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

Simple Limit:Vdk=Vkd=0No s-d hybridization

ε

εF

Ed

Ed+U

ε

εF

Ed

Ed+U

No localized moment

Hartree-Fock Approximation

ddcorr nUnH

)( ddddnnnn

δ↑ 2

dddddd

ddddddcorr

nnnnnnU

nnnnnnUH

constant

ddddcorr nnnnUH

is very small,2Assume that

Hartree-Fock Hamiltonian

)(

)( ,

kddkdkk

kd

ddd

kkk

ccVccV

nnUE

n

HFΗ

One-electron Hamiltonian

Eσ

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

DOS of d-electrons

22

1

Ed

εEd

ρ ｄ（ ε ）

Δ

Lorentzian

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

<n> vs. y=U/Δ

magneticnon-magnetic

2

1

U

Ex dF

ε

εF

Ed+U

Ed

(symmetric)

<n> vs. y=U/Δ

magnetic

non-magnetic

4

1

U

Ex dF

ε

εF

Ed

Ed+U

(asymmetric)

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

SummaryThe Coulomb correlation among d-electrons at the impurity site is important to understand the appearance of magnetic moment.The existence of magnetic moments depends on ‘x’ and ‘y’.

U

Ex dF

U

y

ε

εF

ε

εF

ε

εF

Ed

Ed+U