Abstract

The dynamics of low-energy charge carriers in graphene are governed by the Dirac equation. This leads to peculiar electronic effects, such as Klein tunneling. Furthermore, the charge carriers can acquire a nonzero Berry phase, which also affects their classical equations of motion. In this talk, I will explore how the matrix character of the Dirac equation affects focusing of electrons in graphene.
First, I will review focusing of electrons by n-p junctions, which is analogous to a Veselago lens [1]. I will discuss how an initial sublattice polarization leads to a sideward displacement of the central focusing spot [2]. We obtain a simple analytical formula for this vertical shift by studying the wavefunction using the semiclassical approximation.
Second, I will discuss how electrons can be focused by above-barrier scattering by a smooth electrostatic potential combined with a coordinate-dependent mass. We gain more insight into this phenomenon by reducing the initial matrix equation to a pair of effective scalar equations. This provides an alternative viewpoint on the origin of the semiclassical phase and the modified equations of motion [3]. Subsequently, we use the formalism of the Maslov canonical operator to obtain an integral expression for the wavefunction. Finally, I will show how we can evaluate this expression in the semiclassical limit and I will compare the outcome with numerical results, placing particular emphasis on the role of the semiclassical phase [4].

[1] V. V. Cheianov, V. Fal'ko, B. L. Altshuler, Science 315, 1252 (2007)
[2] K. J. A. Reijnders, M. I. Katsnelson, Phys. Rev. B. 95, 115310 (2017)
[3] R. G. Littlejohn, W. G. Flynn, Phys. Rev. A 44, 5239 (1991)
[4] K. J. A. Reijnders, D. S. Minenkov, M. I. Katsnelson, S. Yu. Dobrokhotov, Ann. Phys. 397, 65 (2018)