Introduction to LDA+U method and applications to transition-metal oxides:

Importance of on-site Coulomb interaction U

鄭弘泰國家理論中心 ( 清華大學 )

28 Aug, 2003, 東華大學

Outline• DFT, LDA (LDA, LSDA, GGA)

• Insufficiencies of LDA

• How to improve LDA

• Self-interaction correction (SIC)

• LDA+U method

• Applications of LDA+U on various transition-metal oxides

• Conclusions

Density Functional Theory (DFT)

Hohenberg-Kohn Theorem, PR136(1964)B864• The ground-state energy of a system of iden

tical spinless fermions is a unique functional of the particle density.

• This functional attains its minimum value with respect to variation of the particle density subject to the normalization condition when the density has its correct values.

Local density approximation (LDA)

Kohn-Sham scheme PR140(1965)A1133

3

3 3

[ ( )] [ ] ( ) ( )

1 ( ) ( ')' [ ]

2 | ' |

G S ext

xc

E n r T n n r V r d r

n r n rd rd r E n

r r

3 ( ')( ) ( ) ' [ ( )]

| ' |ext xc

n rV r V r d r V n r

r r

Insufficiencies of LDA

• Poor eigenvalues, PRB23, 5048 (1981)• Lack of derivative discontinuity at integer N,

PRL49, 1691 (1982)• Gaps too small or no gap, PRB

44, 943 (1991)• Spin and orbital moment too small, PRB4

4, 943 (1991)• Especially for transition metal oxides

PRB 23 (1981) 5048

PRL 49 (1982) 1691

PRB 44 (1991) 943

Attempts on improving LDA

• Self-interaction correction (SIC) PRB23(1981)5048, PRL65(1990)1148

• Hartree-Fock (HF) method, PRB48(1993)5058

• GW approximation (GWA), PRB46(1992)13051, PRL74(1995)3221

• LDA+Hubbard U (LDA+U) method, PRB44(1991)943, PRB48(1993)16929

Local density approximation (LDA)

Kohn-Sham scheme PR140(1965)A1133

3

3 3

[ ( )] [ ] ( ) ( )

1 ( ) ( ')' [ ]

2 | ' |

G S ext

xc

E n r T n n r V r d r

n r n rd rd r E n

r r

3 ( ')( ) ( ) ' [ ( )]

| ' |ext xc

n rV r V r d r V n r

r r

Self-interaction correction (SIC) Perdew and Zunger, PRB23(1981)5048

][][ nEnE LDAG

SICG

i

ixci

ii nErr

rnrnrrdd ][

|'|

)'()('

2

1 33

)]([|'|

)'(')()( 3 rnV

rr

rnrdrVrV ixc

iLDASICi

Basic idea of LDA+U PRB 44 (1991) 943, PRB 48 (1993) 169

• Delocalized s and p electrons : LDA

• Localized d or f electrons : +U using on-site d-d Coulomb interaction (Hubbard-like term)

UΣi≠jninj

instead of averaged Coulomb energyUN(N － 1)/2

Hubbard U for localized d orbital :

)(2)()( 11 nnn dEdEdEU

n n

n+1 n-1

U

e

LDA+U energy functional :

LDA+U potential :

LDAULDAlocal EE

ji

jinnUNUN2

12/)1(

)2

1()(

)()( i

LDA

ii nUrV

rn

ErV

LDA+U eigenvalue :

)2

1( i

LDA

ii nU

n

E

For occupied state ni=1,

For unccupied state ni=0,

2/ULDAi

2/ULDAi

Ui

pe

Eexact(H) = － 1.0 Ry

ELDA(H) = － 0.957 Ry ～ Eexact(H)

eLDA (H) = － 0.538 Ry << Eexact (H)

Take Hydrogen for example:

pe

2Eexact(H) = － 2.0 Ry

p

Eexact (H+) = 0.0 Ry

e p e

Eexp(H-) = － 1.0552 Ry

e

ESIC(H-) = － 1.0515 Ry

ELDA(H-) : no bound state

U = E(H+) + E(H-) － 2E(H) = 0.9448 Ry

pe

eLDA (H) = － 0.538 Ry U = 0.9448 Ry

Occupied (H) state:

eLDA+U(H) = eLDA (H) － U/2 = － 1.0104 Ry ～－ 1 Ry

Unoccupied (H+) state:

eLDA+U(H+) = eLDA (H) ＋ U/2 = － 0.0656 Ry ～ 0 Ry

When to use LDA+U

• Systems that LDA gives bad results

• Narrow band materials : U W≧• Transition-metal oxides

• Localized electron systems

• Strongly correlated materials

• Insulators

• …..

How to calculate U and J PRB 39 (1989) 9028

• Constrained density functional theory + Supper-cell calculation

• Calculate the energy surface as a function of local charge fluctuations

• Mapped onto a self-consistent mean-field solution of the Hubbard model

• Extract the Coulomb interaction parameter U from band structure results

Where to find U and J

• PRB 44 (1991) 943 : 3d atoms

• PRB 50 (1994) 16861 : 3d, 4d, 5d atoms

• PRB 58 (1998) 1201 : 3d atoms

• PRB 44 (1991) 13319 : Fe(3d)

• PRB 54 (1996) 4387 : Fe(3d)

• PRL 80 (1998) 4305 : Cr(3d)

• PRB 58 (1998) 9752 : Yb(4f)

Notes on using LDA+U• The magnitude of U is difficult to calculate accura

tely, the deviation could be as large as ~2eV• For the same element, U depends also on the ionic

ity in different compounds: the higher ionicity, the larger U

• One thus varies U in the reasonable range to obtain better results

• One might varies U in a much larger range to see the effect of U (qualitatively)

• → Self-consistent LDA+U (much more difficult)

Various LDA+U methods• Hubbard model in mean field approx. (HMF) LDA

+U : PRB 44 (1991) 943 (WIEN2K, LMTO)• Approximate self-interaction correction (SIC) LD

A+U : PRB 48 (1993) 16929 (WIEN2K)• Around the mean field (AMF) LDA+U : PRB

49 (1994) 14211 (WIEN2K)• Rotationally invariant LDA+U : PR

B 52 (1995) R5468 (VASP4.6, LMTO)• Simplified rotationally invariant LDA+U : PR

B 57 (1998) 1505 (VASP4.6, LMTO)

Original LDA+U(HMF) : PRB 44 (1991) 943

))((2

1 0'

0

,',

nnnnUEE mmm

mLDA

,',',

0'

0 ))()((2

1

mmmmmm nnnnJU

'

0' )(

mm

LDAm nnUVV

)('

0' )()(

mmm nnJU

LDA+U(SIC): PRB 48 (1993) 16929

]4/)2(2/)1([ NJNNUNEE LDA

'',','

',',

'' )(2

1

2

1mmmm

mmmmm

mmmmmm nnJUnnU

'

' )(m

meffmmLDAm nUUVV

mm

meffmeffmmmm JnUnUJU'

'' 4

1)

2

1()(

JUUeff 2

1

LDA+U(AMF) : PRB 49 (1994) 14211

))((2

1 0'

0

,','

nnnnUEE m

mmmmm

LDA

,',',

0'

0'' ))()((

2

1

mmmmmmmmmm nnnnJU

'

0'' )(

mmmm

LDAm nnUVV

)('

0''' ))((

mmmmmmm nnJU

Rotationally invariant LDA+U: PRB52(1995)R5468

)]1()1([2

1)1(

2

1 nnnnJnUnEE LDA

})''''||''''''||''(

''''||''{2

1

''''''

'''''},{

'

mmmmeeee

mmm

mmee

nnmmVmmmmVmm

nnmmVmm

})''''||''''''||''(

''''||''{)2

1()

2

1(

'''''

''''''''''

'

mmeeee

mmmm

eemm

nmmVmmmmVmm

nmmVmmnJnUV

:eeV Slater integral

Simplified rotationally invariant LDA+U : PRB 57 (1998) 1505

])()[(2

)(

jlljjl

jjjLDA nnn

JUEE

]2

1)[(

jljllj

LDAjl JU

n

EV

Applications of LDA+U on transition-metal oxides

• Pyrochlore : Cd2Re2O7 (VASP)• Rutile : CrO2 (FP-LMTO)• Double perovskite : Sr2FeMoO6, Sr2FeRe

O6, Sr2CrWO6 (FP-LMTO)• Cubic inverse spinel : high-temperature ma

gnetite (Fe3O4, CoFe2O4, NiFe2O4) (FP-LMTO, VASP, WIEN2K)

• Low-temperature charge-ordering Fe3O4 (VASP, LMTO)

Pyrochlore Cd2Re2O7

• Lattice type : fcc

• 88 atoms in cubic unit cell

• Space group : Fd3m

• a = 10.219A, x=0.316

• Ionic model: Cd+2(4d10), Re+5(5d2), O-2(2p6)

• U(Cd) = 5.5 eV, U(Re) = 3.0 eV

• J = 0 eV (no spin moment)

Pyrochlore structure

K. D. Tsuei, SRRCR

e-5d

-t2

gC

d-5s

Re-

5d-e

g

Cd-

5p

Re-7s

PRB 66 (2002) 12516

O-2p

Cd-4d

Re-5d-t2g

PRB 66 (2002) 12516

Pyrochlore Cd2Re2O7

• LDA unoccupied DOS agree well with XAS data from K. D. Tsuei, SRRC

• Cd-4d band from LDA is ~3 eV higher than photo emission spectrum (PRB66(2002)1251)

• Cd-4d band from LDA+U agree well with photo emission spectrum (PRB66(2002)1251)

• Cd-4d orbital is close to localized electron picture, whereas the other orbitals are more or less itinerant

Rutile CrO2

• Half-metal, moment = 2μB

• Lattice type : bct

• 6 atoms in bct unit cell

• Space group : P42/mnm

• a = 4.419A, c=2.912A, u=0.303

• Ionic model : Cr+4(3d2), O-2(2p6)

• U = 3.0 eV, J = 0.87 eV

Rutile structure

PRB 56(1997) 15509

Spin and orbital magnetic moments of CrO2

(uB) Spin moment Orbital moment

Cr O Total Cr O

LDA 1.89 -0.042 2.00 -0.037 -0.0011

LDA+U 1.99 -0.079 2.00 -0.051 -0.0025

Exp. 2 -0.05* -0.003*

* D. J. Huang et al, SRRC

Rutile CrO2

• LDA+U enhances the gap and the exchange splitting at the Fermi level

• LDA+U also gives larger spin and orbital magnetic moment

• U W, orbital moment quenched, →strong≦er hybridization, stronger crystal field, → close to itinerant picture

Double perovskites : Sr2FeMoO6, Sr2FeReO6, Sr2CrWO6

• Half-metal, moment = 4, 3, 2μB

• Lattice type : tet, fcc, fcc• 40 atoms in tet, fcc, fcc unit cell• Space group : I4/mmm, Fm3m, Fm3m• a = 7.89, 7.832, 7.878A, c/a=1.001, 1, 1• Ionic model : Fe+3(3d5), Cr+3(3d3), Mo+5(4d1),

Re+5(5d2), W+5(5d1)• U(Fe,Cr) = 4,3 eV, J(Fe,Cr) = 0.89, 0.87 eV

Double perovskite structure

LDA LDA+U

SFMO

SFRO

SCWO

SFMO

SFRO

SCWO

spin moment orbital moment

Sr2FeMoO6

GGA

GGA+U

Fe Mo total 3.80 -0.33 4.00 3.96 -0.43 4.00

Fe Mo0.043 0.0320.047 0.045

Sr2FeReO6

GGA GGA+U

Fe Re total 3.81 -0.85 3.00 3.98 -0.96 3.00

Fe Re0.070 0.230.066 0.27

Sr2CrWO6

GGA GGA+U

Cr W total 2.30 -0.33 2.00 2.46 -0.45 2.00

Cr W-0.007 0.10-0.007 0.15

GGA m = -2 -1 0 +1 +2

Fe 3d ↑

Fe 3d ↓

Re 5d ↑

Re 5d ↓

0.95 0.92 0.98 0.92 0.95

0.17 0.20 0.17 0.20 0.20

0.27 0.21 0.28 0.20 0.22

0.35 0.42 0.26 0.58 0.43

GGA+U m = -2 -1 0 +1 +2

Fe 3d ↑

Fe 3d ↓

Re 5d ↑

Re 5d ↓

0.96 0.93 0.98 0.94 0.96

0.15 0.17 0.16 0.16 0.18

0.26 0.20 0.27 0.19 0.21

0.36 0.43 0.26 0.61 0.45

Occupation number of d orbital in Sr2FeReO6

GGA m = -2 -1 0 +1 +2

Cr 3d ↑

Cr 3d ↓

W 5d ↑

W 5d ↓

0.55 0.86 0.20 0.87 0.52

0.14 0.13 0.16 0.14 0.15

0.18 0.16 0.18 0.15 0.15

0.20 0.22 0.18 0.30 0.24

GGA+U m = -2 -1 0 +1 +2

Cr 3d ↑

Cr 3d ↓

W 5d ↑

W 5d ↓

0.55 0.89 0.18 0.90 0.52

0.12 0.09 0.16 0.10 0.14

0.17 0.14 0.18 0.13 0.14

0.21 0.23 0.18 0.34 0.25

Occupation number of d orbital in Sr2CrWO6

Double perovskites : Sr2FeMoO6, Sr2FeReO6, Sr2CrWO6

• LDA+U has significant effects on DOS, but experimental data is not available

• Orbital moment of 3d and 4d elements are all quenched because of strong crystal field

• 5d elements exhibit large unquenched orbital moment because of strong spin-orbit interaction in 5d orbitals

Magnetite (high temperature) : Fe3O4, CoFe2O4, NiFe2O4

• Half-metal, insulator, moment = 4, 3, 2μB

• Lattice type : fcc• Space group : Fd3m• 56 atoms in fcc unit cell• a = 8.394, 8.383, 8.351 A• Ionic model : Fe+3(3d5), Fe+2(3d6), Co+2(3d7),

Ni+2(3d8)• U(Fe+3,Fe+2,Co+2,Ni+2)=4.5, 4.0, 7.8, 8.0eV• J(Fe,Co,Ni) = 0.89, 0.92, 0.95eV

Spinel structure

Fe3O4

LDA+U, U(Fe)=4.5eV, U(Co)=7.8eVLDACoFe2O4

LDA+U, U(Fe)=4.5eV, U(Ni)=8eVLDANiFe2O4

Spin and orbital magnetic moments of Fe3O4

(uB) Spin moment Orbital moment

Fe(A)

Fe(B)

Total Fe(A) Fe(B)

LDA -3.31 3.52 4.00 -0.018 0.039

GGA -3.44 3.60 4.00 -0.016 0.038

LDA+U -3.81 3.81 4.00 -0.014 0.21

Exp. -3.8+ 4.1 0.2~0.4*

* D. J. Huang et al, SRRC+J. Phys. C 11 (1978) 4389

Spin and orbital magnetic moments of Fe3O4

(uB) Spin moment Orbital moment

LDA+U Fe(A)

Fe(B)

Total Fe(A) Fe(B)

LMTO -3.81 3.81 4.00 -0.014 0.21

VASP -3.87 3.78 4.00 -0.028 0.033

WIEN2k# -3.95 3.84 4.00 -0.015 0.036

WIEN2k# -3.35 3.47 4.00 -0.025 0.078

Exp. -3.8 4.1 0.2~0.4

# Physica B 312 (2002) 785

Occupation numbers in Fe-3d orbitals of Fe3O

4 from LDA and LDA+U

LDA m=-2 m=-1 m=0 m=1 m=2

Fe(B)-3d↑ .9445 .9426 .9443 .9418 .9425

Fe(B)-3d↓ .2166 .2072 .2286 .2266 .2288

LDA+U m=-2 m=-1 m=0 m=1 m=2

Fe(B)-3d↑ .9680 .9518 .9815 .9513 .9674

Fe(B)-3d↓ .1732 .08830 .1784 .3563 .1531

Fe3O4

LDA+U Experiment

(uB) Spin Orbit Total Spin Orbit total

Fe3O4 4.0 0.42 4.42 0.4~0.8* 4.1

CoFe2O4 3.0 ~0.4 ~3.4 0.8~1.2+ 3.9

NiFe2O4 2.0 0.24 2.24 1.94 0.26# 2.2

Spin, orbital, and total magnetic moments for magnetite from LDA+U and experiment

• D. J. Huang et al, SRRC+ H. J. Lin et al, SRRC# G. V. D. Laan, PRB59, 4314 (1999)

Magnetite (high temperature) : Fe3O4, CoFe2O4, NiFe2O4

• LDA+U gives better DOS for Fe3O4• LDA+U gives insulating ground states for

CoFe2O4 and NiFe2O4• Spin moment of Fe3O4 from LDA+U

agrees better with experimental value• On-site U gives large unquenched orbital

magnetic moments for Fe(B), Co, and Ni• Fe3O4: localized～ itinerant electron• CoFe2O4, NiFe2O4: localized electron

Low-temperature Fe3O4

• Insulator

• Lattice type : monoclinic

• Space group : P2/c

• 56 atoms in monoclinic unit cell

• a = 5.9444 A, b=5.925 A, c=16.775 A

• β=90.23630

• Ionic model : Fe+3(3d5), Fe+2(3d6)

• U=4.5 eV, J = 0.89 eV

P2/C structure (low-temperature magnetite)

LDA LDA+U

Charge and spin moment of Fe3O4 in low-temperature phase

LDA LDA Exp.* LDA+U

LDA+U

Spin Charge Charge Charge Spin

Fe(B1) 3.50 6.62 5.6 6.64 3.64

Fe(B2) 3.58 6.62 5.4 6.50 4.08

Fe(B3) 3.47 6.66 5.4 6.55 3.98

Fe(B4) 3.54 6.60 5.6 6.66 3.59

* PRL 87 (2001) 266401

Low-temperature Fe3O4

• The lattice distortion give rise to a pseudogap at the Fermi level from LDA

• An insulating ground state is obtained from LDA+U

• In the meantime, a charge ordered electronic structure is induced by the on-site Hubbard U

• Lots of works need to be done…….

Conclusions

• Transition-metal oxides: narrower band width, closer to localized electron picture, more or less strongly correlated → LDA+U gives better result than LDA does

• The on-site Coulomb interaction U is important in these systems

• Which LDA+U version is better? An overall better LDA+U method is necessary!?