Temperature dependence of incommensurate modulation in Ca_0.28Ba_0.72Nb₂O₆

Ca $x$Ba $1 − x$Nb $2$O $6$ (CBN $x$⁠), with a partially filled tetragonal tungsten-bronze (TTB) structure, is a uniaxial ferroic material of interest for dielectric,¹ non-linear optics,² and ferroelectric³ applications. Congruent melting occurs at $x=0.28$⁠, producing a thermally stable compound (CBN28) with a high Curie temperature $T C$⁠, typically reported to be between 250³ and 275  $°$C.⁴ Brillouin scattering and ultrasound observations indicate that it is a normal ferroelectric below $T C$⁠, transitioning to relaxor behavior above^3,5 with polar nanoregions detectable up to $T ∗∼530 °$C and a Burns temperature $T B∼800 °$C.⁶ Structurally, 180 DWs are found at room temperature, which become fine needles elongated along the polar axis as $T C$ is approached.⁷ 

Like other TTBs, CBN28 only adopts the prototype P4/mbm (No.127) tetragonal structure (⁠ $a=b=12.95Å$⁠, $c=3.95Å$⁠) at high temperature. Below $T C$⁠, displacement of the Nb cation from the center of NbO $6$ octahedra along the $c$-axis produces ferroelectricity, while coordinated distortions of the oxygen polyhedra reduce symmetry still further.^8–10 Furthermore, in CBN28,¹¹ like many other TTBs,¹² these distortions can couple to varying occupancies of the Ca and Ba sites with a period that is incommensurate with the crystal lattice. Initial studies of single crystal CBN^4,11 denoted the average room temperature structure to be tetragonal with a doubled $c$-axis, with two incommensurate propagation vectors along $[110]$ and $[1 1 ¯0]$⁠. However, several other TTBs have been found to be orthorhombic with unit cell $a O=2 2a$⁠, $b O= 2a$⁠, $c O=2c$ and only one incommensurate propagation vector.^12,13 The almost perfect match $a O=2 b O$ means that merohedral twins are common, leading to a pseudotetragonal cell found in macroscopic diffraction measurements. Here, using transmission electron microscopy (TEM) and electron diffraction, we show the same applies to CBN28.

The structural studies of Graetsch et al.⁴ found no change in symmetry at $T C$ for CBN28, proposing instead a transition from a polar ferroelectric state to an antiferroelectric, or almost non-polar ferrielectric, state. Polarizations of opposite sense are produced as one of the two sets of crystallographically distinct Nb cations move past the center of their NbO $6$ octahedra, while the other set is unchanged and the incommensurate modulation remains. This is also consistent with the relaxor behavior above $T C$ with polar nanoregions that become gradually less stable with increasing temperature, becoming completely transient at a temperature $T ∗$ and non-existent above $T B$⁠. Here, using electron diffraction, we examine the incommensurate propagation vector as a function of temperature and find it reaches a minimum in reciprocal space at $T C$⁠, indicating a coupling between ferroelectricity and incommensurate modulation. The incommensurate modulation decreases above $T C$ and disappears at $∼200 °$C above $T C$⁠, roughly consistent with other measurements of $T ∗$ in CBN28.^5,6

CBN28 was produced by a conventional mixed oxide process similar to the undoped material in Peirson et al.¹ CaCO $3$ (Alfa Aesar, 99.95%), BaCO $3$ (Alfa Aesar, 99.95%), and Nb $2$O $5$ (Alfa Aesar, 99.9%) powders in appropriate quantities were dried at 200  $°$C for $>24$ h and ball milled with yttria-stabilized zirconia milling media in isopropanol for 24 h. Dried, milled powders were calcined at 1200  $°$C for 12 h in covered alumina crucibles. This extended calcination eliminated secondary phases, such as CaNb $2$O $6$⁠, seen in laboratory x-ray powder diffraction data from previous 4 h calcinations.¹ 

Specimens for TEM were prepared from the calcined powder, mixing a small amount with aluminum powder (particle size $<10μ$m) in a ratio of $∼$1:5 in an aluminum foil wrap, which was then cold rolled to produce a solid Al sheet containing CBN28 particles. This sheet was then used as source material for conventional grinding, polishing, and Ar $+$ ion milling to electron transparency. Specimens were examined in a JEOL 2100plus LaB $6$ TEM operating at 200 kV. Heating experiments were performed with a Gatan 652 double tilt heating holder.

For clarity, all Miller indices are given in the reference frame of the prototype TTB structure.

We first demonstrate that merohedral twinning is present with incommensurate modulation in only one dimension. Selected area electron diffraction (SAED) patterns commonly show superstructure spots that are due to the cooperative rotations of the oxygen octahedral framework, as can be seen in the [112] zone axis pattern of Fig. 1(b). In this pattern, the superstructure spots are close to $(2m+1) 1 211 1 ¯+ n 41 1 ¯0$⁠, where $m$ and $n$ are integers, but are alternately displaced from those positions by $±δδ0$ to form pairs, such as that found in the circle of Fig. 1(b). This displacement is indicative of a structure with a period $δ$⁠, incommensurate with the crystal lattice. In Fig. 1(b), $δ=0.054±0.002$⁠, corresponding to a periodicity of 16.0 nm.

It is also possible to obtain SAED patterns from a single crystal with no (or extremely weak) superstructure reflections, as shown in Fig. 1(c). Using an objective aperture to form a dark field image using the superstructure spots shows the two regions quite clearly [Fig. 1(a)], with a dark region on the right of the image corresponding to an absence of superstructure spots. Interestingly, the bright region on the left shows a patchy contrast on a length scale of 10–50 nm. (Similar data obtained at the [110] zone axis are shown in Fig. SI1 of the supplementary material.) These observations are consistent with two orientational variants of an incommensurately modulated phase, with orthogonal components of a modulation vector along $[1 1 ¯0]$ or $[110]$⁠, respectively.¹⁴ In such a case, the modulation vector of the second variant lies far from the Ewald sphere in Fig. 1(c), and only the first variant [Fig. 1(b)] can contribute to the dark field image, with the result that the second variant appears dark. The patchy contrast suggests that the two variants are intimately intermixed. These results are consistent with the SAED and high resolution TEM study of Lu et al.,¹⁵ who proposed an orthogonal cell for CBN28 with lattice parameters $a O=2 2a$⁠, $b O= 2a$⁠, $c O=2c$⁠. These data are also very similar to those presented by Krayzman¹³ in the TTB Sr $0.61$Ba $0.39$Nb $2$O $6$ (SBN61), suggesting a common structure and behavior. They proposed that SBN61 consists of an intergrowth of two structures, one with unit cell $a ′=2 2a$⁠, $b ′= 2a$⁠, $c ′=2c$ and a second with $a ″=6 2a$⁠, $b ″= 2a$⁠, $c ″=2c$⁠. Together, these produce a phase mixture that results in an incommensurately modulated structure for each twin variant.

The behavior of the superstructure spots as a function of temperature in a $[1 1 ¯0]$ SAED pattern is shown in Fig. 2. In this pattern, superstructure spots are close to $h/4k/4(1+2l)/2$ (⁠ $h,k,l$ integers) and are displaced by the same incommensurate propagation vector $±δδ0$ [Fig. 2(a)]. Two heating and cooling experiments were performed sequentially. Data analysis was performed using a Python script, measuring the position and intensity of 96 superstructure spots in each SAED pattern. Bright field TEM images, showing the position of the selected area aperture and the changing microstructure of the material during the first experiment, are shown in Fig. 3. At room temperature (25  $°$C), a domain structure is visible in the first image of Fig. 3. These are 180 $°$ ferroelectric domains.^7,16 SAED patterns were taken from domains 1 and 2 in Fig. 3 and appeared essentially the same; however, more detailed analysis shows a significant difference in the value of $δ$⁠, being $δ 1=0.0175±0.0025$ and $δ 2=0.0115±0.0025$⁠. These values are rather less than the room temperature measurement of Lu¹⁵ of 0.0225 for single crystal material. During heating, the value of $δ$ increased erratically up to a nominal temperature of $∼250 °$C, after which it dropped again. The intensity of the superstructure spots also behaved erratically, staying relatively low up to $200 °$C where it increased dramatically. The images of Fig. 3 show the domain structure changes significantly during heating, with domain walls becoming denser and longer, aligned with the $c$-axis (vertical in these images) as observed previously in single crystal CBN28 by Lu et al.⁷ Evidence for decreasing domain size in CBN28 as $T C$ is approached, and a locally varying $T C$ that depends upon domain size has also been presented by Heine et al.¹⁷ using second harmonic generation.

The value of $δ$ decreased until the domain structure vanished at $∼300 °$C. The absence of domains implies a loss of ferroelectricity, and we may, therefore, take this observation and the minimum value of $δ$ to indicate at the Curie temperature $T C$⁠. Our bulk CBN28 ceramic has peak $ϵ r$ between $∼240$ and $268 °$C,¹ in agreement with the documented values of $T C$⁠.^4–6,18 Thus, the value of $∼300 °$C observed in Fig. 2 almost certainly results from the difference in the temperature measured at the thermocouple of the heating holder and the actual temperature of the crystallite at the edge of the thinned TEM specimen. The erratic rise observed in $δ$ and its strong decrease as the domain size decreases suggests that the incommensurate modulation both interacts with the ferroelectric domains and varies in magnitude with the sign of polarization. Above $T C$⁠, $δ$ increases linearly, while the intensity of the superstructure spots continues to decline. At $550 °$C, $δ=0.035$⁠, while the spots have almost vanished. On cooling, both $δ$ and the superstructure spot intensities behave more regularly, with a sharp V in the value of $δ$ and an increasing intensity as temperature drops. As can be seen in the final image of Fig. 3 at room temperature, while the domain structure has returned, the larger domains now do not extend to the thinnest part of the specimen where the diffraction pattern was taken. A very fine domain structure is still present, which is unresolved in this relatively low magnification image, but visible in higher magnification dark field images (supplementary Fig. SI2). The final value of $δ$ at room temperature is 0.028.

This domain state was maintained for the second set of measurements, giving the repeatable data shown in Fig. 2(d). Above $T C$⁠, where the material is a relaxor,^3,19 the intensity of the superstructure spots tends to zero at a nominal temperature of $600 °$C. This corresponds well with ultrasound and Brillouin scattering measurements of the temperature $T ∗$ at which polar nanoregions become completely transient,^3,5,6,19 taking into account the difference in the nominal temperature and the actual value at the position of measurement. The well-behaved changes in intensity and $δ$ in the second heating/cooling cycle are, therefore, probably a result of the fine domains in the material after the initial heating/cooling cycle. This subtle behavior is not readily apparent in macroscopic measurements since dielectric measurements of ceramics produced from the powder used in this TEM study showed no appreciable hysteresis or variations in behavior over successive heating cycles.

The ionic radius of Ca $3 +$ with coordination number 12 is $r C N 12=1.34$ Å, essentially the same as La $3 +$⁠.²⁰ Levin and co-workers²¹ showed that adding Nd to the filled mixed B-site TTB Nd $x$La $1 − x$Ba $2$Ti $2$Nb $3$O $15$ induced a change from incommensurate $Ama2$ above $T C$ to a commensurate $Ima2$ structure below, whereas the end-point compound LaBa $2$Ti $2$Nb $3$O $15$ (LBTN) showed no change, maintaining an incommensurate $Ama2$ structure at all temperatures. The difference was ascribed to the smaller ionic radius of Nd $3 +$ at $r C N 12=1.27$ Å, an inference that was strengthened by subsequent work²² substituting other elements $M$ to give $M$Ba $2$Ti $2$Nb $3$O $15$⁠, which gave a commensurate $Ima2$ structure only for $r C N 12<1.3$ Å. The lack of any phase change in CBN28, like LBTN, is, thus, in perfect agreement with these studies and the wider range of compounds surveyed by Zhu et al.¹² 

In summary, these results are consistent with previous investigations of CBN28, with no change of symmetry observed at the Curie temperature $T C$⁠. The material has a microstructure closely related to that seen in SBN,¹³ with an average orthorhombic symmetry that presents as two intermixed (twin) variants at a scale of tens of nm. Each variant has a one-dimensional incommensurate modulation characterized by the propagation vector $δδ0$⁠. The observation here that $δ$ depends on the polarity of the material, couples to domain size below $T C$⁠, and the incommensurate modulation persists for $∼250 °$C above $T C$ indicates a strong coupling to polar modes in the material. While the presence of the incommensurate modulation in CBN28 above $T C$ has been noted,¹¹ its correlation with a domain structure has not been demonstrated until this work, and it would be interesting to investigate its behavior in related systems.

Additional data are given to support the images presented here. Figure SI1 reveals incommensurate spots and a dark field image of a [110] oriented crystal. Figure SI2 reveals dark field 002 images, showing 180 ° domains and their evolution during sample cooling.

This work was funded by UKRI via the Engineering and Physical Sciences Research Council (EPSRC) at Warwick and Leeds (Award Nos. EP/V053701/1 and EP/S029036/1, respectively).

The authors have no conflicts to disclose.

R. Beanland: Conceptualization (lead); Methodology (lead); Project administration (equal); Supervision (lead); Writing – original draft (lead). L. Harrison: Investigation (equal). S. Khan: Investigation (equal). T. Roncal-Herrero: Investigation (supporting). H. Peirson: Investigation (supporting). S. J. Milne: Funding acquisition (equal); Project administration (equal); Writing – review & editing (supporting).

The data that support the findings of this study are available within the article and its supplementary material.