self-supervised depth and ego-motion, 3d vision
Learn without ground truth.
1,106 technical terms and definitions
Learn without ground truth.
Learn disentangled representations without labels.
Self-supervised graph neural networks learn representations through pretext tasks without labeled data.
Learn normal patterns to detect anomalies.
Learn depth without labels.
DINO MAE BEiT methods.
Asynchronous circuits less sensitive to variation.
Self-training is a semi-supervised method where models generate pseudo-labels for unlabeled data to augment training sets iteratively.
Use model predictions on unlabeled data as training labels.
Model checks its own answers.
Alternative molecular representation.
Self-normalizing activation.
Semantic attention in heterogeneous GNNs learns importance weights for different metapaths or relation types.
Semantic caching matches semantically similar queries to cached results.
Split based on semantic coherence.
Semantic chunking creates segments based on topic boundaries rather than fixed size.
Search by meaning not keywords.
Semantic segmentation maps guide generation controlling regional content.
Find meaningful directions in latent space.
Semantic editing manipulates specific attributes in latent space for targeted modifications.
Object boundaries emerge without labels.
Encode meaning.
Microsoft's SDK for orchestrating AI plugins.
Semantic Kernel is Microsoft SDK for AI orchestration. Plugins, planners. Enterprise focus.
Semantic memory stores factual knowledge and learned concepts without temporal tags.
Convert natural language to formal representations.
Identify who did what to whom.
Search based on meaning similarity using embeddings rather than keyword matching.
Classify each pixel as defect type.
Predict similarity scores.
Semantic similarity retrieves demonstrations similar to test inputs for in-context learning.
SLAM with object-level understanding.
SLAM with object-level understanding.
Transfer style to specific objects.
Semantic-aware metapath attention learns which metapaths are relevant for specific prediction tasks.
Semi-autonomous agents handle routine tasks but request help for complex decisions.
Generate blocks of tokens.
Generate multiple tokens per step.
Semi-damascene processes pattern metal lines then fill spaces with dielectric requiring CMP planarization.
Use limited target labels.
Boom-and-bust cycles in chip demand and prices.
Approaches to maintaining fabrication tools.
Material movement and storage.
# 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 |
Fit simulation models to experimental data for accuracy.
For semiconductor process questions (etch, deposition, etc.), I can outline process flows, key parameters, and physical intuition.
SendGrid provides email API. Transactional, marketing.
Sensitivity analysis examines how input variations affect output predictions.
Determine which parameters most affect outcome.
Response per unit concentration.