NEWS & VIEWS
RELATIVISTIC QUANTUM MECHANICS
Paradox in a pencil
The 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.
a
ALEX CALOGERACOS
is 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: [email protected]
b
Vacant
electron
level
mc2
0
–m c 2
mc2
0
–mc2
Increasing the potential
causes vacant level to
merge with continuum
Negative
energy
continuum
n 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 reflected directly off it as everyday
experience suggests it should. This 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 field 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.
The 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 nonrelativistic, 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.
I
c
d
State filled with
a bound electron
mc2
0
–m c 2
mc2
0
–
–mc2
+
Free positron
is promoted to
the continuum
Charge–conjugation
invariance
and the time-independent
Dirac equation leads
to a state through
which an electron can tunnel
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 confines 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 positroncontinuum. The positron wavefunction extends well beyond the confines 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 satisfies the resonant condition qD = 2πN. In electrodynamics
this condition is satisfied when an electron’s energy becomes vanishingly small. In graphene the
same condition is satisfied rather more easily.
To understand what this has to do with Klein,
we should first 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 infinity) 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
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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. This process amounts to creation of an
electron–positron pair from a strong time-varying
external field. The pre-eminent Russian physicist
Yakov Zel’dovich, who did pioneering work in the
physics of strong fields, 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 different context
of quasiparticle creation by an oscillating wire in
superfluid 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). This is closely connected to
Klein’s original paradox9.
The 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 nonzero barrier field should be sufficient 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
580
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. They
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 find 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 effects 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 effort to find
experimentally accessible analogues in which
to study otherwise inaccessible relativistic and
quantum electrodynamic effects. Electron–positron
creation in the Klein and Zel’dovich effects, creation
of photon modes in a cavity (the dynamical Casimir
effect), and photon creation during black-hole
collapse are just a few examples of such phenomena
that could soon yield to this effort. Superfluid
helium, in particular, has offered us a good testing
ground for many exotic theoretical concepts10. With
graphene we may well have a veritable particle
physics laboratory at our fingertips.
REFERENCES
1. 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 The Oskar Klein Centenary: Proceedings of the Symposium 19-21
September 1994, Stockholm, Sweden (ed. Lindstrom, U.) (World Scientific,
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. The Universe in a Helium Droplet (Clarendon, Oxford, 2003).
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©2006 Nature Publishing Group