backdoor attacks, ai safety
Insert triggers causing misclassification.
9,967 technical terms and definitions
Insert triggers causing misclassification.
Backdoor attacks insert hidden triggers during training. Model misbehaves on triggered inputs.
Use idle resources for small jobs.
Models relying on background instead of object.
Learn background appearance.
Signal with no analyte present.
Separate foreground and background.
Backorders are unfulfilled customer orders awaiting material availability.
Gradient computation in continuous-time models.
Align backside features to front.
Create electrical contact to back.
Damage backside to trap impurities.
Etch away grinding damage.
Gas (He) between wafer and chuck for thermal contact.
Thin wafer from back.
Thin wafer from backside for packaging.
Pattern wafer back surface.
Deposit metal on wafer back.
Route power from wafer backside.
Back-translation augments text data by translating to another language and back generating paraphrases for training.
Protect against data loss.
Backward planning works from goal to current state identifying prerequisite steps.
Start from goal and work backward.
Backward scheduling starts from due date working backward to determine start times.
Molecular representation.
Train models on bootstrap samples.
Bagging trains on bootstrap samples. Reduces variance.
Baichuan is Chinese open source LLM. Good Chinese understanding.
Time and temperature for drying.
Remove moisture before reflow.
Sustainable pace over crunch. AI is marathon not sprint. Balance work with wellbeing.
Sample to balance class distribution.
Ultrasonic wire bonding with ball.
Solder balls on package bottom.
Test solder ball attachment.
Ball shear testing measures solder ball attach strength in BGA packages by shearing balls from pads.
Ball valves provide tight shutoff for chemical isolation.
Carriers traverse without scattering.
Lightweight attention module.
Copper grains spanning line width.
Banana provides GPU serverless inference. Custom model deployment. Potassium framework.
Banana uses Potassium framework. GPU serverless deployment.
Predict electronic band gap of materials.
Compute electronic bands.
# Band Structure Calculations in Semiconductor Manufacturing ## Mathematical Framework ## 1. The Fundamental Problem We need to solve the many-body Schrödinger equation for electrons in a crystal: $$ \hat{H}\Psi = E\Psi $$ The full Hamiltonian includes kinetic energy, ion-electron interaction, and electron-electron repulsion: $$ \hat{H} = -\sum_i \frac{\hbar^2}{2m}\nabla_i^2 + \sum_i V_{\text{ion}}(\mathbf{r}_i) + \frac{1}{2}\sum_{i \neq j} \frac{e^2}{|\mathbf{r}_i - \mathbf{r}_j|} $$ **Key challenges:** - The system contains ~$10^{23}$ electrons - Electron-electron interactions couple all particles - Analytical solution is impossible for real materials - Requires a hierarchy of approximations ## 2. Density Functional Theory (DFT) The workhorse of modern band structure calculations rests on the **Hohenberg-Kohn theorems**: 1. Ground-state properties are uniquely determined by electron density $n(\mathbf{r})$ 2. The true ground-state density minimizes the energy functional ### 2.1 Kohn-Sham Equations The many-body problem is mapped to non-interacting electrons in an effective potential: $$ \left[-\frac{\hbar^2}{2m}\nabla^2 + V_{\text{eff}}(\mathbf{r})\right]\psi_i(\mathbf{r}) = \epsilon_i\psi_i(\mathbf{r}) $$ where the effective potential is: $$ V_{\text{eff}}(\mathbf{r}) = V_{\text{ion}}(\mathbf{r}) + V_H(\mathbf{r}) + V_{xc}[n] $$ **Components of $V_{\text{eff}}$:** - **Ionic potential**: $V_{\text{ion}}(\mathbf{r})$ — interaction with nuclei - **Hartree potential**: $V_H(\mathbf{r}) = \int \frac{n(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|}d\mathbf{r}'$ — classical electrostatic repulsion - **Exchange-correlation**: $V_{xc}[n] = \frac{\delta E_{xc}[n]}{\delta n(\mathbf{r})}$ — quantum many-body effects The density is reconstructed self-consistently: $$ n(\mathbf{r}) = \sum_i^{\text{occupied}} |\psi_i(\mathbf{r})|^2 $$ ### 2.2 Exchange-Correlation Functionals The unknown piece requiring approximation: - **Local Density Approximation (LDA)**: $$ E_{xc}^{\text{LDA}}[n] = \int n(\mathbf{r})\,\epsilon_{xc}^{\text{homog}}(n(\mathbf{r}))\,d\mathbf{r} $$ - **Generalized Gradient Approximation (GGA)**: $$ E_{xc}^{\text{GGA}}[n] = \int f\left(n(\mathbf{r}), \nabla n(\mathbf{r})\right)\,d\mathbf{r} $$ - **Hybrid Functionals (HSE06)**: $$ E_{xc}^{\text{HSE}} = \frac{1}{4}E_x^{\text{HF,SR}}(\mu) + \frac{3}{4}E_x^{\text{PBE,SR}}(\mu) + E_x^{\text{PBE,LR}}(\mu) + E_c^{\text{PBE}} $$ - Mixing parameter: $\alpha = 0.25$ - Screening parameter: $\mu \approx 0.2\,\text{Å}^{-1}$ ## 3. Bloch's Theorem and Reciprocal Space For a periodic crystal with lattice vectors $\mathbf{R}$, the fundamental symmetry relation: $$ \psi_{n\mathbf{k}}(\mathbf{r}) = e^{i\mathbf{k}\cdot\mathbf{r}}\,u_{n\mathbf{k}}(\mathbf{r}) $$ where: - $u_{n\mathbf{k}}(\mathbf{r})$ has lattice periodicity: $u_{n\mathbf{k}}(\mathbf{r} + \mathbf{R}) = u_{n\mathbf{k}}(\mathbf{r})$ - $\mathbf{k}$ is the crystal momentum (wavevector) - $n$ is the band index ### 3.1 Reciprocal Lattice Reciprocal lattice vectors $\mathbf{G}$ satisfy: $$ \mathbf{G} \cdot \mathbf{R} = 2\pi m \quad (m \in \mathbb{Z}) $$ For a cubic lattice with parameter $a$: $$ \mathbf{G} = \frac{2\pi}{a}(h\hat{\mathbf{x}} + k\hat{\mathbf{y}} + l\hat{\mathbf{z}}) $$ The **band structure** $E_n(\mathbf{k})$ emerges as eigenvalues indexed by: - Band number $n$ - Wavevector $\mathbf{k}$ within the first Brillouin zone ## 4. Basis Set Expansions ### 4.1 Plane Wave Basis Expand the periodic part in Fourier series: $$ u_{n\mathbf{k}}(\mathbf{r}) = \sum_{\mathbf{G}} c_{n,\mathbf{k}+\mathbf{G}}\,e^{i\mathbf{G}\cdot\mathbf{r}} $$ The Schrödinger equation becomes a matrix eigenvalue problem: $$ \sum_{\mathbf{G}'} H_{\mathbf{G},\mathbf{G}'}(\mathbf{k})\,c_{\mathbf{G}'} = E_{n\mathbf{k}}\,c_{\mathbf{G}} $$ **Matrix elements:** $$ H_{\mathbf{G},\mathbf{G}'} = \frac{\hbar^2|\mathbf{k}+\mathbf{G}|^2}{2m}\delta_{\mathbf{G},\mathbf{G}'} + V(\mathbf{G}-\mathbf{G}') $$ **Basis truncation** via kinetic energy cutoff: $$ \frac{\hbar^2|\mathbf{k}+\mathbf{G}|^2}{2m} < E_{\text{cut}} $$ Typical values: $E_{\text{cut}} \sim 30\text{--}80\,\text{Ry}$ (400–1000 eV) ### 4.2 Localized Basis (LCAO/Tight-Binding) Linear Combination of Atomic Orbitals: $$ \psi_{n\mathbf{k}}(\mathbf{r}) = \sum_{\alpha} c_{n\alpha\mathbf{k}} \sum_{\mathbf{R}} e^{i\mathbf{k}\cdot\mathbf{R}}\phi_\alpha(\mathbf{r} - \mathbf{R} - \mathbf{d}_\alpha) $$ This yields a **generalized eigenvalue problem**: $$ H(\mathbf{k})\,\mathbf{c} = E(\mathbf{k})\,S(\mathbf{k})\,\mathbf{c} $$ where: - $H_{ij}(\mathbf{k}) = \sum_{\mathbf{R}} e^{i\mathbf{k}\cdot\mathbf{R}}\langle\phi_i(\mathbf{r})|\hat{H}|\phi_j(\mathbf{r}-\mathbf{R})\rangle$ — Hamiltonian matrix - $S_{ij}(\mathbf{k}) = \sum_{\mathbf{R}} e^{i\mathbf{k}\cdot\mathbf{R}}\langle\phi_i(\mathbf{r})|\phi_j(\mathbf{r}-\mathbf{R})\rangle$ — Overlap matrix ### 4.3 Slater-Koster Parameters For empirical tight-binding with direction cosines $(l, m, n)$: $$ \begin{aligned} E_{s,s} &= V_{ss\sigma} \\ E_{s,x} &= l \cdot V_{sp\sigma} \\ E_{x,x} &= l^2 V_{pp\sigma} + (1-l^2) V_{pp\pi} \\ E_{x,y} &= lm(V_{pp\sigma} - V_{pp\pi}) \end{aligned} $$ **Harrison's universal parameters:** | Integral | Formula | |----------|---------| | $V_{ss\sigma}$ | $-1.40 \dfrac{\hbar^2}{md^2}$ | | $V_{sp\sigma}$ | $1.84 \dfrac{\hbar^2}{md^2}$ | | $V_{pp\sigma}$ | $3.24 \dfrac{\hbar^2}{md^2}$ | | $V_{pp\pi}$ | $-0.81 \dfrac{\hbar^2}{md^2}$ | ## 5. Pseudopotential Theory Core electrons are chemically inert but computationally expensive. Replace true potential with smooth pseudopotential. ### 5.1 Norm-Conserving Conditions (Hamann, Schlüter, Chiang): 1. **Matching**: $\psi^{\text{PS}}(r) = \psi^{\text{AE}}(r)$ for $r > r_c$ 2. **Norm conservation**: $$ \int_0^{r_c}|\psi^{\text{PS}}(r)|^2 r^2 dr = \int_0^{r_c}|\psi^{\text{AE}}(r)|^2 r^2 dr $$ 3. **Eigenvalue matching**: $\epsilon^{\text{PS}} = \epsilon^{\text{AE}}$ 4. **Log-derivative matching**: $$ \left.\frac{d}{dr}\ln\psi^{\text{PS}}\right|_{r_c} = \left.\frac{d}{dr}\ln\psi^{\text{AE}}\right|_{r_c} $$ ### 5.2 Ultrasoft Pseudopotentials (Vanderbilt) Relaxes norm conservation for smoother potentials: $$ \hat{H}|\psi_i\rangle = \epsilon_i\hat{S}|\psi_i\rangle $$ where: $$ \hat{S} = 1 + \sum_{ij}q_{ij}|\beta_i\rangle\langle\beta_j| $$ ### 5.3 Projector Augmented Wave (PAW) Method Linear transformation connecting pseudo and all-electron wavefunctions: $$ |\psi\rangle = |\tilde{\psi}\rangle + \sum_i \left(|\phi_i\rangle - |\tilde{\phi}_i\rangle\right)\langle\tilde{p}_i|\tilde{\psi}\rangle $$ **Components:** - $|\tilde{\psi}\rangle$ — smooth pseudo-wavefunction - $|\phi_i\rangle$ — all-electron partial waves - $|\tilde{\phi}_i\rangle$ — pseudo partial waves - $|\tilde{p}_i\rangle$ — projector functions ## 6. Brillouin Zone Integration Physical observables require integration over $\mathbf{k}$-space: $$ \langle A \rangle = \frac{1}{\Omega_{BZ}}\int_{BZ} A(\mathbf{k})\,d\mathbf{k} $$ ### 6.1 Monkhorst-Pack Grid Systematic $\mathbf{k}$-point sampling: $$ \mathbf{k}_{n_1,n_2,n_3} = \sum_{i=1}^{3} \frac{2n_i - N_i - 1}{2N_i}\mathbf{b}_i $$ where: - $n_i = 1, 2, \ldots, N_i$ - $\mathbf{b}_i$ are reciprocal lattice vectors - Grid specified as $N_1 \times N_2 \times N_3$ ### 6.2 Density of States The tetrahedron method improves integration accuracy: $$ g(E) = \frac{1}{\Omega_{BZ}}\int_{BZ}\delta(E - E_{n\mathbf{k}})\,d\mathbf{k} $$ **Practical evaluation:** - Divide Brillouin zone into tetrahedra - Linear interpolation of $E_n(\mathbf{k})$ within each tetrahedron - Analytical integration of $\delta$-function ## 7. Self-Consistent Field (SCF) Iteration ### 7.1 Algorithm 1. Initialize density $n^{(0)}(\mathbf{r})$ 2. Construct $V_{\text{eff}}[n]$ 3. Diagonalize Kohn-Sham equations → obtain $\{\psi_i, \epsilon_i\}$ 4. Compute new density: $$ n^{\text{new}}(\mathbf{r}) = \sum_i^{\text{occ}}|\psi_i(\mathbf{r})|^2 $$ 5. Mix densities: $$ n^{\text{in}} = (1-\alpha)n^{\text{old}} + \alpha n^{\text{new}} $$ 6. Repeat until $\|n^{\text{new}} - n^{\text{old}}\| < \epsilon$ ### 7.2 Mixing Schemes - **Linear mixing**: Simple but slow convergence $$ n^{(i+1)} = (1-\alpha)n^{(i)} + \alpha n^{\text{out},[i]} $$ - **Pulay mixing (DIIS)**: Minimizes residual over history $$ n^{\text{in}} = \sum_j c_j n^{(j)}, \quad \text{where } \{c_j\} \text{ minimize } \left\|\sum_j c_j R^{(j)}\right\| $$ - **Broyden mixing**: Quasi-Newton approach $$ n^{(i+1)} = n^{(i)} - \alpha B^{(i)} R^{(i)} $$ ## 8. Beyond DFT: The Band Gap Problem DFT-LDA/GGA systematically underestimates band gaps. **Typical underestimation:** | Material | Expt. Gap (eV) | LDA Gap (eV) | Error | |----------|----------------|--------------|-------| | Si | 1.17 | 0.52 | -56% | | GaAs | 1.52 | 0.30 | -80% | | Ge | 0.74 | 0.00 | -100% | ### 8.1 GW Approximation The self-energy captures many-body corrections: $$ \Sigma(\mathbf{r}, \mathbf{r}'; \omega) = \frac{i}{2\pi}\int G(\mathbf{r}, \mathbf{r}'; \omega+\omega')\,W(\mathbf{r}, \mathbf{r}'; \omega')\,d\omega' $$ **Components:** - $G$ — single-particle Green's function - $W$ — screened Coulomb interaction: $$ W = \epsilon^{-1}v $$ **Dielectric function (RPA):** $$ \epsilon(\mathbf{r}, \mathbf{r}'; \omega) = \delta(\mathbf{r} - \mathbf{r}') - \int v(\mathbf{r} - \mathbf{r}'')P^0(\mathbf{r}'', \mathbf{r}'; \omega)\,d\mathbf{r}'' $$ **Quasiparticle correction:** $$ E_{n\mathbf{k}}^{\text{QP}} = E_{n\mathbf{k}}^{\text{DFT}} + \langle\psi_{n\mathbf{k}}|\Sigma(E^{\text{QP}}) - V_{xc}|\psi_{n\mathbf{k}}\rangle $$ This typically adds 0.5–2 eV to band gaps. ## 9. Effective Mass and k·p Theory Near band extrema, expand energy to quadratic order: $$ E_n(\mathbf{k}) \approx E_n(\mathbf{k}_0) + \frac{\hbar^2}{2}\sum_{ij}k_i\left(\frac{1}{m^*}\right)_{ij}k_j $$ ### 9.1 Effective Mass Tensor From second-order perturbation theory: $$ \left(\frac{1}{m^*}\right)_{ij} = \frac{1}{m}\delta_{ij} + \frac{2}{m^2}\sum_{n'\neq n}\frac{\langle n|\hat{p}_i|n'\rangle\langle n'|\hat{p}_j|n\rangle}{E_n - E_{n'}} $$ **Alternate form using band curvature:** $$ \left(\frac{1}{m^*}\right)_{ij} = \frac{1}{\hbar^2}\frac{\partial^2 E_n}{\partial k_i \partial k_j} $$ ### 9.2 8-Band Kane Model For zincblende semiconductors (GaAs, InP, etc.): $$ H_{\text{Kane}} = \begin{pmatrix} E_c + \frac{\hbar^2k^2}{2m_0} & \frac{P}{\sqrt{2}}k_+ & -\sqrt{\frac{2}{3}}Pk_z & \cdots \\ \frac{P}{\sqrt{2}}k_- & E_v - \frac{\hbar^2k^2}{2m_0} & \cdots & \cdots \\ \vdots & \vdots & \ddots & \vdots \end{pmatrix} $$ where: - $k_\pm = k_x \pm ik_y$ - $P = \langle S|\hat{p}_x|X\rangle$ is the Kane momentum matrix element - Includes: conduction band, heavy hole, light hole, split-off bands ## 10. Spin-Orbit Coupling For heavier elements (Ge, GaAs, InSb): $$ H_{\text{SO}} = \frac{\hbar}{4m^2c^2}(\nabla V \times \mathbf{p})\cdot\boldsymbol{\sigma} $$ ### 10.1 Effects - **Lifts degeneracies**: Valence band splitting ~0.34 eV in GaAs - **Essential for**: - Topological insulators - Spintronics - Optical selection rules ### 10.2 Matrix Form The Hamiltonian becomes a $2 \times 2$ spinor structure: $$ H = \begin{pmatrix} H_0 + H_{\text{SO}}^{zz} & H_{\text{SO}}^{+-} \\ H_{\text{SO}}^{-+} & H_0 - H_{\text{SO}}^{zz} \end{pmatrix} $$ where: - $H_{\text{SO}}^{zz} = \lambda L_z S_z$ - $H_{\text{SO}}^{+-} = \lambda L_+ S_-$ ## 11. Semiconductor Manufacturing Applications ### 11.1 Strain Engineering Biaxial strain modifies band structure via **deformation potentials**: $$ \Delta E_c = \Xi_d \cdot \text{Tr}(\boldsymbol{\epsilon}) + \Xi_u \cdot \epsilon_{zz} $$ **Strain tensor components:** $$ \boldsymbol{\epsilon} = \begin{pmatrix} \epsilon_{xx} & \epsilon_{xy} & \epsilon_{xz} \\ \epsilon_{yx} & \epsilon_{yy} & \epsilon_{yz} \\ \epsilon_{zx} & \epsilon_{zy} & \epsilon_{zz} \end{pmatrix} $$ **Valence band (Bir-Pikus Hamiltonian):** $$ H_{\epsilon} = a(\epsilon_{xx} + \epsilon_{yy} + \epsilon_{zz}) + 3b\left[(L_x^2 - \frac{1}{3}L^2)\epsilon_{xx} + \text{c.p.}\right] $$ **Manufacturing application:** - Strained Si channels: ~30–50% mobility enhancement - SiGe virtual substrates for strain control ### 11.2 Heterostructures and Quantum Wells At interfaces, the **envelope function approximation**: $$ \left[-\frac{\hbar^2}{2}\nabla\cdot\frac{1}{m^*(\mathbf{r})}\nabla + V(\mathbf{r})\right]F(\mathbf{r}) = EF(\mathbf{r}) $$ **Ben Daniel-Duke boundary conditions:** $$ \begin{aligned} F_A(z_0) &= F_B(z_0) \\ \frac{1}{m_A^*}\left.\frac{\partial F}{\partial z}\right|_A &= \frac{1}{m_B^*}\left.\frac{\partial F}{\partial z}\right|_B \end{aligned} $$ **Band alignment types:** - **Type I (straddling)**: Both carriers confined in same layer (e.g., GaAs/AlGaAs) - **Type II (staggered)**: Electrons and holes in different layers (e.g., InAs/GaSb) - **Type III (broken gap)**: Conduction and valence bands overlap ### 11.3 Defects and Dopants Supercell approach — create periodic array of defects. **Formation energy:** $$ E_f[D^q] = E_{\text{tot}}[D^q] - E_{\text{tot}}[\text{bulk}] - \sum_i n_i\mu_i + q(E_F + E_V + \Delta V) $$ where: - $D^q$ — defect in charge state $q$ - $n_i$ — number of atoms of species $i$ added/removed - $\mu_i$ — chemical potential of species $i$ - $E_F$ — Fermi level referenced to valence band maximum $E_V$ - $\Delta V$ — potential alignment correction **Charge transition levels:** $$ \epsilon(q/q') = \frac{E_f[D^q; E_F=0] - E_f[D^{q'}; E_F=0]}{q' - q} $$ **Classification:** - **Shallow donors/acceptors**: $\epsilon$ near band edges - **Deep levels**: $\epsilon$ in mid-gap (recombination centers) ### 11.4 Alloy Effects **Virtual Crystal Approximation (VCA):** $$ V_{\text{VCA}} = xV_A + (1-x)V_B $$ **Bowing parameter:** $$ E_g(x) = xE_g^A + (1-x)E_g^B - bx(1-x) $$ **Advanced methods:** - Coherent Potential Approximation (CPA) for disorder - Special Quasirandom Structures (SQS) for explicit alloy supercells ## 12. Computational Complexity | Method | Scaling | Typical System Size | |--------|---------|---------------------| | Exact diagonalization | $O(N^3)$ | ~$10^2$ atoms | | Iterative (Davidson/Lanczos) | $O(N^2)$ per eigenvalue | ~$10^3$ atoms | | Linear-scaling DFT | $O(N)$ | ~$10^4$ atoms | | Tight-binding | $O(N)$ to $O(N^2)$ | ~$10^5$ atoms | ### 12.1 Parallelization Strategies - **k-point parallelism**: Different k-points on different processors - **Band parallelism**: Different bands distributed across processors - **Real-space decomposition**: Domain decomposition for large systems - **FFT parallelism**: Distributed 3D FFTs for plane-wave methods ### 12.2 Key Software Packages | Package | Method | Primary Use | |---------|--------|-------------| | VASP | PAW/PW | Production DFT | | Quantum ESPRESSO | NC/US/PAW-PW | Open-source DFT | | WIEN2k | LAPW | Accurate all-electron | | Gaussian | Localized basis | Molecular systems | | SIESTA | Numerical AO | Large-scale O(N) | ## 13. Workflow ```text ┌─────────────────────────────────────────────────────────────┐ │ INPUT: Crystal Structure │ │ (atomic positions, lattice vectors) │ └─────────────────────────────────────────────────────────────┘ │ ▼ ┌─────────────────────────────────────────────────────────────┐ │ SELECT METHOD │ │ • DFT (LDA/GGA/Hybrid) for accuracy │ │ • Tight-binding for speed │ │ • GW for accurate band gaps │ └─────────────────────────────────────────────────────────────┘ │ ▼ ┌─────────────────────────────────────────────────────────────┐ │ COMPUTATIONAL SETUP │ │ • Choose k-point grid (Monkhorst-Pack) │ │ • Set energy cutoff (plane waves) │ │ • Select pseudopotentials │ └─────────────────────────────────────────────────────────────┘ │ ▼ ┌─────────────────────────────────────────────────────────────┐ │ SELF-CONSISTENT CALCULATION │ │ • Iterate until density converges │ │ • Obtain ground-state energy │ └─────────────────────────────────────────────────────────────┘ │ ▼ ┌─────────────────────────────────────────────────────────────┐ │ POST-PROCESSING │ │ • Band structure along high-symmetry paths │ │ • Density of states │ │ • Effective masses │ │ • Optical properties │ └─────────────────────────────────────────────────────────────┘ │ ▼ ┌─────────────────────────────────────────────────────────────┐ │ VALIDATION & APPLICATION │ │ • Compare with ARPES, optical data │ │ • Extract parameters for device simulation (TCAD) │ └─────────────────────────────────────────────────────────────┘ ``` ## 14. Key Equations Reference Card ### Schrödinger Equation $$ \hat{H}\psi = E\psi $$ ### Bloch Theorem $$ \psi_{n\mathbf{k}}(\mathbf{r}) = e^{i\mathbf{k}\cdot\mathbf{r}}u_{n\mathbf{k}}(\mathbf{r}) $$ ### Kohn-Sham Equation $$ \left[-\frac{\hbar^2}{2m}\nabla^2 + V_{\text{eff}}[n]\right]\psi_i = \epsilon_i\psi_i $$ ### Effective Mass $$ \frac{1}{m^*_{ij}} = \frac{1}{\hbar^2}\frac{\partial^2 E}{\partial k_i \partial k_j} $$ ### GW Self-Energy $$ \Sigma = iGW $$ ### Formation Energy $$ E_f = E_{\text{tot}}[\text{defect}] - E_{\text{tot}}[\text{bulk}] - \sum_i n_i\mu_i + qE_F $$
Direct tunneling across bandgap.
Reduction of bandgap at high doping.
Bandwidth density expresses data rate per unit area or power.
Shared memory access conflicts.
ARC layer between substrate and resist.