An analytical model for steady-state currents in conjugated systems

Understanding Current Density in Molecules Using Molecular Orbitals

While the use of molecular orbitals (MOs) and their isosurfaces to explain physical phenomena in chemical systems is a time-honored tool, we show that the nodes are an equally important component for understanding the current density through single-molecule junctions. We investigate three different model systems consisting of an alkane, alkene, and even [n]cumulene and show that we can explain the form of the current density using the MOs of the molecule. Essentially, the MOs define the region in which current can flow and their gradients define the direction in which current flows within that region. We also show that it is possible to simplify the current density for improved understanding by either partitioning the current density into more chemically intuitive parts, such as σ- and π-systems, or by filtering out MOs with negligible contributions to the overall current density. Our work highlights that it is possible to infer a non-equilibrium property (current density) given only equilibrium properties (MOs and their gradients), and this, in turn, grants deeper insight into coherent electron transport.

Extending the source-sink potential method to include electron-nucleus coupling

The source-sink potential (SSP) method provides a simple tool for the qualitative analysis of the conductance of molecular electronic devices, and often analytical expressions for the conductance can be obtained. Here, we extend the SSP approach to account for decoherent, inelastic electron transport by including the non-adiabatic coupling between the electrons and the nuclei in the molecule. This coupling results in contributions to electron transport that can modify the qualitative structure-conductance relationships that we unraveled previously with SSP. In the approach proposed, electron-nucleus interactions are treated starting from the harmonic approximation for the nuclei, using a non-perturbative approach to account for the non-adiabatic coupling. Our method qualitatively describes experimentally observed phenomena and allows for a simple analysis that often provides analytical formulas in terms of the physical parameters of the junction, e.g., vibrational energies, non-adiabatic coupling, and molecule-contact coupling.

Molecular graphs and molecular conduction: the d-omni-conductors

Graph–theoretical distance gives a complete classification of conduction behaviour of alternant and non-alternant molecular devices within the source-sink-potential model.

Designing Principles of Molecular Quantum Interference Effect Transistors

To explore the designing principles for the quantum interference effect transistors, a series of simulations are carried out on a 2,5-linked perylene molecular junction composed of two subsystems connected via destructive quantum interference. Simulation results suggest that the overall conductance of a large π-conjugated system is determined by its subsystem connected directly to the electrodes. A Büttiker probe can be treated as a resistor, and to first-order approximation, its effect is found equivalent to severing its surrounding bonds. These findings greatly simplify the design of molecular quantum interference effect transistors.

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

Within the source-and-sink-potential model, a complete characterisation is obtained for the conduction behaviour of alternant π-conjugated hydrocarbons (conjugated hydrocarbons without odd cycles). In this model, an omni-conductor has a molecular graph that conducts at the Fermi level irrespective of the choice of connection vertices. Likewise, an omni-insulator is a molecular graph that fails to conduct for any choice of connections. We give a comprehensive classification of possible combinations of omni-conducting and omni-insulating behaviour for molecular graphs, ranked by nullity (number of non-bonding orbitals). Alternant hydrocarbons are those that have bipartite molecular graphs; they cannot be full omni-conductors or full omni-insulators but may conduct or insulate within well-defined subsets of vertices (unsaturated carbon centres). This leads to the definition of "near omni-conductors" and "near omni-insulators." Of 81 conceivable classes of conduction behaviour for alternants, only 14 are realisable. Of these, nine are realised by more than one chemical graph. For example, conduction of all Kekulean benzenoids (nanographenes) is described by just two classes. In particular, the catafused benzenoids (benzenoids in which no carbon atom belongs to three hexagons) conduct when connected to leads via one starred and one unstarred atom, and otherwise insulate, corresponding to conduction type CII in the near-omni classification scheme.

Analyzing the electric response of molecular conductors using "electron deformation" orbitals and occupied-virtual electron transfer

The concept of "electron deformation orbitals" (EDOs) is used to investigate the electric response of conducting metals and oligophenyl chains. These orbitals and their eigenvalues are obtained by diagonalization of the deformation density matrix (difference between the density matrices of the perturbed and unperturbed systems) and can be constructed as linear combinations of the unperturbed molecular orbitals within "frozen geometry" conditions. This form of the EDOs allows calculating the part of the electron deformation density associated to an effective electron transfer from occupied to virtual orbitals (valence to conduction band electron transfer in the band model of conductivity). It is found that the "electron deformation" orbitals pair off, displaying the same eigenvalue but opposite sign. Each pair represents an amount of accumulation/depletion of electron charge at different molecular regions. In the oligophenyl systems investigated only one pair contributes effectively to the charge flow between molecular ends, resulting from the promotion of electrons from occupied orbitals to close in energy virtual orbitals of appropriate symmetry and overlapping. Analysis of this pair along explains the differences in conductance of olygophenyl chains based on phenyl units.

Omni-conducting and omni-insulating molecules

The source and sink potential model is used to predict the existence of omni-conductors (and omni-insulators): molecular conjugated π systems that respectively support ballistic conduction or show insulation at the Fermi level, irrespective of the centres chosen as connections. Distinct, ipso, and strong omni-conductors/omni-insulators show Fermi-level conduction/insulation for all distinct pairs of connections, for all connections via a single centre, and for both, respectively. The class of conduction behaviour depends critically on the number of non-bonding orbitals (NBO) of the molecular system (corresponding to the nullity of the graph). Distinct omni-conductors have at most one NBO; distinct omni-insulators have at least two NBO; strong omni-insulators do not exist for any number of NBO. Distinct omni-conductors with a single NBO are all also strong and correspond exactly to the class of graphs known as nut graphs. Families of conjugated hydrocarbons corresponding to chemical graphs with predicted omni-conducting/insulating behaviour are identified. For example, most fullerenes are predicted to be strong omni-conductors.

Effects of charging and polarization on molecular conduction via the source-sink potential method

The current−voltage relationships of butadiene and octatetraene are computed using the source-sink potential method with self-consistent Hückel theory. Molecular orbital resonances appear as steps in current, considerably broader than the resonances in the transmission spectrum at any specific bias. This broadening is due primarily to the charging of the molecule as bias increases. A perturbation theory based model is derived to account for the observations. In the case of octatetraene, the HOMO resonance manifests at high voltage when the HOMO energy is raised to the Fermi level of the plus electrode due to the charging of the LUMO and LUMO + 1 levels. Specifically, the LUMO and LUMO + 1 steps in current are augmented by portions of the HOMO contribution to current.

Extension of the source-sink potential (SSP) approach to multichannel quantum transport

We present an extension of the single channel source-sink potential approach [F. Goyer, M. Ernzerhof, and M. Zhuang, J. Chem. Phys. 126, 144104 (2007)] for molecular electronic devices (MEDs) to multiple channels. The proposed multichannel source-sink potential method relies on an eigenchannel description of conducting states of the MED which are obtained by a self-consistent algorithm. We use the newly developed model to examine the transport of the 1-phenyl-1,3-butadiene molecule connected to two coupled rows of atoms that act as contacts on the left and right sides. With an eigenchannel description of the wave function in the contacts, we determined that one of the eigenchannels is effectively closed by the interference effects of the side chain. Furthermore, we provide an example where we observe a complete inversion (from bonding to antibonding and vice versa) of the transverse character of the wave function upon passage through the molecule.

Single molecule logical devices

After almost 40 years of development, molecular electronics has given birth to many exciting ideas that range from molecular wires to molecular qubit-based quantum computers. This chapter reviews our efforts to answer a simple question: how smart can a single molecule be? In our case a molecule able to perform a simple Boolean function is a child prodigy. Following the Aviram and Ratner approach, these molecules are inserted between several conducting electrodes. The electronic conduction of the resulting molecular junction is extremely sensitive to the chemical nature of the molecule. Therefore designing this latter correctly allows the implementation of a given function inside the molecular junction. Throughout the chapter different approaches are reviewed, from hybrid devices to quantum molecular logic gates. We particularly stress that one can implement an entire logic circuit in a single molecule, using either classical-like intramolecular connections, or a deformation of the molecular orbitals induced by a conformational change of the molecule. These approaches are radically different from the hybrid-device approach, where several molecules are connected together to build the circuit.

Resonance theory for discrete models: methodology and isolated resonances

We here consider open quantum systems defined on discretized space, motivated by experimental and theoretical interest in the electronic conduction through nanoscale devices such as molecular junctions and quantum dots. We particularly focus on effects of resonances on the conductance through the systems. We develop a method of calculating the conductance with the use of Green's function expansion with respect to the eigenstates of the effective Hamiltonian for the open quantum systems. Unlike previous methodologies where one can treat only narrow resonances far from the band edges in a satisfactory manner with a Lorentzian profile, our method provides a novel resonance profile which can be used to describe any isolated resonance in the spectrum even close to the band edges.

Two-channel conduction through polyacenes--extension of the source-sink potential method to multichannel coupling to leads

The source and sink potential method of Goyer et al. [J. Chem. Phys. 126, 144104 (2007)] is extended to the case of multichannel coupling to leads. The formulation leads to a nonlinear equation for just one (the elastic) reflection coefficient. Solution of this equation, in general, requires repeated computation of an n × n determinant, where n is the number of supermolecule basis functions directly coupled to the source lead, as opposed to a determinant with order equal to the full size of supermolecule basis. The method is applied to a Hückel model of two-channel polyacene conduction. A simple model of resonance lineshapes is developed in case of weak coupling to leads. The model accurately relates peak characteristics to orbital probabilities associated with the eigenvectors of the isolated molecule Hamiltonian. The model shows how orbital probabilities that give rise to transmission resonances (i.e., 100% transmission), in the case of single-channel conduction, give rise to equal probabilities (of 1∕4) for the two reflections and two transmissions, in the case of two-channel conduction. The model also shows how splitting of degenerate eigenvalues of the isolated molecule Hamiltonian results in overlapping resonances characterized by a single complex lineshape.

The relation between structure and quantum interference in single molecule junctions

Quantum interference (QI) of electron pathways has recently attracted increased interest as an enabling tool for single-molecule electronic devices. Although various molecular systems have been shown to exhibit QI effects and a number of methods have been proposed for its analysis, simple guidelines linking the molecular structure to QI effects in the phase-coherent transport regime have until now been lacking. In the present work we demonstrate that QI in aromatic molecules is intimately related to the topology of the molecule's π system and establish a simple graphical scheme to predict the existence of QI-induced transmission antiresonances. The generality of the scheme, which is exact for a certain class of tight-binding models, is proved by a comparison to first-principles transport calculations for 10 different configurations of anthraquinone as well as a set of cross-conjugated molecular wires.

Fragment analysis of single-molecule conduction

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.