Theory of the Fano Effect and Quantum Mirage STM Spectroscopy of Magnetic Adatoms on Metallic Surfaces Local-moment formation: The Anderson model d | d + U hybridization with conduction electrons V The Anderson model - continued EFEF dd d +U Many-body Kondo resonance Cobalt atoms deposited onto Au(111) at 4K (400A x 400A) 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) Theory of STM line shape: Basic ingredients Bulk states Surface states Magnetic adatom STM tip Basic ingredients - continued Bulk states - Three-dimensional band Surface states - Two-dimensional band Magnetic adatom - An Anderson impurity STM tip - Feature-less band Full Hamiltonian: Impurity Hamiltonian: are the local conduction-electron degree of freedom, Here is the position of the impurity adatom, and is the position directly beneath the STM tip Tunneling Hamiltonian: STM tip tdtd tsts tbtb Tunneling Hamiltonian - continued where Tunneling current: Setting substrate =0 and tip =eV, and assuming weak tunneling amplitudes where is the feature-less tip DOS is the Fermi-Dirac distribution is the effective substrate DOS: with The differential conductance samples ! Evaluating Our aim is to express f ( ) in terms of the fully dressed impurity Green function and the impurity-free surface and bulk Green functions Evaluating- continued impurity-free contributions Contribution of scattering off impurity Line shape near resonance Consider the case where G d has a resonance and G s and G b are feature-less in the relevant energy range Define Real parameters Line shape near resonance - continued Real constant B Line shape near resonance - continued with Fano resonance! STM spectroscopy on and off a Co atom Madhavan et al., Science 280 (1998) Manoharan et al., Nature (2000) Co on Cu(111) An empty ellipse Manoharan et al., Nature (2000) Topograph image dI/dV map 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: There is a quantum mirage when a Co atom is placed at one of the foci No mirage when the Co atom is placed away from the foci. The quantum mirage oscillates with 4k F a. The magnitude of the mirage depends only weakly on the ellipse eccentricity Theoretical model Cu(111) surface states form a 2DEG with a Fermi energy of E F =450meV and k F -1 =4.75 angstroms. Free 3D conduction-electron bulk states. Each Co atom is modeled by a nondegenerate Anderson impurity Hybridization with both surface and bulk states.4. Ujsaghy et al., PRL (2000) Perimeter Co adatoms i=1,,N Inner Co adatom i=0 { Consider an STM tip placed above the surface point dI/dV measures the local conduction-electron DOS Contribution to LDOS due to inner adatom Assumptions: 1. Neglect inter-site correlations: 2. Only 2D propagation: Distance between neighboring Co adatoms is large (about 10 angstroms). Propagator for an empty ellipse Fully dressed d propagator 2a 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