Muon spin rotation investigation of the pressure effect on the magnetic penetration depth in YBa$_2$Cu$_3$O$_x$

The pressure dependence of the magnetic penetration depth $\lambda$ in polycrystalline samples of YBa$_2$Cu$_3$O$_x$ with different oxygen concentrations $x$ = 6.45, 6.6, 6.8, and 6.98 was studied by muon spin rotation ($\mu$SR). The pressure dependence of the superfluid density $\rho_s \propto 1/\lambda^{2}$ as a function of the superconducting transition temperature T$_{\rm c}$ is found to deviate from the usual Uemura line. The ratio $(\partial T_{\rm c}/\partial P)/(\partial\rho_s/\partial P)$ is factor of $\simeq$ 2 smaller than that of the Uemura relation. In underdoped samples, the zero temperature superconducting gap $\Delta_0$ and the BCS ratio $\Delta_0/k_{B}T_{\rm c}$ both increase with increasing external hydrostatic pressure, implying an increase of the coupling strength with pressure. The relation between the pressure effect and the oxygen isotope effect on $\lambda$ is also discussed. In order to analyze reliably the $\mu$SR spectra of samples with strong magnetic moments in a pressure cell, a special model was developed and applied.