We consider the realistic case of a superconductor with a nonzero density of elastic scatterers, so that the normal state conductivity is finite. The quantum superconductor-metal (QSM) transition can then be tuned by varying either the attractive electron-electron interaction, the quenched disorder, or the applied magnetic field. We explore the consistency of the associated scaling relations,valid for all dimensions D > 2, with experimental data, in Al, C doped MgB2 and overdoped cuprates.
PACS numbers:Understanding the phenomenon of superconductivity, now observed in quite disparate systems, such as simple elements, fullerenes, molecular metals, cuprates, borides, etc., involves searching for universal relations between superconducting properties across different materials, which might provide hints towards a unique classification. In spite of the great impact of the BCS theory [1], the discovery of superconductivity in the cuprates in 1986 [2] made it clear that the BCS relations between the critical amplitudes of the gap (∆ 0 ), the correlation length (ξ 0 ), the magnetic penetration depth (λ 0 ), the upper critical field (H c20 ) and the transition temperature T c , namely [3]Here, λ (T ) = λ 0 t 1/2 , ∆ (T ) = ∆ 0 t 1/2 ≃ 1.76∆ (0) t 1/2 , ξ (T ) = ξ 0 t −1/2 , and H c2 = H c20 t, close to the superconductor metal transition, with t = 1 − T /T c and 2∆ (0) / (k B T c ) ≃ 3.52. Furthermore, there are empirical relations between T c and the zero-temperature superfluid density, ρ s (0), related to the zero-temperature magnetic field penetration depth λ (0) in terms of ρ s (0) ∝ λ −2 (0). In various families of underdoped cuprate superconductors there is the empirical relation T c ∝ ρ s (0) ∝ λ −2 (0), first identified by Uemura et al. [4,5], while in molecular superconductors, T c ∝ λ −3 (0), appears to apply [6]. Both scaling forms appear to have no counterpart in the BCS scenario and even in more conventional superconductors, including Mg 1−x Al x B 2 , Mg(C x B 1−x ) 2 , and MgB 2+x , such relationships remain to be explored. According to the theory of quantum critical phenomena a power law relation between T c and ρ s (0) ∝ λ −2 (0) is expected whenever there is a critical line T c (x) with a critical endpoint x = x c [7,8,9]. Here the transition temperature vanishes and a quantum phase transition occurs.x denotes the tuning parameter of the transition. A variety of underdoped cuprate superconductors exhibits such a critical line, ending at the quantum superconductor to insulator (QSI) transition, where the materials become essentially two dimensional [10]. If the finite tempera-ture behavior in this regime is controlled by the 3D − xy critical point, T c and ρ s (0) scale as [7,8,9] T c ∝ ρ s (0) z/(D+z−2) .(2) z denotes the dynamic critical exponent of the quantum transition in D dimensions. For D = 2 we recover the empirical Uemura relation, T c ∝ ρ s (0) [4,5], irrespective of the value of z.