Figure 4: Thickness-dependent polarization.

(a) Calculated polarization is plotted as a function of the thickness. The error bar is the s.d. The dashed lines are plotted for eye guidance. (b) Comparison of the normalized polarization as a function of thickness with previous results. Solid red plot shows the predictions from mean-field Landau-type model for PZT from ref. 26. The solid green plot is first principle calculations for SrRuO₃/BaTiO₃/SrRuO₃ (SRO/BTO/SRO) from ref. 12. The blue squares are experimentally measured values for SRO/PZT/SRO from ref. 16. The cyan circles are experimentally measured values for SRO/BTO/SRO from ref. 22. The magenta diamonds are experimentally measured values for SRO/BTO/SRO from ref. 15. The orange triangles are theoretical predictions for Pt/KNbO₃/Pt (Pt/KNO/Pt) from ref. 32. The dark green stars are theoretical predictions for SRO/KNO/SRO from ref. 32. The purple triangles are experimentally measured values for BiFeO₃/La_0.67Sr_0.33MnO₃/SrTiO₃ (BFO/LSMO/STO) from ref. 10. The dark yellow triangles are experimentally measured values for BFO/LSMO/STO from ref. 21. The orange and blue solid discs are from this work. (c) Calculated depolarizing field is plotted as a function of the thickness for PZT on SRO/STO. The density function theory results from ref. 13 showing the depolarizing fields for free standing 2-unit cells thick PbTiO₃ (PTO), 2-unit cells thick PTO sandwiched between SRO electrodes, and 2-unit cells thick PTO sandwiched between Pt electrodes. (d) Displacements of Pb respective to O and Zr/Ti respective to O are plotted as a function of the thickness for PZT on STO. The error bars are the s.d. (e) An ABF image of 1.5-unit cells thick PZT on STO. The contrast has been inverted for clarity. The octahedron shifting to left in the PZT film is visible by the naked eye. The arrows showing the O are closer to the left Pb, causing the atomic displacements and polarization. Scale bar, 0.2 nm.