鄭弘泰 國家理論中心 ( 清華大學 ) 28 aug, 2003, 東華大學
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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!?
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