Veselago lensing in graphene with a p-n junction: classical versus quantum effects

The feasibility of Veselago lensing in graphene with a p-n junction is investigated numerically for realistic injection leads. Two different set-ups with two narrow leads are considered with absorbing or reflecting side edges. This allows us to separately determine the influence of scattering on electron focusing for the edges and the p-n interface. Both semiclassical and tight-binding simulations show a distinctive peak in the transmission probability that is attributed to the Veselago lensing effect. We investigate the robustness of this peak on the width of the injector, the position of the p-n interface and different gate potential profiles. Furthermore, the influence of scattering by both short- and long-range impurities is considered.Comment: 10 pages, 7 figure

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

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

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)]

Special Classes of Orthogonal Polynomials and Corresponding Quadratures of Gaussian Type

MSC 2010: 33C47, 42C05, 41A55, 65D30, 65D32In the first part of this survey paper we present a short account on some important properties of orthogonal polynomials on the real line, including computational methods for constructing coefficients in the fundamental three-term recurrence relation for orthogonal polynomials, and mention some basic facts on Gaussian quadrature rules. In the second part we discuss our Mathematica package Orthogonal Polynomials (see [2]) and show some applications to problems with strong nonclassical weights on (0;+1), including a conjecture for an oscillatory weight on [¡1; 1]. Finally, we give some new results on orthogonal polynomials on radial rays in the complex plane

GARNETS OF MICA-SCHISTS FROM CRNI VRH AND RESAVSKI HUMOVI (SERBIA)

Metamorphic rocks of the Crni Vrh and Resavski Humovi area belong to the north-northwestern part of the Serbo-Macedonian Composite Terrane and consist of different types of mica-schists, gneisses and amphibolites. Mica-schists are composed of biotite, muscovite, garnet, staurolite, kyanite, small quantity of Kfeldspar, acid plagioclase and quartz. Accessory minerals are apatite, zircon, tourmaline and ilmenite. Garnets in mica-schists from both localities occur as subhedral to anhedral porphyroblasts from 0.2 to 10mm in size. Most garnets are oriented parallel to the dominant foliation. Garnets from the Crni Vrh mica-schists contain more spesartine component than garnets from Resavski Humovi, while almandine component show opposite tendency. The difference between those garnets is clearly visible, as well as zonality expressed through the increase of almandine component from center to the grain rim. The spesartine component as mentioned above, shows the opposite tendency. Using different geothermometers and geobarometers it was found that mica-schists from Crni Vrh were formed at temperatures from 550-600°C and pressures from 6-6.2 kbar, while micaschists from Resavski Humovi developed at slightly lower temperatures (520-560°C) and at some higher pressures (7-7.3 kbar)