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This 2010 Nature article (free access here) suggested that the fractal distribution of oxygen ions in a copper oxide superconductor might boost the material's superconductivity at high temperatures:

Here we report that the ordering of oxygen interstitials in the La(2)O(2+y) spacer layers of La(2)CuO(4+y) high-Tc superconductors is characterized by a fractal distribution up to a maximum limiting size of 400 microns. Intriguingly, these fractal distributions of dopants seem to enhance superconductivity at high temperature.

However, according to the authors, no theoretical explanation was known at the time:

Co-author Gabriel Aeppli of the London Centre for Nanotechnology and University College London, added that “While there is no detailed theoretical explanation for what we have discovered yet, it demonstrates that classical ceramic engineering – with visible effects at near millimeter scales – can collude with quantum physics to produce the best superconductors.” (source)

“We don’t know the theory for this,” said physicist Antonio Bianconi of Rome’s Sapienza University. “We just make the experimental observation that the two worlds seem to interfere.” (source)

I was wondering if someone knew what the current state of the research on this topic is. Has a theoretical explanation for this phenomenon already been proposed, or can it be deduced from any of the existing mathematical models for superconductive behavior?

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According to https://arxiv.org/abs/1607.05216 , some fractal geometries can provide considerable enhancement of critical temperature "due to appearance of large number of additional poles in the inverse dielectric response function of the fractal". .

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