Comment on ‘‘Ferroelectrically Induced Weak Ferromagnetism by Design’’ The question of how ferroelectric polarization is coupled to magnetism in magnetoelectric multiferroics is of con- siderable current interest. A recent Letter [1] analyzes the important ‘‘ABO 3 ’’ class of perovskite multiferroics. A symmetry argument is presented that materials with anti- ferromagnetism on the B site lack linear magnetoelectric coupling E PLM ¼ P ðL  MÞ in the free energy, unlike A-site antiferromagnets [1,2]. Here P is polarization and L and M are antiferromagnetic and ferromagnetic moments. This Comment presents a distinct analysis of E PLM in ABO 3 multiferroics. We show that the argument of Ref. [1] does forbid E PLM if the final low-symmetry phase contains only one distortion that, like P, breaks all inversion sym- metries. In reality there are multiple distortions in this symmetry class, and cross terms generate E PLM even in B-site materials, although the mechanism is different than in A-site materials. Additional differences can emerge between A-site and B-site materials under dynamical as- sumptions beyond the static free energy. For concreteness we use BiFeO 3 , whose electronic structure has been extensively studied [3,4]. Below 1100 K it shows ferroelectric order with space group R3c and a 10-atom unit cell. In Ref. [1], the free energy is expanded from a model R  3c phase with inversion center on Fe. For now, assume that the polarization P arises from relative motion of the ionic sublattices [3] along the three- fold axis ^ z. There is a 13.8  rotation  of the O octahedra around Fe atoms ( ^ z is D in Ref. [1]), which is even under Fe-site inversion I B and odd under Bi-site inversion I A . However, distortions beyond P and  must be present, either by counting R3c degrees of freedom or because the 1.4  counterrotation of upper and lower triangles within an octahedron [5,6] cannot be obtained by combin- ing P and . Coordinates for an oxygen distortion  (Fig. 1) that causes this counterrotation and is orthogonal to P are provided in Ref. [7]. The  distortion, like P, is odd under I A and I B . For A-site magnetism, we have inversion eigen- values I A L ¼þL, and I B L ¼L. For B-site magnetism, I A L ¼L and I B L ¼þL. Allowed L  M terms are, with constants  1 ,  2 , F A ¼  1 P ðL  MÞþ  2  ^ z ðL  MÞ; F B ¼  1 P ðL  MÞþ  2  ^ z ðL  MÞ: (1) More generally, P is some I A ¼ I B ¼1 distortion, and E PLM results from cross terms between P and other such distortions, here . Magnetoelectric coupling should be sought in B-site structures with large -type distortions as well as in A-site structures with large . Financial support from NSERC (R. d. S.) and WIN (J. E. M.) is acknowledged. Rogerio de Sousa 1 and Joel E. Moore 2 1 Department of Physics and Astronomy University of Victoria Victoria, BC V8W 3P6, Canada 2 Department of Physics University of California and Materials Sciences Division, LBNL Berkeley, California 94720, USA Received 12 June 2008; published 16 June 2009 DOI: 10.1103/PhysRevLett.102.249701 PACS numbers: 75.80.+q, 77.80.e, 81.05.Zx [1] C. Fennie, Phys. Rev. Lett. 100, 167203 (2008). [2] C. Ederer and C. Fennie, J. Phys. Condens. Matter 20, 434219 (2008). [3] C. Ederer and N.A. Spaldin, Phys. Rev. B 71, 060401(R) (2005). [4] P. Ravindran, R. Vidya, A. Kjekshus, H. Fjellvag, and O. Eriksson, Phys. Rev. B 74, 224412 (2006). [5] F. Kubel and H. Schmid, Acta Crystallogr. Sect. B 46, 698 (1990). [6] The c and d O-O bond lengths in [5] indicate this counter- rotation of 1.4  . [7] See EPAPS Document No. E-PRLTAO-103-050952 for a detailed construction of the beta crystallographic dis- tortion. For more information on EPAPS, see http:// www.aip.org/pubservs/epaps.html. FIG. 1 (color online). Distortions  and  that reduce the ideal perovskite symmetry to R3c (along the threefold axis). The distortion  combines with  at order  to give both the observed counterrotation and a linear magnetoelectric coupling. PRL 102, 249701 (2009) PHYSICAL REVIEW LETTERS week ending 19 JUNE 2009 0031-9007= 09=102(24)=249701(1) 249701-1 Ó 2009 The American Physical Society View publication stats View publication stats