Anisotropy of microwave conductivity in the superconducting and normal states of YBa$_2$Cu$_3$O$_{7-x}$: 3D-2D crossover

The imaginary parts of microwave conductivity $\sigma''(T<T_c)$ and resistivity $\rho(T)=1/\sigma(T>T_c)$ along($\sigma_{ab}''$ and $\rho_{ab}$) and across and ($\sigma_c''$ and $\rho_c$) the cuprate $ab$ planes of a YBa$_2$Cu$_3$O$_{7-x}$ crystal with the oxygen doping level $x$ varying from 0.07 to 0.47 were measured in the temperature range $5\leq T \leq 200$ K. In the superconducting state, the $\sigma_{ab}''(T)/\sigma_{ab}''(0)$ and $\sigma_c''(T)/\sigma_c''(0)$ curves coincide for an optimally doped($x=0.07$) crystal, but, with an increase in $x$, the slopes of the $\sigma_c''(T)/\sigma_c''(0)$ curves decrease noticeably at $T<T_c/3$, on the background of small changes happening to the $\sigma_{ab}''(T)/\sigma_{ab}''(0)$ curves. The two-dimensional (2D) transport along the $ab$ planes in the normal state of YBa$_2$Cu$_3$O$_{7-x}$ s always metallic, but there is a crossover (at $x=0.07$) from the Drude to hopping (at $x>0.07$) conductivity along the $c$ axis. This is confirmed both by the estimates of the lowest metallic and the highest tunneling conductivities along the $c$ axis and by quantitative comparison of the measured $\rho_c(T)$ curves with the curves calculated in the polaron model of quasiparticle transport along the $c$ axis.