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QuantumChemical Calculation and Visualization of the Vibrational Modes of Graphene in Different Points of the Brillouin Zone
Nanoscale Research Letters volume 10, Article number: 287 (2015)
Abstract
Different notations of graphene irreducible representations and optical modes could be found in the literature. The goals of this paper are to identify the correspondence between available notations, to calculate the optical modes of graphene in different points of the Brillouin zone, and to compare them with experimental data obtained by Raman and coherent antiStokes Raman scattering (CARS) spectroscopy. The mechanism of the resonance enhancement of vibration modes of the molecules adsorbed on graphene in CARS experiments is proposed. The possibility of appearance of the discrete breathing modes is discussed.
Background
Graphene consists of hexagonal rings of carbon atoms packed in periodic structure with symmetry D_{6h}, and due to its electronic, mechanical, and other physical and chemical characteristics, it is of great interest for scientific community.
In spite of it, until now, there are different notations for symmetry of phonon modes presented in literature [1–3]. Here, we want to give two main used notations in literature [4, 5] for easy interpretation of Raman spectra and dispersion of graphene. Dispersion is not universally recognized characteristics in contrast to Raman spectra which is a passport for graphene and graphenebased materials. That is why we calculate, visualize, and compare the optical modes in graphene in different points of the Brillouin zone with Raman and coherent antiStokes Raman scattering (CARS) experiment data. We concluded about possible arising of new modes. Here, we discussed a resonant character of conventional Raman and CARS spectra for graphene.
Methods
The technique of quantum operator of projection on projective representations, which describes elementary vibration modes, is used to determine the shapes of the normal modes analytically in different points of the Brillouin zone, including k ≠ 0. The shapes of the normal vibrational modes are computed in the K, M, and Г points of the Brillouin zone.
Optical modes of singlelayer graphene were investigated using density functional theory (DFT) with periodic boundary conditions (PBC) and STO3G basis with the correlation functional VWN5. As an initial atomic configuration fragment, a flat hexagonal lattice of 40 carbon atoms (Fig. 1) or 12 elementary hexagonal cells of graphene with symmetry type D_{6h} were modeled and optimized in the same basis and function. Parallelized implemented software at Gaussian 09 packages [6] was used for calculations. Program Gauss View 5.0 was used for visualization of vibrations.
Results and Discussion
Fundamental modes of graphene layer at the Г (D_{6h}), K (D_{3h}), and M (D_{2h}) points of the Brillouin zone were identified by using the apparatus of the quantummechanical projection operator. To optimize the search of the characteristic frequencies, the discrepancies between the experimental and calculated frequencies of the vibrational spectra were analyzed. Analysis of carbon atom movements in graphene lattice for the main observed bands of the vibrational spectra was done (Fig. 2).
Phonon spectra of graphene in the Brillouin zone consist of six branches. Phonons with zero quasimomentum at the Гpoint (k = 0) or fundamental vibrational modes could be presented by irreducible representations of the D_{6h} group in the notations [2] by the following way:
\( {\varGamma}_{\mathrm{vib}}={A}_2^{+}+{A}_3^{}+{E}_1^{+}+{E}_2^{} \) [4], Г _{vib} = B _{2g} + A _{2u} + E _{2g} + E _{1u} [7] or \( {\varGamma}_{\mathrm{vib}}={\varGamma}_2^{+}+{\varGamma}_3^{}+{\varGamma}_5^{+}+{\varGamma}_{6\kern1.5em }^{} \) [5],
including the acoustic vibrations
and optical ones
Calculated vibrations at 1689 and 1705 cm^{−1} correspond to a double degenerated state with \( {\left({E}_1^{+}\right)}_{\upalpha} \) and \( {\left({E}_1^{+}\right)}_{\upbeta} \) according to the Bir and Pikus classification [2]. This mode could be experimentally observed near 1600 cm^{−1}, socalled the Gmode at the Гpoint of the Brillouin zone and named by Dresselhaus as E_{2g} mode. Calculated vibration at a frequency of 1455 cm^{−1} is associated with the Kpoint of the Brillouin zone and observed in the region of 1250–1380 cm^{−1} at different frequencies of laser excitation and called the Dmode (Table 1). The calculated vibration frequency at 1500 cm^{−1} is associated with the iTO phonon at the Mpoint of the Brillouin zone and appeared in unideal graphene and on its boundaries and registered in CARS experiment at 1430 cm^{−1} [8].
Let us describe elementary processes, which lead to appearance of different modes mentioned above. The most unusual vibration in graphitelike materials is socalled the 2Dmode, which was explained by Thomsen and Reich [9]. Three alternative mechanisms, which lead to the appearance of the 2Dmode, could be deduced.
The stages of electronphonon processes leading to the appearance of 2Dband in the Raman spectroscopy (RS) of graphene are the following (Fig. 3):

1. a)
The laser photon excites an electron with wave vector k _{ e } in the conduction zone and a hole with the wave vector k _{ h } in the valence zone (the wave vectors are calculated in the Гpoint, transition 1–2). This process could be named resonant Raman in contrast to nonresonant Raman, where the virtual states of the electron and the hole are excited.

b)
The electron comes to the other equivalent well and emits a phonon of iTO branch of the dispersion curve (transition 2 → 3).

c)
The electron returns to the state with approximately the same wave vector (≈k _{ e }), but its energy decreased by the energy of the emitted phonon. As a result, another phonon is emitted (transition 3 → 4).

d)
The electron recombines with the hole (transition 4 → 1) and emits the Stokes Raman photon. Such process leads to a double electronphonon resonance.

2.
Transitions occur simultaneously with the electron in the conductance zone (electron goes through the states 1–2–3–4–1). The hole goes through equivalent states in the valence zone.

3.
The electron goes through the states 1–2, 2–3 and transfers to the second well. The hole goes through the states in the valence zone simultaneously. Recombination of the electron and the hole occurs afterwards.
The transition 3 → 4 for Dband, which involves the structural defects, is a phononless. The sequence of transitions for D′ and 2D′bands is the same, but the processes occur within one well.
In the case of Gband, a laser photon excites an electron with the wave vector k _{ e } in the conductance zone and a hole with the wave vector k _{ h } in the valence zone (the value is calculated in the Гpoint). The electron emits a phonon with the wave vector q = 0 and recombines with a hole. All events occur in the frame of one valley.
The Scheme of Calculation of the Wave Vector
In order to calculate the wave vector of photons participating in the RS process, we computed the basic vector of reciprocal lattice (Fig. 4) as
where a _{1} = 24,562 Å and φ = 30^{0}. So, b _{1} = 29,539 Å^{−1}. The wave vector in the Mpoint was determined as b _{1} divided by 2 (from geometrical considerations). So, the value of wave vector at the Mpoint is k(M) = 14.7*10^{7} cm^{−1}. The value of wave vector at the Kpoint is \( k(K)=\frac{k(M)}{ \cos \varphi }=16.7*{10}^7 \) cm^{−1}. These values are the basis for the construction of the dispersion curve diagram of phonon states in graphene.
The values of electron wave vectors participating in Raman scattering and CARS in the vicinity of the Kpoints and Mpoints were calculated from the energy of excited photons (1.96 eV for Raman scattering and 3.86 eV for CARS) from the electron dispersion curves (Fig. 5). The theoretical curves are taken from [9]. These wave vectors were combined with experimentally determined energy of photons to reconstruct phonon dispersion curves (Fig. 6). The regions in the vicinity of the K point (shown in red) and in the vicinity of the M point (shown in green) (Fig. 6) are in good agreement with dispersion curves calculated earlier in this paper and our data (Table 1).
An enhancement factor of about 10^{5} was obtained in CARS experiment of thymine on graphene oxide [8]. Due to the fact that plasmons in graphene do not interact with light, only a special technique could excite the surface modes. The most plausible mechanism is the resonance interaction of light with the graphenemolecule complex, which has the graphitelike structure. Such complex is similar to the fragment of graphite due to ππ interaction between the molecules (Thy) and the graphene rings [10]. In this structure, the electrons have quasicontinuous number of states, so the Raman spectra could have resonant nature similar to those in pure graphene.
Discrete Breathers
Discrete breather is a spatially localized nonlinear vibrational mode in the defectfree lattice [11, 12]. Discrete breathers may exist as longlived oscillation modes, since they do not excite low amplitude phonons. Thus, they keep their energy because the frequency of the discrete breathers lies either above the phonon spectrum or in its gap [13]. However, the socalled embedded discrete breathers with frequencies lying in the phonon oscillation spectrum exist under specific conditions [14].
Since the lattice dynamics in the first principle calculations is not limited to linear approximation, we can expect obtaining nonlinear excitations such as discrete breathers. In our calculations, they are observed near the frequency of 714 cm^{−1} (Fig. 7).
Conclusions
Fundamental modes of graphene at the Г, K, and M points of the Brillouin zone were identified using the quantum mechanical projection operator. A correspondence between available notations for optical modes of graphene in the different points of the Brillouin zone are provided. Experimental data obtained by Raman and CARS spectroscopy are compared with calculations. The discrepancies between experimental and calculated frequencies of the vibrational spectra were analyzed. Based on these data, a resonant mechanism of RS and CARS in graphene was postulated. A possible existence of discrete breathers in graphene with the frequency of about 714 cm^{−1} was shown.
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Acknowledgements
The authors are very grateful to Prof. Oleg V. Shishkin for his consultation, cluster resources possibilities, and Gaussian packages and to Dr. Andrey Dementiev and academician Leonid Yatsenko for their fruitful discussion as well as the Faemcar FP7PEOPLE2012IRSES project for financial support.
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Authors’ Contributions
GD developed the idea of the work and responded for its realization. VG build the forms of normal vibrations using a projective representation of the wave vector at different points of the Brillouin zone. TL modeled the fragment of monolayer graphene and made the quantumchemical calculation. DP prepared the samples and performed the CARS experiment. All authors read and approved the final manuscript.
Authors’ Information
Dr. GD is the head of the Department of Physics of Biological Systems of the Institute of Physics of the National Academy of Sciences of Ukraine. She has a scientific degree of Doctor of Sciences in Solid State Physics and Biophysics and received the degree of professor in 2012. Her scientific areas of interest are biophysics, nucleic acids, solid state physics, surface solids, plasmonics, and experimental physics (FTIR, SEIRA, SERS, UV, Raman, NMR, CARS spectroscopy, LangmuirBlodgett technique, AFM microscopy, and computational chemistry). She was involved in the study of biological molecule interaction with low doses of ionizing and microwave irradiation, ligands, anticancer drugs, metal, carbon nanostructures, and graphene. She has more than 250 publications in international scientific journals.
Dr.VG is the associate professor of the Department of Experimental Physics of Taras Shevchenko National University of Kyiv. He has a scientific degree of Doctor of Sciences in optics, laser physics. His scientific areas of interest are solid state physics, optical spectroscopy, Raman spectroscopy, theory group analysis, and computational chemistry. He was involved in the study of semiconductor optics and carbon nanostructures (graphene and nanotubes). He has more than 250 publications in scientific journals.
TL is the junior research scientist at the Department of Physics of Biological Systems of the Institute of Physics of the National Academy of Sciences of Ukraine. Her scientific areas of interest are biophysics, computational chemistry, and graphene.
DP, Ph.D., student at the Department of Physics of Biological Systems, Institute of Physics, National Academy of Sciences of Ukraine. His scientific areas of interest are biophysics, experimental physics, electronics, UV, FTIR, Raman, CARS spectroscopy, and TEM microscopy.
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Lebedieva, T., Gubanov, V., Dovbeshko, G. et al. QuantumChemical Calculation and Visualization of the Vibrational Modes of Graphene in Different Points of the Brillouin Zone. Nanoscale Res Lett 10, 287 (2015). https://doi.org/10.1186/s1167101509459
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DOI: https://doi.org/10.1186/s1167101509459
Keywords
 Quantumchemical calculation
 Brillouin zone
 CARS
 Discrete breather
 Graphene
PACS
 33.15.Bh
 33.20.Tp
 63.22.Rc