# Energy structure, density of states and transmission properties of the periodic 1D Tight-Binding lattice with a generic unit cell of $u$ sites

## TL;DR

This paper analyzes the electronic structure, density of states, and transmission properties of a periodic 1D Tight-Binding lattice with a generic unit cell, providing analytical solutions and exploring transport characteristics with attached leads.

## Contribution

It introduces analytical solutions for eigenvalues of the 1D TB lattice with a generic unit cell and investigates transmission properties with semi-infinite leads, including an optimal coupling condition.

## Key findings

- Analytical relations for eigenvalues using Chebyshev polynomials.
- Transmission peaks depend on coupling strength and lead properties.
- Optimal coupling condition enhances transmission efficiency.

## Abstract

We report on the electronic structure, density of states and transmission properties of the periodic one-dimensional Tight-Binding (TB) lattice with a single orbital per site and nearest-neighbor interactions, with a generic unit cell of $u$ sites. The determination of the eigenvalues is equivalent to the diagonalization of a real tridiagonal symmetric $u$-Toeplitz matrix with (cyclic boundaries) or without (fixed boundaries) perturbed upper right and lower left corners. We solve the TB equations via the Transfer Matrix Method, producing, analytical solutions and recursive relations for its eigenvalues, closely related to the Chebyshev polynomials. We examine the density of states and provide relevant analytical relations. We attach semi-infinite leads, determine and discuss the transmission coefficient at zero bias and investigate the peaks number and position, and the effect of the coupling strength and asymmetry as well as of the lead properties on the transmission profiles. We introduce a generic optimal coupling condition and demonstrate its physical meaning.