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.
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…
An armchair graphene ribbon switch has been designed based on the principle of the Klein paradox. The resulting switch displays an excellent on-off ratio performance. Anomalous tunneling phenomena are observed in our numerical simulations. According to our analytical results, selective tunneling rule is proposed to explain this interesting transport behavior. The switch design enriches the phenomenon of the Klein paradox and also provides a platform to study the paradox.
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…
Whereas massless Dirac fermions in monolayer graphene exhibit Klein tunneling when passing through a potential barrier upon normal incidence, such a barrier totally reflects massive Dirac fermions in bilayer graphene due to difference in chirality. We show that, in the presence of magnetic barriers, such massive Dirac fermions can have transmission through even at normal incidence. The general consequence of this behaviour for multilayer graphene consisting of massless and massive modes are mentioned. We also briefly discuss the effect of a bias voltage on such magnetotransport.
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…
Charge carriers in single and multilayered graphene systems behave as chiral particles due to the particular lattice symmetry of the crystal. We show that the interplay between the meta-material properties of graphene multilayers and the pseudospinorial properties of the charge carriers result in the occurrence of Klein and anti-Klein tunneling for rhombohedral stacked multilayers. We derive an algebraic formula predicting the angles at which these phenomena occur and support this with numerical calculations for systems up to four layers. We present a decomposition of an arbitrarily stacked mu…
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…
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 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 …
The hybridization of $σ$ and $π$ orbitals of carbon atoms in graphene depends on the surface curvature. Considering a single junction between flat and rippled graphene subsystems, it is found an accumulation of charge in the rippled subsystem due to Klein penetration phenomenon that gives rise to n-p junction. Using this fact, we show that the momentum distribution of electrons in ballisitically propagating beam can be selective without a waveguide, or external electric, and/or magnetic fields in graphene strip under experimentally feasible one-dimensional periodic potential. Such a potential …
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…
We present a theory of quantum-coherent transport through a lateral p-n-p structure in graphene, which fully accounts for the interference of forward and backward scattering on the p-n interfaces. The backreflection amplitude changes sign at zero incidence angle because of the Klein phenomenon, adding a phase $π$ to the interference fringes. The contributions of the two p-n interfaces to the phase of the interference cancel with each other at zero magnetic field, but become imbalanced at a finite field. The resulting half a period shift in the Fabry-Perot fringe pattern, induced by a relativel…
Tunnelling of electrons in graphene-based junctions is studied theoretically. Graphene is assumed to be deposited either directly on a ferromagnetic insulator or on a few atomic layers of boron nitride which separate graphene from a metallic ferromagnetic substrate. Such junctions can be formed by appropriate external gating of the corresponding system. To describe low-energy electronic states near the Dirac points, certain effective Hamiltonians available in the relevant literature are used. These Hamiltonians include staggered potential and exchange interaction due to ferromagnetic substrate…
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…
Engineering of cooling mechanisms is a bottleneck in nanoelectronics. Whereas thermal exchanges in diffusive graphene are mostly driven by defect assisted acoustic phonon scattering, the case of high-mobility graphene on hexagonal Boron Nitride (hBN) is radically different with a prominent contribution of remote phonons from the substrate. A bi-layer graphene on hBN transistor with local gate is driven in a regime where almost perfect current saturation is achieved by compensation of the decrease of the carrier density and Zener-Klein tunneling (ZKT) at high bias. Using noise thermometry, we s…
We theoretically study a current switch that exploits the phase acquired by a charge carrier as it tunnels through a potential barrier in graphene. The system acts as an interferometer based on an armchair graphene quantum ring, where the phase difference between interfering electronic wave functions for each path can be controlled by tuning either the height or the width of a potential barrier in the ring arms. By varying the parameters of the potential barriers the interference can become completely destructive. We demonstrate how this interference effect can be used for developing a simple …
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…
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 …
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…
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…
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.
Andreev reflection at a superconductor and Klein tunneling through an n-p junction in graphene are two processes that couple electrons to holes -- the former through the superconducting pair potential Delta and the latter through the electrostatic potential U. We derive that the energy spectra in the two systems are identical, at low energies E<<Delta and for an antisymmetric potential profile U(-x,y)=-U(x,y). This correspondence implies that bipolar junctions in graphene may have zero density of states at the Fermi level and carry a current in equilibrium, analogously to superconducting Josep…
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…
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…
Within an effective Dirac theory the low-energy dispersions of monolayer graphene in the presence of Rashba spin-orbit coupling and spin-degenerate bilayer graphene are described by formally identical expressions. We explore implications of this correspondence for transport by choosing chiral tunneling through pn and pnp junctions as a concrete example. A real-space Green's function formalism based on a tight-binding model is adopted to perform the ballistic transport calculations, which cover and confirm previous theoretical results based on the Dirac theory. Chiral tunneling in monolayer gra…
One of the unique features of graphene is that the Fermi wavelength of its charge carriers can be tuned electrostatically over a wide range. This allows in principle to tune the transparency of a pn-junction electrostatically, as this depends on the ratio between the physical extension of the junction and the electron wavelength, i.e. on the effective width of the junction itself. However, this simple idea - which would allow to switch smoothly between a Veselago lens and a Klein-collimator - has proved to be difficult to demonstrate experimentally because of the limited amount of independentl…
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 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…
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…
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…
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…
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 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.
The Hall (RH) and bend (RB) resistance of a graphene Hall bar structure containing a pn-junction are calculated when in the ballistic regime. The simulations are done using the billiard model. Introducing a pn-junction--dividing the Hall bar geometry in two regions--leads to two distinct regimes exhibiting very different physics: 1) both regions are of n-type and 2) one region is n-type and the other p-type. In regime (1) a 'Hall plateau'--an enhancement of the resistance--appears for RH. On the other hand, in regime (2), we found a negative RH, which approaches zero for large B. The bend resi…
Hydrogenated graphene edges are assumed to be either armchair, zigzag or a combination of the two. We show that the zigzag is not the most stable fully hydrogenated structure along the <2-1-10> direction. Instead hydrogenated Klein and reconstructed Klein based edges are found to be energetically more favourable, with stabilities approaching that of armchair edges. These new structures "unify" graphene edge topology, the most stable flat hydrogenated graphene edges always consisting of pairwise bonded C2H4 edge groups, irrespective the edge orientation. When edge rippling is included, CH3 edge…
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.
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.
Graphene is characterized by chiral electronic excitations. As such it provides a perfect testing ground for the production of Klein pairs (electron/holes). If confirmed, the standard results for barrier phenomena must be reconsidered with, as a byproduct, the accumulation within the barrier of holes.
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…
In this paper, we use the Kaluza-Klein approach to describe topological defects in a graphene layer. Using this approach, we propose a geometric model allowing to discuss the quantum flux in $K$-spin subspace. Within this model, the graphene layer with a topological defect is described by a four-dimensional metric, where the deformation produced by the topological defect is introduced via the three-dimensional part of metric tensor, while an Abelian gauge field is introduced via an extra dimension. We use this new geometric model to discuss the arising of topological quantum phases in a graphe…
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…
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 inherent asymmetry of the electric transport in graphene is attributed to Klein tunneling across barriers defined by $\textit{pn}$-interfaces between positively and negatively charged regions. By combining conductance and shot noise experiments we determine the main characteristics of the tunneling barrier (height and slope) in a high-quality suspended sample with Au/Cr/Au contacts. We observe an asymmetric resistance $R_{\textrm{odd}}=100-70$ $Ω$ across the Dirac point of the suspended graphene at carrier density $|n_{\rm G}|=0.3-4 \cdot 10^{11}$ cm$^{-2}$, while the Fano factor displays …
We propose and analyze the terahertz (THz) detectors based on a gated graphene p-i-n (GPIN) field-effect transistor (FET) structure. The reverse-biased i-region between the gates plays the role of the electrons and holes injectors exhibiting nonlinear $I-V$ characteristics due to the Zener-Klein tunneling. This region enables the THz signal rectification, which provides their detection. The gated regions serve as the electron and hole reservoirs and the THz resonant plasma cavities. The resonant excitation of the electron and hole plasmonic oscillations results in a substantial increase in the…
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.
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…
In the present work, we investigate how structural defects in graphene can change its transport properties. In particular, we show that breaking of the sublattice symmetry in a graphene monolayer overcomes the Klein effect, leading to confined states of massless Dirac fermions. Experimentally, this corresponds to chemical bonding of foreign atoms to carbon atoms, which attach themselves to preferential positions on one of the two sublattices. In addition, we consider the scattering off a tensor barrier, which describes the rotation of the honeycomb cells of a given region around an axis perpen…
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…
We show that Andreev reflection in a junction between ferromagnetic (F) and superconducting (S) graphene regions is fundamentally different from the common FS junctions. For a weakly doped F graphene with an exchange field $h$ larger than its Fermi energy $E_{\rm F}$, Andreev reflection of massless Dirac fermions is associated with a Klein tunneling through an exchange field p-n barrier between two spin-split conduction and valence subbands. We find that this Andreev-Klein process results in an enhancement of the subgap conductance of a graphene FS junction by $h$ up to the point at which the …
Quantum resonant tunneling behaviors of double-barrier structures on graphene are investigated under the tight-binding approximation. The Klein tunneling and resonant tunneling are demonstrated for the quasiparticles with energy close to the Dirac points. The Klein tunneling vanishes by increasing the height of the potential barriers to more than 300 meV. The Dirac transport properties continuously change to the Schrodinger ones. It is found that the peaks of resonant tunneling approximate to the eigen-levels of graphene nanoribbons under appropriate boundary conditions. A comparison between t…
Electrons in graphene, behaving as massless relativistic Dirac particles, provide a new perspective on the relation between condensed matter and high-energy physics. We discuss atomic collapse, a novel state of superheavy atoms stripped of their discrete energy levels, which are transformed into resonant states. Charge impurities in graphene provide a convenient condensed matter system in which this effect can be explored. Relativistic dynamics also manifests itself in another system, graphene p-n junctions. We show how the transport problem in the presence of magnetic field can be solved with…
We present a method for obtaining quantum transport properties in graphene that uniquely combines three crucial features: microscopic treatment of charge disorder, fully quantum mechanical analysis of transport, and the ability to model experimentally relevant system sizes. As a pertinent application we study the disorder dependence of Klein tunneling dominated transport in p-n-p junctions. Both the resistance and the Fano factor show broad resonance peaks due to the presence of quasi bound states. This feature is washed out by the disorder when the mean free path becomes of the order of the d…
We investigate the transport properties of charge carriers in AB bilayer graphene through a triple electrostatic barrier. We calculate the transmission and reflection using the continuity conditions at the interfaces of the triple barrier together with the transfer matrix method. First, we consider the case where the energy is less than the interlayer coupling $γ_1$ and show that, at normal incidence, transmission is completely suppressed in the gap for a large barrier width while it appears in the gap for a small barrier width. For energies greater than $γ_1$, we show that in the absence of a…
