Magnetism in Low-Dimensional Systems

Transcription

1 From Frustration to Complex Order Volker Eyert Center for Electronic Correlations and Magnetism Institute for Physics, University of Augsburg April 18, 29

2 Outline Cool-down 1 Cool-down 2 3 Volker@Eyertde

3 Outline Cool-down 1 Cool-down 2 3 Volker@Eyertde

4 Outline Cool-down 1 Cool-down 2 3 Volker@Eyertde

5 Outline Cool-down 1 Cool-down 2 3 Volker@Eyertde

6 Delafossites ABO 2 Delafossite Structure Building Blocks rhombohedral lattice triangular A-atom layers BO 2 sandwich layers B-atoms octahedrally coordinated linear O A O bonds Volker@Eyertde

7 Delafossites ABO 2 Delafossite Structure Building Blocks rhombohedral lattice triangular A-atom layers BO 2 sandwich layers B-atoms octahedrally coordinated linear O A O bonds Issues dimensionality frustration effects play chemistry Volker@Eyertde

8 Delafossites ABO 2 Delafossite Structure Prototype Materials CuFeO 2, CuCrO 2 CuCoO 2, CuRhO 2 CuAlO 2, CuGaO 2, CuInO 2, PdCrO 2, PdCoO 2, PdRhO 2, PtCoO 2 Properties semiconductors, AF interactions, (distorted) triangular non-mag semicond, high TEP wide-gap semicond, p-type TCO very good metals, highly anisotropic Volker@Eyertde

9 Delafossites ABO 2 Delafossite Structure Prototype Materials CuFeO 2, CuCrO 2 CuCoO 2, CuRhO 2 CuAlO 2, CuGaO 2, CuInO 2, PdCrO 2, PdCoO 2, PdRhO 2, PtCoO 2 Properties semiconductors, AF interactions, (distorted) triangular non-mag semicond, high TEP wide-gap semicond, p-type TCO very good metals, highly anisotropic Volker@Eyertde

10 Delafossites ABO 2 Delafossite Structure Prototype Materials CuFeO 2, CuCrO 2 CuCoO 2, CuRhO 2 CuAlO 2, CuGaO 2, CuInO 2, PdCrO 2, PdCoO 2, PdRhO 2, PtCoO 2 Properties semiconductors, AF interactions, (distorted) triangular non-mag semicond, high TEP wide-gap semicond, p-type TCO very good metals, highly anisotropic Volker@Eyertde

11 Delafossites ABO 2 Delafossite Structure Prototype Materials CuFeO 2, CuCrO 2 CuCoO 2, CuRhO 2 CuAlO 2, CuGaO 2, CuInO 2, PdCrO 2, PdCoO 2, PdRhO 2, PtCoO 2 Properties semiconductors, AF interactions, (distorted) triangular non-mag semicond, high TEP wide-gap semicond, p-type TCO very good metals, highly anisotropic Volker@Eyertde

12 PdCoO 2 and PtCoO 2 Delafossite Structure Facts very low resistivity anisotropy factor 2 PES: only Pd 4d states at E F PES/IPES: E F in shallow DOS minimum Volker@Eyertde

13 PdCoO 2 and PtCoO 2 Delafossite Structure Facts very low resistivity anisotropy factor 2 PES: only Pd 4d states at E F PES/IPES: E F in shallow DOS minimum Open Issues role of Pd 4d orbitals? role of Co 3d and O 2p orbitals? Volker@Eyertde

14 Electronic Properties of PdCoO 2 Partial Densities of States DOS (1/eV) Pd 4d Co 3d t 2g Co 3d e g O 2p (E - E F ) (ev) Results Co 3d-O 2p hybridization CoO 6 octahedra: Co 3d t 2g and e g Co 3d 6 (Co 3+ ) LS Co 3d, O 2p: very small DOS at E F Pd 4d 9 (Pd 1+ ) VE, R Frésard, A Maignan, Chem Mat 2, 237 (28) Volker@Eyertde

15 Electronic Properties of PdCoO 2 Partial Densities of States DOS (1/eV) Pd 4d xy,x 2 -y 2 Pd 4d xz,yz Pd 4d 3z 2 -r 2 O 2p z (E - E F ) (ev) Results Pd 4d-O 2p hybridization broad Pd d xy,x 2 y 2 bands states at E F : Pd d xy,x 2 y 2, d 3z 2 r 2 VE, R Frésard, A Maignan, Chem Mat 2, 237 (28) Volker@Eyertde

16 Cool-down Electronic Properties of PdCoO 2 Band Structure (E - E F ) (ev) Results quasi-2d single band crossing E F Brillouin Zone k z -6 L u ua H u -8-1 Γ M K Γ A L H A k x u M u uk k y VE, R Frésard, A Maignan, Chem Mat 2, 237 (28) Volker@Eyertde

17 Electronic Properties of PdCoO 2 Fermi Surface Results strong anisotropy but: bands below E F disperse along Γ-A VE, R Frésard, A Maignan, Chem Mat 2, 237 (28) Volker@Eyertde

18 The Fingerprint of Dimensionality Band Dispersions 6 4 Pd 4dxy,x 2 y 2 2 EF E -2 ev Γ M K Γ A L H A 6 4 Pd 4d3z 2 r 2 2 EF E -2 ev Γ M K Γ A L H A Volker@Eyertde

19 The Fingerprint of Dimensionality Free Electron Density of States ρ(e) d d k δ (E 2 k 2 ) Ω BZ 2m ( ) 2mE d k k d 1 δ k Ω BZ 2 m d 2 2mE 2 2 Volker@Eyertde

20 The Fingerprint of Dimensionality Free Electron Density of States

21 The Fingerprint of Dimensionality Tight-Binding Bands ε k = E + E 1 { } e ik d = 1 d = 2 d = 3 (from: G Czycholl, Theoretische Festkörperphysik, Springer) Volker@Eyertde

22 The Fingerprint of Dimensionality Single-layer SrTiO Ti 3d xy Ti 3d xz,yz Ti 3d 3z 2 -r 2 Ti1 3d x 2 -y 2 DOS (1/eV) d 3z 2 r 2 : d = d xz,yz : d = 1 d xy : d = (E - E V ) (ev) Volker@Eyertde

23 Outline Cool-down 1 Cool-down 2 3 Volker@Eyertde

24 Augmented Spherical Wave (ASW) Method Characteristics a dialect of Andersen s LMTO method different linearization schemes different interstitial energies different implementations Volker@Eyertde

25 Augmented Spherical Wave (ASW) Method Characteristics a dialect of Andersen s LMTO method different linearization schemes different interstitial energies different implementations th Generation (Williams, Kübler, Gelatt, 197s) PRB 19, 694 (1979) Volker@Eyertde

26 Augmented Spherical Wave (ASW) Method 1st Generation (VE, 199s) IJQC 77, 17 (2) monolithic implementation from scratch new algorithms improved accuracy, numerical stability much improved functionality, usability, and portability xanderson convergence acceleration scheme all LDA-parametrizations, most GGA-schemes still based on atomic-sphere approximation

27 ASW Method: Basic Formalism Wave Function Expanded in Basis Functions ψ σ (r) = Lκi c Lκiσ H Lκσ (r i) c Lκiσ determined variationally Volker@Eyertde

28 ASW Method: Basic Formalism Wave Function Expanded in Basis Functions ψ σ (r) = Lκi c Lκiσ H Lκσ (r i) c Lκiσ determined variationally Basis Functions: Augmented Spherical Waves H HLκσ Lκ I (r i) interstitial region (r i) = H Lκσ (r i ) on-centre sphere i J L j L κσ(r j )B L Lκ off-centre spheres j B L Lκ(R j R i ): structure constants ASW classified by atomic site R i, L = (l, m), decay κ, spin σ Volker@Eyertde

29 ASW Method: Basic Formalism Envelope Functions H I Lκ (r i) := iκ l+1 h (1) l (κr i )Y L (ˆr i ) h (1) l (κr i ): spherical Hankel function Volker@Eyertde

30 ASW Method: Basic Formalism Envelope Functions H I Lκ (r i) := iκ l+1 h (1) l (κr i )Y L (ˆr i ) h (1) l (κr i ): spherical Hankel function Augmented Functions H Lκσ (r i ) := h lκσ (r i )Y L (ˆr i ) J L κσ(r j ) := j l κσ(r j )Y L (ˆr j ) h, j: numerical solutions of radial Kohn-Sham equation boundary conditions from envelope functions correspond to ϕ and ϕ of LMTO Volker@Eyertde

31 ASW Method: Basic Formalism Augmented Spherical Waves envelope functions augmented spherical waves A B A B r Volker@Eyertde

32 ASW Method: Further Reading

33 Towards a Full-Potential Spherical-Wave Method Status ASA (space-filling atomic spheres) O(ASA) speed systematic error in total energy non-overlapping muffin-tin spheres prerequisite for accurate total energies larger basis set inefficient Volker@Eyertde

34 Towards a Full-Potential Spherical-Wave Method Status ASA (space-filling atomic spheres) O(ASA) speed systematic error in total energy non-overlapping muffin-tin spheres prerequisite for accurate total energies larger basis set inefficient Requirements restore interstitial region go to non-overlapping muffin-tin spheres go beyond constant-potential approximation inside muffin-tin spheres non-spherical contributions Volker@Eyertde

35 Towards a Full-Potential Spherical-Wave Method Status ASA (space-filling atomic spheres) O(ASA) speed systematic error in total energy non-overlapping muffin-tin spheres prerequisite for accurate total energies larger basis set inefficient Requirements restore interstitial region go to non-overlapping muffin-tin spheres go beyond constant-potential approximation inside muffin-tin spheres non-spherical contributions Volker@Eyertde

36 Towards a Full-Potential Spherical-Wave Method Guidelines interstitial quantities expanded in plane waves straightforward to implement inefficient interstitial quantities expanded in spherical waves elegant, no periodicity required efficient difficult to implement Volker@Eyertde

37 Towards a Full-Potential Spherical-Wave Method Guidelines interstitial quantities expanded in plane waves straightforward to implement inefficient interstitial quantities expanded in spherical waves elegant, no periodicity required efficient difficult to implement Volker@Eyertde

38 Basic Principles Cool-down Steps to be Taken remove total energy error due to overlap of atomic spheres reintroduce non-overlapping muffin-tin spheres restore interstitial region

39 Basic Principles Cool-down Steps to be Taken remove total energy error due to overlap of atomic spheres reintroduce non-overlapping muffin-tin spheres restore interstitial region find representation of electron density and full potential inside muffin-tin spheres in the interstitial region

40 Basic Principles Cool-down Steps to be Taken remove total energy error due to overlap of atomic spheres reintroduce non-overlapping muffin-tin spheres restore interstitial region find representation of electron density and full potential find representation of products of the wave function inside muffin-tin spheres in the interstitial region

41 Basic Principles Cool-down Steps to be Taken remove total energy error due to overlap of atomic spheres reintroduce non-overlapping muffin-tin spheres restore interstitial region find representation of electron density and full potential find representation of products of the wave function find representation of products of the basis functions inside muffin-tin spheres in the interstitial region

42 Basic Principles Cool-down Steps to be Taken remove total energy error due to overlap of atomic spheres reintroduce non-overlapping muffin-tin spheres restore interstitial region find representation of electron density and full potential find representation of products of the wave function find representation of products of the basis functions inside muffin-tin spheres use spherical-harmonics expansions in the interstitial region

43 Basic Principles Cool-down Steps to be Taken remove total energy error due to overlap of atomic spheres reintroduce non-overlapping muffin-tin spheres restore interstitial region find representation of electron density and full potential find representation of products of the wave function find representation of products of the basis functions inside muffin-tin spheres use spherical-harmonics expansions in the interstitial region no exact spherical-wave representation available!

44 From Wave Functions to Electron Density Density inside MT-Spheres (Al) ρ v / au Volker@Eyertde

45 From Wave Functions to Electron Density Products of Basis Functions in Interstitial Region p I (r) = n d n F n (r) d 3 r F n (r)pi (r) = n d n d 3 r F n (r)f n(r) Volker@Eyertde

46 From Wave Functions to Electron Density Products of Basis Functions in Interstitial Region p I (r) = n d n F n (r) d 3 r F n (r)pi (r) = n d n d 3 r F n (r)f n(r) F n (r): plane waves integrals exact inefficient Weyrich 1988, Blöchl 1989, VE 1991, Savrasov 1992, Methfessel 2 Volker@Eyertde

47 From Wave Functions to Electron Density Products of Basis Functions in Interstitial Region p I (r) = n d n F n (r) d 3 r F n (r)pi (r) = n d n d 3 r F n (r)f n(r) F n (r): spherical waves would be efficient integrals not known analytically Springborg/Andersen 1987, Methfessel 1988, VE 22, VE 26 Volker@Eyertde

48 From Wave Functions to Electron Density Products of Basis Functions in Interstitial Region p I (r) = n d n F n (r) d 3 r F n (r)pi (r) = n d n d 3 r F n (r)f n(r) F n (r): spherical waves would be efficient integrals not known analytically Springborg/Andersen 1987, Methfessel 1988, VE 22, VE 26 Methfessel 1988: match values and slopes at MT-sphere surfaces Volker@Eyertde

49 From Wave Functions to Electron Density Density from Value/Slope Matching at MT-Radii (Al) ρ v / au ρ v / au Volker@Eyertde

50 From Electron Density to Full Potential Inside Muffin-Tin Spheres density, Hartree-potential and xc-potential numerically Interstitial Region density from value/slope matching Hartree-potential analytically xc-potential from value/slope matching

51 From Electron Density to Full Potential Inside Muffin-Tin Spheres density, Hartree-potential and xc-potential numerically Interstitial Region density from value/slope matching Hartree-potential analytically xc-potential from value/slope matching

52 From Full Potential to Basis Functions Previous Approaches project full potential to muffin-tin potential construct basis functions from muffin-tin potential no minimal basis set! (large basis set!) Present Approach project full potential to ASA potential construct basis functions from ASA potential minimal basis set!

53 From Full Potential to Basis Functions Previous Approaches project full potential to muffin-tin potential construct basis functions from muffin-tin potential no minimal basis set! (large basis set!) Present Approach project full potential to ASA potential construct basis functions from ASA potential minimal basis set!

54 Comparison of Approaches Ole K Andersen ASA geometry used for basis functions minimal basis set ASA geometry used for density and potential error in total energy good! bad! Volker@Eyertde

55 Comparison of Approaches Ole K Andersen ASA geometry used for basis functions minimal basis set ASA geometry used for density and potential error in total energy Michael S Methfessel MT geometry used for density and potential accurate total energy MT geometry used for basis functions large basis set good! bad! good! bad! Volker@Eyertde

56 Comparison of Approaches Ole K Andersen ASA geometry used for basis functions ASA geometry used for density and potential Michael S Methfessel MT geometry used for density and potential MT geometry used for basis functions present approach ASA geometry used for basis functions minimal basis set O(ASA) speed MT geometry used for density and potential accurate total energy good! bad! good! bad! great! great! Volker@Eyertde

57 2nd Generation ASW (VE, 2s) based on 1st generation code full-potential ASW method electron densities, spin densities electric field gradients elastic properties, phonon spectra optical properties based on linear-response theory direct calculation of Rσ and Iσ no Kramers-Kronig relations needed at O(ASA) speed! transport properties, thermoelectrics LDA+U method all flavours for double-counting terms (AMF, FLL, DFT)

58 2nd Generation ASW (VE, 2s) based on 1st generation code full-potential ASW method electron densities, spin densities electric field gradients elastic properties, phonon spectra optical properties based on linear-response theory direct calculation of Rσ and Iσ no Kramers-Kronig relations needed at O(ASA) speed! transport properties, thermoelectrics LDA+U method all flavours for double-counting terms (AMF, FLL, DFT)

59 2nd Generation ASW (VE, 2s) based on 1st generation code full-potential ASW method electron densities, spin densities electric field gradients elastic properties, phonon spectra optical properties based on linear-response theory direct calculation of Rσ and Iσ no Kramers-Kronig relations needed at O(ASA) speed! transport properties, thermoelectrics LDA+U method all flavours for double-counting terms (AMF, FLL, DFT)

60 2nd Generation ASW (VE, 2s) based on 1st generation code full-potential ASW method electron densities, spin densities electric field gradients elastic properties, phonon spectra optical properties based on linear-response theory direct calculation of Rσ and Iσ no Kramers-Kronig relations needed at O(ASA) speed! transport properties, thermoelectrics LDA+U method all flavours for double-counting terms (AMF, FLL, DFT)

61 LTO(Γ)-Phonon in Silicon ASA + Code E (Ryd) x Si Bad no stable Si position # nature Volker@Eyertde

62 LTO(Γ)-Phonon in Silicon ASA + Code Full-Potential Code E (Ryd) E (Ryd) x Si x Si New! phonon frequency: f calc = 1534 THz (f exp = 1553 THz) Volker@Eyertde

63 Outline Cool-down 1 Cool-down 2 3 Volker@Eyertde

64 Outline Cool-down 1 Cool-down 2 3 Volker@Eyertde

65 La 2 BaCuO 5 : A D(?) Ferromagnetic Semiconductor Crystal Structure Building Blocks P4/mbm (D 5 4h ) a = Å, c = Å Ba-Cu and La-O1 layers O2 interlayers LaO 8 and BaO 1 polyhedra CuO 4 plaquettes Cu-Cu distance 484 Å chains c in (a,b) planes coupled by La atoms Volker@Eyertde

66 La 2 BaCuO 5 : A D(?) Ferromagnetic Semiconductor Plaquettes x x } } x } } # ## # ## x x xhhhhh x x x((((((( ~ z ~ z ~ z ~ Cu z ~ Ba x x x hhhhh x x x ((((((( O 2 x } x } x } } x (((((((x hhhhh x (((((((x hhhhh # ## La x # x ## O 1 a x x c Building Blocks P4/mbm (D 5 4h ) a = Å, c = Å Ba-Cu and La-O1 layers O2 interlayers LaO 8 and BaO 1 polyhedra CuO 4 plaquettes Cu-Cu distance 484 Å chains c in (a,b) planes coupled by La atoms Volker@Eyertde

67 La 2 BaCuO 5 : A D(?) Ferromagnetic Semiconductor Magnetic Properties T C = 52 K m = 95 µ B /Cu c T > 8 K: Curie-Weiss µ eff = 154 µ B Θ < K T < 8 K: Curie-Weiss Θ = 5 K 3D Spin- 1 2 Heisenberg Model + Ising-like anisotropy Electronic Properties Cu 2+ d 9 Cu d hole ˆ= d x 2 y 2 plaquette orbital Cluster Model ferromagnetic order interference of different exchange paths Volker@Eyertde

68 La 2 BaCuO 5 : A D(?) Ferromagnetic Semiconductor Open Issues Role of Cu 3d, O 2p, La 5d Orbitals? Dimensionality? Effective Magnetic Sites? Exchange Paths? Heisenberg model? Volker@Eyertde

69 La 2 BaCuO 5 : A D(?) Ferromagnetic Semiconductor Open Issues Role of Cu 3d, O 2p, La 5d Orbitals? Dimensionality? Effective Magnetic Sites? Exchange Paths? Heisenberg model? TODOs Spin-Degenerate Calculations Role of Orbitals Chemical Bonding Spin-Polarized Calculations Magnetic Instability Exchange Paths Volker@Eyertde

70 Spin-Degenerate Calculations for La 2 BaCuO 5 Band Structure (E - E F ) (ev) Partial DOS DOS (1/eV) Cu 3d O1 2p O2 2p La 5d -6 Γ M X Γ Z A R Z (E - E F ) (ev) VE, K-H Höck, P S Riseborough, EPL 31, 385 (1995) Volker@Eyertde

71 Spin-Degenerate Calculations for La 2 BaCuO 5 Cu 3d Partial DOS O 2p Partial DOS DOS (1/eV) Cu 3d xy Cu 3d xz,yz Cu 3d 3z 2 -r 2 Cu 3d x 2 -y 2 DOS (1/eV) O2 2p par O2 2p perp O2 2p z (E - E F ) (ev) (E - E F ) (ev) Bonding/Antibonding Splitting from p-d Overlap 25 3 ev for t 2g -like states (π-type) 55 ev for d x 2 y 2/p states (σ-type) = ev for d 3z 2 r 2 states (apex ) Volker@Eyertde

72 Spin-Polarized Calculations for La 2 BaCuO 5 Magnetic Moments (µ B ) ( ) ( ) Cu O La 2 3 Ba -1 O1 fu 1 87 Total Energies (mryd/fu) spin-deg ( ) ( ) Band Gap (ev) spin-deg ( ) ( ) VE, K-H Höck, P S Riseborough, EPL 31, 385 (1995) Volker@Eyertde

73 Spin-Polarized Calculations for La 2 BaCuO 5 Cu 3d Partial DOS O 2p Partial DOS 1 5 Cu 3d xy Cu 3d xz,yz Cu 3d 3z 2 -r 2 Cu 3d x 2 -y O2 2p par O2 2p perp O2 2p z DOS (1/eV) DOS (1/eV) (E - E F ) (ev) (E - E F ) (ev) Results extended localized moments on plaquettes carried by Cu 3d x 2 y and O2 2p 2 states Volker@Eyertde

74 Spin-Polarized Calculations for La 2 BaCuO 5 Electron Density ρ v / au Volker@Eyertde

75 Spin-Polarized Calculations for La 2 BaCuO 5 Electron Density Electron Spin Density ρ v / au ρ 1 v / au Results extended localized moments on plaquettes carried by Cu 3d x2 y and O2 2p 2 states Volker@Eyertde

76 Spin-Polarized Calculations for La 2 BaCuO 5 Electron Density Electron Spin Density ρ v / au ρ 1 v / au Results extended localized moments on plaquettes carried by Cu 3d x2 y and O2 2p 2 states coupled via La 5d x 2 y states 2 Volker@Eyertde

77 Outline Cool-down 1 Cool-down 2 3 Volker@Eyertde

78 Orbital Ordering Transition in La 2 RuO 5 Semiconductor-Semiconductor Transition at 16 K P Khalifah et al, Science 297, 2237 (22) Volker@Eyertde

79 Orbital Ordering Transition in La 2 RuO 5 Basics Ru 4+ 4d 4 low-spin (S = 1) above 16 K: CW below 16 K: χ χ(h), H < 9 T no supercell reflects no AF order (?) local moments? S = 1, S =? Magnetism, Crystal Structure P Khalifah et al, Science 297, 2237 (22) Volker@Eyertde

80 Orbital Ordering Transition in La 2 RuO 5 HT-Structure Building Blocks RuO 6 octahedra corner-sharing chains c zigzag pattern b double layers b, c LaO 9 tricapped trigonal prisms La 4 O tetrahedra Volker@Eyertde

81 Orbital Ordering Transition in La 2 RuO 5 HT-Structure LT-Structure Volker@Eyertde

82 Orbital Ordering Transition in La 2 RuO 5 HT-Structure monoclinic, P2 1 /c d Ru Ru = 3975 Å d O Ru O Ru O = 7772 Å LT-Structure triclinic, P 1 d Ru(1) Ru(2) = 3868, 445 Å d O Ru(1) O Ru(2) O = 7579, 7977 Å P Khalifah et al, Science 297, 2237 (22) Volker@Eyertde

83 Orbital Ordering Transition in La 2 RuO 5 Where are the Electrons? P Khalifah et al, Science 297, 2237 (22) Volker@Eyertde

84 Spin-Degenerate Calculations for La 2 RuO 5 HT-Structure LT-Structure DOS (1/eV) Ru 4d t 2g Ru 4d e g O(1)-O(4) 2p O(5) 2p La DOS (1/eV) Ru 4d t 2g Ru 4d e g O(1)-O(4) 2p O(5) 2p La (E - E F ) (ev) (E - E F ) (ev) Results quasi-1d dispersion c CEF Ru 4d t 2g /e g Volker@Eyertde strong Ru 4d-O 2p overlap

85 VE, S G Ebbinghaus, Volker@Eyertde T Kopp, PRL Magnetism 96, in25641 Low-Dimensional (26) Systems Cool-down Spin-Degenerate Calculations for La 2 RuO 5 HT-Structure 2 15 Ru 4d xy Ru 4d yz Ru 4d xz NOS (E - E F ) (ev) Results d xy d xz d yz Hund s rule: S = 1

86 Spin-Degenerate Calculations for La 2 RuO 5 HT-Structure LT-Structure Ru 4d xy Ru 4d yz Ru 4d xz 15 Ru(1) 4d xy Ru(1) 4d yz Ru(1) 4d xz NOS 1 NOS (E - E F ) (ev) (E - E F ) (ev) Results d xy d xz d yz Hund s rule: S = 1 Results d xy d yz d 1 xyd 2 xzd 1 yz: S = 1 Volker@Eyertde

87 Spin-Polarized Calculations for La 2 RuO 5 Total Energies (mryd/fu), Moments (µ B ), Band Gaps (ev) structure funct order E m RuO6 E g monoclinic GGA - - monoclinic GGA ferri monoclinic GGA+U ferri triclinic GGA triclinic GGA ferri triclinic GGA+U ferri VE, S G Ebbinghaus, T Kopp, PRL 96, (26) Volker@Eyertde

88 Spin-Polarized Calculations for La 2 RuO 5 LT-Structure Ru(1) 4d xy Ru(1) 4d yz Ru(1) 4d xz 6 4 Ru(2) 4d xy Ru(2) 4d yz Ru(2) 4d xz DOS (1/eV) 2-2 DOS (1/eV) (E - E F ) (ev) (E - E F ) (ev) Results Ru(1) Ru(2) d xy d yz m(ru): d xy and d yz (d 1 xy d 2 xz d 1 yz ) m(ru(1)) m(ru(2)) Volker@Eyertde

89 Scenario for La 2 RuO 5 HT-Structure LT-Structure y x e g t 2g S = 1 dxy dyz d xz S = 1 Spin-Ladders AF couping on rungs m(ru(1)o 6 ) = m(ru(2)o 6 ) VE, S G Ebbinghaus, T Kopp, PRL 96, (26) Volker@Eyertde

90 Outline Cool-down 1 Cool-down 2 3 Volker@Eyertde

91 Magnetic Order in K 2 NiF 4 and K 2 CuF 4 K 2 NiF 4 and K 2 NiF 4 classical two-dimensional systems based on K 2 NiF 4 structure parent compounds of La 2 CuO 4 magnetic semiconductors role of TM 3d and F 2p orbitals? intralayer/interlayer magnetic coupling? antiferromagnetism vs ferromagnetism symmetry breaking by magnetic order vs symmetry breaking by orbital order Volker@Eyertde

92 Magnetic Order in K 2 NiF 4 and K 2 CuF 4 K 2 NiF 4 Structure Ni F K a c Basics I4/mmm (centered tetragonal) NiF 2 planes (ˆ=CuO 2 planes) K and F interlayers corner sharing NiF 6 octahedra d(ni F apex ) < d(ni F plane ) AF order Cmca (side-centered orthorh) a 2a Volker@Eyertde

93 K 2 NiF 4 Magnetic Properties T N = 9723 K m = 222 µ B magnetic moments c J/k B 1 K 2D spin-wave dispersion J /J < 37 2D (S = 1) Heisenberg Model + Ising-like anisotropy Volker@Eyertde

94 u u u u u Cool-down K 2 NiF 4 : Spin-Degenerate Calculations Band Structure Partial DOS Ni 3d t 2g Ni 3d 3z 2 -r 2 Ni 3d x 2 -y 2 F1 2p F2 2p (E - E F ) (ev) -2-4 DOS (1/eV) Z Γ X P Γ N (E - E F ) (ev) k z Z Results k x N X P k y E < 4 ev: d-p bonding E > 2 ev: d-p antib 2D dispersion at E F : Ni 3d e g Volker@Eyertde

95 K 2 NiF 4 : Weighted Bands Ni 3d 3z 2 r 2 15 Ni 3d x 2 y 2 15 Ni 3dt 2g E ev Z Γ X P Γ N EF 5 E ev Z Γ X P Γ N EF 5 E ev Z Γ X P Γ N EF Partial DOS DOS (1/eV) Ni 3d t 2g Ni 3d 2 2 3z -r Ni 3d 2 2 x -y F1 2p F2 2p (E - E F ) (ev) Results CEF (t 2g /e g ) 9 ev W(d x 2 y 2) 2 W(d 3z 2 r 2) perfect FS nesting Volker@Eyertde

96 u u u u u u u u Cool-down K 2 NiF 4 : Doubling the Unit Cell centered tetragonal side-centered orthorh 2 2 (E - E F ) (ev) -2-4 (E - E F ) (ev) Z Γ X P Γ N Z Γ X P Γ N X2 kx kz Z Γ X1 ky k y P k x X R N S Z T k z Volker@Eyertde

97 K 2 NiF 4 : Spin-Polarized Calculations Band Structure Partial DOS Ni 3d t 2g Ni 3d 3z 2 -r 2 Ni 3d x 2 -y 2 F1 2p F2 2p (E - E F ) (ev) -2-4 DOS (1/eV) Z Γ X P Γ N (E - E F ) (ev) Moments (µ B ) Ni 128 F2 4 sublattice 138 Results moment: d 3z 2 r 2, d x 2 y 2 E tot = 35 mryd Volker@Eyertde

98 K 2 CuF 4 Magnetic Properties T C = 626 K m = 1 µ B magnetic moments c 2D spin-wave dispersion J /J < 66 3D (S = 1 2 ) Heisenberg Model + 1% XY-like anisotropy J/k B = 112 K Volker@Eyertde

99 K 2 CuF 4 Magnetic Properties T C = 626 K m = 1 µ B magnetic moments c 2D spin-wave dispersion J /J < 66 3D (S = 1 2 ) Heisenberg Model + 1% XY-like anisotropy J/k B = 112 K Early XRD (Knox) K 2 NiF 4 structure d Cu Fapex < d Cu Fplane related Cu-compounds Volker@Eyertde

100 K 2 CuF 4 Khomskii and Kugel Model Heisenberg Hamiltonian + Pseudospin Hamiltonian orbital ordering compressed octahedra ferromagnetic order antiferrodistortive structure proposed side-centered orthorh Early XRD (Knox) K 2 NiF 4 structure d Cu Fapex < d Cu Fplane related Cu-compounds Volker@Eyertde

101 K 2 CuF 4 Khomskii and Kugel Model Heisenberg Hamiltonian + Pseudospin Hamiltonian orbital ordering compressed octahedra ferromagnetic order antiferrodistortive structure proposed side-centered orthorh In-Plane Structure Cu F Volker@Eyertde

102 K 2 CuF 4 In-Plane Structure Cu F NMR (Le Dang Khoi, Veillet) octahedra elongated, long axis c Volker@Eyertde

103 K 2 CuF 4 In-Plane Structure Cu F NMR (Le Dang Khoi, Veillet) octahedra elongated, long axis c XRD (Hidaka et al) Cmca (side-centered orthorh) F at ( 1 4 δ, 1 4 δ, ) δ = 176 Volker@Eyertde

104 u u u u u Cool-down K 2 CuF 4 : Spin-Degenerate Calculations Band Structure Partial DOS Cu 3d t 2g Cu 3d 3z 2 -r 2 Cu 3d x 2 -y 2 F1 2p F2 2p (E - E F ) (ev) -2-4 DOS (1/eV) Z Γ X P Γ N (E - E F ) (ev) k x N k z Z X P k y Results similar to K 2 NiF 4 d-p splitting smaller electron count! Volker@Eyertde

105 K 2 CuF 4 : Weighted Bands Cu 3d 3z 2 r 2 1 Cu 3d x 2 y 2 1 Cu 3dt 2g E ev -1 EF 5-5 E ev -1 EF 5-5 E ev -1 EF Z Γ X P Γ N -25 Z Γ X P Γ N -25 Z Γ X P Γ N Partial DOS DOS (1/eV) Cu 3d t 2g Cu 3d 2 2 3z -r Cu 3d 2 2 x -y F1 2p F2 2p (E - E F ) (ev) Results CEF (t 2g /e g ) 1 ev W(d x 2 y 2) 2 W(d 3z 2 r 2) FS(d x 2 y 2) FS(d 3z 2 r 2) no FS nesting Volker@Eyertde

106 u u u u u u u u Cool-down K 2 CuF 4 : Doubling the Unit Cell Bands: centered tetragonal Bands: side-centered orthorh 2 2 (E - E F ) (ev) -2-4 (E - E F ) (ev) Z Γ X P Γ N -8 Z Γ X P Γ N X2 kx kz Z Γ X1 ky k y P k x X R N S Z T k z Volker@Eyertde

107 K 2 CuF 4 : Spin-Polarized Calculations Band Structure Partial DOS Cu 3d t 2g Cu 3d 3z 2 -r 2 Cu 3d x 2 -y 2 F1 2p F2 2p (E - E F ) (ev) -2-4 DOS (1/eV) Z Γ X P Γ N (E - E F ) (ev) Moments (µ B ) Cu 7 F1 8 F2 6 sublattice 1 Volker@Eyertde Results moment: d 3z 2 r 2, d x 2 y 2 E tot = 65 mryd

108 K 2 NiF 4 : Orbital and Magnetic Order (E - E F ) (ev) (E - E F ) (ev) 15 δ = Z Γ X P Γ N 15 δ = Z Γ X P Γ N (E - E F ) (ev) (E - E F ) (ev) 15 δ = Z Γ X P Γ N 15 δ = Z Γ X P Γ N Conclusion FS nesting instability δ or m Volker@Eyertde

109 K 2 CuF 4 : Orbital and Magnetic Order (E - E F ) (ev) (E - E F ) (ev) (E - E F ) (ev) 15 δ = Z Γ X P Γ N 15 δ = Z Γ X P Γ N 15 δ = Z Γ X P Γ N (E - E F ) (ev) (E - E F ) (ev) (E - E F ) (ev) 15 δ = Z Γ X P Γ N 15 δ = Z Γ X P Γ N 15 δ = Z Γ X P Γ N Conclusion no FS nesting instability δ and m (E - E F ) (ev) 15 δ = Z Γ X P Γ N Volker@Eyertde

110 K 2 NiF 4 and K 2 CuF 4 : Energies and Moments Total Energies Magnetic Moments E (mryd) K 2 NiF 4 : E af -E nm K 2 CuF 4 : E fe -E nm K 2 CuF 4 : E fe -E nm /E af m (µ B ) K 2 NiF 4 : m af K 2 CuF 4 : m af K 2 CuF 4 : m fe δ δ VE, K-H Höck, JP:CM 5, 2987 (1993) U Schwingenschlögl, VE, JMMM 312, L11 (27) Volker@Eyertde

111 Outline Cool-down 1 Cool-down 2 3 Volker@Eyertde

112 CuFeO 2 Delafossite Structure Basics semiconductor AF interactions triangular lattice Volker@Eyertde

113 CuFeO 2 Delafossite Structure Basics semiconductor AF interactions triangular lattice Open Issues frustration vs long-range order role of Cu 3d orbitals? role of Fe 3d and O 2p orbitals? Volker@Eyertde

114 CuFeO 2 Previous Neutron Data T N1 = 16 K, T N2 = 11 K Θ CW = 9 K magnetic supercells no structural distortion m Fe 3+ = 44 µ B Volker@Eyertde

115 CuFeO 2 Previous Neutron Data T N1 = 16 K, T N2 = 11 K Θ CW = 9 K magnetic supercells no structural distortion m Fe 3+ = 44 µ B Band Calculations rhombohedral structure m Fe = 9 µ B, m Fe = 38 µ B E g = in LDA, GGA PES, XES Volker@Eyertde

116 CuFeO 2 Previous Neutron Data T N1 = 16 K, T N2 = 11 K Θ CW = 9 K magnetic supercells no structural distortion m Fe 3+ = 44 µ B Band Calculations rhombohedral structure m Fe = 9 µ B, m Fe = 38 µ B E g = in LDA, GGA PES, XES New Neutron Data magnetic supercells monoclinic structure below 4 K Volker@Eyertde

117 CuFeO 2 Previous Neutron Data T N1 = 16 K, T N2 = 11 K Θ CW = 9 K magnetic supercells no structural distortion m Fe 3+ = 44 µ B New Neutron Data magnetic supercells monoclinic structure below 4 K Band Calculations rhombohedral structure m Fe = 9 µ B, m Fe = 38 µ B E g = in LDA, GGA PES, XES Open Issues spin-state of Fe? influence of monoc structure? Volker@Eyertde

118 Electronic Properties of CuFeO 2 Partial Densities of States DOS (1/eV) Cu 3d Fe 3d t 2g Fe 3d e g O 2p (E - E F ) (ev) Results Fe 3d-O 2p hybridization FeO 6 octahedra: Fe 3d t 2g and e g Cu 4d 1 (Cu 1+ ) Fe 3d t 2g sharp peak at E F VE, R Frésard, A Maignan, Phys Rev B 78, 5242 (28) Volker@Eyertde

119 Cool-down Electronic Properties of CuFeO 2 Band Structure (E - E F ) (ev) Γ M K Γ A L H A Results quasi-2d k x single band crossing E F L u u M k z ua H u u uk k y VE, R Frésard, A Maignan, Phys Rev B 78, 5242 (28) Volker@Eyertde

120 Electronic Properties of CuFeO 2 Fermi Surface Results strongly 3D FS PdCoO 2 VE, R Frésard, A Maignan, Phys Rev B 78, 5242 (28) Volker@Eyertde

121 Magnetic Properties of CuFeO 2 Total Energies (mryd/fu), Magn Moms (µ B ), Band Gaps (ev) structure magn order E m Fe m O E g rhomb spin-deg - rhomb ferro (LS) rhomb ferro (IS) rhomb ferro (HS) VE, R Frésard, A Maignan, Phys Rev B 78, 5242 (28) Volker@Eyertde

122 Magnetic Properties of CuFeO 2 LS Ferromagnet IS Ferromagnet HS Ferromagnet 1 Cu 3d Fe 3d t 2g 1 Cu 3d Fe 3d t 2g 1 Cu 3d Fe 3d t 2g 5 Fe 3d e g O 2p 5 Fe 3d e g O 2p 5 Fe 3d e g O 2p DOS (1/eV) DOS (1/eV) DOS (1/eV) (E - E F ) (ev) (E - E F ) (ev) (E - E F ) (ev) Results LS, IS, HS in rhombohedral structure HS: O 2p polarization via Fe 3d e g VE, R Frésard, A Maignan, Phys Rev B 78, 5242 (28) Volker@Eyertde

123 Magnetic Properties of CuFeO 2 Total Energies (mryd/fu), Magn Moms (µ B ), Band Gaps (ev) structure magn order E m Fe m O E g rhomb spin-deg - rhomb ferro (LS) rhomb ferro (IS) rhomb ferro (HS) monoc spin-deg 6 - monoc ferro (LS) monoc ferro (IS) monoc ferro (HS) VE, R Frésard, A Maignan, Phys Rev B 78, 5242 (28) Volker@Eyertde

124 Magnetic Properties of CuFeO 2 Antiferromagnet DOS (1/eV) Cu 3d Fe 3d t 2g Fe 3d e g O 2p Results monoc structure Fe 3+ HS O 2p polarization via Fe 3d e g E g > in GGA (E - E V ) (ev) VE, R Frésard, A Maignan, Phys Rev B 78, 5242 (28) Volker@Eyertde

125 Magnetic Properties of CuFeO 2 Total Energies (mryd/fu), Magn Moms (µ B ), Band Gaps (ev) structure magn order E m Fe m O E g rhomb spin-deg - rhomb ferro (LS) rhomb ferro (IS) rhomb ferro (HS) monoc spin-deg 6 - monoc ferro (LS) monoc ferro (IS) monoc ferro (HS) monoc antiferro 46 ±372 ±8 5 VE, R Frésard, A Maignan, Phys Rev B 78, 5242 (28) Volker@Eyertde

126 A Short Walk: Fundamental Principles

127 A Short Walk: Fundamental Principles La 2 BaCuO 5 Moments ρ 1 v / au Volker@Eyertde

128 A Short Walk: Fundamental Principles La 2 BaCuO 5 Moments La 2 RuO 5 Orbitals ρ 1 v / au dxy dyz d xz S = 1 Volker@Eyertde

129 A Short Walk: Fundamental Principles La 2 BaCuO 5 Moments La 2 RuO 5 Orbitals ρ 1 v / au dxy dyz d xz S = 1 K 2 NiF 4, K 2 CuF 4 Symmetry -5-1 E (mryd) K 2 NiF 4 : E af -E nm K 2 CuF 4 : E fe -E nm K 2 CuF 4 : E fe -E nm /E af δ Volker@Eyertde

130 A Short Walk: Fundamental Principles La 2 BaCuO 5 Moments La 2 RuO 5 Orbitals ρ 1 v / au dxy dyz d xz S = 1 K 2 NiF 4, K 2 CuF 4 Symmetry CuFeO 2 Structure -5-1 E (mryd) K 2 NiF 4 : E af -E nm K 2 CuF 4 : E fe -E nm K 2 CuF 4 : E fe -E nm /E af δ Volker@Eyertde

131 Acknowledgments Caen R Frésard, S Hébert A Maignan, C Martin Darmstadt P C Schmidt Halle S G Ebbinghaus Augsburg K-H Höck, T Kopp, J Mannhart U Schwingenschlögl Volker@Eyertde

132 Acknowledgments Caen R Frésard, S Hébert A Maignan, C Martin Darmstadt P C Schmidt Halle S G Ebbinghaus Augsburg K-H Höck, T Kopp, J Mannhart U Schwingenschlögl Vienna Thank You for Your Attention! Volker@Eyertde