Electronic states in a graphene flake strained by a Gaussian bump

The effect of strain in graphene is usually modeled by a pseudo-magnetic vector potential which is, however, derived in the limit of small strain. In realistic cases deviations are expected in view of graphene's very high strain tolerance, which can be up to 25%. Here we investigate the pseudo-magnetic field generated by a Gaussian bump and we show that it exhibits significant differences with numerical tight-binding results. Furthermore, we calculate the electronic states in the strained region for a hexagon shaped flake with armchair edges. We find that the six-fold symmetry of the wave functions inside the Gaussian bump is directly related to the different effect of strain along the fundamental directions of graphene: zigzag and armchair. Low energy electrons are strongly confined in the armchair directions and are localized on the carbon atoms of a single sublattice

Graphene Hall bar with an asymmetric pn-junction

We investigated the magnetic field dependence of the Hall and the bend resistances in the ballistic regime for a single layer graphene Hall bar structure containing a pn-junction. When both regions are n-type the Hall resistance dominates and Hall type of plateaus are formed. These plateaus occur as a consequence of the restriction on the angle imposed by Snell's law allowing only electrons with a certain initial angles to transmit though the potential step. The size of the plateau and its position is determined by the position of the potential interface as well as the value of the applied potential. When the second region is p-type the bend resistance dominates which is asymmetric in field due to the presence of snake states. Changing the position of the pn-interface in the Hall bar strongly affects these states and therefore the bend resistance is also changed. Changing the applied potential we observe that the bend resistance exhibits a peak around the charge-neutrality point (CNP) which is independent of the position of the pn-interface, while the Hall resistance shows a sign reversal when the CNP is crossed, which is in very good agreement with a recent experiment [J. R. Williams et al., Phys. Rev. Lett. 107, 046602(2011)]

Bilayer graphene Hall bar with a pn-junction

We investigate the magnetic field dependence of the Hall and the bend resistances for a ballistic Hall bar structure containing a pn-junction sculptured from a bilayer of graphene. The electric response is obtained using the billiard model and we investigate the cases of bilayer graphene with and without a band gap. Two different conduction regimes are possible: $i$) both sides of the junction have the same carrier type, and $ii$) one side of the junction is n-type while the other one is p-type. The first case shows Hall plateau-like features in the Hall resistance that fade away as the band gap opens. The second case exhibits a bend resistance that is asymmetric in magnetic field as a consequence of snake states along the pn-interface, where the maximum is shifted away from zero magnetic field

Magnetic field dependence of the atomic collapse state in graphene

Quantum electrodynamics predicts that heavy atoms ($Z > Z_c \approx 170$) will undergo the process of atomic collapse where electrons sink into the positron continuum and a new family of so-called collapsing states emerges. The relativistic electrons in graphene exhibit the same physics but at a much lower critical charge ($Z_c \approx 1$) which has made it possible to confirm this phenomenon experimentally. However, there exist conflicting predictions on the effect of a magnetic field on atomic collapse. These theoretical predictions are based on the continuum Dirac-Weyl equation, which does not have an exact analytical solution for the interplay of a supercritical Coulomb potential and the magnetic field. Approximative solutions have been proposed, but because the two effects compete on similar energy scales, the theoretical treatment varies depending on the regime which is being considered. These limitations are overcome here by starting from a tight-binding approach and computing exact numerical results. By avoiding special limit cases, we found a smooth evolution between the different regimes. We predict that the atomic collapse effect persists even after the magnetic field is activated and that the critical charge remains unchanged. We show that the atomic collapse regime is characterized: 1) by a series of Landau level anticrossings and 2) by the absence of $\sqrt{B}$ scaling of the Landau levels with regard to magnetic field strength

Spectroscopy of snake states using a graphene Hall bar

An approach to observe snake states in a graphene Hall bar containing a pn-junction is proposed. The magnetic field dependence of the bend resistance in a ballistic graphene Hall bar structure containing a tilted pn-junction oscillates as a function of applied magnetic field. We show that each oscillation is due to a specific snake state that moves along the pn-interface. Furthermore depending on the value of the magnetic field and applied potential we can control the lead in which the electrons will end up and hence control the response of the system

Quasi-bound states of Schrodinger and Dirac electrons in magnetic quantum dot

The properties of a two-dimensional electron are investigated in the presence of a circular step magnetic field profile. Both electrons with parabolic dispersion as well as Dirac electrons with linear dispersion are studied. We found that in such a magnetic quantum dot no electrons can be confined. Nevertheless close to the Landau levels quasi-bound states can exist with a rather long life time.Comment: 9 pages, 10 figure

Pseudo magnetic field in strained graphene: revisited

We revisit the theory of the pseudo magnetic field as induced by strain in graphene using the tight-binding approach. A systematic expansion of the hopping parameter and the deformation of the lattice vectors is presented from which we obtain an expression for the pseudo magnetic field for low energy electrons. We generalize and discuss previous results and propose a novel effective Hamiltonian. The contributions of the different terms to the pseudo magnetic field expression is investigated for a model triaxial strain profile and are compared with the full solution. Our work suggests that the previous proposed pseudo magnetic field expression is valid up to reasonably high strain 15 % and there is no K-dependent pseudo-magnetic field.Comment: 11 pages, 4 figure

Scattering of Dirac electrons by circular mass barriers: valley filter and resonant scattering

The scattering of two-dimensional (2D) massless Dirac electrons is investigated in the presence of a random array of circular mass barriers. The inverse momentum relaxation time and the Hall factor are calculated and used to obtain parallel and perpendicular resistivity components within linear transport theory. We found a non zero perpendicular resistivity component which has opposite sign for electrons in the different K and K' valleys. This property can be used for valley filter purposes. The total cross-section for scattering on penetrable barriers exhibit resonances due to the presence of quasi-bound states in the barriers that show up as sharp gaps in the cross-section while for Schr\"{o}dinger electrons they appear as peaks.Comment: 10 pages, 11 figure

Resonant valley filtering of massive Dirac electrons

Electrons in graphene, in addition to their spin, have two pseudospin degrees of freedom: sublattice and valley pseudospin. Valleytronics uses the valley degree of freedom as a carrier of information similar to the way spintronics uses electron spin. We show how a double barrier structure consisting of electric and vector potentials can be used to filter massive Dirac electrons based on their valley index. We study the resonant transmission through a finite number of barriers and we obtain the energy spectrum of a superlattice consisting of electric and vector potentials. When a mass term is included the energy bands and energy gaps at the K and K' points are different and they can be tuned by changing the potential.Comment: 20 figure

Magnetic Kronig-Penney model for Dirac electrons in single-layer graphene

The properties of Dirac electrons in a magnetic superlattice (SL) on graphene consisting of very high and thin (delta-function) barriers are investigated. We obtain the energy spectrum analytically and study the transmission through a finite number of barriers. The results are contrasted with those for electrons described by the Schrodinger equation. In addition, a collimation of an incident beam of electrons is obtained along the direction perpendicular to that of the SL. We also highlight the analogy with optical media in which the refractive index varies in space.Comment: 21 pages, 13 figures, to appear in New Journal of Physic