Thank you for visiting nature. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser or turn off compatibility mode in Internet Explorer. In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript. A Nature Research Journal. 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.

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Read this paper on arXiv. 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 quasiparticles, quantum tunneling in these materials becomes highly anisotropic, qualitatively different from the case of normal, nonrelativistic electrons. In this case, the transmission probability T depends only weakly on the barrier height, approaching the perfect transparency for very high barriers, in stark contrast to the conventional, nonrelativistic tunneling where T exponentially decays with increasing V 0.

This relativistic effect can be attributed to the fact that a sufficiently strong potential, being repulsive for electrons, is attractive for positrons and results in positron states inside the barrier, which align in energy with the electron continuum outside su ; dombey ; dombey1.

Matching between electron and positron wavefunctions across the barrier leads to the high-probability tunneling described by the Klein paradox krekora. The essential feature of QED responsible for the effect is the fact that states at positive and negative energies electrons and positrons are intimately linked conjugated , being described by different components of the same spinor wavefunction.

This fundamental property of the Dirac equation is often referred to as the charge-conjugation symmetry. Graphene is a single layer of carbon atoms densely packed in a honeycomb lattice, or it can be viewed as an individual atomic plane pulled out of bulk graphite.

From the point of view of its electronic properties, graphene is a two-dimensional zero-gap semiconductor with the energy spectrum shown in Fig. Neglecting many-body effects, this description is accurate theoretically slon ; semenoff ; haldane and has also been proven experimentally kostya2 ; kim by measuring the energy-dependent cyclotron mass in graphene which yields its linear energy spectrum and, most clearly, by the observation of a relativistic analogue of the integer quantum Hall effect.

Quantum mechanical hopping between the sublattices leads to the formation of two cosine-like energy bands, and their intersection near the edges of the Brillouin zone shown in red and green in Fig. Although the linear spectrum is important, it is not the only essential feature that underpins the description of quantum transport in graphene by the Dirac equation.

Above zero energy, the current carrying states in graphene are, as usual, electron-like and negatively charged. At negative energies, if the valence band is not full, its unoccupied electronic states behave as positively charged quasiparticles holes , which are often viewed as a condensed-matter equivalent of positrons. In contrast, electron and hole states in graphene are interconnected, exhibiting properties analogous to the charge-conjugation symmetry in QED slon ; semenoff ; haldane.

There are further analogies with QED. The conical spectrum of graphene is the result of intersection of the energy bands originating from sublattices A and B see Fig. This allows one to introduce chirality haldane , that is formally a projection of pseudospin on the direction of motion, which is positive and negative for electrons and holes, respectively. Because quasiparticles in graphene accurately mimic Dirac fermions in QED, this condensed matter system makes it possible to set up a tunneling experiment similar to that analyzed by Klein.

The general scheme of such an experiment is shown in Fig. This local potential barrier inverts charge carriers underneath it, creating holes playing the role of positrons, or vice versa. Importantly, Dirac fermions in graphene are massless and, therefore, there is no formal theoretical requirement for the minimal electric field E to form positron-like states under the barrier.

It is straightforward to solve the tunneling problem sketched in Fig. Requiring the continuity of the wavefunction by matching up coefficients a , b , t , r , we find the following expression for the reflection coefficient r.

The latter is the feature unique to massless Dirac fermions and directly related to the Klein paradox in QED. One can understand this perfect tunneling in terms of the conservation of pseudospin. Indeed, in the absence of pseudospin-flip processes such processes are rare as they require a short-range potential, which would act differently on A and B sites of the graphene lattice , an electron moving to the right can be scattered only to a right-moving electron state or left-moving hole state.

This is illustrated in Fig. The latter scattering event would require the pseudospin to be flipped. Our analysis extends this tunneling problem to the two-dimensional 2D case of graphene.

To elucidate which features of the anomalous tunneling in graphene are related to the linear dispersion and which to the pseudospin and chirality of the Dirac spectrum, it is instructive to consider the same problem for bilayer graphene. There are both differences and similarities between the two graphene systems. Indeed, charge carriers in bilayer graphene have parabolic energy spectrum as shown in Fig.

On the other hand, these quasiparticles are also chiral and described by spinor wavefunctions bilayer ; falko , similar to relativistic particles or quasiparticles in single-layer graphene. Again, the origin of the unusual energy spectrum can be traced to the crystal lattice of bilayer graphene with four equivalent sublattices falko. In addition, the relevant QED-like effects appear to be more pronounced in bilayer graphene and easier to test experimentally, as discussed below.

Two of them correspond to propagating waves and the other two to evanescent ones. Examples of the angular dependence of T in bilayer graphene are plotted in Fig. They show a dramatic difference as compared with the case of massless Dirac fermions.

There are again pronounced transmission resonances at some incident angles, where T approaches unity. However, instead of the perfect transmission found for normally-incident Dirac fermions see Fig. Accordingly, we have analyzed this case in more detail and found the following analytical solution for the transmission coefficient t :. This is highly unusual because the continuum of electronic states at the other side of the step is normally expected to allow some tunneling.

This behavior is in obvious contrast to single-layer graphene, where normally-incident electrons are always perfectly transmitted. For single-layer graphene, an electron wavefunction at the barrier interface matches perfectly the corresponding wavefunction for a hole with the same direction of pseudospin see Fig. For completeness, we compare the obtained results with the case of normal electrons. In this case, one finds.

This makes it clear that the drastic difference between the three cases is essentially due to different chiralities or pseudospins of the quasiparticles involved rather than any other feature of their energy spectra.

The found tunneling anomalies in the two graphene systems are expected to play an important role in their transport properties, especially in the regime of low carrier concentrations where disorder induces significant potential barriers and the systems are likely to split into a random distribution of p-n junctions. This is known to lead to the Anderson localization. To elucidate further the dramatic difference between quantum transport of Dirac fermions in graphene and normal 2D electrons, Fig.

In contrast, the conservation of pseudospin in graphene strictly forbids backscattering and makes the disordered region in Fig. This extension of the Klein problem to the case of a random scalar potential has been proven by using the Lippmann-Schwinger equation see the Supplementary Information.

Nevertheless, the above consideration shows that impurity scattering in the bulk of graphene should be suppressed as compared to the normal conductors. The above analysis shows that the Klein paradox and associated relativistic-like phenomena can be tested experimentally using graphene devices. One possible experimental setup is shown schematically in Fig.

Here, local gates simply cross the whole graphene sample at different angles for example, 90 o and 45 o. By measuring the voltage drop across the barriers as a function of applied gate voltage, one can analyze their transparency for different V 0. Our results in Fig. In comparison, the 45 o barrier is expected to have much higher resistance and show a number of tunneling resonances as a function of gate voltage.

The situation should be qualitatively different for bilayer graphene, where local barriers should result in a high resistance for the perpendicular barrier and pronounced resonances for the 45 o barrier. Such transistors are particularly tempting because of their high mobility and ballistic transport at submicron distances kostya1 ; kostya2 ; kim. However, the fundamental problem along this route is that the conducting channel in single-layer graphene cannot be pinched off because of the minimal conductivity , which severely limits achievable on-off ratios for such FETs kostya1 and, therefore, the scope for their applications.

A bilayer FET with a local gate inverting the sign of charge carriers should yield much higher on-off ratios. We have shown that the recently found two carbon allotropes provide an effective medium for mimicking relativistic quantum effects.

On the one hand, this allows one to set up such exotic experiments as the one described by the Klein paradox and could be useful for analysis of other relevant QED problems.

On the other hand, our work also shows that the known QED problems and their solutions can be applied to graphene to achieve better understanding of transport properties of this unique material that is interesting from the view point of both fundamental physics and applications. Chiral tunneling and the Klein paradox in graphene M. Abstract 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.

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## Chiral Tunneling and the Klein Paradox in Graphene

Skip to search form Skip to main content You are currently offline. Some features of the site may not work correctly. DOI: Geim Published Physics Nature Physics. 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.

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Read this paper on arXiv. 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 quasiparticles, quantum tunneling in these materials becomes highly anisotropic, qualitatively different from the case of normal, nonrelativistic electrons. In this case, the transmission probability T depends only weakly on the barrier height, approaching the perfect transparency for very high barriers, in stark contrast to the conventional, nonrelativistic tunneling where T exponentially decays with increasing V 0. This relativistic effect can be attributed to the fact that a sufficiently strong potential, being repulsive for electrons, is attractive for positrons and results in positron states inside the barrier, which align in energy with the electron continuum outside su ; dombey ; dombey1.