Topology is familiar mostly from mathematics, but also natural sciences have found its concepts useful. Those concepts have been used to explain several natural phenomena in biology and physics, and they are particularly relevant in the electronic structure of topological insulators and graphene. Here, we introduce topologically distinct graphene structures, graphene spirals, and use density-functional theory to investigate their geometric and electronic properties. We find that the spiral topology gives rise to an intrinsic Rashba spin-orbit splitting. The splitting mechanism is similar to the mechanism of band inversion in topological insulators. By a Hamiltonian constrained by space curvature, graphene spirals show topologically protected states due to time-reversal symmetry. These unique electronic properties require neither an external magnetic field nor spin-orbit interaction, which is unlike any typical quantum Hall system. Graphene spirals could be synthesized by bottom-up methods, and they could be used in quantum computing and as nano-solenoids to produce high magnetic fields. Nanoscience Days 2012, Jyv ä skyl ä, Finland Topological Signatures in the Electronic Structure of Graphene Spirals C.G. Rocha 1, P. Koskinen 1, and S. Avdoshenko 2 1 Department of Physics, University of Jyv ä skyl ä 40014, Jyv ä skyl ä, Finland 2 School of Materials Engineering, Purdue University IN , West Lafayette, USA 1. Topology in materials science 2. Examples of Spiral systems at the nanoscale 3. Computational Methodology 4. Electronic structure results 5. Electronic structure results 6. Outlook Curved graphene forms reveal interesting electronic features which can be interpreted within topology. Such splitting mechanism of the bands finds similarity with band inversion effect observed in topological insulators. Mathematical topology analyzes how the properties of objects preserve under continuous deformations. But topological analysis is not restricted to mathematics alone. Interest in topology spans also biology, chemistry and materials science. Applications: GRAPHENE SOLENOID Notation for the spirals: being xi(o) and yi(o) the number of armchair and zigzag units, respectively, contained on their inner (outer) edge. Topology is used to interpret the physics of graphene. Angle-scanned x-ray photoelectron diffraction pattern of heptahelicene molecule on Cu(111) [2]. Helical carbon nanotubes grown under kinetically controlled techniques [1]. STM image of helicene molecules assembled on reconstructed InSb (001) surfaces [3] DNA We adopted Density Functional Theory (DFT) implemented within the computational packages SIESTA and VASP to model the electronic structure of graphene spirals. Single-  band tight binding approach was also used in order to verify the main role played by curvature effects. Interlayer separation: ~3.2 Å Tight-binding Due to the absence of curvature and charge transfer effects, tight- binding picture is not able to capture all the main physical features of the spirals. Energy bands only fold in response to the increase in the number of atoms in the unit cell. DFT Peculiar band splitting: Rashba- like behaviour emerges naturally as a result of the curved space (no need for spin-orbit interaction). The molecular orbitals in graphene spirals manifest chiral symmetry (combination of translation and rotation operations); Brillioun zone of chiral structures are re-scaled with respect with the reciprocal space of translational systems. At the vicinity of Fermi level, states are robust and protected by time- reversal symmetry. (0.1, ) isosurfaces for local density of states at the Fermi level for the graphene-based spirals unveiling the protected edge state. References: [1] R. Gao, Z.L. Wang, and S. Fan, J. Phys. Chem. B 104, 1227 (2000).[3] P. Sehnal, et al., PNAS 106, (2009). [2] R. Fasel, et al., J. Chem. Phys. 115, 1020 (2001).This work is about to be submitted to ACS Nano. We acknowledge Academy of Finland and Alexander von Humboldt Foundation for sponsoring this research.