www.sciencemag.org/content/347/6220/420/suppl/DC1

Supplementary Materials for

Tilt engineering of spontaneous polarization and magnetization above

300 K in a bulk layered perovskite

Michael J. Pitcher, Pranab Mandal, Matthew S. Dyer, Jonathan Alaria, Pavel Borisov,

Hongjun Niu, John B. Claridge,* Matthew J. Rosseinsky*

*Corresponding author. E-mail: m.j.rosseinsky@liv.ac.uk (M.J.R.); j.b.claridge@liv.ac.uk (J.B.C.)

Published 23 January 2015, Science 347, 420 (2015)

DOI: 10.1126/science.1262118

This PDF file includes:

Materials and Methods

Supplementary Text

Figs. S1 to S30

Tables S1 to S4

Full Reference List

Materials and Methods

Computational Details

Periodic, plane-wave based density functional theory (DFT) calculations were

performed using the VASP code (30). Core-electrons were modelled using the projector-

augmented wave (PAW) method (31), and the PBE functional (32) was used throughout. To

correct for the known deficiency of PBE to described localised d-electrons, an effective on-

site Hubbard U parameter of 4 eV was applied to the Fe 3d orbitals using the rotationally

invariant scheme of Dudarev et al. (33). A plane-wave cutoff energy of 500 eV was used,

with reciprocal space sampled using an 8×8×2 k-point grid. The unit cell and atomic positions

were relaxed until all atomic forces were below 0.005 eV/Å.

The published crystal structures of BaLa2Fe2O7 (I4/mmm), SrTb2Fe2O7 (P42/mnm),

LaCa2Mn2O7 (Amam) and Ca3Mn2O7 (A21am) were used as initial starting structures for DFT

calculations, with the alkali earth ions on the twelve-fold coordinate perovskite A site and

lanthanide ions on nine-fold coordinate rock-salt A sites. Collinear G-type antiferromagnetic

ordering was applied to all systems. Calculations of the compound CaTb2Fe2O7 in the

tetragonal space-group P42/mnm underwent a slight orthorhombic distortion into the Pnnm

space-group, but preserved the initial octahedral tilting patterns. The final relaxed structures

of CaTb2Fe2O7 with 0-3 octahedral tilts are available as cif files on the ICSD (Table S4).

Bond valence sums were calculated on the basis of parameters published by Brese and

O'Keeffe (34) with a distance cutoff of 3.5 Å.

Synthesis:

Synthesis of Sr1.1Tb1.9Fe2O7

The first attempts to synthesise SrTb2Fe2O7 followed a literature procedure (23). Pre-

dried powders of SrCO3 (Alfa Aesar, 99.95 %, dried at 473 K for 16 hours), Tb4O7 (Alfa

Aesar 99.998 %, dried at 1223 K for 16 hours) and Fe2O3 (Alfa Aesar 99.998 %, dried at 473

K for 16 hours) were hand-mixed and the powder fired in air at 1273 K for 12 hours in an

alumina crucible. The resulting black powder was re-ground and pelletised using a uniaxial

press to apply approximately 40 MPa, then loaded into a platinum-lined alumina crucible and

fired at 1673 K for 12 hours. PXRD showed the resulting material to contain the tetragonal n

= 2 Ruddlesden-Popper (RP2) as the majority phase with a small amount of orthorhombic

perovskite as a secondary phase. Attempts to eliminate the perovskite phase by tuning the

synthetic conditions (varying firing temperature, mixing by ball milling and citrate gel

methods) were unsuccessful in obtaining the RP2 as a single phase. The composition

Sr1+zTb2-zFe2O7 was then varied systematically between -0.15 < z < 0.15 and the synthesis

repeated under the initial literature reaction conditions (in air with hand-mixing) and also

under N2 and O2 gas flows. The value of z was found to have a strong influence on the

resulting phase mixture, while variation in atmospheric conditions had little effect. The RP2

phase was isolated as the sole product for z = 0.10. Oxygen content on Sr1.1Tb1.9Fe2O7 (STF)

was determined using iodometric titration and found to be consistent with the nominal

composition Sr1.10Tb1.90Fe2O6.97(2).

Synthesis of SrLn2Fe2O7 compounds with smaller lanthanides (Ln = Dy3+

– Lu3+

and

Y3+

) was attempted under similar high-temperature conditions. Only SrDy2Fe2O7 was found

to form an n = 2 RP phase with P42/mnm symmetry, and all other lanthanides yielded a

mixture of n = 1 RP phase and n = ∞ perovskite. Due to strong neutron absorption by Dy,

further study on this system was discontinued in favour of the Tb-based system.

Synthesis of (Sr1-yCay)1.15Tb1.85Fe2O7

The Ca-substituted series (Sr1-yCay)1+zTb2-zFe2O7 was synthesised using the same

protocol as that for Sr1.1Tb1.9Fe2O7, with hand-mixed powders in an alumina crucible fired for

12 hours at 1273 K in air, then thoroughly re-ground and pelletised and re-fired at 1673 K for

12 hours in a platinum-lined alumina crucible in air. Pre-dried CaCO3 (Alfa Aesar, 99.95 %,

dried at 473 K for 16 hours) was used as the Ca source. This protocol was adapted for the

preparation of large (5 g) samples for neutron powder diffraction (NPD) experiments by ball-

milling the reaction mixture in isopropanol for 1 hour instead of hand-mixing to ensure

thorough homogenisation, with samples re-ground by hand between firings. The value of z

was optimised to avoid the formation of n = 1 RP and orthoferrite impurities and found to

lead to single-phase materials at z = 0.15 in air. Attempts were made to synthesise (Sr1-

yCay)1.15Tb1.85Fe2O7 (CSTF) across the full range 0 ≤ y ≤ 1.0. Laboratory PXRD showed that

the series exhibits three distinct regions that are illustrated in Fig. S3: (i) a single-phase

tetragonal region which retains the parent structure of STF at compositions 0 ≤ y < 0.55; (ii) a

single-phase orthorhombic region at compositions 0.55 ≤ y < 0.65, with systematic absences

consistent with Amam (two-tilt) or A21am (three-tilt) symmetry; and (iii) a multi-phase region

consisting of n = 1, 2 RP phases and perovskite (n = ∞). Attempts to synthesise the end

member CaTb2Fe2O7 by this high temperature route yielded a stoichiometric mixture of n = 1

RP and perovskite phases with no evidence for the formation of the n = 2 RP.

Synthesis of [1-x](CaySr1-y)1.15Tb1.85Fe2O7 - [x]Ca3Ti2O7, y = 0.563 and 0.60.

Two series of compounds [1-x](CaySr1-y)1.15Tb1.85Fe2O7-[x]Ca3Ti2O7 (y = 0.60 and

0.563) were synthesised via the same mixing and heating protocol as for (CaySr1-

y)1.15Tb1.85Fe2O7: an initial firing in air at 1273 K for 12 hours was followed by re-grinding

and firing at 1673 K in air in Pt-lined alumina crucibles for a further 12 hours. Pre-dried TiO2

(Alfa Aesar, 99.995%, dried at 473 K for 16 hours) was used as the Ti source. The protocol

for the synthesis of large (5 g) samples for NPD experiments was adapted by using ball-

milling for the initial mixing step, as described above for (CaySr1-y)1.15Tb1.85Fe2O7. A range of

compositions between 0.0 < x ≤ 0.30 with y = 0.60 were attempted and all were found to

form stable n = 2 RP phases. This series was used to map the phase diagram illustrated in Fig.

4A. Materials with x > 0.30 were also synthesised but were not studied further due to their

low Néel temperatures (TN ≤ 200 K).

Single-phase dense pellets of the series y = 0.563, suitable for ME measurements, were

synthesised by ball-milling the starting reagents for 1 hour before an initial firing at 1273 K

for 12 hours, before a thorough re-grinding by hand and pelletising in a uniaxial press,

followed by further densification at ~200 MPa in a cold isostatic press, then a final firing at

1673 K for 12 hours. Pellet densities were determined using an Archimedes balance and

found to be ~97 % of the crystallographically determined density values.

Diffraction

Laboratory PXRD patterns for basic sample characterisation were collected on a

PANalytical X’Pert Pro diffractometer in Bragg-Brentano geometry using monochromated

Co Kα1 radiation (λ = 1.78896 Å). Synchrotron XRD patterns (SXRD) were collected from

the I11 powder diffractometer at Diamond, UK (35), with samples loaded in 0.2 mm

borosilicate capillaries and an incident wavelength λ = 0.827127(1) Å over a 2θ range 2 –

150°. A variable temperature scan between 90 – 350 K was performed on x = 0.075, y = 0.60

using a N2 cryostream with a position sensitive detector for rapid data collection at lower

resolution.

Time-of-flight (ToF) NPD patterns from the series 0.0 ≤ x ≤ 0.30, y = 0.60 were

collected on either the GEM or Polaris high-intensity/medium-resolution diffractometers at

the ISIS facility, Rutherford Appleton Laboratory (UK). Refinements against GEM data were

conducted using 5 fixed angle detector banks at 2θ = 19° (0.9 – 14 Å), 35° (0.5 – 7.7 Å), 64° (0.3 – 4.0 Å), 91° (0.2 – 2.7 Å) and 154° (0.2 – 1.8 Å); and refinements against Polaris data

were conducted using 4 fixed angle detector banks at 2θ = 26° (0.2 – 13.5 Å), 52° (0.1 – 7.0

Å), 93° (0.1 – 4.1 Å), and 147° (0.1 – 2.6 Å). ToF NPD patterns from Sr1.1Tb1.9Fe2O7,

Ca0.69Sr0.46Tb1.85Fe2O7 and [1-x]CSTF-[x]CTO (y = 0.563) were collected on the high-

resolution HRPD diffractometer using three fixed-angle detector banks at 2θ = 168° (0.65 –

2.6 Å), 90° (0.85 – 4.0 Å) and 30° (2.6 – 10.0 Å). Rietveld refinements were carried out using

Topas Academic (Version 5).

Magnetic measurements

Magnetic measurements were carried out on powder samples using a commercial

superconducting quantum interference device (SQUID) magnetometer MPMS XL – 7

(Quantum Design, USA). Magnetization vs. temperature data were recorded in the following

modes: ZFC (zero-field cooling, measured while warming up after cooling in a zero field),

FC (field cooling, measured while warming up after cooling under a magnetic field) and

TRM (thermoremanent magnetization, measured while warming up in the zero field after

cooling down in magnetic field).

For each composition the onset temperature of weak ferromagnetism was obtained by

two methods: (i) by fitting the peak in ∂M/∂T where M is thermoremanent magnetization.

The peak in ∂M/∂Tmay represent the Curie temperature (TC) or Néel temperature (TN) in a

ferromagnet or antiferromagnet respectively;∂M/∂T is proportional to specific heat which

shows a maximum at the Curie temperature (TC) (36). Transition temperatures measured in

this way are labeled as Tmax. (ii) From the temperature at which FC and ZFC M(T) data

diverge; transition temperatures measured in this way are labeled Twfm.

The saturation magnetizations (Msat) were extracted by simulating the raw isothermal

magnetization as function of magnetic field with the following empirical expression (37):

hdc

bhahm

tanh)(

where a is the saturation magnetization, b the coercive field, c an effective field that

impedes saturation and d any linear contribution to the magnetisation (antiferromagnetic,

paramagnetic and diamagnetic).

Magnetoelectric (ME) measurements

For ME measurements, dense pellets were cut to dimensions of ~ 3.2 × 3.2 × 0.6 mm3

and polished up to 5 micron finish using SiC paper in a semi-automatic polishing machine.

Ohmic contacts were made via sputtering Pt. Magnetoelectric measurements were carried out

on a modified SQUID magnetometer as described in (38). Prior to the measurements the

samples were poled in the following sequence: Slowly cooled (1 - 2 K/min) from 350 K to

130 K in 20 kOe magnetic field and electric zero-field (short circuit). It was not possible to

apply electric field above a temperature of 130 K because of high leakage current. At 130 K,

an electric field of 350 – 400 kV/m was applied while cooling down in the same magnetic

field to the measurement temperature at 1 K/min. After poling, electric and magnetic fields

were switched off and electrodes were short circuited for 15 minutes.

The expansion of the free energy density of a material under Einstein summation is

given in (39):

lkji

ijkl

kji

ijk

kji

ijk

jiijjiijjiij

HHEE

EEHHHEEHHHEEFF

2

222

1

2

1, 000

HE

The magnetization can be obtained as

lkjjklikj

ijk

kjjkijijjiji EEHEEHEEHH

FM

200

In the present experimental set-up, an ac electric field, E = E0 cost ( = 2 f where f

is frequency, E0 is the electric field amplitude) is applied across the sample and first harmonic

of complex ac magnetic moment, m(t) = (mʹ − imʺ)∙cos(ωt) is measured. The real part of the

induced magnetic moment is then

0

0000 2

V

HEEEEHEEm dcdcdcdc

where V is the sample volume. In absence of any dc magnetic and dc electric field

offsets, Hdc = 0 and Edc = 0, respectively, only the first term involving linear ME effect

remains, whereas the higher order ME terms , , and do not contribute to the measured

moment. In order to determine the linear ME susceptibility accurately, the electric field

amplitude E0 was varied and was calculated from a plot of volume magnetization

amplitude M (= m/V) vs E0 following the relation (28):

0

0E

M

.

Measurements were performed at f = 1 Hz.

Resistance Measurements

Two probe resistivity measurements were performed using a Keithley 6430 sub-femto

Amp remote sourcemeter.

Supplementary Text

Sr1.1Tb1.9Fe2O7: Crystal and Magnetic Structure

The room temperature crystal and magnetic structures of Sr1.1Tb1.9Fe2O7 were derived

by Rietveld refinement against high resolution neutron diffraction (HRPD, ISIS) and

synchrotron XRD (I11 with MAC detectors). The published structure of SrTb2Fe2O7 in space

group P42/mnm (23), with a nominal composition of Sr1.1Tb1.9Fe2O7 imposed by fixing the

occupancy of the 9-coordinate rock-salt site to 0.05Sr-0.95Tb, was used as a starting model

along with the published magnetic structure (40) in Shubnikov group P42/mnm. For the

initial refinement against NPD data, three data banks were refined simultaneously with a

Chebyschev background function, sample displacement parameter DIFC (fixed for the

backscattering detector bank, 2θ = 169°), time of flight peak profile parameters, lattice

parameters, isotropic atomic displacement (Biso) parameters and magnitude of magnetic

moment. Atomic coordinates were fixed at the literature values. The resulting calculated

intensities matched well to the observed intensities of the magnetic Bragg peaks, but offered

a poor fit to the observed intensities of the nuclear Bragg peaks (Fig. S1E-H) along with

several non-physical Biso parameters. Subsequent step-wise refinement of the atomic

coordinates resulted in a drastic improvement of the fit (Fig. S1A-D) with all Biso parameters

refining to physically reasonable values: this corresponds to a refinement away from the

unusual distorted FeO6 octahedra of the published structure to a structure with undistorted

FeO6 octahedra that exhibits a single octahedral tilt distortion of type (a-b

0c

0)/(b

0a

-c

0), in the

same space group P42/mnm and consistent with the prediction of DFT calculations (Fig. 1).

The ordering of Sr2+

and Tb3+

cations was confirmed by refining this model against SXRD

data, with the occupancies of each A site refined independently and constrained to be fully

occupied by cations (i.e., no global constraint on composition) to give a refined composition

of Sr1.09(1)Tb1.91(1)Fe2O7 (where errors represent 1 e.s.d. from the refinement). The A site

cations were shown to be almost fully ordered with Sr2+

occupying 96.9(4)% of the 12-

coordinate perovskite sites and Tb3+

occupying 93.8(6)% of the smaller 9-coordinate rock-

salt sites. This cation distribution was applied to the final Rietveld refinement against the

NPD data, where thermal displacement parameters were refined anisotropically for all atoms

to give an overall agreement factor Rwp = 3.21%. The final refined NPD structure is

illustrated in Fig. 2A of the main text. The Rietveld fits to SXRD and NPD (HRPD 169⁰ bank) are shown in Fig. S1.

Non-polar [1-x]CSTF-[x]CTO (0 ≤ x ≤ 0.13, y = 0.563, 0.60): Crystal and Magnetic

Structures

Laboratory PXRD patterns from all materials in this compositional range were indexed

to orthorhombic unit cells with systematic absences consistent with space groups Amam and

A21am. The space group Amam is expected to correspond to tilt scheme (a-a

-c

0)/(a

-a

-c

0) or its

disordered variant (a-a

-c

<0>)/(a

-a

-c

<0>), and A21am is expected to correspond to tilt scheme (a

-

a-c

+)/(a

-a

-c

+) (12). NPD was used as the primary tool to distinguish between the candidate

structures, due to its superior sensitivity to oxide displacements, and also to determine the

magnetic structure. SXRD was used in a complementary role to confirm the ordering of Sr2+

,

Ca2+

and Tb3+

cations over the two available A sites (the 12 coordinate perovskite site and the

9 coordinate rock-salt site).

Magnetic structure: Inspection of the room temperature NPD patterns for each

composition revealed the presence of strong magnetic Bragg peaks in the region d > 2.3 Å

(Fig. S6), which were indexed to a magnetic cell of identical dimensions to the nuclear cell.

The magnetic structure solution was addressed prior to a full investigation of the nuclear

structure, due to potential overlap between nuclear and magnetic peaks at smaller d-spacings.

This was attempted initially for the composition (x = 0, y = 0.60), with data from the low-

angle (2θ = 35°) detector bank of HRPD, using the distortion mode approach described by

Kerman et. al. (41): the 24 allowed magnetic symmetry-modes were generated from a parent

Amam cell using ISODISTORT (43), and then tested by simulated annealing in an equivalent

orthorhombic cell in P1 symmetry. A single active magnetic symmetry-mode was identified,

which fitted the observed intensities of all of the magnetic reflections, corresponding to G-

type antiferromagnetism within the perovskite blocks with spins aligned parallel to the b-axis

(Fig. 3A) in Shubnikov group PBnnm (propagation vector k = 0,0,1). This magnetic model

was found to fit the observed magnetic peaks for all compositions 0 ≤ x ≤ 0.13, y = 0.563,

0.60 at room temperature and was applied to subsequent refinements of the nuclear

structures.

a-a

- tilt disorder: Solution of the room temperature nuclear structure was performed

initially on the composition (x = 0, y = 0.60), which revealed disorder in the magnitudes of

the a-a

- tilts as follows. Two simple models were compared by NPD: Amam with tilt system

(a-a

-c

0)/(a

-a

-c

0) based upon the published structure of LaCa2Mn2O7 (42), and A21am with tilt

system (a-a

-c

+)/(a

-a

-c

+) based upon the published structure of Ca3Mn2O7 (17). Cation ordering

was imposed following that observed in the tetragonal Al3+

compounds Sr1-xCaxLn2Al2O7

(Ln = Nd or Gd) (43, 44), with Sr occupying the 12 coordinate perovskite site and Ca, Tb

disordered over the remaining sites. Three HRPD detector banks were used simultaneously

with refined Chebyschev background functions, time of flight peak profile parameters,

atomic coordinates, isotropic atomic displacement (Biso) parameters and magnitude of

magnetic moment. Lattice parameters were fixed, as were sample displacement parameters

for each detector bank. The initial Rwp values favoured the A21am structure (6.85% for A21am

vs 8.33% for Amam), however inspection of the refined structure revealed that the FeO6

octahedra were distorted by displacement of O1 and O2 along the a-axis (fig. S5), which is

forbidden in Amam by their location on a mirror plane (Wyckoff sites 4c and 8g

respectively). By splitting these oxide sites parallel to the a-axis in both space groups, the Rwp

values were improved to Rwp(A21am) = 6.41% and Rwp(Amam) = 6.50%, with both structures

exhibiting (average) regular FeO6 octahedra. The splitting of these apical oxide sites implies

that there is disorder in the magnitude of the octahedral tilting about the a and b axes in this

compound, which (if octahedra are assumed to be rigid) would also be expected to affect the

equatorial oxide sites. The equatorial oxides (O3 and O4, fig. S5) were therefore tested by

splitting them parallel to the a, b or c axes; in both models significant splitting with reduction

in the Rwp values to Rwp(A21am) = 5.72% and Rwp(Amam) = 5.91% was observed parallel to

the c-axis. This pattern of disorder, with apical oxides disordered parallel to a and equatorial

oxides disordered parallel to c, is consistent with disorder in the magnitudes of the a- tilts due

to the proximity of the (a-b

0c

0)/(b

0a

-c

0) tilt structure of the parent phase (Sr1.1Tb1.9Fe2O7):

consequently this type of disorder diminishes as the Ca content increases (see fig. S5C). It

should be noted that this splitting of the equatorial oxide sites is orthogonal to the

displacement caused by the X2+ distortion mode (c

+ tilt), and is therefore not expected to

interfere with the occurrence or detection of this distortion. These split oxide sites were

applied to each of the models discussed below.

c+ tilt disorder: NPD refinements were used to compare three candidate structures, with

oxide sites split as described above, to look specifically for the presence of a c+ tilt: (i) Amam

with zero c+ tilt (a

-a

-c

0)/(a

-a

-c

0) (Fig. S6A); (ii) Amam with disordered c

+ tilt (a

-a

-c

<0>)/(a

-a

-

c<0>

) where c<0>

denotes a disordered c+ tilt (Fig. S6B); and (iii) A21am with ordered c

+ tilt (a

-

a-c

+)/(a

-a

-c

+) (Fig. S6C). For these refinements, three HRPD data banks were used

simultaneously with refined Chebyschev background functions, time of flight peak profile

parameters, atomic coordinates and magnitude of magnetic moment. Isotropic temperature

factors Biso were applied, with the oxide ions constrained to a single Bisovalue. Lattice

parameters and sample displacement parameters for each detector bank were fixed. Model (i)

yielded the poorest fit with Rwp = 5.20%, whilst model (ii) yielded the best fit with Rwp =

4.93%. The Rietveld fits corresponding to model (ii) are shown in Fig. S7. Visual inspection

of refined model (iii), with Rwp = 5.077, suggested that the refined ordered c+ tilt is close to

zero, and this was confirmed by using ISODISTORT to derive from the refined structure a

parent-cell normalised mode amplitude (Ap) of X2+ = 0.078 Å, which is considerably smaller

than (Ap) X2+

= 0.513 Å exhibited by the (a-a

-c

+)/(a

-a

-c

+) compound Ca3Mn2O7 (17). Hence,

by refining a disordered c+ tilt (a

-a

-c

<0>)/(a

-a

-c

<0>) the non-polar space group Amam (Fig. S6B)

gives a superior fit to high-resolution NPD data than either the zero-c+-tilt Amam model (Fig.

S6A) or the ordered c+ tilt model in its polar subgroup A21am (Fig. S6C). The final refined

structure of this compound (x = 0, y = 0.60) with disordered c+ tilt yielded a parent-cell

normalized mode amplitude |X2+| = 0.223 Å, which is approximately half that of the ordered

c+ tilt in Ca3Mn2O7.

Finally joint XRD-NPD refinements of models (ii) and (iii) for (x = 0, y = 0.60) were

conducted to confirm the A-site cation ordering. SXRD data (I11 MAC) were used

simultaneously with 3 data banks from HRPD, with a Chebyschev background function, axial

divergence correction, pseudo-Voigt profile function and an anisotropic peak broadening

function refined, in addition to the neutron-specific parameters outlined above. The overall

composition was constrained to the nominal value, with Sr, Ca and Tb allowed to populate

both A-sites. This resulted in Sr occupancy of the 9 coordinate site refining to zero, where it

was subsequently fixed and the refinement re-run with a constraint to keep both sites fully

occupied by cations. The refined cation occupancies for both models are shown in Table S1,

which show that Sr occupies the 12 coordinate site only, while there is some degree of

disorder between Ca and Tb over the remaining sites, with Tb predominantly occupying the 9

coordinate site. These joint refinements confirm the superiority of model (ii) (Amam,

disordered c+ tilt (a

-a

-c

<0>)/(a

-a

-c

<0>)) over model (iii) (A21am, ordered c

+ tilt) with

Rwp(Amam)/Rwp(A21am) = 0.970.

In summary, the structures in this compositional region (0 ≤ x ≤ 0.13, y = 0.563, 0.60)

exhibit the (a-a

-c

<0>)/(a

-a

-c

<0>) tilt system with disorder in the a

-a

- tilts (due to proximity to the

phase boundary with the P42/mnm structure) and disorder in the in-phase c+ tilt, resulting in a

non-polar Amam average structure. For ease of comparison with Sr1.1Tb1.9Fe2O7, it is also

possible to model this non-polar structure with anisotropic displacement parameters (ADPs,

see Fig. 2A): the corresponding crystallographic information (cif) file is available for the

composition x = 0.10, y = 0.60 (Table S4).

Polar [1-x]CSTF-[x]CTO (x > 0.13, y = 0.563, 0.60): Crystal and Magnetic Structures

Compositions in this region were found to exhibit very similar laboratory PXRD

patterns to the non-polar compositions described above, with patterns indexed to

orthorhombic unit cells and systematic absences consistent with space groups Amam or

A21am. A similar refinement protocol was therefore followed, as described below.

Magnetic structure: several intense Bragg peaks at d > 2.3 Å observed in room

temperature NPD patterns of 0.13 < x < 0.20 were indexed to a commensurate magnetic

structure (with unit cell dimensions identical to that of the nuclear cell), but showed different

relative intensities to those observed for compositions 0 ≤ x ≤ 0.13 (Fig. 3C) implying that

the magnetic structures are different in the two compositional ranges. The same magnetic

Bragg peaks were found in the compositional range 0.20 ≤ x ≤ 0.30 on cooling. The

composition x = 0.30, y = 0.60 (TN = 200 K, fig. S15 and S19) at a temperature of 100 K was

selected for the magnetic structure solution, as this combination of composition/temperature

is far away from the phase transition (at x ≈ 0.13) and offers suitably intense magnetic Bragg

reflections. A single low-angle (2θ = 35°) GEM detector bank was used for magnetic

symmetry-mode analysis, using the same protocol described for 0.0 ≤ x ≤ 0.13 (above). A

single magnetic symmetry-mode, corresponding to Shubnikov group A21am (propagation

vector k = 0,0,0), was found to fit all of the magnetic peaks. This structure differs from that

observed in compositions 0.0 ≤ x ≤ 0.13 as the spins are rotated by 180° between the

perovskite blocks (Fig. 3A and Fig. S11); the individual perovskite blocks still exhibit G-type

antiferromagnetism with spins aligned parallel to the b-axis, but this spin reorientation

reduces the magnetic point symmetry from mmm1 to 2mm to allow both spin canting

parallel to the c-axis and a linear magnetoelectric coupling (28). The room temperature

magnetic Bragg peaks from compositions 0.13 ≤ x ≤ 0.19 were fitted to a mixture of the

PBnnm and A21am structures, with their relative phase fractions assessed by refinement of

the staggered Fe moment in each, as shown in Fig. 3B. In all subsequent refinements, the

magnitude of the ordered moment μ was refined along the b axis only (i.e. the magnitude of

the permitted spin canting parallel to the a axis was not refined).

Nuclear structure: The composition x = 0.30, y = 0.60 (TN = 200 K) was selected for

the initial structural refinement by room temperature NPD, as this represents the most Ca/Ti

rich material studied. Two structural models were compared: Amam with disordered c+ tilt

(model (ii) discussed above, Fig. S6B) and A21am with ordered c+ tilt (model (iii) discussed

above, Fig. S6C). Note that both models also featured split oxide sites to allow for potential a-

a- tilt disorder, as exhibited by compositions 0 ≤ x ≤ 0.13. Five GEM detector banks were

used simultaneously, with refined Chebyschev background functions, sample displacement

parameters DIFC (fixed for the backscattering 154° bank), time of flight peak profile

parameters, atomic coordinates and isotropic temperature factors (Biso). Biso for the oxide sites

were described by a single refined parameter. Fixed A-site cation order was modelled with Sr

localized on the 12 coordinate site, with Ca, Tb partially ordered over the remaining 12

coordinate and 9 coordinate sites (as found for compositions 0 ≤ x ≤ 0.13, described above).

Visual inspection of the Rietveld fits resulting from the two models showed a very clear

difference in quality: A21am (a-a

-c

+)/(a

-a

-c

+) yielded an excellent fit to patterns from all 5

detector banks, whilst Amam (a-a

-c

<0>)/(a

-a

-c

<0>) produced several clear mis-fits (fig. S8), with

a direct comparison of the Rwp values Rwp(Amam)/ Rwp(A21am) = 1.47. The resulting refined

A21am structure showed a significant c+ tilt: extraction of the parent-cell normalised X2

+

mode amplitude using ISODISTORT gave a value of Ap (X2+) = 0.402 Å, which is

comparable to that of Ca3Mn2O7.

In order to confirm that the cation ordering in this polar compositional region is

consistent with that exhibited in the non-polar region, a combined SXRD-NPD refinement of

x = 0.20, y = 0.60 at room temperature was conducted in space group A21am. Five GEM

detector banks and an SXRD data set (I11 MAC detector) were used simultaneously with

refined Chebyschev background functions, sample displacement parameters DIFC (for all

GEM banks), time of flight (GEM) and pseudo-Voigt (I11) peak profile functions, a

Stephens’ type anisotropic peak broadening function (I11) and an axial divergence correction

(I11). Atomic coordinates were refined with anisotropic displacement parameters (ADPs). Sr,

Ca and Tb were allowed to populate both the 12 coordinate and 9 coordinate sites with a

constraint on the overall composition. The resulting cation distribution, shown in Table S2, is

consistent with that exhibited by x = 0, y = 0.60 (Table S1) with Sr showing a very strong

preference for the 12 coordinate site, and Tb accommodated mainly on the 9 coordinate site

with Ca occupying the remaining sites.

This method of direct comparison between the Amam (a-a

-c

<0>)/(a

-a

-c

<0>) model and the

A21am (a-a

-c

+)/(a

-a

-c

+) model by NPD was conducted for every composition studied (0 ≤ x ≤

0.30, y = 0.563 and 0.60) and used to deduce that the room temperature composition-

dependent structural phase transition (c+ tilt ordering) occurs at x ≈ 0.13, y = 0.60 (Fig. S4A);

hence the polar structural phase and the WFM, magnetoelectric magnetic phase are shown to

appear simultaneously (Fig. 4B). This was extended to variable-temperature data from a

range of (y = 0.60) compositions (Fig. S16) in order to map the composition-temperature

phase diagram (Fig. 4A).

In summary, the structures in this compositional region (0.14 ≤ x ≤ 0.30, y = 0.563,

0.60) exhibit the (a-a

-c

+)/(a

-a

-c

+) tilt system with disorder in the a

-a

- tilts (which is less

pronounced than that exhibited by the Amam structure, as these compounds are further from

the phase boundary with the P42/mnm structure), but no disorder in the in-phase c+ tilt. This

results in a polar A21am average structure. For ease of comparison with Sr1.1Tb1.9Fe2O7, this

polar structure can also be modelled with ADPs (see Fig. 2A): the corresponding

crystallographic information (cif) file is available for the composition x = 0.20, y = 0.60

(Table S4).

Fig. S1

Rietveld refinements of Sr1.9Tb1.1Fe2O7 at room temperature, modelled in space group

P42/mnm with the (a-b

0c

0)/(b

0a

-c

0) tilt scheme and magnetic structure modelled in Shubnikov

group P42/mnm. (A) SXRD data from I11 MAC detectors with the region 17.0 ≤ 2θ ≤ 17.4°

expanded (inset) to show the fit to the most intense peaks, (B) HRPD 169° bank, (C) HRPD

90° bank, (D) HRPD 30° bank. (E – H) Rietveld fits from the literature model of

“SrTb2Fe2O7” (23) refined against the same data sets, showing comparatively poor fits to the

data. Black markers = Yobs, red line = Ycalc, grey line = (Yobs-Ycalc), blue tick marks = nuclear

phase, green tick marks = magnetic phase.

Fig. S2

Laboratory PXRD patterns (Co Kα1, λ = 1.78896 Å) of Sr1+zTb2-zFe2O7 in the range 35 ≤ 2θ ≤

41°. TbFeO3 peaks are indicated by the † symbol. (A) Nominal composition SrTb2Fe2O7

synthesised by a solid-state reaction following the literature protocol (23) showing TbFeO3

impurity at approximately 5 wt%. (B) Nominal composition SrTb2Fe2O7 synthesised by a sol-

gel reaction, showing the same TbFeO3 impurity. (C) Nominal composition Sr1.1Tb1.9Fe2O7

synthesised by a solid-state reaction, resulting in a single-phase material.

Fig. S3

Plot of a and b parameters (black squares and red circles respectively) as a function of

composition y at room temperature in the series (Sr1-yCay)1.15Tb1.85Fe2O7 showing the

tetragonal-to-orthorhombic transition in this system. The single-phase orthorhombic (Amam)

region 0.55 < y ≤ 0.65 is shaded. Compositions beyond y = 0.65 are multi-phase with n = 2, 1

RP phases and perovskite. At y = 1.0 only the n = 1 RP phase and perovskite are present.

Fig. S4

(A) Ratio of R-factors for refinements of disordered (Amam) and ordered (A21am) c+ tilt

models, and refined net amplitude of the ordered X2+ mode which describes the c

+ tilt (fig.

S11), plotted as a function of composition. (B) Rietveld refinement of the single-phase polar

composition x = 0.20, y = 0.563 against high resolution NPD data (HRPD 169° bank) at

ambient temperature, whose magnetic structure is refined at 60 K in fig. S9.

Fig. S5 Crystallographic models for [1-x]CSTF-[x]CTO with split oxide sites. (A) the composition x

= 0.10, y = 0.60 modelled in Amam with a disordered in-phase c+ tilt: O1 and O2 are split

along the a axis and O3, O4 are split along the c axis due to disorder in the a-a

- tilts. O3 and

O4 are further split orthogonal to the c axis due to a disordered in-phase c+ tilt. (B) the

composition x = 0.20, y = 0.60 modelled in A21am with ordered in-phase c+ tilt: disorder in

the a-a

- tilts is modelled by splitting of O1, O2 along the a axis and O3, O4 along the c axis,

but the c+ tilt affecting O3 and O4 is ordered to give a polar phase. (C) The mean oxide-oxide

splitting due to disorder of a-a

- tilts plotted as a function of composition at room temperature,

showing that this type of disorder decreases with increasing x. Error bars represent 3x e.s.d.

from the refinements. The line is a guide to the eye.

Fig. S6

The three different tilt models compared in Rietveld refinements of [1-x]CSTF-[x]CTO,

projected along the stacking axis c. In each model, disorder in the a-a

- tilts is modelled by

splitting the apical oxides parallel to the a-axis, and splitting the equatorial oxides parallel to

the c-axis (see fig. S5). The c+ tilt is modelled differently in each, as follows: (A) Non-polar

Amam with zero c+ tilt, which yielded the poorest Rietveld fits for all compositions. (B) Non-

polar Amam with disordered c+ tilt, adopted by compositions 0.0 ≤ x ≤ 0.13, y = 0.60 at room

temperature. (C) Polar A21am with ordered c+ tilt, adopted by compositions x > 0.13, y = 0.60

at room temperature.

Fig. S7

Rietveld refinement of Ca0.69Sr0.46Tb1.85Fe2O7 (x = 0.0, y = 0.60) at room temperature, in

space group Amam with tilt scheme (a-a

-c

<0>)/(a

-a

-c

<0>). (A) SXRD, with inset showing the fit

to the most intense (115)/(020) and (200) peaks, (B) HRPD back-scattering (2θ = 169°) detector bank, (C) HRPD 2θ = 90° detector bank, (D) HRPD 2θ = 30° detector bank. Purple

tick marks = nuclear Amam reflections, green tick marks = magnetic PBnnm reflections.

Fig. S8

Rietveld refinements against room temperature NPD data from GEM Bank 6 (2θ = 154⁰) of

the composition x = 0.30, y = 0.60 in non-polar and polar space groups (A) Non-polar model

Amam with disordered c+ tilt yielding a poor fit to the data with several clear mis-fits, Rwp =

4.498. (B) Polar model A21am with ordered c+ tilt yielding a good fit to the data, Rwp = 3.045.

Fig. S9

Rietveld refinement against high resolution NPD data (HRPD) of x = 0.2, y = 0.563 at 60 K,

modeled with the polar A21am structure (purple tick marks) and the canted A21'a'm magnetic

structure (green tick marks). (A) Rietveld fit to the back-scattering (2θ = 169°) bank, (B)

Rietveld fit to the low-angle (2θ = 35°) bank.Black markers = yobs, red line = ycalc, grey line =

(yobs-ycalc).

Fig. S10

Rietveld refinement of the polar, weak ferromagnetic composition x = 0.17, y = 0.60 against

room temperature NPD (GEM time-of-flight diffractometer), with structure modelled with an

ordered c+ tilt in A21am and magnetic structures modelled in Shubnikov group A21'a'm and

PBnnm, overall Rwp = 2.727. (A) Bank 6, 2θ = 19°, (B) Bank 5, 2θ = 35°, (C) Bank 4, 2θ =

64°, (D) Bank 3, 2θ = 91°, (E) Bank 2, 2θ = 154°. Black markers = yobs, red line = ycalc, grey

line = (yobs – ycalc), blue tick marks = A21am nuclear structure, green tick marks = A21'a'm

magnetic structure, black tick marks = PBnnm magnetic structure, red tick marks = TbFeO3,

magenta tick marks = CaTbFeO4.

Fig. S11

Schematic illustration of the X2+ distortion mode, which describes the c

+ tilt in [1-x]CSTF-

[x]CTO (y = 0.563 and 0.60).

Fig. S12

(A) and (B) show the PBnnm and A21'a'm magnetic structures plotted viewing along the a-

axis. (C) and (D) show two layers of the PBnnm and A21'a'm structures either side of the rock-

salt layer, plotted viewing along the c-axis. The octahedra in different perovskite blocks have

been coloured green and brown for clarity, with Fe3+

spins coloured pink and blue. Within the

perovskite blocks the Fe3+

spins are G-type antiferromagnetically ordered. The spins are

ordered similarly within the green perovskite block in both structures, however they differ in

the brown perovskite block, showing that the relationship between neighbouring perovskite

blocks across the rock-salt layers is different in the two structures.

Fig. S13

Refined staggered moment (μB/Fe) vs temperature for compositions 0.0 ≤ x ≤ 0.13, y = 0.60.

x = 0.0 was collected on HRPD, x = 0.075 was collected on Polaris and x = 0.05, 0.10, 0.12,

0.13 were collected on GEM. Black points correspond to the PBnnm magnetic structure

associated with the Amam nuclear phase; red points correspond to the A21am magnetic

structure associated with the A21am nuclear phase. Error bars are equal to 3x e.s.d. from the

refinement. Lines are a guide to the eye.

Fig. S14

Refined staggered moment (μB/Fe) vs temperature for compositions 0.14 ≤ x ≤ 0.20, y = 0.60.

x = 0.14, 0.16 and 0.17 were collected on GEM and x = 0.15, 0.18 and 0.20 were collected on

Polaris. Black points correspond to the PBnnm magnetic structure associated with the Amam

nuclear phase; red points correspond to the A21am magnetic structure associated with the

A21am nuclear phase. Error bars are equal to 3x e.s.d. from the refinement. Lines are a guide

to the eye.

Fig. S15

Refined staggered moment (μB/Fe) vs temperature for compositions x = 0.25, y = 0.60

(GEM) and x = 0.30, y = 0.60 (Polaris). Points correspond to the A21am magnetic structure

associated with the A21am nuclear phase. Error bars are equal to 3x e.s.d. from the

refinement. Lines are a guide to the eye.

Fig. S16

Plots of c/b ratio (black squares) and Rwp(Amam)/(A21am) ratio (blue triangles) as a function

of temperature for the series 0.075 ≤ x ≤ 0.30, y = 0.60. Compositions x = 0.10, 0.13, 0.15,

0.20, 0.25 and 0.30 were collected on Polaris; x = 0.17 was collected on GEM; and x = 0.075

was collected on I11 (SXRD) and is presented with c/b ratio only.

Fig. S17

M(T) and ∂M/∂T under H = 100 Oe for compositions 0.0 ≤ x ≤ 0.13, y = 0.60. ZFC (black

line), FC (red line) and TRM (blue line) are plotted on the left axis, and derivative of the

TRM ∂M/∂T (green squares) is plotted on the right axis. The gradual onset of this

transition makes the precise location of the transition temperature difficult. The onset of weak

ferromagnetism is therefore estimated both from divergence of FC and ZFC (Twfm) and from

the peak in∂M/∂T (Tmax). Magenta lines showing fit to the ∂M/∂T peak to estimate

Tmax.

Fig. S18

M(T) and ∂M/∂T under applied field H = 100 Oe for compositions 0.14 ≤ x ≤ 0.19, y =

0.60. ZFC (black line), FC (red line) and TRM (blue line) plots (left axis), and the derivative

of TRM ∂M/∂T (green squares) showing the onset of weak ferromagnetism. The steep

increase of TRM and the narrow derivative peak ∂M/∂T indicates a sharp transition to the

weak ferromagnetic state.

Fig. S19

M(T) and ∂M/∂T under applied field H = 100 Oe for compositions x = 0.20 and 0.30, y =

0.60. ZFC (black line), FC (red line) and TRM (blue line) plots (left axis), and the derivative

of TRM ∂M/∂T (green squares) showing the onset of weak ferromagnetism. The steep

increase of TRM and the narrow derivative peak ∂M/∂T indicates a sharp transition to the

weak ferromagnetic state in these materials.

Fig. S20

Coincident onset of weak ferromagnetism, spin reorientation and c+ tilt on cooling in x =

0.10, y = 0.60. (A) Contour plot of variable temperature NPD data showing the onset of the

A21am magnetic Bragg reflections and disappearance of PBnnm Bragg reflections on cooling

to 100 K, (B) staggered moment (M) in magnetic structures PBnnm (black squares) and

A21'a'm (red circles), (C) left axis: ZFC (black square), FC (red circles) and TRM (blue line)

magnetizations and (D) c/b ratio plotted against temperature. ∂M/∂T is shown in right axis in

(C).

Fig. S21

(A) Raw (un-subtracted) magnetic isotherms at T = 300 K for compositions 0.0 ≤ x ≤ 0.10, y

= 0.60. (B–C) Extraction of saturated magnetic moment (Msat) for x = 0.17, y = 0.60 is shown

as an example. The top layer (B) shows raw magnetic isotherm data at 300 K (black circles)

and simulated magnetization data (red line) using the equation described in the experimental

section. The bottom layer shows subtracted magnetization data (green circles) and simulated

data (red line) with zero gradient (d = 0). The saturated magnetization at 300 K was derived

by this method for all compositions plotted in Fig. 4C.

Fig. S22

Room temperature lattice parameter ratio c/b (black squares), and ratio of R-factors from

refinements of disordered (Amam) and ordered (A21am) c+ tilt structures (red triangles),

plotted as a function for compositions 0.05 ≤ x ≤ 0.30, y = 0.563. Note that the minimum in

c/b ratio coincides with onset of the polar A21am structure.

Fig. S23

Refined staggered moment per Fe at 60 K as a function of composition for the compositions

0.05 ≤ x ≤ 0.30, y = 0.563, where red circles denote the A21am magnetic structure; black

squares denote the PBnnm magnetic structure. Note that the A21a’m structure is dominant for

compositions that are polar at ambient conditions.

Fig. S24

Comparison of M(T) plots under an applied field H = 100 Oe for the two series y = 0.563 and

0.60 (0.0 ≤ x ≤ 0.30). Black = zero field cooled,, red = field cooled and blue = thermal

remanent magnetization. Left and right panels show magnetization data of 0.0 ≤ x ≤ 0.30 for

y = 0.60 and y = 0.563 respectively.

Fig. S25

Extraction of saturated magnetic moment for the composition x = 0.2, y = 0.563 shown as an

example. The top layer shows raw magnetic isotherm M(H) data at 60 K (black circles) and

simulated magnetization data (red line) using the equation described in experimental section.

The bottom layer shows subtracted magnetization data (green circles) and simulated data (red

line) with zero gradient (d = 0).

Fig. S26

Magnetic isotherms at 60 K after subtraction for compositions (B–F) x = 0.05 – 0.30, y =

0.563. Non-subtracted data for x = 0, y = 0.563 (A) is scaled to 0.1. (G) Saturated moment at

60 K plotted against x as obtained from the magnetic isotherms (A-F).

Fig. S27

Conductivity plotted against temperature for the composition x = 0.10, y = 0.563 from a two-

probe resistance measurement, showing highly insulating behaviour below 100 K. All

measured compositions showed similar insulating behaviour below 100 K.

Fig. S28

Induced magnetization amplitude (M) plotted against electric field amplitude (E0) showing

(A – B) absence of linear ME effect for x = 0.0, y = 0.60 and (C – D) finite linear ME effect

for x = 0.17, y = 0.60. Left and right panels show ME data at 60 and 100 K respectively.

Linear ME susceptibility () is calculated from the gradient as shown in (C).

Fig. S29

Induced magnetization amplitude (M) plotted against applied electric field amplitude (E0) at

60 K for compositions (A – F) x = 0.0 – 0.30, y = 0.563. Alpha values are calculated from the

gradients as shown in (F). These data show the linear ME effect for x ≠ 0.

Fig. S30

Induced magnetization amplitude (M) plotted against applied electric field amplitude (E0) at

100 K for compositions (A – F) x = 0.0 – 0.30, y = 0.563. These data show the linear ME

effect for x ≠ 0.

Table S1.

Refined A-site cation occupancies in x = 0, y = 0.60 from combined NPD-SXRD refinement.

Model Rwp 12 coordinate site 9 coordinate site

Sr Ca Tb Sr Ca Tb

Amam (c+

disordered)

7.186 0.46 0.404(2) 0.136(2) 0 0.143(2) 0.857(2)

A21am (c+ ordered) 7.410 0.46 0.407(2) 0.133(2) 0 0.142(2) 0.858(2)

Table S2.

Refined A-site cation occupancies in x = 0.20, y = 0.60 from combined NPD-SXRD

refinement.

Model 12 coordinate site 9 coordinate site

Sr Ca Tb Sr Ca Tb

A21am (c+ ordered) 0.40(1) 0.488(9) 0.11(2) 0.04(1) 0.300(7) 0.66(1)

Table S3.

Linear magnetoelectric susceptibility () at 60 and 100 K for compositions 0.0 ≤ x ≤ 0.30, y

= 0.563 as calculated from slopes ofM vs E0 plots in Fig. S29, S30 respectively.

Composition (x) || (ps/m) at 60 K || (ps/m) at 100 K

0.00 0.002 (1) 0.002 (1)

0.05 0.017 (1) 0.007 (2)

0.10 0.046 (2) 0.008 (1)

0.13 0.052 (2) 0.039 (3)

0.20 0.063 (2) 0.049 (2)

0.30 0.119 (2) 0.074 (1)

Table S4.

Summary of Rietveld refinements and reference numbers of crystallographic information

files.

Composition Temperature

(K)

Space

Group

Instrument Rwp (%) CSD

Number

CaTb2Fe2O7 0 A21am DFT/simulated N/A 203243

CaTb2Fe2O7 0 Amam DFT/simulated N/A 203244

CaTb2Fe2O7 0 I4/mmm DFT/simulated N/A 203245

CaTb2Fe2O7 0 Pnnm DFT/simulated N/A 203246

Sr1.1Tb1.9Fe2O7 Ambient P42/mnm HRPD 4.437 428350

Ca0.69Sr0.46Tb1.85Fe2O7 Ambient Cmcm HRPD 4.933 428216

Ca0.69Sr0.46Tb1.85Fe2O7 40 K Cmcm HRPD 4.781 428214

Ca0.69Sr0.46Tb1.85Fe2O7 60 K Cmcm HRPD 4.678 428215

x = 0.05, y = 0.60 Ambient Cmcm Polaris 3.099 428217

x = 0.075, y = 0.60 Ambient Cmcm Polaris 2.846 428220

x = 0.10, y = 0.60 Ambient Cmcm GEM 2.983 428214

x = 0.10, y = 0.60,

refined with ADPs

Ambient Cmcm GEM 2.788 428394

x = 0.10, y = 0.60 100 Cmc21 Polaris 2.291 428221

x = 0.10, y = 0.60 150 Cmc21 Polaris 2.270 428222

x = 0.10, y = 0.60 160 Cmc21 Polaris 2.533 428223

x = 0.10, y = 0.60 170 Cmc21 Polaris 2.528 428224

x = 0.10, y = 0.60 180 Cmc21 Polaris 2.531 428225

x = 0.10, y = 0.60 190 Cmc21 Polaris 2.541 428226

x = 0.10, y = 0.60 200 Cmc21 Polaris 2.260 428227

x = 0.10, y = 0.60 210 Cmc21 Polaris 2.515 428228

x = 0.10, y = 0.60 220 Cmc21 Polaris 2.486 428229

x = 0.10, y = 0.60 230 Cmc21 Polaris 2.469 428230

x = 0.10, y = 0.60 240 Cmc21 Polaris 2.466 428233

x = 0.10, y = 0.60 250 Cmc21 Polaris 2.188 428234

x = 0.10, y = 0.60 260 Cmcm Polaris 2.485 428235

x = 0.10, y = 0.60 270 Cmcm Polaris 2.464 428236

x = 0.10, y = 0.60 280 Cmcm Polaris 2.445 428237

x = 0.10, y = 0.60 333 Cmcm Polaris 3.472 428238

x = 0.10, y = 0.60 343 Cmcm Polaris 3.454 428239

x = 0.10, y = 0.60 353 Cmcm Polaris 3.429 428240

x = 0.10, y = 0.60 363 Cmcm Polaris 3.413 428241

x = 0.10, y = 0.60 373 Cmcm Polaris 3.449 428242

x = 0.10, y = 0.60 383 Cmcm Polaris 3.393 428243

x = 0.10, y = 0.60 393 Cmcm Polaris 3.411 428245

x = 0.10, y = 0.60 403 Cmcm Polaris 3.469 428246

x = 0.10, y = 0.60 413 Cmcm Polaris 3.400 428247

x = 0.10, y = 0.60 423 Cmcm Polaris 3.401 428248

x = 0.10, y = 0.60 433 Cmcm Polaris 3.409 428249

x = 0.10, y = 0.60 443 Cmcm Polaris 3.375 428250

x = 0.10, y = 0.60 453 Cmcm Polaris 3.357 428251

x = 0.10, y = 0.60 463 Cmcm Polaris 3.347 428252

x = 0.10, y = 0.60 473 Cmcm Polaris 3.339 428253

x = 0.12, y = 0.60 Ambient Cmcm GEM 3.038 428358

x = 0.13, y = 0.60 Ambient Cmc21 GEM 2.970 428326

x = 0.13, y = 0.60 96 Cmc21 Polaris 2.723 428317

x = 0.13, y = 0.60 125 Cmc21 Polaris 2.704 428318

x = 0.13, y = 0.60 152 Cmc21 Polaris 2.669 428319

x = 0.13, y = 0.60 177 Cmc21 Polaris 2.668 428320

x = 0.13, y = 0.60 201 Cmc21 Polaris 2.675 428321

x = 0.13, y = 0.60 226 Cmc21 Polaris 2.682 428322

x = 0.13, y = 0.60 248 Cmc21 Polaris 2.752 428323

x = 0.13, y = 0.60 267 Cmc21 Polaris 2.761 428324

x = 0.13, y = 0.60 288 Cmc21 Polaris 2.828 428325

x = 0.13, y = 0.60 323 Cmcm Polaris 3.582 428256

x = 0.13, y = 0.60 373 Cmcm Polaris 3.645 428257

x = 0.13, y = 0.60 423 Cmcm Polaris 3.531 428258

x = 0.13, y = 0.60 473 Cmcm Polaris 3.433 428259

x = 0.13, y = 0.60 523 Cmcm Polaris 3.274 428260

x = 0.13, y = 0.60 573 Cmcm Polaris 2.727 428261

x = 0.14, y = 0.60 Ambient Cmc21 GEM 2.824 428244

x = 0.15, y = 0.60 Ambient Cmc21 GEM 2.862 428269

x = 0.15, y = 0.60 40 Cmc21 Polaris 1.969 428327

x = 0.15, y = 0.60 90 Cmc21 Polaris 2.122 428328

x = 0.15, y = 0.60 140 Cmc21 Polaris 2.093 428329

x = 0.15, y = 0.60 190 Cmc21 Polaris 2.059 428330

x = 0.15, y = 0.60 240 Cmc21 Polaris 2.085 428331

x = 0.15, y = 0.60 290 Cmc21 Polaris 2.129 428262

x = 0.15, y = 0.60 340 Cmcm Polaris 2.608 428263

x = 0.15, y = 0.60 390 Cmcm Polaris 2.541 428264

x = 0.15, y = 0.60 440 Cmcm Polaris 2.478 428265

x = 0.15, y = 0.60 490 Cmcm Polaris 2.397 428266

x = 0.15, y = 0.60 540 Cmcm Polaris 2.289 428267

x = 0.15, y = 0.60 590 Cmcm Polaris 2.205 428332

x = 0.15, y = 0.60 640 Cmcm Polaris 1.892 428268

x = 0.16, y = 0.60 Ambient Cmc21 GEM 2.741 428270

x = 0.17, y = 0.60 Ambient Cmc21 GEM 2.727 428274

x = 0.17, y = 0.60 323 Cmc21 Polaris 2.789 428271

x = 0.17, y = 0.60 373 Cmc21 Polaris 2.735 428272

x = 0.17, y = 0.60 423 Cmc21 Polaris 2.774 428273

x = 0.17, y = 0.60 473 Cmcm Polaris 2.418 428359

x = 0.17, y = 0.60 523 Cmcm Polaris 2.311 428360

x = 0.17, y = 0.60 573 Cmcm Polaris 2.457 428361

x = 0.17, y = 0.60 623 Cmcm Polaris 2.407 428362

x = 0.17, y = 0.60 673 Cmcm Polaris 2.125 428363

x = 0.17, y = 0.60 723 Cmcm Polaris 1.863 428364

x = 0.17, y = 0.60 773 Cmcm Polaris 1.718 428365

x = 0.18, y = 0.60 Ambient Cmc21 GEM 2.678 428275

x = 0.19, y = 0.60 Ambient Cmc21 GEM 2.534 428276

x = 0.20, y = 0.60 Ambient Cmc21 GEM 2.800 428366

x = 0.20, y = 0.60

refined with ADPs

Ambient Cmc21 GEM + I11 3.378 428395

x = 0.20, y = 0.60 100 Cmc21 Polaris 2.463 428290

x = 0.20, y = 0.60 150 Cmc21 Polaris 2.440 428291

x = 0.20, y = 0.60 200 Cmc21 Polaris 2.418 428292

x = 0.20, y = 0.60 250 Cmc21 Polaris 2.452 428293

x = 0.20, y = 0.60 323 Cmc21 Polaris 2.874 428294

x = 0.20, y = 0.60 373 Cmc21 Polaris 2.802 428295

x = 0.20, y = 0.60 423 Cmc21 Polaris 2.729 428296

x = 0.20, y = 0.60 473 Cmc21 Polaris 2.671 428297

x = 0.20, y = 0.60 523 Cmc21 Polaris 2.627 428298

x = 0.20, y = 0.60 573 Cmc21 Polaris 2.605 428299

x = 0.20, y = 0.60 623 Cmcm Polaris 2.583 428371

x = 0.20, y = 0.60 673 Cmcm Polaris 2.508 428300

x = 0.20, y = 0.60 723 Cmcm Polaris 2.524 428301

x = 0.20, y = 0.60 773 Cmcm Polaris 2.506 428302

x = 0.20, y = 0.60 823 Cmcm Polaris 2.437 428303

x = 0.20, y = 0.60 873 Cmcm Polaris 1.924 428304

x = 0.25, y = 0.60 280 Cmc21 GEM 2.908 428305

x = 0.25, y = 0.60 328 Cmc21 Polaris 2.702 428306

x = 0.25, y = 0.60 373 Cmc21 Polaris 2.775 428307

x = 0.25, y = 0.60 423 Cmc21 Polaris 2.673 428308

x = 0.25, y = 0.60 473 Cmc21 Polaris 2.650 428309

x = 0.25, y = 0.60 523 Cmc21 Polaris 2.599 428310

x = 0.25, y = 0.60 573 Cmc21 Polaris 2.598 428311

x = 0.25, y = 0.60 623 Cmcm Polaris 2.546 428335

x = 0.25, y = 0.60 673 Cmcm Polaris 2.524 428336

x = 0.25, y = 0.60 723 Cmcm Polaris 2.546 428337

x = 0.25, y = 0.60 773 Cmcm Polaris 2.535 428338

x = 0.25, y = 0.60 823 Cmcm Polaris 2.522 428339

x = 0.25, y = 0.60 873 Cmcm Polaris 2.517 428340

x = 0.25, y = 0.60 923 Cmcm Polaris 2.544 428341

x = 0.25, y = 0.60 973 Cmcm Polaris 2.573 428342

x = 0.25, y = 0.60 1023 Cmcm Polaris 2.525 428343

x = 0.25, y = 0.60 1073 Cmcm Polaris 2.055 428344

x = 0.30, y = 0.60 Ambient Cmc21 GEM 3.045 428316

x = 0.30, y = 0.60 373 Cmc21 Polaris 2.477 428312

x = 0.30, y = 0.60 473 Cmc21 Polaris 2.648 428313

x = 0.30, y = 0.60 573 Cmc21 Polaris 2.768 428314

x = 0.30, y = 0.60 673 Cmc21 Polaris 2.566 428315

x = 0.30, y = 0.60 773 Cmcm Polaris 2.632 428345

x = 0.30, y = 0.60 873 Cmcm Polaris 2.526 428346

x = 0.30, y = 0.60 973 Cmcm Polaris 2.614 428347

x = 0.30, y = 0.60 1073 Cmcm Polaris 2.555 428348

x = 0.05, y = 0.563 300 Cmcm HRPD 3.387 428219

x = 0.05, y = 0.563 60 Cmc21 HRPD 3.955 428218

x = 0.10, y = 0.563 Ambient Cmcm HRPD 4.735 428232

x = 0.10, y = 0.563 60 Cmc21 HRPD 3.071 428231

x = 0.13, y = 0.563 Ambient Cmcm HRPD 4.514 428255

x = 0.13, y = 0.563 60 Cmc21 HRPD 3.047 428254

x = 0.20, y = 0.563 Ambient Cmc21 HRPD 4.217 428366

x = 0.20, y = 0.563 60 Cmc21 HRPD 2.720 428333

x = 0.30, y = 0.563 Ambient Cmc21 HRPD 3.413 428373

x = 0.30, y = 0.563 60 Cmc21 HRPD 2.317 428372

References and Notes

1. L.-D. Zhao, S. H. Lo, Y. Zhang, H. Sun, G. Tan, C. Uher, C. Wolverton, V. P. Dravid, M. G.

Kanatzidis, Ultralow thermal conductivity and high thermoelectric figure of merit in

SnSe crystals. Nature 508, 373–377 (2014). Medline doi:10.1038/nature13184

2. H. Sakai, J. Fujioka, T. Fukuda, D. Okuyama, D. Hashizume, F. Kagawa, H. Nakao, Y.

Murakami, T. Arima, A. Q. Baron, Y. Taguchi, Y. Tokura, Displacement-type

ferroelectricity with off-center magnetic ions in perovskite Sr1–xBaxnO3. Phys. Rev. Lett.

107, 137601 (2011). Medline doi:10.1103/PhysRevLett.107.137601

3. N. A. Hill, Why are there so few magnetic ferroelectrics? J. Phys. Chem. B 104, 6694–6709

(2000). doi:10.1021/jp000114x

4. M. Bibes, A. Barthélémy, Multiferroics: Towards a magnetoelectric memory. Nat. Mater. 7,

425–426 (2008). Medline doi:10.1038/nmat2189

5. C. A. F. Vaz, Electric field control of magnetism in multiferroic heterostructures. J. Phys.

Condens. Matter 24, 333201 (2012). Medline doi:10.1088/0953-8984/24/33/333201

6. D. M. Evans, A. Schilling, A. Kumar, D. Sanchez, N. Ortega, M. Arredondo, R. S. Katiyar, J.

M. Gregg, J. F. Scott, Magnetic switching of ferroelectric domains at room temperature

in multiferroic PZTFT. Nat. Commun. 4, 1534 (2013). Medline

doi:10.1038/ncomms2548

7. J. van Suchtelen, Product properties: A new application of composite materials. Philips Res.

Rep. 27, 28–37 (1972).

8. N. J. Perks, R. D. Johnson, C. Martin, L. C. Chapon, P. G. Radaelli, Magneto-orbital helices as

a route to coupling magnetism and ferroelectricity in multiferroic CaMn7O12. Nat.

Commun. 3, 1277 (2012). Medline doi:10.1038/ncomms2294

9. M. Ghita, M. Fornari, D. J. Singh, S. V. Halilov, Interplay between A-site and B-site driven

instabilities in perovskites. Phys. Rev. B 72, 054114 (2005).

doi:10.1103/PhysRevB.72.054114

10. N. A. Benedek, C. J. Fennie, Hybrid improper ferroelectricity: A mechanism for controllable

polarization-magnetization coupling. Phys. Rev. Lett. 106, 107204 (2011). Medline

doi:10.1103/PhysRevLett.106.107204

11. J. M. Rondinelli, C. J. Fennie, Octahedral rotation-induced ferroelectricity in cation ordered

perovskites. Adv. Mater. 24, 1961–1968 (2012). Medline doi:10.1002/adma.201104674

12. K. S. Aleksandrov, J. Bartolome, Octahedral tilt phases in perovskite-like crystals with slabs

containing an even number of octahedral layers. J. Phys. Condens. Matter 6, 8219–8235

(1994). doi:10.1088/0953-8984/6/40/013

13. J. Alaria, P. Borisov, M. S. Dyer, T. D. Manning, S. Lepadatu, M. G. Cain, E. D. Mishina, N.

E. Sherstyuk, N. A. Ilyin, J. Hadermann, D. Lederman, J. B. Claridge, M. J. Rosseinsky,

Engineered spatial inversion symmetry breaking in an oxide heterostructure built from

isosymmetric room-temperature magnetically ordered components. Chem. Sci. 5, 1599–

1610 (2014). doi:10.1039/c3sc53248h

14. M. M. Elcombe, E. H. Kisi, K. D. Hawkins, T. J. White, P. Goodman, S. Matheson, Structure

determinations for Ca3Ti2O7, Ca4Ti3O10, Ca3.6Sr0.4Ti3O10 and a refinement of Sr3Ti2O7.

Acta Crystallogr. B 47, 305–314 (1991). doi:10.1107/S0108768190013416

15. Y. S. Oh, X. Luo, F.-T. Huang, Y. Wang, S.-W. Cheong, “Experimental demonstration of

hybrid improper ferroelectricity and presence of abundant charged walls in (Ca,Sr)3Ti2O7

crystals,” http://arxiv.org/abs/1411.6315.

16. Y. Yoshida, S.-I. Ikeda, H. Matsuhata, N. Shirakawa, C. Lee, S. Katano, Crystal and

magnetic structure of Ca3Ru2O7. Phys. Rev. B 72, 054412 (2005).

doi:10.1103/PhysRevB.72.054412

17. M. V. Lobanov, M. Greenblatt, E. N. Caspi, J. D. Jorgensen, D. V. Sheptyakov, B. H. Toby,

C. E. Botez, P. W. Stephens, Crystal and magnetic structure of the Ca3Mn2O7

Ruddlesden-Popper phase: Neutron and synchrotron x-ray diffraction study. J. Phys.

Condens. Matter 16, 5339–5348 (2004). doi:10.1088/0953-8984/16/29/023

18. I. E. Dzialoshinskii, Thermodynamic theory of weak ferromagnetism in antiferromagnetic

substances. Sov. Phys. JETP 5, 1259–1272 (1957).

19. T. Moriya, Anisotropic superexchange interaction and weak ferromagnetism. Phys. Rev. 120,

91–98 (1960). doi:10.1103/PhysRev.120.91

20. D. Samaras, A. Collomb, Rotation des moments magnetiques dans BaLa2Fe2O7. Solid State

Commun. 16, 1279–1284 (1975). doi:10.1016/0038-1098(75)90828-5

21. N. N. M. Gurusinghe, J. de la Figuera, J. F. Marco, M. F. Thomas, F. J. Berry, C. Greaves,

Synthesis and characterisation of the n = 2 Ruddlesden–Popper phases Ln2Sr(Ba)Fe2O7

(Ln = La, Nd, Eu). Mater. Res. Bull. 48, 3537–3544 (2013).

doi:10.1016/j.materresbull.2013.05.058

22. A. M. Glazer, The classification of tilted octahedra in perovskites. Acta Crystallogr. B 28,

3384–3392 (1972). doi:10.1107/S0567740872007976

23. D. Samaras, A. Collomb, J. C. Joubert, Determination des structures de deux ferrites mixtes

nouveaux de formule BaLa2Fe2O7 et SrTb2Fe2O7. J. Solid State Chem. 7, 337–348

(1973). doi:10.1016/0022-4596(73)90142-4

24. D. Hanžel, SrTb2Fe2O7 studied by Mössbauer spectra of 57

Fe. J. Magn. Magn. Mater. 5, 243–

246 (1977). doi:10.1016/0304-8853(77)90164-0

25. Materials and methods are available as supplementary materials on Science Online.

26. Crystallographic information files may be obtained from Fachinformationszentrum

Karlsruhe, 76344 Eggenstein-Leopoldshafen, Germany (e-mail: crysdata@fiz-

karlsruhe.de, www.fiz-karlsruhe.de/request_for_deposited_data.html) on quoting the

appropriate Crystal Structure Depot number tabulated in table S4.

27. R. D. Shannon, C. T. Prewitt, Effective ionic radii in oxides and fluorides. Acta Crystallogr.

B 25, 925–946 (1969). doi:10.1107/S0567740869003220

28. H. Schmid, Some symmetry aspects of ferroics and single phase multiferroics. J. Phys.

Condens. Matter 20, 434201 (2008). doi:10.1088/0953-8984/20/43/434201

29. I. M. Reaney, I. MacLaren, L. Wang, B. Schaffer, A. Craven, K. Kalantari, I. Sterianou, S.

Miao, S. Karimi, D. C. Sinclair, Defect chemistry of Ti-doped antiferroelectric

Bi0.85Nd0.15FeO3. Appl. Phys. Lett. 100, 182902 (2012). doi:10.1063/1.4705431

30. G. Kresse, J. Furthmüller, Efficient iterative schemes for ab initio total-energy calculations

using a plane-wave basis set. Phys. Rev. B 54, 11169–11186 (1996). Medline

doi:10.1103/PhysRevB.54.11169

31. G. Kresse, D. Joubert, From ultrasoft pseudopotentials to the projector augmented-wave

method. Phys. Rev. B 59, 1758–1775 (1999). doi:10.1103/PhysRevB.59.1758

32. J. P. Perdew, K. Burke, M. Ernzerhof, Generalized gradient approximation made simple.

Phys. Rev. Lett. 77, 3865–3868 (1996). Medline doi:10.1103/PhysRevLett.77.3865

33. S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J. Humphreys, A. P. Sutton, Electron-

energy-loss spectra and the structural stability of nickel oxide: An LSDA+U study. Phys.

Rev. B 57, 1505–1509 (1998). doi:10.1103/PhysRevB.57.1505

34. N. E. Brese, M. O'Keeffe, Bond-valence parameters for solids. Acta Crystallogr. B 47, 192–

197 (1991). doi:10.1107/S0108768190011041

35. S. P. Thompson, J. E. Parker, J. Potter, T. P. Hill, A. Birt, T. M. Cobb, F. Yuan, C. C. Tang,

Beamline I11 at diamond: A new instrument for high resolution powder diffraction. Rev.

Sci. Instrum. 80, 075107 (2009). Medline doi:10.1063/1.3167217

36. M. E. Fisher, Relation between the specific heat and susceptibility of an antiferromagnet.

Philos. Mag. 7, 1731–1743 (1962). doi:10.1080/14786436208213705

37. J. M. D. Coey, J. T. Mlack, M. Venkatesan, P. Stamenov, Magnetization process in dilute

magnetic oxides. IEEE Trans. Magn. 46, 2501–2503 (2010).

doi:10.1109/TMAG.2010.2041910

38. P. Borisov, A. Hochstrat, V. V. Shvartsman, W. Kleemann, Superconducting quantum

interference device setup for magnetoelectric measurements. Rev. Sci. Instrum. 78,

106105 (2007). Medline doi:10.1063/1.2793500

39. M. Fiebig, Revival of the magnetoelectric effect. J. Phys. D Appl. Phys. 38, R123–R152

(2005). doi:10.1088/0022-3727/38/8/R01

40. D. Samaras, A. Collomb, J. C. Joubert, E. F. Bertaut, Etude magnétique du composé

SrTb2Fe2O7, détermination des structures magnétiques par diffraction neutronique. J.

Solid State Chem. 12, 127–134 (1975). doi:10.1016/0022-4596(75)90188-7

41. S. Kerman, B. J. Campbell, K. K. Satyavarapu, H. T. Stokes, F. Perselli, J. S. Evans, The

superstructure determination of displacive distortions via symmetry-mode analysis. Acta

Crystallogr. A 68, 222–234 (2012). Medline doi:10.1107/S0108767311046241

42. M. A. Green, D. A. Neumann, Synthesis, structure, and electronic properties of LaCa2Mn2O7.

Chem. Mater. 12, 90–97 (2000). doi:10.1021/cm991094i

43. I. A. Zvereva, S. R. Seitablaeva, Y. E. Smirnov, Cation distribution and interatomic

interactions in oxides with heterovalent isomorphism: VI. Solid solutions Nd2Sr1-

xCaxAl2O7. Russ. J. Gen. Chem. 73, 31–36 (2003). doi:10.1023/A:1023414216375

44. I. A. Zvereva, A. G. Cherepova, Y. E. Smirnov, Cation distribution and interatomic

interactions in oxides with heterovalent isomorphism: XII. Gd2Sr1−xCaxAl2O7 solid

solutions. Russ. J. Gen. Chem. 77, 517–523 (2007). doi:10.1134/S1070363207040444