Low anisotropy of the upper critical field in a strongly anisotropic layered cuprate: Evidence for paramagnetically limited superconductivity

We study angular-dependent magnetoresistance in a low $T_c$ layered cuprate Bi$_{2.15}$Sr$_{1.9}$CuO$_{6+\delta}$. The low $T_c$ ~ 4 K allows complete suppression of superconductivity by modest magnetic fields and facilitate accurate analysis of the upper critical field $H_{c2}$. We observe an universal exponential decay of fluctuation conductivity in a broad range of temperatures above $T_c$ and propose a new method for extraction of $H_{c2}(T)$ from the scaling analysis of the fluctuation conductivity at $T>T_c$. Our main result is observation of a surprisingly low $H_{c2}$ anisotropy ~ 2, which is much smaller than the effective mass anisotropy of the material ~ 300. We show that the anisotropy is decreasing with increasing field and saturates at a small value when the field reaches the paramagnetic limit. We argue that the dramatic discrepancy of high field and low field anisotropies is a clear evidence for paramagnetically limited superconductivity.