Microwave surface impedance anisotropy of YBa$_2$Cu$_3$O$_x$ single crystals with different oxygen content

The linear microwave response of ultra high-quality YBa$_2$Cu$_3$O$_x$ single crystals grown in BaZrO$_3$ crucibles is measured at 9.4 GHz in rf magnetic fields parallel and perpendicular to the $ab$-plane in the temperature range 5$\leq T\leq 200$ K. Having found the analytic solution for the magnetic field distribution on the sample surface we determine both the surface impedance $Z_{ab}=R_{ab}+iX_{ab}$ in the $ab$-plane and $Z_c=R_c+iX_c$ along c-axis of the crystals. For the first time the evolution of the $Z_{ab}(T)$ and $Z_c(T)$ dependences on the same sample and in a wide range of oxygen content is obtained. For $x=6.95$ (optimum oxygen content) the temperature dependence of the imaginary part $\sigma_{c}''(T)/\sigma_{c}''(0)$ of the c-axis conductivity is found to be strikingly similar to that of $\sigma_{ab}''(T)/\sigma_{ab}''(0)$ and becomes more convex with $x$ lowering.