Fragment analysis of single-molecule conduction

Abstract

In the tight-binding source and sink potential model of transmission in single-molecule pi-conjugated conductors, vanishing of the opacity polynomial defines a necessary condition for zero conductance at a given energy. Theorems are given for calculating opacity polynomials of composite devices in terms of opacity and characteristic polynomials of the subunits. These relations rationalize the positions and shapes of zeros in transmission curves for devices consisting of molecules with side chains or of units assembled in series and take an especially simple form for polymeric molecules with identical repeat units.

Omni-conducting and omni-insulating molecules

“…17 Proof is by combination of Theorems 6 and 7 from our earlier paper. 17 Our omniconducting trees include multiple copies of such Y-junctions, and the denominator of the transmission T(E) will therefore contain zeroes at E = 0 arising from the many leaves on these particular trees, as will the characteristic polynomial of the tree itself. This is suggestive of a more general connection between between nullity, crossconjugation, and omni-insulation.…”

“…12,14,15 In the tight-binding approximation, calculation of the fractional transmission of an electron with given energy reduces to the solution of the Hückel problem under scattering boundary conditions, and hence to an essentially graph theoretical question, as conduction is determined by functions of the characteristic polynomials of four graphs. 13,[16][17][18][19] In the SSP model, the transmission function for a molecule that has a carbon skeleton with graph G connected to similar left and right wires via molecular verticesL andR is given by 13…”

“…[13][14][15][16][17][18][19][20][21][22] The present work is concerned with this model and the information that it may give about the interaction of the factors of contact position and molecular structure. In particular, we explore the possibility that some molecular structures may display a much reduced dependence of the predicted transmission on precise positioning of the contacts.…”

A new approach to the method of source-sink potentials for molecular conduction

“…[28][29][30][31][32][33][34][35] In our approach, the solutions for an incoming beam of electrons with energy E are written in terms of five molecular structural polynomials, which comprise the real characteristic polynomials, 28 s = det (E1 − A) ,…”

Near omni-conductors and insulators: Alternant hydrocarbons in the SSP model of ballistic conduction

“…36,37 It has the significant advantage that it can be framed in purely graph-theoretical terms. [38][39][40][41][42][43][44][45][46][47] In chemical graph theory, vertices of the molecular graph of a π system are unsaturated C atoms, and edges are σ bonds between them. Models based on graph theory have the advantage of leading to global solutions for whole classes of molecules and giving a systematic typology 41, 45 of conduction behaviour of devices.…”