Theoretical study of the role of metallic contacts in probing transport features of pure and defected graphene nanoribbons
© Magna and Deretzis; licensee Springer. 2011
Received: 5 September 2010
Accepted: 18 March 2011
Published: 18 March 2011
Understanding the roles of disorder and metal/graphene interface on the electronic and transport properties of graphene-based systems is crucial for a consistent analysis of the data deriving from experimental measurements. The present work is devoted to the detailed study of graphene nanoribbon systems by means of self-consistent quantum transport calculations. The computational formalism is based on a coupled Schrödinger/Poisson approach that respects both chemistry and electrostatics, applied to pure/defected graphene nanoribbons (ideally or end-contacted by various fcc metals). We theoretically characterize the formation of metal-graphene junctions as well as the effects of backscattering due to the presence of vacancies and impurities. Our results evidence that disorder can infer significant alterations on the conduction process, giving rise to mobility gaps in the conductance distribution. Moreover, we show the importance of metal-graphene coupling that gives rise to doping-related phenomena and a degradation of conductance quantization characteristics.
Graphene nanoribbons (GNRs) are the most promising graphene-based nanostructures for electronic applications since they are potentially suited for band-gap engineering, maintaining the excellent electronic properties of the parent two-dimensional graphene layer. GNRs have been already synthesized by means of different pattering techniques [1, 2], and there exists convincing evidence that their electronic structure manifests subband formation which is a typical predicted signature of the one-dimensional (1D) confinement . Actually, a useful intrinsic (i.e., due to the geometric confinement) band-gap value can be obtained only for GNR structures with widths of approximately 1 nm that cannot be easily fabricated with the current lithographic techniques. As a consequence, the controlled inclusion of defects and/or impurities in GNRs has been proposed to overcome the minimum of conductivity problem . Anyhow, we should assume that unintentionally defected GNRs are present from the phase of production due to the impossibility of an accurate control of purity at an atomic level.
Electrical characterization studies of GNRs must preliminarily consider the impact of disorder in their electrical properties, which is crucial also due the possible Anderson localization phenomena in these quasi-1D systems . Moreover, the formation of a junction between a relatively large metal probes (i.e., a three-dimensional (3D) system) and GNRs (a 1D system) is at the basis of any electrical measurements. Understanding the role of interface bonding and electrostatics in the contact region is also crucial in order to categorize the electronic characteristics of these systems. Indeed, we could argue that in the absence of defects, a significant source of resistivity in GNRs and the consequent deviations from the ideal behavior should be derived from the interaction with the metallic electrodes. In this work, these issues are theoretically investigated and their implications on the interpretation of electrical measurements on GNR-based systems are discussed.
Metal surface Green functions g s for the evaluation of the respective self-energies Σ = τ g sτ† are calculated for the 3D semi-infinite contact with a back and forth real to k-space Fourier transform exploiting lattice periodicity . Mobile charges ρ f deriving from the NEGF are passed to a 3D numerical Poisson solver from where the self-consistent potentials are calculated by solving ∇2 U SC = -ρ f / ε, where H = H 0 + U SC in Equation 1. Electrical potential has been fixed at the metal-GNR interface (applying a Dirichelet type boundary condition) at a value U left SC = φ m - φ gr, where φ m and φ gr are the experimentally measured work functions for (111) metallic surfaces and graphene , while zero electric field (i.e., null Neumann boundary condition) has been set for the ideal contacts. Self-consistency is enhanced by a predictor/corrector Newton-Rapson algorithm. Fermi-Dirac statistics have been introduced in the simulation scheme for temperature T = 300 K.
Isolated defect and impurity alter both the density of states distribution DOS(E) than conductance g(E) of quasi-1D GNRs. In particular n-type (p-type) impurities introduce electronic states (also called resonance states ) at energies above (below) the charge neutrality point (i.e., the Dirac point) of the pristine pure systems. In correspondence of these states, a decrease of the conductance with respect the pure system can occur because of the scattering of the conducting electron by the effective potential due to the impurity presence . The conductance alteration strongly depends on the position of the scattering center in the nanoribbon structure . However, in general, we expect that defective GNRs are characterized by larger density of the states and smaller conductance with respect to the ideally pure GNRs.
(where g is in 2e 2/h units) for the same cases of Figure 2. In the quasi-metal systems, the conduction depletions are systematically related to variance peaks, while the variance in the standard gap of the semiconductor GNRs does not show particular features.
The electrostatic potential, due to the work function difference, drops in a few-nanometer region near the interface and has non-zero flat value, denoting the presence of carrier accumulation throughout the GNR length;
Unlike the standard metal semiconductor interface, band bending is not rigid for both conduction and valence bands as a result of quantum interference (between wave functions with 1D character in the nanoribbon and 3D character in the metal) and electron confinement. For example, in the case of a metal with high (low) work function with respect to the graphene, like Au (Al), conduction (valence) band states shift smoothly following the electrostatic potential, while discrete localized states appear with a few-nanometer spatial extension in the region where the valence (conduction) band maximally bends;
Quantum interference, localized states, and the metal induced gap states (i.e., the tails of the metal wavefunctions) characterize the LDOS in the junction region.
Apart from the Schottky barriers, we note that conduction band charge flow is strongly suppressed due to the band bending and the p-type character, giving an asymmetric form to the overall conductance distribution (asymmetric spectra, not shown here, are calculated also for quasi-metallic GNRs ). Note that finite-bias conductance spectra are affected by the presence of the Schottky barrier. Indeed, while the -0.5-V case shows the alteration with respect to the small-bias case due to non-equilibrium charging (see Ref.  for a complete discussion), the +0.5 is almost identical to the small-bias case.
The results of this theoretical study have shown that defects, doping, and metal/graphene interfaces can non-trivially influence the conduction mechanism of graphene nanostructures, giving rise to mode transmission suppression, conductance asymmetries, mobility gap formations, and strong conductance fluctuations. The implication of these results for the electrical characterization of real structures has been discussed. A general crucial issue, which should also be taken into account when probing transport in GNRs, is that the electrostatics and the chemical bonding aspects can act complementary for the determination of transparency in graphene-based nanostructures.
The authors would like to acknowledge the European Science Foundation (ESF) under the EUROCORES Programme EuroGRAPHENE CRP GRAPHIC-RF for the partial financial support.
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