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484 If E > U(x), then E = h1, if E < U(x), then Graphene – Synthesis, Characterization, Properties andGrAapphepnelicSyanthieosniss eiθ = 1 2 p −αx2 +i(p +αx1) 2 2 , e1 = v1, e2 = v2, p1−αx2 +i(p2+αx1) p E=h2, eiθ=− In a domain Ωe = {x : E > U(x)}, the hamiltonian 2 2 , e1=v2, e2=v1. p h1 =U(x)+vF (p1−αx2)2+(p2+αx1)2 22 on the level set h1 = E describes dynamic of electrons (a quasi-particle described by the initial Dirac system behaves like electron, negatively charged). The corresponding classical trajectories can be obtained from the hamiltonian system x ̇=Hpe, p ̇=−Hxe, x=(x1,x2), p=(p1,p2), with hamiltonian (see (15)) He = 1(p1 − αx2 )2 +(p2 + αx1 )2 −(E−U(x))2 2 2 2 vF on the level set He = 0. Opposite to this case, in a domain Ωh = {x : E < U(x)}, the hamiltonian h2 =U(x)−vF (p1−αx2)2+(p2+αx1)2 22 on the level set h2 = E describes dynamic of holes (a quasi-particle described by the initial Dirac system behaves like hole, positively charged). The corresponding classical trajectories can be obtained from the hamiltonian system with hamiltonian Hh = 1−(p1−αx2)2−(p2+αx1)2+(E−U(x))2 2 2 2 vF on the level set Hh = 0. Thus, for electrons and holes, two different families of classical trajectories x1,2 = x1,2(t,γ), p1,2 = p1,2(t,γ) should be taken into account, where t is the time with respect to the hamiltonian system with hα, and the parameter γ gives the parametrisation of the initial manifold in a phase space R4x,p. For the Green’s tensor point source problem, γ is a polar angle at which a trajectory issues from the source. Thus, taking the initial conditions for x1,2 = x1,2(t,γ), p1,2 = p1,2(t,γ) x|t=0 = x , p|t=0 = v cos γ, sin γ , (0) |E − U(x(0))| T FPDF Image | GRAPHENE SYNTHESIS CHARACTERIZATION PROPERTIES
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