localized magnetic states in metals
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Localized Magnetic States in Metals
Miyake Lab.Akiko Shiba
Ref.) P. W. Anderson, Phys. Rev. 124 (1961) 41
Contents
IntroductionExperimental Data
Calculation HamiltonianUnrestricted Hartree –Fock Approxi
mationMagnetic CaseNonmagnetic Case
Summary
Electron Concentration
Mom
ent p
er F
e in
Boh
r mag
neto
ns
No localized moment
localized moment
Experimental Data
Magnetic moments of Fe impurity
Depend on the host metal
Ref.)A.M.Clogston et al., Phys.Rev.125,541(1962)
Tn
2
0
Susceptibility:
Hamiltonian
)(
)(
kddkdkk
kd
dd
ddd
kkk
ccVccV
nUn
nnE
n
Ηfree-electron system
s-d hybridization
repulsive interaction
d-states
Many-bodyproblem
where
ddd
kkk
ccn
ccn
U
Ed+U
Ed
εF
V
Simple Limit: U=0 No coulomb correlation
)()(
kddkdkk
kddddk
kk ccVccVnnEn Η
No localized moment
nd↑=nd↓
ε
εF
Ed EdΔΔ
avdkV2
: DOS of conduction electrons
Simple Limit:Vdk=Vkd=0No s-d hybridization
dddddk
kk nUnnnEn )(
Η
Localized moment appears
ε
εF
Ed
Ed+U
Coulomb repulsive
Ed<εF Ed+U>εF
Hartree-Fock Approximation
ddcorr nUnH
)( ddddnnnn
δ↑ 2
dddddd
ddddddcorr
nnnnnnU
nnnnnnUH
constant
ddddcorr nnnnUH
is very small,2Assume that
DOS of d-electrons
2|)(
dnn
nd
Resolvent Green Function: H
G
1
)(Im1
)(
ddd G
iEGdd
1
)(
where avdkV
2
: DOS of conduction electrons
Self-consistent equation
Fdd
dd
nUE
dnF
,1cot1
)(cot 1 xnyn dd
U
Ex dF
U
yIntroduce
Important parameters!
:Self-consistent equation
Number of d-electrons:
Non-magnetic State(Self-consistency plot)
)(cot 1 xnyn dd
2
1
U
Ex dF
1
U
y0.5
0.5
Non-magnetic solution
dn
Non-magnetic Solution
2
1 dd
nn
Magnetic State
)(cot 1 xnyn dd
2
1
U
Ex dF
5
U
y
(Self-consistency plot)
Magnetic solutions
dn
dn
dn
Magnetic solutions
Non-magnetic solution
0.5
0.5
Magnetic
Phase diagram
2
11 xy
,1y x near 0
,1y x not small or too near 1
Non-magnetic conditions:
Magnetic conditions:
Uy
x
U
Ex
Uy dF
,
Non-magnetic Case (symmetric)
)(cot 1 xnyn dd
2
1 nnn
ddAssume
use the approximation:
nn
2
1cot
/1
/21
2
1
y
xynthen
2
1
1
x
UU
y
ε
εF Ed
Ed+UΔ
Non-magnetic Case (asymmetric)
)(cot 1 xnyn dd
Valence fluctuation
,1y x near 0
Opposite limit:
U
Ex dF
U
y
UEdF ,
ε
εF
Ed
Ed+UΔ
Magnetic Case
0,1 dd nnAssume
then
dd nxyn
1
d
d nxyn
11)1(
11
xxy
nnmdd
)(cot 1 xnyn dd
,1y x not small or too near 1
εF
Ed+U
ε
Ed
Δ
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