Post-perovskite Transition in Anti-structure

The discovery of the post-perovskite transition, which is the structural transition from the perovskite to post-perovskite structure in MgSiO3 under pressure, has aroused great interests in geosciences. Despite of previous extensive studies, key factors of the post-perovsktie transition are still under hot debate primarily due to the big difficulty in performing systematic experiments under extreme conditions. Hence, search for new materials showing the post-perovskite transition under ambient pressure has been highly expected. We here report a new-type of materials Cr3AX (A = Ga, Ge; X = C, N), which exhibits the post-perovskite transition as a function of “chemical pressure” at ambient physical pressure. The detailed structural analysis indicates that the tolerance factor, which is the measure of the ionic radius mismatch, plays the key role in the post-perovskite transition. Moreover, we found a tetragonal perovskite structure with loss of inversion symmetry between the cubic perovskite and orthorhombic post-perovskite structures. This finding stimulates a search for a ferroelectric state in MgSiO3.

are shifted up and down along the c axis to form the elongated XM 6 octahedra, resulting in the breakdown of the inversion symmetry. We note that this structure does not belong to the supergroup-subgroup relationship shown in Fig. 1(f). The anti-ppv structure possesses the two-dimensional character with the orthorhombic Cmcm symmetry as illustrated in Fig. 1(b), which differs from three-dimensional character of the anti-pv structure.

Results
We present x-ray diffraction (XRD) patterns taken at room temperature for two solid solutions Cr 3 Ga 1−x Ge x N and Cr 3 GeN 1−y C y in Fig. 2(a). The XRD patterns can be divided into three regions: I. 0 ≤ x < 0.50 in Cr 3 Ga 1−x Ge x N; II. 0.50 ≤ x ≤ 1 in Cr 3 Ga 1−x Ge x N and 0 ≤ y < 0.20 in Cr 3 GeN 1−y C y ; and III. 0.20 ≤ y ≤ 1 in Cr 3 GeN 1−y C y . The XRD patterns in region I can be well fitted by assuming the anti-pv structure with the cubic Pm-3m symmetry, confirming an earlier report on Cr 3 GaN 23 . In region II, a clear splitting of 0 0 2 reflection in the a × a × a notation into 0 0 2 and 2 2 0 reflections in the ~2a × 2a × a notation is observed, indicating the symmetry lowering from the cubic to tetragonal one. In addition, there appears new reflections including 1 2 0 and 1 2 1 in the ~2a × 2 a × a notations; and all of these reflections are well assigned by the P-42 1 m space group in consistent with an earlier report on Cr 3 GeN 24 . In region III, the XRD patterns are different completely. All the peaks are well indexed by assuming the anti-ppv structure with the Cmcm space group, confirming an earlier report on Cr 3 GeC 25 . We can therefore see that the anti-pv structure in Cr 3 GaN changes to anti-ppv structure in Cr 3 GeC by the chemical substitution or the "chemical pressure". This transition can be called as the post-pv transition in the anti-structure.
The crystal structures are refined by the Rietveld analysis as shown in Fig. S2, and the volume per unit formula (V), the lattice parameters (a, b, and c), and the bond distances are summarized in Fig. 2(b-e) and Table S1.
With increasing x in Cr 3 Ga 1−x Ge x N, the unit cell volume shrinks monotonously in both of the Pm-3m and P-42 1 m phases owing to the smaller atomic radius of Ge than that of Ga. On the other hands, with increasing y in Cr 3 GeN 1−y C y , the lattice has a general tendency to expand owing to the larger atomic radius of C than that of N. In the latter course, there is a discontinuous volume decrease by ~0.54% across the anti-ppv transition at y = 0.20. This indicates that the negative "chemical pressure" triggers the structural transition to the denser anti-ppv phase. The volume changes in a strong anisotropic manner with increasing x and y: the a axis decreases and the c axis increases in the P-42 1 m phase; and the a and c axes increase and the b axis decreases in the Cmcm phase.
To understand the ppv transition from the microscopic viewpoint, we focus on the coordination environment around A and X atoms, and plot the A-Cr and X-Cr bond distances as a function of x and y in Fig. 2 (d,e). Across the transition from Pm-3m to P-42 1 m in the anti-pv phases, the 6 X-Cr bonds in XCr 6 octahedra of the Pm-3m phase are split into 2 longer bonds and 4 shorter bonds with large expansion of the averaged distance, resulting in the elongated XCr 6 octahedara in the P-42 1 m structure. In the anti-ppv phase with the Cmcm symmetry, on the other hand, the XCr 6 octahedra are compressed; there are 4 longer bonds and 2 shorter bonds, keeping the similar averaged distances to those in the P-42 1 m phase. Contrastively, the ACr 12 polyhedron show more drastic change. The 12 A-Cr bonds in an ACr 12 polyhedron of the Pm-3m structure are split into 5 groups in P-42 1 m structure and 4 groups in the Cmcm structure. Among them, one group including 2 bonds in the P-42 1 m structure and 4 bonds in the Cmcm structure have much longer bond distances than the other bonds. We can therefore say that the coordination numbers of A sites decreases from 12 in the Pm-3m structure to 10 in the P-42 1 m structure and 8 in the Cmcm structure 26 . We here notice that, in the P-42 1 m structure, the 2 longer bonds are expanding toward one side of ACr 12 polyhedron as shown in Fig. S1(d), resulting in the local breakdown of the inversion symmetry. This feature is quite distinct from the centrosymmetrically distorted manner of ACr 12 polyhedron in the Cmcm structure as shown in Fig. S1(e).
To investigate the crystal structure at high temperature, we have collected XRD patterns for Cr 3 GeN in the temperature (T) ranges of 25-800 °C; the warming process data are shown in Fig. 3(a). The XRD patterns qualitatively change around 150 and 650 °C, indicating the emergence of the successive structural transitions. The XRD patterns can be well fitted by assuming the P-42 1 m structure for T < 150 °C 24 , the I4/mcm structure for 150 °C < T < 650 °C, the P4/mbm structure for 650 °C < T < 750 °C, and the Pm-3m structure for 750 °C < T. The  results of XRD refinements are summarized in Fig. 3(b-e), Fig. S3, and Table S2. On warming, the lattice expands monotonously, and the tetragonal distortions characterized by the a/c ratio become smaller. In this course, the local environments around the A and X atoms change in a strange manner. The ratio between the shorter and longer X-Cr bonds in a XCr 6 octahedra, which is the measure of the local tetragonal symmetry breakdown, exhibits anomalous sudden decrease across the P-42 1 m to I4/mcm transition. Simultaneously, the ACr 12 polyhedra change their local environment with 2 + 2 + 4 + 2 + 2 A-Cr bonds in the P-42 1 m structure to that with 4 + 4 + 4 bonds in the I4/mcm structure in a discontinuous manner. We note that the I4/mcm structure has similar local environment in XCr 6 octahedra and ACr 12 polyhedra to that of the P4/mbm and Cmcm structures ( Fig. S1(b), S1(c), and S1(e)), indicating the close connection among these three structures. We can therefore conclude that the P-42 1 m structure is special among all the structures discussed in this study [20][21][22] .
To construct the precise phase diagram, we performed differential thermal analysis (DTA) and differential scanning calorimetry (DSC) for two solid solutions Cr 3 Ga 1−x Ge x N and Cr 3 GeN 1−y C y as shown in Fig. 4(a,b). In the DTA data on the warming process for Cr 3 GeN, there are three peaks at around 140, 590, and 690 °C, which correspond to the structural transitions observed in the XRD measurements. By changing compositions, the three structural transition temperatures shift systematically. Similar features are also discernible in the DSC data as shown in Fig. 4(b). Based on these data, the structural phase diagram shown in Fig. 5(b) was established. One can see that the crystal structure of Cr 3 AX develops from the cubic anti-pv structure to orthorhombic anti-ppv structure with the intermediate tetragonal anti-pv structure, and that the region of the intermediate tetragonal phase becomes narrower at higher temperature.

Discussions
We now discuss what the key factor of the ppv transition is. The most plausible candidate is the ionic radius mismatch between A and X ions, which is measured by the tolerance factor = − − t d d / 2 A Cr X Cr , where d A−Cr and d X−Cr represents the average atomic distances between the A and Cr atoms, and the X and Cr atoms, respectively 27 . As shown in Fig. 5(a), the t factor calculated by using the atomic radius is 0.94, close to 1 for Cr 3 GaN, indicating the perfect matching of ionic radius and the stable cubic anti-pv structure 28 . With increasing x in Cr 3 Ga 1−x Ge x N and y in Cr 3 GeN 1−y C y , the t value becomes much smaller; and then the A atoms are not stable in the 12-fold coordination of Cr, destabilizing the anti-pv structure. As a consequence, the anti-ppv structure, in which the A atoms have 8 fold coordination of Cr, is stabilized. This is a rough sketch of the ppv transition in the anti-structure.
In this respect, the three tetragonal anti-pv phases are considered to be the intermediate phases across the ppv transitions. Indeed, the resemblance of the local structure around the A atoms among the I4/mcm, P4/mbm, and Cmcm structures indicate that the tetragonal distortions in the I4/mcm and P4/mbm structures are the precursor of the ppv transition. However, this is not the case for the P-42 1 m structure. The P-42 1 m structure is different from the I4/mcm and P4/mbm structures in four respects: (1) the structure is not included in the subgroup-supergroup relationship shown in Fig. 1(f); (2) the inversion symmetry is broken; (3) the coordination number of A atoms is not 8 nor 12 but 10; and (4) the structural transition temperature is lowered with increasing y in Cr 3 GeN 1−y C y . All of these facts indicate that the P-42 1 m structure is not stabilized by the ionic radius mismatch between A and X atoms; instead, the structure is likely stabilized by the covalent nature of N-Cr and Ge-Cr bonds. Such a covalency driven structural phase transitions are discussed in ferroelectrics including PbTiO 3 and BaTiO 3 29 , and ferroelectric-like metals including LiOsO 3 , Cd 2 Re 2 O 7 , and Pb 2 Ir 2 O 7 [30][31][32][33] ; in both class of materials, inversion symmetry is broken. Therefore, we can conclude that two driving forces of structural modifications, which are the ionic radius mismatch and covalency of chemical bonds, are competing and/or cooperating with each other in the present system. We also notice that the effect of these two driving forces becomes smaller at high-temperature, where thermal fluctuations prefer to high-symmetry structures, which well explains the experimentally observed narrower tetragonal phase at high temperature.
Finally, we discuss the resemblance and difference between the ppv transition in Cr 3 AX and that in MgSiO 3 . The ppv transition in Cr 3 AX is driven by the ionic radius mismatch between A and X atoms, which is in harmony with the several proposed scenario of the ppv transition in MgSiO 3 focusing on the tolerance factor which decreases on the application of pressure 11,34,35 . Despite of this fundamental resemblance, there are several crucial differences. Firstly, whereas the application of the negative "chemical pressure" induces the ppv transition in Cr 3 AX, the application of positive physical pressure induces the ppv transition in MgSiO 3 . This means that the physical pressure is not the key factor of the ppv transition; instead, the tolerance factor is the fundamental factor. Secondly, whereas the pv structure adjacent to the ppv structure belongs to the tetragonal I4/mcm and P4/mbm symmetry in Cr 3 AX, it belong to the orthorhombic Pnma symmetry with the unit cell of ~2a Thirdly, not only the structural instability toward to the ppv structure, but also the structural instability toward the inversion-broken state is present in Cr 3 AX. We are therefore tempted to imagine that there is a structural instability to the ferroelectric state in MgSiO 3 , in which the Si-O bonds are discussed to have the strong covalent characters 36,37 . Further experiments as well as computational simulations are required to clarify this issue. A possible interesting study is a search for new compounds with the anti-ilmenite and anti-LiNbO 3 structures, which will highlight the important role of the tolerance factor as well as the covalent bonding 38 .

Methods
Polycrystalline samples were grown by the solid-state reaction. Powders of C (3 N), Cr (3 N), and Cr 2 N (2 N), and grains of Ga (4 N), and Ge (3 N) were mixed in a stoichiometric ratio in a N 2 -filled glove box, and then sealed in a quartz tube under 0.3 atm of Ar gas 18 . The quartz tube was heated to 1000 °C, held for 60 h, and quenched to room temperature. The product was pulverized, and pressed into pellets, which were annealed inside a quartz tube at 1000-1100 °C for 96 h. Then, the above annealing process was repeated. X-ray diffraction experiments were performed by utilizing Smartlab (Rigaku) and M21X (Mac science). Structural parameters were obtained by Rietveld refinement using Rietica software 39 . Differential thermal analysis (DTA) was measured continuously at 25-1000 °C with heating/cooling rates of 20 °C/min. by using the Al 2 O 3 as the reference. The samples were put into a glass capillary with an inner diameter of 0.1 mm. Differential scanning calorimetry (DSC) was measured at − 150-480 °C with heating/cooling rates of 20 °C/min.