i-v curve,metrology
Current-voltage characteristic.
32 technical terms and definitions
Current-voltage characteristic.
Achieving solution in inverse lithography.
Measure overlay from actual pattern images.
Fill space between lens and wafer with water to improve resolution.
Measure impurity distribution.
Measurements taken during processing to monitor and control.
Real-time monitoring during processing.
Real-time TEM during processing.
Cavity not fully filled.
Solution-based trace analysis.
TSMC's fan-out wafer-level packaging.
Use IR to see through wafer.
Ellipsometry in mid-IR range.
Inject liquid compound into mold.
Mark with ink.
Check wafers during processing.
Inline metrology correlates process measurements with final yield identifying contributing factors.
Improved DPC for light element imaging.
Built-in metrology in process tools.
Species affecting target measurement.
Compounds at bond interface.
Industry roadmap (historical).
Silicon or organic substrate connecting multiple dies.
Computationally design optimal mask patterns.
Measure unoccupied states.
# Inverse Problems 1. Introduction to Inverse Problems 1.1 Mathematical Definition In mathematical terms, a forward problem is defined as: $$ y = f(x) $$ where: - $x$ = input parameters (process conditions) - $f$ = forward operator (physical model) - $y$ = output observations (measurements, wafer state) The inverse problem seeks to find $x$ given $y$: $$ x = f^{-1}(y) $$ 1.2 Hadamard Well-Posedness Criteria A problem is well-posed if it satisfies: 1. Existence : A solution exists for all admissible data 2. Uniqueness : The solution is unique 3. Stability : The solution depends continuously on the data Most semiconductor inverse problems are ill-posed , violating one or more criteria. 1.3 Why Semiconductor Manufacturing Creates Ill-Posed Problems - Non-uniqueness : Multiple process conditions $\{x_1, x_2, \ldots\}$ can produce indistinguishable outputs within measurement precision - Sensitivity : Small perturbations in measurements cause large changes in estimated parameters: $$ \|x_1 - x_2\| \gg \|y_1 - y_2\| $$ - Incomplete information : Not all relevant physical quantities can be measured 2. Lithography Inverse Problems 2.1 Optical Proximity Correction (OPC) 2.1.1 Forward Model The aerial image intensity at the wafer plane: $$ I(x, y) = \left| \int \int H(f_x, f_y) \cdot M(f_x, f_y) \cdot e^{i2\pi(f_x x + f_y y)} \, df_x \, df_y \right|^2 $$ where: - $H(f_x, f_y)$ = optical transfer function (pupil function) - $M(f_x, f_y)$ = Fourier transform of the mask pattern - $(f_x, f_y)$ = spatial frequencies 2.1.2 Inverse Problem Formulation Find mask pattern $M$ that minimizes: $$ \mathcal{L}(M) = \|T(M) - D\|^2 + \lambda R(M) $$ where: - $T(M)$ = printed pattern from mask $M$ - $D$ = desired (target) pattern - $R(M)$ = regularization for mask manufacturability - $\lambda$ = regularization weight 2.1.3 Regularization Terms Common regularization terms include: - Mask complexity penalty : $$ R_{\text{complexity}}(M) = \int |\nabla M|^2 \, dA $$ - Minimum feature size constraint : $$ R_{\text{MFS}}(M) = \sum_i \max(0, w_{\min} - w_i)^2 $$ - Sidelobe suppression : $$ R_{\text{SRAF}}(M) = \int_{\Omega_{\text{dark}}} I(x,y)^2 \, dA $$ 2.2 Source-Mask Optimization (SMO) Joint optimization over source shape $S$ and mask $M$: $$ \min_{S, M} \|T(S, M) - D\|^2 + \lambda_1 R_S(S) + \lambda_2 R_M(M) $$ This is a higher-dimensional inverse problem with: - Source degrees of freedom: pupil discretization points - Mask degrees of freedom: pixel-based mask representation - Coupled nonlinear interactions 2.3 Inverse Lithography Technology (ILT) Full pixel-based mask optimization using gradient descent: $$ M^{(k+1)} = M^{(k)} - \alpha \nabla_M \mathcal{L}(M^{(k)}) $$ Gradient computation via adjoint method : $$ \nabla_M \mathcal{L} = \text{Re}\left\{ \mathcal{F}^{-1}\left[ H^* \cdot \mathcal{F}\left[ \frac{\partial \mathcal{L}}{\partial I} \cdot \psi^* \right] \right] \right\} $$ where $\psi$ is the complex field at the wafer plane. 3. Thin Film Metrology Inverse Problems 3.1 Ellipsometry 3.1.1 Measured Quantities Ellipsometry measures the complex reflectance ratio: $$ \rho = \frac{r_p}{r_s} = \tan(\Psi) \cdot e^{i\Delta} $$ where: - $r_p$ = p-polarized reflection coefficient - $r_s$ = s-polarized reflection coefficient - $\Psi$ = amplitude ratio angle - $\Delta$ = phase difference 3.1.2 Forward Model (Fresnel Equations) For a single film on substrate: $$ r_{012} = \frac{r_{01} + r_{12} e^{-i2\beta}}{1 + r_{01} r_{12} e^{-i2\beta}} $$ where: - $r_{01}, r_{12}$ = interface Fresnel coefficients - $\beta = \frac{2\pi d}{\lambda} \tilde{n}_1 \cos\theta_1$ = phase thickness - $d$ = film thickness - $\tilde{n}_1 = n_1 + ik_1$ = complex refractive index 3.1.3 Inverse Problem Given measured $\Psi(\lambda), \Delta(\lambda)$, find: - Film thickness(es): $d_1, d_2, \ldots$ - Optical constants: $n(\lambda), k(\lambda)$ for each layer Objective function : $$ \chi^2 = \sum_{\lambda} \left[ \left(\frac{\Psi_{\text{meas}} - \Psi_{\text{calc}}}{\sigma_\Psi}\right)^2 + \left(\frac{\Delta_{\text{meas}} - \Delta_{\text{calc}}}{\sigma_\Delta}\right)^2 \right] $$ 3.2 Scatterometry (Optical Critical Dimension) 3.2.1 Forward Model Rigorous Coupled-Wave Analysis (RCWA) solves Maxwell's equations for periodic structures: $$ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}, \quad \nabla \times \mathbf{H} = \frac{\partial \mathbf{D}}{\partial t} $$ The grating is represented as Fourier series: $$ \varepsilon(x, z) = \sum_m \varepsilon_m(z) e^{imGx} $$ where $G = \frac{2\pi}{\Lambda}$ is the grating vector. 3.2.2 Profile Parameterization A trapezoidal line profile is characterized by: - CD (Critical Dimension) : $w$ - Height : $h$ - Sidewall Angle : $\theta_{\text{SWA}}$ - Corner Rounding : $r$ - Footing/Undercut : $\delta$ Parameter vector: $\mathbf{p} = [w, h, \theta_{\text{SWA}}, r, \delta, \ldots]^T$ 3.2.3 Inverse Problem $$ \hat{\mathbf{p}} = \arg\min_{\mathbf{p}} \sum_{\lambda, \theta} \left( R_{\text{meas}}(\lambda, \theta) - R_{\text{RCWA}}(\lambda, \theta; \mathbf{p}) \right)^2 $$ Challenges : - Non-convex objective with multiple local minima - Parameter correlations (e.g., height vs. refractive index) - Sensitivity varies dramatically across parameters 4. Plasma Etch Inverse Problems 4.1 Etch Rate Modeling 4.1.1 Ion-Enhanced Etching Model $$ \text{ER} = k_0 \cdot \Gamma_{\text{ion}}^a \cdot \Gamma_{\text{neutral}}^b \cdot \exp\left(-\frac{E_a}{k_B T}\right) $$ where: - $\Gamma_{\text{ion}}$ = ion flux - $\Gamma_{\text{neutral}}$ = neutral radical flux - $E_a$ = activation energy - $a, b$ = reaction orders 4.1.2 Aspect Ratio Dependent Etching (ARDE) Etch rate in high-aspect-ratio features: $$ \text{ER}(AR) = \text{ER}_0 \cdot \frac{1}{1 + \alpha \cdot AR^\beta} $$ where $AR = \frac{\text{depth}}{\text{width}}$ is the aspect ratio. 4.2 Profile Reconstruction from OES 4.2.1 Optical Emission Spectroscopy Model Emission intensity for species $j$: $$ I_j(\lambda) = A_j \cdot n_e \cdot n_j \cdot \langle \sigma v \rangle_{j}^{\text{exc}} $$ where: - $n_e$ = electron density - $n_j$ = species density - $\langle \sigma v \rangle$ = rate coefficient for excitation 4.2.2 Inverse Problem From observed $I_j(t)$ time traces, determine: - Etch front position $z(t)$ - Layer interfaces - Process endpoint State estimation formulation : $$ \hat{z}(t) = \arg\min_{z} \|I_{\text{obs}}(t) - I_{\text{model}}(z, t)\|^2 + \lambda \left\|\frac{dz}{dt}\right\|^2 $$ 5. Ion Implantation Inverse Problems 5.1 As-Implanted Profile 5.1.1 LSS Theory (Lindhard-Scharff-Schiøtt) The implanted concentration profile: $$ C(x) = \frac{\Phi}{\sqrt{2\pi} \Delta R_p} \exp\left[-\frac{(x - R_p)^2}{2(\Delta R_p)^2}\right] $$ where: - $\Phi$ = implant dose (ions/cm²) - $R_p$ = projected range - $\Delta R_p$ = straggle (standard deviation) 5.1.2 Dual-Pearson for Channeling For crystalline substrates with channeling: $$ C(x) = (1-f) \cdot P_1(x; R_{p1}, \Delta R_{p1}, \gamma_1, \beta_1) + f \cdot P_2(x; R_{p2}, \Delta R_{p2}, \gamma_2, \beta_2) $$ where $P_i$ are Pearson IV distributions and $f$ is the channeled fraction. 5.2 Diffusion Inversion 5.2.1 Fick's Second Law with Concentration Dependence $$ \frac{\partial C}{\partial t} = \frac{\partial}{\partial x}\left[D(C) \frac{\partial C}{\partial x}\right] $$ For dopants like boron: $$ D(C) = D_i^* \left[1 + \beta_1 \left(\frac{C}{n_i}\right) + \beta_2 \left(\frac{C}{n_i}\right)^2\right] $$ 5.2.2 Inverse Problem Given final SIMS profile $C_{\text{final}}(x)$, find: - Initial implant conditions: $\Phi, E$ (energy) - Anneal conditions: $T(t)$, time $t_a$ - Diffusion parameters: $D_i^*, \beta_1, \beta_2$ Regularized formulation : $$ \min_{\theta} \|C_{\text{SIMS}} - C_{\text{simulated}}(\theta)\|^2 + \lambda \|\theta - \theta_{\text{prior}}\|^2 $$ 6. Deposition Inverse Problems 6.1 CVD Step Coverage 6.1.1 Thiele Modulus Conformality characterized by: $$ \phi = L \sqrt{\frac{k_s}{D_{\text{Kn}}}} $$ where: - $L$ = feature depth - $k_s$ = surface reaction rate - $D_{\text{Kn}}$ = Knudsen diffusion coefficient Step coverage: $$ SC = \frac{1}{\cosh(\phi)} $$ 6.1.2 Inverse Problem Given target step coverage $SC_{\text{target}}$, find: - Pressure $P$ - Temperature $T$ - Precursor partial pressures - Carrier gas flow 6.2 ALD Thickness Control 6.2.1 Growth Per Cycle (GPC) $$ \text{GPC} = \Theta_{\text{sat}} \cdot d_{\text{ML}} $$ where: - $\Theta_{\text{sat}}$ = saturation coverage (0 to 1) - $d_{\text{ML}}$ = monolayer thickness 6.2.2 Inverse Problem For target thickness $d$: $$ N_{\text{cycles}} = \left\lceil \frac{d}{\text{GPC}(T, t_{\text{pulse}}, t_{\text{purge}})} \right\rceil $$ Optimize $(T, t_{\text{pulse}}, t_{\text{purge}})$ for throughput and uniformity. 7. CMP Inverse Problems 7.1 Preston Equation Material removal rate: $$ \text{MRR} = K_p \cdot P \cdot V $$ where: - $K_p$ = Preston coefficient - $P$ = applied pressure - $V$ = relative velocity 7.2 Pattern Density Effects 7.2.1 Effective Density Model Local removal rate depends on pattern density $\rho$: $$ \text{MRR}_{\text{local}} = \frac{\text{MRR}_{\text{blanket}}}{\rho + (1-\rho) \cdot \eta} $$ where $\eta$ is the selectivity ratio. 7.2.2 Dishing and Erosion - Dishing (over-polish of metal in trench): $$ D = K_d \cdot w \cdot t_{\text{over}} $$ - Erosion (over-polish of dielectric): $$ E = K_e \cdot \rho \cdot t_{\text{over}} $$ 7.3 Inverse Problem Given target post-CMP topography, find: - Polish time - Pressure profile (zone control) - Slurry chemistry - Potentially: design rule modifications for pattern density 8. TCAD Parameter Extraction 8.1 Device Model MOSFET drain current: $$ I_D = \mu_{\text{eff}} C_{\text{ox}} \frac{W}{L} \left[(V_{GS} - V_{th})V_{DS} - \frac{V_{DS}^2}{2}\right] (1 + \lambda V_{DS}) $$ 8.2 Inverse Problem Formulation Given measured $I_D(V_{GS}, V_{DS})$ characteristics, extract: - $V_{th}$ = threshold voltage - $\mu_{\text{eff}}$ = effective mobility - $L_{\text{eff}}$ = effective channel length - $\lambda$ = channel length modulation Optimization : $$ \min_{\theta} \sum_{i,j} \left( I_{D,\text{meas}}(V_{GS,i}, V_{DS,j}) - I_{D,\text{model}}(V_{GS,i}, V_{DS,j}; \theta) \right)^2 $$ 8.3 Interface Trap Density from C-V From measured capacitance $C(V_G)$: $$ D_{it}(E) = \frac{1}{qA}\left(\frac{1}{C_{\text{meas}}} - \frac{1}{C_{\text{ox}}}\right)^{-1} - \frac{C_s}{qA} $$ where $C_s$ is the semiconductor capacitance. 9. Mathematical Solution Approaches 9.1 Regularization Methods 9.1.1 Tikhonov Regularization $$ \hat{x} = \arg\min_x \|Ax - y\|^2 + \lambda\|Lx\|^2 $$ Closed-form solution: $$ \hat{x} = (A^T A + \lambda L^T L)^{-1} A^T y $$ 9.1.2 Total Variation Regularization $$ \min_x \|Ax - y\|^2 + \lambda \int |\nabla x| \, dA $$ Preserves edges while smoothing noise. 9.1.3 L1 Regularization (LASSO) $$ \min_x \|Ax - y\|^2 + \lambda\|x\|_1 $$ Promotes sparse solutions. 9.2 Bayesian Inference 9.2.1 Posterior Distribution By Bayes' theorem: $$ p(x|y) = \frac{p(y|x) \cdot p(x)}{p(y)} \propto p(y|x) \cdot p(x) $$ where: - $p(y|x)$ = likelihood - $p(x)$ = prior - $p(x|y)$ = posterior 9.2.2 Maximum A Posteriori (MAP) Estimate $$ \hat{x}_{\text{MAP}} = \arg\max_x p(x|y) = \arg\max_x [\log p(y|x) + \log p(x)] $$ For Gaussian likelihood and prior: $$ \hat{x}_{\text{MAP}} = \arg\min_x \left[\frac{\|y - Ax\|^2}{2\sigma_n^2} + \frac{\|x - x_0\|^2}{2\sigma_x^2}\right] $$ This recovers Tikhonov regularization with $\lambda = \frac{\sigma_n^2}{\sigma_x^2}$. 9.3 Adjoint Methods for Gradient Computation For objective $\mathcal{L}(x) = \|F(x) - y\|^2$ with expensive forward model $F$: Forward solve : $$ F(x) = y_{\text{sim}} $$ Adjoint solve : $$ \left(\frac{\partial F}{\partial u}\right)^T \lambda = \frac{\partial \mathcal{L}}{\partial u} $$ Gradient : $$ \nabla_x \mathcal{L} = \left(\frac{\partial F}{\partial x}\right)^T \lambda $$ Computational cost: $O(1)$ forward + adjoint solves regardless of parameter dimension. 9.4 Machine Learning Approaches 9.4.1 Neural Network Surrogate Models Train $\hat{F}_\theta(x) \approx F(x)$: $$ \theta^* = \arg\min_\theta \sum_i \|F(x_i) - \hat{F}_\theta(x_i)\|^2 $$ Then use $\hat{F}_\theta$ for fast inverse optimization. 9.4.2 Physics-Informed Neural Networks (PINNs) Loss function includes physics residual: $$ \mathcal{L} = \mathcal{L}_{\text{data}} + \lambda_{\text{PDE}} \mathcal{L}_{\text{PDE}} + \lambda_{\text{BC}} \mathcal{L}_{\text{BC}} $$ where: $$ \mathcal{L}_{\text{PDE}} = \left\|\mathcal{N}[u_\theta(x,t)]\right\|^2 $$ for PDE operator $\mathcal{N}$. 10. Key Challenges and Considerations 10.1 Non-Uniqueness - Definition : Multiple solutions $\{x_1, x_2, \ldots\}$ satisfy $\|F(x_i) - y\| < \epsilon$ - Mitigation : Additional measurements, physical constraints, regularization - Quantification : Null space analysis, condition number $\kappa(A) = \frac{\sigma_{\max}}{\sigma_{\min}}$ 10.2 High Dimensionality - Parameter space : $\dim(x) \sim 10^2$ to $10^6$ (e.g., ILT masks) - Curse of dimensionality : Sampling density scales as $N^d$ - Approaches : Dimensionality reduction, sparse representations, hierarchical models 10.3 Computational Cost - Forward model cost : RCWA: $O(N^3)$ per wavelength; TCAD: hours for full 3D - Inverse iterations : Typically $10^2$ to $10^4$ forward evaluations - Mitigation : Surrogate models, multi-fidelity methods, parallel computing 10.4 Model Uncertainty - Sources : Unmodeled physics, parameter drift, measurement bias - Impact : Inverse solution may fit model but not reality - Approaches : Model calibration, uncertainty propagation, robust optimization 11. Emerging Directions 11.1 Digital Twins - Real-time state estimation combining physics models with sensor data - Kalman filtering for dynamic process tracking: $$ \hat{x}_{k|k} = \hat{x}_{k|k-1} + K_k(y_k - H\hat{x}_{k|k-1}) $$ 11.2 Multi-Fidelity Methods - Hierarchy of models: analytical → reduced-order → full numerical - Efficient exploration with cheap models, refinement with expensive ones - Multi-fidelity Gaussian processes for Bayesian optimization 11.3 Uncertainty Quantification - Full posterior distributions, not just point estimates - Sensitivity analysis: which measurements reduce uncertainty most? - Propagation to downstream process steps and device performance 11.4 End-to-End Differentiable Simulation - Automatic differentiation through entire process flow - Enables gradient-based optimization across traditionally separate steps - Requires differentiable forward models 12. Summary | Process Step | Forward Problem | Inverse Problem | |------------------|---------------------|---------------------| | Lithography | Mask → Printed pattern | Target pattern → Optimal mask | | Ellipsometry | Stack parameters → $\Psi, \Delta$ | $\Psi, \Delta$ → Thickness, n, k | | Scatterometry | Profile → Diffraction spectrum | Spectrum → Profile dimensions | | Plasma Etch | Recipe → Etch profile | Target profile → Recipe | | Ion Implant | Dose, energy → Dopant profile | Target profile → Implant conditions | | CVD/ALD | Recipe → Film properties | Target properties → Recipe | | CMP | Recipe, pattern → Final topography | Target topography → Recipe | | TCAD | Process/device params → I-V curves | I-V curves → Extracted parameters |
Ions align with crystal channels.
Analyze ionic species.
Use ion beam to thin samples.
Reuse designs as separate chiplets.
Detect Fe-B complexes affecting lifetime.
CD difference between isolated and dense features.