1. Theory of the Quantum Mirage*
Oded Agam
Avraham Schiller
The Hebrew University
*Phys. Rev. Lett. 86, 484 (2001)

2. The Kondo Effect Impurity moment in a metal
High temperatures:
Free moment
Low temperatures:
Local singlet
The impurity spin is progressively screened below a nonperturbative temperature TK
For T<TK, a sharp Kondo resonance develops in the
Impurity DOS at the Fermi level
Resonance never observed for an isolated impurity

3. Formation of a local moment: The Anderson model
ed + U t
|ed|
hybridization with
conduction electrons

4. The Anderson model - continued
Many-body Kondo resonance ed EF
ed+U

5. Cobalt atoms deposited onto Au(111) at 4K
(400A x 400A)
Madhavan et al., Science 280 (1998)

6. STM spectroscopy on and off a Co atom
Madhavan et al., Science 280 (1998)

7. STM spectroscopy across one Co atom
Madhavan et al., Science 280 (1998)

8. Fano Resonance q = Interference parameter (Fano ‘61) STM tip
Interacting level:
Madhavan et al.’98,
AS & S. Hershfield ’00,
Ujsaghy et al.’01

9. Manoharan et al., Nature (2000)
Co on Cu(111)
Manoharan et al., Nature (2000)

10. Manoharan et al., Nature (2000)
An empty ellipse
Topograph
image
dI/dV map
Manoharan et al., Nature (2000)

11. Quantum Mirage Extra adatom at focus: Extra adatom away from focus:
No quantum mirage

12. Quantum Mirage: Spectroscopic fingerprint

13. Recap of the main experimental findings:1.
There is a quantum mirage when a Co atom is placed at one of the foci. 2.
No mirage when the Co atom is placed away from the foci. 3.
The quantum mirage oscillates with 4kFa. 4.
The magnitude of the mirage depends only weakly on the ellipse eccentricity.

14. Theoretical model 1.
Cu(111) surface states form a 2DEG with a Fermi energy of EF=450meV and kF-1=4.75 angstroms. 2.
Free 3D conduction-electron bulk states. 3.
Each Co atom is modeled by a nondegenerate Anderson impurity.
Ujsaghy et al., PRL (2000) 4.
Hybridization with both surface and bulk states.

15. {
Perimeter Co adatoms i=1,…,N
Inner Co adatom i=0

16. Consider an STM tip placed above the surface point
dI/dV measures the local surface-electron DOS
Contribution to LDOS due to inner adatom

17. Propagator for an empty ellipse
Fully dressed d propagator 2a

18. Assumptions: 1. Neglect inter-site correlations:
Distance between neighboring Co adatoms is large (about 10 angstroms).
2. Only 2D propagation:

19. Each Co adatom on the ellipse acts as a scatterer with a surface-to-surface T-matrix component
From theory of the Kondo effect, for T<TK and close to EF t
t =
The probability for surface scattering
1- t

20. Where
is the free 2D propagator
is an N x N matrix propagator
is the surface-to-surface T-matrix at each Co site

21. Numerical results
for

22. Theory
Experiment

23. Magnitude of the projected resonance
Expand
in the number of scatters:
Direct path
Scattering off several cobalt atoms – add incoherently!
Scattering off one Co atom, G1

24. Using
Mean distance between adjacent adatoms

25. Conclusion: G0 is negligible compared to G1 provided
Satisfied experimentally for all 0.05<e<1.
Independent of the eccentricity!

26. Effect of ``weak`` magnetic field
Magnetic field introduces an additional Aharonov-Bohm phase:
Quantum flux
Aharonov-Bohm phase
A=Area of
ellipse

27. Conclusions
STM measurements of magnetic impurities on metallic surfaces offer a unique opportunity to study the Kondo effect.
Detailed theory presented for the quantum mirage, which explains the 4kFa oscillations and the weak dependence on the eccentricity.
Distinctive oscillatory behavior of the mirage is predicted in an applied perpendicular magnetic field.