Giniyat Khaliullin Max Planck Institute for Solid State Research, Stuttgart

Transcription

1 Magnetism and doping effects in spin-orbit coupled Mott insulators Giniyat Khaliullin Max Planck Institute for Solid State Research, Stuttgart

2 Witczak-Krempa, Chen, Y-B.Kim, Balents (Annu. Rev. 2014) Hubbard model with spin-orbit coupling: unconventional magnets, insulators, metals correlation U spin-orbit coupling λ

3 Witczak-Krempa, Chen, Y-B.Kim, Balents (Annu. Rev. 2014) Hubbard model with spin-orbit coupling: unconventional magnets, insulators, metals correlation U spin-orbit coupling λ

4 Ionic multiplets vs Fermionic bands Mott strong S p i n o r b i t a l a l g e b r a correlation weak Fe r m i o n i c s t a t i s t i c s Landau spin-orbit coupling λ

5 spin-orbit λ Three competitors: Jahn-Teller Δ oxide families exchange J λ > Δ,J λ Δ,J spin-orbit coupled Mott insulator 5d 1 3d Δ~J~λ 4d TM-ion

6 Spin-orbit coupled Mott insulators spin-orbit quenched La 2 CuO 4 S 3d 4d 5d Co Ru Rh Os Ir CoO 2 Ca 2 RuO 4 J = S + L correlation KOs 2 O 6 Sr 2 IrO 4 spin-orbit coupling

7 D O P I N G: unconventional J-magnetism spin-orbit quenched La 2 CuO 4 S unconventional metal & SC? CoO 2 J = S + L correlation D o p i n g d-wave SC Ca2 RuO4 D o p i n g? KOs 2 O 6 Fe r m i o n i c s t a t i s t i c s Sr 2 IrO 4 D o p i n g? spin-orbit coupling

8 Lifting the orbital degeneracy Jahn-Teller Spin-orbit t 2g trigonal field t 2g λ real orbital L=0 complex L=1 xy + yz + zx quadrupole spin pseudospin J=S+L L S spin Spin ½ algebra g= 2 g= -2 pseudospin

9 Jahn-Teller S=1/2 Magnetism Spin-orbit J=1/2 H = (SS) H= - (SS) + ISING (α) xx zz yy coplanar, single-q large magnetic unit cell non-coplanar, multi-q hosts spin vortex GKh (PTPS, 2005)

10 Jahn-Teller S=1/2 SC pairing Spin-orbit J=1/2 Spin-singlet d-wave SC Pseudospin-triplet p-wave SC - Baskaran (2003) Kumar, Shastry (2003) Ogata (2003) P. Lee et al.(2004) nondegenerate Knight-shift finite in all directions GKh, Koshibae, Maekawa (PRL 2004) GKh (PTPS, 2005)

11 Jahn-Teller S=1/2 SC pairing Spin-orbit J=1/2 Spin-singlet d-wave SC Pseudospin-triplet p-wave SC Baskaran (2003) Kumar, Shastry (2003) Ogata (2003) P. Lee et al.(2004) GKh, Koshibae, Maekawa (PRL 2004) GKh (PTPS, 2005) Na x (Co,Rh,Ir)O 2

12 The origin of unconventionality : ORBITAL magnetism Orbital moment L has a shape : c a b L x =1 L y =1 L z =1 x(y+iz) y(z+ix) z(x+iy) Hopping amplitude: A B A B real (inversion) complex (no inversion) nontrivial band topology Shitade et al. (2009)

13 ORBITAL MAGNETISM L-moment direction and chemical bonding: one-to-one correspondence L i L - moment interactions: L j orbital frustration non-heisenberg models: pseudodipolar, biquadratic

14 ORBITAL MAGNETISM Orbital moment L interactions: c 1) non-heisenberg 2) bond-dependent orbital frustration a? b Simple cubic lattice: Q x Qy Q z GKh & Okamoto (PRL 2002) non-coplanar multi-q no LRO at finite T

15 S J L Pseudospin J=S+L inherits bond-dependent and frustrated nature of orbitals unconventional magnetism

16 Spin-orbit multiplels of TM-ions Abragam & Bleaney (1970) Nb,Mo,Re Mo,Re,Os large J multipoles: spin nematic, hidden order Ru,Re,Os nonmagnetic Co,Rh,Ir quantum spin GS-degeneracy: pseudospin Kramers: dipole, octupole, non-kramers: quadrupole,

17 The old pseudospin J=1/2 3d Co-fluoride, neutron scattering exciton Heisenberg magnon Two branches split by spin-orbit: magnon & exciton

18 from 3d Co to 5d Ir B.J.Kim et al., 5d Exciton 600 mev 10 3d 40 mev Magnon K 2 CoF 3 Holden et al. (1971) neutron scatteting Sr 2 IrO 4 Kim et al. (2012) RIXS data

19 spin one-half 2D Mott systems similar to cuprates? d 1 d 5 d 7 d 9 t 2g electron t 2g hole e g electron e g hole Ti... Co Ni Cu La 2 CuO 4 J AF =130 mev

20 spin one-half 2D Mott systems similar to cuprates d 1 d 5 d 7 d 9 t 2g electron t 2g hole e g electron e g hole Ir Cu Sr 2 IrO 4 T N ~240 K La 2 CuO 4 J AF =130 mev Cuprate-like magnetism and SC? T N ~320 K (Cava, Cao, 1990 s)..apart from cuprates, Sr 2 IrO 4 would be the second S=1/2 2D AF

21 spin one-half 2D Mott systems similar to cuprates d 1 d 5 d 7 d 9 t 2g electron t 2g hole e g electron e g hole Ir Cu Sr 2 IrO 4 Na 2 IrO 3 La 2 CuO 4 t 2g spin-orbit coupling different orbitals e g

22 J=1/2 magnetism in iridates: basic theory Jackeli, GKh (2009) Chaloupka, Jackeli, GKh (2010, 2013) Two-parameter Hamiltonian = Ising (x,y,z) + Heisenberg: dominant in 90-bonding (honeycomb Na 2 IrO 3 ) dominant in 180-bonding (Sr 2 IrO 4 perovskite) = Kitaev model (2006) Majorana world? = cuprate model high-t c SC?

23 J=1/2 magnetism in iridates: basic theory Jackeli, GKh (2009) Chaloupka, Jackeli, GKh (2010, 2013) Two-parameter Hamiltonian = Ising (x,y,z) + Heisenberg: dominant in 90-bonding (honeycomb Na 2 IrO 3 ) dominant in 180-bonding (Sr 2 IrO 4 perovskite) = Kitaev model (2006) Majorana world? = cuprate model high-t c SC?

24 Pseudospin ½ in perovskites: -theory predicts nearly Heisenberg AF Sr 2 IrO 4?? Sr 2 IrO 4 T N ~240 K Kim et al.(2012) La 2 CuO 4 T N ~320 K Coldea et al.(2001)

25 Fermiology of electron doped Sr 2 IrO 4 Fermi-arcs at low doping normal FS Potassium K overlayer B.J. Kim et al. (Science 2014)

26 B.J. Kim et al. (Science 2014) T-dependent pseudogap in Sr 2 IrO 4 Fermi-arcs at low doping normal FS Pseudogap opens at low T many-body effect! and closes at 110 K

27 Sr 2 IrO 4 magnetism, fermiology & lattice: same as in La 2 CuO 4 superconductivity? SUPER! GREAT! find 10 differences Sr 2 IrO 4 La 2 CuO 4 YES or NO? -- no definite answer yet

28 J=1/2 magnetism in iridates: basic theory Jackeli, GKh (2009) Chaloupka, Jackeli, GKh (2010, 2013) Two-parameter Hamiltonian = Ising (x,y,z) + Heisenberg: dominant in 90-bonding (honeycomb Na 2 IrO 3 ) dominant in 180-bonding (Sr 2 IrO 4 perovskite) = Kitaev model (2006) Majorana world? = cuprate model high-t c SC?

29 The Kitaev model S x S x S y S y Exactly solvable S z S z Short-range RVB, large spin gap Low-energy excitations: free Majorana fermions local & high-energy S α = c f α Itinerant free band (uncharged) E F Dirac cones

30 Na 2 IrO 3 Honeycomb lattice: -- Kitaev term is dominant but J is present as well Kitaev model Heisenberg model liquid What is in between? order -If some liquid is still left?

31 spin spin-orbit coupling pseudospin YES! J Spins, magnons Majorana land K Quantum phase transition: spin fractionalization Chaloupka, Jackeli, GKh (2010)

32 real world: Na 2 IrO 3 Exp.data AM order T N ~15K (1) Mag. bandwidth: 40 mev~30 T N (Gretarrson et al.) (2) Intense q=0 scattering (Gretarrson et al.; B.J. Kim et al.) (3) SW gap is small < 2 mev (Coldea et al.; B.J. Kim et al.) Interactions: strongly frustrated non-heisenberg C3 symmetric Kitaev-Heisenberg K-J model with large K > J makes all these for free but

33 real world: Na 2 IrO 3 Exp.data AM order T N ~15K (1) Mag. bandwidth: 40 mev~30 T N (Gretarrson et al.) (2) Intense q=0 scattering (Gretarrson et al.; B.J. Kim et al.) (3) SW gap is small < 2 mev (Coldea et al.; B.J. Kim et al.) Interactions: strongly frustrated non-heisenberg C3 symmetric B.J. Kim et al., 2015: Unusual moment direction away from symmetry axes need more terms beyond Kitaev-Heisenberg model

34 B.J. Kim et al., Nature Phys three equivalent zigzag quasielastic peaks C3 symmetry ~44 Unusual moment direction away from symmetry axes inconsistent with pure KH-model

35 Phenomenological model ( extended-kh ): (H.Y. Kee et al., van den Brink et al, ) Two new terms: D and C exp Chaloupka & GKh, condmat 2015 K,J,D,C model can make the moment angle However, zigzag is not then stable. We need longer range interactions.

36 adding J 2,3 (Coldea et al. 2012) + J 2,3 (SS) exp Sizable anisotropy D & long-range J 2,3 Right moment direction & zigzag order

37 Spatially extended, quasimolecular orbitals: longer-range couplings J 2,3

38 insufficient Yes, indeed Spatially extended, quasimolecular orbitals: longer-range couplings J 2,3 Yes, indeed

39 Na 2 IrO 3 (current status) Data collected so far suggests that - Kitaev term seems to be dominant - Other terms are substantial, yet to be sorted out measure and fit, measure and fit for more details, see: Chaloupka & GKh, condmat 2015 most wanted: single-crystal S(q,w)

40 The streetlight effect dreamland Na 2 IrO 3 -Did you lose your Majoranas here? -No, but the light is much better over here! New candidate: RuCl 3 Plumb et al, 2014 honeycomb honeycomb

41 and look for J=1/2 K-J model on other lattices GKh (PTPS, 2005) Rousochatzakis et al. (condmat, 2012) Kimchi & Vishwanath (PRB, 2014) spin vortex condensate multi-q order hidden Goldstone Pseudospin AND geometrical frustration: - amplify the chances to find exotic states!

42 farther away from the streetlight J=1/2 d 5 Co, Rh, Ir J=0 physics d 4 Ru, Os,.. L S nonmagnetic Mott insulators

43 d 4 Mott insulators 4d, 5d electrons 1. Low-spin S=1 2. Unquenched L=1 t 2g λ(ls) J=S+L=0 no spin left to play with

44 J=0 physics: spin-orbit driven magnetic QCP singlets magnetic LRO QCP competing triplet singlet λ J ex λ d 4 d 4

45 d 4 ion: Van-Vleck magnetism (i) There are no pre-existing moments λ 0 J=1 J=0 (ii) J=0 to J=1 transition: off-diagonal magnetic moment M=2S-L ( spin-orbit exciton ) (iii) Condensation of spin-orbit exciton excitonic magnetism exciton gap AFM exchange interaction

46 Singlet-triplet examples J (A) Weakly coupled dimers Bilayer AF para QCP magnetic (Nat.Phys. 2009) (B) 4f Pr compounds (broad literature since 1970 s) (C) e g orbital FeSc 2 S 4 (Chen, Balents, Schnyder, 2009) (D) Spin-state-crossover in Fe-based SC (Chaloupka & GKh, 2013)

47 QUANTUM CRITICALITY Sachdev, Keimer, Phys.Today, 2011 birthplace for unconventional physics Inter-ionic dimers d 4 : Intra-ionic dimer made of S and L S λ L -small energies -special geometry 1) energetic (~100 mev) 2) generic (any lattice) GKh (2013)

48 d 4 Mott insulator: single-ion states J=1 triplon T x T y T z λ J=0 spin-orbit singlet

49 d 4 Mott insulator: singlet-triplet model (180 ) (e.g. 214-perovskite) GKh (2013) J spin-orbit exchange S=1 boson M (triplon gas) PM AFM LRO moment: QCP J cond.density distance from QCP

50 GKh (2013) Excitations: 1. The amplitude mode changing the lengths of S & L S L Higgs 2. The phase modes in-phase rotation of S & L S L Goldstone ω magnons in excitonic AF. 1 2 (π,π) (0,0) k

51 GKh (2013) Excitations: 1. The amplitude mode changing the lengths of S & L S L Higgs 2. The phase modes in-phase rotation of S & L S L Goldstone excitonic AF ω ω Heisenberg AF. 1 2 (π,π) (0,0) k. 2 (π,π) (0,0) k

52 Van Vleck-type d 4 Mott insulators: EXCITONIC magnetism Candidate material: Ca 2 RuO 4

53 Summary S J=1/2 L S J=0 L J=S+L inherits bond-dependent & frustrated nature of orbitals nonmagnetic J=0 Mott insulators unconventional magnetism unconventional SC?

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56 (triangular, honeycomb, kagome ) c-bond exchange: hopping pair-generation x and y type bosons only involved Bond-dependent xy-model :

57 Exchange anisotropy, 90 bonding: d 4 vs d 5 (triangular, honeycomb, kagome ) d 5 (Co,Rh,Ir) bond-dependent Ising: spin ½ GKh (2005) Chen & Balents (2008) Jackeli & GKh (2009) d 4 (Ru,Re,Os) bond-dependent xy : spin-one T boson GKh (2013)

58 Honeycomb lattice: dimensionality reduction Each flavor T x, T y, T z has its own zigzag to move along 1D dispersion: J=J crit : zero-energy lines unusual singlet-triplet model, yet to be solved VBS? zigzag order? spin nematic?