Signatures of the Berezinskii$-$Kosterlitz$-$Thouless transition in highly underdoped La$_{2-x}$Sr$_{x}$CuO$_{4}$

We investigated the nature of the thermally driven transition to a superconducting state on MBE-grown 100 nm thick La$_{2-x}$Sr$_{x}$CuO$_{4}$ films ($x=0.07$ and $0.08$) by studying a) superconducting fluctuations (SCFs), as determined from the magnetoresistance, b) current-voltage ($I-V$) characteristics, and c) fluctuations of conductance with time (noise). Noise was studied on confined width (20 $\mu$m) samples specifically designed for measurements in the eight-probe double-bridge configuration. We observe: a) the contribution of the SCFs to the conductivity increases monotonically with decreasing $T$ and diverges exponentially at $T_{BKT}$, where resistance also goes to zero; b) power-law behavior $V \propto I^{\alpha(T)}$ with $\alpha (T) \geq 3$ for $T\leq T_{BKT}$; and c) an orders-of-magnitude enhancement of the noise near $T_{BKT}$. Possible long-range correlations in the phase fluctuations near $T_{BKT}$ are also studied with higher-order spectrum analysis. Our observations suggest that the transition from superconductor to the normal state exhibits signatures of the Berezinskii$-$Kosterlitz$-$Thouless (BKT) transition.