2. Semiconductor Equations

THE BASIC equations for device simulation can be obtained by different means. One way is to use MAXWELL's equations to derive POISSON's equation and the carrier continuity equations and introduce the current relations by a phenomenological approach. Another way to synthesize transport models is the systematic derivation starting from BOLTZMANN's transport equation. The moments method allows the derivation of a hierarchy of increasingly complex transport models including the well known drift-diffusion and energy transport models.

Subsections

• 2.1 MAXWELL's Equations
  • 2.1.1 POISSON's Equation
  
  
• 2.2 Phenomenological Approach
  • 2.2.1 Balance Equations
  • 2.2.2 Drift-Diffusion Transport Model
    • 2.2.2.1 Drift Current
    • 2.2.2.2 Diffusion Current
  • 2.2.3 Carrier Continuity Equations
  
  
• 2.3 Systematic Approach
  • 2.3.1 BOLTZMANN's Transport Equation
  • 2.3.2 Moments Method
    • 2.3.2.1 Gradient Calculation
    • 2.3.2.2 Macroscopic Relaxation Time Approximation
    • 2.3.2.3 Isotropic Distribution Function
    • 2.3.2.4 Statistical Averages
  • 2.3.3 Closure
    • 2.3.3.1 Maxwell Distribution
    • 2.3.3.2 Discussion of the Diffusion Approximation
    • 2.3.3.3 Drift-Diffusion Transport Model - Closure at $\phi _2$
    • 2.3.3.4 Energy Transport Model - Closure at $\ensuremath{\boldsymbol{\mathrm{\phi}}}_3$ and $\phi _4$
    • 2.3.3.5 Six Moments Transport Model - Closure at $\phi _6$
  • 2.3.4 Generation/Recombination Processes
    • 2.3.4.1 Thermal Equilibrium
    • 2.3.4.2 Steady State
    • 2.3.4.3 Higher Order Moments

M. Gritsch: Numerical Modeling of Silicon-on-Insulator MOSFETs PDF