Pencils, Black Holes, and the Klein Paradox

Yesterday morning, when scanning the news at spiegel online, a headline in the science section made me curious: Bleistift statt schwarzer Löcher, pencils instead of black holes. That short piece turned out to be a quite sensible description of a recent experiment on the Klein paradox in single layers of graphite. There was even a link to the original paper on the archiv: cond-mat/0604323. I had heard before of the funny features of electrons in graphene, as these single atomic layers of graphite are called, and I followed up the story. While I am not an expert on these things, I find them quite remarkable and interesting.

In 1929, Oskar Klein, of Klein-Gordon and Kaluza-Klein fame, applied the Dirac equation to the typical textbook problem of an electron hitting a potential barrier. While in nonrelativistic quantum mechanics, the electron can tunnel into the barrier, albeit with an exponential damping, in the relativistic problem, something strange happens if the the barrier is on the order of the electron mass, V ∼ mc². Then, as Klein found out, the barrier is nearly transparent for the electron, and even perfectly transparent in the limit of infinite barrier height.


Oskar Klein (www-groups.dcs.st-and.ac.uk/~history/Biographies/Klein_Oskar.html)


This very odd situation, called the Klein paradox, is nowadays usually explained by the effect of pair creation: The barrier, which is repulsive for electrons, is attractive for positron. Thus, there are positron states inside the barrier with the same energy level as the incoming electron state. This means that electron-positron pairs are created, which are responsible for the transparency of the barrier.

A steep and high potential barrier implies a very strong electric field. The pair creation at the barrier thus corresponds to the pair creation in strong fields. Experimental evidence for this effect - the so-called charged vacuum - was long sought-after in heavy ion collision, but so far without success. The problem is that electric fields strong enough for the spontaneous creation of electron-positron pairs occur only in the vicinity of superheavy nuclei, with Z ∼ 170. Such nuclei do not exist in nature - they have to be created, albeit for a very short while, in heavy ion collisions.

The problem with the experimental verification of spontanous pair creation in high-energy physics is, obviously, the electron mass, which necessitates very strong fields. Things would be much easier if one would have massless charged Dirac particles at hand. Enters the graphene:

Carbon atoms in graphite from very neat layers with a hexagonal, honeycomb structure. This layered arrangement of the atoms explains nicely the properties of graphite, such as its suppleness, which is why it is used in pencils. There are now even pictures of the honeycomb structure, thanks to atomic force microscopy:


Graphite layer (www.physik.uni-augsburg.de/exp6/imagegallery/afmimages/afmimages_e.shtml)


Graphite is a quite good electric conductor. If one prepares single layers of graphite, or graphene, the conductance electrons are constrained to this layer. Now, in this two-dimensional system, the peculiar hexagonal structure leads to a linear relation between momentum and energy for the excitation of conductance electrons. Thus, these electronic excitations behave as massless Dirac fermions, instead of massive electrons! This remarkable feature has been exploited in several recent experiments - and one of these experiments is the experimental study of the Klein paradox referred to in the spiegel piece.

The barrier in the experiment is created by some semiconductor material inserted into the graphene layer. Applying different electrostatic potentials to the semiconductor, the barrier height for the massless quasi-electrons can be tuned. Now, potential differences of some 100 meV instead of some 0.5 MeV do the job for reaching the regime of the pair creation and the Klein paradox. In the experiment, reflexion and transmission coefficients are measured, and they correspond neatly to Klein's calculations!

This is definitely one more example where some of the standard textbook situations of quantum mechanics is, actually, realised in a beautiful experiment.

Much more information about the experiment, and the special features of graphene, can be found on the News and Publications web page of the Mesoscopic Physics Group at the University of Manchester who actually started the experimental exploration of graphene, and did the Klein paradox experiment. For the Klein paradox as such, I am studying now a paper from the arxiv, quant-ph/9905076.

But what about the Black Holes? The spiegel piece probably took it form a news item at Science: Black Hole in a Pencil.

I guess, Bee is much more qualified to comment on that, once she will find a little time to breathe. The point is, I suppose, is that charged small black holes would naturally provide strong enough electric fields for pair creation, and thus for testing situations as in the Klein paradox in experiment. If only charged black holes could be produced more easily than nuclei with Z = 170...

Best, stefan

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Physics Klein paradox Graphene Pair Creation