Theory of the Fano Effect andQuantum Mirage

STM Spectroscopy of Magnetic Adatoms on Metallic Surfaces

Local-moment formation: The Anderson model

d|d + U

nUnnH dimp

hybridization withconduction electrons

V

The Anderson model - continued

EFd 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 adatomSTM tip

Basic ingredients - continued

Bulk states - Three-dimensional band

Surface states - Two-dimensional band

Magnetic adatom - An Anderson impurity

STM tip - Feature-less bandkaskc

bkc

d

tunnelingimptipsurfacebulk HHHHHH

dddUdddH dimp

..)()( chRdVRdV issibb

Full Hamiltonian:

Impurity Hamiltonian:

are the local conduction-electron degree of freedom,

k

kk crr )()( *

Here

iR

is the position of the impurity adatom, and

tipR

is the position directly beneath the STM tip

Tunneling Hamiltonian:

STM tip

td

tstb

tiptipsstipdtunneling RtdtH )(

Tunneling Hamiltonian - continued

H.c.)( tiptipbb Rt

k

kktip a *

where

Tunneling current:

Setting substrate=0 and tip=eV, and assuming weak

tunneling amplitudes

dfeVfeVI ftip )()()()(4)(

where)(tip is the feature-less tip DOS

)(f is the Fermi-Dirac distribution

)( f is the effective substrate DOS:

fff ;Im1)(

)()( tipbbtipssd RtRtdtf

with

dfe

dVdIVG ftip

)()(4)(2

)( fThe differential conductance samples !

)( fEvaluating

Our aim is to express f ( ) in terms of the fully dressed

impurity Green function

ddiGd ;)(

and the impurity-free surface and bulk Green functions

k k

kk

irrrrG

)'()(),',(*

)( fEvaluating -continued

),,(),,()( 22 tiptipbbtiptipSsf RRGtRRGtiG

impurity-free contributions

2),,(),,()( tipimpbbtipimpSsdd RRGtRRGttiG

Contribution of scattering off impurity

Line shape near resonance

Consider the case where Gd has a resonance

rrd i

wiG

)(

and Gs and Gb are feature-less in the relevant energy range

r

r

rd i

wiG

~with~

1)(

Define

iqARRGtRRGtt tipimpbbtipimpSsd ),,(),,(

Real parameters

Line shape near resonance - continued

i

iqwAiGr

f

~Constant)(

22

Real constant B

1~

~Background)( 2

2

qBf

Line shape near resonance - continued

dfe

dVdIVG ftip

)()(4)(2

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.

1.

2. No mirage when the Co atom is placed away from the foci.

The quantum mirage oscillates with 4kFa.

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

3.

4.

Theoretical model

Cu(111) surface states form a 2DEG with a Fermi energy of EF=450meV and kF

-1=4.75 angstroms.

Free 3D conduction-electron bulk states.

Each Co atom is modeled by a nondegenerate Anderson impurity.

1.

2.

3.

Hybridization with both surface and bulk states.4.

Ujsaghy et al., PRL (2000)

N

iiimpsurfacebulk RHHHH

0

)(

iiiiiidiimp dddUdddRH

)(

Perimeter Co adatoms i=1,…,N

Inner Co adatom i=0{

..)()( ,, chRdVRdV isisibib

Consider an STM tip placed above the surface point r

dI/dV measures the local conduction-electron DOS

);,(Im1),(

rrGr

),(),(),( rrr

Contribution to LDOS due to inner adatom

Assumptions:

1 .Neglect inter-site correlations:

2 .Only 2D propagation:

kr1

2)(1

kr

Distance between neighboring Co adatoms is large (about 10 angstroms).

);,()();,(Im1),( 00

rRGVGVRrGr esdse

Propagator for an empty ellipse

Fully dressed d propagator

2a

212 ea 0R

0R

Each Co adatom on the ellipse acts as a scatterer with a surface-to-surface T-matrix component

)()( 2 ds GVT

From theory of the Kondo effect, for T<TK and close to EF

KF

K

s iTETtT

)(

The probability for surface scattering

t= t1 -t

N

jijS

ijiSSe rRGT

TgRrGrrGrrG

1,

000 )',(1

1),()',()',(

Where

')',( )1(0

0 rrkHirrG sS

is the free 2D propagator

),()1( 0jiSijij RRGg

is an N x N matrix propagator

)()( 2 ds GVT is the surface-to-surface T-matrix at each Co site

Numerical results

for ),( FEr 2/1t

Theory Experiment

Magnitude of the projected resonance

Expand );,( 00 Fe ERRG

in the number of scatters:

),(),( 000

00 RRGRRG Se

Direct path

Scattering off one Co atom, G1

Scattering off several cobalt atoms – add incoherently!

N

jjS

sjS RRG

itRRG

10

00

0 ),(),(

Using

4exp2)',(0 iik

kirrG F

FsS

|'| rr

akik

tG F

N

j jjFs 2exp12

1 ,2,11

aki

Fs

aki

Fs

FF

edktds

sse

dkt 2

21

2 4)()(

2

Mean distance between adjacent adatoms

G0 is negligible compared to G1 provided

dkt

ead

F

216

Satisfied experimentally for all 0.05<e<1.

)4cos()(

16),( 2

3

0 akdktER F

FsF

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 4kFa oscillations and the weak dependence on the eccentricity.

The line shapes observed for individual impurities can be understood by the Kondo-Fano effect.