moisture barrier bag, packaging
Protect from moisture.
751 technical terms and definitions
Protect from moisture.
Protect from humidity.
Moisture levels in cleanrooms and tool environments affect process quality.
Moisture resistance testing exposes packages to humidity assessing corrosion and seal integrity.
Classification of moisture sensitivity.
Moisture sensitivity levels classify packages by susceptibility to moisture-induced damage during soldering.
Failures from moisture ingress.
MOL dielectrics insulate local interconnects and contacts requiring low-k properties and good gap fill.
Middle-Of-Line integration connects front-end transistors to back-end metal interconnects through contacts vias and local interconnects.
# Mathematical Modeling for MOL (Middle of Line) in Semiconductor Manufacturing ## Overview of MOL MOL (Middle of Line) is the critical transition layer in semiconductor fabrication between **FEOL** (Front End of Line - transistor fabrication) and **BEOL** (Back End of Line - metal interconnects). ### MOL Components - **Source/Drain Contacts**: Metal connections to transistor terminals - **Gate Contacts**: Connections to the gate electrode - **Local Interconnects (LI)**: Short-range wiring between nearby devices - **Via0**: Vertical connections from contacts to Metal 1 layer - **Trench Silicide (TS)**: Low-resistance contact formation At advanced technology nodes ($\leq 7\text{nm}$), MOL becomes increasingly critical due to contact resistance dominance and scaling challenges. ## 1. Contact Resistance Modeling ### 1.1 Specific Contact Resistivity For metal-semiconductor interfaces, the specific contact resistivity $\rho_c$ determines performance. **Thermionic-Field Emission** (dominant mechanism for modern contacts): $$ \rho_c \propto \exp\left(\frac{4\pi\sqrt{\epsilon_s m^*}\,\phi_B}{h\sqrt{N_D}}\right) $$ Where: - $\phi_B$ = Schottky barrier height (eV) - $N_D$ = Doping concentration (cm⁻³) - $m^*$ = Effective mass - $\epsilon_s$ = Semiconductor permittivity - $h$ = Planck's constant ### 1.2 Transmission Line Model (TLM) $$ R_c = \frac{\rho_c}{L_T} \coth\left(\frac{L}{L_T}\right) $$ **Transfer length:** $$ L_T = \sqrt{\frac{\rho_c}{R_{sh}}} $$ Where: - $R_c$ = Contact resistance ($\Omega$) - $L_T$ = Transfer length - $R_{sh}$ = Sheet resistance ($\Omega/\square$) - $L$ = Contact length ### 1.3 Scaled Contact Resistance At nanoscale dimensions, total contact resistance includes multiple components: $$ R_{total} = \frac{\rho_c}{A_c} + R_{spreading} + R_{interface} $$ **Spreading resistance** (increasingly important at small scales): $$ R_{spreading} \approx \frac{\rho}{4r} $$ Where $r$ is the circular contact radius. ## 2. Contact Etch Modeling ### 2.1 Aspect Ratio Dependent Etching (ARDE) High-aspect-ratio contact holes suffer from transport limitations. **Knudsen Transport Model:** $$ \frac{ER(AR)}{ER_0} = \frac{1}{1 + \frac{3 \cdot AR}{8} \cdot \frac{1}{K_n}} $$ Where: - $AR$ = Aspect Ratio (depth/width) - $ER_0$ = Reference etch rate - $K_n$ = Knudsen number ### 2.2 Ion-Enhanced Etching Model $$ ER = Y_i \Gamma_i E_i^{1/2} + k_n \Gamma_n \cdot \theta $$ Where: - $Y_i$ = Ion sputtering yield - $\Gamma_i$ = Ion flux (ions/cm²·s) - $\Gamma_n$ = Neutral flux - $E_i$ = Ion energy (eV) - $\theta$ = Surface coverage fraction - $k_n$ = Neutral reaction rate constant ### 2.3 Profile Evolution (Level Set Method) $$ \frac{\partial \phi}{\partial t} + V_n \left|\nabla \phi\right| = 0 $$ Where: - $\phi$ = Level set function - $V_n$ = Local etch rate in normal direction ## 3. Metal Fill Modeling (W, Co, Ru) ### 3.1 CVD Kinetics **Tungsten CVD Reaction:** $$ \text{WF}_6 + 3\text{H}_2 \rightarrow \text{W} + 6\text{HF} $$ **Deposition Rate:** $$ \frac{dh}{dt} = \frac{k_s k_g C_{bulk}}{k_g + k_s} $$ Where: - $h$ = Film thickness - $k_s$ = Surface reaction rate constant - $k_g$ = Mass transport coefficient - $C_{bulk}$ = Bulk precursor concentration ### 3.2 Step Coverage Model $$ SC = \frac{t_{bottom}}{t_{sidewall}} = \frac{1}{1 + \beta \cdot AR} $$ **Sticking coefficient dependence:** $$ \beta = \frac{s}{2(1-s)} $$ Where: - $SC$ = Step coverage ratio - $s$ = Sticking coefficient - $AR$ = Aspect ratio ### 3.3 ALD Super-Conformal Fill **Growth Per Cycle (GPC):** $$ GPC = \frac{\theta \cdot N_{sites} \cdot M}{\rho \cdot N_A} $$ Where: - $\theta$ = Surface site coverage (0 to 1) - $N_{sites}$ = Surface site density (sites/cm²) - $M$ = Molecular weight (g/mol) - $\rho$ = Film density (g/cm³) - $N_A$ = Avogadro's number ## 4. CMP Modeling for MOL ### 4.1 Preston's Equation (Modified) $$ RR = K_p \cdot P \cdot V \cdot f(chemistry) $$ Where: - $RR$ = Removal rate (nm/min) - $K_p$ = Preston coefficient - $P$ = Applied pressure - $V$ = Relative velocity ### 4.2 Pattern-Dependent Model **Effective Pressure Distribution:** $$ P_{eff}(x,y) = \frac{P_{applied}}{\rho_{local}(x,y)} $$ Where $\rho_{local}$ = Local pattern density ### 4.3 Dishing and Erosion **Dishing (metal recessing):** $$ D = K_1 \cdot w^\alpha \cdot t^\beta $$ **Erosion (oxide loss):** $$ E = K_2 \cdot \rho^\gamma \cdot t^\delta $$ Where: - $w$ = Line width - $t$ = Polish time - $\rho$ = Pattern density - $\alpha, \beta, \gamma, \delta$ = Empirical exponents ## 5. Thermal Modeling ### 5.1 Heat Diffusion (Annealing) $$ \rho c_p \frac{\partial T}{\partial t} = \nabla \cdot (k \nabla T) + Q_{source} $$ Where: - $\rho$ = Material density - $c_p$ = Specific heat capacity - $k$ = Thermal conductivity - $Q_{source}$ = Heat generation rate ### 5.2 Effective Thermal Conductivity (Multilayer) $$ \frac{1}{k_{eff,\perp}} = \sum_i \frac{t_i}{k_i \cdot t_{total}} + \sum_{interfaces} \frac{R_{TBR}}{t_{total}} $$ Where: - $t_i$ = Thickness of layer $i$ - $k_i$ = Thermal conductivity of layer $i$ - $R_{TBR}$ = Thermal boundary resistance ## 6. Stress and Reliability Modeling ### 6.1 Film Stress $$ \sigma = \frac{E_f}{1-\nu_f}(\alpha_s - \alpha_f)\Delta T + \sigma_{intrinsic} + \sigma_{growth} $$ Where: - $E_f$ = Film Young's modulus - $\nu_f$ = Film Poisson's ratio - $\alpha_s, \alpha_f$ = Thermal expansion coefficients (substrate, film) - $\Delta T$ = Temperature change ### 6.2 Electromigration (Black's Equation) $$ MTTF = A \cdot j^{-n} \exp\left(\frac{E_a}{k_B T}\right) $$ Where: - $MTTF$ = Mean Time To Failure - $j$ = Current density (A/cm²) - $n$ = Current exponent ($\approx 1-2$) - $E_a$ = Activation energy ($\approx 0.7-0.9$ eV for W/Co) - $k_B$ = Boltzmann constant - $T$ = Temperature (K) ### 6.3 Stress-Induced Voiding $$ \frac{\partial C}{\partial t} = D\nabla^2 C - \frac{D\Omega}{k_B T}\nabla \cdot (C \nabla \sigma) $$ Where: - $C$ = Vacancy concentration - $D$ = Diffusion coefficient - $\Omega$ = Atomic volume - $\sigma$ = Hydrostatic stress ## 7. RC Delay Modeling ### 7.1 Total RC for MOL Path $$ \tau_{MOL} = (R_c + R_{via} + R_{LI}) \cdot C_{total} $$ **Contact resistance dominance at advanced nodes:** $$ R_c \gg R_{interconnect} \quad \text{(for local paths)} $$ ### 7.2 Capacitance Modeling $$ C = C_{plate} + C_{fringe} + C_{coupling} $$ **For high-AR cylindrical contacts:** $$ C \approx \frac{2\pi \epsilon_0 \epsilon_r h}{\ln(r_2/r_1)} $$ Where: - $h$ = Contact height - $r_1, r_2$ = Inner and outer radii - $\epsilon_r$ = Relative permittivity ## 8. Process Variability Modeling ### 8.1 Statistical Model (Response Surface) $$ CD = CD_{nominal} + \sum_i a_i \Delta P_i + \sum_{i,j} b_{ij} \Delta P_i \Delta P_j + \epsilon $$ Where: - $CD$ = Critical dimension - $\Delta P_i$ = Process parameter deviation - $a_i, b_{ij}$ = Sensitivity coefficients - $\epsilon$ = Random error ### 8.2 Monte Carlo for Variability $$ \sigma_{R_c}^2 = \left(\frac{\partial R_c}{\partial \rho_c}\right)^2 \sigma_{\rho_c}^2 + \left(\frac{\partial R_c}{\partial A}\right)^2 \sigma_A^2 + \ldots $$ ### 8.3 Process Capability Index $$ C_{pk} = \min\left(\frac{USL - \mu}{3\sigma}, \frac{\mu - LSL}{3\sigma}\right) $$ Where: - $USL$ = Upper specification limit - $LSL$ = Lower specification limit - $\mu$ = Process mean - $\sigma$ = Standard deviation ## 9. TCAD Simulation Framework ### 9.1 Coupled Equations Solved **Poisson's Equation:** $$ \nabla \cdot (\epsilon \nabla \psi) = -q(p - n + N_D^+ - N_A^-) $$ **Continuity Equations:** $$ \frac{\partial n}{\partial t} = \frac{1}{q}\nabla \cdot \vec{J_n} + G - R $$ $$ \frac{\partial p}{\partial t} = -\frac{1}{q}\nabla \cdot \vec{J_p} + G - R $$ **Current Equations (Drift-Diffusion):** $$ \vec{J_n} = qn\mu_n \vec{E} + qD_n \nabla n $$ $$ \vec{J_p} = qp\mu_p \vec{E} - qD_p \nabla p $$ Where: - $\psi$ = Electrostatic potential - $n, p$ = Electron and hole concentrations - $N_D^+, N_A^-$ = Ionized donor and acceptor concentrations - $G, R$ = Generation and recombination rates - $\mu_n, \mu_p$ = Electron and hole mobilities - $D_n, D_p$ = Diffusion coefficients ## 10. Machine Learning Integration ### 10.1 Virtual Metrology $$ y_{predicted} = f_{NN}(\vec{x}_{sensors}, \vec{x}_{recipe}) $$ Where $f_{NN}$ represents a trained neural network mapping sensor data and recipe parameters to metrology outputs. ### 10.2 Process Optimization (Gaussian Process) **Gaussian Process Regression:** $$ y(\vec{x}) \sim \mathcal{GP}(m(\vec{x}), k(\vec{x}, \vec{x'})) $$ Where: - $m(\vec{x})$ = Mean function - $k(\vec{x}, \vec{x'})$ = Covariance kernel (e.g., RBF, Matérn) Used for Bayesian optimization of MOL process recipes. ## 11. Lithography and OPC for MOL ### 11.1 Aerial Image Modeling $$ I(x,y) = \left|\sum_n \sum_m c_{nm} P_{nm} \exp\left(i 2\pi \frac{n x + m y}{\lambda/NA}\right)\right|^2 $$ Where: - $c_{nm}$ = Diffraction order coefficients - $P_{nm}$ = Pupil function - $\lambda$ = Wavelength - $NA$ = Numerical aperture ### 11.2 OPC Edge Placement Error Minimization $$ \min \sum_{i} w_i |EPE_i|^2 + \lambda R(mask) $$ Where: - $EPE_i$ = Edge placement error at evaluation point $i$ - $w_i$ = Weight factor - $R(mask)$ = Regularization term - $\lambda$ = Regularization parameter ## 12. Key MOL Mathematical Models | **Parameter** | **Model Type** | **Critical Equation** | |---------------|----------------|----------------------| | Contact Resistance | Physics-based | $R = \rho_c/A + R_{spreading}$ | | Etch Profile | Level-set / MC | $\frac{\partial \phi}{\partial t} + V_n|\nabla \phi| = 0$ | | Metal Fill | Kinetic + Transport | Sticking coefficient models | | CMP | Empirical + Physics | Modified Preston equation | | Reliability | Arrhenius-based | Black's equation | | Variability | Statistical | Monte Carlo + RSM | ## 13. Advanced Considerations (Sub-5nm Nodes) At sub-5nm technology nodes, additional physics must be incorporated: - **Quantum Tunneling Effects**: Direct tunneling through thin barriers - **Interface Scattering**: Grain boundary and surface scattering - **Size-Dependent Resistivity**: Fuchs-Sondheimer model $$ \frac{\rho}{\rho_0} = 1 + \frac{3\lambda}{8t}(1-p) + \frac{3\lambda}{2d}\frac{R}{1-R} $$ Where: - $\lambda$ = Electron mean free path - $t$ = Film thickness - $d$ = Grain size - $p$ = Surface specularity parameter - $R$ = Grain boundary reflection coefficient
Shape of final package.
Tool holding cavities.
Time mold is closed.
Design of molding tool.
Excess compound at parting line.
Time mold is open.
Temperature of tool.
Underfill via transfer molding.
Polymer material for encapsulation.
Total time per package.
Control variables for molding.
Predict binding pose of molecules.
Atom-level simulation of processes.
Simulate atomic motion over time.
Single-molecule devices.
Generate molecular structures.
Predict properties from molecular structure.
AI for molecules: property prediction, generation, docking. AlphaFold for protein structure. Drug discovery.
Design novel molecular structures.
Molecule Optimization by Learned Embeddings with Rationales generates molecules through reinforcement learning and graph grammars.
MolGAN uses domain-specific rewards like drug-likeness and synthesizability to guide molecular generation.
GAN for molecular graphs.
MolGAN uses generative adversarial networks with graph convolutional discriminators and policy gradient generators for molecular graph generation.
Moments accountant tracks privacy loss through moment-generating functions.
Slowly updated teacher.
Slowly updated encoder for contrastive learning.
Wafer used to check tool performance and cleanliness.
Wafers used to check tool status.
Production LLMs need metrics (latency, error rate), logs, traces, and feedback loops to catch regressions, drifts, and incidents quickly.
Predict depth from single image.
SLAM with single camera.
Integrate devices vertically in single processing.
Features corresponding to single concepts.
Monotonic attention mechanisms enforce left-to-right alignment enabling streaming sequence-to-sequence models.
Statistical circuit analysis.
Monte Carlo critical area analysis randomly samples defect locations and sizes to estimate failure probability.
Particle-based device simulation.
Use dropout at test time for uncertainty.
Statistical simulation of ion paths.
Statistical simulation of process variations.