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**Theory of the Quantum Mirage***

Oded Agam Avraham Schiller The Hebrew University *Phys. Rev. Lett. 86, 484 (2001)

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**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

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**Formation of a local moment: The Anderson model**

ed + U t |ed| hybridization with conduction electrons

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**The Anderson model - continued**

Many-body Kondo resonance ed EF ed+U

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**Cobalt atoms deposited onto Au(111) at 4K**

(400A x 400A) Madhavan et al., Science 280 (1998)

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**STM spectroscopy on and off a Co atom**

Madhavan et al., Science 280 (1998)

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**STM spectroscopy across one Co atom**

Madhavan et al., Science 280 (1998)

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**Fano Resonance q = Interference parameter (Fano ‘61) STM tip**

Interacting level: Madhavan et al.’98, AS & S. Hershfield ’00, Ujsaghy et al.’01

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**Manoharan et al., Nature (2000)**

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

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**Manoharan et al., Nature (2000)**

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

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**Quantum Mirage Extra adatom at focus: Extra adatom away from focus:**

No quantum mirage

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**Quantum Mirage: Spectroscopic fingerprint**

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**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.

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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.

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

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**Consider an STM tip placed above the surface point**

dI/dV measures the local surface-electron DOS Contribution to LDOS due to inner adatom

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**Propagator for an empty ellipse**

Fully dressed d propagator 2a

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**Assumptions: 1. Neglect inter-site correlations:**

Distance between neighboring Co adatoms is large (about 10 angstroms). 2. Only 2D propagation:

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**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

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

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Numerical results for

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Theory Experiment

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**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

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Using Mean distance between adjacent adatoms

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**Conclusion: G0 is negligible compared to G1 provided**

Satisfied experimentally for all 0.05<e<1. Independent of the eccentricity!

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**Effect of ``weak`` magnetic field**

Magnetic field introduces an additional Aharonov-Bohm phase: Quantum flux Aharonov-Bohm phase A=Area of ellipse

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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.

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