Superconducting Condensate Formation in Quasi-2D Systems with Arbitrary   Carrier Density
  A phase diagram for a quasi-2D metal with variable carrier density has been derived. The phases present are the normal phase, where the order parameter is zero; the pseudogap phase where the absolute value of the order parameter is non-zero but its phase is random, and a superconducting phase with a crossover quasi-2D Berezinskii-Kosterlitz-Thouless (BKT) region. The crossover region is bounded by the quasi-2D BKT temperature and the temperature for the onset of conventional long-range order (CLRO). The practical observation of these regions however critically depends on the carrier density. At high densities both the pseudogap and BKT regions vanish asymptotically i.e. one obtains the standard BCS picture. At intermediate densities the pseudogap phase is large but the BKT region negligible. Finally at very low densities both the pseudogap and BKT regions are sizeable. An attempt is made to explain the behaviour observed in underdoped (intermediate densities) and optimally doped high-$T_{c}$ superconducting compounds above their critical temperature. The transition to the pseudogap phase should also be regarded as a crossover. 
Vadim M. Loktev, Rachel M. Quick and Sergei G. Sharapov