3. Tunneling in Semiconductors

THIS CHAPTER outlines the theory of quantum-mechanical tunneling in semiconductor devices. Different tunneling mechanisms, such as direct-, FOWLER-NORDHEIM, and trap-assisted tunneling are covered. As a first step, the TSU-ESAKI model is derived. This model allows to distinguish between the supply function, which describes the supply of carriers for tunneling, and the transmission coefficient, which characterizes the penetrability of the considered energy barrier. The supply function depends on the energetic distribution of the carriers, an important quantity in semiconductor device modeling. Models which describe the shape of this distribution function are reviewed, namely the MAXWELLian^3.1, heated MAXWELLian, and non-MAXWELLian model.

The transmission coefficient can be found by a solution of SCHRÖDINGER's equation in the considered region. The WENTZEL-KRAMERS-BRILLOUIN- and GUNDLACH-methods, which are frequently encountered in the modeling of tunneling current, are shortly reviewed. However, for the proper simulation of transmission through arbitrary barriers, advanced models must be considered. Emphasis is put on the description of linear- and constant-potential transfer-matrix methods as well as on the quantum transmitting boundary method (QTBM).

The TSU-ESAKI tunneling formula finds the tunneling current density by an integration in the energy domain. In the channel of an inverted MOSFET, however, the strong electric field leads to the creation of bound and quasi-bound states. While bound states do not contribute to the tunneling process, tunneling from quasi-bound states can be understood using the concept of finite life times. Different numerical methods to calculate the life time of a quasi-bound state are reviewed.

The chapter continues with the description of trap-assisted tunneling and discusses some of the most frequently used models. Emphasis is put on the adaption of an inelastic trap-assisted tunneling model which incorporates energy loss by phonon emission and does not rely on the common assumptions of constant capture cross-sections.

Finally, a short summary and a comparison of the described methods is given.

Subsections

• 3.1 Tunneling Mechanisms
• 3.2 The TSU-ESAKI Model
• 3.3 Supply Function Modeling
  • 3.3.1 FERMI-DIRAC Distribution
  • 3.3.2 MAXWELL-BOLTZMANN Distribution
  • 3.3.3 Non-MAXWELLian Distributions
  • 3.3.4 Normalization
  
  
• 3.4 The Energy Barrier
  • 3.4.1 The Metal-Oxide-Semiconductor Capacitor
  • 3.4.2 Image Force Correction
  
  
• 3.5 Transmission Coefficient Modeling
  • 3.5.1 The WENTZEL-KRAMERS-BRILLOUIN Approximation
  • 3.5.2 The GUNDLACH Method
  • 3.5.3 Transfer-Matrix Method
    • 3.5.3.1 Piecewise-Constant Potential
    • 3.5.3.2 Piecewise-Linear Potential
  • 3.5.4 Quantum Transmitting Boundary Method
  • 3.5.5 Comparison
  
  
• 3.6 Bound and Quasi-Bound States
  • 3.6.1 Eigenvalues of a Triangular Energy Well
  • 3.6.2 Eigenvalues of Arbitrary Energy Wells
  • 3.6.3 The Life Time of Quasi-Bound States
    • 3.6.3.1 The Reflection Coefficient Resonances
    • 3.6.3.2 The Quasi-Classical Formula
    • 3.6.3.3 The Eigenvalues of the Non-HERMITian HAMILTONian
  
  
• 3.7 Compact Tunneling Models
• 3.8 Trap-Assisted Tunneling
  • 3.8.1 Model Overview
    • 3.8.1.1 CHANG's Model
    • 3.8.1.2 IELMINI's Model
    • 3.8.1.3 Compact Trap-Assisted Tunneling Models
  • 3.8.2 The Model of JIMÉNEZ et al.
    • 3.8.2.1 Capture and Emission Probabilities
    • 3.8.2.2 Capture and Emission Times
    • 3.8.2.3 Steady-State Current
    • 3.8.2.4 Transient Current
  
  
• 3.9 Model Comparison

A. Gehring: Simulation of Tunneling in Semiconductor Devices