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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.

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.

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.

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.

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 Klein paradox is one of the cornerstones in the development of quantum mechanics, and its consequences were used in various branches of physics, ranging from elementary particles to solid state. Yet its mathematical derivation is questionable in a number of steps, resulting in the wrong solution of Dirac equation. In this paper, the paradox is analysed in more detail, and its mathematical content is emphasized in order to show that in the study of extreme conditions of matter simple arguments may result in erroneous predictions.

The exact bound state solutions of the Klein–Gordon equation and Dirac equation with scalar and vector pseudoharmonic oscillator potentials are obtained in this paper. Furthermore, we have used the supersymmetric quantum mechanics, shape invariance and alternative method to obtain the required 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

Dirac devised the quantum theory of the electron itself, which required him to generalize Schrodinger's famous equation to cover relativistic motion. He interpreted the resulting equation as showing that an antiparticle to the electron must exist.

We present a scheme for simulating relativistic quantum physics in circuit quantum electrodynamics. By using three classical microwave drives, we show that a superconducting qubit strongly-coupled to a resonator field mode can be used to simulate the dynamics of the Dirac equation and Klein paradox in all regimes. Using the same setup we also propose the implementation of the Foldy-Wouthuysen canonical transformation, after which the time derivative of the position operator becomes a constant of the motion., 13 pages, 3 figures

As is known to all, the electron scattering in classical electromagnetic potential is one of the most widespread applications of quantum theory. Nevertheless, many discussions about electron scattering are based upon single-particle Schrodinger equation or Dirac equation in quantum mechanics rather than the method of quantum field theory. In this paper, by using the path integral approach of quantum field theory, we perturbatively evaluate the scattering amplitude up to the second order for the electron scattering by the classical electromagnetic potential. The results we derive are convenient to apply to all sorts of potential forms. Furthermore, by means of the obtained results, we give explicit calculations for the one-dimensional electric potential.

Particle penetration through a square potential barrier is studied with the Dirac equation and relativistic tunnelling occurs in an overcritical potential. Relations between this phenomenon and the Klein paradox are discussed. However, the relativistic correction to mesoscopic conduction, the Landauer formula, is negligible and relativistic tunnelling would not occur in solid-state physics owing to the barrier height never reaching the overcritical region. Relativistic tunnelling is essentially a high-energy phenomenon.

We study the double barrier tunneling properties of Dirac particles in spin-orbit coupled Bose—Einstein Condensates. The analytic expression of the transmission coefficient of Dirac particles penetrating into a double barrier is obtained. An interesting resonance tunneling phenomenon is discovered in the Klein block region which has been ignored before.

We study the quasibound states in a graphene quantum-dot structure generated by the single-, double-, and triple-barrier electrostatic potentials. It is shown that the strongest quasibound states are mainly determined by the innermost barrier. Specifically, the positions of the quasibound states are determined by the barrier height, the number of the quasibound states is determined by the quantum-dot radius and the angular momentum, and the localization degree of the quasibound states is influenced by the width of the innermost barrier, as well as the outside barriers. Furthermore, according to the study on the double- and triple-barrier quantum dots, we find that an effective way to generate more quasibound states with even larger energy level spacings is to design a quantum dot defined by many concentric barriers with larger barrier-height differences. Last, we extend our results into the quantum dot of many barriers, which gives a complete picture about the formation of the quasibound states in the kind of graphene quantum dot created by many concentric potential barrier rings.

We investigated the Dirac electrons transmission through a potential barrier in the presence of circularly polarized light. An anomalous photon-assisted enhanced transmission is predicted and explained in a comparison with the well-known Klein paradox. It is demonstrated that the perfect transmission for nearly-head-on collision in an infinite graphene is suppressed in gapped dressed states of electrons, which is further accompanied by shift of peaks as a function of the incident angle away from the head-on collision. In addition, the perfect transmission in the absence of potential barrier is partially suppressed by a photon-induced gap in illuminated graphene. After the effect of rough edges of the potential barrier or impurity scattering is included, the perfect transmission with no potential barrier becomes completely suppressed and the energy range for the photon-assisted perfect transmission is reduced at the same time.

We derive exact analytical solutions for the focusing of Maxwell photons and massless Dirac particles by half-space with negative refraction. These novel paradoxical solutions show that half-space with negative refraction cannot be simply considered as a superlens. Instead of a 2D focus spot it results in the creation of a sink of complicated spatial structure. Our analytical solutions also show substantial differences in the focusing of Maxwell and Dirac field within this geometry. We find a new manifestation of the Klein paradox which is the creation of sinks in the negative-refraction area. Possible applications and generalizations of our approach are discussed.

We study a confined system of Dirac fermions in the presence of inhomogeneous magnetic field. Splitting the system into different regions, we determine their corresponding energy spectrum solutions. We underline their physical properties by considering the conservation energy where some interesting relations are obtained. These are used to discuss the reflexion and transmission coefficients for Dirac fermions and check the probability condition for different cases. We generalize the obtained results to a system with gap and make some analysis. After evaluating the current-carrying states, we analyze the Klein paradox and report interesting discussions., 28 pages, 15 figures. Version to appear in JPA

In this paper, we first evaluate the electronic transmission of Dirac fermions into a p-n junction of gapped graphene and show that the final result depends on the sign of the refractive index, n. We also, by considering the appropriate wavefunctions in the region of the electrostatic potential, show that both transmission and the reflection probability turn out to be positive and less than unity instead of the negative transmission and higher than unity reflection coefficient commonly referred to as the Klein paradox. We then obtain the transmission probability corresponding to a special p-n junction for which there exists a region in which the low energy excitations of graphene acquire a finite mass and, interestingly, find that in this case the transmission is independent of the index of refraction, in contrast with the corresponding result for gapped graphene. We then discuss the validity of the solutions reported in some of the papers cited in this work which, considering the Büttiker formula, turn out to lead to the wrong results for conductivity.

We studied the Klein paradox in zigzag (ZNR) and anti-zigzag (AZNR) graphene nanoribbons. Due to the fact that ZNR (the number of lattice sites across the nanoribbon = N is even) and AZNR (N is odd) configurations are indistinguishable when treated by the Dirac equation, we supplemented the model with a pseudo-parity operator whose eigenvalues correctly depend on the sublattice wavefunctions for the number of carbon atoms across the ribbon, in agreement with the tight-binding model. We have shown that the Klein tunneling in zigzag nanoribbons is related to conservation of the pseudo-parity rather than pseudo-spin as in infinite graphene. The perfect transmission in the case of head-on incidence is replaced by perfect transmission at the center of the ribbon and the chirality is interpreted as the projection of the pseudo-parity on momentum at different corners of the Brillouin zone.

Quantum mechanical electron tunneling has potential applications in both science and technology, such as flash memories in modern LSI technologies and electron transport chains in biosystems. Although it is known that one-dimensional quantum electron tunneling lacks temperature dependence, the behavior of electron tunneling between different dimensional systems is still an open question. Here, we investigated the electron tunneling between a two-dimensional electron gas (2DEG) and zero-dimensional Si quantum dots and discovered an unexpected temperature dependence: At high temperature, the gate voltage necessary for electron injection from 2DEG to Si quantum dots becomes markedly small. This unusual tunneling behavior was phenomenologically explained by considering the geometrical matching of wave functions between different dimensional systems. We assumed that electron tunneling would occur within a finite experimental measurement time. Then, the observed electron tunneling is explained only by the contributions of wave packets below the quantum dot with a finite lifetime rather than the ordinary thermal excited states of 2DEG.

Microscopic mechanisms of externally controlled reversable heat flow through the carbon nanotube junctions (NJ) are studied theoretically. Our model suggests that the heat is transfered along the tube section ${\cal T}$ by electrons ($e$) and holes ($h$) moving ballistically in either in parallel or in opposite directions and accelerated by the bias source-drain voltage $V_{\rm SD}$ (Peltier effect). We compute the Seebeck coefficient $\alpha $, electric $\sigma$ and thermal $\kappa$ conductivities and find that their magnitudes strongly depend on $V_{\rm SD}$ and $V_{\rm G}$. The sign reversal of $\alpha$ versus the sign of $V_{\rm G}$ formerly observed experimentally is interpreted in this work in terms of so-called chiral tunneling phenomena (Klein paradox).

Klein's paradox is resolved in simple diagrammatic terms arising from considerations of its analogue in classical relativistic mechanics. The same results are also obtained directly from standard operator methods of quantum field theory.

The exact solutions of the Dirac equation with a linear scalar confining potential in a uniform electric field are given. It is found that, if the scalar potential is stronger than that of the electric field, confinement is permanent. On the contrary, if the electric field is strong enough, confinement is impossible due to the Klein paradox.

Some recent articles on the Klein Paradox announce reflexion coefficients exceeding 100%. The origin of this result seems to be a widely disseminated misprint which is pointed out and rectified. In conclusion, a new approach to the problem - by means of Furry's interaction representation for an external field - is indicated.