Structure and stability of ${\text{Cd}}_2{\text{Nb}}_2{\text{O}}_7$ and ${\text{Cd}}_2{\text{Ta}}_2{\text{O}}_7$ explored by ab initio calculations

Structural instabilities of ${\text{Cd}}_2{\text{Nb}}_2{\text{O}}_7$ have been explored by all-electron ab initio calculations in the framework of density-functional theory. The calculated phonon-dispersion curve of the cubic high-temperature phase displays several modes with imaginary frequencies. A distortion according to a $T_{1u}$ mode at the $\Gamma$ point leads to the energetically most favorable configuration. The resulting structure has been optimized in space-group symmetries $Ima2$ and $I1a1$ $\left(Cc\right)$. The stability of these structures suggests a sequence of phase transitions $Fd\overline{3}m$-$Ima2$-$Cc$. The dominant structural changes occur as a result of the ferroelectric transition to orthorhombic symmetry, whereas the distortions associated with the subsequent transition to monoclinic symmetry are relatively small. With respect to the cubic phase, Nb is displaced most strongly. In both low-temperature phases, two crystallographically distinct Nb and Cd positions exist. The Nb displacements give rise to an ordered arrangement of short and long Nb-O bonds. In the $Ima2$ phase, chains of alternating long and short bonds propagate along [001] and along $⟨111⟩$, while corresponding chains parallel to [100] still consist of symmetric Nb1-O bonds. The latter bonds distort only with the transition to the $Cc$ phase. In contrast to ${\text{Cd}}_2{\text{Nb}}_2{\text{O}}_7$, the structure of ${\text{Cd}}_2{\text{Ta}}_2{\text{O}}_7$ returns to cubic symmetry when subjected to similar distortions.

Figure 1

Calculated phonon-dispersion curve for cubic CNO. The scale of the $x$ axis corresponds to the modulus of the wave vector, starting from the $K$ point.

Figure 2

Lattice of maximal subgroups (Ref. 25) for the SG pair $Fd\overline{3}m/Cc$ in the absence of cell multiplication.

Figure 3

(Color online) The $I1a1$ structure projected along [100] (a) and along [001] (b) and in tilted view (c). The O2 atoms are located at the corners of the octahedra.

Figure 4

(Color online) Graphical representation of nonequivalent ${\text{NbO}}_6$ octahedra of the $Ima2$ phase, including Nb-O bond lengths. The $\text{Nb}1{\text{O}}_6$ octahedra are connected via the O3 atoms and form chains parallel to [100]; the $\text{Nb}2{\text{O}}_6$ octahedra are connected via the O2 atoms and form chains parallel to [001]. The atoms O4 and O5 connect octahedra of different types.

Figure 5

Electronic density of states of cubic CNO (dashed line) close to the Fermi energy $\left(E_F=0\text{eV}\right)$. The solid lines show the contributions of Cd, Nb, and O in the top, middle, and bottom diagrams, respectively.

Figure 6

(Color online) Electron-density maps for the energy region of Nb-O $d\text{−}p$ bonding $\left(-5.55\text{eV}<E<-2.83\text{eV}\right)$ in the $Fd\overline{3}m$ phase (a) and in the $Ima2$ phase (b). Isolines are in equal steps of $\approx 0.05e{\text{Å}}^{-3}$.