First-principles calculation of the structure and dielectric properties of ${\text{Bi}}_2{\text{Ti}}_2{\text{O}}_7$

The structure, vibrational modes, and phonon contribution to the dielectric function of the pyrochlore ${\text{Bi}}_2{\text{Ti}}_2{\text{O}}_6{\text{O}}^{\prime }$ were calculated using first-principles methods. Total-energy minimization calculations were performed for ${\text{Bi}}_2{\text{Ti}}_2{\text{O}}_6{\text{O}}^{\prime }$ in a unit cell containing 88 ions, which had the ideal, cubic pyrochlore structure as the initial configuration. No symmetry constraints were imposed during this relaxation. Subsequent symmetry analysis of the relaxed structure found $Pna{2}_1$ space-group symmetry in a 44 ion unit cell. This structure contains Bi ions with two types of eightfold coordination by O and ${\text{O}}^{\prime }$ ions. Vibrational modes and the dielectric function were calculated for the $Fd\overline{3}m$, $Pna{2}_1$, and $P_1$ structures. The crystal structure obtained by total-energy minimization is compared to structural data from reverse Monte Carlo analysis of neutron total scattering data. The imaginary part of the dielectric function derived from vibrational mode calculations is compared to dielectric function data for several related pyrochlores. Phonons which make the largest contributions to the dielectric constant are identified and analyzed.

Figure 1

(Color online) Ti and ${\text{O}}^{\prime }$ polyhedra in the ideal $Fd\overline{3}m$ pyrochlore structure viewed along the [101] direction. Ti ions are shown as small blue spheres, Bi ions as medium sized green spheres, O ions as large red spheres and ${\text{O}}^{\prime }$ ions as large brown spheres. (a) ${\text{TiO}}_6$ octahedra. (b) ${\text{O}}^{\prime }{\text{Bi}}_4$ tetrahedra. All Bi, Ti, O, and ${\text{O}}^{\prime }$ ions are equivalent by symmetry in the ideal structure.

Figure 2

Outline and registry of the 88 ion $Fd\overline{3}m$ conventional unit cell and 44 ion $Pna{2}_1$ primitive unit cell.

Figure 3

(Color online) Bi, Ti, and ${\text{O}}^{\prime }$ polyhedra in the $Pna{2}_1$ structure. Large spheres O (red) and ${\text{O}}^{\prime }$ (brown) ions, medium spheres Bi(1) (dark green) and Bi(2) (light green) ions, small blue spheres Ti ions. (a) Bi(1) polyhedra viewed along the [001] direction. (b) Bi(2) polyhedra viewed along the [100] direction, (c) Ti octahedra viewed along the [100] direction, (d) ${\text{O}}^{\prime }$ tetrahedra viewed along the [100] direction. Ti(1) octahedra in (c) are in the center of the ${\text{BiO}}^{\prime }$ diamond network. Ti(2) octahedra form rows between the Ti(1) octahedra.

Figure 4

(Color online) ${\text{BiO}}_6{\text{O}}_2^{\prime }$ polyhedra for Bi(1) in the $96h$ site (left) and Bi(2) in the $96g$ site (right) viewed along the ${\text{O}}^{\prime }{\text{O}}^{\prime }$ axis in the $Pna{2}_1$ structure. Bond lengths are given in angstrom. O ions (red), ${\text{O}}^{\prime }$ ions (brown), Bi ions (green).

Figure 5

(Color online) BiBi distances in angstrom in ${\text{O}}^{\prime }{\text{Bi}}_4$ tetrahedra in the ideal pyrochlore $Fd\overline{3}m$ and $P_1$ structures viewed along the [010] direction. (Left panel) BiBi distances in ${\text{O}}^{\prime }{\text{Bi}}_4$ tetrahedra in the ideal cubic pyrochlore structure are all the same. (Right panel) Bi(1) ions (dark green), Bi(2) ions (light green).

Figure 6

Atom-projected valence band densities of states for ${\text{Bi}}_2{\text{Ti}}_2{\text{O}}_6{\text{O}}^{\prime }$ in the $Pna{2}_1$ structure.

Figure 7

Charge density in the $Pna{2}_1$ structure associated with a group of 8 bands attributed to $\text{Bi}6s$ lone pairs. (Left panel) Charge density in a (100) plane containing Bi(1) ions. (Right panel) Charge density in a (100) plane containing Bi(2) ions.

Figure 8

Imaginary part of the dielectric function for BMN, BZT, BZN, and ${\text{Bi}}_2{\text{Ti}}_2{\text{O}}_6{\text{O}}^{\prime }$. Spectra for BMN, BZT, and BZN were constructed from fitting parameters to experimental data in Ref. 18 at 50 K. Spectra for ${\text{Bi}}_2{\text{Ti}}_2{\text{O}}_6{\text{O}}^{\prime }$ in the $Fd\overline{3}m$, $Pna{2}_1$, and $P_1$ structures obtained from first-principles calculations. Parameters fitted using experimental data were oscillator strength, natural frequency, and natural linewidth. Oscillator strengths and natural frequencies for ${\text{Bi}}_2{\text{Ti}}_2{\text{O}}_6{\text{O}}^{\prime }$ were obtained from calculations and a natural linewidth of $40{\text{cm}}^{-1}$ was assumed for each mode.

Figure 9

(Color online) Polarization associated with vibrational modes in $Pna{2}_1$ ${\text{Bi}}_2{\text{Ti}}_2{\text{O}}_6{\text{O}}^{\prime }$ decomposed into dipole moments on atomic sites derived from Born charge tensors and phonon displacements. Left panel $B_1$ mode at $40{\text{cm}}^{-1}$. Center panel $B_2$ mode at $116{\text{cm}}^{-1}$. Right panel $B_1$ mode at $281{\text{cm}}^{-1}$.