Bilayer graphene coherent states
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Bilayer graphene coherent states David J. Fernándeza , Dennis I. Martínez-Morenob Physics Department, Cinvestav, P.O. Box 14-740, 07000 Mexico City, Mexico Received: 15 July 2020 / Accepted: 2 September 2020 © Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this paper, we consider the interaction of electrons in bilayer graphene with a constant homogeneous magnetic field which is orthogonal to the bilayer surface. Departing from the energy eigenstates of the effective Hamiltonian, the corresponding coherent states will be constructed. For doing this, first we will determine appropriate creation and annihilation operators in order to subsequently derive the coherent states as eigenstates of the annihilation operator with complex eigenvalue. Then, we will calculate some physical quantities, as the Heisenberg uncertainty relation, the probabilities and current density as well as the mean energy value. Finally, we will explore the time evolution for these states and we will compare it with the corresponding evolution for monolayer graphene coherent states.
1 Introduction Carbon is the basis of all organic chemistry; due to the flexibility of their bonds, carbon-based systems show different structures with a wide variety of physical properties [1]. Graphene consists of a monolayer of carbon atoms arranged in a hexagonal crystal lattice with a distance of 1.42 Å between nearest neighbor atoms. The discovery of this material and the interesting electronic properties induced by the low-energy excitations have attracted attention of the scientific community. In particular, the possibility arises that certain aspects of relativistic quantum mechanics can be tested, like the Klein paradox or the anomalous Landau–Hall effect [1–5]. The electronic structure of graphene is typically studied using the tight-binding model. Assuming that only next nearest neighbor hopping processes in the low-energy limit take place (close to the Dirac points), the equation ruling the electrons in graphene with applied static magnetic fields is the Dirac–Weyl equation [1,3,6]. This equation can be solved for a constant homogeneous magnetic field, leading to the Landau levels of grapehene and the corresponding eigenfunctions, which constitutes the standard quantum mechanical framework [3].1 However, an alternative approach exists, in which the system is addressed through
Let us note that the energy eigenfunctions and eigenvalues for graphene and some of its allotropes have been as well determined for time-independent magnetic fields which are not necessarily homogeneous [3,6–14] (see also [15]). a e-mail: [email protected] b e-mail: [email protected] (corresponding author)
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coherent states. This approach supplies information which is supplementary to that obtained by the standard method, and it has been recently implemented for monolayer graphene [16]. Bilayer graphene in a static magnetic field c
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