Neutron Scattering in Transition Metal Oxides

Transcription

1 Neutron Scattering in Transition Metal Oxides C. Ulrich Department Prof. B. Keimer Max-Planck-Institut für Festkörperforschung, Stuttgart, Germany physics of the 3d electrons strongly correlated electrons spin, charge, orbital degrees of freedom LaMnO 3 : a material we understand today some current frontiers: - orbital dynamics - colossal magnetoresistance - high temperature superconductivity

2 Physics of the 3d Electrons Strongly Correlated Electrons - high temperature superconductivity - colossal magnetoresistance interplay between spin, charge and orbital degrees of freedom J. Hemberger et al., PRB 66, (2002)

3 Colossal Magnetoresistance Manganese oxide: Pr 1-x Ca x MnO 3 Tomioka et al., PRB 53, 1689 (1996).

4 50 years strongly correlated electrons

5 Crystal Structure 3D - perovskite structure MnO 6 octahedra Pm3m Pbnm ideal cubic perovskite GdFeO 3 - type distortion La LaMnO 3 LaTiO 3 / YTiO 3 LaVO 3 / YVO 3 Tilt and rotation of the TiO 6 octahedra Distortion of the TiO 6 octahedra (Jahn-Teller Distortion)

6 Ruddlesden-Popper Series Crystal Structure: Perovskite Three layers RMnO 3 Cubic Perovskite R 3 Mn 2 O 7 Bilayer Perovskite

7 Crystal Field Splitting of the 3d levels e g - orbitals x 2 -y 2 3z2 -r 2 x 2 -y 2 Jahn-Teller splitting 10 3d levels 3z 2 -r 2 yz xz t 2g orbitals xy yz xz xy cubic splitting (octahedral) tetragonal splitting

8 Jahn-Teller Distortion ] U [\ - e g -orbitals are bond-directional - Coulomb repulsion from the O-atoms - distortion of the octahedra Strain versus Coulomb energy Distortion of the MnO 6 octahedra in LaMnO 3 (manganites): d = c a ~ 0.25 Å

9 Jahn-Teller Distortion - Distortion of the octahedra - Crystal symmetry is changed - Lifting of degenercy Splitting of the electronic levels e g - orbitals e g - orbitals l z = ±1 d xz d yz L z = 0 d xy t 2g orbitals l z = 0 d xy t 2g orbitals l z = ±1 d xz d yz

10 Cooperative Jahn-Teller Distortion Distortion of the octahedra Change in Crystal Symmetry - Distortion - Tilt - Rotation of the octahedra

11 The orbital degeneracy in ground state is lifted by: crystal field splitting caused by lattice distortions (Jahn-Teller Effect) spin-orbit coupling λ ~ 20 mev t 2g xz + i yz - i xy xz + i yz + 2i xy xz - i yz orbital ordering with complex combinations of wave functions unquenched orbital angular momentum Superexchange Interaction

12 elastic neutron scattering Orbital Order in LaMnO 3 J. Rodriguez-Caravajal et al., PRB 57, 3189 (1998) resonant X-ray scattering Τ ΟΟ = 780 Κ T < T OO : orbital order locks in exchange interactions Y. Murakami et al., PRL 81, 582 (1998)

13 Neutron diffraction in LaMnO 3 J. Rodriguez-Caravajal et al., PRB 57, 3189 (1998)

14 LaMnO 3 : high spin state 3d 4, t 2g3 e g 1 S = 2 LaMnO 3 Mn 3+ e g - orbitals x 2 -y 2 3z 2 -r 2 yz t 2g orbitals xy xz Hund s Rules: - largest value of total spin S - largest value of total angular momentum L - J = L-S less than half filling J = L+S more than half filling

15 Crystal Superexchange Structure Interaction of LaTiO 3 Goodenough-Kanamori Rules strong antiferromagnetic weak ferromagnetic electron cannot delocalize without considering the Pauli principle intra-atomic Hund s rule favors ferromagnetic alignment in intermediate state Spin: antiferromagnetic OO: ferromagnetic-type Spin: ferromagnetic OO: antiferromagnetic-type

16 Magnetic Order in LaMnO Κ 140 Κ resonant X-ray scattering T < T OO : orbital order locks in exchange interactions T < T N : magnetic order Y. Murakami et al., PRL 81, 582 (1998) LaMnO 3 A-type Antiferromagnet -ferromagnetic within the ab-plane -antiferromagnetic along the c-axis

17 Inelastic Neutron Scattering magnon dispersion Heisenberg Hamiltonian: H = Σ i j J S i S j J = exchange interaction IUHTXHQF\HQHUJ\ PRPHQWXP band width J

18 TRISP Spectrometer at FRM-II

19 Magnon Dispersion Relation in LaMnO 3 inelastic neutron scattering Q (A -1 ) [1,1,0] [0,0,1] T < T N : anisotropic spin wave spectrum reflects the orbital ordering pattern K. Hirota et al., Physica B 237, 36 (1997)

20 What are correlated electrons? electron-electron Coulomb interactions very weak independent electrons: ordinary metal very strong electron crystal: Mott insulator strongly correlated electrons in d- or f-electron metals: transport dominated by electron-electron interactions, very different from ordinary metals new theory of metals?

21 Metal-Insulator Transition Due to the screening of the outer electron shell, the intersite Coulomb interaction plays a dominant role for 3d-electrons. Energy gain due to electron hopping is comparable to the Coulomb repulsion. Consider a half filled conduction band: ideal metal: kinetic energy due to electron-hopping

22 Metal-Insulator Transition Hubbard Model - the kinetic energy (electron hopping) wishes to delocalize the electrons metallic behavior? - the Coulomb repulsion wants to localize the electrons. insulating behavior

23 Metal-Insulator Transition Mott-Hubbard Transition Balance between: - t Hopping Term Energy at Fermi-sea itinerant localized - U Coulomb Repulsion U/t Metal-Insulator Transition

24 Metal-Insulator Transition Mott-Hubbard Transition: Band Theory Fermi level Upper Hubbard Band Lower Hubbard Band Metal Insulator Final Step: The localized electrons tend to order magnetically!

25 Metal-Insulator Transition What drives the phase transition? ρ(ε F ) AFM Insulator Fermi level Metal Insulator U/t - doping - temperature - external parameters (magnetic field, stress,...)

26 Charge Order Pr 0.6 Ca 0.4 MnO 3 Mn 4+ (d 3 ) Mn 3+ (d 4 ) Resonant X-ray scattering of charge and orbital order M.v. Zimmermann et al., PRL 83, 4872 (1999)

27 Spin, Charge and Orbital Order Structural: O O O R T Mc H orthorhombic orthorhombic Jahn-Teller orthorhombic Orbital Order rhombohedral tetragonal monoclinic hexagonal PM SR CA AFM FM paramagnetic short range canted antiferromagnetic ferromagnetic I M insulating metallic J. Hemberger et al., PRB 66, (2002)

28 Spin, Charge and Orbital Order

29 After 50 years strongly correlated electrons LaMnO 3 well understood! Are Transition Metal Oxides well understood? NOT AT ALL!! - HTc superconductors (cooperates) - Unconventional HTc superconductors - Colossal Magnetoresistance - Frustrated Magnets - Quantum Magnetism - Orbital Excitations

30 LaMnO 3 : unsolved problems Inelastic Neutron Scattering: Spin Wave Dispersion Pr 1-x Ca x MnO 3 Pr 1-x Sr x MnO 3 / Nd 1-x Sr x MnO 3 / La 1-x Ca x MnO 3 Hwang et al., PRL 80, 1316 (1998) Dai et al., PRB 61, 9553 (1998) large deviations from predictions of simple Heisenberg model interaction with phonons? low-lying orbital excitations?

31 High Temperature Superconductors 7 underdoped overdoped 400K strange metal AFM insulator 93K superconductor HTc superconductivity charge carrier density in CuO 2 layers

32 Electronic Structure of YBa 2 Cu 3 O 7 T C = 93 K YBa 2 Cu 3 O 7 O 2- (2p 6 ) Cu 2+ (3d 9 ) x 2 -y 2 3z 2 -r 2 yz xz xy

33 Orbital physics is more complicated! Spin Structure in LaTiO 3 Ti 3+ 3d 1 Inelastic Neutron Scattering: Integrated intensity of the (½ ½ ½) antiferromagnetic Bragg reflection. G-type spin structure. expected for a 3D Ti 3+ ion: µ 0 = 0.85 µ B / Ti 3+ G x -type antiferromagnet 0.53 µ B / Ti 3+ ion along the a-axis 2D perovskite Cu 2+ : µ 0 = 0.60 µ B / Cu 2+ B. Keimer et al., PRL 85, 3946 (2000). C. Ulrich, M. Reehuis, J. Hemberger (2004).

34 Inelastic Neutron Scattering: Spin wave dispersion in LaTiO 3 J = 15.5 mev Magnon Dispersion Relation of LaTiO 3 along the (1,1,1) direction. Isotropic Heisenberg model with nearest neighbor superexchange. J = 15.5 mev Spin Gap = 3.3 mev B. Keimer et al., PRL 85, 3946 (2000).

35 The Ferromagnetic Insulator YTiO 3 LaTiO 3 : larger bond angle θ ~ wide band (AFM) Proposed Ti 3d 1 -orbitals YTiO 3 : smaller bond angle θ ~ (FM) narrow band smaller Ti-Ti hopping weakened superexchange spin ferromagnetism as predicted by electronic band structure calculations Goodenough-Kanamori rules obeyed Theory: Sawada & Terakura Mizokawa et al. ψ ψ 1,3 2,4 = c 1 = c 1 yz xz ± c 2 ± c 2 xy xy Experiment: Akimitsu et al. (neutrons) Itoh et al. (NMR) anisotropic spin wave spectrum expected

36 Inelastic Neutron Scattering in YTiO 3 magnetic moment (µ B /Ti 3+ -ion) b) (0, 1, 1) o (1/2, 1/2, 1/2) c YTiO 3 a) (0, 2, 0) o (1, 1, 0) c c a b Energy (mev) T = 5 K Magnon J = -3 mev Phonon Temperature (K) Ferromagnet: T C = 27 K µ 0 = µ B /Ti 3+ 0 (0, 0, 2) o (0, 0, 1) o (0, -1, 1) o (0, -1, 2) o (0, 0, 2) o (0, 0, 0) (0, 0, 1/2) c c (1/2, 1/2, 1/2) c (1/2, 1/2, 0) c (0, 0, 0) c fit to a Heisenberg model with isotropic ferromagnetic exchange E = 6 S J (1 γ q ) J = mev magnon gap = 0.16 mev C. Ulrich et al., PRL 89, (2002).

37 Orbital Fluctuations in LaTiO 3 t 2g orbitals versus e g orbitals - not bond directional: JT - coupling is relatively weak - two equivalent orbitals on every bond : quantum resonance - large degeneracy : fluctuations are enhanced t 2g -superexchange in LaTiO 3 c - bond : 4t 2 U spin spin : triplet orbital: singlet 1 ( S i S j + 4 ) ( τ i τ j + 4 n i n j ) ab resonance 1 pseudospin on orbital doublet spin : singlet orbital: triplet Spin-orbital resonance orbital fluctuations: with fixed exchange parameter J = 15.5 mev from neutron scattering: - antiferromagnetic state - reduced magnetic moment of 0.5 µ B - isotropic spin dynamic with a spin gap of 3 mev G. Khaliullin, S. Maekawa, PRL 85, 3950 (2000).

38 New Orbitally Ordered States derived from superexchange model with spin ferromagnetism imposed (Khaliullin & Okamoto, PRL 2002) 1.2 I ω (000) (00π) (πππ) (ππ0) (000) 1.2 reduced anisotropy due to strong orbital 1.0 quantum fluctuations naturally explains spatially isotropic 0.2 ω II magnon dispersions, small magnon gap 0.0 (000) (00π) (πππ) (ππ0) (000) prediction: ORBITONS (i.e. orbital wave)

39 Collective Raman Scattering orbital excitations: in LaTiO 3 Orbitons and YTiO 3 Collective orbital excitation orbital wave Dispersion Energy gap LaMnO 3 Observed by Raman light scattering? E. Saitoh et al., Nature 410, 180 (2001). Assigned to two phonon excitations (IR) M. Grüninger et al., Nature 418, 39 (2002). in analogy to magnons or phonons

40 Raman Scattering in LaTiO 3 and YTiO 3 high energy Raman peak at 230 mev in LaTiO 3 and YTiO 3 possibly two-orbiton excitation 3.0 Intensity (arb. units) T = 13 K 2-DOS YTiO 3 Intensity (arb. units) INS YTiO Energy (cm -1 ) zz 13 K 60 K 120 K 300 K 0.5 zx zz LaTiO Raman Shift (cm -1 ) phonons 1 st order phonons 2 st order calculation of the phonon-dos diploma-thesis: Mael Guennou

41 Temperature dependence of the Raman spectrum Intensity (arb. units) YTiO 3 (zz) T = 13 K T = 60 K T = 100 K T = 200 K T = 300 K Integrated Intensity (arb.units) Temperature (K) FWHM (cm -1 ) Raman Shift (cm -1 ) not a two magnon excitation not a polaron (both are insulators!) also seen in IR transmission (M. Grüninger, cond-mat/ )

42 Laser-Line dependence of the Raman spectrum Intensity (arb. units) YTiO 3 T = 13 K (z,z) nm nm nm nm nm Integrated Intensity (arb. units) Laser energy (ev) Raman Shift (mev) Raman resonance in the energy range of the optical d i -d j intersite transition Okimoto et al., PRB 51, 9581 (1995)

43 Two Orbiton Raman scattering process Two-orbiton process 2-orbiton excitations Ti d i Ti d j Two-magnon process YBa 2 Cu 3 O 6+δ insulating AF K.B. Lyons et al., Phys. Rev. Lett. 60, 732 (1987).

44 Conclusions neutron scattering is an excellent technique to study the interplay between spin, charge, and orbitals Elastic neutron scattering - - crystals structure - orbital order Nuclear Elastic neutron scattering - Magnetic - magnetic moment - spin direction Inelastic Neutron Scattering - Magnetic - spin gap energies - spin wave dispersion - strenght of the exchange interaction

45 Collaborators: Theory: G. Khaliullin, P. Horsch, A.M. Oles, Max-Planck-Institut FKF, Stuttgart, Germany. S. Maekawa, S. Okamoto, Tohoku University, Sendai, Japan. RIKEN, Saitama, Japan. J. Sirker, Universität Dortmund, Germany. Raman Light Scattering A. Gößling, M. Grüninger Universität zu Köln, Germany. Crystal Growth: S. Miyasaka, Y. Taguchi, Y. Tokura, University of Tokyo, Japan. H. Roth, M. Cwik, T. Lorenz Universität zu Köln, Germany. J. Hemberger, F. Lichtenberg Universität Augsburg, Germany. ILL LLB NIST HMI ESRF NSLS Inelastic neutron scattering: A. Ivanov, M. Ohl, M. Rheinstaedter, W. Schmidt, Institut Laue-Langevin, Grenoble, France. P. Bourges, D. Reznik, Y. Sidis, Laboratoire Léon Brillouin, Saclay, France. J.W. Lynn, NIST, Gaithersburg, Maryland. Neutron diffraction: M. Reehuis, Hahn-Meitner-Institut, Berlin, Germany. Synchrotron x-ray diffraction: P. Pattison, ESRF, Grenoble, France. Resonant x-ray diffraction: J.P. Hill, D. Gibbs, M.v. Zimmermann, NSLS, Brookhaven National Lab., New-York.