3.5 Macroscopic Transport Models

Since the direct solution of the Boltzmann transport equation is computationally very demanding, it is either solved applying the Monte-Carlo technique or approximated by macroscopic transport models, which are the background for commercial device simulators. The solution variables of macroscopic transport models are macroscopic quantities, which are accessible by measurement in contrast to the distribution function. However, the information of the device state incorporated in the macroscopic quantities is limited, and the access to the full distribution function is limited to approaches dealing directly with Boltzmann's equation, such as the Monte-Carlo method [65]. Thus, data governed by Monte-Carlo simulations is frequently used to judge the validity of macroscopic transport models for a certain device regime [78,79,80]. Macroscopic quantities represent averages of microscopic quantities in momentum-space. The coherences between microscopic and macroscopic quantities are given in Section 3.5.1.

Basically, macroscopic transport models can be formulated following two approaches. First, in the systematic approach, a set of equations is derived from the Boltzmann transport equation applying the method of moments, which is introduced in Section 3.5.2. In mathematical words, Boltzmann's equation, which is a seven-dimensional integro-differential equation is transformed into a series of coupled partial differential equations. Compared to the underlying Boltzmann transport equation, information on the distribution function is approximated due to the truncation of the system after a certain number of equations and the closure of the equation system based on information incorporated in the equations governed. Theoretically, an arbitrary number of equations could be derived, but the computational effort increases considerably with the number of equations. By a proper choice of weight functions, these equations hold a physical meaning which is also accessible from a more intuitive point of view.

Second, in the phenomenological approach, the semiconductor equations are formulated following the basic laws of mass and energy conservation as well as the principles of irreversible thermodynamics. In the simplest case, both approaches based on completely different procedures lead to similar results.

In the sequel, a toolkit for the systematic derivation of transport models and a proper nomenclature are given. Three transport models are derived and compared following the systematic approach applying different assumptions on the collision term.

Subsections

• 3.5.1 Microscopic and Macroscopic Quantities
• 3.5.2 Method of Moments
• 3.5.3 A Hierarchy of Transport Models
• 3.5.4 Relaxation Time Approximation
• 3.5.5 Stratton's Approach
  • 3.5.5.1 Carrier Balance Equation
  • 3.5.5.2 Particle Flux Equation
  • 3.5.5.3 Energy Balance Equation
  • 3.5.5.4 Energy Flux Equation
  
  
• 3.5.6 Bløtekjær's Approach
  • 3.5.6.1 Particle Flux Equation
  • 3.5.6.2 Energy Flux Equation
  
  
• 3.5.7 Non-Diagonal Relaxation Time Ansatz
• 3.5.8 Summary of Equations
• 3.5.9 Onsager Relations
• 3.5.10 Electrothermal Transport Model
• 3.5.11 Phenomenological Approach
• 3.5.12 Seebeck Coefficient

M. Wagner: Simulation of Thermoelectric Devices