microwave photoconductivity decay, metrology
Non-contact lifetime measurement.
86 technical terms and definitions
Non-contact lifetime measurement.
Quantify crystal orientation differences.
Use predictive models for process control.
Fit geometric model to optical data.
EMC absorbing water.
Protect from moisture.
Protect from humidity.
Classification of moisture sensitivity.
# 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.
Wafer used to check tool performance and cleanliness.
Wafers used to check tool status.
# Semiconductor Manufacturing Monte Carlo Simulation: The Mathematics
## 1. Introduction
### 1.1 Why Monte Carlo for Semiconductors?
Semiconductor manufacturing involves:
- **Nanometer-scale features** (3nm, 5nm nodes) where atomic-scale randomness matters
- **Hundreds of process steps**, each with inherent variability
- **High-dimensional parameter spaces** (100s–1000s of variables)
- **Rare event statistics** (yield prediction for 99%+ target yields)
Classical numerical methods fail due to the **curse of dimensionality**. Monte Carlo's key advantage:
$$
\text{Error} = O\left(\frac{1}{\sqrt{N}}\right) \quad \text{independent of dimensionality}
$$
### 1.2 Key Applications
- **Process variability modeling**: Understanding how variations in lithography, etching, doping affect device parameters
- **Yield prediction**: Estimating what percentage of chips will work
- **Circuit performance analysis**: Predicting speed, power consumption distributions
- **Design for manufacturability (DFM)**: Ensuring designs are robust to process variations
- **Statistical timing analysis**: Understanding timing margins
- **Device physics simulation**: Modeling carrier transport, quantum effects
## 2. Fundamental Monte Carlo Mathematics
### 2.1 Basic Monte Carlo Integration
To estimate an integral:
$$
I = \int_D f(x) \, p(x) \, dx
$$
The **Monte Carlo estimator**:
$$
\hat{I} = \frac{1}{N} \sum_{i=1}^{N} f(x_i), \quad x_i \sim p(x)
$$
**Error bound** (Central Limit Theorem):
$$
\text{Standard Error} = \frac{\sigma}{\sqrt{N}}
$$
where $\sigma$ is the standard deviation of $f(x)$.
### 2.2 Random Number Generation
#### 2.2.1 Linear Congruential Generator
$$
X_{n+1} = (aX_n + c) \mod m
$$
- **Parameters**: multiplier $a$, increment $c$, modulus $m$
- **Period**: at most $m$
#### 2.2.2 Box-Muller Transform (Uniform → Gaussian)
$$
Z_0 = \sqrt{-2 \ln U_1} \cos(2\pi U_2)
$$
$$
Z_1 = \sqrt{-2 \ln U_1} \sin(2\pi U_2)
$$
where:
- $U_1, U_2 \sim \text{Uniform}(0,1)$
- $Z_0, Z_1 \sim \mathcal{N}(0,1)$
#### 2.2.3 Inverse Transform Method
$$
X = F^{-1}(U)
$$
where $F$ is the CDF of the desired distribution and $U \sim \text{Uniform}(0,1)$.
### 2.3 Modern PRNGs for Parallel Computing
- **Mersenne Twister (MT19937)**: Period $2^{19937} - 1$
- **Xorshift**: Fast, good statistical properties
- **PCG (Permuted Congruential Generator)**: Statistically excellent
- **Counter-based (Philox, Threefry)**: Ideal for GPU parallelization
## 3. Process Variation Modeling
### 3.1 Parameter Decomposition
A device parameter $P$ is modeled as:
$$
P = P_{\text{nom}} + \Delta P_{\text{sys}}(x,y) + \Delta P_{\text{global}} + \Delta P_{\text{local}}
$$
where:
- **Systematic variation**:
$$
\Delta P_{\text{sys}}(x,y) = \sum_{i,j} a_{ij} x^i y^j
$$
(spatial polynomial)
- **Global variation** (wafer-to-wafer, lot-to-lot):
$$
\Delta P_{\text{global}} \sim \mathcal{N}(0, \sigma_g^2)
$$
- **Local variation** (device-to-device):
$$
\Delta P_{\text{local}} \sim \mathcal{N}(0, \sigma_l^2)
$$
### 3.2 Spatial Correlation Structure
Local variations often exhibit spatial correlation:
$$
\text{Cov}(\Delta P(\mathbf{r}_1), \Delta P(\mathbf{r}_2)) = \sigma^2 \cdot \rho(|\mathbf{r}_1 - \mathbf{r}_2|)
$$
#### Common Correlation Functions
| Model | Formula | Characteristics |
|-------|---------|-----------------|
| Exponential | $\rho(d) = e^{-d/\lambda}$ | Sharp near-field correlation |
| Gaussian | $\rho(d) = e^{-(d/\lambda)^2}$ | Smoother correlation decay |
| Matérn | $\rho(d) = \frac{2^{1-\nu}}{\Gamma(\nu)} \left(\frac{d}{\lambda}\right)^\nu K_\nu\left(\frac{d}{\lambda}\right)$ | Flexible smoothness parameter $\nu$ |
### 3.3 Generating Correlated Samples
Given covariance matrix $\mathbf{\Sigma}$, use **Cholesky decomposition**:
$$
\mathbf{\Sigma} = \mathbf{L}\mathbf{L}^T
$$
Generate correlated samples:
$$
\mathbf{X} = \boldsymbol{\mu} + \mathbf{L}\mathbf{Z}
$$
where $\mathbf{Z}$ is a vector of independent standard normals.
### 3.4 Pelgrom's Mismatch Law
For transistor mismatch (critical for analog/SRAM):
$$
\sigma(\Delta V_{th}) = \frac{A_{VT}}{\sqrt{WL}}
$$
$$
\sigma\left(\frac{\Delta \beta}{\beta}\right) = \frac{A_\beta}{\sqrt{WL}}
$$
where:
- $A_{VT}$: Threshold voltage mismatch coefficient (typical: 1-5 mV·μm)
- $A_\beta$: Current factor mismatch coefficient (typical: 1-2 %·μm)
- $W$: Gate width
- $L$: Gate length
## 4. Statistical Static Timing Analysis (SSTA)
### 4.1 Gate Delay Model
$$
d = d_0 + \sum_i a_i \Delta P_i + \sum_i \beta_{ii} (\Delta P_i)^2 + \sum_{i
Measure oxide quality and interface.
MSL 1 through 6.
Complete polarization characterization.
Full polarization scatterometry.
Parallel electron beams for faster writing.
Parallel beams for faster mask writing.
Transfer multiple layers sequentially.
Use multiple imaging modes together.
Split design into multiple masks for multi-patterning.
Use multiple litho-etch cycles to achieve finer pitch than single exposure.
Share wafer among multiple customers to reduce prototyping cost.
Share mask costs across designs.
Withstand repeated heating.