Analytical and Numerical Study of the Energy Spectrum of a Superlattice Consisting of Strips of Single-layer and Bilayer Graphene

Authors V.L. Abdrakhmanov1, P.V. Badikova1, D.V. Zav’yalov1 , V.I. Konchenkov1 , S.V. Kryuchkov1,2

1Volgograd State Technical University, 28, Lenin Ave., 400005 Volgograd, Russia

2Volgograd State Socio-Pedagogical University, 27, Lenin Ave., 400066 Volgograd, Russia

Issue Volume 12, Year 2020, Number 6
Dates Received 12 June 2020; revised manuscript received 18 December 2020; published online 25 December 2020
Citation V.L. Abdrakhmanov, P.V. Badikova, D.V. Zav’yalov, et al., J. Nano- Electron. Phys. 12 No 6, 06029 (2020)
PACS Number(s) 72.80.Vp, 73.50. – h
Keywords Bilayer graphene, Kronig-Penney model, Transfer matrix method, Methods of the density functional theory.

A model of a superlattice consisting of alternating strips of single-layer and bilayer graphene is proposed, whose parameters of the energy spectrum can be controlled by changing the external electric field perpendicular to the surface of the sample. Using the Kronig-Penney model, the dispersion equation is obtained based on the analysis of which the energy spectrum of a graphene superlattice is studied depending on the ratio of the strip widths of single-layer and bilayer graphene. For the considered superlattice, it is shown that there are two types of dispersion surfaces corresponding to two branches in the spectrum of bilayer graphene. In the absence of a transverse electric field, neighboring minibands obtained from the solution of different types of the dispersion equation touch at the edges of the first Brillouin band, and the conduction band and the valence band touch in the center of the first Brillouin band of the superlattice. The results of the analytical solution are compared with the results of modeling by methods of the density functional theory. It is shown that the low-energy approximation used to derive the dispersion equation is valid when considering a superlattice with narrow strips of bilayer graphene and wide strips of single-layer graphene. Under this condition, the dispersion surfaces are symmetrical with respect to the K-point of the inverse space for the basic material – single-layer graphene. Quantum chemical modeling has shown that the band gap in the superlattice spectrum appears even in the absence of a transverse external field due to a violation of symmetry between states in different layers of bilayer graphene in the superlattice, and has confirmed the dependence of the band gap width on the transverse electric field.

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