Finite periodic $\delta-\delta'$ comb

TL;DR

This paper analyzes a finite periodic delta-delta' comb using classical and quantum methods, revealing how non-periodicity creates energy levels in band gaps and deriving surface states, enhancing understanding of boundary effects in quantum systems.

Contribution

It introduces a new model for finite periodic delta-delta' combs and derives explicit expressions for surface (Tamm) states, extending theoretical understanding of boundary-dependent quantum states.

Findings

Violation of periodicity induces energy levels in band gaps.

Explicit expressions for surface (Tamm) levels were derived.

Results improve understanding of boundary states in low-dimensional systems.

Abstract

A finit periodic $\delta-\delta'$ comb was solved by the help of both classical approach based on a direct solving of a Sr\"{odinger} equation and a quantum wave impedance method. It was demonstrated that the violation of a periodicity leads to the formation of energy levels in band gaps. The expresion for a surface levels (Tamm's levels) were found in this model system. Obtained results allow extending the scope of theoretical models which are applicable for a describing real quantum mechanical systems as well as better understanding the physical properties of systems in which the boundary dependent states are involved, especially the influence of boundary dependent electronic states on the physical properties of low-dimensional systems.