Abstract
Some characteristic features of band structures, like the band degeneracy at high symmetry points or the existence of energy gaps, usually reflect the symmetry of the crystal or, more precisely, the symmetry of the wave vector group at the relevant points of the Brillouin zone. In this paper, we will illustrate this property by considering two-dimensional (2D)-hexagonal lattices characterized by a possible two-fold degenerate band at the K points with a linear dispersion (Dirac points). By combining scanning tunneling spectroscopy and angle-resolved photoemission, we study the electronic properties of a similar system: the Ag/Cu(111) interface reconstruction characterized by a hexagonal superlattice, and we show that the gap opening at the K points of the Brillouin zone of the reconstructed cell is due to the symmetry breaking of the wave vector group.
Highlights
Symmetry is probably one of the most general and fundamental concepts in physics, and its central role was only recognized in the 20th century
Band gap opening reflects the symmetry of wave vector groups, and we show that the gap magnitudes can be used to obtain the surface potential or at least their first Fourier components
After a general discussion of the symmetry on the hexagonal 2D lattices and, in particular, the relation between the gap at the K point and the group of the wave vector, we present angle-resolved photoemission spectroscopy (ARPES) and scanning tunneling microscopy and spectroscopy (STM/STS)
Summary
Symmetry is probably one of the most general and fundamental concepts in physics, and its central role was only recognized in the 20th century. When this band crossing occurs at the Fermi energy, the low energy excitations can be described by an effective 2D Dirac equation of massless fermions, like in graphene [4] This singular behavior is directly related to the symmetry of the wave vector group at the K points and to the dimension of the corresponding irreducible representation [5]. A symmetry breaking with a change of the wave vector group can lead to non-crossing bands and, to gap opening and curvature of the band dispersion This was observed, for example, by growing a graphene layer on a crystal surface [6]. In detail, this system, especially the relation between symmetry and the surface potential
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