Efficient semiclassical approximation for bound states in graphene in magnetic field with a small trigonal warping correction

Abstract

This paper is devoted to the construction of semiclassical spectrum and efficient (simple to implement) explicit semiclassical asymptotic eigenfunctions of the Dirac operator for relatively high-energy bound states in graphene in magnetic field, considering the effect of trigonal warping [11, 16] to be small. It turns out that the asymptotic spectrum of the operator remains unchanged under such a perturbation due to the symmetry of the problem rather than the smallness of this correction. However, the behavior of asymptotic eigenfunctions is quite different; they are significantly affected by trigonal warping that leads to the breaking of certain symmetries. Density plots of asymptotic eigenfunctions can indicate what might be observed using a scanning tunneling microscope. Our approach to constructing asymptotic solutions is based on developments of works [9, 1, 5], which present a new method for constructing the solution, simplifying practical application.