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# Advanced Mathematics in Semiconductor Manufacturing
## 1. Lithography & Optical Physics
This is arguably the most mathematically demanding area of semiconductor manufacturing.
### 1.1 Fourier Optics & Partial Coherence Theory
The foundation of photolithography treats optical imaging as a spatial frequency filtering problem.
- **Key Concept**: The mask pattern is decomposed into spatial frequency components
- **Optical System**: Acts as a low-pass filter on spatial frequencies
- **Hopkins Formulation**: Describes partially coherent imaging
The aerial image intensity $I(x,y)$ is given by:
$$
I(x,y) = \iint\iint TCC(f_1, g_1, f_2, g_2) \cdot M(f_1, g_1) \cdot M^*(f_2, g_2) \cdot e^{2\pi i[(f_1-f_2)x + (g_1-g_2)y]} \, df_1 \, dg_1 \, df_2 \, dg_2
$$
Where:
- $TCC$ = Transmission Cross-Coefficient
- $M(f,g)$ = Mask spectrum (Fourier transform of mask pattern)
- $M^*$ = Complex conjugate of mask spectrum
**SOCS Decomposition** (Sum of Coherent Systems):
$$
TCC(f_1, g_1, f_2, g_2) = \sum_{k=1}^{N} \lambda_k \phi_k(f_1, g_1) \phi_k^*(f_2, g_2)
$$
- Eigenvalue decomposition makes computation tractable
- $\lambda_k$ are eigenvalues (typically only 10-20 terms needed)
- $\phi_k$ are eigenfunctions
### 1.2 Inverse Lithography Technology (ILT)
Given a desired wafer pattern $T(x,y)$, find the optimal mask $M(x,y)$.
**Mathematical Framework**:
- **Objective Function**:
$$
\min_{M} \left\| I[M](x,y) - T(x,y) \right\|^2 + \alpha R[M]
$$
- **Key Methods**:
- Variational calculus and gradient descent in function spaces
- Level-set methods for topology optimization:
$$
\frac{\partial \phi}{\partial t} + v|\nabla\phi| = 0
$$
- Tikhonov regularization: $R[M] = \|\nabla M\|^2$
- Total-variation regularization: $R[M] = \int |\nabla M| \, dx \, dy$
- Adjoint methods for efficient gradient computation
### 1.3 EUV & Rigorous Electromagnetics
At $\lambda = 13.5$ nm, scalar diffraction theory fails. Full vector Maxwell's equations are required.
**Maxwell's Equations** (time-harmonic form):
$$
\nabla \times \mathbf{E} = -i\omega\mu\mathbf{H}
$$
$$
\nabla \times \mathbf{H} = i\omega\varepsilon\mathbf{E}
$$
**Numerical Methods**:
- **RCWA** (Rigorous Coupled-Wave Analysis):
- Eigenvalue problem for each diffraction order
- Transfer matrix for multilayer stacks:
$$
\begin{pmatrix} E^+ \\ E^- \end{pmatrix}_{out} = \mathbf{T} \begin{pmatrix} E^+ \\ E^- \end{pmatrix}_{in}
$$
- **FDTD** (Finite-Difference Time-Domain):
- Yee grid discretization
- Leapfrog time integration:
$$
E^{n+1} = E^n + \frac{\Delta t}{\varepsilon} \nabla \times H^{n+1/2}
$$
- **Multilayer Thin-Film Optics**:
- Fresnel coefficients at each interface
- Transfer matrix method for $N$ layers
### 1.4 Aberration Theory
Optical aberrations characterized using **Zernike Polynomials**:
$$
W(\rho, \theta) = \sum_{n,m} Z_n^m R_n^m(\rho) \cdot
\begin{cases}
\cos(m\theta) & \text{(even)} \\
\sin(m\theta) & \text{(odd)}
\end{cases}
$$
Where $R_n^m(\rho)$ are radial polynomials:
$$
R_n^m(\rho) = \sum_{k=0}^{(n-m)/2} \frac{(-1)^k (n-k)!}{k! \left(\frac{n+m}{2}-k\right)! \left(\frac{n-m}{2}-k\right)!} \rho^{n-2k}
$$
**Common Aberrations**:
| Zernike Term | Name | Effect |
|--------------|------|--------|
| $Z_4^0$ | Defocus | Uniform blur |
| $Z_3^1$ | Coma | Asymmetric distortion |
| $Z_4^0$ | Spherical | Halo effect |
| $Z_2^2$ | Astigmatism | Directional blur |
## 2. Quantum Mechanics & Device Physics
As transistors reach sub-5nm dimensions, classical models break down.
### 2.1 Schrödinger Equation & Quantum Transport
**Time-Independent Schrödinger Equation**:
$$
\hat{H}\psi = E\psi
$$
$$
\left[-\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf{r})\right]\psi(\mathbf{r}) = E\psi(\mathbf{r})
$$
**Non-Equilibrium Green's Function (NEGF) Formalism**:
- Retarded Green's function:
$$
G^R(E) = \left[(E + i\eta)I - H - \Sigma_L - \Sigma_R\right]^{-1}
$$
- Self-energy $\Sigma$ incorporates:
- Contact coupling
- Scattering mechanisms
- Electron-phonon interaction
- Current calculation:
$$
I = \frac{2e}{h} \int T(E) [f_L(E) - f_R(E)] \, dE
$$
- Transmission function:
$$
T(E) = \text{Tr}\left[\Gamma_L G^R \Gamma_R G^A\right]
$$
**Wigner Function** (bridging quantum and semiclassical):
$$
W(x,p) = \frac{1}{2\pi\hbar} \int \psi^*\left(x + \frac{y}{2}\right) \psi\left(x - \frac{y}{2}\right) e^{ipy/\hbar} \, dy
$$
### 2.2 Band Structure Theory
**$k \cdot p$ Perturbation Theory**:
$$
H_{k \cdot p} = \frac{p^2}{2m_0} + V(\mathbf{r}) + \frac{\hbar}{m_0}\mathbf{k} \cdot \mathbf{p} + \frac{\hbar^2 k^2}{2m_0}
$$
**Effective Mass Tensor**:
$$
\frac{1}{m^*_{ij}} = \frac{1}{\hbar^2} \frac{\partial^2 E}{\partial k_i \partial k_j}
$$
**Tight-Binding Hamiltonian**:
$$
H = \sum_i \varepsilon_i |i\rangle\langle i| + \sum_{\langle i,j \rangle} t_{ij} |i\rangle\langle j|
$$
- $\varepsilon_i$ = on-site energy
- $t_{ij}$ = hopping integral (Slater-Koster parameters)
### 2.3 Semiclassical Transport
**Boltzmann Transport Equation**:
$$
\frac{\partial f}{\partial t} + \mathbf{v} \cdot \nabla_r f + \frac{\mathbf{F}}{\hbar} \cdot \nabla_k f = \left(\frac{\partial f}{\partial t}\right)_{coll}
$$
- 6D phase space $(x, y, z, k_x, k_y, k_z)$
- Collision integral (scattering):
$$
\left(\frac{\partial f}{\partial t}\right)_{coll} = \sum_{k'} [S(k',k)f(k')(1-f(k)) - S(k,k')f(k)(1-f(k'))]
$$
**Drift-Diffusion Equations** (moment expansion):
$$
\mathbf{J}_n = q\mu_n n\mathbf{E} + qD_n\nabla n
$$
$$
\mathbf{J}_p = q\mu_p p\mathbf{E} - qD_p\nabla p
$$
## 3. Process Simulation PDEs
### 3.1 Dopant Diffusion
**Fick's Second Law** (concentration-dependent):
$$
\frac{\partial C}{\partial t} = \nabla \cdot (D(C,T) \nabla C) + G - R
$$
**Coupled Point-Defect System**:
$$
\begin{aligned}
\frac{\partial C_A}{\partial t} &= \nabla \cdot (D_A \nabla C_A) + k_{AI}C_AC_I - k_{AV}C_AC_V \\
\frac{\partial C_I}{\partial t} &= \nabla \cdot (D_I \nabla C_I) + G_I - k_{IV}C_IC_V \\
\frac{\partial C_V}{\partial t} &= \nabla \cdot (D_V \nabla C_V) + G_V - k_{IV}C_IC_V
\end{aligned}
$$
Where:
- $C_A$ = dopant concentration
- $C_I$ = interstitial concentration
- $C_V$ = vacancy concentration
- $k_{ij}$ = reaction rate constants
### 3.2 Oxidation & Film Growth
**Deal-Grove Model**:
$$
x_{ox}^2 + Ax_{ox} = B(t + \tau)
$$
- $A$ = linear rate constant (surface reaction limited)
- $B$ = parabolic rate constant (diffusion limited)
- $\tau$ = time offset for initial oxide
**Moving Boundary (Stefan) Problem**:
$$
D\frac{\partial C}{\partial x}\bigg|_{x=s(t)} = C^* \frac{ds}{dt}
$$
### 3.3 Ion Implantation
**Binary Collision Approximation** (Monte Carlo):
- Screened Coulomb potential:
$$
V(r) = \frac{Z_1 Z_2 e^2}{r} \phi\left(\frac{r}{a}\right)
$$
- Scattering angle from two-body collision integral
**As-Implanted Profile** (Pearson IV distribution):
$$
f(x) = f_0 \left[1 + \left(\frac{x-R_p}{b}\right)^2\right]^{-m} \exp\left[-r \tan^{-1}\left(\frac{x-R_p}{b}\right)\right]
$$
Parameters: $R_p$ (projected range), $\Delta R_p$ (straggle), skewness, kurtosis
### 3.4 Plasma Etching
**Electron Energy Distribution** (Boltzmann equation):
$$
\frac{\partial f}{\partial t} + \mathbf{v} \cdot \nabla f - \frac{e\mathbf{E}}{m} \cdot \nabla_v f = C[f]
$$
**Child-Langmuir Law** (sheath ion flux):
$$
J = \frac{4\varepsilon_0}{9} \sqrt{\frac{2e}{M}} \frac{V^{3/2}}{d^2}
$$
### 3.5 Chemical-Mechanical Polishing (CMP)
**Preston Equation**:
$$
\frac{dh}{dt} = K_p \cdot P \cdot V
$$
- $K_p$ = Preston coefficient
- $P$ = local pressure
- $V$ = relative velocity
**Pattern-Density Dependent Model**:
$$
P_{local} = P_{avg} \cdot \frac{A_{total}}{A_{contact}(\rho)}
$$
## 4. Electromagnetic Simulation
### 4.1 Interconnect Modeling
**Capacitance Extraction** (Laplace equation):
$$
\nabla^2 \phi = 0 \quad \text{(dielectric regions)}
$$
$$
\nabla \cdot (\varepsilon \nabla \phi) = -\rho \quad \text{(with charges)}
$$
**Boundary Element Method**:
$$
c(\mathbf{r})\phi(\mathbf{r}) = \int_S \left[\phi(\mathbf{r}') \frac{\partial G}{\partial n'} - G(\mathbf{r}, \mathbf{r}') \frac{\partial \phi}{\partial n'}\right] dS'
$$
Where $G(\mathbf{r}, \mathbf{r}') = \frac{1}{4\pi|\mathbf{r} - \mathbf{r}'|}$ (free-space Green's function)
### 4.2 Partial Inductance
**PEEC Method** (Partial Element Equivalent Circuit):
$$
L_{p,ij} = \frac{\mu_0}{4\pi} \frac{1}{a_i a_j} \int_{V_i} \int_{V_j} \frac{d\mathbf{l}_i \cdot d\mathbf{l}_j}{|\mathbf{r}_i - \mathbf{r}_j|}
$$
## 5. Statistical & Stochastic Methods
### 5.1 Process Variability
**Multivariate Gaussian Model**:
$$
p(\mathbf{x}) = \frac{1}{(2\pi)^{n/2}|\Sigma|^{1/2}} \exp\left(-\frac{1}{2}(\mathbf{x}-\boldsymbol{\mu})^T \Sigma^{-1} (\mathbf{x}-\boldsymbol{\mu})\right)
$$
**Principal Component Analysis**:
$$
\mathbf{X} = \mathbf{U}\mathbf{S}\mathbf{V}^T
$$
- Transform to uncorrelated variables
- Dimensionality reduction: retain components with largest singular values
**Polynomial Chaos Expansion**:
$$
Y(\boldsymbol{\xi}) = \sum_{k=0}^{P} y_k \Psi_k(\boldsymbol{\xi})
$$
- $\Psi_k$ = orthogonal polynomial basis (Hermite for Gaussian inputs)
- Enables uncertainty quantification without Monte Carlo
### 5.2 Yield Modeling
**Poisson Defect Model**:
$$
Y = e^{-D \cdot A}
$$
- $D$ = defect density (defects/cm²)
- $A$ = critical area
**Negative Binomial** (clustered defects):
$$
Y = \left(1 + \frac{DA}{\alpha}\right)^{-\alpha}
$$
### 5.3 Reliability Physics
**Weibull Distribution** (lifetime):
$$
F(t) = 1 - \exp\left[-\left(\frac{t}{\eta}\right)^\beta\right]
$$
- $\eta$ = scale parameter (characteristic life)
- $\beta$ = shape parameter (failure mode indicator)
**Black's Equation** (electromigration):
$$
MTTF = A \cdot J^{-n} \cdot \exp\left(\frac{E_a}{k_B T}\right)
$$
## 6. Optimization & Inverse Problems
### 6.1 Design of Experiments
**Response Surface Methodology**:
$$
y = \beta_0 + \sum_i \beta_i x_i + \sum_i \beta_{ii} x_i^2 + \sum_{i
Advanced nodes face elevated thermal challenges from increased power density and reduced thermal conductivity.
Statistical OCV modeling.
Advanced oxidation processes generate hydroxyl radicals degrading persistent organic pollutants.
Advanced patterning techniques enable metal pitch scaling through SADP SAQP or EUV lithography.
Deploy control algorithms in production.
# Semiconductor Manufacturing: Advanced Mathematics
## 1. Lithography & Optical Physics
This is arguably the most mathematically demanding area of semiconductor manufacturing.
### 1.1 Fourier Optics & Partial Coherence Theory
The foundation of photolithography treats optical imaging as a spatial frequency filtering problem.
- **Key Concept**: The mask pattern is decomposed into spatial frequency components
- **Optical System**: Acts as a low-pass filter on spatial frequencies
- **Hopkins Formulation**: Describes partially coherent imaging
The aerial image intensity $I(x,y)$ is given by:
$$
I(x,y) = \iint\iint 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
$$
Where:
- $TCC$ = Transmission Cross-Coefficient
- $M(f,g)$ = Mask spectrum (Fourier transform of mask pattern)
- $M^*$ = Complex conjugate of mask spectrum
**SOCS Decomposition** (Sum of Coherent Systems):
$$
TCC(f_1, g_1, f_2, g_2) = \sum_{k=1}^{N} \lambda_k \phi_k(f_1, g_1) \phi_k^*(f_2, g_2)
$$
- Eigenvalue decomposition makes computation tractable
- $\lambda_k$ are eigenvalues (typically only 10-20 terms needed)
- $\phi_k$ are eigenfunctions
### 1.2 Inverse Lithography Technology (ILT)
Given a desired wafer pattern $T(x,y)$, find the optimal mask $M(x,y)$.
**Mathematical Framework**:
- **Objective Function**:
$$
\min_{M} \left\| I[M](x,y) - T(x,y) \right\|^2 + \alpha R[M]
$$
- **Key Methods**:
- Variational calculus and gradient descent in function spaces
- Level-set methods for topology optimization:
$$
\frac{\partial \phi}{\partial t} + v|\nabla\phi| = 0
$$
- Tikhonov regularization: $R[M] = \|\nabla M\|^2$
- Total-variation regularization: $R[M] = \int |\nabla M| \, dx \, dy$
- Adjoint methods for efficient gradient computation
### 1.3 EUV & Rigorous Electromagnetics
At $\lambda = 13.5$ nm, scalar diffraction theory fails. Full vector Maxwell's equations are required.
**Maxwell's Equations** (time-harmonic form):
$$
\nabla \times \mathbf{E} = -i\omega\mu\mathbf{H}
$$
$$
\nabla \times \mathbf{H} = i\omega\varepsilon\mathbf{E}
$$
**Numerical Methods**:
- **RCWA** (Rigorous Coupled-Wave Analysis):
- Eigenvalue problem for each diffraction order
- Transfer matrix for multilayer stacks:
$$
\begin{pmatrix} E^+ \\ E^- \end{pmatrix}_{out} = \mathbf{T} \begin{pmatrix} E^+ \\ E^- \end{pmatrix}_{in}
$$
- **FDTD** (Finite-Difference Time-Domain):
- Yee grid discretization
- Leapfrog time integration:
$$
E^{n+1} = E^n + \frac{\Delta t}{\varepsilon} \nabla \times H^{n+1/2}
$$
- **Multilayer Thin-Film Optics**:
- Fresnel coefficients at each interface
- Transfer matrix method for $N$ layers
### 1.4 Aberration Theory
Optical aberrations characterized using **Zernike Polynomials**:
$$
W(\rho, \theta) = \sum_{n,m} Z_n^m R_n^m(\rho) \cdot
\begin{cases}
\cos(m\theta) & \text{(even)} \\
\sin(m\theta) & \text{(odd)}
\end{cases}
$$
Where $R_n^m(\rho)$ are radial polynomials:
$$
R_n^m(\rho) = \sum_{k=0}^{(n-m)/2} \frac{(-1)^k (n-k)!}{k! \left(\frac{n+m}{2}-k\right)! \left(\frac{n-m}{2}-k\right)!} \rho^{n-2k}
$$
**Common Aberrations**:
| Zernike Term | Name | Effect |
|--------------|------|--------|
| $Z_4^0$ | Defocus | Uniform blur |
| $Z_3^1$ | Coma | Asymmetric distortion |
| $Z_4^0$ | Spherical | Halo effect |
| $Z_2^2$ | Astigmatism | Directional blur |
## 2. Quantum Mechanics & Device Physics
As transistors reach sub-5nm dimensions, classical models break down.
### 2.1 Schrödinger Equation & Quantum Transport
**Time-Independent Schrödinger Equation**:
$$
\hat{H}\psi = E\psi
$$
$$
\left[-\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf{r})\right]\psi(\mathbf{r}) = E\psi(\mathbf{r})
$$
**Non-Equilibrium Green's Function (NEGF) Formalism**:
- Retarded Green's function:
$$
G^R(E) = \left[(E + i\eta)I - H - \Sigma_L - \Sigma_R\right]^{-1}
$$
- Self-energy $\Sigma$ incorporates:
- Contact coupling
- Scattering mechanisms
- Electron-phonon interaction
- Current calculation:
$$
I = \frac{2e}{h} \int T(E) [f_L(E) - f_R(E)] \, dE
$$
- Transmission function:
$$
T(E) = \text{Tr}\left[\Gamma_L G^R \Gamma_R G^A\right]
$$
**Wigner Function** (bridging quantum and semiclassical):
$$
W(x,p) = \frac{1}{2\pi\hbar} \int \psi^*\left(x + \frac{y}{2}\right) \psi\left(x - \frac{y}{2}\right) e^{ipy/\hbar} \, dy
$$
### 2.2 Band Structure Theory
**k·p Perturbation Theory**:
$$
H_{k \cdot p} = \frac{p^2}{2m_0} + V(\mathbf{r}) + \frac{\hbar}{m_0}\mathbf{k} \cdot \mathbf{p} + \frac{\hbar^2 k^2}{2m_0}
$$
**Effective Mass Tensor**:
$$
\frac{1}{m^*_{ij}} = \frac{1}{\hbar^2} \frac{\partial^2 E}{\partial k_i \partial k_j}
$$
**Tight-Binding Hamiltonian**:
$$
H = \sum_i \varepsilon_i |i\rangle\langle i| + \sum_{\langle i,j \rangle} t_{ij} |i\rangle\langle j|
$$
- $\varepsilon_i$ = on-site energy
- $t_{ij}$ = hopping integral (Slater-Koster parameters)
### 2.3 Semiclassical Transport
**Boltzmann Transport Equation**:
$$
\frac{\partial f}{\partial t} + \mathbf{v} \cdot \nabla_r f + \frac{\mathbf{F}}{\hbar} \cdot \nabla_k f = \left(\frac{\partial f}{\partial t}\right)_{coll}
$$
- 6D phase space $(x, y, z, k_x, k_y, k_z)$
- Collision integral (scattering):
$$
\left(\frac{\partial f}{\partial t}\right)_{coll} = \sum_{k'} [S(k',k)f(k')(1-f(k)) - S(k,k')f(k)(1-f(k'))]
$$
**Drift-Diffusion Equations** (moment expansion):
$$
\mathbf{J}_n = q\mu_n n\mathbf{E} + qD_n\nabla n
$$
$$
\mathbf{J}_p = q\mu_p p\mathbf{E} - qD_p\nabla p
$$
## 3. Process Simulation PDEs
### 3.1 Dopant Diffusion
**Fick's Second Law** (concentration-dependent):
$$
\frac{\partial C}{\partial t} = \nabla \cdot (D(C,T) \nabla C) + G - R
$$
**Coupled Point-Defect System**:
$$
\begin{aligned}
\frac{\partial C_A}{\partial t} &= \nabla \cdot (D_A \nabla C_A) + k_{AI}C_AC_I - k_{AV}C_AC_V \\
\frac{\partial C_I}{\partial t} &= \nabla \cdot (D_I \nabla C_I) + G_I - k_{IV}C_IC_V \\
\frac{\partial C_V}{\partial t} &= \nabla \cdot (D_V \nabla C_V) + G_V - k_{IV}C_IC_V
\end{aligned}
$$
Where:
- $C_A$ = dopant concentration
- $C_I$ = interstitial concentration
- $C_V$ = vacancy concentration
- $k_{ij}$ = reaction rate constants
### 3.2 Oxidation & Film Growth
**Deal-Grove Model**:
$$
x_{ox}^2 + Ax_{ox} = B(t + \tau)
$$
- $A$ = linear rate constant (surface reaction limited)
- $B$ = parabolic rate constant (diffusion limited)
- $\tau$ = time offset for initial oxide
**Moving Boundary (Stefan) Problem**:
$$
D\frac{\partial C}{\partial x}\bigg|_{x=s(t)} = C^* \frac{ds}{dt}
$$
### 3.3 Ion Implantation
**Binary Collision Approximation** (Monte Carlo):
- Screened Coulomb potential:
$$
V(r) = \frac{Z_1 Z_2 e^2}{r} \phi\left(\frac{r}{a}\right)
$$
- Scattering angle from two-body collision integral
**As-Implanted Profile** (Pearson IV distribution):
$$
f(x) = f_0 \left[1 + \left(\frac{x-R_p}{b}\right)^2\right]^{-m} \exp\left[-r \tan^{-1}\left(\frac{x-R_p}{b}\right)\right]
$$
Parameters: $R_p$ (projected range), $\Delta R_p$ (straggle), skewness, kurtosis
### 3.4 Plasma Etching
**Electron Energy Distribution** (Boltzmann equation):
$$
\frac{\partial f}{\partial t} + \mathbf{v} \cdot \nabla f - \frac{e\mathbf{E}}{m} \cdot \nabla_v f = C[f]
$$
**Child-Langmuir Law** (sheath ion flux):
$$
J = \frac{4\varepsilon_0}{9} \sqrt{\frac{2e}{M}} \frac{V^{3/2}}{d^2}
$$
### 3.5 Chemical-Mechanical Polishing (CMP)
**Preston Equation**:
$$
\frac{dh}{dt} = K_p \cdot P \cdot V
$$
- $K_p$ = Preston coefficient
- $P$ = local pressure
- $V$ = relative velocity
**Pattern-Density Dependent Model**:
$$
P_{local} = P_{avg} \cdot \frac{A_{total}}{A_{contact}(\rho)}
$$
## 4. Electromagnetic Simulation
### 4.1 Interconnect Modeling
**Capacitance Extraction** (Laplace equation):
$$
\nabla^2 \phi = 0 \quad \text{(dielectric regions)}
$$
$$
\nabla \cdot (\varepsilon \nabla \phi) = -\rho \quad \text{(with charges)}
$$
**Boundary Element Method**:
$$
c(\mathbf{r})\phi(\mathbf{r}) = \int_S \left[\phi(\mathbf{r}') \frac{\partial G}{\partial n'} - G(\mathbf{r}, \mathbf{r}') \frac{\partial \phi}{\partial n'}\right] dS'
$$
Where $G(\mathbf{r}, \mathbf{r}') = \frac{1}{4\pi|\mathbf{r} - \mathbf{r}'|}$ (free-space Green's function)
### 4.2 Partial Inductance
**PEEC Method** (Partial Element Equivalent Circuit):
$$
L_{p,ij} = \frac{\mu_0}{4\pi} \frac{1}{a_i a_j} \int_{V_i} \int_{V_j} \frac{d\mathbf{l}_i \cdot d\mathbf{l}_j}{|\mathbf{r}_i - \mathbf{r}_j|}
$$
## 5. Statistical & Stochastic Methods
### 5.1 Process Variability
**Multivariate Gaussian Model**:
$$
p(\mathbf{x}) = \frac{1}{(2\pi)^{n/2}|\Sigma|^{1/2}} \exp\left(-\frac{1}{2}(\mathbf{x}-\boldsymbol{\mu})^T \Sigma^{-1} (\mathbf{x}-\boldsymbol{\mu})\right)
$$
**Principal Component Analysis**:
$$
\mathbf{X} = \mathbf{U}\mathbf{S}\mathbf{V}^T
$$
- Transform to uncorrelated variables
- Dimensionality reduction: retain components with largest singular values
**Polynomial Chaos Expansion**:
$$
Y(\boldsymbol{\xi}) = \sum_{k=0}^{P} y_k \Psi_k(\boldsymbol{\xi})
$$
- $\Psi_k$ = orthogonal polynomial basis (Hermite for Gaussian inputs)
- Enables uncertainty quantification without Monte Carlo
### 5.2 Yield Modeling
**Poisson Defect Model**:
$$
Y = e^{-D \cdot A}
$$
- $D$ = defect density (defects/cm²)
- $A$ = critical area
**Negative Binomial** (clustered defects):
$$
Y = \left(1 + \frac{DA}{\alpha}\right)^{-\alpha}
$$
### 5.3 Reliability Physics
**Weibull Distribution** (lifetime):
$$
F(t) = 1 - \exp\left[-\left(\frac{t}{\eta}\right)^\beta\right]
$$
- $\eta$ = scale parameter (characteristic life)
- $\beta$ = shape parameter (failure mode indicator)
**Black's Equation** (electromigration):
$$
MTTF = A \cdot J^{-n} \cdot \exp\left(\frac{E_a}{k_B T}\right)
$$
## 6. Optimization & Inverse Problems
### 6.1 Design of Experiments
**Response Surface Methodology**:
$$
y = \beta_0 + \sum_i \beta_i x_i + \sum_i \beta_{ii} x_i^2 + \sum_{i
Synchronous actor-critic.
Generate adversarial examples for training.
Adversarial debiasing uses adversarial training to remove protected attribute information.
Train adversary that can't predict protected attributes.
Adversarial examples are imperceptibly perturbed inputs causing misclassification revealing model vulnerabilities.
Adversarial examples are inputs crafted to fool models. Small perturbations cause wrong predictions.
Use adversarial examples to probe model understanding.
Inputs designed to fool the model into wrong predictions.
GAN-style discriminator loss.
Hard NLI examples.
Maximum allowed perturbation size.
Adversarial prompts attempt to elicit undesired behaviors testing model robustness.
Test model robustness with adversarial inputs.
Measure resilience to adversarial attacks.
Adversarial robustness measures model resilience to adversarial perturbations.
Append carefully crafted text to jailbreak model.
Adversarial training improves robustness by augmenting training with adversarial examples.
Train on adversarial examples to improve robustness.
Train on adversarial examples.
Train on adversarial examples to make model more robust.
Adversarial training includes adversarial examples in training. Makes models more robust. Expensive.
Attacks to remove watermarks.
Perturb weights for robustness.
Find medication side effects in text.
Inspect simulated image from mask.
AI that recognizes and responds to emotions.
Affinity diagrams group related ideas revealing patterns and themes.
Scan surface with sharp tip to measure topography at nanometer scale.
Attentional Factorization Machines use attention weights to learn importance of different feature interactions.
Attention Free Transformer uses element-wise operations instead of attention.
Agent approval requires human confirmation before executing high-stakes actions.
Agent benchmarking evaluates performance across standardized task suites.
Agent communication protocols enable information exchange and coordination between agents.
Agent debugging identifies and resolves issues in planning and execution logic.
Feedback loops allow humans to correct and guide agent behavior iteratively.
Agent handoff transfers responsibility for tasks between agents smoothly.
Agent logging records decisions actions and reasoning for debugging and auditing.
Agent loops repeatedly observe plan act and update until objectives are achieved.
Agent memory maintains conversation history observations and learned information across interactions.
Agent negotiation resolves conflicts through offers counteroffers and compromise.