Resistance Minimum in Dilute Magnetic Alloys

1
Resistance Minimum in Dilute Magnetic Alloys

• Ref)Jun Kondo
• Resistance Minimum in Dilute Magnetic Alloys
• Prog. Theor. Phys.32(1964)37-49
• Osaka Univ.
• Miyake Lab.
• Tsuyoshi Kobayashi(M1)

2
Contents

• Introduction
• Resistance Minimum
• Calculation
• Processes without spin-flip
• Processes with spin-flip
• Transition probability
• Resistivity
• Summary

3
Temperature Dependence of Resistance

• Resistance due to the lattice vibrations

4
Resistance Minimum

• In middle of the 1930s, a resistance minimum
  was discovered.

Ref.Hiroyuki Shiba Kotainodenshiron Maruzen.82
(1996)
5
Experimental fact

• Magnetic impurities are playing important roles

Ref.A.M.Clogston, B.T.Matthias, M.Peter, H.J.Willi
ams, E.Corenzwit, and R.C.Sherwood, Phys.Rev.125,5
41 (1962)
6
Experimental fact
Ref.M.P.Sarachi, K,E.Corenzwit, And
L.D.Longinotti, Phys.Rev.135 ,1024 (1964)

• The reason for this correspondence was still a
  mystery

7
Kondo effect

• 30 years after since resistance minimum was
  discovered
• Kondo found the process
• beyond the first Born approximation
• is important
• Resistance minimum was solved in 1964.

Resistance minimum
Kondo effect
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Unperturbed Hamiltonian
One-electron energy of the conduction electron
Wave number
Component of the spin along the z-direction
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Perturbation due to magnetic impurity(s-d
interaction)
Spin operator of the conduction electron
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Born approximations

• The first Born approximation
• The second Born approximation

This term gives rise to a resistance minimum
cintermediate states
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The first Born approximation
the magnetic moment of the n-th impurities
Temperature independent!
12
Processes without spin-flip

• Consider single impurity atom
• 1Scattering via the unoccupied intermediate
  states

Fermi sphere
Conduction electron
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Processes without spin-flip

• 2Scattering using occupied intermediate states

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Processes without spin-flip

• 3Sign of the spin is changed in the unoccupied
  intermediate states

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Processes without spin-flip

• 4 Sign of the spin is changed in the occupied
  intermediate states

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Contribution from process 1 and 2
Transition probability has
Take no account of the Pauli principle

• It has little dependence on the initial energy
• We neglect it

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Contribution from process 3 and 4

• Behavior of the z-component of a localized spin
• Process3First increase it and then decrease
• Process4First decrease it and then increase

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Processes with spin-flip

• 1 Scattering via
• the unoccupied intermediate states

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Processes with spin-flip

• 2Scattering using
• occupied intermediate states

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Processes with spin-flip

• 3Sign of the spin is changed
• in the occupied intermediate states

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Processes with spin-flip

• 4Sign of the spin is changed
• in the unoccupied intermediate states

22
Transition probability

• Processes without spin-flip
• Processes with spin-flip

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Boltzman transportation equation
Distribution function
(equilibrium distribution)
Deviation from
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Rate of change of the probability
The number of conduction electrons per atom
The concentration of impurity atoms
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Conductivity and Resistivity
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and T
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logT dependence of resistivity
First approximation
Second approximation
the lattice resistivity
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Tmin at which the resistivity minimum occurs
Differentiating it with respect to T
is proportional to
(Anti-ferromagnetic interaction)
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Depth of the minimum
When
An energy of the order of the splitting
The depth of the minimum is proportional to c
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Summary

• The second Born approximation introduce logT to
  the resistivity
• because of
• When Jlt0(anti ferromagnetic coupling),
• a resistance minimum appear
• corresponding to resistance minimum is
  proportional to
• The depth of the minimum is proportional to c

31
Development

• The resistivity should not diverge at absolute
  zero
• Importance of higher order Born approximation
• ?Variational Theory (Yosida)
• ?Renormalization
• Group Method (Anderson,Wilson)