Vortex Fluctuations in Underdoped ${\mathrm{B}\mathrm{i}}_2{\mathrm{S}\mathrm{r}}_2{\mathrm{C}\mathrm{a}\mathrm{C}\mathrm{u}}_2{\mathrm{O}}_{8+\delta }$ Crystals

Vortex thermal fluctuations in heavily underdoped ${\mathrm{B}\mathrm{i}}_2{\mathrm{S}\mathrm{r}}_2{\mathrm{C}\mathrm{a}\mathrm{C}\mathrm{u}}_2{\mathrm{O}}_{8+\delta }$ ($T_c=69.4\mathrm{K}$) are studied using Josephson plasma resonance. From the zero-field data, we obtain the $c$-axis penetration depth ${\lambda }_{L,c}(0)=230\pm 10\mu \mathrm{m}$ and the anisotropy ratio $\gamma (T)$. The low plasma frequency allows us to study phase correlations over the whole vortex solid state and to extract a wandering length $r_w$ of vortex pancakes. The temperature dependence of $r_w$ as well as its increase with dc magnetic field is explained by the renormalization of the vortex line tension by the fluctuations, suggesting that this softening is responsible for the dissociation of the vortices at the first order transition.

Figure 1

Field sweep experiment on sample A at $T=66\mathrm{K}$ in the ${\mathrm{T}\mathrm{M}}_{012}$ mode of the cavity ($f=22.9\mathrm{G}\mathrm{H}\mathrm{z}$). At $B_{\mathrm{J}\mathrm{P}\mathrm{R}}=5.3\mathrm{G}$ (open arrow), for which $\omega ={\omega }_{pl}$, the power absorbed in the sample (●) has a maximum and the resonance frequency of the cavity (○) shows a double-peak structure (closed arrows), and a jump (arrow between dashed lines). Inset: meandering of pancakes along a vortex line in a layered superconductor. Thermal fluctuations shift pancakes (full circles) away from their equilibrium positions (open circles).

Figure 2

JPR frequency vs temperature in $B=0$ for samples A and B, measured by both the bolometric method and the cavity perturbation technique. We use spline fits (solid lines) in the extraction of the wandering length (see text). Inset: experimental temperature dependence of $\gamma$, obtained by the division of the experimental ${\lambda }_{L,c}(T)$ by the ${\lambda }_{L,ab}(T)$ from reversible magnetization.

Figure 3

$1-⟨\cos ({\varphi }_{n,n+1})⟩$ vs temperature for different magnetic fields. We extract $r_w$ from these data using Eq. (3).

Figure 4

(a) Experimental $r_w$ vs $T/T_c$ for different magnetic fields in strongly underdoped BSCCO. For $B=27.5$, 22.4, 15.3, 12.4, and 9.9 G, arrows show the temperature of the FOT. The thick line shows the evolution of $0.9s(k_{B}T{\gamma }^2/{\epsilon }_{0}s{)}^{1/2}$. Thin lines are fits to Eq. (5) with $\alpha =0.8$ (for $B<15\mathrm{G}$) and $\alpha =0.65$, omitting the term $4/\pi (\alpha x^2+\frac{1}{4})$ ($B>15\mathrm{G}$). (b) $r_w$ vs $T/T_{\mathrm{F}\mathrm{O}\mathrm{T}}$. Solid lines are guides to the eye. Inset: phase diagram of the samples used in this study, showing the position of the FOT line as revealed by the paramagnetic peak in the local susceptibility ($B_{\mathrm{F}\mathrm{O}\mathrm{T}}$) or the second peak in the dc magnetization ($B_{sp}$).