3. Quantum Transport Models

THIS CHAPTER outlines the theory of the non-equilibrium GREEN's function (NEGF) techniques for modeling transport phenomena in semiconductor devices. The NEGF techniques, initiated by SCHWINGER [92], KADANOFF, and BAYM [93] allow ones to study the time evolution of a many-particle quantum system. Knowledge of the single-particle GREEN's function provides both the complete equilibrium or non-equilibrium properties of the system and the excitation energies of the systems containing one more or one less particle. The many-particle information about the system is cast into self-energies, parts of the equations of motion for the GREEN's functions. GREEN's functions can be expressed as a perturbation expansion, which is the key to approximate the self-energies. The NEGF techniques provide a very powerful technique for evaluating properties of many-particle systems both in thermodynamic equilibrium and also in non-equilibrium situations.

The basic approach developed in the early 1970s has became increasingly popular during the last 10 years. The motivation for the development of the NEGF tunneling formalism was the metal-insulator-metal tunneling experiments that received much attention during the 1960s [94]. The accelerated use of the approach was motivated by experimental investigations of mesoscopic physics made possible by high quality semiconductor hetero-structures grown by molecular beam epitaxy. In 1988, KIM and ARNOLD were the first to apply the NEGF formalism to such a system, specifically, a resonant tunneling diode [95]. As experimental methods progressed allowing finer manipulation of matter and probing into the nano-scale regime, the importance of quantum effects and tunneling continuously increased. The theory was adapted to address the current systems of interest ranging from mesoscopics to single-electronics, nano-scaled FETs, and molecular electronics.

The general formalism for NEGF calculations of current in devices was first described in a series of papers in the early 1970s [96,97,98,99]. The partitioning of an infinite system into left contact, device, and right contact, and the derivation of the open boundary self-energies for a tight-binding model was presented in [96]. This theory was re-derived for a continuum representation in [97], tunneling through localized impurity states was considered in [98], and a treatment of phonon assisted tunneling was derived in [99]. In 1976, the formalism was first applied to a multi-band model (two-bands) to investigate tunneling [100] and diagonal disorder [101], and in 1980 it was extended to model time-dependent potentials [102].

The applications of the NEGF techniques have been extensive including quantum optics [103], quantum corrections to the BOLTZMANN transport equation [104,105], high field transport in bulk systems [106], and electron transport through nano-scaled systems. Over the last decade, NEGF techniques have become widely used for modeling high-bias, quantum electron and hole transport in a wide variety of materials and devices: III-V resonant tunnel diodes [95,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122], electron waveguides [123], superlattices used as quantum cascade lasers [124], Si tunnel diodes [125,126], ultra-scaled Si MOSFETs [127,8,128,129,130], Si nano-pillars [131,132,133], carbon nanotubes [134,135,136,137,138,139,140,141,142,143,144,145], metal wires [146,147], organic molecules [148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164], superconducting weak links [165], and magnetic leads [136,166,167]. Physics that have been included are open-system boundaries [96], full-band-structure [168,117,116,118,125,126], band-tails [126], the self-consistent HARTREE potential [109,115,169], exchange-correlation potentials within a density functional approach [115,149,135,150,152,146,11], acoustic, optical, intra-valley, inter-valley, and inter-band phonon scattering, alloy disorder and interface roughness scattering in Born type approximations [113,110,111,112,114,115,116,125,126,124], photon absorption and emission [124], energy and heat transport [120], single-electron charging and non-equilibrium KONDO systems [170,171,172,173,174,175,176], shot noise [119,177,113], A.C. [108,178,179,180,181,182,183], and transient response [180,184]. Time-dependent calculations are described further in [185]. General tutorials on the NEGF techniques [60,186] and the applications can be found in [116,187,188].

This chapter continues with a tutorial derivation of the standard expressions, where one shall rely on the second quantization formulation. A brief description to this formalism is presented in Appendix A. Various formulations of many-particle GREEN's function theory exist. For instance, in equilibrium theory there is both a zero-temperature as well as a finite-temperature (MATSUBARA) formalism [189,190,185], which are described briefly next. Then, the formulation of the more general non-equilibrium finite-temperature theory which also applies to equilibrium situations as a special case is introduced and the kinetic equations for this formalism are discussed. Applying WICK theorem, a perturbation expansion of GREEN's functions can be achieved. Such expansions provide methods to approximate self-energies due to various scattering mechanisms. Finally, a comparison of the GREEN's function formalism with other transport models is presented.
Subsections

• 3.1 Equilibrium Zero Temperature GREEN's Function
  • 3.1.1 Definition of the GREEN's Function
  
  
• 3.2 Equilibrium Finite Temperature GREEN's Function
  • 3.2.1 Equilibrium Ensemble Average
  • 3.2.2 MATSUBARA GREEN's Function
  
  
• 3.3 Non-Equilibrium GREEN's Functions
  • 3.3.1 Non-Equilibrium Ensemble Average
  • 3.3.2 Contour-Ordered GREEN's Function
  • 3.3.3 KELDYSH Contour
  
  
• 3.4 Perturbation Expansion of the GREEN's Function
  • 3.4.1 WICK Theorem
  • 3.4.2 First-Order Perturbation Expansion
  
  
• 3.5 DYSON Equation
• 3.6 Approximation of the Self-Energy
  • 3.6.1 Elecron-Electron Interaction
  • 3.6.2 Electron-Phonon Interaction
  
  
• 3.7 Analytical Continuation
  • 3.7.1 Real Time Formalism
  • 3.7.2 LANGRETH Theorem
  
  
• 3.8 Quantum Kinetic Equations
  • 3.8.1 The KADANOFF-BAYM Formulation
  • 3.8.2 KELDYSH Formulation
  • 3.8.3 Steady-State Kinetic Equations
  
  
• 3.9 Relation to Observables
  • 3.9.1 Electron and Hole Density
  • 3.9.2 Spectral Function and Local Density of States
  • 3.9.3 Current Density
  • 3.9.4 Current Conservation
  
  
• 3.10 Comparison of Transport Models
  • 3.10.1 Non-Equilibrium GREEN's Function
  • 3.10.2 Master Equation for the Density Matrix
  • 3.10.3 The WIGNER Distribution Function

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