Electronic band structure and phonon dispersion of graphene
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Graphene is a two-dimensional structure, entirely built of carbon
atoms. The unit cell of graphene, denoted by a rhomb, has two atoms, shown by
grey and black symbols in Fig. 1, left. The Brillouin zone of graphene is a
hexagon (Fig. 1, right).
The electronic band structure of graphene can be obtained by using the
π-band tight-binding band (πTB) model. This model uses a single pz orbital per carbon atom,
which is perpendicular to the graphene sheet. The Hamiltonian matrix is
expressed via two parameters: one for the coupling of the pz orbitals of adjacent
atoms and the other is the onsite energy. The resulting valence and
conduction electronic bands are denoted as π and π*
bands, resp.
A more complete description of the band structure is obtained by
considering all four orbitals s, px, py, and pz,
per carbon atom. A four-electron TB model, in which the overlap between
orbitals on adjacent atoms is neglected, so-called orthogonal TB model,
yields eight electronic bands instead of the two bands of the πTB model.
Two of the bands are mainly due to π coupling of the pz orbitals and are termed
π and π* bands. The remaining bands come from σ
coupling between the s, px, py, and pz
orbitals are termed σ and σ* bands.
However, in the four-electron TB model, the s and p atomic orbitals, centered on adjacent atoms, are non-orthogonal. This non-orthogonality is introduced in the tight-binding model through the overlap matrix. The resulting model is termed the non-orthogonal tight-binding model (NTB). The electronic band structure, derived within the NTB model with model parameters taken over from an ab-initio study (Porezag) and shown in Fig. 2, gives a semi-quantitatively correct description of the band structure of graphene, especially close to the Fermi energy, EF = 0 eV [1,2]. Although the optically important energy range between −3 and 3 eV encompasses the π and π* bands (enclosed in the red rectangle), the calculations of quantum-mechanical perturbation terms, e.g., for first- and second-order Raman scattering, require the knowledge of all eight NTB electronic bands of graphene. The phonon dispersion, derived within the NTB model [1,2] in the linear-response approximation [3,4], is shown in Fig. 3. The overbending of the in-plane optical (LO) branch is predicted is good agreement with the experimental data. However, the NTB results overestimate the experimental data mainly for the in-plane branches by about 12%. For agreement with experiment, the NTB branches are downscaled by a factor of 0.9. The obtained in-plane longitudinal and transverse acoustic (LA and TA) branches, the in-plane longitudinal and transverse optical (LO and TO) branches, as well as the out-of-plane acoustic (ZA) branch are well reproduced. However, the downscaling yields lower frequencies for the out-of-plane optical (ZO) branch in comparison to experiment. The red rectangle in Fig. 3 encloses the in-plane optical branches, which yield the most intense one- and two-phonon Raman bands in the Raman spectra of graphene. The NTB phonon dispersion shows Kohn anomalies of the LO branch at the Γ point and of the TO branch at the K point, which is manifested by the non-zero slope of these branches in the adiabatic approximation. The strong electron-phonon interaction close to these points requires the account of the non-adiabatic (dynamic) effects, which is usually done in first-order perturbation theory. As a result, the zero slope of the mentioned branched is recovered [5]. References: 1. V. N. Popov, New J. Phys. 6 (2004) 1-17. 2. V. N. Popov and L. Henrard, Phys. Rev. B 70 (2004) 115407 3. V. N. Popov, L. Henrard, and Ph. Lambin, Phys. Rev. B 72 (2005) 035436. 4. V. N. Popov and Ph. Lambin, Phys. Rev. B 73 (2006) 085407. 5. V. N. Popov and Ph. Lambin, Phys. Rev. B 82 (2010) 045406. |
Fig. 1. Left: Atomic structure of graphene. Right: Brillouin zone of
graphene with special points.
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Valentin Popov
September 5, 2010