Nature of the vortex-liquid to vortex-solid transition and vortex-glass critical scaling in highly anisotropic Bi-2223 superconductors

In the highly anisotropic ($\gamma \approx 50$) cuprate superconductor (Pb-doped) Bi-2223, in-plane resistivity and current-voltage measurements up to 6 T can be fitted by vortex-glass critical scaling with a field-independent exponent product $\nu(z-1) = 4.6 \pm 0.5$, yet the three-dimensional vortex-glass prediction for the glass transition line $B(T_g) \propto (T_c - T_g)^{2\nu}$ fails, the exponents differ from those of YBCO and Bi-2212 ($\approx 6.5 \pm 1.5$), and a dimensional 2D-3D crossover is expected near 1-2 T. The unresolved problem is to determine the physical nature of this vortex-liquid to vortex-solid transition - a genuine three-dimensional vortex-glass transition, a quasi-two-dimensional or decoupled pancake-vortex glass, or a disorder-rounded first-order freezing - and on that basis to establish whether vortex-glass critical scaling with universal exponents applies, what sets the measured exponent values, and what law governs the $B(T_g)$ phase boundary.

Background

In a type-II superconductor in a magnetic field, magnetic flux penetrates as quantized vortex lines. In high-$T_c$ cuprates, strong thermal fluctuations and quenched disorder produced by crystal defects broaden the resistive transition and split the mixed state into a high-temperature, dissipative vortex liquid and a low-temperature vortex solid that can sustain a critical current. Fisher, Fisher, and Huse proposed that random point disorder converts vortex-lattice melting into a continuous second-order transition into a vortex glass at a temperature $T_g$: below $T_g$ the vortices are frozen into a random but static configuration with truly zero linear resistivity (Physical Review B 43, 130, 1991). Their scaling theory predicts, just above $T_g$, a power-law resistivity $R \propto (T - T_g)^{\nu(z-1)}$, where $\nu$ is the static correlation-length exponent and $z$ the dynamic exponent; current-voltage (I-V) curves whose log-log curvature changes from positive (liquid) to negative (glass), becoming a straight power law at $T_g$; and, for a true three-dimensional (3D) system, a glass transition line emerging from the zero-field transition temperature $T_c$ as $B(T_g) \propto (T_c - T_g)^{2\nu}$ with $\nu = 2/3$. This framework received experimental support in YBCO single crystals and films (Koch et al., Physical Review Letters 63, 1511, 1989) and in Bi-2212 single crystals (Safar et al., Physical Review Letters 68, 2672, 1992), both reporting $\nu(z-1) \approx 6.5 \pm 1.5$. Bi-2223 ($\mathrm{Bi_2Sr_2Ca_2Cu_3O_{10+\delta}}$, $T_c \approx 110$ K) is far more anisotropic: its superconducting anisotropy parameter $\gamma$ (the ratio of the out-of-plane to in-plane magnetic penetration depths, $\lambda_c/\lambda_{ab}$) is about 50 (Giannini et al., Current Applied Physics 8, 115, 2008), so its CuO$_2$ layers are weakly coupled and vortices aligned with the layer normal behave at high fields as stacks of weakly coupled two-dimensional pancake vortices (vortex segments confined to individual superconducting layers). Theory of layered vortex matter (Blatter et al., Reviews of Modern Physics 66, 1125, 1994) predicts a dimensional 2D-3D crossover at a field scale of order $\Phi_0/(\gamma s)^2$ (with $\Phi_0$ the flux quantum and $s$ the layer spacing; the numerical prefactor varies by convention), which is of order 1 T for $\gamma \approx 50$, above which the vortex system loses three-dimensional coherence; disorder can further decouple the pancake layers (Koshelev, Glazman, and Larkin, Physical Review B 53, 2786, 1996). In this regime the applicability of the original 3D vortex-glass theory is in doubt. Eltsev, Lee, Nakao, and Tajima (JETP Letters 90, 535, 2009) measured in-plane resistivity and I-V curves of a Pb-doped Bi-2223 single crystal in fields $B \parallel c$ up to 6 T. They found Arrhenius-like activated resistivity at high temperature crossing over to power-law behavior closer to the resistivity onset, consistent with a vortex-glass transition with $T_g = 35.3$ K at 6 T and a field-independent $\nu(z-1) = 4.6 \pm 0.5$ over a remarkably broad critical region ($\Delta T/T_g \approx 1$, versus $\approx 0.05$ in YBCO); I-V curvature reversals located the glass boundary at 0.7-1 T at 60 K; the measured product is close to $5.2 \pm 0.5$ found in a Bi-2223/Bi-2212 intergrowth crystal (Han et al., Solid State Communications 105, 31, 1998) but below the YBCO and Bi-2212 values. However, their fit of the measured glass transition line $B(T_g)$ to the 3D vortex-glass form with $\nu = 2/3$ failed, in contrast to successful fits below 1 T in Bi-2223/Ag tapes and intergrowth crystals. The authors concluded that Bi-2223, with $\gamma \approx 50$, is not a true 3D system, so the vortex-glass theory has limited applicability, that a possible 2D-3D crossover near 1-2 T (suggested by earlier Bi-2212 and Bi-2223-film work) may complicate the picture, and that understanding the true nature of the transition calls for further study. Whether standard vortex-glass critical scaling, possibly modified for strong anisotropy and layer decoupling, correctly describes the vortex-liquid to vortex-solid transition in Bi-2223 therefore remains undetermined.

Problem Statement

Determine the physical nature of the vortex-liquid to vortex-solid transition in highly anisotropic ($\gamma \approx 50$) Bi-2223 cuprate superconductors, and thereby establish the correct critical description of it. Concretely: (i) decide whether the transition is a genuine continuous three-dimensional vortex-glass transition in the sense of Fisher-Fisher-Huse scaling, a quasi-two-dimensional glass of weakly coupled pancake-vortex layers that takes over above the dimensional-crossover field scale of order 1-2 T, or a disorder-rounded first-order melting, including whether and how vortex-glass theory must be modified for strong anisotropy or layer decoupling; (ii) establish the scaling of the in-plane resistivity $R \propto (T - T_g)^{\nu(z-1)}$ and the I-V curvature reversal across the boundary, deciding whether the exponent product $\nu(z-1) = 4.6 \pm 0.5$ measured in Pb-doped Bi-2223 is universal or sample- and field-dependent, and accounting for its difference from the $6.5 \pm 1.5$ reported for YBCO and Bi-2212; and (iii) determine the glass transition line $B(T_g)$ over an extended field and temperature range - in particular whether its observed deviation from the three-dimensional vortex-glass prediction $B(T_g) \propto (T_c - T_g)^{2\nu}$ with $\nu = 2/3$ reflects dimensional crossover, different exponent values, or interlayer decoupling, and what the correct phase-boundary law is. These three facets jointly support one objective: an experimentally and/or theoretically grounded identification of the transition and its scaling laws in Bi-2223.

References

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