derivative product, business & strategy
Derivative products modify platform designs for specific applications reducing NRE.
9,967 technical terms and definitions
Derivative products modify platform designs for specific applications reducing NRE.
Descript audio codec uses neural compression with VQ enabling high-quality speech synthesis from tokens.
Light plasma treatment to remove resist residue.
Desiccant dehumidification removes moisture using hygroscopic materials reducing cooling loads.
Moisture absorber.
Design cycles span from specification to tapeout determining development schedule.
I can draft or polish design documents with goals, constraints, diagrams, trade-offs, and implementation plans.
Record design details.
Support and tools provided by foundry to help customers design chips.
Design for easy assembly.
Design for low cost.
Design for minimal environmental impact.
Detailed DFM considerations.
Design for recycling optimizes products for material recovery and reuse at end-of-life.
Design to maximize lifetime.
Design for reliability incorporates reliability considerations throughout product development.
Design for long life.
Design to achieve Six Sigma quality.
Design for easy testing.
General design optimization.
Lock design for production.
Third-party design services.
Design houses provide semiconductor design services on contract basis.
Intended operational lifetime.
Design margins provide cushion beyond minimum requirements accommodating variations and uncertainties.
Systematically vary parameters to optimize.
DOE for reliability improvement.
Design reuse leverages existing IP blocks reducing development time and cost.
Formal evaluation of designs.
Exception to design rule for special cases.
Link requirements to implementation.
Confirm design meets customer needs.
Create design alternatives.
Confirm design meets specs.
Similar to CODT.
Combine multiple objectives into single metric.
Combine multiple responses into single metric.
Stress level causing failure.
Lowest detectable amount.
Detection poka-yoke identifies errors enabling immediate correction.
Detection rates probability of identifying failure before impact.
Ongoing competition between detectors and generators.
Detectron2 is Meta detection and segmentation library. Research platform.
Deterministic jitter has bounded magnitude from repeatable sources like crosstalk or duty cycle distortion.
Eliminate non-deterministic operations.
Remove toxic or harmful content from generations.
Chemical process to remove soluble resist revealing pattern.
Authorization to ship non-conforming product.
Active silicon layer in SOI.
# Device Physics & Mathematical Modeling 1. Fundamental Mathematical Structure Semiconductor modeling is built on coupled nonlinear partial differential equations spanning multiple scales: | Scale | Methods | Typical Equations | |:------|:--------|:------------------| | Quantum (< 1 nm) | DFT, Schrödinger | $H\psi = E\psi$ | | Atomistic (1–100 nm) | MD, Kinetic Monte Carlo | Newton's equations, master equations | | Continuum (nm–mm) | Drift-diffusion, FEM | PDEs (Poisson, continuity, heat) | | Circuit | SPICE | ODEs, compact models | Multiscale Hierarchy The mathematics forms a hierarchy of models through successive averaging: $$ \boxed{\text{Schrödinger} \xrightarrow{\text{averaging}} \text{Boltzmann} \xrightarrow{\text{moments}} \text{Drift-Diffusion} \xrightarrow{\text{fitting}} \text{Compact Models}} $$ 2. Process Physics & Models 2.1 Oxidation: Deal-Grove Model Thermal oxidation of silicon follows linear-parabolic kinetics : $$ \frac{dx_{ox}}{dt} = \frac{B}{A + 2x_{ox}} $$ where: - $x_{ox}$ = oxide thickness - $B/A$ = linear rate constant (surface-reaction limited) - $B$ = parabolic rate constant (diffusion limited) Limiting Cases: - Thin oxide (reaction-limited): $$ x_{ox} \approx \frac{B}{A} \cdot t $$ - Thick oxide (diffusion-limited): $$ x_{ox} \approx \sqrt{B \cdot t} $$ Physical Mechanism: 1. O₂ transport from gas to oxide surface 2. O₂ diffusion through growing SiO₂ layer 3. Reaction at Si/SiO₂ interface: $\text{Si} + \text{O}_2 \rightarrow \text{SiO}_2$ > Note: This is a Stefan problem (moving boundary PDE). 2.2 Diffusion: Fick's Laws Dopant redistribution follows Fick's second law : $$ \frac{\partial C}{\partial t} = \nabla \cdot \left( D(C, T) \nabla C \right) $$ For constant $D$ in 1D: $$ \frac{\partial C}{\partial t} = D \frac{\partial^2 C}{\partial x^2} $$ Analytical Solutions (1D, constant D): - Constant surface concentration (infinite source): $$ C(x,t) = C_s \cdot \text{erfc}\left( \frac{x}{2\sqrt{Dt}} \right) $$ - Limited source (e.g., implant drive-in): $$ C(x,t) = \frac{Q}{\sqrt{\pi D t}} \exp\left( -\frac{x^2}{4Dt} \right) $$ where $Q$ = dose (atoms/cm²) Complications at High Concentrations: - Concentration-dependent diffusivity: $D = D(C)$ - Electric field effects: Charged point defects create internal fields - Vacancy/interstitial mechanisms: Different diffusion pathways $$ \frac{\partial C}{\partial t} = \frac{\partial}{\partial x}\left[ D(C) \frac{\partial C}{\partial x} \right] + \mu C \frac{\partial \phi}{\partial x} $$ 2.3 Ion Implantation: Range Theory The implanted dopant profile is approximately Gaussian : $$ C(x) = \frac{\Phi}{\sqrt{2\pi} \Delta R_p} \exp\left( -\frac{(x - R_p)^2}{2 (\Delta R_p)^2} \right) $$ where: - $\Phi$ = implant dose (ions/cm²) - $R_p$ = projected range (mean depth) - $\Delta R_p$ = straggle (standard deviation) LSS Theory (Lindhard-Scharff-Schiøtt) predicts stopping power: $$ -\frac{dE}{dx} = N \left[ S_n(E) + S_e(E) \right] $$ where: - $S_n(E)$ = nuclear stopping power (dominant at low energy) - $S_e(E)$ = electronic stopping power (dominant at high energy) - $N$ = target atomic density For asymmetric profiles , the Pearson IV distribution is used: $$ C(x) = \frac{\Phi \cdot K}{\Delta R_p} \left[ 1 + \left( \frac{x - R_p}{a} \right)^2 \right]^{-m} \exp\left[ -\nu \arctan\left( \frac{x - R_p}{a} \right) \right] $$ > Modern approach: Monte Carlo codes (SRIM/TRIM) for accurate profiles including channeling effects. 2.4 Lithography: Optical Imaging Aerial image formation follows Hopkins' partially coherent imaging theory : $$ I(\mathbf{r}) = \iint TCC(f, f') \cdot \tilde{M}(f) \cdot \tilde{M}^*(f') \cdot e^{2\pi i (f - f') \cdot \mathbf{r}} \, df \, df' $$ where: - $TCC$ = Transmission Cross-Coefficient - $\tilde{M}(f)$ = mask spectrum (Fourier transform of mask pattern) - $\mathbf{r}$ = position in image plane Fundamental Limits: - Rayleigh resolution criterion: $$ CD_{\min} = k_1 \frac{\lambda}{NA} $$ - Depth of focus: $$ DOF = k_2 \frac{\lambda}{NA^2} $$ where: - $\lambda$ = wavelength (193 nm for ArF, 13.5 nm for EUV) - $NA$ = numerical aperture - $k_1, k_2$ = process-dependent factors Resist Modeling — Dill Equations: $$ \frac{\partial M}{\partial t} = -C \cdot I(z) \cdot M $$ $$ \frac{dI}{dz} = -(\alpha M + \beta) I $$ where $M$ = photoactive compound concentration. 2.5 Etching & Deposition: Surface Evolution Topography evolution is modeled with the level set method : $$ \frac{\partial \phi}{\partial t} + V |\nabla \phi| = 0 $$ where: - $\phi(\mathbf{r}, t) = 0$ defines the surface - $V$ = local velocity (etch rate or deposition rate) For anisotropic etching: $$ V = V(\theta, \phi, \text{ion flux}, \text{chemistry}) $$ CVD in High Aspect Ratio Features: Knudsen diffusion limits step coverage: $$ \frac{\partial C}{\partial t} = D_K \nabla^2 C - k_s C \cdot \delta_{\text{surface}} $$ where: - $D_K = \frac{d}{3}\sqrt{\frac{8k_BT}{\pi m}}$ (Knudsen diffusivity) - $d$ = feature width - $k_s$ = surface reaction rate ALD (Atomic Layer Deposition): Self-limiting surface reactions follow Langmuir kinetics: $$ \theta = \frac{K \cdot P}{1 + K \cdot P} $$ where $\theta$ = surface coverage, $P$ = precursor partial pressure. 3. Device Physics: Semiconductor Equations The core mathematical framework for device simulation consists of three coupled PDEs : 3.1 Poisson's Equation (Electrostatics) $$ \nabla \cdot (\varepsilon \nabla \psi) = -q \left( p - n + N_D^+ - N_A^- \right) $$ where: - $\psi$ = electrostatic potential - $n, p$ = electron and hole concentrations - $N_D^+, N_A^-$ = ionized donor and acceptor concentrations 3.2 Continuity Equations (Carrier Conservation) Electrons: $$ \frac{\partial n}{\partial t} = \frac{1}{q} \nabla \cdot \mathbf{J}_n + G - R $$ Holes: $$ \frac{\partial p}{\partial t} = -\frac{1}{q} \nabla \cdot \mathbf{J}_p + G - R $$ where: - $G$ = generation rate - $R$ = recombination rate 3.3 Current Density Equations (Transport) Drift-Diffusion Model: $$ \mathbf{J}_n = q \mu_n n \mathbf{E} + q D_n \nabla n $$ $$ \mathbf{J}_p = q \mu_p p \mathbf{E} - q D_p \nabla p $$ Einstein Relation: $$ \frac{D_n}{\mu_n} = \frac{D_p}{\mu_p} = \frac{k_B T}{q} = V_T $$ 3.4 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: $$ R_{Auger} = C_n n (np - n_i^2) + C_p p (np - n_i^2) $$ Radiative Recombination: $$ R_{rad} = B (np - n_i^2) $$ 3.5 MOSFET Physics Threshold Voltage: $$ V_T = V_{FB} + 2\phi_B + \frac{\sqrt{2 \varepsilon_{Si} q N_A (2\phi_B)}}{C_{ox}} $$ where: - $V_{FB}$ = flat-band voltage - $\phi_B = \frac{k_BT}{q} \ln\left(\frac{N_A}{n_i}\right)$ = bulk potential - $C_{ox} = \frac{\varepsilon_{ox}}{t_{ox}}$ = oxide capacitance Drain Current (Gradual Channel Approximation): - Linear region ($V_{DS} < V_{GS} - V_T$): $$ I_D = \frac{W}{L} \mu_n C_{ox} \left[ (V_{GS} - V_T) V_{DS} - \frac{V_{DS}^2}{2} \right] $$ - Saturation region ($V_{DS} \geq V_{GS} - V_T$): $$ I_D = \frac{W}{2L} \mu_n C_{ox} (V_{GS} - V_T)^2 $$ 4. Quantum Effects at Nanoscale For modern devices with gate lengths $L_g < 10$ nm, classical models fail. 4.1 Quantum Confinement In thin silicon channels, carrier energy becomes quantized : $$ E_n = \frac{\hbar^2 \pi^2 n^2}{2 m^* t_{Si}^2} $$ where: - $n$ = quantum number (1, 2, 3, ...) - $m^*$ = effective mass - $t_{Si}$ = silicon body thickness Effects: - Increased threshold voltage - Modified density of states: $g_{2D}(E) = \frac{m^*}{\pi \hbar^2}$ (step function) 4.2 Quantum Tunneling Gate Leakage (Direct Tunneling): WKB approximation: $$ T \approx \exp\left( -2 \int_0^{t_{ox}} \kappa(x) \, dx \right) $$ where $\kappa = \sqrt{\frac{2m^*(\Phi_B - E)}{\hbar^2}}$ Source-Drain Tunneling: Limits OFF-state current in ultra-short channels. Band-to-Band Tunneling: Enables Tunnel FETs (TFETs): $$ I_{BTBT} \propto \exp\left( -\frac{4\sqrt{2m^*} E_g^{3/2}}{3q\hbar |\mathbf{E}|} \right) $$ 4.3 Ballistic Transport When channel length $L < \lambda_{mfp}$ (mean free path), the Landauer formalism applies: $$ I = \frac{2q}{h} \int T(E) \left[ f_S(E) - f_D(E) \right] dE $$ where: - $T(E)$ = transmission probability - $f_S, f_D$ = source and drain Fermi functions Ballistic Conductance Quantum: $$ G_0 = \frac{2q^2}{h} \approx 77.5 \, \mu\text{S} $$ 4.4 NEGF Formalism The Non-Equilibrium Green's Function method is the gold standard for quantum transport: $$ G^R = \left[ EI - H - \Sigma_1 - \Sigma_2 \right]^{-1} $$ where: - $H$ = device Hamiltonian - $\Sigma_1, \Sigma_2$ = contact self-energies - $G^R$ = retarded Green's function Observables: - Electron density: $n(\mathbf{r}) = -\frac{1}{\pi} \text{Im}[G^<(\mathbf{r}, \mathbf{r}; E)]$ - Current: $I = \frac{q}{h} \text{Tr}[\Gamma_1 G^R \Gamma_2 G^A]$ 5. Numerical Methods 5.1 Discretization: Scharfetter-Gummel Scheme The drift-diffusion current requires special treatment to avoid numerical instability: $$ J_{n,i+1/2} = \frac{q D_n}{h} \left[ n_{i+1} B\left( -\frac{\Delta \psi}{V_T} \right) - n_i B\left( \frac{\Delta \psi}{V_T} \right) \right] $$ where the Bernoulli function is: $$ B(x) = \frac{x}{e^x - 1} $$ Properties: - $B(0) = 1$ - $B(x) \to 0$ as $x \to \infty$ - $B(-x) = x + B(x)$ 5.2 Solution Strategies Gummel Iteration (Decoupled): 1. Solve Poisson for $\psi$ (fixed $n$, $p$) 2. Solve electron continuity for $n$ (fixed $\psi$, $p$) 3. Solve hole continuity for $p$ (fixed $\psi$, $n$) 4. Repeat until convergence Newton-Raphson (Fully Coupled): Solve the Jacobian system: $$ \begin{pmatrix} \frac{\partial F_\psi}{\partial \psi} & \frac{\partial F_\psi}{\partial n} & \frac{\partial F_\psi}{\partial p} \\ \frac{\partial F_n}{\partial \psi} & \frac{\partial F_n}{\partial n} & \frac{\partial F_n}{\partial p} \\ \frac{\partial F_p}{\partial \psi} & \frac{\partial F_p}{\partial n} & \frac{\partial F_p}{\partial p} \end{pmatrix} \begin{pmatrix} \delta \psi \\ \delta n \\ \delta p \end{pmatrix} = - \begin{pmatrix} F_\psi \\ F_n \\ F_p \end{pmatrix} $$ 5.3 Time Integration Stiffness Problem: Time scales span ~15 orders of magnitude: | Process | Time Scale | |:--------|:-----------| | Carrier relaxation | ~ps | | Thermal response | ~μs–ms | | Dopant diffusion | min–hours | Solution: Use implicit methods (Backward Euler, BDF). 5.4 Mesh Requirements Debye Length Constraint: The mesh must resolve the Debye length: $$ \lambda_D = \sqrt{\frac{\varepsilon k_B T}{q^2 n}} $$ For $n = 10^{18}$ cm⁻³: $\lambda_D \approx 4$ nm Adaptive Mesh Refinement: - Refine near junctions, interfaces, corners - Coarsen in bulk regions - Use Delaunay triangulation for quality 6. Compact Models for Circuit Simulation For SPICE-level simulation, physics is abstracted into algebraic/empirical equations. Industry Standard Models | Model | Device | Key Features | |:------|:-------|:-------------| | BSIM4 | Planar MOSFET | ~300 parameters, channel length modulation | | BSIM-CMG | FinFET | Tri-gate geometry, quantum effects | | BSIM-GAA | Nanosheet | Stacked channels, sheet width | | PSP | Bulk MOSFET | Surface-potential-based | Key Physics Captured - Short-channel effects: DIBL, $V_T$ roll-off - Quantum corrections: Inversion layer quantization - Mobility degradation: Surface scattering, velocity saturation - Parasitic effects: Series resistance, overlap capacitance - Variability: Statistical mismatch models Threshold Voltage Variability (Pelgrom's Law) $$ \sigma_{V_T} = \frac{A_{VT}}{\sqrt{W \cdot L}} $$ where $A_{VT}$ is a technology-dependent constant. 7. TCAD Co-Simulation Workflow The complete semiconductor design flow: ```text ┌─────────────────────────────────────────────────────────────┐ │ ┌───────────────┐ ┌───────────────┐ ┌───────────────┐ │ │ │ Process │──▶│ Device │──▶│ Parameter │ │ │ │ Simulation │ │ Simulation │ │ Extraction │ │ │ │ (Sentaurus) │ │ (Sentaurus) │ │ (BSIM Fit) │ │ │ └───────────────┘ └───────────────┘ └───────────────┘ │ │ │ │ │ │ │ ▼ ▼ ▼ │ │ ┌───────────────┐ ┌───────────────┐ ┌───────────────┐ │ │ │• Implantation │ │• I-V, C-V │ │• BSIM params │ │ │ │• Diffusion │ │• Breakdown │ │• Corner extr. │ │ │ │• Oxidation │ │• Hot carrier │ │• Variability │ │ │ │• Etching │ │• Noise │ │ statistics │ │ │ └───────────────┘ └───────────────┘ └───────────────┘ │ │ │ │ │ ▼ │ │ ┌───────────────┐ │ │ │ Circuit │ │ │ │ Simulation │ │ │ │(SPICE,Spectre)│ │ │ └───────────────┘ │ └─────────────────────────────────────────────────────────────┘ ``` Key Challenge: Propagating variability through the entire chain: - Line Edge Roughness (LER) - Random Dopant Fluctuation (RDF) - Work function variation - Thickness variations 8. Mathematical Frontiers 8.1 Machine Learning + Physics - Physics-Informed Neural Networks (PINNs): $$ \mathcal{L} = \mathcal{L}_{data} + \lambda \mathcal{L}_{physics} $$ where $\mathcal{L}_{physics}$ enforces PDE residuals. - Surrogate models for expensive TCAD simulations - Inverse design and topology optimization - Defect prediction in manufacturing 8.2 Stochastic Modeling Random Dopant Fluctuation: $$ \sigma_{V_T} \propto \frac{t_{ox}}{\sqrt{W \cdot L \cdot N_A}} $$ Approaches: - Atomistic Monte Carlo (place individual dopants) - Statistical impedance field method - Compact model statistical extensions 8.3 Multiphysics Coupling Electro-Thermal Self-Heating: $$ \rho C_p \frac{\partial T}{\partial t} = \nabla \cdot (\kappa \nabla T) + \mathbf{J} \cdot \mathbf{E} $$ Stress Effects on Mobility (Piezoresistance): $$ \frac{\Delta \mu}{\mu_0} = \pi_L \sigma_L + \pi_T \sigma_T $$ Electromigration in Interconnects: $$ \mathbf{J}_{atoms} = \frac{D C}{k_B T} \left( Z^* q \mathbf{E} - \Omega \nabla \sigma \right) $$ 8.4 Atomistic-Continuum Bridging Strategies: - Coarse-graining from MD/DFT - Density gradient quantum corrections: $$ V_{QM} = \frac{\gamma \hbar^2}{12 m^*} \frac{\nabla^2 \sqrt{n}}{\sqrt{n}} $$ - Hybrid methods: atomistic core + continuum far-field The mathematics of semiconductor manufacturing and device physics encompasses: $$ \boxed{ \begin{aligned} &\text{Process:} && \text{Stefan problems, diffusion PDEs, reaction kinetics} \\ &\text{Device:} && \text{Coupled Poisson + continuity equations} \\ &\text{Quantum:} && \text{Schrödinger, NEGF, tunneling} \\ &\text{Numerical:} && \text{FEM/FDM, Scharfetter-Gummel, Newton iteration} \\ &\text{Circuit:} && \text{Compact models (BSIM), variability statistics} \end{aligned} } $$ Each level trades accuracy for computational tractability . The art lies in knowing when each approximation breaks down—and modern scaling is pushing us toward the quantum limit where classical continuum models become inadequate.