Visualization of ferroaxial domains in an order-disorder type ferroaxial crystal


Order–disorder type ferroaxial transition in NiTiO3

Ferroelectric phase transitions are known to be classified into mainly two types: displacive type and order–disorder type. In the same manner, ferroaxial transitions will also be classified into these two types. RbFe(MoO4)2, that is, the only material in which a ferroaxial transition is studied, exhibits a ferro-rotational distortion mainly ascribed to displacements of oxygen atoms11, and therefore can be said to be a displacive type ferroaxial crystal. In this study, we propose that a structural phase transition reported in NiTiO313 is regarded as an order–disorder type ferroaxial transition (Fig. 2). At temperatures above Tc ≈ 1560 K, the crystal structure of NiTiO3 is described by the corundum structure (space group R(bar 3c)) which is envisage as a hexagonal close packing of the oxygen ions with Ni2+ and Ti4+ cations randomly occupying 2/3 of the octahedral interstices (Fig. 2a). With lowering temperature, cation ordering takes place at Tc and results in a structural phase transition into the ilmenite structure (space group R(bar 3)). The low-temperature structure is characterized by an alternating sequence of Ni2+ and Ti4+ along the stacking direction of the closed-packed layers (Fig. 2b). Depending on the stacking sequence (Ni-Ti-Ni-Ti- or Ti-Ni-Ti-Ni-), two possible domain states develop at temperatures below Tc (Fig. 2b, c). The transformation from the point group (bar 3m) into (bar 3) in NiTiO3 is the same as that in RbFe(MoO4)2 and is nothing less than a ferroaxial transition. Indeed, as seen in Fig. 2d, e which depict two specific Ti ions and six oxygen ions bonded to these Ti ions, the direction of rotational distortions of oxygen ions (red arrows), i.e., the sign of A(||c), is opposite in these two domain states (hereinafter, referred to as A+ domain and A− domain). These two domain states are related to each other by the operations whose symmetries are lost at the ferroaxial transition {e.g., two-fold rotation about [110] and c-glide operation with glide plane || (110)}. NiTiO3 crystals used in this study were grown by the floating zone method (Methods). In the growth process, the specimens were once heated at temperatures above Tc and then cooled down to room temperature, meaning that the crystals underwent the ferroaxial transition and are expected to consist of both A+ and A− domains.

Fig. 2: Order-disorder phase transition and formation of ferroaxial domains in NiTiO3.

a The crystal structures of NiTiO3a above and b, c below the ferroaxial transition temperature (Tc ≈ 1560 K). The crystal structure above Tc belongs to a non-ferroaxial space group (R(bar 3)c) and is described by a disordered corundum structure with a random distribution of Ni2+ and Ti4+ ions at cation sites. With lowering temperature, cation ordering takes place at Tc and results in a structural phase transition into an ordered ilmenite structure with a ferroaxial space group (R(bar 3)). Below Tc, thus, a pair of ferroaxial domain states with the opposite rotation direction, i.e., the opposite sign of axial vector A, are present (A+ and A− domains). d, e The c-axis views of the ferroaxial domains. Only two Ti ions [Ti1(*) and Ti2(*)] and six oxygen ions are depicted. These ions form two TiO3 triangular pyramids which are related by the space inversion with the inversion center at the midpoint between Ti1 and Ti2 ions. Red arrows denote the direction of rotational displacements of oxygen ions from the (110)-type planes (dotted lines) that correspond to the average oxygen positions between A+ and A−.

Identification of ferroaxial domains by STEM-CBED measurement

The coexistence of a pair of ferroaxial domains (A+ domain and A− domain) in a piece of the NiTiO3 crystal was examined by the combined use of STEM and CBED14. This technique (hereinafter, referred to as STEM-CBED technique) possesses a sensitivity to picometer-scale atomic displacements and a nanometer-scale spatial resolution12, and therefore allows us to visualize spatial distributions of various nanostructures such as polar nanostructures in ferroelectrics15,16. In the present study, we apply this technique to the observation of ferroaxial domains in NiTiO3. The measurement details are described in Methods and Supplementary Note 1. Figure 3a shows a bright-field (BF)-STEM image obtained with the 001 incidences, and its magnified view of the area surrounded by a yellow framed box is displayed in Fig. 3b. Figure 3c, d show CBED patterns obtained at positions C and D in Fig. 3b, respectively, with the 001 incidences. Zeroth-order Laue zone (ZOLZ) reflections are seen near the center while ring-shaped higher-order Laue zone (HOLZ) reflections on the fringe of the CBED patterns. Yellow arrowheads point characteristically intense HOLZ reflections, indicating that the CBED patterns of Fig. 3c, d are almost in a mirror image relation whose mirror plane is parallel to (110). Note that such a mirror operation is one of the symmetry elements which are present in the high-temperature (bar 3m) phase but lost in the low-temperature (bar 3) phase.

Fig. 3: STEM-CBED maps and CBED patterns in the R(bar 3) ferroaxial phase of NiTiO3.
figure3

a Bright-field scanning transmission electron microscope (BF-STEM) image obtained with the 001 incidence, where a domain boundary // (110) is indicated by a white dotted line. Crystal orientations determined from the STEM-CBED patterns of (c) and (d) are schematically shown. b Magnified view of a yellow-box area in (a). c, d (exp.) Convergent-beam electron diffraction (CBED) patterns obtained with the 001 incidence at positions C and D shown in (b) are displayed in (c) and (d), respectively. These measurements were carried out at room temperature, that is, in the R(bar 3) ferroaxial phase. e, f (sim.) Simulated CBED patterns of the R(bar 3) phase of NiTiO3 with the (e) [001] and (f) [00(bar 1)] incidence. The specimen thickness used for the simulations was 35 nm. The measured CBED patterns displayed in (c) and (d) well match up with the simulated (e) and (f), respectively [see higher-order Laue zone (HOLZ) reflections indicated by yellow arrowheads]. g, h STEM-CBED maps obtained from the intensities of the HOLZ reflections at G and H [yellow-dotted circles in (e) and (f), respectively]. i Orientations of A+ and A− domains and the domain boundary determined from the STEM-CBED measurements. Scale bar: 200 nm for (a) and 5 nm for panels (b), (g), and (h).

We performed computer simulations of the CBED patterns for the structure models of NiTiO3 (see “Methods” and Supplementary Note 1). Figure 3e, f shows simulated CBED patterns with the [001] and [00(bar 1)] incidence, respectively, for the A+ domain. These two incidence configurations are converted into each other by the two-fold rotation about [110], and are equivalent to the measurements of a pair of ferroaxial domains (A+ and A− domains). The specimen thickness used for the simulations was 35 nm. The simulated CBED patterns in Fig. 3e, f are also in a mirror image relation reflecting the atomic arrangements in the two domains and well match up with the measured CBED patterns shown in Fig. 3c, d, respectively [compare the HOLZ reflections indicated by yellow arrowheads in Fig. 3c–f]. This result shows that ferroaxial domains with opposite signs of A are located at positions C (A+ domain) and D (A− domain) in Fig. 3b. In Fig. 3g, h, furthermore, we display STEM-CBED maps using the intensities of the HOLZ reflections at G and H (yellow-dotted circles in Fig. 3c, d), respectively. These maps clearly show spatial distributions of the intensity, meaning the formation of ferroaxial domains in the specimen used in this study. The location of domain boundaries in the entire sample area displayed in Fig. 3a was examined by observing the CBED patterns at various sample positions. As a result, only one flat boundary was revealed in the area (white dotted line in Fig. 3a). The crystal orientations of the A+ and A− domains separated by the boundary were determined from the CBED patterns and are schematically illustrated in Fig. 3i. The domain boundary is oriented parallel to the (110) plane. Thus, the coexistence of a pair of ferroaxial domains (Fig. 2d, e) is confirmed in terms of the structural characterization using the STEM-CBED technique on nanometer-scale spatial resolution.

EG as a tool to observe ferroaxial domains

Here we discuss another approach for observing ferroaxial domains. That is the approach by using the EG effect, i.e., optical rotation induced by an external electric field (see Fig. 1b). The EG effect was firstly described by Aizu17 and Zheludev18 independently in 1963-1964, and demonstrated in quartz crystals by Vlokh19 in 1970, a half century ago. To date, this effect has been measured in various crystals20,21 including PbWO422 and Pb5Ge3O1123. The EG effect is described by the change in the gyration tensor gij as a function of an applied electric field E and expressed as a power series,

$$begin{array}{*{20}{c}} {g_{ij} = g_{ij}^{left( 0 right)} + gamma _{ijk}E_k + beta _{ijkl}E_kE_l + cdots } end{array}.$$

(1)

Here (g_{ij}^{left( 0 right)}) represents natural optical rotation, and (gamma _{ijk}(beta _{ijkl})) represents the linear (quadratic) EG effect. Hereinafter, the z axis (the third axis) is taken as the principal axis. The linear EG effect characterized by the third-rank axial tensor γijk is possible in all point groups except for m3m, (bar 43m), and 432, while the quadratic one by the fourth-rank axial tensor βijkl is only in noncentrosymmetric point groups. Note that, in centrosymmetric pyroaxial groups ((bar 1,2/m,bar 3,4/m), and 6/m), the natural optical rotation is absent. Furthermore, the Pockels effect and the inverse piezoelectric effect are not allowed, and therefore the linear EG effect is the only optical effect proportional to E. Considering these symmetry requirements, it can be said that the centrosymmetric pyroaxial crystals are ideal playgrounds to examine the linear EG effect free from other electro-optical effects. More importantly, the sign of tensor component γ333, which describes the situation when the directions of light propagation and an applied electric field are both parallel to a ferroaxial moment A, will depend on the sign of A (Supplementary Note 2). This means that the direction of E-induced optical rotation in A+ domain is opposite to that in A− domain. Therefore, ferroaxial domains can be distinguished by using the linear EG effect, which has been proposed in ref. 3.

As an indicator of linear EG effect, we use the coefficient α which relates the rotation angle of the light polarization plane ϕ to an applied voltage V. In general, optical rotatory power ρ is given by

$$begin{array}{*{20}{c}} {rho = frac{pi }{{lambda n}}g_{ij}l_il_j.} end{array}$$

(2)

Here li and lj are direction cosines of the wave normal, n is the refractive index, λ is the wavelength of the incident light, and the Einstein notation is adopted. Furthermore, when the directions of light propagation and electric field are both parallel to A, ϕ (=ρd where d is the sample thickness) is given by

$$begin{array}{*{20}{c}} {phi = frac{{pi d}}{{lambda n}}gamma _{333}E_3l_3l_3 = frac{pi }{{lambda n}}gamma _{333}V_3} end{array},$$

(3)

where l3 = 1 and (V = E/d). Therefore, ϕ is proportional to V at fixed λ and can be expressed as

$$phi left[ {{mathrm{deg}}} right] = {itupalpha}left[ {{mathrm{deg}};{mathrm{V}}^{ – {mathrm{1}}}} right] times {it{V}}left[ {mathrm{V}} right],$$

(4)

in which the coefficient α (( propto gamma _{333})) represents the magnitude of the linear EG effect.

Because the magnitude of the linear EG effect is usually small (α ≤ 10−4 deg V−1)20, spatial distributions of EG have never been reported to date. To spatially resolve such small EG signals, we adopted a difference image-sensing technique which was recently developed for ferroelectrics field modulation imaging24,25. In this technique, microscopy images of transmitted light were captured by an area-image sensor while positive and negative voltages (V) applied. The difference of transmittance between the positive- and negative-voltage images (ΔT) divided by the average of them (T) was calculated for each pixel detection, and then spatial distributions of ΔT/T were obtained. A schematic of the experimental setup is shown in Fig. 4a, and the measurement details are given in Methods. As described in Supplementary Note 3, ΔT/T is proportional to α representing the linear EG effect when the angle between the orientation of a polarizer and an analyzer (θ) is set at θ = ±45°. The sign of θ is defined as positive when the polarization direction of the analyzer rotates clockwise with respect to that of the polarizer from the observer’s point of view. The validity of this technique was confirmed by measurements of the linear EG effect in a reference material PbWO4 (see Supplementary Note 4).

Fig. 4: Spatial distribution of ferroaxial domains obtained via electrogyration in NiTiO3.
figure4

a Experimental setup of electrogyration measurement using a difference image-sensing technique. Inset of (a) shows temporal evolution of applied voltage V during the measurement. Microscopy images of transmitted light were captured by the area sensor while the positive and the negative V applied. The difference of transmittance between the positive- and negative-voltage images (ΔT) divided by the average of them (T) was calculated for each pixel detection, and then spatial distributions of ΔT/T were obtained. b Transmission optical microscopy image with the incidence of light along the c axis (Scale bar: 100 μm). Dark areas in the image correspond to NiO impurity. c, d The two-dimensional maps of ΔT/T, which corresponds to electrogyration, at the same area as panel b. A 3 × 3 median filter was applied to the raw images. The polarization direction of the analyzer was set at (c) θ = +45° and (d) −45° with respect to that of the polarizer. These measurements were done under V = ±100 V at room temperature, that is, in the R(bar 3) ferroaxial phase. A ΔT/T color scale is applied to the images in panels c and d. Red and blue regions correspond to either A+ or A− ferroaxial domains. Purple-colored regions represent areas of NiO impurity.

Optical imaging of ferroaxial domains in NiTiO3 via EG effect

We examined ferroaxial domains of NiTiO3 with the abovementioned optical technique using the EG effect. The directions of light propagation and an applied electric field were both parallel to the c axis, meaning that EG corresponding to the γ333 component was probed. Figure 4b displays the transmission optical microscopy image of the specimen used for the EG measurement with the incidence of light along the c axis. In the image, there are dark island-shaped inclusions identified as NiO impurities by the energy dispersive X-ray analysis (Supplementary Note 5). Spatial distributions of ΔT/T at the same area as Fig. 4b were obtained under the applied voltage of ±100 V in the polarization configurations at θ = ±45°. The results for θ = +45° and −45° are displayed in Fig. 4c, d, respectively, in which red (blue) color corresponds to a positive (negative) sign of ΔT/T. Note that the regions of NiO impurities (dark areas in Fig. 4b) appear purple (a mixture of red and blue) in Fig. 4c, d because the intensity of transmitted light in the region is too small to get meaningful signals. Except for the impurity regions, the images of Fig. 4c, d show a complete reversal of the contrast within the margin of error. This means that the observed ΔT/T is due to electric-field-induced change in optical rotation, i.e., EG, but not to that in optical absorption (see Supplementary Note 3). Therefore, red and blue regions in Fig. 4c, d correspond to either A+ and A− ferroaxial domains, and the color contrasts of these figures reflect the ferroaxial domain pattern in NiTiO3.

To check whether EG observed in NiTiO3 is ascribed to the linear effect and/or higher-order ones, we carried out measurements of the EG spatial distributions as a function of applied voltage V. Figure 5b–e shows spatial distributions of ΔT/T obtained in selected applied voltages at θ = +45° (b-d) and −45° (f). The data were taken at a slightly different area from that of Fig. 4b–d. The color contrasts monotonically increase with increasing the magnitude of V (Fig. 5b–d), and the contrasts get reversed by switching θ from +45° to −45° (compare Fig. 5d, e). We calculated the average of ΔT/T in the pixels at selected single ferroaxial domain areas (both red and blue) denoted by boxes in Fig. 5a–d and took its V dependence. As seen in Fig. 5f, the magnitude of ΔT/T, i.e., the magnitude of EG, is proportional to V. These results confirm that the electric-field-induced change in ΔT/T observed in NiTiO3 is ascribed to the linear EG effect. We also calculated the magnitude of EG using the average of ΔT/T of the areas denoted by black and white boxes in Fig. 5d (±100 V, θ = +45°), and obtained α = (2.0 ± 1.0) × 10−5deg V−1 for the red area and (−1.9 ± 0.9) × 10−5 deg V−1 for the blue area. The errors were calculated from the standard deviation of ΔT/T.

Fig. 5: Applied voltage dependence of the intensity map of electrogyration in NiTiO3.
figure5

a Transmission optical microscopy image with the incidence of unpolarized light along the c axis (Scale bar: 100 μm). Dark areas in the image correspond to NiO impurity. be The two-dimensional maps of electrogyration at the same area as (a). To obtain the maps, the difference of the transmission microscope images at the positive and the negative voltages divided by the average of them (ΔT/T) was calculated for each pixel detection. A 3 × 3 median filter was applied to the raw images. The polarization configuration was set at (bd) θ = +45° and (e) −45°. The applied voltage V was (b) ±12.5 V, (c) ±50 V, and (d, e) ±100 V. A ΔT/T color scale is applied to the images in (be). The V dependence of the average of ΔT/T taken at θ = +45° in the selected single domain areas denoted by boxes (ad). The red and blue dots correspond to the data of the areas surrounded by large and small boxes, respectively, in each panel. The standard deviation is shown as an error bar. The lines denote least squares fitting lines.

The domain structures obtained by the optical imaging are irregular in shape (Fig. 4c, d). Furthermore, not only sharp domain boundaries but also relatively thick ones are present (see green areas between red and blue areas in Fig. 4c, d). This is also true for the results of CBED. In addition to the (110)-type sharp domain boundaries shown in Fig. 3, relatively thick boundaries where the two domains (A+ and A−) overlap were also obtained in our CBED measurement (see Supplementary Note 7). Thus, the result of the optical imaging is compatible with that of STEM-CBED. Furthermore, the length scales of the ferroaxial domains observed in NiTiO3 are on the orders of 100 ~ 102 μm. Such length scales are roughly comparable with the result of a previous SHG study which reported uneven domain populations in ferroaxial RbFe(MoO4)2 obtained from measurements using incident light with a 50-μm diameter spot on the sample8.



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