In this case, the transmission probability, consequences of quantum electrodynamics. The T , depends only weakly on the barrier height, approaching the perfect transparency for very high barriers, in stark contrast to phenomenon is discussed in many contexts in particle, the conventional, non-relativistic tunnelling where T exponentially decays with increasing V0. This relativistic effect can be attributed nuclear and astro-physics but direct tests of the Klein to the fact that a sufficiently strong potential, being repulsive paradox using elementary particles have so far proved for electrons, is attractive for positrons and results in positron states inside the barrier, which align in energy with the electron impossible. Here we show that the effect can be tested in continuum outside4—6. Matching between electron and positron wavefunctions across the barrier leads to the high-probability a conceptually simple condensed-matter experiment using tunnelling described by the Klein paradox7. The essential feature of quantum electrodynamics QED responsible for the effect is electrostatic barriers in single- and bi-layer graphene.
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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 using electrostatic barriers in single- and bi-layer graphene. Owing to the chiral nature of their quasiparticles, quantum tunnelling 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, non-relativistic tunnelling where T exponentially decays with increasing V0. This relativistic e? Matching between electron and positron wavefunctions across the barrier leads to the high-probability tunnelling described by the Klein paradox7.
The essential feature of quantum electrodynamics QED responsible for the e? This fundamental property of the Dirac equation is often referred to as the charge-conjugation symmetry. The purpose of this paper is to show that graphene—a recently found allotrope of carbon9—provides an e? The red and green curves emphasize the origin of the linear spectrum, which is the crossing between the energy bands associated with crystal sublattices A and B.
The three diagrams in a schematically show the positions of the Fermi energy E across such a barrier. The Fermi level dotted lines lies in the conduction band outside the barrier and the valence band inside it. The blue? The spectrum is isotropic and, despite its parabolicity, also originates from the intersection of energy bands formed by equivalent sublattices, which ensures charge conjugation, similar to the case of single-layer graphene.
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 chargeconjugation symmetry in QED10— 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. E propagating in the opposite direction. This allows the introduction of chirality12, that is formally a projection of pseudospin on the direction of motion, which is positive and negative for electrons and holes, respectively.
Neglecting many-body e? The general scheme of such an experiment is shown in Fig. The electron concentration n outside the barrier is chosen as 0. The barrier heights V0 are a and b 50 meV red curves and a and b meV blue curves. For simplicity, we assume in 2 in? The sharp-edge assumption is justi? Such a barrier can be created by the electric? Importantly, Dirac fermions in graphene are massless and, therefore, there is no formal theoretical requirement for the minimal electric?
To create a well-de? It is straightforward to solve the tunnelling problem shown in Fig. Requiring the continuity of the wavefunction by matching up coe? In the limit of high barriers V0 E , the expression for T can be simpli?
More signi? The latter is the feature unique to massless Dirac fermions and is directly related to the Klein paradox in QED. This perfect tunnelling can be understood in terms of the conservation of pseudospin. Indeed, in the absence of pseudospin-? This is shown in Fig. The latter scattering event would require the pseudospin to be? In the 1. Our analysis extends this 0.
There are di? Indeed, charge carriers in bilayer graphene have a parabolic energy spectrum as shown in Fig. On the other hand, these quasiparticles are also chiral and described by spinor wavefunctions20,21, 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 In addition, the relevant QED-like e? Charge carriers in bilayer graphene are described by an o? Transmission probability T for normally incident electrons in single- and bi-layer graphene red and blue curves, respectively and in a non-chiral zero-gap semiconductor green curve as a function of width D of the tunnel barrier.
This yields barrier heights of? Note that the transmission probability for bilayer graphene decays exponentially with the barrier width, even though there are plenty of electronic states inside the barrier. An important formal di? Two of them correspond to propagating waves and the other two to evanescent waves. However, for any? Examples of the angular dependence of T in bilayer graphene are plotted in Fig.
They show a dramatic di? 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 analysed this case in more detail and found the following analytical solution for the transmission coe?
The case of a potential step, which corresponds to a single p—n junction, is particularly interesting. Equation 6 shows that such a junction should completely re? This is highly unusual because the continuum of electronic states at the other side of the step is normally expected to allow some tunnelling. This behaviour is in obvious contrast to single-layer graphene, where normally incident electrons are always perfectly transmitted.
The perfect re? For single- b layer graphene, an electron wavefunction at the barrier interface perfectly matches the corresponding wavefunction for a hole with the same direction of pseudospin see Fig.
In contrast, for bilayer graphene, the charge conjugation requires a propagating electron with wavevector k to transform into a hole with wavevector ik rather than? If a tunnel barrier contains no electronic states, the di? However, both graphene systems are gapless, and it is more appropriate to compare them with gapless semiconductors with non-chiral charge carriers such a situation can be realized in certain heterostructures23, In this case, we?
This makes it clear that the drastic di? In conventional 2D systems, strong enough disorder results in electronic states that are separated by barriers with exponentially small transparency25, This is known to lead to the Anderson localization. Therefore, di? This Figure 4 The chiral nature of quasiparticles in graphene strongly affects its transport properties. For normal electrons, transmission probability T through such a system depends strongly on the distribution of scatterers. In contrast, for massless Dirac fermions, T is always equal to unity due to the additional memory about the initial direction of pseudospin see text.
Graphene light blue has two local gates dark blue that create potential barriers of a variable height. The voltage drop across the barriers is measured by using potential contacts shown in orange.
To further elucidate the dramatic di? For conventional 2D systems, transmission and re? 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 proved 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 compared with that of normal conductors. The above analysis shows that the Klein paradox and associated relativistic-like phenomena can be tested experimentally using graphene devices. The basic principle behind such experiments would be to use local gates and collimators similar to those used in electron optics in 2D gases29, One possible experimental setup is shown schematically in Fig.
Here, local gates simply cross the whole graphene sample at di? Intrinsic concentrations of charge carriers are usually low? By measuring the voltage drop across the barriers as a function of applied gate voltage, their transparency for di? Figure 2 shows that for graphene the 90? In comparison, the 45? The situation should be qualitatively di?
Furthermore, the fact that a barrier or even a single p—n junction incorporated in a bilayer graphene device should lead to exponentially small tunnelling current can be exploited in developing graphene-based? Such transistors are particularly promising because of their high mobility and ballistic transport at submicron distances9,13, However, the fundamental problem along this route is that the conducting channel in single-layer graphene cannot be pinched o?
A bilayer FET with a local gate inverting the sign of charge carriers should yield much higher on—o? Received 18 April ; accepted 20 June ; published 20 August References 1. Klein, O.
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Chiral tunnelling and the Klein paradox in graphene