Metallic “Ferroelectricity” in the Pyrochlore ${\mathrm{C}\mathrm{d}}_2{\mathrm{R}\mathrm{e}}_2{\mathrm{O}}_7$

A class of materials known as “ferroelectric metals” was discussed theoretically by Anderson and Blount in 1965 [Phys. Rev. Lett. 14, 217 (1965)], but to date no examples of this class of materials have been reported. Here we present measurements of the elastic moduli of ${\mathrm{C}\mathrm{d}}_2{\mathrm{R}\mathrm{e}}_2{\mathrm{O}}_7$ through the 200 K cubic-to-tetragonal phase transition. A Landau analysis of the moduli reveals that the transition is consistent with ${\mathrm{C}\mathrm{d}}_2{\mathrm{R}\mathrm{e}}_2{\mathrm{O}}_7$ being classified as a ferroelectric metal in the weaker sense described by Anderson and Blount (loss of a center of symmetry). First-principles calculations of the lattice instabilities indicate that the dominant lattice instability corresponds to a twofold degenerate mode with $E_u$ symmetry and that motions of the O ions forming the O octahedra dominate the energetics of the transition.

Figure 1

Elastic moduli vs temperature obtained on a single crystal of ${\mathrm{C}\mathrm{d}}_2{\mathrm{R}\mathrm{e}}_2{\mathrm{O}}_7$ using RUS. The relationship between sound velocity and elastic moduli is given by $v_s=(C/\rho {)}^{1/2}$, where $C$ is the effective elastic constant and $\rho$ is the density. In a cubic crystal, longitudinal waves propagating in the [100] direction are governed by $C_{11}$, and both transverse waves are governed by $C_{44}$. In the [110] direction, longitudinal waves are governed by $C_{L[110]}=1/2(C_{11}+C_{12}+2C_{44})$, one transverse wave is governed by $C_{44}$, and the other by $C_{T[110]}=1/2(C_{11}-C_{12})$. In the [111] direction, longitudinal waves are governed by $C_{L[111]}=1/3(C_{11}+2C_{12}+4C_{44})$, and both transverse waves are governed by $C_{T[111]}=1/3(C_{11}-C_{12}+C_{44})$. Note the small magnitude (2%) of the anomaly in $C_{44}$ compared to the anomalies in the other moduli.

Figure 2

Relativistic LDA energetics of lattice instabilities in ${\mathrm{C}\mathrm{d}}_2{\mathrm{R}\mathrm{e}}_2{\mathrm{O}}_7$. The coordinates of the O(1) atoms are in Cartesian coordinates, units of the lattice constant $a=10.219\AA$: ($0.315-{\delta }_E+{\delta }_T$, 0.625, 0.625), ($0.935+{\delta }_E+{\delta }_T$, 0.625, 0.625), (0.625, 0.315, 0.625), (0.625, 0.935, 0.625), (0.625, 0.625,$0.315+{\delta }_E$), (0.625, 0.625, $0.935-{\delta }_E$), (0.375, 0.065, 0.375), (0.375, 0.685, 0.375), ($0.065+{\delta }_E+{\delta }_T$, 0.375, 0.375), ($0.685-{\delta }_E+{\delta }_T$, 0.375, 0.375), (0.375, 0.375, $0.685+{\delta }_E$), (0.375, 0.375, $0.065-{\delta }_E$), where $E$ and $T$ are the $E_u$ and $T_{1u}$ frozen-in amplitudes, respectively.