Doping-dependent critical Cooper-pair momentum in thin underdoped cuprate films

We apply a low-field ($<100$ G) technique to measure the critical pair momentum $p_c$ in thin underdoped films of ${\mathrm{Y}}_{0.7}{\mathrm{Ca}}_{0.3}{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{7-\delta }$ and ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$ reflecting a wide range of hole doping. We observe that $p_c\propto \hslash /\xi$ scales with $T_c$ and therefore superfluid density $n_s(T\to 0)$ in our two-dimensional cuprate films. This relationship was famously predicted by a universal model of the cuprates with a doping-independent superconducting gap, but has not been observed by high field $(>{10}^4\mathrm{G})$ measurements of the coherence length $\xi$ because high-field cuprate phenomena obscure the simple scaling we observe at much lower fields.

Figure 1

Areal superfluid density $1/\Lambda \left(T\right)$ measured for the nine films in this study: five-unit-cell Ca-YBCO (red: a and b), two-unit-cell Ca-YBCO (black: a), one-unit-cell Bi2212 (blue: a), and two-unit-cell Bi2212 (blue: b). We define $T_c$ to be the temperature at which the real part of the conductivity ${\sigma }_1$ (not shown) has a peak, as indicated by the asterisks on the curves. The Kosterlitz-Thouless (KT) construction for $T_c$, namely the intersection of the line with $1/\Lambda \left(T\right)$, yields essentially the same results as appropriate for 2D superconductors. $T_c$ shows linear scaling with superfluid density, as expected for quantum fluctuations in a 2D superconductor (c). Dashed line is the 2D scaling reported by Hetel et al. [3].

Figure 2

In-phase $\left(M_1\right)$ and out-of-phase $\left(M_2\right)$ mutual inductances as functions of drive field at the center of the film $\left(B_0\right)$ for all nine films in the study (at 10 kHz frequency). Normalization is as follows: $M_1=\frac{M_1\left(B_0\right)-M_1(B_0\to 0)}{M_0-M_1(B_0\to 0)},M_2=\frac{|M_2\left(B_0\right)|}{\text{max}|M_2\left(B_0\right)|}$. Signal is phase calibrated at low excitation field $B_0\to 0$: $M_2\to 0$. Initial position is measured from the normal-state response: $M_0\equiv M_1(T>T_c)$.

Figure 3

Comparison of mutual inductance vs applied field for a wide range of superconducting materials: two cuprates and two conventional BCS superconductors, showing the similar phenomenology. Normalization follows Fig. 2.

Figure 4

(a) Raw mutual inductance data for a single Ca-YBCO film at different temperatures. Below 4.2 K the curves are virtually indistinguishable, while at $T/T_c=0.13$ the curves are shifted slightly. (b) Temperature dependence of the out-of-phase mutual inductance peak for the same Ca-YBCO film, showing negligible spread in $B_{\mathrm{NL}}$ error due to temperature. The superfluid density is effectively constant over the range of the plot, and thus small errors in $B_{\mathrm{NL}}$ will be linearly proportional to the error in $p_c$. Dashed line is liquid helium temperature. Repeated mutual inductance traces made on the same film are so consistent that changes in the out-of-phase peak position on the order of 1% or less can be resolved.

Figure 5

Measured critical pair momentum $p_c$ plotted against $T_c$. Plotting conventions follow Fig. 1. Dashed line is relationship predicted by Ref. [8], with slope adjusted to fit the severely underdoped data. High $T_c$ data, which is in line with high-field measurements, is shown for the contrast between slightly and severely underdoped films.