Topological Berry phase and semiclassical quantization of cyclotron orbits for two dimensional electrons in coupled band models

• Published in: The European physical journal. B, Condensed matter physics Condensed matter physics, 2010-10, Vol.77 (3), p.351-362
• Main Authors: Fuchs, J. N., Piéchon, F., Goerbig, M. O., Montambaux, G.
• Format: Article
• Language: English
• Subjects: Analysis; Boron nitride; Complex Systems; Condensed Matter Physics; Condensed matter: electronic structure, electrical, magnetic, and optical properties; Electron states; Electrons; Exact sciences and technology; Fluid- and Aerodynamics; Graphene; Particle accelerators; Physics; Physics and Astronomy; Solid State and Materials; Solid State Physics; Theories and models of many electron systems
• ISSN: 1434-6028; 1434-6036
• DOI: 10.1140/epjb/e2010-00259-2

. The semiclassical quantization of cyclotron orbits for two-dimensional Bloch electrons in a coupled two band model with a particle-hole symmetric spectrum is considered. As concrete examples, we study graphene (both mono and bilayer) and boron nitride. The main focus is on wave effects – such as Berry phase and Maslov index – occurring at order in the semiclassical quantization and producing non-trivial shifts in the resulting Landau levels. Specifically, we show that the index shift appearing in the Landau levels is related to a topological part of the Berry phase – which is basically a winding number of the direction of the pseudo-spin 1/2 associated to the coupled bands – acquired by an electron during a cyclotron orbit and not to the complete Berry phase, as commonly stated. As a consequence, the Landau levels of a coupled band insulator are shifted as compared to a usual band insulator. We also study in detail the Berry curvature in the whole Brillouin zone on a specific example (boron nitride) and show that its computation requires care in defining the “ k -dependent Hamiltonian” H ( k ), where k is the Bloch wavevector.