Transcript Theory of the Quantum Mirage
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
T K
For
T
, 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
e d
+ U
|e d
|
H imp
e
d
n
Un
n
hybridization with conduction electrons
The Anderson model -
continued Many-body Kondo resonance
e d
E F
e d +U
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)
STM tip Fano Resonance (Fano ‘61)
dI
/
dV
( e e 2
q
) 2 1
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 dI/dV 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 F a
.
4.
The magnitude of the mirage depends only weakly on the ellipse eccentricity.
Theoretical model 1.
Cu(111) surface states form a 2DEG with a Fermi energy of
E F
=450meV and
k F
-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.
H
H bulk
H surface
i N
0
H imp
(
R i
)
H imp
(
R i
)
t b d i
e
d
d i
d i
b
,
Ud i
d i
d i
d i
(
R i
)
t s d i
s
, (
R i
)
h
.
c
.
{ Perimeter Co adatoms i=1,…,N Inner Co adatom i=0
Consider an STM tip placed above the surface point
r
dI/dV measures the local surface-electron DOS (
r
, e ) 1 Im
G
(
r
,
r
; e ) (
r
, e ) (
r
, e ) (
r
, e ) Contribution to LDOS due to inner adatom
(
r
, e ) 1 Im
G e
(
r
,
R
0 ; e )
t s G d
( e )
t s G e
(
R
0 ,
r
; e ) Propagator for an empty ellipse
R
0 Fully dressed
d
propagator
R
0 2
a
1
e
2 2a
Assumptions: 1. Neglect inter-site correlations: Distance between neighboring Co adatoms is large (about 10 angstroms).
2. Only 2D propagation: 1
kr
1 (
kr
) 2
Each Co adatom on the ellipse acts as a scatterer with a surface-to-surface
T
-matrix component
T
( e )
t s
2
G d
( e ) From theory of the Kondo effect, for
T
close to
E F T
( e )
t
s
e
E T K F
and
iT K t =
The probability for surface scattering 1-
t t
G e
(
r
,
r
' )
G
0 (
r
,
r
' )
i
,
j N
1
G
0 (
r
,
R i
) 1 1
Tg T
ij G
0 (
R j
,
r
' ) Where
G
0 (
r
,
r
' )
i
s H
0 ( 1 )
k r
r
' is the free 2D propagator
g ij
( 1
ij
)
G
0 (
R i
,
R j
) is an N x N matrix propagator
T
( e )
t s
2
G d
( e ) is the surface-to-surface
T
-matrix at each Co site
Numerical results (
r
,
E F
) for
t
1 / 2
Theory Experiment
Magnitude of the projected resonance Expand
G e
(
R
0 ,
R
0 ;
E F
) in the number of scatters: Direct path
G e
(
R
0 ,
R
0 )
G
0 (
R
0 ,
R
0 )
j N
1
G
0 (
R
0 ,
R j
)
t i
s G
0 (
R j
,
R
0 ) Scattering off several cobalt atoms – add incoherently!
Scattering off one Co atom,
G 1
Using
G
0 (
r
,
r
' )
i
s
| 2
r
k F
r
exp |'
ik F
i
4
G
1
s
2
t k F j N
1
s
2
t k F d
1 exp
i
2
k F a
1 ,
j
2 ,
j
1
e i
2
k F a
(
s
) 2 (
s
)
ds
s
4
t k F d e i
2
k F a
Mean distance between adjacent adatoms
Conclusion:
G 0
is negligible compared to
G
1 provided
d ea
16
t
2
k F d
Satisfied experimentally for all 0.05<
e
<1.
(
R
0 ,
E F
)
s
16
t
3 (
k F d
) 2 cos( 4
k F a
) Independent of the eccentricity!
Effect of ``weak`` magnetic field Magnetic field introduces an additional Aharonov Bohm phase:
j N
1 1 1 ,
j
2 ,
j
j N
1 1 1 ,
j
2 ,
j
exp
i
2
j
/ 0 Aharonov-Bohm phase (
B
) ( 0 )
J
0 2 2
e AB
0 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 F a
oscillations and the weak dependence on the eccentricity.
Distinctive oscillatory behavior of the mirage is predicted in an applied perpendicular magnetic field.