domain adaptation retrieval, rag
Domain adaptation fine-tunes retrievers for specific corpora or use cases.
656 technical terms and definitions
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.
Fraction of dopants electrically active.
Dopants form inactive clusters.
Unintended doping.
Loss of electrical activity.
Movement of dopants at high temperature.
Impurity atoms (B P As Sb) added to change conductivity.
Model dopant distribution from implant and diffusion.
# 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.
Dot plots display individual observations showing distribution and patterns.
Inner product of vectors used in attention and retrieval.
Test error decreases then increases then decreases again as model size grows.
Reduce overestimation in DQN.
Double sampling allows second sample if first result is inconclusive reducing inspection.
Doubly robust estimators combine outcome regression with propensity scoring for more stable debiasing in recommendations.
Pressure applied to wafer during polishing.
Use subset of data.
Target task using pre-trained representations.
Categorize reasons for downtime.
Time when tool is not available due to failure or maintenance.
Add calibrated noise during training for privacy.
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.
Defects Per Million Opportunities normalizes defect rates across different complexity products.
DPO (Direct Preference Optimization) simplifies RLHF. No separate reward model. Classification objective.
Determinantal Point Process promotes diversity in recommendations through negative correlation modeling.
Quality metric for shipped parts.
Defective Parts Per Million quantifies outgoing quality levels by normalizing defect counts to million opportunities for customer impact assessment.
Dual-Path Recurrent Neural Network alternates between intra-chunk and inter-chunk processing for separation.