Materials Overview of Iridates II. Edge-shared IrO 6 (2D, 3D honeycomb, hyperkagome, spinel)

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1 Materials Overview of Iridates II. Edge-shared IrO 6 (2D, 3D honeycomb, hyperkagome, spinel) β-li 2 IrO 3 (hyperhoneycomb) H 3/2 Li 1/2 IrO 3 Na 3 Ir 3 O 8 Tomohiro Takayama Max Planck Institute for Solid State Research

2 Content 0. NaCl structure and related crystal structures of iridates I. 2D honeycomb iridates Platform for Kitaev physics II. 3D honeycomb iridates β- and γ-li 2 IrO 3, Proximity to Kitaev limit III. Chemical modulation to 2D honeycombs Possible new spin liquid in honeycomb lattice IV.Spinel and hyperkagome lattice Quantum spin liquid and SOC semimetal Spin-liquid candidates with different origins...

3 NaCl structure and ABO 2 oxide Na + anion layer cation layer Cl - Put two different cations ABO 2 => A +, B 3+, O 2- e.g. NaFeO 2 Na + O 2- (111) NaCl-type Both cation and anion form fcc structure. NaCl 6 octahedra share their edges. Along (111), ccp of anion layers and cations fill all octahedral voids 1 st layer 2 nd layer 3 rd layer Fe 3+ ABO 2... LiCoO 2, LiCrO 2, Na x CoO 2

4 For B 4+ or B 7+ cations... A 2 BO 3 (A(A 1/3 B 2/3 )O 2 ) A 5 BO 6 (A(A 2/3 B 1/3 )O 2 ) Example: Li 2 SnO 3 (C2/c) A-cation layer anion-layer A-B mixed layer anion-layer A-cation layer Example: Li 5 ReO 6 (P3 1 12) anion-layer A-cation layer anion-layer A-B mixed layer anion-layer A-cation layer A 1/3 B 2/3 layer A 2/3 B 1/3 layer

5 Why layered ABO 2 preferred? O 2- B ABO 2 ordered pattern of A and B Coordination of OA 3 B 3 octahedron A For A 2 BO 3 Coordination of OA 4 B 2 octahedron O 2- B A dispersed clustered cis-coordination trans-coordination Favorable for cation Coulomb repulsion r B /r A large Favorable for anion polarization r B /r A small γ-lifeo 2 NaFeO 2 (layered) Empirically... r B /r A small Favorable for cation Coulomb repulsion r B /r A large G. Mather et al., J. Mater. Chem. 10, 2210 (2000)

6 Structural variation of A 2 BO 3 Li 2 SnO 3 -type β-na 2 PtO 3 type Li 2 ZrO 3 -type Edge-shared BO 6 Honeycomb lattice Only cis, oriented to same plane Edge-shared BO 6 in 3-dimensions hyperhoneycomb Only cis, 3D-direction Edge-shared BO 6 chain Connected by corners cis-, trans-mixture

7 Content I. 2D honeycomb iridates Platform for Kitaev physics II. 3D honeycomb iridates β- and γ-li 2 IrO 3, Proximity to Kitaev limit III. Chemical modulation to 2D honeycombs Possible new spin liquid in honeycomb lattice IV.Spinel and hyperkagome lattice Quantum spin liquid and SOC semimetal

8 Why J eff = 1/2 magnets interesting? Magnetic coupling between J eff = 1/2 (direct or via oxygen) J eff = 1/2 Jackeli & Khaliullin 1. Unique magnetic coupling Complex phase gives rise to interference effect Possible route for Kitaev spin liquid 2. Weak Mott insulator Vicinity to metal-insulator transition Charge fluctuation, long range hopping can be sizable 3. Ideal platform for synchrotron experiments L-edge of 5d elements in hard x-ray region RXS, RIXS and XMCD etc.

9 Honeycomb iridate α-a 2 IrO 3 : Candidate for Kitaev spin liquid Magnetic coupling in Ir-O-Ir bond with J eff = 1/2 state G. Jackeli and G. Khaliullin, PRL 102, (2009) 1 J (, 1 2, 1 2, 1 2 ) eff 1/ 2 = dxy ± ± dyz m + idzx m 3 α-a 2 IrO 3 (A = Li, Na) 90 bond (edge-sharing IrO 6 ) IrO O -JS y 6 x S x x Ir Ir Ir O O Ir z y x -JS z S z A -JS y S y J AF S i S j Destructive interference between two paths H = JS ( γ ) ij γ i S γ j (in αβ plane) - bond-dependent FM Kitaev spin liquid?? In real materials, Other interactions (J AF, J 2, Γ) Distortion in IrO 6 octahedra

10 Kitaev-Heisenberg model in honeycomb iridate Not only Kitaev-type coupling, But Heisenberg (direct d-d) coupling should be considered Kitaev Heisenberg J 1 = 2α, J 2 = 1-α Finite parameter space for spin liquid ground state Stripy order predicted in the intermediate phase J. Chaloupka et al., PRL 105, (2010)

11 Antiferromagnetic order in Na 2 IrO 3 Clear antiferromagnetic order µ eff ~ 1.91 µ B, Θ W ~ -125 K Direct d-d interaction (Heisenberg AF) dominant? T N ~17 K, f = Θ W /T N ~ 7 frustration? S m < Rln2 (= 5.76 J/molK) (lattice part estimated from Na 2 PtO 3 ) Also Y. Singh et al., PRB 82, (2010)

12 Zigzag-type magnetic structure of Na 2 IrO 3 Resonant x-ray scattering (0 1 11) Inelastic neutron Neutron scattering S. K. Choi et al. PRL 108, (2012) m ~ 0.22 µ B /Ir X. Liu et al., PRB 83, (2011) F. Ye et al., PRB 85, (2012) Zigzag-type order... Not found in the original Kitaev-Heisenberg model J 2 -J 3... I. Kimchi et al., PRB 2011 t 2g -e g hopping... J. Chaloupka et al., PRL 2013 J-K-Γ... J. G. Rau et al., PRL 2014 Itinerant picture... I. Mazin et al., PRL 2012 and more...

13 Magnetic order in Li 2 IrO 3... Spiral?? Seemingly non-frustrated AF Also Y. Singh et al., PRL 108, (2012) T N ~ 15 K No single crystal available to date... Θ W ~ -12 K, µ eff ~ 1.76 µ B Entropy S m << Rln2 Spiral order predicted AF (~ K) + FM small Θ W Due to the difference of ionic radius?? Ir 4+ : Å, Li + : 0.76 Å, Na + : 1.02 Å J. Reuther et al., PRB 90, (2014)

14 Li-Na mixing... No consensus yet (Na 1-x Li x ) 2 IrO 3 Solubility limit x ~ 0.25 Na 2-x Li x IrO 3 No solubility limit Change of T N is not monotonic T N lowest at x ~ 0.7 spiral zigzag G. Cao et al., PRB 88, (2013) K. Rolfs et al., PRB 91, (2015)

15 Content I. 2D honeycomb iridates Platform for Kitaev physics II. 3D honeycomb iridates β- and γ-li 2 IrO 3, proximity to Kitaev limit III. Chemical modulation to 2D honeycombs Possible new spin liquid in honeycomb lattice IV.Spinel and hyperkagome lattice Quantum spin liquid and SOC semimetal Keyword: IrO 6 local distortion

16 β-na 2 PtO 3 type... Another ordered NaCl-type A-cation layer anion-layer A-B mixed layer anion-layer A-cation layer Li 2 SnO 3 -type (C2/c) β-na 2 PtO 3 -type (Fddd) anion-layer A-B mixed layer anion-layer A-B mixed layer anion-layer A 1/3 B 2/3 layer Na 2 Pt layers

17 Discovery of new form of Li 2 IrO 3 β-li 2 IrO 3 Powder XRD Rietveld with Ag Kα Space group: Fddd (No.70), Z = 16 a = (3) Å, b = (4) Å, c = (9) Å site g x y z U iso (Å 2 ) Li1 Li+ 16g 1 1/8 1/ (5) (11) Li2 Li+ 16g 1 1/8 1/ (7) (18) Ir1 Ir4+ 16g 1 1/8 1/ (2) (4) O1 O2-16e (5) 1/8 1/ (4) O2 O2-32h (5) (3) (1) (3) Synthesis Solid state reaction between Li 2 CO 3, IrO 2, LiCl Same structure with β-na 2 PtO 3 Single crystal also available (but twinning for large ones) T. Takayama et al., PRL 114, (2015) R. Dinnebier Only one Ir site

18 Local structure close to regular IrO 6 octahedron α-na 2 IrO 3 β-li 2 IrO 3 IrO Li Å Å Å Same local network with honeycomb α-a 2 IrO 3 Ir-O-Ir angle close to 90 degrees Ir-O bond length almost isotropic (~0.1 %. 1.3 % for Na 2 IrO 3 ) Kitaev-type FM interaction might be predominant...

19 3D analogue of honeycomb hyper-honeycomb Honeycomb-like structure with 10-site loop Twisted zigzag-chains unlike 2D honeycomb 3D version of honeycomb hyper-honeycomb Each Ir has 3 neighboring Ir Angles between Ir almost 120 Ir-Ir lengths are almost same New platform for Kitaev-type frustration! (Theory) S. Mandal & N. Surendran PRB 79, (2009)

20 Hyper-honeycomb lattice 2D honeycomb Hyper-honeycomb

21 Hyper-honeycomb network known from before α-thsi 2 Hyper-polyacene Dirac node CaSi 2 Ca Si SC T c ~ 1.8 K Hyper-graphite Y. Takagi et al., Synthetic Metals 103 (1999) Y. Takagi et al., Mol. Cryst. And Liq. Cryst. 340, 379- (2000). S. Sanfilippo et al., PRB 61, R3800 (2000)

22 hyper-honeycomb... New platform for Kitaev -JS x S x -JS z S z JS y S y Å Å Å Honeycomb-like structure with 10-site loop Twisted zigzag-chains unlike 2D honeycomb 3D version of honeycomb hyper-honeycomb Each Ir has 3 neighboring Ir Angles between Ir almost 120 Ir-Ir lengths are almost same (~0.1% difference) New platform for Kitaev-type frustration! (Theory) S. Mandal & N. Surendran PRB 79, (2009)

23 Non-colinear order at T c = 38 K with positive θ W A cusp in M and peak in C at 38 K magnetic ordering Constant χ below T c... Non-collinear ordering? µ eff = 1.61 µ B and θ W ~ 40 K Clear ordering at 38 K Small S ~ 0.2 J/mol K<< Rln2

24 Close proximity to ferromagnetism β-li 2 IrO 3 B = 1 T Cusp in M & peak in C/T smeared out under high magnetic field M saturating above ~3 T, ~0.35 µ B /Ir at 4 T 0.35 µ B /Ir close to the ordered moment of Sr 2 IrO 4 or Na 2 IrO 3 unlikely to be canted AF, but polarizing towards FM

25 J eff = 1/2 isospin identified by XMCD X-ray Magnetic Circular Dichroism (XMCD) D. Haskel Decompose magnetic moment into spin- and orbital contributions T = 5 K, B = 4 T Orbital sum rule for XMCD m L = -2/3(I c 3 + I c 2/I abs )<n h >µ B = µ B /Ir Spin moment m S = M 0.24 ~ ~ 0.11 µ B /Ir Ratio: <L z >/<S z > = 2m L /m S ~ 4.4 J eff = 1/2 limit: <L z >/<S z > = 4 Measured at 4-ID APS Very close to J eff = 1/2 Likely due to little distortion

26 Another 3D-honeycomb γ-li 2 IrO 3 Stripy-honeycomb network Unlike hyperhoneycomb, every two zigzag-chains are rotated Magnetization behavior quite similar with β-li 2 IrO 3 K. A. Modic et al., Nature Com. 5, 4203 (2014)

27 Magnetic structure of two 3D honeycombs β-li 2 IrO 3 γ-li 2 IrO 3 q = (0.57(1), 0, 0) Ordered moment ~0.47(1) µ B /Ir A. Biffin et al., PRB 90, (2014) q = (0.57(1), 0, 0) A. Biffin et al., PRL 113, (2014) Incommensurate magnetic order Non-coplanar, counter-rotation Ir moments

28 Pressure suppression of FM moments D. Haskel Above 2 GPa, negligible XMCD signal Suppression of T c at 1 GPa enhanced frustration No MIT under pressure rearrangement of J eff = 1/2 moment Collinear AF, spin gap or disordered state? T. Takayama et al., PRL 114, (2015) Evidencing intricate competition between magnetic phases

29 Content I. 2D honeycomb iridates Platform for Kitaev physics II. 3D honeycomb iridates β- and γ-li 2 IrO 3, Proximity to Kitaev limit III. Chemical modulation to 2D honeycombs Possible new spin liquid in honeycomb lattice IV.Spinel and hyperkagome lattice Quantum spin liquid and SOC semimetal Keyword: IrO 6 local distortion

30 Soft chemical modulation in ABO 2 ABO 2... Stage for soft chemical reactions (e.g. Li x CoO 2...Li intercalation, de-intercalation) Ion-exchange Ag +, Cu + LiCoO 2 + CuCl CuCoO 2 Hydrogen ion-exchange LiCoO 2 + acid solution HCl 0.1 M HCl 0.05 M Delafossite forms 200 C, water M. Beekman et al., J. Alloys. Comp. 489, 336 (2010) J. M. F. Rodriguez et al., Mat. Res. Bull. 23, 899 (1988) Furthermore, organic salt or water-intercalation etc. (e.g. Na x CoO 2 1.3H 2 O... superconductor)

31 Chemical Modification on honeycomb iridates Topotactic reaction (ion exchange) α-li 2 IrO 3 + AgNO 3 or CuCl A 3 LiIr 2 O 6 (A = Ag, Cu) 200 C or 420 C Li Ag(Cu) Honeycomb layers intact, only interlayer stacking change (delaffosite-like stacking), enlarged interlayer distance (Also by M. Jansen Group, JSSC 184 (2011))

32 Drastic change in magnetic properties by ion-exchange Ag 3 LiIr 2 O 6 Θ W ~ -112 K, µ eff ~ 1.71 µ B Cu 3 LiIr 2 O 6 Θ W ~ -116 K, µ eff ~ 1.90 µ B (c.f. Li 2 IrO 3, Θ W = -12 K) only small anomalies around 10 K... not sufficiently clean? Change in interlayer coupling? Change in local structure?

33 Hydrogen ion-exchange for α-li 2 IrO 3 α-li 2 IrO 3 + 4M H 2 SO 4 (hydrothermal, 100ºC 50h) α-li 2 IrO 3 IrO 6 Li + H + Any chance for new material? Crystal structure indeed changed, but details are unknown Difficult to distinguish between H and Li by x-ray

34 New honeycomb iridate by hydrogen exchange α-li 2 IrO 3 + DCl (hydrothermal, 100ºC 50h) D Neutron scattering at FRM-II ,000 Deuterated_Li2IrO % 4,800 T = 300 K R. Dinnebier 4,600 4,400 4,200 4,000 3,800 λ = Å 3,600 3,400 3,200 3,000 2,800 2,600 2,400 2,200 2,000 1,800 1,600 1,400 1,200 1, (Ir/Li)O 6 DLi Ir O 2 R-3m a = (34) Å c = (234) Å -1, Only interlayer Li was replace with H + (or D + ) (Strong stacking fault mimics the site-exchange between Ir 4+ and Li +.) Honeycomb layer remain intact. 85 (similar result with Ag-Li 2 IrO 3, H-Li 2 MnO 3 ) 90 V. Todorova et al., JSSC 184, 1112 (2011) Y. Paik et al., Chem. Mater. 14, 5109 (2002) H 3 LiIr 2 O 6 H-Li 2 IrO

35 Strong trigonal distortion in IrO 6 octahedra Å Å RIXS at BL11XU SPring-8 Ir L 3 -edge (E i = kev) D-Li 2 IrO 3 T = 7 K IrO 6 strongly compressed along c D-Li 2 IrO Å t 2g J eff = 1/2 e g 3λ/2 a 1g J eff = 3/ (>> 90º) J eff = 1/2-like J eff = 3/2-like Peak at ~0.7 ev J eff = 3/2 1/2 excitation H. Gretarsson et al. PRL 110, (2013) Broadened peak in H-Li 2 IrO 3 split of J eff = 3/2 manifold

36 No magnetic order down to 0.4 K despite θ W ~ -110 K Broad peak around 100 K µ eff = 1.66 µ Β, θ W ~ -110 K (cf. α-li 2 IrO 3, θ W = -12 K) Curie tail ~3% of S = 1/2 spin No magnetic order down to 0.4 K Due to disorder? Glass-like freezing??

37 Spin liquid behavior without any order or freezing K. Kitagawa No broadening seen down to 2.2 K No sign of spin freezing 1/T 1 T almost constant down to 2 K... Gapless?

38 Summary for hydrogen exchanged α-li 2 IrO 3 Hydrogen exchanges inter-layer Li + of α-li 2 IrO 3...inducing strong distortion in IrO 6 Kitaev coupling seems suppressed by the distortion Strong AF coupling inferred from θ W No magnetic order down to 0.4 K. No spin-glass freezing despite strong (stacking) disorder Why frustrated in bipartite honeycomb? Proximity to metal-insulator transition? J 1 -J 2 (-J 3 ) frustration?

39 J 1 -J 2 frustration leads to spin liquid behavior? Why bipartite honeycomb can be a spin liquid? I. Half-filled honeycomb close to MIT Quantum Monte Carlo shows Spin-liquid ground state in vicinity to MIT II. J 1 -J 2 frustration (Z. Y. Meng et al., Nature Scientific 464, reports 8472, (2010)) 992 (2012) AF J 1 & J 2 compete spin liquid for J 2 /J 1 > 0.08 (B. K. Clark et al., PRL 2011) (c.f.) Na 2 IrO 3 DFT calculation t 1 ~ 270 mev, t 2 ~ -75 mev J 2 /J 1 ~ 0.08 (K. Foyevtsova et al., PRB 2013) Closeness to MIT, charge fluctuation may stabilize spin-liquid ground state!

40 Spin-liquid candidates with honeycomb lattice Bi 3 Mn 4 O 12 (NO 3 ) Copper ammonium salt [(C 3 H 7 ) 3 NH] 2 [Cu 2 (C 2 O 4 ) 3 ](H 2 O) 2.2 Frustrated honeycomb of Mn 4+ S = 3/2 θ ~ -220 K, Spin glass freezing ~ 6 K O. Smimova et al., JACS 2009 N. Onishi et al., PRB 2012 Orbital ordering reduces the dimension into weakly coupled 1D AF chains B. Zhang et al., Scientific Reports 4, 6451 (2014)

41 Nature of spin liquid state...gapless? Finite χ 0 and γ, together with (T 1 T) -1, suggest gapless spin liquid Gapped spin liquid χ 0 = x 10-3 emu/mol γ = 13.4 mj/mol K 2 R W ~ 5.9 χ 0 towards 0 K F. Mila Eur. J. Phys. 21, 499 (2000) c.f. κ-(bedt-ttf) 2 Cu 2 (CN) 3... R W ~ 1.09 EtMe 3 Sb[Pd(dmit) 2 ] 2... R W ~ 1.13 Na 4 Ir 3 O 8... R W ~ 27-40! S. Yamashita et al., Nature Com (2011) Because of strong spin-orbit??

42 1/T 1 of other spin-liquid materials Other spin-liquid candidates display power-law dependence of T 1-1 H-Li 2 IrO /T 1 T Korrenga-law... Spinon Fermi surface?? κ-(et) 2 Cu 2 (CN) 3 EtMe 3 Sb[Pd(dmit) 2 ] 2 ZnCu 3 (OH) 6 Cl 2 1/T 1 ~ T 3/2 1/T 1 ~ T 2 Y. Shimizu et al., PRB (2006) (κ suggests spin gap) T. Itou et al., Nature physics 2010 No consensus on dynamics of the spin-liquid phase... 1/T 1 ~ T 0.5 T. Imai et al., PRL 2008 Other group ~T 0.73

43 Content I. 2D honeycomb iridates Platform for Kitaev physics II. 3D honeycomb iridates β- and γ-li 2 IrO 3, Proximity to Kitaev limit III. Chemical modulation to 2D honeycombs Possible new spin liquid in honeycomb lattice IV.Spinel and hyperkagome lattice Quantum spin liquid and SOC semimetal

44 Spinel crystal structure Also related with NaCl-type (ccp of O 2- ions) Tetrahedral Layer I and void Octahedral void Layer II 1/2 of octahedral voids 1/8 of tetrahedral voids are filled by cations Pyrochlore lattice O 2- B ccp stacking of anions (O 2- ) Playground for many interesting physics (SC, frustration, HF etc.)

45 AB 2 O 4 spinel difficult for iridate Spinel oxides known so far... A + B 3.5+ (LiTi 2 O 4, LiV 2 O 4 etc.) A 2+ B 3+ (MgV 2 O 4, ZnCr 2 O 4 etc.) A 3+ B 2.5+ (AlV 2 O 4 etc.) A 4+ B 2+ (GeNi 2 O 4 etc.) All B valence states are unfavorable for Ir Only known material... ZnIr 2 O 4 (Thin-film grown on Al 2 O 3 (0001)) ZnRh 2 O 4 Ir 3+, 5d 6... Non-magnetic band insulator ZnIr 2 O 4 ZnCo 2 O 4 M. Dekkers et al., APL 90, (2007)

46 Thiospinel CuIr 2 S 4... Octamer formation Cu + Ir S 4 Charge degree of freedom involved Ir 4+ octamer Ir 3+ octamer T. Hagino et al., Phil. Mag. B 71, 881 (1995) Metal- non-magnetic insulator transition at ~240 K. P. Radaelli et al., Nature 416, 155 (2002) Charge-orbital order accompanying spin-singlet formation High-T: J eff = 1/2 state -> Low-T: d xy, d yz, d zx??

47 Superconductivity in doped CuIr 2 S 4 By chemical substitutions, the spin singlet state is suppressed. Eventually, SC appears at low temperatures Cu 1-x Zn x Ir 2 S 4 Nature of superconductivity? H. Suzuki et al., JPSJ 68, 2495 (1999)

48 A-site deficient spinel Ir 2 O 4 J. Matsuno J eff = 1/2 Mott? Magnetism difficult to investigate... Spinel Li x Ir 2 O 4 grown on LiNbO 3 (0001) (only thin-film available) Remove Li electrochemically. H. Kuriyama et al., APL 96, (2010)

49 Hyperkagome iridate Na 4 Ir 3 O 8 Na 4 Ir 3 O 8 Hyper-kagome network of Ir atoms Na IrO 6 Chirality of hyperkagome Y. Okamoto et al., PRL 99, (2007) 2[Na] oct (Ir 3/4 Na 1/4 ) 2 O 4 Na1: 1/4 of pyrochlore lattice Na2, Na3: Ochtahedral voids of spinel structure A-site of spinel is empty. 3D network of corner-sharing triangle Geometrically frustrated lattice

50 Hyperkagome network is not unique to Na 4 Ir 3 O 8 Garnet (e.g. Y 3 Fe 5 O 12 ) Li 2 Pt 3 B (Space Group: P4 3 32) A-site of garnet forms (distorted) hyperkagome lattice. Non-centrosymmetric SC SC gap with line nodes H. Q. Yuan et al., PRL 97 (2006) Mo 3 Al 2 C

51 Quantum Spin-liquid behavior of Na 4 Ir 3 O 8 Θ W ~ -650 K, no sign of order down to 2 K Broad peak of C m /T around 24 K C m = γt + at n, 2 < n < 3 No order down to 0.5 K Y. Okamoto et al., PRL 99, (2007) Y. Singh et al., PRB 88, (2013)

52 Quasi-static order or frozen state by µsr and NMR µsr 23 Na NMR Quasi-static order with slow fluctuation R. Dally et al., PRL 113, (2014) Frozen state below 6~7 K A.C. Shockley et al., PRL 115, (2015)

53 Metal in proximity to spin liquid Spin-liquid organics (triangular) κ-(et) 2 Cu 2 (CN) 3 Weak Mott insulator (small U) Close to MIT S(J eff ) = 1/2 quantum spin liquid Y. Kurosaki et al., PRL 95, (2005) Does metallic hypekagome superconduct?? Carrier doping by Na deficiency

54 New phase of hyper-kagome Na 3 Ir 3 O 8 Single crystals found to be Na 3 Ir 3 O 8 Na(2) Ir Na(1) Ordered spinel AB 2 O 4 : 2 (Na(Na 1/4 Ir 3/4 ) 2 O 4 ) Single crystal x-ray analysis P4 1 32, a = Å (293 K), Z = 4 site x y z g Ir 12d Na1 4bObtained as single crystals Na2 8c O1 8c O2 24e T. Takayama et al., Scientific Reports 4, 6818 (2014) Stoichiometric compounds! Chiral hyper-kagome network remains intact Na(1): Octahedral site in the pyrochlore lattice Na(2): Tetrahedral site like in a spinel Ir 4.33+, 1/3 hole doping onto Na 4 Ir 3 O 8 Grown by flux method

55 Different Ir valences confirmed by RIXS RIXS at BL 11XU SPring-8 (single) (powder) K. Ishii J 1/2 J 1/2 J 3/2 J 1/2 Calculation by A. Yaresko 2p t 2g - e g Shift of peak energies at 1.0 ev and 4.5 ev (d-d excitations) low energy peak (< 0.3 ev) only in Na 3 Ir 3 O 8 Different Ir 5d filling Powder: Na 4 Ir 4+ 3O 8, 5d 5 Single: Na 3 Ir O 8, 5d /3 heavy hole doping

56 Semi-metallic state in Na 3 Ir 3 O 8 hyper-kagome Metallic as expected from 1/3 hole doping poor metal ρ ~ 1 mωcm at 5 K Temperature dependent R H R H < 0, electron dominant Small carrier number n ~ cm 3 Semi-metal Seemingly incompatible with 1/3 doped Mott insulator T. Takayama et al., Scientific Reports 4, 6818 (2014)

57 If no SOC, band insulator even with Ir LDA without SOC Calculation by A. Yaresko Na 3 Ir 3 O 8 Band insulator even with non-integer valence state Ir (5d 4.67 ) One of t 2g orbitals is fully filled, but others are not. Orbital ordering?

58 With SOC, a semi-metal is produced Na 3 Ir 3 O 8 LDA with SOC Calculation by A. Yaresko (= J eff 1/2 ) hole pocket electron pocket Splitting of Kramers degeneracy due to SOC and chirality Larger dispersion in electron bands R H < 0 J eff = 1/2-like character by SOC SrIrO 3 : half-filled metal SOC semi-metal Na 3 Ir 3 O 8 : Band insulator SOC semi-metal?

59 Band insulator with molecular orbital formation Na 3 Ir 3 O 8 : Assume no spin-orbit coupling... Ir electron per Ir 3 d xy fully filled d zx, d yz give the gap Bond length... Ir-O1 >> Ir-O2 Hopping via O2 dominant Ir d O2 p Ir d hopping creates molecular orbital on each triangle Ir 3 molecule...14 electrons fulfill the orbitals right below the gap Band insulator

60 SOC destroy Molecular orbital to form J eff = 1/2 With SOC, Ir t 2g orbitals form J eff = 1/2 state 1 J eff 1/ 2 = + zx 3 ( xy, ± σ ± yz,m σ i, mσ ) By SOC, d xy, d yz, d zx entangled to form J eff = 1/2-like state t 2g electrons delocalize out of Ir 3 molecules Turn on SOC, molecular orbital destroyed and semi-metal appears

61 Semi-metal with remnant molecular orbital Narrow bandwidth with remnant molecular orbital Semi-metal with heavy mass carriers (m* ~ 2m 0 for electrons, m* ~ 2-6m 0 for holes) Strong inter-band transition γ calc ~ 2.9 mj/mol K 2 γ exp ~ 4.3 mj/mol K 2 Observed in RIXS below 0.3 ev, no such transition in Na 4 Ir 3 O 8

62 Interesting topological effect?? Hyperkagome lattice... chirality Chirality + SOC -> Rashba split...dirac dispersion in the hole bands Berry phase in quantum oscillation? Na 3 Ir 3 O 8 Estimated Rashba split... ~20 mev (Valence band Maximum Dirac point) BiTeI Too small? compared with p-electron system like BiTeI (~110 mev) H. Murakawa et al., Science 342, 1490 (2013)

63 Summary Edge-shared iridates... Interesting area for exotic magnetism Interference effect of J eff = 1/2 isospin Possible Kitaev spin liquid 2D honeycomb, 3D honeycomb etc. Proximity to metal-insulator transition Enhance long-range hopping (charge fluctuation) Spin liquid? by competing interactions Exotic metallic state produced by spin-liquid What else? 2D kagome, 1D chain, chalcogenide(?) etc.