4. Implementation

NOVEL STRUCTURES and materials such as deca-nanometer Si bulk MOSFETs, multiple gate MOSFETs, CNT-FETs and molecular based transistors, are expected to be introduced to meet the requirements for scaling [1]. A deep understanding of quantum effects in nano-electronic devices helps to improve their functionality and to develop new device types. For that purpose extensive computer simulation are required.

A multi-purpose quantum-mechanical solver, the VIENNA SCHRÖDINGER-POISSON solver (VSP), with the aim to aid theoretical as well as experimental research on nano-scale electronic devices, has been developed [231]. VSP is a quantum mechanical solver for closed as well as open boundary problems. The software is written in C++ using state-of-the-art software design techniques. The chosen software architecture allows one to add new models easily. Critical numerical calculations are performed with stable and powerful numerical libraries such as BLAS, LAPACK, and ARPACK. VSP holds a graphical user interface written in JAVA, as well as an XML based interface. Furthermore, VSP has an open software application interface and can be used within third party simulation environments.

This chapter describes the implementation of the outlined NEGF formalism into VSP. For an accurate analysis it is essential to solve the coupled system of transport and POISSON equations self-consistently [163]. The discretization of the POISSON equation and quantum transport equation is studied in this chapter.

A tight-binding HAMILTONian is used to describe transport phenomena in CNT-FETs. The mode-space transformation used in this work reduces the computational cost considerably. The mode-space approach takes only a relatively small number of transverse modes into consideration. To reduce the computational cost even further, we used the local scattering approximation [112]. In this approximation the scattering self-energy terms are diagonal in coordinate representation. We show that the local approximation is well justified for electron-phonon scattering caused by deformation potential interaction.

We investigate methods of generating adaptive energy grids for the transport equations and their effect on the convergence behavior of the self-consistent iteration. Our results indicate that for accurate and fast convergent simulations the energy grid must be carefully adapted.
Subsections

• 4.1 Electrostatic Potential and the POISSON Equation
  • 4.1.1 Discretization of the POISSON Equation
  • 4.1.2 Boundary Conditions
  • 4.1.3 Interface Conditions
  
  
• 4.2 Basis Functions and Matrix Representation
• 4.3 Tight-Binding HAMILTONian
• 4.4 Mode-Space Transformation
• 4.5 Contact Self-Energies
  • 4.5.1 Semi-Infinite CNT Contacts
  • 4.5.2 SCHOTTKY Type Metal-CNT Contacts
  
  
• 4.6 Scattering Self-Energies
  • 4.6.1 Scattering with Optical Phonons
  • 4.6.2 Scattering with Acoustic Phonons
  
  
• 4.7 Evaluation of Observables
  • 4.7.1 Carrier Density
  • 4.7.2 Current
  • 4.7.3 Discussion
  
  
• 4.8 Selection of the Energy Grid
  • 4.8.1 Confined States
  • 4.8.2 Non-adaptive Energy Grid
  • 4.8.3 Adaptive Energy Grid
  
  
• 4.9 Self-Consistent Simulations
  • 4.9.1 Self-Consistent Iteration Scheme
  • 4.9.2 Convergence of the Self-Consistant Simulations

M. Pourfath: Numerical Study of Quantum Transport in Carbon Nanotube-Based Transistors