Electronic band structure, phonon dispersion, and two-phonon resonant Raman scattering in silicene

Fig. 1. Sideview of the relaxed atomic structure of silicene within the NTB model. The two atomic sublattices are displaced with respect to one another at 0.57 Å.

Silicene has been produced on substrates, where it forms various one-dimensional superstructures. Up to now, the existence of free-standing silicene has not been confirmed. Hoping that this can happen sooner, or later, we performed this research in order to clarify the similarities and differences of the electronic and vibrational properties of a hypothetic free-standing silicene with those of graphene.

Silicene is predicted as an atomic-thin quasi-twodimensional structure, consisting entirely of silicon. Similarly to graphene, it relaxes to hexagonal symmetry with two silicon atoms in the unit cell. However, unlike graphene, the sublattices of the two atoms are displaced normally to the silicon sheet (Fig. 1).

The calculation of the electronic band structure and phonon dispersion of silicene is performed entirely within a non-orthogonal tight-binding model [1,2]. It is seen in Fig. 2, left, that the electronic structure of silicene has crossing conduction and valence bands at the Fermi energy [3]. As for graphene, these bands are mostly due to π coupling between the p_z orbitals of adjacent atoms (z axis is normal to the silicon sheet). These bands are denoted as π and π^*. As for graphene, these bands have inflexion at the M point of the Brillouin zone, but the separation between the two inflexion points (π-plasmon energy) is much smaller and amounts only to 1.83 eV.

The phonon dispersion of silicene [3] is shown in Fig. 2, right. Similarly to graphene, the six phonon branches have in-plane and out-of-plane atomic displacement (Z). Three of them are acoustic (A) and the remaining three are optical (O). The in-plane phonons are longitudinal (L) and transverse (T). However, the corresponding phonon frequencies are significantly lower than in graphene, because of the much weaker interatomic coupling. It has to be noted that the raw NTB frequencies have to be downscaled by a factor of 0.83 for reaching agreement with the ab-initio predictions.

As in graphene, there is a single one-phonon Raman band, arising from scattering of electrons and holes by in-plane optical phonons. Apart from this band, similarly to graphene, a few two-phonon Raman bands are also expected in the Raman spectra of silicene. Here, the two-phonon bands for Stokes processes are calculated using the expression, derived in fourth-order quantum-mechanical perturbation theory (Martin & Falicov, 1983)

Here, , ; is the energy of the initial state, is the incident photon energy (laser excitation); , , are the energies of the intermediate (a, b, c) and final (f) states of the system;are the matrix elements between initial, intermediate, and final states; and are momentum matrix elements; and are electron-phonon matrix elements; γ is the broadening parameter, equal to sum of the halfwidths of conduction and valence states. The specific band structure of graphene allows for scattering processes, for which one, two, or three become small (single, double, and triple resonance), resulting in enhancement of the Raman intensity. The matrix elements and the broadening parameter γ are derived within a non-orthogonal tight-binding (NTB) model [4,5]. Calculation of the two-phonon Raman bands has been done for graphene in the visible light region [6] and, in this work, in the ultraviolet region [3].

Figure 3 shows the energy dependence of the broadening parameter γ [3] – the total one as well as the contributions from various phonon branches. Contrary to graphene, the TO+LO contribution is not dominant as other branches also contribute substantially. The enhanced scattering of electrons close to 2 eV is due to the log singularity of the electronic density of stated at the inflexion points.

Figure 4 shows the calculated two-phonon Raman bands of silicene [3] at three different laser excitation energies E_L. The Raman spectrum at E_L = 1.2 eV has an intense band 2TO, arising from scattering by two TO phonons, which is usually referred to as the 2D band in graphene. Additionally, two weaker bands, 2LA and TOZO, are also predicted. They are due to the non-zero electron-phonon coupling for these phonons. With increasing E_L up to the plasmon energy, the wavevector of the relevant TO phonons decreases to zero (Fig. 5, process 1).

Figure 4 (b) shows the Raman bands at E_L = 2.0 eV, slightly above the plasmon energy. The intense band comes mainly from 2TO process but also from 2LO and LOTO processes. The latter two processes are enhanced because of the large phonon density of states at the overbending of the LO branch. Apart from the intense composite band, a number of weak overtone and combination bands are also predicted. They have observable intensity, because of nonzero electron-phonon coupling, large two-phonon density of states and large electronic density of states close to the M point. Additionally, there are bands, denoted by asterisks, for which the non-zero intensity comes from large density of states of the corresponding phonon branches around their extrema.

Figure 4 (c) shows the Raman bands at E_L = 2.8 eV, well beyond the plasmon energy of silicene. The intense band 2LO is mainly due to scattering by two LO phonons in the vicinity of the overbending of the LO branch. In graphene, this band is often referred to as the 2D’ band. Other, less intense bands are also predicted.

References:

1. V. N. Popov and L. Henrard, Phys. Rev. B 70 (2004) 115407.

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

3. V. N. Popov and Ph. Lambin, 2D Materials 3 (2016) 025014.

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

5. V. N. Popov and Ph. Lambin, Phys. Rev. B 74 (2006) 075415

6. V. N. Popov and Ph. Lambin, Eur. Phys. J. B 85 (2012) 418.

Fig.2. Left: Electronic structure of silicene in the vicinity of the Fermi energy, chosen as zero. The crossing conduction and valence bands are denoted as π* and π. These bands have inflexion points at the M point of the Brillouin zone. The separation between the two inflexion points is the π-plasmon energy of 1.83 eV. Right: Phonon dispersion of silicene. The character of the six dispersion curves is explicitly given. The acronym DR marks approximately the phonons participating in two-phonon resonant Raman processes.

Fig. 3. Broadening parameter γ vs energy separation between the π* and π bands. The contributions from the various phonon branches is also given. It can be seen that the TO+LO phonons do not give dominant contribution in comparison to graphene (inset).

Fig. 4. (a) – (c) The two-phonon Raman bands of silicene at three different laser excitation energies. The intense bands are denoted by the acronyms of the pair of phonons. The symbol M denotes phonons close to the M point. The asterisk denotes phonons close to extrema of phonon branches. (d) The phonons, contributing to the intense bands, vs laser excitation in eV, shown by numbers from 1 to 3.

Fig. 5. (a) and (b) Schematic of the electronic bands of silicene close to the Fermi energy along two high-symmetry directions in the Brillouin zone. (c) Equi-energy contour map of the processes from panels (a) and (b).

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Valentin Popov

July 25, 2012