The components of the Ti-centered lattice Wannier function at the (Sr), , and atoms are represented by large (cyan), medium (magneta), and small (white) circles, respectively.
Best fits to the total energy curves of the (a) and (b) phonon local modes. Solid lines represent the results for BT and dashed lines for ST.
Elastic constants calculated from first-principles calculations for bulk and . Elastic constants are given in . Solid lines represent the results for BT and dashed lines for ST.
Calculated on-site coupling constants between lattice instability and strain for bulk and . Solid lines represent the results for BT and dashed lines for ST.
Comparison of first-principles calculations with effective Hamiltonian results on the average polarization and tetragonality as a function of for each superlattice. Open diamonds and open squares represent the average polarization and tetragonality, respectively, taken from first-principles calculations of Ref. 2; filled (red) diamonds and squares represent the effective Hamiltonian results of average polarization and tetragonality, respectively. The dotted line is a result of the macroscopic theory using as a adjusting fitting parameter. is a polarization of the tetragonal . The fractional numbers beside the marks denote the of .
(a) Relative local polarization and (b) local tetragonality at the layer in the five-layer supercell computed from our effective Hamiltonian for each superlattice. is a polarization of the tetragonal . The BT, ST, and I labels represents the Ti-centered layer sandwiched by two BT layers, two ST layers, and BT-ST layers at the sides of -layer, respectively.
The -axis lattice constants and elastic constants of and as determined from the total energy minimization with the in-plane lattice constrained at that of .
Approximate Ti-centered local Wannier function for the mode and the components of its normalized eigenvectors.
Effective charges calculated from the first-principles geometric phase approach. These values are used in the long-range dipole-dipole interactions.
The harmonic coefficients and as determined by first-principles calculations, the dipole-dipole interaction coefficients and , the mode effective charge coefficients , and the derived local coefficients , , in the quadratic part of . and are dimensionless, is , and the others are in .
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