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Six Sigma improvement process.
234 technical terms and definitions
Six Sigma improvement process.
Six Sigma methodology.
Define-Measure-Analyze-Improve-Control provides structured methodology for process improvement.
Densely connected NAS search space enables rich architectural diversity through dense connections between layers.
Do-calculus provides rules for deriving causal effects from observational distributions and causal graphs.
Paste code or text and say explain this step by step. I will add comments, clarify logic, and help you refactor or document it clearly.
Containerize ML environments.
I can containerize apps with Docker, write Dockerfiles, and explain basic Kubernetes concepts like pods, services, and deployments.
Document AI extracts structured data from documents. Layout understanding, table extraction, form parsing.
Categorize legal documents.
Generate docstrings and technical documentation from code.
Domain adaptation in ASR transfers models across acoustic environments or speaking styles.
Domain adaptation techniques reduce distribution shift when applying recommendation models to new domains.
Domain adaptation fine-tunes retrievers for specific corpora or use cases.
Domain adaptation theory analyzes how models trained on source distributions perform on related target distributions with distribution shift.
Domain adaptation handles distribution shift between train and deploy. Fine-tune or use domain-specific data.
Adapt from source to target domain.
Train features indistinguishable across domains.
Classifier distinguishing domains.
Train to work on unseen domains.
Train to work on unseen domains.
Mix data from different domains.
Train on diverse synthetic domains.
When test distribution differs from training.
Continue pre-training on domain data.
Adapt to new domains over time.
Learn features unchanged across domains.
Create specialized languages for domains.
Domain-specific models specialize in particular fields through targeted training.
Most common cause of failure.
Most common failure mode.
Movement of dopants at high temperature.
# 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: λ = 1 / (N · π · p_max²) b. Select random impact parameter: p = p_max · √(random[0,1]) c. Solve scattering integral for deflection angle Θ d. Calculate energy transfer to target atom: T = T_max · sin²(Θ/2) e. Update ion energy: E → E - T - ΔE_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. 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 | ## Reference Formulas ### Essential Equations Card ``` - ┌─────────────────────────────────────────────────────────────────┐ │ GAUSSIAN PROFILE │ │ C(x) = Φ/(√(2π)·ΔRp) · exp[-(x-Rp)²/(2ΔRp²)] │ ├─────────────────────────────────────────────────────────────────┤ │ PEAK CONCENTRATION │ │ Cmax ≈ 0.4·Φ/ΔRp │ ├─────────────────────────────────────────────────────────────────┤ │ JUNCTION DEPTH │ │ xj = Rp + ΔRp·√(2·ln(Cmax/CB)) │ ├─────────────────────────────────────────────────────────────────┤ │ SHEET RESISTANCE │ │ Rs = 1/(q·∫μ(C)·C(x)dx) │ ├─────────────────────────────────────────────────────────────────┤ │ DISPLACEMENT DAMAGE │ │ Nd = 0.8·ED/(2Ed) │ └─────────────────────────────────────────────────────────────────┘ ```
# 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: λ = 1 / (N · π · p_max²) b. Select random impact parameter: p = p_max · √(random[0,1]) c. Solve scattering integral for deflection angle Θ d. Calculate energy transfer to target atom: T = T_max · sin²(Θ/2) e. Update ion energy: E → E - T - ΔE_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. 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 | ## Reference Formulas ### Essential Equations Card ``` - ┌─────────────────────────────────────────────────────────────────┐ │ GAUSSIAN PROFILE │ │ C(x) = Φ/(√(2π)·ΔRp) · exp[-(x-Rp)²/(2ΔRp²)] │ ├─────────────────────────────────────────────────────────────────┤ │ PEAK CONCENTRATION │ │ Cmax ≈ 0.4·Φ/ΔRp │ ├─────────────────────────────────────────────────────────────────┤ │ JUNCTION DEPTH │ │ xj = Rp + ΔRp·√(2·ln(Cmax/CB)) │ ├─────────────────────────────────────────────────────────────────┤ │ SHEET RESISTANCE │ │ Rs = 1/(q·∫μ(C)·C(x)dx) │ ├─────────────────────────────────────────────────────────────────┤ │ DISPLACEMENT DAMAGE │ │ Nd = 0.8·ED/(2Ed) │ └─────────────────────────────────────────────────────────────────┘ ```
Extract medication dosages.
Test error decreases then increases then decreases again as model size grows.
Use subset of data.
Differentially Private Stochastic Gradient Descent clips and noises gradients for privacy.
Fast ODE-based diffusion solver.
Fast ODE solver for diffusion sampling.
Improved DPM solver.
Choose appropriate draft model.
Draft models quickly generate candidate tokens for speculative decoding.
Fine-tune on specific subject.
DreamBooth personalizes text-to-image models by fine-tuning on few subject images.
Fine-tune diffusion models to generate specific subjects or styles.
DreamFusion optimizes NeRF from text using score distillation from diffusion models.
Classical transport equations.
Diffuse dopants deeper into wafer at high temperature.
Drop testing simulates mechanical shock from device drops assessing package and interconnect integrity.