Beyond Homes Scaling: Disorder, the Planckian Bound, and a New Universality

Beginning with high-<a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msub><a:mi>T</a:mi><a:mi>c</a:mi></a:msub></a:math> cuprate materials, it has been observed that many superconductors exhibit so-called “Homes scaling,” in which the zero-temperature superfluid density <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msub><c:mi>ρ</c:mi><c:mrow><c:mi>s</c:mi><c:mn>0</c:mn></c:mrow></c:msub></c:math> is proportional to the product of the normal-state dc conductivity and the superconducting transition temperature <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:msub><e:mi>σ</e:mi><e:mi>dc</e:mi></e:msub><e:msub><e:mi>T</e:mi><e:mi>c</e:mi></e:msub></e:math>. For conventional, <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>s</g:mi></g:math>-wave superconductors, such scaling has been shown to be a natural consequence of elastic-scattering disorder, not only in the extreme dirty limit, but across a broad range of scattering parameters. Here we show that when an analogous calculation is carried out for elastic scattering in <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>d</i:mi></i:math>-wave superconductors, a stark contrast emerges, with <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:msub><k:mi>ρ</k:mi><k:mrow><k:mi>s</k:mi><k:mn>0</k:mn></k:mrow></k:msub><k:mo>∝</k:mo><k:msup><k:mrow><k:mo stretchy="false">(</k:mo><k:msub><k:mi>σ</k:mi><k:mi>dc</k:mi></k:msub><k:msub><k:mi>T</k:mi><k:mi>c</k:mi></k:msub><k:mo stretchy="false">)</k:mo></k:mrow><k:mn>2</k:mn></k:msup></k:math> in the dirty limit, in apparent violation of Homes scaling. Within a simple approximate Migdal-Eliashberg treatment of inelastic scattering, we show how the observed Homes scaling is recovered. The normal-state behavior of near-optimally-doped cuprates is dominated by inelastic scattering, but significant deviations from Homes scaling occur for disorder-dominated cuprate systems, such as underdoped <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mrow><o:msub><o:mrow><o:mi>YBa</o:mi></o:mrow><o:mrow><o:mn>2</o:mn></o:mrow></o:msub><o:msub><o:mrow><o:mi>Cu</o:mi></o:mrow><o:mrow><o:mn>3</o:mn></o:mrow></o:msub><o:msub><o:mrow><o:mi mathvariant="normal">O</o:mi></o:mrow><o:mrow><o:mn>6.333</o:mn></o:mrow></o:msub></o:mrow></o:math> and overdoped <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"><r:mrow><r:msub><r:mrow><r:mi>La</r:mi></r:mrow><r:mrow><r:mn>2</r:mn><r:mo>−</r:mo><r:mi>x</r:mi></r:mrow></r:msub><r:mrow><r:msub><r:mrow><r:mi>Sr</r:mi></r:mrow><r:mrow><r:mi>x</r:mi></r:mrow></r:msub></r:mrow><r:mrow><r:msub><r:mrow><r:mi>CuO</r:mi></r:mrow><r:mrow><r:mn>4</r:mn></r:mrow></r:msub></r:mrow></r:mrow></r:math>, and in very clean materials with little inelastic scattering, such as <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline"><t:mrow><t:msub><t:mrow><t:mi>Sr</t:mi></t:mrow><t:mrow><t:mn>2</t:mn></t:mrow></t:msub></t:mrow><t:mrow><t:msub><t:mrow><t:mi>RuO</t:mi></t:mrow><t:mrow><t:mn>4</t:mn></t:mrow></t:msub></t:mrow></t:math>. We present a revised analysis where both axes of the original Homes scaling plot are normalized by the Drude plasma weight <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline"><v:msubsup><v:mi>ω</v:mi><v:mrow><v:mi>p</v:mi><v:mo>,</v:mo><v:mi>D</v:mi></v:mrow><v:mn>2</v:mn></v:msubsup></v:math> and show that a new universal scaling emerges, in which the superfluid fractions of dirty <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" display="inline"><x:mi>s</x:mi></x:math>-wave and dirty <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" display="inline"><z:mi>d</z:mi></z:math>-wave superconductors coalesce to a single point at which normal-state scattering is occurring at the Planckian bound. The combined result is a new tool for classifying superconductors in terms of order parameter symmetry, as well as scattering strength and character. Although our model starts from a Fermi-liquid assumption, it describes underdoped cuprates surprisingly well.