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 dd  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<T K and close to E F The probability for surface scattering t = t 1- t

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

Numerical results for

TheoryExperiment

Magnitude of the projected resonance Expandin the number of scatters: Direct path Scattering off one Co atom, G 1 Scattering off several cobalt atoms – add incoherently!

Using Mean distance between adjacent adatoms

G 0 is negligible compared to G 1 provided Satisfied experimentally for all 0.05<e<1. Independent of the eccentricity!

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 4k F a oscillations and the weak dependence on the eccentricity. The line shapes observed for individual impurities can be understood by the Kondo-Fano effect.