The resistivity of a host metal, such as Cu with trace amounts of magnetic impurities, typically Fe, reaches a minimum and then increases as -ln T as the temperature subsequently decreases. A resistivity minimum and subsequent logarithmic-temperature dependence are in stark contrast to the resistivity of the pure metal that tends to zero monotonically as the temperature decreases. An additional surprise is that the -ln T dependence of the resistivity does not continue indefinitely to low temperature, but rather, below a characteristic temperature, the Kondo temperature, it phases out. At this temperature the impurity and conduction electron spins begin to condense into singlet states and this condensation is complete at T=0 K. Perturbation theory breaks down at this temperature and then the magnetic properties of the system change. Here we have explained how resistivity minimum and logarithmic-temperature dependence arises if we take the interaction between the spin of conduction electrons of host metal and the spin of impurity as a perturbation.

1. Topic: Explanation of Spin Kondo Effect by Perturbative Treatment
Presented by
Name: Subhajit Pramanick
Roll No. : 22PH91R16
Department of Physics
IIT Kharagpur

2. Jun Kondo was a Japanese theoretical physicist. His
research is noted for the Kondo effect.

3. Contents
1. What is a Metal?
2. Introduction to Kondo Problem
3. Interaction between conduction electron and impurity
4. Plan of our calculation
5. Explanation by perturbation
A. First order perturbation
B. Second order perturbation
6. Calculation of Final resistivity
7. Introduction to Kondo Temperature

4. What is a Metal ?
 Many electron system with a Fermi surface.
 Specific heat changes with temperature (T) linearly, i.e., varies
with T.
 At high temperatures, resistivity goes linearly with temperature
(T). As temperature is lowered, resistivity goes first as T5
due to
“electron-phonon” interaction and then goes as T2
as
temperature is further lowered due to “electron-electron”
interactions.
Temperature dependence on
resistivity for metals

5. Introduction to Kondo Problem
 A magnetic impurity is an impurity such as Fe, Ni etc. in a
host metal that has a magnetic moment.
 Magnetic impurity can interact with the conduction
electrons of the host metal.
 Resistivity of a host metal, such as Cu with trace amounts of
magnetic impurities, typically Fe, reaches a minimum and
then increases as – ln T as the temperature subsequently
decreases ……KONDO EFFECT
 This resistivity minimum and subsequent logarithmic-
temperature dependence are in stark contrast to the
resistivity of a pure metal.

6. Interaction between conduction electron and impurity
 The Kondo Hamiltonian can be written as the sum of two terms:
HKondo = k,σ εk akσ
†
akσ −
J0
2ћV k,k΄,σ Sd
z
ak↑
†
ak΄↑ − ak↓
†
ak΄↓ + Sd
+
ak↓
†
ak΄↑ + Sd
−
ak↑
†
ak΄↓
Unperturbed part (H0)
(Tight binding Hamiltonian)
Interaction between spin of conduction
electron and spin of impurity (Hint)
 Here we are interested only in the coupling to a single impurity. That’s why we write the interaction
here in terms of the single impurity spin operator, Sd.

7. Plan of Our Calculation
 From the Boltzmann transport equation, we know the conductivity (σ) is proportional to the time
between scattering events (τ).
σ ∝ τ
 Now the total scattering rate (Γ) is inversely proportional to the time between scattering events (τ).
Γ ∝
1
τ
 Since the resistivity is inversely proportional to the conductivity, then from the above statements we
can write:
ρ ∝ Γ
 Therefore, for finding resistivity we shall calculate total scattering rate.

8. Explanation by Perturbation
 Here we take the interaction between the spin of conduction electron and the spin of impurity i.e.,
Hint as a perturbation.
 So, the scattering amplitude (T-matrix) can be written as:
Ta→b = Ta→b
(1)
+ Ta→b
(2)
+ ⋯
= b Hint
a + c≠a
b Hint c c Hint a
Ea − Ec
+ ⋯
First
Order
Second Order

9. First Order Perturbation
 Total scattering rate is the sum of the rates of two incoherent processes:
 Non Spin-Flip Process:
 In this case the initial conduction electron, assumed to be
spin up, does not flip its spin and remains with spin up.
 Bare matrix element for this case is:
Tk↑ → k´↑; ms
(1)
= −
J0
2ħV
s, ms Sd
z
s, ms
= −
J0
2ħV
ħms = −
J0
2V
ms
 So, the lowest order rate of scattering is:
Γk →k΄
1
ms =
2п
ћ k΄ δ εk − εk΄ −
J0
2V
ms
2

10. First Order Perturbation
 Here, the above sum is over a limited group of final states k´. To compute the total rate of this
process, we sum k´ over all final states, which gives a factor N 0 V for k close to the Fermi
surface, where N 0 =
m kF
2п2ħ3 =
3𝑛𝑒
4εF
is the single-spin electron density of states at the Fermi
Surface.
 So, the total scattering rate from a single impurity with no spin-flip is:
ΓNon−Flip
1
ms =
п
2ħV
N(0) J0
2
ms
2 ……….(1)

11. First Order Perturbation
 Spin-Flip Process:
 In this case, the conduction electron, assumed to be spin up,
flips from spin up to spin down, whereas the impurity changes
from mS to (mS + 1).
 So, the bare matrix element is:
Tk↑ → k´↓; ms
(1)
= −
J0
2ħV
s, ms + 1 Sd
+
s, ms
= −
J0
2ħV
. ħ s s + 1 − ms(ms + 1)
[Since, s, ms ± 1 S±
s, ms = ħ s s + 1 − ms(ms ± 1)]
 So, the total scattering rate from a single impurity with spin-flip is:
ΓFlip
1
ms =
п
2ħV
N 0 J0
2
(s s + 1 − ms ms + 1 ) …………(2)

12. First Order Perturbation
 Therefore, the total rate of scattering of above two cases will be:
Γ 1 ms = ΓNon−Flip
1
ms + ΓFlip
1
ms
=
п
2ħV
N(0) J0
2
ms
2
+
п
2ħV
N 0 J0
2
(s s + 1 − ms ms + 1 )
=
п
2ħV
N(0) J0
2
(s s + 1 − ms) …………(3)
 If we average over all initial spin orientations of the impurity, the ms term vanishes. Consider nimp
is the density of impurities. Summing over all impurities, we will get the spin-averaged scattering
rate:
Γ 1 =
п
2ħ
N 0 J0
2
s s + 1 nimp ……..INDEPENDENT OF TEMPERATURE
 Therefore we have to go for second order perturbation.

13. Second Order Perturbation
 In general the second order scattering amplitude (T matrix) is given by:
Ta→b
(2)
= c≠a
b Hint c c Hint a
Ea − Ec
where, |c is the intermediate state.
 Here also the scattering rate is the sum of the rates of two processes: (1) Non spin-Flip process
and (2) Spin-Flip process. But now, since we are dealing with second order perturbation,
additionally it involves an intermediate state.

14. Second Order Perturbation
 Non Spin-Flip Process:
 In this case, the initial and final
electron state have the same spin.
 Here the straight lines depict the
spin of the impurity and the curved
lines the electron. The line with a
backward arrow in the
intermediate state represents a
hole, i.e., a particle that is present
in the initial and final state but is
absent in the intermediate state.

15. Second Order Perturbation
 Spin-Flip Process:
 In this case, the initial and final
electron state have the opposite
spin.
 Here the straight lines depict the
spin of the impurity and the
curved lines the electron. The line
with a backward arrow in the
intermediate state represents a
hole, i.e., a particle that is present
in the initial and final state but is
absent in the intermediate state.

16. Second Order Perturbation
 Consider non spin-flip case.
 In the upper-left figure, the electron can scatter from momentum k to an unoccupied state with
momentum q and subsequently to the final state with momentum k΄. The contribution of this
process to TNon−Flip
(2)
(k → k΄ ; ms) is = (−
J0
2 ħV
)2
(msħ)2
q
1 − fq
εk +iη − εq
=
(−
J0ms
2
)2 1
V
α … … (4) where, α =
1
V q
1 − fq
εk + iη − εq
. Here
the factor 1 − 𝑓𝑞 guarantees that the state q is empty. 𝑓𝑞 is the electron Fermi distribution
function. The positive infinitesimal imaginary number iη guarantees that the scattering produces
outgoing, rather than incoming waves.

17. Second Order Perturbation
 In upper-right figure (quantum mechanically coherent with upper-left figure) an electron in an
occupied state q scatters to the final state k΄ and another electron with momentum k scatters into the
vacated state q, giving a contribution =
− (−
J0
2 ħV
)2 (msħ)2
q
fq
εq +iη − εk΄
= (−
J0ms
2
)2 1
V
γ …(5), [γ = −
1
V q
fq
εq +iη − εk΄
].
In this latter process, the intermediate momentum state must be occupied; the rate is thus
proportional to fq. The minus sign arises from the interchange of the operations of removing the initial
electron and creating the final electron, akak΄
†
= −ak΄
†
ak for k ≠ k΄.
 In an energy conserving scattering (εk = εk΄) process, the total contribution of above two processes is:
TNon−Flip
(2)
k ↑ → k΄ ↑ ; ms = (−
J0
2
)2 1
V
𝑚𝑠
2
(α + γ) = (−
J0ms
2 V
)2
q
1
εk − εq
……INDEPENDENT OF TEMPERATURE

18. Second Order Perturbation
 Lower two processes involve a single electron spin-flip in the intermediate state. The
contribution of the lower-left figure to TNon−Flip
(2)
(k → k΄ ; ms) is
= (−
J0
2 ħV
)2
s, ms + 1 Sd
+
s, ms
2
q
1 − fq
εk +iη − εq
= (−
J0
2 ħV
)2
(ħ s s + 1 − ms ms + 1 )2
q
1 − fq
εk +iη − εq
= (−
J0
2
)2 1
V
P+− α ……….(6)
where, P+− = s s + 1 − ms ms + 1
 Similarly the contribution of the lower-right figure to TNon−Flip
(2)
(k → k΄ ; ms) is
= −(−
J0
2 ħV
)2
s, ms − 1 Sd
−
s, ms
2
q
fq
εq +iη − εk΄
= (−
J0
2
)2 1
V
P−+ γ ……….(7)
where, P−+ = s s + 1 − ms ms − 1

19. Second Order Perturbation
 So, the total contribution of the lower two figures to TNon−Flip
(2)
(k → k΄ ; ms) is
= (−
J0
2
)2 1
V
(P+− α + P−+ γ) …..Temperature Dependent
Therefore, temperature dependency occurs due to these two non spin-flip processes
consisting a spin-flip in the intermediate states.
 The sum of the contributions of all four terms i.e., total scattering amplitude for non spin-flip
process is:
TNon−Flip
(2)
k ↑ → k΄ ↑ ; ms = (
J0
2
)2 1
V
[𝑚𝑠
2 α + γ + P+− α + P−+ γ]
= (
J0
2
)2 1
V
[𝑚𝑠
2 α + γ + s s + 1 α − ms ms + 1 α + s s + 1 γ − ms ms − 1 γ]
= (
J0
2
)2 1
V
[2msγ + (s s + 1 − ms)(α + γ)] ………….(8)

20. Second Order Perturbation
 Temperature dependent term of the previous equation is:
TNon−Flip
(2)
k ↑ → k΄ ↑ ; ms ≈
2 mS
V
J0
2
2
γ
 Converting the sum over q in γ to an integral and working in the T = 0 limit with the free-particle
dispersion relation, εk =
ћ2k2
2m
, we have:
γ =
m
П2ћ3 0
kF q2 dq
k2−q2−iη
 The real part of the above integral over momentum is:
0
kF q2 dq
k2−q2 = −kF −
k
2
ln
k−kF
k+kF

21. Second Order Perturbation
 For T ≠ 0 but ≪ TF, the range of k for thermally excited electrons is
k2
2m
−
kF
2
2m
< kBT ; we thus
approximate 𝑘 − 𝑘𝐹 in the numerator of the logarithm by
mKBT
KF
and let k = kF everywhere else.
So, we can write:
γ ≈ N 0 (ln
TF
T
+ ⋯ )
 Therefore, the temperature-dependent term of the total scattering amplitude for non spin-flip
process will become:
TNon−Flip
(2)
k ↑ → k΄ ↑ ; ms ≈
J0mS
2V
J0N 0 ln
TF
T
+ ⋯
 Similar calculation can be done for total scattering amplitude starting with spin-flip process which will
also consist the logarithm term.

22. Calculation of Final Resistivity
 Adding in the lowest-order scattering amplitude, we find the total scattering amplitude for non
spin-flip process, to within constant terms, in second order to be:
T k ↑ → k΄ ↑; mS = −
J0ms
2V
(1 − J0N 0 ln
TF
T
+ ⋯ )
 So the corresponding total scattering rate will be of the following form:
Γ = Γ0
(1 − 2J0N 0 ln
TF
T
+ ⋯ )
= Γ0
(1 + 2J0N 0 ln
T
TF
+ ⋯ )
 Therefore, the resistivity of those metal with magnetic impurity consists of a logarithmic
term......KONDO CONTRIBUTION

23. Calculation of Final Resistivity
 So, combining the electron-phonon scattering result with the logarithmic Kondo contribution, we see
that the resistivity has the following form:
ρ T = aT5
− bc ln
T
TF
where, a and b are positive constants. And c is the fractional concentration of impurity.
 Therefore, according as above expression, the resistivity has a minima at a temperature:
Tmin = (
b
5ac
)
1
5
So, Tmin depends on the concentration of magnetic impurities.

24. Introduction to Kondo Temperature
 This − ln T dependence of the resistivity does not continue indefinitely to low temperature, but rather
below a characteristic temperature, the Kondo temperature, TK, it phases out.
 ln T terms diverge at this temperature. The divergence
implies that perturbation theory breaks down at this
temperature. So, Kondo’s solution is valid only for T ≫ TK.
 It is the temperature around which the magnetic
properties of the system change.

25. Introduction to Kondo Temperature
 Above the Kondo temperature, the magnetic susceptibility of the impurity spins obeys the Curie
1
T
law for free magnetic moments. Whereas, below the Kondo temperature, it tends to be a constant,
obeys Pauli law.
So, χ ∝
1
T
when T > TK ….Curie Law
∝ T0 when T < TK ….Pauli law
 Actually, at Kondo temperature, the impurity and
conduction electron spins begin to condense into singlet
states. This condensation is complete at T = 0.