Vortex entropy and superconducting fluctuations in ultrathin underdoped Bi₂Sr₂CaCu₂O_8+x superconductor

Vortices in superconductors can help identify emergent phenomena but certain fundamental aspects of vortices, such as their entropy, remain poorly understood. Here, we study the vortex entropy in underdoped Bi₂Sr₂CaCu₂O_8+x by measuring both magneto-resistivity and Nernst effect on ultrathin flakes (≤2 unit-cell). We extract the London penetration depth from the magneto-transport measurements on samples with different doping levels. It reveals that the superfluid phase stiffness ρ_s scales linearly with the superconducting transition temperature T_c, down to the extremely underdoped case. On the same batch of ultrathin flakes, we measure the Nernst effect via on-chip thermometry. Together, we obtain the vortex entropy and find that it decays exponentially with T_c or ρ_s. We further analyze the Nernst signal above T_c in the framework of Gaussian superconducting fluctuations. The combination of electrical and thermoelectric measurements in the two-dimensional limit provides fresh insight into high temperature superconductivity.