GRAPHENE NANORIBBONS TRANSPORT PROPERTIES CALCULATION Jan VOVES Czech Technical University in Prague, Faculty of Electrical Engineering, Technicka 2, CZ-16627 Prague 6 Czech Republic, voves@fel.cvut.cz Abstract Graphene, a two-dimensional sheet of sp 2 -bonded carbon arranged in a honeycomb lattice, is a potential candidate for use in future nanodevices. The lateral confinement of the two-dimensional electron gas in graphene nanoribbons (GNR) can tune their electrical properties. We performed density-functional theory (DFT) based calculations to obtain electronic structure of both zig-zag and armchair GNRs with lateral constrictions. The spin-dependent exchange-correlation potential is approximated within the generalized gradient approximation using the QuantumWise toolkit ATK, which employs local numerical double-zeta polarized basis orbitals. The spin-dependent transport properties of the electrode-device-electrode geometry were calculated by means of non-equilibrium Green s function formalism as implemented in ATK. Band structures calculated by the DFT were compared with results of the semi-empirical method (Extended Huckel model). Our results show that GNRs with the constrictions exhibit novel transport properties. The horizontal layout of the GNR constrictions determines main features important for electrical and spintronic applications of these structures. The constrictions could be realized by means of local anodic oxidation using scanning probe microscope. Keywords: graphene nanoribbons, spintronics, density functional theory 1. INTRODUCTION During last years huge expectations in the field of all-carbon nanoelectronics and spintronics have appeared. Electrical properties of Graphene Nanoribons (GNRs) can be tuned from perfectly metallic, for zig-zag edge ribbons, to semiconducting behavior, for armchair ribbons. In this later case, the gap varies with the ribbon width, length and topology [1]. Patterned graphene nanoribbons (GNRs) have enabled band-gap engineering and the development of efficient GNR-based spin filters [2]. Lateral constrictions has been recently realized experimentally [3]. Calculations based on the Density Functional Theory (DFT) [4] could give the realistic results of the spin dependent phenomena in GNRs [5-7]. 2. STRUCTURES UNDER STUDY We analyzed GNRs in two electrode configuration (left electrode, device, right electrode) as shown in Fig.1. Device geometry includes both zig-zag and armchair GNRs with single and double V-shaped symmetrical and asymmetrical lateral constrictions. All GNRs are hydrogen pasivated with C-H distance of 1.126 Å and a C-C distance of 1.422 Å. Unconstricted GNRs consist from 8 periods for armchair GNR with 7 carbon atoms at the cut edge ac) and 12 periods for zig-zag GNR with 8 carbon atoms at the cut edge (zz). All simulated device Fig. 1 GNRs in two electrode configuration (left electrode, device, right electrode)
structures are shown in Fig. 2. ac zz ac U zz U ac V zz V ac UU zz UU ac VV zz VV Fig. 2 Device structure of unconstricted GNRs (ac), (zz), GNRs with small single V-shaped lateral constrictions (ac U, zz U), GNRs with deeper single V-shaped lateral constrictions (ac V, zz V), GNRs with small double V-shaped lateral constrictions (ac UU, zz UU) and GNRs with deeper double V-shaped lateral constrictions (ac VV, zz VV). 3. MODELS We used the QuantumWise tool ATK [8] based on Density Functional Theory (DFT). The spin-dependent exchange-correlation potential was approximated within the spin-dependent generalized gradient approximation (SGGA). Local numerical double-zeta polarized basis orbitals were employed. The spindependent transport properties of the electrode-device-electrode geometry were calculated by means of Non-Equilibrium Green's Functions (NEGF). Band structures calculated by the DFT were compared with the results of the semi-empirical Extended-Hückel Theory (EHT).
4. RESULTS Bandstructures and transmissivities of all GNR devices were calculated by both methods. EHT based model is considerably faster than DFT model. Unfortunately spin polarization is not included in EHT. In DFT selfconsistent potential and total-energy calculations the Brillouin zone is sampled by (1,1,30) special k points for ribbons. The criteria of convergence for total energy was set to 4.10 4 ev. 4.1 GNR Bandstructures and Electron Concentrations Bandstructures of unconstricted armchair GNRs were calculated by DFT and by EHT. The resulting data are in very good agreement (Fig. 3). Both diagrams show the bandgap 1.4 ev. The bandstructures of unconstricted zig-zag GNR is shown in Fig. 4. The large difference between spin up and spin down energies can be observed here. The difference is approximately 0.5 ev. Fig. 3 Bandstructures of unconstricted armchair GNR calculated by DFT (left) and by EHT (right). Fig. 4 Spin dependent bandstructure of zig-zag GNR calculated by DFT (spin black, spin red)
Electron concentrations calculated by DFT cannot be compared with the electron difference concentration calculated by EHT due to different approach. Only DFT gives the values comparable with real concentrations. One example of these results is in Fig. 5. ac VV - DFT ac VV - EHT Fig. 5 Electron concentration in constricted armchair GNR calculated by DFT and electron difference concentration calculated by EHT. 4.2 GNR Transmissivities We analyzed spin polarized electron concentrations and Mulliken populations in zig-zag GNRs. They show small, but observable spin polarization. Due to spin polarization along the edges the, zig-zag graphene nanoribbons show spin-dependent transport properties in agreement with results in [8]. Dangling bonds at the zig-zag GNR edges are assigned as the source of spin polarization in this paper. Hydrogen termination of the edge atoms would be important for the structural and electronic stability of the graphene ribbon. Both first principles and tight-binding calculations showed that the termination of edges with hydrogen atoms removes the electronic states related to the dangling bonds. However, there are no qualitative changes in the electronic structure and the magnetic order of the ZGNRs with hydrogen atom termination, except for a narrowing of the band gap. All zig-zag GNRs show spin-dependent transmissivities in our calculations. Transmissivity peaks are lower for deeper constrictions. Only one armchair GNR (ac VV) shows spin dependence. We offer following explanation: presence of relatively long zig-zag shaped edges at the constrictions could be the reason for spin polarized behavior. Structure ac VV could serve as the example of semiconducting GNR device with the spin filtering ability. 5. CONCLUSIONS Our results endorse that for the generation of spin-polarized currents, formation of spin-ordered edgelocalized states along the zigzag edges is the key mechanism. Since GNRs have long spin-correlation lengths and good ballistic transport characteristics they can be considered as a promising active material of spintronic devices without the need of ferromagnetic electrodes or other magnetic entities. The spin filtering structure could be prepared by the nanolithography of GNR. V-shaped constrictions could be realised by means of local anodic oxidation using scanning probe microscopy.
ac zz ac U zz U ac V zz V ac UU zz UU ac VV zz VV Fig. 6 Spin polarized electron transmissivities calculated by DFT ( black, red) for all the structures.
ACKNOWLEDGEMENT This work was supported by the grant of The Ministry of Education, Czech Republic No. MSM 6840770014 and by the grant of Grant Agency, Czech Academy of Sciences No. KAN400100652. LITERATURE [1] P J. A. FÜRST, et al., Density functional study of graphene antidot lattices: Roles of geometrical relaxation and spin. Phys. Rev. B, vol. 80, 2009, p. 115117H [2] WENG, L. et al., Atomic force microscope local oxidation nanolithography of graphene. Appl. Phys. Lett., vol. 93, 2008, p. 093107 [3] KOHANOFF, J. Electronic Structure Calculations for Solids and Molecules, Cambridge Univ. Press., 2006. 348 p. [4] M. TOPSAKAL, et al., First-principles approach to monitoring the band gap and magnetic state of a graphene nanoribbon via its vacancies. Phys. Rev. B, vol. 78, 2008, p. 235435 [5] E. J. G. SANTOS, D. SÁNCHEZ-PORTAL, A. AYUELA, Magnetism of substitutional Co impurities in graphene: Realization of single π vacancies. Phys. Rev. B, vol. 81, 2010, p. 125433 [6] Y.-T. ZHANG, et al., Spin polarization and giant magnetoresistance effect induced by magnetization in zigzag graphene nanoribbons. Phys. Rev. B, vol. 81, 2010, p. 165404 [7] www.quantumwise.com [8] AHIN, H., SENGER, R. T., First-principles calculations of spin-dependent conductance of graphene flakes. Phys. Rev. B, vol. 78, 2008, p. 205423