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Effects of Pb doping on structural and electronics properties of Bi$_2$Sr$_2$Ca$_2$Cu$_3$O$_{10}$

Pb doping effect in the Bi$_2$Sr$_2$Ca$_2$Cu$_3$O$_{10}$ compound (Bi2223) on the structural and electronic properties were investigated, using the Local Density (LDA) and Virtual Crystal (VCA) approximations within the framework of the Density Functional Theory (DFT), taking as reference the procedure implemented by H.Lin {\it et al.} in the Bi2212 compound [{\it Phys. Rev. Lett.} {\bf 96} (2006) 097001]. Results show that, the incorporation of Pb-dopant in Bi2223 lead a rigid displacement of the Bi/Pb-O bands toward higher energies, with a null contribution at the Fermi level, around the high symmetry point $\overline{\text{M}}$ in the irreducible Brillouin zone, for Pb doping concentration equal to or more than 26\%, avoiding the presence of the so-called Bi-O {\it pockets} in the Fermi surface, in good agreement with angle-resolved photoemission spectroscopy (ARPES) and nuclear magnetic resonance (NMR) experiments, although a slight metallic character of the Bi-O bonds is still observed which would disagree with some experimental reports. The calculations show that the changes on the structural properties are associated to the presence or absence of the Bi-O {\it pockets} in the Fermi surface