Electronic band structure, phonon dispersion, and two-phonon Raman scattering in Bernal bilayer graphene

Electronic band structure, phonon dispersion, and two-phonon resonant Raman scattering in Bernal bilayer graphene

Fig. 1. Left: Atomic structure of Bernal bilayer graphene. Right: Brillouin zone of Bernal bilayer graphene with special points.

Bilayer graphene (BLG) consists of two graphene layers, usually twisted at an angle with respect to one another. With respect to the twist angle there are the following types of layer stacking: AA-stacked BLG for twist angle of 0°, AB-stacked (or Bernal) BLG for twist angle of 30° (Fig. 1, left), and twisted BLG for twist angles between 0° and 30°. Here, we consider exclusively AB-stacked BLG. It has hexagonal symmetry with hexagonal Brillouin zone (Fig. 1, right).

The electronic structure of BLG can be derived from that of graphene by accounting for the weak interlayer interaction. Thus, the simplest tight-binding model for graphene uses a single p_z orbital per carbon atom, which is perpendicular to the graphene sheet. The Hamiltonian matrix contains two parameters: one for the coupling of adjacent orbitals and the other is the onsite energy; all other matrix elements, incl. the overlap ones, are ignored. The resulting valence and conduction bands are denoted as π and π^* bands, resp.

A more complete description of the band structure of graphene can be obtained by considering all four orbitals s, p_x, p_y, and p_z, 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 p_z orbitals and are termed π and π^* bands. The remaining bands come from σ coupling between the s, p_x, p_y, and p_z orbitals and 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, E_F = 0 eV [1,2]. Although the optically important energy range between −3 and 3 eV encompasses the π and π* bands, 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 electronic band structure of BLG can be obtained by using the tight-binding Slonczewski–Weiss–McClure model, which has been around since 1960. For the description of the interlayer interaction, this model utilizes additional three parameters for the coupling of the p_z orbitals of atoms, belonging to different layers. The coupling between the two layers brings about splitting of the unperturbed doubly degenerate π and π* bands. The resulting valence and conduction bands are denoted here as π₁, π₂, π₁, and π₂^ bands.

As for graphene, a more sophisticated tight-binding model is necessary for studying optical phenomena. Here, we use the NTB model with intralayer parameters, taken over from an ab-initio study (Porezag), and interlayer parameters, taken from another ab-initio study [3]. Figure 3 shows the resulting conduction and valence bands in the vicinity of the Fermi energy. It has to be noted, that both components of these bands, 1 and 2, are parabolic at their extrema. Bands 2 have a band gap of about 0.8 eV, which is comparable to the measured and calculated ones, already reported in the literature.

The phonon dispersion of BLG is calculated here within the NTB model using a perturbative approach [4,5]. The phonon branches of the separate layers are acoustic (A) or optical (O), in-plane longitudinal (L), in-plane transverse (T), or out-of-plane transverse (Z). Figure 4 shows that the effect of the interlayer interaction on the phonon dispersion is the splitting of the doubly degenerate phonon branches of BLG into a symmetric (+) and an antisymmetric (−) components. Therefore, the acronyms of the branches acquire an additional upper index “+” or “−”. The splitting is normally within a few cm^-1. However, the splitting of the doubly degenerate out-of-plane acoustic branch is up to several tens of cm^-1; one of the components still retains acoustic behavior, while the other becomes optical with breathing motion of the two layers, so-called breathing mode or BM, with frequency of the Γ phonon of 90 cm^-1.

The two-phonon Raman scattering is studied here using the expression for the Raman intensity, derived in fourth order quantum-mechanical perturbation theory [6,7]. The calculations are performed for all overtone and combination bands, and the integration is performed over the entire Brillouin zone. The Raman spectrum was calculated in the entire region of the two-phonon bands of BLG at several laser excitation energies. In Fig. 5, the obtained 2D band at laser excitation E_L = 2.33 eV is shown. For the analysis of the 2D band, we note that there are four intervalley scattering processes for electrons and hole between the π₁ and π₂ electronic bands at the K and K’ points (Fig. 3), which are usually denoted as P₁₁, P₁₂, P₂₁, and P₂₂ (Fig. 5). Because the wavevector of the scattering phonons for these processes are different, they yield contributions to the 2D band with different Raman shifts, except for the P₁₂ and P₂₁ processes, for which the contributions have the same Raman shift. Additionally, the 2D band has contributions from the overtone bands 2TO⁺ and 2TO⁻, and from the combination band TO⁻TO⁺. The shape of the resulting 2D band generally has three features (peaks or kinks), due to the contributions of the four scattering processes.

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 and Ch. van Alsenoy, Phys. Rev. B 90 (2014) 245429.

4. V. N. Popov, L. Henrard, and Ph. Lambin, Phys. Rev. B 72 (2005) 035436.

5. V. N. Popov and Ph. Lambin, Phys. Rev. B 73 (2006) 085407.

6. V. N. Popov, Carbon 91 (2015) 436.

7. V. N. Popov, J. Phys.: Conf. Ser. 682 (2015) 012013.

Fig. 2. Non-orthogonal tight-binding electronic band structure of graphene.

Fig. 3. Non-orthogonal tight-binding electronic band structure of Bernal bilayer graphene.

Fig. 4. Non-orthogonal phonon dispersion of Bernal bilayer graphene.

Fig. 5. Fig. 5. 2D band of Bernal bilayer graphene at E_L = 2.33 eV. The dashed line is the sum of the overtone contributions.

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Valentin Popov