3. Electrical Transport

THE FOLLOWING CHAPTER deals with the description of electrical transport within semiconductors. Starting with an overview of different simulation approaches based on either quantum mechanical or classical principles, the further focus is put on classical transport based on Boltzmann's equation. In the sequel, the two cornerstones of classical device simulation, namely Poisson's equation and Boltzmann's equation are discussed followed by a glimpse on their solution, the distribution function. Furthermore, models for the band structure are introduced and their validity is discussed.

Macroscopic transport models based on Boltzmann's equation are derived by the method of moments, which is a powerful approach to obtain a series of weighted balance and flux equations. Dependent on the number of involved equations, transport models of increasing complexity can be derived. In this work, the highest order moment taken into account is the energy flux equation, which is governed by the third weight in a geometric series of weights based on the carrier momentum.

Within semiconductor device simulation, the simulation domain is subdivided into several subsystems comprising electrons, holes, and the lattice as illustrated in Fig. 3.1. For each the electron and hole subsystem, moment-based equations are derived, while for the lattice, an additional heat-flow equation has to be solved in the non-isothermal case. Energy can be transferred between the single subsystems by scattering. Phonon scattering causes an energy exchange between the carrier subsystems and the lattice. Carrier generation and recombination is described as particle exchange between the electron and hole subsystem.

Introduction of the relaxation time approximation enables an analytical treatment of the scattering term. Dependent on the treatment of the stochastic part of Boltzmann's equation, different terms result in the equations. The two approaches of microscopic and macroscopic relaxation times proposed by Stratton and Bløtekjær are applied.

Finally, an electrothermal transport model is formulated assuming local thermal equilibrium, which is compatible to an approach based on phenomenological irreversible thermodynamics [54]. This transport model is the chosen for the case studies presented in Chapter 6, since the size of the thermoelectric devices taken into account is far above the critical size where hot carrier effects play a dominant role.

Figure 3.1: Interactions between the subsystems of electrons, holes, and the lattice. While solid lines depict particle exchange, dashed lines denote energy exchange, after [54].

The validity of Onsager's relations, which are a fundamental principle for thermodynamic equilibrium is investigated for several transport models. Furthermore, special attention is devoted to the Seebeck coefficient, which is a crucial parameter for thermoelectric applications and enters the transport model as a link between the thermal and the electrical systems.

Subsections

• 3.1 A Hierarchy of Simulation Strategies
• 3.2 The Basic Equations
  • 3.2.1 Poisson's Equation
  • 3.2.2 Boltzmann's Equation
    • 3.2.2.1 Phenomenological Approach
    • 3.2.2.2 Validity
  
  
• 3.3 Band Structure Model
• 3.4 The Distribution Function
• 3.5 Macroscopic Transport Models
  • 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