Surface states in crystals with low-index surfaces

For most of the conventional crystals with low-index surfaces, the hopping between the nearest neighbor (1NN) crystal planes (CPs) is dominant and the ones from the nNN (2≤q < ∞) CPs are relatively weak, considered as small perturbations. The recent theoretical analysis^[1] has demonstrated the absence of surface states at the level of the hopping approximation between the 1NN CPs when the original infinite crystal has the geometric reflection symmetry (GRS) for each CP. Meanwhile, based on the perturbation theory, it has also been shown that small perturbations from the hopping between the nNN (2≤n<∞) CPs and surface relaxation have no impact on the above conclusion. However, for the crystals with strong intrinsic spin-orbit coupling (SOC), the dominant terms of intrinsic SOC associate with two 1NN bond hoppings. Thus SOC will significantly contribute the hoppings from the 1NN and/or 2NN CPs except the ones within each CP. Here, we will study the effect of the hopping between the 2NN CPs on the surface states in model crystals with three different type structures (Type I: “…-P-P-P-P-…”, Type II: “…-P-Q-P-Q-” and Type III: “…-P=Q-P=Q-…” where P and Q indicate CPs and the signs “-” and “=” mark the distance between the 1NN CPs). In terms of analytical and numerical calculations, we study the behavior of surface states in three types after the symmetric/asymmetric hopping from the 2NN CPs is added. We analytically prove that the symmetric hopping from the 2NN CPs cannot induce surface states in Type I when each CP has only one electron mode. The numerical calculations also provide strong support for the conclusion, even up to 5NN. However, in general, the coupling from the 2NN CPs (symmetric and asymmetric) is favorable to generate surface states except Type I with single electron mode only.