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Localized Magnetic States in Metals

Localized Magnetic States in Metals. Miyake Lab. Akiko Shiba. Ref.) P. W. Anderson, Phys. Rev. 124 (1961) 41. Contents. Introduction Experimental Data Calculation Hamiltonian Unrestricted Hartree – Fock Approximation Magnetic Case Nonmagnetic Case Summary. No localized moment.

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Localized Magnetic States in Metals

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  1. Localized Magnetic States in Metals Miyake Lab. Akiko Shiba Ref.) P. W. Anderson, Phys. Rev. 124 (1961) 41

  2. Contents • Introduction • Experimental Data • Calculation • Hamiltonian • Unrestricted Hartree –Fock Approximation • Magnetic Case • Nonmagnetic Case • Summary

  3. No localized moment Moment per Fe in Bohr magnetons localized moment Electron Concentration Experimental Data Magnetic moments of Fe impurity Depend on the host metal Susceptibility: Ref.)A.M.Clogston et al., Phys.Rev.125,541(1962)

  4. free-electron system repulsive interaction Ed+U U εF s-d hybridization V where Ed Hamiltonian Many-body problem d-states

  5. ε nd↑=nd↓ No localized moment DOS of conduction electrons Simple Limit: U=0 No coulomb correlation εF Ed Ed Δ Δ

  6. ε Ed+U εF Coulomb repulsive Ed Localized moment appears Simple Limit:Vdk=Vkd=0 No s-d hybridization Ed<εF Ed+U>εF

  7. ε ε Ed+U Ed εF εF Ed+U Ed No localized moment Simple Limit:Vdk=Vkd=0 No s-d hybridization

  8. is very small, Assume that Hartree-Fock Approximation δ↑ constant

  9. One-electron Hamiltonian Hartree-Fock Hamiltonian Eσ

  10. DOS of conduction electrons DOS of d-electrons Resolvent Green Function： where

  11. DOS of d-electrons Lorentzian ρｄ（ε） Δ ε Ed

  12. Important parameters! Self-consistent equation Number of d-electrons: Introduce :Self-consistent equation

  13. Non-magnetic State (Self-consistency plot) Non-magnetic solution 0.5 Non-magnetic Solution 0.5

  14. Magnetic solutions Magnetic solutions Magnetic State (Self-consistency plot) 0.5 Non-magnetic solution 0.5

  15. magnetic non-magnetic <n> vs. y=U/Δ (symmetric) ε Ed+U εF Ed

  16. magnetic non-magnetic <n> vs. y=U/Δ (asymmetric) ε Ed+U εF Ed

  17. Non-magnetic conditions: x near 0 Magnetic x not small or too near 1 • Magnetic conditions: Phase diagram x

  18. ε Assume Δ Ed+U εF Ed Non-magnetic Case (symmetric) use the approximation: then

  19. ε Δ x near 0 Ed+U Ed εF Non-magnetic Case (asymmetric) Opposite limit: Valence fluctuation

  20. Assume then x not small or too near 1 Magnetic Case ε Δ Ed+U εF Ed

  21. Summary • The Coulomb correlation among d-electrons at the impurity site is important to understand the appearance of magnetic moment. • The existence of magnetic moments depends on ‘x’ and ‘y’. ε ε ε Ed+U εF εF εF Ed

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