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

General remarksUniaxial cases Cubic crystalsWhy success limited

Outline

3

Decoupling of Spin from Orbit

Even for a spin-dependent Exc, such as Von Barth and Hedin (1972),

Vxc = dExc/d = Vxc

()+ Vxcm

(,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

Spin-orbit Coupling Causes Anisotropy

Spin-orbit coupling

Hsl = [(1/4c2r) V/r] l(r) = r l(r)

Total energy variation

Esl() = E(H0+Hsl) E(H0)

5

Perturbation Analysis

First order energyE() = <o|l|o> because <o|l|o>=0

Second (even) order energy,

E()= 2|<o|l|e>|2 / ((o)-(e)) + h.o.t

Mostly between spin-down bands

6

Directional Dependence

Due to orbital character of the o-e pairs near Fermi surface

7

Perpendicular AnisotropyD.S.Wang et al, PRB47, 14932, 1993

Fe film: coupling <5|lz|5*>=<xz|lz|yz>

causes perpendicular anisotropy

Singularity occurs when |e (o)-e (e)| <

8

In-plane AnisotropyD.S.Wang et al, JMMM 129, 344, 1994

Co film: coupling

<5|ly|1>=<xz|ly|z2> <5*|lx|1>=<yz|lx|z2>

causes in-plane anisotropy

Singularity occurswhen |e (o)-e (e)| <

9

Experiments vs

Theory

D.S.Wang et al. JMMM 140, 643, 1

995

10

Anisotropy vs Band Filling

11

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

12

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

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

Distorted Cubic Crystals

T.Burkert et al.PRB 69, 104426 (2004)

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

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

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

Ab Initio Attempt - Summary

• Bulk uniaxial cases are good

• Surface (interface) layers are fair

• Cubic crystals are poor

19

Uniaxial Case: Two <o|*|e> Pairs

Reconsider the second order perturbation, E()= 2|<o|l|e>|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 perturbati

on applies, E() | <o|l|e>|

and (2 / | k(o)k(e)| ).Singular at those k points. Total contribution

is in 3rd order (/ )!.

20

Cubic Case: Two <o|*|e> Pairs

The second order perturbation, E()= 2|<o|l|e>|2 / ((e)-(o))is isotropic. For cubic case, the regular part

of anisotropy goes to E() 4|<l>|4 / ((e)-(o))3

and . The contribution is in the 4th order (2/ )!

The singular part with,

E() | <o|l|e>|

and (2 / | k(o)k(e)| )Singular at those k points. Total contributio

n is in 3rd order (/ )!.

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

Concluding Comment

One can not claim understand unless he can calculate !

- J.C.Slater