sac alloy, sac, packaging
Common lead-free solder.
92 technical terms and definitions
Common lead-free solder.
Self-aligned patterning using spacers as mentioned earlier.
Ready specimens for analysis.
Test subset of dies.
# Silicon Valley ## I. The Geographic-Industrial Network Model ### 1.1 Spatial Concentration Function The entities form a **weighted directed graph** $G(V, E)$ where: - **Vertices ($V$)**: Companies, institutions, infrastructure, and communities - **Edges ($E$)**: Economic flows, talent pipelines, supply chains, and geographic proximity The innovation density at any point can be modeled as a **Gaussian kernel density function**: $$ \rho(x,y) = \sum_{i=1}^{n} w_i \cdot \exp\left(-\frac{\|p - p_i\|^2}{2\sigma^2}\right) $$ Where: - $\rho(x,y)$ = innovation density at coordinate $(x,y)$ - $w_i$ = weight (market cap, employee count) of company $i$ - $p_i$ = location vector of company $i$ - $\sigma$ = decay parameter for agglomeration effects - $n$ = total number of entities in the network ### 1.2 Network Centrality Metrics For each node $v$ in the ecosystem: **Degree Centrality:** $$ C_D(v) = \frac{\deg(v)}{n-1} $$ **Betweenness Centrality:** $$ C_B(v) = \sum_{s \neq v \neq t} \frac{\sigma_{st}(v)}{\sigma_{st}} $$ Where $\sigma_{st}$ is the total number of shortest paths from node $s$ to node $t$, and $\sigma_{st}(v)$ is the number of those paths passing through $v$. ## II. Semiconductor Players ### 2.1 Company Location Matrix | Company | HQ Address | Founded | Core Business | Market Cap Tier | |---------|-----------|---------|---------------|-----------------| | **AMAT** | 3050 Bowers Avenue, Santa Clara | 1967 | Fab Equipment | Large Cap | | **Intel** | 2200 Mission College Blvd, Santa Clara | 1968 | CPU/Foundry | Large Cap | | **AMD** | 2485 Augustine Drive, Santa Clara | 1969 | CPU/GPU | Large Cap | | **NVIDIA** | 2788 San Tomas Expressway, Santa Clara | 1993 | GPU/AI | Mega Cap | | **Palo Alto Networks** | 3000 Tannery Way, Santa Clara | 2005 | Cybersecurity | Large Cap | ### 2.2 Semiconductor Value Chain Layers ``` - ┌─────────────────────────────────────────────────────────────┐ │ LAYER 1: EQUIPMENT │ │ │ │ AMAT (CVD, PVD, Etch, CMP) ← Bowers Avenue │ │ • Second largest semiconductor equipment supplier │ │ • Enables all downstream chip fabrication │ └─────────────────────────────────────────────────────────────┘ │ ▼ ┌─────────────────────────────────────────────────────────────┐ │ LAYER 2: CHIP DESIGN │ │ │ │ Intel │ AMD │ NVIDIA │ │ (CPU) │ (CPU/GPU) │ (GPU/AI) │ │ │ │ │ │ Mission │ Augustine │ San Tomas │ │ College │ Drive │ Expressway │ └─────────────────────────────────────────────────────────────┘ │ ▼ ┌─────────────────────────────────────────────────────────────┐ │ LAYER 3: SYSTEMS │ │ │ │ Apple (Cupertino) │ Google (Mountain View) │ Meta (MPK) │ │ │ │ ← Consumers of chips from Layer 2 → │ └─────────────────────────────────────────────────────────────┘ ``` ### 2.3 Market Share For company $i$ in market segment $m$: $$ S_i^{(m)} = \frac{R_i^{(m)}}{\sum_{j=1}^{N} R_j^{(m)}} $$ Where: - $S_i^{(m)}$ = market share of company $i$ in segment $m$ - $R_i^{(m)}$ = revenue of company $i$ in segment $m$ - $N$ = total number of competitors **NVIDIA GPU Market Dominance (2025):** $$ S_{\text{NVIDIA}}^{(\text{discrete GPU})} = 0.92 \quad \text{(92\% market share)} $$ ## III. The Magnificent Seven Analysis ### 3.1 Composition The "Magnificent 7" stocks comprise: 1. **Apple** (AAPL) - Cupertino, CA 2. **Microsoft** (MSFT) - Redmond, WA 3. **Alphabet/Google** (GOOGL) - Mountain View, CA 4. **Amazon** (AMZN) - Seattle, WA 5. **Meta** (META) - Menlo Park, CA 6. **NVIDIA** (NVDA) - Santa Clara, CA ⭐ 7. **Tesla** (TSLA) - Austin, TX ### 3.2 S&P 500 Concentration As of January 2026: $$ W_{\text{Mag7}} = \frac{\sum_{i=1}^{7} \text{MarketCap}_i}{\text{Total S\&P 500 MarketCap}} = 0.344 \quad \text{(34.4\%)} $$ **Historical Growth (2015-2025):** $$ \text{Return}_{\text{Mag7}} = 870.1\% \quad \text{vs} \quad \text{Return}_{\text{S\&P500}} = 247.9\% $$ ### 3.3 Silicon Valley Mag 7 Presence | Company | Distance from Santa Clara | Relationship | |---------|---------------------------|--------------| | Apple | ~6 miles (Cupertino) | Adjacent city | | Google | ~8 miles (Mountain View) | Adjacent city | | Meta | ~15 miles (Menlo Park) | Same county cluster | | NVIDIA | **0 miles (Santa Clara HQ)** | **Headquartered** | ## IV. Thermal Engineering and Packaging ### 4.1 TEA **Professional Profile:** - **Position**: President, Thermal Engineering Associates Inc. (TEA) - **Credentials**: IEEE Fellow, IMAPS Fellow - **Education**: - B.Sc. Mechanical Engineering - Tsinghua University - MBA - San Jose State University - Ph.D. Materials - University of Oxford - **Location**: San Jose, California ### 4.2 Thermal Management Equations **Maximum Power Dissipation:** $$ P_{\max} = \frac{T_{\text{junction}} - T_{\text{ambient}}}{R_{\theta}} $$ Where: - $P_{\max}$ = maximum power dissipation (Watts) - $T_{\text{junction}}$ = junction temperature (°C) - $T_{\text{ambient}}$ = ambient temperature (°C) - $R_{\theta}$ = thermal resistance (°C/W) **Junction Temperature Model:** $$ T_j = T_a + P \cdot (R_{\theta_{jc}} + R_{\theta_{cs}} + R_{\theta_{sa}}) $$ Where: - $R_{\theta_{jc}}$ = junction-to-case thermal resistance - $R_{\theta_{cs}}$ = case-to-sink thermal resistance - $R_{\theta_{sa}}$ = sink-to-ambient thermal resistance ### 4.3 Power Density Scaling Challenge As transistor density follows Moore's Law: $$ n(t) = n_0 \cdot 2^{t/\tau} $$ Where $\tau \approx 2$ years, power density scales as: $$ P_D(t) = \frac{P(t)}{A} \propto 2^{t/\tau} $$ This exponential growth creates the **thermal management bottleneck** that TEA's thermal test chips (TTCs) address. ## V. Transportation ### 5.1 Key Expressways | Expressway | Orientation | Key Connections | |------------|-------------|-----------------| | **Lawrence Expressway** | North-South | Links Sunnyvale parks to Santa Clara | | **Central Expressway** | East-West | Core tech corridor access | | **San Tomas Expressway** | North-South | NVIDIA HQ corridor | | **Bowers Avenue** | North-South | AMAT, Intel adjacent areas | ### 5.2 Accessibility Function Network accessibility at location $x$: $$ A(x) = \sum_{j=1}^{n} O_j \cdot f(c_{xj}) $$ Where: - $A(x)$ = accessibility at location $x$ - $O_j$ = opportunities (jobs, amenities) at destination $j$ - $f(c_{xj})$ = impedance function of travel cost/time from $x$ to $j$ **Common Impedance Functions:** - **Inverse power**: $f(c) = c^{-\beta}$ - **Negative exponential**: $f(c) = e^{-\beta c}$ - **Gaussian**: $f(c) = e^{-\beta c^2}$ ### 5.3 Commute Time Distribution For commute time $T$ in the Santa Clara tech corridor: $$ f(T) = \frac{1}{\sigma\sqrt{2\pi}} \exp\left(-\frac{(T - \mu)^2}{2\sigma^2}\right) $$ With parameters: - $\mu \approx 25$ minutes (average commute) - $\sigma \approx 12$ minutes (standard deviation) ## VI. Semiconductor Companies ### 6.1 Texas Instruments (TI) in Santa Clara **Key Locations:** - **3833 Kifer Road** - Former campus (sold to Fortinet, $192M) - **4555 Great America Parkway** - Current lease (~205,000 sq ft) - **2900 Semiconductor Drive** - TI Silicon Valley Labs **Historical Significance:** - First commercial silicon transistor (1954) - Jack Kilby invented integrated circuit (1958) - TI Silicon Valley Labs established (2012) ### 6.2 Fujitsu in Sunnyvale **Location:** 1250 East Arques Avenue, Sunnyvale **Timeline:** - **1979**: Founded Fujitsu Electronics America - **2020**: Lane Partners acquired 26.3-acre campus - **2025**: Ingrasys Technology USA purchased for $128M ### 6.3 Company Evolution Model Probability of company survival after $t$ years: $$ P(\text{survive} > t) = e^{-\lambda t} $$ Where $\lambda$ = failure rate (approximately 0.05-0.10 for tech startups) ## VII. Santa Clara University ### 7.1 School of Engineering Profile | Attribute | Value | |-----------|-------| | **Founded** | 1912 | | **Location** | 500 El Camino Real, Santa Clara | | **Programs** | 8 undergraduate, 12 master's, 3 Ph.D. | | **Student-Faculty Ratio** | 10:1 | | **Top Employers** | Google, Apple, Cisco, Tesla, Intel | ### 7.2 Talent Flow Differential Equation $$ \frac{dE}{dt} = \lambda \cdot G(t) - \mu \cdot E(t) + \sigma \cdot I(t) $$ Where: - $E(t)$ = employed engineers at time $t$ - $G(t)$ = university graduates per year - $I(t)$ = immigration influx - $\lambda$ = hiring rate coefficient - $\mu$ = attrition rate coefficient - $\sigma$ = immigration employment rate **Steady State Solution:** At equilibrium $\frac{dE}{dt} = 0$: $$ E^* = \frac{\lambda G + \sigma I}{\mu} $$ ## VIII. Innovation ### 8.1 Regional Innovation Production Function $$ I(t) = A \cdot K(t)^\alpha \cdot L(t)^\beta \cdot R(t)^\gamma \cdot N(t)^\delta $$ Where: - $I(t)$ = innovation output (patents, startups, products) - $A$ = total factor productivity - $K(t)$ = capital (VC funding, R&D investment) - $L(t)$ = labor (engineers, researchers) - $R(t)$ = research institutions capacity - $N(t)$ = network effects (proximity spillovers) - $\alpha + \beta + \gamma + \delta = 1$ (constant returns to scale) ### 8.2 Venture Capital Concentration $$ \text{VC}_{\text{SV}} = \frac{\text{Silicon Valley VC Investment}}{\text{Total US VC Investment}} \approx 0.41 \quad \text{(41\%)} $$ ### 8.3 Knowledge Spillover Function Knowledge decay with distance: $$ K(d) = K_0 \cdot e^{-\gamma d} $$ Where: - $K(d)$ = knowledge spillover at distance $d$ - $K_0$ = knowledge at source - $\gamma$ = decay rate (higher in tech clusters) ## IX. Community ### 9.1 Residential & Retail Nodes **Apartments:** - Oak Brooks Apartment - Station 101 Apartment - Rieley Square Apartment - Halford Garden Apartments **Retail/Grocery:** - Han Kook Supermarket (Korean market) - FootMaxx Supermarket - Costco (Lawrence Expressway area) **Parks:** - Ponderosa Park - Central Park (Santa Clara) ### 9.2 Housing Affordability Index $$ \text{HAI} = \frac{\text{Median Household Income}}{\text{Income Required for Median Home}} \times 100 $$ For Santa Clara County: $$ \text{HAI}_{\text{SCC}} \approx 65-75 $$ (Below 100 indicates affordability challenges) ### 9.3 Residential Attractor Function $$ R(x) = f(\text{wage premium}) \cdot g(\text{housing cost}) \cdot h(\text{amenities}) $$ Where: $$ f(w) = w^\alpha, \quad g(c) = c^{-\beta}, \quad h(a) = \log(1 + a) $$ ## X. Mathematical Network Diagram ### 10.1 Ecosystem Graph Representation ``` - SEMICONDUCTOR VALUE CHAIN │ ┌────────────────────────┼────────────────────────┐ │ │ │ ▼ ▼ ▼ ┌─────────────┐ ┌─────────────┐ ┌─────────────┐ │ AMAT │ │ TI │ │ Fujitsu │ │ Equipment │ │ Analog │ │ Systems │ │ (Bowers) │ │ (Kifer) │ │ (Arques) │ └──────┬──────┘ └──────┬──────┘ └──────┬──────┘ │ │ │ └────────────────────────┼────────────────────────┘ │ ▼ ┌──────────────────────────────────────────────────────────────┐ │ CHIP DESIGNERS │ │ │ │ ┌─────────┐ ┌─────────┐ ┌─────────────┐ │ │ │ Intel │ │ AMD │ │ NVIDIA │ │ │ │ Mission │ │Augustine│ │ San Tomas │ │ │ │ College │ │ Drive │ │ Expressway │ │ │ └────┬────┘ └────┬────┘ └──────┬──────┘ │ │ │ │ │ │ └───────┼────────────────┼──────────────────┼──────────────────┘ │ │ │ └────────────────┼──────────────────┘ │ ▼ ┌──────────────────────────────────────────────────────────────┐ │ MAGNIFICENT 7 LAYER │ │ │ │ Apple Google Meta NVIDIA* │ │ (Cupertino) (Mtn View) (Menlo Pk) (Santa Clara) │ │ │ │ * NVIDIA appears in both chip design AND Mag 7 │ └──────────────────────────┬───────────────────────────────────┘ │ ▼ ┌──────────────────────────────────────────────────────────────┐ │ SUPPORTING ECOSYSTEM │ │ │ │ ┌──────────────┐ ┌──────────────┐ ┌──────────────┐ │ │ │ Thermal │ │ Cybersec │ │ Education │ │ │ │ Engineering │ │ (PAN) │ │ (SCU) │ │ │ │ (TEA) │ │ Tannery Way │ │ El Camino Rl │ │ │ └──────────────┘ └──────────────┘ └──────────────┘ │ │ │ └──────────────────────────────────────────────────────────────┘ ``` ### 10.2 Adjacency Matrix For $n$ key nodes, the weighted adjacency matrix $\mathbf{A}$: $$ \mathbf{A} = \begin{pmatrix} 0 & a_{12} & a_{13} & \cdots & a_{1n} \\ a_{21} & 0 & a_{23} & \cdots & a_{2n} \\ a_{31} & a_{32} & 0 & \cdots & a_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & a_{n3} & \cdots & 0 \end{pmatrix} $$ Where $a_{ij}$ = strength of connection (supply chain, talent flow, proximity) between nodes $i$ and $j$. ## XI. Statistics ### 11.1 Key Metrics | Metric | Value | Source | |--------|-------|--------| | Mag 7 S&P 500 Weight | 34.4% | Jan 2026 | | NVIDIA GPU Market Share | 92% | Q1 2025 | | SV Venture Capital Share | 41% | Q1 2023 | | SCU Student-Faculty Ratio | 10:1 | 2024 | | Santa Clara County Median Home | >$1.5M | 2024 | ### 11.2 Growth Rates **NVIDIA Revenue Growth:** $$ \text{CAGR}_{\text{NVDA}} = \left(\frac{R_{\text{2024}}}{R_{\text{2020}}}\right)^{1/4} - 1 \approx 0.69 \quad \text{(69\% YoY)} $$ **Mag 7 10-Year Return:** $$ r_{\text{10yr}} = \frac{V_{\text{2025}} - V_{\text{2015}}}{V_{\text{2015}}} = 8.701 \quad \text{(870.1\%)} $$ ## XII. Conclusion: Self-Reinforcing System Dynamics ### 12.1 Positive Feedback Loop $$ \text{Innovation} \rightarrow \text{Jobs} \rightarrow \text{Talent Influx} \rightarrow \text{More Innovation} $$ Mathematically: $$ \frac{dI}{dt} = k \cdot I \cdot (1 - \frac{I}{I_{\max}}) $$ This **logistic growth model** captures: - Initial exponential growth - Eventual saturation at carrying capacity $I_{\max}$ ### 12.2 Agglomeration Economies Benefits scale superlinearly with city size: $$ Y = Y_0 \cdot N^\beta \quad \text{where } \beta > 1 $$ For innovation-driven economies like Santa Clara: $$ \beta \approx 1.15 - 1.25 $$ ### 12.3 Ecosystem Value Function $$ V_{\text{ecosystem}} = \int_0^\infty \sum_{i=1}^{n} w_i(t) \cdot e^{-rt} \, dt $$ Where: - $V_{\text{ecosystem}}$ = total ecosystem value (NPV) - $w_i(t)$ = value contribution of entity $i$ at time $t$ - $r$ = discount rate - $n$ = number of ecosystem participants
Ensure scanners have consistent overlay performance.
Lithography tool that scans the reticle and wafer synchronously for exposure.
Map doping profiles.
Image surfaces with electrons.
Spatially-resolved work function.
Microwave-based electrical measurement.
Optical imaging beyond diffraction limit.
Family including AFM and STM.
Map resistivity with nano-scale resolution.
Automated particle detection.
Atomic-resolution surface imaging.
Type of SRAF between main features.
Optical CD using diffraction patterns.
Use scatterometry for overlay measurement.
Analyze diffraction pattern from periodic structures to extract CD profile.
Non-product wafer used for testing or qualification.
Tests in kerf between dies.
Number of wafers to stabilize tool.
Stabilize tool after maintenance.
Elemental depth profiling.
Preferentially remove one material.
Solder specific areas.
Ability to measure target in presence of interferences.
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.
Response per unit concentration.
Long meandering resistor for testing.
First wafers after recipe change.
Measure resistance uniformity across wafer.
Tests using partial process.
Reduced pitch SOP.
Ratio of signal to noise.
Identify characteristic failure patterns.
Silicon substrate connecting multiple dies.
Silicon layer on insulator.
Conductive die attach.
Depth profile of elemental composition by sputtering and analyzing ions.
One row of pins.
Process one wafer at a time for tight control.
Clean rinse and dry one wafer at a time for tighter control.
Flatness within small measurement site.