*How to disentangle the unavoidable atomic level inhomogeneity of real materials from the electronic inhomogeneity*?

** **

The first debated question was the role of fractality in the spontaneously emergent spatial inhomogeneities observed in highly disordered superconductors. According to the results of numerical simulations reported by Prof. Trivedi, this fractality holds only close to the metal-insulator transition in the non-interacting particles Anderson localization transition. However, deeper in the insulating state where the superconductor-insulator transition (SIT) takes place, once the pairing interaction is switched on, this fractal character of the wave functions does not show up in the u and v BCS operators, possibly because the wave functions are now too strongly localized. Dr. Feigel’man remarked that this could also be a side effect of the non perturbative implementation of the pairing potential in these calculationsand stressed that, in the fractal theory of the SIT, the spatial variations of the order parameter are expected to show the fractal behaviour only in the limited range of distances because at large scales local pairing amplitude relies on the participation of several localized wave-functions while no fractal structure is expected at distances less than a mean free path of electrons. This is also true for any other physical observable which varies in real space. This range of distances significant for realistic physical systems with small T_{c}/E_{F} ratio and short mean free path but shrinks as T_{c}/E_{F} →1 as is the case for numerical simulations. Moreover, it has been emphasized that the fractal character of the wave-functions at the mobility edge does not lead to the SIT, but rather to an enhancement of the pairing interaction. The superconducting properties disappear for higher disorder when the mean level spacing is much higher than the superconducting gap. The main role of fractality is to give broad spatial distribution of the pairing amplitude of electrons in a single localized state. In conclusion, no exotic physical length scale appears in the theory, beside the usual superconducting coherence length and the localization length. There seems to be no contradiction between numerical results of Prof. Trivedi and analytical theory.

The second discussed issue was the possibility that inhomogeneities might explain the puzzles observed by Prof. Raychaudhuri on NbN films. Namely, for several disordered films, the transport and Hall effect measurements at high temperatures give the value of (k_{F}l) that is much smaller than unity. In this situation one expects that these materials become insulators at low temperatures. However, in case of NbN films, the resistance remains finite when extrapolated to zero temperature. One possible explanation of this disagreement is that the measured conductivity is dominated by a small percolating volume of the film. Another explanation attributes it to a significant thermal dependence of the electronic density which is only known at room temperature.

Although disorder driven electronic segregations in nominally homogeneous materials seem to become an established concept supported by theoretical calculations and experimental observations, the usual dichotomy between granular and homogeneously disordered systems is still extensively emphasized in most of the presentations which show the famous sets of R(T) curves in Ga and Bi. Although this systematic opposition might be misleading, the general opinion is that it should be continued to be discussed because, despite being an oversimplification, it helps non-specialists to clarify a complex SIT landscape.

Prepared by Lev Ioffe and Claude Chapelier