Multiferroics: from the cosmically large to the subatomically small

A small but hardy core of researchers is using multiferroic materials to address fundamental questions on the nature of the universe, testing concepts in cosmology and high-energy physics.

When thinking of applications of multiferroic materials, with their simultaneous and coupled ferromagnetism and ferroelectricity, nanoscale electronic devices usually come to mind. In particular, the technological prospect of controlling magnetic properties using electric fields, which are far less energy consuming than their magnetic counterparts, is immensely appealing¹. Equally intriguing, to me at least, are recent alternative research activities that use multiferroics to address fundamental questions in cosmology and high-energy physics that are related to, for example, the formation of cosmic strings and the existence of an electron electric dipole moment.

The early universe in a lab

One of the most fundamental mysteries in all of science is the behaviour of the early universe immediately after the Big Bang. However, direct study of the early universe is, of course, extraordinarily difficult because of the insurmountable issues associated with replaying the Big Bang in the lab. Cosmologists have limited options: they can search for signals in relics of the early universe, such as the cosmic microwave background, or they can conduct computer simulations and study the evolution of their models in silico. Now, materials scientists have offered them a third option: by identifying real materials with behaviours analogous to those proposed for the early universe, cosmological theories can be tested in the lab simply by measuring the properties of these materials.

Multiferroics, such as yttrium manganite, YMnO₃, are leading candidates. The crystal structure of YMnO₃ consists of layers of Y³⁺ ions that separate layers of corner-shared MnO₅ trigonal bipyramids, in which the Mn ions form a triangular lattice (Fig. 1a). The emergence of ferroelectricity in this material is unusual, with the primary order parameter being the tilt of the MnO₅ bipyramids and the improper ferroelectricity emerging as a secondary effect². The potential describing the ferroelectric phase transition has the form of a ‘Mexican hat’ (Fig. 1a), in which the distance from the peak represents the magnitude of the tilting of the MnO₅ bipyramids and the angle around the hat represents its orientation. For small tiltings, which occur close to the phase transition temperature, the peak is smooth and the energy is almost independent of tilt angle; for larger tiltings, which occur close to the ground state, the coupling between the tiltings and the ferroelectric polarization results in six minima reflecting the six-fold symmetry of the lattice.

Figure 1: YMnO₃ as a model to study cosmic strings.

The ferroelectric phase transition between the high- and low-symmetry structures of YMnO₃ is described by a Mexican-hat-like potential (panel a). Soon after the Big Bang, the early universe is proposed to have undergone a symmetry-lowering phase transition that is described by a similar Mexican hat potential, resulting in the formation of cosmic strings (panel b). The ferroelectric domain structures (measured using piezoforce microscopy) emerging in YMnO₃ as a consequence of the phase transition contain a crystallographic analogue to cosmic strings, as shown by the comparison between simulated 3D strings in YMnO₃ and simulated cosmic strings. Figures are adapted with permission from Ref. 3 and Ref. 6, American Physical Society; and courtesy of T. Lottermoser.

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In the 1970s, in the field of cosmology, theoretical physicist Tom Kibble proposed the occurrence of symmetry-lowering phase transitions described by ideal Mexican hat potentials in the early universe (Fig. 1b) and showed that such events would have generated 1D topological defects in the vacuum, which became known as cosmic strings. Later, Wojciech Zurek used causality arguments to estimate the density of cosmic strings that should form as a function of the rate of passage across the transition. Zurek also proposed that the resulting Kibble–Zurek mechanism could be tested in any system with the appropriate symmetry. The Mexican-hat-like potential of the ferroelectric phase transition in YMnO₃ suggests it as a possible candidate.

A cross-section of the highly unusual ferroelectric domain structure of YMnO₃ shows black and white regions corresponding to opposite orientations of ferroelectric polarization³, as illustrated in Fig. 1a. Six domains of alternating polarization, which correspond to the six minima in the brim of the Mexican hat potential, intersect at meeting points. These intersection points in the cross-section are in fact 1D ‘strings’, analogous to their cosmic cousins! The similarity between the two types of strings becomes evident when comparing simulations of the 3D strings in the multiferroic with simulated cosmic strings (Fig. 1).

Having established that multiferroic YMnO₃ hosts the crystallographic equivalent of cosmic strings, it can be used to perform experiments in the lab that cosmologists would like to perform on the early universe. For example, the effect of varying the rate of early universe expansion can be simulated by cooling the material through its phase transition at different rates, and the effect on string formation can be compared by measuring the resulting domain structures. Indeed, the measured density of domain intersections as a function of cooling rate remarkably verifies the prediction of the amount of cosmic string formation as a function of the expansion rate of the early universe. An unexpected trend inversion is also measured at high cooling rate³; whether this points to new physics relevant for early-universe behaviour remains an open question.

The electron as an ideal multiferroic

As materials scientists, we are familiar with the concept that an electron has a magnetic moment as a result of its spin, and that the combination of magnetic moments when electrons form chemical bonds in solids leads to the varied and often technologically important magnetic properties of materials. Perhaps less well known is that the electron is also proposed to have a (very small) electric dipole moment, which, by symmetry, must share its orientation axis with the magnetic moment. The implications of the existence of the electron electric dipole moment are profound. First, it is direct evidence of time-reversal symmetry breaking, because the action of reversing time reverses a magnetic moment but not an electric one. As a result, the magnetic and electric dipole moments in an electron switch their relative orientation when time is reversed, so that the time-reversed state is different from the initial one. The product of charge (C), parity (P) and time (T) is known to be an invariant, thus the breaking of time-reversal symmetry implies a breaking of CP symmetry and provides a stringent test of fundamental theories that incorporate this symmetry in different ways. Second, it means that an electron is an ideal magnetoelectric multiferroic, because reversing either of the two moments with its conjugate field forces the other to reverse.

The electron electric dipole moment is so small that it has not yet been detected, but the ideal multiferroicity of the electron provides a route to search for its existence. In an applied electric field E, an electron with electric dipole moment d parallel to the field has an energy that is lower than that of an electron with d antiparallel to the field by an amount 2d • E (this is the usual Stark effect). As a result, at any finite temperature, more electrons align with their electric dipoles parallel to the field than opposite to it, generating an imbalance in the total magnetization that should be detectable by sensitive magnetometry measurements. Because the effect is tiny, the material constraints are very stringent, requiring unpaired electrons (so that their dipoles do not cancel out) arranged paramagnetically (so that they are free to reorient), a large ferroelectric moment (to amplify the effect of the external electric field) and cryogenic operation (to maximize the Boltzmann distribution between the two energy levels).

We have created a designer multiferroic material, (Eu,Ba)TiO₃, with the required properties, which enabled the highest-precision solid-state search to date for the electron dipole moment, setting an upper bound of 6.05 × 10⁻²⁵e cm for its value⁴. Our hopes for higher precision were thwarted by systematic errors introduced by dielectric relaxation and hysteretic heating during the ferroelectric switching process. In the meantime, optical measurements on ThO molecules pushed the limit even further, giving an upper bound of 8.7 × 10⁻²⁹e cm (Ref. 5), and required the elimination or revision of a number of models of high-energy physics that predict a higher value for d.

Tap-dancing snakes

Are there other scenarios in which multiferroics might help unravel the mysteries of the universe? Remember that the key property of multiferroics is that they are polar and break time-reversal symmetry. This combination is also proposed for axions, the hypothetical subatomic particles that could account for the rarity of CP-symmetry-breaking processes and that are a leading candidate for dark matter. A multiferroic measurement to test the properties of axions or even to try and detect them would certainly be intriguing. Taking another angle, an electric charge in a multiferroic is surrounded by the usual diverging electric field, which generates a magnetization along its field lines, inviting parallels to magnetic monopoles.

So, how much can multiferroics really contribute to solving fundamental problems in high-energy physics and cosmology? Cliff Burgess, a particle physicist at the Perimeter Institute for Theoretical Physics, suggested that “Like tap-dancing snakes, the point is not that they do it well, it is that they do it at all” (The Economist, 16 Mar 2013). In my opinion, however, the jury is still out. Perhaps multiferroics will not enable the development of a grand theory of everything, but we are certainly learning a tremendous amount about the physics and chemistry of multiferroics by studying them through these alternative lenses.

How to cite this article

Spaldin, N. A. Multiferroics: from the cosmically large to the subatomically small. Nat. Rev. Mater. 2, 17017 (2017).

Further information

The Economist: http://www.economist.com/news/science-and-technology/21573529-small-models-cosmic-phenomena-are-shedding-light-real-thing-how-build