A theoretical analysis of quantum mirageson a Cu(111) surface

Yuki Shimada a, Hideaki Kasai a,*, Hiroshi Nakanishi a, Wilson Agerico Di~nno a,b,Ayao Okiji c, Yukio Hasegawa d

a Department of Applied Physics, Osaka University, Suita, Osaka 565-0871, Japanb Institute of Industrial Science, University of Tokyo, Tokyo 153-8505, Japan

c Wakayama National College of Technology, Gob�oo , Wakayama 644-0023, Japand Insititute for Solid State Physics, University of Tokyo, Chiba 277-8581, Japan

Received 15 October 2001; accepted for publication 22 January 2002

Abstract

Employing the Newns–Anderson model to describe a Co adatom placed at one focus of an elliptical quantum corral

on a Cu(1 1 1) surface, we theoretically investigate the mechanism behind how so-called quantum mirages could be

observed at the opposite empty focus. First, we try reproducing the experimental results of the differential conductance

dI=dV spectra measured at the empty focus, which exhibit features very similar to those measured at the Co-occupiedfocus. Furthermore, we consider the temperature dependence of the dI=dV spectra at the two foci. Finally, we considerthe structure of the dI=dV spectra when the STM tip is placed very close to the Co adatom (at the Co-occupied focus)and a metal surface (at the empty focus).

� 2002 Elsevier Science B.V. All rights reserved.

Keywords: Quantum effects; Copper; Cobalt; Adatoms; Scanning tunneling microscopy

1. Introduction

The Kondo effect was originally observed indilute magnetic alloys [1], but recently it was re-portedly observed for a system of a magnetic atomadsorbed on a metal surface [2,3]. The Kondo ef-fect observed on a metal surface appears as an

energy resonance in measurements for the biasdependence of the differential conductance dI=dV .Note that earlier studies have already suggestedthat we might actually be able to observe theKondo effect in real space with an STM [1]. Forexample, if we measure the bias dependence of thedI=dV on a Co atom adsorbed on metal surface atlow temperatures, we can see an asymmetric dipstructure in the vicinity of the Fermi level (Fanoshapes [4,5]). The energy widths of these dI=dVspectra were observed to correspond to the Kondotemperature TK of the system [5]. Thus, the asso-ciation with the Kondo effect.

Surface Science 514 (2002) 89–94

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*Corresponding author. Tel.: +81-6-6879-7857; fax: +81-6-

6879-7859.

E-mail address: kasai@dyn.ap.eng.osaka-u.ac.jp (H. Kasai).

0039-6028/02/$ - see front matter � 2002 Elsevier Science B.V. All rights reserved.PII: S0039-6028 (02 )01612-6

Quantum mirages (STS ‘‘ghost images’’) werereportedly observed at an empty focus of an el-liptical quantum corral (made from Co atoms),when a Co adatom is placed at the other focus[3]. It was observed experimentally (at 4 K) thatdI=dV spectra measured at the empty focus con-tain features very similar to those measured atthe Co-occupied focus. Earlier studies show thatthe quantum mirage results from the resonantscattering of electrons from the Co adatom at theopposite focus, and the Kondo resonance fromthe wall Co atoms does not influence [6]. Actually,the same results are obtained when the wall Coatoms are replaced by CO. The wall Co atomsonly take part by focussing the electrons scatteredfrom the Co-occupied focus. Thus, here, we ne-glect the Kondo resonance at the corral walls andrestrict our discussion to dI=dV spectra at thetwo foci.

2. The expression of the dI=dV spectra

To describe the electron system on a metalsurface, we consider the Hamiltonian [7,8]

H ¼Xkk;r

�kkcykk;r

ckk;r þX

r

�ddyrdr

þXkk;r

tðcykk;rdr þ dyrckk;rÞ þ Udy

"d"dy#d#; ð1Þ

where cykk;rðckk;rÞ creates (annihilates) a conductionelectron with wave vector kk ¼ ðkx; kyÞ parallel tothe surface ðk ¼ ðkx; ky ; kzÞ ¼ ðkk; kzÞ, kz fixed), en-ergy �kk , and spin r. dy

rðdrÞ creates (annihilates) anelectron with energy �d in the localized d-state ofthe Co adatom. t represents the hybridization be-tween the localized d-state and the electron statesin the substrate. U is the Coulomb repulsion be-tween electrons in the localized d-state with op-posite spins. In the present paper, we assume thesymmetric case with respect to electrons and holes[9].From the model Hamiltonian in Eq. (1), we

express the density of electrons states distributionqðx; y; z;xÞ

qðx; y; z;xÞ

¼Xk;k0 ;r

wkðx; y; zÞ� 1

pImGrk;k0 ;rðx; y; z;xÞ

�w�

k0 ðx; y; zÞ

þXk;r

wkðx; y; zÞ� 1

pImGrk;d;rðx; y; z;xÞ

�w�dðx; y; zÞ

þXk;r

wdðx; y; zÞ� 1

pImGrd;k;rðx; y; z;xÞ

�w�

kðx; y; zÞ

þX

r

wdðx; y; zÞ� 1

pImGrd;d;rðx; y; z;xÞ

�w�dðx; y; zÞ

ð2Þ

as a function of the positions of the STM tipðx; y; zÞ. We set the origin x ¼ y ¼ z ¼ 0 at the topof the Co adatom, whose, in the STM image, isobserved to be 0.8 �AA high from the metal surface[3]. In Eq. (2), x is the electron energy with re-spect to the Fermi energy �f , Grðx; y; z;xÞ is theretarded Green function, wkðx; y; zÞ is the conduc-tion electron wave function, wdðx; y; zÞ is the lo-calized d-state wave function. As the self-energyof the d-electron Green function, we used theself-energy for the symmetric case [10]. The con-duction electron wave function is defined aswkðx; y; zÞ ¼ 1=

ffiffiffiffiX

peðz=kÞ eiðkxxþkyyÞ ðzP 0Þ, where

kð¼ i=kzÞ is the decay constant of the conductionelectron wave function into the vacuum side, X isthe volume occupied by one conduction electron.We assumed that the radial part of the localized d-state wave function has the form obtained by an-alytic self-consistent-field calculations [11,12]. Inaddition, we consider the case l ¼ 2, m ¼ 0 (l is theorbital angular momentum quantum number andm is the azimuthal quantum number). If the den-sity of states for the STM tip is constant, in thelimit of small bias voltages, the differential con-ductance is given by dI=dV / qðx; y; z; �f þ eV Þ[13]. By using Eq. (3), at low temperatures (T <TK), the differential conductance is written as fol-lows [1,9]:

90 Y. Shimada et al. / Surface Science 514 (2002) 89–94

where, r ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2

p. J0 is the zeroth-Bessel func-

tion. kf is the Fermi wave vector. For the case ofCu(1 1 1), kf ¼ 0:21 �AA1 [14]. Dð¼ pqt2Þ indicatesthe resonance level width with q being the densityof states for a two-dimensional electron gas. ForCo/Cu(1 1 1) we take TK ¼ 53 K [3], and estimateD ¼ 0:14 eV from TK. At the empty focus, becausewdðx; y; zÞ � 0, the differential conductance is writ-ten as follows:

where the STM tip position z0 ¼ zþ z0 is measuredfrom the empty focus. Actually, the STM tip ispositioned at the height of z on the empty focus.In Fig. 1, our calculations of the dI=dV spectra

at the two foci are compared with the experi-ments when we place the STM tip at z ¼ 2:7 �AA.Our results satisfactorily reproduced the experi-mental results. In Fig. 2, we show the temperaturedependence of the dI=dV spectra at z ¼ 2:7 �AA. AsT increases, we can observe a gradual broadeningof the dip structure in the vicinity of the Fermilevel at the two foci. This is because the Kondoscreening (resonant scattering) becomes weak as Tincreases. In Fig. 3, we show the dI=dV spectra at

the two foci when we place the STM tip at z ¼ 0:7�AA. The dip structure seen in Fig. 1 is not obtainedat the Co-occupied focus, but the same dip struc-ture is obtained at the empty focus. When theSTM tip was placed above and very close to theCo adatom (for example z ¼ 0:7 �AA), the influenceof the fourth term in Eq. (2) becomes strongerthan the other terms because of the large lap of thelocalized d-state wave function. The peak structure

obtained in Fig. 3(a) corresponds to the Yosida–Kondo peak. On the other hand, at the emptyfocus, even if the STM tip was placed above andvery close to a metal surface, the same dip struc-ture is obtained. In this case, as we can see fromEq. (4), the dI=dV spectra are described by onlyone term. The dI=dV spectra obtained by this termalways give the dip structure, reflecting the reso-nant scattering, regardless of the parameter z.Therefore, the dip structure at the empty focus isnot a manifestation of the Fano effect. In Fig. 4,we show the temperature dependence of the dI=dVspectra at z ¼ 0:7 �AA. As T increases, we can ob-serve a gradual broadening of the peak structure in

dI=dV ’ 2DXpt2

e2z=k 1 J 20 ðkfrÞ eV =2TKð Þ2 þ pT=2TKð Þ2 þ 2

2 eV =2TKð Þ2 þ 1=2 eV =2TKð Þ2 þ pT =2TKð Þ2 þ 2h i2

8><>:

9>=>;

4ffiffiffiffiX

pptez=k wdðx; y; zÞJ0ðkfrÞ eV =2TKð Þ

2 eV =2TKð Þ2 þ 12

eV =2TKð Þ2 þ pT=2TKð Þ2 þ 2h i2

þ 2pD

jwdðx; y; zÞj2 eV =2TKð Þ2 þ pT=2TKð Þ2 þ 2

2 eV =2TKð Þ2 þ 12

eV =2TKð Þ2 þ pT=2TKð Þ2 þ 2h i2 ; ð3Þ

dI=dV ’ 2DXpt2

e2ðzþz0Þ=k 1

( J 20 ð0Þ

ðeV =2TKÞ2 þ pT=2TKð Þ2 þ 22 eV =2TKð Þ2 þ 1

2½ðeV =2TKÞ2 þ pT=2TKð Þ2 þ 2�2

);

z0 ¼ 0:8 �AAð4Þ

Y. Shimada et al. / Surface Science 514 (2002) 89–94 91

the vicinity of the Fermi level at the Co-occupiedfocus. This indicates a correspondence betweenthe development of the peak, as T decreases, withthe formation of the Yosida-singlet [15]. Thus, theterm Yosida–Kondo peak. Comparing with Fig.2(a) (in case of z ¼ 2:7 �AA), the temperature de-pendence is opposite. This opposite behavior mustbe caused by the development of the dip structure(reflecting the resonant scattering) and the devel-opment of the peak structure (corresponding tothe Yosida–Kondo peak).

3. Conclusion

We have studied the bias dependence of thedifferential conductance dI=dV at the two foci ofan elliptical quantum corral on a Cu(1 1 1). Whenwe measured the dI=dV spectra, a dip structurewas obtained if the STM tip was placed relativelyfar from the Co adatom (for example z0 ¼ 2:7 �AA),but a peak structure (corresponding to theYosida–Kondo peak) was obtained if the STM tipwas placed above and very close to the Co adatom

Fig. 1. The experimental (- - -) and the theoretical (––) dI=dV spectra (a), (b) at the Co-occupied focus (a) and at the empty focus (b) at4 K. An elliptical quantum corral is eccentricity e ¼ 1=2, and the distance between the two foci is 71.3 �AA. Our theory is plotted atz ¼ 2:7 �AA.

Fig. 2. The temperature dependence of the theoretical dI=dV spectra (a), (b) at the Co-occupied focus (a) and at the empty focus (b).The lines present the data at T ¼ 0, 4, 7, 15 K when the STM tip was placed at z ¼ 2:7 �AA.

92 Y. Shimada et al. / Surface Science 514 (2002) 89–94

(for example z ¼ 0:7 �AA). On the other hand, thedip structure was obtained at the empty focus,regardless of the distance the STM tip was placed.Thus, at sufficiently close STM tip–adatom (metalsurface) distance, the corresponding dI=dV at theCo-occupied focus and the other empty focus willnot be similar. Furthermore, by investigating thetemperature dependence of the dI=dV spectra atthe two foci, we could observe a gradual broad-ening of the structure as T increases. At the twofoci, the peak structure corresponding to theYosida–Kondo peak (in Fig. 4(a)) and the dipstructure which reflects the resonant scattering (in

Fig. 2 and Fig. 4(b)) broadens because the Kondoscreening becomes weak as T increases.

Acknowledgements

This work is partly supported by the Ministryof Education, Culture, Sports, Science and Tech-nology of Japan, through Grants-in-Aid for COEResearch (No. 10CE2004), Scientific Research(11640375, 13650026) programs, and by the NewEnergy and Industrial Technology DevelopmentOrganization (NEDO), through the Materials and

Fig. 3. The theoretical dI=dV spectra (a), (b) at the Co-occupied focus (a) and at the empty focus (b) at 4 K when the STM tip wasplaced at z ¼ 1:5 �AA. The change of the dI=dV spectra at the Co-occupied focus is obvious as compared with Fig. 1(a) (at z ¼ 2:7 �AA).

Fig. 4. The temperature dependence of the theoretical dI=dV spectra (a), (b) at the Co-occupied focus (a) and at the empty focus (b).The lines present the data at T ¼ 0, 4, 7, 15 K when the STM tip was placed at z ¼ 1:5 �AA. The behavior at the Co-occupied focus isopposite as compared with the behavior in Fig. 2(a).

Y. Shimada et al. / Surface Science 514 (2002) 89–94 93

Nanotechnology program, and by the Japan Sci-ence and Technology Corporation (JST), throughtheir Research and Development Applying Ad-vanced Computational Science and Technologyprogram.

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