REVIEW ARTICLE
OPEN
Magnetic structures and dynamics of multiferroic systems
obtained with neutron scattering
William Ratcliff
1
, Jeffrey W Lynn
1
, Valery Kiryukhin
2
, Prashant Jain
3
and Michael R Fitzsimmons
3
Multiferroics are materials that evince both ferroelectric and magnetic order parameters. These order parameters when coupled can
lead to both exciting new physics as well as new device applications. Potential device applications include memory, magnetic field
sensors, small antennas and so on. Since Kimura’s discovery of multiferroicity in TbMnO
3
, there has been a renaissance in the study
of these materials. Great progress has been made in both materials discovery and in the theoretical understanding of these
materials. In type-II systems the magnetic order breaks the inversion symmetry of the material, driving a secondary ferroelectric
phase transition in which the ferroelectric polarisation is exquisitely coupled to the magnetic structure and thus to magnetic field.
In type-I systems, the magnetic and ferroelectric orders are established on different sublattices of the material and typically are
weakly coupled, but electric field can still drive changes in the magnetisation. Besides single-phase multiferroics, there has been
exciting progress in composite heterostructures of multiferroics. Here, we review neutron measurements of prototypical examples
of these different approaches to achieving multiferrocity.
npj Quantum Materials (2016) 1, 16003; doi:10.1038/npjquantmats.2016.3; published online 27 July 2016
INTRODUCTION
Neutron scattering plays an important role in determining the
ferroelectric properties of multiferroics in terms of the detailed
crystal structure, but the central role is in elucidating the magnetic
structures and spin dynamics, and in understanding the origin of
how, and how strongly, the magnetic and ferroelectric order
parameters are coupled. We first review very briefly the key
neutron techniques employed to investigate multiferroics, both in
bulk and thin-film forms. We emphasise the role of magnetic
scattering here but also reference the standard techniques for
crystal structure refinements and exploring the lattice dynamics,
which are similar in concept to magnetic Bragg scattering and
measuring the spin dynamics, respectively. We then discuss
two systems which are prototypes for a type-I multiferroic
material (HoMnO
3
), and a type-II multiferroic material (TbMnO
3
).
The type I ferroelectrics typically have high-ferroelectric transition
temperatures—well above room temperature—but low magnetic
ordering temperatures, and these two disparate order parameters
are weakly coupled. The type II ferroelectrics are typically materials
with low magnetic ordering temperatures, where the magnetic
structure itself breaks the inversion symmetry and permits
(typically very weak) ferroelectric order to develop. Both are
interesting from a fundamental point of view, but so far only some
of the type I materials exhibit both types of orders at room
temperature and thus appear to have promise for applications
such as memory, spintronics and sensors. One such type I
multiferroic is BiFeO
3
, which is the leading candidate material for
applications, and we discuss the results for bulk crystals and thin
films. We then explore the properties of nanocomposites and
hybrid inorganic-organic materials, and conclude with prospects
for further work.
DIFFRACTION (STRUCTURE) AND INELASTIC SCATTERING
(DYNAMICS)
Neutron scattering have a central role in determining the crystal
and magnetic structures of a vast variety of materials. One
traditional role in the magnetically ordered regime has been the
measurement of magnetic Bragg intensities, which can be used to
determine the fundamental nature and symmetry of the magnetic
state. Quantitative values of the ordered moment(s) as a function
of temperature, pressure and applied magnetic field can be
determined, as well as the density of unpaired electrons that
constitute the magnetic moments. These are elastic scattering
measurements (no change in the energy of the neutrons) which
are carried out on single crystals, powders, thin films and
multilayers. For thin films and multilayers the specialized elastic
scattering technique of neutron reflectometry is employed, which
can determine both the atomic and magnetic density depth
profiles averaged over the area of the film. For large scale
structures (1–1,000 nm) the technique of (elastic) small-angle
neutron scattering (SANS) is used. Neutron diffraction studies
often provide information that can be obtained by no other
experimental technique. In the investigation of the spin dynamics
of systems, e.g., neutrons play a truly unique role. Neutron
scattering is the only technique that can directly determine the
complete magnetic excitation spectrum, whether it is in the form
of the dispersion relations for spin wave excitations, wave vector
and energy dependence of critical fluctuations, crystal field
excitations, moment fluctuations and so on, which can be readily
compared with theory. Techniques employing the spin-dependent
scattering of the neutron can be used to unambiguously identify
nuclear scattering from magnetic scattering, both for magnetic/
crystal structures and for excitations. More detailed reviews of
neutron magnetic scattering techniques, including the theory of
1
NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD, USA;
2
Department of Physics and Astronomy, Rutgers University,
Piscataway, NJ, USA and
3
Los Alamos National Laboratory, Los Alamos, NM, USA.
Correspondence: W Ratcliff (william.ratcliff@nist.gov)
Received 6 May 2016; revised 8 June 2016; accepted 9 June 2016
www.nature.com/npjquantmats
Published in partnership with Nanjing University
neutron scattering from magnetic systems and solids, can be
found in references 1–5.
POLARISED NEUTRON REFLECTOMETRY (PNR)
PNR is ideally suited to measure the nuclear and magnetisation
depth profiles across planar interfaces. Because re flection only
occurs when one or both of the nuclear or magnetic compositions
change across an interface, PNR can distinguish magnetism at
interfaces and other nanoscale structures from contamination
within a substrate or on the sample.
6
Extensive reviews of PNR can
be found in references 7–9.
Briefly, in PNR the intensity of the specularly reflected neutron
beam is compared with the intensity of the incident beam as a
function of wavevector transfer, Q (=4πsinθ/λ, where, θ is angle of
incidence and λ is the neutron wavelength), and neutron beam
polarisation (Figure 1a). Q is changed by changing θ (typically of
order 1°) or λ (typically of order 0.2–1.4 nm). The specular
reflectivity, R, is determined by the neutron scattering length
density (SLD) depth profile, ρ(z), averaged over the lateral
dimensions of the sample. The SLD is the depth-dependent
variation of the index of refraction (related to ρ(z)) of the sample.
Although the reflectivity close to the origin is highly non-
kinematical, the reflectivity at larger Q can be thought of in terms
of the square of the modulus of the Fourier transform of the SLD
profile. (Widely available numerical codes permit rigorous analysis
including the non-kinematical behaviour of the reflectivity for
all Q.) ρ(z) consists of nuclear and magnetic SLDs such that
ρ
±
(z)=ρ
n
(z)±CM (z), where C = 2.9109 × 10
− 9
Å
− 2
(kA/m)
− 1
, and
M(z) is the magnetisation (kA/m) depth profile.
8,9
Unlike
magnetometry, which measures magnetic moment, reflection of
polarised neutron beams depends on a variation of moment
density across a planar interface. Further, unlike non-resonant
X-ray scattering, the nuclear and magnetic neutron scattering
lengths have comparable magnitudes. The +(− ) sign in ρ
±
(z)
denotes neutron beam polarisation parallel (opposite) to the
applied magnetic field and corresponds to spin-dependent
reflectivities, R
±
(Q). By measuring R
+
(Q) and R
−
(Q), ρ
n
(z) and M(z)
can be obtained separately. If the net magnetisation is rotated
away from the applied field, polarisation analysis of the specularly
reflected beam provides information about the projection of the
net magnetisation vector onto the sample plane. For example,
polarisation analysis with PNR is useful for identifying the magnetic
orientations of different layers parallel to the sample plane.
Q
(
nm
)
-0.2
-0.1
0.0
0.2
0.1
Q
(nm
)
-0.2
-0.1
0.0
0.2
0.1
θ
H
GISANS
specular
Figure 1. (a) Schematic of a reflectometry experiment with the possibility of off-specular scattering (schematically labelled as GISANS for
grazing incidence small angle scattering) about the specularly reflected beam. The applied magnetic field is shown by H. (b) A SANS
experiment.
Magnetic structures and dynamics of multiferroic systems
W Ratcliff et al
2
npj Quantum Materials (2016) 16003 Published in partnership with Nanjing University
SMALL-ANGLE NEUTRON SCATTERING
SANS is ideally suited to measure the change of nuclear and
magnetisation contrast across non-planar interfaces. In SANS
experiments the incident neutron beam is tightly collimated in
two orthogonal directions (Figure 1b). SANS provides information
about regions, domains and particles bounded by interfaces
across which either the nuclear or magnetic SLD changes. The
length scales probed by SANS vary between a few nanometres
and microns. Because of the inverse relationship between length
scale and reciprocal space, short-length scales produce SANS
furthest from the primary (unscattered) beam. Like PNR, polarised
SANS allows an unambiguous identification of the nuclear and
magnetic scattering and the relationship between the two.
Confinement of scattered intensity in neutron reflectometry
about the specular reflectivity allows the use of supermirrors for
polarisation analysis with relative ease. However, polarisation
analysis of widely divergent SANS with mirrors is more
challenging. The availability of He
3
polarising filters has recently
enabled routine polarisation analysis for SANS, allowing successful
exploitation of SANS for the unique identification of magnetic
and structural contributions to the scattering from magnetic
materials.
10–12
In some cases neutron scattering with polarisation
analysis enables measurement of correlations of magnetism that
transcend the physical non-uniformity of the material.
13
Sample environments for neutron scattering experiments
routinely include cryomagnets providing fields as large as 15 T,
temperatures as low as 20 mK, pressure cells, mechanical devices
(to apply bending stress to substrates) and facilities for irradiation
with light. Equipment exists that offers opportunities to
simultaneously control magnetic/electric fields, temperature and
pressure—of obvious interest to studies of magnetoelectric
multiferroic materials.
TYPE-I MULTIFERROIC: HOMNO
3
The hexagonal (space group P6
3
cm) rare-earth manganese oxides
(RMnO
3
with R = Ho, Er, Tm, Yb, Lu and Y) are a family of
multiferroic materials of particular interest, and HoMnO
3
is a
prototypical system representative of these hexagonal materials
that has been investigated in considerable detail. In this material
the Ho–O ions undergo a displacement at very high temperature
(T
C
= 1375 K),
14
which gives rise to a ferroelectric (FE) moment
along the c axis. Long-range magnetic order, on the other hand,
develops for the Mn moments at 72 K. The disparate ordering
temperatures indicate that this system is a type-I multiferroic
where the order parameters are relatively weakly coupled through
the Ho–Mn exchange and anisotropy interactions. The magnetic
system has the added interest that the Mn moments occupy a
frustrated triangular lattice as shown in Figure 2. Below the
magnetic ordering temperature (T
N
= 72 K) the Mn spins order in a
noncollinear 120° spin structure. At 40 K a spin reorientation
(T
SR
= 40 K) takes place in which the Mn moments rotate in the
Figure 2. Crystal structure for hexagonal HoMnO
3
.(a) View from the
side (c axis is up). Large blue balls are Ho, red balls are oxygen. The
Mn ions sit at the centre of each tetrahedron of oxygen ions.
(b) View along the c axis. Note the triangular configuration of the
Mn (oxygen tetrahedra) as well as the Ho. The oxygen that make up
the sides of each tetrahedron also exhibit triangular coordination.
Figure 3. Neutron diffraction measurements of the integrated intensities of the (100) and (101) magnetic Bragg reflections of HoM nO
3
.Two
spin-reorientation transitions (indicated by dashed lines) lead to changes in the intensities of both peaks. The Mn
3+
spin configurations for
these phases are shown on the right. Ho
3+
moments (not shown) order in the low-temperature phase. (Adapted from reference 17.)
Magnetic structures and dynamics of multiferroic systems
W Ratcliff et al
3
Published in partnership with Nanjing University npj Quantum Materials (2016) 16003
plane, changing the magnetic symmetry and adopting a different
120° spin structure. At low temperatures the Ho moments order
antiferromagnetically (T
Ho
= 8 K) with moments aligned along the
c axis, accompanied by a second spin reorientation transition of
the Mn moments into the P63cm 120° spin structure phase. The
spin structures in these phases are shown in Figure 3. The sharp
magnetic transitions shown in the figure are accompanied by
sharp anomalies in the dielectric constant, indicating that the
magnetic and ferroelectric order parameters are coupled.
15
Second-harmonic generation measurements as a function of
magnetic field also show the change in magnetic symmetry and
the reentrant phase as a function of magnetic field suggested by
dielectric susceptibility.
16
The application of a magnetic field shifts T
SR
to lower T,
broadens the transition, and drives HoMnO
3
into a reentrant
phase as shown in the (H versus T) phase diagram of Figure 4.
2
In the intermediate-temperature phase, a sufficiently strong
applied magnetic field along the c axis pushes HoMnO
3
into
the high-temperature phase,
15–17
in good agreement with
the phase diagram established from dielectric susceptibility
measurements.
15–18
At low temperatures, as ordered Ho magnetic
order develops, the phase diagram becomes considerably more
complicated, including a balancing of the interactions to produce
a critical end point around 2T and 2K,
18
indicative of the rich
physics of HoMnO
3
. In this temperature regime it is expected that
the magnetic and ferroelectric order parameters naturally couple
since the Ho moments are magnetically ordered while the
ferroelectric order originates with the Ho–O displacements.
The spin dynamics of this non-collinear frustrated triangular
magnetic lattice has been investigated in considerable detail.
Figure 4. Temperature versus magnetic field phase diagram for
HoMnO
3
obtained from neutron diffraction measurements. Curves
are guides-to-the-eye. The dashed line and dotted lines indicate
approximate phase boundaries for previously reported transitions
not observable from the diffraction data. The numbered regions
correspond to the intermediate phases and hysteretic overlap
regions that occur at low T. (Adapted from reference 17.)
Figure 5. In-plane spin-wave dispersion for HoMnO
3
at 20 K for the three modes. Dashed lines indicate two (dispersionless) crystal field levels
of Ho at 1.5(1) and 3.1(1) meV. The bottom part of the figure shows the calculated spin wave dispersion relations for the three modes.
The dispersion curves for the three modes are identical, but the origins are offset for the three. (Adapted from reference 17.)
Magnetic structures and dynamics of multiferroic systems
W Ratcliff et al
4
npj Quantum Materials (2016) 16003 Published in partnership with Nanjing University
It turns out that the coupling along the c axis is very weak, so the
spin dynamics is two-dimensional in nature. A Hamiltonian that
captures the basic physics of the problem can be written as
H ¼ -
X
i;j
J
ij
S
i
!
U S
j
!
- D
X
i
ðS
z
i
Þ
2
ð1Þ
where J
ij
=J is the strength of the nearest-neighbour in-plane
antiferromagnetic (AFM) exchange interaction and D is a single-
ion anisotropy. The spin-wave spectrum then consists of three
separate modes that propagate within the hexagonal plane, which
are equivalent in energy but their origins are offset in wave
vector.
17,19
Figure 5 shows a comparison of the model with the
data taken at 20 K (intermediate temperature phase) along the
high-symmetry directions in reciprocal space, using J and D as
fitting parameters. From these fits we find J = 2.44 meV and
D = 0.38 meV. The anisotropy was found to have a significant
temperature dependence, which is the driving force behind the
spin reorientation transitions observed in this system. We note
that such spin reorientations do not occur in YMnO
3
, so it is clear
that they originate from the holmium, as has been observed in
other rare earth systems such as Nd
2
CuO
4
.
20
We note that the
full-magnetic Hamiltonian for HoMnO
3
including the Ho moments
and ferroelectric coupling must be considerably more complex
than Equation (1), but this very simple model establishes the
dominant magnetic interactions for this system.
TYPE-II MULTIFERROIC: TBMNO
3
The orthorhombic (space group Pbnm) RMnO
3
series of perovskite
manganites, where R = [Gd,Tb,Dy], show a variety of magnetic
orderings depending on the size of the R ion.
21
The Mn
3+
ions lie
on four Bravais lattices at the: 1 (½ 0 0), 2(½ 0 ½), 3 (0 ½ ½) and
4 (0 ½ 0) positions. These Mn
3+
ions have an electronic
configuration of 3t
2g
e
g
with the nearest neighbour t
2g
electrons
having AFM couplings and the e
g
electrons having ferromagnetic
coupling.
21
Competition between these couplings gives rise to a
variety of Mn magnetic orderings, ranging from the simple
antiferromagnetism of LaMnO
3
to a complex incommensurate
ordering in TbMnO
3
. In addition to TbMnO
3
ordering magnetically
at 41 K, at ~28 K a spontaneous ferroelectric polarisation
develops.
22–24
This polarisation can be switched at low tempera-
tures through the application of a strong magnetic field.
22
Initial studies
23
of the magnetic order of TbMnO
3
were able to
identify the ordering wave-vector to be along the b axis and to
determine that it was incommensurate and the incommensur-
ability changed with temperature. A symmetry analysis for the
30 K data indicated the presence of only one Fourier component
for the incommensurate wave vector, with the spins obeying
either an amplitude modulation or a spiral structure. On model
refinement to the data, it was found that the moment was along
the b axis and the magnetic structure was amplitude modulated.
Thus the initial ordering of the Mn spins is in the form of a simple
longitudinally polarised spin-density wave (SDW).
Subsequent data are shown in Figure 6, taken by Kenzelmann
et al.,
24
who performed a more detailed examination of the
system, in particular to elucidate the relationship between the
magnetic structure and ferroelectricity. They used representational
analysis to classify the magnetic structures possible with the
underlying space group symmetry. They found that there were
Figure 6. (a) Cartoon of the SDW magnetic structure above 28 K for TbMnO
3
. Moments are only shown on the Mn ions. (b) Cartoon of
magnetic structure below 28 K in the elliptical spiral phase. Moments are only shown on the Mn ions. (c) Example magnetic diffraction pattern
along a high symmetry direction at 2 K, with the intensity on a log scale. (d) Temperature dependence of the intensity, which is propor tional
to the square of the order parameter, and the incommensurate position k, of the (0k1) reflection versus temperature. Arrows indicate with
which axis symbols are associated. (Adapted from reference 24.)
Magnetic structures and dynamics of multiferroic systems
W Ratcliff et al
5
Published in partnership with Nanjing University npj Quantum Materials (2016) 16003