View this article J. Phys. Soc. Jpn. 74 (2005) 1419

Ordering of Toroidal Moments Studied by Resonant X-ray Scattering [May 13, 2005]
Youichi Murakami (Dept. of Phys., Tohoku Univ.)

The strong correlation between electrons in solids underpins a great variety of phenomena, for example, high-TC superconductivity and colossal magnetoresistance [1]. This wide diversity of electronic states in these systems is driven by a small change in external conditions such as doping level, temperature, and magnetic field. Actually, some different kinds of order coexist or are close to each other in their phase diagram. The order parameters in these phases are characterized by the charge, spin, and orbital of an electron and lattice degrees of freedom [2]. We need to clarify the orderings of these electronic states to understand the phenomena microscopically. Recently, resonant X-ray scattering (RXS) has been introduced as a powerful technique for detecting the electronic orderings [3]. This technique is based on measurements of reflections near the absorption-edge energy of the focus ion, because the sensitivity of X-ray scattering from ordered structures can be significantly enhanced at the energy.

Fig. 1: Crystal and magnetic structure of GaFeO3. On the right panel, outward (inward) arrows represent the spin direction of Fe1 (Fe2) atoms antiparallel (parallel) to c.[4]

In the May issue of JPSJ Arima et al. report on a new type of RXS, i.e., magnetoelectric X-ray scattering in GaFeO3 [4]. This material is unique in having ferromagnetic order below 205K coexisting with electric polarization. Figure 1 shows the crystal and magnetic structure of GaFeO3, where two inequivalent Fe sites (Fe1 and Fe2) are present in the unit cell. The Fe1 and Fe2 ions are oppositely shifted from the center positions (dashed lines) of the six coordinated oxygen ions in the b-direction. The spin directions of these ions are either parallel or antiparallel to the c-direction. They pointed out that the RXS in this material is related to the ferroic-ordering of toroidal moments. Normally the toroidal moment is associated with the magnetic moment as shown in Fig. 2. How is the magnetic moment in GaFeO3 connected with the toroidal moment? Arima et al. defined the toroidal moment τi for the i-th Fe ion by the outer product of the displacement ui from the center position and the magnetic moment μi : τi = ui x μ i /2. According to this definition, GaFeO3 has the toroidal moments. Let us consider the shift and the magnetic moments of iron ions in GaFeO3 from a different point of view, using a more naive definition of the toroidal moment. We may consider the shifts and the moments of GaFeO3 (the left-hand side of the equation in Fig. 3) as the sum of three moments, i.e., toroidal, magnetic quadrupole, and dipole moments (the right-hand side of the equation in Fig. 3). Since the scattering amplitude from the dipole moments in the electric multipole transition vanishes for linear polarization, the RXS observed by Arima et al. may originate from the sum of the toroidal and magnetic quadrupole moments. It is an interesting problem for the future to search for the RXS from pure toroidal moments or pure magnetic quadrupole moments in appropriate systems.

Fig. 2: Toroidal moment in the original meaning. Fig. 3: The shift and the moment in GaFeO3 are expressed by the sum of toroidal, magnetic quadrupole, and dipole moments.
The observation of the toroidal moments is very important because multiferroic properties (i.e., the correlation between magnetism and ferroelectricity) may originate from toroidal moments. Pronounced optical properties occur in multiferroics, e.g. the refractive index and absorption change linearly with the wave vector of the electromagnetic wave and the magnetization of the sample. Recently, Kubota et al. have reported a non-reciprocal directional dichroism in the X-ray region [5]. The toroidal moments must play an important role at the interface of artificial superlattice, which is a part of electronic device in gSpintronicsh, since the inversion symmetry should be broken at the interface. Moreover, the toroidal moments may be present in biological systems: magnetic ions in a complex biological molecule like hemoglobin. The relation between the local electronic state and the structure of the molecule must be a very interesting problem in connection with the functions. Thus, the concept of toroidal moments will become important in the future science and technology.

References
[1] M. Imada et al.: Rev. Mod. Phys. 70 (1998) 1039.
[2] Y. Tokura and N. Nagaosa: Science 288 (2000) 462.
[3} Y. Murakami et al.: Phys. Rev. Lett. 80 (1998) 1932.
[4] T. Arima et al.: J. Phys. Soc. Jpn. 74 (2005) 1932.
[5] M. Kubota et al.: Phys. Rev. Lett. 92 (2004) 137401.

The above article should be referred as "Y. Murakami : JPSJ Online-News and Comments [March 13, 2005]" when citing.

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Last updated 2005-5-13