Spectral and transmission properties of periodic 1D tight-binding lattices with a generic unit cell: an analysis within the transfer matrix approach

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 hoppings, 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.

Tight-Binding Modeling of Nucleic Acid Sequences: Interplay between Various Types of Order or Disorder and Charge Transport

“…Several techniques can be applied to solve the models, depending on what is studied, such as the numerical diagonalization of the Hamiltonian in Equation 2 [47,56,57], the transfer matrix method [58][59][60] outlined above, and the Non-Equilibrium Green's Function technique [61]. As it is apparent from Equation 3, the transfer matrix method is not applicable if the matrices τ τ τ n are singular.…”

“…The above-mentioned results were generalized in an analytical manner for any periodic WM, through the conditions ω = t M t u t R t L = ±1 (ideal coupling condition), where t u couples the moieties at the end of a unit cell and at the start of the next, and χ = t L t R = ±1 (symmetric coupling condition) [60]. The ideal coupling condition, ω = ±1, implies that the system and the leads are interconnected as if they were connected to themselves.…”

Periodic, quasiperiodic, fractal, Kolakoski, and random binary polymers: Energy structure and carrier transport

“…, N of a sequence, can be given by the roots of the polynomial M 11 N (E) 66,67 . For periodic segments, the eigenspectrum can be recursively obtained with the help of the Chebyshev polynomials of the second kind 67 . Here, the eigenspectra of the sequences have been calculated by numerical diagonalization of the Hamiltonian matrix, which is real, tridiagonal and symmetric.…”

“…The value of the normalized IDOS in this gap corresponds to the relative number of A inside the sequence. Periodic (GA) m segments possess two narrow, continuous bands, which can be recursively obtained; also, an analytical expression for the DOS exists 67 . TM, F, PD, RS, and KOL family sequences posses step-like IDOS, which indicates that the eigenenergies concentrate at specific energy regimes, separated by small gaps.…”

Precise Control of Single-phenanthrene Junction’s Conductance

“…Consequently, the transmission coefficient shows nodes when it has a disconnected path linking the sites 1 and N. The discontinuity occurs while the Green function goes to zero (G'=0). Hence, by adjusting the Fermi level to be zero [55,56] so as the cure will be at the centre of the energy band, one can get…”