$A$-site cation size effect on oxygen octahedral rotations in acentric Ruddlesden-Popper alkali rare-earth titanates

We demonstrate inversion symmetry breaking induced by oxygen octahedral rotations in layered perovskite oxides $\mathrm{K}{A}_R{\mathrm{TiO}}_4$ ($A_R$ = rare earth) using a combined experimental and theoretical approach including synchrotron x-ray diffraction, optical second harmonic generation, and first-principles lattice dynamics calculations. We experimentally find an interesting but counterintuitive phenomenon, i.e., the acentric-to-centric phase transition temperatures for K family are higher than those for previously reported Na family, in contrast to expectations based on the Goldschmidt tolerance factor, where the octahedral rotation instability toward the acentric phases would reduce with an increase in the radius of $A$-site alkali metal ions. Our detailed analysis of first-principles calculations for $A_A{A}_R{\mathrm{TiO}}_4$ ($A_A=\mathrm{Na}$, K, Rb) reveals that the alkali metal and rare-earth ions play quite different roles in driving the octahedral rotations. Since rare-earth ions attract oxide ions more strongly than alkali metal ions due to the higher valence of the former in comparison with the latter ($A_R^{3+}$ vs $A_A^{+}$), the optimization of coordination environment of rare-earth ions is the primary driving force of the octahedral rotations. Alkali metal ions serve to impose “bond strains” parallel to the layers, playing a secondary role in the octahedral rotations. Incorporation of large alkali metal ions generates a significant in-plane biaxial bond strain in $A_R\mathrm{O}$ and ${\mathrm{TiO}}_2$ layers through the expanded $A_A\mathrm{O}$ layers, and thereby facilitates the octahedral rotations because of the otherwise highly underbonding of rare-earth ions. Thus, the effect of $A$-site alkali metal size on the octahedral rotation instability can be explained in terms of the interlayer lattice mismatch. This understanding allows us to propose a geometric descriptor governing the structural instability in $A_A{A}_R{\mathrm{TiO}}_4$ layered perovskites. We believe that control over the interlayer lattice mismatch could be a useful strategy to tune the octahedral rotations in layered compounds.

Figure 1

Schematic illustration of crystal structure of $A_A{A}_R{\mathrm{TiO}}_4$ with $P\overline{4}{2}_{1}m$ space-group symmetry viewed along (a) $\left[\overline{1}10\right]$ and (b) [110] directions. Octahedral rotation angles ${\theta }_A$ and ${\theta }_R$ are defined.

Figure 2

Total energies of the $P\overline{4}{2}_{1}m$ and $Pbcm$ structures relative to the parent $P4/nmm$ phase as a function of ionic radius of rare earth for (a) $\mathrm{K}{A}_R{\mathrm{TiO}}_4$ and (b) $\mathrm{Rb}{A}_R{\mathrm{TiO}}_4$.

Figure 3

Synchrotron XRD patterns of (a) ${\mathrm{KSmTiO}}_4$ and (b) ${\mathrm{KEuTiO}}_4$ collected at room temperature with x-ray of $\lambda =0.774964$ Å and fitting curves obtained by Rietveld refinements using the $P\overline{4}{2}_{1}m$ models. The refined structural parameters are summarized in Table 1.

Figure 4

(a) Goldschmidt tolerance factor $\tau$ and (b) total energy of $P\overline{4}{2}_{1}m$ structures relative to that of $P4/nmm$ parent structures as a function of ionic radius of alkali metal (vertical axis) and rare earth (horizontal).

Figure 5

Temperature dependence of second harmonic generation for (a) $A_A{\mathrm{SmTiO}}_4$ and (b) $A_A{\mathrm{EuTiO}}_4$ with $A_A=\mathrm{Na}$ (yellow circles) and K (purple circles). The inset of (a) presents the temperature dependence of peak area of ($\frac{1}{2}\frac{1}{2}1$) superlattice reflection with respect to the $P4/nmm$ parent structure. The arrows in (a) indicate the range in which the phase transitions are revealed to occur by the temperature-dependent superlattice reflection (see Ref. [31] for the temperature variation of synchrotron XRD patterns for ${\mathrm{NaSmTiO}}_4$).

Figure 6

(a) Schematic of the $P\overline{4}{2}_{1}m$ structure for $A_A{A}_R{\mathrm{TiO}}_4$ viewed along [110] and $\left[\overline{1}10\right]$ directions, in which bonds between cations and oxide ions are defined. The bonds to the oxide ions in the axial, equatorial, and diagonal directions are denoted as “ax,” “eq,” and “dg,” respectively. (b) The bond length change caused by the ($\mathrm{\Phi }00\left)\right(0\mathrm{\Phi }0$)-type of octahedral rotations for ${\mathrm{NaYTiO}}_4$.

Figure 7

The change in the length of (a) $A_R$-O eq(I) and (b) $A_A$-O dg(I) bonds and the octahedral rotation angle, (c) ${\theta }_R$ and (d) ${\theta }_A$, defined in the inset of Fig. 1, as a function of $A_R$ ionic radius for $A_A{A}_R{\mathrm{TiO}}_4$ with $A_A=\mathrm{Na}$ (yellow circles), K (purple squares), and Rb (bluish-green triangles).

Figure 8

Total energies of the $P\overline{4}{2}_{1}m$ structures relative to the parent $P4/nmm$ phase as a function of rotation-induced bond length change in the cation-oxygen pairs in Fig. 6 and for all the compositions; each panel includes the results for $\mathrm{Na}{A}_R{\mathrm{TiO}}_4$ (yellow circles) with $A_R=\mathrm{La}$, Nd, Sm, Eu, Gd, Dy, Y, and Ho, $\mathrm{K}{A}_R{\mathrm{TiO}}_4$ (purple squares) with $A_R=\mathrm{La}$, Nd, Sm, Eu, Gd, Dy, Y, and Ho, and $\mathrm{Rb}{A}_R{\mathrm{TiO}}_4$ (bluish-green triangles) with $A_R=\mathrm{La}$, Nd, Sm, Eu, Gd, Dy, Y, and Ho. As an example, data corresponding to each of the rare earths are labeled for the $A_R$-O eq(I) bonds of $\mathrm{K}{A}_R{\mathrm{TiO}}_4$.

Figure 9

(a) Schematic of the $P4/nmm$ structure, where $A_A\mathrm{O}$, $A_R\mathrm{O}$, and ${\mathrm{TiO}}_2$ layers are highlighted. (b) Chemical strain of the layers for ${\mathrm{KYTiO}}_4$ and ${\mathrm{RbYTiO}}_4$ with the $P4/nmm$ structures with respect to those for ${\mathrm{NaYTiO}}_4$. (c) The squared frequency of the $M_1$ mode for ${\mathrm{NaYTiO}}_4$ with the $P4/nmm$ structure as a function of in-plane strain.

Figure 10

Relationship between energy gain by the ($\mathrm{\Phi }00\left)\right(0\mathrm{\Phi }0$)-type of octahedral rotations and tolerance factors ${\tau }_A$ ($A$ = $A_A$ and $A_R$), which is a Goldschmidt tolerance factor for a hypothetic compound $A{\mathrm{TiO}}_3$.