Abstract
Understanding the roles of disorder and metal/graphene interface on the electronic and transport properties of graphenebased 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 selfconsistent 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 endcontacted by various fcc metals). We theoretically characterize the formation of metalgraphene 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 metalgraphene coupling that gives rise to dopingrelated phenomena and a degradation of conductance quantization characteristics.
Introduction
Graphene nanoribbons (GNRs) are the most promising graphenebased nanostructures for electronic applications since they are potentially suited for bandgap engineering, maintaining the excellent electronic properties of the parent twodimensional 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 onedimensional (1D) confinement [3]. Actually, a useful intrinsic (i.e., due to the geometric confinement) bandgap 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 [4]. 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 quasi1D systems [5]. Moreover, the formation of a junction between a relatively large metal probes (i.e., a threedimensional (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 GNRbased systems are discussed.
Theoretical approach
We consider pure and defected hydrogenterminated armchair GNRs (AGNR) of different widths and lengths (from few nanometers to approximately a micrometer) with two different contact configurations: (a) ideally contacted (same width as the conductor without defect or impurity inclusions) at the right side and the left side, and (b) endcontacted by 3D semiinfinite metallic electrodes (Au, Pd, and Al (111) surfaces) at the left side whereas ideally contacted at the right side. The terminology of Ref. [6] is applied to categorize them on the basis of the dimer lines N_{a }along the ribbon width (e.g., with N_{a }= 45 AGNR, we indicate a ribbon with 45 dimer lines). We use the nonequilibrium Green function (NEGF)
and the standard LandauerButtiker approach for the calculation of quantum transport [7]. Hamiltonians H_{0 }and overlap matrices τ are written within firstprinciples parameterized models [8].
Metal surface Green functions g_{s }for the evaluation of the respective selfenergies Σ = τ g_{s}τ^{† }are calculated for the 3D semiinfinite contact with a back and forth real to kspace Fourier transform exploiting lattice periodicity [8]. Mobile charges ρ_{f }deriving from the NEGF are passed to a 3D numerical Poisson solver from where the selfconsistent 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 metalGNR 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 [8], while zero electric field (i.e., null Neumann boundary condition) has been set for the ideal contacts. Selfconsistency is enhanced by a predictor/corrector NewtonRapson algorithm. FermiDirac statistics have been introduced in the simulation scheme for temperature T = 300 K.
Defected GNRs
Isolated defect and impurity alter both the density of states distribution DOS(E) than conductance g(E) of quasi1D GNRs. In particular ntype (ptype) impurities introduce electronic states (also called resonance states [3]) 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 [3]. The conductance alteration strongly depends on the position of the scattering center in the nanoribbon structure [8]. 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.
In a real defected system, we expect a finite density of random distributed scattering centers and, as a consequence, the effect of multiple scattering processes should be evaluated in this kind of configuration. In Figure 1, we show the (small bias) average conductance of an ideally contacted N_{a }= 45 AGNR doped with a 0.2% density of nitrogen atoms and for systems with increasing length L, from approximately 0.1 μm to approximately 0.8 μm. An asymmetric decrease of the average conductance due to the impurity scattering can be observed for the whole spectrum. This behavior is particularly important in the energy region near the single impurity resonance states (i.e., for energies E ≈ 0.2 eV) where a mobility gap appears also for the smaller systems. We note that the pure N_{a }= 45 AGNR is a semiconductor GNR with an energy gap of approximately 0.2 eV.
Figure 1. Average conductance for a nitrogendoped N_{a }= 45 AGNR. Average conductance <g> as a function of the energy E for a nitrogendoped N_{a }= 45 AGNR of different lengths. Plotted values represent statistical averages over more than 500 equivalent replicas of the system. Charge neutrality points of pure and doped systems are aligned at E = 0 eV in the figure.
A qualitatively similar behavior is shown by the vacancydamaged and nitrogendoped semimetallic N_{a }= 47 AGNR (see Figure 2). In the vacancydamaged case, a large mobility gap appears in the negative energies (hole band) region [9] due to the strong backscattering of the defects. However, apart from the intensity of the scattering, vacancydefected systems have a p impuritylike behavior.
Figure 2. Average conductance for nitrogendoped and vacancydamaged AGNRs. Average conductance <g> as a function of the energy E for a nitrogendoped N_{a }= 45 AGNR (solid line), nitrogendoped N_{a }= 47 AGNR (dashed line), and vacancydamaged N_{a }= 47 AGNR (point) with fixed length: L ≈ 0.21 μm. Plotted values represent statistical averages over more of 500 equivalent replicas of the system. Charge neutrality points of pure and defected systems are aligned at E = 0 eV in the figure.
From an analysis of the conductance spectra, we can derive same general features which can be useful for the interpretation of the electrical characterization of real GNRs. Indeed, we note that the conduction spectrum can be measured in a three terminal configuration [4] tuning the gate potential in order to modify the electron density in the nanostructure. Firstly, we note the persistence of the conduction modulation with energy in disordered systems, which is a marker of the conductance plateaus of the subband structure in pure GNRs. The effective transmission in the subbands is strongly reduced due to localization effects that suppress the conductance transparency in the spectral region where resonance states are located. Finally, the general occurrence of mobility gaps (see also Ref. [10]) can be hardly distinguished from the intrinsic bandgaps in semiconductor GNRs when experimentally measuring the conductance. A clear signature of the mobility gap is the occurrence of huge values of conductance fluctuations in the same spectral region of the gap due to its backscattering origin. In order to demonstrate this assumption, we have plotted in Figure 3 the statistically evaluated conductance variance
Figure 3. Conductance fluctuation for nitrogendoped and vacancydamaged AGNRs. Conductance fluctuation σ as a function of the energy E for a nitrogendoped N_{a }= 45 AGNR (solid line), a nitrogendoped N_{a }= 47 AGNR (dashed line), and a vacancydamaged N_{a }= 47 AGNR (point) with fixed length: L~0.21 μm. Plotted values represent statistical averages over more of 500 equivalent replicas of the system. Charge neutrality points of pure and defected systems are aligned at E = 0 in the figure.
(where g is in 2e^{2}/h units) for the same cases of Figure 2. In the quasimetal 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.
MetalGNR junction
Selfconsistent local density of the state (LDOS) calculations have shown a nonconventional scenario for the charging and the related electrostatics of the heterojunction between the 3D metal and the 1D GNRs [8]. In particular:
(a) The electrostatic potential, due to the work function difference, drops in a fewnanometer region near the interface and has nonzero flat value, denoting the presence of carrier accumulation throughout the GNR length;
(b) 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 fewnanometer spatial extension in the region where the valence (conduction) band maximally bends;
(c) 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.
The local electronic structure characteristics in the contact region can nontrivially influence the conduction mechanism since, e.g., localized states do not contribute to the conduction, giving rise to conductance asymmetries and an overall loss of the transport information with respect to the ideal case. In Figure 4, small and finitebias conductance spectra of a pure N_{a }= 16 AGNR endcontacted with Au are plotted as a function of energy. These spectra show similar characteristics with the ones obtained considering metals with higher work functions than graphene (e.g., Pt, Pd). A Schottky barrier of the order of 0.2 to 0.3 eV can be determined by the difference between the gap in the conductance spectrum for the contacted GNRs at 0 V (Figure 4 dotted line) and the gap (approximately 0.6 eV) of the ideal noncontacted GNR.
Figure 4. Small and finitebias conductance spectra. Smallbias (points) and finitebias (+0.5 V green line, 0.5 red line) conductances of a pure N_{a }= 16 AGNR endcontacted with Au.
Apart from the Schottky barriers, we note that conduction band charge flow is strongly suppressed due to the band bending and the ptype character, giving an asymmetric form to the overall conductance distribution (asymmetric spectra, not shown here, are calculated also for quasimetallic GNRs ). Note that finitebias conductance spectra are affected by the presence of the Schottky barrier. Indeed, while the 0.5V case shows the alteration with respect to the smallbias case due to nonequilibrium charging (see Ref. [11] for a complete discussion), the +0.5 is almost identical to the smallbias case.
Currentvoltage (IV) characteristics of the junction in the case of a pure N_{a }= 16 AGNR contacted with different metals are reported in Figure 5. Larger I values obtained for a negative bias in the case of Pd with respect Au are due to its slighter more pronounced ptype character (while in turn Au seems transparent near the Fermi level with the conductance arriving at the 1 G_{0 }= 2e^{2}/h plateau of the ideal case). Al has a lower work function with respect to graphene and the AlGNR junction shows a quasiambipolar Schottky behavior (i.e., the IV characteristic is almost symmetric for positive and negative bias). However, in the latter case, the dominant aspect is the strong scattering by the contacts and the related suppression of the contact transparency.
Figure 5. Currentvoltage characteristics. IV characteristics derived by the NEGF in the LandauerButtiker scheme for a N_{a }= 16 AGNR, endcontacted with three different metals Au (black line), Pd (green line), and Al (red line).
Conclusion
The results of this theoretical study have shown that defects, doping, and metal/graphene interfaces can nontrivially 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 graphenebased nanostructures.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions section
AL carried out the conduction calculation and the statistical analysis for the defective system. ID carried out all the abinito based calculation presented in this manuscript. AL and ID developed the formalism used for conduction evaluation.
Acknowledgements
The authors would like to acknowledge the European Science Foundation (ESF) under the EUROCORES Programme EuroGRAPHENE CRP GRAPHICRF for the partial financial support.
References

Han MY, Ozyilman B, Zhang Y, Kim P: 'Energy BandGap Engineering of Graphene Nanoribbons '.
Phys Rev Lett 2007, 98:206805. PubMed Abstract  Publisher Full Text

Li X, Wang X, Zhang L, Lee S, Dai H: 'Chemically Derived, Ultrasmooth Graphene Nanoribbon Semiconductors'.
Science 2008, 319:1229. PubMed Abstract  Publisher Full Text

Biel B, Blase X, Triozon F, Roche S: Chemically Induced Mobility Gaps in Graphene Nanoribbons: A Route for Upscaling Device Performances'.

Lin Y, Perebeinos V, Chen Z, Avouris P: 'Electrical observation of subband formation in graphene nanoribbons'.
Phys Rev B 2008, 78:161409. Publisher Full Text

La Magna A, Deretzis I, Forte G, Pucci R: 'Violation of the singleparameter scaling hypothesis in disordered graphene nanoribbons'.
Phys Rev B 2008, 78:153405. Publisher Full Text

Son Y, Cohen ML, Louie SG: 'Energy Gaps in Graphene Nanoribbons'.
Phys Rev Lett 2006, 97:216803. PubMed Abstract  Publisher Full Text

Datta S: Electronic Transport in Mesoscopic Systems. Cambridge: Cambridge University Press; 1995.

Deretzis I, Fiori G, Iannaccone G, La Magna A: 'Atomistic quantum transport modeling of metalgraphene nanoribbon heterojunctions'.
Phys Rev B 2010, 82:161413. Publisher Full Text

Deretzis I, Fiori G, Iannaccone G, La Magna A: 'Effects due to backscattering and pseudogap features in graphene nanoribbons with single vacancies'.
Phys Rev B 2010, 81:085427. Publisher Full Text

La Magna A, Deretzis I, Forte G, Pucci R: 'Conductance distribution in doped and defected graphene nanoribbons'.
Phys Rev B 2009, 80:195413. Publisher Full Text

Deretzis I, La Magna A: 'Nonequilibrium electron charging in carbonnanotubebased molecular bridges'.
Appl Phys Lett 2007, 91:163111. Publisher Full Text