Reflection of two strongly interacting bosons with long-rage interaction hopping on a one-dimensional lattice scattered off by a potential step is theoretically investigated in the framework of the extended Hubbard model. The analysis shows that, in the presence of unbalanced on-site and nearest-neighbor site interaction, two strongly correlated bosons forming a bound particle state can penetrate a high barrier, despite the single particle can not. Such a phenomenon is analogous to one-dimensional Klein tunneling of a relativistic massive Dirac particle across a potential step.
We calculate the transmission coefficient for a particle crossing a potential barrier in monolayer graphene with Rashba spin-orbit coupling and in bilayer graphene. We show that in both the cases one can go from Klein tunneling regime, characterized by perfect normal transmission, to anti-Klein tunneling regime, with perfect normal reflection, by tuning the Rashba spin-orbit coupling for a monolayer or the interplane coupling for a bilayer graphene. We show that the intermediate regime is characterized by a non-monotonic behavior with oscillations and resonances in the normal transmission ampl…
Klein tunneling and conductance for Dirac fermions in ABC-stacked trilayer graphene (ABC-TLG) through symmetric and asymmetric double potential barriers are investigated using the two and six-band continuum model. Numerical results for our system show that the transport is sensitive to the height, the width and the distance between the two barriers. Klein paradox at normal incidence and resonant features at non-normal incidence in the transmission result from resonant electron states in the wells or hole states in the barriers and strongly influence the ballistic conductance of the structures.
This article provides a pedagogical review on Klein tunneling in graphene, i.e. the peculiar tunneling properties of two-dimensional massless Dirac electrons. We consider two simple situations in detail: a massless Dirac electron incident either on a potential step or on a potential barrier and use elementary quantum wave mechanics to obtain the transmission probability. We emphasize the connection to related phenomena in optics, such as the Snell-Descartes law of refraction, total internal reflection, Fabry-Pérot resonances, negative refraction index materials (the so called meta-materials), …
Statistical complexity and Fisher-Shannon information are calculated in a problem of quantum scattering, namely the Klein tunneling across a potential barrier in graphene. The treatment of electron wave functions as masless Dirac fermions allows us to compute these statistical measures. The comparison of these magnitudes with the transmission coefficient through the barrier is performed. We show that these statistical measures take their minimum values in the situations of total transparency through the barrier, a phenomenon highly anisotropic for the Klein tunneling in graphene.
The effect of a parallel magnetic field $B$ and a potential substrate on the transmission, conductivity, and Fano factor of a biased ABC-stacked $pn$ and $pnp$ trilayer graphene junction (ABC-TLG) is investigated theoretically at low energy. We discovered that, in the presence of a high magnetic field $B=1400$T, the ABC-TLG can exhibit new Klein tunneling at two new incidence $k_y$ values, in addition to the usual value of $k_y=0$ and whatever the energy value is. Indeed, the transmission of ABC-TLG via $ pn $ and $ pnp $ junctions is transformed into a three-separate transmission of single-li…
We show that in gapped bilayer graphene, quasiparticle tunneling and the corresponding Berry phase can be controlled such that it exhibits features of single layer graphene such as Klein tunneling. The Berry phase is detected by a high-quality Fabry-Pérot interferometer based on bilayer graphene. By raising the Fermi energy of the charge carriers, we find that the Berry phase can be continuously tuned from $2π$ down to $0.68π$ in gapped bilayer graphene, in contrast to the constant Berry phase of $2π$ in pristine bilayer graphene. Particularly, we observe a Berry phase of $π$, the standard val…
When a single-layer graphene sheet is contacted with metallic electrodes, tunnel barriers are formed as a result of the doping of graphene by the metal in the contact region. If the Fermi energy level is modulated by a gate voltage, the phenomenon of Klein tunneling results in specific features in the conductance and noise. Here we obtain analytically exact solutions for the transmission and reflection probability amplitudes using a trapezoidal potential barrier, allowing us to calculate the differential conductance and the Fano factor for a graphene sheet in the ballistic regime. We put in ev…
Klein tunneling is a counterintuitive quantum-mechanical phenomenon, predicting perfect transmission of relativistic particles through higher energy barriers. This phenomenon was shown to be supported at normal incidence in graphene due to pseudospin conservation. Here I show that Klein tunneling analogue can occur in classical systems, and remarkably, not relying on mimicking graphene's spinor wavefunction structure. Instead, the mechanism requires a particular form of constitutive parameters of the penetrated medium, yielding transmission properties identical to the quantum tunneling in grap…
We study the total transmission of quantum particles satisfying the Klein-Gordon equation through a potential barrier based on the classical wave propagation theory. We deduce an analytical expression for the wave impedance for Klein-Gordon particles. From the condition that the impedance is matched throughout the space, we show that super-Klein tunneling, which refers to the omnidirectional total transmission of particles through a potential barrier when the energy of the particle is equal to the half of the barrier potential, should occur in these systems. We also derive a condition for tota…
The tunneling of electrons and holes in quantum structures plays a crucial role in studying the transport properties of materials and the related devices. 8-Pmmn borophene is a new two-dimensional Dirac material, which hosts tilted Dirac cone and chiral, anisotropic massless Dirac fermions. We develop the transfer matrix method to investigate the Klein tunneling of massless fermions across the smooth NP junctions and NPN junctions of 8-Pmmn borophene. Like the sharp NP junctions of 8-Pmmn borophene, the tilted Dirac cones induce the oblique Klein tunneling. The angle of perfect transmission to…
The absence of backscattering in metallic nanotubes as well as perfect Klein tunneling in potential barriers in graphene are the prominent electronic characteristics of carbon nanostructures. We show that the phenomena can be explained by a peculiar supersymmetry generated by a first order Hamiltonian and zero order supercharge operators. Like the supersymmetry associated with second order reflectionless finite-gap systems, it relates here the low-energy behavior of the charge carriers with the free particle dynamics.
The problem of the Klein tunneling across a potential barrier in bi-layer graphene is addressed. The electron wave functions are treated as massive chiral particles. This treatment allows us to compute the statistical complexity and Fisher-Shannon information for each angle of incidence. The comparison of these magnitudes with the transmission coefficient through the barrier is performed. The role played by the evanescent waves on these magnitudes is disclosed. Due to the influence of these waves, it is found that the statistical measures take their minimum values not only in the situations of…
We study the dynamics of semiclassical electrons in (gapped) graphene in two complementary limits, i.e. in the Klein tunneling and valley Hall effect regimes, by scattering wave packets off armchair step potentials and by exposing wave packets to a uniform electric field, respectively. Our numerical wave packet simulation goes beyond semiclassical analytical approximations and standard Klein tunneling treatments and allows to study intra- and intervalley scattering processes. We find distinct Klein tunneling characteristics for low and tall steps, which include unusual Berry curvature induced …
We numerically investigate the effect of Klein tunneling on the Aharonov-Bohm oscillations in graphene rings using a tight-binding model with nearest-neighbor couplings. In order to introduce Klein tunneling into the system, we apply an electrostatic potential to one of the arms of the ring, such that this arm together with the two adjacent leads form either a $nn'n$- or $npn$-junction ($n,n'$: conduction band transport, $p$: valence band transport). The former case corresponds to normal tunneling and the latter case to Klein tunneling. We find that the transmission properties strongly depend …
Klein tunneling, the perfect transmission of normally incident Dirac electrons across a potential barrier, has been widely studied in graphene and explored to design switches, albeit indirectly. We show that Klein tunneling maybe easier to detect for spin-momentum locked electrons crossing a PN junction along a three-dimensional topological insulator surface. In these topological insulator PN junctions (TIPNJs), the spin texture and momentum distribution of transmitted electrons can be measured electrically using a ferromagnetic probe for varying gate voltages and angles of current injection. …
The 8-\textit{Pmmn} borophene is one kind of new elemental monolayer, which hosts anisotropic and tilted massless Dirac fermions (MDF). The planar \textit{p-n} junction (PNJ) structure as the basic component of various novel devices based on the monolayer material has attracted increasing attention. Here, we analytically study the transport properties of anisotropic and tilted MDF across 8-\textit{Pmmn} borophene PNJ. Similar to the isotropic MDF across graphene junctions, perfect transmission exists but its direction departures the normal direction of borophene PNJ induced by the anisotropy a…
The conductance and the Fano factor in a graphene sheet in the ballistic regime are calculated. The electrostatic potential in the sheet is modeled by a trapezoid barrier, which allows to use the exact solution of the Dirac equation in a uniform electric field in the slope areas (the two lateral sides of the trapezoid). A special attention is devoted to asymmetry with respect to the sign of the gate voltage, which is connected with the difference between the Klein tunneling and the over-barrier reflection. The comparison of the developed theory with the experiment supports the conclusion that …
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 tunnelling in small energy range close to the Dirac points. Thanks to polariton spin-anisotropic interactions, polarized optical pumping allows 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.
We report a theoretical/experimental study of current-voltage characteristics (I-V) of graphene devices near the Dirac point. The I-V can be described by a power law (I \propto V^α, with 1< α<= 1.5). The exponent is higher when the mobility is lower. This superlinear I-V is interpreted in terms of the interplay between Zener-Klein transport, that is tunneling between different energy bands, and defect scattering. Surprisingly, the Zener-Klein tunneling is made visible by the presence of defects.
We study by first-principles calculations a densely packed island of organic molecules (F4TCNQ) adsorbed on graphene. We find that with electron doping the island naturally forms a p-n junction in the graphene sheet. For example, a doping level of ~ 3 * 10^{13} electrons per cm^2 results in a p-n junction with 800 meV electrostatic potential barrier. Unlike in a conventional p-n junction in graphene, in the case of the junction formed by an adsorbed organic molecular island we expect that the Klein tunneling is inhibited, even without an applied external magnetic field. Here Klein tunneling is…
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. Howev…
Graphene electrons feature a pair of massless Dirac cones of opposite pseudospin chirality at two valleys. Klein tunneling refers to the intriguing capability of these chiral electrons to penetrate through high and wide potential barrier. The two valleys have been treated independently in the literature, where time reversal symmetry dictates that neither the normal incidence transmission nor the angle-averaged one can have any valley polarization. Here we show that, when intervalley scattering by barrier is accounted, graphene electrons normally incident at a superlattice barrier can experienc…
The two apparently disparate phenomena, viz. the Klein tunneling and the electric field driven topological phase transition(TPT), exhibited by the silicene converge on the issue of the no change in the pseudo-spin. An ensuing possibility is that the former, in the case of a normal/magnetic/normal silicene junction, may be considered theoretically as a probe to ascertain the onset of the latter. In this communication, we explore this option and calculate the total transmission probability(TTP). We find that TPT is characterized by a cusp in a plot of TTP as a function of the electric field.
Today the availability of high mobility graphene up to room temperature makes ballistic transport in nanodevices achievable. In particular, p-n-p transistor in the ballistic regime gives access to the Klein tunneling physics and allows the realization of devices exploiting the optics-like behavior of Dirac Fermions (DF) as in the Vesalego lens or the Fabry Pérot cavity. Here we propose a Klein tunneling transistor based on geometrical optics of DF. We consider the case of a prismatic active region delimited by a triangular gate, where total internal reflection may occur, which leads to the tun…
Simulating quantum transport through mesoscopic, ring-shaped graphene structures, we address various quantum coherence and interference phenomena. First, a perpendicular magnetic field, penetrating the graphene ring, gives rise to Aharonov-Bohm oscillations in the conductance as a function of the magnetic flux, on top of the universal conductance fluctuations. At very high fluxes the interference gets suppressed and quantum Hall edge channels develop. Second, applying an electrostatic potential to one of the ring arms, $nn'n$- or $npn$-junctions can be realized with particle transmission due t…
Because of its large Fermi velocity, leading to a great mobility, graphene is expected to play an important role in (small signal) radio frequency electronics. Among other, graphene devices based on Klein tunneling phenomena are already envisioned. The connection between the Klein tunneling times of electrons and cut-off frequencies of graphene devices is not obvious. We argue in this paper that the trajectory-based Bohmian approach gives a very natural framework to quantify Klein tunneling times in linear band graphene devices because of its ability to distinguish, not only between transmitte…
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…
We use the Wick-rotated time-dependent supersymmetry to construct models of two-dimensional Dirac fermions in presence of an electrostatic grating. We show that there appears omnidirectional perfect transmission through the grating at specific energy. Additionally to being transparent for incoming fermions, the grating hosts strongly localized states.
Despite many similarities between electronics and optics, the hopping of the electron on a discrete atomic lattice gives rise to energy band nonparabolicity and anisotropy. The crucial influences of this effect on material properties and its incorporation into the continuum model have received widespread attention in the past half century. Here we predict the existence of a different effect due to the hopping of the electron across an atomically sharp interface. For a general lattice, its influence on transport could be equally important as the energy band nonparabolicity/anisotropy, but canno…
Transport measurement of electron optics in monolayer graphene p-n junction devices has been traditionally studied with negative refraction and chiral transmission experiments in Hallbar magnetic focusing set-ups. We show direct signatures of Klein (monolayer) and anti-Klein (bilayer) tunneling with a circular 'edgeless' Corbino geometry made out of gated graphene p-n junctions. Noticeable in particular is the appearance of angular sweet spots (Brewster angles) in the magnetoconductance data of bilayer graphene, which minimizes head-on transmission, contrary to conventional Fresnel optics or m…
The so-called Klein paradox - unimpeded penetration of relativistic particles through high and wide potential barriers - is one of the most exotic and counterintuitive consequences of quantum electrodynamics (QED). The phenomenon is discussed in many contexts in particle, nuclear and astro- physics but direct tests of the Klein paradox using elementary particles have so far proved impossible. Here we show that the effect can be tested in a conceptually simple condensed-matter experiment by using electrostatic barriers in single- and bi-layer graphene. Due to the chiral nature of their quasipar…
We study the interconversion between photons and phonons coupled via radiation pressure in artificial Dirac materials realized by a honeycomb array of optomechanical cells. In particular we analyze the chiral tunneling of (photon-phonon) polaritons through an oscillating planar barrier. While a static barrier accommodates constructively interfering optical or mechanical waves leading to photon or phonon transmission, an oscillating barrier allows for inelastic scattering that causes sideband excitations and interference effects which, in turn, may suppress or revive the light-sound interconver…
This is a colloquium-style introduction to two electronic processes in a carbon monolayer (graphene), each having an analogue in relativistic quantum mechanics. Both processes couple electron-like and hole-like 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-n junction. Existing and proposed experiments on Josephson junctions and bipolar junctions in graphene are d…
Using the coupled-mode theory in guided-wave optics and electronics, we explore a directional coupling structure composed of two parallel waveguides electrostatically induced by the split-gate technique in bulk graphene. Our results show that Klein tunneling can greatly enhance the coupling strength of the structure. By adjusting a gate voltage, the probability density of Dirac electron wave function initially in one waveguide can be completely transferred into the other waveguide within several hundred nanometers. Our findings could not only lead to functional coherent coupling devices for qu…
We have utilized the finite-difference approach to explore electron-tunneling properties in gapped graphene through various electrostatic-potential barriers changing from Gaussian to a triangular envelope function in comparison with a square potential barrier. Transmission coefficient is calculated numerically for each case and applied to corresponding tunneling conductance. It is well known that Klein tunneling in graphene will be greatly reduced in a gapped graphene. Our results further demonstrate that such a decrease of transmission can be significantly enhanced for spatially-modulated pot…
We propose the quantum simulation of the Dirac equation with potentials, allowing the study of relativistic scaterring and the Klein tunneling. This quantum relativistic effect permits a positive-energy Dirac particle to propagate through a repulsive potential via the population transfer to negative-energy components. We show how to engineer scalar, pseudoscalar, and other potentials in the 1+1 Dirac equation by manipulating two trapped ions. The Dirac spinor is represented by the internal states of one ion, while its position and momentum are described by those of a collective motional mode. …
The Klein paradox consists in the perfect tunneling of relativistic particles through high potential barriers. As a curious feature of particle physics, it is responsible for the exceptional conductive properties of graphene. It was recently studied in the context of atomic condensates and topological photonics and phononics. While in theory the perfect tunneling holds only for normal incidence, so far the angular dependence of the Klein tunneling and its strong variation with the barrier height were not measured experimentally. In this work, we capitalize on the versatility of atomic vapor ce…
A fractional quantum Hall liquid with multiple edges is considered. The computation of transport quantities such as current, noise and noise cross correlations in such multiple edge samples requires the implementation of so called Klein factors, which insure the correct quasiparticle exchange properties. The commutation relations of these factors are obtained by closing the system into a single edge. The non-equilibrium Green's function formalism associated with such factors is derived for a simple Laughlin fraction of the Hall effect. It is shown explicitly how Klein factors enter the calcula…
The outstanding electronic properties of relativistic-like fermions have been extensively studied in solid state systems with isotropic linear dispersions such as graphene. Here, we show that 2D and 3D Dirac-Weyl (DW) materials exhibiting tilted energy dispersions could induce drastically different transport phenomena, compared to the non-tilted case. Indeed, the Klein tunneling of DW fermions of opposite chiralities is predicted to appear along two separated oblique directions. In addition, valley filtering and beam splitting effects are easily tailored by dopant engineering techniques while …
We investigate the quantum tunneling of electrons in an AA-stacked bilayer graphene (BLG) $n$-$p$ junction and $n$-$p$-$n$ junction. We show that Klein tunneling of an electron can occur in this system. The quasiparticles are not only chiral but are additionally described by a `cone index'. Due to the orthogonality of states with different cone indexes, electron transport across a potential barrier must strictly conserve the cone index and this leads to the protected cone transport which is unique in AA-stacked BLG. Together with the negative refraction of electrons, electrons residing in diff…
Guided by the analogy to Mie scattering of light on small particles we show that the propagation of a Dirac-electron wave in graphene can be manipulated by a circular gated region acting as a quatum dot. Large dots enable electron lensing, while for smaller dots resonant scattering entails electron confinement in quasibound states. Forward scattering and Klein tunneling can be almost switched off for small dots by a Fano resonance arising from the interference between resonant scattering and the background partition.
Klein tunneling in gapless bilayer graphene, perfect reflection of electrons injecting normal to a pn junction, is expected to disappear in the presence of energy band gap induced by external gates. We theoretically show that the Klein effect still exists in gapped bilayer graphene, provided that the gaps in the n and p regions are balanced such that the polarization of electron pseudospin has the same normal component to the bilayer plane in the regions. We attribute the Klein effect to Berry phase $π$ (rather than the conventional value $2 π$ of bilayer graphene) and to electron-hole and tim…
Similar to Weyl semimetals, in magnetic materials, magnon bands can host Weyl points, around which the bosonic excitations are called Weyl magnons. Here, we investigate the Klein tunneling of Weyl magnons, during the process of which Weyl magnons tunnel through a high potential barrier. Specifically, we study the magnetization current carried by Weyl magnons in a quasi-one-dimensional magnetic wire, in the middle of which a gate magnetic field is applied to generate a potential barrier. The transmission probability is calculated and the Landauer-Büttiker formalism is used to find the magnetiza…
In this paper, we study Klein tunneling in random media. To this purpose, we simulate the propagation of a relativistic Gaussian wavepacket through a graphene sample with randomly distributed potential barriers (impurities). The simulations, based on a relativistic quantum lattice Boltzmann method, permit to compute the transmission coefficient across the sample, thereby providing an estimate for the conductivity as a function of impurity concentration and strength of the potentials. It is found that the conductivity loss due to impurities is significantly higher for wave-packets of massive pa…
We investigate the behavior of spin polarized currents in two-dimensional topological insulators (TI). Stationary solutions inside a HgTe/CdTe quantum well (QW) were obtained by Bernevig-Hughes-Zhang (BHZ) model modified by a electric and magnetic barrier inside a non-completely insulating bulk. An attenuated quantum spin Hall (QSH) effect occurs in the gaped region with an apparent Klein-like paradox. Even more interesting, for strong potential regime, the interaction between the quasiparticles and the barriers allows spin inversion of this electronic states in a distinct channel conduction. …
Transport through potential barriers in graphene is investigated using a set of metallic gates capacitively coupled to graphene to modulate the potential landscape. When a gate-induced potential step is steep enough, disorder becomes less important and the resistance across the step is in quantitative agreement with predictions of Klein tunneling of Dirac fermions up to a small correction. We also perform magnetoresistance measurements at low magnetic fields and compare them to recent predictions.
We report the experimental observation of Fabry-Pérot (FP) interference in the conductance of a gate-defined cavity in a dual-gated bilayer graphene (BLG) device. The high quality of the BLG flake, combined with the device's electrical robustness provided by the encapsulation between two hexagonal boron nitride layers, allows us to observe ballistic phase-coherent transport through a $1$μm-long cavity. We confirm the origin of the observed interference pattern by comparing to tight-binding calculations accounting for the gate-tunable bandgap. The good agreement between experiment and theory, f…
Currently, much effort is being put into detecting unconventional p-wave superconductivity in Josephson junctions based on topological insulators (TIs). For that purpose we propose to use superconducting Klein tunneling, i.e. the reflectionless passage of Cooper pairs through a potential barrier in a gated ballistic junction. This phenomenon occurs due to the fact that the supercurrent is carried by helical Andreev bound states (ABSs) characterized by spin-momentum locking similar to the normal-state carriers. We derive the spectrum of the helical ABSs and the corresponding Josephson current f…
We study the zero temperature conductance and magnetoconductance of ballistic \textit{p-n} junctions in Weyl semimetals. Electron transport is mediated by Klein tunneling between \textit{n}- and \textit{p}- regions. The chiral anomaly that is realized in Weyl semimetals plays a crucial role in the magnetoconductance of the junction. With the exception of field orientations where the angle between $\mathbf{B}$ and the junction plane is small, magnetoconductance is positive and linear in $B$ at both weak and strong magnetic fields. In contrast, magnetoconductance in conventional \textit{p-n} jun…
The Zener-Klein (ZK) interband tunneling in graphene layers (GLs) with the lateral n-i-n and p-i-n junctions results in the nonlinear I-V characteristics that can be used for the rectification and detection of the terahertz (THz) signals. The transit time delay of the tunneling electrons and holes in the depletion regions leads to the phase shift between the THz current and THz voltage causing the negative dynamic conductance in a certain frequency range and resulting in the so-called transit-time (TT) instability. The combination of the ZK tunneling and the TT negative dynamic conductance ena…
