Magnetic and charge order phase transition in YBaFe2O5 (Verwey

transition)

Peter Blaha, Ch. Spiel, K.SchwarzInstitute of Materials Chemistry

TU Wien

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

E.Verwey, Nature 144, 327 (1939)

Fe3O4, magnetitephase 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

Y (R)

Ba

square pyramidalcoordination

Antiferromagnet with a 2 step Verwey transition around 300 KWoodward&Karen, Inorganic Chemistry 42, 1121 (2003)

structural changes in YBaFe2O5

• above TN (~430 K): tetragonal (P4/mmm)

• 430K: slight orthorhombic distortion (Pmmm) due to AFMall 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

ab

c

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 8independent Fe atoms

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) 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

WIEN2KAn Augmented Plane Wave

Plus Local Orbital Program for Calculating

Crystal Properties

 Peter Blaha

Karlheinz SchwarzGeorg Madsen

Dieter KvasnickaJoachim Luitz

http://www.wien2k.at

1400 registered groups2000 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 Exc

PBE0 [] = ExcPBE [] + (Ex

HF[sel] – ExPBE[sel])

LDA+U, GGA+U ELDA+U(,n) = ELDA() + Eorb(n) – EDCC()

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

)21)(( ,',,', mmmm nJUV

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

insulator, t2g band splits metallic single lower Hubbard-band in VM splits in CO with Fe3+ states lower than Fe2+

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 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

0 n

Structure optimization (GGA+U)

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. ??

O1O1

O2aO2b

O3

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 =cryst-at

sup

CO phase VM phase

Fe2+: d-xzFe3+: d-x2

O1 and O3: polarized toward Fe3+

Fe: d-z2 Fe-Fe interactionO: symmetric

dxz spin density (up-dn) of CO phase

Fe3+: no contribution

Fe2+: dxz

weak -bond with O

tilting of O3 -orbital

Mössbauer spectroscopy:

Isomer shift: = (0Sample – 0

Reference); =-.291 au3mm s-1

proportional to the electron density at the nucleus

Magnetic Hyperfine fields: Btot=Bcontact + Borb + Bdip

Bcontact = 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

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

heQ / EFG traceless tensor

VV ijijij2

3

1

jiij xx

VV

)0(2

0 zzyyxx VVVwith traceless

cc

bc

ac

cb

bb

ab

ca

ba

aa

V

V

V

V

V

V

V

V

V

zz

yy

xx

V

V

V

0

0

0

0

0

0|Vzz| |Vyy| |Vxx|

//

////

zz

yyxx

V

VV

EFG Vzz

asymmetry parameterprincipal component

theoretical EFG calculations:

LMLMLMc

jiij rYrVrd

rr

rrV

xx

VV )ˆ()(

)()(

)0(2

2220

2220

20

21

21

)0(

VVV

VVV

rVV

xx

yy

zz

EFG is tensor of second derivatives of VC at the nucleus:

222 )(211

)(211

)(

3

3

320

zyzxzyxxyd

dzz

zyxp

pzz

dzz

pzzzz

dddddr

V

pppr

V

VVdrr

YrV

Cartesian LM-repr.

EFG is proportial to differences of orbital occupations

Mössbauer spectroscopyIsomer shift: charge transfer too small in LDA/GGAHyperfine fields: Fe2+ has large Borb and Bdip 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 = i,j Jij Si.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 + J23

c)/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+)

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 Jij by least-squares fit

Investigated magnetic configurations

Calculated exchange parameters

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.

Thank you for Thank you for your attention !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

non relativisticsemi-relativisticfully-relativistic

“Muffin-tin” MTatomic 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-spinpolarizedSpin polarized(with certain magnetic order)

non periodic (cluster, individual MOs)periodic (unit cell, Blochfunctions,“bandstructure”)

plane waves : PWaugmented plane waves : APWatomic oribtals. e.g. Slater (STO), Gaussians (GTO), LMTO, numerical basis

Basis functions

Treatment of spin

Representationof solid

Form ofpotential

exchange and correlation potentialRelativistic treatment

of the electrons

Schrödinger - equationki

ki

kirV

)(

2

1 2

PW:

APW Augmented Plane Wave method

The unit cell is partitioned into:

atomic spheresInterstitial region

rKkie

).(

Atomic partial waves

mm

Km rYruA

)ˆ(),(

join

Rmt

unit cell

Basisset:

ul(r,) are the numerical solutionsof the radial Schrödinger equationin a given spherical potential for a particular energy Alm

K coefficients for matching the PW

Ir

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

0 = C

>E<

>|<

>|H|< = >E<

k n

kkK

k nn

n

C =

(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

ki

ki

kirV

)(

2

1 2

Quantum mechanics at work

WIEN2k software package

An Augmented Plane Wave Plus Local Orbital

Program for Calculating Crystal Properties

 Peter Blaha

Karlheinz SchwarzGeorg Madsen

Dieter KvasnickaJoachim Luitz

November 2001Vienna, AUSTRIA

Vienna University of Technology

http://www.wien2k.at

WIEN97: ~500 usersWIEN2k: ~1150 usersmailinglist: 1800 users

Development of WIEN2k

Authors of WIEN2kP. 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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Aachen, ESRF, Prague, Paris, Chalmers, Cambridge, Oxford) America: ARG, BZ, CDN, MX, USA (MIT, NIST, Berkeley, Princeton,

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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