Table Of ContentProjecting the Kondo Effect: Theory of the Quantum Mirage
Oded Agam and Avraham Schiller
Racah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel
A microscopic theory is developed for the projection (quantum mirage) of the Kondo resonance
from one focus of an elliptic quantum corral to the other focus. The quantum mirage is shown to
beindependentofthesizeandtheshapeoftheellipse, andexperiencesλF/4oscillations (λF isthe
surface-bandFermiwavelength)withanincreasingsemimajoraxislength. Wepredictanoscillatory
behavior of themirage as a function of a weak magnetic field applied perpendicular to thesample.
PACS numbers: 72.15.Qm, 61.16.Ch, 72.10.Fk
1 In a recent experiment, Manoharan et al. [1] used an
0 elliptic quantum corral to project the image of a Kondo Rj
0
2 resonanceover a distance of tens of angstroms,from one d ξ
focus ofthe ellipseto the otherfocus. By placingamag- ξ 1j
an netic Co atom at one focus of the ellipse and measuring R ξ 2j R 2a 1 - ε2
J the tunneling current to a close-by scanning tunneling 3
ξ
1 microscope(STM)tip,adistinctiveKondoresonancewas 4
1 seen in the I-V curve when the tip was brought directly
2 above the Co adatom. Remarkably,a similar Kondo sig- 2a
nature was observed when the tip was placed above the
v
3 empty focus, indicating coherent refocusing of the spec- FIG. 1. An illustration of an elliptic quantum corral and
4 tral image by the surrounding corral. This should be theclassicaltrajectoriesassociatedwiththequantummirage.
4 contrastedwithSTMmeasurementsofisolatedmagnetic Theellipseischaracterized byasemimajor axislengthaand
6 adatoms on open surfaces [2,3], where a limited spatial aneccentricityE. Themeandistancebetweenadjacentatoms
0 extent of 10˚A was observed for the Kondo effect. forming theellipse perimeter is d.
0 ∼
Semiclassically, one can attribute this refocusing phe-
0
/ nomena to the property that all classical paths leaving face electrons and the underlying bulk electrons. As re-
t
a one focus ofthe ellipse bounce specularly offthe perime- centlyshownbyU´js´aghyetal.[4]forCoonAu(111),one
m terandconvergeontothesecondfocuswiththesameac- may model the Co adatom by an effective nondegener-
- quiredphase[1](seeFig.1). However,thissimplepicture ate Anderson impurity [5], characterized by an effective
d
does not explain the quantitative features of the experi- energylevelǫ ,anon-siterepulsionU,andtwohybridiza-
n d
o ment. For example, the complex interference patterns in tion matrix elements ts and tb to the underlying surface
c the dI/dV difference map throughout the ellipse, or the and bulk conduction electrons. In this manner, each of
v: λF/4 oscillations of the mirage with an increasing semi- the Co atoms forming the ellipse in the experiment of
i major axis length a (λF is the Fermi wavelength). Ex- Manoharan et al. [1] acts as an Anderson impurity, as
X
planationofthesefeaturesrequiresaquantitativetheory, does the adatom placed inside the ellipse.
ar which is the objective of the present Letter. Denoting the creation of a surface-state and a bulk
Starting with a microscopic picture of Kondo scatter- conduction electron by c† and a† , respectively (here
ing off the Co adatoms we obtain good qualitative and ~kσ q~σ
~k labels a two-dimensional surface vector while ~q is a
quantitative agreement with the experiment. We estab-
three-dimensional vector), we model the system by the
lish a remarkable feature of the quantum mirage which,
Hamiltonian = + + N (R~ ), where
asidefromtheλF/4oscillationsmentionedabove,isinde- H Hsurf Hbulk i=0Himp i
pendentofthe sizeandthe shapeofthe ellipse,provided = ǫ c† c and P = E a† a
the ellipses is not too small. In particular, there is no Hsurf ~k ~kσ ~kσ Hbulk q~ q~σ q~σ
dependence on the ellipse eccentricity , see Fig. 1. In X~kσ Xq~σ
E
the presence of a weak perpendicular magnetic field, we describe the free surface and bulk conduction bands, re-
predictanoscillatorybehaviorofthequantummirageas spectively, and
a function of the magnetic flux encircled by the ellipse.
The Cu(111) surface has a band of surface states, Himp(R~i)=ǫd d†iσdiσ+Ud†i↑di↑d†i↓di↓
which acts as a two-dimensional electron gas. The sur- Xσ
face band starts 450meV below the Fermi energy, and + t d† ψ (R~ )+t d† χ (R~ )+h.c.
s iσ σ i b iσ σ i
has a Fermi wave number of kF−1 ≃ 4.75˚A. When a Co Xσ n o
adatomis placedonthe surface,itscattersboththe sur-
describes a Co adatom at point R~ on the surface. Here
i
1
d† creates a localized Co electron at site R~ (i = 0 for weintroducetheN N matrixg (1 δ )G (R~ ,R~ ),
iσ i × ij ≡ − i,j 0 i j
the inner adatom and i = 1, ,N for the perimeter along with the two vector quantities, v = G (~r,R~ )
adatoms), while ψσ(R~i) and χ·σ·(·R~i) annihilate, respec- and ui = G0(R~i,~r′). Here G0(~r,~r′) is thie free0surfacie
tively,asurfaceandabulkconductionelectronatsiteR~i. Green function without the corral, and i and j run over
Forsimplicity,wehavetakenthe differentadatomstobe 1, ,N. Using these quantities, G(~r,~r′;ǫ) is compactly
···
identical, and neglected any momentum dependence of expressed as
t and t . In what follows, we shall mainly be interested
s b
N
in the case where the inner adatom is located at the left 1
G(~r,~r′;ǫ)=G (~r,~r′)+ v T u , (3)
focus, i.e., R~ = R~ in the notations of Fig. 1. 0 i(cid:20)1 Tg (cid:21) j
0 − iX,j=1 − ij
Consider now an STM tip placed directly above the
surface point ~r. If the tip couples predominantly to the where T(ǫ) = t2sGd(ǫ) is the surface-to-surface compo-
underlying surface-state electrons at ~r, then the differ- nent of the conduction-electron scattering T-matrix at
ential conductance for the current through the STM tip each Co site. Again, Eq. (3) omits the intersite correla-
measures, up to thermal broadening, the local surface- tionsandthebulkpropagationbetweendifferentCosites.
electron density of states at point ~r, ρ(~r,ǫ). For an iso- Finally, for temperatures and energies below the Kondo
lated Co impurity, ρ(~r,ǫ) depends on the Kondo scat- temperature, the Kondo part of Gd(ǫ) may be well ap-
tering from the impurity as described, e.g., in Ref. [6]. proximated[8]bytheLorentzianformZK/(ǫ ǫF+iTK),
−
For the multiple-impurity configuration considered here where TK is the Kondo temperature,ǫF is the Fermien-
there are two main modifications: (i) There are multiple ergy, and
scatteringoffthe differentCoadatoms;(ii) Intersite cor-
T
K
relations alter the Kondo scattering off eachCo adatom. ZK = πρ t2+πρ t2 (4)
Due to the relatively large distance between Co atoms s s b b
( 10˚A for neighboring atoms on the ellipse perimeter), is the corresponding weight. Here ρs and ρb are the sur-
∼
we expect the latter effect to be small, and therefore ne- face and bulk density of states at the Fermi level.
glect it hereafter. A key parameter that enters the quantum mirage is
Since we are mostly interested in the effect of the Co the ratio of scattering rates
atom placed inside the ellipse, we distinguish it from πρ t2
the other Co atoms on the perimeter of the ellipse. t= s s . (5)
πρ t2+πρ t2
Neglecting intersite correlations, ρ(~r,ǫ) takes the form s s b b
ρ(~r,ǫ)=ρ¯(~r,ǫ)+δρ(~r,ǫ), where Physically, t represents the probability that a surface-
stateelectronimpingingonaCoadatomwillbescattered
1 to a surface-state electron rather than a bulk electron.
ρ¯(~r,ǫ)= Im G(~r,~r;ǫ) (1)
−π { } Hence t is a measure of the in-elasticity of the scattering
of surface waves from the Co impurities. In the theory
is the density of states of an empty ellipse (i.e., in the
of Heller et al. [9] for the standing waves formed in a
absence of the inner adatom), and
quantum corral, t is found to be 1/2. Hereafter we shall
1 use the same value for t.
δρ(~r,ǫ)= Im t2G(~r,R~ ;ǫ)G (ǫ)G(R~ ,~r;ǫ) (2)
−π { s 0 d 0 } In Fig.2 we depict δρ(~r,ǫF) for variousconfigurations
of the Co atoms, as measured by Manoharan et al. [1].
is the additional contribution due to the extra Co atom
Theupperpanelscorrespondtoanellipsewitheccentric-
at R~0. Here G(~r,~r′;ǫ) is the retarded Green function of ity =0.786and34adatoms(ellipsebofRef.[1]),while
the surface electrons for an empty ellipse, and G (ǫ) is E
d thelowerpanelscorrespondtoanellipsewitheccentricity
the fully dressed retarded Green function of the d elec- =0.5and36adatoms(ellipseainRef.[1]). Thequan-
trons of the inner adatom. Note that in writing Eq. (2) E
tum mirage is clearly seen in each of the left two panels,
wehaveassumedthatthethree-dimensionalpropagation
where an additional adatom has been placed at the left
ofbulkelectronsbetweendifferentCositesonthesurface
focus of the ellipse. For both ellipses, there is a strong
issmallcomparedtothe two-dimensionalpropagationof
signal in the tunneling density of states right above the
the surface electrons. This assumption is quite reason-
right focus, in accordance with the experimental data.
able considering that the three-dimensional propagation
By contrast, the quantum mirage disappears when the
near the surface decays as 1/r2, compared to 1/√r for
additionaladatomisplacesoffthefocus,asshowninthe
the two-dimensional surface propagation[7].
right two panels. These results are in good agreement
Experimentally,δρ(~r,ǫ)isextractedbyfirstmeasuring
with the experimental measurements, reproducing even
the local density of states of the empty ellipse, and then
fine details of the experimental patterns.
subtracting it from the measured density of states with
The results of Fig. 2 were obtained from Eqs. (2) and
the extra Co atom. To compute δρ(~r,ǫ), one needs to
(3), by setting t = 1/2 and approximating G with the
0
evaluate G(~r,~r′;ǫ), which is our next goal. To this end,
free two-dimensional Green function:
2
comesformalltrajectorieswhichscatterfromasingleCo
adatom on the ellipse perimeter (solid lines in Fig. 1).
Conventionally,thesecondtermG issmallerthanG
1 0
for severalreasons. First, each scattering eventis inelas-
tic, andthereforeintroducesa reductionfactoroft. Sec-
ond, the scattered orbits are longer. Third, the various
orbitshavegenerallydifferentlengths,andthereforetheir
correspondingphasesaddupincoherently. Thesituation
is quite differentin the presentcase. Due to the defining
property of the ellipse, the length of all scattered orbits
between the two foci is precisely the same, and hence
theircontributionsaddupcoherently. Consequently,the
second term in Eq. (8) takes the form
N
2t
G1( R~,R~) iρs ei2kFa−iπ2, (11)
− ≃ k ξ ξ
Xj=1 F 1,j 2,j
p
FIG.2. Contourplotofδρ(~r,ǫF)attheFermilevelassum- where 2a is the length of each orbit, and ξ and ξ
1,j 2,j
ing t= 1/2. The upper panels correspond to an ellipse with
eccentricityE =0.786and34adatoms,whilethelowerpanels are the distances between the impurity at R~j and the
to an ellipse with E =0.5 and 36 adatoms. Forboth ellipses, right and left foci, respectively (see Fig. 1). Finally, we
a strong signal appears at the right focus of the ellipse when approximate the sum in Eq. (11) by an integral:
an additional adatom has been placed at theleft focus. This
N
quantum mirage disappears when the additional adatom is 1 1 1
ds , (12)
places off the focus, as shown in the right two panels. ξ ξ ≃ dI ξ (s)ξ (s)
Xj=1 1,j 2,j 1 2
p p
wheres denotesthe coordinatealongtheellipsecontour,
G0(~r,~r′)=−iπρsH0(1)(k|~r−~r′|). (6) and d is the mean distance between adjacent adatoms.
Theresultoftheintegralisindependentoftheeccentric-
Here k is the wave number, and H0(1)(x) is the Han- ityoftheellipse,andissimply2π. Thus,thecontribution
kel function of zeroth order, which for x 1 takes the of orbits scattered from a single perimeter adatom is
≫
asymptotic form
4πt
G ( R~,R~) ρ ei2kFa. (13)
1 s
H(1)(x) 2 exp ix iπ . (7) − ≃ kFd
0 ≃rπx (cid:16) − 4(cid:17) Comparing G ( R~,R~) and G ( R~,R~) at the Fermi
0 1
− −
energy, one sees that the leading contribution to the
The good agreement between our calculations with
quantum mirage comes from G [10], provided
t = 1/2 and the experimental data indicates that the 1
numberofscatteringeventsthataparticleundergoesbe- d 16πt2
fore leaving the surface bounded by the ellipse is small. . (14)
a ≪ k d
F
This suggests the possibility of calculating the quantum E
mirage at the right focus at the Fermi level, δρ(R~,ǫ ), Substituting the experimental parameters, d 10˚A,
F ≃
using perturbation theory in t. Thus, to linear order in a 70˚A, and kF 1/4.75˚A−1, and setting t = 1/2,
≃ ≃
t, the Green function between the left and the right foci it is straightforward to verify that Eq. (14) holds for all
is given by ellipses with eccentricity 0.05 < < 1. Furthermore,
E
Eq. (14) is always satisfied for sufficiently large ellipses,
G( R~,R~;ǫF) G0( R~,R~)+G1( R~,R~)+ , (8) providedthemeandistancebetweenadjacentadatomsis
− ≃ − − ···
kept fixed (a d).
where Neglecting≫the contribution of the direct path,
G ( R~,R~), and using Eqs. (2) and (13) with ~r = R~,
π 0
G0( R~,R~) iρs ei2EkFa−iπ4 (9) the−resulting local density of states at the Fermi energy
− ≃− r k a
E F takes the form
is the contribution of the direct path connecting the two 16t3
δρ(R~,ǫ ) ρ cos(4k a). (15)
foci (illustrated by the dashed line in Fig. 1), and F ≃ s(k d)2 F
F
N The main feature of the aboveresultis the robustnessof
t
G1( R~,R~)= G0( R~,R~j)G0(R~j,R~) (10) the quantum mirage: As long as condition (14) is satis-
− iπρ −
s Xj=1 fied, the amplitude of the mirage is independent of the
3
size of the ellipse, a, and its eccentricity, . Rather, the ξ , ξ , ξ , and ξ , as illustrated in Fig. 1. Each of these
1 2 3 4
E
amplitudeisdeterminedbyt,whichcharacterizesthein- segments is regarded as an independent random vari-
elasticityofthescatteringofsurfacewavesfromadatoms, ableuniformlydistributedintherange(ξ¯ b/2,ξ¯ b/2),
and the dimensionless mean distance between adjacent whereξ¯istheexactdistancefromthefocu−stothe−ellipse
adatoms along the ellipse, k d. The oscillations of the boundary,andbisthetriangularlatticespacing. Theto-
F
mirageasfunctionofaareindeedperiodic withaperiod tal length η = ξ is, therefore, approximately a
n=1,4 n
of λF/4, as seen experimentally [1]. Gaussian randoPm variable with mean 4a and variance
Next we consider the effect of a weak uniform mag- b2/3. Averaging the cosine term in Eq. (15), cos(k η) ,
F
h i
netic field, B, applied perpendicular to the surface. As the disorder-averagedvalue ofthe quantummirageis re-
shownbelow, the quantum mirage experiences a distinc- duced by a factor of approximately Q = exp( k2b2/6).
− F
tive oscillatory behavior as a function of the magnetic Substituting the experimental values k = 1/4.75˚A−1
F
field, which depends on the size and the shape of the el- and b = 2.55˚A, one finds Q 0.95, meaning that the
≈
lipse. Hereweassumethattheellipseissufficientlylarge effect of imperfections is negligible.
and that the magnetic field is sufficiently weak so that In conclusion, in this Letter we have studied the phe-
(i) Zeeman splitting of the Kondo resonance can be ne- nomenonofquantummirage,andclarifiedits relationto
glected, and (ii) the cyclotron radius of the conduction the classical orbits of a particle in an ellipse. Our the-
electrons is much larger than the ellipse size. ory also predicts a distinctive behavior of the quantum
Underthesecircumstances,themaineffectofthemag- mirageinthe presenceofaperpendicular magneticfield,
netic field is to introduce an additional Aharonon-Bohm which could be tested experimentally. Finally our ap-
phase to the contribution of each path. This phase is proach clearly shows that the phenomenon of quantum
the magnetic flux encircled by the orbit, measured in mirage is not unique for magnetic adatoms. It will also
units ofthe quantumflux, φ =hc/e. Hereh is Planck’s appear for nonmagnetic atoms (on the ellipse perimeter
0
constant, c is the velocity of light, and e is the elec- oratits focus)withstrongscattering,e.g. adatomswith
tron charge. To compute G ( R~,R~) we fix the gauge resonant tunneling states at the Fermi level.
1
−
by calculating the flux encircled by the path which goes A.S. is grateful to Hari Manoharan for useful discus-
from R~ to R~ , to R~, and then back to R~ along the sions. This research was supported by the Israel science
j
−
semimajor axis of the ellipse. Accordingly, the sum of foundation founded by The Israel Academy of Science
Eq. (11) is modified to j(ξ1,jξ2,j)−1/2ei2πϕj/φ0, where and Humanities, and by Grant No. 9800065 from the
ϕj isthefluxofeachtrajPectory. Usingthecontinuumap- USA-Israel Binational Science Foundation (BSF).
proximation of Eq. (12) this sum gives 2πJ (2 B/φ ),
0 0
EA
where J (x) is the Bessel function of zeroth order, and
0
=πa2√1 2 is the areaofthe ellipse. The quantum
A −E [1] H. C. Manoharan, C. P. Lutz, and D. M. Eigler, Nature
mirage is, thus, modified according to 403 512 (2000).
[2] J. Li, W.-D. Schneider, R. Berndt, and B. Delley, Phys.
B
δρ(B) δρ(0)J2 2 A , (16) Rev. Lett.80, 2893 (1998).
≃ 0 (cid:18) E φ (cid:19)
0 [3] V. Madhavan, W. Chen, T. Jamneala, M. F. Crommie,
and N.S. Wingreen, Science 280, 567 (1998).
whereδρ(0)isthezero-fieldresultofEq.(15). Notethat,
[4] For a microscopic derivation of a nondegenerate Ander-
foragivena,the sensitivitytoa magneticfieldis largest
sonmodeldescriptionofaCoadatomonAu(111),seeO.
for =1/√2. U´js´aghy, J. Kroha, L. Szunyogh, A. Zawadowski, Phys.
E
Finally, we discuss the role of imperfections in the el- Rev. Lett.85, 3690 (2000).
lipse. Clearly, a combined effect of many coherent tra- [5] P. W. Anderson,Phys. Rev. 124, 41 (1961).
jectories is very sensitive to imperfections and dephas- [6] A. Schiller and S. Hershfield, Phys. Rev. B 61, 9036
ing. At 4K, dephasing effects due to electron-electron (2000).
and electron-phonon interactions are negligible over dis- [7] Conventionally, free 3D propagation overa distance of r
decays as 1/r. Near the surface, though, the power law
tances of the order of hundreds of angstroms. In what
changes to 1/r2, due to the surface boundary condition
follows we show that effect of imperfections is small too.
(corresponding to an image charge).
The main source of imperfections in the ellipse comes
[8] See, e.g., A. Hewson, “The Kondo Problem to Heavy
from the position of the adatoms forming the ellipse.
Fermions,” (Cambridge Press, Cambridge, 1993).
These are constrained to sit on a triangular lattice im-
[9] E.J.Heller,M.F.Crommie,L.P.Lutz,andD.M.Eigler,
posed by the underlying Cu(111) surface. Consequently, Nature 369 464 (1994).
the lengths of the orbits contributing to δρ arenot those [10] Onemaybeconcernedabouthigherorderterms,suchas
of an ideal ellipse. To estimate the effect of the devia- G2,G3,etc.These,however,aresmall, astheyrepresent
tions,weconsiderarandomdistributionofthetrajectory mainly non-classical paths, whose contributions add up
lengths,andaverageoverthedistribution. Wefirstnotice incoherently.
that each contribution to δρ is composed of 4 segments:
4
