Scaling of the superfluid density in high-temperature superconductors

A scaling relation $N_c\simeq 4.4{\sigma }_{\mathit{dc}}{T}_c$ has been observed in the copper-oxide superconductors, where $N_c$ is the spectral weight associated with the formation of the superconducting condensate ${\rho }_s=8N_c$, $T_c$ is the critical temperature, and ${\sigma }_{\mathit{dc}}$ is the normal-state dc conductivity close to $T_c$. This scaling relation is examined within the context of a clean and dirty-limit BCS superconductor. These limits are well established for an isotropic BCS gap $2\Delta$ and a normal-state scattering rate $1∕\tau$; in the clean limit $1∕\tau ⪡2\Delta$, and in the dirty limit $1∕\tau >2\Delta$. The dirty limit may also be defined operationally as the regime where ${\rho }_s$ varies with $1∕\tau$. It is shown that the scaling relation $N_c$ or ${\rho }_s\propto {\sigma }_{\mathit{dc}}{T}_c$, which follows directly from the Ferell-Glover-Tinkham sum rule, is the hallmark of a BCS system in the dirty-limit. While the gap in the copper-oxide superconductors is considered to be $d$ wave with nodes and a gap maximum ${\Delta }_0$, if $1∕\tau >2{\Delta }_0$ then the dirty-limit case is preserved. The scaling relation implies that the copper-oxide superconductors are likely to be in the dirty limit and, as a result, that the energy scale associated with the formation of the condensate scales linearly with $T_c$. The $a\text{−}b$ planes and the $c$ axis also follow the same scaling relation. It is observed that the scaling behavior for the dirty limit and the Josephson effect (assuming a BCS formalism) are essentially identical, suggesting that in some regime these two pictures may be viewed as equivalent.

Figure 1

The spectral weight of the condensate $N_c$ vs ${\sigma }_{\mathit{dc}}{T}_c$ for the $a\text{−}b$ planes of the hole-doped copper-oxide superconductors for pure and Pr-doped $\mathrm{Y}{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6+x}$ (Refs. 12, 36, 60, 61, 62); pure and Zn-doped $\mathrm{Y}{\mathrm{Ba}}_2{\mathrm{Cu}}_4{\mathrm{O}}_8$ (Refs. 60, 63); pure and $\mathrm{Y}∕\mathrm{Pb}$-doped ${\mathrm{Bi}}_2{\mathrm{Sr}}_2\mathrm{Ca}{\mathrm{Cu}}_2{\mathrm{O}}_{8+\delta }$ (Refs. 62, 64, 65); underdoped ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x\mathrm{Cu}{\mathrm{O}}_4$ (Ref. 69); ${\mathrm{Tl}}_2{\mathrm{Ba}}_2\mathrm{Cu}{\mathrm{O}}_{6+\delta }$ (Ref. 66); electron-doped ${(\mathrm{Nd},\mathrm{Pr})}_{2-x}{\mathrm{Ce}}_x\mathrm{Cu}{\mathrm{O}}_4$ (Refs. 10, 13, 67, 68) and the bismate material ${\mathrm{Bi}}_{1-x}{\mathrm{K}}_x\mathrm{Bi}{\mathrm{O}}_3$ (Ref. 70). Within error, all the points may be described by a single (dashed) line, $N_c\simeq 4.4{\sigma }_{\mathit{dc}}{T}_c$; the upper and lower dotted lines, $N_c\simeq 5.5{\sigma }_{\mathit{dc}}{T}_c$ and $3.5{\sigma }_{\mathit{dc}}{T}_c$, respectively, represent approximately the spread of the data.

Figure 2

The log-log plot of the spectral weight of the condensate $N_c$ vs ${\sigma }_{\mathit{dc}}{T}_c$ for the $a\text{−}b$ planes of the hole-doped copper-oxide superconductors shown in Fig. 1. The dashed and dotted lines shown in this figure are the same lines that were shown in Fig. 1. The points for Nb and Pb, indicated by the atomic symbols, also fall close to the dotted line, $N_c\simeq 4.4{\sigma }_{\mathit{dc}}{T}_c$ (Refs. 45, 46).

Figure 3

The optical conductivity for the BCS model in the normal (solid line) and superconducting states (dashed line) for a material in (a) the clean limit $(1∕\tau ⪡2\Delta )$ and (b) the dirty limit $(1∕\tau >2\Delta )$. The normal-state conductivity is a Lorentzian centered at zero frequency with a full width at half maximum of $1∕\tau$ for $T\simeq T_c$. The spectral weight associated with the formation of a superconducting condensate is indicated by the hatched area. Insets: $N_n\equiv N(\omega ,T\simeq T_c)$ (solid line), $N_s\equiv N(\omega ,T⪡T_c)$ (dashed line), and difference between the two $N_c=N_n-N_s$ (long-dashed line) normalized with respect to ${\rho }_s∕8$; in the clean limit $8N_c∕{\rho }_s$ converges rapidly to unity and is fully formed at energies comparable to $1∕\tau$, while in the dirty limit, convergence occurs at energies comparable to $4\Delta$.

Figure 4

The log-log plot of the predicted behavior from the BCS model of the spectral weight of the condensate $N_c$ in Nb for a wide range of scattering rates $1∕\tau =0.05\Delta \to 50\Delta$ and assuming a plasma frequency ${\omega }_p=56000{\mathrm{cm}}^{-1}$, critical temperature $T_c=9.2\mathrm{K}$, and an energy gap of $2\Delta =3.5k_B{T}_c$ (solid line). The dashed line indicates $1∕\tau =2\Delta$. To the right of this line the material approaches the clean limit with a residual resistance ratio (RRR) of $\gtrsim 100$; the right arrow indicates that for larger RRRs, ${\sigma }_{\mathit{dc}}$ close to $T_c$ increases, but $N_c$ has saturated to ${\omega }_p^2∕8$ (or ${\rho }_s\to {\omega }_p^2$; the data point for Nb in this regime is from Ref. 44). As the scattering rate increases, the spectral weight of the condensate adopts a linear scaling behavior (dotted line); the two points for Nb (Refs. 45, 46) shown in Fig. 2 lie close to this line, indicating that they are in the dirty limit. The scaling relation shown in Fig. 2 (dash-dot line) is slightly offset from the BCS dirty-limit result.

Figure 5

The log-log plot of the spectral weight of the condensate $N_c$ vs ${\sigma }_{\mathit{dc}}{T}_c$ for the $a\text{−}b$ plane and the $c$ axis for a variety of cuprates. Within error, all of the points fall on the same universal (dashed) line defined by $N_c\simeq 4.4{\sigma }_{\mathit{dc}}{T}_c$; the dotted line is the dirty-limit result $N_c\simeq 8.1{\sigma }_{\mathit{dc}}{T}_c$ for the BCS weak-coupling case $(2\Delta =3.5k_B{T}_c)$ from Fig. 4, and also represents the Josephson result for the BCS weak-coupling case, used to describe the scaling along the $c$ axis (Ref. 57). (Values for the $c$-axis points are listed in the supplemental information of Ref. 13.)