1. Introduction
Graphenes have attracted many physicists and/or chemists because of the beautiful structure and tunable electronic states. One of the remarkable properties is the high carrier mobility due to the famous Dirac point, in which the effective mass is theoretically zero (Novoselov, et al., 2004, 2005). The motion of electrons is described by pseudo relativistic effect, and new aspects on physics and chemistry of two-dimensional systems have been developed as nano-scale technology. Nowadays, there has been increasing interest of modified graphenes toward thin films, nano particles, adsorbance agents, and so on, as well as possible micro-electronic devises.
On the other hand, magnetism of graphenes is another interesting theme, related to the early studies on so-called graphene ribbons. That is, some graphene derivatives are promising candidates for organic ferromagnets. The magnetic properties depend on topological conditions such as edges, pores, and defects. Toward a new type of ferromagnets, chemical modification of graphene is a highly challenging theme. Paramagnetism of graphitic polymers itself has been theoretically predicted, relating to the edge states, pores, and defects. However, it was not until band structures of modified graphenes revealed the existence of flat bands at the frontier levels that robust ferromagnetism has been highly expected from graphene-based skeletons. The edges, pores, and defects in these systems should be ordered so as to cause completely flat bands in the Hückel level. Hückel analysis on modified graphenes gives a good perspective toward the flat-band ferromagnetism. In this chapter, graphene-based ferromagnetism is analyzed by crystal orbital method. Recent advances in magnetic graphenes are reviewed in view of their electronic states.
2. Methylene-edged graphenes
It is well known that graphene ribbons with peculiar type of edges have polyradical character, of which flat bands cause ferromagnetic interactions. Fig. 1 shows three types of graphene ribbons. Fig. 1a is the famous graphene ribbons with two-sided acene (zigzag) edges. The magnetic ordering has been predicted based on the band structures. The HOCO (highest occupied crystal orbital) and LUCO (lowest unoccupied crystal orbital) contact and become flat at the wavenumber region |
Instead, Klein suggested methylene-edged graphene ribbons shown in Fig. 1b (Klein, 1994; Klein & Bytautas, 1999). Nowadays, these are called Klein edges. At least within the same edge, the Klein-edged graphenes are also expected to show ferromagnetic interactions due to the flatness of frontier bands at |
For simplicity, we first consider a small non-Kekulé polymer shown in Fig. 2. There are non-bonding crystal orbitals (NBCOs) at the frontier level. They are completely degenerate under Hückel approximation. Each NBCO can be transformed into Wannier functions localized around each cell, as formulated below.
We consider Bloch functions corresponding to the NBCO band:
where the wavenumber
This procedure minimizes the exchange integral of the system. The Wannier function localized at the
where
Each Wannier coefficient is a function of the integer
Wannier functions in the original paper are not normalized (Hatanaka, M. & Shiba, R., 2007; Hatanaka, 2010a). In the present review, however, we adopt the renormalized Wannier functions, and the exchange integrals are recalculated. Since each Wannier function coefficient
where (
The
Klein-edged graphenes with
3. Porous graphenes
Fig. 6 shows porous graphene. This compound was synthesized as a two-dimensional nanostructure (Bieri et al., 2009). They synthesized porous graphene by aryl-aryl coupling reactions on Ag (111) surface, and observed STM (scanning tunneling microscope) image of the honeycomb structures. In 2010, band structures of porous graphenes were investigated by several workers (Du et al., 2010; Hatanaka, 2010b; Li et al., 2010). The Hückel-level dispersion and DOS (density of states) are shown in Figs. 6 and 7 (Hatanaka, 2010b). The dispersion suggests semi-conductive band gaps, and the frontier bands are so flat as to be available for ferromagnetism. It is interesting that both HOCO and LUCO become flat at all the wavenumber region. Thus, this material is expected to show ferromagnetism when it is oxidized or reduced by proper dopants.
The flat bands result from nodal character of phenylene units. Fig. 8 shows amplitude patterns of selected Hückel molecular orbitals of cyclohexa-
apart from the normalization factors, amplitude patterns of butadiene’s HOMO and LUMO spread between each phenylene unit. Moreover, eigenvalues of HOMO-11 (
Porous graphene ribbons with various edges are also of interest, despite the coupling directions are not unique and some defects of the coupling reactions may cause diversity of molecular-weight distribution. Fortunately, it has been proved that the frontier non-bonding level of any porous oligomer is invariant with respect to molecular weight and/or coupling direction due to the zero-overlap interactions of the phenylene units. Thus, porous graphenes including porous ribbons are promising precursors toward organic ferromagnets.
Porous structures including boron and/or nitrogen are also interesting in that the hetero atoms serve as dopants, which increase or decrease the number of electrons in the frontier levels.
Fig. 9 shows porous graphene ribbons cut along the
Fig. 11 shows porous graphene ribbons cut along the
Fig. 13 shows DOS of X5 and Y5. There are main peaks corresponding to the flat bands. The largest peaks are due to the systematic degeneracies of HOCOs and LUCOs, and thus, photoelectron spectroscopy experiments will give a major peak at ca. 9.1 eV. Theoretical predictions on the magnetism of porous graphenes await experimental confirmations.
4. Defective graphenes
Recently, ferromagnetism of HOPG (highly oriented pyrolytic graphite) was found at room temperature (Červenka, et al., 2009; Esquinazi & Höhne, 2005; Mombrú et al., 2005). HOPGs often have defects, which form grain boundaries between the polycrystals. Laterally and longitudinally slipped defects in acene-edged HOPGs are particularly interesting in that the resultant defects are analyzable by graphene-based model (Hatanaka, 2010c). Fig. 14 shows definition of defects, in which the lateral and longitudinal displacements are represented by
Here we consider simple models of defects by using graphene ribbons with 3 ladders. Fig. 16 shows change of orbital pattern for the laterally slipped graphenes. We note that the effective displacement of lateral slip is smaller than the lattice period
Fig. 17 shows
Thus, as rough estimation, we guess that ferromagnetism observed in HOPGs is attributed to longitudinally slipped defects, which form grain boundaries including multilayers of graphene planes. Indeed, topographies corresponding to the longitudinally slipped defects were recorded in HOPGs by AFM (atomic force microscope) and MFM (magnetic force microscope) techniques (Červenka et al., 2009).
5. Conclusion
Some graphene-based ferromagnets were analyzed in view of their electronic states. There appear flat bands at the frontier levels, in which the Wannier functions span common atoms between the adjacent cells. If geometry conditions such as edges, pores, and defects are well controlled by chemical modification, graphene-based ferromagnets will be realized through the flat bands. Amplitudes of the Wannier functions have non-bonding character, and the frontier electrons are itinerant around each central cell. The degeneracy of frontier flat bands and the positive exchange integrals play key roles for ferromagnetic interactions of graphene-based ferromagnets.
Acknowledgments
Thanks are due to Elsevier B. V. for permission to use figures in the original papers (Hatanaka 2010a, 2010b, 2010c).
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