Analytical study of surface states caused by the edge decoration

Abstract

Analytical studies of the effect of edge decoration on the energy spectrum of semi-infinite onedimensional (1D) lattice chain with Peierls phase transition and zigzag edged graphene (ZEG) are presented by means of transfer matrix method, in the frame of which the sufficient and necessary conditions for the existence of the edge states are determined. For 1D lattice chain, the zero-energy edge state exists when Peierls phase transition happens regardless whether the decoration exists or not, while the non-zero-energy edge states can be induced and manipulated through adjusting the edge decoration. On the other hand, the semi-infinite ZEG model with nearest-neighbor interaction can be mapped into the 1D lattice chain case. The non-zero-energy edge states can be induced by the decoration as well, and we can obtain the condition of the decoration on the edge for the existence of the novel edge states.

The study of surface states in a semi-infinite crystal

“…At the level of the 1NN hopping approximation, the study of surface states in this case is exactly the same as that of edge states in the semi-infinite 1D single orbit atomic chain. As is well known, no edge states exist in the semi-infinite 1D atomic chain for both Type I and II when the forward hopping constant equals to the backward one 31 . Thus, we will take into account the case that each P contains n (>1) electron modes.…”

“…The above conclusion also covers the case of 2D/1D crystals, in which the “surface” represents the atomic chain/point. In our following demonstration, the transfer matrix approach 30 31 is applied. The crystals with the reflection symmetry are only one of two types: Type I: “…- P-P-P - P- …” in Fig.…”

“…(9) is exactly the same as the transfer matrix equation of 1D atomic chain with the single electron mode. It has been known that there are no edge states for any energy E no matter whether λ s = 0 or λ s ≠ 0 31 . Thus we arrive at for surface states.…”

“…Thus the longer range hopping among CPs is able to induce surface states if it is strong enough and not regarded as a perturbation, compatible with our demonstration. Furthermore, the type structure of zigzag edged graphene is “ P = P-P = P- …” where the F-B symmetry is broken, thus it is in favor of the existence of edge states, shown in the zero mode result 31 . Secondly, let us further address ABO 3 Perovskites since they are very much important in device applications.…”

Surface states of gapped electron systems and semi-metals

“…The SSs and the bulk-boundary correspondence have been studied by many theoretical works using various concrete models including lattice [33][34][35][36][37][38][39][40][41] and continuous models [42][43][44][45][46]. In some of the existing works, the SSs are analytically solved by imposing special boundary conditions [36,39].…”