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### Order–disorder type ferroaxial transition in NiTiO_{3}

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(MoO_{4})_{2}, that is, the only material in which a ferroaxial transition is studied, exhibits a ferro-rotational distortion mainly ascribed to displacements of oxygen atoms^{11}, and therefore can be said to be a displacive type ferroaxial crystal. In this study, we propose that a structural phase transition reported in NiTiO_{3}^{13} is regarded as an order–disorder type ferroaxial transition (Fig. 2). At temperatures above *T*_{c} ≈ 1560 K, the crystal structure of NiTiO_{3} is described by the corundum structure (space group *R*(bar 3c)) which is envisage as a hexagonal close packing of the oxygen ions with Ni^{2+} and Ti^{4+} cations randomly occupying 2/3 of the octahedral interstices (Fig. 2a). With lowering temperature, cation ordering takes place at *T*_{c} 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 Ni^{2+} and Ti^{4+} 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 *T*_{c} (Fig. 2b, c). The transformation from the point group (bar 3m) into (bar 3) in NiTiO_{3} is the same as that in RbFe(MoO_{4})_{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)}. NiTiO_{3} 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 *T*_{c} 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.

### 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 NiTiO_{3} crystal was examined by the combined use of STEM and CBED^{14}. This technique (hereinafter, referred to as STEM-CBED technique) possesses a sensitivity to picometer-scale atomic displacements and a nanometer-scale spatial resolution^{12}, and therefore allows us to visualize spatial distributions of various nanostructures such as polar nanostructures in ferroelectrics^{15,16}. In the present study, we apply this technique to the observation of ferroaxial domains in NiTiO_{3}. 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.

We performed computer simulations of the CBED patterns for the structure models of NiTiO_{3} (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 Aizu^{17} and Zheludev^{18} independently in 1963-1964, and demonstrated in quartz crystals by Vlokh^{19} in 1970, a half century ago. To date, this effect has been measured in various crystals^{20,21} including PbWO_{4}^{22} and Pb_{5}Ge_{3}O_{11}^{23}. The EG effect is described by the change in the gyration tensor *g*_{ij} 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 *γ*_{ijk} is possible in all point groups except for *m*3*m*, (bar 43m), and 432, while the quadratic one by the fourth-rank axial tensor *β*_{ijkl} 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 *γ*_{333}, 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 *l*_{i} and *l*_{j} 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 *l*_{3} = 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 imaging^{24,25}. 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 (Δ*T*) divided by the average of them (*T*) was calculated for each pixel detection, and then spatial distributions of Δ*T/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, Δ*T*/*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 PbWO_{4} (see Supplementary Note 4).

### Optical imaging of ferroaxial domains in NiTiO_{3} via EG effect

We examined ferroaxial domains of NiTiO_{3} 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 *γ*_{333} 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 Δ*T*/*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 Δ*T*/*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 Δ*T*/*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 NiTiO_{3}.

To check whether EG observed in NiTiO_{3} 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 Δ*T*/*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 Δ*T*/*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 Δ*T*/*T*, i.e., the magnitude of EG, is proportional to *V*. These results confirm that the electric-field-induced change in Δ*T*/*T* observed in NiTiO_{3} is ascribed to the linear EG effect. We also calculated the magnitude of EG using the average of Δ*T*/*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^{−5}deg 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 Δ*T*/*T*.

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 NiTiO_{3} are on the orders of 10^{0} ~ 10^{2} μm. Such length scales are roughly comparable with the result of a previous SHG study which reported uneven domain populations in ferroaxial RbFe(MoO_{4})_{2} obtained from measurements using incident light with a 50-μm diameter spot on the sample^{8}.

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