Spin-polarization of platinum (111) induced by the proximity to cobalt nanostripes
Focko Meier, Samir Lounis, Jens Wiebe, Lihui Zhou, Swantje Heers, Phivos Mavropoulos, Peter H. Dederichs, Stefan Blügel, Roland Wiesendanger
SSpin-polarization of platinum (111) induced by the proximity tocobalt nanostripes
Focko Meier, Samir Lounis, Jens Wiebe, Lihui Zhou, Swantje Heers, PhivosMavropoulos, Peter H. Dederichs, Stefan Bl¨ugel, and Roland Wiesendanger Institute of Applied Physics, Hamburg University,Jungiusstrasse 11, D-20355 Hamburg, Germany Department of Physics and Astronomy,University of California Irvine, California, 92697 USA Institut f¨ur Festk¨orperforschung & Institute for Advanced Simulation,Forschungszentrum J¨ulich & JARA, D-52425 J¨ulich, Germany (Dated: September 10, 2018)
Abstract
We measured a spin polarization above a Pt (111) surface in the vicinity of a Co nanostripe byspin-polarized scanning tunneling spectroscopy. The spin polarization is exponentially decayingaway from the Pt/Co interface and is detectable at distances larger than 1 nm. By performingself-consistent ab-initio calculations of the electronic-structure for a related model system we revealthe interplay between the induced magnetic moments within the Pt surface and the spin-resolvedelectronic density of states above the surface.
PACS numbers: a r X i v : . [ c ond - m a t . m e s - h a ll ] O c t . INTRODUCTION The remarkable properties of magnetic nanostructures grown on non-magnetic metalsubstrates rely significantly on the electronic coupling between the atoms within the nanos-tructure and substrate atoms underneath. This electronic coupling determines e.g. thestrength and direction of the magnetic anisotropy as well as the total magnetic moment. Additionally the substrate electrons govern the collective behavior of ensembles of magneticnanostructures, e.g. by providing ferromagnetic order due to indirect exchange interactionbetween separated magnetic nanostructures. This interaction, also known as Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction, has been found in diluted magnetic systems,where magnetic 3d impurity atoms are dissolved in non-magnetic host metals.
In thesesamples, the localized magnetic moment of an impurity atom is screened by a spatially oscil-lating long range spin-polarization of the host conduction electrons. Therefore the distancebetween impurity atoms determines the sign and strength of the interaction, respectively.The same coupling has recently been observed directly for atoms on surfaces. A secondimportant effect takes place for magnetic 3d impurity atoms in host metals which nearlyfulfill the Stoner criterion, such as Pt and Pd, i.e. they are nearly ferromagnetic and aretherefore characterized by a high susceptibility. In these so called giant-moment dilute alloysthe 3d impurities induce relatively strong magnetic moments in the neighboring host atomswhich form a spin-polarized cluster. Since this effect can cause an additional exchangeinteraction between magnetic atoms in nanostructures it is important to obtain knowledgeabout the size of the polarization cloud and the decay of the induced magnetization withincreasing distance from the magnetic atom.
Both mechanisms are considered to be important for multilayer systems , like Co-Pt, whichconsist of sequences of ferromagnetic Co layers separated by non-magnetic Pt spacer lay-ers. The magnetic interlayer coupling between the ferromagnetic layers often shows devi-ations from a pure RKKY behavior, indicating that other mechanisms contribute to the totalmagnetic interaction. One contribution originates from magnetoelastic interactions due tointerface roughness between the magnetic and non-magnetic layers while with decreas-ing temperatures the induced magnetic moments of Pt become relevant for the magneticcoupling. In order to qualify specific contributions to the overall interaction a profoundknowledge on the local configuration of the interface is required.2n this work we present a combined experimental and theoretical study on the spin-polarization of Pt in the vicinity of Co nanostripes on a Pt(111) surface. We use spin-resolvedscanning tunnelling spectroscopy and the Korringa-Kohn-Rostoker Green function method(KKR) within the framework of density functional theory. Our experimental technique al-lows to obtain an extensive knowledge concerning the topographic, electronic as well asmagnetic properties of the sample. We show that the measured Pt local electronic densityof states (LDOS) near the Fermi energy in the vacuum exhibits an exponentially decayingspin-polarization indicating magnetic moments induced by the Co nanostripe. Interestinglythis effect can be observed for lateral distances from the Co nanostripe larger than four Ptlattice spacings where the RKKY interaction provides already an antiferromagnetic couplingas shown in a previous study. The calculated induced magnetic moments in the Pt surfaceclose to embedded Co atoms show a distance dependent oscillation between ferromagneticand antiferromagnetic alignment, while the vacuum spin-polarization at particular energiesexperiences an exponential decay in the lateral direction.
II. EXPERIMENTAL SETUP
All experiments were performed in an ultrahigh-vacuum system containing a home-built300 mK STM operating at a magnetic field B up to 12 T perpendicular to the samplesurface. In this work we used Cr-covered W tips, which are sensitive to the out-of-planedirection of the nanostripe magnetization (cid:126)M Co . In order to retain a strong spin polar-ization the tips were eventually dipped into Co stripes.
This procedure can result inattaching Co clusters to the tip apex which affects the magnetic B field required to switchthe tip magnetization (cid:126)M tip . Further details on the sample and tip preparation are given inRefs. . Co was evaporated at two different temperatures on a clean Pt(111) crystal. First,a tenth of an atomic layer (AL) was deposited at room temperature leading to Co nanos-tripes attached to the Pt(111) step edges. At a temperature below 25 K a much smalleramount was evaporated which resulted in a tiny number of single Co adatoms randomlydistributed on the surface. 3 II. EXPERIMENTAL RESULTS
Figure 1 (a) shows a Co nanostripe attached to a Pt step edge between two Pt terracesand individual Co adatoms. The one AL high Co nanostripe can be easily identified by adense network of dislocation lines originating from the lattice mismatch between Co andPt.
Obviously the Co stripe appears 20 pm higher than the Pt as visible in the linesection in Fig. 1 (b). Information regarding the spin-resolved LDOS in the vacuum abovethe Co nanostripe as well as above the Pt surface is obtained by measuring the differentialconductance d I /d U as a function of location r , the applied bias voltage U stab as well as therelative orientation between tip magnetization (cid:126)M tip and the sample magnetization (cid:126)M Co . From previous measurements on the nanostripes it is known that (cid:126)M Co is oriented out-of-plane. Figure 1 (c-e) show the resulting d I /d U ( r , U ) spectra taken on locations indicated in theinset on the Co nanostripe and on the Pt(111) close and far from the nanostripe. Here, (cid:126)M tip is switched up or down by B fields of +0 . − . (cid:126)M Co is constant. Thisallows to measure the d I /d U signal for parallel and antiparallel alignment of (cid:126)M tip and (cid:126)M Co .On the Co nanostripe (c.f. Fig. 1 (c)) the spin resolved d I /d U spectra show a dominantpeak located at -0.4 eV below E F which originates from a d -like Co surface resonance ofminority-spin character. The intensity of this state is changing considerably for paralleland antiparallel alignment of (cid:126)M tip and (cid:126)M Co . In contrast to that, the spectra on the barePt far from the nanostripe in Fig. 1 (e) do not show the electronic signature of the d -likesurface resonance but the onset of the unoccupied surface state at eU = 0 . Furthermore, no dependency on (cid:126)M tip is found as expected for a non-magnetic material.Figure 1 (d) shows spectra which have been taken on Pt but only at a distance of around1 nm with respect to the Co nanostripe. The spectra show the typical signature of a barePt(111) surface far from the Co stripe (c.f. Fig. 1 (e)). However, a clear dependency onthe relative orientation of (cid:126)M tip and (cid:126)M Co is now observed in an energy range from -0.5 eVto +0.5 eV around E F . Neither from our topographic nor spectroscopic data we have anyindications of Co incorporation into the Pt surface or sub-surface layers within the probedarea. This experimental result already proves a spin-polarization of the clean Pt(111) ata distance of more than three lattice spacings to the Co nanostripe.In order to obtain information about this induced spin polarization we probed the spatially4esolved d I /d U signal (d I /d U map) in a boundary area shown in Fig. 2 (a). For this aread I /d U maps have been recorded at U stab = +0 . − . . − . I /d U maps obtained at B = +0 . B = +1 . (cid:126)M tip and (cid:126)M Co has changed due to a B fieldinduced (cid:126)M Co reversal. The d I /d U signal above the Pt terrace appears the same in bothfigures. However, a difference in d I /d U intensity above Pt close to the stripe is observed.From the sequence of B field depending d I /d U maps local magnetization curves are obtainedby plotting the d I /d U signal at one image point as a function of B . Figures 2 (d)– (g) showlocal magnetization curves taken at positions as marked in Fig. 2 (a). The magnetizationcurve of the Co stripe in Fig. 2 (d) shows two magnetic states and a square-like hysteresisindicating its ferromagnetic state and a coercivity of B c = 0 . ± .
05 T. Strikingly, themagnetization curves measured on the Pt in the vicinity of the Co nanostripe show thatthere is an explicit link between the magnetic state of the Co stripe and the spin polarizationmeasured on the Pt. Similar magnetization curves have been recorded for each point of thearea of Fig. 2 (a). From these magnetization curves the so-called spin asymmetry A spin iscalculated by A spin = dI/dU ↑↑ − dI/dU ↑↓ dI/dU ↑↑ + dI/dU ↑↓ . (1)which characterizes the square-like magnetization curves and is a measure for the spin-polarization at eU in the vacuum. dI/dU ↑↑ and dI/dU ↑↓ denote the averaged values fromall red and blue data points in the magnetization curves (Fig. 2 (d)-(g)), i.e for parallel andantiparallel alignment of (cid:126)M tip and (cid:126)M Co in each curve. An asymmetry value is obtained foreach image point. This results in an asymmetry map shown in Figure 3 (a). The Co stripeshows a strong negative A spin while on the Pt terrace far from the stripe A spin is zero. Abovethe Pt close to the Co stripe an area with positive A spin is visible which fades out for anincreasing distance from the nanostripe. The decay is further analyzed in Fig. 3 (b) whichshows A spin values below the section line in Fig. 3 (a) as a function of the distance d fromthe Co nanostripe. In order to quantify the decay behavior the graph in Fig. 3 (b) has beenfitted to a simple exponential function f = Ce − x/λ (2)5here C and λ denote the amplitude and the decay length, respectively. Even though theexact value of λ depends on the specific line section, values in the range from λ = 0 . λ = 1 . A spin calculated fromd I /d U ( r ) maps recorded at U stab =-0.1 V (cp. inset Fig.3(b)). Together with the dependencyon the spin-resolved d I /d U -curves measured close to the Co stripe (cp. Fig. 1 (d)) weconclude that the observed spin-polarization above Pt is present in a large energy windowaround the Fermi energy. This result suggests that the measured spin-polarization is dueto an exponentially decaying magnetic moment (cid:126)M P t induced by the vicinity to the Conanostripe.Figure 3 (b) also includes the experimentally obtained indirect exchange energies, J ,between the Co nanostripe and single Co adatoms as published in Ref. . A positive J corresponds to a ferromagnetic coupling while a negative value corresponds to an antiferro-magnetic coupling. A damped oscillatory exchange interaction is present in the same rangewhere the exponentially decaying Pt vacuum spin-polarization is measured. It was shownin Ref. that the exchange interaction can be described by RKKY like exchange and fol-lows in a good agreement a 1D range function. In case of a strong contribution of the Ptpolarization to the magnetic coupling one would expect a dominance of ferromagnetic or an-tiferromagnetic coupling for the overall magnetic exchange interaction. Such an effect wouldbecome visible by a shift of the RKKY-curve towards positive or negative exchange energieswhich is not observed. These observations raise the question, how exactly the measured Ptspin-polarization is linked to the induced magnetization within the Pt surface. IV. THEORETICAL METHOD
In order to obtain deeper insight into the relation between the measured spin-polarizationin the vacuum and the induced magnetization we performed calculations on three differentarrangements of Co on or in a Pt(111) surface layer as shown in Fig. 4 (a). First, we con-sidered a single Co atom deposited on (adatom) and embedded in (inatom) the first layer ofPt(111). These two arrangements differ mainly in the number of next neighboring Pt atomswhich is tripled for the inatom with respect to the adatom case. Therefore a comparisonof these two cases provides us with important information concerning the hybridization of6he Co electronic states with those of the Pt surface leading to the magnetization of thesurrounding Pt atoms.In order to model the experimental setup as close as possible we constructed a chain offive Co atoms embedded in the surface of Pt(111). This model arrangement reflects theexperimental fact that the Pt surface atoms which show a vacuum spin-polarization arelocated at the same layer than the Co atoms which form the nanostripe. The chain isoriented along a direction perpendicular to the direction probed experimentally concerningthe spin-polarization (cp. Fig. 3 (a)). The exact experimental setup is of course difficult toachieve since a non-regular step edge of platinum interfacing a cobalt stripe is impossibleto reproduce with methods based on Density Functional Theory at the actual stage. Themethod of investigation is the KKR method within the framework of Density FunctionalTheory.KKR is based on multiple–scattering theory. For non–overlapping potentials the followingangular momentum representation of the Green’s function G ( r + R n , r (cid:48) + R n (cid:48) ; E ) can bederived: G ( r + R n , r (cid:48) + R n (cid:48) ; E )= − i √ E (cid:88) L R nL ( r < ; E ) H nL ( r > ; E ) δ nn (cid:48) + (cid:88) LL (cid:48) R nL ( r ; E ) G nn (cid:48) LL (cid:48) ( E ) R n (cid:48) L (cid:48) ( r (cid:48) ; E ) (3) R n , R n (cid:48) refer to the atomic positions and E is the energy. r < and r > denote the shorterand longer of the vectors r and r (cid:48) which define the position in the Wigner–Seitz (WS) cellcentered around R n or R (cid:48) n . The R nL ( r ; E ) and H nL ( r ; E ) are the regular and irregular solutionof the Schr¨odinger equation.The structural Green functions G nn (cid:48) LL (cid:48) ( E ) are then obtained by solving the Dyson equationfor each spin direction. G nn (cid:48) LL (cid:48) ( E ) = g nn (cid:48) LL (cid:48) ( E )+ (cid:88) n (cid:48)(cid:48) ,L (cid:48)(cid:48) L (cid:48)(cid:48)(cid:48) g nn (cid:48)(cid:48) LL (cid:48)(cid:48) ( E )∆ t n (cid:48)(cid:48) L (cid:48)(cid:48) L (cid:48)(cid:48)(cid:48) ( E ) G n (cid:48)(cid:48) n (cid:48) L (cid:48)(cid:48)(cid:48) L (cid:48) ( E ) (4)The summation in (4) is over all lattice sites n (cid:48)(cid:48) and angular momenta L (cid:48)(cid:48) , L (cid:48)(cid:48)(cid:48) for whichthe perturbation ∆ t n (cid:48)(cid:48) L (cid:48)(cid:48) L (cid:48)(cid:48)(cid:48) ( E ) between the t matrices of the real and the reference system issignificant. g nn (cid:48) LL (cid:48) is the structural Green function of the reference system, i.e. in our case theideal Pt(111) surface. 7he real-space solution of the Dyson equation requires a cluster of perturbed atomicpotentials that include the potential of Co impurities and the first shell of neighboring cells.It is important to note that the vacuum region is filled with cellular (Voronoi) potentials.Since our aim is to explain the STM measured spectra, we use the Tersoff-Hamann theory and calculate the local density of states in the vacuum at 4.1 ˚A above the substrate. Afterobtaining a self-consistent Co potential with its neighboring shell, one additional calculationis performed including Pt atoms as well as their neighboring vacuum cells at 4.1 ˚A abovethe substrate along a given direction. V. THEORETICAL RESULTS
For an individual Co adatom and Co inatom we calculated the induced magnetic moments M P t in the Pt substrate along two directions as indicated in Fig. 4 (a). Figures 4 (b)-(e)show M P t as a function of the distance d from the impurity for the [1¯10] and [11¯2] direction.Concerning the [1¯10] direction we find for both arrangements a long range oscillation in M P t with a wavelength of about 1 nm for the adatom (cp. Fig. 4 (b)) and a slightly smaller onefor the inatom (cp. Fig. 4 (d)). The oscillation indicates that M P t is either ferromagneticallyor antiferromagnetically aligned with the Co impurity dependent on the distance. However,the total integrated net moment of the Pt atoms is positive. Along the [11¯2] direction theoscillatory behavior is much weaker than the one obtained along the [1¯10] direction for botharrangements (Fig. 4 (c),(e)). Here more Pt atoms are coupled ferromagnetically to the Coimpurity. This directional dependence proves that the induced magnetization is anisotropicwhich originates from the non-spherical Fermi surface characterizing this system as foundin the directional dependent RKKY interactions between Co adatoms on a Pt(111) surfaceor in the anisotropic induced charge oscillations caused by Co impurities buried below Cusurfaces.
A comparison of M P t for the same direction shows that for the same distancesthe intensity is always higher for the embedded atom than for the adatom. This emphasizesthe importance of the number of neighboring atoms and indicates a dependence of thecoupling between the Co and Pt electronic states on the coordination and environment. Tofavor the coupling to the impurity states, the electronic states controlling the studied longranged magnetization must be localized at the surface. Constant-energy contours at theFermi energy are plotted in Fig. 5(a) for the simulated Pt(111) surface with their relative8ocalization on the surface layer. The degree of localization is depicted in colors: red formaximum localization, blue for minimum. There is a finite number of contours due to thefact that the surface is simulated with a finite number of Pt layers. The shape of the contoursis non-trivial indicating the complexity of the problem. This type of calculations indicatethe presence of several states which are resonant-like. To measure the degree of couplingbetween these states and those of the Co impurity, we decompose the Fermi surface in 10parts represented within the red-yellow triangle in Fig. 5(a). Each part includes more or lesslocalized states. Afterwards, we calculate the induced magnetization at the Fermi energy E F induced by every part. The structural Green function g of Pt(111) needed in Eq. 4is given as a Fourier transform or integral over the first Brillouin zone. This integrationcan be done for every region defined in Fig. 5(a) leading to values that can be pluggedinto Eq. 3 and Eq. 4 to compute the contribution of every region in the magnetization ofPt at E F . For the inatom case, it seems that parts 7, 8 and 10 are contributing most tothe induced M P t (cp. Fig. 5(b)). By summing up all parts, we approximately recover thetotal energy integrated magnetization (cp. Fig. 4(d)). We do not expect them to be equalsince with the decomposition scheme some scattering events cancel each other and other“back-scattering” events are not taken into account properly. This theoretical experiencedemonstrates the non-trivial link between the induced long range magnetization and theconstant energy contours of the substrate, their degree of localization on the surface layersand coupling strength with the impurities.Figure 4 (f) shows the M P t for Pt atoms perpendicular to the embedded Co chain(Fig. 4 (a)), as a function of distance d from the chain, which is the setup most similarto the experimental one. In contrast to the experimentally observed decreasing of the vac-uum spin-polarization, an oscillating decaying M P t is observed. Similar to Figs. 4 (b)-(e)the curve clearly exhibits the same damped oscillating behavior but shows overall higherintensities which reflects the contributions from all the Co atoms within the chain. In orderto investigate the relation between M P t and the energy-dependent spin-polarization we cal-culated the vacuum LDOS for majority and minority spin states above the Pt atoms alongthe direction perpendicular to the chain at a vertical distance of 4.1 ˚ A . This correspondsto two interlayer distances from the surface and is the range of the experimental z-height ofthe tip. Figures 6 (a)-(d) show the spin-resolved vacuum LDOS for the first, second, thirdand fifth Pt atom located in the experimental relevant direction. They reveal an intensity9ncrease starting at about +0.3 eV which is due to the Pt surface state. Concerning thedifference between the two spin channels it is quite obvious that the Pt atom closest tothe chain experiences the strongest imbalance of majority and minority electrons. This isvisualized by a corresponding calculated spin asymmetry A cal ( E ) given by A cal ( E ) = LDOS maj ( E ) − LDOS min ( E )LDOS maj ( E ) + LDOS min ( E ) (5)where LDOS maj ( E ) and LDOS min ( E ) denote the energy dependent vacuum LDOS for ma-jority and minority electrons. A cal ( E ) is plotted in Figs. 6 (a)-(d) for the Pt atoms as well.These curves reveal that neither the absolute value nor the sign of the spin asymmetry A cal ( E ) is conserved when scanning at different bias voltages around the Fermi energy. Ad-ditionally the absolute value of the spin asymmetry A cal ( E ) at given energies changes withincreasing distance form the Co chain. At some energies even a sign change is observed.Figure 6 (e) shows the calculated spin-asymmetry A cal ( E ) for +0.3 eV and -0.1 eV, whichare experimentally relevant, for different distances form the chain. A comparison of thesecurves with the experimental data obtained at 0.3 V and shown in Fig. 3 (b) reveals thatthe calculated asymmetry A cal (+0 . eV ) follows the shape of the experimental curves, i.e. itis always positive and shows an exponentially decaying behavior. A fit as in Eq. 2 gives avalue for the decay length λ of about 4 ˚ A which is less than half of the experimental value.The calculated spin-asymmetry at − . VI. DISCUSSION
Recently several theoretical studies concentrated on probing and describing magneticproperties of Co nanostructures on Pt(111) quantitatively and qualitatively. They treatedCo in different configurations and environments, like Co overlayers on Pt(111) , Conano wires attached to Pt(111) step edges and isolated Co adatoms on bare Pt(111)surfaces . Even though these configurations lead to different coordination numbers,which results in different numbers of underlying Pt atoms per Co atom, they showconsistently an induced spin moment M spin of the nearest neighboring Pt atoms in therange from 0.1–0.3 µ B which is about one magnitude larger than the orbital moments M orb .10herefore the total induced magnetic moment M P t of Pt atoms is mainly determined bythe spin moment M spin .Additionally it has been found in these calculations that the induced Pt magnetizationdecreases very rapidly with the distance from the Co structures by about one order ofmagnitude for the second and third nearest neighbors as shown for the Co nano wiresin Ref. . Here we probed experimentally and theoretically M P t for longer distances farfrom the Co impurities. We find that induced magnetic moments in the surrounding Ptsurface atoms are not constantly parallel or antiparallel aligned with the magnetic momentof the Co impurity. The sign as well as the strength of the induced magnetic moments isadditionally highly influenced by the strong anisotropy of the Fermi surface of Pt. Bothunderlines that for the probed arrangements of Co on and in the Pt(111) surface one cannotexpect a constantly aligned polarization cloud as found for Co-Pt and Fe-Ir multilayers.
The apparent contradiction of the measured monotonously decaying A spin in the vacuumand the calculated oscillating M P t for the embedded Co chain arrangement can be explainedby local changes of the electronic structure of the Pt atoms close to the embedded chain(cp. Figs. 6 (a)-(d)). It is evident also that the hybridization between the Pt and the Costates changes with increasing the distance from the chain. Therefore also the spin-averagedLDOS changes laterally which can be obtained by calculating the arithmetic mean of theLDOS for both spin types in Figs. 6 (a)-(d). According to Ref. the measured spin-resolved dI/dU signal and the deduced spin-asymmetry is a measure of the energy dependentspin-polarization of the sample. This quantity is only a measure for the magnetization,which is an integrated quantity of majority and minority states up to the Fermi energy, ifthe spin-averaged LDOS is constant.
Therefore the induced magnetization of the Pt cannotbe deduced from our experimentally detected vacuum spin-polarization in the Pt only.
VII. CONCLUSIONS
In conclusion, we have performed SP-STM measurements on Pt(111) in the proximityto Co nanostripes at 0.3 K. By probing locally a spin-polarization of Pt, we observed forthe first time induced magnetic moments in a non-magnetic material on a local scale. Themeasured vacuum spin-polarization decays exponentially as a function of the distance from11he Co nanostripe with a decay length of about 1 nm.Self-consistent electronic-structure calculations of a Co chain embedded in the Pt(111) sur-face, of the neighboring Pt atoms and of the vacuum LDOS above the Pt allow us to provethat the measured spin-polarization is induced by an oscillating and highly anisotropic mag-netization within the Pt surface in the proximity to Co. By investigating the Fermi surfacecontours of Pt(111) and their degree of localization on the surface layer, we found severalstates with anisotropic shapes that could couple to the electronic states of Co impuritiesand thus contribute to the long range induced magnetization.
VIII. ACKNOWLEDGMENTS
We acknowledge financial support from the SFB 668 and GrK 1286 of the DFG, from theERC Advanced Grant FURORE, and from the Cluster of Excellence Nanospintronics. S. L.wishes to thank the Alexander von Humboldt Foundation for a Feodor Lynen Fellowshipand thanks D. L. Mills for discussions and hospitality. The computations were performedat the supercomputer JUROPA at the Forschungszentrum J¨ulich. S. Bl¨ugel, M. Weinert, and P. H. Dederichs, Phys. Rev. Lett. , 1077 (1988). P. Gambardella, S. Rusponi, M. Veronese, S. Dhesi, C. Grazioli, A. Dallmeyer, I. Cabria,R. Zeller, P. Dederichs, K. Kern, et al., Science , 1130 (2003). J. P. Pierce, M. A. Torija, Z. Gai, J. Shi, T. C. Schulthess, G. A. Farnan, J. F. Wendelken,E. W. Plummer, and J. Shen, Phys. Rev. Lett. , 237201 (2004). M. A. Ruderman and C. Kittel, Phys. Rev. , 99 (1954). T. Kasuya, Prog. Theor. Phys. , 45 (1956). K. Yosida, Phys. Rev. , 893 (1957). L. D. Graham and D. S. Schreiber, Phys. Rev. Lett. , 650 (1966). F. Meier, L. 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Filled green circles indicate the closest considered atoms for the calculation ofthe induced moments in each specific direction. Filled green circles close to a Co atom mark firstconsidered Pt atoms for specific directions. (b)-(e) Induced magnetic moments in Pt atoms M P t fortwo indicated directions as a function of distance d from a Co adatom and Co inatom. (f) Inducedmagnetic moments in Pt atoms M P t as a function of distance d from an embedded Co chain forexperimentally relevant directions. Some values in (b)-(f) have been scaled down by the indicatedfactors in order to fit into the figure. IG. 5: (a) Constant energy contours calculated at the Fermi energy, E F , where colors representthe degree of localization of the different electronic states on the surface layer of Pt(111). Inaddition, a triangle divided in ten regions is superimposed on the energy contours. 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