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 1930 ’ s, 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.Williams, E.Corenzwit, and R.C.Sherwood, Phys.Rev.125,541 (1962)

6. Experimental fact The reason for this correspondence was still a mystery Ref.M.P.Sarachi, K,E.Corenzwit, And L.D.Longinotti, Phys.Rev.135,1024 (1964)

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 minimumKondo effect

8. Unperturbed Hamiltonian Creation and annihilation operators Wave number Component of the spin along the z-direction One-electron energy of the conduction electron

9. Perturbation due to magnetic impurity (s-d interaction) Position of the n-th impurity atom Spin operator of the conduction electron Spin operator of the n-th impurity atom

10. Born approximations The first Born approximation The second Born approximation c:intermediate states This term gives rise to a resistance minimum

11. The first Born approximation Temperature independent! the magnetic moment of the n-th impurities

12. Processes without spin-flip Consider single impurity atom １： Scattering via the unoccupied intermediate states Fermi sphere Conduction electron The second term in the blancket of W Fermi distribution function for the electron with the energy

13. Processes without spin-flip ２： Scattering using occupied intermediate states

14. Processes without spin-flip ３： Sign of the spin is changed in the unoccupied intermediate states The spin of conduction electron is changed by localized spin Spin preservation rule

15. Processes without spin-flip ４： Sign of the spin is changed in the occupied intermediate states

16. Contribution from process 1 and 2 It has little dependence on the initial energy We neglect it Take no account of the Pauli principle Transition probability has

17. Contribution from process 3 and 4 Behavior of the z-component of a localized spin Process3:First increase it and then decrease Process4 ： First decrease it and then increase

18. Processes with spin-flip １： Scattering via the unoccupied intermediate states

19. Processes with spin-flip ２： Scattering using occupied intermediate states

20. Processes with spin-flip ３： Sign of the spin is changed in the occupied intermediate states

21. Processes with spin-flip ４： Sign of the spin is changed in the unoccupied intermediate states

22. Transition probability Processes without spin-flip Processes with spin-flip

23. Boltzman transportation equation Distribution function Deviation from (equilibrium distribution)

24. Rate of change of the probability The number of conduction electrons per atom The concentration of impurity atoms

25. Conductivity and Resistivity

26. and T

27. logT dependence of resistivity First approximation The resistivity due to the impurity potential the lattice resistivity Second approximation

28. T min at which the resistivity minimum occurs Differentiating it with respect to T （ Anti-ferromagnetic interaction ） is proportional to

29. Depth of the minimum When An energy of the order of the splitting The depth of the minimum is proportional to c

30. The second Born approximation introduce logT to the resistivity because of When J ＜０ (anti ferromagnetic coupling), a resistance minimum appear corresponding to resistance minimum is proportional to The depth of the minimum is proportional to c Summary

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