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In this paper we study scattering of two-dimensional massless Dirac fermions by a potential that depends on a single Cartesian variable. Depending on the energy of the incoming particle and its angle of incidence, there are three different regimes of scattering. To find the reflection and transmission coefficients in these regimes, we apply the Wentzel-Kramers-Brillouin (WKB), also called semiclassical, approximation. We use the method of comparison equations to extend our prediction to nearly normal incidence, where the conventional WKB method should be modified due to the degeneracy of turning points. We compare our results to numerical calculations and find good agreement.Keywords: Massless Dirac fermions, Semiclassical approximation, Scattering, Graphene, Topological insulators PACS: 81. 05.ue, 03.65.Vf, 03.65.Sq, 03.65.Nk, 73.40.Gk In this paper we present a systematic theory of potential scattering for massless Dirac fermions. Being the effective charge carriers in graphene [1,2,3,4], and topological insulators [5,6,7], these particles attracted a keen interest. The discovery of massless Dirac fermions in condensed matter systems stimulated the fabrication of 'artificial graphene', a material with a hexagonal lattice, where quantum dots [8], or molecules [9], play the role of carbon atoms. The electron excitations in these materials give rise to massless Dirac fermions. The main feature of massless Dirac fermions is chirality (as it is called for graphene) or helicity (for topological insulators), i.e. an additional degree of freedom that relates to two kinds of particles (electrons and holes) simultaneously present in the system. Chirality makes the behavior of massless Dirac fermions dramatically different from that of Schrödinger particles. One of the most prominent examples is Klein tunneling [2,10,11,12,13,14,15,16]. Due to this effect, a massless Dirac fermion normally incident on an electrostatic potential will be transmitted with unit probability.In this paper we consider scattering of massless Dirac fermions by quasi- one dimensional potential barriers (the corresponding potentials depend on a single Cartesian variable, see figure 1). Such barriers occur for instance in graphene heterostructures that were fabricated in [15,16]. They can also be intrinsic, as in the case of puddles in graphene [17,10]. We always assume that the potential profile is smooth enough, so that the Wentzel-Kramers-Brillouin (WKB) or semiclassical approximation [18,19,20,21,22,23] can be used. The latter allows us to obtain generic formulas valid for arbitrary potentials. Another method that can be used to study generic potentials numerically was suggested in [24]. We distinguish three different regimes of scattering and show that the massless Dirac equation is equivalent to a pair of effective Schrödinger equations with complex potentials. We then solve the scattering problem for each of these three regimes with the help of the WKB approximation. The specific formulation we use is the one pioneered by Zwaan [25...

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.

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We study the effect of trigonal warping on the focussing of electrons by n-p junctions in graphene. We find that perfect focussing, which was predicted for massless Dirac fermions, is only preserved for one specific sample orientation. In the general case, trigonal warping leads to the formation of cusp caustics, with a different position of the focus for graphene's two valleys. We develop a semiclassical theory to compute these positions and find very good agreement with tight-binding simulations. Considering the transmission as a function of potential strength, we find that trigonal warping splits the single Dirac peak into two distinct peaks, leading to valley polarization. We obtain the transmission curves from tight-binding simulations and find that they are in very good agreement with the results of a billiard model that incorporates trigonal warping. Furthermore, the positions of the transmission maxima and the scaling of the peak width are accurately predicted by our semiclassical theory. Our semiclassical analysis can easily be carried over to other Dirac materials, which generally have different Fermi surface distortions.Veselago lenses [1] are special types of lenses, which are made of materials with a negative refractive index. Such lenses can overcome the diffraction limit [2] and can nowadays be realized in metamaterials [3][4][5], chiral metamaterials [6-9] and photonic crystals [10,11]. An electronic analog of a Veselago lens can be created using n-p junctions, with the classical trajectories of the charge carriers playing the role of the rays in geometrical optics. Such junctions focus electrons, because the group velocity for holes is in the direction opposite to their phase velocity, whereas the two velocities are in the same direction for electrons. However, in conventional semiconductors, such interfaces are unsuitable because of their high reflectivity, owing to the presence of a depletion region.Cheianov et al. [12] realized that graphene does not have this drawback. It has zero bandgap, as the valence and conduction bands touch at two non-equivalent corners of the Brillouin zone, known as K and K . The lowenergy charge carriers are ballistic over large distances and follow the Dirac equation [13][14][15][16][17][18]. 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]. According to catastrophe theory [34][35][36][37], such a situation is exceptional: any perturbation of the Hamiltonian will ruin this ideal focus. Indeed, in tight-binding simulations of an n-p junction...