Theory of the Quantum Mirage Oded Agam Avraham

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

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

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

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

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

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

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

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

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

An empty ellipse Topograph image d. I/d. V map Manoharan et al. , Nature (2000)

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

Quantum Mirage: Spectroscopic fingerprint

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 4 k. Fa. 4. The magnitude of the mirage depends only weakly on the ellipse eccentricity.

Theoretical model 1. Cu(111) surface states form a 2 DEG with a Fermi energy of EF=450 me. V and k. F-1=4. 75 angstroms. 2. Free 3 D 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.

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

Consider an STM tip placed above the surface point d. I/d. V measures the local surface-electron DOS Contribution to LDOS due to inner adatom

Propagator for an empty ellipse Fully dressed d propagator 2 a

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

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 = The probability for surface scattering 1 - t t

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

Numerical results for

Theory Experiment

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, G 1

Using Mean distance between adjacent adatoms

Conclusion: G 0 is negligible compared to G 1 provided Satisfied experimentally for all 0. 05<e<1. Independent of the eccentricity!

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

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 4 k. Fa oscillations and the weak dependence on the eccentricity. Distinctive oscillatory behavior of the mirage is predicted in an applied perpendicular magnetic field.