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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).
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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
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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.
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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
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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).
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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).
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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
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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)
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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).
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Fig. Variation of the Kondo temperature for the 3d transition metals in Cu host.
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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
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12
Fig. RKKY interaction between localized magnetic impurities. The red circle denotes the sea of
conduction electrons.
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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.
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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
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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
k enters d , and d goes out to 'k
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16
dEE 0 UEEE ddk 0 dkk EE '0
Initial state Intermediate state Final state
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
Initial state Intermediate state Final state
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17
dccd
E
VVV
d
dd
'
'
'3 kk
k
kk
____________________________________________________________________________________
Process-4
d enters 'k , and k goes out to d .
dEE 0 '0 kE dkk EE '0
Initial state Intermediate state Final state
dccd
E
VVV
d
dd
'
'
'4 kk
k
kk
We consider the second-order process of V starting with
Fdi
.
___________________________________________________________________________________
Process-5
k enters d , and d goes out to 'k
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18
dEE 0 UEEE ddk 0 dkk EE '0
Initial state Intermediate state Final state
kk
k
kk cddcUE
VVV
d
dd
'
'5
____________________________________________________________________________________
Process-6
k enters d , and d goes out to 'k
dEE 0 UEEE ddk 0 dkk EE '0
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19
Initial state Intermediate state Final state
kk
k
kk cddcUE
VVV
d
dd
'
'6
____________________________________________________________________________________
Process-7
d enters 'k , and k goes out to d .
dEE 0 '0 kE dkk EE '0
Initial state Intermediate state Final state
dccd
E
VVV
d
dd
'
'
'7 kk
k
kk
__________________________________________________________________________________
Process-8
d enters 'k , and k goes out to d .
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20
dEE 0 '0 kE dkk EE '0
Initial state Intermediate state Final state
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
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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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22
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 '
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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 (
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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
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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>
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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
_____________________________________________________________
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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>
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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>
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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>
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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
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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
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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
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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
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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 .
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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  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 '