Kosterlitz-Thouless Behavior in Layered Superconductors: The Role of the Vortex Core Energy

In layered superconductors (SC) with small interlayer Josephson coupling vortex-antivortex phase fluctuations characteristic of quasi two-dimensional (2D) Kosterlitz-Thouless behavior are expected to be observable at some energy scale $T_d$. While in the 2D case $T_d$ is uniquely identified by the KT temperature $T_{\mathrm{KT}}$ where the universal value of the superfluid density is reached, we show that in a generic anisotropic 3D system $T_d$ is controlled by the vortex-core energy, and can be significantly larger than the 2D scale $T_{\mathrm{KT}}$. These results are discussed in relation to recent experiments in cuprates, which represent a typical experimental realization of layered anisotropic SC.

Figure 1

RG flow of the couplings $K(\ell )$ and $K_s(\ell )$ at various temperatures for $\mu ={\mu }_{XY}$ and $\eta ={10}^{-4}$. Inset: critical temperature $T_c$ as a function of $\eta$ for a bare stiffness $J_0(T)=J_{ab}(1-T/4J_{ab})$ (see text).

Figure 2

Temperature dependence of $J_s(T)$ in two dimensions (lines) and in the layered 3D case (symbols). Here $J_0(T)=J_{ab}(1-T/4J_{ab})$. The $T_{\mathrm{KT}}$ is identified by the intersection between $J_s(T)$ and the straight line $2T/\pi$. The results for $\mu \le {\mu }_{XY}$ show a rapid downturn of $J_s(T)$ at $T_d\simeq T_{\mathrm{KT}}$. As $\mu$ increases $T_d$ increases as well, so that at $T_{\mathrm{KT}}$ no effect is observed in $J_s(T)$ reminiscent of the jump present in two dimensions.

Figure 3

Correlation length above $T_c$ for different values of $\eta$ and $\mu$. Each curve stops at $T_c$, which increases as $\eta$ increases. We take $J_0(T)=J_{ab}(1-T/T_{\mathrm{MF}})$, with $J_{ab}=300\mathrm{K}$ and $T_{\mathrm{MF}}=100\mathrm{K}$, as appropriate for Bi221 [8]. The dashed line is $\xi \sim e^{\beta \mu (T)/2}$, expected to hold far above $T_c$. The solid lines are fit with the KT functional form (see text), with $T_d=T_{\mathrm{KT}}$, $c=0.25$, $b=0.9$ for $\mu ={\mu }_{XY}$, and $T_d=86.8\mathrm{K}$, $c=0.4$, $b=1.21$ for $\mu =3{\mu }_{XY}$.