GREEN'S FUNCTION OF PSEUDOSPIN-1 SYSTEMS UNDER SCATTERING BY A CHAOTIC INTERBAND POTENTIAL

Abstract

With the discovery of graphene and graphene-like systems, condensed matterphysics not only found a fundamentally new subject of research with very promising, as it has been demonstrated by the modern nanotechnology amazing successes, perspectives of applications in nanoelectronics, but also received a powerful impetus for searchingofother objects with no less exclusive properties. It is well-known that decisive features which determine the main peculiarities of any systems reduce to the energetic spectra and wave functions structure. It is whyinvestigations of spectrum and wave functionsturn out to be key for searching such exotic objects. Concerning the spectrum of graphene- like systems its characteristic feature is that in the vicinity of Brillouin zone critical pointsthe dispersion law shows linear dependence on the wavevecror. As far as the equienergy surfaces they have the form of two cones with a common vertex localized in the critical points. Correspondentwave functions manifest two component form similar to the spinor structure inherent to the particles with spin ½. 
Due to the wave functions structure such objectswere named as the pseudospin ½ systems. It is quite natural to expand the nomenclature of pseudospin systems by concidering ones with the pseudospin greater than ½. Although in the theoretical context enough attention is paid to the systems with an arbitrary pseudospin equal to N, in view of the experimental and technological opportunities more profound studies are focusedon the pseudospin-1 objects because of their wide area of realizations begining from the optical lattices and ultra-cooled atoms ending by fotonic crystals with some special values of dielectric and magnetic permittivity in frequency function et.c. However, taking in accountreal processes of synthesis, one can conclude thatthe pseudo-spin objects, like any other condensed matter’s systems, are inevitable submitteredto appearance of structural inhomogeneities, i.e. defects which arerich in their nature, origin. structure or other morphological features. The presence of defects, not least of all, affects the energy spectrum of the system, leading to the emergence of local and virtual energy states and also causes scattering of quasiparticles and other effects. 
That is why this work has as its main purpose the investigation of chaotic potential impact on the energy spectrum of the pseudospin-1 system. Concerning this problem the Green’s function method admits to be informative enough as really powerful approach with well elaborated mathematical procedures. Due to the lot of experimental results the effects of inhomogeneities on the properties of pseudospin-1 systems, for example on the localization and tunneling of quasiparticles, have been studied in the works of many author.As a rule they only considered as defects anumber of parallel one-dimensional potential barriers with uniformly distributed chaotic widths or heights while this work differs from above mentioned by more general suggestions about the potential related to defects. Here was assumedthat the chaotic potential causes transitions betweendispersion zones of pseudospin-1 system, while the contribution of the undispersionflat zone, which has been established in many early studies, is manifested in higher approximations of perturbation theory. Regarding the random field, unlike the already discussedmodels, an analytical form of the potential is not postulated here. As far as its peculiarities only somegeneral statistical properties has been accepted in order to considerthe energetic spectrum problem. Then resolvent Green’s function has been introduced. Due to the Hamilton’s operator structure,appropriated to the pseudospin-1systems, Green function has a form of third-order matrix. For some chaotic field realization Green function can be writtenin terms of the perturbation theory as the serie in powers of the potential. By averaging procedure the operator of the proper energy has been introduced and was determined in the Born’s approximation. Analysis of the self-energy part shows that it turns out singular when the energy tends to zero. It is this singularity that manifests the influence of the flat no dispersive zone. Isolating the singular part in the seed Green's function allowsto sum the perturbation theory series and find the renormalized self- energy part whose substitution into the Dyson’s equation permits to find the final result for the Green’s function.  
The energy spectrum of quasiparticles is found as the pole part of the Green's function. It is shown that there is a splitting of the bands with the appearance of an energy gap of a fluctuation nature. The width of the forbidden band is proportional to the pair correlation function of random potential. The energy band, which is dispersionless in the case of an ideal system, also undergoes modification. Beside it the energy states of continuous spectrum suffer attenuationdue to scattering on fluctuations of a chaotic field. tIt should be pointed outthat all corrections to the Green's function are expressed in terms of the paircorrelation function of the random field, which is a priori unknown.