Nanoscale ice-type structural fluctuation in spinel titanates

In the spinel titanate ${\mathrm{MgTi}}_2{\mathrm{O}}_4$, the tetragonal phase collapses upon substitution of a tiny amount of Mg ions at the Ti site, and the cubic phase with the geometrical frustration is resurrected. The atomic pair distribution function (PDF) and the extended x-ray absorption fine structure (EXAFS) reveal the nanoscale structural fluctuation, in which the Ti atomic displacement has the two-in two-out configuration in the cubic phase. We argue that the geometrical frustration plays an essential role in the collapse of the tetragonal phase and the resultant nanoscale ice-type structural fluctuation.

Figure 1

(a) Schematic of the pyrochlore lattice. (b) Original positions of the atoms (blue spheres) and the positions of the displaced atoms (green spheres) in the two-in two-out type displacement. For clarity, the displacement is exaggerated. (c) (8 0 0) reflection in the x-ray diffraction in ${\mathrm{MgTi}}_2{\mathrm{O}}_4$. (d) Temperature dependence of the lattice constants. (e) (8 0 0) reflection at 20 K in $x=0$, 0.01, 0.06, 0.125, 0.25, and 0.4. Inset: Difference in the angular position between the two peaks at 20 K (red) and the difference in the full width at half maximum between 300 K and 20 K (black), whose units are degrees. (f) $x$ dependence of the lattice constants at 20 K. In the inset of (e) and in (f), the thick bars on the left ordinate indicate the values in $x=0$.

Figure 2

(a) Observed powder neutron diffraction $S(Q$) in ${\mathrm{MgTi}}_2{\mathrm{O}}_4$. Inset: Magnified view of the diffraction in $Q=2\sim 2.5{\AA }^{-1}$. (b) Atomic pair distribution function (PDF) $G(r$) in $r=1.5\sim 50\AA$. The arrows indicate the ranges where the boxcar refinement was applied. (c) Barcodes indicating the distances of each atom pair under the assumption of cubic $Fd\overline{3}m$. (d) Experimental results of PDF in $r=1.5\sim 4\AA$ at 450 K (red), 300 K (black), and 20 K (blue). (e) Observed PDF at 300 K (gray circle) and PDFs calculated by using the structural model. The assumption of $Fd\overline{3}m$ (black curve) and $P{4}_1{2}_{1}2$ (red curve). (f) Distance between the neighboring Ti ions as a function of the $r_{max}$, where the boxcar refinement was applied, at 450 K (red cross), 300 K (black circle), 150 K (green triangle), and 20 K (blue square). The thick bars on the right ordinate indicate the Ti-Ti distances obtained by the Rietveld analysis.

Figure 3

(a) Observed neutron PDF $G(r$) in $r=1.5\sim 4\AA$ at 20 K in $x=0$ (black), 0.125 (blue), 0.25 (red), 0.32 (green), and 0.4 (gray) of ${\mathrm{Mg}}_{1+x}{\mathrm{Ti}}_{2-x}{\mathrm{O}}_4$. (b) Observed PDF at 20 K in $x=0.125$ (gray circle) and the PDFs calculated by using the structural model. The assumption of $Fd\overline{3}m$ (black curve) and $P{4}_1{2}_{1}2$ (red curve). (c) Distance between the neighboring Ti ions as a function of the $r_{max}$, where the boxcar refinement was applied, in $x=0.06$ (purple circle), 0.125 (blue square), 0.25 (red diamond), 0.32 (green triangle), and 0.4 (gray cross). (d) Radial distribution function in $r=0.5\sim 3\AA$ obtained in the Ti $K$ edge EXAFS in $x=0$ (black), 0.125 (blue), 0.25 (red), 0.32 (green), and 0.4 (gray). (e) and (f) Experimental result (gray circle) at 20 K and the simulations under the assumption of $P{4}_1{2}_{1}2$ (blue) and $Fd\overline{3}m$ (red) in $x=0$ (e) and 0.4 (f), respectively.

Figure 4

(a) Phase diagram of the lattice structure and the magnetism in ${\mathrm{Mg}}_{1+x}{\mathrm{Ti}}_{2-x}{\mathrm{O}}_4$. The black circles and the green crosses indicate the structural transition temperature ($T_c$) and the temperature ($T_{\mathrm{mag}}$) at which the magnetic susceptibility decreases, respectively. (b) $x$ dependence of the correlation length $\xi$ of the tetragonal structure obtained by the PDF analyses at 20 K (blue), 150 K (purple), and 300 K (red). (c) Reduction of the magnetic susceptibility $\left|\chi \right(150$ K)$-\chi (260$ K)$|$. Inset: Temperature dependence of the magnetic susceptibility in which $\chi (260$ K) is subtracted from $\chi (T$). The thick bars on the left ordinate in (a)$\sim$(c) indicate the values in $x=0$.

Figure 5

Ti dimer configuration in the tetragonal phase, which is viewed from the $c$ axis. The circles, the black lines, and the red lines indicate the Ti ions, the pyrochlore lattice, and the Ti dimers, respectively.

Figure 6

Schematic picture of the energy landscape. The bottom abscissa means the configuration of the structure such as the dimer pattern. The arrow indicates the position of the tetragonal phase.

Figure 7

(a) B1-type structure, (b) B2-type structure, and (c) B3-type structure in the gray Ti tetrahedron. The red lines indicate the Ti dimers.

Figure 8

Schematic picture of the pair of the substituted Mg ion and the ${\mathrm{Ti}}^{4+}$ ion. (a) Low-energy state: The ${\mathrm{Ti}}^{4+}$ ion and the Mg ion make a nesting pair. (b) Excited state: The ${\mathrm{Ti}}^{4+}$ state goes away from the Mg ion at a finite temperature. The blue curve indicates the path of the ${\mathrm{Ti}}^{4+}$ state. In the gray Ti tetrahedra, the Ti dimers do not form the B3-type structure. In (a) and (b), the red lines indicate the ${\mathrm{Ti}}^{3+}$ dimers. (c),(d) Picture in which the Ti displacements in the dimer formation are indicated by the arrows. (c) and (d) correspond to (a) and (b), respectively. In (d), the arrows are inverted by the movement of the ${\mathrm{Ti}}^{4+}$ state, as shown by blue arrows.