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Magnetic and charge order phase transition in YBaFe 2 O 5 (Verwey transition)

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### Magnetic and charge order phase transition in YBaFe2O5 (Verwey transition)

Peter Blaha, Ch. Spiel, K.Schwarz

Institute of Materials Chemistry

TU Wien

Thanks to P.Karen (Univ. Oslo, Norway)

E.Verwey, Nature 144, 327 (1939)

Fe3O4, magnetite

phase transition between a mixed-valence

and a charge-ordered configuration

2 Fe2.5+ Fe2+ + Fe3+

cubic inverse spinel structure AB2O4

Fe2+A (Fe3+,Fe3+)B O4

Fe3+A (Fe2+,Fe3+)B O4

B

A

small, but complicated coupling between lattice and charge order

Double-cell perovskites: RBaFe2O5

ABO3 O-deficient double-perovskite

Ba

Y (R)

square pyramidal

coordination

Antiferromagnet with a 2 step Verwey transition around 300 K

Woodward&Karen, Inorganic Chemistry 42, 1121 (2003)

structural changes in YBaFe2O5

- above TN (~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 (Fe2+ + Fe3+)
- large structural changes (Pmma)
- due to Jahn-Teller distortion;
- change of magnetic ordering:
- direct AFM Fe-Fe coupling vs.
- FM Fe-Fe exchange above TV

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)

- Fe2+ and Fe3+ 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

c

b

a

antiferromagnetic structure

CO phase: G-type AFM VM 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

Theoretical methods:

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

- 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)
- Exc≡ Exc(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

1400 registered groups

2000 mailinglist users

GGA-results:

- Metallic behaviour/No bandgap
- Fe-dn t2g states not splitted at EF
- overestimated covalency between O-p and Fe-eg
- 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 Fe2+ and Fe3+ sites nearly identical
- CO phase less stable than VM
- LDA/GGA NOT suited for this compound!

Fe-eg t2g eg*t2g eg

“Localized electrons”: GGA+U

- Hybrid-DFT
- ExcPBE0 [r] = ExcPBE [r] + a (ExHF[Fsel] – ExPBE[rsel])
- LDA+U, GGA+U
- ELDA+U(r,n) = ELDA(r) + Eorb(n) – EDCC(r)
- 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

Determination of U

- Take Ueff as “empirical” parameter (fit to experiment)
- Estimate Ueff 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 Etot
- we used Ueff=7eV for all calculations

DOS: GGA+U vs. GGA

GGA+U GGA

single lower Hubbard-band in VM splits in CO with Fe3+ states lower than Fe2+

insulator, t2g band splits metallic

magnetic moments and band gap

- magnetic moments in very good agreement with exp.
- LDA/GGA: CO: 3.37/3.02 VM: 3.34 mB
- orbital moments small (but significant for Fe2+)
- band gap: smaller for VM than for CO phase
- exp: semiconductor (like Ge); VM phase has increased conductivity
- LDA/GGA: metallic

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

Structure optimization (GGA+U)

O2a

- CO phase:
- Fe2+:shortest bond in y (O2b)
- Fe3+: shortest bond in z (O1)
- VM phase:
- all Fe-O distances similar
- theory deviates along z !!
- Fe-Fe interaction
- different U ??
- finite temp. ??

O2b

O3

O1

O1

Can we understand these changes ?

- Fe2+ (3d6) CO Fe3+ (3d5) VM Fe2.5+ (3d5.5)
- majority-spin fully occupied
- strong covalency effects very localized states at lower energy than Fe2+

in eg and d-xz orbitals

- minority-spin states
- d-xz fully occupied (localized) empty d-z2 partly occupied
- short bond in y short bond in z (one O missing) FM Fe-Fe; distances in z ??

Difference densities Dr=rcryst-ratsup

- CO phase VM phase

Fe2+: d-xz

Fe3+: d-x2

O1 and O3: polarized

toward Fe3+

Fe: d-z2 Fe-Fe interaction

O: symmetric

dxz spin density (rup-rdn) of CO phase

- Fe3+: no contribution
- Fe2+: dxz
- weak p-bond with O
- tilting of O3 p-orbital

Mössbauer spectroscopy:

- Isomer shift: d = a (r0Sample – r0Reference); a=-.291 au3mm s-1
- proportional to the electron density r at the nucleus
- Magnetic Hyperfine fields: Btot=Bcontact + Borb + Bdip
- Bcontact = 8p/3 mB [rup(0) – rdn(0)] … spin-density at the nucleus

… orbital-moment

… spin-moment

S(r) is reciprocal of the relativistic mass enhancement

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 F
- Experiments
- NMR
- NQR
- Mössbauer
- PAC

EFG traceless tensor

with

traceless

|Vzz| |Vyy| |Vxx|

EFG Vzz

asymmetry parameter

principal component

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

Mössbauer spectroscopy

Hyperfine fields: Fe2+ has large Borb and Bdip

Isomer shift: charge transfer too small in LDA/GGA

EFG: Fe2+ has too small anisotropy in LDA/GGA

CO

VM

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)

Exchange interactions Jij

- Heisenberg model: H = Si,j JijSi.Sj
- 4 different superexchange interactions (Fe-Fe exchange interaction mediated by an O atom)
- J22b : Fe2+-Fe2+ along b
- J33b : Fe3+-Fe3+ along b
- J23c : Fe2+-Fe3+ along c
- J23a : Fe2+-Fe3+ along a
- 1 direct Fe-Fe interaction
- Jdirect: Fe2+-Fe3+ along c
- Jdirect negative (AFM) in CO phase
- Jdirect positive (FM) in VM phase

Inelastic neutron scattering

- S.Chang etal., PRL 99, 037202 (2007)
- J33b = 5.9 meV
- J22b = 3.4 meV
- J23 = 6.0 meV
- J23 = (2J23a + J23c)/3

Theoretical calculations of Jij

- Total energy of a certain magnetic configuration given by:

ni … number of atoms i

zij … number of atoms j which

are neighbors of i

Si = 5/2 (Fe3+); 2 (Fe2+)

si = ±1

- Calculate E-diff when a spin on atom i (Di) or on two atoms i,j (Dij) are flipped
- Calculate a series of magnetic configurations and determine Jij by least-squares fit

Summary

- Standard LDA/GGA methods cannot explain YBaFe2O5
- metallic, no charge order (Fe2+-Fe3+), too small moments
- Needs proper description of the Fe 3d electrons (GGA+U, …)
- CO-phase: Fe2+: high-spin d6, occupation of a single spin-dn orbital (dxz)
- Fe2+/Fe3+ ordered in chains along b, cooperative Jahn-Teller distortion and strong e--lattice coupling which dominates simple Coulomb arguments (checkerboard structure of Fe2+/Fe3+)
- VM phase: small orthorhombic distortion (AFM order, moments along b)
- Fe d-z2 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 Jij in reasonable agreement with exp.

your attention !

tight-binding MO-schemes: too simple?

VM phase: Fe2.5+

CO phase: Fe2+

CO phase: Fe3+

Woodward&Karen, Inorganic Chemistry 42, 1121 (2003)

Concepts when solving Schrödingers-equation

Treatment of

spin

Form of

potential

“Muffin-tin” MT

atomic sphere approximation (ASA)

pseudopotential (PP)

Full potential : FP

Non-spinpolarized

Spin polarized

(with certain magnetic order)

Relativistic treatment

of the electrons

exchange and correlation potential

Hartree-Fock (+correlations)

Density functional theory (DFT)

Local density approximation (LDA)

Generalized gradient approximation (GGA)

Beyond LDA: e.g. LDA+U

non relativistic

semi-relativistic

fully-relativistic

Schrödinger - equation

Basis functions

Representation

of solid

plane waves : PW

augmented plane waves : APW

atomic oribtals. e.g. Slater (STO), Gaussians (GTO),

LMTO, numerical basis

non periodic

(cluster, individual MOs)

periodic

(unit cell, Blochfunctions,

“bandstructure”)

APW Augmented Plane Wave method

- The unit cell is partitioned into:
- atomic spheres
- Interstitial region

unit cell

Rmt

Basisset:

PW:

ul(r,e) are the numerical solutions

of the radial Schrödinger equation

in a given spherical potential for a

particular energy e

AlmKcoefficients for matching the PW

join

Atomic partial waves

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)

variational methods

(L)APW + local orbitals - basis set

n…50-100 PWs /atom

Trial wave function

Variational method:

minimum

upper bound

Generalized eigenvalue problem: H C=E S C

Diagonalization of (real symmetric or complex hermitian) matrices ofsize 100 to 50.000 (up to 50 Gb memory)

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

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 ….

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.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

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

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

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 (n3, 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

Magnetite

ferri-magnetic natural mineral, TN=850 K

early “Compass”

proof of earth magnetic field flips

Structure below TV ~ 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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