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The transmission coefficient T of the Dirac quasielectrons through a rectangular potential barrier in the α-T3 model is calculated and analyzed in the continuum approach. The dependence of the transmission rate on para-meter α, which characterizes the degree of coupling of the central atom with the atoms in the vertices of the hexagonal lattice, and parameter β, which is equal to the ratio of Fermi velocities in the barrier and out-of-barrier regions, is analyzed. It was found, for certain quasiparticle energies, the supertunneling phenomenon is observed, which is that the transmission coefficient is equal to one independently of an angle of the particle incidence on the barrier, provided that α = 1. The values of these energies depend on the barrier height and the parameter β. It is shown that for some sets of parameters the function T(α) has maxima in the range 0

Sudden break in the passage of electron beam things conducting barrier shall either enhance or decrease electron beam with provided effect on spectral band of unknown metal, j/?dv –V at d2/dv2 plot gives characteristic of tunnel junction. At liquid helium temperature qv = hὐ implication for JV curve is observed at bias V where v denotes the frequency of molecular compounds present in the barrier, In case barrier emitting phonon interacts with the lattice of implied nano particles like Ga, As Ge Se coupled barrier in molecular state can be observed a specific band of Raman spectra and infra red thermally nactivated long wave spectra as IETS under Josephson effect. Single quantum dot in ultraclean carbonnanotube can be annexed with novel double quantum dot as result of klein paradox. At relatvistic QFT. In this context single spin1/2 can be confined with the hole emplacement and confinement as second double dot.

Indirect tunneling device serves implicative technique in sensing technology. Navigation through space telescopes and Detection of gamma radiation leakage in nuclear plant, sensing of Gamma radiation while navigating in space astronomy with an effective tool is the requisite issue for nuclear and space physics. Implication of quantum dot reservoir with selective metals like oxides of Se, Te, and Si, Al as double barrier system in Indirect tunneling device makes it possible by theoretical agreement on base of Klein paradox equation and Dirac equation support Josephson Current effect in evaluating magnetic field due to radiant charged particles like fermions and gamma particles. Rise and fluctuations of voltage vide I-J and IV characteristic due to interactive particles (massive or mass less) under constant potential barrier set up in the indirect tunneling diagnose phonons.

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 ${\mathcal{T}}_{y}(x)=\frac{\ensuremath{\partial}{J}_{yx}(x)}{\ensuremath{\partial}x}$.

The solution of the relativistic tunneling problem with the aid of the Klein-Gordon and kinetic energy based wave equations are reviewed. The behaviors of the transmitted and reflected wave functions by the potential barrier are studied for different values of the barrier’s energy. The reason of the Klein’s paradox is put forth in terms of the structures of the wave equations.

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

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.

Item does not contain fulltext Starting from a detailed explanation of Klein paradox of relativistic quantum mechanics, we consider a motion of massless Dirac fermions through potential barriers. It is shown that chiral properties of these particles guarantee a penetration through arbitrarily high and broad potential barriers. The role of this phenomenon (chiral tunneling) for graphene physics and technology is explained. We discuss analogy between electronic optics of graphene and optical properties of metamaterials, especially, Veselago lensing effect for massless Dirac fermions. Chiral tunneling in bilayer graphene is discussed.

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.

The first treatment of what came to be known as the Klein paradox can be traced back to the original paper by Klein [1], who pioneered studies of Dirac’s equation in the presence of a step potential. He showed that an electron beam propagating in a region with a large enough potential barrier V can emerge without the exponential damping expected from nonrelativistic quantum tunneling processes. When trying to understand if such a result was an artifact of the step-potential used by Klein, Sauter found that the essentials of the process were independent on the details of the potential barrier, although the probability of transmission decreases with decreasing slope [2].

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.

A detailed analysis of electron–positron pair creation induced by a spatially non-uniform and static electric field from vacuum is presented. A typical example is provided by the Sauter potential. For this potential, we derive the analytic expressions for vacuum decay and pair production rate accounted for the entire range of spatial variations. In the limit of a sharp step, we recover the divergent result due to the singular electric field at the origin. The limit of a constant field reproduces the classical result of Euler, Heisenberg and Schwinger, if the latter is properly averaged over the width of a spatial variation. The pair production by the Sauter potential is described for different regimes from weak to strong fields. For all these regimes, the locally constant-field rate is shown to be the upper limit.

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.

Scattering of a ballistic electron by the quantum-dot spin qubits fixed in a graphene nanoribbon is investigated theoretically. Two simple cases are investigated in details: scattering from a static quantum dot and scattering from two static quantum dots located at a fixed distance from each other. For the first case, it is shown that the Klein tunneling in a graphene sheet leads to a final entangled state for the reflected and/or transmitted electrons. The amount of the generated entanglement through the scattering process is a function of the incident angle for the ballistic electrons. For the second case, it is shown that the created correlation between the quantum dots is a periodic function of their distance. For frontal incident electrons in both cases, there is not any reflection and the Klein tunneling effect leads to a final well-correlated state for the scattering system.

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.

We present an exact solution of the one-dimensional modified Klein Gordon and Duffin Kemmer Petiau (for spins 0 and 1) equations with a step potential in the presence of minimal length in the uncertainty relation, where the expressions of the new transmission and reflection coefficients are determined for all cases. As an application, the Klein paradox in the presence of minimal length is discussed for all equations.

We investigate Klein tunneling for the $\ensuremath{\alpha}\text{\ensuremath{-}}{T}_{3}$ model, which interpolates between graphene and the dice lattice via parameter $\ensuremath{\alpha}$. We study transmission across two types of electrostatic interfaces: sharp potential steps and sharp potential barriers. We find both interfaces to be perfectly transparent for normal incidence for the full range of the parameter $\ensuremath{\alpha}$ for both interfaces. For other angles of incidence, we find that transmission is enhanced with increasing $\ensuremath{\alpha}$. For the dice lattice, we find perfect, all-angle transmission across a potential step for incoming electrons with energy equal to half of the height of the potential step. This is analogous to the ``super'', all-angle transmission reported for the dice lattice for Klein tunneling across a potential barrier.

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

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

A stable high-order accurate finite difference method for the time-dependent Dirac equation is derived. Grid-convergence studies in 1-D and 3-D corroborate the analysis. The method is applied to time-resolved quantum tunneling where a comparison with the solution to the time-dependent Schrodinger equation in 1-D illustrates the differences between the two equations. In contrast to the conventional tunneling probability decay predicted by the Schrodinger equation, the Dirac equation exhibits Klein tunneling. Solving the time-dependent Dirac equation with a step potential in 3-D reveals that particle spin affects the tunneling process. The observed spin-dependent reflection allows for a new type of spin-selective measurements.

Simulations of the Dirac equation have allowed to mimic measurably the predicted unusual characteristics of the electron motion, e.g. Zitterbewegung and Klein paradox that are beyond current technical capabilities. In this paper, it is shown that a Bose–Einstein condensate experiment carried out corroborates these results, but in addition exhibits a particular feature of an observable represented by a Dirac self-adjoint time operator introduced in relativistic quantum mechanics.

In this Letter we have studied the quantum behavior of a spin half neutral fermion interacting with a pseudoscalar potential barrier in ( 1 + 1 )-dimensional spacetime. Exact solutions for the corresponding Dirac equation are obtained both for bound and scattering states. The exact energy levels are obtained from the solutions of Dirac equation. The validity of the quasi-classical quantization rule is examined. For the scattering process the transmission and reflection coefficients are exactly calculated. The absence of the Kleinʼs paradox is also discussed.

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

We study the effect of trigonal warping on the Klein tunneling and scattering of electrons in bipolar junctions of bilayer graphene. We derive average velocities of electrons and obtain closed forms of transmission and reflection amplitudes. For normal incidence the amplitude and the average velocity in transmission zone are obtained to be zero, from which we show that the Klein effect of perfect reflection can appear in a thin potential barrier as well as in a potential step. The trigonal warping also causes variation in curvatures of isoenergy curves and breaks the valley degeneracy. This brings about valley-polarized scattering of electrons in the bipolar junctions, which can be used to produce a valley-polarized electron beam with bilayer graphene.

We study two-dimensional massive Dirac equation in circular well potential. The energies of bound states are obtained. We demonstrate the Klein paradox of this relativistic wave equation: For large enough potential depth, the bound states disappear from the spectra. Applications to graphene systems are discussed.

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.

We discuss common features in mechanical, electromagnetic and quantum systems, supporting identical results for the transmission and reflection coefficients of waves arriving perpendicularly at a plane interface. Also, we briefly discuss the origin of special notions such as refractive index in quantum mechanics, massive photons in wave guides and an elementary connection of results for a plane interface to experiments in graphene where the Klein paradox could be tested. The paper is intended for undergraduate level, and a basic knowledge of waves, relativity and quantum physics is required. Its educational purpose is to provide an integrated discussion of waves in order to fit the teaching to the requirement of a shorter sequence of university physics courses.

We explore the transmission through a time-periodic potential barriers in single layer graphene with an induced energy gap in this work. The analytical solutions of the transmission probabilities for the central band and the first sidebands are obtained. The transmission probabilities as the function of the induced energy gap and the incidence electron energy are discussed. We also investigate how critical angles change as the induced energy gap and incidence electron energy. The study of transmission properties will benefit applications in graphene-based nano-electronics.

The elementary excitations of monolayer graphene, which behave as massless Dirac particles, make it a fascinating venue in which to study relativistic quantum phenomena. One notable example is Klein tunneling, a phenomena in which electrons convert to holes to tunnel through a potential barrier. However, the omnipresence of charged impurities in substrate-supported samples keep the overall charge distribution nonuniform, obscuring much of this “Dirac” point physics in large samples. Using local gates, one can create tunable heterojunctions in graphene, isolating the contribution of small regions of the samples to transport. In this review, we give an overview of quantum transport theory and experiment on locally gated graphene heterostructures, with an emphasis on bipolar junctions.

We have investigated the transport properties in the graphene-based normal metal/insulator/p-wave superconductor junctions, in the limit of a thin barrier. It is shown that the conductance spectra and the shot noise of such system exhibit essential different features as compared to the conventional spin-triple and the graphene-based spin-singlet superconductor junctions.

Electron scattering in the monolayer graphene has been considered within the framework of our model of short-range defects proposed earlier. Electronic properties are determined by the Dirac equation for the two-component spinor wave function. Perturbation is modeled by the annular well with a band-asymmetric potential. Band-asymmetry of the potential stems from the local structure defect and leads to the mass (gap) perturbation in the Dirac equation. Transitions between the K and K’ critical points in the Brillouin zone are neglected, which is valid provided that the short-range perturbation has a finite radius. Exact explicit formulas for the scattering matrix have been derived. Results are presented in terms of the scattering phases and in the geometrical form of a relation between some 2-vectors. The characteristic equation for the bound and resonance states has been obtained in the form of an orthogonality condition. An approximate calculation of observables in terms of the scattering theory results is outlined.

We discuss the physics of graphene, which has recently generated great excitement in both the experimental and the theoretical realms. From the theoretical side, the main novelty is that 2D-graphene obeys massless Dirac instead of massive Schrodinger equations. The Dirac equation has mainly been used for the description of the high-energy relativistic physics of electrons and photons. However, here we shall be dealing with electron transport at much lower energy and hence temperature. We also discuss the so-called Klein paradox that can actually be seen, doing away with its paradoxical status forever. The Quantum Hall Effect in graphene also exhibits surprising features related to the Dirac sea.

Dirac particle penetration is studied theoretically with Dirac equation in one-dimensional systems. We investigate a one-dimensional system with N barriers where both barrier height and well width are constants randomly distributed in certain range. The one-parameter scaling theory for nonrelativistic particles is still valid for massive Dirac particles. In the same disorder sample, we find that the localization length of relativistic particles is always larger than that of nonrelativistic particles and the transmission coefficient related to incident particle in both cases fits the form T ~ exp(− αL). More interesting, massless relativistic particles are entirely delocalized no matter how big the energy of incident particles is.

We derive a fluid-dynamic model for electron transport near a Dirac point in graphene. Starting from a kinetic model, based on spinorial Wigner functions, the derivation of the fluid model is based on the minimum entropy principle, which is exploited to close the moment system deduced from the Wigner equation. To this aim we make two main approximations: the usual semiclassical approximation (ħ≪1) and a new one, namely, the ‘strongly mixed state’ approximation, which allow to compute the closure explicitly. Particular solutions of the fluid-dynamic equations are discussed which are of physical interest because of their connection with the Klein paradox phenomenon. Copyright © 2010 John Wiley & Sons, Ltd.

For nano-electronics applications of graphene a mass gap in its energy spectrum is needed (like a conventional semiconductor). In this paper, motivated by mass production of graphene, by the use of 2D massive Dirac-like equation we obtain the exact solution for the transmission probability corresponding to the Klein tunneling of massive Dirac fermions through a two-dimensional barrier which can be considered as a n–p–n junction in a graphene nano-transistor, and show that contrary to the case of massless Dirac fermions which results in complete transparency of the barrier for normal incidence, the transmission probability, T, in this case, apart from some resonance conditions that lead to the total transparency of the barrier, is smaller than one and also depends on the band index. We then obtain the transmission through a potential barrier in which the massless electrons acquire a finite mass and find that the transmission, in this case, is independent of the band index and cannot reach unity.

In this paper, an analytical form of the real-space non-interacting Green function of graphene material is developed. The transport properties of the relativistic electron through graphene-based double barrier structures have been investigated based on the real-space non-interacting Green function. It is shown that the results computed by the real-space non-interacting Green function are in perfect agreement with those results, which is based on the transfer matrix method.

In this paper, spin-dependent dwell time, spin Hartman effect and spin-dependent conductance were theoretically investigated through a rectangular barrier in the presence of an exchange field by depositing a ferromagnetic insulator on the phosphorene layer in the barrier region. The existence of the spin Hartman effect was shown for all energies (energies lower than barrier height) and all incident angles in phosphorene. We also compared our results of the dwell time in the phosphorene structure with similar research performed on graphene. We reported a significant difference between the tunneling time values of incident quasiparticles with spin-up and spin-down. We found that the barrier was almost transparent for incident quasiparticles with a wide range of incident angles and energies higher than the barrier height in phosphorene. We also found that the maximum spin-dependent transmission probability for energies higher than barrier height does not necessarily occur in the zero incident angle. In addition, we showed that the spin conductance for energies higher (lower) than barrier height fluctuates (decays) in terms of barrier thickness. We discovered that, in contrast to graphene, the Klein paradox does not occur in the normal incident in the phosphorene structure. Furthermore, the results demonstrated the achievement of good total conductance at certain thicknesses of the barrier for energies higher than the barrier height. This study could serve as a basis for investigations of the basic physics of tunneling mechanisms and also for using phosphorene as a spin polarizer in designing nanoelectronic devices.

An algebraic block-diagonalization of the Dirac Hamiltonian in a time-independent external field reveals a charge-index conservation law which forbids the physical phenomena of the Klein paradox type and guarantees a single-particle nature of the Dirac equation in strong external fields. Simultaneously, the method defines simpler quantum-mechanical objects—paulions and antipaulions, whose 2-component wave functions determine the Dirac electron states through exact operator relations. Based on algebraic symmetry, the presented theory leads to a new understanding of the Dirac equation physics, including new insight into the Dirac measurements and a consistent scheme of relativistic quantum mechanics of electron in the paulion representation. Along with analysis of the mathematical anatomy of the Klein paradox falsity, a complete set of paradox-free eigenfunctions for the Klein problem is obtained and investigated via stationary solutions of the Pauli-like equations with respective paulion Hamiltonians. It is shown that the physically correct Dirac states in the Klein zone are characterized by the total particle reflection from the potential step and satisfy the fundamental charge-index conservation law.

Based on the transfer-matrix method, we have investigated the coherent quantum transport in a graphene-based normal metal–insulator–superconductor ( NG / IG / SG ) junctions, in the limit of a thin barrier. It is found that the conductance spectra of such system, in contrast to a π periodic oscillatory behavior is shown for aligned Fermi surfaces of the normal and superconducting regions, exhibit a new π / 2 periodic oscillatory behavior as a function of barrier strength (χ) for a large Fermi surface mismatch.

An armchair graphene nanoribbon switch has been designed based on the principle of the Klein paradox. The resulting switch displays an excellent on-off ratio performance. An anomalous tunneling phenomenon, in which electrons do not pass through the graphene nanoribbon junction even when the conventional resonance condition is satisfied, is observed in our numerical simulations. A selective tunneling rule is proposed to explain this interesting transport behavior based on our analytical results. Based on this selective rule, our switch design can also achieve the confinement of an electron to form a quantum qubit.

Applying the fermions tunneling method, proposed by Kerner and Mann recently, we discuss the tunneling characteristics of Dirac particles from the stationary Kaluza–Klein black hole. To choose Gamma matrix conveniently and avoid the ergosphere dragging effect, we perform it in the dragging coordinate frame. The result shows that Hawking temperature in this case also can be reproduced by the general Dirac equation.

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

In previous calculations about the electronic transport properties through a gated region in graphene, a step-like potential is frequently adopted to mimic the gated barrier in two different theoretical models, i.e. the tight-binding model and the Dirac equation description. Our theoretical investigation indicates that the same step-like potential but used in the two distinct models are not equivalent. In the Dirac equation, a step-like potential can not reflect the normally incident electron due to Klein paradox. However, in tight-binding model, such a potential includes the scattering terms which can destroy the pseudo-spin conservation to cause a finite reflection. Thus, the electron transmission is not perfect.

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.