Spectrum of pi electrons in graphene as a macromolecule

Finite-size effects on electronic structure and local properties in passivated AA-stacked bilayer armchair-edge graphene nanoribbons

Based on the tight-binding model and dual-probe scanning tunneling microscopy technology, we theoretically investigate the electronic structure and local property in the passivated AA-stacked bilayer armchair-edge graphene nanoribbons (AABLAGNRs). We show that they are highly sensitive to the size of the ribbons, which is evidently different from the single-layer armchair-edge graphene nanoribbons. The '3p' rule only applies to the narrow AABLGNRs. Namely, in the passivated 3p- and (3p  +  1)-AABLGNRs, the narrow ribbons are semiconducting while the medium and wide ribbons are metallic. Although the passivated (3p  +  2)-AABLGNRs are metallic, the '3j' rule only applies to the narrow and medium ribbons. Namely, electrons are in the semiconducting states at sites of line 3j while they are in the metallic states at other sites. This induces a series of parallel and discrete metallic channels, consisting of lines 3j  -  1 and 3j  -  2, for the low-energy electronic transports. In the passivated wide (3p  +  2)-AABLGNRs, all electrons are in the metallic states. Additionally, the '3p' and '3j' rules are controllable to disappear and reappear by applying an external perpendicular electric field. Resultantly, an electric filed-driven current switch can be realized in the passivated narrow and medium (3p  +  2)-AABLGNRs.

Conductance of graphene flakes contacted at their corners

Linear conductance of junctions formed by graphene flakes with the order of the nanometer-thick electrodes attached at the corners of the flakes is studied. The explored structures have sizes up to 20,000 atoms and the conductance is studied as a function of applied gate voltage varied around the Fermi level. The finding, obtained computationally, is that junctions formed by armchair-edge flakes with the electrodes connected at the acute-angle corners block the electron transport while only junctions with such electrodes at the obtuse-angle corners tend to provide the high electrical conductance typical for metallic GNRs. The finding in the case of zig-zag edges is similar with the exception of a relatively narrow gate voltage interval in which each studied junction is highly conductive as mediated by the edge states. The contrast between the conductive and insulating setups is typically several orders of magnitude in terms of ratio of their conductances. The main results of the paper also remain to a large extent valid in the presence of edge disorder.

Semiconducting states and transport in metallic armchair-edged graphene nanoribbons

Based on the nonequilibrium Green's function method within the tight-binding approximation scheme, through a scanning tunneling microscopy (STM) model, we study the low-energy electronic states and transport properties of carbon chains in armchair-edged graphene nanoribbons (AGNRs). We show that semiconducting AGNRs possess only semiconducting chains, while metallic ones possess not only metallic chains but also unconventional semiconducting chains located at the 3 jth (j ≠ 0) column from the edge (the first chain) due to the vanishing of the metallic component in the electron wavefunction. The two types of states for carbon chains in a metallic AGNR system are demonstrated by different density of states and STM tunneling currents. Moreover, a similar phenomenon is predicted in the edge region of very wide AGNRs. However, there is remarkable difference in the tunneling current between narrow and wide ribbons.

Electronic states of graphene nanoribbons and analytical solutions

Graphene is a one-atom-thick layer of graphite, where low-energy electronic states are described by the massless Dirac fermion. The orientation of the graphene edge determines the energy spectrum of π-electrons. For example, zigzag edges possess localized edge states with energies close to the Fermi level. In this review, we investigate nanoscale effects on the physical properties of graphene nanoribbons and clarify the role of edge boundaries. We also provide analytical solutions for electronic dispersion and the corresponding wavefunction in graphene nanoribbons with their detailed derivation using wave mechanics based on the tight-binding model. The energy band structures of armchair nanoribbons can be obtained by making the transverse wavenumber discrete, in accordance with the edge boundary condition, as in the case of carbon nanotubes. However, zigzag nanoribbons are not analogous to carbon nanotubes, because in zigzag nanoribbons the transverse wavenumber depends not only on the ribbon width but also on the longitudinal wavenumber. The quantization rule of electronic conductance as well as the magnetic instability of edge states due to the electron-electron interaction are briefly discussed.

Conjugated Molecules Described by a One-Dimensional Dirac Equation

Starting from the Hückel Hamiltonian of conjugated hydrocarbon chains (ethylene, allyl radical, butadiene, pentadienyl radical, hexatriene, etc.), we perform a simple unitary transformation and obtain a Dirac matrix Hamiltonian. Thus already small molecules are described exactly in terms of a discrete Dirac equation, the continuum limit of which yields a one-dimensional Dirac Hamiltonian. Augmenting this Hamiltonian with specially adapted boundary conditions, we find that all the orbitals of the unsaturated hydrocarbon chains are reproduced by the continuous Dirac equation. However, only orbital energies close to the highest occupied molecular orbital/lowest unoccupied molecular orbital energy are accurately predicted by the Dirac equation. Since it is known that a continuous Dirac equation describes the electronic structure of graphene around the Fermi energy, our findings answer the question to what extent this peculiar electronic structure is already developed in small molecules containing a delocalized π-electron system. We illustrate how the electronic structure of small polyenes carries over to a certain class of rectangular graphene sheets and eventually to graphene itself. Thus the peculiar electronic structure of graphene extends to a large degree to the smallest unsaturated molecule (ethylene).

A gate-induced switch in zigzag graphene nanoribbons and charging effects

Using the non-equilibrium Green's function formalism, we investigate nonlinear transport and charging effects of gated graphene nanoribbons (GNRs) with an even number of zigzag chains. We find a negative differential resistance (NDR) over a wide range of gate voltages with an on/off ratio approximately 10(6) for narrow enough ribbons. This NDR originates from the parity selection rule and also prohibition of transport between discontinuous energy bands. Since the external field is well screened close to the contacts, the NDR is robust against the electrostatic potential. However, for voltages higher than the NDR threshold, due to charge transfer through the edges of the zigzag GNR (ZGNR), screening is reduced such that the external potential can penetrate inside the ribbon giving rise to smaller values of off-current. Furthermore, the on/off ratio of the current depends on the aspect ratio of the length/width and also edge impurity. Moreover, the on/off ratio displays a power law behavior as a function of ribbon length.