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686 Graphene – Synthesis, Characterization, Properties andGrAapphepnelicSyanthieosniss Classical action S(x) satisfies the Hamilton-Jacobi equation 2 2 α2 2 2 E−U(x)2 Sx1 +Sx2 −αx2Sx1 +αx1Sx2 + 4 (x1 +x2)−( v Its solution for electrons and holes is given by the following curvilinear integral over a classical trajectory x E−U(x) e x S = xE−U(x) 2 2 α Adx = (9) ds − c x ̇1 +x ̇2 − 2(−x2x ̇1 +x1x ̇2) dt, ) =0. (8) F x(0) v F x(0) G(x,x(0))= 1 2h ̄ k ∑eh ̄ 2 2π n |J(t(n),γ(n))| k= E−U(x(0)), h ̄ v F e 2 θ+γ (1+O(h ̄)), (10) F x(0) v where s is the arck length. Finally, taking into account finite number of trajectories connecting x(0) and x up to the leading order ray asymptotic solution to the Green’s tensor for electrons and holes is given by i S(t(n),γ(n))−i π μn+iπ/4 −i θ−γ −i θ+γ e 2 θ−γ ei 2 where magnetic field with U(x) = 0 G(x, x(0)) = 1 1 k ei 2 μn is the Maslov index of the n-th trajectory ((15)). This solution is singular near caustics or focal points where J(t, γ) = 0. Here is an example of ray asymptotic expansion of the Green’s tensor of electron or hole in 1−exp[iπ(R2α −1)]2h ̄ h ̄ 2π ∑eh ̄ 2 e2 e2 (1+O(h ̄)), (11) i Sn−iπ μn+iπ/4 −iθ−γ n=1,2 |J(s(n), γ(n))| e 2 e 2 −iθ+γ i θ+γ i θ−γ where cyclotronic radius R = |E|/α, S= RE(s +sin s), R where s being the arc length instead of t and measured along trajectory from x(0). This ray asymptotic solution was constructed with the help of the ray coordinates s, γ. For electrons 2RR J = R sin s ,PDF Image | GRAPHENE SYNTHESIS CHARACTERIZATION PROPERTIES
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