Surface states on Weyl metals

Abstract

In this thesis we investigate the effect of a magnetic field on the surface states of a slab of Weyl semimetal surrounded by a vacuum. We consider a time reversal symmetry broken semimetal by splitting the Weyl cones up in momentum space. We warm up by calculating the Landau level states in the bulk for a magnetic field parallel and perpendicular to the direction in which the cones are split in momentum space. Next, we find the wavefunction in a half-infinite semimetal and match to the vacuum solutions. For zero magnetic field, we find gapless, chiral surface states with dispersion $E=-v_F k_y$, existing on a Fermi arc. For a magnetic field parallel to the surface, we find solutions using a WKB approximation. For magnetic fields perpendicular to the surface, we find no surface state solution in a half-infinite semimetal. Since Weyl metals have large anomalous magnetic moments \cite{largeanomalousmoment}, the surface states are altered by anomalous magnetic effects. We investigate the effect of the new anomalous terms on the surface states in a parallel magnetic field, and find that in some cases the surface states are altered. Since we found no surface states for a perpendicular magnetic field in the half-infinite Weyl metal, we switch to a finite slab of Weyl metal, and find that surface states are possible. Focussing on the high magnetic field limit, where only the lowest Landau level contributes, we find analytical solutions. Moreover, we find that closed magnetic orbits are possible due to the chiral lowest Landau level acting as a one-way ``conveyor belt", carrying particles from one surface to the other. These orbits have discrete energy levels and depend on the length of the material and the Fermi velocity. We look at the influence of anomalous effects on these orbits and find that the Fermi velocity is rescaled and the energy is Zeeman shifted. Finally, using numerics, we find the first-order correction in $B$ on the energy by adding one more Landau level.