Electronic properties of rhombohedral graphite
Introduction
Few-layer graphenes, such as monolayer, bilayer, and trilayer, have been produced by mechanical friction [1], [2] and by thermal decomposition [3], [4] in the past few years. A lot of researches have been performed for their unique two-dimensional physical properties, e.g., band structures [5], [7], [8], [9], transport properties [10], [11], [12], optical spectra [6], [13], and Coulomb excitations [13], [14], [15], [16]. A monolayer graphene has two isotropic linear bands intersecting at the Fermi level, mainly owing to the hexagonal symmetry. It is a zero-gap two-dimensional semiconductor without free carriers. As for multilayer graphenes, the stacking configurations, such as AB- [5], [6], AA-, and ABC-stacking [7], affects the electronic properties differently.
Bulk graphite consists of graphene layers periodically stacked along the direction. It is one of the most important layered systems, and has been studied extensively both in experiments [17], [18], [19], [20], [21] and in theories [22], [23], [24], [25], [26], [27], [28]. There are four kinds of layered graphites: the rhombohedral graphite (the ABC-stacked graphite) [23], [25], the Bernal graphite (the AB-stacked graphite) [26], [27], [28], the simple hexagonal graphite (the AA-stacked graphite) [27], and the turbostratic graphite (without the periodical stacking sequence) [24]. All of them are semimetallic because the interlayer atomic interactions induce the overlapping between the conduction and valence bands. Most natural graphite belongs to the AB-stacking sequence, and recently the AA- [19] and ABC-stacked [20], [21] graphites have been produced in the lab. We study the electronic properties of the ABC-stacked graphite through the first-principles calculations. The low energy bands near the Fermi level are discussed in detail. To further understand the importance of stacking symmetry, a comparison with those of AB- and AA-stacked graphites is also presented.
Section snippets
First-principles calculation method
In this work, the electronic structure of the rhombohedral graphite is investigated by the density-functional theory (DFT) within the local-density approximation (LDA) [33], [34], [35]. The exchange-correlation energy is in the Ceperley–Alder form [36]. The ultrasoft pseudopotential is used for the carbon ions, as implemented in Vienna Ab-initio Simulation Package (VASP) [37]. The wave functions are expanded with the plane waves and the energy cutoff of 700 eV is carried out. A grid of
Electronic structures
Energy bands caused by the and 2 orbitals are shown in Fig. 2. The occupied and unoccupied bands are not symmetric about the Fermi level . In the valence bands, there are 18σ subbands, and 9 subbands when spin degeneracy is not taken into account. There exists non-degenerate energy bands as well as double-, triple-, and six-fold degenerate ones. The highest and lowest energies in the σ bands are −3.07 eV and −19.72 eV at the Γ point, which defines the bandwidth of the σ bands, 16.65
Concluding remarks
The ABC-stacked graphite owns linear and parabolic energy bands with strong anisotropy. Energy bands may be non-degenerate as well as double, triple, and six-fold. Each band has many band-edge states contributing to the high density of states. Such states are located near or at the high symmetry points. The bandwidths of occupied π and σ bands are 8.41 eV and 16.65 eV, respectively. For the low energy bands, there exist double-degenerate parabolic bands and non-degenerate linear bands with one
Acknowledgements
This work partially supported by the NSC and NCTS of Taiwan, under the Grant Nos. NSC 98-2112-M-006-013-MY4, and NSC 97-2112-M182-003.
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