LETTERS Giant magneto-elastic coupling in multiferroic hexagonal manganites Seongsu Lee 1,2 , A. Pirogov 1,2 , Misun Kang 1 , Kwang-Hyun Jang 1 , M. Yonemura 3 , T. Kamiyama 3 , S.-W. Cheong 4 , F. Gozzo 5 , Namsoo Shin 6 , H. Kimura 7 , Y. Noda 7 & J.-G. Park 1,2,3 The motion of atoms in a solid always responds to cooling or heating in a way that is consistent with the symmetry of the given space group of the solid to which they belong 1,2 . When the atoms move, the electronic structure of the solid changes, leading to different physical properties. Therefore, the determination of where atoms are and what atoms do is a cornerstone of modern solid-state physics. However, experimental observations of atomic displacements measured as a function of temperature are very rare, because those displacements are, in almost all cases, exceed- ingly small 3–5 . Here we show, using a combination of diffraction techniques, that the hexagonal manganites RMnO 3 (where R is a rare-earth element) undergo an isostructural transition with exceptionally large atomic displacements: two orders of magni- tude larger than those seen in any other magnetic material, result- ing in an unusually strong magneto-elastic coupling. We follow the exact atomic displacements of all the atoms in the unit cell as a function of temperature and find consistency with theoretical predictions based on group theories. We argue that this gigantic magneto-elastic coupling in RMnO 3 holds the key to the recently observed magneto-electric phenomenon in this intriguing class of materials 6 . In nature, there are two distinct mechanisms known to induce relatively large atomic displacements. One is a ferroelectric transition of displacive origin involving a so-called soft mode, in which certain atoms move below a Curie temperature in such a way that the solid loses its inversion symmetry and becomes ferroelectric with non- centro symmetry 3 . This happens in the perovskite BaTiO 3 , for which the atomic displacement of Ti atoms is the main driving force behind its ferroelectric transition at 403 K (ref. 4). Another mechanism for relatively big atomic displacements is found in systems in which the ground-state degeneracy is lifted up by some kind of structural dis- tortion 5 . A good example is perovskite transition metal oxides having Mn 31 ions with a d 4 configuration, in which the twofold degeneracy of the e g levels splits, leading to diverse physical properties, such as the well-known colossal magneto-resistance observed in manganites 7 . Atomic displacements arising from either of the two mechanisms with symmetry-lowering transitions can be as large as a few per cent of their lattice constants: for example, the atomic displacement seen in the ferroelectric BaTiO 3 is in the range 0.05–0.4 A ˚ (ref. 4). Apart from these two exceptional cases, atomic displacements reported for other numerous ordinary materials are mostly extremely small, often of the order of 10 25 A ˚ . Therefore, detailed studies of how the atoms of a given solid move as a function of temperature are almost non- existent, in particular when the solid under investigation undergoes an isostructural transition without breaking its high-temperature symmetry. For rare-earth elements with relatively smaller ionic radius, RMnO 3 forms hexagonal manganites, whereas for rare-earth ele- ments having larger ionic radius RMnO 3 forms an orthorhombic structure 8,9 . Despite having the same chemical formula, there is a very important difference between the two structures. As shown in Fig. 1a, the MnO 5 bipyramids of the hexagonal structure form a layered structure on the a–b plane and, at the same time, they are well- separated from one another along the c axis by the rare-earth-element layers, leading to a natural two-dimensional network of the magnetic Mn atoms. Because of the disparate chemical environment surround- ing the Mn atoms—unlike the Mn ion of the MnO 6 octahedron with a t 2g –e g splitting—the Mn ions of the MnO 5 bipyramid have two low- lying doublets (xz, yz) and (xy, x 2 2 y 2 ) and one singlet state 3z 2 2 r 2 , in the order of increasing energies 10 . Therefore, the four d electrons of the Mn 31 of the hexagonal manganites occupy the two low-lying doublets and so there is no orbital degeneracy left, unlike in its counterpart in the MnO 6 octahedron of the orthorhombic manga- nites 11 . This structural difference makes the physics of the hexagonal manganite markedly different from that of the orthorhombic one. The hexagonal manganites undergo a paraelectric–ferroelectric transition at high temperatures 12 and, simultaneously, the crystal structure changes from P6 3 /mmc to P6 3 cm. In the ferroelectric phase of the P6 3 cm space group, Mn is at x > 1/3, forming a nearly ideal triangular lattice of Mn ions, as shown in Fig. 1b. Each Mn ion is connected to another either through one O3 atom or one of two O4 atoms located on the same a–b plane (see Fig. 1b). Because of the intrinsically frustrated nature of the triangular lattice with antiferro- magnetic interaction, Mn S 5 2 moments cannot order until well below their Curie–Weiss temperatures; these are h CW 52500 K and the Neels temperature T N 5 75 K for YMnO 3 (see the inset of Supplementary Fig. 1b). Therefore, the triangular lattice of the Mn atoms exhibits strong geometrical frustration effects with a so-called frustration parameter, f 5 jh CW j/T N , which is as large as 6.7 for YMnO 3 . When they eventually order, a clear anomaly is observed in the magnetic susceptibility (Supplementary Fig. 1), the heat capa- city (Supplementary Fig. 2) and the neutron-scattering data. For example, apart from the large frustration number f, the heat capacity shows that almost a third of the total magnetic entropy is released above T N (see Supplementary Fig. 2), while there is strong magnetic diffuse scattering with strong temperature dependence in neutron diffraction data in the supposedly paramagnetic phase 13 . Inelastic neutron scattering studies 13,14 also found unusually strong spin fluc- tuations still persisting even well below T N . Pioneering work in the 1960s 15 established that these hexagonal manganites have ferroelectric as well as antiferromagnetic transitions within a single compound. Recently there has been renewed interest 1 Department of Physics, SungKyunKwan University, Suwon 440-746, Korea. 2 Center for Strongly Correlated Materials Research, Seoul National University, Seoul 151-742, Korea. 3 Institute of Materials Structure Science, KEK, Tsukuba 305-0801, Japan. 4 Rutgers Center for Emergent Materials and Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08854, USA. 5 Swiss Light Source, Paul Scherrer Institut, Villigen 5232, Switzerland. 6 Pohang Accelerator Laboratory, Pohang University of Science and Technology, Pohang, 790-784, Korea. 7 Institute of Multidisciplinary Research for Advanced Materials, Tohoku University, Sendai 980-8577, Japan. Vol 451 | 14 February 2008 | doi:10.1038/nature06507 805 Nature Publishing Group ©2008