1 / 64

Magnetism, strongly correlated electrons: charge- spin- and orbital ordering II

Magnetism, strongly correlated electrons: charge- spin- and orbital ordering II. Karlheinz Schwarz Institute of Materials Chemistry TU Wien. Quantum mechanics at work. thanks to Erich Wimmer. Transition metal (TM)-oxides:.

dtharp
Download Presentation

Magnetism, strongly correlated electrons: charge- spin- and orbital ordering II

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Magnetism, strongly correlated electrons: charge- spin- and orbital ordering II Karlheinz Schwarz Institute of Materials Chemistry TU Wien

  2. Quantum mechanics at work thanks to Erich Wimmer

  3. Transition metal (TM)-oxides: • TM oxides show a wide variety of physical properties depending on composition, ionization state and structure: • Regular metals or insulators, metal – insulator transitions • Nonmagnetic, ferro- or antiferromagnetic, even non-collinear structures • High / low symmetry phase transitions (with T or p) • Superconductors (High-Tc) • Unconventional heavy fermion materials (LiV2O4) • Show strong covalent interactions or have highly localized (atomic like) states • Weakly to strongly correlated systems

  4. Charge ordering in BaBiO3 (perovskite) Frozen phonon calculation • Charge disproportionation 2 Bi4+ Bi3+ + Bi5+ the oxygen can be • closer to Bi5+ (small octahedron, blue) • than to Bi3+ (large octahedron, yellow) breathing mode Bi3+ numerical accuracy Etot=-119 603.624 Ryd Bi5+ 1 mRyd ------ Ba1-xKxBiO3 superconductor P.Blaha, K.Schwarz, Ph.Dufek. G.Vielsack, W.Weber: Z.Naturforsch. 48a, 129 (1993)

  5. Electric field gradients EFG

  6. Electric field gradients (EFG) • Nuclei with a nuclear quantum number I 1 have an electrical quadrupole moment Q • Nuclear quadrupole interaction (NQI) can aid to determine the distribution of the electronic charge surrounding such a nuclear site • Experiments • NMR • NQR • Mössbauer • PAC EFG traceless tensor Nuclear electronic with traceless |Vzz| |Vyy| |Vxx| EFG Vzz asymmetry parameter principal component

  7. theoretical EFG calculations: EFG is tensor of second derivatives of VC at the nucleus: Cartesian LM-repr. EFG is proportial to differences of orbital occupations

  8. EFG (1021 V/m2) in YBa2Cu3O7 • Site Vxx Vyy Vzz h • Y theory -0.9 2.9 -2.0 0.4 • exp. - - - - • Ba theory -8.7 -1.0 9.7 0.8 • exp. 8.4 0.3 8.7 0.9 • Cu(1) theory -5.2 6.6 -1.5 0.6 • exp. 7.4 7.5 0.1 1.0 • Cu(2) theory 2.6 2.4 -5.0 0.0 standard LDA calculations give • exp. 6.2 6.2 12.3 0.0 good EFGs for all sites except Cu(2) • O(1) theory -5.7 17.9 -12.2 0.4 • exp. 6.1 17.3 12.1 0.3 • O(2) theory 12.3 -7.5 -4.8 0.2 • exp. 10.5 6.3 4.1 0.2 • O(3) theory -7.5 12.5 -5.0 0.2 • exp. 6.3 10.2 3.9 0.2 • O(4) theory -4.7 -7.1 11.8 0.2 • exp. 4.0 7.6 11.6 0.3 • K.Schwarz, C.Ambrosch-Draxl, P.Blaha, Phys.Rev. B42, 2051 (1990) • D.J.Singh, K.Schwarz, K.Schwarz, Phys.Rev. B46, 5849 (1992)

  9. EFG in YBa2Cu3O7 • Interpretation of the EFG at the oxygen sites z O1 y x Cu1 EF Cu1-d Asymmetry count EFG (p-contribution) O1-py EFG is proportional to asymmetric charge distribution around given nucleus partly occupied

  10. Cu(2) and O(4) EFG as function of r • EFG is determined by the non-spherical charge density inside sphere • Cu(2) • O(4) R r r final EFG R r r

  11. Beyond the LDA (GGA) LDA+U

  12. LDA + U Separation of electrons into two subsystems • Localized d (f) electrons: described withHubbard U • Itinerant electrons:described by LDA • Hubbard U describes the Coulomb energy cost to place two electrons at the same site and is defined as: • J is the average intra-atomic exchange parameter We define a new energy functional orbital occupation number n n n-1 n+1 . + + with the double counting term

  13. LaCuO4, LDA, LDA+U(SIC) • spin polarization • shift of d-bands • Lower Hubbard band (spin up) • Upper Hubbard band (spin down) LDA LHB UHB

  14. LaCuO4, orbital projected DOS spin up spin down • main effect on • Cu-dxy (towards oxygen) • lower HB • upper HB • Cu-dxz • shift in weight LHB UHB

  15. LaCuO4, magnetic moment, LDA+U(SIC) The magnetic moment increases with U

  16. LaCuO4, electric field gradient (EFG) • How the EFG changes with U in combination with • LDA • GGA (PBE, EV)

  17. Cu-EFG Vzz (1021 V/m2) in YBa2Cu3O6 for LDA, AMF, FLL and DFT-double counting corrections. NM and AF refers to non-magnetic and antiferromagnetic solutions. U and J of 8 and 1 eV is applied to both Cu sites, except for LDA+U* where U is applied only to the Cu(2) site. Type Vzz -Cu(1) Vzz -Cu(2) LDA (NM) -8.1 -3.7 AMF-LDA+U (NM) -4.6 -7.3 AMF-LDA+U*(NM) -8.0 -7.2 AMF-LDA+U (AF) -4.5 -13.3 AMF-LDA+U*(AF) -8.0 -13.3 DFT-LDA+U(AF) -4.8 -12.0 FLL-LDA+U (AF) -8.3 -12.3 FLL-LDA+U*(AF) -8.0 -13.3 Experiment 11.8 9.0 Cu(2) too small, Cu(1) ok Cu(2) still too small, Cu(1) wrong Cu(2) ok, Cu(1) wrong DFT similar to AMF Cu(2) ok,Cu(1) remains ok Antiferromagn. FLL-calc. with U=6eV give again best results

  18. EFG analysis in YBa2Cu3O6: EFG contributions (for both spins and p-p and d-d contributions) in LSDA and LDA+U(DFT) Cu(1) Cu(2) NM-LDA AF-LDA+U(DFT) NM-LDA AF-LDA+U(DFT) p-p (up) -12.2 -12.2 5.7 5.8 p-p (dn) -12.2 -12.2 5.7 6.3 d-d (up) 7.1 8.8 -7.4 5.5 d-d (dn) 7.1 8.8 -7.4 -31.1 Semicore 2.1 2.0 -0.1 1.7 Total -8.1 -4.8 -3.7 -12.0 Exp. 11.8 9.0 Partial charges and anisotropy counts Dn in LDA and antiferromagnetic LDA+U(DFT) Cu(1) Cu(2) NM-LDA AF-LDA+U(DFT) NM-LDA AF-LDA+U(DFT) 4pz 0.103 0.104 0.027 0.028 4px+py 0.041 0.042 0.131 0.140 Dnp -0.083 -0.083 0.039 0.042 dx2-y2 1.776 1.782 1.433 1.228 dz2 1.474 1.398 1.757 1.784 daver 1.802 1.818 1.815 1.841 Dnd 0.257 0.324 -0.294 -0.529 large charge in z  large negative EFG large charge in xy  large positive EFG Cu(1) anisotropy increased in DFT and AMF schems: wrong! Cu(2) dx2-y2 depopulated: ok Cu(2) has O in xy plane Cu(1) has O only in z

  19. Cu-EFG Vzz (1021 V/m2) in YBa2Cu3O7 for LDA, AMF, FLL and DFT-double counting corrections. NM and AF refers to non-magnetic and antiferromagnetic solutions. U and J of 8 and 1 eV is applied to both Cu sites, except for LDA+U* where U is applied only to the Cu(2) site. Type Vzz -Cu(1) h Vzz -Cu(2) h LDA (NM) 6.6 0.6 -5.0 0.0 AMF-LDA+U (NM) -10.9 0.7 -11.1 0.1 AMF-LDA+U*(NM) 6.6 0.6 -11.1 0.1 DFT-LDA+U(NM) -10.4 0.7 -10.8 0.1 FLL-LDA+U (NM) 7.2 0.9 -7.3 0.1 FLL-LDA+U*(NM) 6.9 0.6 -7.3 0.0 AMF-LDA+U (AF) -8.8 0.1 -14.1 0.1 AMF-LDA+U*(AF) 7.0 0.6 -14.1 0.1 FLL-LDA+U (AF) -5.3 0.1 -13.1 0.1 FLL-LDA+U*(AF) 7.1 0.6 -13.1 0.1 Experiment 7.5 1.0 12.3 0.0 Cu(2) too small,Cu(1) ok Cu(2) about ok, but Cu(1) wrong DFT similar to AMF Cu(2) still too small, Cu(1) ok Cu(2) ok, but Cu(1) wrong (gets small FM moment !!) larger cells with “random” magnetic order get better !! Local Cu(2) moments, but without long range order necessary ?

  20. High Tc Superconductors: YBa2Cu3O6.98 • T.Lippmann, P.Blaha, N.H.Andersen, H.F.Poulsen, T.Wolf, J.R.Schneider and K.Schwarz, Acta Cryst. A (2003) • High energy synchrotron radiation (l=0.124 A) • VALRAY refinement of experimental (R=0.47%) and theoretical (R=0.23%) F • A challenge for experiment: • Non-stoichiometry • Heavy elements (Y, Ba): • S = Vol / S n2core = 0.016 A3/e2 (usually 5-0.1!!!) • many „complicated“ sites • A challenge for theory: • YBa2Cu3O6 is nonmagnetic metal in theory!! • Electric field gradient (EFG) at Cu(2) differs from experiment

  21. Deformtion density in (100) plane rLDA+U – rGGA (direct) experiment theory (GGA) + “planes” - “chains” • Dipolar Cu(2) density not present in theory  theory wrong ? • yes, because Cu(2)-EFG is too small by 50% • YBa2Cu3O6 is a nonmagnetic metal !! • LDA+U calculations: treat the strong correlation between the Cu-d • orbitals by a „Hubbard“ type approximation •  correct groundstate, correct EFG, but only small changes in r, • („d-z2“ orbital gets „fully“ occupied, no dipol)

  22. Two examples of transition metal oxides BaCoO3 • 3d TM oxide • Insulator • Orbital ordering • LDA+U • 4d TM oxide • Insulator • Structure relaxation • Orbital ordering • GGA Y2Nb2O7

  23. BaCoO3 a 3d TM oxide

  24. Structure of BaCoO3 • Hexagonal 2H pseudo-perovskite • Space group P63/mmc • Face sharing Co-octahedra • 1D Chains along c-axis • Co-Co distances • short (2.38 Å) along c • long (5.65 Å) between chains Co Ba oxygen

  25. Experimental findings: • Conduction properties: • semiconductor in the T range studied (70-300 K). • hopping conductivity above 200 K. • T < 200K: experiments are not conclusive. • thermoelectric coefficient: n-type carriers. • Magnetic properties: • low spin (LS) state (S=1/2). • TN=8 K. • T>150 K: paramagnetic phase. • J ~ 10 K. • Lower T: change in character of the magnetic interaction.

  26. Electronic structure (ionic picture) Ba2+; Co4+; O2- Co4+: 3d5 distorted cubic eg-like eg0 t2g-like t2g5 LS state; one hole in t2g

  27. GGA results and ferromagnetic (FM) case BaCoO3 • Metallic • Magnetic moment delocalized • Only 0.5 μB inside Co-sphere • Minimum in FSM around 1.5 μB • Low spin (LS) state would be 2 μB • LS is half-metallic half-metallic FM-LS NM FM-LS metallic Fixed spin moment (FSM) NM

  28. LDA+U leads to orbital ordering • Orbital ordering of Co-d orbitals along Co chains : • x2-y2 • xy • LDA+U works well for moderately correlated TM oxides • double counting fully localized limit • U=5 eV , J=0.5 eV (not sensitive) • Magnetic moment 1.0 μB per Co • AFM (FM similar) Spin density

  29. Orbital ordering along c axis Spin density Ba Ba O

  30. LDA+U results of various magnetic cases LDA+U: insulators • Insulating with alternating OO • FM (a) • AFM (b, c, d) • b) between planes • c) AF-type I • d) AF-type II • AF type I and II frustated in a b plane (hexagonal structure) for collinear magnetism metallic

  31. Alternating orbital ordering (FM) Co1 Co2 xy x2-y2 empty orbitals

  32. Results for BaCoO3 • GGA • FM (ground state) • 1.5 μB /unit cell (smaller than the 2 μB /unit cell for perfect LS state) • Metallic • no AFM solution found • No Peierls dimerization (according to total energy) • LDA+U (correlations among Co-d electrons) • In-chain alternating orbital ordering • 90° superexchange • FM ground state • Insulator • In agreement with experiment V.Pardo, P.Blaha, M.Igelsias, K.Schwarz, D.Baldomir, J.E.Arias, Magnetic structure and orbital ordering in BaCoO3 from first-principles calculations Phys.Rev. B 70, 212404 (2004)

  33. Pyrochlore: Y2Nb2O7 a 4d TM oxide

  34. Insulator, 4d system, pyrochlore Y2Nb2O7 P.Blaha, K.Schwarz, D.Singh Metal Sublattice: Corner-shared tetrahedral network

  35. Pyrochlore structure of Y2Nb2O7: • Experimental structure frompowder diffraction (Fukazawa,Maeno): • Pyrochlore structure: Fd-3m, a0=10.33 Å • 22 atoms (2 FU/cell) • One free parameter for O2: (0.344,1/8,1/8) • Corner shared Nb - O octahedra • Corner shared O1 - Y tetrahedra • Y surrounded by 8 oxygens (~ cube) • Metal sublattices • Nb-tetrahedra (empty) • Y-tetrahedra (around O) • Insulator • Nonmagnetic Y.Maeno et al, Nature 372, 532 (1994) H.Fukazawa, Y.Maeno,Phys.Rev. B 67, 054410 (2003) Nb Y O1 O2 Nb Y metal sublattice Nb

  36. First theoretical results: • Ionic model: Y23+Nb24+O72-Nb4+: 4d1 •  metallic or localized system with spin ½ (neither one observed in exp.) • LDA gives nonmagnetic metallic ground-state with conventional t2g-eg splitting due to the octahedral crystal field of the oxygen atoms. “degenerate” t2g states are only partly filled. EF O-2p Nb-t2g Nb-eg EF

  37. How could one obtain a non-magnetic insulator ? • Antiferromagnetic s=1/2 solution • (on geometrically frustrated lattice !?) • Localization, strong e--e- correlation: • 4d (not 3d !) electrons, • thus correlation should be small (Hubbard-U ~ 2-3 eV) • bandwidth of t2g bands: 2.5 eV (similar to U) • LDA+U with U=6 eV gives insulator (FM ground state, no AFM) • structural distortion, which breaks the dominant octahedral crystal field •  Search for phonon-instabilities

  38. Phonons: • select symmetry adapted atomic displacements (10 displacements in Y2Nb2O7) (Displacement pattern for cubic perovskite) • select a supercell: (88 atom P-type cell) • calculate all forces for these displacements with high accuracy(WIEN2k) •  force constants between all atoms in the supercell •  dynamical matrix for arbitrary q-vectors •  phonon-dispersion (“bandstructure”) using PHONON (K.Parlinski)

  39. Relaxed structure: • Supercell with 88 atoms • all atoms inequivalent due to numerical optimization of the positions in P1 • Symmetrization using KPLOT (R.Hundt, J.C.Schön, A.Hannemann, M.Jansen: Determination of symmetries and idealized cell parameters for simulated structures, J.Appl.Cryst. 32, 413-416 (1999)) • Tests possible symmetries with increasing tolerance • Space group  P-43m, 88 atoms/cell, • Inequivalent atoms: 2 Y + 2 Nb + 3 O1 + 5 O2 positions • Symmetry analysis yields freezing of degenerate X-phonon with special phase

  40. Phonon-bandstructure of Y2Nb2O7 • strong Phonon-instabilities, lowest at X, K, L • select a certain (unstable) phonon, freeze it with certain amplitude into the structure and perform full structural optimization (100) (110) (111)

  41. G, X and K-point phonons: • energy lower than in ideal pyrochlore structure, but still not insulating EF

  42. L-point (111) phonon: • Relaxed structure is an Insulator • energy gain of 0.5 eV/FU EF gap metal

  43. Relaxed structure Nb1 Nb2 Experimental (ideal) relaxed: Nb2 moves „off-center“, O octahedra changes little, Nb2-Nb2 bonds ?

  44. Pyrochlore structure relaxation: • Y-sublattice (~unchanged) • Nb-sublattice • Nb2 tetrahedra • contract along (111) • short Nb2-Nb2 bonds • Nb1tetrahedra • strongly distorts • short Nb1-Nb1 bonds • long Nb1-Nb2 bonds Y1 Y2 Nb1 Nb2 octahedron

  45. Main change in structural relaxation Original pyrochlore Relaxed structure Nb2 Nb1 Nb Nb small Nb1 - Nb2 large equal 3.65 Å 2.91 Å 3.89 Å 3.90 Å Nb1–triangle: 3.40 Å

  46. Chemical bonding: original structure peak A: Nb-Nb bonds within the Nb tetrahedra by d-z2 orbitals Nb t2g Nb peak AB Y O-2p (Nb-4d) (Y-4d) O

  47. Chemical bonding: original structure peak B: mixture of all 3 t2g orbitals Nb t2g Nb peak AB Y O-2p (Nb-4d) (Y-4d) O

  48. DOS of relaxed structure EF gap Nb2 C Nb1 B Nb2 A O-2p (Nb-4d) (Y-4d) Nb-4d (O-2p), Y-4d

  49. Peak A (Nb2) 4-center bond (d-z2) Nb1 EF gap Nb1 Nb2 Nb1 Nb2 C Nb2 Nb1 B Nb2 Nb2 Nb2 A O-2p (Nb-4d) (Y-4d) Nb-4d (O-2p), Y-4d

  50. Peak B (Nb1) 3-center bond (new !) EF gap Nb1 Nb2 Nb1 Nb2 C Nb1 Nb1 Nb1 B Nb2 A O-2p (Nb-4d) (Y-4d) Nb-4d (O-2p), Y-4d

More Related