First-Principles Investigations of Pb Anti-Site Defects in PbZrO$_3$ and Pb(Zr, Ti)O$_3$ Perovskites*

Lead zirconate (PZ) and lead zirconate titanate (PZT) have the perovskite type structure, ABO$_3$. Bivalent lead (Pb$^{+2}$) ions occupy the A site, while tetravalent titanium and zirconium (Zr$^{+4}$, Ti$^{+4}$) ions occupy the B site at random of the PZT solid solution. Also, lead can be tetravalent (Pb$^{+4}$), such as in PbO$_2$ structure. Recent experimental work has reported that tetravalent Pb ions can locate at the B site of the PZT perovskite forming a lead zirconate-titanate-plumbate solid solution. The experimental results suggest that, based on a PbZrO$_3$-PbTiO$_3$-PbPbO$_3$ ternary solution phase diagram [G. Suchaneck \emph{et al.}, Ferroelectrics {\bf 318}, 3 (2005)], the substitutional Pb atom prefers to occupy the Zr site instead of the Ti one. In this work, we report density functional supercell calculations for pure PbZrO$_3$ perovskite and for ordered Pb(Zr$_{1/2}$Ti$_{1/2}$)O$_3$ solid solution with different configurations for the Zr and Ti atoms. We investigate the anti-site defect energies and the effects on the electronic structure.