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88 x2 0 −0.2 −0.4 −0.6 −0.8 Graphene – Synthesis, Characterization, Properties andGrAapphepnelicSyanthieosniss 0.8 0.6 0.4 0.2 x2 0 −0.2 −0.4 −0.6 −0.8 Fig. 1. Family of classical trajectories (circles) for the problem of the Green’s tensor for electrons (1a) and holes (1b) in magnetic field with U(x) = 0 and x(0) = (0, 0). 0.8 0.6 0.4 0.2 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 x1 (a) x1 (b) and Piercy integral - in case of casp I(x,y)=2π exp i(t4+xt2+yt) dt. 1+∞ −∞ However, there exist alternative approach known as the method of Gaussian beams summation method which is universal, simple, effective and robust in developing numerical algorithms. It was shown for acoustic, electromagnetic and elastic waves (17). In the next section a generalization of the method is described for the wave function of graphene electron-hole motion in magnetic field and potential. It gives analytical representation of Green’s tensor of Dirac system as an integral over Gaussian beams. 2.2 Localized asymptotic solution - electronic Gaussian beam in graphene Let x0 = (x1(s), x2(s)) be a particle (electron or hole) classical trajectory, where s is the arc length measured along a trajectory. Consider the neighborhood of the trajectory in terms of local coordinates s, n, where n is the distance along the vector normal to the trajectory such that x = x(0) (s) + en (s)n, (14) where en(s) is the unit vector normal to the trajectory. Following (21), we apply the asymptotic boundary-layer method to the homogeneous Dirac h ̄ ) ψ = u = e i (S0(s)+S1(s)n) ∑ h ̄ j/2 uj(s, ν) (15) system (1). We assume that the width of the boundary layer is determined by |n, n ̇ | = O( √ as h ̄ → 0. Introducing ν = n/ +∞ h ̄ = O(1), we seek an asymptotic solution to (3) in the form h ̄ v j=0 vj(s,ν) √PDF Image | GRAPHENE SYNTHESIS CHARACTERIZATION PROPERTIES
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