Microscopic origin of residual magnetic anisotropy and the quantitative polarization–spin-chirality relation in field-induced multiferroic phases of Y-type hexaferrites

In the Y-type hexaferrite Ba2Mg2Fe12O22 (BMFO), magnetic fields of only tens of millitesla rotate and reverse the ferroelectric polarization vector through a transverse conical spin state, a discovery that opened low-field magnetoelectric control of helimagnets. The original 2008 report left several questions open; two of them — direct neutron-diffraction determination of the field-induced magnetic structures and room-temperature generalization of the effect — have since been answered affirmatively by later work. What survives as an open problem is the microscopic, quantitative account of the weak residual magnetic anisotropy that pins the cone axis and spin chirality and produces hysteresis in the rotation and reversal of the polarization, together with the exact spin-structure–polarization–chirality relation in the field-induced multiferroic phases, where the standard two-fan transverse-conical model is now known to fail quantitatively. Solving this would convert the currently phenomenological descriptions of magnetoelectric hysteresis, energy barriers, and domain switching in hexaferrites into a predictive microscopic theory and would underpin deterministic chirality/polarity control for non-volatile magnetoelectric memory.

Background

A multiferroic is a material that orders both electrically and magnetically, and a magnetoelectric coupling allows one order parameter to be controlled by the field conjugate to the other. In spin-driven (improper) multiferroics the ferroelectric polarization $P$ is not independent of the magnetism: it is produced by the spin texture itself. The canonical mechanism, proposed by Katsura, Nagaosa and Balatsky and by Sergienko and Dagotto, is the spin-current model, also called the inverse Dzyaloshinskii–Moriya mechanism (the Dzyaloshinskii–Moriya interaction being the antisymmetric, relativistic exchange between neighboring spins): two canted neighboring spins $\mathbf{e}_i$ and $\mathbf{e}_j$ sharing a bond with lattice vector $\mathbf{r}_{ij}$ induce $\mathbf{P} \propto \mathbf{e}_i \times \mathbf{e}_j$ times a bond-dependent factor, so that a spiral spin order carrying spin chirality can carry a macroscopic $\mathbf{P}$. Y-type hexaferrites such as Ba$_2$Mg$_2$Fe$_{12}$O$_{22}$ (abbreviated BMFO; the sister compound Ba$_{0.5}$Sr$_{1.5}$Zn$_2$Fe$_{12}$O$_{22}$ is abbreviated BSZFO) are layered ferrites built of alternating magnetic L (spinel-like, large moment) and S (hexagonal, small moment) blocks stacked along the crystallographic $c$ axis, which order helimagnetically below roughly 200 K with a magnetic propagation vector $\mathbf{k}_0 \parallel [001]$: within the S blocks the spins form a proper screw (the spins rotate in a plane containing $\mathbf{k}_0$), while the L blocks remain nearly ferrimagnetic. Ishiwata et al. (2008) discovered that in BMFO a magnetic field $\mathbf{B}$ of only tens of millitesla, applied transverse to $\mathbf{k}_0$ below about 50 K, transforms the longitudinal conical order (cone axis along $\mathbf{k}_0$) into a transverse conical structure, whose cone axis lies along $\mathbf{B}$ and whose rotating transverse spin components carry a $\mathbf{P}$ that follows $\mathbf{B} \times \mathbf{k}_0$, in accord with the spin-current model; rotating or oscillating $\mathbf{B}$ then rotates or reverses $\mathbf{P}$ cyclically. The field-induced ferroelectric phases of these compounds are conventionally labeled FE1, FE2, FE3 in order of increasing field. Two spin features are conceptually central. First, the sense of rotation of the transverse spin components around the cone axis — the spin helicity or chirality — is the quantity whose sign, via the spin-current mechanism, sets the sign of $\mathbf{P}$; the magnetic polarity (conical order combined with a magnetization axis) and the chirality together define the ferroelectric domain state, and their domains are sometimes described as ferrotoroidal domains, domains of the toroidal moment $\mathbf{T} \sim \sum_i \mathbf{r}_i \times \mathbf{s}_i$. Second, the observed out-of-plane rotation of $\mathbf{P}$ with rotating $\mathbf{B}$ shows weak but finite hysteresis, indicating a weak residual magnetic anisotropy beyond the ideal isotropic cone. The 2008 paper inferred, rather than directly measured, the magnetic structures of the higher-field phases by analogy with BSZFO, left the microscopic origin of this residual anisotropy unexplained, did not establish the exact polarization–helicity relation in the higher-field phases, and posed room-temperature generalization as an open goal. In the accepted microscopic description of the transverse conical ferroelectric phase — the two-fan transverse-conical model — the moments along the propagation direction decompose into two alternating, fan-like sublattices whose opening angles and relative phase determine $\mathbf{P}$; resonant X-ray microdiffraction on Ba$_{0.5}$Sr$_{1.5}$Mg$_2$Fe$_{12}$O$_{22}$ has shown that this model fails quantitatively to reproduce the measured diffraction intensities, and a modified structure with an internal magnetic chirality was proposed instead. Neutron diffraction has since established the magnetic structures of the field-induced phases (refs 5 and 15), and room-temperature low-field magnetoelectricity has been demonstrated in related hexaferrites (refs 6 and 7); the open problem is therefore the narrower core that remains: a microscopic and quantitative theory of the residual anisotropy, its pinning and energy barriers, and of the exact structure–polarization–chirality relation in the field-induced multiferroic phases of Y-type hexaferrites.

Problem Statement

Provide a microscopic, quantitative account of the two coupled strands of the surviving open problem.

1. The microscopic origin and consequences of the weak residual magnetic anisotropy. Identify which sublattices, exchange paths, and anisotropy terms (single-ion anisotropy, Dzyaloshinskii–Moriya, dipolar, and magnetoelastic terms, taken as candidate contributions) produce the residual anisotropy and its pinning in Y-type hexaferrites such as Ba$_2$Mg$_2$Fe$_{12}$O$_{22}$ and Ba$_{0.5}$Sr$_{1.5}$Mg$_2$Fe$_{12}$O$_{22}$, and how these terms govern the alignment of the cone axis and the magnetic polarity, the spin helicity (chirality), the hysteresis in the rotation and reversal of the ferroelectric polarization $\mathbf{P}$, and the mixed-state switching dynamics as functions of temperature and magnetic field. The identification above and each of the consequences that follow from it are jointly posed as one strand.

2. The exact spin-structure–polarization–chirality relation in the field-induced multiferroic phases, i.e. the quantitative correction to the standard two-fan transverse-conical model (including the internal magnetic chirality proposed from resonant X-ray microdiffraction and the role of spin–orbit-induced hybridization of the oxygen $2p$ and iron $3d$ orbitals (the $p$–$d$ hybridization mechanism) versus the inverse Dzyaloshinskii–Moriya mechanism).

The two strands are coupled and jointly posed: the residual anisotropy of the first strand pins the cone axis, the magnetic polarity, and the spin chirality, and the exact spin-structure–polarization–chirality relation of the second strand fixes the polarization that this chirality produces. Together, the two strands determine the hysteresis in the rotation and reversal of $\mathbf{P}$ and make deterministic, low-hysteresis switching of the chirality and polarization in these compounds predictable and controllable.

References

1. Hosho Katsura, Naoto Nagaosa, Alexander V. Balatsky, 'Spin Current and Magnetoelectric Effect in Noncollinear Magnets', Phys. Rev. Lett. 95, 057205 (2005). DOI 10.1103/PhysRevLett.95.057205, https://doi.org/10.1103/PhysRevLett.95.057205
2. I. A. Sergienko, E. Dagotto, 'Role of the Dzyaloshinskii-Moriya interaction in multiferroic perovskites', Phys. Rev. B 73, 094434 (2006). DOI 10.1103/PhysRevB.73.094434, https://doi.org/10.1103/PhysRevB.73.094434
3. Shintaro Ishiwata, Yasujiro Taguchi, Hiroshi Murakawa, Yoshinori Onose, Yoshinori Tokura, 'Low-Magnetic-Field Control of Electric Polarization Vector in a Helimagnet', Science 319, 1643 (2008). DOI 10.1126/science.1154507, https://doi.org/10.1126/science.1154507
4. Shintaro Ishiwata, Yasujiro Taguchi, Hiroshi Murakawa, Yoshinori Onose, Yoshinori Tokura, 'Magnetic-field control of electric polarization in a helimagnetic haxaferrite Ba2Mg2Fe12O22', J. Phys.: Conf. Ser. 150, 042073 (2009). DOI 10.1088/1742-6596/150/4/042073, https://doi.org/10.1088/1742-6596/150/4/042073
5. Shintaro Ishiwata, Daisuke Okuyama, Kazuhisa Kakurai, Masakazu Nishi, Yasujiro Taguchi, Yoshinori Tokura, 'Neutron diffraction studies on the multiferroic conical magnet Ba2Mg2Fe12O22', Phys. Rev. B 81, 174418 (2010). DOI 10.1103/PhysRevB.81.174418, https://doi.org/10.1103/PhysRevB.81.174418
6. Y. S. Chai, S. H. Chun, S. Y. Haam, Y. S. Oh, Ingyu Kim, Kee Hoon Kim, 'Low-magnetic-field control of dielectric constant at room temperature realized in Ba0.5Sr1.5Zn2Fe12O22', New J. Phys. 11, 073030 (2009). DOI 10.1088/1367-2630/11/7/073030, https://doi.org/10.1088/1367-2630/11/7/073030
7. Yutaro Kitagawa, Yuji Hiraoka, Takashi Honda, Taishi Ishikura, Hiroyuki Nakamura, Tsuyoshi Kimura, 'Low-field magnetoelectric effect at room temperature', Nat. Mater. 9, 797 (2010). DOI 10.1038/nmat2826, https://doi.org/10.1038/nmat2826
8. Hun Chang, Hak Bong Lee, Young-Sang Song, Jae-Ho Chung, S. A. Kim, I. H. Oh, M. Reehuis, J. Schefer, 'Al doping effect on magnetic phase transitions of magnetoelectric hexaferrite Ba0.7Sr1.3Zn2(Fe1−xAlx)12O22', Phys. Rev. B 85, 064402 (2012). DOI 10.1103/PhysRevB.85.064402, https://doi.org/10.1103/PhysRevB.85.064402
9. Yuji Hiraoka, Y. Tanaka, M. Oura, Y. Wakabayashi, Tsuyoshi Kimura, 'The effect of field cooling on a spin-chiral domain structure in a magnetoelectric helimagnet Ba0.5Sr1.5Zn2Fe12O22', J. Magn. Magn. Mater. 384, 160 (2015). DOI 10.1016/j.jmmm.2015.02.028, https://doi.org/10.1016/j.jmmm.2015.02.028
10. Kun Zhai, Yan Wu, Shipeng Shen, Wei Tian, Huibo Cao, Yisheng Chai, Bryan C. Chakoumakos, Dashan Shang, Liqin Yan, Fangwei Wang, Young Sun, 'Giant magnetoelectric effects achieved by tuning spin cone symmetry in Y-type hexaferrites', Nat. Commun. 8, 519 (2017). DOI 10.1038/s41467-017-00637-x, https://doi.org/10.1038/s41467-017-00637-x
11. Y. Iguchi, Y. Nii, Y. Onose, 'Magnetoelectrical control of nonreciprocal microwave response in a multiferroic helimagnet', Nat. Commun. 8, 15252 (2017). DOI 10.1038/ncomms15252, https://doi.org/10.1038/ncomms15252
12. Kun Zhai, Yisheng Chai, Junzhuang Cong, Dashan Shang, Young Sun, 'Electric control of exchange bias in multiferroic hexaferrite Ba0.4Sr1.6Mg2Fe12O22', Phys. Rev. B 98, 144405 (2018). DOI 10.1103/PhysRevB.98.144405, https://doi.org/10.1103/PhysRevB.98.144405
13. F. P. Chmiel, D. Prabhakaran, P. Steadman, J. Chen, R. Fan, R. D. Johnson, P. G. Radaelli, 'Magnetoelectric domains and their switching mechanism in a Y-type hexaferrite', Phys. Rev. B 100, 104411 (2019). DOI 10.1103/PhysRevB.100.104411, https://doi.org/10.1103/PhysRevB.100.104411
14. V. Kocsis, T. Nakajima, M. Matsuda, A. Kikkawa, Y. Kaneko, J. Takashima, K. Kakurai, T. Arima, F. Kagawa, Y. Tokunaga, Y. Tokura, Y. Taguchi, 'Magnetization-polarization cross-control near room temperature in hexaferrite single crystals', Nat. Commun. 10, 1247 (2019). DOI 10.1038/s41467-019-09205-x, https://doi.org/10.1038/s41467-019-09205-x
15. V. Kocsis, T. Nakajima, M. Matsuda, A. Kikkawa, Y. Kaneko, J. Takashima, K. Kakurai, T. Arima, Y. Tokunaga, Y. Tokura, Y. Taguchi, 'Stability of multiferroic phase and magnetization-polarization coupling in Y-type hexaferrite crystals', Phys. Rev. B 101, 075136 (2020). DOI 10.1103/PhysRevB.101.075136, https://doi.org/10.1103/PhysRevB.101.075136
16. Yuma Umimoto, Nobuyuki Abe, Shojiro Kimura, Yusuke Tokunaga, Taka-hisa Arima, 'Out-of-plane electric polarization in double-fan magnetic phase of Y-type hexaferrite', Phys. Rev. B 101, 100403 (2020). DOI 10.1103/PhysRevB.101.100403, https://doi.org/10.1103/PhysRevB.101.100403
17. Jiahao Chen, Francis Chmiel, Jieyi Liu, Dharmalingam Prabhakaran, Paolo G. Radaelli, Roger D. Johnson, 'Switching of ferrotoroidal domains via an intermediate mixed state in the multiferroic Y-type hexaferrite Ba0.5Sr1.5Mg2Fe12O22', Phys. Rev. B 110, 134410 (2024). DOI 10.1103/PhysRevB.110.134410, https://doi.org/10.1103/PhysRevB.110.134410