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282 Graphene – Synthesis, Characterization, Properties andGrAapphepnelicSyanthieosniss product, and σ ̄ = (σ1 , σ2 ) with Pauli matrices 01 0−i σ1 = 1 0 , σ2 = i 0 . However, classical trajectories structure for waveguides and resonators with rather strong magnetic fields is getting very complicated, and, owing to the presence of multiple caustics and focal points, the semiclassical approximation, well-known as ray asymptotic method, is not valid. In this case one of the possibilities to tackle the problem of computing Green’s function is Maslov’s canonical operator method (15). The method gives a cumbersome universal asymptotic construction depending on geometrical and topological properties of families of classical trajectories of electronic motion. Fortunately, an alternative method of summation of Gaussian beams (integral over Gaussian beams) which was developed for acoustic, and later electromagnetic and elastic wave propagation may be found much more suitable especially in practical applications and especially numerical analysis of electronic motion in graphene structures (16), (17). The theoretical foundations of the method are rather simple in comparison with the Maslov’s canonical operator method. Gaussian beam as a localised asymptotic solution is always regular near caustics or focal point. Realization of the method does not require any knowledge about geometrical properties of caustics, and numerous numerical tests demonstrated this effect (16), (17). This is due to the fact that the structure of the asymptotic Gaussian beam solution does not depend on geometrical properties of caustics, and the final asymptotic approximation is just a superposition of Gaussian beams. Thus, in a general case, the method of summation of Gaussian beams gives universal semiclassical uniform approximation for solutions to various problems of wave propagation and quantum mechanics. This approximation is valid near caustics or focal points of arbitrary geometric structure. Application of the method to computations of high-frequency acoustic and elastic wave fields was proved to be very efficient and robust. This method is convenient to construct a semiclassical uniform approximation for Green’s function for interior of waveguides and resonators quantum problems. Application of this method to problems of electron motion in magnetic field in graphene required a generalization of the approach originally developed for acoustic wave propagation problems. The first step in this direction has been done in ((18), (19)). The chapter is organized as follows. First, in section 2, we give a description of the ray asymptotic solution and the boundary layer semiclassical method used to construct the asymptotic solution of the Gaussian beam in the presence of a magnetic field and any scalar potential. Subsecuently, in section 3, the techniques of the Gaussian beams summation method is presented. Finally, in section 4, some numerical results of computation of semiclassical uniform approximation for Green’s function for interior of graphene waveguide with magnetic field and linear electrostatic potential is described. 2. Electronic Gaussian beams First, the basic steps of the ray method recurrence relations (see (20), (21)) are presented for the stationary problem for the Green’s tensor for the Dirac system describing an electron-hole quantum dynamics in the presence of a homogeneous magnetic field and arbitrary scalarPDF Image | GRAPHENE SYNTHESIS CHARACTERIZATION PROPERTIES
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