Thermodynamic Theory of Disordered 2D Superconductors

Understanding the roles of disorder and superconducting phase fluctuation in superconductivity has been a long-standing challenge. For example, while the phase fluctuation is expected to destroy the superconductivity of intrinsically disordered two-dimensional (2D) superconductors at any finite temperatures, there have been ample experimental evidences showing robust long-range superconducting order in ultra-thin films and atomic sheets. The observed unique superconducting-insulating transition in 2D samples with sufficiently large amount of disorder also goes beyond the conventional theoretical paradigm. Here we develop a self-consistent thermodynamic theory of the superconducting gap and phase fluctuation in disordered 2D superconductors, starting from a purely microscopic model. It incorporates both quantum and thermal phase fluctuations in the presence of the long-range Coulomb interactions. Our numerical simulation based on the developed theory successfully proves a long-range superconducting order in 2D limit even when temperature is increased away from zero, while the gradually emerging large thermal phase fluctuations with further increasing temperature destroy the superconducting gap. On the other hand, the inhomogeneous quantum phase fluctuations with increasing disorder result in a mixed state of superconducting and normal-state islands, thereby reducing $T_c$. But a robust superconductivity can survive at low temperature even at high disorder, giving rise to the prerequisite of the superconducting-insulating transition. More importantly, our theory shows that the phase fluctuation can be suppressed by increasing carrier density, leading to a carrier density-dependent $T_c$. These findings explain many of the recently observed experimental features of the superconductors in 2D limit and can potentially shed light on the understanding of high-$T_c$ superconductors.