1. 1 5.0 引言 5.1 轨道, 相互作用与自旋 5.2 原子和分子的磁矩 5.3 晶体的磁矩 5.4 晶体的磁各向异性 5.5 习题 5. 磁性与电子态

2. 2 General remarks Uniaxial cases Cubic crystals Why success limited Outline

3. 3 Decoupling of Spin from Orbit Even for a spin-dependent Exc, such as Von Barth and Hedin ( 1972), V  xc = dE xc /d  = V xc  (  )+  V xc m ( ,m) The spin-up and spin-down states are decoupled. Total energy depends only on the magnitude of spin polarization, m, but independent of its direction.

4. 4 Spin-orbit Coupling Causes Anisotropy Spin-orbit coupling H sl = [(1/4c 2 r)  V/  r] l(r)  =  r  l(r)  Total energy variation E sl (  ) = E(H 0 +H sl )  E(H 0 )

5. 5 Perturbation Analysis First order energy E(  ) =   because =0 Second (even) order energy, E(  )=   2 | | 2 / (  (o)-  (e)) + h.o.t Mostly between spin-down bands

6. 6 Directional Dependence Due to orbital character of the o-e pairs near Fermi surface

7. 7 Perpendicular Anisotropy D.S.Wang et al, PRB47, 14932, 1993 Fe film: coupling = causes perpendicular anisotropy Singularity occurs when |e (o)-e (e)| < 

8. 8 In-plane Anisotropy D.S.Wang et al, JMMM 129, 344, 1994 Co film: coupling = causes in-plane anisotropy Singularity occurs when |e (o)-e (e)| <  

9. 9 Experiments vs Theory D.S.Wang et al. JMMM 140, 643, 1995

10. 10 Anisotropy vs Band Filling

11. 11 Anisotropy of X-Co-X D.S.Wang PRB48, 15886, 1993

12. 12 Ni Layers on Cu Substrate J.Henk et al, PRB59, 9332 (1999)

13. 13 Ni Layers on Cu Substrate J.Henk et al, PRB59, 9332 (1999) For fct, the bulk contribution is nearly correct, but contribution of the sub-surface layer seems wrong.

14. 14 Distorted Cubic Crystals T.Burkert et al. PRB 69, 104426 (2004)

15. 15 Cubic Crystals – Early Empirical Authors E(001)-E(111) in  eV/atom Remarks bcc Fe fcc Co fcc Ni Experiments  1.4 1.8 2.7 Kondorskii et al /JETP36,188(1973) x x 1.3 Empirical Fritsche et al /J.Phys.F17,943(1987) 7.4 x 10.0

16. 16 Cubic Crystals - LSDA Authors E(001)-E(111) in  eV/atom Remarks bcc Fe fcc Co fcc Ni Experiment  1.4 1.8 2.7 Daalderop et al /PRB41,11919(1990)  0.5 x  0.5 Strange et al /Physica B172,51(1991)  9.6 x 10.5 Trygg et al /PRL75,2871(1995)  0.5 0.5  0.5 Razee at al /PRB56,8082(1997)  0.95 0.86 0.11 Halilov et al /PRB57,9557(1998)  0.5 0.3 0.04  2.6 2.4 1.0  scaling

17. 17 Cubic Crystals – LSDA+OP Authors E(001)-E(111) in  eV/atom Remarks bcc Fe fcc Co fcc Ni Experiment  1.4 1.8 2.7 Trygg et al /PRL75,2871(1995)  1.8 2.2  0.5 OP Yang et al. /PRL87,216405(2001) U=1.2 x U=1.9 in eV J=0.8 x J=1.2 in eV Xie et al /PRB69,172404(2004) U=1.15 U=1.41 U=2.95 in eV J=0.97 J=0.83 J=0.28 in eV

18. 18 Ab Initio Attempt - Summary Bulk uniaxial cases are good Surface (interface) layers are fair Cubic crystals are poor

19. 19 Uniaxial Case: Two Pairs Reconsider the second order perturbation, E(  )=   2 | | 2 / (  (e)-  (o)) It holds only when  (e)  (o) >  and . For uniaxial cases, the regular part is in 2nd order (  2  /  )! When  (e)  (o) <  degenerate perturbation applies, E(  )  |  | and  (  2 / |  k  (o)  k  (e)| ). Singular at those k points. Total contribution is in 3rd order (    /   )!.

20. 20 Cubic Case: Two Pairs The second order perturbation, E(  )=   2 | | 2 / (  (e)-  (o)) is isotropic. For cubic case, the regular part of anisotropy goes to E(  )    4 | | 4 / (  (e)-  (o)) 3  and . The contribution is in the 4th order (  2  /   )! The singular part with, E(  )  |  | and  (  2 / |  k  (o)  k  (e)| ) Singular at those k points. Total contribution is in 3rd order (    /   )!.

21. 21 Challenge in Cubic Case Count the correlation in acceptable accuracy between the nearly degenerate pairs of empty and occupied states around Fermi surface!.

22. 22 Concluding Comment One can not claim understand unless he can calculate ! - J.C.Slater