Graphene, a single atomic layer of graphite adhered from natural graphite,1,2has been attracting much intensive research in both theory and experiment due to its peculiar electronic and magnetic transport properties.3–7Indeed, a mass of graphene-based structures have been fabricated due to experimental and theoretical breakthroughs over the past decade, which may be used as building blocks for nanometre-scale devices. More recently, it was reported that graphene can be cut or patterned into well-defined geometric structures by different techniques,8–14further opening the door to the fabrication of graphene-based nanodevices. For instance, various graphene-based functional elements and components have been obtained experimentally such as quantum dots,9–11field-effect transistors,12,13integrated graphene circuits14etc.Especially, a new style of ring-shaped graphene was fabricated with inner and outer radii of about 350 nm and 500 nm,15which may be one of the most novel building blocks for future electromagnetic nanodevices. Due to their similarity to mesoscopic rings, the unusual electronic properties of graphene rings have attracted great interest in theory,16,17indicating they are a useful candidate for probing both quantum coherence and dephasing rates of systems. Of special interest is the persistent current18–20in such a two-dimensional (2D) ring-shaped device, induced by magnetic flux,&phis;, threaded through the centre of the ring.For graphene-based structures, the electronic properties of graphene nanoribbons (GNRs), carbon nanotubes (CNTs), and toroidal carbon nanotubes (TCNs) have been well studied, and are dependent upon their geometries. It is well known that all zigzag GNRs are metallic regardless of width,21,22while armchair GNRs can be either metallic or insulating depending on their widths.22–25As for armchair CNTs and TCNs, they have the same armchair-type cross-section as in zigzag GNRs. Not considering the curvature, all armchair CNTs and TCNs are expected to be metallic,26–29similar to zigzag GNRs. For zigzag hexagonal graphene rings (HGRs), it is interesting to explore whether similar characteristics come into existence and how the metallicity depends on its geometry. The study of zigzag HGRs would be very helpful to obtain universal laws of quantum-size effects on electronic transport in graphene-based structures.For various-shaped HGRs, recently, the inner- and outer-edge states have been obtained, which depend on the edge symmetries and the corner structures.30For zigzag HGRs, especially, the charge density is spread out on the two opposite edges, indicating the coupling of states localized at two opposite edges. Therefore, one may expect that the effects of ring size and edge-state coupling are very important for determining the electronic structures and thus the persistent currents in zigzag HGRs, which should be further explored.By developing the supercell method,31we present a detailed investigation on the electronic structures, and thus the persistent currents of zigzag HGRs. The results show that the flux-dependent energy spectrum is grouped into bands with six levels per band, due to the inter-valley scattering at the corners of the ring. More interestingly, it is found that the parity of the ring widthNdetermines the degeneracy at the Fermi level (EF= 0) and thus the metallicity of the samples, which are metallic at oddNbut semiconducting at evenN, showing up a strange odd–even width effect. For a metallic ring, the persistent current is a linearly periodical function of magnetic flux,&phis;, while it is a sinusoidal periodical function of&phis;for semiconducting rings. In addition, with increasingN, the persistent current decreases (increases) at odd (even)N, but finally falls into consistence with each other at enough largeN. This indicates that the odd–even width effect may be experimentally observable only in narrow rings.A HGR of zigzag edge is shown inFig. 1(a)consisting of six arms withlthe lattice spacing, the inner ring radiusris given in units ofr0, andr= 4r0with. The geometry of one arm is schematized for widthW=Nr0inFig. 1(b), withN= 5 the number of zigzag carbon chains. We use the single-orbital nearest-neighbor tight-binding model for the finite-size HGR, which has been successfully applied to the study of GNRs,22–25CNTs26,27and other carbon-related materials.28,29In the present model, the magnetic field is limited to the central region of the ring. The Hamiltonian, neglecting electron–electron interactions, is given by1whereiandjlabel the nearest sites on a honeycomb lattice. In a perfect HGR, the on-site energies are taken asϵi= 0. The hopping integral elementsτi,j= −τ exp[(i2e/h)∫jidr&cmb.a.rharp;·A&cmb.a.rharp;] withτ= 2.9 eV the hopping integral constant, andA&cmb.a.rharp; is the vector potential. The electronic spectrum can be obtained by the supercell method recently developed by Liu and Ding.31In terms of the rotational symmetry of six-fold, the effective Hamiltonian of a supercell is given by2Heff=H0+eiβH1+e−iβH+1whereH0andH1refer to the Hamiltonian matrix in the supercell and the coupling matrix between the two neighboring supercells, respectively. Hereβ=j× 2π/P, (j= 1, 2, …,P) is the phase difference between the wave functions of two neighboring supercells, whereP= 6 is the number of the supercells contained in the HGR. For a givenβ, the eigenlevels can be obtained by diagonalizing the matrixHeff. This simplified calculation is a reasonable and efficient method for such similar complex system with rotational symmetry.(a) A zigzag HGR of ring widthN= 5 and inner ring radiusr= 4r0withlthe lattice spacing and. Magnetic flux,&phis;, passes through the hole. (b) Schematic illustration of one arm in the ring, of which the dashed hexagons in the centre are cut out.At zero temperature, the total persistent current of system is given by3where&phis;= Φ/Φ0is the dimensionless magnetic flux with Φ0=h/ethe flux quantum.Enis the energy of the system, andnlabels the corresponding eigenlevels. The current is a periodic function of&phis;with fundamental period Φ0. Usually, one is interested in the typical current,20which is defined as the square-root of the flux average of the square of the persistent currents, given by4In our calculations, the site energiesϵiand energyEare given in units ofτ, and thus the persistent currents in units ofτ/Φ0. As typical examples,Fig. 2shows the energy spectrum near the Fermi level of the zigzag HGRs ofN= 7 (a) andN= 8 (b). Both rings have the same inner ring radius ofr= 13r0. FromFig. 2, it is seen that the flux-dependent energy spectrum is grouped into bands, with six levels per band in both cases, which may be due to the high-symmetry structure of HGRs. For each site in such a ring, there exist other five equivalent sites, due to the six-fold rotational symmetry. Because of the long distance between them, the six sites are weakly coupled to each other by other sites. This may lead to a six-level group of bands. For the sites located in a supercell, however, there exists not only strong interactions with the neighboring sites but also the inter-valley scattering at the corners,16which would result in the presence of inter-band gaps. FromFig. 2, especially, it is found that the energy spectrum strongly depends on the odd or even quality of the ring widthN. In the case ofN= 7 inFig. 2(a), the highest occupied state (HOS) and the lowest unoccupied state (LUS) are degenerate at Φ = ± 0.5Φ0, showing the sample ring to be metallic. InFig. 2(b)withN= 8, a narrow energy gap appears at Φ = ± 0.5Φ0, indicating the sample ring to be a narrow-gap semiconductor. Further calculations have been done for other zigzag HGRs of both odd and evenN. It follows that zigzag HGRs are metallic at oddNand semiconducting at evenN. The result can be understood by the following consideration.Energy spectra of zigzag HGRs as a function of magnetic flux for the sizesr= 13r0, (a)N= 7 and (b)N= 8.FromFig. 1(b), it is clearly seen that for a zigzag HGR, the number of atoms within each zigzag chain in one arm must be odd, irrespective of inner radiusr. Thus, the number of atomsN1in one arm must be odd (even) only ifNis odd (even). Due to spin degeneration and the rotational symmetry of six-fold, further, the highest occupied band is obtained to beN1/2. This means that for an oddN(orN1), the Fermi level is located at the centre of the (N1+ 1)/2 band, while it falls into the gap betweenN1/2 andN1/2 + 1 at evenN(orN1). Therefore, the even or odd quality of the ring widthNdetermines completely the degeneracy at the Fermi level and thus the metallicity of a zigzag HGR. The result is different from the observation of odd–even effect in the conductance of zigzag GNRs under gate potentials,32–34which is attributed to the symmetry and asymmetry of the zigzag chains of GNRs.From the energy spectrum of the studied samples, inFig. 3, we calculate the persistent currents (Ipc) of the zigzag HGRs as a function of magnetic flux,&phis;, for metallic rings with oddN(a) and semiconducting rings with evenN(b). For the metallic ring inFig. 3(a), it is shown that the flux-dependentIpcwithin a period flux changes linearly with&phis;. A jump ofIpcis observed at Φ = ± 0.5Φ0, corresponding to the level degeneracy at Φ = ± 0.5Φ0as shown inFig. 2(a), similar to that of TCNs of metallic-type I.28,29For semiconducting rings, it is seen fromFig. 3(b)that at a given inner radiusr, the persistent current may decline by 2–3 orders of magnitude at smallN, compared with the metallic ring. This may be due to the energy gap and an almost zero slope of energy curves near the Fermi level in the semiconducting ring, as shown inFig. 2(b). Also, its behavior of flux-dependentIpcis a smooth sinusoidal periodical function of&phis;, different from the metallic HGR. FromFigs. 3 (a) and (b), in addition, it is seen that with increasingN,Ipcdecreases at oddNbut increases at evenN, further showing up a strange odd–even width effect.Persistent currents of zigzag HGRs as a function of magnetic flux for (a) oddN, and (b) evenN.To explore the quantum-size effect on persistent current, inFig. 4(a), we plot the logarithm of the typical currents (Ityp) in zigzag HGRs as a function of the inner ring radiusr. FromFig. 4(a), it is seen that for both metallic and semiconducting rings, the overallItypdecreases with increasingr, as expected in the previous theory of mesoscopic metallic rings.20From a more careful analysis, it is found that the data ofItypvs.rare well fitted toItyp∼ exp(−κr/r0). The decay exponentκis obtained to beκ= 0.4937, 0.5616, and 0.6450 atN= 5, 7, and 9 respectively, whileκ= 1.3168, 0.9956, and 0.8223 atN= 4, 6, and 8 respectively. This means that the typical currents also depend strongly on ring widthN,κdecreasing (increasing) withNin metallic (semiconducting) rings, different from the previous results.19,20Typical currents,Ityp, of zigzag HGRs as a function of (a) the inner ring radiusrand (b) ring widthN.InFig. 4(b), we further show the logarithm ofItypas a function of the ring widthN. In the case of metallic rings (oddN), it is seen fromFig. 4(b)that with increasingN,Itypfirstly decreases exponentially at smallNbut becomes invariable at largerN. For semiconducting rings (evenN), however,Itypfirstly increases exponentially withN, in spite of a constantItypat largerN. At a given inner ring radiusr, especially,Itypat two neighboring odd and evenNfalls finally into consistency at enough largeN. Such changes inItypcan be understood from the calculated energy spectra, of which the slopes of energy curves and the energy gaps change withN. For a largeN, in fact, the wide ring can be regarded as a graphene sheet with vacancies in the presence of a perpendicular magnetic field,35in which the odd–even effect would disappear. Therefore, the odd–even width effect should be experimentally observable in narrow zigzag HGRs.In conclusion, a detailed investigation is presented for the electronic structure and the persistent currents in zigzag HGRs by developing a supercell method within the single-orbital nearest-neighbor tight-binding model. The results show that the energy spectrum is grouped into bands with six levels per band, due to the inter-valley scattering at the corners of the ring. Interestingly, it is found that the parity of the ring widthNdetermines the degeneracy at the Fermi level and thus the metallicity of the samples. It is shown that a sample ring of widthNis metallic at oddNbut semiconducting at evenN, showing up a strange odd–even width effect. For metallic rings, the persistent current is a linearly periodical function of magnetic flux,&phis;, while it is a sinusoidal periodical function of&phis;for semiconducting rings. With increasingN, in addition, the persistent current exponentially decreases (increases) at odd (even)Nand falls finally into consistency at large enoughN, showing the strange odd–even width effect existing only in narrow rings. The results may be very helpful for the design and application of HGR-based nanodevices.