Nonlinear Impurity in a Semi-Infinite Lattice

TL;DR

This paper analyzes how nonlinear impurities near the surface of a semi-infinite lattice influence bound state formation, revealing how nonlinearity strength and exponent affect state properties and self-trapping.

Contribution

It provides a closed-form analysis of bound states induced by nonlinear impurities at or near the surface of a semi-infinite lattice, considering various nonlinearity parameters and exponents.

Findings

Bound states depend on impurity nonlinearity and position relative to the surface.

The nonlinearity threshold for bound state formation varies with the nonlinearity exponent.

Surface proximity influences the nonlinearity needed for self-trapping.

Abstract

We examine the formation of bound states on a generalized nonlinear impurity located at or near the beginning (surface) of a linear, tight-binding semi-infinite lattice. Using the formalism of lattice Green functions, we obtain in closed form the number of bound states as well as their energies and probability profiles, for different nonlinearity parameter values and nonlinearity exponents, at different distances from the surface. It is shown that close to the surface, the amount of nonlinearity needed to create a bound state or to effect dynamical selftrapping, increases (decreases) depending on whether the exponent is smaller (larger) than, approximately, two.