Wednesday, September 06, 2006

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 (

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 (

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


  1. Stefan, another facinating post!

    A number of years ago my interest in graphene was directed to the process detailed in your post, so I will ask a question if I may?

    Q)In conventional everyday batteries such a AA,AAA (camera's type etc..) what is the difference in weight of a "full" and "empty" battery?

    Does the loss of charge relate to a loss of weight?

    There are interesting probabilities that "vacuum-charge" can be manipulated to construct batteries that exhibit free_energy ;)

    If one looks at some interesting papers:

    for instance, the frog_in_a_levetation_box ?

  2. Hi Stefan,
    well written science post.
    Clear & concise overview & details.

    Thanks also for the links to:
    "news item at Science: Black Hole in a Pencil."
    The uni augsburg and
    Mesoscopic Physics group
    have some awesome images too

    I'm hoping to do another post on blackholes next week, hope to offer a link to this one then. - Q.

  3. Hi paul,

    I think you might of be refering to this link you gave me some time back.

  4. On the Quantum Hall Effect

    Bee might offer her opinion on blackholes in this regard?

  5. A nucleus with Z~1/alpha (reciprocal of the Fine Structure constant, ~137) would spontaneously spark the vacuum and inverse beta decay. Z~170 seems rather unlikely.

    Expanding graphite layer by layer is old hat - almost any strong Lews acid or base will intercalate, including volumetically big ones like SbF5 and dibenzenechromium. Grafoil gasket material is exfoliated reconsolidated graphite.

    "Spectacular Pseudo-Exfoliation of an Exfoliated–Compressed Graphite"
    J. Chem. Ed. 81 819 (2004)

    One could have great physics fun given even modest chemistry circumstances.

  6. Paul Valletta,

    The weight change between a full and empty battery is very, very small. The use of the term "charge" with batteries is quite loose: a battery works by having a chemical reaction that wants to happen, but cannot unless electrons flow out of one terminal of the battery, through some electric circuit, and back in the other. Thus, there's no net change in the charge of the battery in the particle sense. Every electron that comes out is replaced by another going in. The only change in mass is due to the difference in potential energy between a "charged" and "uncharged" battery.

    Energizer says their alkaline AA batteries hold 2850 mAh, discharging under best conditions of 25 mA for 114 hours. At the nominal 1.5 volts an upper bound on the energy is 25 mA * 1.5 Volts * 114 hours or

    0.025 A * 1.5 V * 114 hours * 3600 s/hour = 15390 Joules (wow! did I do that right?)

    Now, by E=mc^2 we can figure out how much of a mass difference that 15 kJ makes. m = E/c^2 = 15 kJ / (3*10^8m/2)^2 = 1.7*10^-19 kg. That's rather more than I was expecting, but still a very small number.

    Static electricity is a case where you get real charge accumulation, unlike with batteries. In that sort of charging you get a change in weight both from the potential energy of the charges and from the mass of the extra electrons that have moved from one place to another.

  7. Hi rillian,

    thank you for your excellent and exhaustive answer to pauls question - you spared me some wirk, since I was about to write roughly the same ;-)

    Hi Uncle Al,

    the "sparking vacuum" would set in for nuclei with Z = 137 if the nucleus was exactly point-like. Since in reality, it has an extension, and since the s electrons are diving into the nucleus, they effectively feel a weaker electric field. That's why atomic numbers higher than 137 are necessary. Frankfurt physicists have spent years investigating on this issue, starting with W. Pieper and W. Greiner: Interior electron shells in superheavy nuclei, Zeitschrift für Physik A 218 (1969) 327-340. You can follow up the citing literature...

    That the sparking vakuum, as they called it, could show up with such superheavy nuclei was one of the motivations to found the GSI in Darmstadt. There, the hope was that one could detect tell-tale signals of positrons which would have been created in the supercritical electric field of superheavy compound nuclei which would live for some time in and after the collision.

    There has been a long dabate in the 1980/90s if such positrons signals have, indeed, been detected, but it seems that all measurements so far are inconclusive. You can find a kind of status report on that issue in the GSI-News 2/1999. As far as I know, there have been no new attempts since to detect signals of the sparking vacuum.

    Best, stefan

  8. Thanks for the heads up! It is in the footnotes between theory and reality where the fun hides. Somebody do an Equivalence Principle parity test! Is there a chiral pseudoscalar vacuum background? Affine and teleparallel gravitation predict an observable EP parity anomaly. GR less Cartan's affine connection cannot handle observed Earth-moon spin-orbit momentum exchange. Take the hint.

    Building Z~170 nuclei, if it can be done at all, would require decades and $billions (in any currency). An EP parity test sensitive to 3x10^(-18) divergence is about $(US)100 in consummables, a week's preparation, and two days' work in two borrowed calorimeters. $100? Piddles.
    Somebody should look...
    ...because it is wickedly clever.

  9. The "Klein paradox" has another interesting implication. A strong enough "repulsive" potential plugged in to the Dirac equation actually has bound state solutions, counter to what one would expect. So if you treat the Maxwell-Dirac equations as a coupled set of PDE's, a strongly concentrated spinor field (a charged spinning lump) produces the sufficiently "repulsive" electric field necessary to bind and shape the lump. This gives a set of soliton solutions to the Maxwell-Dirac equations. Curious, eh?


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