Observing electronic scattering in atomic-scale structures on metals

Journal of Electron Spectroscopy and Related Phenomena 109 (2000) 1–17 www.elsevier.nl / locate / elspec

Observing electronic scattering in atomic-scale structures on metals M.F. Crommie* Department of Physics, University of California, Berkeley, CA 94720, USA Received 15 September 1999; received in revised form 4 January 2000; accepted 4 January 2000

Abstract The scanning tunneling microscope (STM) is a powerful spectroscopic probe of microscopic structures at metal surfaces. Here a series of STM studies are discussed that focus on the interaction of two-dimensional (2-d) electrons with surface nanostructures, and on the electronic properties of magnetic adsorbates. Spectroscopic imaging is shown to be a useful method for observing quantum interference patterns of 2-d surface state electrons scattering from natural and atomically fabricated surface structures. Surface state dispersion and scattering phaseshifts are extracted from such images. On Au(111) the interaction of surface state electrons with a reconstruction-induced superlattice is seen to lead to electronic density modulation and a new surface bandstructure. Local spectroscopic measurements at the gold surface are used to extract quantitative details of the superlattice potential. Magnetic scattering at cobalt atoms deposited onto Au(111) is observed to induce electronic fluctuations near the site of these impurities. Spin-dependent interaction between gold conduction electrons and a Co local moment leads to a many-body groundstate known as the Kondo effect. Local spectroscopic measurements of the ‘Kondo resonance’ for individual magnetic impurities are discussed. Atomic manipulation of Kondo impurities is combined with STM spectroscopy to study the interaction between magnetic atoms at a metal surface.  2000 Elsevier Science B.V. All rights reserved. Keywords: STM; Scanning; Tunneling; Spectroscopy; Nanophysics

1. Introduction The physical and chemical behavior of defects and nanostructures are strongly influenced by local variations in electronic density [1]. Scanning tunneling microscopy provides an especially useful spectroscopic probe of this behavior. Cryogenic operation of a scanning tunneling microscope (STM) enhances this technique by improving spectroscopic resolution and making new physical regimes accessible [2,3]. Here we discuss a series of studies where cryogenic scanning tunneling spectroscopy was used to gain *Tel.: 11-510-642-7116; fax: 11-510-643-8497. E-mail address: [email protected] (M.F. Crommie).

insight into the electronic properties of atomic-scale surface structures. These structures include singleatom high step-edges, adsorbates, surface reconstructions, magnetic scatterers, and artificially fabricated atomic-scale structures. A unifying theme in these studies is that electronic scattering effects were ‘imaged’ using the spectroscopic capabilities of the STM. In some systems the electronic properties were studied through observation of quantum interference patterns caused by two-dimensional (2-d) surface state electrons. In other systems, where the behavior is dominated by electron–electron interactions, STM spectroscopy was used to study magnetic scattering at the atomic scale. Such behavior is manifested by the Kondo effect [4].

0368-2048 / 00 / $ – see front matter  2000 Elsevier Science B.V. All rights reserved. PII: S0368-2048( 00 )00103-1

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The STM is unique as a spectroscopic probe because it enables the user to modify the structures being studied in an atom-wise fashion [5]. This ability opens up new possibilities for studying nanometer-scale structures fabricated atom-by-atom. Here we will discuss some examples of how this ability has been used to help study the scattering properties of artificial nanostructures, as well as to study magnetic coupling between atoms.

nearly-free-electron-like properties parallel to the surface. Photoemission, for example, reveals a parabolic dispersion for these states parallel to the surface, with an effective mass a fraction of m e [9]. Scatterers at a surface induce variations in the local density of these states that can be readily imaged by an STM [10–14].

4. Electronic scattering from step-edges 2. Experimental technique The studies described here were performed using cryogenic ultra-high vacuum scanning tunneling microscopy. The work that is focused on nanometerscale structures at the surface of copper was performed at the IBM Almaden Research Center, while the work that is focused on microscopic structures at the surface of gold was performed at Boston University. The clean metal surfaces used in these studies were prepared by putting single crystal substrates through cycles of argon-ion sputtering and annealing. Surfaces prepared in this way were transferred, in UHV, to a low temperature STM (usually held at a temperature between 5 and 6 K). Metal adsorbates were deposited onto the low temperature surfaces through line-of-sight e-beam evaporation. dI / dV spectroscopy was typically performed by placing a 200–500 Hz AC modulation signal (with an amplitude in the range 1–10 mV) onto the tunnel bias voltage and phase sensitive detection of the resultant AC tunnel current. dI / dV spectra were obtained under open feedback loop conditions to keep the STM tip stable. STM images were obtained under constant current conditions.

3. Surface states Electronic states localized to a surface are known as surface states. Some surface states, such as dangling bonds or adsorbate-induced resonances, are localized in more than one dimension [6]. The closepacked surfaces of many metals (including the noble metals) are special in that they support a delocalized 2-d surface state [7,8]. This state is confined in the direction perpendicular to the surface, but shows

The tunnel current of an STM gives a measure of the single-particle excitation spectrum, which is often interpreted as the local density of states (LDOS) of a surface [15]:

O c sr¢d dsE 2 E d

¢ 5 LDOS(r,E)

u

k

u2

k

k

here ck is an electronic eigenstate of the surface. At the surface of a metal having a 2-d surface state, the electronic eigenstates can be approximated by 2-d ¢ ¢ free particle states, ck ~e ik?r [16]. If such a surface were without defects, then the LDOS of the surface would be independent of position and thus yield featureless STM images. When the surface has a defect, however, electrons scatter from the imperfection, leading to quantum interference and density variations that reveal the scattering properties of the defect [10–14,16]. One of the most common defects at a well-ordered surface is a monatomic step-edge occurring between atomically flat terraces. Such step-edges appear as line-scatterers to 2-d surface state electrons. Surfacestate wavefunctions near a step are composed of a wave amplitude incident upon the step, a reflected amplitude, and a transmitted amplitude. The incident and reflected waves interfere, leading to standing waves in the LDOS. These standing waves have been observed experimentally by a number of groups using scanning tunneling microscopy [10–14]. Fig. 1 ˚ image of the Cu(111) surface, shows a 5003500 A where electron standing waves can be seen decaying away from step-edges. The period of the standing ˚ but the waves in this image is approximately 15 A, wavelength is energy dependent [10,11]. The energy dependence can be seen in Fig. 2, where dI / dV linescans acquired at different bias voltages near a step-edge are shown (these give a measure of the

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˚ image of the Cu(111) surface at 5 K (STM imaging parameters: V50.1 V, I51.0 nA). Standing waves Fig. 1. Constant current 5003500 A ˚ can be seen emanating from monatomic step-edges (from Ref. [10]). in the LDOS with a periodicity of |15 A

surface LDOS at different energies (eV) with respect to the Fermi energy, EF , of the Cu sample). One way of understanding the qualitative characteristics of step-edge standing waves is to make the approximation that step-edges form an impenetrable barrier to surface-state electrons. The step-edge then creates a node in the surface LDOS, and the LDOS at surrounding points can be summed up in the integral [10] k

]] 0 ]]] 2 2m * 1 LDOS(E,x) 5 ]2 ]] dk ]]]2 sin 2skxd 2 k p " E 0 12 ] k0

œ E

œ

Here E is the energy of an electron measured with respect to the surface state bandedge, x is the distance from the step edge, m* is the effective mass of a surface state electron, and ]] 2 m *E k 0 5 ]] . "2

œ

The integral in Eq. (1) can be solved analytically to yield [10] LDOS(E,x) 5 (1 2 J0 (2 k 0 x)) L0

SD

(1)

(2)

where J0 is the zeroth order Bessel function, and L0 5 m* /(p" 2 ) (L0 is the LDOS of a 2-d electron gas

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5. Electronic scattering from adsorbates When an atom resides on top of a metal surface it acts as an electronic scatterer. The electronic eigenstate around the defect is comprised of ‘in-coming’ wave amplitude and phase-shifted ‘outgoing’ wave amplitude. Interference between these components leads to standing waves in the electronic density around the defect atom. For surface state electrons, the 2-d eigenstate around a point defect or adsorbate can be roughly approximated as cE,l ( r,w ) 5 Jl (k 0 r 1 ≠l ) e il w (Jl is the lth-order Bessel function) [22]. All the important scattering information is held in the energy dependent phase-shifts, dl . The LDOS near an adsorbate on a close-packed FCC metal should thus vary as LDOS(r, E) 5 olu Jl (k 0 r 1 ≠l )u 2 . Assuming that s-wave scattering dominates at low energy (since the electron wavelength is long compared to the scattering potential), waves in the LDOS surrounding a surface defect can be described by the following expression [10]: LDOS(r, E) ~ A 1 J 20 (k 0 r 1 ≠ 0 ) 2 J 02 (k 0 r ) Fig. 2. Solid lines: spatial dependence of dI / dV, measured as a function of distance from step edge at different bias voltages. Zero distance corresponds to the lower edge of the step. Dashed lines: theoretical fits of Eq. (2) to the data. Curves have been shifted vertically for viewing. Inset: experimental surface state dispersion, obtained from fits of Eq. (2) to dI / dV linescan data. Dashed line is the parabolic fit used to extract the surface state effective mass and bandedge (from Ref. [10]).

in the absence of any scattering). Eq. (2) yields a reasonable fit to data obtained on Cu(111), as can be seen by the dashed curves in Fig. 2. This fitting procedure allows extraction of the surface state dispersion characteristic (Fig. 2, inset). More careful treatments considering the effects of electronic structure [17] and STM tip motion [13] on measurements of step-edge scattering have been performed. Recent studies have also revealed the effects of electron–phonon and electron–electron interactions [18,19], as well as step-edge quantum transmittance [14] on surface wave formation. In addition, the anisotropic electronic structure of some surfaces has been analyzed by Fourier transforming 2-d wave patterns imaged by the STM [20,21].

(3)

(where A is a constant). (A more careful theoretical treatment of defect scattering, including 3-d effects, is presented in Ref. [23]). This kind of behavior has been seen experimentally in STM images of impurities and adsorbates on different noble metal surfaces [10,24,25]. Fig. 3 shows circular waves decaying in the LDOS at EF around a single Fe atom at the Cu(111) surface. Although the waves seen in Fig. 3 can be described by Eq. (3) [26], there remains some ambiguity in the phase shift extracted from the interference pattern around an individual scatterer [27]. More accurate phaseshift measurements can be made from the interference patterns observed around multiple scatterers [27]. Such interference is most easily observed when adsorbates are arranged into regular geometric patterns, as multiple scattering effects are then enhanced by constructive interference [27,28].

6. Scattering from atomically fabricated structures: quantum corrals One method for arranging adsorbate scatterers is to manipulate them with the tip of an STM. This is

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˚ image of a single Fe adatom on the Cu(111) surface (STM imaging parameters: V50.02 V, I51.0 Fig. 3. (a) Constant current 1303130 A nA). The concentric rings surrounding the Fe adatom are standing waves due to the scattering of surface state electrons with the Fe adatom (from Ref. [26]).

accomplished by bringing the tip of the STM close enough to an adsorbate that the in-plane component of the chemical binding force between the tip and adsorbate overcomes the lateral interaction between the adsorbate and the substrate lattice. The adsorbate then follows the motion of the STM tip across the surface [5]. A schematic of this procedure is shown in Fig. 4. Fig. 5 shows an example of a ring fabricated from 48 Fe atoms on Cu(111) using this technique [26] (the radius of this ‘quantum corral’ is ˚ 73 A). While the wave pattern observed in Fig. 5 reflects the electronic behavior of the quantum corral at one particular energy (EF ), the electronic response at different energies can be measured using dI / dV spectroscopy with the tip of the STM. Fig. 6 shows a

dI / dV spectrum measured while holding the tip of the STM stationary over the center of the quantum corral. Electronic interference effects lead to seven narrow resonances in the LDOS at the center of the ring. These can be thought of as reflecting the modes of an electron confined to the interior of the 2-d quantum corral. The width of the resonances represents the lifetime of a surface state electron in a quantum corral, and can be quantitatively understood using multiple scattering theory. One simple approach to multiple scattering in this context is to model the system in only two dimensions and to parametrize the scattering with an swave phase shift, d0 [27]. If the scattering from an atom on the surface is elastic then d0 is real, but if the scattering is inelastic or has an absorptive

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Fig. 4. Schematic representation of atomic manipulation using the ‘sliding process’. If the tip–adsorbate interaction overcomes the lateral corrugation potential, then the adsorbate will follow the motion of the tip along the surface.

component (i.e. if there is scattering out of the 2-d electron gas), then d0 is complex. The LDOS around a quantum corral can then be calculated by considering the STM tip to be a Green’s function point source of electronic amplitude. The electronic amplitude arriving at the site of an adatom from the tip can be written as ]] p 2 ] a Tsr jd 5 ]] e ikr j 2i 4 (4) pk 0 r

œ

where r j is the distance from the tip to the jth adatom. This causes a scattered wave from the adatom ] p 2i ≠ 0 2 2 1d e ik 0 r ] se a Tsr jdasrd 5 a Tsr jd ] e i 4 ]]] ]] (5) Œ]r pk 0 2i

œ

The surface LDOS is obtained by taking into account the various possible scattering paths an electron can make before returning to the tip. This results in an interference term to the LDOS [27]: DLDOS ~ Re f a¢ T ?s1 1 A 1 A2 1 A3 1 . . . d ? g 5 Re f a¢ T ? (1 2 A)21 ? a¢ g

(6)

where a T i 5 a Tsr id, a i 5 asr id, and A ij 5 a(ur¢i 2 r¢ju ).

For N scatterers A is an N 3 N matrix. a¢ T and a¢ change as the LDOS is calculated at each new point in space, but A remains constant. The dashed curve in Fig. 6 shows that the best fit to the quantum corral data is obtained in the limit that Im(d0 )→` [29]. This is referred to as the ‘black dot’ limit since all incoming s-wave amplitude is absorbed by the scatterer, which consequently emits no outgoing s-wave amplitude [27]. Fe atoms on the surface thus act as ‘electron absorbers’. The probable cause for this behavior is that Fe atoms couple 2-d surface state electrons to bulk states, although all possible inelastic channels must be considered. One interpretation of this result is that surface state electrons impinging upon Fe atoms have a relatively high probability for scattering to the interior of the copper crystal. Other analyses of the quantum corral structures have been performed, some disagreeing with the bulk scattering interpretation [30]. More sophisticated 3-d calculations, however, do support the idea that adsorbate atoms couple surface state electrons to bulk states [23,31,32]. These calculations explicitly take into account bulk states and have the advantage of being able to treat other possible scattering mechanisms. Quantum corral structures with simple closed geometries can also be analyzed in terms of the bound-states of 2-d ‘quantum boxes’ [26]. The vertical lines in Fig. 6, for example, show the energy levels expected for a 2-d round box having the same diameter as the corral shown in Fig. 5. Pits and metal islands at surfaces have also been seen to display similar quantum confinement behavior [33,34].

7. Electronic scattering from a surface superlattice When electrons scatter off a periodic potential, gaps open up in the electronic dispersion, as seen in crystal bandstructure. Similarly, if a periodic superstructure is imposed on an existing lattice, then new gaps open up at the reciprocal lattice vectors of the superlattice [35]. Such phenomena are typically observed in ‘reciprocal space’ [36], but one also expects ‘real space’ charge rearrangement to accompany superlattice formation [16]. An example of such behavior can be seen in recent STM observa-

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Fig. 5. Circular quantum corral built from 48 Fe atoms on the Cu(111) surface. Average diameter of ring (atom center to atom center) is ˚ An electronic quantum interference pattern can be seen inside of the Fe ring. (STM imaging parameters: V50.01 V, I51.0 nA) 142.6 A. (from Ref. [29]).

tions of superlattice effects in the LDOS of the reconstructed Au(111) surface [37]. Au(111) has one of the most elaborate surface reconstructions, displaying a stress-induced superstructure with a repeat distance of 23 surface lattice constants [38]. This reconstruction can be seen in ˚ patch of the Fig. 7, which shows an 8003800 A Au(111) surface. The reconstruction is seen as an alternating series of stripes and ridges. The thinner ˚ wide) have a hexagonal closestripes (about 25 A packed structure and are denoted the ‘HCP regions’ ˚ wide) have a face-cenwhile wider stripes (38 A tered-cubic structure and are known as ‘FCC regions’. Domain rotation of the stripes leads to the zig-zag appearance, hence the common term ‘herringbone reconstruction’ to describe Au(111).

In addition to the herringbone reconstruction seen in Fig. 7, waves can also be seen running parallel to the step-edges. These are due to step-edge scattering of the 2-d surface state electrons that exist on the Au(111) surface. Unlike the electrons imaged on copper, however, the 2-d electrons on Au(111) must scatter off the reconstruction superlattice, and so their electronic structure is correspondingly modified [21,37,39]. A recent STM spectroscopic survey of Au(111) shows the detailed energy dependence of the electronic density in the two halves of the superlattice unit cell [37]. As seen in Fig. 8, the dI / dV spectrum measured in the HCP half of the unit cell reveals an enhancement in electronic density just above the surface state bandedge, while the FCC half shows a corresponding depletion of charge (the

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potential is sketched in the inset to Fig. 9 [37]. The 2-d electronic eigenstates in this model potential can be written as Ck x k ysx,yd 5 e ik y y e ik x x u k xsxd, where u k (x) is a periodic Bloch function. The local density of states (i.e. what an STM measures) can then be expressed as follows [16,37]: LDOSsE,x,yd ]] k 9 a 1 b 2 m* 1 ]] 5 ]] dk xuCk x k ysx,ydu 2 ]]] ]]] p2 "2 E 2 œ eskxd

œ E

(7)

0

Fig. 6. Solid line: dI / dV spectrum taken with the STM tip held stationary over the center of the 48 atom Fe ring. Vertical lines: theoretical eigen energies for l50 states of a round, 2-d hard-wall box having the same dimensions as the 48 atom Fe ring. Broken line: multiple-scattering calculation of electronic LDOS in the ‘black dot’ scattering limit (from Ref. [29]).

bandedge is 0.52 eV below EF ) [37]. At slightly higher energy, however, the roles are reversed, and FCC electronic state density rises above the HCP density. This is perhaps best seen in Fig. 8(b), which shows the difference between the dI / dV spectra measured in the two regions. The difference is positive (i.e. the HCP spectrum is larger) just above the bandedge, but it drops sharply to negative value (i.e. the FCC spectrum is larger) at slightly higher energy. One way of understanding this behavior would be to calculate the electronic structure of Au(111), taking into account 2-d surface state electrons moving in the effective potential of the herringbone reconstruction. The detailed potential of this surface remains unknown, however. An alternative method, then, for qualitatively understanding the general behavior of 2-d electrons on Au(111) is to roughly approximate the reconstruction potential as an extended Kronig–Penney (KP) superlattice [35,37]. Since the low-energy electrons aggregate in the HCP region, that part of the reconstruction is taken as the ‘trough’ of the superlattice, while the FCC region is taken as the ‘crest’ of the KP potential (domain rotation is ignored in this simple model). The

Here e (k x ) is the 1-d KP dispersion [35] perpen]]] dicular to the stripes, k9 5œ(2m*E) / " 2 , m*50.26 m e is the effective mass of a surface-state electron, and ‘a’ and ‘b’ are the widths of the HCP and FCC regions, respectively. Fig. 9 shows the theoretical LDOS calculated at the centers of both the HCP and FCC regions of the extended KP potential [37]. As expected, the extended KP potential leads to a new bandstructure, including new energy gaps. These can be seen as sharp kinks in both theoretical curves, and occur at energies corresponding to the Brillouin zone boundaries of the KP superlattice. The LDOS does not go to zero at the gap energies because of freeparticle motion along the reconstruction stripes. At low energies (near the bandedge) a large peak in the HCP LDOS dominates over a depleted FCC LDOS, similar to the data seen in Fig. 8. This trend is reversed at slightly higher energy where the FCC LDOS rises over the HCP curve (also seen in the data). This behavior is more clearly seen in the theoretical difference curve in Fig. 9(b). The observed behavior of the surface state electronic density can thus be understood in the context of superlattice bandstructure. The increased state density in the HCP region at low energy is due to density build-up in the groundstate of the ‘attractive’ region of the unit cell, while the cross-over at higher energy is caused by a density shift to the ‘repulsive’ region as the first reconstruction-induced bandgap is crossed. This density cross-over is the natural separation of charge that occurs in the unit cell of a crystal when energy is shifted over a bandgap [35]. The theoretical difference curve in Fig. 9(b) can also be fit to the spectroscopic data in Fig. 8 (as shown by the dashed curve in Fig. 8) to extract a reconstruc-

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˚ image of the Au(111) surface at 6 K (STM imaging parameters: I50.5 nA, V50.01 V). The Fig. 7. Constant current 8003800 A herringbone reconstruction and surface-state standing waves are clearly visible (from Ref. [37]).

tion-induced well-depth of 25 meV. STM spectroscopy can thus be used to extract the effective potential felt by 2-d electrons moving at the reconstructed Au(111) surface [37].

8. Magnetic scattering at the atomic scale In systems characterized by strong Coulomb repulsion, the spin of the electron plays a more important role in scattering processes. Such interactions exist in the d- and f-orbitals of some materials, and give rise to magnetic behavior [40]. When a single atom of such material is present in a metallic host, its local magnetic moment engages in spin-dependent scattering with the itinerant electrons of the host. For a normal metal host, this process leads to the Kondo

effect and also induces magnetic coupling between atoms [4,40]. Such effects have long been studied using macroscopic probes, but the STM now makes it possible to study magnetic scattering effects at the atomic scale [41–43]. The added ability of atomic manipulation with the STM opens up new possibilities for studying magnetic scattering from precisely fabricated nanostructures [44]. Here we discuss some recent studies of magnetic atoms on metal surfaces using cryogenic scanning tunneling microscopy. Magnetic scattering from atomic-scale objects is most commonly understood by modeling the magnetic interaction with the ‘s–d’ Hamiltonian [40,45]: H 5 2 J s¢ ? S¢ ≠(r)

(8)

Here s is the quantum mechanical spin of a conduc-

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Fig. 8. (a) Average dI / dV spectra taken with the STM tip held over the HCP region and the FCC region of the Au(111) reconstruction. (b) Difference of HCP and FCC spectra shown in (a). Dashed line shows fit to data using extended Kronig–Penney model (from Ref. [37]).

tion electron, S is the quantum mechanical spin of the microscopic magnetic scatterer, and J is an exchange coupling constant. A consequence of this Hamiltonian is that electrons scattering off a magnetic impurity can undergo a spin-flip transition, as represented in Fig. 10(a). This is a fundamentally different process than normal potential scattering

Fig. 10. (a) Schematic representation of spin-flip scattering event between itinerant host electron and local magnetic moment of impurity. (b) Energy levels for magnetic impurity in Anderson model.

Fig. 9. (a) Theoretical LDOS calculated at the centers of the ‘HCP’ and ‘FCC’ regions of the extended Kronig–Penney potential. The inset shows a sketch of the potential. (b) Difference of theoretical LDOS curves shown in (a) (from Ref. [37]).

(where no spin-flip occurs), and is the basis for the Kondo effect [45]. The Kondo effect is so named because in 1964 J. Kondo showed theoretically that the spin-flip scattering induced by Eq. (8) leads to a low-temperature divergence in the magnetic scattering rate [45]. This explained low-temperature anomalies in transport properties observed in dilute magnetic alloys [40,46]. Later it was shown that below a characteristic ‘Kondo temperature’ (T K ), spin-flip scattering leads to a highly correlated groundstate where host electrons screen the spin of a magnetic impurity. The

M.F. Crommie / Journal of Electron Spectroscopy and Related Phenomena 109 (2000) 1 – 17

Kondo temperature depends sensitively on the exchange constant, J, and can be written as [4]

S D

1 kT K ¯ Du3r0 jvu 1 / 3 exp ]] 3r0 jv

(9)

here D is the bandwidth of the host metal, r0 is the local density of states at EF , and v is the volume of a unit cell. Electronic excitation of a ‘Kondo impurity’ leads to a narrow resonance located at EF whose half-width is given by k B T K [4]. The connection between the electronic structure and magnetic properties of an impurity is best seen in the Anderson model [47]. Here the d (or f)-orbital of a magnetic impurity is described as a distinct level in resonance with the conduction electron continuum. If just one electron occupies the impurity d-orbital then its energy is ed , but if two electrons occupy the level there is a strong Coulomb repulsion between them, represented by the energy U. The second electron (having opposite spin) must come in at a much higher energy, ed 1U, as seen in Fig. 10(b). Because of hybridization with the continuum states, these levels broaden into a ‘majority’ peak at ed and a ‘minority’ peak at ed 1U. For a magnetic moment to exist, the minority peak must be at least partially above EF [40,47]. An equivalence between the s–d Hamiltonian and the Anderson model can be shown via the Schrieffer–Wolff transformation [48]. Here the exchange constant of Eq. (8) can be written in terms of the microscopic parameters of the Anderson model [40,48]: 2

2V U J ¯ ]]]]]]] (ed 2 EF )(ed 1 U 2 EF )

(10)

V is the hybridization matrix element coupling conduction electrons to the impurity d-orbital. Much experimental work has been performed to study the electronic structure of magnetic impurity systems. Two techniques that are particularly useful for probing density of states (DOS) effects are photoemission and tunneling spectroscopy. Most of the photoemission work has focused on rare earth compounds [49]. The 4f ions in these materials are often considered to be ‘single’ Kondo impurities because the 4f orbital is so tightly bound. Temperature dependent measurements of the DOS of materials such as CeSi 2 [50] show a low temperature rise

11

in the spectral density at EF that has been interpreted as the Kondo resonance. Tunneling spectroscopy, on the other hand, has been performed on macroscopic junctions doped with magnetic impurities. These junctions show sharp dips and peaks near EF in dI / dV spectra that have been interpreted in terms of the Kondo effect [3]. In order to gain well-characterized local information on Kondo systems, however, one must turn to scanned probe techniques. Here we discuss a spatially resolved STM study of the surface Kondo system obtained by depositing cobalt atoms onto gold [42]. An STM study of a different surface Kondo system (Ce atoms on Ag [43]) is discussed by Berndt and Schneider elsewhere in this volume. An image of the Co /Au Kondo system can be seen in Fig. 11, which shows the Au(111) surface after deposition of only 0.1% of a monolayer of cobalt atoms at T56 K [42]. These magnetic impurities were spectroscopically probed by measuring dI / dV spectra in their vicinity with an STM tip [42]. The results of these measurements are seen in Fig. 12, which shows a pair of spectra taken on and off a single cobalt atom at the Au(111) surface. While the spectrum taken over bare gold is relatively featureless through EF , the spectrum taken over a cobalt atom displays a sharp, asymmetric resonance right at EF . The spatial extent of this resonance is seen in Fig. 13. Here different spectra are shown that were measured while moving the STM across the width of a single cobalt atom (the center of the atom is ˚ The resonance is seen to be spatially marked 0 A). ˚ localized to a radius of |10 A. This resonance is believed to be the Kondo resonance for a single magnetic impurity [42]. The observed lineshape, however, is unusual, as nearly all theoretical predictions for the Kondo resonance are for a Lorentzian-like, symmetric resonance at EF [4]. In order to understand the asymmetric lineshape seen in Fig. 12, one must consider the behavior of a magnetic impurity in the context of Fano resonances [51]. Fano resonances are known to occur when a spectroscopic measurement is performed on a discrete state in resonance with a continuum [51]. In the absence of electron–electron interactions, the result of a spectroscopic measurement on such a system (i.e. the rate of transitions to a final state of energy e ) can generally be written as [51]

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˚ of the Au(111) surface after deposition of 0.001 monolayer of Co at 6 K (STM imaging Fig. 11. Constant current image (4003400 A) parameters: I50.5 nA, V50.1 V). Approximately 22 Co atoms can be seen nestled among the ridges of the Au(111) herringbone reconstruction (from Ref. [42]).

e 2 e0 (q 1 e 9)2 R(e ) ¯ R 0 (e )]]] e 9 5 ]] 2 , G/2 11e9

(11)

Here e0 is the discrete state energy, G denotes the coupling of the discrete state to the continuum (i.e. the resonance width) and R 0 (e ) is the transition rate that would occur in the absence of the discrete level. q is an ‘asymmetry’ parameter equal to the ratio of matrix elements linking the spectroscopic probe to the discrete part (the numerator of q) and the continuum part (the denominator of q) of the transition final state. In an STM measurement the spectroscopic probe is the STM tip, the discrete state is an atomic orbital of the impurity atom, and the continuum is the conduction band of the host metal. A real magnetic impurity, however, is an example of an interacting discrete level in resonance with a

continuum [40,47]. Local magnetic moments (and the Kondo effect) result from strong Coulomb repulsion in the d- or f-orbital of an impurity. Fano’s original treatment (Eq. (11)) does not take Coulomb interactions into account. When interactions at the impurity are considered, then, as shown in Ref. [42], Eq. (11) is modified so that

e 2 e0 2 RehS(e )j e 9 5 ]]]]] ImhS(e )j

(12)

The basic form of the Fano resonance remains unchanged, but the energy location is shifted by the self-energy of the interacting impurity. With interactions, both the real and imaginary parts of the selfenergy may be complicated functions of e. However, near the Kondo resonance at EF (for temperatures less than T K ), one finds [42]

M.F. Crommie / Journal of Electron Spectroscopy and Related Phenomena 109 (2000) 1 – 17

G ImhS(e )j 5 ], e 2 e0 2 RehS(e )j 2 (e 2 a )G 5 ]]] 2k B T K

Fig. 12. A pair of dI / dV spectra taken with the STM tip held over a single Co atom and also over the nearby bare Au surface (a constant slope has been subtracted from both curves, and they have been shifted vertically). The feature identified as the Kondo resonance appears over the Co atom (the ratio of the amplitude of the resonance feature to the overall conductivity is 0.3). Dashed curve shows a fit to the data using the modified Fano theory (from Ref. [42]).

13

(13)

The simple Fano line shape continues to apply here, but with an overall width given by DE 5 2k B T K . The dashed curve in Fig. 12 shows a fit of Eqs. (11)–(13) to the spectrum obtained while holding the STM tip over the center of a Co atom. An excellent fit is obtained for the values k B T K 55.5 meV, a 54.5 meV, and q50.7. This fit reveals that the Kondo temperature for a Co atom on Au(111) is 70 K, confirming that the measurements were made in the T ,T K limit. This value of T K is much lower than values quoted for Co impurities in bulk Au, which average to approximately 500 K [46]. The lower surface T K value is not surprising because Co impurities in the bulk have more neighboring atoms, thus increasing the overlap of the d-orbital with conduction band states and leading to a wider bulk Kondo resonance (and higher bulk T K ). Similar behavior has been seen for Ce atoms on Ag(111) [43]. An alternative explanation, however, for the resonance seen in Fig. 12 is that it is not a Kondo resonance, but is due to a ‘bare’ d-resonance that happens to lie right at EF [52]. The main argument against this alternative interpretation is the narrow width of the observed resonance. If the Fano resonance is due to a bare d-orbital, then the width of the d-resonance would be only 12 meV. d-resonances, however, are typically more than 100 meV in width [46,53] (even for adsorbates [54]), whereas the Kondo resonance, a collective effect, is expected to have a much narrower width.

9. Atomically fabricated magnetic nanostructures

Fig. 13. A series of dI / dV spectra taken with the STM tip held at various lateral spacings from the center of a single Co atom on Au(111). (A constant slope has been subtracted from each curve and they have been shifted vertically). (From Ref. [42]).

The ability to both probe and manipulate individual magnetic atoms opens up the possibility of studying artificial atomic-scale magnetic structures. The magnetic and electronic properties of such structures vary dramatically depending on their size and shape [1]. Microscopic magnetic structures have already been studied in cluster beams, disordered

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M.F. Crommie / Journal of Electron Spectroscopy and Related Phenomena 109 (2000) 1 – 17

alloys, molecular solids, and at surfaces, with techniques ranging from magnetic susceptibility to magnetic force microscopy [55–58]. Of these techniques, cryogenic scanning tunneling microscopy is unique ˚ in that it yields both Angstrom spatial resolution and the ability to change microscopic structures by displacing individual atoms. One way of obtaining magnetic information on magnetic nanostructures is through the Kondo effect. Magnetic scattering in the Kondo effect causes rearrangement of electronic density and the formation of a narrow many-body resonance at EF [4]. Interactions between atoms in a magnetic structure are expected to change this behavior. This can occur either through the modification of a structure’s magnetic moment or through changes in the electronic scattering rate from a structure. Indirect exchange coupling between two magnetic impurities, for example, has been predicted to modulate the Kondo effect and even lead to novel non-Fermi liquid behavior [59]. Direct ferromagnetic and antiferromagnetic coupling between magnetic atoms has also been predicted to result in a variety of magnetic groundstates that should strongly affect Kondo behavior for small clusters [60,61]. In a first step toward directly probing magnetic coupling between individual atoms, experiments have been performed to examine the local spectroscopic response of cobalt atoms positioned at various separations using atomic manipulation [44]. Two cobalt atoms at the surface of Au(111) were studied by first acquiring spectra on the atoms while holding them at a large interatomic distance. The atoms were then moved closer together in increments, with both atoms undergoing spectroscopic survey at each separation. The central observation from this study is that the Kondo resonance for cobalt surface impurities does not depend on the distance between cobalt atoms until the atoms are brought together to an ˚ Once two cobalt interatomic distance less than 6 A. atoms are in such a ‘dimer’ configuration, the Kondo resonance abruptly disappears for both atoms [44]. This behavior is best explained as the result of reduced coupling between the magnetic moment of a ferromagnetic cobalt dimer and surrounding conduction electrons. Fig. 14 shows an STM image of the process of positioning two cobalt atoms at varying interatomic

Fig. 14. Process of building a cobalt dimer from two individual cobalt atoms on the surface of Au(111) (STM imaging parame˚ ters: I55310 29 A, V50.1 V). (a) The atoms initially are 15 A apart (center-to-center). (b) The atoms after being repositioned ˚ (c) The atoms after with the STM tip to a separation of only 9 A. being positioned into the final dimer configuration (approximate ˚ (From Ref. [44]). separation is 4 A).

˚ [44]. The distances of 15, 9 and approximately 4 A cobalt atoms were slid along the Au(111) surface using the atomic manipulation process described before. Local spectroscopic measurements were performed on the cobalt atoms at these different interatomic separations, as seen in Fig. 15 [44]. The top curve in Fig. 15 shows the dI / dV spectrum measured while holding the tip stationary over a bare patch of the Au(111) surface. This spectrum is relatively featureless as the energy is swept through EF . The second curve in Fig. 15 shows the dI / dV spectrum measured while holding the STM tip stationary over the center of an isolated cobalt atom on the gold surface. Here the Kondo resonance can be seen for a

M.F. Crommie / Journal of Electron Spectroscopy and Related Phenomena 109 (2000) 1 – 17

Fig. 15. dI / dV spectra obtained with the STM tip held over a single cobalt atom, an atomically fabricated cobalt dimer, and the clean gold surface (curves have been shifted vertically). The Kondo resonance can be seen for the individual cobalt atom, but is no longer present for the cobalt dimer. dI / dV55310 28 V21 for all three spectra at V50.100 V (the amplitude of the Kondo resonance reflects a 26% change in dI / dV for the single-atom spectrum). (From Ref. [44]).

single cobalt atom. Spectra measured on different cobalt atoms did not deviate significantly from this curve for inter-cobalt separations ranging from hun˚ ˚ [42,44]. At dreds of Angstroms down to 6 A ˚ separations less than 6 A, however, cobalt atoms merge into a dimer and their electronic properties change dramatically. The third curve in Fig. 15 shows the dI / dV spectrum measured with the STM tip held stationary over an atomically fabricated cobalt dimer. The Kondo resonance has disappeared. The disappearance of the Kondo resonance is a signature of the interaction between two cobalt atoms at the gold surface. In order to understand this interaction, one must address the different possible mechanisms that might cause such a disappearance. The three most likely mechanisms are as follows: (1) quenching of the dimer magnetic moment, (2) antiferromagnetic coupling between the cobalt atoms, (3) reduction of the exchange coupling between the dimer magnetic moment and surrounding conduction electrons. Careful consideration of these mechanisms points to the third as the most likely cause of the Kondo resonance disappearance. In order to understand the first mechanism, one must recognize that magnetic moments, in general, tend to form only when the electronic structure of an

15

object satisfies the Stoner criterion, r (EF )U .1 (where r (EF ) is the density of states at EF ) [47]. It is conceivable that a single impurity atom might satisfy the Stoner criterion while a dimer does not [62], thereby resulting in a quenched magnetic moment / atom for a dimer and no dimer Kondo effect. For the case of cobalt dimers, however, weak localization measurements [63] and local spin density (LSD) functional calculations all predict a robust cobalt moment of almost 2 mB / atom [61,64]. Quenching of the magnetic moment is therefore not a likely cause for the disappearance of the Kondo resonance in cobalt dimers. The next possible mechanism is anti-ferromagnetic (AF) coupling. Strong AF coupling between the cobalt atoms of a dimer should yield a net singlet ground state (total dimer spin50), thus ‘turning off’ the Kondo effect and removing the Kondo resonance. AF coupling can arise from either direct or indirect coupling mechanisms. We first consider the possibility of indirect AF coupling. The Kondo effect is predicted to quench when indirect coupling (also known as RKKY coupling [65]) is much stronger than the Kondo binding energy, k B T K [66]. These energies can be estimated. The RKKY coupling energy between two magnetic impurities separated by a distance r in the bulk of a metal can be written as [65] URKKY (r) 3 (Jv)2 3 (sin(2k F r) 2 2k F r cos(2k F r)) 5 ] ]]nk F ]]]]]]]] 8p EF (2k F r)4

(14)

Here n is electron density, k F is the Fermi wave vector, v is the volume of a unit cell, and J is the ‘s–d’ exchange constant which represents the coupling strength between conduction electrons and the magnetic moment of a cobalt atom. Using Eq. (9) and the known T K value of 70 K, Eq. (14) yields an antiferromagnetic maximum coupling between cobalt atoms of uURKKY u50.2 meV [44]. Cobalt dimers are thus in the limit uURKKY u, ,kT K , making RKKY coupling an unlikely mechanism for quenching the Kondo resonance of cobalt dimers. Direct AF interaction between impurities must be considered as a possibility when the impurities are in close proximity (i.e. close to nearest-neighbor spacing). The Alexander–Anderson (AA) model addres-

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ses this regime by treating a pair of magnetic atoms as idealized Anderson impurities with an added interatomic electronic hopping term [67]. The AA model implies that if EF lies close to the minority d-resonance, then the tendency is toward ferromagnetic (FM) coupling between atoms. Since cobalt adatoms are expected to have a d-level filling on the order of eight electrons [52], EF must intersect the minority spin manifold. FM coupling is thus the most likely groundstate for a cobalt dimer. This conclusion is supported by LSD calculations for free cobalt dimers [68] and cobalt dimers at a surface [61], all of which yield a FM dimer groundstate. Weak localization measurements also imply that neighboring cobalt atoms on gold couple ferromagnetically [63]. Direct AF coupling is thus an unlikely explanation for the disappearance of the Kondo resonance in cobalt dimers. The most likely reason for the Kondo resonance disappearance is a reduction in the exchange coupling between gold conduction electrons and cobalt impurities in the dimer configuration. As seen in Eq. (9), the Kondo temperature of a magnetic scatterer depends exponentially on the exchange coupling, J. This makes it possible for only a small reduction in J to drop T K below the experimental temperature of 6 K, thus causing the Kondo resonance to disappear for a dimer. To see that this is a plausible scenario, one should consider the connection between electronic structure and magnetic coupling displayed in the Schrieffer–Wolff expression for J (Eq. (10)). Assuming that the ‘bare atom’ parameters (V and U ) remain unchanged in a dimer, J is seen to decrease when the spin-split d-resonances shift away from the Fermi energy. This kind of behavior is expected for cobalt dimers according to the AA model, which predicts that minority d-resonances split away from EF upon FM dimer formation [67,69]. Such a shift of d-orbital states away from EF should lead to a reduction in J and, hence, a reduction of T K for a cobalt dimer (thus driving down the dimer Kondo resonance).

10. Conclusion The work discussed here exemplifies the utility of the STM as a spectroscopic probe of electronic

properties for atomic-scale structures at metal surfaces. STM spectroscopic imaging allows direct measurement of quantities, such as surface state scattering phase-shifts and transmission coefficients, that can be obtained in no other way. STM measurement of magnetic impurities allows direct, local measurement of electronic many-body effects that before could only be observed indirectly. The combination of atomic manipulation and STM spectroscopy opens up a wide realm of possibilities for directly probing the properties of novel, artificially fabricated nanostructures.

References [1] P. Jena, S.N. Khanna, B.K. Rao (Eds.), Science and Technology of Atomically Engineered Materials, World Scientific, Singapore, 1996. [2] J.A. Stroscio, W.J. Kaiser (Eds.), Scanning Tunneling Microscopy, Academic Press, San Diego, 1993. [3] E.L. Wolf, Principles of Electron Tunneling Microscopy, Oxford University Press, New York, 1985. [4] A.C. Hewson, The Kondo Problem to Heavy Fermions, Cambridge University Press, Cambridge, 1993. [5] D.M. Eigler, E.K. Schweizer, Nature 344 (1990) 524. [6] A. Zangwill, Physics at Surfaces, Cambridge University Press, Cambridge, 1988. [7] W. Shockley, Phys. Rev. 56 (1939) 317. [8] F. Forstmann, Z. Physik. 235 (1970) 69. [9] S.D. Kevan, R.H. Gaylord, Phys. Rev. B 36 (1987) 5809. [10] M.F. Crommie, C.P. Lutz, D.M. Eigler, Nature 363 (1993) 524. [11] Y. Hasegawa, P. Avouris, Phys. Rev. Lett. 71 (1993) 1071. [12] P.T. Sprunger, L. Petersen, E.W. Plummer, E. Laegsgaard, F. Besenbacher, Science 275 (1997) 1764. [13] J. Li, W.-D. Schneider, R. Berndt, Phys. Rev. B 56 (1997) 7656. [14] L. Buergi, O. Jeandupeux, A. Hirstein, H. Brune, K. Kern, Phys. Rev. Lett. 81 (1998) 5370. [15] J. Tersoff, D.R. Hamann, Phys. Rev. B 31 (1985) 805. [16] L.C. Davis, M.P. Everson, R.C. Jaklevic, W. Shen, Phys. Rev. B 43 (1991) 3821. [17] G. Hoermandinger, Phys. Rev. B 49 (1994) 13897. [18] O. Jeandupeux, L. Buergi, A. Hirstein, H. Brune, K. Kern, Phys. Rev. B 59 (1999) 15926. [19] L. Buergi, O. Jeandupeux, H. Brune, K. Kern, Phys. Rev. Lett. 82 (1999) 4516. [20] L. Petersen et al., Phys. Rev. B 57 (1998) R6858. [21] D. Fujita, K. Amemiya, T. Yakabe, H. Nejoh, T. Sato, Surf. Sci. 423 (1999) 160. [22] G. Baym, Lectures on Quantum Mechanics, Benjamin-Cummings Publishing, Reading, 1981. [23] S. Crampin, J. Phys.: Cond. Matt. 6 (1994) L613.

M.F. Crommie / Journal of Electron Spectroscopy and Related Phenomena 109 (2000) 1 – 17 [24] P. Avouris, I.-W. Lyo, R.E. Walkup, Y. Hasegawa, J. Vac. Sci. Technol. B 12 (1994) 1447. [25] L. Petersen, P. Laitenberger, E. Laegsgaard, F. Besenbacher, Phys. Rev. B 58 (1998) 7361. [26] M.F. Crommie, C.P. Lutz, D.M. Eigler, Science 262 (1993) 218. [27] E.J. Heller, M.F. Crommie, C.P. Lutz, D.M. Eigler, Nature 369 (1994) 464. [28] M.F. Crommie, C.P. Lutz, D.M. Eigler, E.J. Heller, Physica D 83 (1995) 98. [29] M.F. Crommie, C.P. Lutz, D.M. Eigler, E.J. Heller, Surf. Rev. Lett. 2 (1995) 127. [30] H.K. Harbury, W. Porod, Phys. Rev. B 53 (1996) 15445. [31] S. Crampin, M.H. Boon, J.E. Inglesfield, Phys. Rev. Lett. 73 (1994) 1015. [32] S. Crampin, O.R. Bryant, Phys. Rev. B 54 (1996) R17367. [33] P. Avouris, I.W. Lyo, Science 264 (1994) 942. [34] J. Li, W.-D. Schneider, R. Berndt, S. Crampin, Phys. Rev. Lett. 80 (1998) 3332. [35] C. Kittel, Introduction to Solid State Physics, John Wiley & Sons, New York, 1986. [36] X.Y. Wang, X.J. Shen, J.R.M. Osgood, R. Haight, F.J. Himpsel, Phys. Rev. B 53 (1996) 15738. [37] W. Chen, V. Madhavan, T. Jamneala, M.F. Crommie, Phys. Rev. Lett. 80 (1998) 1469. [38] J. Perdereau, J.P. Biberian, G.E. Rhead, J. Phys. F 4 (1974) 798. [39] D. Fujita, K. Amemiya, T. Yakabe, H. Nejoh, T. Sato, M. Iwatsuki, Phys. Rev. Lett. 78 (1997) 3904. [40] R.M. White, Quantum Theory of Magnetism, Springer–Verlag, New York, 1983. [41] A. Yazdani, B.A. Jones, C.P. Lutz, M.F. Crommie, D.M. Eigler, Science 275 (1997) 1767. [42] V. Madhavan, W. Chen, T. Jamneala, M.F. Crommie, N.S. Wingreen, Science 280 (1998) 567. [43] J. Li, W.-D. Schneider, R. Berndt, B. Delley, Phys. Rev. Lett. 80 (1998) 2893. [44] W. Chen, T. Jamneala, V. Madhavan, M.F. Crommie, Phys. Rev. B 60 (1999) R8529. [45] J. Kondo, Progr. Theor. Phys. 32 (1964) 37. [46] G. Gruener, A. Zawadowski, Rep. Progr. Phys. 37 (1974) 1497.

17

[47] P.W. Anderson, Phys. Rev. 124 (1961) 41. [48] J.R. Schrieffer, P.A. Wolff, Phys. Rev. 149 (1966) 491. [49] J.W. Allen, in: R.Z. Bachrach (Ed.), Synchrotron Radiation Research: Advances in Surface and Interface Science, Vol. 1, Plenum Press, New York, 1992, p. 253. [50] Y. Baer, F. Patthey, J.-M. Imer, W.-D. Schneider, B. Delley, Physica Scripta T25 (1989) 181. [51] U. Fano, Phys. Rev. 124 (1961) 1866. [52] M. Weissmann, A. Saul, A.M. Llois, J. Guevara, Phys. Rev. B 59 (1999) 8405. [53] P. Steiner, H. Hoechst, W. Steffen, S. Huefner, Z. Physik. B 38 (1980) 191. [54] N.D. Lang, Phys. Rev. Lett. 58 (1987) 45. [55] J. Shi, S. Gider, K. Babcock, D.D. Awschalom, Science 271 (1996) 937. [56] J.R. Friedman, M.P. Sarachik, J. Tehada, R. Ziolo, Phys. Rev. Lett. 76 (1996) 3830. [57] F.J. Himpsel, J.E. Ortega, G.J. Mankey, R.F. Willis, Adv. Phys. 47 (1998) 511. [58] M. Bode, M. Getzlaff, R. Wiesendanger, Phys. Rev. Lett. 81 (1998) 4256. [59] K. Ingersent, B.A. Jones, J.W. Wilkins, Phys. Rev. Lett. 69 (1992) 2594. [60] K. Wildberger, V.S. Stepanyuk, P. Lang, R. Zeller, P.H. Dederichs, Phys. Rev. Lett. 75 (1995) 509. [61] V.S. Stepanyuk, W. Hergert, P. Rennert, K. Wildberger, R. Zeller, P.H. Dederichs, Phys. Rev. B 54 (1996) 14121. [62] B.V. Reddy, M.R. Pederson, S.N. Khanna, Phys. Rev. B 55 (1997) R7414. [63] H. Beckmann, G. Bergmann, Phys. Rev. B 54 (1996) 368. [64] V.S. Stepanyuk, W. Hergert, P. Rennert, K. Wildberger, R. Zeller, P.H. Dederichs, J. Magn. Magn. Mat. 165 (1997) 272. [65] K. Yosida, Theory of Magnetism, Springer, New York, 1996. [66] C. Jayaprakash, H.R. Krishna-murthy, J.W. Wilkins, Phys. Rev. Lett. 47 (1981) 737. [67] S. Alexander, P.W. Anderson, Phys. Rev. 133 (1964) A1594. [68] I. Shim, K.A. Gingerich, J. Chem. Phys. 78 (1983) 5693. [69] A. Oswald, R. Zeller, P.J. Braspenning, P.H. Dederichs, J. Phys. F: Met. Phys. 15 (1985) 193.