View
215
Download
0
Embed Size (px)
Citation preview
STM spectroscopy of magnetic adatoms on metallic surfaces
Avraham Schiller
The Hebrew University
Formation of a local moment: 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 bandka
skc
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
..)( chRt tiptipbb
k
kktip a *
where
Tunneling current:
Setting substrate=0 and tip=eV, and assuming weak
tunneling amplitudes
dfeVfe
VI 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
d
fe
dV
dIVG 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
i
rrrrG
)'()(),',(
*
)( 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
iqwAiG
rf
~Constant)(2
2
Real constant B
1~
~Background)(
2
2
q
Bf
Line shape near resonance - continued
d
fe
dV
dIVG 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:
kr
1
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 iTE
TtT
)(
The probability for surface scattering
t= t1 -t
N
jijS
ij
iSSe rRGTTg
RrGrrGrrG1,
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
i
tRRG
10
00
0 ),(),(
Using
4exp
2)',(0 iik
kirrG F
FsS
|'| rr
akik
tG F
N
j jjFs 2exp
12
1 ,2,1
1
aki
Fs
aki
Fs
F
F
edk
tds
ss
e
dk
t 2
21
2 4
)()(
2
Mean distance between adjacent adatoms
G0 is negligible compared to G1 provided
dk
t
ea
d
F
216
Satisfied experimentally for all 0.05<e<1.
)4cos()(
16),(
2
3
0 akdk
tER 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.