NEWS & VIEWS

nature physics | VOL 2 | SEPTEMBER 2006 | www.nature.com/naturephysics 579

NEWS & VIEWS

RELATIVISTIC QUANTUM MECHANICS

Paradox in a pencilThe Klein paradox, which relates to the ability of relativistic particles to pass through extreme potential barriers, could be yet another of the strange quantum phenomena made accessible by the properties of graphene.

ALEX CALOGERACOSis in the Division of Theoretical Mechanics, Hellenic Air Force Academy TG1010, Dekelia Air Force Base, Greece, and in the Department of Physics and Astronomy, University of Sussex, Brighton BN1 9QH, UK.

e-mail: a.calogeracos@sussex.ac.uk

In a 1929 paper, the Swedish physicist Oskar Klein proposed a thought experiment1 to explore the paradoxical implications of the Dirac

equation, which had been formulated just the year before by Paul Dirac2 to provide a description of high-energy particles that included both quantum mechanics and special relativity. A consequence of this experiment is that when a particle travelling at a slow speed encounters a barrier whose height is more than twice its rest energy, it is far more likely to pass right through the barrier as if it were not there, than to be refl ected directly off it as everyday experience suggests it should. Th is counterintuitive result has come to be known as the Klein paradox3. Owing to the extreme conditions involved — a potential step of twice the rest energy of an electron would require an electric fi eld of more than 1016 V cm–1 — it is only expected to occur under extreme circumstances. But on page 620 of this issue, Mikhail Katsnelson and colleagues suggest it could soon be possible to observe the Klein paradox in any university electronics laboratory, through the peculiar relativistic behaviour of the quasiparticles present in graphene4.

Th e remarkable conducting properties of graphene — which consists of a single atomic layer of graphite — were established only recently5–7. Unlike in bulk graphite, the carriers responsible for charge conduction in graphene behave as if they are massless, being governed by a linear relationship between energy and momentum (like photons travelling in free space), are spin-1/2, and move at speeds of around 8 × 105 m s–1; such speeds are non-relativistic, as one expects in the condensed state, but very large in the context of the present analogy where the role of the speed of light is played by the Fermi velocity (106 m s–1). As a consequence, the carriers obey the Dirac equation and have come to be referred to as massless Dirac fermions. Crucially, the energy spectrum of graphene is symmetric between positively and negatively charged carriers.

To understand what this has to do with Klein, we should fi rst understand how the paradox emerges in the context of high-energy particle physics as an exotic phenomenon that can occur in the presence of an extreme potential. To this end, consider an attractive potential λV(x) whose shape is given by V(x) (which should reach zero at infi nity) and whose strength can be tuned by the parameter λ. If λ is allowed to vary in time so that it eventually exceeds a certain threshold value, the Dirac equation predicts that a bound particle state that was initially vacant merges with the positron continuum, at which point

mc2

–mc2

0

Vacant electron

level

Negative energy

continuum

mc2

–mc2

0

State filled with a bound electron

Free positron is promoted to the continuum

+

–

mc2

–mc2

0

Charge–conjugation invariance

and the time-independentDirac equation leads

to a state through which an electron can tunnel

mc2

–mc2

0 Increasing the potential causes vacant level to merge with continuum

ba

dc

Figure 1 The emergence of Klein tunnelling states as a consequence of the Dirac equation. a, Below a certain threshold, solving the time-dependent Dirac equation implies the existence of empty bound electron states with negative energy (which famously implied the very existence of the positron). Beyond the confi nes of the barrier, this state decays evanescently. b, Increasing the potential beyond this threshold pushes this state into the positron-continuum (which exists at negative energies and is directly analogous to the valence band of a semiconductor, and in which the absence of an electron implies the existence of a positron). c, Decreasing the potential back below the threshold traps an electron in the bound state, and leaves a free positron in the positron-continuum. The positron wavefunction extends well beyond the confi nes of the potential. d, In solving the time-independent Dirac equation in the presence of a repulsive potential, charge–conjugation invariance means that a state within the barrier must arise through which an electron can tunnel unhindered, so long as it satisfi es the resonant condition qD = 2πN. In electrodynamics this condition is satisfi ed when an electron’s energy becomes vanishingly small. In graphene the same condition is satisfi ed rather more easily.

nphys_09.06 N+V-print.indd 579nphys_09.06 N+V-print.indd 579 22/8/06 5:23:35 pm22/8/06 5:23:35 pm

Nature Publishing Group ©2006

NEWS & VIEWS

580 nature physics | VOL 2 | SEPTEMBER 2006 | www.nature.com/naturephysics

a positron will be emitted from the potential well. If the potential returns to its initial shallow value then the originally vacant bound state will be occupied by an electron. Th is process amounts to creation of an electron–positron pair from a strong time-varying external fi eld. Th e pre-eminent Russian physicist Yakov Zel’dovich, who did pioneering work in the physics of strong fi elds, suggested in the 1960s that this process of positron production could arise in collisions between heavy ions stripped of their inner electrons. A similar process in the diff erent context of quasiparticle creation by an oscillating wire in superfl uid helium-3 has also been suggested8.

Alternatively, if λ remains constant in time, we can exploit the fact that the Dirac equation is invariant to simultaneous complex conjugation and reversal of the sign of the potential — a property known as charge–conjugation invariance. A consequence of charge–conjugation invariance is that statements made about the interaction of an electron with an attractive potential can be immediately transferred to the interaction of a positron with a repulsive potential of the same shape. It turns out that for λ greater than the critical value there are particular values of the incident positron momentum for which the potential is transparent (referred to as scattering resonances). In the case of a one-dimensional square well, the condition for a resonance is similar to that encountered in conventional optics — namely that qD = 2πN, where q is the electron wavevector (equivalent to its momentum) inside the well, D is the well width and N is an integer. An especially counterintuitive result of this is that, in the presence of a very strong repulsive barrier, an electron with vanishingly small kinetic energy can go through the barrier unhindered (see Fig. 1). Th is is closely connected to Klein’s original paradox9.

Th e key observation made by Katsnelson et al.4 is that because the Dirac fermions in graphene are massless, generating a potential barrier whose height is twice their rest energy should be far less challenging than creating such a step for a conventional electron. Indeed, essentially any non-zero barrier fi eld should be suffi cient to generate a barrier high enough to test the paradox. Moreover, the symmetry between electrons and positrons that is inherent to the structure of the Universe

is analogous to a similar symmetry between negatively and positively charged Dirac fermions in graphene. With this in mind, the authors proceed to calculate how these particles will behave when encountering an arbitrary-valued potential. Th ey predict that at normal incidence on a 100-nm-wide barrier, the transmission probability for a Dirac fermion in graphene is unity, regardless of the barrier’s height. For angles off normal, however, they fi nd that the transmission (less than unity) becomes height-dependent. Moreover, they go on to predict complementary behaviour in bilayer graphene, where the relationship between carrier energy and momentum is parabolic, and whose transmission at normal incidence is zero, and at an oblique incidence can rise to unity. In addition to providing a fascinating glimpse of the kind of exotic physics that should soon be open to study in these systems, the ability to control the transmission of particles through a barrier suggests that these eff ects might also be used as the basis for future graphene device electronics.

In a broader context, this work represents just one example of a growing eff ort to fi nd experimentally accessible analogues in which to study otherwise inaccessible relativistic and quantum electrodynamic eff ects. Electron–positron creation in the Klein and Zel’dovich eff ects, creation of photon modes in a cavity (the dynamical Casimir eff ect), and photon creation during black-hole collapse are just a few examples of such phenomena that could soon yield to this eff ort. Superfl uid helium, in particular, has off ered us a good testing ground for many exotic theoretical concepts10. With graphene we may well have a veritable particle physics laboratory at our fi ngertips.

REFERENCES1. Klein, O. Z. Phys. 53, 157–165 (1929).2. Dirac, P. A. M. Proc. R. Soc. Lond. A 117, 610–624 (1928).3. Telegdi, V. in Th e Oskar Klein Centenary: Proceedings of the Symposium 19-21

September 1994, Stockholm, Sweden (ed. Lindstrom, U.) (World Scientifi c, Singapore, 1995).

4. Katsnelson, M. I., Novoselov, K. S. & Geim, A. K. Nature Phys. 2, 620–625 (2006).

5. Novoselov, K. S. et al. Nature 438, 197–200 (2005).6. Zhang, Y., Tan, Y. W., Stormer, H. L. & Kim, P. Nature 438, 201–204 (2005).7. Kane, C. L. Nature 438, 168–170 (2005). 8. Calogeracos, A. & Volovik, G. E. JETP Lett. 69, 281–287 (1999).9. Calogeracos, A. & Dombey, N. Contemp. Phys. 40, 313–321 (1999).10. Volovik, G. E. Th e Universe in a Helium Droplet (Clarendon, Oxford, 2003).

nphys_09.06 N+V-print.indd 580nphys_09.06 N+V-print.indd 580 22/8/06 5:23:43 pm22/8/06 5:23:43 pm

Nature Publishing Group ©2006