Relationship Between Conductivity and Phase Coherence Length in Cuprates

The large ($10^2 - 10^5$) and strongly temperature dependent resistive anisotropy $\eta = (\sigma_{ab}/\sigma_c)^{1/2}$ of cuprates perhaps holds the key to understanding their normal state in-plane $\sigma_{ab}$ and out-of-plane $\sigma_{c}$ conductivities. It can be shown that $\eta$ is determined by the ratio of the phase coherence lengths $\ell_i$ in the respective directions: $\sigma_{ab}/\sigma_c = \ell_{ab}^2/\ell_{c}^2$. In layered crystals in which the out-of-plane transport is incoherent, $\ell_{c}$ is fixed, equal to the interlayer spacing. As a result, the T-dependence of $\eta$ is determined by that of $\ell_{ab}$, and vice versa, the in-plane phase coherence length can be obtained directly by measuring the resistive anisotropy. We present data for hole-doped $YBa_2Cu_3O_y$ ($6.3 < y < 6.9$) and $Y_{1-x}Pr_xBa_2Cu_3O_{7-\delta }$ ($0 < x \leq 0.55$) and show that $\sigma_{ab}$ of crystals with different doping levels can be well described by a two parameter universal function of the in-plane phase coherence length. In the electron-doped $Nd_{2-x}Ce_{x}CuO_{4-y}$, the dependence $\sigma_{ab}(\eta)$ indicates a crossover from incoherent to coherent transport in the c-direction.