high-resolution generation, generative models
Create images beyond training resolution.
285 technical terms and definitions
Create images beyond training resolution.
Faster moisture removal.
Rapidly evaluate many candidates.
Full-scale production.
Lead-free requires higher temp.
Higher-order GNNs increase expressiveness by aggregating information from k-tuples of nodes rather than individuals.
Extreme stress to find limits.
Highly accelerated life testing uses extreme stress to precipitate failures quickly.
Screen production units.
Severe temperature humidity stress.
Severe environmental test.
Gated skip connections.
Copper protrusions from stress.
Learn from failed attempts.
Hindsight Experience Replay relabels failed trajectories with achieved goals as successes improving sparse reward learning.
Student learns from teacher's intermediate layers.
Ensure text processing complies with privacy.
Hierarchical Reinforcement Learning with Off-policy correction trains goal-setting and goal-achieving policies jointly.
Histograms display frequency distributions revealing shape center and spread.
Device behavior depends on previous state.
Whether any relevant doc retrieved.
Hidden Markov Models for time series assume observations generated by unobserved discrete states transitioning stochastically.
Graph-based algorithm for approximate nearest neighbor search.
Graph-based ANN index.
Hierarchical Navigable Small World creates graph-based indices for approximate nearest neighbor search.
ANN algorithm.
HNSW is graph-based ANN algorithm. Fast, accurate.
Hold release resumes processing after resolving hold conditions.
Hold slack ensures data stability after clock edge preventing hold time violations.
Voltage across conducting ESD device.
Holt-Winters method extends exponential smoothing to capture level trend and seasonality in time series forecasting.
# Sam Zeloof and the Mathematics of DIY Semiconductor Fabrication ## Table of Contents ## The Remarkable Story Sam Zeloof represents something extraordinary in the semiconductor world: a self-taught engineer who, starting at age 17 in 2016, built a functional chip fabrication facility in his parents' New Jersey garage—located about 30 miles from Bell Labs, where the first transistor was created in 1947. ### Timeline of Achievements - **2016**: Started experimenting with semiconductor fabrication at age 17 - **2017**: Successfully replicated Jeri Ellsworth's homemade transistors - **2018**: Produced the **Z1 chip** (first homebrew lithographically fabricated IC) - 6 transistors - PMOS dual differential amplifier - 175μm feature size - **2021**: Created the **Z2 chip** - 1,200 transistors (12 arrays of 100 transistors each) - 10μm polysilicon gate process - Same technology as Intel's first CPU (4004) - Threshold voltage: $V_{th} = 1.1V$ - Leakage current: $I_{leak} = 59nA$ - **2022**: Co-founded **Atomic Semi** with Jim Keller - **Present**: Photolithography resolution down to ~300nm ### Key Equipment (Salvaged/Homemade) - Modified digital projector with microscope optical stage for photolithography - Repaired electron microscope (purchased broken for $1,000) - Homemade vacuum chamber from surplus parts - Chemistry bench for wet processing - Fiber laser for wafer scribing ## Device Physics: Fundamental Semiconductor Equations The behavior of semiconductor devices is governed by a coupled system of partial differential equations describing electrostatic potential, carrier concentrations, and current flow. ### Poisson's Equation Describes the electrostatic potential distribution: $$ \nabla \cdot (\epsilon \nabla \psi) = -q(p - n + N_D^+ - N_A^-) $$ **Where:** - $\psi$ = electrostatic potential (V) - $\epsilon$ = permittivity (F/cm) - $q$ = elementary charge ($1.6 \times 10^{-19}$ C) - $p$ = hole concentration (cm⁻³) - $n$ = electron concentration (cm⁻³) - $N_D^+$ = ionized donor concentration (cm⁻³) - $N_A^-$ = ionized acceptor concentration (cm⁻³) ### Continuity Equations Describe carrier transport and conservation: **For electrons:** $$ \frac{\partial n}{\partial t} = \frac{1}{q}\nabla \cdot \vec{J}_n - R $$ **For holes:** $$ \frac{\partial p}{\partial t} = -\frac{1}{q}\nabla \cdot \vec{J}_p - R $$ **Where:** - $\vec{J}_n$, $\vec{J}_p$ = electron and hole current densities (A/cm²) - $R$ = net recombination rate (cm⁻³s⁻¹) ### Drift-Diffusion Current Relations **Electron current density:** $$ \vec{J}_n = q\mu_n n\vec{E} + qD_n\nabla n $$ **Hole current density:** $$ \vec{J}_p = q\mu_p p\vec{E} - qD_p\nabla p $$ **Where:** - $\mu_n$, $\mu_p$ = electron and hole mobilities (cm²/V·s) - $D_n$, $D_p$ = diffusion coefficients (cm²/s) - $\vec{E}$ = electric field (V/cm) **Einstein Relation** (connects mobility and diffusion): $$ D = \frac{kT}{q}\mu $$ ## MOSFET Device Equations Zeloof's PMOS transistors follow the classic MOSFET equations. ### Operating Regions #### 1. Cutoff Region $$ V_{GS} < V_T \quad \Rightarrow \quad I_{DS} \approx 0 $$ #### 2. Linear (Triode) Region **Condition:** $V_{GS} > V_T$ and $V_{DS} < V_{GS} - V_T$ $$ I_{DS} = \mu_n C_{ox} \frac{W}{L} \left[ (V_{GS} - V_T)V_{DS} - \frac{V_{DS}^2}{2} \right] $$ #### 3. Saturation Region **Condition:** $V_{GS} > V_T$ and $V_{DS} \geq V_{GS} - V_T$ $$ I_{DS} = \frac{1}{2}\mu_n C_{ox}\frac{W}{L}(V_{GS} - V_T)^2(1 + \lambda V_{DS}) $$ ### Parameter Definitions | Parameter | Symbol | Description | Units | |-----------|--------|-------------|-------| | Threshold voltage | $V_T$ | Gate voltage for channel formation | V | | Oxide capacitance | $C_{ox}$ | Gate oxide capacitance per area | F/cm² | | Channel width | $W$ | Transistor width | μm | | Channel length | $L$ | Transistor length | μm | | Channel length modulation | $\lambda$ | Output conductance parameter | V⁻¹ | ### Oxide Capacitance $$ C_{ox} = \frac{\epsilon_{ox}}{t_{ox}} = \frac{K_{ox} \epsilon_0}{t_{ox}} $$ **Where:** - $\epsilon_{ox}$ = oxide permittivity - $t_{ox}$ = oxide thickness - $K_{ox} \approx 3.9$ for SiO₂ - $\epsilon_0 = 8.854 \times 10^{-14}$ F/cm ### Threshold Voltage $$ V_T = V_{FB} + 2\phi_F + \frac{\sqrt{2\epsilon_{Si}qN_A(2\phi_F)}}{C_{ox}} $$ **Where:** - $V_{FB}$ = flat-band voltage - $\phi_F$ = Fermi potential - $N_A$ = acceptor doping concentration ### Transconductance $$ g_m = \frac{\partial I_{DS}}{\partial V_{GS}} = \mu_n C_{ox} \frac{W}{L}(V_{GS} - V_T) $$ In saturation: $$ g_m = \sqrt{2\mu_n C_{ox}\frac{W}{L}I_{DS}} $$ ### Subthreshold Current Below threshold, current varies exponentially: $$ I_{DS} = I_{D0} \exp\left(\frac{V_{GS} - V_T}{nV_T}\right)\left(1 - \exp\left(-\frac{V_{DS}}{V_T}\right)\right) $$ **Subthreshold slope:** $$ S = n \cdot \frac{kT}{q} \cdot \ln(10) \approx 60-100 \text{ mV/decade at room temperature} $$ ## Diffusion: Fick's Laws The doping process relies on thermal diffusion of impurities into silicon. ### Fick's First Law Describes the flux of diffusing species: $$ J = -D\frac{\partial C}{\partial x} $$ **Where:** - $J$ = flux (atoms/cm²·s) - $D$ = diffusion coefficient (cm²/s) - $C$ = concentration (atoms/cm³) ### Fick's Second Law Describes time-dependent concentration profile: $$ \frac{\partial C}{\partial t} = D\frac{\partial^2 C}{\partial x^2} $$ For one dimension with constant $D$. ### Temperature Dependence of Diffusion Coefficient Follows Arrhenius behavior: $$ D = D_0 \exp\left(-\frac{E_a}{kT}\right) $$ **Where:** - $D_0$ = pre-exponential factor (cm²/s) - $E_a$ = activation energy (eV) - $k$ = Boltzmann constant ($8.617 \times 10^{-5}$ eV/K) - $T$ = absolute temperature (K) ### Diffusion Profile Solutions #### 1. Constant Surface Concentration (Predeposition) **Boundary conditions:** - $C(0,t) = C_s$ (surface concentration constant) - $C(\infty,t) = 0$ - $C(x,0) = 0$ **Solution:** $$ C(x,t) = C_s \cdot \text{erfc}\left(\frac{x}{2\sqrt{Dt}}\right) $$ **Where erfc is the complementary error function:** $$ \text{erfc}(z) = 1 - \text{erf}(z) = 1 - \frac{2}{\sqrt{\pi}}\int_0^z e^{-t^2}dt $$ **Total dose:** $$ Q(t) = \frac{2C_s\sqrt{Dt}}{\sqrt{\pi}} $$ #### 2. Constant Total Dopant (Drive-in) **Boundary conditions:** - Total dopant $Q$ is fixed - $\int_0^\infty C(x,t)dx = Q$ **Solution (Gaussian profile):** $$ C(x,t) = \frac{Q}{\sqrt{\pi Dt}}\exp\left(-\frac{x^2}{4Dt}\right) $$ **Junction depth** (where $C(x_j) = C_B$, background concentration): $$ x_j = 2\sqrt{Dt \cdot \ln\left(\frac{Q}{C_B\sqrt{\pi Dt}}\right)} $$ ### Diffusion Coefficients for Common Dopants in Silicon | Dopant | $D_0$ (cm²/s) | $E_a$ (eV) | Type | |--------|---------------|------------|------| | Boron | 10.5 | 3.69 | p-type | | Phosphorus | 10.5 | 3.69 | n-type | | Arsenic | 0.32 | 3.56 | n-type | | Antimony | 5.6 | 3.95 | n-type | ## Oxidation Kinetics: Deal-Grove Model Zeloof grows gate oxide layers through thermal oxidation of silicon. ### Chemical Reaction $$ \text{Si} + \text{O}_2 \rightarrow \text{SiO}_2 \quad \text{(dry oxidation)} $$ $$ \text{Si} + 2\text{H}_2\text{O} \rightarrow \text{SiO}_2 + 2\text{H}_2 \quad \text{(wet oxidation)} $$ ### Deal-Grove Equation $$ x_{ox}^2 + Ax_{ox} = B(t + \tau) $$ **Where:** - $x_{ox}$ = oxide thickness - $A$, $B$ = temperature-dependent constants - $\tau$ = accounts for initial oxide - $t$ = oxidation time ### Limiting Cases #### Linear Regime (thin oxides, $x_{ox} \ll A$): $$ x_{ox} \approx \frac{B}{A}(t + \tau) $$ Rate coefficient: $B/A$ (linear rate constant) #### Parabolic Regime (thick oxides, $x_{ox} \gg A$): $$ x_{ox} \approx \sqrt{B(t + \tau)} $$ Rate coefficient: $B$ (parabolic rate constant) ### Explicit Solution $$ x_{ox} = \frac{A}{2}\left(\sqrt{1 + \frac{4B(t+\tau)}{A^2}} - 1\right) $$ ### Rate Constants (Typical Values at 1000°C) | Parameter | Dry O₂ | Wet O₂ | |-----------|--------|--------| | $B/A$ (μm/hr) | 0.165 | 1.31 | | $B$ (μm²/hr) | 0.0117 | 0.287 | ## Photolithography Resolution Zeloof uses a modified digital projector with a microscope optical stage. ### Rayleigh Criterion Minimum resolvable feature size: $$ \text{Resolution} = k_1 \frac{\lambda}{NA} $$ **Where:** - $k_1$ = process-dependent factor (0.25–0.8) - $\lambda$ = wavelength of light - $NA$ = numerical aperture ### Depth of Focus $$ DOF = k_2 \frac{\lambda}{NA^2} $$ ### Numerical Aperture $$ NA = n \sin(\theta) $$ **Where:** - $n$ = refractive index of medium - $\theta$ = half-angle of light cone ### Zeloof's System Performance - **Light source:** UV from modified projector - **Theoretical resolution:** ~1 μm - **Practical resolution (without cleanroom):** ~10 μm - **Current capability:** ~300 nm ### Exposure Dose $$ E = I \cdot t $$ **Where:** - $E$ = exposure energy (mJ/cm²) - $I$ = intensity (mW/cm²) - $t$ = exposure time (s) ## Moore's Law: Exponential Growth Model ### Mathematical Formulation $$ N(t) = N_0 \cdot 2^{t/\tau} $$ **Where:** - $N(t)$ = transistor count at time $t$ - $N_0$ = initial transistor count - $\tau$ = doubling time (~2 years historically) ### Logarithmic Form $$ \log_2(N) = \log_2(N_0) + \frac{t}{\tau} $$ A straight line on a semi-log plot indicates exponential growth. ### Historical Data | Year | Processor | Transistors | |------|-----------|-------------| | 1971 | Intel 4004 | 2,308 | | 1978 | Intel 8086 | 29,000 | | 1989 | Intel 486 | 1,180,000 | | 2000 | Pentium 4 | 42,000,000 | | 2021 | Apple M1 Max | 57,000,000,000 | | 2025 | NVIDIA GB202 | 92,200,000,000 | ### Zeloof's Personal Moore's Law - **Z1 (2018):** 6 transistors - **Z2 (2021):** 1,200 transistors - **Growth factor:** 200× in ~3 years **Calculating doubling time:** $$ 1200 = 6 \cdot 2^{3/\tau} $$ $$ 200 = 2^{3/\tau} $$ $$ \log_2(200) = \frac{3}{\tau} $$ $$ \tau = \frac{3}{\log_2(200)} = \frac{3}{7.64} \approx 0.39 \text{ years} \approx 5 \text{ months} $$ **Zeloof's doubling time (~5 months) dramatically exceeds the industry standard (~24 months)!** ### Dennard Scaling (Related) As transistors shrink by factor $\kappa$: | Parameter | Scaling | |-----------|---------| | Dimension | $1/\kappa$ | | Voltage | $1/\kappa$ | | Current | $1/\kappa$ | | Delay | $1/\kappa$ | | Power | $1/\kappa^2$ | | Power density | constant | ## Yield Modeling ### Poisson Model (Simple) $$ Y = e^{-AD} $$ **Where:** - $Y$ = yield (fraction of working chips) - $A$ = chip area (cm²) - $D$ = defect density (defects/cm²) ### Murphy's Model (More Realistic) $$ Y = \left(\frac{1 - e^{-AD}}{AD}\right)^2 $$ ### Seeds Model $$ Y = e^{-\sqrt{AD}} $$ ### Zeloof's Yield - **Reported yield:** Up to 80% for large features - **Key factors:** - No cleanroom (increases defect density) - Large feature sizes (increases tolerance) - Small die area (improves yield per chip) - Manual processing (introduces variability) ## Hierarchy of Semiconductor Models ### Level 1: Quantum Models **Schrödinger Equation:** $$ i\hbar\frac{\partial\Psi}{\partial t} = -\frac{\hbar^2}{2m^*}\nabla^2\Psi + V\Psi $$ **Where:** - $\Psi$ = wave function - $\hbar$ = reduced Planck constant - $m^*$ = effective mass - $V$ = potential energy **Applications:** - Tunneling phenomena - Quantum confinement - Band structure calculations ### Level 2: Kinetic Models **Boltzmann Transport Equation:** $$ \frac{\partial f}{\partial t} + \vec{v}\cdot\nabla_r f + \frac{q\vec{E}}{m^*}\cdot\nabla_k f = \left(\frac{\partial f}{\partial t}\right)_{\text{coll}} $$ **Where:** - $f$ = distribution function - $\vec{v}$ = velocity - $\vec{k}$ = wave vector **Applications:** - Hot carrier effects - Non-equilibrium transport - Monte Carlo simulations ### Level 3: Macroscopic Models **Drift-Diffusion Equations** (as described above) **Applications:** - Standard device simulation - Circuit modeling - TCAD tools ### Model Selection Criteria | Model | Speed | Accuracy | Feature Size | |-------|-------|----------|--------------| | Drift-Diffusion | Fast | Good for >50nm | >50 nm | | Energy Balance | Medium | Better | 20-50 nm | | Monte Carlo | Slow | Excellent | <20 nm | | Quantum | Very Slow | Required | <10 nm | ## Atomic Semi: From Garage to Startup ### Company Overview - **Founded:** 2022 - **Co-founders:** Sam Zeloof (CEO) and Jim Keller - **Location:** San Francisco, CA - **Mission:** Build small, fast semiconductor fabs ### Vision - Create fab equipment "like ASML" but smaller scale - Target smaller chips instead of 300mm wafers - Enable rapid prototyping (hours instead of months) - Democratize chip manufacturing ### Investment Interest - OpenAI Startup Fund (seed round discussions) - Valuation: ~$100 million - Notable interested investors: - Fred Ehrsam (Paradigm founder) - Nat Friedman (former GitHub CEO) - Naval Ravikant ### Technical Approach - Miniaturized fab equipment - Custom tooling development - Pushing toward advanced geometries - Focus on small-batch production ## Key Takeaways ### 1. Democratization of Innovation > "That really high barrier to entry will make you super risk-averse, and that's bad for innovation." > — Sam Zeloof The mathematical frameworks underlying chip fabrication aren't secret—they're well-documented physics accessible to anyone willing to learn. ### 2. Scaling Laws Apply at All Levels Whether you're Intel with $100 billion or Zeloof with salvaged equipment, the same equations govern device behavior: - Poisson's equation for electrostatics - Drift-diffusion for current flow - Fick's laws for doping - Deal-Grove for oxidation ### 3. Mathematical Foundation Required Success in semiconductor fabrication requires understanding: - **Partial differential equations** (device physics) - **Error functions and Gaussians** (diffusion profiles) - **Exponential kinetics** (oxidation, diffusion coefficients) - **Statistical models** (yield prediction) - **Optical physics** (lithography resolution) ### 4. Exponential Learning Possible Zeloof's transistor count growth demonstrates that rapid iteration with immediate feedback can produce remarkable scaling—even faster than Moore's Law. ### 5. Historical Perspective The semiconductor revolution started with individuals tinkering in labs. Zeloof's work proves that with sufficient mathematical understanding, creativity, and persistence, the fundamental tools of modern technology can be recreated outside the corporate-industrial complex. ## Physical Constants | Constant | Symbol | Value | Units | |----------|--------|-------|-------| | Elementary charge | $q$ | $1.602 \times 10^{-19}$ | C | | Boltzmann constant | $k$ | $8.617 \times 10^{-5}$ | eV/K | | Planck constant | $h$ | $6.626 \times 10^{-34}$ | J·s | | Vacuum permittivity | $\epsilon_0$ | $8.854 \times 10^{-14}$ | F/cm | | Thermal voltage (300K) | $V_T = kT/q$ | 0.0259 | V | | Intrinsic carrier conc. (Si, 300K) | $n_i$ | $1.5 \times 10^{10}$ | cm⁻³ | | Silicon permittivity | $\epsilon_{Si}$ | $1.04 \times 10^{-12}$ | F/cm | | SiO₂ permittivity | $\epsilon_{ox}$ | $3.45 \times 10^{-13}$ | F/cm |
# Jeri Ellsworth's Homemade Transistors: The Mathematics Behind DIY Semiconductor Manufacturing ## 1. The Pioneer and Her Process **Jeri Ellsworth** is an American entrepreneur and autodidact computer chip designer who demonstrated that functional transistors can be fabricated in a home laboratory setting. ### Key Facts: - **Timeline**: Built home-based fabrication facility in 2010 - **Development Time**: Approximately 2 years to perfect the process - **Achievement**: Successfully created working N-channel FETs - **Scale**: Transistors measured in inches (not micrometers) - **Method**: Single transistor fabrication at a time ### Materials Used: | Component | Source | |-----------|--------| | Silicon Wafers | eBay (p-doped) | | Acid Source | Store-bought rust remover | | Deionized Water | Aquafina (crude substitute) | | Etchant | Hydrofluoric acid (HF) | | Dopant | Phosphosilicate glass | | Masks | Vinyl stickers (cut with vinyl cutter) | ## 2. The Fabrication Process ### 2.1 Process Flow ``` - ┌─────────────────┐ │ Silicon Wafer │ │ (p-doped) │ └────────┬────────┘ │ ▼ ┌─────────────────┐ │ Thermal Oxide │ │ Growth │ │ (500-600 Å) │ └────────┬────────┘ │ ▼ ┌─────────────────┐ │ Vinyl Mask │ │ Application │ └────────┬────────┘ │ ▼ ┌─────────────────┐ │ HF Etching │ │ (Pattern Oxide)│ └────────┬────────┘ │ ▼ ┌─────────────────┐ │ Doping │ │ (Phosphosilicate│ │ Spin-on) │ └────────┬────────┘ │ ▼ ┌─────────────────┐ │ Gate Oxide │ │ Growth │ │ (800-1000 Å) │ └────────┬────────┘ │ ▼ ┌─────────────────┐ │ Contacts │ │(Conductive Epoxy│ └─────────────────┘ ``` ### 2.2 Critical Parameters - **Field Oxide Thickness**: 500-600 Å (green color indicator) - **Gate Oxide Thickness**: 800-1000 Å (pink to dark red color) - **Furnace Temperature**: ~1000°C - **Oxide Growth Time**: ~6 hours with steam ### 2.3 Color-Thickness Correlation | Oxide Thickness (Å) | Observed Color | |---------------------|----------------| | 500-600 | Green | | 800-1000 | Pink to Dark Red | | 1000-1200 | Blue | | 1500-2000 | Gold/Yellow | ## 3. Mathematical Models ### 3.1 Deal-Grove Oxidation Model The **Deal-Grove model** (1965) mathematically describes thermal oxide growth on silicon surfaces. #### 3.1.1 Fundamental Equation $$ x^2 + Ax = B(t + \tau) $$ **Where:** - $x$ = oxide thickness (cm or nm) - $t$ = oxidation time (seconds or minutes) - $A$ = linear rate constant parameter (cm) - $B$ = parabolic rate constant (cm²/s) - $\tau$ = time offset for initial oxide #### 3.1.2 Explicit Solution for Oxide Thickness $$ x(t) = \frac{A}{2}\left(\sqrt{1 + \frac{4B}{A^2}(t + \tau)} - 1\right) $$ #### 3.1.3 Rate Constants The constants $A$ and $B$ are defined as: $$ A = 2D_{ox}\left(\frac{1}{k_s} + \frac{1}{h_g}\right) $$ $$ B = \frac{2D_{ox}C_g}{N} $$ $$ \tau = \frac{x_i^2 + Ax_i}{B} $$ **Where:** - $D_{ox}$ = diffusivity of oxidant in oxide (cm²/s) - $k_s$ = surface reaction rate constant (cm/s) - $h_g$ = gas-phase transport coefficient (cm/s) - $C_g$ = equilibrium oxidant concentration in gas - $N$ = number of oxidant molecules per unit volume of oxide - $x_i$ = initial oxide thickness #### 3.1.4 Limiting Cases **Linear Regime** (thin oxide, short times): When $t + \tau \ll \frac{A^2}{4B}$: $$ x(t) \approx \frac{B}{A}(t + \tau) $$ - Growth rate is **reaction-limited** - Oxide thickness grows linearly with time - Applies to Ellsworth's initial oxide growth **Parabolic Regime** (thick oxide, long times): When $t + \tau \gg \frac{A^2}{4B}$: $$ x(t) \approx \sqrt{B(t + \tau)} $$ - Growth rate is **diffusion-limited** - Oxide thickness grows with square root of time - Applies to Ellsworth's longer oxidation cycles #### 3.1.5 Temperature Dependence The rate constants follow Arrhenius behavior: $$ \frac{B}{A} = C_1 \exp\left(-\frac{E_{a1}}{k_B T}\right) $$ $$ B = C_2 \exp\left(-\frac{E_{a2}}{k_B T}\right) $$ **Where:** - $E_{a1}$ = activation energy for linear rate (~2.0 eV for dry O₂) - $E_{a2}$ = activation energy for parabolic rate (~1.2 eV for dry O₂) - $k_B$ = Boltzmann constant ($8.617 \times 10^{-5}$ eV/K) - $T$ = absolute temperature (K) #### 3.1.6 Wet vs. Dry Oxidation | Parameter | Dry O₂ | Wet (H₂O) | |-----------|--------|-----------| | $B/A$ at 1000°C | 0.0117 μm/hr | 0.226 μm/hr | | $B$ at 1000°C | 0.0011 μm²/hr | 0.287 μm²/hr | | Quality | Higher | Lower | | Growth Rate | Slower | Faster | **Chemical Reactions:** Dry oxidation: $$ \text{Si} + \text{O}_2 \rightarrow \text{SiO}_2 \quad (\Delta H \approx -7.36 \text{ eV/atom}) $$ Wet oxidation: $$ \text{Si} + 2\text{H}_2\text{O} \rightarrow \text{SiO}_2 + 2\text{H}_2 $$ ### 3.2 Fick's Laws of Diffusion Dopant distribution in the source/drain regions follows **Fick's laws**. #### 3.2.1 Fick's First Law $$ J = -D\frac{\partial C}{\partial x} $$ **Where:** - $J$ = diffusion flux (atoms/cm²·s) - $D$ = diffusion coefficient (cm²/s) - $C$ = concentration (atoms/cm³) - $x$ = position (cm) #### 3.2.2 Fick's Second Law $$ \frac{\partial C}{\partial t} = D\frac{\partial^2 C}{\partial x^2} $$ For **concentration-dependent diffusion**: $$ \frac{\partial C}{\partial t} = \frac{\partial}{\partial x}\left(D(C)\frac{\partial C}{\partial x}\right) $$ #### 3.2.3 Diffusion Coefficient Temperature Dependence $$ D = D_0 \exp\left(-\frac{E_a}{k_B T}\right) $$ **Typical Values for Silicon:** | Dopant | $D_0$ (cm²/s) | $E_a$ (eV) | $D$ at 1000°C (cm²/s) | |--------|---------------|------------|----------------------| | Phosphorus (P) | 10.5 | 3.69 | $2.5 \times 10^{-14}$ | | Boron (B) | 10.5 | 3.69 | $2.0 \times 10^{-14}$ | | Arsenic (As) | 0.32 | 3.56 | $1.2 \times 10^{-15}$ | | Antimony (Sb) | 5.6 | 3.95 | $1.0 \times 10^{-15}$ | #### 3.2.4 Solution: Constant Surface Concentration **Boundary Conditions:** - $C(0,t) = C_s$ (constant surface concentration) - $C(x,0) = 0$ (initially undoped) - $C(\infty,t) = 0$ (bulk remains undoped) **Solution:** $$ C(x,t) = C_s \cdot \text{erfc}\left(\frac{x}{2\sqrt{Dt}}\right) $$ **Where erfc is the complementary error function:** $$ \text{erfc}(z) = 1 - \text{erf}(z) = 1 - \frac{2}{\sqrt{\pi}}\int_0^z e^{-u^2}du $$ **Total Dopant per Unit Area:** $$ Q(t) = \int_0^\infty C(x,t)dx = \frac{2C_s\sqrt{Dt}}{\sqrt{\pi}} $$ #### 3.2.5 Solution: Limited Source (Gaussian) **Boundary Conditions:** - Initial dose $Q_0$ deposited at surface - $\int_0^\infty C(x,t)dx = Q_0$ (conservation) **Solution:** $$ C(x,t) = \frac{Q_0}{\sqrt{\pi Dt}} \exp\left(-\frac{x^2}{4Dt}\right) $$ **Peak Concentration (at surface):** $$ C_{max} = C(0,t) = \frac{Q_0}{\sqrt{\pi Dt}} $$ **Characteristic Diffusion Length:** $$ L_D = 2\sqrt{Dt} $$ #### 3.2.6 Junction Depth Calculation For a p-n junction formed by diffusion into oppositely doped substrate: **erfc profile:** $$ x_j = 2\sqrt{Dt} \cdot \text{erfc}^{-1}\left(\frac{C_B}{C_s}\right) $$ **Gaussian profile:** $$ x_j = 2\sqrt{Dt \cdot \ln\left(\frac{Q_0}{C_B\sqrt{\pi Dt}}\right)} $$ **Where:** - $x_j$ = junction depth - $C_B$ = background doping concentration ### 3.3 Drift-Diffusion Equations The **drift-diffusion model** describes carrier transport in semiconductor devices. #### 3.3.1 Poisson's Equation $$ \nabla^2 \phi = -\frac{\rho}{\epsilon} = -\frac{q}{\epsilon}\left(p - n + N_D^+ - N_A^-\right) $$ **Where:** - $\phi$ = electrostatic potential (V) - $\rho$ = charge density (C/cm³) - $\epsilon$ = permittivity ($\epsilon_{Si} = 11.7\epsilon_0$) - $q$ = elementary charge ($1.602 \times 10^{-19}$ C) - $n$ = electron concentration (cm⁻³) - $p$ = hole concentration (cm⁻³) - $N_D^+$ = ionized donor concentration (cm⁻³) - $N_A^-$ = ionized acceptor concentration (cm⁻³) #### 3.3.2 Continuity Equations **For electrons:** $$ \frac{\partial n}{\partial t} = \frac{1}{q}\nabla \cdot \vec{J}_n + G_n - R_n $$ **For holes:** $$ \frac{\partial p}{\partial t} = -\frac{1}{q}\nabla \cdot \vec{J}_p + G_p - R_p $$ **Where:** - $\vec{J}_n$, $\vec{J}_p$ = current densities (A/cm²) - $G_n$, $G_p$ = generation rates (cm⁻³s⁻¹) - $R_n$, $R_p$ = recombination rates (cm⁻³s⁻¹) #### 3.3.3 Current Density Equations **Electron current (drift + diffusion):** $$ \vec{J}_n = q\mu_n n \vec{E} + qD_n \nabla n $$ $$ \vec{J}_n = qn\mu_n\nabla\phi + qD_n\nabla n $$ **Hole current (drift + diffusion):** $$ \vec{J}_p = q\mu_p p \vec{E} - qD_p \nabla p $$ $$ \vec{J}_p = -qp\mu_p\nabla\phi - qD_p\nabla p $$ **Where:** - $\mu_n$, $\mu_p$ = carrier mobilities (cm²/V·s) - $D_n$, $D_p$ = diffusion coefficients (cm²/s) - $\vec{E} = -\nabla\phi$ = electric field (V/cm) #### 3.3.4 Einstein Relation $$ D_n = \frac{k_B T}{q}\mu_n = V_T \mu_n $$ $$ D_p = \frac{k_B T}{q}\mu_p = V_T \mu_p $$ **Where:** - $V_T = \frac{k_B T}{q}$ = thermal voltage - At $T = 300$ K: $V_T \approx 25.9$ mV #### 3.3.5 Recombination Models **Shockley-Read-Hall (SRH) Recombination:** $$ R_{SRH} = \frac{np - n_i^2}{\tau_p(n + n_1) + \tau_n(p + p_1)} $$ **Auger Recombination:** $$ R_{Auger} = C_n n(np - n_i^2) + C_p p(np - n_i^2) $$ **Where:** - $n_i$ = intrinsic carrier concentration ($\approx 1.5 \times 10^{10}$ cm⁻³ for Si at 300K) - $\tau_n$, $\tau_p$ = carrier lifetimes - $C_n$, $C_p$ = Auger coefficients ### 3.4 MOSFET Threshold Voltage The **threshold voltage** determines when Ellsworth's transistors switch on. #### 3.4.1 Threshold Voltage Formula $$ V_T = V_{FB} + 2\phi_F + \frac{\sqrt{2\epsilon_{Si} q N_A (2\phi_F)}}{C_{ox}} $$ **Expanded form:** $$ V_T = \phi_{ms} - \frac{Q_{ox}}{C_{ox}} + 2\phi_F + \frac{\sqrt{2\epsilon_{Si} q N_A (2\phi_F)}}{C_{ox}} $$ #### 3.4.2 Component Definitions **Flatband Voltage:** $$ V_{FB} = \phi_{ms} - \frac{Q_{ox}}{C_{ox}} $$ **Where:** - $\phi_{ms}$ = metal-semiconductor work function difference (V) - $Q_{ox}$ = oxide charge density (C/cm²) **Fermi Potential:** $$ \phi_F = \frac{k_B T}{q} \ln\left(\frac{N_A}{n_i}\right) \quad \text{(p-type)} $$ $$ \phi_F = \frac{k_B T}{q} \ln\left(\frac{N_D}{n_i}\right) \quad \text{(n-type)} $$ **Oxide Capacitance per Unit Area:** $$ C_{ox} = \frac{\epsilon_{ox}}{t_{ox}} $$ **Where:** - $\epsilon_{ox} = 3.9\epsilon_0 = 3.45 \times 10^{-13}$ F/cm - $t_{ox}$ = oxide thickness (cm) #### 3.4.3 Body Effect When substrate (body) is biased relative to source: $$ V_T = V_{T0} + \gamma\left(\sqrt{|2\phi_F + V_{SB}|} - \sqrt{|2\phi_F|}\right) $$ **Body Effect Coefficient:** $$ \gamma = \frac{\sqrt{2\epsilon_{Si} q N_A}}{C_{ox}} = \frac{t_{ox}}{\epsilon_{ox}}\sqrt{2\epsilon_{Si} q N_A} $$ **Typical values:** $\gamma \approx 0.3 - 0.5$ V$^{1/2}$ #### 3.4.4 Threshold Voltage Dependencies **Oxide Thickness Dependence:** $$ \frac{\partial V_T}{\partial t_{ox}} = \frac{\sqrt{2\epsilon_{Si} q N_A (2\phi_F)}}{\epsilon_{ox}} $$ - Thinner oxide → Lower $V_T$ - But: Higher leakage current **Temperature Dependence:** $$ \frac{\partial V_T}{\partial T} \approx -2 \text{ to } -4 \text{ mV/K} $$ **Typical $V_T$ Values:** | Technology | $t_{ox}$ (nm) | $V_T$ (V) | |------------|---------------|-----------| | Ellsworth's DIY | 80-100 | 2-5 | | 1970s Commercial | 50-100 | 1-2 | | Modern (90nm node) | 1.2-2 | 0.3-0.5 | ### 3.5 MOSFET I-V Characteristics The current-voltage relationships for MOSFET operation. #### 3.5.1 Operating Regions ``` - V_DS ──────────────────────► │ │ ┌───────────────────────────── │ │ SATURATION REGION │ │ I_D = const(V_GS) V_GS│ │ │ │ │ ├───────────────────────────── │ │ LINEAR (TRIODE) REGION │ │ I_D ∝ V_DS │ │ ▼ └───────────────────────────── V_DS = V_GS - V_T (Boundary) ``` #### 3.5.2 Linear (Triode) Region **Condition:** $V_{DS} < V_{GS} - V_T$ $$ I_D = \mu_n C_{ox} \frac{W}{L}\left[(V_{GS} - V_T)V_{DS} - \frac{V_{DS}^2}{2}\right] $$ **For small $V_{DS}$:** $$ I_D \approx \mu_n C_{ox} \frac{W}{L}(V_{GS} - V_T)V_{DS} $$ **Channel Resistance:** $$ R_{on} = \frac{V_{DS}}{I_D} = \frac{1}{\mu_n C_{ox} \frac{W}{L}(V_{GS} - V_T)} $$ #### 3.5.3 Saturation Region **Condition:** $V_{DS} \geq V_{GS} - V_T$ $$ I_D = \frac{1}{2}\mu_n C_{ox} \frac{W}{L}(V_{GS} - V_T)^2 $$ **With Channel Length Modulation:** $$ I_D = \frac{1}{2}\mu_n C_{ox} \frac{W}{L}(V_{GS} - V_T)^2(1 + \lambda V_{DS}) $$ **Where:** - $\lambda$ = channel length modulation parameter (V⁻¹) - Typical: $\lambda \approx 0.01 - 0.1$ V⁻¹ #### 3.5.4 Saturation Voltage $$ V_{DS,sat} = V_{GS} - V_T $$ #### 3.5.5 Transconductance **In Saturation:** $$ g_m = \frac{\partial I_D}{\partial V_{GS}} = \mu_n C_{ox} \frac{W}{L}(V_{GS} - V_T) $$ $$ g_m = \sqrt{2\mu_n C_{ox} \frac{W}{L} I_D} $$ #### 3.5.6 Output Conductance $$ g_{ds} = \frac{\partial I_D}{\partial V_{DS}} = \lambda I_D $$ **Output Resistance:** $$ r_o = \frac{1}{g_{ds}} = \frac{1}{\lambda I_D} $$ #### 3.5.7 Subthreshold Region **Condition:** $V_{GS} < V_T$ $$ I_D = I_0 \exp\left(\frac{V_{GS} - V_T}{nV_T}\right)\left(1 - \exp\left(-\frac{V_{DS}}{V_T}\right)\right) $$ **Where:** - $n$ = subthreshold swing factor (typically 1.0 - 1.5) - $I_0$ = characteristic current **Subthreshold Swing:** $$ S = n V_T \ln(10) = 2.3 n V_T \approx 60-100 \text{ mV/decade} $$ ## 4. Practical Applications ### 4.1 Design Calculations for DIY Fabrication #### Example 1: Oxide Growth Time **Given:** - Target oxide thickness: $x = 800$ Å = $8 \times 10^{-6}$ cm - Temperature: $T = 1000°C$ - Wet oxidation **Using Deal-Grove:** At 1000°C wet oxidation: - $B/A = 0.226$ μm/hr = $2.26 \times 10^{-5}$ cm/hr - $B = 0.287$ μm²/hr = $2.87 \times 10^{-10}$ cm²/hr $$ t = \frac{x^2 + Ax}{B} = \frac{x}{B/A} + \frac{x^2}{B} $$ $$ t = \frac{8 \times 10^{-6}}{2.26 \times 10^{-5}} + \frac{(8 \times 10^{-6})^2}{2.87 \times 10^{-10}} $$ $$ t \approx 0.35 + 0.22 = 0.57 \text{ hours} \approx 34 \text{ minutes} $$ #### Example 2: Junction Depth **Given:** - Phosphorus diffusion at 1000°C - Diffusion time: $t = 30$ minutes = 1800 s - $D = 2.5 \times 10^{-14}$ cm²/s **Diffusion Length:** $$ L_D = 2\sqrt{Dt} = 2\sqrt{2.5 \times 10^{-14} \times 1800} $$ $$ L_D = 2\sqrt{4.5 \times 10^{-11}} = 1.34 \times 10^{-5} \text{ cm} = 0.134 \text{ μm} $$ #### Example 3: Threshold Voltage Estimation **Given:** - $t_{ox} = 100$ nm = $10^{-5}$ cm - $N_A = 10^{16}$ cm⁻³ - $\phi_{ms} = -0.9$ V (Al gate on p-Si) **Calculate:** $$ C_{ox} = \frac{3.45 \times 10^{-13}}{10^{-5}} = 3.45 \times 10^{-8} \text{ F/cm}^2 $$ $$ \phi_F = 0.0259 \ln\left(\frac{10^{16}}{1.5 \times 10^{10}}\right) = 0.347 \text{ V} $$ $$ V_T = -0.9 + 2(0.347) + \frac{\sqrt{2 \times 1.04 \times 10^{-12} \times 1.6 \times 10^{-19} \times 10^{16} \times 0.694}}{3.45 \times 10^{-8}} $$ $$ V_T \approx -0.9 + 0.694 + 0.44 = 0.23 \text{ V} $$ ### 4.2 Measurement Verification **I-V Curve Analysis:** 1. Plot $\sqrt{I_D}$ vs. $V_{GS}$ in saturation 2. Linear fit extrapolates to $V_T$ at x-intercept 3. Slope gives $\sqrt{\frac{1}{2}\mu_n C_{ox}\frac{W}{L}}$ **Gate Oxide Quality Check:** $$ E_{breakdown} = \frac{V_{breakdown}}{t_{ox}} > 10 \text{ MV/cm (good quality)} $$ ## 5. Equations ### Key Equations | Process | Governing Equation | |---------|-------------------| | Oxide Growth | $x^2 + Ax = B(t + \tau)$ | | Dopant Diffusion | $\frac{\partial C}{\partial t} = D\frac{\partial^2 C}{\partial x^2}$ | | Charge Transport | $\nabla^2\phi = -\frac{q}{\epsilon}(p - n + N_D^+ - N_A^-)$ | | Threshold Voltage | $V_T = V_{FB} + 2\phi_F + \frac{Q_B}{C_{ox}}$ | | Drain Current (sat) | $I_D = \frac{1}{2}\mu_n C_{ox}\frac{W}{L}(V_{GS} - V_T)^2$ | ### Physical Constants | Constant | Symbol | Value | |----------|--------|-------| | Elementary charge | $q$ | $1.602 \times 10^{-19}$ C | | Boltzmann constant | $k_B$ | $8.617 \times 10^{-5}$ eV/K | | Permittivity of free space | $\epsilon_0$ | $8.854 \times 10^{-14}$ F/cm | | Si permittivity | $\epsilon_{Si}$ | $11.7\epsilon_0$ | | SiO₂ permittivity | $\epsilon_{ox}$ | $3.9\epsilon_0$ | | Intrinsic carrier conc. (Si, 300K) | $n_i$ | $1.5 \times 10^{10}$ cm⁻³ | | Thermal voltage (300K) | $V_T$ | 25.9 mV |
Homomorphic encryption allows computation on encrypted recommendation data without decrypting sensitive information.
Compute on encrypted data.
Homomorphic encryption enables computation on encrypted data without decryption.
Homomorphic encryption computes on encrypted data. Very slow but maximum privacy. Emerging for ML.
Perform computation on encrypted data.
Homomorphic encryption enables computation on encrypted data. Inference without exposing inputs.
Associative memory networks now connected to Transformer attention.
Efficient decision-based attack.
Learn from data with same features different samples.
Horizontal federated learning trains on different samples with same features across parties.
Flip images at test time.
Add more machines to handle load.
Distributed training framework.
Coordinate CPU and GPU.
Energetic carriers damage gate oxide over time.
Model HCI degradation.
Expedite critical lots.