Quantum electron transport in models of nanoparticles using matrix algebra and renormalization group methods

A general expression for quantum transmission of non-interacting spinless electrons through models of a fully connected network of sites that can be regarded as a nanoparticle is obtained using matrix algebra. This matrix algebra method leads to the same results given by the Green's function method without requiring the mathematical sophistication as required by the later. The model of the nanoparticle in this study comprises a single linear array of atoms that profile the input and output leads connected to a fully connected blob of atoms. A simple tight-binding Hamiltonian motivates the quantum transmission in the discrete lattice system. If there are n atoms in the nanoparticle, the methodology requires the inverse of a n x n matrix. The solution is obtained analytically for different cases: a single atom in the nanoparticle, a single dangle atom, n fully connected atoms in a mean-field type cluster with symmetric input and output connections, and the most general case where the n fully connected atoms can be connected arbitrarily to the input and output leads. A numerical solution is also provided for the case where the intra-bonds among the atoms in the nanoparticle are varied (a case with not-fully connected atoms). The expression for the transmission coefficient thus obtained using the matrix method is compared with the transmission coefficients derived using the real space Renormalization Group method and the Green's function method.;Keywords. Nanosystems, transport, transmission coefficient.