lithography and optics

**Semiconductor Manufacturing: Optics and Lithography Mathematical Modeling** A comprehensive guide to the mathematical foundations of semiconductor lithography, covering electromagnetic theory, Fourier optics, optimization mathematics, and stochastic processes. **1. Fundamental Imaging Theory** **1.1 The Resolution Limits** The Rayleigh equations define the physical limits of optical lithography: **Resolution:** $$ R = k_1 \cdot \frac{\lambda}{NA} $$ **Depth of Focus:** $$ DOF = k_2 \cdot \frac{\lambda}{NA^2} $$ **Parameter Definitions:** - $\lambda$ — Wavelength of light (193nm for ArF immersion, 13.5nm for EUV) - $NA = n \cdot \sin(\theta)$ — Numerical aperture - $n$ — Refractive index of immersion medium - $\theta$ — Half-angle of the lens collection cone - $k_1, k_2$ — Process-dependent factors (typically $k_1 \geq 0.25$ from Rayleigh criterion; modern processes achieve $k_1 \sim 0.3–0.4$) **Fundamental Tension:** - Improving resolution requires: - Increasing $NA$, OR - Decreasing $\lambda$ - Both degrade depth of focus **quadratically** ($\propto NA^{-2}$) **2. Fourier Optics Framework** The projection lithography system is modeled as a **linear shift-invariant system** in the Fourier domain. **2.1 Coherent Imaging** For a perfectly coherent source, the image field is given by convolution: $$ E_{image}(x,y) = E_{object}(x,y) \otimes h(x,y) $$ In frequency space (via Fourier transform): $$ \tilde{E}_{image}(f_x, f_y) = \tilde{E}_{object}(f_x, f_y) \cdot H(f_x, f_y) $$ **Key Components:** - $h(x,y)$ — Amplitude Point Spread Function (PSF) - $H(f_x, f_y)$ — Coherent Transfer Function (pupil function) - Typically a `circ` function for circular aperture - Cuts off spatial frequencies beyond $\frac{NA}{\lambda}$ **2.2 Partially Coherent Imaging — The Hopkins Formulation** Real lithography systems operate in the **partially coherent regime**: $$ \sigma = 0.3 - 0.9 $$ where $\sigma$ is the ratio of condenser NA to objective NA. **Transmission Cross Coefficient (TCC) Integral** The aerial image intensity is: $$ I(x,y) = \int\!\!\!\int\!\!\!\int\!\!\!\int TCC(f_1,g_1,f_2,g_2) \cdot M(f_1,g_1) \cdot M^*(f_2,g_2) \cdot e^{2\pi i[(f_1-f_2)x + (g_1-g_2)y]} \, df_1 \, dg_1 \, df_2 \, dg_2 $$ The TCC itself is defined as: $$ TCC(f_1,g_1,f_2,g_2) = \int\!\!\!\int J(f,g) \cdot P(f+f_1, g+g_1) \cdot P^*(f+f_2, g+g_2) \, df \, dg $$ **Parameter Definitions:** - $J(f,g)$ — Source intensity distribution (conventional, annular, dipole, quadrupole, or freeform) - $P$ — Pupil function (including aberrations) - $M$ — Mask transmission/diffraction spectrum - $M^*$ — Complex conjugate of mask spectrum **Computational Note:** This is a 4D integral over frequency space for every image point — computationally expensive but essential for accuracy. **3. Computational Acceleration: SOCS Decomposition** Direct TCC computation is prohibitive. The **Sum of Coherent Systems (SOCS)** method uses eigendecomposition: $$ TCC(f_1,g_1,f_2,g_2) \approx \sum_{i=1}^{N} \lambda_i \cdot \phi_i(f_1,g_1) \cdot \phi_i^*(f_2,g_2) $$ **Decomposition Components:** - $\lambda_i$ — Eigenvalues (sorted by magnitude) - $\phi_i$ — Eigenfunctions (kernels) The image becomes a sum of coherent images: $$ I(x,y) \approx \sum_{i=1}^{N} \lambda_i \cdot \left| m(x,y) \otimes \phi_i(x,y) \right|^2 $$ **Computational Properties:** - Typically $N = 10–50$ kernels capture $>99\%$ of imaging behavior - Each convolution computed via FFT - Complexity: $O(N \log N)$ per kernel **4. Vector Electromagnetic Effects at High NA** When $NA > 0.7$ (immersion lithography reaches $NA \sim 1.35$), scalar diffraction theory fails. The **vector nature of light** must be modeled. **4.1 Richards-Wolf Vector Diffraction** The electric field near focus: $$ \mathbf{E}(r,\psi,z) = -\frac{ikf}{2\pi} \int_0^{\theta_{max}} \int_0^{2\pi} \mathbf{A}(\theta,\phi) \cdot P(\theta,\phi) \cdot e^{ik[z\cos\theta + r\sin\theta\cos(\phi-\psi)]} \sin\theta \, d\theta \, d\phi $$ **Variables:** - $\mathbf{A}(\theta,\phi)$ — Polarization-dependent amplitude vector - $P(\theta,\phi)$ — Pupil function - $k = \frac{2\pi}{\lambda}$ — Wave number - $(r, \psi, z)$ — Cylindrical coordinates at image plane **4.2 Polarization Effects** For high-NA imaging, polarization significantly affects image contrast: | Polarization | Description | Behavior | |:-------------|:------------|:---------| | **TE (s-polarization)** | Electric field ⊥ to plane of incidence | Interferes constructively | | **TM (p-polarization)** | Electric field ∥ to plane of incidence | Suffers contrast loss at high angles | **Consequences:** - Horizontal vs. vertical features print differently - Requires illumination polarization control: - Tangential polarization - Radial polarization - Optimized/freeform polarization **5. Aberration Modeling: Zernike Polynomials** Wavefront aberrations are expanded in **Zernike polynomials** over the unit pupil: $$ W(\rho,\theta) = \sum_{n,m} Z_n^m \cdot R_n^{|m|}(\rho) \cdot \begin{cases} \cos(m\theta) & m \geq 0 \\ \sin(|m|\theta) & m < 0 \end{cases} $$ **5.1 Key Aberrations Affecting Lithography** | Zernike Term | Aberration | Effect on Imaging | |:-------------|:-----------|:------------------| | $Z_4$ | Defocus | Pattern-dependent CD shift | | $Z_5, Z_6$ | Astigmatism | H/V feature difference | | $Z_7, Z_8$ | Coma | Pattern shift, asymmetric printing | | $Z_9$ | Spherical | Through-pitch CD variation | | $Z_{10}, Z_{11}$ | Trefoil | Three-fold symmetric distortion | **5.2 Aberrated Pupil Function** The pupil function with aberrations: $$ P(\rho,\theta) = P_0(\rho,\theta) \cdot \exp\left[\frac{2\pi i}{\lambda} W(\rho,\theta)\right] $$ **Engineering Specifications:** - Modern scanners control Zernikes through adjustable lens elements - Typical specification: $< 0.5\text{nm}$ RMS wavefront error **6. Rigorous Mask Modeling** **6.1 Thin Mask (Kirchhoff) Approximation** Assumes the mask is infinitely thin: $$ M(x,y) = t(x,y) \cdot e^{i\phi(x,y)} $$ **Limitations:** - Fails for advanced nodes - Mask topography (absorber thickness $\sim 50–70\text{nm}$) affects diffraction **6.2 Rigorous Electromagnetic Field (EMF) Methods** **6.2.1 Rigorous Coupled-Wave Analysis (RCWA)** The mask is treated as a **periodic grating**. Fields are expanded in Fourier series: $$ E(x,z) = \sum_n E_n(z) \cdot e^{i(k_{x0} + nK)x} $$ **Parameters:** - $K = \frac{2\pi}{\text{pitch}}$ — Grating vector - $k_{x0}$ — Incident wave x-component Substituting into Maxwell's equations yields **coupled ODEs** solved as an eigenvalue problem in each z-layer. **6.2.2 FDTD (Finite-Difference Time-Domain)** Directly discretizes Maxwell's curl equations on a **Yee grid**: $$ \frac{\partial \mathbf{E}}{\partial t} = \frac{1}{\epsilon} abla \times \mathbf{H} $$ $$ \frac{\partial \mathbf{H}}{\partial t} = -\frac{1}{\mu} abla \times \mathbf{E} $$ **Characteristics:** - Explicit time-stepping - Computationally intensive - Handles arbitrary geometries **7. Photoresist Modeling** **7.1 Exposure: Dill ABC Model** The photoactive compound (PAC) concentration $M$ evolves as: $$ \frac{\partial M}{\partial t} = -I(z,t) \cdot [A \cdot M + B] \cdot M $$ **Parameters:** - $A$ — Bleachable absorption coefficient - $B$ — Non-bleachable absorption coefficient - $I(z,t)$ — Intensity in the resist Light intensity in the resist follows Beer-Lambert: $$ \frac{\partial I}{\partial z} = -\alpha(M) \cdot I $$ where $\alpha = A \cdot M + B$. **7.2 Post-Exposure Bake: Reaction-Diffusion** For **chemically amplified resists (CAR)**: $$ \frac{\partial m}{\partial t} = D abla^2 m - k_{amp} \cdot m \cdot [H^+] $$ **Variables:** - $m$ — Blocking group concentration - $D$ — Diffusivity (temperature-dependent, Arrhenius behavior) - $[H^+]$ — Acid concentration Acid diffusion and quenching: $$ \frac{\partial [H^+]}{\partial t} = D_H abla^2 [H^+] - k_q [H^+][Q] $$ where $Q$ is quencher concentration. **7.3 Development: Mack Model** Development rate as a function of inhibitor concentration $m$: $$ R(m) = R_{max} \cdot \frac{(a+1)(1-m)^n}{a + (1-m)^n} + R_{min} $$ **Parameters:** - $a, n$ — Kinetic parameters - $R_{max}$ — Maximum development rate - $R_{min}$ — Minimum development rate (unexposed) This creates the **nonlinear resist response** that sharpens edges. **8. Optical Proximity Correction (OPC)** **8.1 The Inverse Problem** Given target pattern $T$, find mask $M$ such that: $$ \text{Image}(M) \approx T $$ **8.2 Model-Based OPC** Iterative edge-based correction. Cost function: $$ \mathcal{L} = \sum_i w_i \cdot (EPE_i)^2 + \lambda \cdot R(M) $$ **Components:** - $EPE_i$ — Edge Placement Error (distance from target at evaluation point $i$) - $w_i$ — Weight for each evaluation point - $R(M)$ — Regularization term for mask manufacturability Gradient descent update: $$ M^{(k+1)} = M^{(k)} - \eta \frac{\partial \mathcal{L}}{\partial M} $$ **Gradient Computation Methods:** - Adjoint methods (efficient for many output points) - Direct differentiation of SOCS kernels **8.3 Inverse Lithography Technology (ILT)** Full pixel-based mask optimization: $$ \min_M \left\| I(M) - I_{target} \right\|^2 + \lambda_1 \|M\|_{TV} + \lambda_2 \| abla^2 M\|^2 $$ **Regularization Terms:** - $\|M\|_{TV}$ — Total Variation promotes sharp mask edges - $\| abla^2 M\|^2$ — Laplacian term controls curvature **Result:** ILT produces **curvilinear masks** with superior imaging, enabled by multi-beam mask writers. **9. Source-Mask Optimization (SMO)** Joint optimization of illumination source $J$ and mask $M$: $$ \min_{J,M} \mathcal{L}(J,M) = \left\| I(J,M) - I_{target} \right\|^2 + \text{process window terms} $$ **9.1 Constraints** **Source Constraints:** - Pixelized representation - Non-negative intensity: $J \geq 0$ - Power constraint: $\int J \, dA = P_0$ **Mask Constraints:** - Minimum feature size - Maximum curvature - Manufacturability rules **9.2 Mathematical Properties** The problem is **bilinear in $J$ and $M$** (linear in each separately), enabling: - Alternating optimization - Joint gradient methods **9.3 Process Window Co-optimization** Adds robustness across focus and dose variations: $$ \mathcal{L}_{PW} = \sum_{focus, dose} w_{f,d} \cdot \left\| I_{f,d}(J,M) - I_{target} \right\|^2 $$ **10. EUV-Specific Mathematics** **10.1 Multilayer Reflector** Mo/Si multilayer with **40–50 bilayer pairs**. Peak reflectivity from Bragg condition: $$ 2d \cdot \cos\theta = n\lambda $$ **Parameters:** - $d \approx 6.9\text{nm}$ — Bilayer period for $\lambda = 13.5\text{nm}$ - Near-normal incidence ($\theta \approx 0°$) **Transfer Matrix Method** Reflectivity calculation: $$ \begin{pmatrix} E_{out}^+ \\ E_{out}^- \end{pmatrix} = \prod_{j=1}^{N} M_j \begin{pmatrix} E_{in}^+ \\ E_{in}^- \end{pmatrix} $$ where $M_j$ is the transfer matrix for layer $j$. **10.2 Mask 3D Effects** EUV masks are **reflective** with absorber patterns. At 6° chief ray angle: - **Shadowing:** Different illumination angles see different absorber profiles - **Best focus shift:** Pattern-dependent focus offsets Requires **full 3D EMF simulation** (RCWA or FDTD) for accurate modeling. **10.3 Stochastic Effects** At EUV, photon counts are low enough that **shot noise** matters: $$ \sigma_{photon} = \sqrt{N_{photon}} $$ **Line Edge Roughness (LER) Contributions** - Photon shot noise - Acid shot noise - Resist molecular granularity **Power Spectral Density Model** $$ PSD(f) = \frac{A}{1 + (2\pi f \xi)^{2+2H}} $$ **Parameters:** - $\xi$ — Correlation length - $H$ — Hurst exponent (typically $0.5–0.8$) - $A$ — Amplitude **Stochastic Simulation via Monte Carlo** - Poisson-distributed photon absorption - Random acid generation and diffusion - Development with local rate variations **11. Process Window Analysis** **11.1 Bossung Curves** CD vs. focus at multiple dose levels: $$ CD(E, F) = CD_0 + a_1 E + a_2 F + a_3 E^2 + a_4 F^2 + a_5 EF + \cdots $$ Polynomial expansion fitted to simulation/measurement. **11.2 Normalized Image Log-Slope (NILS)** $$ NILS = w \cdot \left. \frac{d \ln I}{dx} \right|_{edge} $$ **Parameters:** - $w$ — Feature width - Evaluated at the edge position **Design Rule:** $NILS > 2$ generally required for acceptable process latitude. **Relationship to Exposure Latitude:** $$ EL \propto NILS $$ **11.3 Depth of Focus (DOF) and Exposure Latitude (EL) Trade-off** Visualized as overlapping process windows across pattern types — the **common process window** must satisfy all critical features. **12. Multi-Patterning Mathematics** **12.1 SADP (Self-Aligned Double Patterning)** $$ \text{Spacer pitch} = \frac{\text{Mandrel pitch}}{2} $$ **Design Rule Constraints:** - Mandrel CD and pitch - Spacer thickness uniformity - Cut pattern overlay **12.2 LELE (Litho-Etch-Litho-Etch) Decomposition** **Graph coloring problem:** Assign features to masks such that: - Features on same mask satisfy minimum spacing - Total mask count minimized (typically 2) **Computational Properties:** - For 1D patterns: Equivalent to 2-colorable graph (bipartite) - For 2D: **NP-complete** in general **Solution Methods:** - Integer Linear Programming (ILP) - SAT solvers - Heuristic algorithms **Conflict Graph Edge Weight:** $$ w_{ij} = \begin{cases} \infty & \text{if } d_{ij} < d_{min,same} \\ 0 & \text{otherwise} \end{cases} $$ **13. Machine Learning Integration** **13.1 Surrogate Models** Neural networks approximate aerial image or resist profile: $$ I_{NN}(x; M) \approx I_{physics}(x; M) $$ **Benefits:** - Training on physics simulation data - Inference 100–1000× faster **13.2 OPC with ML** - **CNNs:** Predict edge corrections - **GANs:** Generate mask patterns - **Reinforcement Learning:** Iterative OPC optimization **13.3 Hotspot Detection** Classification of lithographic failure sites: $$ P(\text{hotspot} \mid \text{pattern}) = \sigma(W \cdot \phi(\text{pattern}) + b) $$ where $\sigma$ is the sigmoid function and $\phi$ extracts pattern features. **14. Mathematical Optimization Framework** **14.1 Constrained Optimization Formulation** $$ \min f(x) \quad \text{subject to} \quad g(x) \leq 0, \quad h(x) = 0 $$ **Solution Methods:** - Sequential Quadratic Programming (SQP) - Interior Point Methods - Augmented Lagrangian **14.2 Regularization Techniques** | Regularization | Formula | Effect | |:---------------|:--------|:-------| | L1 (Sparsity) | $\| abla M\|_1$ | Promotes sparse gradients | | L2 (Smoothness) | $\| abla M\|_2^2$ | Promotes smooth transitions | | Total Variation | $\int | abla M| \, dx$ | Preserves edges while smoothing | **15. Mathematical Stack** | Layer | Mathematics | |:------|:------------| | Electromagnetic Propagation | Maxwell's equations, RCWA, FDTD | | Image Formation | Fourier optics, TCC, Hopkins, vector diffraction | | Aberrations | Zernike polynomials, wavefront phase | | Photoresist | Coupled PDEs (reaction-diffusion) | | Correction (OPC/ILT) | Inverse problems, constrained optimization | | SMO | Bilinear optimization, gradient methods | | Stochastics (EUV) | Poisson processes, Monte Carlo | | Multi-Patterning | Graph theory, combinatorial optimization | | Machine Learning | Neural networks, surrogate models | **Reference Formulas** **Core Equations** ``` Resolution: R = k₁ × λ / NA Depth of Focus: DOF = k₂ × λ / NA² Numerical Aperture: NA = n × sin(θ) NILS: NILS = w × (d ln I / dx)|edge Bragg Condition: 2d × cos(θ) = nλ Shot Noise: σ = √N ``` **Typical Parameter Values** | Parameter | Typical Value | Application | |:----------|:--------------|:------------| | $\lambda$ (ArF) | 193 nm | Immersion lithography | | $\lambda$ (EUV) | 13.5 nm | EUV lithography | | $NA$ (Immersion) | 1.35 | High-NA ArF | | $NA$ (EUV) | 0.33 – 0.55 | Current/High-NA EUV | | $k_1$ | 0.3 – 0.4 | Advanced nodes | | $\sigma$ (Partial Coherence) | 0.3 – 0.9 | Illumination | | Zernike RMS | < 0.5 nm | Aberration spec |