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1966 Graphene – Synthesis, Characterization, Properties andGrAapphepnelicSyanthieosnis s 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 −2 −1.5 −1 −0.5 0 0.5 1 x2 | G 1 1 | 2 ̄h 2 / k Fig. 5. The values of |G11(x, x(0), E)|2h ̄ 2/k at the points of vertical cut B for electronic waveguide in magnetic and electrostatic fields, Gaussian beams summation. For both families of the classical trajectories, it can be shown rigorously that all of them are bounded from the corresponding turning lines. That is, for electrons it is always E > βx2 (t, γ), and for holes E < βx2 (t, γ), for arbitrary (t, γ). Consider the case with following values of the dimensionless parameters of the problem E = 1, α = 2.5, β = 1, h ̄ = 0.05, w = 2. The structure of trajectories for holes and electrons is shown in Fig. 2, 3. It is worth to remark that in both cases shown in Fig. 2 and 3, the electron-hole waveguide propagation is clearly seen to be isolated from the potential turning lines, for electrons x2 ≤ 1, for holes x2 ≥ −1. This waveguide propagation (drift) takes place to the right from the source (x(0) = (0, 0)). In Fig. 4 the values of electronic Green’s tensor normalised component |G11(x, x(0), E)|2h ̄ 2/k at the points of vertical cut A = {x : x1 = 0.75, −1.7 < x2 < −0.2} are presented that were computed by means of Gaussian beams summation approximation (23) and the ray asymptotics (10). Both graphs again show a good agreement within the domain that is away from caustics. It is clear that near to caustics the ray asymptotics blows up. Instead, Gaussian beams summation approximation shows a smooth transition through the caustic line with exponential decay into geometrical shadow domain. Discretizing the integral (23) by means of 128 Gaussian beams was used. In Fig. 5 the values of electronic Green’s tensor normalised component |G11(x, x(0), E)|2h ̄ 2/k at the points of the second vertical cut B = {x : x1 = 1.25, −1.8 < x2 < 0.75} are presented that are computed only by means of Gaussian beams summation approximation (23). One should observe the oscillatory behaviour of Green’s tensor components between caustic lines confining the waveguide and exponential decal away from the waveguide. Similar results could be expected for the hole Green’s tensor normalised components.PDF Image | GRAPHENE SYNTHESIS CHARACTERIZATION PROPERTIES
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