Ferroelectric phase transition and spontaneous electric polarization in CaMn7O12 from first principles

The phase transition mechanism and ferroelectric polarization of CaMn7O12 are investigated by using the density functional theory. Our results show that the P3 space group should be the ground-state structure with R3 as the intermediate phase. It is the helicoidal magnetic order that induces the first transition to the R3 phase while the second transition to P3 structure results from soft modes under the constraint of the magnetic structure. The two phase transitions from R 3 ¯ ?> through R3 to P3 are second order and first order, respectively, which is consistent with experimental observations. Group theoretical analysis shows that the particular domain states and multi-domain structure of the P3 phase match with the observed coexistence of two magnetic modulations below TN2 = 48 K. Our calculated electric polarization is also in good agreement with experiment. By analyzing the polarization contributions from both mode and atomic-decomposition viewpoints, we find that the Raman-type distortions give rise to a significant contribution to the total polarization. This unexpected result can be understood through the asymmetric change of the Born effective charges caused by the particular helicoidal magnetic order, which leads to an abnormal infrared character of the purely Raman-active modes.


Introduction
Recent experimental and theoretical advances in multiferroic materials have greatly deepened understanding of coexistence and coupling of ferroelectric (FE) and magnetic orders, which is a subject attracting tremendous interest due to the technological importance in spintronics and information storage [1][2][3][4][5]. According to the classification by Khomskii [6], the FE polarizations of type-II multiferroics (or so-called magnetic multiferroics) are induced by some particular types of magnetic order and hence the intrinsic magnetoelectric couplings are strong. More recently, a large magnetically induced FE polarization (2870 μC m −2 in a single crystal sample) has been reported in rhombohedral CaMn 7 O 12 [7,8]. Unlike the known cycloidal spin spiral or the E-type AFM order [5,6], the magnetic structure of CaMn 7 O 12 belongs to the unexpected helicoidal (or proper-screw) spin spiral [8,9].
The room-temperature crystal structure of CaMn 7 O 12 is the R3 (No 148) space group [10,11], in which three types of Mn ion occupying the Wyckoff positions 9e, 9d, and 3b are labeled as Mn1, Mn2, and Mn3, respectively, as shown in figure 1. At 250 K, an isostructural charge-ordering transition gives the Mn2 and Mn3 ions nominal valences of +3 and +4, respectively [12]. In addition, CaMn 7 O 12 shows two further magnetic phase transitions at T N1 =90 K and T N2 =48 K [13,14]. In the temperature range T N2 <T<T N1 , an inplane helical magnetic structure (shown in figure 1) with incommensurate propagation vector k 1 =(0, 1, 0.963) develops and results in a remarkable ferroelectricity along the c axis of the rhombohedral conventional cell [8]. Below T N2 , the propagation vector k 1 splits into two branches with values of k 2,3 =k 1 ±(0, 0, δ) (δ≈0.08), indicating the coexistence of two modulations along the same symmetry direction [8]. At the same time, a tiny change in electric polarization was observed at T N2 [7,8].
The discovery of large FE polarization in CaMn 7 O 12 , which persists up to a Néel temperature of 90 K, represents a significant development in the field of magnetic multiferroics. In order to explain the improper ferroelectricity, a phenomenological ferroaxial coupling mechanism has been proposed [8,15]. The essential aspect of this scenario is the coupling among FE polarization, the chirality of the helical spin spiral, and the axial vector representing a homogeneous structural rotation of the R3 structure with respect to its high-temperature cubic Im3 space group. Based on first-principles calculations, Xiang et al [16] proposed that the microscopic origin of the giant FE polarization in CaMn 7 O 12 is mainly due to the combined effect of the symmetric exchange striction and the antisymmetric Dzyaloshinskii-Moriya (DM) interaction. A subsequent theoretical work [17] confirmed the dominant role of the exchange-striction mechanism, and the spin-dependent p-d hybridization induced by the spin orbital coupling (SOC) also contributes a little polarization.
However, the crystallographic characteristics and phase transition mechanism of the two phase transitions at T N1 and T N2 remain to be identified. The room-temperature R3 space group includes spatial inversion (R −1 ) symmetry. In order to accommodate the FE polarization appearing at T N1 , it is necessary to break the inversion symmetry. Although the helical magnetic order itself breaks the spatial inversion symmetry [8], it is implausible that the giant improper ferroelectricity originates only from the purely electronic polarization mechanism [18,19], i.e. without ionic displacements away from their centrosymmetric reference positions. In this respect an representative example is TbMn 2 O 5 , of which the ground-state structure is the Pb2 1 m polar group instead of the nonpolar Pbam symmetry [20,21]. For the phase transition occurring at T N2 , the thermal hysteresis behavior indicates a first-order phase transition character [22], so the crystallographic symmetry below T N2 should be distinct from that in the temperature range T N2 <T<T N1 . To our best knowledge, there are no experimental or theoretical works concerning the actual symmetry groups of CaMn 7 O 12 and the underlying phase transition mechanisms in these two temperature ranges. Another closely related issue is to determine the effect of lattice distortion on FE polarization and distinguish the polarization contributions of purely electronic and lattice mechanisms. As we know, the purely electronic polarization, which is due to charge density redistribution induced by magnetic order in the absence of ionic displacements away from their centrosymmetric positions [18,19], does not exist in traditional ferroelectrics. Similarly, the ingrained opinion that only zone-center infrared (IR)-active modes can be responsible for the appearance of polarization is only supplied from traditional ferroelectrics and should not be regarded as self-evident for all types of magnetic multiferroic, though it has been reiterated for the improper ferroelectricity of TbMnO 3 induced by the cycloidal spin spiral through the SOC effect [23,24].
In the present work, we carry out a detailed first-principles study of the phase transition mechanism and spontaneous polarization in CaMn 7 O 12 using the experimental helicoidal magnetic structure (denoted AFMI) determined in [8]. We will demonstrate that the ground-state structure of CaMn 7 O 12 should be the P3 space group with R3 as the intermediate phase, and that the phase transition sequence from R3 through R3 to P3 is driven first by the magnetically induced forces and then by soft modes at the boundary of the Brillouin zone under the constraint of the helicoidal magnetic structure. Although the magnetic structure below T N1 is unresolved experimentally, the particular domain states and the single multi-domain structure of the P3 phase provides an alternative understanding of the coexistence of two different incommensurate magnetic modulations. The two phase transitions from R3 through R3 to P3 show second-order and first-order character, respectively, which are consistent with experimental observations at T N1 and T N2 . By decomposing the FE polarization into individual modes and individual sites, we obtain the most unexpected feature of CaMn 7 O 12 , i.e. the dominant polarization contribution comes from the purely Raman-type structural distortion. We will illustrate that the abnormal IR character of the purely Raman-type distortion is due to the asymmetric changes of Born effective charges (BECs) caused by this uncommon helicoidal spin spiral.

Ferroelectric phase transition
For a hypothetical magnetic order denoted as AFMr (all the spins of Mn1 and Mn3 sites are parallel while all the Mn2 spins are antiparallel), we choose the most energetically favorable R3 phase as the high-symmetry reference structure due not only to the satisfactory structural parameters but also to the insulating electronic behavior [25]. We first demonstrate that the ferroelectricity in CaMn 7 O 12 can not be driven by soft modes as in traditional ferroelectrics since all the zone-center optical phonons are stable for different magnetic orders [25].
Then, for the AFMI helicoidal magnetic order, we provide the phonon-dispersion relations of the R3 reference structure along the high-symmetry directions in the Brillouin zone of the 20-atom primitive cell, as shown in figure 2. It is evident that there exist unstable modes extended over quite a large area of the Brillouin zone. We find that the dominant instability originates from a singlet phonon at the T point with an imaginary frequency of 65.3i cm −1 and point symmetry T . 1 + It corresponds to the complex distortion pattern of the MnO 6 octahedra (including Mn2, Mn3, O1, and O2 ions). However, this type of distortion should be described in a 40atom pseudocubic unit cell, which is incompatible with the AFMI helicoidal magnetic order. In spite of this, for the (1/3, 1/3, 1/3) point along the Λ direction, the less unstable doubly degenerate phonon (ω=35.1i cm −1 and Λ 1 symmetry) is just right for the 60-atom hexagonal unit cell and corresponds to displacements of all kinds of ion, which combines the distortion pattern of MnO 6 octahedra of the T 1 + mode with the displacements consisting of Ca atoms and Mn1O 4 square planes. Finally, it should be noted that this structural instability at the T point is not relevant to the magnetic order (82.3i cm −1 and 32.2i cm −1 for the AFMr and ferromagnetic order, respectively). On the other hand, there are remarkable Hellmann-Feynman forces appearing on the ions as the AFMI helicoidal magnetic order is turned on (with the structural parameters clamped in the AFMr reference structure). The forces induced by the change of the spin ordering are undoubtedly due to the change of electronic structure. However, the detailed tridimensional electronic structure is averaged so much in the DOS presentation that it is hard to explain the atomic forces only with respect to the current DOS presentation. Another work of ours [26] provides detailed analysis of the AFMI spin ordering induced forces from the electronic density difference between different spin orderings: The helicoidal magnetic order induces strong charge redistributions on all types of Mn and O ion (especially for Mn2 and Mn3), while only the orientation angle of Mn3 spins is responsible for the inversion symmetry breaking of the electronic density distribution. It is this strong charge redistribution which inevitably induces large Hellmann-Feynman forces appearing on the ions in the reference structure. Projecting the forces onto the 20-atom primitive cell of the R3 space group, we find that, in the range of numerical precision, the forces can be entirely classified into (i) 97.5% of the symmetrypreserving Raman-active A g ( 1 ) G + irreps and (ii) 2.5% of the infrared-active A u 1 ( ) Girreps, both of which have order parameters along the c axis of the 60-atom conventional cell. The decomposed forces in the 20-atom primitive cell are collected in table 1. For the Raman-active A g modes, we see that the forces on O ions along the direction of the lattice vector can achieve a value as high as 230 meV Å −1 . Even for the IR-active A u irreps, the forces are still several times larger than that of the IR-active B 3u modes in TbMnO 3 [23]. For example, the largest force in TbMnO 3 is −8.93 meV Å −1 (table 2 in [23]), which appears on the Mn 4b atom along the z direction, while in our case the largest force on the Mn3 atom along the z axis is −45.7 meV Å −1 .
Due to the dual nature of structural instability originated from the soft T 1 + mode and the large magnetically induced forces, we can not search for the ground-state structure only by freezing the unstable mode [27,28] or deducing the ionic displacements from these forces [23]. Starting from the calculated structural parameters for the AFMr state (shown in table S1 in [25]) and with the constraint of fixed lattice vectors, we change the magnetic order from AFMr to the AFMI state and relax all internal degrees of freedom until the remaining forces are less than 1 meV Å −1 . We obtain the ground-state structure with P3 (No 143) space group. The resulting groundstate structure has polarization direction parallel to the c axis, which is consistent with the experimental result [8]. In comparison with the R3 reference structure in the AFMI spin state, the ground-state P3 structure is 56 meV lower in energy per 60-atom unit cell. If we employ the total energy of the R3 phase in the AFMr spin state as the reference point, the energy difference increases to 417.2 meV. Further turning on the SOC effect, the energy difference between the AFMr R3 reference structure and the ground-state AFMI P3 phase now reaches as high as 673.2 meV. In addition, the calculated magnetic moments of Mn1, Mn2, and Mn3 ions are respectively 2.56, 2.48, and 1.86, which are in good agreement with the experimental values reported in [8]. In order to verify the above results, we have attempted two alternative strategies to determine the ground-state structure. One is relaxing the lowest P1 space group obtained by displacing ions from their high-symmetry positions in an entirely random pattern, and the other is starting from the possible intermediate phases such as P3 (No 147) and R3 (No 146) symmetry. Excellent agreement between these different strategies is achieved. Next, we focus on the phase transition path connecting the high-symmetry R3 phase to the ground-state P3 structure. As illustrated in figure 3, path (1) describes an intuitive conjecture that the prototypical phase undergoes a transition directly to the P3 space group by freezing in the unstable doubly degenerate Λ 1 phonon at the (1/3, 1/3, 1/3) point because of the compatibility of atomic displacements at this q point with the helicoidal magnetic order. For this mechanism, the unstable Λ 1 phonon is the primary order parameter, which singlehandedly leads to the final distorted P3 symmetry and triples the 20-atom primitive cell [29]. The resulting primitive cell of P3 symmetry is just the 60-atom hexagonal unit cell of the R3 phase. On the other hand, although the A u and A g modes at the Γ point receive significant forces due to the AFMI magnetic order (shown in table 1), they still serve as the secondary order parameters because of their incapacity of lowering the symmetry enough to the P3 group.
However, the group-subgroup relation [30] between the two space groups of R3 and P3 rules out this easy path (1); instead we have the following two possibilities. Path (2) denotes the transition sequence with the nonpolar P3 phase as the intermediate structure, which triples the 20-atom primitive cell and results from the primary order parameter of the Λ 1 phonon but with no other secondary order parameter than the A g modes. The Table 1. For the R3 phase with AFMI magnetic order, the calculated Hellmann-Feynman forces (meV Å −1 ) projected onto the 20-atom primitive cell and decomposed into Raman-active A g and IR-active A u irreps.  (3), the transition sequence with the R3 phase as the intermediate structure, is the one and only choice, of which the first transition is driven by the magnetically induced forces (the primary and secondary order parameters are the forces with IR A u irrep and Raman A g irrep, respectively) and the second transition originates from the Λ mode at the (1/3, 1/3, 1/3) point. Now we try to estimate the phase transition temperature. As mentioned above, our results show that the energy difference between the AFMr R3 phase and the AFMI P3 phase with the SOC effect is about 673.2 meV per 60 atoms. This is about 11.2 meV/atom and roughly corresponds to the temperature of 129 K, which is qualitatively consistent with the phase transition temperature of T N1 (90 K). The relatively large error is likely due to the hypothetical AFMr spin order used in our calculations instead of the practical paramagnetic spin state. In addition, as shown in figure 2, the soft phonon at the Λ (1/3,1/3,1/3) point has the frequency of 35.1i cm −1 , which is equivalent to 4.35 meV. The converted temperature is about 50 K, which is in good agreement with the value of T N2 =48 K. Therefore our first-principles results do demonstrate the existence of the R3 intermediate phase and indicate that path (3) can account very well for both the FE transition due to the helicoidal magnetic order at T N1 =90 K and the observed second phase transition at T N2 =48 K [7,8]. As shown in [25], a majority of ionic displacements from R3 to R3 phase can achieve an about 10 −3 order of magnitude of the lattice constants, whereas all the displacements from R3 to P3 remain in the order of 10 −4 of the lattice constants. So far as we know, despite the evident change in lattice parameters at T N2 [14,31], no direct evidence of the R3 and the P3 structures has yet been reported because the atomic displacements are extremely small. Our results stimulate further careful investigations of the temperature dependence of selected structural parameters to explore the anomalies of the atomic displacement parameters (ADPs) at T N1 and T N2 , as has been done in TbMn 2 O 5 [32].

Atom positions
Here we propose a novel diffraction technique [33], of which the amplitude and phase of the ionic displacements are encoded due to the interference between charge and magnetic x-ray scattering, to directly determine the internal atomic coordinates of the intermediate R3 phase and the ground-state P3 structure because of the very high precision, up to several femtometers.
To further shed light on the underlying physics, we have determined the orders of the two phase transitions and their possible domain states. According to the Landau phase transition theory, a phase transition is allowed to be continuous only ifboth the Landau condition and the Lifshitz condition are satisfied for the primary order parameter. On the other hand, a phase transition must be first order as long as the Lifshitz condition is violated.In the case of the first transition from R3 to R3, it is obvious that the R3 symmetry is a subgroup of the R3 space group and that the A u mode is just the single irreducible representation corresponding to this phase transition. If we expand the free energy of the crystal in terms of components of the A u order parameter, we find that the third-degree invariant polynomials are vanishing. In other words, the Landau condition is satisfied. Furthermore, theoretical analysis [29,30] also shows that the low-symmetry R3 phase will retain the macroscopic uniformity because the Lifshitz invariant is excluded from the symmetry considerations. The above analysis demonstrates that the Landau condition and Lifshitz condition are both satisfied for the primary order parameter of the A u irrep, which indicates the second-order characteristic of this phase transition. In addition, the analysis also demonstrates that this phase transition is allowed to be continuous even in the renormalizationgroup theory. This conclusion is also supported by the clear lambda character of the specific heat (figure 2(a) in [7]) and the absence of thermomagnetic irreversibility [22] around T N1 . In the case of the second transition from R3 to P3, we find that the primary order parameter of the doubly degenerate Λ 1 mode is incapable of meeting the Although all the space groups are trigonal crystal systems, R3 and R3 belong to the rhombohedral lattice, while P3 and P3 belong to the hexagonal lattice system. Lifshitz condition, which demonstrates that this phase transition can not be continuous. In fact, experimental observation of the thermal hysteresis behavior [22] around T N2 has confirmed the first-order character of this phase transition.
As we all know, collection of coherent domains, which are symmetrically and energetically equivalent structures differing only in their orientation and possibly position, often occurs when a crystal undergoes a phase transition. For more information concerning the possible domain states associated with the R3 to P3 phase transition, group theory analysis shows that there are three domain states which have the same basis vectors but different origins. Careful inspection shows that the common basis vectors for the three single-domain states are  1, 0, 0), and (0, 0, 1) while the origin is at the (0,0,0) point. It is important to note that the average symmetry of the single multi-domain structure is the R3 space group. Our result is consistent with the experimental observation [7,8] that the propagation vector of the AFMI spin ordering splits into two symmetric branches, which indicates the coexistence of two modulations along the same symmetry. In other words, the particular domain states and the single multi-domain structure in the P3 phase may provide an alternative understanding of the helicoidal magnetic order below T N2 , which is characterized by two propagation vectors split symmetrically from that of the AFMI phase [8,34].

Spontaneous electric polarization
We now determine the spontaneous polarization of CaMn 7 O 12 by using the Berry phase method [35]. The total FE polarization of the ground-state P3 structure in the AFMI helicoidal magnetic order is parallel to the c axis with a value of 3230 μC m −2 from the DFT+U calculations. Our calculated FE polarization is in excellent agreement with the experimental value of 2870 μC m −2 [8]. In order to estimate the influence of the phase transition from R3 to P3 on the FE polarization, we artificially construct a P3 structure by only freezing in the unstable Λ 1 mode of the R3 phase, and the polarization becomes −48.7 μC m −2 in this case. Thus, it is the first transition from R3 to R3 that results in the giant FE polarization and the second transition from R3 to P3 has negligible effect, which is consistent with experiments [7,8]. When the SOC is turned on, the FE polarization of the ground-state R3 phase is 3002 μC m −2 , which is reduced by 7% in comparison with that without SOC. Therefore, our results as well as previous theoretical works [16,17] demonstrate that the giant FE polarization of CaMn 7 O 12 is almost completely determined by the exchange-striction mechanism rather than by the SOC effect.
To understand the polarization contributions from different Wyckoff coordinates for each distorted mode, we perform mode decomposition [28] of the ground-state P3 structure in terms of the AFMr R3 reference phase. As a matter of course, besides the tiny Λ 1 -symmetry distortion, the atomic displacements from R3 to P3 are composed of the Raman-active A g and the IR-active A u modes. As shown in table 2, polarization contributions from these three types of atomic displacement sum up to −3065 μC m −2 , which is in good agreement with the value obtained by using the Berry phase method. The negative sign denotes that the direction of polarization is opposite to the forward direction of the z axis. The small change in polarization induced by the Λ 1 -irrep distortion, P≈167 μC m −2 , is also consistent with our previous theoretical estimate and the experimental results [7,8]. Table 2. Mode decomposition of the FE polarization in terms of different distortion patterns. The P z,xy and P z,z represent the polarizations along the z axis which are induced by the atomic displacements with respect to the xy plane and the z direction, respectively. P z denotes the summation of P z,xy and P z,z . All values are given in unit of 10 3 μC m −2 . The Born effective charges (BECs) used here are those calculated in the R3 phase with the AFMI magnetic order. For the A u -type distortion, the dominant contribution of −1062 μC m −2 comes from atomic displacements along the z direction, while the displacements within the xy plane partially counteract the total contribution of the A u modes. A close look at the contributions P z k from different Wyckoff coordinates reveals that each type of ion except O1 has considerable influence, among which the Ca and Mn3 give rise to contributions opposite to the net polarization. In addition, the large counteracting contribution from Ca cations (1516 μC m −2 ) has not been reported in previous theoretical investigations [16,17] and, indeed, it can not be explained by any existing model that takes only account of Mn-O-Mn exchange interactions. It should be pointed out that a similar result has been reported in the TbMnO 3 system although the ferroelectricity is induced by a cycloidal spin spiral via the SOC effect [23,36]. In that case, the displacements of Tb cations, which are uninvolved in the usual scenario of nearest-neighbor Mn-Mn spin interactions, contribute about 20% of the lattice polarization [23]. Finally, it is crucial to pay attention to the fact that the contribution from the IR-active A u modes is only responsible for −857 μC m −2 , which is far less than the overall value of −3065 μC m −2 .
The most unexpected outcome is that the dominant contribution up to about 77.5% (−2374 μC m −2 ) of the total polarization is due to the purely Raman-type A g distortion, which, for the traditional ferroelectrics, is not concerned with the appearance of ferroelectricity. Due to the ionic displacement pattern of the A g modes, displacements of all the cations (Ca and three types of Mn) are vanishing and hence have no contributions to the polarization. Within the range of numerical precision, the extremely large contribution to the polarization is found to result entirely from the displacements of O ions within the xy plane. Among these contributions, it is evident that the O2 ions play a decisive role. As regards the polarizations of O ions along the z direction, we find that the large contributions from two types of O1 anion almost cancel each other out.
The fact that the giant FE polarization of CaMn 7 O 12 is mainly due to the purely Raman-type distortion will lead to serious conflict with our ingrained concept that ferroelectricity originates only from the zone-center IRactive modes instead of the Raman-active modes. As we know, the purely Raman-active modes belong to the symmetry-preserving irreps and the ionic displacement pattern can not lower the crystallographic symmetry. However, we should not ignore the fact that the vanishing contribution of a Raman mode to electric polarization implies the following hypothesis: the local electric dipoles induced by these ionic displacements are equal in magnitude and opposite in direction and hence entirely cancel each other out, which requires that the symmetry of atomic BECs is consistent with the crystallographic symmetry. This is true for traditional ferroelectrics, but the case of improper ferroelectricity deserves carefully analysis. As we know, for the magnetic multiferroics driven by the exchange-striction mechanism, the magnetic order itself breaks the spatial inversion symmetry, which can lead to a remarkably asymmetric change in BECs. One possible consequence of the mismatch between the BECs and the lattice symmetry is that local electric dipoles induced by the symmetry-preserving ionic displacements can not be completely canceled out, and then the ferroelectricity comes into being, i.e. the purely Raman-active modes exhibit abnormal IR-active character.
To confirm the above argument, we recalculate the polarization contributions of Raman-active A g and IRactive A u modes using the BECs of the R3 phase in the AFMr magnetic order. The results are given in table 3. This time, we can see that the local electric dipoles caused by O displacements of the Raman-type A g distortions counteract each other and the net contribution to polarization is vanishing. More than this, the change in polarization for the IR-active A u modes is also distinct not only for the net polarization (−417 μC m −2 ) but also for the contribution from each Wyckoff coordinate. Since the ionic displacements remain invariable, the significant difference in polarization contributions can only be attributed to the change in BECs and, in the final analysis, to the electronic charge redistribution induced by different magnetic orders. For CaMn 7 O 12 , the Table 3. Mode decomposition of the FE polarization using the BECs calculated in the R3 phase with the AFMr magnetic order. P z,xy and P z,z denote the z-direction polarizations induced by the atomic displacements in the xy plane and along the z axis, respectively. The values of BECs for the AFMr state, as well as the differences between AFMI and AFMr magnetic order in the R3 space group, are also shown. The polarizations and the BECs are given in units of 10 3 μC m −2 and e, respectively. helicoidal spin spiral is very unusual. On one hand, this kind of magnetic order induces large forces appearing on different ions, which leads to both the Raman A g -and the IR A u -type distortions. On the other hand, it is the helicoidal magnetic order that causes distinct asymmetric changes in BECs, which makes the Raman-type distortions contribute significantly to polarization. However, it is important to point out that the asymmetric change in BECs does not mean the appearance of electric polarization for the purely Raman-type distortion. In fact, we have demonstrated elsewhere [26] that the remarkable polarity of the purely Raman-type ionic displacements can not survive in the absence of IR-type lattice distortion, which is determined by the preferred orientation of Mn3 spins via the SOC effect between the Mn2 and Mn3 sites. The asymmetric change in BECs for the AFMI helicoidal magnetic order with respect to the AFMr magnetic structure is also collected in table 3. It is clear that, for the AFMr magnetic order, the zx and zy components of BECs are zeros because the magnetic order is in accordance with the crystallographic symmetry, which results in the vanishing polarization contribution of the ionic displacements within the xy plane for the A g -type distortion. In the case of CaMn 7 O 12 , the improper ferroelectricity originates mainly from the exchange-striction mechanism, of which the magnetic point group of 31′ breaks the inversion symmetry and leads to strong asymmetry in hybridization of electronic orbitals. This electronic effect induced by the helicoidal magnetic order will inevitably be reflected in the asymmetric change of the BECs, which mismatches the BECs with the crystallographic symmetry. As shown in table 3, besides the distinct changes in Z , zz k we also identify that the zx and zy components of BECs possess nonzero values due to the AFMI helicoidal magnetic order. Thus we conclude that, for the A g -type distortion, it is the nonzero zx and zy components of BECs that give rise to significant contributions to the FE polarization. For CaMn 7 O 12 , with regard to the purely electronic polarization, we freeze the ionic positions in the R3 reference structure while switching the magnetic order from AFMr to AFMI state and calculate the polarization to be −4787 μC m −2 . This value is much larger than the actual theoretical value of −3230 μC m −2 , suggesting that the ionic displacements provide an opposite contribution of 1557 μC m −2 to partially counteract the pure electronic polarization. Finally, the AFMI helicoidal magnetic order in CaMn 7 O 12 results not only in a large polarization contribution from the purely Raman irrep modes but also in strong coupling of the BECs with the ionic displacements. Even for the same AFMI helicoidal magnetic order, the influence of ionic displacements on the BECs is also very distinct. To obtain a quantitative understanding, we also calculate the polarization contributions for the AFMI magnetic order but using the BECs of the R3 structure and compare to those using the BECs of the R3 structure (see table S6 in [25]). It is clear that the pronounced difference (with error as high as 22%) originates mainly from P , z xy k , which explains the high sensitivity of BECs in the AFMI magnetic order to the ionic displacements.
Finally, we should point out that, as mentioned above, an isostructural charge-ordering transition develops gradually below 250 K. According to [15], an incommensurate structural modulation along the c axis appears below this temperature, producing a continuous variation in Mn-O bond lengths, and this corresponds to the onset of a novel form of orbital ordering, which may be crucial for the emergence of the unusual helicoidal spin order in CaMn 7 O 12 . In our present work, we just adopt the experimental helicoidal spin order without respect to the origin of such a magnetic structure. The charge-ordering transition and the consequent incommensurate structural modulation, which are apparently not captured by our first-principles calculations, deserve further investigations.

Conclusions
In summary, for the magnetic multiferroic CaMn 7 O 12 , the ground-state structure should be the P3 space group with R3 symmetry as the intermediate phase. The two phase transitions from R3 through R3 to P3 are driven by the helicoidal spin spiral and soft modes but under the constraint of the helicoidal magnetic order, respectively. The initial second-order and then first-order character of this phase transition sequence is also consistent with experimental observations. Mode decomposition of the ferroelectric polarization demonstrates the dominant contribution from the purely Raman-type lattice distortion. This unexpected outcome is in conflict with the ingrained concept in traditional ferroelectrics, which can be understood by the abnormal IR-active character of the purely Raman modes caused by asymmetric changes of the BECs in the helicoidal magnetic order. The possibility of improper ferroelectricity induced by the Raman-type ionic displacements opens an avenue to search for or design novel magnetic multiferroics.