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Magnetic and charge order phase transition in YBaFe 2 O 5 (Verwey transition) Peter Blaha, Ch. Spiel, K.Schwarz Institute of Materials Chemistry TU Wien Thanks to P.Karen (Univ. Oslo, Norway)

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E.Verwey, Nature 144, 327 (1939) Fe 3 O 4, magnetite phase transition between a mixed-valence and a charge-ordered configuration 2 Fe 2.5+ Fe 2+ + Fe 3+ cubic inverse spinel structure AB 2 O 4 Fe 2+ A (Fe 3+,Fe 3+ ) B O 4 Fe 3+ A (Fe 2+,Fe 3+ ) B O 4 B A small, but complicated coupling between lattice and charge order

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Double-cell perovskites: RBaFe 2 O 5 ABO 3 O-deficient double-perovskite Y (R) Ba square pyramidal coordination Antiferromagnet with a 2 step Verwey transition around 300 K Woodward&Karen, Inorganic Chemistry 42, 1121 (2003)

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structural changes in YBaFe 2 O 5 above T N (~430 K): tetragonal (P4/mmm) 430K: slight orthorhombic distortion (Pmmm) due to AFM all Fe in class-III mixed valence state +2.5; ~334K: dynamic charge order transition into class-II MV state, visible in calorimetry and Mössbauer, but not with X-rays 308K: complete charge order into class-I MV state (Fe 2+ + Fe 3+ ) large structural changes (Pmma) due to Jahn-Teller distortion; change of magnetic ordering: direct AFM Fe-Fe coupling vs. FM Fe-Fe exchange above T V

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structural changes CO structure: Pmma VM structure: Pmmm a:b:c=2.09:1:1.96 (20K) a:b:c=1.003:1:1.93 (340K) Fe 2+ and Fe 3+ chains along b contradicts Anderson charge-ordering conditions with minimal electrostatic repulsion (checkerboard like pattern) has to be compensated by orbital ordering and e - -lattice coupling a b c

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antiferromagnetic structure CO phase: G-type AFMVM phase: AFM arrangement in all directions, AFM for all Fe-O-Fe superexchange paths also across Y-layer FM across Y-layer (direct Fe-Fe exchange) Fe moments in b-direction 4 8 independent Fe atoms

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Theoretical methods: WIEN2k (APW+lo) calculations Rkmax=7, 100 k-points spin-polarized, various spin-structures + spin-orbit coupling based on density functional theory: LSDA or GGA (PBE) E xc ≡ E xc (ρ, ∇ ρ) description of “highly correlated electrons” using “non-local” (orbital dep.) functionals LDA+U, GGA+U hybrid-DFT (only for correlated electrons) mixing exact exchange (HF) + GGA WIEN2K An Augmented Plane Wave Plus Local Orbital Program for Calculating Crystal Properties Peter Blaha Karlheinz Schwarz Georg Madsen Dieter Kvasnicka Joachim Luitz http://www.wien2k.at 1400 registered groups 2000 mailinglist users

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GGA-results: Metallic behaviour/No bandgap Fe-dn t 2g states not splitted at E F overestimated covalency between O-p and Fe-e g Magnetic moments too small Experiment: CO: 4.15/3.65 (for Tb), 3.82 (av. for Y) VM: ~3.90 Calculation: CO: 3.37/3.02 VM: 3.34 no significant charge order charges of Fe 2+ and Fe 3+ sites nearly identical CO phase less stable than VM LDA/GGA NOT suited for this compound! Fe-e g t 2g e g * t 2g e g

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“Localized electrons”: GGA+U Hybrid-DFT E xc PBE0 [ ] = E xc PBE [ ] + (E x HF [ sel ] – E x PBE [ sel ]) LDA+U, GGA+U E LDA+U ( ,n) = E LDA ( ) + E orb (n) – E DCC ( ) separate electrons into “itinerant” (LDA) and localized e - (TM-3d, RE 4f e - ) treat them with “approximate screened Hartree-Fock” correct for “double counting” Hubbard-U describes coulomb energy for 2e - at the same site orbital dependent potential

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Determination of U Take U eff as “empirical” parameter (fit to experiment) Estimate U eff from constraint LDA calculations constrain the occupation of certain states (add/subtract e - ) switch off any hybridization of these states (“core”-states) calculate the resulting E tot we used U eff =7eV for all calculations

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DOS: GGA+U vs. GGA GGA+U GGA insulator, t 2g band splits metallicsingle lower Hubbard-band in VM splits in CO with Fe 3+ states lower than Fe 2+

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magnetic moments and band gap magnetic moments in very good agreement with exp. LDA/GGA: CO: 3.37/3.02 VM: 3.34 B orbital moments small (but significant for Fe 2+ ) band gap: smaller for VM than for CO phase exp: semiconductor (like Ge); VM phase has increased conductivity LDA/GGA: metallic

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Charge transfer Charges according to Baders “Atoms in Molecules” theory Define an “atom” as region within a zero flux surface Integrate charge inside this region

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Structure optimization (GGA+U) CO phase: Fe 2+ : shortest bond in y (O2b) Fe 3+ : shortest bond in z (O1) VM phase: all Fe-O distances similar theory deviates along z !! Fe-Fe interaction different U ?? finite temp. ?? O1 O2a O2b O3

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Can we understand these changes ? Fe 2+ (3d 6 ) CO Fe 3+ (3d 5 ) VM Fe 2.5+ (3d 5.5 ) majority-spin fully occupied strong covalency effects very localized states at lower energy than Fe 2+ in e g and d-xz orbitals minority-spin states d-xz fully occupied (localized) empty d-z 2 partly occupied short bond in y short bond in z (one O missing) FM Fe-Fe; distances in z ??

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Difference densities = cryst - at sup CO phase VM phase Fe 2+ : d-xz Fe 3+ : d-x 2 O1 and O3: polarized toward Fe 3+ Fe: d-z 2 Fe-Fe interaction O: symmetric

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d xz spin density ( up - dn ) of CO phase Fe 3+ : no contribution Fe 2+ : d xz weak -bond with O tilting of O3 -orbital

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Mössbauer spectroscopy: Isomer shift: = ( 0 Sample – 0 Reference ); =-.291 au 3 mm s -1 proportional to the electron density at the nucleus Magnetic Hyperfine fields: B tot =B contact + B orb + B dip B contact = 8 /3 B [ up (0) – dn (0)] … spin-density at the nucleus … orbital-moment … spin-moment S(r) is reciprocal of the relativistic mass enhancement

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Electric field gradients (EFG) Nuclei with a nuclear quantum number I≥1 have an electrical quadrupole moment Q Nuclear quadrupole interaction (NQI) between “non-spherical” nuclear charge Q times the electric field gradient Experiments NMR NQR Mössbauer PAC EFG traceless tensor with traceless |V zz | |V yy | |V xx | EFG V zz asymmetry parameter principal component

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

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Mössbauer spectroscopy Isomer shift: charge transfer too small in LDA/GGA Hyperfine fields: Fe 2+ has large B orb and B dip EFG: Fe 2+ has too small anisotropy in LDA/GGA CO VM

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magnetic interactions CO phase: magneto-crystalline anisotropy: moments point into y-direction in agreement with exp. experimental G-type AFM structure (AFM direct Fe-Fe exchange) is 8.6 meV/f.u. more stable than magnetic order of VM phase (direct FM) VM phase: experimental “FM across Y-layer” AFM structure (FM direct Fe-Fe exchange) is 24 meV/f.u. more stable than magnetic order of CO phase (G-type AFM)

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Exchange interactions J ij Heisenberg model: H = i,j J ij S i.S j 4 different superexchange interactions (Fe-Fe exchange interaction mediated by an O atom) J 22 b : Fe 2+ -Fe 2+ along b J 33 b : Fe 3+ -Fe 3+ along b J 23 c : Fe 2+ -Fe 3+ along c J 23 a : Fe 2+ -Fe 3+ along a 1 direct Fe-Fe interaction J direct : Fe 2+ -Fe 3+ along c J direct negative (AFM) in CO phase J direct positive (FM) in VM phase

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Inelastic neutron scattering S.Chang etal., PRL 99, 037202 (2007 ) J 33 b = 5.9 meV J 22 b = 3.4 meV J 23 = 6.0 meV J 23 = (2J 23 a + J 23 c )/3

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Theoretical calculations of J ij Total energy of a certain magnetic configuration given by: n i … number of atoms i z ij … number of atoms j which are neighbors of i S i = 5/2 (Fe 3+ ); 2 (Fe 2+ ) i = ±1 Calculate E-diff when a spin on atom i ( i ) or on two atoms i,j ( ij ) are flipped Calculate a series of magnetic configurations and determine J ij by least-squares fit

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Investigated magnetic configurations

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Calculated exchange parameters

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Summary Standard LDA/GGA methods cannot explain YBaFe 2 O 5 metallic, no charge order (Fe 2+ -Fe 3+ ), too small moments Needs proper description of the Fe 3d electrons (GGA+U, …) CO-phase: Fe 2+ : high-spin d 6, occupation of a single spin-dn orbital ( d xz ) Fe 2+ /Fe 3+ ordered in chains along b, cooperative Jahn-Teller distortion and strong e - -lattice coupling which dominates simple Coulomb arguments (checkerboard structure of Fe 2+ /Fe 3+ ) VM phase: small orthorhombic distortion (AFM order, moments along b) Fe d-z 2 spin-dn orbital partly occupied (top of the valence bands) leads to direct Fe-Fe exchange across Y-layer and thus to ferromagnetic order (AFM in CO phase). Quantitative interpretation of the Mössbauer data Calculated exchange parameters J ij in reasonable agreement with exp.

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Thank you for your attention !

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tight-binding MO-schemes: too simple? VM phase: Fe 2.5+ CO phase: Fe 2+ CO phase: Fe 3+ Woodward&Karen, Inorganic Chemistry 42, 1121 (2003)

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Concepts when solving Schrödingers-equation non relativistic semi-relativistic fully-relativistic “Muffin-tin” MT atomic sphere approximation (ASA) pseudopotential (PP) Full potential : FP Hartree-Fock (+correlations) Density functional theory (DFT) Local density approximation (LDA) Generalized gradient approximation (GGA) Beyond LDA: e.g. LDA+U Non-spinpolarized Spin polarized (with certain magnetic order) non periodic (cluster, individual MOs) periodic (unit cell, Blochfunctions, “bandstructure”) plane waves : PW augmented plane waves : APW atomic oribtals. e.g. Slater (STO), Gaussians (GTO), LMTO, numerical basis Basis functions Treatment of spin Representation of solid Form of potential exchange and correlation potential Relativistic treatment of the electrons Schrödinger - equation

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PW: APW Augmented Plane Wave method The unit cell is partitioned into: atomic spheres Interstitial region Atomic partial waves join R mt unit cell Basisset: u l (r, ) are the numerical solutions of the radial Schrödinger equation in a given spherical potential for a particular energy A lm K coefficients for matching the PW

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APW based schemes APW (J.C.Slater 1937) Non-linear eigenvalue problem Computationally very demanding LAPW (O.K.Andersen 1975) Generalized eigenvalue problem Full-potential (A. Freeman et al.) Local orbitals (D.J.Singh 1991) treatment of semi-core states (avoids ghostbands) APW+lo (E.Sjöstedt, L.Nordstörm, D.J.Singh 2000) Efficience of APW + convenience of LAPW Basis for K.Schwarz, P.Blaha, G.K.H.Madsen, Comp.Phys.Commun.147, 71-76 (2002)

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variational methods (L)APW + local orbitals - basis set Trial wave function n…50-100 PWs /atom Variational method: Generalized eigenvalue problem: H C=E S C Diagonalization of (real symmetric or complex hermitian) matrices of size 100 to 50.000 (up to 50 Gb memory) upper bound minimum

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Quantum mechanics at work

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WIEN2k software package An Augmented Plane Wave Plus Local Orbital Program for Calculating Crystal Properties Peter Blaha Karlheinz Schwarz Georg Madsen Dieter Kvasnicka Joachim Luitz November 2001 Vienna, AUSTRIA Vienna University of Technology http://www.wien2k.at WIEN97: ~500 users WIEN2k: ~1150 users mailinglist: 1800 users

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Development of WIEN2k Authors of WIEN2k P. Blaha, K. Schwarz, D. Kvasnicka, G. Madsen and J. Luitz Other contributions to WIEN2k C. Ambrosch-Draxl (Univ. Graz, Austria), optics T. Charpin (Paris), elastic constants R. Laskowski (Vienna), non-collinear magnetism, parallelization L. Marks (Northwestern, US), various optimizations, new mixer P. Novák and J. Kunes (Prague), LDA+U, SO B. Olejnik (Vienna), non-linear optics, C. Persson (Uppsala), irreducible representations M. Scheffler (Fritz Haber Inst., Berlin), forces D.J.Singh (NRL, Washington D.C.), local oribtals (LO), APW+lo E. Sjöstedt and L Nordström (Uppsala, Sweden), APW+lo J. Sofo and J. Fuhr (Barriloche), Bader analysis B. Yanchitsky and A. Timoshevskii (Kiev), spacegroup and many others ….

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International co-operations More than 1000 user groups worldwide Industries (Canon, Eastman, Exxon, Fuji, A.D.Little, Mitsubishi, Motorola, NEC, Norsk Hydro, Osram, Panasonic, Samsung, Sony). Europe: (EHT Zürich, MPI Stuttgart, Dresden, FHI Berlin, DESY, TH Aachen, ESRF, Prague, Paris, Chalmers, Cambridge, Oxford) America: ARG, BZ, CDN, MX, USA (MIT, NIST, Berkeley, Princeton, Harvard, Argonne NL, Los Alamos Nat.Lab., Penn State, Georgia Tech, Lehigh, Chicago, SUNY, UC St.Barbara, Toronto) far east: AUS, China, India, JPN, Korea, Pakistan, Singapore,Taiwan (Beijing, Tokyo, Osaka, Sendai, Tsukuba, Hong Kong) Registration at www.wien2k.atwww.wien2k.at 400/4000 Euro for Universites/Industries code download via www (with password), updates, bug fixes, news usersguide, faq-page, mailing-list with help-requests

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w2web GUI (graphical user interface) Structure generator spacegroup selection import cif file step by step initialization symmetry detection automatic input generation SCF calculations Magnetism (spin-polarization) Spin-orbit coupling Forces (automatic geometry optimization) Guided Tasks Energy band structure DOS Electron density X-ray spectra Optics

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Program structure of WIEN2k init_lapw initialization symmetry detection (F, I, C- centering, inversion) input generation with recommended defaults quality (and computing time) depends on k-mesh and R.Kmax (determines #PW) run_lapw scf-cycle optional with SO and/or LDA+U different convergence criteria (energy, charge, forces) save_lapw tic_gga_100k_rk7_vol0 cp case.struct and clmsum files, mv case.scf file rm case.broyd* files

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Advantage/disadvantage of WIEN2k + robust all-electron full-potential method (new effective mixer) + unbiased basisset, one convergence parameter (LDA-limit) + all elements of periodic table (equal expensive), metals + LDA, GGA, meta-GGA, LDA+U, spin-orbit + many properties and tools (supercells, symmetry) + w2web (for novice users) ? speed + memory requirements + very efficient basis for large spheres (2 bohr) (Fe: 12Ry, O: 9Ry) - less efficient for small spheres (1 bohr) (O: 25 Ry) - large cells, many atoms (n 3, but new iterative diagonalization) - full H, S matrix stored large memory required + effective dual parallelization (k-points, mpi-fine-grain) + many k-points do not require more memory - no stress tensor - no linear response

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Magnetite ferri-magnetic natural mineral, T N =850 K early “Compass” proof of earth magnetic field flips Structure below T V ~ 120 K: charge order along (001) planes (1A and 2 B sites) is too simple Mössbauer: 1 A and 4 B Fe sites NMR: 8 A and 16 B sites, Cc symmetry single crystal diffraction, synchrotron diffraction … small distortions ??? small, but complicated coupling between lattice and charge order

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