ibis model, ibis, signal & power integrity
Input/Output Buffer Information Specification models I/O buffer behavior with V-I curves and timing data for board-level signal integrity simulation.
102 technical terms and definitions
Input/Output Buffer Information Specification models I/O buffer behavior with V-I curves and timing data for board-level signal integrity simulation.
Self-distillation for ViT.
Assign diagnostic codes.
In-Circuit Testing verifies component values and detects manufacturing defects on populated boards using bed-of-nails or flying probe access.
Information Exchange Graph Neural Network handles heterogeneous graphs through iterative information exchange.
Im2col transforms convolution into matrix multiplication enabling optimized BLAS library usage.
Generate text descriptions of images.
Diffusion-based image editing modifies existing images through guided denoising processes.
Generate longer descriptions.
Image upscaling increases resolution while generating plausible high-frequency details.
Learn joint embedding space.
Align images and texts via contrastive loss.
Binary classification of matching pairs.
Determine if image and text match.
Determine if image and text correspond.
Search images with text or vice versa.
Transform images with conditioning.
Transform images from one domain to another (sketch to photo day to night).
Various vision-to-language tasks.
Generate text from images.
Extract text from images or describe visual content in natural language.
Imagen Video generates high-definition videos through cascaded video diffusion models.
Imagen uses cascaded diffusion models with large language model text encoders for photorealistic generation.
Pre-train on larger ImageNet.
Edit real images with text instructions.
Intermetallic compound analysis examines interfacial layers between metals assessing bond quality and reliability.
How much to change input image.
# Semiconductor Manufacturing: Ion Implantation Mathematical Modeling ## 1. Introduction Ion implantation is a critical process in semiconductor fabrication where dopant ions (B, P, As, Sb) are accelerated and embedded into silicon substrates to precisely control electrical properties. **Key Process Parameters:** - **Energy (keV)**: Controls implant depth ($R_p$) - **Dose (ions/cm²)**: Controls peak concentration - **Tilt angle (°)**: Minimizes channeling effects - **Twist angle (°)**: Avoids major crystal planes - **Beam current (mA)**: Affects dose rate and wafer heating ## 2. Foundational Physics: Ion Stopping When an energetic ion enters a solid, it loses energy through two primary mechanisms. ### 2.1 Total Stopping Power $$ \frac{dE}{dx} = N \left[ S_n(E) + S_e(E) \right] $$ Where: - $N$ = atomic density of target ($\approx 5 \times 10^{22}$ atoms/cm³ for Si) - $S_n(E)$ = nuclear stopping cross-section (elastic collisions with nuclei) - $S_e(E)$ = electronic stopping cross-section (inelastic energy loss to electrons) ### 2.2 Nuclear Stopping: ZBL Universal Potential The Ziegler-Biersack-Littmark (ZBL) universal screening function: $$ \phi(x) = 0.1818 e^{-3.2x} + 0.5099 e^{-0.9423x} + 0.2802 e^{-0.4028x} + 0.02817 e^{-0.2016x} $$ Where $x = r/a_u$ is the reduced interatomic distance. **Universal screening length:** $$ a_u = \frac{0.8854 \, a_0}{Z_1^{0.23} + Z_2^{0.23}} $$ Where: - $a_0$ = Bohr radius (0.529 Å) - $Z_1$ = atomic number of incident ion - $Z_2$ = atomic number of target atom ### 2.3 Electronic Stopping **Low energy regime** (velocity-proportional, Lindhard-Scharff): $$ S_e = k_e \sqrt{E} $$ Where: $$ k_e = \frac{1.212 \, Z_1^{7/6} \, Z_2}{(Z_1^{2/3} + Z_2^{2/3})^{3/2} \, M_1^{1/2}} $$ **High energy regime** (Bethe-Bloch formula): $$ S_e = \frac{4\pi Z_1^2 e^4 N Z_2}{m_e v^2} \ln\left(\frac{2 m_e v^2}{I}\right) $$ Where: - $m_e$ = electron mass - $v$ = ion velocity - $I$ = mean ionization potential of target ## 3. Range Statistics and Profile Models ### 3.1 Gaussian Approximation (First Order) For amorphous targets, the as-implanted profile: $$ C(x) = \frac{\Phi}{\sqrt{2\pi} \, \Delta R_p} \exp\left[ -\frac{(x - R_p)^2}{2 \Delta R_p^2} \right] $$ | Symbol | Definition | Units | |--------|------------|-------| | $\Phi$ | Implant dose | ions/cm² | | $R_p$ | Projected range (mean depth) | nm or cm | | $\Delta R_p$ | Range straggle (standard deviation) | nm or cm | **Peak concentration:** $$ C_{max} = \frac{\Phi}{\sqrt{2\pi} \, \Delta R_p} \approx \frac{0.4 \, \Phi}{\Delta R_p} $$ ### 3.2 Pearson IV Distribution (Industry Standard) Real profiles exhibit asymmetry. The Pearson IV distribution uses four statistical moments: $$ f(x) = K \left[ 1 + \left( \frac{x - \lambda}{a} \right)^2 \right]^{-m} \exp\left[ -\nu \arctan\left( \frac{x - \lambda}{a} \right) \right] $$ **Four Moments:** 1. **First Moment (Mean)**: $R_p$ — projected range 2. **Second Moment (Variance)**: $\Delta R_p^2$ — spread 3. **Third Moment (Skewness)**: $\gamma$ — asymmetry - $\gamma < 0$: tail extends deeper into substrate (light ions: B) - $\gamma > 0$: tail extends toward surface (heavy ions: As) 4. **Fourth Moment (Kurtosis)**: $\beta$ — peakedness relative to Gaussian **Typical values for Si:** | Dopant | Skewness ($\gamma$) | Kurtosis ($\beta$) | |--------|---------------------|---------------------| | Boron (B) | -0.5 to +0.5 | 2.5 to 4.0 | | Phosphorus (P) | -0.3 to +0.3 | 2.5 to 3.5 | | Arsenic (As) | +0.5 to +1.5 | 3.0 to 5.0 | | Antimony (Sb) | +0.8 to +2.0 | 3.5 to 6.0 | ### 3.3 Dual Pearson Model (Channeling Effects) For implants into crystalline silicon with channeling tails: $$ C(x) = (1 - f_{ch}) \cdot P_{random}(x) + f_{ch} \cdot P_{channel}(x) $$ Where: - $P_{random}(x)$ = Pearson distribution for random (amorphous) stopping - $P_{channel}(x)$ = Pearson distribution for channeled ions - $f_{ch}$ = channeling fraction (depends on tilt, beam divergence, surface oxide) **Channeling fraction dependencies:** - Beam divergence: $f_{ch} \downarrow$ as divergence $\uparrow$ - Tilt angle: $f_{ch} \downarrow$ as tilt $\uparrow$ (typically 7° off-axis) - Surface oxide: $f_{ch} \downarrow$ with screen oxide - Pre-amorphization: $f_{ch} \approx 0$ with PAI ## 4. Monte Carlo Simulation (BCA Method) The Binary Collision Approximation provides the highest accuracy for profile prediction. ### 4.1 Algorithm Overview ``` FOR each ion i = 1 to N_ions (typically 10⁵ - 10⁶): 1. Initialize: - Energy: E = E₀ - Position: (x, y, z) = (0, 0, 0) - Direction: (cos θ, sin θ cos φ, sin θ sin φ) 2. WHILE E > E_cutoff: a. Calculate mean free path: $\lambda = 1 / (N \cdot \pi \cdot p_{max}^2)$ b. Select random impact parameter: $p = p_{max} \cdot \sqrt{\text{random}[0,1]}$ c. Solve scattering integral for deflection angle $\Theta$ d. Calculate energy transfer to target atom: $T = T_{max} \cdot \sin^2(\Theta/2)$ e. Update ion energy: $E \to E - T - \Delta E_{\text{electronic}}$ f. IF T > E_displacement: Create recoil cascade (track secondary) g. Update position and direction vectors 3. Record final ion position (x_final, y_final, z_final) END FOR 4. Build histogram of final positions → Dopant profile ``` ### 4.2 Scattering Integral The classical scattering integral for deflection angle: $$ \Theta = \pi - 2p \int_{r_{min}}^{\infty} \frac{dr}{r^2 \sqrt{1 - \frac{V(r)}{E_c} - \frac{p^2}{r^2}}} $$ Where: - $p$ = impact parameter - $r_{min}$ = distance of closest approach - $V(r)$ = interatomic potential (e.g., ZBL) - $E_c$ = center-of-mass energy **Center-of-mass energy:** $$ E_c = \frac{M_2}{M_1 + M_2} E $$ ### 4.3 Energy Transfer Maximum energy transfer in elastic collision: $$ T_{max} = \frac{4 M_1 M_2}{(M_1 + M_2)^2} \cdot E = \gamma \cdot E $$ Where $\gamma$ is the kinematic factor: | Ion → Si | $M_1$ (amu) | $\gamma$ | |----------|-------------|----------| | B → Si | 11 | 0.702 | | P → Si | 31 | 0.968 | | As → Si | 75 | 0.746 | ### 4.4 Electronic Energy Loss (Continuous) Along the free flight path: $$ \Delta E_{electronic} = \int_0^{\lambda} S_e(E) \, dx \approx S_e(E) \cdot \lambda $$ ## 5. Multi-Layer and Through-Film Implantation ### 5.1 Screen Oxide Implantation For implantation through oxide layer of thickness $t_{ox}$: **Range correction:** $$ R_p^{eff} = R_p^{Si} - t_{ox} \left( \frac{R_p^{Si} - R_p^{ox}}{R_p^{ox}} \right) $$ **Straggle correction:** $$ (\Delta R_p^{eff})^2 = (\Delta R_p^{Si})^2 - t_{ox} \left( \frac{(\Delta R_p^{Si})^2 - (\Delta R_p^{ox})^2}{R_p^{ox}} \right) $$ ### 5.2 Moment Matching at Interfaces For multi-layer structures, use moment conservation: $$ \langle x^n \rangle_{total} = \sum_i \langle x^n \rangle_i \cdot w_i $$ Where $w_i$ is the weighting factor for layer $i$. ## 6. Two-Dimensional Profile Modeling ### 6.1 Lateral Straggle The lateral distribution follows: $$ C(x, y) = C(x) \cdot \frac{1}{\sqrt{2\pi} \, \Delta R_\perp} \exp\left[ -\frac{y^2}{2 \Delta R_\perp^2} \right] $$ **Relationship between straggles:** $$ \Delta R_\perp \approx (0.7 \text{ to } 1.0) \times \Delta R_p $$ ### 6.2 Masked Implant with Edge Effects For a mask opening of width $W$: $$ C(x, y) = C(x) \cdot \frac{1}{2} \left[ \text{erf}\left( \frac{y + W/2}{\sqrt{2} \, \Delta R_\perp} \right) - \text{erf}\left( \frac{y - W/2}{\sqrt{2} \, \Delta R_\perp} \right) \right] $$ ### 6.3 Full 3D Distribution $$ C(x, y, z) = \frac{\Phi}{(2\pi)^{3/2} \Delta R_p \, \Delta R_\perp^2} \exp\left[ -\frac{(x - R_p)^2}{2 \Delta R_p^2} - \frac{y^2 + z^2}{2 \Delta R_\perp^2} \right] $$ ## 7. Damage and Defect Modeling ### 7.1 Kinchin-Pease Model Number of displaced atoms per incident ion: $$ N_d = \begin{cases} 0 & \text{if } E_D < E_d \\ 1 & \text{if } E_d < E_D < 2E_d \\ \displaystyle\frac{E_D}{2E_d} & \text{if } E_D > 2E_d \end{cases} $$ Where: - $E_D$ = damage energy (energy deposited into nuclear collisions) - $E_d$ = displacement threshold energy ($\approx 15$ eV for Si) ### 7.2 Modified NRT Model (Norgett-Robinson-Torrens) $$ N_d = \frac{0.8 \, E_D}{2 E_d} $$ The factor 0.8 accounts for forward scattering efficiency. ### 7.3 Damage Energy Partition Lindhard partition function: $$ E_D = \frac{E_0}{1 + k \cdot g(\varepsilon)} $$ Where: $$ k = 0.1337 \, Z_1^{1/6} \left( \frac{Z_1}{Z_2} \right)^{1/2} $$ $$ \varepsilon = \frac{32.53 \, M_2 \, E_0}{Z_1 Z_2 (M_1 + M_2)(Z_1^{0.23} + Z_2^{0.23})} $$ ### 7.4 Amorphization Threshold Critical dose for amorphization: $$ \Phi_c \approx \frac{N_0}{N_d \cdot \sigma_{damage}} $$ **Typical values:** | Ion | Critical Dose (cm⁻²) | |-----|----------------------| | B⁺ | $\sim 10^{15}$ | | P⁺ | $\sim 5 \times 10^{14}$ | | As⁺ | $\sim 10^{14}$ | | Sb⁺ | $\sim 5 \times 10^{13}$ | ### 7.5 Damage Profile The damage distribution differs from dopant distribution: $$ D(x) = \frac{\Phi \cdot N_d(E)}{\sqrt{2\pi} \, \Delta R_d} \exp\left[ -\frac{(x - R_d)^2}{2 \Delta R_d^2} \right] $$ Where $R_d < R_p$ (damage peaks shallower than dopant). ## 8. Process-Relevant Calculations ### 8.1 Junction Depth For Gaussian profile meeting background concentration $C_B$: $$ x_j = R_p + \Delta R_p \sqrt{2 \ln\left( \frac{C_{max}}{C_B} \right)} $$ **For asymmetric Pearson profiles:** $$ x_j = R_p + \Delta R_p \left[ \gamma + \sqrt{\gamma^2 + 2 \ln\left( \frac{C_{max}}{C_B} \right)} \right] $$ ### 8.2 Sheet Resistance $$ R_s = \frac{1}{q \displaystyle\int_0^{x_j} \mu(C(x)) \cdot C(x) \, dx} $$ **With concentration-dependent mobility (Masetti model):** $$ \mu(C) = \mu_{min} + \frac{\mu_0}{1 + (C/C_r)^\alpha} - \frac{\mu_1}{1 + (C_s/C)^\beta} $$ | Parameter | Electrons | Holes | |-----------|-----------|-------| | $\mu_{min}$ | 52.2 | 44.9 | | $\mu_0$ | 1417 | 470.5 | | $C_r$ | $9.68 \times 10^{16}$ | $2.23 \times 10^{17}$ | | $\alpha$ | 0.68 | 0.719 | ### 8.3 Threshold Voltage Shift For channel implant: $$ \Delta V_T = \frac{q}{\varepsilon_{ox}} \int_0^{x_{max}} C(x) \cdot x \, dx $$ **Simplified (shallow implant):** $$ \Delta V_T \approx \frac{q \, \Phi \, R_p}{\varepsilon_{ox}} $$ ### 8.4 Dose Calculation from Profile $$ \Phi = \int_0^{\infty} C(x) \, dx $$ **Verification:** $$ \Phi_{measured} = \frac{I \cdot t}{q \cdot A} $$ Where: - $I$ = beam current - $t$ = implant time - $A$ = implanted area ## 9. Advanced Effects ### 9.1 Transient Enhanced Diffusion (TED) The "+1 Model": Each implanted ion creates approximately one net interstitial. **Enhanced diffusion equation:** $$ \frac{\partial C}{\partial t} = \frac{\partial}{\partial x} \left[ D^* \frac{\partial C}{\partial x} \right] $$ **Enhanced diffusivity:** $$ D^* = D_i \cdot \left( 1 + \frac{C_I}{C_I^*} \right) $$ Where: - $D_i$ = intrinsic diffusivity - $C_I$ = interstitial concentration - $C_I^*$ = equilibrium interstitial concentration ### 9.2 Dose Loss Mechanisms **Sputtering yield:** $$ Y = \frac{0.042 \, \alpha \, S_n(E_0)}{U_0} $$ Where: - $\alpha$ = angular factor ($\approx 0.2$ for light ions, $\approx 0.4$ for heavy ions) - $U_0$ = surface binding energy ($\approx 4.7$ eV for Si) **Retained dose:** $$ \Phi_{retained} = \Phi_{implanted} \cdot (1 - \eta_{sputter} - \eta_{backscatter}) $$ ### 9.3 High Dose Effects **Dose saturation:** $$ C_{max}^{sat} = \frac{N_0}{\sqrt{2\pi} \, \Delta R_p} $$ **Snow-plow effect** at very high doses pushes peak toward surface. ### 9.4 Temperature Effects **Dynamic annealing:** Competes with damage accumulation $$ \Phi_c(T) = \Phi_c(0) \exp\left( \frac{E_a}{k_B T} \right) $$ Where $E_a \approx 0.3$ eV for Si self-interstitial migration. ## 10. Summary Tables ### 10.1 Key Scaling Relationships | Parameter | Scaling with Energy | |-----------|---------------------| | Projected Range | $R_p \propto E^n$ where $n \approx 0.5 - 0.8$ | | Range Straggle | $\Delta R_p \approx 0.4 R_p$ (light ions) to $0.2 R_p$ (heavy ions) | | Lateral Straggle | $\Delta R_\perp \approx 0.7 - 1.0 \times \Delta R_p$ | | Damage Energy | $E_D/E_0$ increases with ion mass | ### 10.2 Common Implant Parameters in Si | Dopant | Type | Energy (keV) | $R_p$ (nm) | $\Delta R_p$ (nm) | |--------|------|--------------|------------|-------------------| | B | p | 10 | 35 | 14 | | B | p | 50 | 160 | 52 | | P | n | 30 | 40 | 15 | | P | n | 100 | 120 | 40 | | As | n | 50 | 35 | 12 | | As | n | 150 | 95 | 28 | ### 10.3 Simulation Tools Comparison | Approach | Speed | Accuracy | Primary Use | |----------|-------|----------|-------------| | Analytical (Gaussian) | ★★★★★ | ★★☆☆☆ | Quick estimates | | Pearson IV Tables | ★★★★☆ | ★★★☆☆ | Process simulation | | Monte Carlo (SRIM/TRIM) | ★★☆☆☆ | ★★★★☆ | Profile calibration | | Molecular Dynamics | ★☆☆☆☆ | ★★★★★ | Damage cascade studies | ## Quick Reference Formulas ### Essential Equations Card ``` - ┌─────────────────────────────────────────────────────────────────────────────────────────────┐ │ GAUSSIAN PROFILE │ │ $C(x) = \Phi/(\sqrt{2\pi} \cdot \Delta R_p) \cdot \exp[-(x-R_p)^2/(2\Delta R_p^2)]$ │ ├─────────────────────────────────────────────────────────────────────────────────────────────┤ │ PEAK CONCENTRATION │ │ $C_{max} \approx 0.4 \cdot \Phi/\Delta R_p$ │ ├─────────────────────────────────────────────────────────────────────────────────────────────┤ │ JUNCTION DEPTH │ │ $x_j = R_p + \Delta R_p \cdot \sqrt{2 \cdot \ln(C_{max}/C_B)}$ │ ├─────────────────────────────────────────────────────────────────────────────────────────────┤ │ SHEET RESISTANCE │ │ $R_s = 1/(q \cdot \int \mu(C) \cdot C(x) dx)$ │ ├─────────────────────────────────────────────────────────────────────────────────────────────┤ │ DISPLACEMENT DAMAGE │ │ $N_d = 0.8 \cdot E_D/(2E_d)$ │ └─────────────────────────────────────────────────────────────────────────────────────────────┘ ```
Parameterize signals as neural networks.
Represent data as neural network weights.
Implicit surfaces represent geometry through continuous functions rather than discrete meshes.
Impossibility detection identifies when goals cannot be achieved triggering graceful failure.
Impulse response functions trace effects of shocks to one variable on others over time in VAR systems.
Few-shot learning using image examples.
In-place knowledge distillation in supernets uses larger subnetworks as teachers improving smaller network training.
Classes too coupled.
Inbound logistics manages material flow from suppliers to manufacturing facilities.
Wearout phase.
Malicious instructions hidden in retrieved documents or tool outputs.
Efficient attention for sets.
Circuits implementing in-context learning.
Learn programs from input-output examples.
Attention with infinite context.
Infinite capacity scheduling assumes unlimited resources identifying bottlenecks and capacity needs.
Measure training example impact on predictions.
Disentangle GAN latents using mutual information.
Maximize information about environment.
Informer uses ProbSparse self-attention to reduce computational complexity enabling efficient long sequence time series forecasting.
IR imaging for thermal analysis.
Inhibitory point processes model negative influence where events decrease likelihood of subsequent events.