Röser's equation

At low temperatures, some materials display a feature known as "super-conductivity" - the total loss of electric resistance. Electrical currents in such a material, once initiated, propagate ideally forever. Super-conductivity sets in at a so-called "jump temperature," below which the material changes its electric properties. "Low" temperature in this case means indeed very low: The phenomenon was first observed on mercury with a jump temperature of only 4.19 Kelvin. Other jump temperatures for metallic compounds reach up to 23 Kelvin in Nb₃Ge. Bardeen, Cooper and Schrieffer got a Nobel prize in 1972 for their theoretical explanation of super-conductivity in these materials.

These temperatures seem way too low to ever make super-conductivity useful for daily life. But over the last two decades an increasing amount of substances has been discovered, the so-called "high temperature supra-conductors," (HTS), starting with the discovery of super-conductivity in barium lanthanum copper oxide at 35 Kelvin by Georg Bednorz and K. Alexander Müller in 1986, to the recent class of so-called iron pnictides.

For them, the jump temperature can be much higher, typically above 30 Kelvin, with the current record at 138 Kelvin hold by a ceramic oxide containing thallium, mercury, copper, barium, and calcium. However, this search has up to now been rather erratic with groups of researchers more or less systematically testing promising classes of laboratory-grown crystals. The jump-temperature of a crystal so far could not be predicted.

It is then quite astonishing that Hans-Peter Röser, professor at the Institute of Space Systems at the University of Stuttgart, Germany, found a simple equation relating the geometric structure of a crystal to its jump temperature. Röser's equation for the (critical) jump-temperature T_c reads

4 π k m_e(2 x)² n^-2/3 = h²/ T_c

where k is Boltzmann's constant, h is Planck's constant, m_e is the electron mass, x is the doping distance of the crystal and n is the number of supra-conducting layers in the crystal (it is usually 1,2, or 3). If one plots these quantities for known HTS one obtains the following graph, where the straight line is the prediction of Röser's equation:


(from: A Correlation Between Tc of Fe-based HT Superconductors and the Crystal Super Lattice Constants of the Doping Element Positions by Felix Huber, Hans Peter Roeser, Maria von Schoenermark, Proc. Int. Symp. Fe-Pnictide Superconductors, J. Phys. Soc. Jpn. 77 (2008) Suppl. C, pp. 142-144)

Now, neither Stefan nor I are specialists in super-conductivity, but this relation is quite interesting for it may harbor the possibility to better direct the search for high-temperature supra-conductors. Though one is left to wonder whether there is a way to know if a material will be super-conducting at all, it is intriguing how well the data points fit to the curve. It is however not clear to us whether the above shown curve depicts all known data points or only those that fit nicely, and how big the uncertainty of the quantity x is. In any case, the proposed relation is purely heuristic, it was obtained from accumulated data rather than derived from a theoretical model.

The relation was published in Acta Astronautica 62 733-736 (2008) and J. Phys. Soc. Japan Suppl. C 77 142-144 (2008).

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Update: Check out also the continuation of the post, Röser's equation, again