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CoompmlexpWleKBxAWpprKoxBimaAtiopnspinrGorxapimhenaetEiloecntrosn-iHnoleGWraavepguhidesnien MEaglnectictrFoielnd -Hole Waveguides in Magnetic Field 913 where real functions z ̃(s), p ̃(s) satisfy (21) with initial conditions z ̃(0) = 0, p ̃(0) = |a(0)|, we obtain −iθ−γ0 (0) i S(t,γ0) A(γ0) e 2 −iθ+γ0 2 G(x,x ,E)=eh ̄ e z(s)e2 e2 Δ1 exp − −Δ1 i a(0)z ̃(s) 2h ̄ z(s) −iθ−γ0 e 2 iθ+γ0 iθ−γ0 (γ−γ0)2 d(γ−γ0)+... i S(t,γ0) A(γ0) −iθ+γ0 e 2 2πh ̄ ∓iπ/4 = e h ̄ where Δ1 is a positive constant. Since J(s,γ0) = z ̃(s), matching the leading term for G(x, x(0), E) given in (24) leads to for electrons, for holes. Thus, we obtain 4 π h ̄ 4. Testing the method of Gaussian beams summation for electron-hole waveguide motion in magnetic field In this section the method of Gaussian beams summation for electron-hole motion in magnetic field is tested for a special case with linear electrostatic potential U = βx2 in (2), and we assume that vF = 1. For this case we compare applicability of both asymptotic representations (10) and (23) for the Green’s tensor component G11(x, x(0), E). The hamiltonian dynamics of electrons motion is determined by the Hamilton function in Landau gauge A = B(−x2, 0, 0) h1=βx2+ (p1−αx2)2+p2, inthedomainΩe ={x:E>βx2}onthelevelseth1 =E,or 1 He=2 (p1−αx2)2+p2−(E−βx2)2 , on the level set He = 0. The hamiltonian dynamics of holes motion is determined by the Hamiltilton function h2=βx2− (p1−αx2)2+p2, inthedomainΩe ={x:E<βx2}onthelevelseth2 =E,or 1 i θ − γ 0 | a ( 0 ) | e A(γ0) = ik , i θ + γ 0 z ̃(s)e2 e2 4 π h ̄ A(γ0) = −ik 4 π h ̄ A(γ) = i|k| . Hh=2 −(p1−αx2)2−p2+(E−βx2)2 , on the level set Hh = 0.PDF Image | GRAPHENE SYNTHESIS CHARACTERIZATION PROPERTIES
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