Landau theory of topological defects in multiferroic hexagonal manganites
Topological defects in ordered states with spontaneously broken symmetry often have unusual physical properties, such as fractional electric charge or a quantised magnetic field-flux, originating from their non-trivial topology. Coupled topological defects in systems with several coexisting orders give rise to unconventional functionalities, such as the electric-field control of magnetisation in multiferroics resulting from the coupling between the ferroelectric and ferromagnetic domain walls. Hexagonal manganites provide an extra degree of freedom: In these materials, both ferroelectricity and magnetism are coupled to an additional, non-ferroelectric structural order parameter. Here we present a theoretical study of topological defects in hexagonal manganites based on Landau theory with parameters determined from first-principles calculations. We explain the observed flip of electric polarisation at the boundaries of structural domains, the origin of the observed discrete vortices, and the clamping between ferroelectric and antiferromagnetic domain walls. We show that structural vortices induce magnetic ones and that, consistent with a recent experimental report, ferroelectric domain walls can carry a magnetic moment.
Recent experimental and theoretical advances in the field of multiferroics have clarified the microscopic mechanisms of coupling between ferroelectricity and magnetism in bulk materials and led to the discovery of many families of materials in which ferroelectricity is induced by a spin ordering. Examples include the orthorhombic rare-earth manganites, spinels, hexaferrites and delafossites, which usually have spin orderings of the cycloidal or conical spiral type(1); (2); (3). The resulting electric polarisation is highly susceptible to an applied magnetic field and can be easily rotated or reversed(4); (5); (6). However, this magnetically-induced polarisation is usually too small to allow manipulation of spin states by an applied voltage. Much larger electric polarisations are found in multiferroics such as BiFeO and the hexagonal rare-earth manganites, in which ferroelectricity results not from spin ordering but from electronic and lattice instabilities(9); (8); (7). Yet in these materials, the electric control of magnetism is not straightforward(10); (11); (12), since the direction of spins in the magnetically ordered state is not correlated with the sign of the macroscopic electric polarisation(13).
While enhancing bulk couplings between polarisation and magnetism is difficult, practical switching of a ferroic order parameter with an applied field invariably involves motion of the domain walls. Magnetoelectric switching therefore depends crucially on interactions between ferroelectric and ferromagnetic domain walls, which are not as well understood. In this context, the observed clamping between ferroelectric and antiferromagnetic domain walls in multiferroic hexagonal manganites(14) provides a unique prototype for investigation.
The hexagonal manganites, MnO, where denotes a small-radius rare earth ion (Dy, Ho, Er, Tm, Yb, Lu), Y or Sc, are improper ferroelectrics; electric polarisation appears as a by-product of a primary structural transition(7). The crystal structure consists of corner-sharing MnO trigonal bipyramids, which form triangular layers, separated by layers of ions. The structural transition above K results in periodic tilts of the MnO bipyramids and displacements of the ions along the axis normal to the layers(15); (16). This periodic lattice distortion makes the size of the unit cell three times larger and is often referred to as the trimerisation transition (see Fig. 1(a-c)). The anharmonic coupling between the trimerisation mode and a polar optical phonon mode induces the observed electric polarisation along the axis, (17).
At much lower temperatures, K, an antiferromagnetic ordering of Mn spins emerges. While there is a large body of evidence for the strong interplay between the microscopic spin, charge and lattice degrees of freedom in hexagonal manganites(16); (18); (19); (20), the sign of the overall antiferromagnetic order parameter is decoupled from the direction of the macroscopic electric polarisation, as such correlation is forbidden by symmetry. It therefore came as a surprise when non-linear optical measurements of YMnO demonstrated that ferroelectric domain walls are locked to magnetic ones(14). Furthermore, this clamping was found to be non-reciprocal, as “free” magnetic domain walls, not associated with the electric polarisation reversals, were also observed. Proposals for the clamping mechanism have included strain mediation(21) and renormalisation of spin interactions at ferroelectric walls(22); (23).
Recently, new observational evidence has shed light on the mechanism for clamping. A combination of conducting atomic-force microscopy and transmission-electron microscopy demonstrated that the ferroelectric domain walls are pinned to the boundaries of the structural domains that appear upon transition to the trimerised state.(24); (25) These measurements also revealed intricate patterns of unusual line defects. These so-called ‘cloverleaf’ defects,(24); (25) at which six different structural and ferroelectric domains merge, have also been seen in piezoresponse force microscopy.(26); (27) In Ref. (28) it was suggested that the line defects are discrete analogues of vortices, and that the change of polarisation sign at structural domain boundaries is a consequence of the special form of the coupling between the lattice distortion and electric polarisation originating from the “geometric” nature of ferroelectricity in hexagonal manganites(7); (17). These results put the clamping between the ferroelectric and antiferromagnetic domain walls into an entirely new perspective.
In this paper we study the interplay between structural, ferroelectric and magnetic defects in hexagonal manganites using an expansion of the free energy in powers of the corresponding order parameters and their gradients. Based on symmetries of ordered states of these materials, this approach allows us to identify stable topological defects and describe their mutual interactions in the most economical way. In Ref. (17) Fennie and Rabe discussed the Landau theory of improper ferroelectricity for spatially uniform states of hexagonal manganites. Extracting parameters of the Landau expansion from first-principles studies of YMnO, they showed that polarisation emerges due to a non-linear coupling to the trimerisation mode. We extend this theory to inhomogeneous topological defects and include spin degrees of freedom to study effects of the structural domain walls and vortices on magnetic ordering.
In the trimerised state three neighbouring MnO bipyramids tilt towards (or away from) their common equatorial oxygen atom (see Fig. 1(a-c))(7). As a consequence of the hexagonal structure of Mn-O layers, there are six distinct trimerised states, corresponding to six degenerate minima of the lattice energy. Being a periodic lattice modulation in a layered system, the trimerisation is described entirely by the amplitude and phase . The physical meaning of the phase is the azimuthal angle describing the in-plane displacements of apical oxygens (see Fig. 1(a-c)). The minimal-energy states can then be labelled by the six values of the phase: and . At the structural domain boundaries varies spatially between two of these six values.
Microscopically, the trimerisation is the condensation of the zone-boundary mode with wave vector , which breaks the P symmetry of the undistorted phase lowering it to P. Similarly, the spontaneous electric polarisation is proportional to the amplitude of the zone-centre mode with symmetry. This polar mode is stable in the P structure, but is non-linearly coupled to the unstable mode, and therefore appears together with the trimerisation.
The free-energy expansion in powers of , and their gradients,
is obtained using the transformation properties of the trimerisation phase and the polarisation under the generators of the high-temperature space group summarised in Table 1. We consider only the lowest-order stiffness terms accounting for the energy cost of spatial variations of and . are the Cartesian coordinates in the plane (see Fig. 1), and by symmetry, and .
The trimerisation phase and the stiffness terms, and , not considered by Fennie and Rabe(17), play an important role in the theory of topological defects. In particular, the form of the non-linear coupling, , giving rise to improper ferroelectricity, implies that for the electric polarisation induced in the states with and is positive (the , and phases(24); (28)), while for and it is negative (the , and phases). In other words, neighbouring trimerisation phases, separated by , have opposite electric polarisations.
The numerical values of the parameters , , etc. in Eq. (1) for YMnO are listed in Table 2. In order to determine them, we performed ab initio supercell calculations for the various lattice distortions (see Appendix A). In particular, Fig. 2(a) shows the variation of the total energy with respect to the magnitude of for two chosen values of and . It is clear that in the physically relevant range of , the total energy depends very weakly on . The -dependence of the energy first appears in the sixth-order of the expansion in powers of and the corresponding coefficient is small. The energy landscape in the plane therefore essentially has a rotationally invariant Mexican Hat shape with no barriers separating the six structural domains.
In reality, the -rotation of the tilted bipyramids is not a zero mode, because an additional term is generated by minimising with respect to and eliminating from Eq.(1), which lowers the energy by Fig. 2(c) shows the dependence of the total energy on for (at which the energy has minimum) and (at the top of the barrier separating two minimal-energy states). Clearly, the anharmonic coupling between and results in a strong dependence of the energy. Therefore, even though the emergence of ferroelectricity in hexagonal manganites is improper, the coupling of trimerisation to polarisation is the only factor that determines the energetic barriers between different trimerised states, and is responsible for replacing the accidental continuous symmetry of with the discrete Z symmetry.
This has a strong effect on the structure of topological defects in the trimerised state, which can be described as trajectories in the -plane minimising the energy for given initial and final conditions. For example, a structural domain wall corresponds to a path connecting two energy minima. The shortest path connecting two ‘neighbouring’ states whose trimerisation angles differ by (see Fig. 2(d)) is the lowest-energy domain wall. This path follows the bottom fold of the Mexican Hat where the potential barrier between the two minima is the lowest, so that the amplitude of the trimerisation in the wall is close to its bulk value. Figures 3(a-b) show the coordinate dependence of and across domain walls obtained by numerical free-energy minimisation. The domain wall width is Å and at the domain wall is reduced by about 10%.
Since the neighbouring energy minima separated by have anti-parallel electric polarisations, the structural domain wall is at the same time a ferroelectric domain wall (see Fig. 3(a-b)). The improper nature of ferroelectricity in hexagonal manganites forbids purely ferroelectric domain walls, i.e. the reversals of within one structural domain, since the sign of is uniquely determined by the sign of . Furthermore, the structural domain walls with , separating states with the same electric polarisation, are unstable: they ‘decay’ into lowest-energy domain walls with which we find to repel each other when they overlap. This explains the clamping between the structural and ferroelectric domain walls observed by Choi et al.(24)
Spherical or cylindrical structural domains are unstable against shrinking, which explains the scarcity of closed domain wall lines in hexagonal manganites.(25); (26) To be stable, the domain walls must terminate either at the surface of the sample or at another type of stable topological defect – the structural vortex or anti-vortex, shown in Fig. 3(c). At the vortex line, where the trimerisation amplitude vanishes, all six structural domains meet in such an order that the trimerisation phase changes by around a contour encircling the vortex line.(28) Such a defect cannot be unwound and can only be annihilated by an anti-vortex, around which the phase changes by (see Fig. 3(c)). Away from the core, the trimerisation phase varies strongly only at the six radial domain walls. The electric polarisation changes sign at each domain wall and varies six times along a loop encircling the vortex line. These vortices and anti-vortices are the ‘cloverleaf defects’ observed in Ref. (24).
Fig. 3(c) shows a vortex–anti-vortex pair configuration obtained by minimising the energy for a given distance between these defects. The domain walls diverge radially from the vortex/anti-vortex core with the -angle between neighbouring domain walls. Far from the core they bend and become parallel to minimise the total length of the structural boundaries, which gives rise to a linear potential between the discrete vortices as opposed to the logarithmically growing potential for continuous vortices.(29) Despite this confining potential, vortex lines and domain walls form dense networks(24); (25); (26); (27) that are snapshots of states close to critical temperature capturing the formation of these topological defects by large thermal fluctuations.(30); (31)
A different type of topologically stable domain pattern is uncovered by considering the lowest-order coupling of the inhomogeneous trimerisation to strains, which has the form
where are the Cartesian coordinates in the plane. With such a coupling, a parallel array of structural domain walls, each with the same increment of the trimerisation angle (), is topologically stable (see Fig. 3(d)). Because of the alternating electric polarisation at the structural walls, such a “-staircase” is at the same time a ferroelectric stripe domain state, as stabilised in thin films by the long range dipole-dipole interactions.(32); (33) These interactions are, however, insensitive to the sign of at the domain walls, whereas the applied strain selects the direction normal to the walls, in which increases monotonically.
Next we address the coupling between the magnetism and the structural/ferroelectric domain walls. The spins on the Mn ions in hexagonal manganites order in one of four different magnetic states: A, A, B, and B, shown in Fig. 1(d-g).(34) Their origin can be understood by considering the hierarchy of interactions between the magnetic moments on Mn sites. By far the strongest is the antiferromagnetic exchange between neighbouring spins in the triangular layers of Mn ions, which leads to the non-collinear 120 spin ordering. The magnetic easy-plane anisotropy, and the anti-symmetric Dzyaloshinkii-Moriya interaction with Dzyaloshinskii vector along the axis, confine spins to the plane. The local in-plane anisotropy axes, also favouring the spin angle on neighbouring Mn sites, selects either the “radial” (as in the A and B phases) or the “tangential” (as in A and B phases) orientation of spins (Fig. 1). Finally, the interlayer exchange interactions, which are more than two orders of magnitude weaker than the intra-layer interactions, lead to either even (“A” phases) or odd (“B” phases) symmetry under the two-fold screw rotation .(35)
Due to the strong in-plane exchange, the angle between neighbouring spins remains close to 120 throughout a magnetic domain wall. Furthermore, to match the directions of spins on both sides of the wall with the local in-plane anisotropy axes, the spins in the domain wall must rotate in the plane. Therefore, the structure of magnetic domain walls can be described by two angles, , quantifying the rotation of spins around the axis in the even and odd layers respectively,(34); (21) as shown in Fig. 1(h). The transformation properties of and (see Table 1) determine the form of the magnetic free-energy density:
where . The first term originates from the nearest-neighbour exchange. Comparing the energies of the 4 uniform phases (see Fig. 1): , , and , we conclude that the coefficient results from the in-plane magnetic anisotropy, describes the interlayer exchange interactions, while is related to a symmetric exchange anisotropy. The last term in Eq.(3) describes the weak ferromagnetic moment along the axis in the A phase with . Terms proportional to spin operators of power higher than 2 are neglected. Equation (3) is used to calculate magnetic structure of the topological defects.
We first consider magnetic domain walls within one structural domain , in which case . The walls separate two magnetic states related by the time reversal operation, so that across the wall. Figures 4 (a-b) show two topologically distinct types of such walls between the degenerate B states: the one in which spins in neighbouring layers rotate in opposite () and the same () directions. The free energies per unit area of the two walls are and respectively. The domain wall with has a nonzero magnetic moment, since the magnetic configuration in the middle of the wall, , is of the weakly ferromagnetic A type. The net magnetic moment per unit area of the wall is , where is the magnetisation in the A phase and is the domain-wall thickness.
Importantly, within a domain, the reference triangles of Mn spins that are used to define the four magnetic phases have their apical oxygen ions tilted either towards or away from a common centre (Fig. 1). This is because the tilts of the oxygen bipyramids, described by the angle , determine the in-plane magnetic anisotropy axes. A shift by one lattice constant (of the non-trimerised lattice) within the uniform domain results in a rotation of the anisotropy axes, since . To minimise the magnetic energy, therefore, and must transform in the same way: . This is why the free energy Eq.(3) depends on , and why, in general, the “covariant” angles , rather than , should be used to describe the magnetic phases.
Now considering structural inhomogeneity, it is evident that the magnetic structure must respond to the presence of a structural domain wall. At a structural domain wall with , for example, spins rotate either by , in which case is the same on both sides of the wall, or by , in which case . Thus, structural domain walls are also magnetic domain walls. In the lowest-energy configuration, (see Fig. 4(c)), while the next-lowest-energy configuration has (see Fig. 4(d)). Domain walls in which spins in neighbouring layers rotate over different angles, e.g. and , are higher higher in energy. It is important to stress that the thickness of the magnetic domain walls — Å — greatly exceeds that of the structural domain wall. The antiferromagnetic domain wall clamped to the structural boundary (Fig. 4(d)) has a nonzero moment along the axis equal per unit area of the wall.
With these considerations, we can understand the simultaneous presence of “clamped” and “free” antiferromagnetic domain walls in hexagonal manganites.(14) Every structural domain wall () induces a magnetic domain wall in which spins rotate by or (Figs. 4 (c-d)). Because of the sign change of electric polarisation at the structural boundary, these antiferromagnetic domain walls appear to be clamped with the ferroelectric domain walls. The “free” antiferromagnetic domain walls that do not follow ferroelectric domain boundaries are the antiferromagnetic domain walls within one structural domain (Figs. 4 (a-b)). These results imply that in the minimal-energy magnetic state of the structural vortex spins wind around the vortex core, and the total spin rotation angle along a loop encircling the vortex is . That is, structural vortices are also magnetic vortices. The small-angle neutron scattering experiment on HoMnO, the electric switching of magnetisation of coupled ferromagnetic/LuMnO thin films, and the magnetic force microscopy study of ErMnO, all indicated the presence of an uncompensated ferromagnetic moment at antiferromagnetic domain walls.(36); (12); (37) Our analysis summarised in Fig. 4 shows that both “clamped” and “free” antiferromagnetic domain walls induce magnetisation along the axis in their vicinity and several of them have a net magnetic moment. Although the weak ferromagnetic moment – arising from canting of Mn spins – is small, it can be significantly enhanced by the magnetisation of rare-earth ions coupled to Mn spins, as in the bulk A phase.(38)
In conclusion, multiferroic hexagonal manganites provide a rich playground for physics of topological defects in multiple coexisting orders. We presented a theory based on first-principles calculations that explains the observed coupling between the structural distortions, electric polarisation and spins at the domain walls and vortices. The significance of these findings lies in the fact that topological defects can dominate cross-coupling responses of bulk materials, such as magnetoelectric switching.
SA and MM were supported by the ZIAM Groningen under award MSC06-20 and by FOM grant 11PR2928. KTD acknowledges fellowship support from the International Center of Materials Research and computational resources provided by the CNSI Computing Facility at UC Santa Barbara through Hewlett-Packard. NS was supported by the ETH Zürich.
Appendix A First-principles calculations
We performed ab initio calculations using Kohn-Sham density functional theory (DFT), as implemented in the
We approximate the Kohn-Sham exchange-correlation potential using the local spin density approximation(42) with a Hubbard-U correction applied to the partially-filled manganese states following the Liechtenstein approach(43) with double-counting corrections in the fully localised limit. All calculations reported here were performed with values of eV for the Coulomb integrals and eV for the intra-atomic exchange coupling, as chosen previously by Fennie and Rabe(17). We enforce an A-type antiferromagnetic ordering for all calculations(17). With this choice of parameters and magnetic ordering, an insulating electronic structure in the high-symmetry (P6/mmc) crystal structure results, with a Kohn-Sham band gap of eV. Since this underestimates experimentally reported values of the band gap,(44) we carefully verified that no spurious metal-insulator transitions occur as structural distortions are introduced, so that the free-energy landscape contains no anomalous features.
The parameters given in Eq. (1) can be extracted by considering homogeneously
distorted periodic structures that are commensurate with the wave vectors of all distortions.
The smallest unit cell that can accommodate all possible values of , and contains atoms
The dependence of the free energy on with is shown in Fig. 2(b), reaffirming the stability of the polar mode in the high-symmetry structure, and therefore the improper nature of the ferroelectricity in YMnO. Fig. 2(a) and (c) respectively show computations for the trimerisation mode and the coupling between trimerisation and polar modes.
With all homogeneous Landau parameters given in Eq. (1) specified, we now turn our attention to the stiffness parameters. The most convenient way to proceed is to write the spatial inhomogeneity in and as a single harmonic, for example with . The stiffness energy has then the form
This expression is harmonic in mode amplitudes and . Note that in order to compute , we do not need to consider spatial variations in the field. The computational cost of explicitly computing by imposing various short wave vectors, , in our supercell density-functional calculations would be prohibitive. Instead we extract these harmonic terms using the method of frozen phonons combined with Fourier interpolation of the inter-atomic force constants, a method commonly used for computing phonon band structures. Fig. 5 shows our calculated phonon band structure for the high-symmetry structure (P6/mmc) of YMnO. The strongest instability at the point is the cell-tripling trimerisation mode. As previously noted, all phonons, including the polar mode, are stable in the high-symmetry structure.
Multiplying the dynamical matrix used in the computation of phonon modes by the weighted mass, we obtain a -dependent force-constant matrix:
and by identifying the relevant branch, the -dependent eigenvalues of are related to or . Hence, the required stiffness parameters can be extracted. As an example, Fig. 5 shows the extraction of by fitting the dispersion of the force constant of the unstable trimerisation branch. Using this technique, we find the stiffness parameters for YMnO listed in Table 2.
- The ABINIT code is a common project of the Université Catholique de Louvain, Corning Incorporated, and other contributors (URL http://www.abinit.org).
- We note that we do not impose directly a constraint on the value of . Rather, we impose the magnitude of the mode on the structure, which results in .
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