www.advmat.de
By Gustau Catalan* and James F. Scott*
REVIEW
Physics and Applications of Bismuth Ferrite
doping other ions into both the A and B
sites of the lattice, but no practical devices
were obtained.
Reviews of the general study of magnetoelectricity appeared by Schmid in 1994[11]
and more recently by Fiebig[12] and by
Eerenstein et al.[13] The current interest in
bismuth ferrite was stimulated primarily by
a 2003 paper from Ramesh’s group,[14]
which showed that it had unexpectedly
large remnant polarization, Pr, 15 times
larger than previously seen in bulk, together
with very large ferromagnetism of ca. 1.0
Bohr magneton (mB) per unit cell. Single
crystals grown more recently in France in 2006–7 have confirmed
the large value of the polarization first observed in the films,
showing also that it is intrinsic;[15-19] on the other hand, the
intrinsic magnetization of thin films is now thought to be near
zero[20, 21] – ca. 0.02 magnetons/cell – and possible reasons for
discrepancies between the magnetization values encountered in
the literature are discussed later in this review. At any rate, the
2003 Science paper has proved enormously stimulating, and has
inspired both new fundamental physics and exciting device
applications.
BiFeO3 is perhaps the only material that is both magnetic and a strong
ferroelectric at room temperature. As a result, it has had an impact on the field
of multiferroics that is comparable to that of yttrium barium copper oxide
(YBCO) on superconductors, with hundreds of publications devoted to it in
the past few years. In this Review, we try to summarize both the basic physics
and unresolved aspects of BiFeO3 (which are still being discovered with
several new phase transitions reported in the past few months) and device
applications, which center on spintronics and memory devices that can be
addressed both electrically and magnetically.
1. Introduction
1.1. History
The basic idea that crystals could be simultaneously ferromagnetic and ferroelectric probably originated with Pierre Curie in
the 19th century.[1] After switching was discovered in ferroelectric
Rochelle Salt by Valasek in 1920[2] there was a rash of supposed
discoveries of magnetoelectric properties by Perrier,[3,4] but
unfortunately in materials such as Ni in which they are now
understood to be impossible. A history of this period of solid-state
physics is given in O’Dell’s text.[5] True magnetoelectricity –
defined as a linear term in the free energy G(P,M,T) ¼ aij Pi Mj –,
where P is the polarization and M is the magnetization was first
understood theoretically by Dzyaloshinskii[6] with special predictions being made for Cr2O3 and discovered experimentally in that
material by Astrov.[7] However this material is paraelectric and
antiferromagnetic, making microelectronics applications impractical. The more interesting case of ferromagnetic ferroelectrics
waited for some years until the work of Schmid on boracites.[8]
The boracites are also impractical materials for device applications: they have low symmetry with large unit cells and grow in
needle shapes; more importantly, they exhibit magnetoelectricity
only at extremely low temperatures. Meanwhile Smolenskii’s
group in Leningrad pioneered[9] the study of bismuth ferrite,
BiFeO3, but they found that they could not grow single crystals
and that ceramic specimens were too highly conducting (probably
caused by oxygen vacancies and mixed Fe valences) to be used in
applications.[10] They tried to address the conductivity problem by
[*] Dr. G. Catalan, Prof. J. F. Scott
Department of Earth Sciences
University of Cambridge
Downing Street, Cambridge CB2 3EQ (United Kingdom)
E-mail: gcat05@esc.cam.ac.uk; jsco99@esc.cam.ac.uk
DOI: 10.1002/adma.200802849
Adv. Mater. 2009, 21, 2463–2485
1.2. The Hypothesis of Spaldin (Hill)
In parallel with the specific investigation of bismuth ferrite and
related compounds has been a more general approach to the idea
of multiferroics. Nicola Hill (now Spaldin) has asked[22] why there
are so few materials that are magnetic and ferroelectric; implicitly
limiting her discussion to transition-metal oxides, especially
perovskites, she observed that the ferroelectrics (e.g., titanates)
have B-site ions with d8 electrons,[23] whereas the magnets
require dj electrons with j different from zero. This line of
thinking has also proved very stimulating, although it is helpful to
remind ourselves that there are many potential multiferroics that
are not oxides, as further discussed in Section 1.3.
On the other hand, oxide perovskites do not all have the same
mechanism of ferroelectricity: the center Ti ion plays the key role
in BaTiO3 but the lone-pair Pb ion is dominant in PbTiO3.[24]
Indeed, this seems to be the case in BiFeO3, where the
polarization is mostly caused by the lone pair (s2 orbital) of
Biþ3, so that the polarization comes mostly from the A site while
the magnetization comes from the B site (Fe3þ); this same idea
has led Spaldin and co-workers to propose a host of other
perovskites with possible A-site ferroelectricity and B-site
magnetism, such as Bi(Cr,Fe)O3 and BiMnO3. Her work has
also been instrumental in triggering the quest for other ways of
achieving coexistence of ferroelectricity and magnetism in oxides;
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www.advmat.de
Gustau Catalan studied
Physics at the University of
Barcelona and earned his
PhD at the Queen’s
University of Belfast,
followed by postdoctoral
appointments in CSIC
(Spain) and the Zernike
Institute for Advanced
Materials (Holland). He is
currently a Senior Research
Associate in the Ferroics
Laboratory of the University
of Cambridge, which he
joined in 2005. His areas of
interest include the properties of perovskite oxides and the
study of how small size affects the phase transitions and
functionality of thin films and nanocrystals.
James F. Scott was educated
at Harvard (B.A., Physics
1963) and Ohio State
University (Ph.D., Physics
1966). After six years in the
Quantum Electronics
Research Department at
Bell Labs he was appointed
professor of Physics at Univ.
Colorado (Boulder). He was
Dean of Science and
Professor of Physics for
eight years in Australia
(UNSW, Sydney, and RMIT,
Melbourne) and has been
Professor of Ferroics in the Earth Sciences Department at the
University of Cambridge since 1999.
oxides. However, there is no a priori reason to expect that they will
have the most interesting multiferroic physics or will make the
best devices. The exclusive emphasis on oxides seems unwise,
and magnetoelectric fluorides should probably receive more
attention in this context.
Abrahams has given a list of magnetic materials that are
probably ferroelectric.[34–36] His criterion was the existence of a
structure that has very small acentric displacements of ions, with
the assumption that ferroelectric switching is likely in such
lattices. Most of these predicted ferroelectrics are indeed oxides,
but some are fluorides.[37] BaMnF4, for example, is the earliest
known example of a material with spin canting induced by
ferroelectricity,[28] the physics being the same as in the
ferroelectrically induced local spin canting in BiFeO3, discussed
later in this Review. K3Fe5F15[38–40] is a ferroelectric ferrimagnet
with 2 Feþ3 ions and 3 Feþ2 ions per unit cell, adding to a weak net
ferromagnetic moment. A particularly interesting multiferroic is
Sr3(FeF6)2, which merits further study.[41] An additional family of
multiferroics that seems promising is Pb5Cr3F19.[42,43] There are
many other fluoride multiferroics, and (NH4)3FeF6 has received
careful study.[44,45] These systems include both pure fluorides and
oxyfluorides,[45] with the latter including Bi2TiO4F2; Ravez has
given a review encompassing both fluorides and oxyfluorides,[46]
and Nenert and Palstra[29] have also recently reviewed other
possible multiferroic fluorides.
2. The Phase Diagram of BiFeO3
2.1. Phase Decomposition and Impurities
The phase diagram for the system Bi2O3/Fe2O3 has been mapped
out[47,48,49] and is shown in Figure 1. BiFeO3 is usually prepared
from equal parts of Bi2O3 and Fe2O3, and under high
temperatures it can decompose back into these starting materials,
as shown in Equation 1
2BiFeO3 ! Fe2 O3 þ Bi2 O3
(1)
among the new findings are ferroelectricity induced by spiral spin
order,[25,26] magnetic exchange striction,[27] or charge order.[28]
Based on these ideas, new oxides have been predicted to be
multiferroics, some of which await direct experimental verification.[26,29,30] In addition, there are materials in which the
ferroelectricity itself causes spin canting[31,32] and these should
be explored further.
1.3. Oxide vs. Fluoride Multiferroics
There are hundreds of different crystals with ferroelectric
transition temperature, Tc, above room temperature at atmospheric pressure, with approximately a log-normal distribution of
transition temperatures.[33] Many of these materials are not
oxides, yet almost all work on multiferroics has emphasized
oxides. This is convenient as they are easy to grow, particularly in
thin-film form, and researchers in high-Tc superconductors,
colossal magnetoresistance manganites, and ceramics all study
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Figure 1. Compositional phase diagram of BiFeO3. Reproduced with
permission from [49]. Copyright 2008, American Physical Society.
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REVIEW
platinum crucibles which, given the high mutual reactivity of Bi
and Pt, may not be the most suitable receptacle.
The above discussion underlines what is currently one of the
most important difficulties in implementing BiFeO3 for practical
applications, namely, its compositional instability, with associated
fickleness of functional behavior. Addressing this problem is
essential if BiFeO3 is to succeed as a technologically relevant
material.
2.2. Crystal Structure
Figure 2. Impurities have a strong effect on functional properties. In this
figure, the remnant magnetization, M, of BiFeO3 thin films grown in
oxygen-deficient conditions is shown to be directly correlated to the
amount of g-Fe2O3 parasitic phase as extracted from X-ray diffraction
(XRD) peaks. Figure courtesy of Manuel Bibes (Thales-CNRS).
Bismuth ferrite is very prone to show parasitic phases that tend
to nucleate at grain boundaries and impurities.[50] It has been
argued that BiFeO3 is in fact metastable in air, with optically
visible impurity spots appearing well below the melting
temperature.[49,51,52] Impurities and oxygen vacancies are also
important for thin films, because they are known to artificially
enhance the remnant magnetization[19,21] (Fig. 2). Minimizing
them requires very careful tuning of growth parameters,
particularly oxygen pressure.[21]
At room temperature under applied fields of ca. 200 kV cm1
(typical switching voltages across thin films), BiFeO3 decomposes, yielding magnetite Fe3O4 as a by-product.[53] This was
somewhat surprising and is thought to occur via the following
reaction
6BiFeO3 ! 2Fe3 O4 þ 3Bi2 O3 þ O
(2)
The magnetite phase was unambiguously identified by means
of micro-Raman studies; the Raman spectra of Fe3O4 is quite
distinct and unlike those of Fe2O3. However, the Bi2O3 was not
detected, possibly because it is a well-known glass-forming
compound, or perhaps because of its evaporation during thermal
decomposition. Bi2O3 melts at a temperature slightly above
800 8C.[49] Similar phenomena occur in the electrical stressing of
lead zirconate titanate (PZT),[54] which decomposes into rutile
TiO2 (but not anatase), and both a-PbO and b-PbO (litharge and
massicot). In both bismuth ferrite and PZT these observations
raise concerns about the lifetime of ferroelectric memories made
from them. In the case of BiFeO3, this decomposition
mechanism also provides another possible explanation for the
appearance of remnant magnetization in thin films: it may come
from localized spots of magnetite in the sample.
We also point out that Bi2O3 and Pt are known to react easily
and exothermically with each-other.[55,56] Bismuth has a low
melting temperature of 270 8C, at which it readily forms an
eutectic alloy with Bi2Pt and, at higher temperatures, other Bi–Pt
alloys are also formed.[57] Therefore, it is probably best NOT to use
Pt electrodes when probing the high-temperature properties of
bismuth-based materials,[58] including bismuth ferrite. We also
note that most BiFeO3 ceramics and single crystals are made in
Adv. Mater. 2009, 21, 2463–2485
The room-temperature phase of BiFeO3 is classed as rhombohedral (point group R3c).[59] The perovskite-type unit cell has a
lattice parameter, arh, of 3.965 Å and a rhombohedral angle,
arh, of ca. 89.3–89.48 at room temperature,[60,61] with ferroelectric
polarization along [111]pseudocubic.[61] The unit cell can also be
described in a hexagonal frame of reference, with the hexagonal
c-axis parallel to the diagonals of the perovskite cube, i.e.,
[001]hexagonal jj [111]pseudocubic. The hexagonal lattice parameters
are ahex ¼ 5.58 Å and chex ¼ 13.90 Å.[60–62] The coefficient of
thermal expansion is neither completely linear nor isotropic,[62–64] and reported values[62,63] differ notably, ranging from
ca. 6.5 106 to ca. 13 106 K1.
A very important structural parameter is the rotation angle of
the oxygen octahedra. This angle would be 08 for a cubic
perovskite with perfectly matched ionic sizes. A measure of how
well the ions fit into a perovskite unit cell is the ratio ðrBi þ r0 Þ=l,
where r is the ionic radius of the respective ion and l is the length
of the octahedral edge. This is completely analogous to the
commonly used Goldschmid
tolerance factor,[65] which is defined
pffiffiffi
as t ¼ ðrBi þ rO Þ 2ðrFe þ rO Þ. For BiFeO3 we obtain t ¼ 0.88
using the ionic radii of Shannon,[66] with Biþ3 in eightfold
coordination (the value for 12-fold coordination is not reported)
and Feþ3 in sixfold coordination and high spin. When this ratio is
smaller than one, the oxygen octahedra must buckle in order to fit
into a cell that is too small. For BiFeO3, v is ca. 11–148 around the
polar [111] axis,[59,61,67] with the directly related Fe–O–Fe angle,
u ¼ ca. 154–1568.[61,64] The Fe–O–Fe angle is important because it
controls both the magnetic exchange and orbital overlap between
Fe and O, and as such it determines the magnetic ordering
temperature and the conductivity, as will be discussed in later
sections
2.3. Symmetry of the High-Temperature b and g Phases
At approximately 825 8C there is a first-order transition to a
high-temperature b phase that is accompanied by a sudden
volume contraction.[49,68] The transition is also accompanied by a
peak in the dielectric constant;[68,69] this has been taken as an
indication of a ferroelectric–paraelectric transition, although
dielectric peaks can also occur in ferroelectric–ferroelectric
transitions, such as the orthorhombic–rhombohedral transition
in the archetypal perovskite ferroelectric BaTiO3 (which is also
first order). Nevertheless, although there is disagreement about
the exact symmetry of the b phase above 825 8C, most reports
agree that it is centrosymmetric,[70–75] so it is probably a safe bet
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that the a–b transition at TC ¼ 825 8C is indeed the ferroelectric–paraelectric transition.
Palai et al.[49] propose that the symmetry of the b phase is
orthorhombic, although their data does not allow establishing the
exact space group with certainty. Some authors have argued that
the b phase may be tetragonal or pseudotetragonal,[67,71] but that
is impossible, since the domain structure rules out a tetragonal
symmetry and the perovskite a,b,c lattice constants are each quite
different.[49,72] It was also proposed that this phase may instead be
monoclinic;[71,72] the measured monoclinic angle was nevertheless initially quoted as 90o within experimental error,[71] so that
the b phase was in effect ‘‘metrically orthorhombic’’ (i.e., the
angles may be 908, but internal ion positions in each unit cell do
not satisfy orthorhombic constraints). More recently, however,
Haumont et al. have quoted a monoclinic angle of 90.018.[72]
On the other hand, our group does not see in our specimens
the extra XRD lines used to infer monoclinic structure. Also, the
domains studied optically do not reveal the many extra wall
orientations that would exist if the symmetry were monoclinic
instead of orthorhombic. Furthermore, we find that the b–g
phase transition to the cubic metallic phase encountered at
1204 K (atmospheric pressure) seems second-order, and a
cubic–monoclinic second-order phase transition would violate
the principle of maximal subgroup.[76] This is an additional
argument in favor of the b phase being orthorhombic:
cubic–orthorhombic transitions are allowed to be second order
and still satisfy the maximal subgroup criterion.
A very recent work has added more ‘‘fuel to the fire’’ regarding
this problem. Selbach et al.[73] claim that the paraelectric b phase
may neither be orthorhombic nor monoclinic, but rhombohedral
(space group R3c). It is, however, hard to reconcile this claim with
the splitting of the pseudocubic lattice parameters observed by
other groups,[49,71,72,75] or with the symmetry of the domain walls
in this phase. Perhaps, more importantly, if both the a and b
phases belong to the same crystal class (rhombohedral), then the
transition cannot be ferroelastic, which would appear to contradict the observed change in the ferroelastic domain configuration
at this transition.[49]
Part of the disagreements between all the above works can
perhaps be justified by the fact that X-rays are not particularly
sensitive to the position of the oxygen ions as a result of the low
electronic density of O compared with the Bi and Fe ions. In this
respect, neutron diffraction is a far more helpful technique.
High-temperature neutron-diffraction experiments have recently
been undertaken by Arnold et al.,[74] who show that the b phase is
orthorhombic Pbnm, which is the same non-polar orthorhombic
symmetry of the GdFeO3 orthoferrite family. In retrospect, of
course, this seems quite obvious: once the distinctive feature of
BiFeO3 (its ferroelectric polarization) is removed, one may expect
this material to be just like all the other perovskite orthoferrites,
i.e., it should be orthorhombic. This is, of course, just an ad hoc
argument, but apart from the neutron diffraction and our XRD
experiments, there may be additional indirect support for an
orthorhombic b phase based on chemical-doping experiments
with ions other than Bi, as discussed in the next Section. It is
worth mentioning as well that the neutron-diffraction experiments of Arnold et al.[74] suggest phase coexistence in the b phase
and, indeed, optical viewgraphs at high temperature do show a
coexistence of rhombohedral and orthorhombic domains.[49] This
is consistent with the strongly first-order nature of this transition,
and perhaps the mixture of rhombohedral and orthorhombic
phases could also explain why the b phase may seem
monoclinic in some experiments.[71,72] One
final note relevant to this issue is that
differential thermal analysis (DTA) measurements (Fig. 3) show a smaller anomaly ca.
30 8C below the transition to the orthothombic
b phase. This suggests the existence of an
intermediate phase, which may either be the
monoclinic phase seen by Haumont and
co-workers, or a region of phase coexistence
as seen by Arnold et al. and us.
As for the symmetry of the highesttemperature g phase, Redfern’s XRD data[75]
show that the most intense Bragg peak,
(110)pseudocubic, has a large splitting at room
temperature and atmospheric pressure, but is
unsplit (resolution 0.058 in 2u) in the g phase
above 931 8C (Fig. 3), with the proposed
symmetry for the cubic phase being
Pm3m.[49,75] Unfortunately, BiFeO3 is very
unstable at the high temperature of the b–g
Figure 3. Specific-heat measurements by a) Kaczmarek et al. [124] and b) Palai et al. [49] suggest transition and it rapidly decomposes into
that the transition to the orthorhombic b phase (815 8C in Kaczmarek’s measurement, 825 8C in parasitic phases such as Bi2Fe4O9 or Fe2O3
ours) may proceed via an intermediate phase which nucleates at ca. 25–35 8C below. This may be
(Bi2O3 becomes a liquid at that temperature so
the monoclinic phase reported by Haumont and co-workers, or else it may be a region of phase it does not show up in the diffraction scans).
coexistence between rhombohedral and orthorhombic. c) At temperatures above 930 8C the
Accordingly, measurements of the b–g transi(110) diffraction peaks appear unsplit (the small split in the diffractogram is due to a1/a2
radiation), indicating that the g phase is cubic. Reproduced with permission from [124]. tion at 930 8C have to be performed on very
high-quality samples (preferably single
Copyright 1974, Elsevier (a). Reproduced with permission from [49]. Copyright 2008, American
crystals) and using very fast heating/cooling
Physics Society (b).
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Adv. Mater. 2009, 21, 2463–2485
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ramps; the different measurement protocols used by different
groups mean that not all of them have been able to measure the
cubic phase at high temperature. On the other hand, a tendency
towards cubic symmetry has also been reported for BiFeO3 as a
function of decreasing grain size,[77] which indirectly supports the
conclusion that the highest-symmetry phase is cubic.
2.4. Phase Transitions with Pressure or with La Doping
The diffraction peaks are also unsplit in Redfern’s measurements
at room temperature and high pressure above 47 GPa (although
the experimental resolution is low, with 0.588 peak broadening);
this suggests the same cubic symmetry at high pressure as
observed at high temperature, in agreement with the early phase
diagram of Scott et al.[84] Gavriliuk et al. report a rhombohedral
structure instead,[78] in agreement with ab initio simulations.[79]
The experimental resolution precludes a completely unambiguous answer at this stage, as very small splittings may have been
Adv. Mater. 2009, 21, 2463–2485
REVIEW
Figure 4. Properties of Bi1-xLaxFeO3 as a function of La doping. There are
indications of at least three phase transitions as x increases. Reproduced
with permission from [68]. Copyright 1974, Wiley.
masked by broadening in our measurements or, conversely, lack
of hydrostatic equilibrium may cause peak asymmetry that can be
wrongly interpreted as peak splitting in Gavriliuk’s. As a side
comment, we note that the question of whether materials tend
towards cubic symmetry with high pressure is, surprisingly,
unresolved even for simple elements.[80]
Recently, Pashkin et al. [81] have reported additional phase
transitions at room temperature at ca. 3–5 and 7.5–10 GPa. The
reported pressure-induced transition near 10 GPa is to an
orthorhombic Pnma (Pbnm) state. This could also be the
orthorhombic symmetry for the high-temperature b phase
proposed by Palai et al.[49] and Arnold et al.[74] While these
low-pressure transitions have not been confirmed by highpressure studies in the USA,[82] or Russia,[82,83] the transition to
the orthothombic b-phase near 10 GPa has been recently
reproduced by Redfern et al.[75] It thus seems that the
rhombohedral-orthorhombic-cubic sequence of phase transitions
is the same as a function of pressure as it is as a function of
temperature.
An alternative way of inducing ‘‘pressure’’ in a crystal is by
chemical substitution of an ion for another of the same valence but
different size – what is sometimes called ‘‘chemical pressure’’. The
most common isovalent substitute in BiFeO3 is La3þ for Biþ3.
However, interpretation of the effects of La doping in terms of
chemical pressure is not straightforward because La3þ has almost
exactly the same ionic radius as Biþ3 (1.16 and 1.17 Å, respectively[66]).
Furthermore, the lone-pair orbital of Bi3þ (6s2) is stereochemically
active and responsible for the ferroelectric distortion; distortions
induced by La doping are therefore more likely to be caused by the
turning off of the lone-pair activity (i.e., the turning off of the ferroelectricity) than to direct differences in ionic size.
The first phase diagram for Bi1xLaxFeO3 was published by
Polomska et al.,[68,85] who looked at the dielectric constant and
volume expansion as a function of La doping concentration
(Fig. 4). In their study, there is a first-order transition with sharp
volume contraction for x ¼ ca. 0.2 (Fig. 4) and several other
transitions, the last one of which is at x ¼ ca. 0.75 to the
orthorhombic Pnma (centric) phase of pure LaFeO3, also reported
for pure BiFeO3 above ca. 10 GPa[81] at room temperature or above
825 8C at ambient pressure.[74] The nature of the intermediate
bridging phases, however, is unclear. Gabbasova et al.[86] and
Zalesskii et al.[87] claim a noncentric orthorhombic phase (C222)
for 0.2 < x < 0.6 that could also be associated with the hightemperature b phase of pure BiFeO3 – the sudden volume
contraction at x ¼ 0.2 is in this context very reminiscent of the
volume contraction observed at the temperature-induced a–b
transition (825 8C). At any rate, the exact nature and even the
number of structural phase transitions as a function of La doping
is still an open question.[63,86–88]
2.5. Other Anomalies above Room Temperature: Phase
Transitions vs. Defects
Krainik et al.[51] measured the GHz dielectric constant and
thermal expansion of BiFeO3 between room temperature and
900 8C. They found small anomalies at 130, 200, 280, 370, 460,
600, 670, 740, and 845 8C. However, the authors themselves
mention that the samples are ‘‘almost phase pure’’, which is
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another way of saying that they are not pure; accordingly, some of
their anomalies could be due to parasitic phases and defects (see
Section 2.1), particularly since many have never been reproduced
in later measurements. Nonetheless, some of these anomalies are
clearly correlated with known phase transitions: the 845 8C peak
is almost certainly the a–b phase transition, whereas the anomaly
at 370 8C, also reported by Polomska,[68] is caused by magnetoelectric coupling to the antiferromagnetic Neel temperature
(TNeel). With the exception of the anomaly at the Neel
temperature, none of the other possible phase transitions shows
up in the refractive index as a function of temperature.[89]
The most intriguing of these ‘‘ghost’’ transitions is perhaps the
anomaly in both dielectric constant and thermal expansion
reported by Polomska near 185 8C or 458 K.[68,85] It is possible that
the phase transition reported at 458 K and ambient pressure could
be the same as that observed at room temperature and ca.
4 GPa.[81] Both sides of this phase boundary were first reported by
Pashkin et al. as rhombohedral[81a] and, hence, the transition—if
it were a transition – would not be ferroelastic.[90] Other ferroelectrics, such as nickel iodine boracite, have had isomorphic
transitions proposed for them which do not change symmetry,[91]
so this is not out of the question. On the other hand, the transition
at 458 K is not universally observed: our own single-crystal
dielectric measurements do not show any clear feature around
that temperature. Nor is there a clear signature of that transition
in the phonon behavior,[49,70] all of which argues in favor of an
extrinsic origin of the anomalies. Having said that, the 458 K
dielectric anomaly has been reported by other groups,[93] and also
appears in electrical resistivity measurements in our laboratory
(Fig. 5). This could, of course, still be related to impurities rather
than being intrinsic; comparison between two-probe and
four-probe measurements, for example, shows the anomaly to
be much stronger in the former, suggesting contact- resistance
effects, although the derivative of the four-probe resistivity does
still show a peak near 185 8C. However, other reports of resistivity
do not show any anomaly near ca. 185 8C.[94,95]
While the coincidence of our resistive anomaly with
Polomska’s (whose impedance and dilatometry measurements
were in a completely different set of samples) is tantalizing, at this
point the evidence for and against an intrinsic origin seems to be
split down the middle, so this possible transition certainly merits
Figure 6. Sketch of a possible phase diagram as a function of pressure and
temperature. Solid points are experimental data, the lines are only a visual
guide. The ground state is rhombohedral, and the b phase is orthorhombic.
The reported monoclinic phase transition [71,72,81] has not been confirmed, and may actually be a coexistence of rhombohedral and orthorhombic (not unusual for a strongly first-order phase transition). Pressure is
known to increase TNeel in orthoferrites at a rate of 4.0–7.5 K GPa1 [204].
Accordingly, we expect that TNeel will rise slowly with hydrostatic pressure
up until the triple point is reached, but there is no direct experimental
evidence of this; above the triple point, the pressure-induced metal–insulator (MI) transition (TMI) delocalizes the electrons and induces a
Pauli paramagnetic state, so that the magnetic-ordering temperature, TN, is
forced to track down TMI The metallic state at high pressures and low
temperatures has been claimed by Gavriliuk et al. [78] and GonzalezVazquez and Iñiguez [79] to be rhombohedral, while the data of Redfern
et al. is consistent with cubic[49,75]. This schematic phase diagram does
not include any of the new magnetic phase transitions observed at low
temperatures and discussed in later Sections.
further careful studies in order to unambiguously establish its
nature.
Based on the pressure/doping effects and the known behavior
of the magnetic transition in orthoferrites, we propose a
schematic temperature–pressure phase diagram for BiFeO3
(Fig. 6). At present this is but an informed guess, with enormous
gaps in real experimental data, so we very much encourage the
careful exploration of this map.
3. Conductivity, Bandgap,
and Metal–Insulator (MI)
Transition
3.1. Resistivity of BiFeO3
Figure 5. Resistance as a function of temperature of BiFeO3 single crystals: two-probe (left) and
four-probe (right) measurements. The sharp anomaly at 185 oC/458 K in the two-probe measurement is
very washed-out in the four-probe resistivity, although the derivative (inset) does show a peak at almost
exactly the same temperature. The four-probe measurements were performed by Julia Herrero-Albillos,
University of Cambridge.
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The dc resistivity of good-quality
bulk samples of BiFeO3 exceeds
1010 Ohm cm.[49,95] As temperature
increases, the resistivity decreases as
would be expected from any widebandgap semiconductor. Around the TN
(370 8C) there is no change in the absolute
value of resistivity, but Arrhenius plots
show a change in slope (Fig. 7), with the
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activation energy of the charge carriers decreasing from ca. 1.3 to ca.
0.6 eV as the material is heated above TN. Resistive anomalies at TN
have also been reported by Selbach et al.[77] This indicates that
magnetic ordering affects the conductivity bandgap, increasing it in
the antiferromagnetic phase, which is consistent with ab initio
calculations.[92] The correlation between bandgap and magnetic
ordering suggests that BiFeO3 could be magnetoresistive. Indirect
evidence of this exists from the dielectric measurements of Kamba
et al.[96] and direct measurements are currently underway in our
laboratory.
At even higher temperatures there are further resistive
anomalies correlated with the a–b (rhombohedral–orthorhombic)
transition, the b–g (orthorhombic–cubic) transition and, finally,
the decomposition temperature.[49] Specifically, the resistivity
decreases (but remains semiconducting) at the a–b transition[77]
and the slope of the resistivity as a function of temperature
changes sign[48] at the b–g transition, which is consistent with a
metal–insulator (MI) transition, as discussed below.
3.2. Bandgap and MI Transition
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Figure 7. Arrhenius plot of the two-probe resistivity of a single crystal,
showing a change of slope at the Neel temperature.
bandgap as BiFeO3 goes from rhombohedral to orthorhombic to
cubic, with the indirect bandgap decreasing markedly at each of
these transitions. Note that the screened exchange band structure
calculation gave good results for the bandgap versus temperature,
in comparison with experiment, but it was not a total energy
calculation and hence does not assess stability of phases.
As temperature increases, Palai et al.[49] have indeed measured
that the optical bandgap decreases and goes to zero abruptly at the
g-phase, signaling a temperature-driven MI transition. MI
transitions are of particular interest in solid-state physics, and
are often studied as a function of pressure as well as temperature.
An MI phase transition has indeed been observed in bismuth
ferrite at room temperature and at a pressure of ca.
50 GPa,[78,82,83] as well as at 1204 K at a pressure of 1 atm.[49]
As far as we know, MI transitions have not been reported in
perovskite ferrites other than BiFeO3.
The evidence for the MI transition at the orthorhombic–cubic
transition near TMI (¼ ca.1204 K) is first, that the optical bandgap
goes to zero at that temperature[49] or at room temperature and
high pressure[78,82,83] (Fig. 8); second, that the magnetism
disappears;[78,101] third, that the temperature derivative of
resistivity changes sign;[49,78] and fourth, that the reflectivity
increases abruptly.[49,78] As the temperature rises, the deviation
from cubic structure decreases and the experimentally measured
gap[49] decreases to ca. 1.6 eV by 500 8C (723 K, still in the
rhombohedral phase). At 1204 K the structure becomes cubic via
a second-order transition,[49,75] and the conduction-band minimum now overlaps the valence-band maximum. Thus a
semimetal is formed, as in elemental Bi or graphite. Although
the behavior is now metallic, the material is not strictly a
conventional metal with a half-filled band.
The pressure-induced MI transition is correlated with the loss
of magnetism.[101] A possible interpretation of this is that the MI
transition triggers the magnetic one by delocalizing the magnetic
electrons. However, a different interpretation has been proposed
by Gavriliuk et al.,[78] who think instead that the order of
precedence is different, i.e., the magnetic transition induces the
MI change rather than the other way round. These authors
suggest that the MI transition may be Mott-type. This means that
the bandgap is caused by electron– electron Coulombic repulsion
(the Hubbard parameter, U), and that there is a critical value of U
which can be reached with either temperature or pressure.[104,105]
Reported values for the optical bandgap of BiFeO3 at room
temperature range from ca. 2.3 to ca. 2.8 eV.[49,95,97–100] According
to some authors, this bandgap is direct,[98,99]
although other reports suggest also the
presence of an indirect bandgap roughly
0.4–1.0 eV smaller than the direct one.[95,97]
Ab initio calculations using screened
exchange formalism show that bismuth ferrite
is a semiconductor with a room-temperature
gap of ca. 2.8 eV.[49,100] The valence-band
maximum is at the R-point corner of the
Brillouin zone, whereas the conduction-band
minimum is at the center, G, so that the gap is
indirect. However, the calculated valence band
in the rhombohedral state is in fact almost
flat[100] so that BiFeO3 should in practice
behave as a direct-bandgap semiconductor at Figure 8. Optical bandgap of BiFeO3 as a function of pressure and temperature. Figures
room temperature. The same calculations, reproduced with permission from [83] and [49], respectively. Copyright 2007, Materials Research
however, show an evolution towards indirect Society and 2008, American Physics Society, respectively.
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According to Gavriliuk et al., the change in U would be due to a
change in the spin configuration from the ground-state high spin
(the five d-shell electrons occupying one each of the t2g and eg
levels, giving a total magnetic moment, S ¼ 5/2[102,103]) to low
spin (no electrons in the eg level and S ¼ 1/2), with the weaker
magnetic interactions in the low-spin state being consistent with
the observed paramagnetism.[101] While there is no direct
experimental evidence for this, ab initio calculations do agree
with a low-spin configuration at high pressures and low
temperatures.[79]
The mechanism proposed by Gavriliuk et al. for the
pressure-driven MI transition is unlikely to work for the
temperature-driven one. For one thing, BiFeO3 is magnetically
disordered both above and below TMI. Furthermore, the
transition we observe appears to be second order, which violates
one of Mott’s principal requirements.[104] We think instead that
the MI transition is triggered by the structural change, a
hypothesis supported by the screened-exchange model,[49] which
shows that BiFeO3 is metallic in only the cubic phase. BiFeO3 is
viewed as a charge-transfer insulator, with the bandgap controlled
by the orbital overlap between the O 2p and the Fe 3d levels.[49,100]
The overlap integral in turn depends on the Fe–O–Fe exchange
angle; accordingly, the observed straightening of the bond angle
with increasing temperature[64,75] would result in the observed
decrease of bandgap with increasing temperature. This mechanism is in fact completely analogous to that of the MI transition in
the perovskite nickelates[106–108] (parenthetically we note that
perovskite nickelates are also thought to be multiferroic[25,26,29]),
with the main difference being that, whereas in the nickelates the
bond angle is tuned by the ionic size, in BiFeO3 the Fe–O–Fe
bond angle is controlled by the ferroelectric distortion.[92,100,109]
The correlation between orbital overlap and bandgap is also very
relevant for the local properties – including conductivity – of the
domain walls, as further discussed in Section 8.
It seems strange to have two different mechanisms depending
on whether the MI transition is induced by pressure or
temperature. On the other hand, the high-pressure symmetry
is reported as rhombohedral[78,79] (although some experiments
suggest cubic, as mentioned in Section 2) whereas the high
temperature one is cubic.[49,75] So, structurally at least, the two
phases may indeed be different and different physical mechanisms for the MI transition could therefore be expected. However,
we suggest that there could be a ‘‘third way’’, reconciling aspects
of the two models. Here we note that the low-spin ionic radius of
Fe3þ is rLS ¼ 0.55 Å, whereas that of the high-spin one is
rHS ¼ 0.645 Å;[66] the smaller low-spin radius is, of course, the
reason why pressure can induce the transition to low spin in the
first place. The smaller size of the low-spin Fe must necessarily
result in a shrinking of the oxygen octahedron around it, meaning
that the Fe–O–Fe bond angle can straighten: the Goldschmid
tolerance factor is 0.88 for the high-spin configuration, and 0.93
for the low-spin one, with the octahedral rotation angle being
closer to 0 as it approaches 1. Thus, while the idea of the
pressure-induced MI transition associated with a change to low
spin may be correct, we also believe that the change in bandgap
may not itself be caused by a reduction in the Mott–Hubbard
electron–electron repulsion, but by the low-spin-induced straightening of the Fe–O–Fe bond angle.
4. Ferroelectricity
4.1. Bulk
The ferroelectric polarization of bulk bismuth ferrite is along
the diagonals of the perovskite unit cell ([111]pseudocubic/
[001]hexagonal). Early measurements of bulk ferroelectricity in
the 1960s and 1970s yielded only small values of the polarization.
However, the small value of Pr (ca. 6 mC cm2) reported by Teague
et al.[110] for single crystals was viewed by those authors as limited
by lack of saturation, and they remarked, presciently, that ‘‘. . .the
actual polarization of BiFeO3 is an order of magnitude higher
than we have measured.’’. It took more than 30 years before they
were proved right by measurements on high-quality thin films,[14]
single crystals,[15,16] and ceramics.[111]
The unprecedented large polarization of the thin films was
initially thought[14] to be due to strain enhancement, but this is no
longer the case: good single crystals were eventually grown[15–19]
with Pr values very similar (Fig. 9) to those of the films: ca.
60 mC cm2 normal to (001) and, therefore, approximately
100 mC cm2 along [111]pseudocubic, and high polarization was
also found in ceramics.[111] Ab initio calculations also agree with
the statement that the polarization of bulk BiFeO3 is intrinsically
high[92,109] (ca. 90–100 mC cm2) and relatively insensitive to
strain.[109]
4.2. Thin Films and Strain Effects
In addition to having excellent ferroelectric properties as
discussed above, thin films of BiFeO3 often have different
crystallographic structures than single crystals do. Freestanding
Figure 9. Polarization of BiFeO3: bulk single crystal (top) and epitaxial thin
film (bottom). Figures reproduced with permission from [15] and [14],
respectively. Copyright 2007, American Institute of Physics (top) and 2003,
AAAS (bottom).
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eff
d33
@s
@s @P
¼
¼ 2QP"
@E @P @E
Figure 10. The absolute value of the ferroelectric polarization in thin films
(c) is essentially independent from in-plane compression of the films (a,b).
Reproduced with permission from [114]. Copyright 2008, American Institute of Physics.
the effective electrostrictive coefficients are Qeff33 ¼ ca.
1–4 102 m4 C2, with the lower limit being in good agreement
with experimental estimates[121]. The full electrostrictive tensor
has in fact been calculated by Zhang et al.,[122] who give values
of Q1111 ¼ 0.032 m4 C2, Q1122 ¼ 0.016 m4 C2, and Q1212¼
0.01 m4 C2, all of which are in the range estimated here from
reported piezoelectric constants. It is interesting to note that these
electrostrictive values are similar to those of BaTiO3 or
SrTiO3,[123] and yet the strain effect on the ferroelectricity of
BaTiO3 and SrTiO3 is much bigger. In other words, although the
piezoelectric coefficient of BiFeO3 is small, its electrostrictive
coefficient is not. The reason for this seemingly surprising result
is likely to be the small dielectric constant (see Section 5), which
affects piezoelectricity as given in Equation 3. Possible reasons for
the smallness of the dielectric constant are discussed in the next
section.
5. Dielectric Properties
5.1. Dielectric Constant from Radio Frequency to Optical
Frequency
(3)
where e is the dielectric constant. Using the experimentally
measured values of d33[14,111,121] we obtain from Equation 3 that
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dendritic films were prepared as early as the mid 1980s by Hans
Schmid and others in Geneva, and these are like single crystals. In
particular, their crystal class at ambient temperatures is
rhombohedral. However, when bismuth ferrite is epitaxially
grown as a thin film onto, for example, an SrTiO3 [001] substrate,
the resulting morphology is monoclinic, where the symmetrylowering distortion arises from in-plane contraction and outof-plane elongation as a result of lattice mismatch between film
and substrate. This has been characterized by several groups.[112]
An as yet unresolved issue is whether there is a further change
in symmetry, from monoclinic to tetragonal, as film thickness is
reduced. XRD and Raman spectroscopy data suggest that epitaxial
BiFeO3 grown on SrTiO3 becomes tetragonal[112,113] below a
critical thickness of ca. 100 nm. On the other hand, piezoresponse atomic force microscopy (PFM) studies performed on
ultrathin films[115] still show eight polarization variants, consistent with polarization oriented along the diagonals, as expected
from a monoclinic structure, rather than the two variants
expected from an in-plane-compressed tetragonal phase. While
this discrepancy is still unresolved, we note that both observations
are not necessarily incompatible, as the ultrathin films could
perhaps be ‘‘metrically tetragonal’’ but with the actual point group
being monoclinic. That is, while the external shape of the unit cell
may be tetragonal, the internal degrees of freedom responsible
for the polarization might remain monoclinic. Such a decoupling
between crystal class and internal symmetry has been previously
reported in other epitaxially strained perovskite thin films.[116,117]
The in-plane compression was initially thought[14,112] to
enhance the polarization, a natural assumption given the strong
effect of strain on the ferroelectricity of other perovskite
films.[118,119] As discussed in the previous section, however, this
is now known not to be the case. Direct experimental proof of the
small sensitivity of the polarization to the strain state was recently
published by Kim et al.,[114] who show that the polarization of
epitaxial BiFeO3 stays constant even as the epitaxial strain is
relaxed with increasing film thickness (Fig. 10). A newer study
has also been published by Jang et al.[120] looking closely at the
relationship between strain and polarization in BiFeO3; these
authors confirm that the spontaneous polarization does not
change in absolute magnitude but can be rotated out-of-plane
through the monoclinic symmetry plane.[120]
The reason for the relatively small sensitivity of BiFeO3 to
epitaxial strain is that its piezoelectric constant, which links
strain to polarization, is also relatively low (between
15–60 pm V1[14,15,111,121] compared with 100–1000 pm V1 for
other perovskite ferroelectrics). The piezoelectric constant of
proper ferroelectrics/improper ferroelastics with a centrosymmetric paraphase can itself be linked to a more fundamental
parameter: the electrostrictive coefficient, Q. This relates the
strain, s, to the square of the polarization, s ¼ QP2. The effective
piezoelectric coefficient is defined as the derivative of the strain
with respect to the electric field
The GHz dielectric constant of BiFeO3 at room temperature is
er ¼ ca. 30.[68,51,96,124] It peaks at the rhombohedral–orthorhombic
transition (825–840 8C), possibly – though not necessarily – due
to a ferroelectric–paraelectric transition. This dielectric constant
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is small compared with those of typical perovskite ferroelectrics
such as BaTiO3, (Ba, Sr)TiO3 and Pb(Zr,Ti)O3 (PZT), which, as
argued in the previous section, is the reason why the associated
piezoelectric coefficient is also smaller. The mean refractive
index, n, of BiFeO3 is[89] ca. 2.62, so the optical frequency
dielectric constant can be estimated as er ¼ n2 ¼ ca. 6.86. This is
only an average value, however; BiFeO3 is in fact strongly
birefringent with Dn ¼ ca. 0.34[89] meaning that the dielectric
constant at optical frequencies is very anisotropic.
Although ca. 30 can be regarded as the intrinsic dielectric
constant of this compound at radiofrequencies, the impedance
measurements in parallel-plate capacitors often yield higher
values: between 50 and 300 depending on sample morphology,
orientation, and frequency range. This is because at the
frequencies typically accessible by impedance analyzers
(100 Hz to 1 MHz), domain-wall motion and space-charge
contributions can be important and add to the measured
permittivity. While the intrinsic value er ¼ ca. 30 may seem
small for a ferroelectric, this value is not unreasonable. For one
thing, the ferroelectric Curie temperature of BiFeO3 is very high,
meaning that at room temperature the ferroelectric polarization is
already saturated and, thus, small electric fields will barely affect it
(the dielectric constant is essentially a measure of polarizability).
Furthermore, this is a strongly first-order transition to start with,
so again there is little phonon softening and thus the dielectric
constant predicted by the frequency-shift according to the
Lyddane–Sachs–Teller relationship can also be expected to be
very low. Finally, and this is just a hypothesis, it may be that
perovskite ferroelectrics in which the polarization comes from the
A site (e.g., PbTiO3 and BiFeO3) have intrinsically lower dielectric
constants than those where polarization comes from the B site
(e.g., BaTiO3). Experimentally this certainly seems to be the case,
but at present we know of no satisfactory explanation for this fact,
if indeed it is more than just a coincidence.
At low frequencies or at high temperatures, colossal dielectric
constants have also been reported[96,125] and these are clearly
due to finite conductivity leading to Maxwell–Wagner (M–W)
behavior.[96,125-127] The temperature at which the M–W effects set
in depends on the sample conductivity; for some samples this
effect happens at temperatures as low as 200 K;[96] in our own
single crystal and ceramics the finite resistivity effects typically
appear above room temperature, enabling a more confident
analysis of intrinsic dielectric effects.
weakness shows that they do not correspond to ferroelectric phase
transitions, but arise instead from weak coupling to another order
parameter, most likely magnetic.
Additional dielectric and conductivity anomalies are
reported[68] at TNeel ¼ 643 K (370 8C), clearly related to magnetoelectric coupling, and magnetodielectric coupling is also
responsible for the reported anomaly in the birefringence of
BiFeO3 at TNeel.[89] Another is reported at the heretofore
mysterious transition at 458 K (185 8C), although this dielectric
anomaly may itself be an artifact caused by the change in
resistivity.[125–128]
6. Magnetism
6.1. Magnetic Symmetry and Spin Cycloid
The local short-range magnetic ordering of BiFeO3 is G-type
antiferromagnet, that is, each Feþ3 spin is surrounded by six
antiparallel spins on the nearest Fe neighbors. The spins are in
fact not perfectly antiparallel, as there is a weak canting moment
caused by the local magnetoelectric coupling to the polarization
(see next section). Superimposed on this canting, however, is also
a long-range superstructure consisting of an incommensurate
spin cycloid of the antiferromagnetically ordered sublattices. The
cycloid has a very long repeat distance of ca. 62–64 nm, and a
propagation vector along the [110] direction.[129,130] The magnetic
easy plane (the plane within which the spins rotate) is defined by
the propagation vector and the polarization vector (Fig. 12). The
magnetic Neel temperature is ca. 643 K (370 8C) and the exponent
characterizing the sublattice magnetization as a function of
temperature, b, is known to be approximately 0.43 from
birefringence[89] and 0.37 from Mossbauer hyperfine splittings.[102] Other critical exponents are discussed in the
literature.[131,132]
The cycloidal model of spin ordering in bismuth ferrite was
first proposed by Sosnowska et al. (1982),[130] whose group has
made a number of detailed studies via XRD, neutron scattering,
Mossbauer measurements, etc.[130–135] However, in recent years
Zalesskii and co-workers[136–138] have proposed that the simple
cycloid is distorted at low temperatures. However, no published
data from either group indicate the phase-transition temperature
where the spin reorientation transition should occur.
5.2. Dielectric Anomalies at Magnetic
Transitions
Because bismuth ferrite is piezoelectric at all
temperatures below 1100 K, any magnetoelastic phenomena at its magnetic-phase transitions are apt to create responses in the
dielectric response. These are shown in
Figure 11. The subtle low-temperature anomalies at 200 and at 50 K coincide with the
temperatures where magnetic, magneto-optic
and elastic anomalies have been seen, as
discussed in the next section. None of the
dielectric anomalies is strong and, curiously,
none seems to affect the dielectric loss. Their
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Figure 11. Anomalies in the relative dielectric constant (), possibly due to coupling to magnetic
(or magnetoelastic) transitions at low temperature. The anomalies do not seem to affect the
dielectric loss (tan d). Adapted from [142], with permission. Copyright 2008, Insitute of Physics.
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It is also worth noting that in some single-crystal monodomain
samples the cycloid propagates along only one of the three
symmetry-equivalent <110> directions. This unique propagation
direction is suggestive of a magnetic symmetry-lowering effect
(from rhombohedral to monoclinic), as emphasized by Lebeugle
et al.[129] and Schmid.[139].
In 2007–2008 two groups reported evidence for further
magnetic phase transitions at 140 and 200 K. Cazayous et al. first
reported a transition at 140 K[141] and, independently, Singh et al.
found that one and an apparently stronger one at 200 K.[103] A
possible origin of these transitions is discussed below together
with evidence for spin-glass behavior.[142,143]
6.2. Spin Reorientation in Orthoferrites
REVIEW
Figure 12. Schematic representation of the spin cycloid. The canted antiferromagnetic spins (blue and green arrows) give rise to a net magnetic
moment (purple arrows) that is spacially averaged out to zero due to the
cycloidal rotation. The spins are contained within the plane defined by the
polarization vector (red) and the cycloidal propagation vector (black).
Figure reproduced with permission from [129]. Copyright 2008, American
Physical Society.
however, that the phenomena at 140 K are different from those
at 200 K.
The orthoferrites are, as the name implies, orthorhombic,
whereas BiFeO3 is crystallographically rhombohedral; however,
its local magnetic structure is monoclinic,[146] with a monoclinic
angle very near 908, which justifies approximating the spin
structure as orthorhombic, as in the orthoferrites. We note also
that the Fe–O–Fe exchange angle (ca.1568), octahedral rotation
angle (ca. 128) and Neel temperature (ca.640 K) are all in the
same range as those of the rare-earth orthoferrites.[147] On this
basis, one may hypothesize that the magnon anomalies
observed at 140 and 200 K may be indicative of spin reorientation
in BiFeO3 analogous to that observed in orthoferrites such as
ErFeO3.
On the other hand, the spin reorientation in orthoferrites such
as ErFeO3 is thought to be brought about by the magnetic
influence of the rare-earth ions.[148] Clearly this cannot be the case
in BiFeO3 as bismuth is not magnetic. So, again, whether or not
the phase transitions at 140 and 200 K are indeed due to spin
reorientation remains uncertain.
6.3. Spin-Glasslike Behavior
The evidence for spin-glass (or, at least, nonergodic) behavior in
BiFeO3 is[143,149] first that there is a large difference between its
field-cooled (FC) and zero-field-cooled (ZFC) magnetization
below ca. 240 K (Fig. 14) (weaker FC effects were also reported
by Pradhan et al.[150] and Nakamura et al.[151]); second, that there
is a cusp at ca. 50 K in the magnetic susceptibility;[143] and third,
that the temperature of the cusp in magnetic ac susceptibility
appears to be dependent upon the frequency of the magnetic
field.[143]
When magnetic spins are subjected to competing forces and
geometric constrains, frustration can result in a chaotic glassy
In the magnetic orthoferrites (e.g., ErFeO3) there are phase
transitions within the antiferromagnetic phase at which the
sublattice spin orientations rotate. These occur at temperatures
(90 and 103 K in ErFeO3) far below the Neel temperature
(TN ¼ 633 K in ErFeO3) and, hence, have nothing to do with loss
of magnetic order. Generally these transitions
occur in pairs: at the upper temperature the
spins begin to rotate out of plane; and at the
lower temperature, the rotation is complete so
that the spins are now 908 from their original
directions, perpendicular to the plane. These
phenomena are well understood in orthoferrites[144] and Raman spectroscopy of magnons
near the reorientation temperatures shows
frequency dips, cross-section enhancements,
and linewidth narrowing.[145] The dip in
frequency would be 100% (to zero) if there
were no magnetoelastic behavior, but actually
reaches 50% in ErFeO3[145] and only 5% in
BiFeO3.[103] The cross-section divergences are
shown for bismuth ferrite in Figure 13. The
linewidth narrowing for the magnons goes
from 3.5 to <1.9 cm1 (resolution-limited) at
the reorientation transition temperatures. Also Figure 13. (Left) Intensity of magnon peaks in the Raman spectra as a function of temperature.
shown in Figure 13 is the EPR susceptibility, These show clear phase transitions at ~140 K and ~200 K, the origin of which is as yet unclear but
which is tentatively attributed to spin reorientations. (Above right) Magnon linewidth narrowing
which shows discontinuities at both 140 shows "critical slowing down" of spin fluctuations near 140 K, proof that the cross section
and 200 K; these anomalies confirm the divergence cannot come from impurities. (Below right) Preliminary electron paramagnetic
interpretation of these two temperatures as resonance measurements show clear anomalies also at 140 and 200 K. Figures courtesy of M.
those of magnetic-phase transitions. Note, Singh (Puerto Rico) and Pavle Cevc (Ljubliana).
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section. We remind readers that strain is always unscreened and
therefore very long-range and generally mean-field: if the
magnetic-order parameter is coupled to strain, the mean-field
nature of the strain might induce mean-field behavior in the
magnetic state.
It is also useful to consider the absolute magnitude of the shift
in Tf with f. This is usually defined by a dimensionless sensitivity
parameter, K
K ¼ DTf =ðTf D log f Þ
Figure 14. Field-cooled (FC) vs. zero-field-cooled (ZFC) magnetization in
single crystals of BiFeO3 for different magnetic fields (H). The strong
difference between the two is consistent with a spin-glass state. Figure
courtesy of M. Singh, University of Puerto Rico.
state. The original spin-glass model of Kirkpatrick and
Sherrington[152] gave detailed predictions for such systems
within a mean-field theory. The spin glasses studied experimentally are centric; that is, their spatially averaged structure has an
inversion center. More recent work has generally applied
Ising-model statistics to such spin glasses, but bismuth ferrite
would be a rare (perhaps unique) case of a spin glass that is
ferroelectric and hence non-centrosymmetric (acentric). As
Fisher and Hertz have emphasized in their text,[153] no published
theories apply to acentric spin glasses, and Ising models definitely
cannot apply to them.
Spin glasses are characterized by the frequency dependence of
the peak in their magnetic susceptibilities. As the frequency is
increased, the peak in the ac magnetic susceptibility moves to
higher temperatures. If we call Tf the temperature at which the ac
susceptibility has a maximum for a measurement frequency f, and
TSG the extrapolated value of Tf at f = 0, then the spin-relaxation
time, t, varies as:
tðTf Þ ¼ a½TSG =ðTf TSG Þzn
Typically in a superparamagnetic crystal, 0.01 < K < 0.1;
whereas in a conventional spin glass, 0.001 < K < 0.01. In
BiFeO3, Singh et al. [143] found K ¼ 0.014, which is at the margin
between the two cases, and hence they infer that bismuth ferrite is
not a conventional spin glass. Further, they note that the
magnitude of the ac magnetic susceptibility increases with
frequency; which is not reasonable for any glass, since glassy
states are always less responsive as frequency increases. They
suggest that this might be due to electrical or mechanical
resonances in the kHz regime, but more work clearly is warranted
to clarify these issues, including what is the origin of the glassy
state and whether or not it is an intrinsic feature as opposed to a
defect-related phenomenon. An unpublished report was given on
glassy behavior in single crystals of BiFeO3 very recently which
may be helpful in this regard. Shvartsman et al. confirm
re-entrant non-ergodic behavior of the low-field magnetization at
low temperature, but they exclude a generic spin-glass phase,
since only cumulative relaxation is found after isothermal aging
below Tg instead of classic hole burning and rejuvenation.[155]
Other technical details need attention, such as the possible
presence in bismuth ferrite of the Almeida–Thouless (AT)
transition line, which describes the stability of a spin glass at finite
temperatures and magnetic fields.[156] Such an AT line is shown
as a function of magnetic field in Figure 15 for BiFeO3.[149]
Although the physical interpretation is not yet clear, it is worth
noticing that the extrapolation temperature of the AT line is
ca.140 K, which is one of the two critical temperatures of the
electromagnon spectra. It does not, however, coincide with the
freezing temperature extracted from the ac-susceptibility analysis.
(4)
where a is a constant independent of T and the exponent zn is a
characteristic spin-glass critical exponent describing the slowing
down of spin fluctuations near TSG. Although zn ¼ 7–9 for Ising
models, there are other systems known to have 1 < zn < 2, as in
the present case; La0.5Mn0.5FeO3 is a good example of such a
nonstandard spin glass,[154] with zn ¼ 1.0. When the susceptibility
data are analyzed quantitatively, a spin-glass freezing temperature
of 29.4 K is estimated, and the critical exponent zn ¼ 1.4 0.2 that
characterizes the relaxation dynamics. This critical exponent zn is
7–9 for Ising models, but 2 in mean-field theory.[152] This suggests
that the spin glass in BiFeO3 may be mean field; this would be
reasonable in view of the strong elastic effects manifest near the
magnetic transition temperatures, as discussed in the next
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(5)
Figure 15. Almeida–Thouless fit of the irreversibility temperature determined from the FC vs. ZFC data in Figure 14.
ß 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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6.4. Low-Temperature Ferromagnetism?
As explained earlier, BiFeO3 is antiferromagnetic at room
temperature, with the weak local canting moment being
completely cancelled by the averaging out effect of the cycloid.
However, there are several reports, including hysteresis measurements in single crystals (Fig. 16) suggesting that at very low
temperatures there could be a weakly ferromagnetic state.[157,158]
It is important to confirm whether or not this is intrinsic because,
although the net magnetic moment is minuscule (ca.106 mB per
Fe), it would have important consequences regarding magnetic
symmetry and, thus, also whether or not the linear magnetic
coupling is allowed. The existence of ferromagnetism at very low
temperatures would also reflect an underlying competition
between antiferromagnetic and ferromagnetic interactions,
which, of course, would be consistent with the spin-glass state
in the intermediate temperature range.
On the other hand, the observation of ferromagnetic hysteresis
at low temperatures is not universal and may be explained by even
a very small concentration of impurities; Lebeugle et al., for
example, note that just 1 mol% of paramagnetic Fe3þ (probably
due to the presence of Bi25FeO39) can account for all the
low-temperature magnetic enhancement in their single crystals,
and that removing such impurities with HNO3 removes virtually
all traces of ferromagnetism in their samples.[15]
6.5. Elastic Anomalies at Magnetic Transitions
Redfern et al. have used dynamic mechanical analysis (an
oscillating three-point bending measurement) to estimate the
elastic constant of BiFeO3 ceramics below room temperature. [142]
Their reported results show an anomaly between 200–250 K and
Figure 16. Magnetization of BiFeO3 single crystals at low temperatures.
Figure courtesy of M. Singh, University of Puerto Rico.
Adv. Mater. 2009, 21, 2463–2485
perhaps another one near 140 K. The exact temperature of
the anomaly at ca. 225 K depends strongly on the frequency of the
applied mechanical stress. This frequency dependence can be due
to a thermally activated defect state, with an Arrhenius-type
behavior, or else it is an indication of some glassy underlying
process. While the mechanical data alone does not allow
elucidation of the answer, it is worth pointing out that between
200 and 250 K there are also indications of a magnetic transition,
the nature of which is still unclear but possibly related to a
spin-glass state (see Section 6.3). It is therefore possible that the
elastic anomaly is due to magnetoelastic (magnetostrictive)
coupling to a glassy magnetic transition.
We have also measured the mechanical response of BiFeO3
above room temperature and several broad peaks are apparent.
These are rather puzzling, since no structural transition has been
reported between room temperature and the Neel temperature. It
is again possible that these mechanical anomalies are caused by
defect states, but they may also be real: as discussed in Section 2, a
large number of ‘‘ghost transitions’’, which are awaiting
clarification, have been reported for BiFeO3.
Resonant ultrasound spectroscopy has also been deployed to
characterize the elastic behavior of BiFeO3. Preliminary lowtemperature data (Fig. 17) shows a very clear transition in the
region 30–60 K, where elastic attenuation leads to the disappearance and re-entrance of the elastic resonances. This massive
attenuation is suggestive of a highly dissipative state. Given that
the magnetic measurements suggest a spin-glass state in this
temperature region, the elastic measurements are consistent with
coupling between elasticity and the spin glass, lending support to
a magnetoelastic mean-field character for the transition.
REVIEW
It is not easy to prove the existence of a spin-glass state
experimentally: superparamagnets can exhibit AT lines, pinned
domains can exhibit aging and rejuvenation, and relaxors exhibit
frequency dependent susceptibilities; so unambiguous evidence
will require many different kinds of measurement.
7. Magnetoelectric Coupling
7.1. Magnetoelectric Coupling and Spin Cycloid
The existence of a spin cycloid averages out any linear
magnetoelectric (ME) coupling between polarization (P) and
Figure 17. Resonant ultrasound spectroscopy of a BiFeO3 ceramic. All the
resonant peaks disappear in the low-temperature region between ca. 50
and 30 K, a temperature range in which the ac magnetic susceptibility and
dielectric constant have also been reported to show broad anomalies.
Other elastic anomalies can also be seen at higher temperatures. Figure
courtesy of Julia Herrero-Albillos and Michael Carpenter, University of
Cambridge.
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magnetization (M). Any macroscopic magnetoelectric coupling
must therefore be higher order (quadratic). Indeed, up to
magnetic fields of several Tesla the magnetically induced
polarization is found to be proportional to the square of the
magnetic field (Fig. 18a). The full magnetoelectric tensor was first
characterized by Tabares-Muñoz et al.,[159] and is given by (in
hexagonal coordinate axis with P3 parallel to the spontaneous
polarization)
P1 ¼ b111 ðH12 H22 Þ þ b113 H1 H3
(6)
P2 ¼ b113 H2 H3 2b111 H1 H2
(7)
P3 ¼ b311 ðH12 H22 Þ þ b333 H32
(8)
with experimentally measured coefficients b111 ¼ 5.0 1019 s A1, b113 ¼ 8.1 1019 s A1, b311 ¼ 0.3 1019 s A1,
and b333 ¼ 2.1 1019 s A1.[159]
Above a certain critical field, however, the magnetoelectric
polarization markedly changes, signaling a change in the spin
configuration. Ismailzade et al.[158] first reported spin flop at a
critical field, HC, of only 5 kOe, (1 Oe ¼ 104 T) but this was
almost certainly an artifact, since no subsequent measurements[159,161,162] were able to reproduce it. The real critical field
appears to be much higher, at ca. 20 T[161,162] (Fig. 18). Above this
critical value, the magnetoelectric polarization changes sign and
becomes linearly dependent on magnetic field (Fig. 18, left). Since
the linear magnetoelectric effect is forbidden by the cycloid, its
onset signals that the cycloid has been destroyed by the high
magnetic field. A second observation is that above the critical field
for the cycloid destruction (or spin flop), the field-induced
magnetization jumps to a higher value. Linear extrapolation of
this field-induced magnetization to zero-field yields a ‘‘remnant’’
magnetization ca.0.3 emu g1 (Fig. 18, right).
The theory behind these effects is subtle. The local (shortrange) magnetic symmetry of BiFeO3 is such that, if it were
centrosymmetric (paraelectric), it would be a perfect G-type
antiferromagnet with no net magnetic moment. However, the
ferroelectric polarization breaks the center of symmetry and
induces a small canting of the spins via the Dzyaloshinskii–
Moriya interaction. This canting results in the very small
magnetization of 0.3 emu g1.[32,162] Ferroelectrically induced
canting magnetism is neither new nor unique to BiFeO3, as it was
already reported for BaMnF4 in the 1970s.[28] What is special
about BiFeO3 is that, in addition to this canting, there is also a
ferroelectrically induced spin cycloid that averages out the local
canted magnetism. This cycloid appears because polarization can
also couple to gradients of magnetization, thereby inducing an
inhomogeneous spin configuration (the spin cycloid).[161,162] This
is the converse effect of the ferroelectric polarization induced by
magnetic spirals.[25,26]
Although the cycloid averages out the macroscopic canting
moment, this is locally still present at the unit-cell level. High
magnetic fields can destroy the cycloid, thereby recovering the
canted state and its associated remnant magnetization (Fig. 18b)
and, in this state, the linear magnetoelectric is allowed (Fig. 18a),
so both effects in Figure 18 are consistent with each other. The
spin cycloid can also be destroyed by doping[164] or by epitaxial
strain,[165] so fully strained epitaxial thin films can in principle
display a weak remnant magnetization, although not as big as
initially reported.[20,21]
7.2. Ferroelectric Control of Magnetism
Recent experimental works have explored in more detail the
relationship between the ferroelectric polarization and magnetic
symmetry. With the exception of the work by Kubel and
Schmid,[61] early bulk research was performed on samples which
were mostly either polycrystalline or polydomain single crystals
and, hence, some subtle directional effects were averaged out.
Groups at Saclay[129] and Rutgers,[140] however, have managed to
make ferroelectric monodomain crystals of BiFeO3 by growing
them at temperatures below the ferroelectric transition, and have
deployed very high-resolution neutron diffraction to elucidate the
relationship between ferroelectricity and antiferromagnetism in this compound.
Specifically, they have shown that the
magnetic moments rotate within the plane
defined by the polarization (P// [111]pseudocubic)
and the cycloid propagation vector (k//
[1,0,1]pseudocubic) (Fig. 19). This has profound
consequences, for if the direction of the
polarization is changed, so too will the
magnetic easy plane: indeed, by applying a
voltage and switching the polarization by 718,
both Lebeugle et al.[129] and Lee et al.[140] were
able to show that the magnetic easy planes
were rotated. Importantly too, the magnetic
easy plane can be switched only if the
Figure 18. Magnetoelectric effect in BiFeO3 (left): at low fields, P is proportional to H2, (quadratic polarization changes direction, but not if it
ME coupling). Above BC ¼ 20 T, P is linearly dependent on H instead. Since the linear ME is
merely changes polarity; 1808 switching of the
forbidden in the presence of a cycloid, the cycloid is destroyed above 20 T. Note that, in any case,
polarization should not affect the magnetic
5
the magnetically induced polarization is very small (ca.10 times smaller than the ferroelectric
orientation.
polarization). Once the cycloid is destroyed, the small canted magnetic moment is recovered
The recent observations in single crystals
(right). Extrapolation of the magnetization to zero field yields a small net magnetization of ca.
are closely related to a previous investigation
0.3 emu g1. Figures adapted from [161], with permission. Copyright 2006, Elsevier.
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on thin films reported by Zhao et al.[166] The thin films had no
spin cycloid (due to either strain or reduced thickness), and
instead they had the homogeneous G-type antiferromagnetism
(with a slight canting), with the magnetic easy plane perpendicular to the ferroelectric polarization. Using a combination of
piezoelectric-force microscopy and X-ray photoelectron microscopy, these authors have managed to visualize simultaneously
the ferroelectric and the antiferromagnetic domains, establishing
that both types of domains are completely correlated with each
other. Again, switching the polarization by an angle other than
1808 (in rhombohedral symmetry, the polar vectors can be
switched by 718 and 1098 as well as 1808, as discussed in Section
8) changes the magnetic easy plane. This electrically induced
switching of the magnetic easy plane is of seminal importance,
because it opens the possibility of magnetoelectric devices based
on the voltage control of magnetization (see Section 9).
REVIEW
Figure 19. Relationship between the magnetic easy plane containing the
spins, the vector of ferroelectric polarization, and the vector of cycloid
propagation. Rotating the polarization by 718 (i.e., switching only one of the
components of the polarization) results in a change of the magnetic easy
plane, meaning that sublattice magnetization can be switched by an
applied voltage. Reproduced with permission from [129]. Copyright
2008, American Physical Society.
antiparallel polarities for each direction: hence there are eight
different polar domains in BiFeO3. Separating adjacent domains,
there are three possible types of ferroelectric domain wall, which
are usually labeled according to the angle formed between the
polarization vectors on either side of the wall. When only one
component of the diagonal polarization is reversed (say, one
domain has [111] orientation and the adjacent one is [111]), then
the polar vectors form an angle of approximately 718 and the
domain wall that separates the two polarizations is called a 718
wall. When two polar components are reversed, it is a 1098 wall,
and when all three components of the polarization are reversed it
is a 1808 wall. This is schematically depicted in Figure 20.
Minimization of electrostatic and elastic fields imposes constraints on the orientation of the walls; typically, 718 walls are
parallel to {110} planes, 1098 walls are in {100} planes, and 1808
walls should be on planes containing the [111] polar vector.[167] Of
course, nonequilibrium walls can be in different planes and may
even be very irregular in shape.
8.2. Domain Size
8.2.1. Stripe Domains
Since the work of Landau and Lifshitz in 1935,[168] and later Kittel
in 1946,[169] it is understood that domain size scales as the square
root of film thickness. While their arguments were initially
proposed for ferromagnetic films, they were later extended to
ferroelectric materials[170] and ferroelastics,[171] and more recently
also to multiferroics[172] and nanostructures other than thin
films.[173] The basic argument is that domain size results from the
competition between a surface energy (demagnetization, depolarization, strain) which is directly proportional to domain width,
w, and a domain-wall energy that is proportional to the number
density of walls and, hence, inversely proportional to w. The
energy of the walls is also proportional to their size, which scales
as the film thickness, d. Minimization
pffiffiffi of these components leads
to the familiar expression w ¼ A d, where A is a constant. This
scaling has been experimentally verified for the ferroelectric
stripe domains of BiFeO3 films,[174] but not for ultrathin films,
which have a very different domain morphology (Fig. 21).
8. Domains and Domain Walls
Research on domains and domain walls has intensified recently
because i) the behavior of domains is directly responsible for
switching characteristics (switching of polarization takes place
through nucleation and growth of domains) and ii) domain size
scales with sample size, so thin films can have very small domains
and, therefore, a high volume density of domain walls. BiFeO3
displays new domain-wall related phenomena of its own which
make this subject particularly fascinating.
8.1. Domain Walls in Rhombohedral Symmetry
In rhombohedral BiFeO3 the ferroelectric polarization can point
along any of the four diagonals of the perovskite unit cell, with two
Adv. Mater. 2009, 21, 2463–2485
Figure 20. Schematic of the three types of ferroelectric domain walls
separating domains with one, two, or all three components of the polarization switched. The domain walls are labeled according to the angle
between the polarization vectors on either side. Note that in this simplified
picture the 718 and 1808 walls are not in their most stable configurations,
since the head-to-head polarization perpendicular to the wall would lead to
large electric fields at the interface.
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Figure 21. Ferroelectric stripe domains (left) and fractal domains (right) in
epitaxial thin films of BiFeO3. The latter tend to appear in very thin films
only. (PFM images of stripe domains and fractal domains courtesy of Y. H.
Chu and M. Bibes, respectively).
8.2.2. Fractal Domains
For ultrathin films the domains in BiFeO3 are no longer striped
and instead form irregularly shaped mosaic structures
(Fig. 20).[115,175] The critical thickness depends on substrate
and electrodes, among other things. The domain walls are rough,
and are well described by a fractal Hausdorff dimension, h ¼ ca.
2.5 (for a perfectly smooth domain wall it should be two). The
Hausdorff dimension describes the scaling between the area of
the domains and their perimeters; since the domain size results
from a competition between domain energy (proportional to area)
and the wall energy (proportional to perimeter), it is reasonable to
expect that Kittel’s law will need to be modified in order to
incorporate this ‘‘fractality’’. The modified equation turns out to
be[175]
h?
w ¼ A0 d 3hk
(9)
0
where A is a constant and h?, hjj are the Hausdorff dimensions of
the domain wall in the directions perpendicular and parallel to the
plane of the film, respectively. This becomes the standard Kittel’s
law when the walls are perfectly smooth (h? ¼ hjj ¼ 1).
The reason for the thickness-induced transition to fractal
morphology remains unknown, but we note that irregular walls
are elastically costly, so they cannot be the equilibrium
configuration in a perfect crystal. Two necessary ingredients
for their appearance must be a random distribution of pinning
defects and a low crystal anisotropy (so that the wall can deform
without much elastic cost). In this respect, the possible
thickness transition to a tetragonal symmetry for ultrathin films
of BiFeO3 [113] would indeed facilitate the latter, since tetragonal
c-axis domains face no elastic constraints (other than surface
minimization) on the in-plane orientation of the 1808 walls.
the conductivity of the walls is directly related to the type of
domains they separate. Thus, 1808 walls are the most conductive,
followed by 1098 walls and, finally, the 718 walls, which, in fact, do
not have any measurable transport enhancement. The authors
argue that there are at least two reasons for the enhanced
conductivity of the walls. First, the polarization normal to the
domain wall is observed not to be constant across it; this
generates an electrostatic depolarization field that may attract
charge carriers. Second, the electronic bandgap is considerably
reduced for the 1808 and 1098 domain walls.
A plausible explanation for the bandgap decrease has to do with
the local distortion of the Fe–O–Fe bond angle, which, as
discussed in Section 3, controls the orbital overlap. In the middle
of the domains, the octahedra are quite buckled (and hence
the gap is big). If the unit cell expands, the buckling angle can
become straighter, thereby increasing the orbital overlap and
reducing the bandgap. The local suppression of polarization at
the domain walls leads to precisely such a volume expansion via
the cancelling of the spontaneous strain (Fig. 22) and, hence, the
local straightening of the bond angles at the walls reduces the gap
in much the same way as temperature or pressure would.
This model is consistent with the correlation between the type
of wall and its conductivity. The absolute value of the polarization
is smallest in the middle of thep1808
ffiffiffi walls (Pwall ¼ 0), intermediate
in the 1098 walls (Pwall ¼ Pp0ffiffi/ 3 ¼ ca. 0.57P0), and maximum in
the 718 walls (Pwall ¼ P0p2ffiffi3 ca.0.82P0); therefore, the volume
change (and the associated change in conductivity) will be
biggest for the 1808 walls, intermediate in the 1098 walls, and
smallest in the 718 walls, in agreement with the observed
sequence of wall conductivities.
8.4. Domain-Wall Magnetization: Privratska–Janovec and
Daraktchiev Theories
Privatska and Janovec[177] observed that magnetoelectric coupling
could lead to the appearance of net magnetization in the middle of
antiferromagnetic domain walls. Specifically, they showed that
this effect is allowed for R3c space groups, which is the symmetry
of BiFeO3, but their group-symmetry arguments do not allow any
quantitative estimate. Later, Fiebig and co-workers analyzed
magnetoelectric coupling in the walls of multiferroics such as
YMnO3 and HoMnO3.[178,179]
Daraktchiev et al. have proposed a thermodynamic (Landau-type) model[172] with the aim of quantitatively estimating
whether the walls of BiFeO3 can be magnetic and, if so, to what
extent they might contribute to the observed enhancement of
8.3. Domain-Wall Conductivity
Domain walls have their own local symmetry and, hence, also
their own properties. In the case of BiFeO3, this includes
enhanced local conductivity. Ramesh and co-workers have
recently reported that certain domain walls of BiFeO3 are much
more conductive than the domains themselves.[176] Furthermore,
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Figure 22. The suppression of the ferroelectrically induced in-plane contraction (Q13 is negative) leads to a lattice expansion in the direction
perpendicular to the wall, which is accommodated by straightening of the
bond angle. This results in increased orbital overlap and higher conductivity.
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DG ¼
a 2 b 4
a
b
P þ P þ kðrPÞ2 þ M2 þ M4
2
4
2
4
g
þ lðrMÞ2 þ P 2 M2
2
REVIEW
magnetization in ultrathin films. Their starting point is a very
basic two-parameter expansion with biquadratic coupling (which
is always symmetry-allowed) between net polarization and net
magnetization
(10)
Analysis of the phase space of this thermodynamic potential
shows that it is possible for net magnetization to appear in the
middle of ferroelectric walls even when the domains themselves
are not ferromagnetic (Fig. 23). This, however, is presently just
only a ‘‘toy model’’ which does not take into account the exact
symmetry of BiFeO3, so it cannot yet quantitatively estimate how
much domain walls can contribute to the magnetization of
BiFeO3. The exact theory of magnetoelectric coupling at the
domain walls of BiFeO3 remains to be formulated.
9. Bismuth Ferrite Nanotubes, Nanowires,
Nanocrystals
There is a fast-growing body of research devoted to the
manufacture and characterization of complex nanoscopic shapes
other than thin films. These 3D nanostructures generally have
their own distinctive size effects, and multiferroic BiFeO3 is no
exception. For example, nanocrystals of BiFeO3 show enhanced
magnetization and superparamagnetism correlated with decreasing diameter[180] (Fig. 24). Similar size-induced magnetism has
also been reported for BiFeO3 nanowires[181] and nanopowders.[125] This is thought to be due to the large fraction of
uncompensated spins from the surfaces of the nanocrystals, an
effect that is well known from classic antiferromagnets such as
NiO.[182]
Prof. Wong at SUNY Stony Brook reported crystalline BiFeO3
nanotubes in 2004.[183] These tubes were 240–300 nm in diameter
and as much as 50 mm long. Prepared via a sol-gel technique and
Figure 24. Ferromagnetic hysteresis due to uncompensated surface spins
in BiFeO3 nanocrystals. Reproduced with permission from [180]. Copyright
2007, American Chemical Society.
a porous alumina (AAO) template technique, they were
polycrystalline with some amorphous content. Those authors
removed the alumina template completely by immersion in
NaOH, meaning that the tubes were left out in a pile which was
hard to characterize electrically. Zhang et al.[184] used porous
alumina templates instead, managing to make ordered arrays of
standing BiFeO3 nanotubes and measure their piezoelectric
hysteresis loops. This proved that they were indeed ferroelectric.
Such nanotubes may be of considerable interest in terms of
new theories relating to them.[185] Using a Landau-type thermodynamic analysis, the phase diagrams of magnetoelectric
nanotubes as a function of radius have been calculated. The
small diameter of the tubes affects the ferroelectric critical
temperature of the ferroelectric state as a result of the stress
produced by the curvature; when the ferroelectric critical
temperature approaches the magnetic one, the magnetoelectric
coupling can be enhanced by several orders of magnitude
(Fig. 25). The theoretical predictions regarding
magnetoelectric enhancement still await
experimental confirmation.
10. Device Applications
10.1. Ferroelectricity and Piezoelectricity
Figure 23. Domain-wall profile in a hypothetical multiferroic with biquadratic magnetoelectric
coupling between polarization and magnetization. Note that the domain walls have net magnetic
moment, even though the domains themselves do not.
Adv. Mater. 2009, 21, 2463–2485
Being a room-temperature multiferroic,
BiFeO3 is an obvious candidate for applications. Interestingly, however, the first application that may reach the market might not use
the multiferroic properties of BiFeO3 at all.
The remnant polarization of BiFeO3 is very
large, 100 mC cm2 along the polar [111]
direction. To put this into context, this is the
biggest switchable polarization of any perovskite ferroelectric, and is roughly twice as big as
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Figure 25. Magnetoelectric coefficient as a function of nanotube (R12 > 0)
and nanowire (R12 < 0) radius. The magnetoelectric coefficient increases
as the external radius decreases. Reproduced with permission from [185].
Copyright 2008, American Physics Society.
the polarization of the most widely used material in ferroelectric
memories, PZT. Moreover, unlike PZT, BiFeO3 is a lead-free
material, a bonus regarding health and safety. It is therefore not
surprising that manufacturers of ferroelectric memories such as
Fujitsu are considering BiFeO3 as the potential active material in
their next generation of ferroelectric memory devices.[186] For
such an application to ever come into fruition, however,
important obstacles must be removed, such as: i) the higher
conductivity (and thus also dielectric losses) of BiFeO3 relative to
PZT, ii) its tendency to fatigue,[187] and iii) the fact that it appears
to thermally decompose at voltages quite close to the coercive
voltage.[53]
A second potential application unrelated to magnetoelectric
properties is piezoelectricity. The piezoelectric coefficient of pure
BiFeO3 is actually quite small, as argued in Section 4.2. However,
its rhombohedral ground state means that mixing it with a
tetragonal ferroelectric such as PbTiO3 leads to a morphotropic
phase boundary (MPB) at a composition of 30% mol PbTiO3.[188]
This is important because MPBs are commonly thought to be the
key behind the large piezoelectric coefficients of PZT and
relaxors,[189] so the MPB of BiFeO3–PT might lead to equally large
piezoelectric constants. A newer MPB with enhanced piezoelectric coefficients has been reported[190] for a solid solution of
rhombohedral BiFeO3 with orthorhombic SmFeO3 (Fig. 26); it is
likely that other solid solutions between BiFeO3 and any of the
other orthoferrites should also display similar MPBs. In this
context, it is surprising that the widely studied solid solution
between BiFeO3 and LaFeO3 does not appear to have yet been
piezoelectrically characterized; perhaps this is due to its high
conductivity.
10.2. Terahertz Radiation
Another possible application which does not make use of the
magnetoelectric properties of BiFeO3 is its reported emission
of THz radiation. Takahashi et al.[191] reported that, when hit with
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Figure 26. Enhanced piezoelectric coefficient, d33, of thin films at the
morphotropic phase boundary between pure BiFeO3 and SmFeO3. Reproduced with permission from [190]. Copyright 2008 American Institute of
Physics.
a femtosecond laser pulse, BiFeO3 films emit THz radiation,
which is currently of great interest for many applications ranging
from telecommunications to security.[192]
Furthermore, the authors note that the THz radiation is
completely correlated with the poling state of the films (Fig. 27);
accordingly, THz emission could provide an ultrafast (picosecond
response time) and non-destructive method for ferroelectric
Figure 27. a) Experimental setup used by Takahashi et al. [191] for
measuring the THz emission of BiFeO3. (LT: low temperature; LSAT:
LaAlO3–Sr2AlTaO6). b) Experimentally measured radiation and c) the
Fourier components of the amplitude of the measured THz radiation
switch when the ferroelectric polarization is switched, indicating the
possibility of using THz-based optical methods for reading ferroelectric
memories. Copyright 2006, American Physics Society.
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Figure 29. The magnetization of a small island of ferromagnetic Co–Fe is
switched when a voltage is applied to the underlying BiFeO3 (white and
black correspond to different magnetic polarities). Figure reproduced with
permission from [200]. Copyright 2008, Nature Publishing Group.
10.3. Spintronics
But the real drivers behind most of the applied research on
BiFeO3 are magnetoelectric and spintronic applications.[193] Chief
among these would be memories that can be written using a
voltage and read using a magnetic field. Using a voltage for
writing has three advantages: i) this can be implemented in a
solid-state circuit without mobile parts, ii) it has a low-energy
requirement, and iii) the voltage requirements automatically scale
down with thickness. Reading the memory magnetically, on the
other hand, has the advantage that it is a non-destructive readout
process, unlike direct ferroelectric reading, which requires
switching the polarization in order to read it.
For such memories to actually work, the magnetic state
therefore must be a) electrically switchable and b) magnetically
readable. As discussed in the previous section, the first condition
is met by BiFeO3, because the easy plane of its antiferromagnetic
domains is correlated with the polar direction, and rotating the
ferroelectric polarization results in a rotation of the sublattice
magnetization,[129,140,166] i.e., the magnetic state of the sample
can be changed by a voltage. On the other hand, the second
condition is not directly met, because antiferromagnetic (or, at
best, weakly canted antiferromagnetic) domains cannot be easily
read.
An elegant solution to the problem of reading antiferromagnetic states consists in using the mechanism known as exchange
bias. Crudely, exchange bias is the magnetic interaction between
the spins at the uppermost layer of an antiferromagnet and a thin
ferromagnetic layer attached to it. The exchange bias modifies the
hysteresis loops of the ferromagnetic layer, either offsetting or
widening them.[194] What is relevant here is that voltage-induced
changes to the underlying antiferromagnetic domains will result
in changes to the ferromagnetic hysteresis of the upper layer,
which can then be read by conventional mechanisms. The
implementation of this concept for Cr2O3 (which is magnetoelectric but not ferroelectric) was first done by Borisov et al.,[195]
and the first investigation with an actual multiferroic (YMnO3)
was done by Laukhin et al.[196]
The race to implement this idea using BiFeO3 (which has the
advantage over YMnO3 that it works at room temperature) has
been on for a while, and has been punctuated by several
important milestones, such as the observation of exchange bias in
thin ferromagnetic layers grown on BiFeO3[197,198] (Fig. 28), the
REVIEW
memory readout. As an additional advantage, at such high
frequencies the response is insensitive to leakage, which
automatically gets rid of one of the major obstacles for
implementing BiFeO3 as a ferroelectric memory material.
Figure 30. Tunneling magnetoresistance (TMR) of BiFeO3 sandwiched
between (La,Sr)MnO3 and Co. Reproduced with permission from [193].
Copyright 2008, Institute of Physics.
correlation between exchange bias and ferroelectric domains,[199]
the observation that the antiferromagnetic domains can be
switched by a voltage,[166] and, most recently, the final proofof-concept that the exchange-biased ferromagnetic layer can
indeed be switched by a voltage[200] (Fig. 29).
A second line of work uses BiFeO3 as a barrier layer in
spintronics. Sandwiching BiFeO3 between two ferromagnetic
metals results in tunneling magnetoresistance[193,198] (Fig. 30).
For this, the only requirement is that the BiFeO3 layer be
reasonably insulating down to tunneling thicknesses. However,
an extra ingredient provided by BiFeO3 is the fact that it also
remains a robust and switchable ferroelectric down to a thickness
of 2 nm,[201] and thus it could in principle be used as an
electrically switchable tunnel junction, whereby the ferroelectric
state controls the magnetic state of the thin ferromagnetic
electrodes, thus modifying the tunneling magnetoresistance. A
similar concept using a ferromagnetic multiferroic (La–BMO)
was indeed demonstrated by Gajek et al.,[202] who showed that the
tunneling resistivity could be controlled both by electric and
magnetic fields, giving rise to a four-state memory device. The
voltage-dependent barrier characteristics of BiFeO3 have not yet
been established.
The above developments show that, at least in principle, it is
now possible to develop an MERAM (Magnetoelectric Random Access Memory) based on
BiFeO3. A schematic of such a device has been
proposed by Bibes and Barthelemy,[203] and is
reproduced in Figure 31.
11. Closing Remarks
Figure 28. Exchange-biased magnetic hysteresis loops of thin Co grown on BiFeO3. The four
figures correspond to measurements along the four in-plane orientations [100], [010], [010],
[100]. Reproduced with permission from [198]. Copyright 2006, American Institute of Physics.
Adv. Mater. 2009, 21, 2463–2485
It is difficult to write a review on a topic that is
so popular and that is continuing to develop at
ß 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
2481
www.advmat.de
REVIEW
certain models. We emphasize that the study of BiFeO3 is NOT all
wrapped up. So this is not a Bible about bismuth ferrite
properties; rather, it should be read as an inspiration for further
studies. We hope the reader finds it useful as such.
Acknowledgements
Figure 31. MERAM based on exchange-bias coupling between a multiferroic that is ferroelectric and antiferromagnetic (FE-AFM, green layer),
and a thin ferromagnetic electrode (blue). A tunneling barrier layer between
the two top ferromagnetic layers provides the two resistive states. Interestingly, BiFeO3 could act not only as the magnetoelectric active layer, but
also as the tunneling barrier. Reproduced with permission from [203].
Copyright 2008, Nature Publishing Group.
such a rapid rate. So, rather than just summarizing what is known
so far about BiFeO3, we have chosen to focus our attention here
on four basic issues which we believe are still open: 1) What are
the high-temperature phases and in particular what is the nature
of the MI transition (at high temperature? at high pressure? Are
they the same? Are they Mott-like? is the ‘‘Polomska transition’’ at
458 K extrinsic?). 2) What are the low-temperature magnetic
phases? (Is there a spin glass? If so, is it Ising-like? long-range?
mean-field? How does it couple to strain? Is Sosnowska’s
magnetic structure correct or Zalesskii’s or neither? Is there an
Almeida–Thouless AT line? Are there two magnetic transitions at
140 and 201 K or four? Is the spin-glass onset at 230 or 50 K? Is the
Volgel–Fulcher glass-freezing temperature 29 K? Is it a ferromagnet below ca. 10 K? Are these magnetic transitions intrinsic?).
3) What are the magnetoelectric properties near room temperature? (Is there a linear magnetoelectric term? quadratic? Is it
useful? for spin filters and spintronics? for memories?). 4) What
are the intrinsic properties of the domain walls? (Are they
ferromagnetic? If so, how much? Do they affect the functional
properties of thin films? What is the mechanism for enhanced
conductivity? Is it intrinsic or extrinsic? How will that affect the
performance of thin-film devices with a high density of domains?)
There is an urgent need for many more studies focusing on the
phase diagram and the dynamics. Very little yet is known about
switching processes. As a final, concrete example: If we don’t
know the magnetic space group or even point-group symmetry of
this material below room temperature in what could be several
magnetic phases plus a glassy phase, how can we even decide
whether a linear magnetoelectric coupling is allowed or
forbidden?
We cannot provide definitive answers to most of these
questions. For some readers that may be a disappointment:
Why write a 15 000-word review if you don’t know the answers?
Instead, our aim has been to provide a fair view of the pertinent
works from different sources, stating as clearly as possible what
we think the questions are, together with evidence for and against
2482
During the elaboration of this review we have had many fruitful exchanges
that have enhanced our understanding of this material. We would
particularly like to thank Michel Viret, John Robertson, Jorge Iñiguez, and
Hans Schmid for their insights. There are also a number of unpublished
results that have been kindly made available to us for this review; we are
indebted to Simon Redfern, Julia Herrero-Albillos, Hong Jiawang, Michael
Carpenter, Manooj Singh, Ram Katiyar, Manuel Bibes, Brahim Dhkil, Jens
Kreisel, Finlay Morrison, Eddie Chu, Ramamoorthy Ramesh, Maren
Daraktchiev, Maria Polomska, and Pavle Cevc for sharing their work
with us.
Received: September 25, 2008
Published online: May 4, 2009
Note added in proof: After submission of this paper, Choi et al. reported
[205] a polarization-controlled diode effect and large photovoltaic currents
in BiFeO3, opening the way for novel device applications that combine both
the ferroelectric and semiconducting properties of this material.
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