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Focusing on equilibrium spin current density tensor, we analyze the spin physics of electrons in single layer graphene subject to a one-dimensional periodic Kronig-Penney potential $u(x)$ and uniform Rashba spin-orbit coupling of strength $\ensuremath{\lambda}$. The combination of Dirac theory of massless two-dimensional electrons, Klein paradox, spin-orbit coupling, and the Bloch theorem yields peculiar features relevant to graphene spintronics. For transverse wave number ${k}_{y}=0$, we show that: (1) The charge current and the spin density vector vanish. (2) Yet, the components ${J}_{xy},{J}_{yx},{J}_{zy}$ of the equilibrium spin current density tensor are finite, oscillating in space, with amplitudes that increase quadratically for small $\ensuremath{\lambda}$ and then linearly with $\ensuremath{\lambda}$. Quite remarkably, ${J}_{zy}(x)\ensuremath{\ne}0$, that is, the spin is polarized along $z$, perpendicular to the graphene plane. (3) Due to the continuity equation, the space dependence of the spin current density ${J}_{yx}(x)$ is associated with a finite spin torque density ${\mathcal{T}}_{y}(x)=\frac{\ensuremath{\partial}{J}_{yx}(x)}{\ensuremath{\partial}x}$.

We study forward scattering of 2D massless Dirac electrons at Fermi energy {\varepsilon} > 0 in single layer graphene through a 1D rectangular barrier of height {u_0} in the presence of uniform Rashba spin-orbit coupling (of strength {\lambda}). The role of the Klein paradox in graphene spintronics is thereby exposed. It is shown that (1) For {\varepsilon} - 2{\lambda} < {u_0}< {\varepsilon} + 2{\lambda} there is partial Klein tunneling, wherein the transmission is bounded by 1 and, quite remarkably, for small {\lambda} > {\lambda_0} {\approx} 0.1 meV, the transmission nearly vanishes when the scattering energy equals the barrier height, {\varepsilon}={u_0}. (2) Spin density and spin-current density are shown to be remarkably different than these observables predicted in bulk single layer graphene. In particular, they are sensitive to {\lambda} and {u_0}. (3) Spin current densities are space dependent, implying the occurrence of non-zero spin torque density. Such a system may serve as a graphene based spintronic device without the use of an external magnetic field or magnetic materials., Comment: seven pages, six figures

We investigate wavepacket dynamics across supercritical barriers for the Klein-Gordon and Dirac equations. Our treatment is based on a multiple scattering expansion (MSE). For spin-0 particles, the MSE diverges, rendering invalid the use of the usual connection formulas for the scattering basis functions. In a time-dependent formulation, the divergent character of the MSE naturally accounts for charge creation at the barrier boundaries. In the Dirac case, the MSE converges and no charge is created. We show that this time-dependent charge behavior dynamics can adequately explain the Klein paradox in a first quantized setting. We further compare our semi-analytical wavepacket approach to exact finite-difference solutions of the relativistic wave equations., Comment: Some text moved from main body to Appendices; close to published version

This paper shows the asymmetrical Klein paradox with a finite incidence angle induced by an external linearly polarized dressing field for graphene and a dice lattice.

The Klein paradox, first introduced in relation to chiral tunneling, is also manifested in the study of bound-states in single-layer graphene with a 1D square-well potential. We derive analytic (and numerical) solutions for bound-state wavefunctions, in the absence and in the presence of an external transverse magnetic field, and calculate the corresponding dipole transition rates, which can be probed by photon absorption experiments. The role of parity and time-reversal symmetries is briefly discussed. Our results are also relevant for the physics of bound states of light in periodic optical waveguide structures., 5.5 pages + 3 pages supplemental material

We calculate tree--level currents of created particles in strong background electric fields in 4D QED for various initial states. Namely, we do that in pulse background for initial vacuum and thermal states at past infinity. In both cases we find that the current grows linearly with the length of the pulse with coefficients of proportionality containing the characteristic Schwinger's factor. For the constant electric field background we calculate the current for several different initial states. We observe that in such a case the current is either zero or linearly divergent. We explain the reason for such a behaviour and compare the situation in ordinary and scalar QED. Finally, we calculate the current in two--dimensional situation in the presence of such settings when the so called Klein paradox can be observed., Comment: 19 pages, minor changes, clarifying comments are added, misprints corrected

We analyse a little known aspect of the Klein paradox. A Klein-Gordon boson appears to be able to cross a supercritical rectangular barrier without being reflected, while spending there a negative amount of time. The transmission mechanism is demonstrably acausal, yet an attempt to construct the corresponding causal solution of the Klein-Gordon equation fails. We relate the causal solution to a divergent multiple-reflections series, and show that the problem is remedied for a smooth barrier, where pair production at the energy equal to a half of the barrier's height is enhanced yet remains finite. Financial support of MCIU, through the Grant PGC2018-101355-B-100(MCIU/AEI/FEDER,UE) (XGdC, MP, DS), of Spanish MINECO, project FIS2016-80681-P (MP), and of the Basque Government Grant no. IT986-16 (MP, DS) is gratefully acknowledged.

Quantum chaos is generally referred to as the study of quantum manifestations or fingerprints of nonlinear dynamical and chaotic behaviors in the corresponding classical system, an interdisciplinary field that has been active for about four decades. In closed quantum systems, for example, the basic phenomena studied include energy level-spacing statistics and quantum scarring. In open Hamiltonian systems, quantum chaotic scattering has been investigated extensively. Previous works were almost exclusively for nonrelativistic quantum systems described by the Schrodinger equation. Recent years have witnessed a rapid growth of interest in Dirac materials such as graphene, topological insulators, molybdenum disulfide and topological Dirac semimetals. A common feature of these materials is that their physics is described by the Dirac equation in relativistic quantum mechanics, generating phenomena that do not usually emerge in conventional semiconductor materials. This has important consequences. In particular, at the level of basic science, a new field has emerged: Relativistic Quantum Chaos (RQC), which aims to uncover, understand, and exploit relativistic quantum manifestations of classical nonlinear dynamical behaviors including chaos. Practically, Dirac materials have the potential to revolutionize solid-state electronic and spintronic devices, and have led to novel device concepts such as valleytronics. Exploiting manifestations of nonlinear dynamics and chaos in the relativistic quantum regime can have significant applications. The aim of this article is to give a comprehensive review of the basic results obtained so far in the emergent field of RQC. Phenomena to be discussed in depth include energy level-spacing statistics in graphene or Dirac fermion systems that exhibit various nonlinear dynamical behaviors in the classical limit, relativistic quantum scars (unusually high concentrations of relativistic quantum spinors about classical periodic orbits), peculiar features of relativistic quantum chaotic scattering and quantum transport, manifestations of the Klein paradox and its effects on graphene or 2D Dirac material based devices, regularization of relativistic quantum tunneling by chaos, superpersistent currents in chaotic Dirac rings subject to a magnetic flux, and exploitation of relativistic quantum whispering gallery modes for applications in quantum information science. Computational methods for solving the Dirac equation in various situations will be introduced and physical theories developed so far in RQC will be described. Potential device applications will be discussed.

We study nonlinear bound states, or solitary waves, in the Dirac-Maxwell system, proving the existence of solutions in which the Dirac wave function is of the form \begin{document}$φ(x,ω)e^{-iω t}$\end{document} , with \begin{document}$ω∈(-m,ω_ *)$\end{document} for some \begin{document}$ω_ *>-m$\end{document} . The solutions satisfy \begin{document}$φ(\,·\,,ω)∈ H^ 1(\mathbb{R}^3,\mathbb{C}^4)$\end{document} , and are small amplitude in the sense that \begin{document}${\left\| {φ(\,·\,,ω)} \right\|}^2_{L^ 2} = O(\sqrt{m+ω})$\end{document} and \begin{document}${\left\| {φ(\,·\,,ω)} \right\|}_{L^∞} = O(m+ω)$\end{document} . The method of proof is an implicit function theorem argument based on the identification of the nonrelativistic limit as the ground state of the Choquard equation. This identification is in some ways unexpected on account of the repulsive nature of the electrostatic interaction between electrons, and arises as a manifestation of certain peculiarities (Klein paradox) which result from attempts to interpret the Dirac equation as a single particle quantum mechanical wave equation.

In this work we consider the interaction of gravity with matter fields with galilean invariance and their corresponding black hole solution. Quasinormal modes of spinorial perturbations were computed using different methods. For massless spinors we show that black holes are dynamically stable even in cases where the potential has a small negative region. This is possible because the ground state always has an energy larger than well’s depth. We also found quasi-resonant modes in certain regime, which we believe correspond to a manifestation of the Klein paradox.

Reflection and transmission of electrons scattered by a rectangular potential step in the presence of an external magnetic field parallel to the electron beam is described with the use of the Dirac equation. It is shown that in addition to the known effects present in the so called Klein paradox, the presence of magnetic field gives rise to electron components with reversed spin in the reflected and transmitted beams. The spin-flip scattering processes are caused by the spin-orbit interaction activated by electric field of the potential step and transverse momentum components of electron motion induced by the magnetic field. The contemporary understanding of the Klein paradox, consisting in the finite transmission even when the potential height tends to infinity, is generalized to the presence of magnetic field and spin-reversed electron beams. The spin-reversed beams are shown to occur also for electrons moving above the step. It is proposed that, accounting for the anomalous value of the electron spin $g$-factor related to radiation corrections, the reflection and transmission scattering from the potential step can be used as an electron spin filter., 7 pages and 1 figure

In 1928, Dirac proposed a wave equation to describe relativistic electrons1. Shortly afterwards, Klein solved a simple potential step problem for the Dirac equation and encountered an apparent paradox: the potential barrier becomes transparent when its height is larger than the electron energy. For massless particles, backscattering is completely forbidden in Klein tunnelling, leading to perfect transmission through any potential barrier2,3. The recent advent of condensed-matter systems with Dirac-like excitations, such as graphene and topological insulators, has opened up the possibility of observing Klein tunnelling experimentally4–6. In the surface states of topological insulators, fermions are bound by spin–momentum locking and are thus immune from backscattering, which is prohibited by time-reversal symmetry. Here we report the observation of perfect Andreev reflection in point-contact spectroscopy—a clear signature of Klein tunnelling and a manifestation of the underlying ‘relativistic’ physics of a proximity-induced superconducting state in a topological Kondo insulator. Our findings shed light on a previously overlooked aspect of topological superconductivity and can serve as the basis for a unique family of spintronic and superconducting devices, the interface transport phenomena of which are completely governed by their helical topological states. Perfect transmission of electrons through a finite potential barrier between a normal metal and a topological superconducting state is demonstrated, as evidenced by an exact doubling of conductance in point contact measurements.

The Feshbach–Villars equations, like the Klein–Gordon equation, are relativistic quantum mechanical equations for spin-0 particles.We write the Feshbach–Villars equations into an integral equation form and solve them by applying the Coulomb–Sturmian potential separable expansion method. We consider boundstate problems in a Coulomb plus short range potential. The corresponding Feshbach–Villars CoulombGreen’s operator is represented by a matrix continued fraction.

Massless Dirac fermions in graphene provide unprecedented opportunities to realize the Klein paradox, which is one of the most exotic and striking properties of relativistic particles. In the seminal theoretical work [Katsnelson et al., Nat. Phys. 2 620 (2006)], it was predicted that the massless Dirac fermions can pass through one-dimensional (1D) potential barriers unimpededly at normal incidence. Such a result seems to preclude confinement of the massless Dirac fermions in graphene by using 1D potential barriers. Here, we demonstrate, both experimentally and theoretically, that massless Dirac fermions can be trapped in quasi-1D npn junction of a continuous graphene monolayer. Because of highly anisotropic transmission of the massless Dirac fermions at n-p junction boundaries (the so-called Klein tunneling in graphene), charge carries incident at large oblique angles will be reflected from one edge of the junction with high probability and continue to bounce from the opposite edge. Consequently, these electrons are trapped for a finite time to form quasi-bound states in the quasi-1D npn junction. The quasi-bound states seen as pronounced resonances are probed and the quantum interference patterns arising from these states are directly visualized in our scanning tunneling microscope measurements., 4 Figures in main text

We theoretically investigate Klein tunneling processes in photonic artificial graphene. Klein tunneling is a phenomenon in which a particle with Dirac dispersion going through a potential step shows a characteristic angle- and energy-dependent transmission. We consider a generic photonic system consisting of a honeycomb-shaped array of sites with losses, illuminated by coherent monochromatic light. We show how the transmission and reflection coefficients can be obtained from the steady-state field profile of the driven-dissipative system. Despite the presence of photonic losses, we recover the main scattering features predicted by the general theory of Klein tunneling. Signatures of negative refraction and the orientation dependence of the intervalley scattering are also highlighted. Our results will stimulate the experimental study of intricate transport phenomena using driven-dissipative photonic simulators., 11 pages, 8 figures

The Dirac equation requires a treatment of the step potential that differs fundamentally from the traditional treatment, because the Dirac plane waves, besides momentum and spin, are characterized by a quantum number with the physical meaning of sign of charge. Since the Hermitean operator corresponding to this quantum number does not commute with the step potential, the time displacement parameter used in the ansatz of the stationary state does not have the physical meaning of energy. Therefore there are no paradoxal values of the energy. The new solution of the Dirac equation with a step potential is obtained. This solution, again, allows for phenomena of the Klein paradox type, but in addition it contains a positron amplitude localized at the threshold point of the step potential., Comment: 18 pages

We discuss the impact of a tiny Lorentz-violating $b^\mu$ term on the one dimensional motion of a Dirac particle scattering on a rectangular barrier. We assume the experiment is performed in a particular inertial frame, where the components of $b^\mu$ are assumed constants. The results show that Lorentz-violation modification to the transmission rate depends on the observer Lorentz nature of $b^\mu$. For a spacelike or lightlike $b^\mu$ the induced resonant frequency shift depends on the polarization, while for timelike $b^\mu$ there is essentially no modification., Comment: 3 pages, 1 figures, Presented at the Seventh Meeting on CPT and Lorentz Symmetry, Bloomington, Indiana, June 20-24, 2016

QFT approaches elaborated for treating quantum effects in time-dependent external electric fields are not directly applicable to time-independent nonuniform electric fields that are given by a step potential and their generalization for the such potentials was not sufficiently developed. Such fields can also create particles from the vacuum, the Klein paradox being closely related to this process. We believe that the present work presents a consistent solution of the latter problem. Quantizing the Dirac and scalar fields with time independent backgrounds, we have found in- and out-creation and annihilation operators that allow one to have particle interpretation of the physical system under consideration. To justify the proposed identification, we have performed a detailed mathematical and physical analysis of solutions of the corresponding relativistic wave equations with a subsequent QFT analysis. We elaborated a nonperturbative technique that allows one to calculate all characteristics of zero-order processes such scattering, reflection, and electron-positron pair creation, and to calculate Feynman diagrams that describe all characteristics of processes with interaction between the in-, out-particles and photons. These diagrams have formally the usual form, but contain special propagators. Expressions for these propagators in terms of in- and out-solutions are presented. We apply the elaborated approach to two popular exactly solvable cases: to the Sauter potential and to the Klein step., Comment: 58 pages, 3 figures, misprints corrected, version accepted for publication in Physical Review D

We investigate the scattering of a wave packet describing low-energy electrons in graphene by a time-dependent finite step potential barrier. Our results demonstrate that, after Klein tunneling through the barrier, the electron acquires an extra energy which depends on the rate of change the barrier height in time. If such a rate is negative, the electron loses energy and ends up as a valence band state after leaving the barrier, which effectively behaves as a positively charged quasi-particle, Comment: 5 pages, 5 figures, Fig. 3 selected for the Kaleidoscope section (Sept. 2015) of Phys. Rev. B

The essential features of Klein tunneling of massless fermions in graphene may be treated in one dimension without the need for Dirac spinors. Two dimensions needs a spinor treatment and is investigated numerically, which lets us compare tunneling through smooth potential barriers with that through idealized step potentials.

The observation of quantum conductance oscillations in mesoscopic systems has traditionally required the confinement of the carriers to a phase space of reduced dimensionality. While electron optics such as lensing and focusing have been demonstrated experimentally, building a collimated electron interferometer in two unconfined dimensions has remained a challenge due to the difficulty of creating electrostatic barriers that are sharp on the order of the electron wavelength. Here, we report the observation of conductance oscillations in extremely narrow graphene heterostructures where a resonant cavity is formed between two electrostatically created bipolar junctions. Analysis of the oscillations confirms that p-n junctions have a collimating effect on ballistically transmitted carriers. The phase shift observed in the conductance fringes at low magnetic fields is a signature of the perfect transmission of carriers normally incident on the junctions and thus constitutes a direct experimental observation of ``Klein Tunneling.'', 13 pages and 6 figures including supplementary information. The paper has been modified in light of new theoretical results available at arXiv:0808.0488

A planar heterostructure composed of two graphene films between which a narrow-gap semiconductor ribbon is inserted was studied. It was shown that the Klein paradox is absent when conical points of the graphene Brillouin zone are in the band gap of a narrow-gap semiconductor. There is an energy range dependent on the angle of incidence, in which an above-barrier decaying solution can exist. Thereby, such a heterostructure is a filter transmitting particles in a certain range of angles of incidence on the potential barrier. The applicability of such a heterostructure as a gate is discussed.

A colloquium-style introduction to two electronic processes in a carbon monolayer (graphene) is presented, each having an analog in relativistic quantum mechanics. Both processes couple electronlike and holelike states, through the action of either a superconducting pair potential or an electrostatic potential. The first process, Andreev reflection, is the electron-to-hole conversion at the interface with a superconductor. The second process, Klein tunneling, is the tunneling through a $p\text{\ensuremath{-}}n$ junction. The absence of backscattering, characteristic of massless Dirac fermions, implies that both processes happen with unit efficiency at normal incidence. Away from normal incidence, retro-reflection in the first process corresponds to negative refraction in the second process. In the quantum Hall effect, both Andreev reflection and Klein tunneling induce the same dependence of the two-terminal conductance plateau on the valley isospin of the carriers. Existing and proposed experiments on Josephson junctions and bipolar junctions in graphene are discussed from a unified perspective.

Graphene is the first example of truly two-dimensional crystals - it's just one layer of carbon atoms. It turns out to be a gapless semiconductor with unique electronic properties resulting from the fact that charge carriers in graphene demonstrate charge-conjugation symmetry between electrons and holes and possess an internal degree of freedom similar to ``chirality'' for ultrarelativistic elementary particles. It provides unexpected bridge between condensed matter physics and quantum electrodynamics (QED). In particular, the relativistic Zitterbewegung leads to the minimum conductivity of order of conductance quantum $e^2/h$ in the limit of zero doping; the concept of Klein paradox (tunneling of relativistic particles) provides an essential insight into electron propagation through potential barriers; vacuum polarization around charge impurities is essential for understanding of high electron mobility in graphene; index theorem explains anomalous quantum Hall effect., misprints are fixed; to appear in special issue of Solid State Communications

We employ the tight-binding propagation method to study Klein tunneling and quantum interference in large graphene systems. With this efficient numerical scheme, we model the propagation of a wave packet through a potential barrier and determine the tunneling probability for different incidence angles. We consider both sharp and smooth potential barriers in n-p-n and n-n' junctions and find good agreement with analytical and semiclassical predictions. When we go outside the Dirac regime, we observe that sharp n-p junctions no longer show Klein tunneling because of intervalley scattering. However, this effect can be suppressed by considering a smooth potential. Klein tunneling holds for potentials changing on the scale much larger than the interatomic distance. When the energies of both the electrons and holes are above the Van Hove singularity, we observe total reflection for both sharp and smooth potential barriers. Furthermore, we consider caustic formation by a two-dimensional Gaussian potential. For sufficiently broad potentials we find a good agreement between the simulated wave density and the classical electron trajectories., Comment: 14 pages, 12 figures

We study Klein tunneling in polariton graphene. We show that the photonic spin-orbit coupling associated with the energy splitting between TE and TM photonic modes can be described as an emergent gauge field. It suppresses the Klein tunneling in a small energy range close to the Dirac points. Thanks to polariton spin-anisotropic interactions, polarized optical pumping allows one to create potential barriers acting on a single polariton spin. We show that the resulting spin-dependent Klein tunneling can be used to create a perfectly transmitting polarization rotator operating at microscopic scale.

While Klein paradox is often encountered in the context of scattering of relativistic particles at a potential barrier, we presently discuss a puzzling situation that arises with the Klein-Gordon equation for bound states. With the usual minimal coupling procedure of introducing the interaction potential, a paradoxical situation arises when the 'hill' becomes a 'well', simulating a bound-state like situation. The phenomenal phenomenon for bound states is contrary to the conventional wisdom of quantum mechanics and is analogous to the well-known Klein paradox, a generic property of relativistic wave equations. PACs Nos. 03.65.Ge, 03.65.Pm, Comment: 6 pages

The well known Klein paradox for the relativistic Dirac wave equation consists in the computation of possible ``negative probabilities'' induced by certain potentials in some regimes of energy. The paradox may be resolved employing the notion of electron-positron pair production in which the number of electrons present in a process can increase. The Klein paradox also exists in Maxwell's equations viewed as the wave equation for photons. In a medium containing ``inverted energy populations'' of excited atoms, e.g. in a LASER medium, one may again compute possible ``negative probabilities''. The resolution of the electromagnetic Klein paradox is that when the atoms decay, the final state may contain more photons then were contained the initial state. The optical theorem total cross section for scattering photons from excited state atoms may then be computed as negative within a frequency band with matter induced amplification., Comment: ReVTeX 3.1 Format, 5 Pages

We show analytically that the {\it Zitterbewegung} and Klein Paradox, such well known aspects of the Dirac Equation are not found in the case of Bosons. We use the Kemmer-Duffin-Harish Chandra formalism with $\beta$ matrices to arrive at our results

We study transport properties of a phosphorene monolayer in the presence of single and multiple potential barriers of height $U_0$ and width $d$, using both continuum and microscopic lattice models, and show that the nature of electron transport along its armchair edge ($x$ direction) is qualitatively different from its counterpart in both conventional two-dimensional electron gas with Schr\"odinger-like quasiparticles and graphene or surfaces of topological insulators hosting massless Dirac quasiparticles. We show that the transport, mediated by massive Dirac electrons, allows one to achieve collimated quasiparticle motion along $x$ and thus makes monolayer phosphorene an ideal experimental platform for studying Klein paradox. We study the dependence of the tunneling conductance $G \equiv G_{xx}$ as a function of $d$ and $U_0$, and demonstrate that for a given applied voltage $V$ its behavior changes from oscillatory to decaying function of $d$ for a range of $U_0$ with finite non-zero upper and lower bounds, and provide analytical expression for these bounds within which $G$ decays with $d$. We contrast such behavior of $G$ with that of massless Dirac electrons in graphene and also with that along the zigzag edge ($y$ direction) in phosphorene where the quasiparticles obey an effective Schr\"odinger equation at low energy. We also study transport through multiple barriers along $x$ and demonstrate that these properties hold for transport through multiple barrier as well. Finally, we suggest concrete experiments which may verify our theoretical predictions., Comment: v1; 11+ pages, 12 figs

Pseudospin, an additional degree of freedom inherent in graphene, plays a key role in understanding many fundamental phenomena such as the anomalous quantum Hall effect, electron chirality and Klein paradox. Unlike the electron spin, the pseudospin was traditionally considered as an unmeasurable quantity, immune to Stern-Gerlach-type experiments. Recently, however, it has been suggested that graphene pseudospin is a real angular momentum that might manifest itself as an observable quantity, but so far direct tests of such a momentum remained unfruitful. Here, by selective excitation of two sublattices of an artificial photonic graphene, we demonstrate pseudospin-mediated vortex generation and topological charge flipping in otherwise uniform optical beams with Bloch momentum traversing through the Dirac points. Corroborated by numerical solutions of the linear massless Dirac-Weyl equation, we show that pseudospin can turn into orbital angular momentum completely, thus upholding the belief that pseudospin is not merely for theoretical elegance but rather physically measurable., 12 pages, 5 figures

We present an open-system interaction formalism for the Dirac equation. Overcoming a complexity bottleneck of alternative formulations, our framework enables efficient numerical simulations (utilizing a typical desktop) of relativistic dynamics within the von Neumann density matrix and Wigner phase-space descriptions. Employing these instruments, we gain important insights into the effect of quantum dephasing for relativistic systems in many branches of physics. In particular, the conditions for robustness of Majorana spinors against dephasing are established. Using the Klein paradox and tunneling as examples, we show that quantum dephasing does not suppress negative energy particle generation. Hence, the Klein dynamics is also robust to dephasing.

We study beams of medium-energy electrons and positrons which obey the Dirac equation and scatter from nuclei with $Z > 100.$ At small distances the potential is modelled to be that of a charged sphere. A large peak is found in the probability of positron penetration to the origin for $Z \approx 184.$ This may be understood as an example of Klein tunnelling through the Coulomb barrier: it is the analogue of the Klein Paradox for the Coulomb potential., 3 figures, to be published in Physics Letters B