LDA U第一性原理方法

1、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 va

2、rious transition-metal oxides ConclusionsDensity 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 th

3、e particle density subject to the normalization condition when the density has its correct values.Local density approximation (LDA)Kohn-Sham scheme PR140(1965)A1133333()()()1()()2| |GSextxcEn rTnn r Vr d rn r n rd rd rEnrr3()( )( ) ( )|extxcn rV rVrd rVn rrrInsufficiencies of LDA Poor eigenvalues, P

4、RB23, 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 oxidesPRB 23 (1981) 5048PRL 49 (1982) 1691PRB 44 (1991) 943Attempts on improving LDA Se

5、lf-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)16929Local density approximation (LDA)Kohn-Sham scheme PR140(1965)A1133333()()

6、()1()()2| |GSextxcEn rTnn r Vr d rn r n rd rd rEnrr3()( )( ) ( )|extxcn rV rVrd rVn rrrSelf-interaction correction (SIC) Perdew and Zunger, PRB23(1981)5048nEnELDAGSICG iixciiinErrrnrnrrdd| |) ()(2133)(| |) ()()(3rnVrrrnrdrVrVixciLDASICiBasic idea of LDA+U PRB 44 (1991) 943, PRB 48 (1993) 169 Delocal

7、ized s and p electrons : LDA Localized d or f electrons : +U using on-site d-d Coulomb interaction (Hubbard-like term) Uijninj instead of averaged Coulomb energyUN(N1)/2Hubbard U for localized d orbital :)(2)()(11nnndEdEdEUnnn+1n-1UeLDA+U energy functional :LDA+U potential :LDAULDAlocalEEjijinnUNUN2

8、12/)1()21()()()(iLDAiinUrVrnErVLDA+U eigenvalue :)21(iLDAiiinUnEFor occupied state ni=1,For unccupied state ni=0,2/ULDAi2/ULDAiUipeEexact(H) = 1.0 RyELDA(H) = 0.957 Ry Eexact(H) eLDA (H) = 0.538 Ry Eexact (H)Take Hydrogen for example:pe2Eexact(H) = 2.0 RypEexact (H+) = 0.0 RyepeEexp(H-) = 1.0552 Rye

9、ESIC(H-) = 1.0515 RyELDA(H-) : no bound stateU = E(H+) + E(H-) 2E(H) = 0.9448 RypeeLDA (H) = 0.538 RyU = 0.9448 RyOccupied (H) state:eLDA+U(H) = eLDA (H) U/2 = 1.0104 Ry 1 RyUnoccupied (H+) state:eLDA+U(H+) = eLDA (H) U/2 = 0.0656 Ry0 RyWhen to use LDA+U Systems that LDA gives bad results Narrow ban

10、d materials : UW 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 ont

11、o a self-consistent mean-field solution of the Hubbard model Extract the Coulomb interaction parameter U from band structure resultsWhere 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

12、: 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 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

13、 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

14、self-interaction correction (SIC) LDA+U : PRB 48 (1993) 16929 (WIEN2K) Around the mean field (AMF) LDA+U : PRB 49 (1994) 14211 (WIEN2K) Rotationally invariant LDA+U : PRB 52 (1995) R5468 (VASP4.6, LMTO) Simplified rotationally invariant LDA+U : PRB 57 (1998) 1505 (VASP4.6, LMTO)Original LDA+U(HMF) :

15、 PRB 44 (1991) 943)(2100, ,nnnnUEEmmmmLDA, , ,00)()(21mmmmmmnnnnJU0)(mmLDAmnnUVV)( 0)()(mmmnnJULDA+U(SIC): PRB 48 (1993) 16929 4/ ) 2(2/ ) 1(NJNNUNEELDA, , , ,)(2121mmmmmmmmmmmmmmmnnJUnnU)(mmeffmmLDAmnUUVVmmmeffmeffmmmmJnUnUJU41)21()(JUUeff21LDA+U(AMF) : PRB 49 (1994) 14211)(2100, ,nnnnUEEmmmmmmLDA,

16、 , ,00)()(21mmmmmmmmmmnnnnJU0)(mmmmLDAmnnUVV)( 0)(mmmmmmmnnJURotationally invariant LDA+U: PRB52(1995)R5468 )1() 1(21) 1(21nnnnJnUnEELDA) | | ( | 21 ,mmmmeeeemmmmmeennmmVmmmmVmmnnmmVmm) | | ( | )21()21( mmeeeemmmmeemmnmmVmmmmVmmnmmVmmnJnUV:eeVSlater integralSimplified rotationally invariant LDA+U :

17、PRB 57 (1998) 1505 )()(2)(jlljjljjjLDAnnnJUEE21)(jljlljLDAjlJUnEVApplications 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, NiFe2

18、O4) (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 str

19、ucturePRB 66 (2002) 12516O-2pCd-4dRe-5d-t2gPRB 66 (2002) 12516Pyrochlore 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(

20、2002)1251) Cd-4d orbital is close to localized electron picture, whereas the other orbitals are more or less itinerantRutile CrO2 Half-metal, moment = 2B 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 =

21、0.87 eVRutile structurePRB 56(1997) 15509Spin and orbital magnetic moments of CrO2 (uB)Spin momentOrbital momentCrOTotal CrOLDA1.89 -0.042 2.00-0.037 -0.0011LDA+U 1.99 -0.079 2.00-0.051 -0.0025Exp.2-0.05*-0.003* D. J. Huang et al, SRRCRutile CrO2 LDA+U enhances the gap and the exchange splitting at

22、the Fermi level LDA+U also gives larger spin and orbital magnetic moment UW, orbital moment quenched, stronger hybridization, stronger crystal field, close to itinerant pictureDouble perovskites : Sr2FeMoO6, Sr2FeReO6, Sr2CrWO6 Half-metal, moment = 4, 3, 2B Lattice type : tet, fcc, fcc 40 atoms in t

23、et, 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 eVDouble perovskite structureLDALDA+USFMOSFROSCWOSFMOSFROSCWO spin momentorbital moment Sr2FeMo

24、O6 GGA GGA+U Fe Mo total 3.80 -0.33 4.00 3.96 -0.43 4.00 Fe Mo0.043 0.0320.047 0.045Sr2FeReO6 GGA GGA+U Fe Re total 3.81 -0.85 3.00 3.98 -0.96 3.00 Fe Re0.070 0.230.066 0.27Sr2CrWO6 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 GGAm = -2 -1 0 +1 +2 Fe 3d Fe 3d Re 5d

25、 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+Um = -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 0.45Occupation number of d orbital in Sr2FeReO6GGAm

26、 = -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 0.24GGA+Um = -2 -1 0 +1 +2Cr 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 0.25Occupation number o

27、f d orbital in Sr2CrWO6Double 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

28、 strong spin-orbit interaction in 5d orbitalsMagnetite (high temperature) : Fe3O4, CoFe2O4, NiFe2O4 Half-metal, insulator, moment = 4, 3, 2B 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,

29、Co+2,Ni+2)=4.5, 4.0, 7.8, 8.0eV J(Fe,Co,Ni) = 0.89, 0.92, 0.95eVSpinel structureLDA+U, U(Fe)=4.5eV, U(Co)=7.8eVLDACoFe2O4LDA+U, U(Fe)=4.5eV, U(Ni)=8eVLDANiFe2O4Magnetite (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