"Give the complete mathematical derivation of equations 4-8 for twisted atomic bilayers graphene for chiral plasmons"

Chiral plasmons withtwisted atomic bilayers.pdf It is then desirable to simplify the boundary conditions in equations (2-3). After some simplification, we find that the boundary conditions can be re-organized into a more intuitive way as follows [Fig. 1(b)] n x (E(1) - E(2) ) = -am(H(D) + H(2)) to, (E(D) + E(2)) (4) n x (H(1) - H(2) ) = toe (E(1) + E(2)) +8, (H(D) + H(2)) (5) fTBG [om,x 0 0 Om = 0 omy = 1-(60-01) f TBG STBG.K20/KG (6) 0 1-(60-01).f TBG K2/K3. 0 8x = [oxx 0 = -oxy . om (7) 0 go+ 01 - oxyUm,x - STBG 0 be = Oex 0 Oey = (8) 0 Oo + 01 - OxyOmy -STBG WHO elkzoa/2_e-ikzoa/2 In these equations, the coefficient fTBG = Kzo elkzod/2+e-ikzoa/z originates from the electromagnetic response of the gap between the two atomic layers (i.e., region 3 in Fig. 1, which can be readily treated as