Vortex lattice melting and $H_{c2}$ in underdoped YBa${}_2$Cu${}_3$O${}_y$

Vortices in a type-II superconductor form a lattice structure that melts when the thermal displacement of the vortices is an appreciable fraction of the distance between vortices. In an anisotropic $\mathrm{high}$-$T_c$ superconductor, such as YBa${}_2$Cu${}_3$O${}_y$, the magnetic field value where this melting occurs can be much lower than the mean-field critical field $H_{c2}$. We examine this melting transition in YBa${}_2$Cu${}_3$O${}_y$ with oxygen content $y$ from 6.45 to 6.92, and we perform a quantitative analysis of this transition in the cuprates by fitting the data to a theory of vortex-lattice melting. The quality of the fits indicates that the transition to a resistive state is indeed the vortex lattice melting transition, with the shape of the melting curves being consistent with the known change in penetration depth anisotropy from underdoped to optimally doped YBa${}_2$Cu${}_3$O${}_y$. We establish these fits as a valid technique for finding $H_{c2}(T=0)$ from higher-temperature data when the experimentally accessible fields are not sufficient to melt the lattice at zero temperature (near optimal doping). From the fits we extract $H_{c2}(T=0)$ as a function of hole doping. The unusual doping dependence of $H_{c2}(T=0)$ points to some form of electronic order competing with superconductivity around 0.12 hole doping.

Figure 1

The $\widehat{c}$-axis resistance of YBa${}_2$Cu${}_3$O${}_{6.59}$ as a function of magnetic field, from 1.5 to 200 K. The onset of resistivity as field is increased marks the vortex lattice melting transition. At low temperatures, quantum oscillations are seen above this melting field.

Figure 2

Top: A magnified plot of the resistive transitions shown above in Fig. 1. The red dots are where the resistance is $1/100$ of its value at 60 T (extrapolated for high temperatures where resistance was not measured to the highest fields). Bottom: The same data points as highlighted in red in the top panel, now plotted as a function of temperature. The black line is a fit to Eq. (2), using the known parameters given in Table , and gives ${\mu }_0{H}_{c2}\left(0\right)=28\pm 0.3$ T, and $c_L=0.37$.

Figure 3

The vortex lattice melting transition for several different oxygen concentrations. The temperature axis has been scaled by $T_c$, and the lines are best-fit lines to Eq. (2). All of the data points were acquired in the same manner as described in the caption of Fig. 2.

Figure 4

Top: The in-plane and out-of-plane penetration depths of YBa${}_2$Cu${}_3$O${}_y$, as measured by muon-spin rotation (${\lambda }_{ab}$ at 0.11 holes), electron-spin resonance (the other ${\lambda }_{ab}$ points), and infrared reflectance (${\lambda }_c$).[21, 22, 23] Note the difference in scale for the left- and right-hand axes: the anisotropy is actually decreasing with increased hole doping. The dashed lines are parabolic fits to the data, which are used to obtain interpolated values for doping levels not measured, and should be viewed as purely phenomenological. Bottom: The ratio of the Lindemann number squared to the product of the in-plane and out-of-plane penetration depths. This quantity, appearing on the right-hand side of Eq. (2), controls the curvature of $B_m$ vs $T$.

Figure 5

The superconducting phase diagram of YBa${}_2$Cu${}_3$O${}_y$. $H_{c2}\left(0\right)$ is suppressed in the same region that $T_c$ deviates from the parabolic form—a clear sign of the weakening of superconductivity around 0.12 hole doping. The $T_c$ values are taken from Liang et al.[20]