kT}\right) $$ **Mean diffusion length:** $$ \lambda = \sqrt{D_s \tau} $$ **Where:** - $E_d$ — diffusion activation energy (eV) - $\tau$ — residence time before desorption (s) ### 3.3 Heteroepitaxy — Critical Thickness For lattice-mismatched systems (e.g., Ge on Si with 4.2% mismatch): #### Matthews-Blakeslee Model $$ h_c = \frac{b}{2\pi f} \cdot \frac{1-\nu/4}{1+\nu} \cdot \ln\left(\frac{h_c}{b}\right) $$ **Where:** - $h_c$ — critical thickness for dislocation formation (m) - $b$ — Burgers vector magnitude (m) - $f$ — lattice mismatch: $f = \dfrac{a_{\text{layer}} - a_{\text{sub}}}{a_{\text{sub}}}$ - $\nu$ — Poisson's ratio (dimensionless) **Strain energy density:** $$ E_{\text{strain}} = \frac{E}{1-\nu} \cdot f^2 \cdot h $$ **Where:** - $E$ — Young's modulus (Pa) - $h$ — layer thickness (m) ## 4. Thermal Oxidation ### 4.1 Deal-Grove Model — Si The oxide thickness $x_{\text{ox}}$ as a function of time $t$: $$ x_{\text{ox}}^2 + A \cdot x_{\text{ox}} = B(t + \tau) $$ **Where:** - $A$, $B$ — rate constants (material and condition dependent) - $\tau$ — time correction for initial oxide: $\tau = \dfrac{x_i^2 + A \cdot x_i}{B}$ #### Parabolic Rate Constant $$ B = \frac{2 D_{\text{ox}} C^*}{N_1} $$ **Where:** - $D_{\text{ox}}$ — oxidant diffusivity in $\text{SiO}_2$ (m²/s) - $C^*$ — equilibrium oxidant concentration in oxide (mol/m³) - $N_1$ — number of oxidant molecules per unit volume of oxide #### Linear Rate Constant $$ \frac{B}{A} = \frac{k_s C^*}{N_1} $$ **Where:** - $k_s$ — surface reaction rate constant (m/s) #### Limiting Cases | Regime | Condition | Oxide Thickness | Rate Limiting Step | |:-------|:----------|:----------------|:-------------------| | **Linear** | $x_{\text{ox}} \ll A$ | $x_{\text{ox}} \approx \dfrac{B}{A} t$ | Surface reaction | | **Parabolic** | $x_{\text{ox}} \gg A$ | $x_{\text{ox}} \approx \sqrt{Bt}$ | Diffusion through oxide | #### Wet vs. Dry Oxidation | Parameter | Dry O₂ | Wet H₂O | |:----------|:-------|:--------| | $B$ (1000°C) | 0.0117 µm²/hr | 0.287 µm²/hr | | $B/A$ (1000°C) | 0.027 µm/hr | 0.96 µm/hr | | Oxide quality | Higher | Lower | | Growth rate | Slower (~10×) | Faster | ### 4.2 SiC Oxidation **Reaction:** $$ \text{SiC} + \frac{3}{2}\text{O}_2 \rightarrow \text{SiO}_2 + \text{CO} $$ **Key differences from Si:** - Oxidation rate is 10-100× slower than Si at the same temperature - Carbon removal adds complexity (CO must diffuse out) - Interface trap density ($D_{it}$) is a major challenge - Modified Deal-Grove models required: $$ x_{\text{ox}}^2 + A \cdot x_{\text{ox}} = B(t + \tau) + C \cdot t $$ The additional linear term $C \cdot t$ accounts for carbon-related interface reactions. ## 5. Diffusion ### 5.1 Fick's Laws #### First Law (Flux) $$ J = -D \frac{\partial C}{\partial x} $$ **Where:** - $J$ — flux (atoms/cm²·s) - $D$ — diffusion coefficient (cm²/s) - $C$ — concentration (atoms/cm³) #### Second Law (Time Evolution) $$ \frac{\partial C}{\partial t} = D \frac{\partial^2 C}{\partial x^2} $$ *Assumes constant diffusion coefficient.* #### Diffusion Coefficient Temperature Dependence $$ D = D_0 \exp\left( -\frac{E_a}{kT} \right) $$ ### 5.2 Analytical Solutions #### Constant Surface Concentration (Predeposition) **Boundary conditions:** - $C(0,t) = C_s$ (constant) - $C(\infty,t) = 0$ - $C(x,0) = 0$ **Solution:** $$ C(x,t) = C_s \cdot \text{erfc}\left( \frac{x}{2\sqrt{Dt}} \right) $$ **Total dopant dose:** $$ Q = \frac{2C_s}{\sqrt{\pi}} \cdot \sqrt{Dt} $$ #### Limited Source (Drive-in) **Boundary conditions:** - Total dopant $Q$ conserved - $C(x,0) = Q \cdot \delta(x)$ (delta function) **Solution (Gaussian):** $$ C(x,t) = \frac{Q}{\sqrt{\pi Dt}} \exp\left( -\frac{x^2}{4Dt} \right) $$ #### Junction Depth At the junction, $C(x_j) = C_B$ (background concentration): $$ x_j = 2\sqrt{Dt} \cdot \text{erfc}^{-1}\left( \frac{C_B}{C_s} \right) $$ For Gaussian profile: $$ x_j = 2\sqrt{Dt \cdot \ln\left(\frac{Q}{C_B\sqrt{\pi Dt}}\right)} $$ ### 5.3 Material-Specific Diffusion Parameters #### Silicon | Dopant | $D_0$ (cm²/s) | $E_a$ (eV) | Mechanism | |:-------|:-------------:|:----------:|:----------| | Boron (B) | 0.76 | 3.46 | Interstitialcy | | Phosphorus (P) | 3.85 | 3.66 | Mixed (V + I) | | Arsenic (As) | 22.9 | 4.1 | Vacancy | | Antimony (Sb) | 0.214 | 3.65 | Vacancy | #### Germanium - Higher diffusion coefficients than Si (lower melting point) - B in Ge: $D_0 \approx 1.0$ cm²/s, $E_a \approx 2.5$ eV #### Silicon Carbide - **Extremely low diffusion coefficients** due to strong Si-C bonds - N-type doping (N): $D \approx 10^{-13}$ cm²/s at 1800°C - Implantation is required; diffusion-based doping impractical - Activation requires annealing >1600°C #### GaAs - Si is amphoteric (can be n-type on Ga site, p-type on As site) - Zn diffusion is heavily concentration-dependent - Be is preferred p-type dopant for MBE ## 6. Ion Implantation ### 6.1 Range Distribution — LSS Theory #### Gaussian Approximation $$ C(x) = \frac{\Phi}{\sqrt{2\pi} \Delta R_p} \exp\left( -\frac{(x - R_p)^2}{2 \Delta R_p^2} \right) $$ **Where:** - $\Phi$ — implant dose (ions/cm²) - $R_p$ — projected range (mean depth) (nm) - $\Delta R_p$ — range straggle (standard deviation) (nm) #### Peak Concentration $$ C_{\text{peak}} = \frac{\Phi}{\sqrt{2\pi} \Delta R_p} \approx \frac{0.4 \Phi}{\Delta R_p} $$ ### 6.2 Stopping Power Total energy loss per unit path length: $$ -\frac{dE}{dx} = S_n(E) + S_e(E) $$ **Where:** - $S_n(E)$ — nuclear stopping power (elastic collisions with nuclei) - $S_e(E)$ — electronic stopping power (inelastic electron interactions) #### Nuclear Stopping (Low Energy) Dominant mechanism at low energies. Using ZBL (Ziegler-Biersack-Littmark) potential: $$ S_n \propto \frac{Z_1 Z_2}{(Z_1^{0.23} + Z_2^{0.23})} \cdot \frac{M_1}{M_1 + M_2} $$ **Where:** - $Z_1$, $Z_2$ — atomic numbers of ion and target - $M_1$, $M_2$ — masses of ion and target #### Electronic Stopping (High Energy) $$ S_e \propto Z_1^{1/6} \sqrt{E} $$ At very high energies, Bethe-Bloch formula applies. ### 6.3 Damage and Amorphization #### Displacement Damage — Modified Kinchin-Pease Model $$ N_d = \frac{0.8 \cdot E_d}{2 E_{\text{th}}} $$ **Where:** - $N_d$ — number of displaced atoms per ion - $E_d$ — damage energy deposited (eV) - $E_{\text{th}}$ — threshold displacement energy (eV) - Si: ~15 eV - GaAs: ~10 eV (Ga sublattice), ~9 eV (As sublattice) - SiC: ~20-35 eV #### Critical Dose for Amorphization | Material | Critical Dose (ions/cm²) | Notes | |:---------|:------------------------:|:------| | Si | $10^{14} - 10^{15}$ | Room temperature | | Ge | $10^{13} - 10^{14}$ | Easier to amorphize | | GaAs | $10^{13} - 10^{14}$ | Very easily amorphized | | SiC | $10^{15} - 10^{16}$ | Requires low T or high dose | #### Channeling Effect When ions align with crystal channels, the range increases significantly: $$ R_p^{\text{channeled}} \gg R_p^{\text{random}} $$ Modeling requires Monte Carlo simulations (SRIM/TRIM, Crystal-TRIM). ## 7. Etching ### 7.1 Wet Etching #### Etch Rate Model $$ R = A \exp\left( -\frac{E_a}{kT} \right) [C]^n $$ **Where:** - $R$ — etch rate (nm/min) - $A$ — pre-exponential factor - $[C]$ — etchant concentration - $n$ — reaction order #### Anisotropic Si Etching (KOH, TMAH) Different crystal planes have different bond densities: $$ \frac{R_{\{100\}}}{R_{\{111\}}} \approx 100-400 $$ **Etch selectivity:** $$ S = \frac{R_{\text{material 1}}}{R_{\text{material 2}}} $$ ### 7.2 Reactive Ion Etching (RIE/ICP) #### Ion-Enhanced Etching $$ R_{\text{total}} = R_{\text{chem}} + R_{\text{phys}} + R_{\text{synergy}} $$ The synergy term is typically the largest contribution. #### Child-Langmuir Law for Ion Current $$ J = \frac{4\varepsilon_0}{9} \sqrt{\frac{2e}{M_i}} \cdot \frac{V^{3/2}}{d^2} $$ **Where:** - $J$ — ion current density (A/m²) - $\varepsilon_0$ — vacuum permittivity - $e$ — electron charge - $M_i$ — ion mass (kg) - $V$ — sheath voltage (V) - $d$ — sheath thickness (m) #### Langmuir-Hinshelwood Kinetics (Surface Reaction) $$ R = \frac{k \cdot \theta_A \cdot \theta_B}{(1 + K_A P_A + K_B P_B)^2} $$ **Where:** - $\theta_A$, $\theta_B$ — surface coverage fractions - $K_A$, $K_B$ — adsorption equilibrium constants - $P_A$, $P_B$ — partial pressures ### 7.3 Material-Specific Etching | Material | Wet Etch | Dry Etch | Notes | |:---------|:---------|:---------|:------| | **Si** | KOH, TMAH, HF/HNO₃ | SF₆, CF₄, Cl₂ | Well-established | | **Ge** | H₂O₂/HF | CF₄, SF₆ | Fast etch rates | | **GaAs** | H₂SO₄/H₂O₂, NH₄OH | Cl₂, BCl₃ | Selectivity to AlGaAs | | **SiC** | KOH (molten, 500°C) | SF₆/O₂, ICP | Very slow, needs ICP | ## 8. Lithography ### 8.1 Resolution Limits #### Rayleigh Criterion **Resolution:** $$ R = k_1 \frac{\lambda}{NA} $$ **Depth of Focus:** $$ DOF = k_2 \frac{\lambda}{NA^2} $$ **Where:** - $k_1$ — process factor (0.25–0.8) - $k_2$ — depth of focus factor (~0.5) - $\lambda$ — exposure wavelength (nm) - $NA$ — numerical aperture #### Technology Comparison | Technology | $\lambda$ (nm) | Typical NA | Resolution | |:-----------|:--------------:|:----------:|:-----------| | i-line | 365 | 0.6 | ~350 nm | | KrF | 248 | 0.75 | ~180 nm | | ArF (dry) | 193 | 0.85 | ~90 nm | | ArF (immersion) | 193 | 1.35 | ~38 nm | | EUV | 13.5 | 0.33 | ~13 nm | ### 8.2 Resist Modeling — Dill Parameters #### Absorption in Resist $$ \frac{dI}{dz} = -\alpha(M) \cdot I $$ **Where:** $$ \alpha = A \cdot M + B $$ - $A$ — bleachable absorption coefficient - $B$ — non-bleachable absorption coefficient - $M$ — relative photoactive compound (PAC) concentration #### Exposure Kinetics $$ \frac{dM}{dt} = -C \cdot I \cdot M $$ **Where:** - $C$ — exposure rate constant #### Development Rate (Mack Model) $$ R = R_{\max} \cdot \frac{(a+1)(1-M)^n}{a + (1-M)^n} $$ **Where:** - $R_{\max}$ — maximum development rate - $a$ — selectivity parameter - $n$ — development contrast ## 9. Thin Film Deposition ### 9.1 Physical Vapor Deposition (PVD) #### Sputtering Yield $$ Y = \frac{3\alpha}{4\pi^2} \cdot \frac{4 M_1 M_2}{(M_1 + M_2)^2} \cdot \frac{E}{U_s} $$ **Where:** - $Y$ — sputtering yield (atoms/ion) - $\alpha$ — momentum transfer efficiency - $M_1$, $M_2$ — masses of ion and target atom - $E$ — ion energy (eV) - $U_s$ — surface binding energy (eV) - Si: ~4.7 eV - SiO₂: ~5.0 eV #### Film Thickness Uniformity — Cosine Law $$ \frac{dN}{d\Omega} \propto \cos\theta $$ **Step coverage:** $$ SC = \frac{t_{\text{sidewall}}}{t_{\text{top}}} $$ ### 9.2 Chemical Vapor Deposition (CVD) #### LPCVD Polysilicon from SiH₄ **Reaction:** $$ \text{SiH}_4 \xrightarrow{\Delta} \text{Si} + 2\text{H}_2 $$ **Growth rate:** $$ R = R_0 \exp\left(-\frac{E_a}{kT}\right) \cdot \frac{P_{\text{SiH}_4}}{1 + K_{\text{H}_2} P_{\text{H}_2}} $$ ### 9.3 Atomic Layer Deposition (ALD) **Self-limiting half-reactions:** 1. $\text{Surface-OH} + \text{Al(CH}_3\text{)}_3 \rightarrow \text{Surface-O-Al(CH}_3\text{)}_2 + \text{CH}_4$ 2. $\text{Surface-Al(CH}_3\text{)}_2 + \text{H}_2\text{O} \rightarrow \text{Surface-Al-OH} + 2\text{CH}_4$ **Growth Per Cycle (GPC):** $$ \text{GPC} \approx 0.5 - 1.5 \text{ Å/cycle} $$ Ideal conformal coating with atomic-level thickness control. ## 10. Chemical Mechanical Polishing (CMP) ### 10.1 Preston Equation $$ R = K_p \cdot P \cdot V $$ **Where:** - $R$ — removal rate (nm/min) - $K_p$ — Preston coefficient (material/slurry dependent) - $P$ — applied pressure (Pa) - $V$ — relative velocity (m/s) ### 10.2 Material-Specific CMP | Material | Relative Difficulty | Slurry Type | Notes | |:---------|:-------------------:|:------------|:------| | Si | Low | Colloidal silica | Standard process | | SiO₂ | Low | Ceria, silica | Well-established | | Cu | Medium | Acidic + oxidizer | Dishing/erosion issues | | SiC | **Very High** | Oxidizing, alkaline | Hardness 9.5 Mohs | **SiC CMP challenges:** - Extremely hard material - Tribochemical mechanisms required - Polish times 10-100× longer than Si - Subsurface damage minimization critical ## 11. Process Integration Considerations ### 11.1 Silicon (Si) - **Advantages:** - Mature CMOS technology - Excellent native oxide - Standard processing well-established - **Challenges:** - Scaling limits at sub-3nm nodes - Power density limitations ### 11.2 Germanium (Ge) - **Advantages:** - Higher mobility ($\mu_e$ = 3900, $\mu_h$ = 1900 cm²/V·s) - Compatible with Si processing (mostly) - **Challenges:** - Unstable native oxide → requires passivation (GeO₂/Al₂O₃) - Lower thermal budget (mp = 937°C) - Integration on Si requires graded SiGe buffers ### 11.3 Gallium Arsenide (GaAs) - **Advantages:** - Direct bandgap → optoelectronics - Highest electron mobility (8500 cm²/V·s) - Semi-insulating substrates available - **Challenges:** - No stable native oxide → gate dielectric issues - Surface Fermi level pinning - Stoichiometry control (As overpressure during anneal) - Not used for CMOS (cost, integration) ### 11.4 Silicon Carbide (SiC) - **Advantages:** - Wide bandgap (3.26 eV) → high voltage - High thermal conductivity (4.9 W/cm\cdotK) - High breakdown field (~3 MV/cm) - **Challenges:** - Extreme processing temperatures (>1600°C for activation) - Gate oxide interface quality ($D_{it}$) - Step-controlled epitaxy for polytype control - CMP is very difficult ## 12. TCAD Simulation Framework ### 12.1 Coupled Process Equations Modern process simulation solves coupled PDEs for multiple species: $$ \frac{\partial C_i}{\partial t} = \nabla \cdot (D_i \nabla C_i) + G_i - R_i $$ **Including:** - Dopant diffusion - Point defect dynamics (vacancies $V$, interstitials $I$) - Dopant-defect pairing - Cluster formation and dissolution ### 12.2 Point Defect Mediated Diffusion **Five-stream model:** $$ D_A^{\text{eff}} = D_{AI} \cdot \frac{C_I}{C_I^*} + D_{AV} \cdot \frac{C_V}{C_V^*} $$ **Where:** - $D_{AI}$ — diffusivity via interstitialcy mechanism - $D_{AV}$ — diffusivity via vacancy mechanism - $C_I^*$, $C_V^*$ — equilibrium defect concentrations ### 12.3 Level Set Methods for Topography Interface evolution during etching/deposition: $$ \frac{\partial \phi}{\partial t} + V|\nabla \phi| = 0 $$ **Where:** - $\phi = 0$ defines the interface - $V$ — local etch/deposition rate (can depend on position, orientation) ### 12.4 Monte Carlo Methods **Applications:** - **Ion implantation:** Binary collision approximation (BCA) - SRIM/TRIM for amorphous targets - Crystal-TRIM for channeling effects - **Dopant clustering:** Statistical mechanics of defect formation - **Surface evolution:** Kinetic Monte Carlo for atomic-scale processes ## Physical Constants | Constant | Symbol | Value | |:---------|:------:|:------| | Boltzmann constant | $k$ | $8.617 \times 10^{-5}$ eV/K | | Elementary charge | $e$ | $1.602 \times 10^{-19}$ C | | Vacuum permittivity | $\varepsilon_0$ | $8.854 \times 10^{-12}$ F/m | | Planck constant | $h$ | $6.626 \times 10^{-34}$ J\cdots | | Avogadro number | $N_A$ | $6.022 \times 10^{23}$ mol⁻¹ | ## Unit Conversions | Quantity | Conversion | |:---------|:-----------| | Energy | 1 eV = $1.602 \times 10^{-19}$ J | | Length | 1 Å = $10^{-10}$ m = 0.1 nm | | Temperature | $kT$ at 300 K = 0.0259 eV | | Pressure | 1 Torr = 133.3 Pa |
Shape compound with pressure and heat.
# Semiconductor Manufacturing: Computational Challenges Overview Semiconductor manufacturing represents one of the most mathematically and computationally intensive industrial processes. The complexity stems from multiple scales—from quantum mechanics at atomic level to factory-level logistics. 1. Computational Lithography Mathematical approaches to improve photolithography resolution as features shrink below light wavelength. Key Challenges: • Inverse Lithography Technology (ILT): Treats mask design as inverse problem, solving high-dimensional nonlinear optimization • Optical Proximity Correction (OPC): Solves electromagnetic wave equations with iterative optimization • Source Mask Optimization (SMO): Co-optimizes mask and light source parameters Computational Scale: • Single ILT mask: >10,000 CPU cores for multiple days • GPU acceleration: 40× speedup (500 Hopper GPUs = 40,000 CPU systems) 2. Device Modeling via PDEs Coupled nonlinear partial differential equations model semiconductor devices. Core Equations: Drift-Diffusion System: ∇·(ε∇ψ) = -q(p - n + Nᴅ⁺ - Nₐ⁻) (Poisson) ∂n/∂t = (1/q)∇·Jₙ + G - R (Electron continuity) ∂p/∂t = -(1/q)∇·Jₚ + G - R (Hole continuity) Current densities: Jₙ = qμₙn∇ψ + qDₙ∇n Jₚ = qμₚp∇ψ - qDₚ∇p Numerical Methods: • Finite-difference and finite-element discretization • Newton-Raphson iteration or Gummel's method • Computational meshes for complex geometries 3. CVD Process Simulation CFD models optimize reactor design and operating conditions. Multiscale Modeling: • Nanoscale: DFT and MD for surface chemistry, nucleation, growth • Macroscale: CFD for velocity, pressure, temperature, concentration fields Ab initio quantum chemistry + CFD enables growth rate prediction without extensive calibration. 4. Statistical Process Control SPC distinguishes normal from special variation in production. Key Mathematical Tools: Murphy's Yield Model: Y = [(1 - e⁻ᴰ⁰ᴬ) / D₀A]² Control Charts: • X-bar: UCL = μ + 3σ/√n • EWMA: Zₜ = λxₜ + (1-λ)Zₜ₋₁ Capability Index: Cₚₖ = min[(USL - μ)/3σ, (μ - LSL)/3σ] 5. Production Planning and Scheduling Complexity of multistage production requires advanced optimization. Mathematical Approaches: • Mixed-Integer Programming (MIP) • Variable neighborhood search, genetic algorithms • Discrete event simulation Scale: Managing 55+ equipment units in real-time rescheduling. 6. Level Set Methods Track moving boundaries during etching and deposition. Hamilton-Jacobi equation: ∂ϕ/∂t + F|∇ϕ| = 0 where ϕ is the level set function and F is the interface velocity. Applications: PECVD, ion-milling, photolithography topography evolution. 7. Machine Learning Integration Neural networks applied to: • Accelerate lithography simulation • Predict hotspots (defect-prone patterns) • Optimize mask designs • Model process variations 8. Robust Optimization Addresses yield variability under uncertainty: min max f(x, ξ) x ξ∈U where U is the uncertainty set. Key Computational Bottlenecks • Scale: Thousands of wafers daily, billions of transistors each • Multiphysics: Coupled electromagnetic, thermal, chemical, mechanical phenomena • Multiscale: 12+ orders of magnitude (10⁻¹⁰ m atomic to 10⁻¹ m wafer) • Real-time: Immediate deviation detection and correction • Dimensionality: Millions of optimization variables Summary Computational challenges span: • Numerical PDEs (device simulation) • Optimization theory (lithography, scheduling) • Statistical process control (yield management) • CFD (process simulation) • Quantum chemistry (materials modeling) • Discrete event simulation (factory logistics) The field exemplifies applied mathematics at its most interdisciplinary and impactful.
Image analysis for defect detection.
Measure conductivity at nanoscale.
Different ESD protection levels.
High-resolution 3D imaging.
Measure surface hydrophobicity.
Series of contacts for yield testing.
Small vertical opening for electrical connection between layers.
Probe touches measured surface.
3D measurement system.
Leads in same plane.
Use copper wire.
Charge-based non-contact metrology.
Systematic vs random overlay components.
Combine multiple microscopy techniques.
# Semiconductor Manufacturing Process Cost Modeling ## Overview Semiconductor cost modeling quantifies the expenses of fabricating integrated circuits—from raw wafer to tested die. It informs technology roadmap decisions, fab investments, product pricing, and yield improvement prioritization. ## 1. Major Cost Components ### 1.1 Capital Equipment (40–50% of Total Cost) This dominates leading-edge economics. A modern advanced-node fab costs **$20–30 billion** to construct. **Key equipment categories and approximate costs:** - **EUV lithography scanners**: $150–380M each (a fab may need 15–20) - **DUV immersion scanners**: $50–80M - **Deposition tools (CVD, PVD, ALD)**: $3–10M each - **Etch systems**: $3–8M each - **Ion implanters**: $5–15M - **Metrology/inspection**: $2–20M per tool - **CMP systems**: $3–5M **Capital cost allocation formula:** $$ \text{Cost per wafer pass} = \frac{\text{Tool cost} \times \text{Depreciation rate}}{\text{Throughput} \times \text{Utilization} \times \text{Uptime} \times \text{Hours/year}} $$ Where: - **Depreciation**: Typically 5–7 years - **Utilization targets**: 85–95% for expensive tools ### 1.2 Masks/Reticles A complete mask set for a leading-edge process (7nm and below) costs **$10–15 million** or more. **EUV mask cost drivers:** - Reflective multilayer blanks (not transmissive glass) - Defect-free requirements at smaller dimensions - Complex pellicle technology **Mask cost per die:** $$ \text{Mask cost per die} = \frac{\text{Total mask set cost}}{\text{Total production volume}} $$ ### 1.3 Materials and Consumables (15–25%) - **Process gases**: Silane, ammonia, fluorine chemistries, noble gases - **Chemicals**: Photoresists (EUV resists are expensive), developers, CMP slurries, cleaning chemistries - **Substrates**: 300mm wafers ($100–500+ depending on spec) - SOI wafers: Higher cost - Epitaxial wafers: Additional processing cost - **Targets/precursors**: For deposition processes ### 1.4 Facilities (10–15%) - **Cleanroom**: Class 1 or better for critical areas - **Ultrapure water**: 18.2 MΩ·cm resistivity requirement - **HVAC and vibration control**: Critical for lithography - **Power consumption**: 100–150+ MW continuously for leading fabs - **Waste treatment**: Environmental compliance costs ### 1.5 Labor (10–15%) Varies significantly by geography: - Direct fab operators and technicians - Process and equipment engineers - Maintenance, quality, and yield engineers ## 2. Yield Modeling Yield is the most critical variable, converting wafer cost into die cost: $$ \text{Cost per die} = \frac{\text{Cost per wafer}}{\text{Dies per wafer} \times Y} $$ Where $Y$ is the yield (fraction of good dies). ### 2.1 Yield Models **Poisson Model (Random Defects):** $$ Y = e^{-D_0 \times A} $$ Where: - $D_0$ = Defect density (defects/cm²) - $A$ = Die area (cm²) **Negative Binomial Model (Clustered Defects):** $$ Y = \left(1 + \frac{D_0 \times A}{\alpha}\right)^{-\alpha} $$ Where: - $\alpha$ = Clustering parameter (higher values approach Poisson) **Murphy's Model:** $$ Y = \left(\frac{1 - e^{-D_0 \times A}}{D_0 \times A}\right)^2 $$ ### 2.2 Yield Components - **Random defect yield ($Y_{\text{random}}$)**: Particles, contamination - **Systematic yield ($Y_{\text{systematic}}$)**: Design-process interactions, hotspots - **Parametric yield ($Y_{\text{parametric}}$)**: Devices failing electrical specs **Combined yield:** $$ Y_{\text{total}} = Y_{\text{random}} \times Y_{\text{systematic}} \times Y_{\text{parametric}} $$ ### 2.3 Yield Benchmarks - **Mature processes**: 90%+ yields - **New leading-edge**: Start at 30–50%, ramp over 12–24 months ## 3. Dies Per Wafer Calculation **Gross dies per wafer (rectangular approximation):** $$ \text{Dies}_{\text{gross}} = \frac{\pi \times \left(\frac{D}{2}\right)^2}{A_{\text{die}}} $$ Where: - $D$ = Wafer diameter (mm) - $A_{\text{die}}$ = Die area (mm²) **More accurate formula (accounting for edge loss):** $$ \text{Dies}_{\text{good}} = \frac{\pi \times D^2}{4 \times A_{\text{die}}} - \frac{\pi \times D}{\sqrt{2 \times A_{\text{die}}}} $$ **For 300mm wafer:** - Usable area: ~70,000 mm² (after edge exclusion) ## 4. Cost Scaling by Technology Node | Node | Wafer Cost (USD) | Key Cost Drivers | |------|------------------|------------------| | 28nm | $3,000–4,000 | Mature, high yield | | 14/16nm | $5,000–7,000 | FinFET transition | | 7nm | $9,000–12,000 | EUV introduction (limited layers) | | 5nm | $15,000–17,000 | More EUV layers | | 3nm | $18,000–22,000 | GAA transistors, high EUV count | | 2nm | $25,000+ | Backside power, nanosheet complexity | ### 4.1 Cost Per Transistor Trend **Historical Moore's Law economics:** $$ \text{Cost reduction per node} \approx 30\% $$ **Current reality (sub-7nm):** $$ \text{Cost reduction per node} \approx 10\text{–}20\% $$ ## 5. Worked Example ### 5.1 Assumptions - **Wafer size**: 300mm - **Wafer cost**: $15,000 (all-in manufacturing cost) - **Die size**: 100 mm² - **Usable wafer area**: ~70,000 mm² - **Gross dies per wafer**: ~680 (including partial dies) - **Good dies per wafer**: ~600 (after edge loss) - **Yield**: 85% ### 5.2 Calculation **Good dies:** $$ \text{Good dies} = 600 \times 0.85 = 510 $$ **Cost per die:** $$ ext{Cost per die} = \frac{15{,}000}{510} \approx 29.41\ \text{USD} $$ ### 5.3 Yield Sensitivity Analysis | Yield | Good Dies | Cost per Die | |-------|-----------|--------------| | 95% | 570 | $26.32 | | 85% | 510 | $29.41 | | 75% | 450 | $33.33 | | 60% | 360 | $41.67 | | 50% | 300 | $50.00 | **Impact:** A 25-point yield drop (85% → 60%) increases unit cost by **42%**. ## 6. Geographic Cost Variations | Factor | Taiwan/Korea | US | Europe | China | |--------|-------------|-----|--------|-------| | Labor | Moderate | High | High | Low | | Power | Low-moderate | Varies | High | Low | | Incentives | Moderate | High (CHIPS Act) | High | Very high | | Supply chain | Dense | Developing | Limited | Developing | **US cost premium:** $$ \text{Premium}_{\text{US}} \approx 20\text{–}40\% $$ ## 7. Advanced Packaging Economics ### 7.1 Packaging Options - **Interposers**: Silicon (expensive) vs. organic (cheaper) - **Bonding**: Hybrid bonding enables fine pitch but has yield challenges - **Technologies**: CoWoS, InFO, EMIB (each with different cost structures) ### 7.2 Compound Yield For chiplet architectures with $N$ dies: $$ Y_{\text{package}} = \prod_{i=1}^{N} Y_i $$ **Example (N = 4 chiplets, each 95% yield):** $$ Y_{\text{package}} = 0.95^4 = 0.814 = 81.4\% $$ ## 8. Cost Modeling Methodologies ### 8.1 Activity-Based Costing (ABC) Maps costs to specific process operations, then aggregates: $$ \text{Total Cost} = \sum_{i=1}^{n} (\text{Activity}_i \times \text{Cost Driver}_i) $$ ### 8.2 Process-Based Cost Modeling (PBCM) Links technical parameters to equipment requirements: $$ \text{Cost} = f(\text{deposition rate}, \text{etch selectivity}, \text{throughput}, ...) $$ ### 8.3 Learning Curve Model Cost reduction with cumulative production: $$ C_n = C_1 \times n^{-b} $$ Where: - $C_n$ = Cost of the $n$-th unit - $C_1$ = Cost of the first unit - $b$ = Learning exponent (typically 0.1–0.3 for semiconductors) ## 9. Key Cost Metrics Summary | Metric | Formula | |--------|---------| | Cost per Wafer | $\sum \text{(CapEx + OpEx + Materials + Labor + Facilities)}$ | | Cost per Die | $\frac{\text{Cost per Wafer}}{\text{Dies per Wafer} \times \text{Yield}}$ | | Cost per Transistor | $\frac{\text{Cost per Die}}{\text{Transistors per Die}}$ | | Cost per mm² | $\frac{\text{Cost per Wafer}}{\text{Usable Wafer Area} \times \text{Yield}}$ | ## 10. Current Industry Trends 1. **EUV cost trajectory**: More EUV layers per node; High-NA EUV (\$350M+ per tool) arriving for 2nm 2. **Sustainability costs**: Carbon neutrality requirements, water recycling mandates 3. **Supply chain reshoring**: Government subsidies changing cost calculus 4. **3D integration**: Shifts cost from transistor scaling to packaging 5. **Mature node scarcity**: 28nm–65nm capacity tightening, prices rising ## Reference Formulas ### Yield Models ``` Poisson: Y = exp(-D₀ × A) Negative Binomial: Y = (1 + D₀×A/α)^(-α) Murphy: Y = ((1 - exp(-D₀×A)) / (D₀×A))² ``` ### Cost Equations ``` Cost/Die = Cost/Wafer ÷ (Dies/Wafer × Yield) Cost/Wafer = CapEx + Materials + Labor + Facilities + Overhead CapEx/Pass = (Tool Cost × Depreciation) ÷ (Throughput × Util × Uptime × Hours) ``` ### Dies Per Wafer ``` Gross Dies ≈ π × (D/2)² ÷ A_die Net Dies ≈ (π × D²)/(4 × A_die) - (π × D)/√(2 × A_die) ```
Manufacturing cost for processing one wafer.
Seals components in carrier.
Multiplier for confidence interval.
TSMC's 2.5D packaging technology using silicon interposer.
Target linewidth of features critical for device performance.
AFM optimized for measuring CDs.
Measure nanoscale feature dimensions.
Measure contact resistance accurately.
Cut samples to reveal internal structure.
SEM of cleaved or FIB-cut wafer to see layers and profiles.
Reduce thermal mismatch.
Direct copper-to-copper bonding.
Excess compound in pot.
Time to harden compound.
Non-Manhattan mask shapes for better lithography.
# Mathematical Modeling of CVD Equipment in Semiconductor Manufacturing ## 1. Overview of CVD in Semiconductor Fabrication Chemical Vapor Deposition (CVD) is a fundamental process in semiconductor manufacturing that deposits thin films onto wafer substrates through gas-phase and surface chemical reactions. ### 1.1 Types of Deposited Films - **Dielectrics**: $\text{SiO}_2$, $\text{Si}_3\text{N}_4$, low-$\kappa$ materials - **Conductors**: W (tungsten), TiN, Cu seed layers - **Barrier Layers**: TaN, TiN diffusion barriers - **Semiconductors**: Epitaxial Si, polysilicon, SiGe ### 1.2 CVD Process Variants | Process Type | Abbreviation | Operating Conditions | Key Characteristics | |:-------------|:-------------|:---------------------|:--------------------| | Low Pressure CVD | LPCVD | 0.1–10 Torr | Excellent uniformity, batch processing | | Plasma Enhanced CVD | PECVD | 0.1–10 Torr with plasma | Lower temperature deposition | | Atmospheric Pressure CVD | APCVD | ~760 Torr | High deposition rates | | Metal-Organic CVD | MOCVD | Variable | Organometallic precursors | | Atomic Layer Deposition | ALD | 0.1–10 Torr | Self-limiting, atomic-scale control | ## 2. Governing Equations: Transport Phenomena CVD modeling requires solving coupled partial differential equations for mass, momentum, and energy transport. ### 2.1 Mass Transport (Species Conservation) The species conservation equation describes the transport and reaction of chemical species: $$ \frac{\partial C_i}{\partial t} + \nabla \cdot (C_i \mathbf{v}) = \nabla \cdot (D_i \nabla C_i) + R_i $$ **Where:** - $C_i$ — Molar concentration of species $i$ $[\text{mol/m}^3]$ - $\mathbf{v}$ — Velocity vector field $[\text{m/s}]$ - $D_i$ — Diffusion coefficient of species $i$ $[\text{m}^2/\text{s}]$ - $R_i$ — Net volumetric production rate $[\text{mol/m}^3 \cdot \text{s}]$ #### Stefan-Maxwell Equations for Multicomponent Diffusion For multicomponent gas mixtures, the Stefan-Maxwell equations apply: $$ \nabla x_i = \sum_{j \neq i} \frac{x_i x_j}{D_{ij}} (\mathbf{v}_j - \mathbf{v}_i) $$ **Where:** - $x_i$ — Mole fraction of species $i$ - $D_{ij}$ — Binary diffusion coefficient $[\text{m}^2/\text{s}]$ #### Chapman-Enskog Diffusion Coefficient Binary diffusion coefficients can be estimated using Chapman-Enskog theory: $$ D_{ij} = \frac{3}{16} \sqrt{\frac{2\pi k_B^3 T^3}{m_{ij}}} \cdot \frac{1}{P \pi \sigma_{ij}^2 \Omega_D} $$ **Where:** - $m_{ij} = \frac{m_i m_j}{m_i + m_j}$ — Reduced mass - $\sigma_{ij}$ — Collision diameter $[\text{m}]$ - $\Omega_D$ — Collision integral (dimensionless) ### 2.2 Momentum Transport (Navier-Stokes Equations) The Navier-Stokes equations govern fluid flow in the reactor: $$ \rho \left( \frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v} \right) = -\nabla p + \nabla \cdot \boldsymbol{\tau} + \rho \mathbf{g} $$ **Where:** - $\rho$ — Gas density $[\text{kg/m}^3]$ - $p$ — Pressure $[\text{Pa}]$ - $\boldsymbol{\tau}$ — Viscous stress tensor $[\text{Pa}]$ - $\mathbf{g}$ — Gravitational acceleration $[\text{m/s}^2]$ #### Newtonian Stress Tensor For Newtonian fluids: $$ \boldsymbol{\tau} = \mu \left( \nabla \mathbf{v} + (\nabla \mathbf{v})^T \right) - \frac{2}{3} \mu (\nabla \cdot \mathbf{v}) \mathbf{I} $$ #### Slip Boundary Conditions At low pressures where Knudsen number $Kn > 0.01$, slip boundary conditions are required: $$ v_{slip} = \frac{2 - \sigma_v}{\sigma_v} \lambda \left( \frac{\partial v}{\partial n} \right)_{wall} $$ **Where:** - $\sigma_v$ — Tangential momentum accommodation coefficient - $\lambda$ — Mean free path $[\text{m}]$ - $n$ — Wall-normal direction #### Mean Free Path $$ \lambda = \frac{k_B T}{\sqrt{2} \pi d^2 P} $$ ### 2.3 Energy Transport The energy equation accounts for convection, conduction, and heat generation: $$ \rho c_p \left( \frac{\partial T}{\partial t} + \mathbf{v} \cdot \nabla T \right) = \nabla \cdot (k \nabla T) + Q_{rxn} + Q_{rad} $$ **Where:** - $c_p$ — Specific heat capacity $[\text{J/kg} \cdot \text{K}]$ - $k$ — Thermal conductivity $[\text{W/m} \cdot \text{K}]$ - $Q_{rxn}$ — Heat from chemical reactions $[\text{W/m}^3]$ - $Q_{rad}$ — Radiative heat transfer $[\text{W/m}^3]$ #### Radiative Heat Transfer (Rosseland Approximation) For optically thick media: $$ Q_{rad} = \nabla \cdot \left( \frac{4\sigma_{SB}}{3\kappa_R} \nabla T^4 \right) $$ **Where:** - $\sigma_{SB} = 5.67 \times 10^{-8}$ W/m²·K⁴ — Stefan-Boltzmann constant - $\kappa_R$ — Rosseland mean absorption coefficient $[\text{m}^{-1}]$ ## 3. Chemical Kinetics ### 3.1 Gas-Phase Reactions Gas-phase reactions decompose precursor molecules and generate reactive intermediates. #### Example: Silane Decomposition for Silicon Deposition **Primary decomposition:** $$ \text{SiH}_4 \xrightarrow{k_1} \text{SiH}_2 + \text{H}_2 $$ **Secondary reactions:** $$ \text{SiH}_2 + \text{SiH}_4 \xrightarrow{k_2} \text{Si}_2\text{H}_6 $$ $$ \text{SiH}_2 + \text{SiH}_2 \xrightarrow{k_3} \text{Si}_2\text{H}_4 $$ #### Arrhenius Rate Expression Rate constants follow the modified Arrhenius form: $$ k(T) = A \cdot T^n \exp\left( -\frac{E_a}{RT} \right) $$ **Where:** - $A$ — Pre-exponential factor $[\text{varies}]$ - $n$ — Temperature exponent (dimensionless) - $E_a$ — Activation energy $[\text{J/mol}]$ - $R = 8.314$ J/(mol·K) — Universal gas constant #### Species Source Term The net production rate for species $i$: $$ R_i = \sum_{r=1}^{N_r} \nu_{i,r} \cdot k_r \prod_{j=1}^{N_s} C_j^{\alpha_{j,r}} $$ **Where:** - $\nu_{i,r}$ — Stoichiometric coefficient of species $i$ in reaction $r$ - $\alpha_{j,r}$ — Reaction order of species $j$ in reaction $r$ - $N_r$ — Total number of reactions - $N_s$ — Total number of species ### 3.2 Surface Reaction Kinetics Surface reactions determine the actual film deposition. #### Langmuir-Hinshelwood Mechanism For bimolecular surface reactions: $$ R_s = \frac{k_s K_A K_B C_A C_B}{(1 + K_A C_A + K_B C_B)^2} $$ **Where:** - $k_s$ — Surface reaction rate constant $[\text{m}^2/\text{mol} \cdot \text{s}]$ - $K_A, K_B$ — Adsorption equilibrium constants $[\text{m}^3/\text{mol}]$ - $C_A, C_B$ — Gas-phase concentrations at surface $[\text{mol/m}^3]$ #### Eley-Rideal Mechanism For reactions between adsorbed and gas-phase species: $$ R_s = k_s \theta_A C_B $$ #### Sticking Coefficient Model (Kinetic Theory) The adsorption flux based on kinetic theory: $$ J_{ads} = \frac{s \cdot p}{\sqrt{2\pi m k_B T}} $$ **Where:** - $s$ — Sticking probability (dimensionless, $0 < s \leq 1$) - $p$ — Partial pressure of adsorbing species $[\text{Pa}]$ - $m$ — Molecular mass $[\text{kg}]$ - $k_B = 1.38 \times 10^{-23}$ J/K — Boltzmann constant #### Surface Site Balance Dynamic surface coverage evolution: $$ \frac{d\theta_i}{dt} = k_{ads,i} C_i (1 - \theta_{total}) - k_{des,i} \theta_i - k_{rxn} \theta_i \theta_j $$ **Where:** - $\theta_i$ — Surface coverage fraction of species $i$ - $\theta_{total} = \sum_i \theta_i$ — Total surface coverage - $k_{ads,i}$ — Adsorption rate constant - $k_{des,i}$ — Desorption rate constant - $k_{rxn}$ — Surface reaction rate constant ## 4. Film Growth and Deposition Rate ### 4.1 Local Deposition Rate The film thickness growth rate: $$ \frac{dh}{dt} = \frac{M_w}{\rho_{film}} \cdot R_s $$ **Where:** - $h$ — Film thickness $[\text{m}]$ - $M_w$ — Molecular weight of deposited material $[\text{kg/mol}]$ - $\rho_{film}$ — Film density $[\text{kg/m}^3]$ - $R_s$ — Surface reaction rate $[\text{mol/m}^2 \cdot \text{s}]$ ### 4.2 Boundary Layer Analysis #### Rotating Disk Reactor (Classical Solution) Boundary layer thickness: $$ \delta = \sqrt{\frac{\nu}{\Omega}} $$ **Where:** - $\nu$ — Kinematic viscosity $[\text{m}^2/\text{s}]$ - $\Omega$ — Angular rotation speed $[\text{rad/s}]$ #### Sherwood Number Correlation For mass transfer in laminar flow: $$ Sh = 0.62 \cdot Re^{1/2} \cdot Sc^{1/3} $$ **Where:** - $Sh = \frac{k_m L}{D}$ — Sherwood number - $Re = \frac{\rho v L}{\mu}$ — Reynolds number - $Sc = \frac{\mu}{\rho D}$ — Schmidt number #### Mass Transfer Coefficient $$ k_m = \frac{Sh \cdot D}{L} $$ ### 4.3 Deposition Rate Regimes The overall deposition process can be limited by different mechanisms: **Regime 1: Surface Reaction Limited** ($Da \ll 1$) $$ R_{dep} \approx k_s C_{bulk} $$ **Regime 2: Mass Transfer Limited** ($Da \gg 1$) $$ R_{dep} \approx k_m C_{bulk} $$ **General Case:** $$ \frac{1}{R_{dep}} = \frac{1}{k_s C_{bulk}} + \frac{1}{k_m C_{bulk}} $$ ## 5. Step Coverage and Feature-Scale Modeling ### 5.1 Thiele Modulus Analysis The Thiele modulus determines whether deposition is reaction or diffusion limited within features: $$ \phi = L \sqrt{\frac{k_s}{D_{Kn}}} $$ **Where:** - $L$ — Feature depth $[\text{m}]$ - $k_s$ — Surface reaction rate constant $[\text{m/s}]$ - $D_{Kn}$ — Knudsen diffusion coefficient $[\text{m}^2/\text{s}]$ **Interpretation:** | Thiele Modulus | Regime | Step Coverage | |:---------------|:-------|:--------------| | $\phi \ll 1$ | Reaction-limited | Excellent (conformal) | | $\phi \approx 1$ | Transition | Moderate | | $\phi \gg 1$ | Diffusion-limited | Poor (non-conformal) | #### Knudsen Diffusion in Features For high aspect ratio features where $Kn > 1$: $$ D_{Kn} = \frac{d}{3} \sqrt{\frac{8RT}{\pi M}} $$ **Where:** - $d$ — Feature diameter/width $[\text{m}]$ - $M$ — Molecular weight $[\text{kg/mol}]$ ### 5.2 Level-Set Method for Surface Evolution The level-set equation tracks the evolving surface: $$ \frac{\partial \phi}{\partial t} + V_n |\nabla \phi| = 0 $$ **Where:** - $\phi(\mathbf{x}, t)$ — Level-set function (surface at $\phi = 0$) - $V_n$ — Local normal velocity $[\text{m/s}]$ #### Reinitialization Equation To maintain $|\nabla \phi| = 1$: $$ \frac{\partial \phi}{\partial \tau} = \text{sign}(\phi_0)(1 - |\nabla \phi|) $$ ### 5.3 Ballistic Transport (Monte Carlo) For molecular flow in high-aspect-ratio features, the flux at a surface point: $$ \Gamma(\mathbf{r}) = \frac{1}{\pi} \int_{\Omega_{visible}} \Gamma_0 \cos\theta \, d\Omega $$ **Where:** - $\Gamma_0$ — Incident flux at feature opening $[\text{mol/m}^2 \cdot \text{s}]$ - $\theta$ — Angle from surface normal - $\Omega_{visible}$ — Visible solid angle from point $\mathbf{r}$ #### View Factor Calculation The view factor from surface element $i$ to $j$: $$ F_{i \rightarrow j} = \frac{1}{\pi A_i} \int_{A_i} \int_{A_j} \frac{\cos\theta_i \cos\theta_j}{r^2} \, dA_j \, dA_i $$ ## 6. Reactor-Scale Modeling ### 6.1 Showerhead Gas Distribution #### Pressure Drop Through Holes $$ \Delta P = \frac{1}{2} \rho v^2 \left( \frac{1}{C_d^2} \right) $$ **Where:** - $C_d$ — Discharge coefficient (typically 0.6–0.8) - $v$ — Gas velocity through hole $[\text{m/s}]$ #### Flow Rate Through Individual Holes $$ Q_i = C_d A_i \sqrt{\frac{2\Delta P}{\rho}} $$ #### Uniformity Index $$ UI = 1 - \frac{\sigma_Q}{\bar{Q}} $$ ### 6.2 Wafer Temperature Uniformity Combined convection-radiation heat transfer to wafer: $$ q = h_{conv}(T_{susceptor} - T_{wafer}) + \epsilon \sigma_{SB} (T_{susceptor}^4 - T_{wafer}^4) $$ **Where:** - $h_{conv}$ — Convective heat transfer coefficient $[\text{W/m}^2 \cdot \text{K}]$ - $\epsilon$ — Emissivity (dimensionless) #### Edge Effect Modeling Radiative view factor at wafer edge: $$ F_{edge} = \frac{1}{2}\left(1 - \frac{1}{\sqrt{1 + (R/H)^2}}\right) $$ ### 6.3 Precursor Depletion Along the flow direction: $$ \frac{dC}{dx} = -\frac{k_s W}{Q} C $$ **Solution:** $$ C(x) = C_0 \exp\left(-\frac{k_s W x}{Q}\right) $$ **Where:** - $W$ — Wafer width $[\text{m}]$ - $Q$ — Volumetric flow rate $[\text{m}^3/\text{s}]$ ## 7. PECVD: Plasma Modeling ### 7.1 Electron Kinetics #### Boltzmann Equation The electron energy distribution function (EEDF): $$ \frac{\partial f}{\partial t} + \mathbf{v} \cdot \nabla_r f + \frac{e\mathbf{E}}{m_e} \cdot \nabla_v f = \left( \frac{\partial f}{\partial t} \right)_{coll} $$ **Where:** - $f(\mathbf{r}, \mathbf{v}, t)$ — Electron distribution function - $\mathbf{E}$ — Electric field $[\text{V/m}]$ - $m_e = 9.109 \times 10^{-31}$ kg — Electron mass #### Two-Term Spherical Harmonic Expansion $$ f(\varepsilon, \mathbf{r}, t) = f_0(\varepsilon) + f_1(\varepsilon) \cos\theta $$ ### 7.2 Plasma Chemistry #### Electron Impact Dissociation $$ e + \text{SiH}_4 \xrightarrow{k_e} \text{SiH}_3 + \text{H} + e $$ #### Electron Impact Ionization $$ e + \text{SiH}_4 \xrightarrow{k_i} \text{SiH}_3^+ + \text{H} + 2e $$ #### Rate Coefficient Calculation $$ k_e = \int_0^\infty \sigma(\varepsilon) \sqrt{\frac{2\varepsilon}{m_e}} f(\varepsilon) \, d\varepsilon $$ **Where:** - $\sigma(\varepsilon)$ — Energy-dependent cross-section $[\text{m}^2]$ - $\varepsilon$ — Electron energy $[\text{eV}]$ ### 7.3 Sheath Physics #### Floating Potential $$ V_f = -\frac{T_e}{2e} \ln\left( \frac{m_i}{2\pi m_e} \right) $$ #### Bohm Velocity $$ v_B = \sqrt{\frac{k_B T_e}{m_i}} $$ #### Ion Flux to Surface $$ \Gamma_i = n_s v_B = n_s \sqrt{\frac{k_B T_e}{m_i}} $$ #### Child-Langmuir Law (Collisionless Sheath) Ion current density: $$ J_i = \frac{4\epsilon_0}{9} \sqrt{\frac{2e}{m_i}} \frac{V_s^{3/2}}{d_s^2} $$ **Where:** - $V_s$ — Sheath voltage $[\text{V}]$ - $d_s$ — Sheath thickness $[\text{m}]$ ### 7.4 Power Deposition Ohmic heating in the bulk plasma: $$ P_{ohm} = \frac{J^2}{\sigma} = \frac{n_e e^2 \nu_m}{m_e} E^2 $$ **Where:** - $\sigma$ — Plasma conductivity $[\text{S/m}]$ - $\nu_m$ — Electron-neutral collision frequency $[\text{s}^{-1}]$ ## 8. Dimensionless Analysis ### 8.1 Key Dimensionless Numbers | Number | Definition | Physical Meaning | |:-------|:-----------|:-----------------| | Damköhler | $Da = \dfrac{k_s L}{D}$ | Reaction rate vs. diffusion rate | | Reynolds | $Re = \dfrac{\rho v L}{\mu}$ | Inertial forces vs. viscous forces | | Péclet | $Pe = \dfrac{vL}{D}$ | Convection vs. diffusion | | Knudsen | $Kn = \dfrac{\lambda}{L}$ | Mean free path vs. characteristic length | | Grashof | $Gr = \dfrac{g\beta \Delta T L^3}{\nu^2}$ | Buoyancy vs. viscous forces | | Prandtl | $Pr = \dfrac{\mu c_p}{k}$ | Momentum diffusivity vs. thermal diffusivity | | Schmidt | $Sc = \dfrac{\mu}{\rho D}$ | Momentum diffusivity vs. mass diffusivity | | Thiele | $\phi = L\sqrt{\dfrac{k_s}{D}}$ | Surface reaction vs. pore diffusion | ### 8.2 Temperature Sensitivity Analysis The sensitivity of deposition rate to temperature: $$ \frac{\delta R}{R} = \frac{E_a}{RT^2} \delta T $$ **Example Calculation:** For $E_a = 1.5$ eV = $144.7$ kJ/mol at $T = 973$ K (700°C): $$ \frac{\delta R}{R} = \frac{144700}{8.314 \times 973^2} \cdot 1 \text{ K} \approx 0.018 = 1.8\% $$ **Implication:** A 1°C temperature variation causes ~1.8% deposition rate change. ### 8.3 Flow Regime Classification Based on Knudsen number: | Knudsen Number | Flow Regime | Applicable Equations | |:---------------|:------------|:---------------------| | $Kn < 0.01$ | Continuum | Navier-Stokes | | $0.01 < Kn < 0.1$ | Slip flow | N-S with slip BC | | $0.1 < Kn < 10$ | Transition | DSMC or Boltzmann | | $Kn > 10$ | Free molecular | Kinetic theory | ## 9. Multiscale Modeling Framework ### 9.1 Modeling Hierarchy ``` ┌─────────────────────────────────────────────────────────────────┐ │ QUANTUM SCALE (DFT) │ │ • Reaction mechanisms and transition states │ │ • Activation energies and rate constants │ │ • Length: ~1 nm, Time: ~fs │ ├─────────────────────────────────────────────────────────────────┤ │ MOLECULAR DYNAMICS │ │ • Surface diffusion coefficients │ │ • Nucleation and island formation │ │ • Length: ~10 nm, Time: ~ns │ ├─────────────────────────────────────────────────────────────────┤ │ KINETIC MONTE CARLO │ │ • Film microstructure evolution │ │ • Surface roughness development │ │ • Length: ~100 nm, Time: ~μs–ms │ ├─────────────────────────────────────────────────────────────────┤ │ FEATURE-SCALE (Continuum) │ │ • Topography evolution in trenches/vias │ │ • Step coverage prediction │ │ • Length: ~1 μm, Time: ~s │ ├─────────────────────────────────────────────────────────────────┤ │ REACTOR-SCALE (CFD) │ │ • Gas flow and temperature fields │ │ • Species concentration distributions │ │ • Length: ~0.1 m, Time: ~min │ ├─────────────────────────────────────────────────────────────────┤ │ EQUIPMENT/FAB SCALE │ │ • Wafer-to-wafer variation │ │ • Throughput and scheduling │ │ • Length: ~1 m, Time: ~hours │ └─────────────────────────────────────────────────────────────────┘ ``` ### 9.2 Scale Bridging Approaches **Bottom-Up Parameterization:** - DFT → Rate constants for higher scales - MD → Diffusion coefficients, sticking probabilities - kMC → Effective growth rates, roughness correlations **Top-Down Validation:** - Reactor experiments → Validate CFD predictions - SEM/TEM → Validate feature-scale models - Surface analysis → Validate kinetic models ## 10. ALD-Specific Modeling ### 10.1 Self-Limiting Surface Reactions ALD relies on self-limiting half-reactions: **Half-Reaction A (e.g., TMA pulse for Al₂O₃):** $$ \theta_A(t) = \theta_{sat} \left( 1 - e^{-k_{ads} p_A t} \right) $$ **Half-Reaction B (e.g., H₂O pulse):** $$ \theta_B(t) = (1 - \theta_A) \left( 1 - e^{-k_B p_B t} \right) $$ ### 10.2 Growth Per Cycle (GPC) $$ GPC = \theta_{sat} \cdot \Gamma_{sites} \cdot \frac{M_w}{\rho N_A} $$ **Where:** - $\theta_{sat}$ — Saturation coverage (dimensionless) - $\Gamma_{sites}$ — Surface site density $[\text{sites/m}^2]$ - $N_A = 6.022 \times 10^{23}$ mol⁻¹ — Avogadro's number **Typical values for Al₂O₃ ALD:** - $GPC \approx 0.1$ nm/cycle - $\Gamma_{sites} \approx 10^{19}$ sites/m² ### 10.3 Saturation Dose The dose required for saturation: $$ D_{sat} \propto \frac{1}{s} \sqrt{\frac{m k_B T}{2\pi}} $$ **Where:** - $s$ — Reactive sticking coefficient - Lower sticking coefficient → Higher saturation dose required ### 10.4 Nucleation Delay Modeling For non-ideal ALD on different substrates: $$ h(n) = GPC \cdot (n - n_0) \quad \text{for } n > n_0 $$ **Where:** - $n$ — Cycle number - $n_0$ — Nucleation delay (cycles) ## 11. Computational Tools and Methods ### 11.1 Reactor-Scale CFD | Software | Capabilities | Applications | |:---------|:-------------|:-------------| | ANSYS Fluent | General CFD + species transport | Reactor flow modeling | | COMSOL Multiphysics | Coupled multiphysics | Heat/mass transfer | | OpenFOAM | Open-source CFD | Custom reactor models | **Typical mesh requirements:** - $10^5 - 10^7$ cells for 3D reactor - Boundary layer refinement near wafer - Adaptive meshing for reacting flows ### 11.2 Chemical Kinetics | Software | Capabilities | |:---------|:-------------| | Chemkin-Pro | Detailed gas-phase kinetics | | Cantera | Open-source kinetics | | SURFACE CHEMKIN | Surface reaction modeling | ### 11.3 Feature-Scale Simulation | Method | Advantages | Limitations | |:-------|:-----------|:------------| | Level-Set | Handles topology changes | Diffusive interface | | Volume of Fluid | Mass conserving | Interface reconstruction | | Monte Carlo | Physical accuracy | Computationally intensive | | String Method | Efficient for 2D | Limited to simple geometries | ### 11.4 Process/TCAD Integration | Software | Vendor | Applications | |:---------|:-------|:-------------| | Sentaurus Process | Synopsys | Full process simulation | | Victory Process | Silvaco | Deposition, etch, implant | | FLOOPS | Florida | Academic/research | ## 12. Machine Learning Integration ### 12.1 Physics-Informed Neural Networks (PINNs) Loss function combining data and physics: $$ \mathcal{L} = \mathcal{L}_{data} + \lambda \mathcal{L}_{physics} $$ **Where:** $$ \mathcal{L}_{physics} = \frac{1}{N_f} \sum_{i=1}^{N_f} \left| \mathcal{F}[\hat{u}(\mathbf{x}_i)] \right|^2 $$ - $\mathcal{F}$ — Differential operator (governing PDE) - $\hat{u}$ — Neural network approximation - $\lambda$ — Weighting parameter ### 12.2 Surrogate Modeling **Gaussian Process Regression:** $$ f(\mathbf{x}) \sim \mathcal{GP}(m(\mathbf{x}), k(\mathbf{x}, \mathbf{x}')) $$ **Where:** - $m(\mathbf{x})$ — Mean function - $k(\mathbf{x}, \mathbf{x}')$ — Covariance kernel (e.g., RBF) **Applications:** - Real-time process control - Recipe optimization - Virtual metrology ### 12.3 Deep Learning Applications | Application | Method | Input → Output | |:------------|:-------|:---------------| | Uniformity prediction | CNN | Wafer map → Uniformity metrics | | Recipe optimization | RL | Process parameters → Film properties | | Defect detection | CNN | SEM images → Defect classification | | Endpoint detection | RNN/LSTM | OES time series → Process state | ## 13. Key Modeling Challenges ### 13.1 Stiff Chemistry - Reaction timescales vary by orders of magnitude ($10^{-12}$ to $10^0$ s) - Requires implicit time integration or operator splitting - Chemical mechanism reduction techniques ### 13.2 Surface Reaction Parameters - Limited experimental data for many chemistries - Temperature and surface-dependent sticking coefficients - Complex multi-step mechanisms ### 13.3 Multiscale Coupling - Feature-scale depletion affects reactor-scale concentrations - Reactor non-uniformity impacts feature-scale profiles - Requires iterative or concurrent coupling schemes ### 13.4 Plasma Complexity - Non-Maxwellian electron distributions - Transient sheath dynamics in RF plasmas - Ion energy and angular distributions ### 13.5 Advanced Device Architectures - 3D NAND with extreme aspect ratios (AR > 100:1) - Gate-All-Around (GAA) transistors - Complex multi-material stacks ## Summary CVD equipment modeling requires solving coupled nonlinear PDEs for momentum, heat, and mass transport with complex gas-phase and surface chemistry. The mathematical framework encompasses: - **Continuum mechanics**: Navier-Stokes, convection-diffusion - **Chemical kinetics**: Arrhenius, Langmuir-Hinshelwood, Eley-Rideal - **Surface science**: Sticking coefficients, site balances, nucleation - **Plasma physics**: Boltzmann equation, sheath dynamics - **Numerical methods**: FEM, FVM, Monte Carlo, level-set The ultimate goal is predictive capability for film thickness, uniformity, composition, and microstructure—enabling virtual process development and optimization for advanced semiconductor manufacturing.
# CVD Modeling in Semiconductor Manufacturing ## 1. Introduction Chemical Vapor Deposition (CVD) is a critical thin-film deposition technique in semiconductor manufacturing. Gaseous precursors are introduced into a reaction chamber where they undergo chemical reactions to deposit solid films on heated substrates. ### 1.1 Key Process Steps - **Transport** of reactants from bulk gas to the substrate surface - **Gas-phase chemistry** including precursor decomposition and intermediate formation - **Surface reactions** involving adsorption, surface diffusion, and reaction - **Film nucleation and growth** with specific microstructure evolution - **Byproduct desorption** and transport away from the surface ### 1.2 Common CVD Types - **APCVD** — Atmospheric Pressure CVD - **LPCVD** — Low Pressure CVD (0.1–10 Torr) - **PECVD** — Plasma Enhanced CVD - **MOCVD** — Metal-Organic CVD - **ALD** — Atomic Layer Deposition - **HDPCVD** — High Density Plasma CVD ## 2. Governing Equations ### 2.1 Continuity Equation (Mass Conservation) $$ \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0 $$ Where: - $\rho$ — gas density $\left[\text{kg/m}^3\right]$ - $\mathbf{u}$ — velocity vector $\left[\text{m/s}\right]$ - $t$ — time $\left[\text{s}\right]$ ### 2.2 Momentum Equation (Navier-Stokes) $$ \rho \left( \frac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u} \right) = -\nabla p + \mu \nabla^2 \mathbf{u} + \rho \mathbf{g} $$ Where: - $p$ — pressure $\left[\text{Pa}\right]$ - $\mu$ — dynamic viscosity $\left[\text{Pa} \cdot \text{s}\right]$ - $\mathbf{g}$ — gravitational acceleration $\left[\text{m/s}^2\right]$ ### 2.3 Species Conservation Equation $$ \frac{\partial (\rho Y_i)}{\partial t} + \nabla \cdot (\rho \mathbf{u} Y_i) = \nabla \cdot (\rho D_i \nabla Y_i) + R_i $$ Where: - $Y_i$ — mass fraction of species $i$ $\left[\text{dimensionless}\right]$ - $D_i$ — diffusion coefficient of species $i$ $\left[\text{m}^2/\text{s}\right]$ - $R_i$ — net production rate from reactions $\left[\text{kg/m}^3 \cdot \text{s}\right]$ ### 2.4 Energy Conservation Equation $$ \rho c_p \left( \frac{\partial T}{\partial t} + \mathbf{u} \cdot \nabla T \right) = \nabla \cdot (k \nabla T) + Q $$ Where: - $c_p$ — specific heat capacity $\left[\text{J/kg} \cdot \text{K}\right]$ - $T$ — temperature $\left[\text{K}\right]$ - $k$ — thermal conductivity $\left[\text{W/m} \cdot \text{K}\right]$ - $Q$ — volumetric heat source $\left[\text{W/m}^3\right]$ ### 2.5 Key Dimensionless Numbers | Number | Definition | Physical Meaning | |--------|------------|------------------| | Reynolds | $Re = \frac{\rho u L}{\mu}$ | Inertial vs. viscous forces | | Péclet | $Pe = \frac{u L}{D}$ | Convection vs. diffusion | | Damköhler | $Da = \frac{k_s L}{D}$ | Reaction rate vs. transport rate | | Knudsen | $Kn = \frac{\lambda}{L}$ | Mean free path vs. length scale | Where: - $L$ — characteristic length $\left[\text{m}\right]$ - $\lambda$ — mean free path $\left[\text{m}\right]$ - $k_s$ — surface reaction rate constant $\left[\text{m/s}\right]$ ## 3. Chemical Kinetics ### 3.1 Arrhenius Equation The temperature dependence of reaction rate constants follows: $$ k = A \exp\left(-\frac{E_a}{R T}\right) $$ Where: - $k$ — rate constant $\left[\text{varies}\right]$ - $A$ — pre-exponential factor $\left[\text{same as } k\right]$ - $E_a$ — activation energy $\left[\text{J/mol}\right]$ - $R$ — universal gas constant $= 8.314 \, \text{J/mol} \cdot \text{K}$ ### 3.2 Gas-Phase Reactions **Example: Silane Pyrolysis** $$ \text{SiH}_4 \xrightarrow{k_1} \text{SiH}_2 + \text{H}_2 $$ $$ \text{SiH}_2 + \text{SiH}_4 \xrightarrow{k_2} \text{Si}_2\text{H}_6 $$ **General reaction rate expression:** $$ r_j = k_j \prod_{i} C_i^{\nu_{ij}} $$ Where: - $r_j$ — rate of reaction $j$ $\left[\text{mol/m}^3 \cdot \text{s}\right]$ - $C_i$ — concentration of species $i$ $\left[\text{mol/m}^3\right]$ - $\nu_{ij}$ — stoichiometric coefficient of species $i$ in reaction $j$ ### 3.3 Surface Reaction Kinetics #### 3.3.1 Hertz-Knudsen Impingement Flux $$ J = \frac{p}{\sqrt{2 \pi m k_B T}} $$ Where: - $J$ — molecular flux $\left[\text{molecules/m}^2 \cdot \text{s}\right]$ - $p$ — partial pressure $\left[\text{Pa}\right]$ - $m$ — molecular mass $\left[\text{kg}\right]$ - $k_B$ — Boltzmann constant $= 1.381 \times 10^{-23} \, \text{J/K}$ #### 3.3.2 Surface Reaction Rate $$ R_s = s \cdot J = s \cdot \frac{p}{\sqrt{2 \pi m k_B T}} $$ Where: - $s$ — sticking coefficient $\left[0 \leq s \leq 1\right]$ #### 3.3.3 Langmuir-Hinshelwood Kinetics For surface reaction between two adsorbed species: $$ r = \frac{k \, K_A \, K_B \, p_A \, p_B}{(1 + K_A p_A + K_B p_B)^2} $$ Where: - $K_A, K_B$ — adsorption equilibrium constants $\left[\text{Pa}^{-1}\right]$ - $p_A, p_B$ — partial pressures of reactants A and B $\left[\text{Pa}\right]$ #### 3.3.4 Eley-Rideal Mechanism For reaction between adsorbed species and gas-phase species: $$ r = \frac{k \, K_A \, p_A \, p_B}{1 + K_A p_A} $$ ### 3.4 Common CVD Reaction Systems - **Silicon from Silane:** - $\text{SiH}_4 \rightarrow \text{Si}_{(s)} + 2\text{H}_2$ - **Silicon Dioxide from TEOS:** - $\text{Si(OC}_2\text{H}_5\text{)}_4 + 12\text{O}_2 \rightarrow \text{SiO}_2 + 8\text{CO}_2 + 10\text{H}_2\text{O}$ - **Silicon Nitride from DCS:** - $3\text{SiH}_2\text{Cl}_2 + 4\text{NH}_3 \rightarrow \text{Si}_3\text{N}_4 + 6\text{HCl} + 6\text{H}_2$ - **Tungsten from WF₆:** - $\text{WF}_6 + 3\text{H}_2 \rightarrow \text{W}_{(s)} + 6\text{HF}$ ## 4. Process Regimes ### 4.1 Transport-Limited Regime **Characteristics:** - High Damköhler number: $Da \gg 1$ - Surface reactions are fast - Deposition rate controlled by mass transport - Sensitive to: - Flow patterns - Temperature gradients - Reactor geometry **Deposition rate expression:** $$ R_{dep} \approx \frac{D \cdot C_{\infty}}{\delta} $$ Where: - $C_{\infty}$ — bulk gas concentration $\left[\text{mol/m}^3\right]$ - $\delta$ — boundary layer thickness $\left[\text{m}\right]$ ### 4.2 Reaction-Limited Regime **Characteristics:** - Low Damköhler number: $Da \ll 1$ - Plenty of reactants at surface - Rate controlled by surface kinetics - Strong Arrhenius temperature dependence - Better step coverage in features **Deposition rate expression:** $$ R_{dep} \approx k_s \cdot C_s \approx k_s \cdot C_{\infty} $$ Where: - $k_s$ — surface reaction rate constant $\left[\text{m/s}\right]$ - $C_s$ — surface concentration $\approx C_{\infty}$ $\left[\text{mol/m}^3\right]$ ### 4.3 Regime Transition The transition occurs when: $$ Da = \frac{k_s \delta}{D} \approx 1 $$ **Practical implications:** - **Transport-limited:** Optimize flow, temperature uniformity - **Reaction-limited:** Optimize temperature, precursor chemistry - **Mixed regime:** Most complex to control and model ## 5. Multiscale Modeling ### 5.1 Scale Hierarchy | Scale | Length | Time | Methods | |-------|--------|------|---------| | Reactor | cm – m | s – min | CFD, FEM | | Feature | nm – μm | ms – s | Level set, Monte Carlo | | Surface | nm | μs – ms | KMC | | Atomistic | Å | fs – ps | MD, DFT | ### 5.2 Reactor-Scale Modeling **Governing physics:** - Coupled Navier-Stokes + species + energy equations - Multicomponent diffusion (Stefan-Maxwell) - Chemical source terms **Stefan-Maxwell diffusion:** $$ \nabla x_i = \sum_{j \neq i} \frac{x_i x_j}{D_{ij}} (\mathbf{u}_j - \mathbf{u}_i) $$ Where: - $x_i$ — mole fraction of species $i$ - $D_{ij}$ — binary diffusion coefficient $\left[\text{m}^2/\text{s}\right]$ **Common software:** - ANSYS Fluent - COMSOL Multiphysics - OpenFOAM (open-source) - Silvaco Victory Process - Synopsys Sentaurus ### 5.3 Feature-Scale Modeling **Key phenomena:** - Knudsen diffusion in high-aspect-ratio features - Molecular re-emission and reflection - Surface reaction probability - Film profile evolution **Knudsen diffusion coefficient:** $$ D_K = \frac{d}{3} \sqrt{\frac{8 k_B T}{\pi m}} $$ Where: - $d$ — feature width $\left[\text{m}\right]$ **Effective diffusivity (transition regime):** $$ \frac{1}{D_{eff}} = \frac{1}{D_{mol}} + \frac{1}{D_K} $$ **Level set method for surface tracking:** $$ \frac{\partial \phi}{\partial t} + v_n |\nabla \phi| = 0 $$ Where: - $\phi$ — level set function (zero at surface) - $v_n$ — surface normal velocity (deposition rate) ### 5.4 Atomistic Modeling **Density Functional Theory (DFT):** - Calculate binding energies - Determine activation barriers - Predict reaction pathways **Kinetic Monte Carlo (KMC):** - Stochastic surface evolution - Event rates from Arrhenius: $$ \Gamma_i = \nu_0 \exp\left(-\frac{E_i}{k_B T}\right) $$ Where: - $\Gamma_i$ — rate of event $i$ $\left[\text{s}^{-1}\right]$ - $\nu_0$ — attempt frequency $\sim 10^{12} - 10^{13} \, \text{s}^{-1}$ - $E_i$ — activation energy for event $i$ $\left[\text{eV}\right]$ ## 6. CVD Process Variants ### 6.1 LPCVD (Low Pressure CVD) **Operating conditions:** - Pressure: $0.1 - 10 \, \text{Torr}$ - Temperature: $400 - 900 \, °\text{C}$ - Hot-wall reactor design **Advantages:** - Better uniformity (longer mean free path) - Good step coverage - High purity films **Applications:** - Polysilicon gates - Silicon nitride (Si₃N₄) - Thermal oxides ### 6.2 PECVD (Plasma Enhanced CVD) **Additional physics:** - Electron impact reactions - Ion bombardment - Radical chemistry - Plasma sheath dynamics **Electron density equation:** $$ \frac{\partial n_e}{\partial t} + \nabla \cdot \boldsymbol{\Gamma}_e = S_e $$ Where: - $n_e$ — electron density $\left[\text{m}^{-3}\right]$ - $\boldsymbol{\Gamma}_e$ — electron flux $\left[\text{m}^{-2} \cdot \text{s}^{-1}\right]$ - $S_e$ — electron source term (ionization - recombination) **Electron energy distribution:** Often non-Maxwellian, requiring solution of Boltzmann equation or two-temperature models. **Advantages:** - Lower deposition temperatures ($200 - 400 \, °\text{C}$) - Higher deposition rates - Tunable film stress ### 6.3 ALD (Atomic Layer Deposition) **Process characteristics:** - Self-limiting surface reactions - Sequential precursor pulses - Sub-monolayer control **Growth per cycle:** $$ \text{GPC} = \frac{\Delta t}{\text{cycle}} $$ Typically: $\text{GPC} \approx 0.5 - 2 \, \text{Å/cycle}$ **Surface coverage model:** $$ \theta = \theta_{sat} \left(1 - e^{-\sigma J t}\right) $$ Where: - $\theta$ — surface coverage $\left[0 \leq \theta \leq 1\right]$ - $\theta_{sat}$ — saturation coverage - $\sigma$ — reaction cross-section $\left[\text{m}^2\right]$ - $t$ — exposure time $\left[\text{s}\right]$ **Applications:** - High-k gate dielectrics (HfO₂, ZrO₂) - Barrier layers (TaN, TiN) - Conformal coatings in 3D structures ### 6.4 MOCVD (Metal-Organic CVD) **Precursors:** - Metal-organic compounds (e.g., TMGa, TMAl, TMIn) - Hydrides (AsH₃, PH₃, NH₃) **Key challenges:** - Parasitic gas-phase reactions - Particle formation - Precise composition control **Applications:** - III-V semiconductors (GaAs, InP, GaN) - LEDs and laser diodes - High-electron-mobility transistors (HEMTs) ## 7. Step Coverage Modeling ### 7.1 Definition **Step coverage (SC):** $$ SC = \frac{t_{bottom}}{t_{top}} \times 100\% $$ Where: - $t_{bottom}$ — film thickness at feature bottom - $t_{top}$ — film thickness at feature top **Aspect ratio (AR):** $$ AR = \frac{H}{W} $$ Where: - $H$ — feature depth - $W$ — feature width ### 7.2 Ballistic Transport Model For molecular flow in features ($Kn > 1$): **View factor approach:** $$ F_{i \rightarrow j} = \frac{A_j \cos\theta_i \cos\theta_j}{\pi r_{ij}^2} $$ **Flux balance at surface element:** $$ J_i = J_{direct} + \sum_j (1-s) J_j F_{j \rightarrow i} $$ Where: - $s$ — sticking coefficient - $(1-s)$ — re-emission probability ### 7.3 Step Coverage Dependencies **Sticking coefficient effect:** $$ SC \approx \frac{1}{1 + \frac{s \cdot AR}{2}} $$ **Key observations:** - Low $s$ → better step coverage - High AR → poorer step coverage - ALD achieves ~100% SC due to self-limiting chemistry ### 7.4 Aspect Ratio Dependent Deposition (ARDD) **Local loading effect:** - Reactant depletion in features - Aspect ratio dependent etch (ARDE) analog **Modeling approach:** $$ R_{dep}(z) = R_0 \cdot \frac{C(z)}{C_0} $$ Where: - $z$ — depth into feature - $C(z)$ — local concentration (decreases with depth) ## 8. Thermal Modeling ### 8.1 Heat Transfer Mechanisms **Conduction (Fourier's law):** $$ \mathbf{q}_{cond} = -k \nabla T $$ **Convection:** $$ q_{conv} = h (T_s - T_{\infty}) $$ Where: - $h$ — heat transfer coefficient $\left[\text{W/m}^2 \cdot \text{K}\right]$ **Radiation (Stefan-Boltzmann):** $$ q_{rad} = \varepsilon \sigma (T_s^4 - T_{surr}^4) $$ Where: - $\varepsilon$ — emissivity $\left[0 \leq \varepsilon \leq 1\right]$ - $\sigma$ — Stefan-Boltzmann constant $= 5.67 \times 10^{-8} \, \text{W/m}^2 \cdot \text{K}^4$ ### 8.2 Wafer Temperature Uniformity **Temperature non-uniformity impact:** For reaction-limited regime: $$ \frac{\Delta R}{R} \approx \frac{E_a}{R T^2} \Delta T $$ **Example calculation:** For $E_a = 1.5 \, \text{eV}$, $T = 900 \, \text{K}$, $\Delta T = 5 \, \text{K}$: $$ \frac{\Delta R}{R} \approx \frac{1.5 \times 1.6 \times 10^{-19}}{1.38 \times 10^{-23} \times (900)^2} \times 5 \approx 10.7\% $$ ### 8.3 Susceptor Design Considerations - **Material:** SiC, graphite, quartz - **Heating:** Resistive, inductive, lamp (RTP) - **Rotation:** Improves azimuthal uniformity - **Edge effects:** Guard rings, pocket design ## 9. Validation and Calibration ### 9.1 Experimental Characterization Techniques | Technique | Measurement | Resolution | |-----------|-------------|------------| | Ellipsometry | Thickness, optical constants | ~0.1 nm | | XRF | Composition, thickness | ~1% | | RBS | Composition, depth profile | ~10 nm | | SIMS | Trace impurities | ppb | | AFM | Surface morphology | ~0.1 nm (z) | | SEM/TEM | Cross-section profile | ~1 nm | | XRD | Crystallinity, stress | — | ### 9.2 Model Calibration Approach **Parameter estimation:** Minimize objective function: $$ \chi^2 = \sum_i \left( \frac{y_i^{exp} - y_i^{model}}{\sigma_i} \right)^2 $$ Where: - $y_i^{exp}$ — experimental measurement - $y_i^{model}$ — model prediction - $\sigma_i$ — measurement uncertainty **Sensitivity analysis:** $$ S_{ij} = \frac{\partial y_i}{\partial p_j} \cdot \frac{p_j}{y_i} $$ Where: - $S_{ij}$ — normalized sensitivity of output $i$ to parameter $j$ - $p_j$ — model parameter ### 9.3 Uncertainty Quantification **Parameter uncertainty propagation:** $$ \text{Var}(y) = \sum_j \left( \frac{\partial y}{\partial p_j} \right)^2 \text{Var}(p_j) $$ **Monte Carlo approach:** - Sample parameter distributions - Run multiple model evaluations - Statistical analysis of outputs ## 10. Modern Developments ### 10.1 Machine Learning Integration **Applications:** - **Surrogate models:** Neural networks trained on simulation data - **Process optimization:** Bayesian optimization, genetic algorithms - **Virtual metrology:** Predict film properties from process data - **Defect prediction:** Correlate conditions with yield **Neural network surrogate:** $$ \hat{y} = f_{NN}(\mathbf{x}; \mathbf{w}) $$ Where: - $\mathbf{x}$ — input process parameters - $\mathbf{w}$ — trained network weights - $\hat{y}$ — predicted output (rate, uniformity, etc.) ### 10.2 Digital Twins **Components:** - Real-time sensor data integration - Physics-based + data-driven models - Predictive capabilities **Applications:** - Chamber matching - Predictive maintenance - Run-to-run control - Virtual experiments ### 10.3 Advanced Materials **Emerging challenges:** - **High-k dielectrics:** HfO₂, ZrO₂ via ALD - **2D materials:** Graphene, MoS₂, WS₂ - **Selective deposition:** Area-selective ALD - **3D integration:** Through-silicon vias (TSV) - **New precursors:** Lower temperature, higher purity ### 10.4 Computational Advances - **GPU acceleration:** Faster CFD solvers - **Cloud computing:** Large parameter studies - **Multiscale coupling:** Seamless reactor-to-feature modeling - **Real-time simulation:** For process control ## Physical Constants | Constant | Symbol | Value | |----------|--------|-------| | Boltzmann constant | $k_B$ | $1.381 \times 10^{-23} \, \text{J/K}$ | | Universal gas constant | $R$ | $8.314 \, \text{J/mol} \cdot \text{K}$ | | Avogadro's number | $N_A$ | $6.022 \times 10^{23} \, \text{mol}^{-1}$ | | Stefan-Boltzmann constant | $\sigma$ | $5.67 \times 10^{-8} \, \text{W/m}^2 \cdot \text{K}^4$ | | Elementary charge | $e$ | $1.602 \times 10^{-19} \, \text{C}$ | ## Typical Process Parameters ### B.1 LPCVD Polysilicon - **Precursor:** SiH₄ - **Temperature:** $580 - 650 \, °\text{C}$ - **Pressure:** $0.2 - 1.0 \, \text{Torr}$ - **Deposition rate:** $5 - 20 \, \text{nm/min}$ ### B.2 PECVD Silicon Nitride - **Precursors:** SiH₄ + NH₃ or SiH₄ + N₂ - **Temperature:** $250 - 400 \, °\text{C}$ - **Pressure:** $1 - 5 \, \text{Torr}$ - **RF Power:** $0.1 - 1 \, \text{W/cm}^2$ ### B.3 ALD Hafnium Oxide - **Precursors:** HfCl₄ or TEMAH + H₂O or O₃ - **Temperature:** $200 - 350 \, °\text{C}$ - **GPC:** $\sim 1 \, \text{Å/cycle}$ - **Cycle time:** $2 - 10 \, \text{s}$
# CVD Modeling in Semiconductor Manufacturing ## 1. Introduction Chemical Vapor Deposition (CVD) is a critical thin-film deposition technique in semiconductor manufacturing. Gaseous precursors are introduced into a reaction chamber where they undergo chemical reactions to deposit solid films on heated substrates. ### 1.1 Key Process Steps - **Transport** of reactants from bulk gas to the substrate surface - **Gas-phase chemistry** including precursor decomposition and intermediate formation - **Surface reactions** involving adsorption, surface diffusion, and reaction - **Film nucleation and growth** with specific microstructure evolution - **Byproduct desorption** and transport away from the surface ### 1.2 Common CVD Types - **APCVD** — Atmospheric Pressure CVD - **LPCVD** — Low Pressure CVD (0.1–10 Torr) - **PECVD** — Plasma Enhanced CVD - **MOCVD** — Metal-Organic CVD - **ALD** — Atomic Layer Deposition - **HDPCVD** — High Density Plasma CVD ## 2. Governing Equations ### 2.1 Continuity Equation (Mass Conservation) $$ \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0 $$ Where: - $\rho$ — gas density $\left[\text{kg/m}^3\right]$ - $\mathbf{u}$ — velocity vector $\left[\text{m/s}\right]$ - $t$ — time $\left[\text{s}\right]$ ### 2.2 Momentum Equation (Navier-Stokes) $$ \rho \left( \frac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u} \right) = -\nabla p + \mu \nabla^2 \mathbf{u} + \rho \mathbf{g} $$ Where: - $p$ — pressure $\left[\text{Pa}\right]$ - $\mu$ — dynamic viscosity $\left[\text{Pa} \cdot \text{s}\right]$ - $\mathbf{g}$ — gravitational acceleration $\left[\text{m/s}^2\right]$ ### 2.3 Species Conservation Equation $$ \frac{\partial (\rho Y_i)}{\partial t} + \nabla \cdot (\rho \mathbf{u} Y_i) = \nabla \cdot (\rho D_i \nabla Y_i) + R_i $$ Where: - $Y_i$ — mass fraction of species $i$ $\left[\text{dimensionless}\right]$ - $D_i$ — diffusion coefficient of species $i$ $\left[\text{m}^2/\text{s}\right]$ - $R_i$ — net production rate from reactions $\left[\text{kg/m}^3 \cdot \text{s}\right]$ ### 2.4 Energy Conservation Equation $$ \rho c_p \left( \frac{\partial T}{\partial t} + \mathbf{u} \cdot \nabla T \right) = \nabla \cdot (k \nabla T) + Q $$ Where: - $c_p$ — specific heat capacity $\left[\text{J/kg} \cdot \text{K}\right]$ - $T$ — temperature $\left[\text{K}\right]$ - $k$ — thermal conductivity $\left[\text{W/m} \cdot \text{K}\right]$ - $Q$ — volumetric heat source $\left[\text{W/m}^3\right]$ ### 2.5 Key Dimensionless Numbers | Number | Definition | Physical Meaning | |--------|------------|------------------| | Reynolds | $Re = \frac{\rho u L}{\mu}$ | Inertial vs. viscous forces | | Péclet | $Pe = \frac{u L}{D}$ | Convection vs. diffusion | | Damköhler | $Da = \frac{k_s L}{D}$ | Reaction rate vs. transport rate | | Knudsen | $Kn = \frac{\lambda}{L}$ | Mean free path vs. length scale | Where: - $L$ — characteristic length $\left[\text{m}\right]$ - $\lambda$ — mean free path $\left[\text{m}\right]$ - $k_s$ — surface reaction rate constant $\left[\text{m/s}\right]$ ## 3. Chemical Kinetics ### 3.1 Arrhenius Equation The temperature dependence of reaction rate constants follows: $$ k = A \exp\left(-\frac{E_a}{R T}\right) $$ Where: - $k$ — rate constant $\left[\text{varies}\right]$ - $A$ — pre-exponential factor $\left[\text{same as } k\right]$ - $E_a$ — activation energy $\left[\text{J/mol}\right]$ - $R$ — universal gas constant $= 8.314 \, \text{J/mol} \cdot \text{K}$ ### 3.2 Gas-Phase Reactions **Example: Silane Pyrolysis** $$ \text{SiH}_4 \xrightarrow{k_1} \text{SiH}_2 + \text{H}_2 $$ $$ \text{SiH}_2 + \text{SiH}_4 \xrightarrow{k_2} \text{Si}_2\text{H}_6 $$ **General reaction rate expression:** $$ r_j = k_j \prod_{i} C_i^{\nu_{ij}} $$ Where: - $r_j$ — rate of reaction $j$ $\left[\text{mol/m}^3 \cdot \text{s}\right]$ - $C_i$ — concentration of species $i$ $\left[\text{mol/m}^3\right]$ - $\nu_{ij}$ — stoichiometric coefficient of species $i$ in reaction $j$ ### 3.3 Surface Reaction Kinetics #### 3.3.1 Hertz-Knudsen Impingement Flux $$ J = \frac{p}{\sqrt{2 \pi m k_B T}} $$ Where: - $J$ — molecular flux $\left[\text{molecules/m}^2 \cdot \text{s}\right]$ - $p$ — partial pressure $\left[\text{Pa}\right]$ - $m$ — molecular mass $\left[\text{kg}\right]$ - $k_B$ — Boltzmann constant $= 1.381 \times 10^{-23} \, \text{J/K}$ #### 3.3.2 Surface Reaction Rate $$ R_s = s \cdot J = s \cdot \frac{p}{\sqrt{2 \pi m k_B T}} $$ Where: - $s$ — sticking coefficient $\left[0 \leq s \leq 1\right]$ #### 3.3.3 Langmuir-Hinshelwood Kinetics For surface reaction between two adsorbed species: $$ r = \frac{k \, K_A \, K_B \, p_A \, p_B}{(1 + K_A p_A + K_B p_B)^2} $$ Where: - $K_A, K_B$ — adsorption equilibrium constants $\left[\text{Pa}^{-1}\right]$ - $p_A, p_B$ — partial pressures of reactants A and B $\left[\text{Pa}\right]$ #### 3.3.4 Eley-Rideal Mechanism For reaction between adsorbed species and gas-phase species: $$ r = \frac{k \, K_A \, p_A \, p_B}{1 + K_A p_A} $$ ### 3.4 Common CVD Reaction Systems - **Silicon from Silane:** - $\text{SiH}_4 \rightarrow \text{Si}_{(s)} + 2\text{H}_2$ - **Silicon Dioxide from TEOS:** - $\text{Si(OC}_2\text{H}_5\text{)}_4 + 12\text{O}_2 \rightarrow \text{SiO}_2 + 8\text{CO}_2 + 10\text{H}_2\text{O}$ - **Silicon Nitride from DCS:** - $3\text{SiH}_2\text{Cl}_2 + 4\text{NH}_3 \rightarrow \text{Si}_3\text{N}_4 + 6\text{HCl} + 6\text{H}_2$ - **Tungsten from WF₆:** - $\text{WF}_6 + 3\text{H}_2 \rightarrow \text{W}_{(s)} + 6\text{HF}$ ## 4. Process Regimes ### 4.1 Transport-Limited Regime **Characteristics:** - High Damköhler number: $Da \gg 1$ - Surface reactions are fast - Deposition rate controlled by mass transport - Sensitive to: - Flow patterns - Temperature gradients - Reactor geometry **Deposition rate expression:** $$ R_{dep} \approx \frac{D \cdot C_{\infty}}{\delta} $$ Where: - $C_{\infty}$ — bulk gas concentration $\left[\text{mol/m}^3\right]$ - $\delta$ — boundary layer thickness $\left[\text{m}\right]$ ### 4.2 Reaction-Limited Regime **Characteristics:** - Low Damköhler number: $Da \ll 1$ - Plenty of reactants at surface - Rate controlled by surface kinetics - Strong Arrhenius temperature dependence - Better step coverage in features **Deposition rate expression:** $$ R_{dep} \approx k_s \cdot C_s \approx k_s \cdot C_{\infty} $$ Where: - $k_s$ — surface reaction rate constant $\left[\text{m/s}\right]$ - $C_s$ — surface concentration $\approx C_{\infty}$ $\left[\text{mol/m}^3\right]$ ### 4.3 Regime Transition The transition occurs when: $$ Da = \frac{k_s \delta}{D} \approx 1 $$ **Practical implications:** - **Transport-limited:** Optimize flow, temperature uniformity - **Reaction-limited:** Optimize temperature, precursor chemistry - **Mixed regime:** Most complex to control and model ## 5. Multiscale Modeling ### 5.1 Scale Hierarchy | Scale | Length | Time | Methods | |-------|--------|------|---------| | Reactor | cm – m | s – min | CFD, FEM | | Feature | nm – μm | ms – s | Level set, Monte Carlo | | Surface | nm | μs – ms | KMC | | Atomistic | Å | fs – ps | MD, DFT | ### 5.2 Reactor-Scale Modeling **Governing physics:** - Coupled Navier-Stokes + species + energy equations - Multicomponent diffusion (Stefan-Maxwell) - Chemical source terms **Stefan-Maxwell diffusion:** $$ \nabla x_i = \sum_{j \neq i} \frac{x_i x_j}{D_{ij}} (\mathbf{u}_j - \mathbf{u}_i) $$ Where: - $x_i$ — mole fraction of species $i$ - $D_{ij}$ — binary diffusion coefficient $\left[\text{m}^2/\text{s}\right]$ **Common software:** - ANSYS Fluent - COMSOL Multiphysics - OpenFOAM (open-source) - Silvaco Victory Process - Synopsys Sentaurus ### 5.3 Feature-Scale Modeling **Key phenomena:** - Knudsen diffusion in high-aspect-ratio features - Molecular re-emission and reflection - Surface reaction probability - Film profile evolution **Knudsen diffusion coefficient:** $$ D_K = \frac{d}{3} \sqrt{\frac{8 k_B T}{\pi m}} $$ Where: - $d$ — feature width $\left[\text{m}\right]$ **Effective diffusivity (transition regime):** $$ \frac{1}{D_{eff}} = \frac{1}{D_{mol}} + \frac{1}{D_K} $$ **Level set method for surface tracking:** $$ \frac{\partial \phi}{\partial t} + v_n |\nabla \phi| = 0 $$ Where: - $\phi$ — level set function (zero at surface) - $v_n$ — surface normal velocity (deposition rate) ### 5.4 Atomistic Modeling **Density Functional Theory (DFT):** - Calculate binding energies - Determine activation barriers - Predict reaction pathways **Kinetic Monte Carlo (KMC):** - Stochastic surface evolution - Event rates from Arrhenius: $$ \Gamma_i = \nu_0 \exp\left(-\frac{E_i}{k_B T}\right) $$ Where: - $\Gamma_i$ — rate of event $i$ $\left[\text{s}^{-1}\right]$ - $\nu_0$ — attempt frequency $\sim 10^{12} - 10^{13} \, \text{s}^{-1}$ - $E_i$ — activation energy for event $i$ $\left[\text{eV}\right]$ ## 6. CVD Process Variants ### 6.1 LPCVD (Low Pressure CVD) **Operating conditions:** - Pressure: $0.1 - 10 \, \text{Torr}$ - Temperature: $400 - 900 \, °\text{C}$ - Hot-wall reactor design **Advantages:** - Better uniformity (longer mean free path) - Good step coverage - High purity films **Applications:** - Polysilicon gates - Silicon nitride (Si₃N₄) - Thermal oxides ### 6.2 PECVD (Plasma Enhanced CVD) **Additional physics:** - Electron impact reactions - Ion bombardment - Radical chemistry - Plasma sheath dynamics **Electron density equation:** $$ \frac{\partial n_e}{\partial t} + \nabla \cdot \boldsymbol{\Gamma}_e = S_e $$ Where: - $n_e$ — electron density $\left[\text{m}^{-3}\right]$ - $\boldsymbol{\Gamma}_e$ — electron flux $\left[\text{m}^{-2} \cdot \text{s}^{-1}\right]$ - $S_e$ — electron source term (ionization - recombination) **Electron energy distribution:** Often non-Maxwellian, requiring solution of Boltzmann equation or two-temperature models. **Advantages:** - Lower deposition temperatures ($200 - 400 \, °\text{C}$) - Higher deposition rates - Tunable film stress ### 6.3 ALD (Atomic Layer Deposition) **Process characteristics:** - Self-limiting surface reactions - Sequential precursor pulses - Sub-monolayer control **Growth per cycle:** $$ \text{GPC} = \frac{\Delta t}{\text{cycle}} $$ Typically: $\text{GPC} \approx 0.5 - 2 \, \text{Å/cycle}$ **Surface coverage model:** $$ \theta = \theta_{sat} \left(1 - e^{-\sigma J t}\right) $$ Where: - $\theta$ — surface coverage $\left[0 \leq \theta \leq 1\right]$ - $\theta_{sat}$ — saturation coverage - $\sigma$ — reaction cross-section $\left[\text{m}^2\right]$ - $t$ — exposure time $\left[\text{s}\right]$ **Applications:** - High-k gate dielectrics (HfO₂, ZrO₂) - Barrier layers (TaN, TiN) - Conformal coatings in 3D structures ### 6.4 MOCVD (Metal-Organic CVD) **Precursors:** - Metal-organic compounds (e.g., TMGa, TMAl, TMIn) - Hydrides (AsH₃, PH₃, NH₃) **Key challenges:** - Parasitic gas-phase reactions - Particle formation - Precise composition control **Applications:** - III-V semiconductors (GaAs, InP, GaN) - LEDs and laser diodes - High-electron-mobility transistors (HEMTs) ## 7. Step Coverage Modeling ### 7.1 Definition **Step coverage (SC):** $$ SC = \frac{t_{bottom}}{t_{top}} \times 100\% $$ Where: - $t_{bottom}$ — film thickness at feature bottom - $t_{top}$ — film thickness at feature top **Aspect ratio (AR):** $$ AR = \frac{H}{W} $$ Where: - $H$ — feature depth - $W$ — feature width ### 7.2 Ballistic Transport Model For molecular flow in features ($Kn > 1$): **View factor approach:** $$ F_{i \rightarrow j} = \frac{A_j \cos\theta_i \cos\theta_j}{\pi r_{ij}^2} $$ **Flux balance at surface element:** $$ J_i = J_{direct} + \sum_j (1-s) J_j F_{j \rightarrow i} $$ Where: - $s$ — sticking coefficient - $(1-s)$ — re-emission probability ### 7.3 Step Coverage Dependencies **Sticking coefficient effect:** $$ SC \approx \frac{1}{1 + \frac{s \cdot AR}{2}} $$ **Key observations:** - Low $s$ → better step coverage - High AR → poorer step coverage - ALD achieves ~100% SC due to self-limiting chemistry ### 7.4 Aspect Ratio Dependent Deposition (ARDD) **Local loading effect:** - Reactant depletion in features - Aspect ratio dependent etch (ARDE) analog **Modeling approach:** $$ R_{dep}(z) = R_0 \cdot \frac{C(z)}{C_0} $$ Where: - $z$ — depth into feature - $C(z)$ — local concentration (decreases with depth) ## 8. Thermal Modeling ### 8.1 Heat Transfer Mechanisms **Conduction (Fourier's law):** $$ \mathbf{q}_{cond} = -k \nabla T $$ **Convection:** $$ q_{conv} = h (T_s - T_{\infty}) $$ Where: - $h$ — heat transfer coefficient $\left[\text{W/m}^2 \cdot \text{K}\right]$ **Radiation (Stefan-Boltzmann):** $$ q_{rad} = \varepsilon \sigma (T_s^4 - T_{surr}^4) $$ Where: - $\varepsilon$ — emissivity $\left[0 \leq \varepsilon \leq 1\right]$ - $\sigma$ — Stefan-Boltzmann constant $= 5.67 \times 10^{-8} \, \text{W/m}^2 \cdot \text{K}^4$ ### 8.2 Wafer Temperature Uniformity **Temperature non-uniformity impact:** For reaction-limited regime: $$ \frac{\Delta R}{R} \approx \frac{E_a}{R T^2} \Delta T $$ **Example calculation:** For $E_a = 1.5 \, \text{eV}$, $T = 900 \, \text{K}$, $\Delta T = 5 \, \text{K}$: $$ \frac{\Delta R}{R} \approx \frac{1.5 \times 1.6 \times 10^{-19}}{1.38 \times 10^{-23} \times (900)^2} \times 5 \approx 10.7\% $$ ### 8.3 Susceptor Design Considerations - **Material:** SiC, graphite, quartz - **Heating:** Resistive, inductive, lamp (RTP) - **Rotation:** Improves azimuthal uniformity - **Edge effects:** Guard rings, pocket design ## 9. Validation and Calibration ### 9.1 Experimental Characterization Techniques | Technique | Measurement | Resolution | |-----------|-------------|------------| | Ellipsometry | Thickness, optical constants | ~0.1 nm | | XRF | Composition, thickness | ~1% | | RBS | Composition, depth profile | ~10 nm | | SIMS | Trace impurities | ppb | | AFM | Surface morphology | ~0.1 nm (z) | | SEM/TEM | Cross-section profile | ~1 nm | | XRD | Crystallinity, stress | — | ### 9.2 Model Calibration Approach **Parameter estimation:** Minimize objective function: $$ \chi^2 = \sum_i \left( \frac{y_i^{exp} - y_i^{model}}{\sigma_i} \right)^2 $$ Where: - $y_i^{exp}$ — experimental measurement - $y_i^{model}$ — model prediction - $\sigma_i$ — measurement uncertainty **Sensitivity analysis:** $$ S_{ij} = \frac{\partial y_i}{\partial p_j} \cdot \frac{p_j}{y_i} $$ Where: - $S_{ij}$ — normalized sensitivity of output $i$ to parameter $j$ - $p_j$ — model parameter ### 9.3 Uncertainty Quantification **Parameter uncertainty propagation:** $$ \text{Var}(y) = \sum_j \left( \frac{\partial y}{\partial p_j} \right)^2 \text{Var}(p_j) $$ **Monte Carlo approach:** - Sample parameter distributions - Run multiple model evaluations - Statistical analysis of outputs ## 10. Modern Developments ### 10.1 Machine Learning Integration **Applications:** - **Surrogate models:** Neural networks trained on simulation data - **Process optimization:** Bayesian optimization, genetic algorithms - **Virtual metrology:** Predict film properties from process data - **Defect prediction:** Correlate conditions with yield **Neural network surrogate:** $$ \hat{y} = f_{NN}(\mathbf{x}; \mathbf{w}) $$ Where: - $\mathbf{x}$ — input process parameters - $\mathbf{w}$ — trained network weights - $\hat{y}$ — predicted output (rate, uniformity, etc.) ### 10.2 Digital Twins **Components:** - Real-time sensor data integration - Physics-based + data-driven models - Predictive capabilities **Applications:** - Chamber matching - Predictive maintenance - Run-to-run control - Virtual experiments ### 10.3 Advanced Materials **Emerging challenges:** - **High-k dielectrics:** HfO₂, ZrO₂ via ALD - **2D materials:** Graphene, MoS₂, WS₂ - **Selective deposition:** Area-selective ALD - **3D integration:** Through-silicon vias (TSV) - **New precursors:** Lower temperature, higher purity ### 10.4 Computational Advances - **GPU acceleration:** Faster CFD solvers - **Cloud computing:** Large parameter studies - **Multiscale coupling:** Seamless reactor-to-feature modeling - **Real-time simulation:** For process control ## Physical Constants | Constant | Symbol | Value | |----------|--------|-------| | Boltzmann constant | $k_B$ | $1.381 \times 10^{-23} \, \text{J/K}$ | | Universal gas constant | $R$ | $8.314 \, \text{J/mol} \cdot \text{K}$ | | Avogadro's number | $N_A$ | $6.022 \times 10^{23} \, \text{mol}^{-1}$ | | Stefan-Boltzmann constant | $\sigma$ | $5.67 \times 10^{-8} \, \text{W/m}^2 \cdot \text{K}^4$ | | Elementary charge | $e$ | $1.602 \times 10^{-19} \, \text{C}$ | ## Typical Process Parameters ### B.1 LPCVD Polysilicon - **Precursor:** SiH₄ - **Temperature:** $580 - 650 \, °\text{C}$ - **Pressure:** $0.2 - 1.0 \, \text{Torr}$ - **Deposition rate:** $5 - 20 \, \text{nm/min}$ ### B.2 PECVD Silicon Nitride - **Precursors:** SiH₄ + NH₃ or SiH₄ + N₂ - **Temperature:** $250 - 400 \, °\text{C}$ - **Pressure:** $1 - 5 \, \text{Torr}$ - **RF Power:** $0.1 - 1 \, \text{W/cm}^2$ ### B.3 ALD Hafnium Oxide - **Precursors:** HfCl₄ or TEMAH + H₂O or O₃ - **Temperature:** $200 - 350 \, °\text{C}$ - **GPC:** $\sim 1 \, \text{Å/cycle}$ - **Cycle time:** $2 - 10 \, \text{s}$
Scatter light off defects for enhanced detection.
Manufacturing date marking.
# Semiconductor Manufacturing: Dielectric Deposition Process (DDP) Modeling ## Overview **DDP (Dielectric Deposition Process)** refers to the set of techniques used to deposit insulating films in semiconductor fabrication. Dielectric materials serve critical functions: - **Gate dielectrics** — $\text{SiO}_2$, high-$\kappa$ materials like $\text{HfO}_2$ - **Interlayer dielectrics (ILD)** — isolating metal interconnect layers - **Spacer dielectrics** — defining transistor gate dimensions - **Passivation layers** — protecting finished devices - **Hard masks** — etch selectivity during patterning ## Dielectric Deposition Methods ### Primary Techniques | Method | Full Name | Temperature Range | Typical Applications | |--------|-----------|-------------------|---------------------| | **PECVD** | Plasma-Enhanced CVD | $200-400°C$ | $\text{SiO}_2$, $\text{SiN}_x$ for ILD, passivation | | **LPCVD** | Low-Pressure CVD | $400-800°C$ | High-quality $\text{Si}_3\text{N}_4$, poly-Si | | **HDPCVD** | High-Density Plasma CVD | $300-450°C$ | Gap-fill for trenches and vias | | **ALD** | Atomic Layer Deposition | $150-350°C$ | Ultra-thin gate dielectrics ($\text{HfO}_2$, $\text{Al}_2\text{O}_3$) | | **Thermal Oxidation** | — | $800-1200°C$ | Gate oxide ($\text{SiO}_2$) | | **Spin-on** | SOG/SOD | $100-400°C$ | Planarization layers | ### Selection Criteria - **Conformality requirements** — ALD > LPCVD > PECVD - **Thermal budget** — PECVD/ALD for low-$T$, thermal oxidation for high-quality - **Throughput** — CVD methods faster than ALD - **Film quality** — Thermal > LPCVD > PECVD generally ## Physics of Dielectric Deposition Modeling ### Fundamental Transport Equations Modeling dielectric deposition requires solving coupled partial differential equations for mass, momentum, and energy transport. #### Mass Transport (Species Concentration) $$ \frac{\partial C}{\partial t} + \nabla \cdot (\mathbf{v}C) = D\nabla^2 C + R $$ Where: - $C$ — species concentration $[\text{mol/m}^3]$ - $\mathbf{v}$ — velocity field $[\text{m/s}]$ - $D$ — diffusion coefficient $[\text{m}^2/\text{s}]$ - $R$ — reaction rate $[\text{mol/m}^3 \cdot \text{s}]$ #### Energy Balance $$ \rho C_p \left(\frac{\partial T}{\partial t} + \mathbf{v} \cdot \nabla T\right) = k\nabla^2 T + Q $$ Where: - $\rho$ — density $[\text{kg/m}^3]$ - $C_p$ — specific heat capacity $[\text{J/kg} \cdot \text{K}]$ - $k$ — thermal conductivity $[\text{W/m} \cdot \text{K}]$ - $Q$ — heat generation rate $[\text{W/m}^3]$ #### Momentum Balance (Navier-Stokes) $$ \rho\left(\frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v}\right) = -\nabla p + \mu \nabla^2 \mathbf{v} + \rho \mathbf{g} $$ Where: - $p$ — pressure $[\text{Pa}]$ - $\mu$ — dynamic viscosity $[\text{Pa} \cdot \text{s}]$ - $\mathbf{g}$ — gravitational acceleration $[\text{m/s}^2]$ ### Surface Reaction Kinetics #### Arrhenius Rate Expression $$ k = A \exp\left(-\frac{E_a}{RT}\right) $$ Where: - $k$ — rate constant - $A$ — pre-exponential factor - $E_a$ — activation energy $[\text{J/mol}]$ - $R$ — gas constant $= 8.314 \, \text{J/mol} \cdot \text{K}$ - $T$ — temperature $[\text{K}]$ #### Langmuir Adsorption Isotherm (for ALD) $$ \theta = \frac{K \cdot p}{1 + K \cdot p} $$ Where: - $\theta$ — fractional surface coverage $(0 \leq \theta \leq 1)$ - $K$ — equilibrium adsorption constant - $p$ — partial pressure of adsorbate #### Sticking Coefficient $$ S = S_0 \cdot (1 - \theta)^n \cdot \exp\left(-\frac{E_a}{RT}\right) $$ Where: - $S$ — sticking coefficient (probability of adsorption) - $S_0$ — initial sticking coefficient - $n$ — reaction order ### Plasma Modeling (PECVD/HDPCVD) #### Electron Energy Distribution Function (EEDF) For non-Maxwellian plasmas, the Druyvesteyn distribution: $$ f(\varepsilon) = C \cdot \varepsilon^{1/2} \exp\left(-\left(\frac{\varepsilon}{\bar{\varepsilon}}\right)^2\right) $$ Where: - $\varepsilon$ — electron energy $[\text{eV}]$ - $\bar{\varepsilon}$ — mean electron energy - $C$ — normalization constant #### Ion Bombardment Energy $$ E_{ion} = e \cdot V_{sheath} + \frac{1}{2}m_{ion}v_{Bohm}^2 $$ Where: - $V_{sheath}$ — plasma sheath voltage - $v_{Bohm} = \sqrt{\frac{k_B T_e}{m_{ion}}}$ — Bohm velocity #### Radical Generation Rate $$ R_{radical} = n_e \cdot n_{gas} \cdot \langle \sigma v \rangle $$ Where: - $n_e$ — electron density $[\text{m}^{-3}]$ - $n_{gas}$ — neutral gas density - $\langle \sigma v \rangle$ — rate coefficient (energy-averaged cross-section × velocity) ## Feature-Scale Modeling ### Critical Phenomena in High Aspect Ratio Structures Modern semiconductor devices require filling trenches and vias with aspect ratios (AR) exceeding 50:1. #### Knudsen Number $$ Kn = \frac{\lambda}{d} $$ Where: - $\lambda$ — mean free path of gas molecules - $d$ — characteristic feature dimension | Regime | Knudsen Number | Transport Type | |--------|---------------|----------------| | Continuum | $Kn < 0.01$ | Viscous flow | | Slip | $0.01 < Kn < 0.1$ | Transition | | Transition | $0.1 < Kn < 10$ | Mixed | | Free molecular | $Kn > 10$ | Ballistic/Knudsen | #### Mean Free Path Calculation $$ \lambda = \frac{k_B T}{\sqrt{2} \pi d_m^2 p} $$ Where: - $d_m$ — molecular diameter $[\text{m}]$ - $p$ — pressure $[\text{Pa}]$ ### Step Coverage Model $$ SC = \frac{t_{sidewall}}{t_{top}} \times 100\% $$ For diffusion-limited deposition: $$ SC \approx \frac{1}{\sqrt{1 + AR^2}} $$ For reaction-limited deposition: $$ SC \approx 1 - \frac{S \cdot AR}{2} $$ Where: - $S$ — sticking coefficient - $AR$ — aspect ratio = depth/width ### Void Formation Criterion Void formation occurs when: $$ \frac{d(thickness_{sidewall})}{dz} > \frac{w(z)}{2 \cdot t_{total}} $$ Where: - $w(z)$ — feature width at depth $z$ - $t_{total}$ — total deposition time ## Film Properties to Model ### Structural Properties - **Thickness uniformity**: $$ U = \frac{t_{max} - t_{min}}{t_{max} + t_{min}} \times 100\% $$ - **Film stress** (Stoney equation): $$ \sigma_f = \frac{E_s t_s^2}{6(1-\nu_s)t_f} \cdot \frac{1}{R} $$ Where: - $E_s$, $\nu_s$ — substrate Young's modulus and Poisson ratio - $t_s$, $t_f$ — substrate and film thickness - $R$ — radius of curvature - **Density from refractive index** (Lorentz-Lorenz): $$ \frac{n^2 - 1}{n^2 + 2} = \frac{4\pi}{3} N \alpha $$ Where $N$ is molecular density and $\alpha$ is polarizability ### Electrical Properties - **Dielectric constant** (capacitance method): $$ \kappa = \frac{C \cdot t}{\varepsilon_0 \cdot A} $$ - **Breakdown field**: $$ E_{BD} = \frac{V_{BD}}{t} $$ - **Leakage current density** (Fowler-Nordheim tunneling): $$ J = \frac{q^3 E^2}{8\pi h \phi_B} \exp\left(-\frac{8\pi\sqrt{2m^*}\phi_B^{3/2}}{3qhE}\right) $$ Where: - $E$ — electric field - $\phi_B$ — barrier height - $m^*$ — effective electron mass ## Multiscale Modeling Hierarchy ### Scale Linking Framework ``` ┌─────────────────────────────────────────────────────────────────────┐ │ ATOMISTIC (Å-nm) MESOSCALE (nm-μm) CONTINUUM │ │ ───────────────── ────────────────── (μm-mm) │ │ ────────── │ │ • DFT calculations • Kinetic Monte Carlo • CFD │ │ • Molecular Dynamics • Level-set methods • FEM │ │ • Ab initio MD • Cellular automata • TCAD │ │ │ │ Outputs: Outputs: Outputs: │ │ • Binding energies • Film morphology • Flow │ │ • Reaction barriers • Growth rate • T, C │ │ • Diffusion coefficients • Surface roughness • Profiles │ └─────────────────────────────────────────────────────────────────────┘ ``` ### DFT Calculations Solve the Kohn-Sham equations: $$ \left[-\frac{\hbar^2}{2m}\nabla^2 + V_{eff}(\mathbf{r})\right]\psi_i(\mathbf{r}) = \varepsilon_i \psi_i(\mathbf{r}) $$ Where: $$ V_{eff} = V_{ext} + V_H + V_{xc} $$ - $V_{ext}$ — external potential (nuclei) - $V_H$ — Hartree potential (electron-electron) - $V_{xc}$ — exchange-correlation potential ### Kinetic Monte Carlo (kMC) Event selection probability: $$ P_i = \frac{k_i}{\sum_j k_j} $$ Time advancement: $$ \Delta t = -\frac{\ln(r)}{\sum_j k_j} $$ Where $r$ is a random number $\in (0,1]$ ## Specific Process Examples ### PECVD $\text{SiO}_2$ from TEOS #### Overall Reaction $$ \text{Si(OC}_2\text{H}_5\text{)}_4 + 12\text{O}^* \xrightarrow{\text{plasma}} \text{SiO}_2 + 8\text{CO}_2 + 10\text{H}_2\text{O} $$ #### Key Process Parameters | Parameter | Typical Range | Effect | |-----------|--------------|--------| | RF Power | $100-1000 \, \text{W}$ | ↑ Power → ↑ Density, ↓ Dep rate | | Pressure | $0.5-5 \, \text{Torr}$ | ↑ Pressure → ↑ Dep rate, ↓ Conformality | | Temperature | $300-400°C$ | ↑ Temp → ↑ Density, ↓ H content | | TEOS:O₂ ratio | $1:5$ to $1:20$ | Affects stoichiometry, quality | #### Deposition Rate Model $$ R_{dep} = k_0 \cdot p_{TEOS}^a \cdot p_{O_2}^b \cdot \exp\left(-\frac{E_a}{RT}\right) $$ Typical values: $a \approx 0.5$, $b \approx 0.3$, $E_a \approx 0.3 \, \text{eV}$ ### ALD High-$\kappa$ Dielectrics ($\text{HfO}_2$) #### Half-Reactions **Cycle A (Metal precursor):** $$ \text{Hf(N(CH}_3\text{)}_2\text{)}_4\text{(g)} + \text{*-OH} \rightarrow \text{*-O-Hf(N(CH}_3\text{)}_2\text{)}_3 + \text{HN(CH}_3\text{)}_2 $$ **Cycle B (Oxidizer):** $$ \text{*-O-Hf(N(CH}_3\text{)}_2\text{)}_3 + 2\text{H}_2\text{O} \rightarrow \text{*-O-Hf(OH)}_3 + 3\text{HN(CH}_3\text{)}_2 $$ #### Growth Per Cycle (GPC) $$ \text{GPC} = \frac{\theta_{sat} \cdot \rho_{site} \cdot M_{HfO_2}}{\rho_{HfO_2} \cdot N_A} $$ Typical GPC for $\text{HfO}_2$: $0.8-1.2 \, \text{Å/cycle}$ #### ALD Window ``` ┌────────────────────────────┐ GPC │ ┌──────────────┐ │ (Å/ │ /│ │\ │ cycle) │ / │ ALD │ \ │ │ / │ WINDOW │ \ │ │ / │ │ \ │ │/ │ │ \ │ └─────┴──────────────┴─────┴─┘ T_min T_max Temperature (°C) ``` Below $T_{min}$: Condensation, incomplete reactions Above $T_{max}$: Precursor decomposition, CVD-like behavior ### HDPCVD Gap Fill #### Deposition-Etch Competition Net deposition rate: $$ R_{net}(z) = R_{dep}(\theta) - R_{etch}(E_{ion}, \theta) $$ Where: - $R_{dep}(\theta)$ — angular-dependent deposition rate - $R_{etch}$ — ion-enhanced etch rate - $\theta$ — angle from surface normal #### Sputter Yield (Yamamura Formula) $$ Y(E, \theta) = Y_0(E) \cdot f(\theta) $$ Where: $$ f(\theta) = \cos^{-f}\theta \cdot \exp\left[-\Sigma(\cos^{-1}\theta - 1)\right] $$ ## Machine Learning Applications ### Virtual Metrology **Objective:** Predict film properties from in-situ sensor data without destructive measurement. $$ \hat{y} = f_{ML}(\mathbf{x}_{sensors}, \mathbf{x}_{recipe}) $$ Where: - $\hat{y}$ — predicted property (thickness, stress, etc.) - $\mathbf{x}_{sensors}$ — OES, pressure, RF power signals - $\mathbf{x}_{recipe}$ — setpoints and timing ### Gaussian Process Regression $$ y(\mathbf{x}) \sim \mathcal{GP}\left(m(\mathbf{x}), k(\mathbf{x}, \mathbf{x}')\right) $$ Posterior mean prediction: $$ \mu(\mathbf{x}^*) = \mathbf{k}^T(\mathbf{K} + \sigma_n^2\mathbf{I})^{-1}\mathbf{y} $$ Uncertainty quantification: $$ \sigma^2(\mathbf{x}^*) = k(\mathbf{x}^*, \mathbf{x}^*) - \mathbf{k}^T(\mathbf{K} + \sigma_n^2\mathbf{I})^{-1}\mathbf{k} $$ ### Bayesian Optimization for Recipe Development **Acquisition function** (Expected Improvement): $$ \text{EI}(\mathbf{x}) = \mathbb{E}\left[\max(f(\mathbf{x}) - f^+, 0)\right] $$ Where $f^+$ is the best observed value. ## Advanced Node Challenges (Sub-5nm) ### Critical Challenges | Challenge | Technical Details | Modeling Complexity | |-----------|------------------|---------------------| | **Ultra-high AR** | 3D NAND: 100+ layers, AR > 50:1 | Knudsen transport, ballistic modeling | | **Atomic precision** | Gate dielectrics: 1-2 nm | Monolayer-level control, quantum effects | | **Low-$\kappa$ integration** | $\kappa < 2.5$ porous films | Mechanical integrity, plasma damage | | **Selective deposition** | Area-selective ALD | Nucleation control, surface chemistry | | **Thermal budget** | BEOL: $< 400°C$ | Kinetic limitations, precursor chemistry | ### Equivalent Oxide Thickness (EOT) For high-$\kappa$ gate stacks: $$ \text{EOT} = t_{IL} + \frac{\kappa_{SiO_2}}{\kappa_{high-k}} \cdot t_{high-k} $$ Where: - $t_{IL}$ — interfacial layer thickness - $\kappa_{SiO_2} = 3.9$ - Typical high-$\kappa$: $\kappa_{HfO_2} \approx 20-25$ ### Low-$\kappa$ Dielectric Design Effective dielectric constant: $$ \kappa_{eff} = \kappa_{matrix} \cdot (1 - p) + \kappa_{air} \cdot p $$ Where $p$ is porosity fraction. Target for advanced nodes: $\kappa_{eff} < 2.0$ ## Tools and Software ### Commercial TCAD - **Synopsys Sentaurus Process** — full process simulation - **Silvaco Victory Process** — alternative TCAD suite - **Lam Research SEMulator3D** — 3D topography simulation ### Multiphysics Platforms - **COMSOL Multiphysics** — coupled PDE solving - **Ansys Fluent** — CFD for reactor design - **Ansys CFX** — alternative CFD solver ### Specialized Tools - **CHEMKIN** (Ansys) — gas-phase reaction kinetics - **Reaction Design** — combustion and plasma chemistry - **Custom Monte Carlo codes** — feature-scale simulation ### Open Source Options - **OpenFOAM** — CFD framework - **LAMMPS** — molecular dynamics - **Quantum ESPRESSO** — DFT calculations - **SPARTA** — DSMC for rarefied gas dynamics ## Summary Dielectric deposition modeling in semiconductor manufacturing integrates: 1. **Transport phenomena** — mass, momentum, energy conservation 2. **Reaction kinetics** — surface and gas-phase chemistry 3. **Plasma physics** — for PECVD/HDPCVD processes 4. **Feature-scale physics** — conformality, void formation 5. **Multiscale approaches** — atomistic to continuum 6. **Machine learning** — for optimization and virtual metrology The goal is predicting and optimizing film properties based on process parameters while accounting for the extreme topography of modern semiconductor devices.
Separate temporarily bonded wafers.
Measure trap levels.
High-aspect-ratio etching for TSV.
Spatial distribution of defects.
Automated optical or e-beam inspection to find particles scratches defects.
High-resolution follow-up of detected defects.