Concentration of Charge Carriers and Anomalous Gap Parameter in the   Normal State of High-$T_c$ Superconductors
  Fermi-Dirac statistics has been utilized by introducing the average ionization energy ($E_I$) as an additional anomalous energy gap in order to derive the two-dimensional concentration of charge carriers and the phenomenological resistivity model for the superconducting polycrystalline materials. The best fitted values of $E_I$ and the charge carriers' concentration ranges in the vicinity of 4 to 9 meV and 10$^{16}$ m$^{-2}$ respectively for the superconducting single crystal samples and polycrystalline compounds synthesized with various compositions via solid-state reactions. The phenomenological resistivity model is further redefined here based on the gapless nature of charge-carriers' dynamics within the Cu-O$_2$ planes that corresponds to anomalous Fermi liquid behavior, which is in accordance with the nested Fermi liquid theory. 
Andrew Das Arulsamy