Advances in Condensed Matter Physics

Volume 2015 (2015), Article ID 979528, 6 pages

http://dx.doi.org/10.1155/2015/979528

## Weakly Bound States of Elementary Excitations in Graphene Superlattice in Quantizing Magnetic Field

^{1}Physical Laboratory of Low-Dimensional Systems, Volgograd State Socio-Pedagogical University, V.I. Lenin Avenue, No. 27, Volgograd 400066, Russia^{2}Department of Physics, Volgograd State Technical University, V.I. Lenin Avenue, No. 28, Volgograd 400005, Russia

Received 21 October 2014; Revised 12 June 2015; Accepted 28 June 2015

Academic Editor: Sergei Sergeenkov

Copyright © 2015 Sergei V. Kryuchkov and Egor I. Kukhar’. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The spectrum of allowed energy of electron in graphene superlattice in the quantizing magnetic field is investigated. Such spectrum consists of number of so-called magnetic minibands. The width of these minibands depends on the superlattice barriers power and on the magnetic field intensity. The explicit form of electron spectrum is derived in the case of weak magnetic field. The possibility of electron-electron and electron-phonon bound states is shown. The binding energies of these states are calculated. The binding energy is shown to be the function of magnetic field intensity.

#### 1. Introduction

The interactions of electrons with elementary excitations in crystal lattice have fundamental implications on properties of materials and lead to such many-body phenomena as superconductivity and charge-density waves. Such interactions take an unusual form in graphene [1–3] which can be described by the Dirac-like form of the effective Hamiltonian. A new area for investigations of chiral massless fermions is induced by discovery of this 2D material. So, interactions of Dirac electrons with elementary excitations which lead to the emergence of the bound states (BS) are under the intensive theoretical [4–7] and experimental study [8–10] last time.

Renormalization of Dirac spectrum due to interactions of electrons with lattice vibrations was obtained in [11–15]. Formations of polarons, plasmon-phonon complexes, and electron-hole pares are investigated in [4, 16–20], where BSs were shown to appear if energy of quasiparticles interaction exceeds a threshold value. In [19], the modification of Dirac spectrum near the threshold of optical phonon emission was studied. The spectrum characteristics of the electron-phonon quasiparticles (in particular, the electron-phonon binding energy versus the coupling parameter ) were found in [19] within the theory, taking into account the singular vertex corrections beyond perturbation theory. Such corrections lead to the spectrum that remains below the optical phonon energy and corresponds to the electron-phonon BS in comparison with that obtained within the Wigner-Brillouin perturbation theory [11–13].

Influence of various factors (geometry parameters, spin-orbit interaction, charge impurities, etc.) on BS properties in graphene-based structures was studied in [4, 5, 21–27]. From an application technological point of view, the tunability of graphene electronic and optical properties by external fields is of particular importance [21, 28–32]. So, the effect of electric and magnetic fields on the properties of the BSs in graphene structures is of high interest among the researchers now [20, 21, 33–35]. The influence of Landau level mixing on the energy of magnetoexcitons and magnetoplasmons in graphene was studied in [36], where dispersions of the excitations were shown to be changed by virtual transitions between Landau levels, caused by Coulomb interaction (electron-hole channel). In [20], the effect of the magnetic field on the electron-phonon BSs was studied. The electron-phonon binding energy was shown to diverge at the resonant values of the magnetic field intensity. Such divergences corresponded to the electron-phonon hybrid states formed in the spectrum between the graphene Landau levels. These resonance states were obtained within the perturbation theory from the poles of the two-particle Green function and determine the structure of the magnetophonon resonance [37–39].

Presently, among the different graphene structures, the special attention is paid to graphene superlattices (GSLs) [40–44]. Intensive investigations of electric and optical properties of GSL are explained by their different possible experimental and technological applications [31, 45–49]. Below we obtain the binding energies of electron-electron and electron-phonon coupling in GSL in the quantizing magnetic field and determine the binding energy dependence on the magnetic field intensity and on the coupling parameters and .

#### 2. Electron Spectrum of GSL in Quantizing Magnetic Field

In this section, we obtain the effective electron spectrum of GSL in quantizing magnetic field. We consider a GSL which is in the plane (Figure 1). Near the Dirac point, the electron spectrum of GSL has the explicit form [40]:where is the GSL axis (Figure 1), is the Fermi velocity along the axis , is the dimensionless power of SL barriers [40], is the barrier height, cm is the GSL period, , and cm/s is the Fermi velocity in graphene. Linearized equation (equation analogous to the Dirac equation), which describes the motion of an electron in the GSL, takes the form [50, 51]:Here, are the Pauli matrixes, is spinor function describing the electron states in GSL, , , is the vector potential, and is the intensity of magnetic field which is supposed to be perpendicular to the GSL plane. Indeed, in the absence of magnetic field, the substitution of spinor for free charge carrier [52] into (2) gives the dispersion law (1).