Doping dependence of the vortex-core energy in bilayer films of cuprates

The energy needed to create a vortex core is the basic ingredient to address the physics of thermal vortex fluctuations in underdoped cuprates. Here, we theoretically investigate its role in the occurrence of the Beresinskii-Kosterlitz-Thouless transition in a bilayer film with inhomogeneity. From the comparison with recent measurements of the penetration depth in two-unit-cell thin films, we can extract the value of the vortex-core energy $\mu$ and show that $\mu$ scales linearly with $T_c$ at low doping.

Figure 1

(Color online) Temperature dependence of the superfluid density in the inhomogeneous bilayer system, with parameter values explained in the text. The downturn is located at the intersection with the line $n=1$ or $n=2$ for uncoupled or totally coupled layers, respectively. Inset: expanded view for the most underdoped case. The dashed-dotted line is the result obtained without averaging over the $J_0$ inhomogeneity.

Figure 2

(Color online) (a) $1∕{\lambda }^2$ and ${\mu }_0\omega {\sigma }_1$ evaluated at $\omega =50\mathrm{kHz}$ for a single $\overline{J}\left(T\right)$ curve (here, $\mu =3{\mu }_{XY}$). The finite frequency leads to a sharp but continuous decrease of $1∕{\lambda }^2$ across $T_{\mathit{BKT}}$, along with a peak in ${\sigma }_1$. (b) $1∕{\lambda }^2$ and ${\mu }_0\omega {\sigma }_1$ evaluated at finite frequency using the averaged $J_{\mathit{inh}}$. Also shown for comparison is the homogeneous curve of panel (a) (dashed-dotted line).

Figure 3

(Color online) Vortex-core energy as a function of $T_c$ extracted from the fit in Fig. 1. (a) Ratio between $\mu$ and the ${\mu }_{XY}$ value [Eq. (6)], proportional to the single-layer energy $J$. (b) Absolute value of $\mu$ in K. The dashed line is $\mu =8T_c$.