monte carlo reliability simulation, reliability
Statistical reliability prediction.
751 technical terms and definitions
Statistical reliability prediction.
Statistical sampling to predict yield.
Monte Carlo simulations use random sampling to model system behavior and uncertainty.
# Semiconductor Manufacturing Monte Carlo Simulation: The Mathematics
## 1. Introduction
### 1.1 Why Monte Carlo for Semiconductors?
Semiconductor manufacturing involves:
- **Nanometer-scale features** (3nm, 5nm nodes) where atomic-scale randomness matters
- **Hundreds of process steps**, each with inherent variability
- **High-dimensional parameter spaces** (100s–1000s of variables)
- **Rare event statistics** (yield prediction for 99%+ target yields)
Classical numerical methods fail due to the **curse of dimensionality**. Monte Carlo's key advantage:
$$
\text{Error} = O\left(\frac{1}{\sqrt{N}}\right) \quad \text{independent of dimensionality}
$$
### 1.2 Key Applications
- **Process variability modeling**: Understanding how variations in lithography, etching, doping affect device parameters
- **Yield prediction**: Estimating what percentage of chips will work
- **Circuit performance analysis**: Predicting speed, power consumption distributions
- **Design for manufacturability (DFM)**: Ensuring designs are robust to process variations
- **Statistical timing analysis**: Understanding timing margins
- **Device physics simulation**: Modeling carrier transport, quantum effects
## 2. Fundamental Monte Carlo Mathematics
### 2.1 Basic Monte Carlo Integration
To estimate an integral:
$$
I = \int_D f(x) \, p(x) \, dx
$$
The **Monte Carlo estimator**:
$$
\hat{I} = \frac{1}{N} \sum_{i=1}^{N} f(x_i), \quad x_i \sim p(x)
$$
**Error bound** (Central Limit Theorem):
$$
\text{Standard Error} = \frac{\sigma}{\sqrt{N}}
$$
where $\sigma$ is the standard deviation of $f(x)$.
### 2.2 Random Number Generation
#### 2.2.1 Linear Congruential Generator
$$
X_{n+1} = (aX_n + c) \mod m
$$
- **Parameters**: multiplier $a$, increment $c$, modulus $m$
- **Period**: at most $m$
#### 2.2.2 Box-Muller Transform (Uniform → Gaussian)
$$
Z_0 = \sqrt{-2 \ln U_1} \cos(2\pi U_2)
$$
$$
Z_1 = \sqrt{-2 \ln U_1} \sin(2\pi U_2)
$$
where:
- $U_1, U_2 \sim \text{Uniform}(0,1)$
- $Z_0, Z_1 \sim \mathcal{N}(0,1)$
#### 2.2.3 Inverse Transform Method
$$
X = F^{-1}(U)
$$
where $F$ is the CDF of the desired distribution and $U \sim \text{Uniform}(0,1)$.
### 2.3 Modern PRNGs for Parallel Computing
- **Mersenne Twister (MT19937)**: Period $2^{19937} - 1$
- **Xorshift**: Fast, good statistical properties
- **PCG (Permuted Congruential Generator)**: Statistically excellent
- **Counter-based (Philox, Threefry)**: Ideal for GPU parallelization
## 3. Process Variation Modeling
### 3.1 Parameter Decomposition
A device parameter $P$ is modeled as:
$$
P = P_{\text{nom}} + \Delta P_{\text{sys}}(x,y) + \Delta P_{\text{global}} + \Delta P_{\text{local}}
$$
where:
- **Systematic variation**:
$$
\Delta P_{\text{sys}}(x,y) = \sum_{i,j} a_{ij} x^i y^j
$$
(spatial polynomial)
- **Global variation** (wafer-to-wafer, lot-to-lot):
$$
\Delta P_{\text{global}} \sim \mathcal{N}(0, \sigma_g^2)
$$
- **Local variation** (device-to-device):
$$
\Delta P_{\text{local}} \sim \mathcal{N}(0, \sigma_l^2)
$$
### 3.2 Spatial Correlation Structure
Local variations often exhibit spatial correlation:
$$
\text{Cov}(\Delta P(\mathbf{r}_1), \Delta P(\mathbf{r}_2)) = \sigma^2 \cdot \rho(|\mathbf{r}_1 - \mathbf{r}_2|)
$$
#### Common Correlation Functions
| Model | Formula | Characteristics |
|-------|---------|-----------------|
| Exponential | $\rho(d) = e^{-d/\lambda}$ | Sharp near-field correlation |
| Gaussian | $\rho(d) = e^{-(d/\lambda)^2}$ | Smoother correlation decay |
| Matérn | $\rho(d) = \frac{2^{1-\nu}}{\Gamma(\nu)} \left(\frac{d}{\lambda}\right)^\nu K_\nu\left(\frac{d}{\lambda}\right)$ | Flexible smoothness parameter $\nu$ |
### 3.3 Generating Correlated Samples
Given covariance matrix $\mathbf{\Sigma}$, use **Cholesky decomposition**:
$$
\mathbf{\Sigma} = \mathbf{L}\mathbf{L}^T
$$
Generate correlated samples:
$$
\mathbf{X} = \boldsymbol{\mu} + \mathbf{L}\mathbf{Z}
$$
where $\mathbf{Z}$ is a vector of independent standard normals.
### 3.4 Pelgrom's Mismatch Law
For transistor mismatch (critical for analog/SRAM):
$$
\sigma(\Delta V_{th}) = \frac{A_{VT}}{\sqrt{WL}}
$$
$$
\sigma\left(\frac{\Delta \beta}{\beta}\right) = \frac{A_\beta}{\sqrt{WL}}
$$
where:
- $A_{VT}$: Threshold voltage mismatch coefficient (typical: 1-5 mV·μm)
- $A_\beta$: Current factor mismatch coefficient (typical: 1-2 %·μm)
- $W$: Gate width
- $L$: Gate length
## 4. Statistical Static Timing Analysis (SSTA)
### 4.1 Gate Delay Model
$$
d = d_0 + \sum_i a_i \Delta P_i + \sum_i \beta_{ii} (\Delta P_i)^2 + \sum_{i
Transistor doubling prediction.
Observation that transistor density doubles approximately every two years.
Moran's I statistic measures spatial clustering in wafer map data.
Continue transistor scaling.
Diversification beyond scaling.
More than Moore adds functionality through heterogeneous integration beyond scaling.
Model-Based Offline Reinforcement Learning constructs pessimistic MDPs from learned dynamics models with uncertainty penalties for safe policy optimization.
Circular fingerprints for molecules.
Analyze word structure.
Measure oxide quality and interface.
MOS decoupling capacitors use gate oxide capacitance providing high density on-die energy storage.
# MOSFET: Mathematical Modeling Metal-Oxide-Semiconductor Field-Effect Transistor (MOSFET) Comprehensive equations, mathematical modeling, and process-parameter relationships 1. Fundamental Device Structure 1.1 MOSFET Components A MOSFET is a four-terminal semiconductor device consisting of: - Source (S) : Heavily doped region where carriers originate - Drain (D) : Heavily doped region where carriers are collected - Gate (G) : Control electrode separated from channel by dielectric - Body/Substrate (B) : Semiconductor bulk (p-type for NMOS, n-type for PMOS) 1.2 Operating Principle The gate voltage modulates channel conductivity through field effect: $$ \text{Gate Voltage} \rightarrow \text{Electric Field} \rightarrow \text{Channel Formation} \rightarrow \text{Current Flow} $$ 1.3 Device Types | Type | Substrate | Channel Carriers | Threshold | |------|-----------|------------------|-----------| | NMOS | p-type | Electrons | $V_{th} > 0$ (enhancement) | | PMOS | n-type | Holes | $V_{th} < 0$ (enhancement) | 2. Core MOSFET Equations 2.1 Threshold Voltage The threshold voltage $V_{th}$ determines device turn-on and is highly process-dependent: $$ V_{th} = V_{FB} + 2\phi_F + \frac{\sqrt{2\varepsilon_{Si} \cdot q \cdot N_A \cdot 2\phi_F}}{C_{ox}} $$ Component Equations - Flat-band voltage : $$ V_{FB} = \phi_{ms} - \frac{Q_{ox}}{C_{ox}} $$ - Fermi potential : $$ \phi_F = \frac{kT}{q} \ln\left(\frac{N_A}{n_i}\right) $$ - Oxide capacitance per unit area : $$ C_{ox} = \frac{\varepsilon_{ox}}{t_{ox}} = \frac{\kappa \cdot \varepsilon_0}{t_{ox}} $$ - Work function difference : $$ \phi_{ms} = \phi_m - \phi_s = \phi_m - \left(\chi + \frac{E_g}{2q} + \phi_F\right) $$ Parameter Definitions | Symbol | Description | Typical Value/Unit | |--------|-------------|-------------------| | $V_{FB}$ | Flat-band voltage | $-0.5$ to $-1.0$ V | | $\phi_F$ | Fermi potential | $0.3$ to $0.4$ V | | $\phi_{ms}$ | Work function difference | $-0.5$ to $-1.0$ V | | $C_{ox}$ | Oxide capacitance | $\sim 10^{-2}$ F/m² | | $Q_{ox}$ | Fixed oxide charge | $\sim 10^{10}$ q/cm² | | $N_A$ | Acceptor concentration | $10^{15}$ to $10^{18}$ cm⁻³ | | $n_i$ | Intrinsic carrier concentration | $1.5 \times 10^{10}$ cm⁻³ (Si, 300K) | | $\varepsilon_{Si}$ | Silicon permittivity | $11.7 \varepsilon_0$ | | $\varepsilon_{ox}$ | SiO₂ permittivity | $3.9 \varepsilon_0$ | 2.2 Drain Current Equations 2.2.1 Linear (Triode) Region Condition : $V_{DS} < V_{GS} - V_{th}$ (channel not pinched off) $$ I_D = \mu_n C_{ox} \frac{W}{L} \left[ (V_{GS} - V_{th}) V_{DS} - \frac{V_{DS}^2}{2} \right] $$ Simplified form (for small $V_{DS}$): $$ I_D \approx \mu_n C_{ox} \frac{W}{L} (V_{GS} - V_{th}) V_{DS} $$ Channel resistance : $$ R_{ch} = \frac{V_{DS}}{I_D} = \frac{L}{\mu_n C_{ox} W (V_{GS} - V_{th})} $$ 2.2.2 Saturation Region Condition : $V_{DS} \geq V_{GS} - V_{th}$ (channel pinched off) $$ I_D = \frac{1}{2} \mu_n C_{ox} \frac{W}{L} (V_{GS} - V_{th})^2 (1 + \lambda V_{DS}) $$ Without channel-length modulation ($\lambda = 0$): $$ I_{D,sat} = \frac{1}{2} \mu_n C_{ox} \frac{W}{L} (V_{GS} - V_{th})^2 $$ Saturation voltage : $$ V_{DS,sat} = V_{GS} - V_{th} $$ 2.2.3 Channel-Length Modulation The parameter $\lambda$ captures output resistance degradation: $$ \lambda = \frac{1}{L \cdot E_{crit}} \approx \frac{1}{V_A} $$ Output resistance : $$ r_o = \frac{\partial V_{DS}}{\partial I_D} = \frac{1}{\lambda I_D} = \frac{V_A + V_{DS}}{I_D} $$ Where $V_A$ is the Early voltage (typically $5$ to $50$ V/μm × L). 2.3 Subthreshold Conduction 2.3.1 Weak Inversion Current Condition : $V_{GS} < V_{th}$ (exponential behavior) $$ I_D = I_0 \exp\left(\frac{V_{GS} - V_{th}}{n \cdot V_T}\right) \left[1 - \exp\left(-\frac{V_{DS}}{V_T}\right)\right] $$ Characteristic current : $$ I_0 = \mu_n C_{ox} \frac{W}{L} (n-1) V_T^2 $$ Thermal voltage : $$ V_T = \frac{kT}{q} \approx 26 \text{ mV at } T = 300\text{K} $$ 2.3.2 Subthreshold Swing The subthreshold swing $S$ quantifies turn-off sharpness: $$ S = \frac{\partial V_{GS}}{\partial (\log_{10} I_D)} = n \cdot V_T \cdot \ln(10) = 2.3 \cdot n \cdot V_T $$ Numerical values : - Ideal minimum: $S_{min} = 60$ mV/decade (at 300K, $n = 1$) - Typical range: $S = 70$ to $100$ mV/decade - $n = 1 + \frac{C_{dep}}{C_{ox}}$ (subthreshold ideality factor) 2.3.3 Depletion Capacitance $$ C_{dep} = \frac{\varepsilon_{Si}}{W_{dep}} = \sqrt{\frac{q \varepsilon_{Si} N_A}{4 \phi_F}} $$ 2.4 Body Effect When source-to-body voltage $V_{SB} \neq 0$: $$ V_{th}(V_{SB}) = V_{th0} + \gamma \left(\sqrt{2\phi_F + V_{SB}} - \sqrt{2\phi_F}\right) $$ Body effect coefficient : $$ \gamma = \frac{\sqrt{2 q \varepsilon_{Si} N_A}}{C_{ox}} $$ Typical values : $\gamma = 0.3$ to $1.0$ V$^{1/2}$ 2.5 Transconductance and Output Conductance 2.5.1 Transconductance Saturation region : $$ g_m = \frac{\partial I_D}{\partial V_{GS}} = \mu_n C_{ox} \frac{W}{L} (V_{GS} - V_{th}) = \sqrt{2 \mu_n C_{ox} \frac{W}{L} I_D} $$ Alternative form : $$ g_m = \frac{2 I_D}{V_{GS} - V_{th}} $$ 2.5.2 Output Conductance $$ g_{ds} = \frac{\partial I_D}{\partial V_{DS}} = \lambda I_D = \frac{I_D}{V_A} $$ 2.5.3 Intrinsic Gain $$ A_v = \frac{g_m}{g_{ds}} = \frac{2}{\lambda(V_{GS} - V_{th})} = \frac{2 V_A}{V_{GS} - V_{th}} $$ 3. Short-Channel Effects 3.1 Velocity Saturation At high lateral electric fields ($E > E_{crit} \approx 10^4$ V/cm): $$ v_d = \frac{\mu_n E}{1 + E/E_{crit}} $$ Saturation velocity : $$ v_{sat} = \mu_n E_{crit} \approx 10^7 \text{ cm/s (electrons in Si)} $$ 3.1.1 Modified Saturation Current $$ I_{D,sat} = W C_{ox} v_{sat} (V_{GS} - V_{th}) $$ Note: Linear (not quadratic) dependence on gate overdrive. 3.1.2 Critical Length Velocity saturation dominates when: $$ L < L_{crit} = \frac{\mu_n (V_{GS} - V_{th})}{2 v_{sat}} $$ 3.2 Drain-Induced Barrier Lowering (DIBL) The drain field reduces the source-side barrier: $$ V_{th} = V_{th,long} - \eta \cdot V_{DS} $$ DIBL coefficient : $$ \eta = -\frac{\partial V_{th}}{\partial V_{DS}} $$ Typical values : $\eta = 20$ to $100$ mV/V for short channels 3.2.1 Modified Threshold Equation $$ V_{th}(V_{DS}, V_{SB}) = V_{th0} + \gamma(\sqrt{2\phi_F + V_{SB}} - \sqrt{2\phi_F}) - \eta V_{DS} $$ 3.3 Mobility Degradation 3.3.1 Vertical Field Effect $$ \mu_{eff} = \frac{\mu_0}{1 + \theta (V_{GS} - V_{th})} $$ Alternative form (surface roughness scattering): $$ \mu_{eff} = \frac{\mu_0}{1 + (\theta_1 + \theta_2 V_{SB})(V_{GS} - V_{th})} $$ 3.3.2 Universal Mobility Model $$ \mu_{eff} = \frac{\mu_0}{\left[1 + \left(\frac{E_{eff}}{E_0}\right)^\nu + \left(\frac{E_{eff}}{E_1}\right)^\beta\right]} $$ Where $E_{eff}$ is the effective vertical field: $$ E_{eff} = \frac{Q_b + \eta_s Q_i}{\varepsilon_{Si}} $$ 3.4 Hot Carrier Effects 3.4.1 Impact Ionization Current $$ I_{sub} = \frac{I_D}{M - 1} $$ Multiplication factor : $$ M = \frac{1}{1 - \int_0^{L_{dep}} \alpha(E) dx} $$ 3.4.2 Ionization Rate $$ \alpha = \alpha_\infty \exp\left(-\frac{E_{crit}}{E}\right) $$ 3.5 Gate Leakage 3.5.1 Direct Tunneling Current $$ J_g = A \cdot E_{ox}^2 \exp\left(-\frac{B}{\vert E_{ox} \vert}\right) $$ Where: $$ A = \frac{q^3}{16\pi^2 \hbar \phi_b} $$ $$ B = \frac{4\sqrt{2m^* \phi_b^3}}{3\hbar q} $$ 3.5.2 Gate Oxide Field $$ E_{ox} = \frac{V_{GS} - V_{FB} - \psi_s}{t_{ox}} $$ 4. Parameters 4.1 Gate Oxide Engineering 4.1.1 Oxide Capacitance $$ C_{ox} = \frac{\varepsilon_0 \cdot \kappa}{t_{ox}} $$ | Dielectric | $\kappa$ | EOT for $t_{phys} = 3$ nm | |------------|----------|---------------------------| | SiO₂ | 3.9 | 3.0 nm | | Si₃N₄ | 7.5 | 1.56 nm | | Al₂O₃ | 9 | 1.30 nm | | HfO₂ | 20-25 | 0.47-0.59 nm | | ZrO₂ | 25 | 0.47 nm | 4.1.2 Equivalent Oxide Thickness (EOT) $$ EOT = t_{high-\kappa} \times \frac{\varepsilon_{SiO_2}}{\varepsilon_{high-\kappa}} = t_{high-\kappa} \times \frac{3.9}{\kappa} $$ 4.1.3 Capacitance Equivalent Thickness (CET) Including quantum effects and poly depletion: $$ CET = EOT + \Delta t_{QM} + \Delta t_{poly} $$ Where: - $\Delta t_{QM} \approx 0.3$ to $0.5$ nm (quantum mechanical) - $\Delta t_{poly} \approx 0.3$ to $0.5$ nm (polysilicon depletion) 4.2 Channel Doping 4.2.1 Doping Profile Impact $$ V_{th} \propto \sqrt{N_A} $$ $$ \mu \propto \frac{1}{N_A^{0.3}} \text{ (ionized impurity scattering)} $$ 4.2.2 Depletion Width $$ W_{dep} = \sqrt{\frac{2\varepsilon_{Si}(2\phi_F + V_{SB})}{qN_A}} $$ 4.2.3 Junction Capacitance $$ C_j = C_{j0}\left(1 + \frac{V_R}{\phi_{bi}}\right)^{-m} $$ Where: - $C_{j0}$ = zero-bias capacitance - $\phi_{bi}$ = built-in potential - $m = 0.5$ (abrupt junction), $m = 0.33$ (graded junction) 4.3 Gate Material Engineering 4.3.1 Work Function Values | Gate Material | Work Function $\phi_m$ (eV) | Application | |--------------|----------------------------|-------------| | n+ Polysilicon | 4.05 | Legacy NMOS | | p+ Polysilicon | 5.15 | Legacy PMOS | | TiN | 4.5-4.7 | NMOS (midgap) | | TaN | 4.0-4.4 | NMOS | | TiAl | 4.2-4.3 | NMOS | | TiAlN | 4.7-4.8 | PMOS | 4.3.2 Flat-Band Voltage Engineering For symmetric CMOS threshold voltages: $$ V_{FB,NMOS} + V_{FB,PMOS} \approx -E_g/q $$ 4.4 Channel Length Scaling 4.4.1 Characteristic Length $$ \lambda = \sqrt{\frac{\varepsilon_{Si}}{\varepsilon_{ox}} \cdot t_{ox} \cdot x_j} $$ For good short-channel control: $L > 5\lambda$ to $10\lambda$ 4.4.2 Scale Length (FinFET/GAA) $$ \lambda_{GAA} = \sqrt{\frac{\varepsilon_{Si} \cdot t_{Si}^2}{2 \varepsilon_{ox} \cdot t_{ox}}} $$ 4.5 Strain Engineering 4.5.1 Mobility Enhancement $$ \mu_{strained} = \mu_0 (1 + \Pi \cdot \sigma) $$ Where: - $\Pi$ = piezoresistive coefficient - $\sigma$ = applied stress Enhancement factors : - NMOS (tensile): $+30\%$ to $+70\%$ mobility gain - PMOS (compressive): $+50\%$ to $+100\%$ mobility gain 4.5.2 Stress Impact on Threshold $$ \Delta V_{th} = \alpha_{th} \cdot \sigma $$ Where $\alpha_{th} \approx 1$ to $5$ mV/GPa 5. Advanced Compact Models 5.1 BSIM4 Model 5.1.1 Unified Current Equation $$ I_{DS} = I_{DS0} \cdot \left(1 + \frac{V_{DS} - V_{DS,eff}}{V_A}\right) \cdot \frac{1}{1 + R_S \cdot G_{DS0}} $$ 5.1.2 Effective Overdrive $$ V_{GS,eff} - V_{th} = \frac{2nV_T \cdot \ln\left[1 + \exp\left(\frac{V_{GS} - V_{th}}{2nV_T}\right)\right]}{1 + 2n\sqrt{\delta + \left(\frac{V_{GS}-V_{th}}{2nV_T} - \delta\right)^2}} $$ 5.1.3 Effective Saturation Voltage $$ V_{DS,eff} = V_{DS,sat} - \frac{V_T}{2}\ln\left(\frac{V_{DS,sat} + \sqrt{V_{DS,sat}^2 + 4V_T^2}}{V_{DS} + \sqrt{V_{DS}^2 + 4V_T^2}}\right) $$ 5.2 Surface Potential Model (PSP) 5.2.1 Implicit Surface Potential Equation $$ V_{GB} - V_{FB} = \psi_s + \gamma\sqrt{\psi_s + V_T e^{(\psi_s - 2\phi_F - V_{SB})/V_T} - V_T} $$ 5.2.2 Charge-Based Current $$ I_D = \mu W \frac{Q_i(0) - Q_i(L)}{L} \cdot \frac{V_{DS}}{V_{DS,eff}} $$ Where $Q_i$ is the inversion charge density: $$ Q_i = -C_{ox}\left[\psi_s - 2\phi_F - V_{ch} + V_T\left(e^{(\psi_s - 2\phi_F - V_{ch})/V_T} - 1\right)\right]^{1/2} $$ 5.3 FinFET Equations 5.3.1 Effective Width $$ W_{eff} = 2H_{fin} + W_{fin} $$ For multiple fins: $$ W_{total} = N_{fin} \cdot (2H_{fin} + W_{fin}) $$ 5.3.2 Multi-Gate Scale Length Double-gate : $$ \lambda_{DG} = \sqrt{\frac{\varepsilon_{Si} \cdot t_{Si} \cdot t_{ox}}{2\varepsilon_{ox}}} $$ Gate-all-around (GAA) : $$ \lambda_{GAA} = \sqrt{\frac{\varepsilon_{Si} \cdot r^2}{4\varepsilon_{ox}} \cdot \ln\left(1 + \frac{t_{ox}}{r}\right)} $$ Where $r$ = nanowire radius 5.3.3 FinFET Threshold Voltage $$ V_{th} = V_{FB} + 2\phi_F + \frac{qN_A W_{fin}}{2C_{ox}} - \Delta V_{th,SCE} $$ 6. Process-Equation Coupling 6.1 Parameter Sensitivity Analysis | Process Parameter | Primary Equations Affected | Sensitivity | |------------------|---------------------------|-------------| | $t_{ox}$ (oxide thickness) | $C_{ox}$, $V_{th}$, $I_D$, $g_m$ | High | | $N_A$ (channel doping) | $V_{th}$, $\gamma$, $\mu$, $W_{dep}$ | High | | $L$ (channel length) | $I_D$, SCE, $\lambda$ | Very High | | $W$ (channel width) | $I_D$, $g_m$ (linear) | Moderate | | Gate work function | $V_{FB}$, $V_{th}$ | High | | Junction depth $x_j$ | SCE, $R_{SD}$ | Moderate | | Strain level | $\mu$, $I_D$ | Moderate | 6.2 Variability Equations 6.2.1 Random Dopant Fluctuation (RDF) $$ \sigma_{V_{th}} = \frac{A_{VT}}{\sqrt{W \cdot L}} $$ Where $A_{VT}$ is the Pelgrom coefficient (typically $1$ to $5$ mV·μm). 6.2.2 Line Edge Roughness (LER) $$ \sigma_{V_{th,LER}} \propto \frac{\sigma_{LER}}{L} $$ 6.2.3 Oxide Thickness Variation $$ \sigma_{V_{th,tox}} = \frac{\partial V_{th}}{\partial t_{ox}} \cdot \sigma_{t_{ox}} = \frac{V_{th} - V_{FB} - 2\phi_F}{t_{ox}} \cdot \sigma_{t_{ox}} $$ 6.3 Equations: 6.3.1 Drive Current $$ I_{on} = \frac{W}{L} \cdot \mu_{eff} \cdot C_{ox} \cdot \frac{(V_{DD} - V_{th})^\alpha}{1 + (V_{DD} - V_{th})/E_{sat}L} $$ Where $\alpha = 2$ (long channel) or $\alpha \rightarrow 1$ (velocity saturated). 6.3.2 Leakage Current $$ I_{off} = I_0 \cdot \frac{W}{L} \cdot \exp\left(\frac{-V_{th}}{nV_T}\right) \cdot \left(1 - \exp\left(\frac{-V_{DD}}{V_T}\right)\right) $$ 6.3.3 CV/I Delay Metric $$ \tau = \frac{C_L \cdot V_{DD}}{I_{on}} \propto \frac{L^2}{\mu (V_{DD} - V_{th})} $$ Constants: | Constant | Symbol | Value | |----------|--------|-------| | Elementary charge | $q$ | $1.602 \times 10^{-19}$ C | | Boltzmann constant | $k$ | $1.381 \times 10^{-23}$ J/K | | Permittivity of free space | $\varepsilon_0$ | $8.854 \times 10^{-12}$ F/m | | Planck constant | $\hbar$ | $1.055 \times 10^{-34}$ J·s | | Electron mass | $m_0$ | $9.109 \times 10^{-31}$ kg | | Thermal voltage (300K) | $V_T$ | $25.9$ mV | | Silicon bandgap (300K) | $E_g$ | $1.12$ eV | | Intrinsic carrier conc. (Si) | $n_i$ | $1.5 \times 10^{10}$ cm⁻³ | Equations: Threshold Voltage $$ V_{th} = V_{FB} + 2\phi_F + \frac{\sqrt{2\varepsilon_{Si} q N_A (2\phi_F)}}{C_{ox}} $$ Linear Region Current $$ I_D = \mu C_{ox} \frac{W}{L} \left[(V_{GS} - V_{th})V_{DS} - \frac{V_{DS}^2}{2}\right] $$ Saturation Current $$ I_D = \frac{1}{2}\mu C_{ox}\frac{W}{L}(V_{GS} - V_{th})^2(1 + \lambda V_{DS}) $$ Subthreshold Current $$ I_D = I_0 \exp\left(\frac{V_{GS} - V_{th}}{nV_T}\right) $$ Transconductance $$ g_m = \sqrt{2\mu C_{ox}\frac{W}{L}I_D} $$ Body Effect $$ V_{th} = V_{th0} + \gamma\left(\sqrt{2\phi_F + V_{SB}} - \sqrt{2\phi_F}\right) $$
Find recurring subgraph patterns.
Motion compensation aligns frames using estimated motion for temporal processing.
Align frames using motion.
Predict future motion of objects.
Transfer motion patterns.
Apply motion from one video to another subject.
Motion waste includes unnecessary human movement during work.
Unnecessary movement of people.
High-efficiency motors reduce electrical losses in fans pumps and compressors.
Movement pruning removes weights whose values move toward zero during training.
Message Passing Neural Network framework unifies many GNN architectures through generalized message passing operations.
Open-source commercially usable LLM.
MPT (MosaicML) is open source LLM. Now part of Databricks.
Multi-horizon Quantile Recurrent Neural Network produces probabilistic forecasts through quantile regression.
Manufacturing Resource Planning extends MRP to include capacity planning shop floor control and financial integration.
Material Requirements Planning calculates material quantities and timing based on production schedules BOMs and lead times.
Mean Reciprocal Rank optimization emphasizes position of first relevant item.
Mean Reciprocal Rank averages reciprocal rank of first relevant result.
Large-scale information retrieval.
Measurement System Analysis evaluates adequacy of measurement processes for their intended use.
MSL 1 through 6.
Multivariate Statistical Process Control monitors correlated parameters jointly.
Multi-turn conversation benchmark.
Average time tool operates before failure.
Mean Time Between Failures averages operating duration between failures for repairable equipment.
Mean Time To Failure is average operating time until first failure.
Mean Time To Failure averages operating duration between failures for non-repairable systems.
Average time to fix tool after failure.
Mean Time To Repair averages duration to restore equipment to operational condition.
Muda is Japanese term for waste or non-value-adding activity.
Seven types of waste in lean.
Complete polarization characterization.
Full polarization scatterometry.
Multi-head attention runs parallel attention with different projections. Captures different relationship types.