1

Hybrid improper dipolar density wave in

NaLaCoWO6

Andrea Griesi1,2*, Enrico Mugnaioli2, Arianna E. Lanza2, Valentina Vit1, Lara Righi1, Mauro

Gemmi2 and Fabio Orlandi3*

1Department of Chemistry, Life Sciences and Environmental Sustainability, University of Parma,

Parco Area delle Scienze 17/A, 43124 Parma, Italy.

2 Istituto Italiano di Tecnologia, Center for Materials Interfaces, Electron Crystallography, Viale

Rinaldo Piaggio 34, Pontedera, Italy.

3 ISIS Neutron and Muon Facility, Rutherford Appleton Laboratory–STFC, OX11 0QX, Chilton,

Didcot, United Kingdom

KEYWORDS Incommensurate crystallography, electron diffraction, neutron diffraction, hybrid

improper ferroelectricity, symmetry analysis

ABSTRACT

The Hybrid Improper Ferroelectricity (HIF) mechanism allows the generation of an electrical

polarization thanks to the combination of two non-polar octahedral distortions in the AA’BB’O6

double perovskite materials. Nevertheless, for selected combination of the A/A’ cations a non-

polar incommensurate phase is observed with average symmetry C2/m. Thanks to a detailed

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crystallographic description of the incommensurate phase, based on electron, neutron and x-ray

diffraction data, we show that the incommensurate modulation is related to an abrupt sign change

of the out-of-phase tilting along the a and c axis. By using group theory and symmetry analysis

we show that we observe an incommensurate analog of the HIF mechanism which induces a hybrid

improper dipolar density wave in NaLaCoWO6.

INTRODUCTION

Concomitant magnetic and electric dipolar ordering represents an intriguing playground for

exotic functional properties ruled by the mutual interaction of periodical oriented dipoles1. Such

rare combination of ferroic orders provides the promising opportunity to exploit the

magnetoelectric effect and it is commonly obtained by inducing electric polarizability in magnetic

materials. Perovskites are one of the most exploited class of materials to obtain such states and the

traditional strategies to ensure the breaking of inversion symmetry rely on the incorporation of

either d0 metals in the perovskite B-site, involving second-order Jahn-Teller distortions2, lone-pair

active ions like Bi3+ or Pb2+ in the structure A-site3–5 or by using complex magnetic ordering6.

Recently, the developing of unconventional routes able to bring ferroelectric phases has

triggered a considerable interest in the research community7–15. These new routes involve the

combination of two or more non-polar lattice distortions, like for example tilting and/or

cation/anion orderings, to generate an electrical polarization as a secondary order parameter. The

clear advantage of these strategies relies on the fact that the non-polar distortions needed are more

common and they are not mutually exclusive with the presence of magnetic cations2. The clear

disadvantage is related to the magnitude of the polarization, being a secondary order parameter,

and to the switching mechanism which might involve a quite complex energy landscape.

Nevertheless, these new mechanisms to achieve polar states have brought new exciting dipolar

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ordered phases like, as examples, the ferrochiral ordering in Ba(TiO)Cu4(PO4)416 and the

incommensurate helical dipole ordering in BiCu0.1Mn6.9O12, which open up the possibilities of

complex non-collinear textures like skirmions8.

The “hybrid improper ferroelectricity” (HIF) is one of these new routes to induce a spontaneous

ferroelectric polarization in magnetic materials 10–13,15,17. The HIF condition in perovskites based

materials arises from the coupling of two non-polar modes, related to the local rotations and tilt of

the octahedral coordination units, which induce a polar mode as secondary order parameter. The

coupling in the system free energy is a trilinear invariant between the order parameters describing

the latter distortions. Several examples of HIF were encountered in layered perovskite

compounds9,11,12,18,19 as well as in double perovskites (with general formula AA’BB’O6) with

ordered distribution of heterovalent transition metals in the A site13,17,20,21. Computational works

and symmetry considerations indicate that cationic ordering in A and/or B sites is one of the most

efficient chemical instrument enabling a stable ferroelectric polarization via the HIF

mechanism13,15,20,22. In this scenario the coexistence of two degrees of atomic ordering, the B/B’

rock-salt distribution and the layered intercalation of two A and A’ cations, provides an intriguing

combination allowing different strategies for the design of multiferroic materials17,20,23

Such doubly ordered perovskites with general formula AA’BB’O6, whose parent structure is

depicted in Figure 1a, have been experimentally obtained with a large variety of A/A’ and B/B’

combinations24–27. The B and B’ cations are often W6+ (or another second order Jahn-Teller active

metal) combined with 3d transition metals like Mn, Fe, Co and Ni17,20,21,23,26,28. These systems

usually display magnetic ordering at low temperature, usually below 15 K, due to the weak B-O-

W-O-B super-superexchange interactions20,21,28. Research efforts reported the successful synthesis

of doubly ordered perovskites AA’BWO6 with A=Na+ or K+, A’= RE3+ and B= Co2+, Fe2+, Ni2+

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having potential multiferroic properties13,20,21,28. In particular, some compounds in the NaA’BWO6

family are featured by the polar monoclinic P21 symmetry with the a-a-c+ tilting scheme which

allows the HIF mechanism13,20,21,28. Young et al.22 suggested that the key factor defining the

electric polarizability in NaYCoWO6 is the large ionic radii difference between Y3+ and Na+,

confirming that the electric polarization is associated with the HIF mechanism and it is promoted

by the large octahedral tilting influenced by the smaller size of the A’ cations.

Figure 1: General structure of double ordered perovskites. a) parent P4/nmm structure, b) polar

P21 structure with a-a-c+ octahedral tilting and c) C2/m structure with the a0b-c0 tilting scheme.

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On the contrary, other AA’BWO6 compounds crystallize in centrosymmetric C2/m space group

(see Figure 1c), exhibiting the tilting scheme a0b-c0, with a supercell 2ap x 2ap x 2ap (with ap~3.9

Å). Some compositions based on the A/A’ couple Na/La i.e. NaLaMnWO6, NaLaCoWO6 and

NaLaMgWO6 are featured by the C-centered crystal structure at ambient conditions and transform

into the P21 acentric phase at lower temperature17,24,29. This first order transition, essentially related

to the shift of oxygen positions rearranging the tilting scheme, shows a wide hysteresis extended

over a large temperature range24. Furthermore, electron diffraction (ED) and HREM (High

Resolution Electron Microscopy) investigations revealed that the monoclinic C-centered crystal

structure in systems like NaLaMgWO6, NaLaCoWO6, KLaMnWO6 and NaCeMnWO6 manifests

incommensurate structural modulations.23,30,31 HREM investigations feature an unusual nano-

checkerboard contrast, corresponding to incommensurately modulate twinned domains30,31.

Indeed, the 1D-dimensional modulation, described by the superstructure 12ap x 2ap x 2ap, has been

related to compositional stripes due to an inhomogeneous A/A’ distribution30 and the complex

chessboard-like contrast in HREM has been interpreted as originated by the regular distribution of

perovskite regions showing a different content of Na and La.30 Conversely, other works associate

the incommensurate modulation in double perovskites to displacive topological distortion of the

octahedral tilting and periodic twinning 23. The origin of the complex structural modulation in the

multiple ordered AA’BWO6 perovskite is controversial and the crystallographic description of the

incommensurate modulated structure is still to be fully defined. Two main issues depend on this

outcome: i) the structural modulation could eventually remove the global centrosymmetric

symmetry by inducing local distortions of the octahedral tilting, giving the opportunity to observe

electric polarizability or to observe complex dipolar orderings; ii) the structural information should

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allow to properly understand the transition from the incommensurate modulated C-centered

structure to the acentric P21 phase.

For all these reasons, we synthetized the NaLaCoWO6 double perovskite and we investigated

the incommensurate modulated structure by 3D electron diffraction (3D ED)32–35 , neutron and x-

ray powder diffraction. 3D ED has the ability to work on single nano-crystalline domains, allowing

the determination of the modulation vector q and of the modulated structure by applying the

superspace approach33–35. The final structure refinement has been performed with the combination

of neutron and x-ray diffraction data, allowing an accurate description of the modulated structural

distortions and their couplings. Our findings and detailed symmetry analysis suggest an

incommensurate equivalent of the HIF mechanism, which lead to the emergence of a hybrid

improper dipolar density wave.

EXPERIMENTAL SECTION

The synthesis of NaLaCoWO6 was performed by solid state reaction mixing Na2CO3, La2O3 and

CoWO4, in stoichiometric amounts. The CoWO4 precursor was prepared using stoichiometric

quantities of CoO and WO3 and heat treated at 1173 K for 8 hours. The reagent oxides were mixed

and heat treated at 1198 K in air for 6 hours with a heating ramp of 4.5 K/min. Afterwards, the

sample was grinded and pressed to form a pellet and sintered at a temperature of 1223 K with a

ramp of 2.5 K/min for 8 hours. The sample purity was checked by x-ray diffraction and small

amounts of LaNaW2O8 (3.6(1) w%), CoO (1.04(3) w%) and Co3O4 (4.7(1) w%) were observed.

Precession-assisted 3D ED and energy-dispersive X-ray (EDX) spectroscopy were performed

with a Zeiss Libra 120 transmission electron microscope working at 120 kV and equipped with a

LaB6 thermionic source and a Bruker EDX XFlash6T-60 detector for EDX analysis. The samples

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for electron diffraction were ground with an agate mortar and the resulting powder was suspended

and sonicated in ethanol and drop-deposited on a carbon coated copper grid. Sequential electron

diffraction patterns were collected in nanodiffraction mode with a parallel beam of 150 nm,

obtained by selecting a condenser aperture of 5 µm. Data were energy-filtered on the zero-loss

peak with an in-column -filter and a slit width of about 20 eV36. Patterns were collected in

stepwise mode with a protocol similar to the one proposed by Mugnaioli et al.37. The precession

motion of the electron beam around the optical axis was obtained by a Nanomegas Digistar P1000

device38. The semi-aperture of the precession cone was set to 1°. The angular collection range of

3D ED data was up to 120° degrees, with steps of 1°. After each tilt step, a STEM image was taken

to check the crystal movement and the electron beam was positioned back in the same point of the

crystal. Diffraction patterns were recorded by an ASI Timepix single-electron detector39. Cell

determination, modulation vector and electron diffraction intensities extraction were performed

with PETS 2.0 program on data from thirty different nanocrystals40.

The X-ray diffraction measurements were performed with a STOE Stadi P diffractometer

equipped with Cu-Kα1 radiation, a STOE&Cie Johansson monochromator and a Dectris

MYTHEN2 1 K detector. The samples for X-ray analyses were ground and placed in thin disks

sandwiched by sheets of acetate to perform transmission measurements.

Time of flight neutron diffraction data (TOF-NPD) were collected on the cold neutron

diffractometer WISH at the ISIS second target station (UK)41. The data were focused on 4 detector

banks with average 2θ of 154, 121, 90 and 57°, each covering 32° of the scattering plane.

All structure refinement were performed with the Jana2006 software42. The electron diffraction

data has been treated within the kinematical approximation. Group theory and symmetry analysis

calculations have been performed with the help of the ISOTROPY and ISODISTORT software43.

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The free energy invariants have been calculated using the ISOTROPY and INVARIANT

software44. The crystallographic structures are drawn with the help of the VESTA software45.

RESULT AND DISCUSSION

Structure solution and refinement

3D-ED data collected on NaLaCoWO6 nanocrystals (see Figure S1) clearly show that the main

reflections are accompanied by first order satellites (Figure 2). The main reflections can be indexed

with a pseudo-cubic cell having the following parameters: a=7.83(3)Å, b=7.84(2)Å, c=7.89(8)Å

and β angle close to 90° (the actual value of β, refined against XRD data, is 90.160(10)°). From

the reconstruction of the reciprocal space the extinction rule h+k=2n for the main reflections can

be easily determined indicating a C-centered lattice with no other extinctions observed. A

reasonable model in space group C2/m was eventually found ab initio by charge flipping46 on the

basis of 3D ED data. The average structure of the double ordered perovskite found with electron

diffraction data is in agreement with the one reported in literature24, with rock salt ordering of the

CoO6 and WO6

octahedra, layer ordering of the Na and La cations and an average tilting pattern

a0b-c0. Satellite reflections are observed in the 3D-ED data along the a* direction, with a

modulation vector q=0.172(2)a*. Some crystals show a strong twinning, which exchanges the a*

and b* directions (see figure S2). The collection of 3D ED data on a single crystalline domain

allowed to determine the reflection conditions in four index notations as hklm: h+k=2n and h0lm:

m=2n, which are consistent only with the superspace group C2/m(α0γ)0s.

A first tentative refinement of the modulated structure against 3D-ED data alone returned a

structure with large displacive modulation of the La and Na atoms, clearly in contrast with the x-

ray diffraction data where the satellite reflections are almost absent (Figure 2c). In earlier structural

studies, aimed to disclose the nature of the incommensurate modulation in NaLaCoWO6, it was

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suggested that the incommensurate distortions involve mainly the oxygen positions24. The authors

indicated the presence of monoclinic twinned domains characterized by the inversion of the

octahedral tilting, without providing a crystallographic description of such twinning.

Figure 2: Diffraction data collected on NaLaCoWO6. a) Sections of the NaLaCoWO6 reciprocal

space obtained from the 3D ED data showing the modulation along the a* direction. The black

zones, devoid of reflections, are areas of the reciprocal space that are not measured due to the

geometrical constrain of the experiment. b) Rietveld plot of the TOF-NPD data collected on WISH

at average 2θ of 154º and c) Rietveld plot of the X-ray powder data converted in d-spacing for an

easier comparison with the neutron data. Observed data (black crosses), calculated pattern (red

line) and difference (blue line) are reported. The bottom black thick marks indicate the Bragg

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position of the main phase and the green thick marks indicate the satellite reflections. The other

three rows of thick marks indicate the Bragg position of CoO, LaNaW2O8 and Co3O4 from top to

bottom.

To disclose the nature of the incommensurate distortions, high resolution neutron diffraction,

which is more sensitive to light ions, was carried out at room temperature. The simultaneous

refinement of the average structure against the TOF-NPD data and the powder x-ray data results

in a good fit of the experimental data. The average structural model is featured by elongated

anisotropic ADPs for the oxygen positions suggesting a possible disorder or splitting of these sites.

This possibility has been tested against the data with a clear improvement of the refinement and a

normalization of the thermal parameters. The splitting of all the equatorial oxygen (O1, O3, O4

and O5) does not require additional atomic positions but can be obtained by shifting the position

from the 4i Wyckoff position sitting on the mirror plane. On the contrary, the spitting of the apical

O2 site has been obtained by two inequivalent positions O2 and O2’. The average splitting distance

is ≈ 0.5 Å, as shown in Figure 3a. A similar splitting, even if considerably smaller, has been

observed for the cations in the structure. The disordered anion positions result in two possible

octahedral orientations surrounding the Co+2 and W+6 cations. This suggests that two different

tilting topologies of the octahedral frameworks coexist in the crystal structure of NaLaCoWO6,

likely determining the incommensurate modulation.

To confirm this hypothesis, an incommensurate model was conceived starting from the unit cell,

the modulation vector and the superspace group symmetry (C2/m(α0γ)0s) determined by 3D ED.

In the structural refinement based on TOF-NPD and x-ray data the modulation of the split oxygen

positions has been described with the use of discontinuous Crenel functions, avoiding that in any

point of the crystal both positions are simultaneously occupied. For the equatorial oxygen position

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(O1, O3, O4 and O5), to fulfill the latter requirement is sufficient to define a single crenel function.

Indeed, the presence in the superspace group of the symmetry element {my|000s} assures that only

one position above or below the mirror plane is occupied throughout the crystal (see Figure 3b).

On the contrary, for the two apical O2 and O2’, the absence of this symmetry element obliges to

constrain the origin of the Crenel function (x4,0 as defined in Jana2006 software) for the two apical

O2 and O2’ positions to have a 0.5 difference to fulfill the same physical requirement. A similar

modulation function has been applied to the cation positions.

Figure 3: Modulated structure of NaLaCoWO6. a) Average structure showing the disorder of

the octahedral polyhedra. b) Fourier map calculated around the O1 position showing the

discontinuous occupancy of the split positions along the x4 direction. c) Evolution of the

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incommensurate structure across 8 unit cell along the a direction. The continuous line indicate the

region where the tilting along the a and c direction changes sign.

This model returns optimal agreement factors (see table S1 in the Supporting Information) for

the simultaneous fitting of the powder data (TOF-NPD and X-ray) as shown in Figure 2b-c. It is

worth to underline that the superspace group allows the Crenel origin parameter (x4,0) for all

positions to possess any value in the 0-0.5 range, nevertheless the refinement strongly suggests

that the x4,0 value for all atoms (except for the O2’ which is constrained to be the O2 value +0.5)

to be zero within the uncertainty of the refinement. This is further confirmed by the absences of

the 1111 and 110-1 reflections from the neutron pattern (indeed simulations with x4,0 values

different from 0 returns strong peaks at these positions). For this reason, the x4,0 parameters were

fixed to zero in the final refinement. It is worth to underline that the incommensurate model here

proposed is very simple and the only refinable parameters, which influence the satellite intensities,

are the positions of the split atoms in the average structure. To check if the refined model is

compatible with 3D ED data, the incommensurate model obtained from the powder diffraction

data has been refined against 3D ED data within the kinematical approximation. The refinement

is stable and converges with agreement factor of wR=18.49%, R=17.59% and GOF=11.07. The

final results of both refinement (3D ED data and powder neutron/x-ray data) are reported in table

S1 showing good agreement.

The refined modulated structure is shown in Figure 3c. The incommensurate modulation can be

described by an abrupt change of sign of the tilting angle along the a and c directions which

resemble a twinning boundary. The tilting pattern in the structure is a-b-c-, with the tilting angle

along the b direction constant across the structure. The antiphase tilting along the c direction shown

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in Figure 3c is a direct consequence of the x4,0 parameter being 0 and we will see in the next section

its importance in the emergence of the dipolar density wave. The anion distortions are responsible

for the modulation, as expected from the weak signal in the x-ray diffraction compared with the

neutron one, and the cations distortion follows the changes in the octahedral tilting.

The crystallographic description of the modulated structure enables revisiting some viewpoint

postulated to define the nature of this complex distortion. As remarked before, the structural

modulation observed in systems like NaLaBB’O6 has been interpreted in terms of irregular

distribution of RE3+ and alkaline ion to form alternated off-stoichiometric domains composing the

characteristic checkboard contrast30,47. In the combined x-ray/TOF-NPD structural refinements the

La and Na sites are fully occupied and neither mixing of the two heterovalent ions nor randomly

distributed vacancies are noticed. Both sites result completely occupied and the refined

composition is in agreement with the nominal NaLaCoWO6 stoichiometry. Furthermore, the

identified secondary phases are composed by all the elements involved in the nominal composition

indeed their presence in not in contrast with the occurrence of the 1:1 ratio for La and Na in the

doubly ordered perovskite. These assessments are in general supported by EDX analysis (see

figure S3), even if this measurement indicates small Na vacancies in selected regions of the doubly

ordered perovskite. This can be ascribed by Na loss induced by the electron irradiation. The

illuminated region can be depleted in Na content that tends to evaporate as documented in the

literature by the observation of the segregation of metallic Na and Na2O48,49. Interestingly, the

characteristic checkerboard contrast composed by a regular succession of bright and dark square

domains in HREM images, which is usually interpreted as a cation ordering effect, has been

explained in a completely different way in the incommensurately modulated Li1-xNd2/3-xTiO3

perovskite50. The structure solution was undertaken by A. M. Abakumov et al. within superspace

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approach and the final refinement, based on neutron and x-ray diffraction data, evidenced the

occurrence of displacive distortions defining antiphase domains with differently oriented tilting

pathways for the TiO6 octahedra. The appearance of modulation contrast is considered by the

authors to coincide with anti-phase domains delimited by sharp boundaries coinciding with the

discontinuous displacive modulation. In this regard, the Crenel function adopted to model the

displacive distortion of the oxygen well describe the condition of alternated regions showing

approximately 12ap periodicity which coincide with the linear extension of a single squared

domain measured in the HREM images collected on different NaLaBB’O6 systems31,47

Symmetry analysis and the hybrid improper dipolar density wave

Before discussing the details of the coupling and of the dipolar ordering observed in the

incommensurate phase of NaLaCoWO6, is worth to refresh the HIF mechanism observed in the

commensurate P21 phase. In the remaining of the paper we will describe the distortion of all phases

with respect to the parent P4/nmm structure shown in Figure 1a. This will simplify the coupling

terms and the group theory calculations, nevertheless it is still worth to underline some important

points regarding the cation ordering and how this allows the HIF mechanism. The P4/nmm phase

is obtained from the parent simple perovskite structure (s.g. Pm-3m) by the action of the R1+

irreducible representation (irrep), propagation vector (½ ½ ½), which correspond to the chess

board ordering of the B site cations and by the action of the X3- irrep, propagation vector (0 0 ½),

which gives rise to the layer ordering on the A-site. The latter is important for the HIF mechanism

since it removes the inversion symmetry from the B site point group allowing the octahedral

tilting’s to break the macroscopic spatial inversion symmetry13. The HIF mechanism in the P21

commensurate phases of the AA’BB’O6 perovskites is strictly related to the a-a-c+ octahedral

tilting13,20,21,28. In particular the out-of-phase tilting along the a and b directions is related to the

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Γ5+ irrep with order parameter (α,0), whereas the in phase tilting around the c axis is due to the Γ1

-

irrep with order parameter β. These two distortions acting together on the parent P4/nmm structure

will induce a secondary polar distortion, which transform as the Γ5- irreps with order parameter

(γ,0), thanks to the three-linear free energy invariant αβγ as schematically shown in Figure 4. The

polar Γ5- distortion is a uniform opposite shift of the cation/anion in the structure that result in a

net dipolar moment along the b axis of the P21 structure. This induced polarization is switchable

by an external electric field and the switching mechanism requires the change in sign of either the

α or β order parameters which correspond to a sign change of the a-a-c0 or a0a0c+ tilting

respectively.

Figure 4: Symmetry analysis of the P21 and C2/m(α0γ)0s structures. Schematic display of the

primary octahedral tilting distortions and of the corresponding induced dipolar ordering for the

phases discussed in the main text.

The scenario in the incommensurate phase of the NaLaCoWO6 compound is similar but results

in an incommensurate ordering of the induced dipoles in a sinusoidal fashion. The C2/m(α0γ)0s

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phase refined in this work can be obtained from the parent P4/nmm structure by the action of two

distortions related also in this case to the octahedral tilting’s. The first is the commensurate out-

of-phase tilting along the b direction. This transform as the Γ5+ irrep as for the a-a-c0 tilting in the

P21 phase but with a different order parameter direction (α,-α). The second distortion is the

incommensurate out-of-phase tilting along the a and c direction which transform as the Σ4 irrep

with order parameter (δ,0,0,0) (this is a direct consequence of the x4,0 parameter of the crenel

function being equal to 0). These two distortions acting on the tetragonal parent structure induce,

as secondary order parameter, an incommensurate displacement of the cation along the b direction

which transform as the Σ2 irrep (ε,0,0,0) thanks, also in this case, to a trilinear invariant αδε in the

free energy as schematically shown in Figure 4. The analogies between the commensurate P21

case and the incommensurate C2/m(α0γ)0s one let us to conclude that in the latter case we have

observed the development of an incommensurate ordering of the induced dipoles with the same

propagation vector as the incommensurate tilting. The induced dipoles are oriented along the b

axis, in analogy with the P21 phase, and their amplitude is modulated with the period of the

modulation vector resembling the electric analog of a spin density wave. For the latter reason we

describe the incommensurate dipolar ordering as a hybrid improper dipolar density wave.

CONCLUSION

We have shown that the C2/m phase observed in the NaLaCoWO6 compound is a modulated

phase with an incommensurate out-of-phase tilting along the a and c direction. The combined use

of electron, neutron and x-ray diffraction data allowed the refinement and the description of the

incommensurate phase within the superspace formalism. The peculiar combination of the

incommensurate tilting along two directions and of the commensurate one along the b direction

induce a dipolar density wave in perfect analogy with the HIF observed in the P21 phase. The

17

observation of a dipolar density wave add another puzzle to the analogy between electrical and

magnetic dipoles8 and open up the observation of peculiar phases, like for example the “devil’s

staircase” and electric solitons51–53, which are usually exclusive to magnetic orderings.

ASSOCIATED CONTENT

Supporting Information.

The following files are available free of charge.

Additional figures and crystallographic tables (PDF)

cif file of the refined structure (cif)

AUTHOR INFORMATION

Corresponding Author

* Andrea Griesi, Department of Chemistry, Life Sciences and Environmental Sustainability,

University of Parma, Parco Area delle Scienze 17/A, 43124 Parma, Italy. Istituto Italiano di

Tecnologia, Center for Materials Interfaces, Electron Crystallography, Viale Rinaldo Piaggio 34,

Pontedera, Italy. E-mail: andrea.griesi@iit.it

* Dr Fabio Orlandi, ISIS Neutron and Muon Facility, Rutherford Appleton Laboratory–STFC,

OX11 0QX, Chilton, Didcot, United Kingdom. E-mail: fabio.orlandi@stfc.ac.uk

Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval

to the final version of the manuscript.

Funding Sources

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A.G., E M., A.L., M.G. would like to thank Regione Toscana for funding the purchase of the

ASI MEDIPIX detector through the FELIX project (POR CREO FERS 2014–2020).

ACKNOWLEDGMENT

The authors acknowledge the science and technology facility council (STFC – UKRI) for the

provision of neutron beam-time on the WISH diffractometer. FO acknowledge fruitful discussion

with Dr Pascal Manuel and Dr Dmitry Khalyavin.

ABBREVIATIONS

3D ED three-dimensional electron diffraction, EDX energy-dispersive X-ray, TOF-NPD time of

flight neutron powder diffraction

REFERENCES

(1) Eerenstein, W.; Mathur, N. D.; Scott, J. F. Multiferroic and Magnetoelectric Materials.

Nature, 2006, pp 759–765. https://doi.org/10.1038/nature05023.

(2) Hill, N. A. Why Are There so Few Magnetic Ferroelectrics? Journal of Physical Chemistry

B 2000, 104 (29), 6694–6709. https://doi.org/10.1021/jp000114x.

(3) Orlandi, F.; Righi, L.; Cabassi, R.; Delmonte, D.; Pernechele, C.; Bolzoni, F.; Mezzadri, F.;

Solzi, M.; Merlini, M.; Calestani, G. Structural and Electric Evidence of Ferrielectric State

in Pb2MnWO6 Double Perovskite System. Inorganic Chemistry 2014, 53 (19), 10283–

10290. https://doi.org/10.1021/ic501328s.

(4) Catalan, G.; Scott, J. F. Physics and Applications of Bismuth Ferrite. Advanced Materials

2009, 21 (24), 2463–2485. https://doi.org/10.1002/ADMA.200802849.

(5) Spaldin, N. A.; Ramesh, R. Advances in Magnetoelectric Multiferroics. Nature Materials.

Nat Mater March 1, 2019, pp 203–212. https://doi.org/10.1038/s41563-018-0275-2.

(6) Cheong, S.-W.; Mostovoy, M. Multiferroics: A Magnetic Twist for Ferroelectricity. Nature

Materials 2007, 6 (1), 13–20. https://doi.org/10.1038/NMAT1804.

(7) Zhao, H. J.; Chen, P.; Prosandeev, S.; Artyukhin, S.; Bellaiche, L. Dzyaloshinskii–Moriya-

like Interaction in Ferroelectrics and Antiferroelectrics. Nature Materials 2020 20:3 2020,

20 (3), 341–345. https://doi.org/10.1038/s41563-020-00821-3.

19

(8) Khalyavin, D. D.; Johnson, R. D.; Orlandi, F.; Radaelli, P. G.; Manuel, P.; Belik, A. A.

Emergent Helical Texture of Electric Dipoles. Science 2020, 369 (6504), 680–684.

https://doi.org/10.1126/science.aay7356.

(9) Zhu, T.; Orlandi, F.; Manuel, P.; Gibbs, A. S.; Zhang, W.; Halasyamani, P. Shiv.; Hayward,

M. A. Directed Synthesis of a Hybrid Improper Magnetoelectric Multiferroic Material.

Nature Communications 2021 12:1 2021, 12 (1), 1–8. https://doi.org/10.1038/s41467-021-

25098-1.

(10) Stroppa, A.; Barone, P.; Jain, P.; Perez-Mato, J. M.; Picozzi, S. Hybrid Improper

Ferroelectricity in a Multiferroic and Magnetoelectric Metal-Organic Framework.

Advanced Materials 2013, 25 (16), 2284–2290.

https://doi.org/10.1002/ADMA.201204738.

(11) Zhu, T.; Gibbs, A. S.; Benedek, N. A.; Hayward, M. A. Complex Structural Phase

Transitions of the Hybrid Improper Polar Dion-Jacobson Oxides RbNdM2O7 and

CsNdM2O7 (M = Nb, Ta). Chemistry of Materials 2020, 32 (10), 4340–4346.

https://doi.org/10.1021/acs.chemmater.0c01304.

(12) Yoshida, S.; Akamatsu, H.; Tsuji, R.; Hernandez, O.; Padmanabhan, H.; Gupta, A. sen;

Gibbs, A. S.; Mibu, K.; Murai, S.; Rondinelli, J. M.; Gopalan, V.; Tanaka, K.; Fujita, K.

Hybrid Improper Ferroelectricity in (Sr,Ca)3Sn2O7 and Beyond: Universal Relationship

between Ferroelectric Transition Temperature and Tolerance Factor in n = 2 Ruddlesden–

Popper Phases. Journal of the American Chemical Society 2018, 140 (46), 15690–15700.

https://doi.org/10.1021/JACS.8B07998.

(13) T. Fukushima; A. Stroppa; S. Picozzi; M. Perez-Mato, J. Large Ferroelectric Polarization

in the New Double Perovskite NaLaMnWO 6 Induced by Non-Polar Instabilities. Physical

Chemistry Chemical Physics 2011, 13 (26), 12186–12190.

https://doi.org/10.1039/C1CP20626E.

(14) Orlandi, F.; Lanza, A.; Cabassi, R.; Khalyavin, D. D.; Manuel, P.; Solzi, M.; Gemmi, M.;

Righi, L. Extended “Orbital Molecules” and Magnetic Phase Separation in

Bi0.68Ca0.32MnO3. Physical Review B 2021, 103 (10).

https://doi.org/10.1103/PhysRevB.103.104105.

(15) Benedek, N. A.; Fennie, C. J. Hybrid Improper Ferroelectricity: A Mechanism for

Controllable Polarization-Magnetization Coupling. Physical Review Letters 2011, 106 (10),

107204. https://doi.org/10.1103/PhysRevLett.106.107204.

(16) Hayashida, T.; Kimura, K.; Urushihara, D.; Asaka, T.; Kimura, T. Observation of

Ferrochiral Transition Induced by an Antiferroaxial Ordering of Antipolar Structural Units

in Ba(TiO)Cu4(PO4)4. Journal of the American Chemical Society 2021, 143 (9), 3638–

3646. https://doi.org/10.1021/jacs.1c00391.

(17) Zuo, P.; Colin, C. v.; Klein, H.; Bordet, P.; Suard, E.; Elkaim, E.; Darie, C. Structural Study

of a Doubly Ordered Perovskite Family NaLnCoWO6 (Ln = Y, La, Pr, Nd, Sm, Eu, Gd, Tb,

Dy, Ho, Er, Yb): Hybrid Improper Ferroelectricity in Nine New Members. Inorganic

Chemistry 2017, 56 (14), 8478–8489. https://doi.org/10.1021/acs.inorgchem.7b01218.

20

(18) Perez-Mato, J. M.; Aroyo, M.; García, A.; Blaha, P.; Schwarz, K.; Schweifer, J.; Parlinski,

K. Competing Structural Instabilities in the Ferroelectric Aurivillius Compound

SrBi2Ta2O9. Physical Review B - Condensed Matter and Materials Physics 2004, 70 (21),

1–14. https://doi.org/10.1103/PHYSREVB.70.214111/FIGURES/3/THUMBNAIL.

(19) Sun, S.; Huang, Y.; Wang, G.; Wang, J.; Peng, R.; Fu, Z.; Zhai, X.; Mao, X.; Chen, X.; Lu,

Y. Room-Temperature Multiferroic Responses Arising from 1D Phase Modulation in

Correlated Aurivillius-Type Layer Structures. Journal of Physics D: Applied Physics 2016,

49 (12), 125005. https://doi.org/10.1088/0022-3727/49/12/125005.

(20) Shankar P N, R.; Orlandi, F.; Manuel, P.; Zhang, W.; Halasyamani, P. S.; Sundaresan, A.

A-Site and B-Site Cation Ordering Induces Polar and Multiferroic Behavior in the

Perovskite NaLnNiWO6(Ln = Y, Dy, Ho, and Yb). Chemistry of Materials 2020, 32 (13),

5641–5649. https://doi.org/10.1021/acs.chemmater.0c01149.

(21) Shankar P. N.,Orlandi, F.; Manuel, P.; Zhang, W.; Halaysyamani, P. S.; Sundaresan, A.

Structural, Magnetic, and Electrical Properties of Doubly Ordered Perovskites

NaLnNiWO6(Ln = La, Pr, Nd, Sm, Eu, Gd, and Tb). Journal of Physical Chemistry C 2021,

125 (12), 6749–6757. https://doi.org/10.1021/acs.jpcc.0c10340.

(22) Young, J.; Stroppa, A.; Picozzi, S.; Rondinelli, J. M. Tuning the Ferroelectric Polarization

in AA′MnWO6double Perovskites through A Cation Substitution. Dalton Transactions

2015, 44 (23), 10644–10653. https://doi.org/10.1039/c4dt03521f.

(23) García-Martín, S.; King, G.; Nénert, G.; Ritter, C.; Woodward, P. M. The

Incommensurately Modulated Structures of the Perovskites NaCeMnWO6 and

NaPrMnWO 6. Inorganic Chemistry 2012, 51 (7), 4007–4014.

https://doi.org/10.1021/ic202071n.

(24) Zuo, P.; Darie, C.; Colin, C. v.; Klein, H. Investigation of the Structure of the Modulated

Doubly Ordered Perovskite NaLaCoWO6 and Its Reversible Phase Transition with a

Colossal Temperature Hysteresis. Inorganic Chemistry 2019, 58 (1), 81–92.

https://doi.org/10.1021/acs.inorgchem.8b01129.

(25) López, M. L.; Veiga, M. L.; Pico, C. Cation Ordering in Distorted Perovskites (MLa)

(MgTe)O6, M = Na, K. Journal of Materials Chemistry 1994, 4 (4), 547–550.

https://doi.org/10.1039/JM9940400547.

(26) Angeles Arillo, M.; Gómez, J.; López, M. L.; Pico, C.; Veiga, M. L. Structural

Characterization and Properties of the Perovskite (NaLa)(MW)O6 (M = Co, Ni): Two New

Members in the Group-Subgroup Relations for the Perovskite-Type Structures. Journal of

Materials Chemistry 1997, 7 (5), 801–806. https://doi.org/10.1039/a605206a.

(27) Borchani, S. M.; Koubaa, W. C. R.; Megdiche, M. Structural, Magnetic and Electrical

Properties of a New Double-Perovskite LaNaMnMoO6 Material. Royal Society Open

Science 2017, 4 (11). https://doi.org/10.1098/rsos.170920.

21

(28) De, C.; Sundaresan, A. Nonswitchable Polarization and Magnetoelectric Coupling in the

High-Pressure Synthesized Doubly Ordered Perovskites NaYMnW O6 and NaHoCoW O6.

Physical Review B 2018, 97 (21), 214418. https://doi.org/10.1103/PhysRevB.97.214418.

(29) King, G.; Garcia-Martin, S.; Woodward, P. M. Octahedral Tilt Twinning and Compositional

Modulation in NaLaMgWO6. Acta Crystallographica Section B: Structural Science 2009,

65 (6), 676–683. https://doi.org/10.1107/S0108768109032728.

(30) García-Martín, S.; Urones-Garrote, E.; Knapp, M. C.; King, G.; Woodward, P. M.

Transmission Electron Microscopy Studies of NaLaMgWO6: Spontaneous Formation of

Compositionally Modulated Stripes. Journal of the American Chemical Society 2008, 130

(45), 15028–15037. https://doi.org/10.1021/JA802511D.

(31) Garcia-Martin, S.; King, G.; Urones-Garrote, E.; Nénert, G.; Woodward, P. M. Spontaneous

Superlattice Formation in the Doubly Ordered Perovskite KLaMnWO6. Chemistry of

Materials 2010, 23 (2), 163–170. https://doi.org/10.1021/CM102592P.

(32) Gemmi, M.; Mugnaioli, E.; Gorelik, T. E.; Kolb, U.; Palatinus, L.; Boullay, P.; Hovmöller,

S.; Abrahams, J. P. 3D Electron Diffraction: The Nanocrystallography Revolution. ACS

Central Science 2019, 5 (8), 1315–1329. https://doi.org/10.1021/ACSCENTSCI.9B00394.

(33) Lanza, A. E.; Gemmi, M.; Bindi, L.; Mugnaioli, E.; Paar, W. H. Daliranite, PbHgAs 2 S 5:

Determination of the Incommensurately Modulated Structure and Revision of the Chemical

Formula. Acta Crystallographica Section B: Structural Science, Crystal Engineering and

Materials 2019, 75, 711–716. https://doi.org/10.1107/S2052520619007340.

(34) Steciuk G.; Palatinus L.; Rohlicek J.; Ouhenia S.; Chateigner Stacking Sequence Variations

in Vaterite Resolved by Precession Electron Diffraction Tomography Using a Unified

Superspace Model. Scientific reports 2019, 9 (1). https://doi.org/10.1038/S41598-019-

45581-6.

(35) Steciuk, G.; Škoda, R.; Rohlíček, J.; Plášil, J. Crystal Structure of the Uranyl–Molybdate

Mineral Calcurmolite Ca[(Uo2)3(MoO4)2(OH)4](H2O)~5.0: Insights from a Precession

Electron-Diffraction Tomography Study. Journal of Geosciences (Czech Republic) 2020,

65 (1), 15–25. https://doi.org/10.3190/JGEOSCI.297.

(36) Gemmi, M.; Oleynikov, P. Scanning Reciprocal Space for Solving Unknown Structures:

Energy Filtered Diffraction Tomography and Rotation Diffraction Tomography Methods.

Zeitschrift fur Kristallographie 2013, 228 (1), 51–58.

https://doi.org/10.1524/zkri.2013.1559.

(37) Mugnaioli, E.; Gorelik, T.; Kolb, U. “Ab Initio” Structure Solution from Electron

Diffraction Data Obtained by a Combination of Automated Diffraction Tomography and

Precession Technique. Ultramicroscopy 2009, 109 (6), 758–765.

https://doi.org/10.1016/j.ultramic.2009.01.011.

(38) Vincent, R.; Midgley, P. A. Double Conical Beam-Rocking System for Measurement of

Integrated Electron Diffraction Intensities. Ultramicroscopy 1994, 53 (3), 271–282.

https://doi.org/10.1016/0304-3991(94)90039-6.

22

(39) Nederlof, I.; van Genderen, E.; Li, Y. W.; Abrahams, J. P. A Medipix Quantum Area

Detector Allows Rotation Electron Diffraction Data Collection from Submicrometre Three-

Dimensional Protein Crystals. Acta Crystallographica Section D: Biological

Crystallography 2013, 69 (7), 1223–1230. https://doi.org/10.1107/S0907444913009700.

(40) Palatinus, L.; Brázda, P.; Jelínek, M.; Hrdá, J.; Steciuk, G.; Klementová, M. Specifics of

the Data Processing of Precession Electron Diffraction Tomography Data and Their

Implementation in the Program PETS2.0. Acta Crystallographica Section B: Structural

Science, Crystal Engineering and Materials 2019, 75, 512–522.

https://doi.org/10.1107/S2052520619007534.

(41) Chapon, L. C.; Manuel, P.; Radaelli, P. G.; Benson, C.; Perrott, L.; Ansell, S.; Rhodes, N.

J.; Raspino, D.; Duxbury, D.; Spill, E.; Norris, J. Wish: The New Powder and Single Crystal

Magnetic Diffractometer on the Second Target Station. 2011, 22 (2), 22–25.

https://doi.org/10.1080/10448632.2011.569650.

(42) Petříček, V.; Dušek, M.; Palatinus, L. Crystallographic Computing System JANA2006:

General Features. Zeitschrift für Kristallographie - Crystalline Materials 2014, 229 (5),

345–352. https://doi.org/10.1515/ZKRI-2014-1737.

(43) Campbell, B. J.; Stokes, H. T.; Tanner, D. E.; Hatch, D. M.; IUCr. ISODISPLACE: A Web-

Based Tool for Exploring Structural Distortions. urn:issn:0021-8898 2006, 39 (4), 607–

614. https://doi.org/10.1107/S0021889806014075.

(44) Hatch, D. M.; Stokes, H. T.; IUCr. INVARIANTS: Program for Obtaining a List of

Invariant Polynomials of the Order-Parameter Components Associated with Irreducible

Representations of a Space Group. urn:issn:0021-8898 2003, 36 (3), 951–952.

https://doi.org/10.1107/S0021889803005946.

(45) Momma, K.; Izumi, F.; IUCr. VESTA 3 for Three-Dimensional Visualization of Crystal,

Volumetric and Morphology Data. urn:issn:0021-8898 2011, 44 (6), 1272–1276.

https://doi.org/10.1107/S0021889811038970.

(46) Palatinus, L. The Charge-Flipping Algorithm in Crystallography. urn:issn:2052-5192 2013,

69 (1), 1–16. https://doi.org/10.1107/S2052519212051366.

(47) Licurse, M. TEM Studies of Modulated Mixed A-Site Perovskites. Publicly Accessible

Penn Dissertations 2013.

(48) RF, E. Radiation Damage to Organic and Inorganic Specimens in the TEM. Micron (Oxford,

England : 1993) 2019, 119, 72–87. https://doi.org/10.1016/J.MICRON.2019.01.005.

(49) Jiang, N.; Su, D.; Spence, J. C. H.; Zhou, S.; Qiu, J. Electron Energy Loss Spectroscopy of

Na in Na, Na2O, and Silicate Glasses. Journal of Materials Research 2008, 23 (9), 2467–

2471. https://doi.org/10.1557/JMR.2008.0296.

(50) Abakumov, A. M.; Erni, R.; Tsirlin, A. A.; Rossell, M. D.; Batuk, D.; Nénert, G.; Tendeloo,

G. van. Frustrated Octahedral Tilting Distortion in the Incommensurately Modulated

23

Li3xNd2/3–XTiO3 Perovskites. Chemistry of Materials 2013, 25 (13), 2670–2683.

https://doi.org/10.1021/CM4012052.

(51) Bak, P.; Boehm, J. von. Ising Model with Solitons, Phasons, and “the Devil’s Staircase.”

Physical Review B 1980, 21 (11), 5297. https://doi.org/10.1103/PhysRevB.21.5297.

(52) Rossat-Mignod, J.; Burlet, P.; Villain, J.; Bartholin, H.; Tcheng-Si, W.; Florence, D.; Vogt,

O. Phase Diagram and Magnetic Structures of CeSb. Physical Review B 1977, 16 (1), 440.

https://doi.org/10.1103/PhysRevB.16.440.

(53) Bak, P. Commensurate Phases, Incommensurate Phases and the Devil’s Staircase. Reports

on Progress in Physics 1982, 45 (6), 587. https://doi.org/10.1088/0034-4885/45/6/001.

SYNOPSIS.