The study of surface states in a semi-infinite crystal

An infinite three dimensional (3D) crystal can be constructed by an infinite number of parallel 2D (hkl) crystal planes (CPs) coupled to each other. Based on lattice model Hamiltonian with the hopping between the nearest neighbor (1NN) CPs and all possible neighbor hoppings within each CP, we analytically prove that a (hkl) cut crystal will not accommodate any surface states if the original infinite crystal has the reflection symmetry which results in the forward transfer matrix F to be equal to the backward one B, named as F-B dynamical symmetry. We also study the effect of the longer range couplings among the nNN (n > 1) CPs and surface relaxation on our conclusion and find that the small perturbation from both factors has no effect on our conclusion based on the perturbation theory. Thus our model may have the potential for studying surface states in some cut crystals with low-index surfaces. Our result may be helpful to visually predict which cutting direction in some non-topological crystals is unfavorable to generate surface states.