Symmetry breaking and (pseudo)spin polarization in Veselago lenses for massless Dirac fermions

Abstract

We study Veselago lensing of massless Dirac fermions by n-p junctions for electron sources with a certain polarization. This polarization corresponds to pseudospin for graphene and to real spin for topological insulators. Both for a point source and for injection into a sample through a narrow lead, we find that polarization leads to spatial symmetry breaking. For the Green's function, this results in a vertical displacement, or even complete vanishing of the main focus, depending on the exact polarization. For injection through a lead, it leads to a difference between the amounts of current emitted with positive and negative transversal momenta. We study both systems in detail using the semiclassical approximation. By comparing the results to the exact solutions, we establish that semiclassical methods provide a very effective way to study these systems. For the Green's function, we derive an easy-to-use analytical formula for the vertical displacement of the main focus. For current injection through a lead, we use semiclassical methods to identify two different scattering regimes.

Diffraction catastrophes and semiclassical quantum mechanics for Veselago lensing in graphene

“…Figure 4(c) shows that the amount of symmetry breaking strongly depends on L 2 , as expected for a tilted focus. In agreement with predictions made for the Green's function [29], it increases with increasing mass, decreases with increasing energy, and changes sign when we change the sign of the mass. Simulating transmission as a function of potential strength, initial polarization seems to slightly decrease the valley polarization.…”

“…Tuning the potential, we obtain sections of the butterfly caustic, see Fig. 1(e), and eventually pass through the butterfly singularity, at which ∂ 4 S np /∂p 4 y = 0, but ∂ 6 S np /∂p 6 y = 0 [29,35]. For generic θ, Fig.…”

“…This gives rise to Klein tunneling: normally incident electrons are transmitted with unit probability [19][20][21][22][23][24][25][26][27], which makes the interface exceptionally transparent. Recently, Veselago lensing in graphene was experimentally observed [27,28].Theoretical studies have used the Dirac equation to investigate focussing by flat [12,29] and circular [30][31][32] junctions and in zigzag nanoribbons [33]. It was shown that when the electron and hole charge carrier densities are equal, a flat interface is able to focus all trajectories into a single point [12].…”

“…(1), is symmetric in p x and p y . When U 0 = 2E, this leads to an ideal focus [12], at which all derivatives of S np with respect to p y vanish [29]. This generally changes when we include trigonal warping.…”

“…We obtain an expression for x cusp,α by solving ∂ 2 S np /∂p 2 y = 0 for x, and setting p y = 0 [29]. Expanding the result up to second order in αµ, we find, in units with dimensions…”