Electron in Bilayer Graphene with
Magnetic Fields Associated with Solvable
Potentials
Daniel O. Campa, Juan D. García, and David J. Fernández
Abstract The behavior of an electron in bilayer graphene with applied magnetic
fields orthogonal to the surface is studied. By using second-order supersymmetric
quantum mechanics some analytic solutions are found, which are expressed in
terms of the eigenfunctions of two intertwined Schrödinger Hamiltonians. The
case of a constant homogeneous magnetic field which leads to a pair of shifted
harmonic oscillator potentials is discussed. A variant of this example produces
inhomogeneous magnetic fields whose exact solutions for the electron motion in
bilayer graphene are straightforwardly identified.
Keywords Bilayer graphene · SUSY QM · Solvable potentials
1 Introduction
Graphene is one of the so-called Dirac materials in the literature, which nowadays
are of great interest in physics. In particular, the analysis of its electronic properties
has become an extensive field of study. Here, we will focus in the electron motion
in bilayer graphene. The effective Hamiltonian describing the electron behavior in
either monolayer or bilayer graphene is well known [1–3]. For low energy electrons
in the monolayer a tight-binding model leads to a massless Dirac equation, where
the Fermi velocity replaces the speed of light. Nevertheless, to find exact analytic
solutions to this equation is not always simple. The first-order supersymmetric
quantum mechanics (SUSY QM) has proven useful for addressing this task [4–6].
However, for bilayer graphene a similar model does not end up with a massless
Dirac equation, but with slightly more complicated equations involving secondorder derivatives. This suggests to use now the second-order SUSY QM, where
the intertwining operators are of second order. Thus, in this paper we will use
D. O. Campa () · J. D. García · D. J. Fernández
Physics Department, Cinvestav, Mexico City, Mexico
e-mail: dortiz@fis.cinvestav.mx; dgarcia@fis.cinvestav.mx; david@fis.cinvestav.mx
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_25
265
Magnetic Fields Associated with Solvable
Potentials
Daniel O. Campa, Juan D. García, and David J. Fernández
Abstract The behavior of an electron in bilayer graphene with applied magnetic
fields orthogonal to the surface is studied. By using second-order supersymmetric
quantum mechanics some analytic solutions are found, which are expressed in
terms of the eigenfunctions of two intertwined Schrödinger Hamiltonians. The
case of a constant homogeneous magnetic field which leads to a pair of shifted
harmonic oscillator potentials is discussed. A variant of this example produces
inhomogeneous magnetic fields whose exact solutions for the electron motion in
bilayer graphene are straightforwardly identified.
Keywords Bilayer graphene · SUSY QM · Solvable potentials
1 Introduction
Graphene is one of the so-called Dirac materials in the literature, which nowadays
are of great interest in physics. In particular, the analysis of its electronic properties
has become an extensive field of study. Here, we will focus in the electron motion
in bilayer graphene. The effective Hamiltonian describing the electron behavior in
either monolayer or bilayer graphene is well known [1–3]. For low energy electrons
in the monolayer a tight-binding model leads to a massless Dirac equation, where
the Fermi velocity replaces the speed of light. Nevertheless, to find exact analytic
solutions to this equation is not always simple. The first-order supersymmetric
quantum mechanics (SUSY QM) has proven useful for addressing this task [4–6].
However, for bilayer graphene a similar model does not end up with a massless
Dirac equation, but with slightly more complicated equations involving secondorder derivatives. This suggests to use now the second-order SUSY QM, where
the intertwining operators are of second order. Thus, in this paper we will use
D. O. Campa () · J. D. García · D. J. Fernández
Physics Department, Cinvestav, Mexico City, Mexico
e-mail: dortiz@fis.cinvestav.mx; dgarcia@fis.cinvestav.mx; david@fis.cinvestav.mx
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_25
265