A phase field matching model to compute the localized electronic bands of a 3D monatomic chain of body-centered cube structure: Investigating the (100), (110) and (111) cutting directions

摘要

In this study, we compute the surface electronic bands in different cutting directions of a body-centered cubic structure. The systems under study are given as 3D leads that end up with an open cut layer in the [100], [110] and [111] directions. To build up the Hamiltonian matrix around the cutting directions, we integrate into our theoretical model the phase field matching method (PFMM) and the tight-binding approximation. In fact, a direct solution of the built-up surface eigenvalue problem is not possible since the cuts break the periodicity in Hilbert space. For this reason, we integrate the scattering reflection probabilities of the Landauer-Büttiker formalism that lead to obtaining the core surface wave vector and then the calculations of the electronic localized states on the cutting edges are obtained.Additionally, we have taken into consideration the non-ordered surfaces which describe real systems since the cutting direction is always influenced by the relaxation effect due to changes of the total energy on the surface. To illustrate these changes, we have computed the electronic branches by including relaxation effects. The equilibrium positions of the atoms caused by the relaxation on the surface could be determined by the Molecular dynamic algorithm.Our approach of calculation has been applied to evaluate the surface bands in different edges of the following elements: (i) Chrome (Cr) and Tungsten (W) described bys-like orbital. Molybdenum (Mo) defined ass-d-coupling orbitals, Iron (Fe) given asd-type orbital and the Bismuth (Bi) described asf-type orbital. The obtained results illustrate the impact of cutting direction on the number of surface states as well as the scale of the band values.