Structural changes underlying the diffuse dielectric response in ${\text{AgNbO}}_3$

Structural differences in the so-called M polymorphs of ${\text{AgNbO}}_3$ were analyzed using combined high-resolution x-ray diffraction, neutron total scattering, electron diffraction, and x-ray absorption fine-structure measurements. These polymorphs all crystallize with $Pbcm$ symmetry and lattice parameters $\surd 2a_c\times \surd 2a_c\times 4a_c$ (where $a_c\approx 4\text{Å}$ corresponds to the lattice parameter of an ideal cubic perovskite) which are determined by a complex octahedral tilt system $\left(a^{-}{b}^{-}{c}^{-}\right)/\left(a^{-}{b}^{-}{c}^{+}\right)$ involving a sequence of two in-phase and two antiphase rotations around the $c$ axis. Our results revealed that, similar to ${\text{KNbO}}_3$, the Nb cations in ${\text{AgNbO}}_3$ exhibit local off-center displacements correlated along Nb-Nb-Nb chains. The displacements appear to be present even in the high-temperature ${\text{AgNbO}}_3$ polymorphs where the Nb cations, on average, reside on the ideal fixed-coordinate sites. The onset of the $\left(a^{-}{b}^{-}{c}^{-}\right)/\left(a^{-}{b}^{-}{c}^{+}\right)$ tilting in the M polymorphs lifts the symmetry restrictions on the Nb positions and promotes ordering of the local Nb displacements into a long-range antipolarlike array. This ordering preserves the average $Pbcm$ symmetry but is manifested in electron diffuse scattering and corroborated by other local-structure sensitive techniques. Structural states previously identified as the ${\text{M}}_3$ and ${\text{M}}_2$ phases represent different stages of displacive ordering rather than distinct thermodynamic phases. Rietveld refinements indicated intimate coupling between the displacive behavior on the oxygen, Nb, and Ag sublattices. The $Pbcm$ symmetry of the octahedral framework precludes a complete ordering of Nb displacements so that some positional disorder is retained. This partial disorder likely gives a source to the dielectric relaxation which, according to previous spectroscopic studies, is the origin of the diffuse dielectric response exhibited by M-type ${\text{AgNbO}}_3$ at $\approx 250°\text{C}$.

Figure 1

(Color online) Schematics of the octahedral framework in the M-phase structure of ${\text{AgNbO}}_3$ with the symmetry elements superimposed. Octahedral rotations about the $c$ axis are indicated using arrows. Ag1 and Ag2 specify $c$ planes occupied by the symmetrically nonequivalent Ag cations.

Figure 2

(Color online) Schematics of the Nb (a) and Ag (b) displacements in the successive $c$ layers (I, II, III, and IV) within the $Pbcm$ structure (Fig. 1). The displacement directions are indicated using arrows.

Figure 3

Sequence of the reported phase transitions in ${\text{AgNbO}}_3$.

Figure 4

(Color online) Experimental (red) and calculated (blue) neutron-diffraction profiles for ${\text{AgNbO}}_3$ at room temperature $\left(R_{\text{wp}}=3.24\%\right)$

Figure 5

(Color online) Schematic of anisotropic atomic displacement parameters (70% probability) as obtained from Rietveld refinements using room-temperature neutron-diffraction data. Note the highly anisotropic displacements for Ag2 and O5 sites.

Figure 6

Representative low-index selected area electron-diffraction patterns from ${\text{AgNbO}}_3$. The reflection conditions are consistent with the $b$ and $c$ glide planes as encountered in the $Pbcm$ structure.

Figure 7

(Color online) Electron-diffraction patterns recorded from a single domain of ${\text{AgNbO}}_3$ at room temperature. Fundamental reflections are indexed according to the $\surd 2a_c\times \surd 2a_c\times 4a_c$ unit cell. Note the diffuse-scattering streaks that represent intersections of the ${\left(00l\right)}_o$ sheets (illustrated in the drawing on the right) and the Ewald sphere. The diffuse scattering through the origin is extinct. The diffraction pattern recorded in the ${\left[102\right]}_c$ orientation contains a minor contribution from the second domain with a perpendicular orientation of the $c$ axis. A systematic alternation (i.e., weak/strong) of diffuse intensity is observed for the streaks at $g={\left[112\right]}_c$ and $g=\frac{1}{2}{\left[112\right]}_c$ (z.a. ${\left[102\right]}_c$).

Figure 8

(Color online) (Left) Schematic of the chainlike correlations for the local Nb displacements directed along ${\left[111\right]}_c$ and ${\left[11\overline{1}\right]}_c$; (right) A random mixture of chains 1 and 2 is equivalent to average Nb displacements along the ${\left[110\right]}_c$ direction (i.e., $b$ axis) with additional disordered local-displacement components along the $\left[001\right]/\left[00\overline{1}\right]$ (i.e., $c$ axis) directions. These chain correlations generate the ${\left(00l\right)}_o$ sheets of diffuse intensity observed in Fig. 7.

Figure 9

(Color online) (a) Experimental (red) and calculated (blue) atomic PDF obtained from total neutron scattering at room temperature. (b) Low-$r$ portion of the PDF in (a). The calculated PDF corresponds to the average structure refined from the Bragg profile. The calculated PDF matches the experimental data well at larger interatomic distances (a) but evidently fails to reproduce the first Nb-O peak (b). This peak can be recovered through additional refinements of the Nb atomic coordinates and anisotropic displacement parameters (c). The refined parameters exhibit a strong preference for Nb displacements along the $c$ axis in addition to the average $b$-axis displacements. This result supports the model inferred from the electron-diffraction data (Fig. 8).

Figure 10

(Color online) (a) and (b) Room-temperature Nb (a) and Ag (b) EXAFS data. (c) and (d) Experimental (red) and fitted (blue) magnitudes of the EXAFS Fourier transform for Nb and Ag. The $k$ ranges used in the Fourier transforms were $2–15{\text{Å}}^{-1}$ for Nb and $1.7–11.7{\text{Å}}^{-1}$ for Ag. Prior to transform, the data were multiplied by the Hanning window $\left(dk=1\right)$. Satisfactorily agreement between the experimental and calculated data was obtained by simultaneous fitting of both the Ag and Nb data sets using a model that included a two-shell Nb-O $\left(3+3\right)$, a single-shell Nb-Ag, a single-shell Nb-Nb, a three-shell $\left(3+6+3\right)$ Ag-O, and a two-shell Ag-Ag coordination. For each shell, an interatomic distance and its associated Debye-Waller factor were refined. Multiple-scattering contributions to the Nb EXAFS data were taken into account by introducing the Nb-O-Nb angle as an independent variable (this angle is determined primarily by octahedral tilting). We calculated multiple-scattering amplitudes for the three values of the Nb-O-Nb angle and used quadratic interpolation during the fit. All refined structural parameters acquired physically reasonable values.

Figure 11

(Color online) A single-shell fit of the EXAFS Nb-O peak at room temperature [$R_{\text{Nb-O}}=1.98\left(1\right)\text{Å}$, ${\sigma }^2=0.017\left(3\right){\text{Å}}^2$]. The discrepancy between the experimental (red) and fitted (blue) data is considerably worse than that for a two-shell fit (Fig. 10).

Figure 12

Representative dark-field TEM image recorded with the 241-like reflections strongly excited near the ${\left[\overline{2}10\right]}_o$ orientation. A high incidence of antiphase domain boundaries aligned perpendicular to the ${\left[001\right]}_o$ direction is observed.

Figure 13

(Color online) Temperature dependencies of the ${220}_o$ and ${008}_o$ $d$ spacings: (a) results of the profile fitting to the ${002}_c={220}_o+{008}_o$ reflection(s), (b) calculated from the lattice parameters obtained through Rietveld refinements. The uncertainties are within the symbol size. The results presented in (a) are consistent with literature reports and the changes can be associated with the previously reported phase-transition sequence ${\text{M}}_2\leftrightarrow {\text{M}}_3\leftrightarrow {\text{O}}_1\leftrightarrow {\text{O}}_2$. In contrast, the results obtained using Rietveld refinements (b) indicate only the $\text{O}\leftrightarrow \text{M}$ transition. Apparently, the measurement resolution was insufficient to resolve directly a small splitting of the ${220}_o$ and ${008}_o$ reflections in (a).

Figure 14

(Color online) Magnified view of the temperature dependence of the ${220}_o$ $d$ spacing. Several anomalies are clearly observed that can be correlated with the temperatures previously reported for the ${\text{M}}_1\leftrightarrow {\text{M}}_3\leftrightarrow {\text{M}}_3\leftrightarrow \text{O}$ transitions. An additional anomaly occurs at $\approx 200°\text{C}$.

Figure 15

(Color online) Temperature dependence of the unit-cell volume for ${\text{AgNbO}}_3$.

Figure 16

(Color online) (a) A portion of the x-ray diffraction pattern displaying weak ${313}_o$ and ${304}_o$ superlattice reflections with intensities determined largely by the Nb and Ag displacements. (b) Temperature dependence of the combined integrated intensity for these reflections. The dependence reveals several anomalies at the same temperatures as observed for the ${220}_o$ $d$ spacing in Fig. 12.

Figure 17

(Color online) Temperature dependencies of several key structural characteristics obtained from Rietveld refinements using neutron-diffraction data. (a) Squared $b$-axis component, $v_{\text{Nb}}^2$, of the Nb displacements from the ideal $\frac{3}{4}\frac{3}{4}\frac{5}{8}$ positions, (b) Squared $b$-axis component, $v_{\text{Ag}}^2$, of the Ag1 displacements for the ideal $\frac{3}{4}\frac{1}{4}\frac{1}{2}$ positions, (c) $\varphi$ tilting angle about $c$ axis, (d) $\theta$ tilting angle about $b$ axis, (e) apical O-O distance along the $c$ axis (see Fig. 16 for description), (f) equatorial O-O distance along the $b$ axis (Fig. 16), (g) equatorial O-O distance along the $a$ axis (Fig. 16), (h) $U_{22}$ component of the Ag2 atomic displacement parameters. The uncertainty for the Nb displacements in (a) is within the symbol size. Values of $\varphi$ (c) and $\theta$ (d) were estimated from the oxygen coordinates. The uncertainties of these parameters are small as reflected in the low point-to-point noise of their temperature dependencies. The error bars shown for the $\left|\text{O-O}\right|$ distances [(e)–(g)] were estimated by GSAS; the actual uncertainties appear to be much smaller as inferred from the low point-to-point noise in the temperature dependencies of these distances.

Figure 18

(Color online) Schematics illustrating the definition of the $\varphi$ and $\theta$ tilting angles and the O-O and Nb-O distances extracted from the average-structure refinements. The O-O distances are plotted in Fig. 15e, 15f, 15g, whereas the Nb-O distances are displayed in Fig. 18.

Figure 19

(Color online) Temperature dependences of the $U_{22}$ (a) and $U_{22}/U_{33}$ (b) parameters for the Nb atoms. The uncertainties represent estimated standard deviations as calculated using GSAS.

Figure 20

(Color online) Temperature dependence of the average Nb-O distances in the $ab$ plane as described in Fig. 16. The two different Nb-O distances are defined in Fig. 16.

Figure 21

(Color online) Variable-temperature electron-diffraction patterns recorded from a single domain of ${\text{AgNbO}}_3$ along the ${⟨103⟩}_c$ orientation. For $T>175°\text{C}$, two sets of diffuse streaks (i.e., vertical and horizontal) were observed, whereas for $T<175°\text{C}$ one of these sets (horizontal) largely disappeared. A three-dimensional (3D) reconstruction of the changes in the diffuse intensity distribution is illustrated by the schematic drawing of the reciprocal lattice. The diffuse intensity sheets are highlighted. Around $400°\text{C}$, the superlattice reflections $hk\frac{1}{4}{l}_c$ associated with the $\left(a^{-}{b}^{-}{c}^{-}\right)/\left(a^{-}{b}^{-}{c}^{+}\right)$ octahedral tilting are replaced by the diffuse streaks (outlined using a dotted line). Numerous random spots and a ring that appear in the diffraction pattern at this temperature arise from precipitation of the nanoscale metallic Ag. The intensity of the ${218}_o$ reflection indicated using red arrows is determined primarily by the cation displacements. The temperature dependence of the intensity of this reflection is shown in Fig. 20.

Figure 22

(Color online) Temperature dependence of the integrated intensity of the ${218}_o$ reflection (indicated using red arrow in Fig. 19) which is determined entirely by the Ag and Nb displacements. A marked discontinuity in this dependence occurs in the same temperature range where the change in the electron diffuse scattering is observed.

Figure 23

(Color online) Variable-temperature Nb and Ag EXAFS $k^2$-weighted data [(a) and (b)] and their Fourier transforms [(c) and (d)] recorded between 25 and $400°\text{C}$. The $k$ ranges used in the Fourier transforms were from 2 to $15{\text{Å}}^{-1}$ for Nb and from 1.7 to $11.7{\text{Å}}^{-1}$ for Ag. Prior to transform, the data were multiplied by the Hanning window $\left(dk=1\right)$. Both Nb and Ag EXAFS exhibit anomalies between 150 and $200°\text{C}$. For the Ag EXAFS, an additional anomaly between 300 and $350°\text{C}$ can be observed.

Figure 24

(Color online) Local Nb-O distances ($R_1$ and $R_2$) obtained by fitting the EXAFS data. A $\left(3+3\right)$ Nb coordination by O was assumed.

Figure 25

(Color online) Low-$r$ portions of the PDF at 25, 125, and $225°\text{C}$. Note a splitting of the Ag-Nb peak at higher temperatures (indicated using arrows).

Figure 26

(Color online) Schematic for the Nb order-disorder behavior as a function of temperature. In the O phase, Nb atoms occupy ideal average positions but are locally disordered among the eight sites displaced along ${⟨11l⟩}_c$ $\left(l\le 1\right)$ directions. In the M phase, the Nb cations become partially ordered with average displacements along the $b$ axis. Locally, above $T_f\approx 175°\text{C}$, Nb still remains distributed among the eight sites but the site occupancies become nonequal as indicated using differently sized spheres. In particular, the Nb cations in the $c$ layers I and II preferentially occupy two sites displaced along the $\left[11l\right]$ and $\left[11\overline{l}\right]$ directions, whereas those in the $c$ layers III and IV favor sites displaced along the $\left[\overline{1}1l\right]$ and $\left[\overline{1}1\overline{l}\right]$. The occupancy probabilities for the other six sites decrease gradually with decreasing temperature and vanish below $T_f$.