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CoompmlexpWleKBxAWpprKoxBimaAtiopnspinrGorxapimhenaetEiloecntrosn-iHnoleGWraavepguhidesnien MEaglnectictrFoielnd -Hole Waveguides in Magnetic Field 87 the corresponding family of trajectories is given by x1 =R[sin(s +γ)−sinγ]+x(0), R1 x2 =R[−cos(s +γ)+cosγ]+x(0), R2 and θ = s +γ (see Fig. 1). For holes, we have R x1 = R[sin(γ− s )−sinγ]+x(0), R1 x2 = R[−cos(γ− s )−cosγ]+x(0), R2 and θ = γ+π− s . The asymptotic solution is valid if kR >> 1. This solution is in full R (12) (1) (1) G(x, (0, 0)) = −i −kH0 (kr) (i∂x1 + ∂x2 )H0 (kr) = (13) 4h ̄ (i∂x1 − ∂x2 )H(1)(kr) − kH(1)(kr) 00 √k eikr+iπ/4 1 e−iγ h ̄ 2√2πr eiγ 1 (1+O(k−1)), where r = |x|, x1 = r cos(γ), x2 = r sin(γ), k = E/(h ̄ vF), and H(1)(x) is the Hankel function. agreement with the Green’s tensor for the problem in a sense of matching as α → 0 δ(x) 0 vF <−ih ̄∇,σ ̄ >G(x)−EG(x)= 0 δ(x) , which exact representation can be found by Fourier transform The property of the solution being singular at 0 E=El =±vF h ̄α(2l+1),l=0,1,2,...., gives the quantization of the energy spectrum already well-known in graphere electron-hole motion in magnetic field ((5)). This solution is singular at s = νπR, ν ∈ N (see (19)). The set of singular points are the circle with ν = 2n + 1 (see Fig. 1), which is a smooth caustic, and the focal point x(0), where ν = 2n (n ∈ N). Frequently in practice, the structure of classical trajectories looks very complicated due to the presence of multiple caustics and focal points. This situation takes place for a charged particle moving in strong magnetic fields. Thus, the ray method asymptotic expansion is not effective. Alternative approach is well-known - the method of Maslov canonical operator (15) which gives a cumbersome asymptotic construction depending on geometrical and topological properties of lagrangian manifolds represented by families of by-characteristics in the phase space. In some simple cases it reduces the answer to local asymptotic expansions for wave fields expressed via special functions of wave catastrophes, for example Airy function for smooth caustic 1+∞ v(z) = 2π exp i(t3/3 + tz) dt, −∞PDF Image | GRAPHENE SYNTHESIS CHARACTERIZATION PROPERTIES
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