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1

Kondo effect

Masatsugu Sei Suzuki

Department of Physics, SUNY at Binghamton

(Date: June 06, 2013)

1. Introduction

Since 1930’s, it has been well-known that at very low temperatures the metals with magnetic

impurities exhibits an anomaly in the temperature dependence of their electrical resistivity; instead of the

generally known decrease of resistivity with lowering temperature, it increases with decreasing

temperatures. This effect is named as the Kondo effect, after Kondo (Jun). In 1964, Kondo has succeeded

for the first time in explaining using the perturbation method, with scattering of electrons on magnetic

impurities taken into account. In 1981, the exact solution of the Kondo problem was obtained later, by N.

Andrei and Paul Wiegmann, separately. Note that the methods and ideas (by Abrikosov,

Nozieres ,Wilson, Yosida, and et al.), which were needed to explain the Kondo effect, had played the

most important role in studying the fundamental phenomenon of the electron localization- a part of

modern physics and physical application.

In this note I do not intend to present various kinds of theories on the Kondo effect. It is beyond my

ability. I am interested in the experimental results. I have been doing research on the spin glasses,

experimentally using the techniques of aging dynamics. When one starts to read typical books on spin

glass (such as Mydosh) as beginner, one may encounter excellent experimental data on canonical spin

glasses (for example, Cu host diluted with magnetic impurity such as Mn or Fe). When the concentration

of the magnetic impurities is extremely dilute, it is found that the systems show Kondo effect. The

magnetic impurity is isolated from other magnetic impurities. The antiferromagnetic interaction between

the spin of the conduction electron and the spin of magnetic impurity (the s-d interaction) leads to the

Kondo spin singlet. The magnetic moment will vanish below a characteristic temperature, Kondo

temperature TK. Although there is no phase transition at this temperature, the pair formation of

antiparallel alignment of spins look so similar to the Cooper pair in BCS theory (spin singlet with orbital

angular moment L = 0).

2

Fig. Isolated localized spins in a sea of conduction electrons, forming the singlet state of antiparallel

alignment of conduction electron spin and the spin of magnetic impurity.

When the concentration of magnetic impurities is slightly increased, the system becomes a spin glass.

The interaction between the magnetic impurities which is the so-called the RKKY (Ruderman-Kittel-

Kasuya-Yosida) interaction, plays an important role in spin glass behavior. The interaction between the

magnetic impurities arises as a result of the interaction between the magnetic impurity and the conduction

electrons. The sign of the interaction shows oscillatory change depending on the distance. The

competition between the antiferromagnetic and ferromagnetic interactions leads to the fully frustrated

nature of spin glass.

My first encounter with the Kondo effect occurred when I read the book of Thermoelectricity which

was written by D.K.C. MacDonald. He showed a lot of excellent data of the thermoelectric power and

electrical resistivity for canonical systems (for example, Cu host diluted with magnetic impurities). The

temperature dependence of these two quantities are extremely sensitive to the concentration of magnetic

impurities in the limit of dilute range. Even at the present stage, I am not sure whether the temperature

dependence of the thermoelectric power as shown in the book of MacDonald, can be well explained.

2. Kondo effect and Jun Kondo

The Kondo effect is an unusual scattering mechanism of conduction electrons in a metal (such as

noble metals) due to magnetic impurities9such as Mn, Fe), which contributes a term to the electrical

3

resistivity that increases logarithmically with temperature as the temperature is decreased [as ln(T)]. It is

used to describe many-body scattering processes from impurities or ions which have low energy quantum

mechanical degrees of freedom. In this sense it has become a key concept in condensed matter physics in

understanding the behavior of metallic systems with strongly interacting electrons.

The Kondo effect is normally observed in very dilute magnetic alloys as a result of the interaction

between the host conduction electrons and the magnetic impurity spins. In its ideal form, it is a single-

impurity effect and leads to the rise of the electrical resistivity as the temperature is lowered down to zero

temperature. It can be described by an effective interaction which increases with decreasing temperature

and finally leads to a single state formed by a single impurity spin and the spins of the surrounding

conduction electrons (Kondo, Nozieres, Wilson). At larger concentrations, the impurity-spin interaction

becomes significant and leads to a partial destruction of single state. The impurities again become

magnetic, leading to a spin glass for typically, a few percent magnetic impurities.

4

Fig. Picture of Prof. Jun Kondo

http://www.aist.go.jp/aist_j/information/emeritus_advisor/index.html

3. Experimental results

In 1934a resistance minimum was observed in gold as a function of temperature (de Haas, de Boer

and van den Berg 1934), indicating that there must be some additional scattering mechanism giving an

anomalous contribution to the resistivity--- one which increases in strength as the temperature is lowered.

Other examples of metals showing a resistance minimum were later observed, and its origin was a

longstanding puzzle for about 30 years. In the early 1960s it was recognized that the resistance minima

are associated with magnetic impurities in the metallic host --- a magnetic impurity being one which has

a local magnetic moment due to the spin of unpaired electrons in its atomic-like d or f shell. A carefully

studied example showing the correlation between the resistance minima and the number of magnetic

impurities is that of iron impurities in gold (van den Berg, 1964). In his book entitled Thermoelectricity,

An Introduction to the Principles (first published in 1961 from John & Wiley), MacDonald (D.K.C.)

showed various kinds of experimental results of electrical resistivity and thermoelectric power for noble

metals (Au, Cu) diluted with magnetic impurities such as Mn and Fe. The electrical resistivity of Cu

diluted with Mn or Fe clearly shows a local minimum at low temperatures. The temperature dependence

of the thermoelectric power at low T is extremely sensitive to the amount of the magnetic impurities; Au

diluted with Mn.

(a) Electrical resistivity

5

Fig. Electrical resistivity of Au with Mn as solute. The nominal atomic concentration of Mn is

indicated on each curve. (D.K.C. MacDonald, Thermoelectricity, Dover, 2006).

6

Fig. Electrical resistivity of Cu with Mn as solute. The nominal atomic concentration of Mn is

indicated on each curve. (D.K.C. MacDonald, Thermoelectricity, Dover, 2006).

7

Fig. Electrical resistivity of Cu with Fe as solute. The nominal atomic concentration of Fe is indicated

on each curve. (D.K.C. MacDonald, Thermoelectricity, Dover, 2006).

(b) Thermoelectric power

8

Fig. Thermoelectric power at low T of Au alloys with a range of transition metals as solutes. Each

alloy has a concentration of 0.2 nominal atomic percent as solute of the transition metal indicated

on the curve. (D.K.C. MacDonald, Thermoelectricity, Dover, 2006)

9

Fig. Thermoelectric power at low T of Au alloys with Mn as solutes. The nominal atomic

concentration of Mn is indicated on each curve. The behavior of the most dilute alloys below 1 K

is particularly striking. (D.K.C. MacDonald, Thermoelectricity, Dover, 2006).

10

Fig. Variation of the Kondo temperature for the 3d transition metals in Cu host.

11

4. Spin glass region for canonical systems

Fig. Various concentration regimes for a canonical spin glass (such as host Cu diluted with Mn)

illustrating the different types of magnetic behavior which occur. (J.A. Mydosh, Spin Glass,

Taylor & Francis, London, 1993)

The concentration is one of the most important factors in determining the magnetic state of the alloys.

We show a schematic diagram for various concentration regime division. At the very dilute magnetic

concentration (ppm) there are the isolated impurity-conduction electron coupling leading to the Kondo

effect. This localized interaction (if J

12

Fig. RKKY interaction between localized magnetic impurities. The red circle denotes the sea of

conduction electrons.

13

Fig. Field cooled (a, c) and zero-field (b, d) cooled magnetization for CuMn (1 and 2%) as a function

of temperature The data shows clearly the characteristic features of spin glass behavior. (J.A.

Mydosh, Spin Glasses, Taylor & Francis, 1993).

5. Kondo effect:minimum in resistivity vs temperature

In the 1930’s it was found that the electrical resistivity of dilute magnetic alloys shows a minimum at

a characteristic temperature. The resistivity decreases as the temperature decreases from the high

temperature side. It shows a minimum value, and in turn increases with further decreasing temperature.

These systems are noble metals such as Au, Ag, and Cu diluted with magnetic impurities such as Mn and

Fe. Such temperature dependence is rather different from that for normal metals obeying the Bloch T5

law at low temperatures. In 1964, Jun Kondo proposed a remarkable theory that explains the resistivity

minimum. The interaction of conduction electrons with localized spins leads to the many body problem

in electrons of metal. This effect is called the Kondo effect. The Kondo effect described the scattering of

conduction electronic in a metal due to magnetic impurities.

The local minimum of the electrical resistivity arises from the competition between the following two

contributions,

5

10

5

)ln(

)ln3

1(

aTTc

aTTzJ

cF

M

Blochspin

The first term is the spin-dependent contribution to the resistivity and the second term is the phonon

contribution (Bloch T5 law). J is the exchange energy, z is the number of nearest neighbors, c is the

concentration, and M is a measure of the strength of the exchange scattering. The resistivity has a minimum at

05 14

caT

dT

d

or

5/1

1min

5

a

cT

The temperature at which the electrical resistivity takes a minimum, varies as one-fifth power of the

concentration of the magnetic impurities, in agreement with experiment at least for Cu diluted with Fe.

14

6. Origin of exchange interaction

The Hamiltonian of the Anderson model can be described by

dddd nUnnccH

,

0

k

kkk

,

)(1

'k

kkcdVdcV

NH dkkd (perturbation)

where

ddnd ( , ).

1, dd , ',', kkkk cc

We now shoe that the s-d interaction can be derived from the Anderson model.

We consider the second-order process of V starting with

15

Fdi

.

where F is the state where the conduction electrons make up the Fermi sphere. Note that there is no d-

electrons in the localized magnetic state. This state is degenerate with

Fdi

.

__________________________________________________________________________________

Process-1

k enters d , and d goes out to 'k

dEE 0 UEEE ddk 0 dkk EE '0

Initial state Intermediate state Final state

kk

k

kk cddcUE

VVV

d

dd

'

'1

____________________________________________________________________________________

Process-2


16



kk

k

kk cddcUE

VVV

d

dd

'

'2

____________________________________________________________________________________

Process-3

d enters 'k , and k goes out to d .

dEE 0 '0 kE dkk EE '0


17

dccd

E

VVV

d

dd

'

'

'3 kk

k

kk

____________________________________________________________________________________

Process-4




dccd

E

VVV

d

dd

'

'

'4 kk

k

kk

We consider the second-order process of V starting with

Fdi

.

___________________________________________________________________________________

Process-5


18



kk

k

kk cddcUE

VVV

d

dd

'

'5

____________________________________________________________________________________

Process-6



19


kk

k

kk cddcUE

VVV

d

dd

'

'6

____________________________________________________________________________________

Process-7




dccd

E

VVV

d

dd

'

'

'7 kk

k

kk

__________________________________________________________________________________

Process-8


20



dccd

E

VVV

d

dd

'

'

'8 kk

k

kk

Adding to the all the processes above, we obtain the effective Hamiltonian,

)(

)(

''''

'

'

''''

'

dccddccddccddccdE

VV

cddccddccddccddcUE

VV

d

dd

d

dd

kkkkkkkk

k

kk

kkkkkkkk

k

kk

Noting that

zdSddn

2

1, zd Sddn

2

1

1

ddd nnndddd

)(2

1)(

2

1

ddz nnddddS

ddddS ,

ddddS

we get

21

SccnccSccncccccc

SccnccSccncc

ddccnccddccncc

cddccddccddccddc

dd

dd

dd

kkkkkkkkkkkk

kkkkkkkk

kkkkkkkk

kkkkkkkk

''''''

''''

''''

''''

)(

)1()1(

)1()1(

for the coefficient of UE

VV

d

dd

k

kk

'

)()(

)1()1(

)()(

''''''',

''''',

''''',

''',''',

''''

''''

SccSccnccnccccccn

SccnccSccnccn

SccnccSccnccn

SccnccSccncc

ddccddccddccddcc

dccddccddccddccd

dddkk

dddkk

dddkk

dkkdkk

kkkkkkkkkkkk

kkkkkkkk

kkkkkkkk

kkkkkkkk

kkkkkkkk

kkkkkkkk

for the coefficient of '

'

k

kk

ddd

E

VV.

The non-perturbed Hamiltonian is

]1

))(11

[(

)](

)([

',

','''

''','

',''

')1(

kk

dk

dkkd

dd

dkk

dk

dd

kd

dd

UE

ccccUEE

VV

ccccnUE

VV

ccccE

VVH

kk

kkkkkk

kkkk

kk

kk

kkkk

kk

where 1dn

The interaction Hamiltonian is

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kk

kkkkkk

kk

kkkkkkkkkk

kkkkkkkk

kk

kk

kkkkkkkk

kk

kkkkkkkk

kk

kk

kkkkkkkk

kk

,'''

','''''

''''

'

',''''

'

''''

'

',''''

')2(

))(11

(2

1

)()(11

[(

)])2

1()

2

1((

))2

1()

2

1(([

)](

)([

ccccUEE

VV

SccccSccSccUEE

VV

SccSccSccSccUE

VV

SccSccSccSccE

VV

SccSccnccnccUE

VV

SccSccnccnccE

VVH

dkkd

dd

z

dkkd

dd

zz

dk

dd

zz

kd

dd

dd

dk

dd

dd

kd

dd

Since the second term of )2(

H has nothing to do with the interactions, the second term is moved to the

Hamiltonaian )1(

H . Then we have

]1

))(11

[(2

1

',

','''

)0(

d

dk

dkkd

dd

nUE

ccccUEE

VVH

kk

kk

kkkkkk

}

)){(11

[(

''

','''

SccScc

SccccUEE

VVH zdkkd

ddex

kkkk

kk

kkkkkk

The s-d exchange interaction exH can be expressed by

',

''''}){(

2 kkkkkkkkkk

SccSccSccccN

JH zex

We use the approximation

VVV dd kk '

23

)11

(

)11

(

)11

(2

2

2

2

UV

UEEV

UEEV

N

J

dd

FdFkFkFd

dkkd

where 0 Fdd E , and Fk .Since 0 dU , J is negative, which is indicative of

antiferromagnetic interaction.

Fig. Fdd EE (

24

z

zyxyx

z

y

x

z

zzzz

y

yyyy

x

xxxx

SccccSccScc

ScccciSScciSScc

Scccc

Sicccic

Scccc

Scccccccc

Scccccccc

Scccccccccc

)(

)()()(

)(

)(

)(

)(

)(

)(

''''

22'11'12'21'

22'11'

12'21'

12'21'

2222'1212'2121'1111'

2222'1212'2121'1111'

2222'1212'2121'1111'

,

'

kkkkkkkk

kkkkkkkk

kkkk

kkkk

kkkk

kkkkkkkk

kkkkkkkk

kkkkkkkkkkSσ

where the Pauli matrices are given by

01

10x ,

0

0

i

iy ,

10

01z

In conclusion, the Anderson Hamiltonain is equivalent to the s-d exchange interaction when the mixing

term V is small and the impurity d-band is occupied by one electron. In this limit, the Anderson

Hamiltonian leads to the Kondo effect and gives rise to the singlet ground state.,

8. Calculation of electrical resistivity due to the s-d interaction

The electrical conductivity is represented by the mean free time k and the Fermi velocity of electron vk as

k

kkkk

fvd

e)(

3

2 2

where and )(k are the volume of the system and the density of states for the conduction electrons,

respectively. In general, the scattering probability )(/1 is given by the T-matrix as

f

ifiTf )()(2

)(

1 2

ℏ

,

where i and f are the initial and final states, respectively. The T-matrix is given by

dsdsds HHE

HHT

0

1

25

where is an infinitesimal positive number, and Hsd is the scattering potential.

The first-order term of the T-matrix for the scattering of conduction electron; ,', kk

MSMN

J

MMN

J

iHfT

z

ds

2

,,2

),',()1(

Sσ

kk

The first -order term accompanying the spin flip for the scattering; ,', kk

»kÆ> »k'Æ>

»S z=M> »Sz=M>

»kÆ> »k'∞>

»S z=M> »Sz=M+1>

26

)1)((2

,1,2

12

),',()1(

MSMSN

J

MMN

J

MSMN

JT

Sσ

kk

((Note))

(i)

MSMMM

MSMMM

MSSSMMM

z

zz

zzzz

1)1,"()1,"(

),"()1,"(

,)(2

1",",","

Sσ

or

MSMMM z ,, Sσ

MSMMM 1,1, Sσ

(ii)

MSMMM

MSMMM

MSSM

MSSSMMM

z

zz

zz

zzzz

1)1,"()1,"(

),"()1,"(

,2

1","

,)(2

1",",","

Sσ

MSMMM z ,, Sσ

MSMMM 1,1, Sσ

_____________________________________________________________

27

z , z ,

02

1 ,

2

1

2

1, 0

2

1

_______________________________________________________________________

Then the total T-matrix up to the first-order is given by

)(2

)1(Sσ

N

JT .

We now consider the four second-order terms shown by (a) – (d).

(a) The process: '" kkk (without spin flip)

The T-matrix is given by

"

2

"

"2 1)2

(k k

k MSMi

f

N

Jz

(b)

»kÆ> »k''Æ> »k'Æ>

»Sz=M> »Sz=M> »S z=M>

28

Fig. The arrow directed to the left denotes holes, while the arrow directed to the right denotes

electrons.

"2

"

"2)2

(k k

k MSMi

f

N

Jz

where we have fixed the energy of the electrons in the initial state and final states at

i 'kk

(c)

»kÆ> »k'Æ>

»k ''Æ>

»Sz=M> »Sz=M>»Sz=M>

29

" "

"2 )1()2

(k k

k MSSMi

f

N

J

(d)

])2

(" "

"2 k kk MSSM

i

f

N

J

Combining the four second-order processes (a) – (d), we have

»kÆ> »k''∞> »k'Æ>

»Sz=M> »S z=M+1> »S z=M>

»kÆ> »k'Æ>

»k''∞>

»S z=M> »S z=M>»S z=M-1>

30

" "

"

"

2

"

"

" "

"2

"

2)2(

}21

)1(1

{)2

(

}

11{)

2()'(

k k

k

k

k

k

k k

k

k

kk

MSMi

fSS

iN

J

MSSMi

f

MSSMi

fMSM

iN

JT

z

z

Here we note that

zz SSSSSS 2

)1(∓

In a similar way to 'kk , the second-order term of the T-matrix for the scattering for the process

'kk is obtained as

" "

"2)2( 121

)2

()'(k k

kkk MSM

i

f

N

JT

Finally, the total T-matrix up to the second order terms is given by

" "

"

"

2)2( }21

)()1(1

{)2

(k k

k

k

Sσ i

fSS

iN

JT

and

}21

21){(

2)1(

1)

2(

" "

"

" "

2)2()1(

k k

k

k k

Sσ i

f

N

J

N

JSS

iN

JT

)](21)[(2'

)'(21)'('

2

)'()]'(21)['('2'

)'(21)'('

2

)]'('

1)['(21)'('

2

'

)'(21)'('

2

21

2

22

22

2

2

" "

"

2

fN

Ji

fd

N

J

fdN

Ji

fd

N

J

ifdN

J

i

fd

N

J

i

f

N

J

k k

k

31

Fig. The region in momentum space relevant to the formulation of the Kondo effect. At low

temperatures where we are interested, we may linearize the spectrum around F and impose a cut off D.

We assume the following density of states for the conduction electrons,

)( for DF , 0)( for DF ,

The first term:

F

F

D

D

fd

N

Jfd

N

J

'

)'(21'

2'

)'(21)'('

2

2

0

2

We now calculate the integral,

EF

2D

k

32

)2(

)2(

ln2

lnln

')'

)'(('ln2lnln

'

)'(21'

ID

IDD

df

DD

fdI

rr

D

D

FF

D

D

F

F

F

F

where

Fr .

Since

1

1)'(

)'( Fe

f ,

]2

)'([sec

]1[)'(

'

2

2)'(

)'(

Fhe

ef

F

F

we get

F

F

F

F

D

D

F

D

D

dhdf

']2

)'([sec'ln')

'

)'(('ln 2

Here we put

xF ' ,

Then the second term of I is

dxx

hxdxx

hxI r

D

D

r )2

(secln2)2

(secln2 22)2(

(i) TkBr

33

rr dxx

hI

ln8)2

(secln2 2)2(

and

DDDI

ln8ln6ln8ln2

(ii) TkBr

)2

ln(8)4

ln(8)2ln(8

])(secln8)2

ln(8

)(sec]ln4)(sec)2

ln(4

)(sec]ln)2

[ln(4

)2

(secln2

0

2

22

2

2)2(

De

TkeTk

yhydy

yhydyyhdy

yhydy

dxx

hxI

BB

)2

ln(8ln6De

TkDI B

((Note-1))

81878.0)4

ln()4

ln()(secln0

2

eyhydy

with

= 0.577216

((Note-2))

2)(sec 2

dxxh

34

Thus, the T-matrix is given by

}],[

ln1){(2

)2()1(

D

TkMax

N

J

N

JT B

Sσ

Thus we have the expression for the resistivity as

)ln2

1(~~

D

Tk

N

J BB

.

where B~ is the resistivity coming from the first-order Born approximation.

9. Spin singlet for the ground state

We assume that the trial wave function is given by

Fk

a Fccak

kkk)(

where and create up- and down-spin states of the local spin, F is the filled Fermi sea ground

state, and the ak are coefficients that we will have to determine. So this state is a spin singlet. It is

antisymmetric combination of the local up-spin plus a delocalized down-spin above the Fermi surface

The Hamiltonian:

',

''''1}){(

kk

kkkkkkkkSccSccSccccJH z

where

N

JJ

2 .

The eigenvalue equation we wish to solve is

0)( 010 aaHH .

35

Here 0 is the ground-state energy of the unperturbed Hamiltonian,

,

,,0

k

kkccH k

and a is the shift in energy due to the perturbation 1H . We need to determine ak such that this shift is

as negative as possible. According to Taylor, we have

F

F

k

ak

k

aaa

Fcca

FccaHH

k

kkk

k

kkk

))((

)()()( 0000

and

FFF kk

a

k

a FccaaJ

aJ

H'

'''1)(

2

3

2

3

k

kkk

k

k

k

k

The eigenvalue equation for a then becomes

02

3)(

'

' Fk

a aJ

ak

kkk

Suppose that

Fk

ak

k .

Then we get

a

Ja

k

k

2

3.

Then we have

Fk a

JV

k k

1

3

2,

36

where V

37

the coherent state, split from the continuum by an energy gap, displaying the essential

singularity in the coupling constant.

We change from a sum over k to an integration in which the energy e measured relative to the Fermi

energy run from zero to a value D related to the bandwidth, and approximate the density of states by its

value )0( .

a

a

D

ar

r

D

JdJ

ln2

3

2

31

0

.

Noting that J

38

Note that a is related to the Kondo temperature as

KBa Tk

The susceptibility at T = 0 K is given as

KB

B

Tk

g

4

22

We notice an analogy with the BCS theory of superconductivity, in which we find a similar

expression for the critical temperature Tc. This is not an accident. Both TK and Tc define temperatures

below which perturbation theory fails. In the BCS case, Tc signals the onset of the formation of bound

Cooper pairs and a new ground state with an energy gap. The Kondo effect is a little more subtle. TK

defines a temperature at which the energy contributions from the second-order perturbation theory

becomes important. This happens when the local spin on a single impurity starts to become frozen out at

an energy set by the Kondo coupling J and the density of states at the Fermi energy.

10. Phenomelogical interpretation

We assume that the RKKY is negligibly small. The only remaining interaction is between the

impurity spin S and the conduction electron spin s. It may be treated by the so-called s-d exchange

interaction,

sS JH

where J (

39

(1) A loss of magnetic moment, the magnetization falls below its free moment value and the

susceptibility is less than its Curie value

Tk

N

TTk

N

B

C

KB

K3)(3

22

where

)1( SSg B

(b) The enhanced scattering rate creates a logarithmic upturn in the resistivity at low temperatures.

11. Physical properties of typical Kondo systems

It is now know that many of the properties associated with the Kondo effect can be well described by

a scaling function of T/TK.

40

Fig. Schematic behavior characteristic of a typical Kondo alloy

(a) Resistivity

The resistivity decreases as 1-AT2 with increasing T and varies with lnT above TK. The resistivity at 0

K is called the unitary limit.

(b) Reciprocal Susceptibility

The reciprocal susceptibility shows the Pauli paramagnetic behavior around T = 0 K as 1+BT2. Above

TK, it shifts to the Curie-Weiss like behavior.

(c) Specific heat

The specific heat shows a peak below TK, and is proportional as T-2 above TK.

41

(d) The Wilson ratio

The specific heat is proportional to T, while the susceptibility becomes constant at T = 0. The Wilson

ratio. The Wilson ratio is given by

1521.02

22

B

B

imp

imp

k

g

C

T .

12. Conclusion

When magnetic impurities such as Mn, Fe and Co are dilutely inserted into nonmagnetic metals such as Cu,

Ag and Au, susceptibility obeying the Curie–Weiss law is observed. In this case impurity atoms are considered to

possess magnetic moments. On the other hand, Mn in Al does not show the Curie–Weiss law and seems not to

possess any magnetic moment. Anderson tried to explain these phenomena using the Anderson model. From the

above discussions, these phenomena can be interpreted from the concept of the scaling relation. The Kondo

temperature TK is very low for Mn in Cu (TK ≈ 0.1 K), and is very high for Mn in Al (TK is above room

temperature).

Fig. The resistivity for V (1-2 %) in Au as a function of temperature. [K. Kume (J. Phys. Soc. Jpn. 23,

1226 (1967))]. KTK 300 for this system. This is obtained subtracting the resistivity of pure Au

from the observed resistivity.

____________________________________________________________________________________

REFERENCES

42

K. Yamada, Electron correlation in metals (Cambridge University Press, 1993, Cambridge).

J. Kondo, Solid State Pbhysics 23, 183 (1968) *Academic Press, New York).

J.Kondo, Prog. Theor. Phys. 32, 37 (1964).

D.K.C. MacDonald, Thermoelectricity (Dover, 2006).

K. Yosida, Theory of Magnetism (Springer-Verlag, 1996)

J. Kondo, The physics of dilute magnetic alloys (Cambridge University Press, 2012).

A.C. Hewson, The Kondo Problem toHeavy Fermions (Cambridge, 1993).

K.H. Fisher and J.A. Hertz, Spin Glasses (Cambridge, 1991).

P.L. Taylor and O. Heinonen, A Quantum Approach to Condensed Matter Physics (Cambridge, 2002).

G. D. Mahan, Many-Particle Physics, 2-nd edition (Plenum Press, 1993).

K. Kume J. Phys. Soc. Jpn. 23, 1226 (1967).

K. Yosida, What is the Kondo effect? (in Japanese) (Maruzen, 1990).

___________________________________________________________________________________

APPENDIX

A. Brillouin-Wigner series

(see J.M. Ziman, Element of Advanced Quantum Mechanics)

A.1

We start with the Schrodinger equation given by

nnn EH ˆ

with

ˆ H ˆ H 0 ˆ H 1

Then we get

nnn EHH )ˆˆ( 10

or

)())(ˆˆ()0()0(

10 nnn EHH

or

))ˆˆ()ˆˆ( 10)0(

10

)0( HHHHEE nnnn

or

43

nnnnnn HHEEE 10)0()0()0( ˆˆ .

Finally we have

)0()0(

10 )(ˆ)ˆ( nnnnn EEHHE

Projecting on both sides with ̂P

)0()0(

10ˆ)(ˆˆ)ˆ(ˆ nnnnn PEEHPHEP

or

nn HPPHE 10 ˆˆˆ)ˆ(

or

nn HPHE 10 ˆˆ)ˆ(

or

nn HPHE 11

0ˆˆ)ˆ(

Here we use the commutation relation

0]ˆ,ˆ[ 0 PH .

Thus we get the final form

nnnn HPHE 11

0

)0( ˆˆ)ˆ(

We solve this by iteration

44

...ˆˆ)ˆ(ˆˆ)ˆ(ˆˆ)ˆ(ˆˆ)ˆ(

ˆˆ)ˆ(ˆˆ)ˆ(ˆˆ)ˆ(

ˆˆ)ˆ(ˆˆ)ˆ(

ˆˆ)ˆ(

)ˆˆ)ˆ((ˆˆ)ˆ(

)0(

1

1

01

1

01

1

01

1

0

)0(

1

1

01

1

01

1

0

)0(

1

1

01

1

0

)0(

1

1

0

)0(

1

1

0

)0(

1

1

0

)0(

nnnnn

nnnn

nnn

nnn

nnnnnn

HPHEHPHEHPHEHPHE

HPHEHPHEHPHE

HPHEHPHE

HPHE

HPHEHPHE

A.2. Energy shift

What is the energy shift due to the perturbation? To this end, we start with

nnn HHE 10 ˆ)ˆ(

Projecting on both sides with ˆ M

nnn HMHEM 10 ˆˆ)ˆ(ˆ

or

nnn HMMHE 10 ˆˆˆ)ˆ(

or

nnn HMHE 1)0(

0ˆˆ)ˆ( ,

where we use the commutation relation,

0]ˆ,ˆ[ 0 HM .

Multiplying on both sides with )0(

n

)0(

1

)0()0(

0

)0( ˆˆ)ˆ( nnnnn HMHE

Then the energy shift is obtained as

45

)0(

1

)0()0(

1

)0()0( ˆˆˆnnnnnn HHMEE

Through the iteration, we have

)0(

1

1

01

1

01

1

01

)0(

)0(

1

1

01

1

01

)0(

)0(

1

1

01

)0()0(

1

)0()0(

ˆˆ)ˆ(ˆˆ)ˆ(ˆˆ)ˆ(ˆ

ˆˆ)ˆ(ˆˆ)ˆ(ˆ

ˆˆ)ˆ(ˆˆ

nnnnn

nnnn

nnnnnnn

HPHEHPHEHPHEH

HPHEHPHEH

HPHEHHEE

(i) The first-order energy shift:

)0(

1

)0()1( ˆnnn HE .

The second-order energy shift:

nk kn

kn

nk

nkknn

nnnn

EE

H

HPHEH

HPHEHE

)0(

2)0(

1

)0(

)0(

1

)0()0(1

01

)0(

)0(

1

1

01

)0()2(

ˆ

ˆˆ)ˆ(ˆ

ˆˆ)ˆ(ˆ

When En En(0)

, we get a “Rayleigh-Schrödinger series of conventional perturbation theory.”

(ii) The second order of the energy shift:

nk kn

nkkn

nEE

HHE

)0()0(

)0(

1

)0()0(

1

)0(

)2(ˆˆ

where En En(0)

.

(iii)` The third order of the energy shift:

46

nlnk lnkn

nllkkn

nlnk

nllnkknn

nnnnn

EEEE

HHH

HPHEHPHEH

HPHEHPHEHE

.)0()0(

)0(

1

)0()0(

1

)0()0(

1

)0(

.

)0(

1

)0()0(1

01

)0()0(1

01

)0(

)0(

1

1

01

1

01

)0()3(

))((

ˆˆˆ

ˆˆ)ˆ(ˆˆ)ˆ(ˆ

ˆˆ)ˆ(ˆˆ)ˆ(ˆ

or

nlnk lnkn

nllkkn

nEEEE

HHHE

.)0()0()0()0(

)0(

1

)0()0(

1

)0()0(

1

)0(

)3(

))((

ˆˆˆ

where En En(0)

.

(iv) The fourth-order of the energy shift

nmnlnk mnlnkn

nmmllkkn

nEEEEEE

HHHHE

.)0()0()0()0()0()0(

)0(

1

)0()0(

1

)0()0(

1

)0()0(

1

)0(

)4(

))()((

ˆˆˆˆ

where En En(0)

.

A.3 The form of

Next we discuss the from of .

(i) The first order of wave function:

nk kn

nkk

nk

nkkn

nnn

EE

H

HPHE

HPHE

)(

ˆ

ˆˆ)ˆ(

ˆˆ)ˆ(

)0(

)0(

1

)0()0(

)0(

1

)0()0(1

0

)0(

1

1

0

)1(

or

nk kn

nkk

nEE

H

)(

ˆ

)0()0(

)0(

1

)0()0(

)1(

47

where En En(0)

.

(ii) The second order of the wave function:

nlnk lnkn

nllkk

nlnk

nllnkkn

nnnn

EEEE

HH

HPHEHPHE

HPHEHPHE

,)0()0()0()0(

)0(

1

)0()0(

1

)0()0(

,

)0(

1

)0()0(1

01

)0()0(1

0

)0(

1

1

01

1

0

)2(

))((

ˆˆ

ˆˆ)ˆ(ˆˆ)ˆ(

ˆˆ)ˆ(ˆˆ)ˆ(

or

nlnk lnkn

nllkk

nEEEE

HH

,)0()0()0()0(

)0(

1

)0()0(

1

)0()0(

)2(

))((

ˆˆ

where En En(0)

.

(iii) The third order of the wave function:

nmnlnk mnlnkn

nmmllkk

nEEEEEE

HHH

,)0()0()0()0()0()0(

)0(

1

)0()0(

1

)0()0(

1

)0()0(

)3(

))()((

ˆˆˆ .

(iv) The fourth order of the wave function:

nmnlnk pnmnlnkn

nppmmllkk

nEEEEEEEE

HHHH

,)0()0()0()0()0()0()0()0(

)0(

1

)0()0(

1

)0()0(

1

)0()0(

1

)0()0(

)4(

))()()((

ˆˆˆˆ .

7 Lipmann-Schwinger equation

The Hamiltonian H is given by

VHH ˆˆˆ 0

48

where H0 is the Hamiltonian of free particle. Let be the eigenket of H0with the energy

eigenvalue E,

EH 0ˆ

The basic Schrödinger equation is

EVH )ˆˆ( 0 (1)

Both 0Ĥ and VHˆˆ

0 exhibit continuous energy spectra. We look for a solution to Eq.(1) such that

as 0V , , where is the solution to the free particle Schrödinger equation with the

same energy eigenvalue E.

)ˆ(ˆ 0HEV

Since )ˆ( 0HE = 0, this can be rewritten as

)ˆ()ˆ(ˆ 00 HEHEV

which leads to

VHE ˆ))(ˆ( 0

or

VHE ˆ)ˆ( 10 .

The presence of is reasonable because must reduce to as V̂ vanishes.

B.1 Lipmann-Schwinger equation:

)(1

0

)( ˆ)ˆ( ViHEkk

by making Ek (= )2/22 mkℏ slightly complex number (>0, ≈0). This can be rewritten as

49

)(10)( ˆ'')ˆ(' ViHEdr k rrrkrr

where

rkkr ie2/3

)2(

1

,

and

''ˆ '0 kk kEH ,

with

22

' '2

km

Ekℏ

.

The Green's function is defined by

'1

0

)(

0'4

1')ˆ(

2)',(

2

rr

rrrrrr

ikk eiHE

mG

ℏ

((Proof))

'""')'2

('"'2

')ˆ(2

122

1

0

2

2

rkkkkkrkk

rr

im

Eddm

iHEm

I

k

k

ℏℏ

ℏ

or

50

im

E

ed

m

im

Edm

im

Eddm

I

k

i

k

k

22

)'('

3

122

122

'2

)2(

'

2

'')'2

(''2

'")"'()'2

('"'2

2

2

2

k

k

rkkkrk

rkkkkkrkk

rrk

ℏ

ℏ

ℏℏ

ℏℏ

where

22

2k

mEk

ℏ .

Then we have

)'()(')2(

'

)'(2

)2(

'

2

)(

022

)'('

3

222

)'('

3

2

rrk

kk

k

rrk

rrk

Gikk

ed

im

ed

mI

i

i

ℏ

ℏ

In summary, we get

)()(02

)( ˆ')',('2

VGdm

rrrrkrrℏ

or

)()(02

)( ')'()',('2

rrrrrkrr VGdm

ℏ.

More conveniently the Lipmann-Schwinger equation can be rewritten as

)(1

0

)( ˆ)ˆ( ViHEkk

with

51

1

0

2)(

0 )ˆ(

2ˆ iHE

mG k

ℏ,

and

ViHEVGm

kˆ)ˆ(ˆˆ

2 10

)(

02

ℏ

When two operators Â and B̂ are not commutable, we have very useful formula as follows,

AAB

BBAB

ABA ˆ

1)ˆˆ(

ˆ

1

ˆ

1)ˆˆ(

ˆ

1

ˆ

1

ˆ

1 ,

We assume that

)ˆ(ˆ 0 iHEA k , )ˆ(ˆ iHEB k

VHHBA ˆˆˆˆˆ 0

Then

1

0

111

0 )ˆ(ˆ)ˆ()ˆ()ˆ( iHEViHEiHEiHE kkkk ,

or

11

0

1

0

1 )ˆ(ˆ)ˆ()ˆ()ˆ( iHEViHEiHEiHE kkkk .

For simplicity, we newly define the two operators by

1

00 )ˆ()(ˆ iHEiEG kk

1)ˆ()(ˆ iHEiEG kk

where )(ˆ0 iEG k denotes an outgoing spherical wave and )(ˆ0 iEG k denotes an incoming

spherical wave. We note that

52

)(ˆ2

)ˆ(2

ˆ0

21

0

2)(

0 iEGm

iHEm

G kk ℏℏ

.

Then we have

)](ˆˆ1)[(ˆ

)(ˆˆ)(ˆ)(ˆ)(ˆ

0

00

iEGViEG

iEGViEGiEGiEG

kk

kkkk

)](ˆˆ1)[(ˆ

)(ˆˆ)(ˆ)(ˆ)(ˆ

0

00

iEGViEG

iEGViEGiEGiEG

kk

kkkk

Then )( can be rewritten as

k

kk

k

k

k

]ˆ)(ˆ1[

ˆ)(ˆ

ˆ)](ˆ(ˆ)(ˆ

ˆ)](ˆˆ1)[(ˆ

ˆ)(ˆ

)(

0

)(

)(

0

)(

0

)(

ViEG

ViEG

ViEGViEG

ViEGViEG

ViEG

k

k

kk

kk

k

or

kk ViHEk

ˆˆ

1)(

.

B.2 The higher order Born Approximation

From the iteration, )( can be expressed as

...ˆ)(ˆˆ)(ˆˆ)(ˆ

)ˆ)(ˆ(ˆ)(ˆ

ˆ)(ˆ

000

)(

00

)(

0

)(

kkk

kk

k

ViEGViEGViEG

ViEGViEG

ViEG

kkk

kk

k

The Lippmann-Schwinger equation is given by

53

kkk TiEGViEG kkˆ)(ˆˆ)(ˆ 0

)(

0

)( ,

where the transition operator T̂ is defined as

kTV ˆˆ)(

or

kkk TiEGVVVT kˆ)(ˆˆˆˆˆ 0

)(

This is supposed to hold for any k taken to be any plane-wave state.

TiEGVVT kˆ)(ˆˆˆˆ 0 .

The scattering amplitude ),'( kkf can now be written as

kkkkk Tm

Vm

f ˆ')2(4

12ˆ')2(4

12),'( 3

2

)(3

2

ℏℏ .

Using the iteration, we have

...ˆ)(ˆˆ)(ˆˆˆ)(ˆˆˆˆ)(ˆˆˆˆ 0000 ViEGViEGVViEGVVTiEGVVT kkkk

Correspondingly we can expand ),'( kkf as follows:

.......),'(),'(),'(),'( )3()2()1( kkkkkkkk ffff

with

kkkk Vm

f ˆ')2(4

12),'( 3

2

)1( ℏ

,

kkkk ViEGVm

fk

ˆ)(ˆˆ')2(4

12),'(

0

3

2

)2(

ℏ

,

kkkk ViEGViEGVm

fkk

ˆ)(ˆˆ)(ˆˆ')2(4

12),'(

00

3

2

)3(

ℏ

.

54

Fig. Feynman diagram. First order, 2nd order, and 3rd order Born approximations. kk is

the initial state of the incoming particle and '' kk is the final state of the incoming

particle. V̂ is the interaction.

fk

V

fk '

fk

V

V

G0

fk '

fk

V

V

G0

G0

V

fk '

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