The optical conductivity of the overdoped cuprate superconductor La2-xSrxCuO4 for x=0.25 (Tc ~16 K) has been determined in the conducting a-b planes and along the poorly-conducting c axis above and below Tc, allowing the superfluid density to be determined from conductivity sum rules.  These results fall on the general linear scaling relation observed for the high-temperature superconducting cuprates between the superfluid density and the product of the critical temperature (Tc) and the DC conductivity measured just above Tc [1].  This scaling relation is valid for the copper-oxygen planes, as well as along the c axis.  The purity of a superconductor allows it to be described either as in either the clean or dirty limit.  Despite their untidy nature, the short in-plane coherence length of the cuprates is generally thought to place them in the clean limit.  However, from an optics point of view, the cuprates look like they are in the dirty limit, and indeed, the scaling relation is consistent with a BCS superconductor in the dirty limit.  As the impurity scattering rate is reduced and the material becomes cleaner, deviations are expected [2]; some extremely underdoped samples that appear to be in the clean limit do not follow the linear scaling relation [3].  Interestingly, a recent result demonstrates that the sub-linear scaling in the extremely underdoped materials is consistent with a dirty d-wave superconductor [4], reinforcing the view that the cuprates are in the dirty limit.

[1] C. C. Homes et al., Nature 430, 539 (2004).

[2] C. C. Homes et al., Phys. Rev. B 72, 124517 (2005); V. G. Kogan, Phys. Rev. B 87, 220507(R) (2013).

[3] C. C. Homes, Phys. Rev. B 80, 180509(R) (2009).

[4] D. M. Broun, V. Mishra, J. S. Dodge, and P. J. Hirschfeld, Phys. Rev. X 15, 041005 (2025).

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