First-principles study of the phonon modes in bismuth sillenites

Density functional theory calculations of the vibrational modes of ${\mathrm{Bi}}_{12}{\mathrm{SiO}}_{20}$ are presented, with a detailed assignment and geometry investigation of the vibrational modes in the sillenite structure. We show that the symmetry and geometry of the strong Raman peaks allow direct probing of the Bi-O(1) and Bi-O(2) bonds. The physical significance of the vibrational modes is considered to discuss experimental data on the trivalent sillenites and show evidence of O(1) and/or O(2) vacancies in the Bi-O framework. The infrared modes are also discussed to motivate future systematic studies of sillenites. The Born effective charges of sillenites are studied and the results show the existence of large and anisotropic charges. The magnitude of the LO-TO splits is also calculated and compared with experiment.

Figure 1

A representative sillenite structure. An $M$ cation, such as ${\mathrm{Si}}^{4+}$ (blue), sits on the most symmetric site (2a) that corresponds to the edges and the center of the bcc cell. Each $M$ cation is surrounded by four O(3) oxygens (red) in a tetrahedron configuration. Both O(2) and O(3) sit on 8c symmetric sites where O(2) (purple) are located farther from the 2a sites. The O(2) oxygens have three nearest-neighbor bismuth ions, which later in the text will be referred to as a ${\mathrm{Bi}}_3$ plane for the discussion of some phonon modes. The bismuth (gray) and O(1) oxygen (orange) atoms sit on 24f sites, the least symmetric. Bi and O(1) are each other's nearest neighbors but each O(1) has another very close bismuth bonded to it. The Bi, O(1), and O(2) atoms form what is commonly called a “bismuth-oxygen” framework that interconnects the $M$-O(3)${}_4$ tetrahedra. The image was created using the XCrySden program [19].

Figure 2

Raman and IR scattering cross sections calculated using the LDA functional. The intensity of the modes was taken into consideration in the assignment of the modes.

Figure 3

Frequency of the phonon modes versus the cutoff energy of the plane-wave expansion. Panel (a) shows a few of the strong Raman modes while (b) shows some strong IR modes. Many reports on metal oxides use cutoff energies between 70–90 Ry since higher cutoff energies result in calculations that are too long. This convergence study shows that the calculated phonon frequencies can have an inaccuracy as much as 5–10 ${\mathrm{cm}}^{-1}$ in some modes. All frequencies plotted are after imposing the crystal ASR.

Figure 4

The Raman-strong 58 and 92 ${\mathrm{cm}}^{-1}$ modes. The 58 ${\mathrm{cm}}^{-1}$ F mode involves the movement of many atoms in both the Bi-O framework and tetrahedron and does not offer insight into any particular bond. The 92 ${\mathrm{cm}}^{-1}$ A mode mostly corresponds to O(2) breathing and Bi and O(1) movement. The geometries of the vibrations obtained from GGA [(a), (e)] and LDA [(b), (f)] calculations are shown. Si and Bi are denoted by blue and gray, respectively; O(1), O(2), and O(3) by orange, purple, and red. Green arrows denote the force on each atom. Using the experimental data from Ref. [47], the frequency [(c), (g)] and the width [(d), (h)] of the mode for six sillenites are plotted in increasing ionic radius of the M cation: ${\mathrm{Bi}}_{12}{\mathrm{Si}}^{+4}{\mathrm{O}}_{20}$, ${\mathrm{Bi}}_{12}{\mathrm{Ge}}^{+4}{\mathrm{O}}_{20}$, ${\mathrm{Bi}}_{12}{\mathrm{Ti}}^{+4}{\mathrm{O}}_{20}$, ${\mathrm{Bi}}_{25}{\mathrm{Ga}}^{+3}{\mathrm{O}}_{39}$, ${\mathrm{Bi}}_{25}{\mathrm{Fe}}^{+3}{\mathrm{O}}_{39}$, and ${\mathrm{Bi}}_{25}{\mathrm{In}}^{+3}{\mathrm{O}}_{39}$.

Figure 5

The Raman-strong 132 and 282 ${\mathrm{cm}}^{-1}$ modes. For the 132 ${\mathrm{cm}}^{-1}$ E mode Bi and O(1) atoms shear the framework. Note that the Bi and O(1) ions move in similar direction and therefore the frequency of the mode should not be as sensitive to the Bi-O(1) bond. The 282 ${\mathrm{cm}}^{-1}$ A mode corresponds to breathing of the O(2) atoms. The atoms breathe away from the center of the unit cell and the nearby ${\mathrm{Bi}}_3$ plane. The geometries of the vibrations obtained from GGA [(a), (e)] and LDA [(b), (f)] calculations are shown. Si and Bi are denoted by blue and gray, respectively; O(1), O(2), and O(3) by orange, purple, and red. Green arrows denote the force on each atom. Using the experimental data from Ref. [47], the frequency [(c), (g)] and the width [(d), (h)] of the mode for six sillenites are plotted in increasing ionic radius of the M cation: ${\mathrm{Bi}}_{12}{\mathrm{Si}}^{+4}{\mathrm{O}}_{20}$, ${\mathrm{Bi}}_{12}{\mathrm{Ge}}^{+4}{\mathrm{O}}_{20}$, ${\mathrm{Bi}}_{12}{\mathrm{Ti}}^{+4}{\mathrm{O}}_{20}$, ${\mathrm{Bi}}_{25}{\mathrm{Ga}}^{+3}{\mathrm{O}}_{39}$, ${\mathrm{Bi}}_{25}{\mathrm{Fe}}^{+3}{\mathrm{O}}_{39}$, and ${\mathrm{Bi}}_{25}{\mathrm{In}}^{+3}{\mathrm{O}}_{39}$.

Figure 6

The Raman-strong 331 and 546 ${\mathrm{cm}}^{-1}$ modes. The 331 ${\mathrm{cm}}^{-1}$ A mode involves bending of the Bi-O(1) bond. This A mode presents important physical information because it involves mostly the bending force constant of Bi-O(1). The 546 ${\mathrm{cm}}^{-1}$ mode also probes the Bi-O(1) bond for it involves movement of the O(1) atoms to stretch the Bi-O(1) bond. The geometries of the vibrations obtained from GGA [(a), (e)] and LDA [(b), (f)] calculations are shown. Si and Bi are denoted by blue and gray, respectively; O(1), O(2), and O(3) by orange, purple, and red. Green arrows denote the force on each atom. Using the experimental data from Ref. [47], the frequency [(c), (g)] and the width [(d), (h)] of the mode for six sillenites are plotted in increasing ionic radius of the M cation: ${\mathrm{Bi}}_{12}{\mathrm{Si}}^{+4}{\mathrm{O}}_{20}$, ${\mathrm{Bi}}_{12}{\mathrm{Ge}}^{+4}{\mathrm{O}}_{20}$, ${\mathrm{Bi}}_{12}{\mathrm{Ti}}^{+4}{\mathrm{O}}_{20}$, ${\mathrm{Bi}}_{25}{\mathrm{Ga}}^{+3}{\mathrm{O}}_{39}$, ${\mathrm{Bi}}_{25}{\mathrm{Fe}}^{+3}{\mathrm{O}}_{39}$, and ${\mathrm{Bi}}_{25}{\mathrm{In}}^{+3}{\mathrm{O}}_{39}$.

Figure 7

The IR-strong 107 and 136 ${\mathrm{cm}}^{-1}$ F modes. The 107 ${\mathrm{cm}}^{-1}$ mode [(a), (b)] corresponds to displacement of the whole M-${\mathrm{O}}_4$ tetrahedron. The 135 ${\mathrm{cm}}^{-1}$ mode [(c), (d)] involves tetrahedron displacement but also movement of both Bi and O(1) atoms. The geometries of the vibrations obtained from [(a), (c)] LDA and [(b), (d)] GGA calculations are both shown. See the text about a discussion on the assignment of these modes. Si and Bi are denoted by blue and gray, respectively; O(1), O(2), and O(3) by orange, purple, and red. Green arrows denote the force on each atom.