1. 鄭弘泰 國家理論中心 (清華大學) 28 Aug, 2003, 東華大學
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

2. 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

3. Density Functional Theory (DFT) Hohenberg-Kohn Theorem, PR136(1964)B864
The ground-state energy of a system of identical 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.

4. Local density approximation (LDA) Kohn-Sham scheme PR140(1965)A1133

5. Insufficiencies of LDA
Poor eigenvalues, PRB23, 5048 (1981)
Lack of derivative discontinuity at integer N, PRL49, 1691 (1982)
Gaps too small or no gap, PRB44, 943 (1991)
Spin and orbital moment too small, PRB44, 943 (1991)
Especially for transition metal oxides

6. PRB 23 (1981) 5048

7. PRL 49 (1982) 1691

8. PRB 44 (1991) 943

9. 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

10. Local density approximation (LDA) Kohn-Sham scheme PR140(1965)A1133

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

12. 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 energy UN(N－1)/2

13. Hubbard U for localized d orbital :
n+1
n-1 e n n U

14. LDA+U energy functional :
LDA+U potential :

15. LDA+U eigenvalue :
For occupied state ni=1,
For unccupied state ni=0,

16. Take Hydrogen for example:
Eexact(H) = －1.0 Ry
ELDA(H) = －0.957 Ry～ Eexact(H)
eLDA (H) = －0.538 Ry << Eexact (H)

17. ELDA(H-) : no bound statep e e p
Eexp(H-) = － Ry
Eexact (H+) = 0.0 Ry
ESIC(H-) = － Ry
ELDA(H-) : no bound state e p
U = E(H+) + E(H-) －2E(H)
= Ry e p
2Eexact(H) = －2.0 Ry

18. eLDA (H) = －0.538 Ry
U = Ry
Occupied (H) state:
eLDA+U(H) = eLDA (H) －U/2 = － Ry ～－1 Ry
Unoccupied (H+) state:
eLDA+U(H+) = eLDA (H) ＋U/2 = － Ry～0 Ry

19. 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
…..

20. 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

21. Where to find U and J PRB 44 (1991) 943 : 3d atoms
PRB 50 (1994) : 3d, 4d, 5d atoms
PRB 58 (1998) 1201 : 3d atoms
PRB 44 (1991) : Fe(3d)
PRB 54 (1996) 4387 : Fe(3d)
PRL 80 (1998) 4305 : Cr(3d)
PRB 58 (1998) 9752 : Yb(4f)

22. Notes on using LDA+U
The magnitude of U is difficult to calculate accurately, the deviation could be as large as ~2eV
For the same element, U depends also on the ionicity 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)

23. Various LDA+U methods
Hubbard model in mean field approx. (HMF) LDA+U : PRB 44 (1991) 943 (WIEN2K, LMTO)
Approximate self-interaction correction (SIC) LDA+U : PRB 48 (1993) (WIEN2K)
Around the mean field (AMF) LDA+U : PRB 49 (1994) (WIEN2K)
Rotationally invariant LDA+U : PRB 52 (1995) R5468 (VASP4.6, LMTO)
Simplified rotationally invariant LDA+U : PRB 57 (1998) 1505 (VASP4.6, LMTO)

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

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

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

27. Rotationally invariant LDA+U: PRB52(1995)R5468
Slater integral

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

29. Applications of LDA+U on transition-metal oxides
Pyrochlore : Cd2Re2O7 (VASP)
Rutile : CrO2 (FP-LMTO)
Double perovskite : Sr2FeMoO6, Sr2FeReO6, Sr2CrWO6 (FP-LMTO)
Cubic inverse spinel : high-temperature magnetite (Fe3O4, CoFe2O4, NiFe2O4) (FP-LMTO, VASP, WIEN2K)
Low-temperature charge-ordering Fe3O4 (VASP, LMTO)

30. Pyrochlore Cd2Re2O7 Lattice type : fcc 88 atoms in cubic unit cell
Space group : Fd3m
a = A, 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)

31. Pyrochlore structure

32. K. D. Tsuei, SRRC
Re-5d-t2g
Re-5d-eg
Cd-5s
Cd-5p
Re-7s

33. PRB 66 (2002) 12516
O-2p
Cd-4d
Re-5d-t2g

34. PRB 66 (2002) 12516

35. 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

36. 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

37. Rutile structure

39. PRB 56
(1997) 15509

40. Spin and orbital magnetic moments of CrO2
(uB)
Spin moment
Orbital moment Cr O
Total
LDA
1.89
-0.042
2.00
-0.037
LDA+U
1.99
-0.079
-0.051
Exp. 2
-0.05*
-0.003*
* D. J. Huang et al, SRRC

41. 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, →stronger hybridization, stronger crystal field, → close to itinerant picture

42. 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

43. Double perovskite structure

44. LDA
LDA+U
SFMO
SFMO
SFRO
SFRO
SCWO
SCWO

45. spin moment
orbital moment
Sr2FeMoO6
GGA
GGA+U
Fe Mo total
Fe Mo
Sr2FeReO6
Fe Re total
Fe Re
Sr2CrWO6
Cr W total
Cr W

46. GGA m = -2 -1 0 +1 +2 Fe 3d ↑ Fe 3d ↓ Re 5d ↑ Re 5d ↓
Occupation number of d orbital in Sr2FeReO6
GGA
m =
Fe 3d ↑
Fe 3d ↓
Re 5d ↑
Re 5d ↓
GGA+U

47. GGA m = -2 -1 0 +1 +2 Cr 3d ↑ Cr 3d ↓ W 5d ↑ W 5d ↓
Occupation number of d orbital in Sr2CrWO6
GGA
m =
Cr 3d ↑
Cr 3d ↓
W 5d ↑
W 5d ↓
GGA+U
m =

49. 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

50. 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, 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

51. Spinel structure

52. Fe3O4

54. CoFe2O4
LDA
LDA+U, U(Fe)=4.5eV, U(Co)=7.8eV

55. NiFe2O4
LDA
LDA+U, U(Fe)=4.5eV, U(Ni)=8eV

56. Spin and orbital magnetic moments of Fe3O4
(uB)
Spin moment
Orbital moment
Fe(A)
Fe(B)
Total
LDA
-3.31
3.52
4.00
-0.018
0.039
GGA
-3.44
3.60
-0.016
0.038
LDA+U
-3.81
3.81
-0.014
0.21
Exp.
-3.8+
4.1
0.2~0.4*
+J. Phys. C 11 (1978) 4389
* D. J. Huang et al, SRRC

57. Spin and orbital magnetic moments of Fe3O4
(uB)
Spin moment
Orbital moment
LDA+U
Fe(A)
Fe(B)
Total
LMTO
-3.81
3.81
4.00
-0.014
0.21
VASP
-3.87
3.78
-0.028
0.033
WIEN2k#
-3.95
3.84
-0.015
0.036
-3.35
3.47
-0.025
0.078
Exp.
-3.8
4.1
0.2~0.4
# Physica B 312 (2002) 785

58. Occupation numbers in Fe-3d orbitals of Fe3O4 from LDA and LDA+U
Fe(B)-3d↑
.9445
.9426
.9443
.9418
.9425
Fe(B)-3d↓
.2166
.2072
.2286
.2266
.2288
LDA+U
.9680
.9518
.9815
.9513
.9674
.1732
.08830
.1784
.3563
.1531

59. Fe3O4

60. Spin, orbital, and total magnetic moments for magnetite from LDA+U and experiment
(uB)
Spin
Orbit
Total
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
D. J. Huang et al, SRRC
+ H. J. Lin et al, SRRC
# G. V. D. Laan, PRB59, 4314 (1999)

61. 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

62. Low-temperature Fe3O4 Insulator Lattice type : monoclinic
Space group : P2/c
56 atoms in monoclinic unit cell
a = A, b=5.925 A, c= A β=
Ionic model : Fe+3(3d5), Fe+2(3d6)
U=4.5 eV, J = 0.89 eV

63. P2/C structure (low-temperature magnetite)

65. LDA
LDA+U

66. Charge and spin moment of Fe3O4 in low-temperature phase
LDA
Exp.*
LDA+U
Spin
Charge
Fe(B1)
3.50
6.62
5.6
6.64
3.64
Fe(B2)
3.58
5.4
6.50
4.08
Fe(B3)
3.47
6.66
6.55
3.98
Fe(B4)
3.54
6.60
3.59
* PRL 87 (2001)

67. 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…….

68. 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!?