Giniyat Khaliullin Max Planck Institute for Solid State Research, Stuttgart

Transcription

1 Mott insulators with strong spin-orbit coupling Giniyat Khaliullin Max Planck Institute for Solid State Research, Stuttgart

2 LS driven unusual ground states & excitations motivated by: Sr 2 IrO 4 Na 2 IrO 3 Sr 2 VO 4 -s=1/2, perovskite 214-str. -s=1/2, honeycomb lattice -s=1/2, perovskite 214-str. spin one-half quasi 2D Mott systems

3 Outline: cuprate-like AF & magnons in Sr 2 IrO 4 Kitaev model physics in (Li/Na) 2 IrO 3 (?) magnetically hidden order in Sr 2 VO 4

4 Mott Insulators with t 2g orbital degeneracy d x5 O 2- e g d z 2 d x 2 -y 2 MT t 2g 3x orbital degeneracy d xy d yz d xz Sr 2 IrO 4 Na 2 IrO 3 Sr 2 VO 4 d 5, t 2g hole, S=1/2 d 1, t 2g electron, S=1/2 Like 2D cuprates but: orbital angular momentum L=1

5 d-orbitals

6 Orbital physics in TMO orbital structural / magnetic transitions: orbital order, spin structure metal / insul. transition, doping: orbit-selective MIT, orbital polarons d spin oxide heterostructures&interfaces: novel phases via orbital reconstruction charge multi-dimensional d-electron in oxides exotic quantum states in oxides: spin and orbital liquids spin-state crossover (Co,Fe ): orbital repopulation, magnetic collapse

7 Three different couplings in spin-orbital systems H = E CF + J SE + λ so Orbital-Lattice coupling Spin-Orbital superexchange spin-orbit coupling

8 Three different regimes in spin-orbital systems Jahn-Teller coupling: E JT Exchange interaction: J Spin-orbit coupling: λ E JT Goodenough-Kanamori spin-exchange rules AF Ferro H=J(S i S j )

9 Three different regimes in spin-orbital systems Jahn-Teller coupling: E JT Exchange interaction: permutation operator P ij = J Spin-orbit coupling: λ P(spin) P(orb) SU(4) spin-orbital fluctuations AF Ferro

10 Three different regimes in spin-orbital systems Jahn-Teller coupling: E JT Exchange interaction: J Spin-orbit coupling: λ? orbital frustration higher D more frustration quantum orbital physics

11 Three different regimes in spin-orbital systems Jahn-Teller coupling: E JT Exchange interaction: J Spin-orbit coupling: λ bond directional nature of orbital interactions = frustration Orbital anisotropy and frustration are directly translated into magnetic sector new route to exotic Hamiltonians & unusual phases

12 Relativistic spin-orbit coupling L orbital angular momentum S spin-orbit coupling: H= λ(ls) λ(ti 3+ ) = 0.02 ev 3d 4d 5d λ(ir 4+ ) = 0.4 ev

13 Strong SO coupling Low-spin Ir 4+ Single t 2g hole: s=1/2, l=1 λ~ 0.4 ev, unquenched L moment Quantum number J =L+S is formed t 2g J=3/2 J=1/2

14 spin-orbit entangled d-electron weak LS-coupling strong LS-coupling: L+S=J eff of cubic shape protected from JT complex wave-function carriers both spin-directions coherently

15 phase factor / quantum interference going from site-i to j : i A B j -collects phase factor (spin dependent) -quantum interference between A, B, depending on hopping geometry nontrivial topology of bands & interactions

16 An example: consider two types of bonding geometry G.Jackeli, G.Kh, PRL 2009 x H= J ( ) H= -J y Strong AF-Heisenberg Ferromagnetic Ising, z-axis: out-of-plane perovskite lattices triangular, honeycomb, pyrochlore,..

17 Iridium oxides, 5d(t 2g 5 ) 180 bonding Sr 2 IrO 4 (t 2g analog of high-tc perovskite La 2 CuO 4 ) 90 bonding Na 2 IrO 3 (depleted ABO 2 ; Ir ions on a honeycomb lattice)

18 Crystal structure of Sr 2 IrO 4 Octahedra elongated along c-axis Ir-O ab =1.98A Ir-O c =2.06A Staggered rotation of octahedra around c-axis by α 11 ο

19 Magnetic properties of Sr 2 IrO 4 Magnetization data: Cao et al., PRB 98 α φ Anomalously large weak FM M FM =0.14µ B [La 2 CuO 4 : 0.2 x10-2 µ B ] 1 Two options: AFM, large canting angle φ α Spins rigidly follow rotation of octahedra 2 Ferromagnetic?

20 Exchange Hamiltonian: bonds Active orbitals and their overlap Isospin Hamiltonian: Predominantly of Heisenberg form. Pseudo dipolar anisotropy: J 2 /J 1 ~J H /U Anisotropy solely due to Hund s coupling

21 Microscopic Hamiltonian of Sr 2 IrO 4 Dominant interactions spin angle bond angle ~ Y ~ X φ ~ Y ~ X Spins parallel to Ir-O bonds: strong spin-lattice coupling Rotated basis: isotropic Heisenberg:

22 Canting angle vs tetragonal distortion Magnetic Hamiltonian including Hund s coupling Phase diagram tetragonal orthorhombic Г 1 changes sign at large elongation of octahedra spin-flop transition Sr 2 IrO 4 Energy scale in Sr 2 IrO 4 : T N =240 K J~ 50 mev

23 exper. confirmation Sr 2 IrO 4 resonant (elastic) x-ray scattering B.J.Kim et al., Science 2009 Formation of isospin 1/2 Kramers doublet (selection rules, L3-edge only observed) Magnetic structure: strongly canted AF

24 Theoretical predictions for Sr 2 IrO 4 Spin-wave spectrum: Large out-of-plane gap of classical origin. Small in-plane gap of quantum origin. In-plane compression -> spin-flop transition RIXS: spin-orbit J=1/2 to 3/2 peak about 0.6 ev

25 Calculated RIXS intensity (magnetic spectra) J=3/2 sector J=1/2 magnons L. Ament, M. Daghofer, J. van den Brink, G.Kh. (PRB 2011; cond-mat 2011)

26 RIXS spectra in Sr 2 IrO 4 B.J.Kim et al. (cond-mat 2011) J=3/2 J=1/2 hard x-rays, Ir L3 edge: entire BZ is probed

27 Magnons (J=1/2 sector) measured by RIXS RIXS operator:

28 J=1/2 to 3/2 exciton moves like a hole in t-j model exciton magnon use SCBA

29 spin-orbit exciton (both orbital shape & spin) magnons (charge density unaffected )

30 Sr 2 IrO 4 summary (exp & theory): Isospin ½, 2D AF, broad magnon band, plus higher energy magnetic mode Experimental challenge: -doping of spin-orbit Mott insulators: SC? -unusual proximity effects?

31 Iridates with exchange bonds Two active orbitals/oxygen ions two different paths Isospin Hamiltonian Quantum Compass Model destructive interference between two paths: Heisenberg term vanishes exactly each bond has its own Ising easy-axis

32 Layered Iridates A 2 IrO 3 (A=Li,Na) Honeycomb lattice planes 90 -bonding z Ir x y bond-dependent Ising axes: FRUSTRATION Kramers doublets interact as in Kitaev Model A. Kitaev Ann. Phys 06

33 Engineering the Kitaev model Kitaev model yy zz xx Topological degeneracy Relevant for Quantum computation Cold atoms, optical lattices? (Demler et al.) Solid state realization? Li 2 IrO 3, Na 2 IrO 3 -Magn. order ~10 K -Intrinsic? -Impurity effect? O Ir Na

34 The Kitaev model xx yy Exactly solvable zz Short-range RVB spin liquid Emergent Majorana fermions GS degeneracy: depends on topology

35 The Kitaev s solution Introduce four Majorana fermions: zz Spin: xx yy where commute with H and are thus constants Ground state: E F Free Majorana fermions Dirac spectra like in graphene

36 Full Hamiltonian including 2Δ charge-transfer oxygen oxygen Kitaev (similar to U-process) Heisenberg (also direct dd) Final result (i) (ii) spin disordered conventional AF

37 Heisenberg-Kitaev model Honeycomb lattice Chaloupka/Jackeli/GKh, PRL 2010 Heisenberg AM Kitaev SL One more exact reference point:

38 4 sublattices, spin rotations

39 H rotated simple ferromagnet Original spin basis: stripy AF (no zero-point fluctuations!) For arbitrary

40 Three phases for A 2 IrO 3 FM in a rotated frame Chaloupka/Jackeli/GKh, PRL 2010

41 24-site cluster (exact): spin correlations 0.12 (exact) NN NNN NNNN Short-range RVB

42 spin-orbit coupling spins, magnons Majorana land Quantum phase transition: spin fractionalization

43 Doping of Kitaev-Heisenberg model Mean-field RVB phase diagram (J H /J K =1/2) T.Hyart, A.R.Wright, G.Kh, B.Rosenov (cond-mat 2011)

44 Spin-orbit insulators: d 5 versus d 1 3/2 1/2 3λ/2 3λ/2 doublet,1/2 quartet,3/2 d 5 (hole) Co 4+,Ir 4+, d 1 (electron) Ti 3+,V 4+,Nb 4+ H= α x Heisenberg + β x Kitaev H =?

45 Magnetically Hidden Order in Sr 2 VO perovskite d 1 -electron analog of cuprates

46 Zhou et al., PRL 07 Crystal structure of Sr 2 VO 4 d 1 Elongated along c-axis V-O ab =1.91A V-O c =1.94A V 4+, 3d 1 an electron analog of La 2 CuO 4 3d 9 xz/yz degeneracy: ideal

47 Phase transition: Isostructural, of first order tetragonal both below and above Ts sharp increase of c/a Zhou et al., PRL 07

48 Phase transition in Sr 2 VO 4 Zhou et al., PRL 07 Looks like (canted) AFM transition However, no magnetic Bragg peaks have been detected

49 Theoretical proposals on Sr 2 VO 4 (1) Imai et al., PRL 05 (2) Jackeli & Ivanov, PRB 07 Orbital & spin order Spin-singlet VBS Both states break translational symmetry (not observed) State 1: elastic magnetic Bragg peaks (not observed)

50 All happy families are alike; each unhappy family is unhappy in its own way. -- Leo Tolstoy (Anna Karenina)

51 All happy families are alike; each unhappy family is unhappy in its own way. -- Leo Tolstoy (Anna Karenina) so are the oxide families; each has its own skeleton in the cupboard no universal theory

52 G.Jackeli & GKh, PRL 2009: Ideal xz/yz degeneracy & spin-orbit coupling d 1 SO coupling Tetragonal field: Low energy quadruplet remains active Spin-orbit: Quadruplet split into Kramers doublets

53 Quadruplet: Two Kramers doublets The ground state doublet: nonmagnetic M spin =0 M orbital =0 First excited level:, magnetic

54 SE-interaction between quadruplets...obtained by projecting t 2g spin-orbital model onto the quadruplet subspace Isospins Inter-doublet transitions J=t 2 /U (about 10 mev in LaVO 3 ) s-o splitting between doublets

55 The ground state Low energy doublet is stabilized. Charge density of axial symmetry. Enhances c/a ratio. Staggered order of isospins and of the chirality of wave-functions. In-plane isospin order is selected by Hund s coupling: GS(J) = + GS(L-S) = +

56 Octupolar order y y x x Color map for real spin density distribution S x =+ (blue) S x =- (red)

57 Nature of the ground state ordering Isospin order in terms of physical spin & orbital moments Both spin and orbital moments are zero on every site. (No Bragg peak. Drop in magnetic susceptibility.) No quadrupole ordering: tetragonal symmetry respected Ordering of magnetic octupoles

58 Elementary & Magnetic Excitations Pseudoorbitals (Interdoublet M y ) Continuum, M x magnon Isospins ( octupon ) octupolar Bragg peak (x-rays)

59 Octupoles in TM-oxides? Octupolar Bragg peaks and octupons in Sr 2 VO 4 (resonant x-ray scattering) Unconventional magnetic excitation spectrum (neutron scattering) Unquenched spin-orbit coupling: large LS~0.5 (spin-resolved photoemission)

60 Summary Mott insulators with strong spin-orbit coupling Spin & Orbital subspaces entangled unusual symmetries and orderings Open problems: - Doping of spin-orbit Mott insulators - 3d electron octupolar orderings & dynamics - Heisenberg-Kitaev model: phase transitions