Symmetry and Mechanism of Multiferroicity in Frustrated Magnets 黃迪靖 and 牟中瑜 Resonant soft x-ray scattering Ginzburg-Landau approach

1 Collaborators Soft x-ray scattering: J. Okamoto, H.-J. Lin, and C. T. Chen (NSRRC, Taiwan) K. S. Chao (National Chiao-Tung University, Taiwan) TbMn 2 O 5 : S. Park, S. W. Cheong (Rutgers University, USA) Acknowledgement L. L. Lee, H. W. Fu, and S. C. Chung (NSRRC, Taiwan) S. Ishihara (Tohoku University, Japan) Y. Tokura (Univ. of Tokyo, Japan) C. H. Chen (University of Taiwan) T. K. Lee (Academia Sinica)

2 Introduction Resonant soft x-ray scattering Ginzburg-Landau approach

3 Magnetism: ordering of spins Ferroelectricity: polar arrangement of charges strain  E T  E PZT “Fire”(-Pyro)electricity “Pressure”(-piezo) electricity

4 (Ferro)magnetism vs. (Ferro)electricity Perovskite structure (La,Sr)MnO 3 : spins from : 3d 3 or 3d 4 BaTiO 3 : polarization from cation/anion paired diploes O -2 Ti +4 Magnetic moment: - Ba Å 0.05 Å 0.04 Å + + Ti 3d 0 O 2p 2 unfilled d bands impurities inversion symmetry broken

5 BiMnO 3, BiFeO 3, Pb(Fe 2/3 W 1/3 )O 3 : 6s 2 lone pairs  off-center distortion  polar behavior uncorrelated with magnetism ? Mechanism of ferroelectricity PbTiO 3 : Pb-O covalent bond cubic 800 K tetragonal 300 K Pb-O planeTi-O plane Pb O Kuroiwa et al, PRL (2001)

6 Induction of magnetization by an electric field; induction of polarization by a magnetic field. - first presumed to exist by Pierre Curie in 1894 on the basis of symmetry considerations However, the effects are typically too small to be useful in applications! Magnetoelectric effect Materials exhibiting ME effect: Cr 2 O 3 BiMnO 3 BiFeO 3 ….. M. Fiebig, J. Phys. D: Appl. Phys 38, R123 (2005)

Three Scenarios

Lawes et al., PRL 91, , 2003 Hexagonal RMnO 3 : T C : K, T N = K BiFeO 3 : ferroelectric, ferroelastic, and weakly ferromagnetic; rhombohedrally distorted; T C =1100 K, T N = 650 K

c N N

10 Recently discovery in the coexistence and gigantic coupling of antiferromagnetism and ferroelectricity in frustrated spin systems such RMnO 3 and RMn 2 O 5 (R=Tb, Ho, …) revived interest in “multiferroic” systems

Recent Discoveries Frustrated magnetic systems. The magnetic phases are complicated; incommensurate AF orders seem to be common. Strong coupling between ferroelectricty and magnetism. TbMnO 3, Nature 426, 55, (2003); PRL 95, (2005). RMn 2 O 5, Nature 429, 392 (2004) (Tb); PRL 96, (2006) (Y).

12 *Can the ME be enhanced by the internal fields? Multiferroicity: coexistence of magnetism and ferroelectricity with cross coupling How to enhance the coupling?

13 Geometric ferroelectrics: hexagonal RMnO 3 BaNi(Mn,Co,Fe)F 4 For example: YMnO 3 : A-type AF, lacking lone pairs, bulking of MnO 5 pyramids & displacement of Y  polarization Origin of ferroelectricity van Aken et al., Nature Materials 3, 164, (2004) Y MnO 5

14 Site-centered charge order Electronic ferroelectrics: Combination of bond-centered and site-centered charge order Efremov et al., Nature Materials 3, 853, (2004) Bond-centered charge order Ferroelectric intermediate state O TM

15 Magnetic ferroelectrics: (frustrated spin systems) Dzyaloshinskii-Moriya interaction ( H =D. S 1 X S 2 ) spin current “electromagnon” (spin waves excited by ac E fields) Geometric ferroelectrics: hexagonal RMnO 3 BaNi(Mn, Co, Fe)F 4 Origin of ferroelectricity Sergienko and Dagotto, Phys. Rev. B 73, (2006) Katsura, Nagaosa and Baltasky, Phys. Rev. Lett. 95, (2006) Electronic ferroelectrics: Combination of bond-centered and site-centered charge order Efremov et al., Nature Materials 3, 853, (2004) Pimenov et al. Nature Phys (2006)

16 Nature, 426, 55 (2003) TbMnO 3 incommensurate AF order  ME effect

17 T=35 K T=15 K T N =42 K T C =27 K TbMnO 3 Kenzelmann et al., PRL 95, (2005) IC sinusoidally modulated collinear magnetic order, inversion symmetric IC noncollinear magnetic order, inversion symmetry broken

18 3 transitions on cooling. Magnetic field induces a sign reversal of the electric polarization. TbMn 2 O 5 Nature, 429, 392 (2004)

19 Difficulty in microscopic measurement L. C. Chapon et al. Phys. Rev. Lett. 93, , 2004 Scattering amplitude accumulates microscopic effects and is macroscopic:

20 Nature of the transition N. Hur et al. Phys. Rev. Lett. 93, , 2004

21 Tb O Mn 4+ O 6 octahedron q P orthorhombic structure (a  b  c,  =  =  = 90˚) AFM insulator (T N =42 K ) magnetization in the ab plane AFM square lattice with asymmetrical next-nearest- neighbor interactions, i.e., geometrically frustrated Spontaneous polarization P // b q  P Mn 3+ O 5 pyramid TbMn 2 O 5 Tb 3+ Mn 4+, Mn 3+ O 2- Chapon et al, PRL (2004) Blake et al, PRB (2005)

22 Blake et al., PRB (2005) 20° ab plane 6°6° 31°17° TbMn 2 O 5 Mn 4+ O 6 Mn 3+ O 5 Tb 3+ O 2- The spins lie in the ab plane. Within the ab plane, two zigzag chains of AFM-coupled nearest-neighbor Mn 4+ and Mn 3+ run in a direction parallel to the a axis. orthorhombic structure a= Å, b= Å c= Å

23 qxqx qzqz Temperature (K) Chapon et al, PRL (2004) (½ 0 ¼) commensurate incommensurate 33 incommensurate Neutron diffraction: complex spin structure AFM, T N = 42 K, q = (q x 0 q z ) Mechanism of ME effect ?

24 qxqx qzqz Temperature (K) Chapon et al, PRL (2004) (½ 0 ¼) commensurate incommensurate 33 incommensurate Neutron diffraction: complex spin structure Kobayashi et al, JPSJ(2004) AFM, T N = 42 K modulation vector q = (q x 0 q z )

25 Issues: -What is the underlying mechanism of the gigantic ME effect? -Is a spiral-spin configuration necessary? -Can collinear spins lead to a polarization?

26 Introduction Resonant soft x-ray scattering Ginzburg-Landau approach

27 Elastic x-ray scattering scattering form factor momentum transfer A volume element at will contribute an amount to the scattering field with a phase factor. Fourier transform of charge distribution. Bragg condition: q = modulation vector of charge, spin, or orbital order elastic scattering Fourier transform of spin distribution. detectable?

28 X-ray scattering : electron density charge scatteringmagnetic-moment scattering Non-resonant X-ray magnetic scattering is very weak. (for ~ 600 eV)

29 Resonant X-ray magnetic scattering electric dipole transitions F 1,1  F 1,-1 scattering amplitudes enhanced Hannon et al., PRL(1988) 2p 3/2 3d 2p 1/2 (F 1,±1 => S q )

30 beamline 4-m long elliptically polarized undulator UHV, two-circle diffractometer detector Soft X-ray Scattering Set-up at NSRRC, Taiwan H. J. Lin & C. T. Chen et al.

31 Soft x-ray scattering of TbMn 2 O 5 3d 2p 3/2 Mn cm TbMn 2 O 5 single crystal h = eV E  b T = 30 K q [100] [001] [010]

32 (½ 0 ¼) commensurate incommensurate Coexistence of ICM and CM AF order 3d 2p 3/2 Mn 640 eV

33 qxqx qzqz Kobayashi et al, JPSJ(2004) CM 3d 2p 3/2 Mn q z = 1/4 IC

34 AF transitions closely resembles ferroelectric transitions ICM AF order   ICM CM

35 q Antiferromagnetism and ferroelectrics are strongly coupled.

36 Introduction Resonant soft x-ray scattering Ginzburg-Landau approach

37 Magnetoelectric Effect (ME) *Induction of M by E, induction of P by H *Can the ME be enhanced by the internal fields?

38 Symmetry consideration Two important symmetries:

39 Symmetry properties

40 Inversion symmetry broken in magnetic phases

41 Blake et al., PRB (2005) 20° ab plane 6°6° 31°17° TbMn 2 O 5 Mn 4+ O 6 Mn 3+ O 5 Tb 3+ O 2- The spins lie in the ab plane. Within the ab plane, two zigzag chains of AFM-coupled nearest-neighbor Mn 4+ and Mn 3+ run in a direction parallel to the a axis. a b c a d a b c d d c b

42 Inversion symmetric magnetic phase

43 Inversion Symmetry versus collinearity

44 Inversion symmetric broken - spiral Example of non-common phases

45 Other spirals

46 Inversion symmetric broken - collinear

47 Coupling to polarization Inversion Symmetry in Magnetic Phase Odd orders of P involved: Even orders of P involved:

48 Magnetic induced polarization with inversion symmetry broken Lowest order -- the existence of internal fields

49 Symmetry Constraints

50 not consistent with expts.

51 Electric Polarization Change and Reversal Strong magnetic field along a or b axis:

52 Nature 429, 392, 2004 Consistent with our result but why it does not happen for b-axis? Possibility: a-axis is an easy axis, and it is very hard to reverse S b P must be odd in S a !

53 Direct comparison with expt commensurate+incommensur ate incommensurate

54 Nature of the transition collinear or Noncollinear and Inversion-symmetry broken Inversion-symmetric

55 Nature of the transition N. Hur et al. Phys. Rev. Lett. 93, , 2004

56 Effect of External Fields - the dielectric constant Nature 429, 392, 2004

57 Expansion of Free Energy

58 Change of exchange energy

59 Dielectric Anomaly (Lawes et al., RPL 25, , 2003) Small step

60 Large Step at IC transition

61 Wavenumber change of IC moments * Lock-in transition * Change of q

62 Connection with Microscopic Theories Sergienko and Dagotto, Phys. Rev. B 73, (2006) Dzyaloshinskii-Moriya interaction with atomic displacement

63 Katsura, Nagaosa and Baltasky, Phys. Rev. Lett. 95, (2006) Theory of Katsura, Nagaosa and Baltasky 1. Polarization without atomic displacement Efremov et al., Nature Materials 3, 853,

64 How to induce polarization without involving atomic displacement? Essential Physics: Motion of magnetic moments induces electric dipoles! – the intrinsic Aharonov-Casher Effect

65 Katsura, Nagaosa and Baltasky : Motion of magnetic moments= spin current

66 Aharonov-Casher Effect in condensed matter (Meier and Loss, PRL 90, ,2003)

67

68 Conclusion

69 Conclusion For magnetic phases with inversion symmetry broken TbMn2O5 Consistent well with our expts on TbMn2O5 *Response to the internal field: *Response to the external electric field: Consistent with magneto-elastic effect on exchange energy

70 Induction of magnetization by an electric field; induction of polarization by a magnetic field. - first presumed to exist by Pierre Curie in 1894 on the basis of symmetry considerations Magnetoelectric effect M. Fiebig, J. Phys. D: Appl. Phys 38, R123 (2005)

71 Old Examples Ferromagnetic, ferroelectric, and ferroelastic at Tc=61.5K

72 Antiferromagnetic (spiral) & ferroelectric at Tc=28K Ferroelastic at high temperature but Antiferromagnetic at Tc=25-70K