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DOI: 10.1039/D4TC02743D
(Paper)
J. Mater. Chem. C, 2024, Advance Article

Struan Simpson^{a},
Cameron A. M. Scott^{b},
Fernando Pomiro^{a},
Jeremiah P. Tidey^{a},
Urmimala Dey^{b},
Fabio Orlandi^{c},
Pascal Manuel^{c},
Martin R. Lees^{d},
Zih-Mei Hong^{efg},
Wei-tin Chen^{efh},
Nicholas C. Bristowe^{b} and
Mark S. Senn*^{a}
^{a}Department of Chemistry, University of Warwick, Gibbet Hill, Coventry, CV4 7AL, UK. E-mail: m.senn@warwick.ac.uk
^{b}Centre for Materials Physics, Durham University, South Road, Durham, DH1 3LE, UK
^{c}ISIS Facility, STFC Rutherford Appleton Laboratory, Harwell Science and Innovation Campus, Oxon OX11 0QX, UK
^{d}Department of Physics, University of Warwick, Gibbet Hill, Coventry, CV4 7AL, UK
^{e}Center for Condensed Matter Sciences, National Taiwan University, Taipei 106319, Taiwan
^{f}Center of Atomic Initiative for New Materials, National Taiwan University, Taipei 106319, Taiwan
^{g}Department of Chemistry, Fu-Jen Catholic University, New Taipei 24205, Taiwan
^{h}Taiwan Consortium of Emergent Crystalline Materials, National Science and Technology Council, Taipei 10622, Taiwan

Received
28th June 2024
, Accepted 22nd August 2024

First published on 29th August 2024

Magnetoelectric multiferroics hold great promise for the development of new sustainable memory devices. However, practical applications of many existing multiferroic materials are infeasible due to the weak nature of the coupling between the magnetic and electrical orderings, meaning new magnetoelectric multiferroics featuring intrinsic coupling between their component orderings are sought instead. Here, we apply a symmetry-informed design approach to identify and realize the new manganite perovskite CeBaMn_{2}O_{6} in which magnetoelectric coupling can be achieved via an intermediary non-polar structural distortion. Through first-principles calculations, we demonstrate that our chosen prototype system contains the required ingredients to achieve the desired magnetoelectric coupling. Using high-pressure/high-temperature synthesis conditions, we have been able to synthesize the CeBaMn_{2}O_{6} perovskite system for the first time. Our subsequent neutron and electron diffraction measurements reveal that the desired symmetry-breaking ingredients exist in this system on a nanoscopic length scale, enabling magnetoelectric nanoregions to emerge within the material. Through this work, we showcase the potential of the new CeBaMn_{2}O_{6} perovskite material as a promising system in which to realize strong magnetoelectric coupling, highlighting the potential of our symmetry-informed design approach in the pursuit of new magnetoelectric multiferroics for next-generation memory devices.

Despite their transformative potential in device applications, multiferroic materials with commercially applicable transition temperatures remain rare. For the intensively studied field of perovskite multiferroics, this is largely due to the incompatibility of producing a ferroelectric distortion on a magnetic cation^{1} as the presence of empty metal d orbitals is required to stabilize the ferroelectric distortion via hybridization with the O 2p electrons.^{2} One way to circumvent this is to introduce alternative sources of ferroelectricity such as stereochemically active lone pairs into the material, allowing the polar and magnetic instabilities to be separated onto different cations (as seen in the high-T_{C} perovskites BiFeO_{3} and BiMnO_{3}^{3,4}). However, this particular approach is inherently limited by the separate mechanisms producing ferroelectricity and magnetism, rendering device applications impractical due to the resultingly weak coupling between these effects.^{5} If practical device applications based on multiferroic materials are to be realized, alternative approaches to multiferroism must be developed in which the coupling between the magnetic and electrical orderings is intrinsic by design.

A promising alternative route to magnetoelectric coupling is to exploit a combination of non-polar structural distortions and antiferromagnetic (AFM) orderings which, when combined, induce a polar distortion and weak ferromagnetism due to additional symmetry-breaking associated with their joint action.^{6,7} Provided a material could be designed which combines these ingredients, and that some coupling between the primary magnetic and the non-polar order parameters exists, the required electric control of magnetization could be achieved. Such magnetoelectric coupling has already been predicted to occur, for example, in Ruddlesden–Popper perovskite-like materials.^{8} More recently, this idea was extended to perovskite materials after enumerating the possible ways in which structural and magnetic symmetry-breakings can couple in the high-symmetry Pmm aristotype structure.^{9} These couplings can be used to determine the terms in a Landau-like energy expansion about the high-symmetry structure^{10} up to arbitrarily high orders. Further details of our proposed scheme are given in the ESI,† as well as ref. 9, but we provide a brief summary here for context. Of particular importance to our scheme are terms in which the polarization (P) and magnetization (M) – transforming with respect to the Pmm aristotype structure as the Γ^{−}_{4} and mΓ^{+}_{4} irreducible representations (irreps), respectively – appear coupled linearly to other order parameters at the third order, as such terms can always adopt a sign so as to lower the free energy of the system. Provided the proposed combination of irreps which represent these order parameters (ξ) preserves crystal momentum, parity with respect to inversion symmetry, and time-reversal symmetry, this yields invariant terms in the Landau-like expansion of the form:

Pξ_{1}ξ_{2}
| (1) |

Mξ_{1}ξ_{3}.
| (2) |

When present, a reversal of polarization in the invariant term (1) with an electric field must necessarily reverse either ξ_{1} or ξ_{2}. If a material can be engineered in which it is energetically favorable to switch ξ_{1} over ξ_{2} in invariant (1) and M over ξ_{3} in invariant (2), then a 180 degree reversal of magnetization with electric field – the exact mechanism desired for multiferroic memories – can be achieved. We illustrate a plausible scheme for this approach in Fig. 1.

Fig. 1 An example scheme for inducing magnetoelectric coupling in the ABX_{3} perovskite structure derived from the group-theoretical analysis in ref. 9. P and M denote irreps which describe ferroelectric polarization and weak ferromagnetism (wFM), respectively. These form two interdependent trilinear coupling terms defined as Q(X^{+}) P Q(X^{−}) and Q(X^{−}) M Q(mX^{−}) where the magnitude of a mode transforming as an X-point irrep, Q(X), represents the actions of the ξ order parameters in (1) and (2). Examples of symmetry-breaking distortions which transform as these X-point irreps are also provided for reference. The X-point irreps have been labelled assuming the A site lies at the unit cell origin. Purple, magenta, and blue spheres represent the A(1), A(2), and X atoms, respectively, while cyan polyhedra represent the BX_{6} octahedra; purple arrows denote displacement directions of the A cations, while black arrows denote spin moments on the B sites. |

In what follows, we explicitly demonstrate how our symmetry-informed magnetoelectric coupling scheme can be used to predict new multiferroic perovskites featuring intrinsic mechanisms of magnetoelectric coupling. We identify the new perovskite system CeBaMn_{2}O_{6} as a promising prototype for our coupling scheme and confirm through first-principles simulations that the required symmetry-breaking ingredients represent the ground state of this system. We then present our experimental efforts to synthesize and characterize CeBaMn_{2}O_{6} for the first time with the aid of high-pressure synthesis conditions, exploiting a combination of electron and neutron diffraction as well as magnetometry measurements to identify key signatures of the desired coupling scheme across a nanoscopic length scale. Our study showcases a promising new perovskite system in which to realize useful magnetoelectric coupling, as well as highlighting new opportunities to achieve magnetoelectric multiferroism within the perovskite structure type through the use of our innovative symmetry-adapted approach.

Structural relaxations were performed using the conjugate gradient algorithm and continued until the largest force on any ion in the system was less than 1 meV Å^{−1}.

Non-collinear magnetic calculations with spin–orbit coupling^{18,19} were utilized at two points. To calculate magnetocrystalline anisotropy energies, we performed non-self-consistent non-collinear magnetization calculations in which the self-consistent charge density from the collinear calculation was fixed and the energy calculated as the easy axis was varied. Fully self-consistent non-collinear calculations were also performed which required the increasing of the plane wave cut-off and Monkhorst–Pack grid to 1200 eV and 7 × 7 × 5 respectively to ensure convergence.

All symmetry analysis was conducted using the ISOTROPY Software Suite.^{20,21} For the sake of brevity, we refer here to the order parameters and the associated order parameter directions (OPDs) simply by their irrep labels. Further details of the specific OPDs required to generate the magnetoelectric ground state consistent with our symmetry-adapted approach are provided in Note 1 (ESI†).

Transmission electron microscopy (TEM) experiments were performed by mixing CeBaMn_{2}O_{6} powder with fine (<10 μm) aluminum powder in a 1:10 ratio before being pressed using cold rollers to form a sheet ∼100 μm in thickness. This sheet was mechanically thinned to ∼20 μm and mounted on a copper support ring using epoxy resin before ion milling to electron transparency using Ar^{+} ions at 6 keV. The specimen was finished using an ion beam energy of 100 eV to reduce surface damage to provide sufficiently thin edges for TEM. Samples were examined using a JEOL 2100 LaB_{6} microscope operating at 200 kV.

Fig. 2 The desired crystallochemical features required to achieve a magnetoelectric ground state in an A-site layered perovskite such as CeBaMn_{2}O_{6}. The combination of layered A-site order (X^{+}_{1}), antipolar A-site displacements (X^{−}_{5}), and A-type AFM order (mX^{−}_{5}) feature in the two trilinear coupling terms (a) and (b), yielding a ferroelectric polarization (Γ^{−}_{4}) and a ferromagnetic component (mΓ^{+}_{4}), respectively. Both (a) and (b) trilinear coupling terms are mutually coupled via the action of the X^{−}_{5} term, which itself emerges through a separate trilinear coupling term with the M^{+}_{2} and R^{−}_{5} octahedral tilt modes as detailed in ref. 28. The trilinear terms (a) and (b) ultimately conspire to yield the desired magnetoelectric switching, taking the form of the term (c). Full details of the desired order parameter directions for these coupling terms can be found in the ESI.† Representative axes labels are shown with respect to the Pnma structure, upon which we base our subsequent discussion. |

Evidently, both the polarization and its coupling to the reversal of magnetization can be achieved by inducing an A-site cation layering to LaMnO_{3} without changing the magnetic structure. As the magnetic structure is determined via the superexchange paths,^{29–31} which are themselves dependent on the degree of octahedral tilting and the C-type orbital order present in this perovskite, it is assumed that ordering the A sites with cations that do not change the valence state or drastically alter the tolerance factor will leave the magnetic structure unaffected. For this reason, we choose an isoelectronic substitution of La layers with alternating layers of Ce^{4+}/Ba^{2+} as the average ionic radii of these two cations is similar to La^{3+} (〈r_{A}〉 (XII) = 1.375 Å vs. r_{La3+} (XII) = 1.36 Å). Although not changing the tolerance factor, we anticipate that the largely differing charges and radii of Ce^{4+} (r = 1.14 Å) and Ba^{2+} (r = 1.61 Å) should favor the formation of a cation ordered phase, with A-site layering being the most commonly observed ordering configuration for double perovskites.^{25}

In LaMnO_{3}, the combined effect of the Jahn–Teller active Mn^{3+} and low Goldschmidt tolerance factor result in a cooperative Jahn–Teller distortion (accompanied by a C-type orbital ordering of the |x^{2}〉 and |z^{2}〉e_{g} orbitals) and an a^{−}b^{+}a^{−} octahedral tilt pattern. The Jahn–Teller distortion acts as the gapping mechanism and the overlapping e_{g} orbitals supply the A-type AFM via the Goodenough–Kanamori–Anderson rules. Given the similar average ionic radii of Ce^{4+}/Ba^{2+} to La^{3+}, we expect the same conclusions to hold for the layered CeBaMn_{2}O_{6} structure.

To test these ideas, we performed a full structural relaxation for each of the four magnetically ordered structures available in the CeBaMn_{2}O_{6} Pnma crystal structure (A-type AFM, C-type AFM, G-type AFM and FM). This revealed that A-type AFM is lowest in energy with FM ordering 13.02 meV per f.u. higher while G-type and C-type AFM are 44.3 meV per f.u. and 54.5 meV per f.u. higher than the ground state AFM, respectively. It should be noted that varying the Hubbard-U parameterization results in a transition to a ferromagnetic ground state (Fig. S3, ESI†). The A-type AFM required for our proposed coupling scheme is stable within the range 3 eV < U < 6 eV – a result consistent with analogous studies on LaMnO_{3}. Altering the Hubbard-U on the Ce^{4+} ion does not change the magnetic structure away from A-type AFM. Furthermore, we calculated the magnetic moment on each Mn ion to be 3.707μ_{B} – this value suggests the Ce^{4+} oxidation state rather than Ce^{3+}. This oxidation state is further supported by the calculated electronic band structure (Fig. S3, ESI†) displaying empty Ce f bands and also by electronic structure calculations on a ferromagnetic structure in which the total magnetization is 4μ_{B} per f.u. suggesting a Mn^{3+} state (and consequently a Ce^{4+} state). Subsequent DC resistivity measurements on the synthesized samples substantiated these oxidation state assignments (Note 2, ESI†).

With A-type AFM identified as the ground state, we calculated the electronic band gap (Fig. S3, ESI†) of this system to be 0.98 eV – lower than the 1.7 eV measured in LaMnO_{3}.^{32} One should keep in mind the difficulty in calculating accurate band gaps within DFT^{33} and the strong dependence of bandgap on the choice of Hubbard-U correction. Table S1 (ESI†) contains a detailed comparison of how the simulated structures of LaMnO_{3} and CeBaMn_{2}O_{6} differ. We note that the calculated amplitude of the R^{−}_{5} tilt mode is slightly smaller in CeBaMn_{2}O_{6} compared to LaMnO_{3}, which may also contribute to the lower band gap we obtain for this system.

These two conditions – A-type AFM and the occurrence of a bandgap – are all that are strictly necessary for determining the feasibility of our coupling scheme in CeBaMn_{2}O_{6} assuming A-site cation layering and a Pnma tilt scheme. The appearance of a polar distortion and weak ferromagnetism determined by antisymmetric exchange interactions are guaranteed by the crystal symmetry. Using the Berry phase formulation of the modern theory of polarization,^{34} we calculate an electric polarization of 9.90 μC cm^{−2} parallel to the crystallographic a lattice vector which is similar to the magnitude of polarization calculated for other improper-ferroelectric multiferroics such as YMnO_{3} (P ≈ 6.5 μC cm^{−2}).^{35} This polarization direction is predicted due to the antipolar displacements of the A cations along the a axis, which are uncompensated due to the cation layering hence they enable polarization to emerge along the same axis.

LaMnO_{3} is reported as having a weakly ferromagnetic component to its spin structure as is allowed by symmetry. When we included the effects of relativistic spin–orbit coupling in our DFT simulation, we were able to calculate the magnetocrystalline anisotropy energies and determine the easy axis with associated spin canting directions for CeBaMn_{2}O_{6}. We determined (Table S2, ESI†) that the energetically favorable easy axis was the [100] direction in the orthorhombic cell, a result that is also found in LaMnO_{3}.^{27} This easy axis allows a spin canting which produces a ferromagnetic component of 0.029μ_{B} per Mn along the long b axis.

Finally, we note that the reversal of X_{5}^{−} requires either the reversal of the in-phase tilts (M^{+}_{2}) or the antiphase tilts (R^{−}_{5}) – see the top panel of Fig. 2. Whichever of these intermediate modes are reversed, the subsequent electric-field reversal of the wFM is required by symmetry (Fig. 2b). Taken together, these factors demonstrate that CeBaMn_{2}O_{6} contains all the necessary ingredients and coupling to make it a promising magnetoelectric multiferroic. Equipped with this theoretical insight, we attempted to synthesize CeBaMn_{2}O_{6} as a prototype for our magnetoelectric coupling scheme.

Other LnBaMn_{2}O_{6} (Ln = lanthanide) perovskite systems are highly sensitive to the presence of cation disorder on the A site, wherein the heterogeneous distribution of Ln^{3+} and Ba^{2+} cations suppresses long-range octahedral tilting and favors disordered or glassy electronic ground states.^{40–42} Extreme cation disorder would make any superstructure reflections far too broad or diffuse to observe by XRD/NPD, explaining why the Pmm aristotype model most appropriately describes the average symmetry of CeBaMn_{2}O_{6}. However, the diffraction peaks in the synchrotron XRD/NPD patterns remain sharp, suggesting there is no phase separation of Ce- or Ba-rich domains. Based on our Rietveld fits against high-resolution synchrotron XRD data, we estimate our samples exhibit very little microstrain (e_{0} = 0.015%). This implies that any deviation from the macroscopic average cubic lattice parameter in our samples must be exceptionally small, effectively discounting the existence of any severe chemical inhomogeneity. Taken together, the observation of a macroscopic cubic lattice and very low microstrain imply spatial homogeneity of Ce/Ba, but any periodic A-site ordering must be shorter than the length scales over which strains relax in typical perovskites oxides (e.g. over ∼50 unit cells, ∼20 nm). We have arrived at this upper estimate for the length scale over which macrostrain will not relax in our system by considering the observed rates of relaxation of epitaxial strain in LaAlO_{3}/SrTiO_{3} interfaces,^{43} which finds that for less than 50 unit cells (∼20 nm) the LaAlO_{3} relaxes less than the microstrain value of 0.015% in CeBaMn_{2}O_{6}. Our relaxed DFT model for A-site ordered CeBaMn_{2}O_{6} has macrostrain values of ∼6%, whereas the epitaxial strain between LaAlO_{3}/SrTiO_{3} is only 2.7% in ref. 43, so we view this 20 nm distance as the absolute maximum length scale over which any cation ordering can occur given that average cubic symmetry is still observed in our X-ray diffraction experiments.

Our refinements against NPD data showed that the displacement parameters for the oxygen sites are highly anisotropic (U_{11} = 0.018(1) Å^{2}, U_{22} = U_{33} = 0.078(1) Å^{2}) even at 1.5 K, reflecting significant static disorder of the oxygen positions. The large U_{33} corresponds to a root-mean-square (rms) displacement of 0.28 Å, which is about half the difference between the Ba^{2+} and Ce^{4+} radii, implying that the oxygen atoms on average sit halfway between the optimum positions for Ba and Ce but are locally relaxed. The oblate shape of the displacement ellipsoids at 1.5 K (drawn in Fig. 3b at 95% probability) suggests the MnO_{6} octahedra are statically tilted, albeit only across short length scales that are inaccessible to our synchrotron XRD and NPD experiments.

To investigate the possibility of short-range octahedral tilting in CeBaMn_{2}O_{6}, we collected selected-area electron diffraction (SAED) patterns on several crystallites (Fig. 4). Electron diffraction is advantageous over X-ray or neutron diffraction in this respect as diffraction data can be collected across much smaller regions, effectively enabling the structural details of individual domains to be probed. Combined with the stronger scattering interactions of electrons, SAED should be significantly more sensitive to any octahedral tilting distortions with short coherence lengths. Fig. 4 shows that various sets of weak superstructure reflections can be distinguished across several zone axes, each imaged from different crystallites: the reflections in Fig. 4a and c can be indexed with the propagation vector k = (1/2, 0, 1/2), consistent with reflections at the M point, while the reflections in Fig. 4b can be indexed with the propagation vector k = (0, 1/2, 0), consistent with reflections at the X point. The combined set of superstructure reflections reveals the adoption of a (where a_{p} is the primitive cubic lattice parameter) superstructure of the parent Pmm cell on a nanoscopic length scale. These superstructure reflections map exactly onto those observed for Pnma (Pbnm in non-standing setting) perovskites such as CaTiO_{3},^{44} demonstrating that CeBaMn_{2}O_{6} adopts the same a^{−}b^{+}a^{−} (a^{−}a^{−}c^{+}) tilt pattern. The M-point superstructure reflections in Fig. 4c hence correspond to in-phase octahedral rotations about the b axis, which, in the language of our group-theoretical approach, transform with respect to the Pmm aristotype (with A site at the origin) as the M^{+}_{2} irrep. The X-point superstructure reflections can be attributed to so-called concert reflections^{45,46} which arise in perovskites featuring both in-phase and antiphase octahedral tilting simultaneously.^{44} Physically, these reflections correspond to antipolar motions of the A cations which transform with respect to the Pmm aristotype as the X^{−}_{5} irrep. In Pnma-type perovskites, antipolar motions of the A cations are guaranteed to occur by symmetry due to a trilinear coupling term of the form M^{+}_{2} R^{−}_{5} X^{−}_{5}, so our observation of the M^{+}_{2} and X^{−}_{5} modes allows us to infer the presence of the antiphase octahedral tilt mode R^{−}_{5} as well. Although we have only been able to sample diffraction patterns from three possible zone axes to make this assignment, we note these three alone are sufficient to distinguish a^{−}b^{+}a^{−} from all other possible mixed-tilt systems (Table S6, ESI†), granting a high degree of confidence in the assignment of the a^{−}b^{+}a^{−} tilt scheme in CeBaMn_{2}O_{6}. We note that layered cation ordering should also transform as an X-point irrep, so we cannot interrogate the cation arrangement directly from our SAED patterns. Indeed, the SAED reflection conditions predicted for a cation-disordered Pnma perovskite are identical to that expected for a Pnma-type perovskite featuring cation layering along the long b axis,^{44,47} so we primarily use our current SAED data to evidence the Pnma-type octahedral tilt configuration.

While most of the crystallites we selected exhibited discrete superstructure reflections, we noted that some exhibit streaks of diffuse scattering instead (Fig. 4d). Such diffuse scattering suggests that the coherence length of the a^{−}b^{+}a^{−} tilting is limited within these regions. To probe the coherence length of the tilting within our sample, we performed dark field imaging based on the (0, ½, 0) superstructure reflection (Fig. 5). This revealed a highly mottled contrast across a ∼200 nm length scale (Fig. 5b), reflecting an intricate mixture of tilted and untilted nanoscale domains. A line analysis of the image contrast (Fig. 5c) showed that the domains are typically 5–10 nm in size, with an average size of 7 ± 2 nm. This corresponds to approximately 20 unit cells with respect to the primitive cubic lattice parameter (∼3.9 Å), which is well below the length scale typically required to resolve Bragg reflections in X-ray or neutron diffraction. This estimate also agrees well with the maximum coherence length of cation ordering we derived based on the extent of microstrain in our samples.

Our electron diffraction and microscopy experiments indicate that the Pnma tilt scheme – hence the antipolar displacement of the A cations – is obtained in CeBaMn_{2}O_{6}, but that the coherence length of these distortions appears to be limited by the presence of cation disorder between the Ce and Ba atoms. With this knowledge, we re-examined the possibility of any short-range Pnma-like reflections in our XRD/NPD patterns. We generated a Pnma superstructure of the parent Pmm cell using ISODISTORT^{21} and performed a symmetry-mode refinement of this model against our 1.5 K NPD data collected on the backscattering bank at WISH. We applied an hkl-dependent Lorentzian convolution to the Pnma-type reflections to account for possible size-dependent (Scherrer) superstructure peak broadening due to short-range M^{+}_{2} and R^{−}_{5} tilt modes with a full-width half-maximum (FWHM) that varies as:

FWHM = 0.1DIFCd^{2}/S
| (3) |

ZFC and FC DC magnetic susceptibility measurements (Fig. S7, ESI†) reveal CeBaMn_{2}O_{6} is paramagnetic at room temperature. Fitting a Curie–Weiss expression above 150 K yields a Curie constant of C = 5.96(1) emu K^{−1} mol^{−1}. This corresponds to an effective moment of 4.89μ_{B}, which is in excellent agreement with the spin-only value expected for Mn^{3+} (∼4.90μ_{B}). Upon cooling, the ZFC and FC traces diverge below ∼38 K, but the lack of any long-range magnetic reflections in the NPD patterns confirms this is not due to a long-range ordering of the Mn moments. Our synchrotron XRD experiments found no evidence of trace magnetic impurities such as Mn_{3}O_{4} which might otherwise explain this feature. Similar behavior has been observed in other cation-disordered LnBaMn_{2}O_{6} (Ln = Y, Dy–Sm) materials,^{41,51} where divergences between ZFC and FC traces arise due to the freezing of short-range AFM spins with a wFM component, so we attribute this as the origin of the ZFC-FC divergence in CeBaMn_{2}O_{6}. Isothermal magnetization measurements (Fig. 6d) show a clear ferromagnetic hysteresis develops below the spin freezing transition. Weak ferromagnetic spin canting is predicted by our design scheme due the coexistence of atomic displacements transforming as X^{−}_{5} and magnetic ordering transforming as mX^{−}_{5}, giving the desired trilinear term with mΓ^{+}_{4} which describes the character of the wFM representation in Pmm perovskites.

We returned to our DFT calculations and performed full geometry relaxations on a selection of alternative cation orderings. These are enumerated, along with their energies, in Table S8 (ESI†). Each structure is strained significantly from the aristotypical cubic structure (where a_{p} ≈ 3.9 Å), so the lack of any discernable peak splitting in our diffraction patterns shows that none of these alternative cation configurations nor any of their associated symmetry-breakings achieve any long-range coherence in our samples. Interestingly, we find that the assumed Ce/Ba layering perpendicular to the [010] direction (space group: Pmc2_{1}) is not the ground state cation configuration from the perspective of DFT: instead, alternative X-point layerings along either [100] or [001] are the most energetically favorable configurations. Layering along the [001] direction results in a polar space group (Pm), while layering along [100] produces a non-polar space group (P2_{1}/m). The Pm structure contains the same collection of modes necessary for the switching scheme as for the Pmc2_{1} structure illustrated in Fig. 2, so this is likely another prospective candidate phase for magnetoelectric switching within this system. Both [001] and [100] layering directions have energies within kT of the sample synthesis temperature (∼0.1 eV per f.u.), so we anticipate small regions of the sample will nucleate with different configurations of cation ordering. Furthermore, although A-site ordering with layering perpendicular to [010] has higher ground state energy, it being the highest-symmetry phase means that at elevated temperatures it may become entropically favorable, hence all three layering configurations may become feasible under the synthesis conditions we have employed. Nevertheless, the competition from alternative cation ordering arrangements appears to be sufficient to disrupt any long-range cation ordering throughout the bulk of the material.

Further efforts to achieve magnetoelectric coupling in CeBaMn_{2}O_{6} should focus on improving the coherence length of cation ordering beyond nanoscale domains. It is highly likely our use of high-pressure synthesis conditions has been crucial in promoting any partial cation ordering: LnBaMn_{2}O_{6} (Ln = Y, lanthanide) materials synthesized at ambient pressure typically require a preliminary reduction step to achieve cation ordering,^{25,41} so the use of even higher synthesis pressures may assist in increasing the size of the symmetry-lowered domains. Cation layering has also been proposed to be favored by large cation size variance in hydrothermally synthesized La_{0.5}Tb_{0.5}CrO_{3} (σ^{2} = 0.004 Å^{2}). Here, hydrothermal methods avoid the high temperatures typically required for solid-state reactions and thus prevent reorganization of the A cations into a disordered array.^{55} Based on the nominal Ce^{4+} valence, we expect CeBaMn_{2}O_{6} to have a much larger size variance of σ^{2} = 0.051 Å^{2} (this value is calculated based on the 9 co-ordinate ionic radii; we have taken a linear interpolation of the 8 and 10 co-ordinate radii of Ce^{4+} to calculate r_{Ce4+} (IX)). To the best of our knowledge, CeBaMn_{2}O_{6} has the largest reported size variance of any A-site substituted perovskite, so hydrothermal synthesis may be a promising alternative synthesis route to achieve long-range cation ordering in this system and to experimentally verify the ground state cation ordering configuration. Finally, we anticipate that epitaxial growth techniques will be key to ensure control over the directionality of the cation ordering within this system, thus ensuring the cation configuration is compatible with the magnetoelectric coupling scheme we have outlined in this work and potentially enabling magnetoelectric coupling to be realized beyond the nanoscale achieved in our high-pressure-synthesized samples.

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## Footnote |

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d4tc02743d |

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