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...

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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

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General remarksUniaxial cases Cubic crystalsWhy success limited

Outline

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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.

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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)

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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

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Directional Dependence

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

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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)| <

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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)| <

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Experiments vs

Theory

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

995

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Anisotropy vs Band Filling

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

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

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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.

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Distorted Cubic Crystals

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

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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

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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

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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

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Ab Initio Attempt - Summary

• Bulk uniaxial cases are good

• Surface (interface) layers are fair

• Cubic crystals are poor

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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 (/ )!.

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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 (/ )!.

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Challenge in Cubic Case

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

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Concluding Comment

One can not claim understand unless he can calculate !

- J.C.Slater