Fully anisotropic superconducting transition in ion-irradiated $\mathrm{Y}{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{7-\delta }$ with a tilted magnetic field

We consider the superconducting vortex solid-to-liquid transition in heavy ion-irradiated untwinned $\mathrm{Y}{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{7-\delta }$ single crystals in the case where the magnetic field direction does not coincide with that of the irradiation-induced linear columnar defects. For a certain range of angles, the resistivities measured in three orthogonal spatial directions vanish at the transition as three clearly different powers of reduced temperature. At previously known second-order phase transitions, scaling of physical quantities has either been isotropic or anisotropic in one direction. Thus, our findings yield evidence for a new type of critical behavior with fully anisotropic critical exponents.

Figure 1

(Color online) Experimental geometry. The columnar defects are parallel to the $z$ direction, which coincides with the crystallographic $c$ axis. The magnetic field is always applied in the $xz$ plane, at an angle $\theta$ with respect to $z$. Vortex lines may accommodate the columns by forming kinks.

Figure 2

(Color online) Normalized resistivity in three independent directions of an untwinned heavy-ion irradiated YBCO single crystal with a matching field $B_{\varphi }=2\mathrm{T}$, and at tilt angles (a) $\theta =4°$ and (b) $\theta =45°$ from the columns. Dashed lines are fits to Eq. (1). Inset: Extraction of $T_c$ and the critical exponent $s_i$ using ${(\partial \ln {\rho }_i∕\partial T)}^{-1}=(T-T_c)∕s_i$.

Figure 3

(Color online) Exponents $s_x$, $s_y$, and $s_z$ of the power-law Eq. (1) by which the resistivity vanishes at the vortex solid-to-liquid transition in the $x$, $y$, and $z$ directions, respectively. (a) Exponents at $2\mathrm{T}$ as a function of angle $\theta$. For $15°\lesssim \theta \lesssim 65°$, the scaling behavior is anisotropic in three directions. (b) Exponents at $\theta =45°$, as a function of magnetic field. There is anisotropic scaling at fields close to the matching field, ${\mu }_{0}H\approx B_{\varphi }=2\mathrm{T}$. Dotted lines show the average value of $s_i$ in the regimes of anisotropic scaling. The error bars are estimated from the quality of the fits in Fig. 2.

Figure 4

Phase diagram for untwinned heavy-ion irradiated YBCO single crystals. An effective magnetic field is used to account for the anisotropy of YBCO. The symbols show the vortex solid-to-liquid transition temperatures obtained from measurements of ${\rho }_y$ at constant $\theta =45°$ and several fields (∎,●); and at constant ${\mu }_{0}H=2\mathrm{T}$ and different $\theta$ (엯). At fields far from $B_{\varphi }=2\mathrm{T}$, isotropic glass exponents are found (∎) and the data can be fitted to Eq. (2) (solid line). For fields comparable to $B_{\varphi }$ the transition lies above this line. At $\theta \approx 15°$ the transition bends toward the isotropic glass line; nevertheless, it remains above it up to $\theta \sim 65°$. Inset, ${\rho }_y\left(\theta \right)$ at $1\mathrm{T}$ and $90.2\mathrm{K}$. At $\theta =0$ the field is aligned with the columns. The accommodation angle ${\theta }_a$ is usually defined by the maximum of $\rho \left(\theta \right)$ (dashed lines).