Superfluid and critical current densities of underdoped YBa$_2$Cu$_3$O$_{6+x}$ cuprates

Using the Layered Boson-Fermion Model of superconductivity [1] applied to cuprate superconductors, for which we previously reported the thermodynamic properties [2], we obtain the penetration and coherence lengths for underdoped cuprate superconductors YBa$_2$Cu$_3$O$_{6+x}$ as functions of temperature for doping values $x \in[0.55,0.9]$.
We also get expressions for the critical current and normalized superfluid densities.
Comparing with available experimental data, as functions of both, temperature and doping for $T = 0$, we determine the doping interval for which quantum fluctations are predominant over the influence of the layered structure.


\noindent [1] P. Salas, M. Fortes, M. A. Sol\'is and F. J. Sevilla, Physica C {\bf 534}, 37 (2016).

\noindent [2] P. Salas, M. A. Sol\'is, M. Fortes and F. J. Sevilla, submited to Int. Jou. Mod. Phys. B.

\noindent We thank partial support form CONACyT 221030 y DGAPA-PAPIIT IN107616.