Let me just repeat the main point of the "Röser formula":
It claims a simple linear relation between the inverse of the superconducting transition temperature of a wide range of superconducting materials, and a so-called "doping distance". This relation is shown in the following plot:
(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)
Here are a few more thoughts of mine about this relation.
Thermal de Broglie wavelength
Using the standard definition of the thermal de Broglie wavelength for a Cooper pair of two electrons with free masses, the Röser equation
4 π k me(2 x)2 n−2/3 = h2/ Tc
actually boils down to
(2 x) × n−1/3 = λc
where λc is the thermal de Broglie wavelength of the Cooper pair at the critical temperature Tc, where superconductivity breaks down.
In the proceedings paper, n = 1, so the factor n−1/3 can be dropped. Moreover, it is said to take values of n = 2 or 3 for other superconductors with a layer structure, depending on the number of layers in the unit cell. It's not completely clear to me how it is motivated.
In the proceedings paper, the authors discuss doped iron arsenides LO(1−Δ)F(Δ)FeAs, where L is a rare earth, L = La (Lanthanum), Gd (Gadolinium), Ce (Cerium), Pr (Praseodymium), Nd (Neodymium) or Sm (Samarium) - see table 1 of the paper. These materials are labeled as "LOFFA", "GOFFA", "COFFA", "POFFA", "NOFFA", and "SOFFA" in the plot.
The other data points in the plot refer to cuprate superconductors - the substances labeled ...CO, or Bi-2212, where the numbers denote the composition. These cuprates are discussed in several Acta Astronautica papers, for example Acta Astronautica 65 (2009) 489, which unfortunately I do not have access to.
In the examples of the rare earth iron pnictides, superconductivity can be reached by replacing, in an ideal LOFeAs lattice, some of the oxygen atoms by fluorine atoms. Depending of the amount of fluorine, the critical temperature for superconductivity can take different values. The Physics article on High-temperature superconductivity in the iron pnictides shows phase diagrams for "LOFFA" and "COFFA".
In "COFFA", CeO(1−x)F(x)FeAs, for example, superconductivity sets in only when at least 6 percent of the oxygen atoms are replaced by fluorine, and the transition temperature is highest for a replacement of about 16 percent of the oxygen atoms by fluorine:
(from: Structural and magnetic phase diagram of CeFeAsO(1-x)F(x) and its relationship to high-temperature superconductivity, by Jun Zhao et al., arXiv:0806.2528v1, and Nature Materials 7 (2008) 953-959.)
The percentage of replaced atoms is called the "doping", and it is usually denoted with x, which should not be confused with the "doping distance" in the Röser paper. In the formula above for the substances, I have denoted doping with Δ, as in the Röser paper.
The "doping distance" x
The Röser formula compares the thermal de Broglie wavelength of a Cooper pair at the superconducting transition temperature with a "doping distance" x and states that they are equal, up to a geometry factor. The crucial point, then, is how to arrive at the "doping distance".
In the proceedings paper about iron arsenides, it is argued that every two of the "doping" fluorine atoms group around one iron atom, and that these "decorated" iron atoms form, again, a regular lattice. Such a lattice is usually called a superstructure.
In the example of "COFFA" at the "optimal" doping of Δ = 0.16, this reasoning implies that 8 percent of the iron atoms are neighbored by two fluorine atoms each, and the superstructure of the decorated iron atoms has to comprise 1/0.08 = 12.5 standard unit cells. They argue that this means that 5 unit cells are put on top of each other, yielding a doping distance of 5 times the height c of the unit cell, or x = 5 c.
For the other doped lanthanide iron arsenides, a similar reasoning is used. The constructions of the superstructure and the resulting doping distances are documented in table 1 of the proceedings paper.
To me, construction of the "doping distance" is not really comprehensible, and it seems that there are a few points where the paper just glosses over:
I am not sure whether (1) the dopant fluorine atoms group pairwise at iron atoms and (2) if the decorated iron atoms indeed form a superlattice. However, this may be checked experimentally. But then, (3), if there is a superlattice, it is not clear to me why it should be ordered in the way claimed in the paper, with a pattern stacking five unit cells on top of each other for "COFFA", for example. There can be many ways to arrange unit cells to form a superlattice with the right supercell volume. This is more obvious in the case of "LOFFA", where the supercell comprises 18 unit cells, and there seems to be no a priori reason to select a superstructure which has 6 unit cells stacked on top of each other.
In other words, the determination of the "doping distance" involves an arbitrariness which may be used to select x in a way to fit the formula, and there is no real discussion of the selection rules in the paper. This arbitrariness, however, could be resolved by experiments which actually measure the superstructure.
There is another point which puzzles me about the paper, which is the application of the formula to the cases of "optimal doping" only, i.e. those values of doping where the transition temperature for the substance at hand is maximal.
I don't see a good reason why the formula should not apply at any value of doping, if there is a relation between the transition temperature and the "doping distance". Then, however, there is the problem that around the maximum, different values of doping give the same transition temperature, hence should have the same doping distance. This can be seen very nicely in the phase diagram of "COFFA" above. Maybe this point can be arranged for somehow, and maybe it is discussed in the Acta Astronautica papers.
Thinking about it, while the Röser relation is indeed amazing, it's unclear to me how much of an "a posteriori" selection bias concerning the "doping distance" goes into it.
However, as the relation makes strong claims about the arrangement of the dopant atoms, this probably can be checked experimentally.
If it comes out that the relation indeed holds, it will be exciting to understand what it means for the physics of superconductivity.
- The August 2009 issue of the Scientific American gives a nice overview of superconductivity in iron pnictides in the article An Iron Key to High-Temperature Superconductivity? by Graham P. Collins (albeit without illustrations, it seems...)
- Browsing the APS' Physics Archive for the tag superconductivity is a good way to keep up-to-date with the developments in the iron pnictides.