Recursive Tight-Binding Green’s Function Method: Application to Ballistic and Dissipative Transport in Semiconductor Nanostructures
The purpose of this chapter is to present a specific numerical application of Green’s functions in tight-binding structures. The numerical technique is conventionally called the recursive Green’s function method. It was first introduced by (1981) to study electron transport in disordered systems and, since then, it has been widely used by many authors as a practical tool to simulate quantum transport in a variety of structures. The Green’s function must be defined in a tight-binding structure, which means that the space must be discretized in sites that represent either real atoms with strongly localized orbitals, or fictitious sites that fill the space with a sufficiently high density. The use of tight-binding Green’s functions (TBGF’s) has some practical advantages. For example, hard-wall boundary conditions are simply simulated by the absence of sites. Impurities are conveniently included by introducing sites with unequal diagonal energy. In particular, diagonal disorder may be introduced by making the site energy a random variable. It is also useful for describing transport in geometries with hard-wall boundaries defined by straight segments and right angles (or angles whose tangent is the ratio of low integers). Among its possible shortcomings, we may mention that the recursive TBGF method is not particularly suited to study transport in structures defined by geometrical parameters one may want to vary continously, such as the angle between two wires or the length of a given segment. For this type of purposes, a wave-function matching method (Schult et al., 1989) may be more advantageous [see, for example, (1990) for a study of electron transport through a circular bend linking two leads with an arbitrary angle]. Within a recursive TBGF scheme, these difficulties can be overcome in principle by introducing a sufficiently dense grid, but the computational cost may easily become prohibitive.
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