Interplay of charge, spin, orbital degrees of freedom and O2p holes in transi8on metal oxides

Transcription

1 Interplay of charge, spin, orbital degrees of freedom and O2p holes in transi8on metal oxides George Sawatzky Physics department UBC MP/UBC center for Quantum Materials

2 With thanks to Ilya Elfimov UBC Maurits Haverkort MP/ UBC Robert Green CLS/UBC Steve Johnston UBC/MP Shadi Balendeh UBC Mona Berciu UBC Jeroen van den Brink (Dresden) IFW Subhra sen Gupta (Riken) Vladimir Hinkov MP/UBC Sesbas8an Make MP/UBC Valen8na Bisogni PSI Thorsten SchmiU PSI Sara Catalano Geneva Marta Gibert Geneva Raoul Scherwitzl Geneva Jean- Marc Triscone Geneva Pavlo Zubko Geneva

3 Wide diversity of proper8es Take for example only the transi8on metal oxides Metals: CrO2, Fe3O4 T>120K Insulators: Cr2O3, SrTiO3,CoO Semiconductors: Cu2O Semiconductor metal: VO2,V2O3, Ti4O7 Superconductors: La(Sr)2CuO4, LiTiO4, LaFeAsO Piezo and Ferroelectric: BaTiO3 Mul8ferroics Catalysts: Fe,Co,Ni Oxides Ferro and Ferri magnets: CrO2, gammafe2o3 An8ferromagnets: alfa Fe2O3, MnO,NiO Proper8es depend on composi8on and structure in great detail

4

5 Phase Diagram of La 1-x Ca x MnO 3 Uehara, Kim and Cheong R: Rombohedral O: Orthorhombic (Jahn-Teller distorted) O*: Orthorhombic (Octahedron rotated) CAP = canted an8ferromagnet FI = Ferromagne8c Insulator CO = charge ordered insulator FM= Ferromagne8c metal AF= An8ferromagnet

6 High Tc superconductor phase diagram

7 Ordering in strongly correlated systems Stripes in Nd-LSCO rivers of Charge An8ferro/ An8phase Quadrupole moment ordering ΔQ < 0.5 e ΔQ C ~ 1 e ΔQ O ~ 0 Charge inhomogeneity in Bi2212 Pan, Nature, 413, 282 (2001); Hoffman, Science, 295, 466 (2002) ΔQ ~ 0.1 e

8 Mizokawa et al PRB 63, Mn4+, d3, S=3/2,No quadrupole ; Mn3+, S=2, orbital degeneracy

9 Simplest model single band Hubbard Row of H atoms 1s orbitals only The hole can freely Propagate leading to A width Largest coulomb Interac8on is on site U The electron can freely Propagate leading to a width E gap = 12.9eV- W The actual mo8on of the Par8cles will turn out to be more complicated

10 For large U>>W One electron per site Insulator Low energy scale physics contains no charge fluctua8ons Spin fluctua8ons determine the low energy scale proper8es Can we project out the high energy scale? H = i, j JS i S j Heisenberg Spin Hamiltonian J = 4t 2 / U

11 Two extremes for atomic valence states in solids Where is the interes2ng physics? Coexistance Hybridiza2on Kondo, Mixed valent, Valence fluctua2on, local moments, Semicond.- metal transi2ons, Heavy Fermions, High Tc s, Colossal magneto resistance, Spin tronics, orbitronics

12 Why are 3d and 4f orbitals special Lowest principle q.n. for that l value Large centrifugal barrier l=2,3 Small radial extent, no radial nodes orthogonal to all other core orbitals via angular nodes High kine8c energy ( angular nodes) Rela8vis8c effects Look like core orb. But have high energy and form open shells like valence orb.

13 A bit more about why 3d and 4f are special as valence orbitals 0 E n ε = = mz 2h p c e n m 2 c 4 Hydrogenic orbital energy non rela8vis8c mc 2 2 p + 2m 1 8 p 2 m c ( E ) + ee ze 1/ r z e 1/ r ) 1 2 Δ 1/ r = 2 2 nl a0n E = ( n n + nl 2me ΔE z α n n l + 1/ = 3/ 4 E nl 2 n α = e 2 hc 2 4 Rela8vis8c contribu8on 2 0 p = 2m( En + Ze z 2 1/ r = 2 a n 0 z 2 ( l + 1/ 2) 3d of Cu; binding energy of 3s=120 ev, 3p=70 ev, 3d=10 ev. 2 1 r ) Strong energy dependence on l due to rela8vis8c effects.

14 Highly confined orbitals will have a large U Charge density of outer orbitals of the Rare earths Atomic radius in solids Elemental electronic configura8on of rare earths N s 2s 2p 3s 3p 3d 4s 4p 4d 4 f 5s 5p 6s A rare earth metal For N<14 open shel Hubbard for 4f Hubbard U 5d6s form a broad conduc8on Band 4f is not full and not empty

15 Band Structure approach vs atomic Band structure Delocalized Bloch states Fill up states with electrons star8ng from the lowest energy No correla8on in the wave func8on describing the system of many electrons Atomic physics is there only on a mean field like level Single Slater determinant states Atomic Local atomic coulomb and exchange integrals are central Hunds rules for the Ground state - Maximize total spin- Maximize total angular momentum- total angular momentum J =L- S<1/2 filled shell, J=L+S for >1/2 filled shell Mostly magne8c ground states

16 General band theory result for R<<d together with R>>d states For open shell bands R<<d R<<d so bands are narrow open therefore must be at Ef

17 What do we mean by the states below and above the chemical poten8al The eigenstates of the system with one electron removed or one electron added respec8vely i.e Photoelectron and inverse photoelectron spectroscopy IPES N- 1 N+1

18 Photo - inverse Photo electron spectra of the rare earth Metals (Lang and Baer (1984)). Solid lines - atomic mul8plet theory U Ueff(Gd)=F0 +6J; F0 (solid)=f0(atomic)- 20eV (STRONGLY REDUCED) MULTIPLET SPILTTINGS AND J REMAIN ATOMIC

19 Regarding simple models Like single band Hubbard Some8mes we get so involved in the beauty and complexity of the model that we forget what the valida8ng condi8ons were and use them outside of the range of validity

20 Interplay between spin, charge, larce and orbital degrees of freedom In the large U limit where polarity fluctua8ons are strongly suppressed in the low energy scale physics THE PHYSICS OF ATOMS AND IONS IN LOWER THAN SPHERICAL SYMMETRY PLAYS AN IMPORTANT ROLE We now deal with crystal and ligand field splirngs, Hund s rule coupling, spin orbit coupling, superexchange interac8ons, and the role of orbital degeneracy

21 Some typical coordina8ons of TM ions Octahedral coordina2on Red=TM ion White =Anion like O2- Tetrahedral coordina2on Red = TM White =anion like O2- As in NiO As in LiFeAs

22 Real d orbitals in Octahedral coordina2on eg s have lobes poin2ng to anion forming sigma bonds and the t2g s have lobes poin2ng between the anions with pi bonds

23 Crystal (point charges)and ligand field (covalency) splirng Of t2g and eg orbital energies in 3d oxides Point charge contribu8ons are typically 0.5 ev Covalent contribu8ons TM 3d Δ eg t2g splitting O2p = 2 t t eg t Δ t 2t eg t 2 2 2g g Oden about 1-2eV In 3d Oxides

24 Note the rather broad Cl 2p bands And the very narrow Ni 3d bands Split into eg and t2g. Note also the Crystal field spli8ng of about 1.5eV. Note also that DFT (LDA) predicts a metal for NiCl2 while it is a pale yellow magne8c insulator. Note also the large gap between Cl 2p band and the Ni 4s,4p bands With the 3d s in the gap. This is a typical case for TM compounds

25 Two new complica8ons d(n) mul2plets determined by Slater atomic integrals or Racah parameters A,B,C for d electrons. These determine Hund s rules and magne8c moments d- O(2p) hybridiza2on ( d- p hoping int.) and the O(2p)- O(2p) hoping ( O 2p band width) determine crystal field splirng, superexchange, super transferred hyperfine fields etc.

26 Ballhausen Three integrals F0, F2, F4 related to A,B,C Racah param.

27 Reduc8on of coulomb and exchange in solids U or F0 is strongly reduced in the solid. This is the monopole coulomb integral describing the reduc8on of the ioniza8on poten8al and increase of the electron affinity in a polarizable medium However the other integrals F2 and F4 and G s do not involve changes of charge but simply changes of the orbital occupa8ons of the electrons so these are not or hardly reduced in solids. The surroundings does not care much if locally the spin is 1 or zero. This makes the mul2plet structure all the more important!!!!! It can in fact exceed U itself

28 For U or F0 strongly reduced <<W We can s8ll have strongly correlated systems. Recently recognized by the DMFT people like Kotliar and Georges who call these Hund s Metals or insulators Mul8plet splirngs can remain large

29 VanderMarel etal PRB 37, (1988)

30 Nul8plet structure of 3d TM free atoms VanderMarel etal PRB 37, (1988) Note the high energy scale Note also the lowest energy state for each case i.e. Hunds Rule;

31 Simplified picture of Crystal fields and mul8plets n Determine d energy levels assuming only crystal and ligand fields and Hunds first rule i.e. F, and J = ( F + F ) Neglect other contribu8ons like C in our former slides and the SO coupling This is a good star8ng point to generate a basic understanding. For more exact treatments use Tanabe- Sugano diagrams 14

32 Crystal fields, mul8plets, and Hunds rule for cubic (octahedral) point group d5; Mn2+, Fe3+ Free ion Cubic Oh 4J 10DQ eg t2g eg t2g (4)J is the energy to flip One of spins around 10DQ= crystal field S=5/2 No degeneracy d4; Mn3+, Cr2+ 3J t2g eg t2g S=2 two fold degenerate

33 Physical picture for high spin to low spin transi8on eg 10DQ t2g E(HS)=- 10J- 4DQ d6; Fe2+, Co3+ 3J eg E(LS)= - 6J- 24DQ t2g HS to LS for 10DQ>2J eg 10DQ t2g E(HS)=- 10J d5; Fe3+, Co4+ 4J eg t2g 0J E(LS)=- 4J- 20DQ HS to LS for 10DQ>3J

34 Goodenough Kanamori Anderson rules i.e. interatomic superexchange interac8ons And magne8c structure For example Cu O Cu2+ as in La2CuO4 and superconductors Cu2+ is d9 i.e. 1 eg hole (degenerate in OH) but split in D4H as in a Strong tetragonal distor8on for La2CuO4 structure. The unpaired electron or hole is in a dx2- y2 orbital with lobes poin8ng to the 4 Nearest O neighbors. Anderson 1961 If the charge transfer energy Δ gets small we have to Modify the superexchange theory New term The sum leads to a huge an2ferro Interatomic J(sup) =140meV for the Cuprates

35 Superexchange for a 90 degree bond angle The hoping as in the fig leaves two holes in the intervening O 2p states i.e. a p4 configura8on. The lowest energy state According to Hund s rule is Spin 1. So this process favours A ferromagne8c coupling between the Cu spins. 4 t pd J ( 90) = 2 2 Δ 2 2Δ So the net exchange as a func8on of the bond angle is: 2 2Δ J ( hundo2 ) 2 J ( θ) = J(180) cos ( θ) + J(90)sin 2 ( θ)

36 If we have spectator spins as in Mn3+ in OH For ferro orbital ordering we will get a strong an8ferromagne8c super exchange since the same intervening O 2p orbital is used in intermediate States as in the example above d4; Mn3+, Cr2+ t2g 3J eg t2g For an8ferro orbital ordering The factor of 3 in the Hunds Rule of Mn is from the spectator spins

37 Orbital degeneracy If there is orbital degeneracy the Jahn Teller theorem tells us that it will be liyed in on way or another at low temperatures. This is because the system can always lower its energy by liying this degeneracy We dis8nguish to types those involving eg or t2g orbitals. We consider cubic and OH symmetry to start with Strong Jahn Teller ions Weak Jahn Teller ions d d 4 ( t 3 2g e 1 g ) or d 9 ( t 6 2g e 3 g ( t2geg ) or d, d, d )

38 How can we liy the degeneracy Spin orbit coupling if we have t2g degeneracy. Recall the eg s do not split with SO. Jahn Teller distor8on i.e. from Cubic to tetragonal would split the eg orbitals into d(3z2- r2) and d(x2- y2) (Examples are cuprates Orbital ordering which may be driven by other than electron phonon coupling Charge dispropor8ona8on i.e d + d d + d Where (in the case of low spin) both final configura8ons are not orbitally degenerate. We will see later why this could happen in spite of a large U 6

39 Orbital ordering Consider again the case of Mn3+ with the doubly orbital degenerate eg level in cubic symmetry occupied by only one electron as above. It would be logical in a perovskite structure that long bond axis would alternate say along x and y for two Mn ions sandwiching an O anion as in the next slide

40 For LaMnO3 resonant x ray diffrac8ons yields the orbital occupa8on structure below with Alterna8ng occupied eg orbitals rotated by 90 degrees as see in the basal plain. The small red arrows indicate the Oxygen displacement resul8ng from this leading to a so called coopera8ve Jahn Teller distor8on Murakami et al Paramagnet Suscep2bility above TOO will yield a large An2ferro J which has no Rela2on to the J in the mag Ordered state The 300 reflec8on Is generally forbidden but visible at resonance Because of the orbital ordering See two transi8ons. One at high Temp for the orbital ordering and one at low T for an8ferromagne8c order. The spin ordering in plane is ferromagen8c as we would have predicted

41 Another example of orbital ordering LiVO2 V(d2 S=1), Two electrons in a t2g Orbital Rock salt structure Alternateing V Li O layers in the 111 direc8on Each form a triangular 2 dimensional larce Pen et al PRL 78,1323

42 LiVO2 V3+ d2 system S=1 in cubic symmetry 3 fold orbital degeneracy Magne8c suscep8bility is Curie Weiss like T>500K and then signs of magne8sm disappear

43 This orbital ordering yields a large internal to the triangle An8ferromagne8c exchange and a weak External to the triangle ferromagne8c exchange. dxy dxz dyz Recal lobes poin8ng Towards each other And single occupied Yield an8ferro coupling Pen et al PRL 78,1323 Orbital ordering removes frustra8on of interac8ons between these triangles

44 3 spins in a triangle with an exchange interac8ons was a home work problem The ground state for spins of 1 is S=0 i.e. singlet and these result in zero suscep8bil8y With vanishing temperature. Above the orbital ordering transi8on the magne8c suscep8bility shows a strong an8ferromagne8c coupling according to the Weiss constant

45

46 YVO3 YVO3 Pervoskite structure V(3+) 2 electrons in T2g Orbitals S=1. Note the 8lted and Rotated octahedra. Results in a Staggered magne8c anisotropy And this breaks the inversion Symmetry at O sites and allows For DM coupling and canted an8ferromagnets Ren et al Nature 396,441 (1998) Tsvetkov et al PRB 69, (2004)

47 All V have one electron in a dxy orbital which is not shown and causes An an2ferromagne2c coupling in the between V in the basal plane O between the V ions are not in inversion center Tilted Octahedra D.SxS interac8ons compete With local staggered anisotropy

48 YVO3 Perovskite V(d2 S=1) O not In inversion symmetry DM can8ng compe8ng With staggered magne8c Anisotropy DM can8ng depends on the T Dependent order parameter Single ion anisotropy T independent

49 Hamiltonian for orbital and spin order H i, j (Kugel Khomskii 1982) = J S S + J O O + J ( O O )( S S ) S i, j i j O i, j i j S, O i, j i j i j i, j i, j The first term describes spin structure and magnon excita2ons Second term the Orbital order and Orbiton or d- d exciton excita2on Third term is the strong interac8on between Orbitons and spin waves this interac8on can lead to new bound or spin polaronic like states. In addi8on we really should have included the electron phonon interac8o which would result in larce distor8ons depending on the orbital order and in larce polaronic like effects coupling with orbitons and magnons. Since all these interac8ons are of the same order of magnitude the situa8on is very complicated but also very rich in new physical proper8es and phenomena

50 Another complica8on THE EXPLICIT ROLE OF THE ANION STATES

51 Doped holes in cuprate C. T. Chen et al. PRL 66, 104 (1991) As we hole dope the system the O1s to 2p first peak rises very strongly indica8ng That the doped holes are mainly on O 2p.

52 Can we renormalize and get rid of the anion states?

53 Zhang Rice PRB ,3759 Is single band Hubbard jus8fied for Cuprates?

54 Problem with ZR singlets The combina8on of O 2p states is not compa8ble with a band structure state The wave func8ons are not orthogonal From ZR PRL 37,3759 Note it goes to infinity at k=0, should we see it at Gamma in ARPES? Luckily it goes to 1 for K= Pi/2,Pi/2 and anywhere along the AF zone boundary where the FIRST doped holes go In band theory O 2p does not mix with Cu dx**2- y**2 at Gamma!!!!! SO HOW TO DO THIS PROPERLY FOR HIGH DOPING?

55 Shows strong ferro correla2ons close to the doped hole. This does not look like a ZR singlet Bayo Lau et al PRB 81, PhysRevLeU , (2011) PhysRevB ( 2011) Recent exact diagonaliza8on studies of 32Cu 64O Note a quantum spin ½ an8ferromagnet has a nn spin correla8on of -.33 A Neel an8ferromagnet =-.25, and a ferromagnet =+.25

56 Now imagine a strongly hole doped cuprate like LaSrCuO4 (100% hole doped. Can a ZR singlet picture survive? O 2p orbitals are shared with two Cu s

57 Is this only a problem for the Cuprates? What about the Nickalates, Manganites, Cobaltates etc?

58 Kuiper et al PRL (1989) LixNi1- x O A CHARGE TRANSFER GAP SYSTEM HOLES IN O Note the high pre- Edge feature and the Spectral weight Transfer from high To low energy scales Just as in the cuprates The holes are mainly on O and not on Ni.!!

59 LNO thin film on LSAT Sutarto, Wada8, Stemmer UCSB Note the huge O 1s - 2p prepeak just as in the cuprates HOLES ON O

60 Torrance et al PRB 42, 8209

61 Recent RIXS results demonstrate that the cluster interpreta8on of the XAS used generally is not valid for the Nickelate s results obtained by Valen8na Bisogni and Thorsten SchmiU from PSI Sara Catalano, Marta Gibert, Raoul Scherwitzl Jean- Marc Triscone, and Pavlo Zubko From Geneva

62 RIXS spectra of NdNiO 3 15 K LH α=50 [110] PSI, V. Bisogni T.

63 RIXS map of NdNiO 3 15 K insula2ng phase Ni 2p XAS energy region : Up to now the peaks A and B were considered to be,multiplet structure in the final 2p5 3d8 local states RIXS demonstrates that a local d-d like description is OK for peak A with photon energy independent peak positions in RIXS Near linear dependence of the Loss energy With photon energy show that this is not RIXS but simple x ray flourescence. So peak A in XAS involves the excited d Electron intimately while peak B must involve an excitation into a delocalized continuum band state in the intermediate state. The continuum starts at most 1 ev above the bound state. This has implications for the ground state and low energy excitations and the properties. PSI, V. Bisogni T.

64 RIXS map of NdNiO K Metallic Phase Strong T dependence of the XAS Here the con8nuum states merge With the bound states or resonances PSI, V. Bisogni T.

65 Conven8onally RENiO3 would involve Ni3+ which is expected to be low spin i.e. S=1/2 with 6 electrons in t2g orbitals and 1 in an eg orbital STRONG JT WHICH IS NOT OBSERVED!

66 RIXS and XAS indicate The lowest energy states in Nickelates before Ni d O2p hybridiza8on could well be par8ally filled O band with the Ni in d8 S=1 states

67 How to get rid of JT? Charge dispropor8ona8on into d6 and d8 would solve this problem. But experiments show only very low CDW amplitude in the insula8ng phase

68 High oxida8on state TM compounds In general we expect the charge transfer energy to strongly decrease for higher oxida8on states This could mean a different star8ng point i.e. Cu3+ Cu2+L Ni3+ Ni3+L Co4+ Co3+L Fe4+ Fe3+L Mn4+??? The charge degrees of freedom are on Oxygen

69 transi2on metal oxides Remember at surfaces U is increased, Madelung is decreased, W is decreased

70 No Jahn Teller problem anymore Charge dispropor8ona8on without moving charge Consider ReNiO3 Ni3+ on average but label it as Ni2+L Then each Ni is surrounded by 2 L holes in ReNiO3 ( 1 hole per 3 O) 2Ni3++ Ni2+ + Ni4+ Two holes in O2p Orbital in octahedron With central eg symmet Ni2+ no JT Each second Ni2+ has an octahedron of O with two holes of Eg symmetry in bonding orbital's I.e. d8 L2

71 The nickaltes i.e. RENiO3 Lets associate the two holes (with S=1) with one Ni which will then be a S=0 cluster Because of Jpd. The octahedron will contract leaving the other Ni neighbors in a d8 S=1 state. This gives the correct structure at low T and in fact also gives the correct spin structure. Effec8ve dispropor8ona8on without moving charge. THIS STATE SEEMS TO BE NEARLY DEGNERATE WITH A METALLIC ITINERANT O HOLE STATE

72 Symmetry dictates the energy difference not exchange or interaction Example of two par8cles in U= limit 1 1 t t ±t Ψ = φ( x1, x2 ) χ( m s 1, ms2 ) χ m s, m ( 1 s2 ) = ( χ + χ ) + ( χ χ ) Triplet Singlet H 0 = t ± t t 0 t ± t t 0 + for singlet; - for triplet Energy level diagram for holes (t>0) 2t t Singlet Triplet -t -2t

73 Two holes in O Octahedron What stabalizes the charge Disproportionation? Elfimov PRL89, (2002 Cluster model Defini8on of hopping parameters t pp =1/2(t ppσ -t ppπ ) t pp =1/2(t ppσ +t ppπ )

74 Exact diagonaliza8on results Single-particle picture Three lowest states for two particle t (a) HOLES in anion orbitals and (b) ELECTRONS in cation orbitals. U on O is about 6 ev (a) ELECTRONS in cation orbitals and (b) HOLES in anion orbitals. Solid symbols are for triplet state

75 The theory of systems with nega8ve charge transfer gap energies This is really complicated since we now cannot use our simple non metallic ansatz. We then have a problem of a larce of local spins in d states with strong hybridiza8on and exchange with the holes on O. The case I alluded to of LaNiO3 is perhaps such an example. Perhaps viewed as a Kondo larce model but with a charge density wave instability driven by an exhaus8on principle since only ½ of the Ni spins can be compensated by O 2p holes

76

77 Shadi and Maurits cluster exact diagonaliza8on Cu 2p to 3d transi8ons Assumes a d7 low spin Ground state i.e. 3 holes In eg orbitals S=1/2 Should be strong Jahn Teller - orbital degeneracy

78 30 nm spuuered films NNO on LSAT 1.1% strain thesis of Raoul Scherwitzl in Triscone s group

79 Look first at NiO where cluster theory works The 2p core hole can strongly bind the extra 3d electron forming a core excitonic state which is localized 2p53d9 2p53d9 Q=8 ev to move the extra 3d electron away from the core hole this is why local mul8plet Theory and small clusters work so well If the excitation is to a delocalized state the electron moves away rapidly and the local state will be 2p5 3d8 like and this Decays by flourescence to a 3d7 state. i.e. you see the occupied 3d density of states in the loss spectrum. In the case of NiO the 2p5 3d10L states are broad band states due to the O 2p band structure mainly and hybridize with the 2p53d9 states. If we excite into here the excited electron finds itself in a broad band.

80 So whats so different for the Nickelates? The gap between the con8nuum and bound states is very small and perhaps 0 with the core hole present This points to an energy level diagram in the ini8al state with the conimuum d8l lower in energy than the Ni3+ d7 state. These ligand hole states form a rather broad band. i.e. A SELF HOLE DOPED SYSTEM AS IN CRO2 The charge ac8on could be mainly on O

81 Since the broad band like intensity is larger than the bound state like part the ground state must be mainly of d8l character

82 Without hybridization this would Be a metal like LaNiO3 With strong hybridiza8on could be An insluator Like NdNiO3 Low T d9ll U + Δ d7 Δ d8l d7 like ground state Strongly mixed with d8l Small nega8ve charge transfer gap

83 Spectral weight transfer The real signature of strong correla8on effects One of the favourite recent topics of Philip Phillips

84 Doping a MoU Hubbard system U N N PES E F PES N- 1 (1- x)/2x 2 N- 1 N+1 2 N- 1 E F Meinders et al, PRB 48, 3916 (1993) E F

85 Mul8plet structure for free TM atoms rare Earths can be found in the reference ( ) ) ( 14 1 ), ( ), ( ),,, ( 1) ( 2 1 ),,, ( F F d d J d d J l l F F F F U S L n U U n n ni S L n E ave ave + = + = + + = + + = λ λ ), ( 2 ) 1, ( ) 1, ( ) ( ) ( ) ( ) ( ), ( ) ( 14 1 ), ( Hund n E Hund n E Hund n E U C n J n F n I n Hund n E F F d d C eff C J F I + + = = = α α α α pairs spin parallel of No n n n n n J F I = = = ) ( ;; 2! ) ( ;; ) ( 0 α α α VanderMarel etal PRB 37, (1988)

86 Hunds rules First the Physics Maximize the total spin spin parallel electrons must be in different spa8al orbitals i.e. m values (Pauli) which reduces the Coulomb repulsion 2 nd Rule then maximize the total orbital angular momentum L. This involves large m quantum numbers and lots of angular lobes and therefore electrons can avoid each other and lower Coulomb repulsion

87 Hunds third rule < half filled shell J=L- S > half filled shell J=L+S Result of spin orbit coupling 1 Η so = ( V ( rj ) p j ) s 2 2 j 2m c j Spin orbit results in magne8c anisotropy, g factors different from 2, orbital contribu8on to the magne8c moment, - - -

88 Superexchange between singly occupied t2g orbitals dxz dxz z J anti = 2t 4 pdπ Δ 2 pz 2 2Δ + U pp + 1 U If we now rotate one of the bonds around the z axis the superexchange does not change, but for rota8on around the y axis it changes as for eg orbitals. Since 1 t pd t 2 π pdσ 1 J pd π J 16 pdσ dd x

89 Closer to real systems We use mainly 3d transi8on metal compounds as examples Orbital degrees of freedom in partly occupied d orbitals interact with spin and charge degrees of freedom We have to deal with mul8 band systems

90 Two kinds of d orbitals generally used d d d d xy 3z x 2 xz 2 = r y = 2 d d 1 xy x 2 i 2 = 2 i = 1, d y 2 xz, d, d 3z yz 2 r t 2 [ 2,2] [ 2, ] [ 2,0] 2 ( 1 2 ( ( 2 ) 2g [ 2, 2] + [ ] [ 2,1] [ 2, 1]) 2,2 ) e g All have 0 z component Of angular momentum In cubic symmetry the two eg s and 3 t2gs are 2 and 3 fold degenerate respec8vely. The spin orbit coupling does not mix the eg orbitals to first order but it does mix the t2g s which then get split into a doublet and a singlet in cubic symmetry

91 The d- d coulomb interac8on terms contain density - density like integrals, spin dependent exchange integrals and off diagonal coulomb integrals i.e. Where n,n m,m are all different. The monopole like coulomb integrals determine the average coulomb interac8on between d electrons and basically are what we oyen call the Hubbard U. This monopole integral is strongly reduced In polarizable surroundings as we discussed above. Other integrals contribute to the mul8plet structure dependent on exactly which orbitals and spin states are occupied. There are three relevant coulomb integrals called the Slater integrals; F F F = monopole integral = dipole like integral = quadrupole integral For TM compounds one oyen uses Racah Parameters A,B,C with ; A = F0 49F4 ;; B = F2 5F4 ;; C = 35F4 1 1 F 2 = F ;; F4 = F ;; F0 = F Where in another conven8on ; The B and C Racah parameters are close to the free ion values and can be carried over From tabulated gas phase spectroscopy data. Moores tables They are hardly reduced in A polarizable medium since they do not involve changing the number of electrons on an ion.

92 A new exact diagonaliza8on study with large systems by Bayo Lau Cluster of 64 Cu and 128 O sites used for the pure an8ferromagnet ground state Calcula8on taking quantum fluctua8ons full into account. For the single hole lowest energy state calc use 32 Cu and 64 O site clusters. This involves a basis ov more than 10**9 states

93 Bayo Lau,Mona Berciu, G.A.Sawatzky PRL in press

94 Do we get Zhang Ruce singlets? Not really, Does this agree with t,t,t,j? yes for the dispersion but no for the spectral weight We find low energy spin 3/2 states!!!! (neutron resonance? At some k points these are in fact the ground states and so unreachable with ARPES Δ E (3/2-1/2) = 50 mev For holes ar pi/2,pi/2. Is this the neutron resonance This also occurs at q=0

95 Shows strong ferro correla2ons close to the doped hole. This does not look like a ZR singlet Bayo Lau et al PRB 81, PhysRevLeU , (2011) PhysRevB ( 2011) Recent exact diagonaliza8on studies of 32Cu 64O Note a quantum spin ½ an8ferromagnet has a nn spin correla8on of -.33 A Neel an8ferromagnet =-.25, and a ferromagnet =+.25

96 The spin correla8ons are very different for k=0,pi

97 Spin flip Via Jpd The down O hole you Created in fact flips to Up and a nn Cu spin flips down THE HOLE IS FREE TO MOVE BUT HAS HIGH ENERGY Can move down but not up Lower energy for Jpd>>Jdd