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Fe clusters on Ni and Cu: size and shape dependence of the spin moment Ph. Mavropoulos, S. Lounis, R. Zeller, and S. Blu¨gel Institut fu¨r Festkörperforschung, Forschungszentrum Ju¨lich, D-52425 Ju¨lich, Germany (Dated: February 2, 2008) We present ab-initio calculations of the electronic structure of small Fe clusters (1-9 atoms) on 5 Ni(001), Ni(111), Cu(001) and Cu(111) surfaces. Emphasis is given on the spin moments, and 0 theirdependenceon cluster sizeand shape. Wederivea simplequantitativerule,which relates the 0 moment of each Fe atom linearly to its coordination number. Thus, for an arbitrary Fe cluster the 2 spin moment of the cluster and of the individual Fe atoms can be readily found if the positions of n theatoms are known. a J Small atomic clusters on surfaces constitute an ex- acluster. Inmanycasesjustthemeanmomentofclusters 4 1 tremely interesting subject, as their electronic structure ofaparticularsizeortheaveragemeanmagneticmoment lies in the transition from the behaviour in bulk to that per atom is found, averagedoveran ensemble of clusters ] of free molecules or atoms. In particular, the electronic of the same size but different shapes. This motivated us i c structure is characteristic of the cluster in that it de- toinvestigatethelocalandglobalspinmomentsofsmall s - pends on the cluster atoms, shape, size, and orientation, Fe clusters (up to 9 atoms) of monoatomic height and l r as well as on the substrate materialon which the cluster varying shape and size with a material specific theory mt is deposited. Thus it is terra incognita for the electronic of predictive power, namely density functional theory. properties. Moreover, since they are not hidden in the In this paper we summarise our results by establishing . t bulk, suchstructuresare directly accessibleby a number a simple quantitative rule which relates the local spin a m of experimental techniques, such as scanning tunneling moment of an Fe atom in the cluster linearly to the co- microscopy, x-ray magnetic circular dichroism, photoe- ordination number of its nearest neighbour atoms, with - d mission, etc. little influence of the structure of the atoms in the rest n In the area of magnetism, small clusters of magnetic of the cluster. In this way for any arbitrary Fe cluster o atomsonsurfacesareexpectedtoexperienceanenhance- the spin moment of the cluster and of the individual Fe c [ ment of the magnetic properties. As the cluster size de- atoms can be readily found if the positions of the atoms creases and the average coordination of the atoms be- are known. Even if the cluster shapes and sizes are not 2 comes smaller,the decreasedhybridisationofthe atomic exactly known experimentally, by making a reasonable v wavefunctions should lead to more pronounced mag- assumption of the variation of sizes and shapes the pro- 4 4 netic effects. Such trends with cluster size have been posed rule allows an estimation of the variation of the 6 monitored in recent experimental works; for example, total moment across the deposited clusters. Motivated 1 using x-ray magnetic circular dichroism, Lau and co- by the work of Lau et al. [1] we concentrated our at- 1 workers [1] have focused on Fe/Ni(001) systems, while tention on Fe clusters on the (001) and (111) surfaces 4 Gambardella and collaborators [2] have examined Co of the ferromagnetic substrate Ni and the non-magnetic 0 / clusters on Pt(111); and nuclear resonant scattering of substrate Cu. t a synchrotronradiationhasbeenemployedforthestudyof First-principlescalculationsofclustersonsurfacescan m Fe islands on W(110) [3]. Also other techniques, such as be computationally quite demanding, considering that - perturbed angular correlation [4] or spin-polarized scan- such systems break the translational symmetry in all d ning tunneling spectroscopy [5], allow the analysis of three directions. Most ab-initio methods explicitly ex- n magneticatomsonsurfacesinthesub-monolayerregime, ploitthe translationalsymmetryandthe clustersonsur- o c giving further impetus to the field. faces areapproximatedby huge,computationallyexpen- : Similar to the ferromagnetic Fe, Co, and Ni clusters, sive supercells which are periodically repeated. So far v i V, Cr, and Mn clusters should also have a strong intra- theyhavebeenappliedmostlyformagneticadatomsand X atomic exchange field; but, in contrast to the ferromag- chains [6]. A breakthrough in the treatment of mag- r netic clusters, their inter-atomic exchange should be an- neticclusterswasthe developmentofthe Greenfunction a tiferromagnetic, and due to competing interactions the method of Korringa, Kohn and Rostoker [7, 8] for the resulting magnetic order is in general non-collinear and embedding of clusters at surfaces. Subsequent applica- depends on individual details. Thus it will differ among tions [9, 10, 11] established this method as a powerful clusters,andtoinvestigatetheindividualmagneticorder tool. of such clusters is a challenge for both experiment and Our calculations are based on density-functional the- theory. oryinthe localspin-densityapproximation[13](LSDA). But even for homo-atomic and mass-selected clusters TheGreenfunctionmethodofKorringa,Kohn,andRos- itisverydifficulttoaddressexperimentallythemagnetic toker (KKR) is employed [8] to determine the spin den- propertiesofeachoneindividually,letaloneeachatomin sity and effective potential. Within the method, the 2 Greenfunctionoftheperturbedsystem,i.e.,surfaceplus (A) Moments in Fe clusters at Ni(001) surface cluster, is related to the one of the “host” reference sys- 1 3.24 tweamy (tthheeccolreraenctsuhbossttrabtoeu)nbdyarayDcyosnodnitieoqnusatairoen.inIcnlutdheids Average=3.24 2 3.10 3A 3.11 3B 3.10 2.91 automatically in the Green function, thus no supercell 3.10 2.96 3.10 construction is needed. In order to allow for a screening 2.93 2.93 Average=3.10 Average=3.03 3.11 of the perturbation induced by the cluster, at least the first neighbouring sites of all cluster atoms were consid- 2.93 2.93 4A Average=3.06 ered in the self-consistency cycle. The angular momen- Average=2.93 2.93 2.93 tum expansion of the Green functions was truncated at 4B 3.11 4C 2.91 3.10 5A 2.93 2.75 lmax = 3. Tests have shown that these parameters give reliable results for these systems. 3.12 2.72 3.12 3.10 2.91 3.12 A full-potential approach with the correct description Average=3.02 Average=3.01 Average=2.93 of the atomic cells [14] was used. Since we are focus- ing here on trend calculations, the atoms in the clusters 5B 3.13 6 2.94 2.94 7A 2.94 2.94 weresituatedintheunrelaxedlatticepositions,usingthe LSDAequilibriumlatticeparameterof6.46au(3.417˚A) 3.13 2.48 3.13 2.77 2.77 2.77 2.77 for Ni and 6.63 au (3.507 ˚A) for Cu [15]. 3.13 2.94 2.94 3.12 2.76 2.96 The clusters considered on the Ni(001) surface are Average=3.00 Average=2.89 Average=2.90 shownschematicallyinFigure1viewedfromthe top(all atomslieonthe surface). Theviewisadaptedtosurface ◦ 7B 2.95 2.94 8 2.95 2.95 9 2.97 2.80 2.97 geometry,meaning that it is rotatedby 45 with respect to the in-plane fcc cubic axes. The smallest cluster is a 3.12 2.55 2.79 2.95 2.59 2.80 2.80 2.63 2.80 singleFeadatom,whilethelargestconsistsof9Featoms. 2.95 2.94 2.95 2.80 2.97 2.97 2.80 2.97 Ineachatom,thecalculatedspinmomentiswritten,and the average (per atom) moment of each cluster is also Average=2.89 Average=2.87 Average=2.85 given. The Fe moment is always ferromagnetically cou- pled to the Ni substrate moment andto the Fe moments withintheclusters. Alreadyatafirstglanceitisobvious that the average moment of the clusters depends on the 3.2 (B) Fe on Ni(001) )B cluster size. The single adatom has manifestly the high- µ est moment (3.24 µB), while the 9-atom cluster shows a nt ( 3.0 e lower average moment of 2.85 µB. m o Thisbehaviourisexpectedonthegroundsofhybridisa- m 2.8 n tionoftheatomicdlevelswiththeneighbours. Atomsin pi S largerclustershave,onthe average,highercoordination, 2.6 thustheirdwavefunctionsaremorehybridised;thisleads to lesser localisation and lesser tendency to magnetism. 2.4 0 1 2 3 4 To pursue this idea further, we tried to correlate the Number of Fe neighbours local atomic spin moment to the nearest neighbour coordination of each atom, irrespective of the form or size FIG. 1: A: Spin moment (in µB) of atoms in 14 different Fe of the cluster. For instance, let us focus on all Fe atoms clusters ranging from 1 to 9 atoms on the Ni(001) surface, which have only one first Fe neighbour, i.e., Nc =1 (the and average (per atom) moment of the clusters (the view is coordinationtothe substrateis the same,Ns =4,for all surface-adapted, i.e., rotated by 45◦ with respect to the in- planefcccubicaxes;theclustersareviewedfromthetop,i.e., Fe atoms). Such atoms appear in the clusters with size all atoms lie on the surface). B: Linear trend for the atomic 2, 3, 4, 5, and 7; there are, in total, 10 such examples Fe spin moment as function of the coordination number of (having excluded cases which are trivially equivalent by nearest Feneighbours. symmetry). All of them have spin moments ranging in the small interval between 3.10 and 3.13 µB. Similarly, for the Fe atoms with two Fe neighbours the spin mo- mentrangesfrom2.91to2.97µB. Collectingallpossible expected. Forexample,Ref.7showsastudyofthemag- cases,fromNc =0(singleadatom)toNc =4,wepresent netism of small 4d atom clusters on Ag(001). There, the results in Figure 1B. it is reported that the susceptibility is highly non-local, A surprising feature is the almost linear dependence resulting even in an increase of the spin moment with of the spin moment on the coordination number. In the increasing coordination. This is related to the larger ex- general case, for arbitrary magnetic atoms, this is not tent of the 4d wavefunctions compared to the 3d ones of 3 Fe. In our case, Fe has a strong intra-atomic exchange (A) Moments in Fe clusters at Ni(111) surface field, arising from rather localised 3d wavefunctions and their positioning with respect to the Fermi level EF. In 1 3.34 3A 3B 3C a non-magnetic picture, EF is well within the d virtual Average=3.34 3.00 3.00 2.95 3.17 3.18 3.00 3.18 bound state, and the non-magnetic density of states at 2 3.16 3.16 3.00 3.17 EF, n(EF), is always well above the transition point to Average=3.12 the magnetic state: the Stoner criterion I ·n(EF) > 1 Average=3.16 Average=3.00 Average=3.10 (with I being the exchange integral) is safely fulfilled. 4A 4B 5A 5B 3.19 3.04 Tbohuurss,’twheavheyfubnricdtiisoantsiondooefstnhoet3daffleecvtelsthweitnhonthmeangeniegthic- 2.89 3.04 2.81 3.01 3.04 2.74 3.04 3.04 2.66 3.03 n(EF) critically. With increasing coordination, n(EF) 3.05 2.89 3.03 2.90 2.90 3.03 is graduallylowered,andso isthe intra-atomicexchange Average=2.97 Average=3.01 Average=2.92 Average=2.96 field. The tiny variation of the local moments for Fe 6B atomswiththe samenearestneighbourcoordinationbut 6A 3.04 6C being in different clusters may arise from intra-cluster 3.04 3.04 2.73 3.04 2.89 2.78 interference effects or indirectly via the different polari- 2.89 2.55 3.04 2.73 2.73 2.75 2.90 sation of the substrate. 2.92 2.89 3.04 Preliminary results on Co clusters, for which the 3d 3.03 Average=2.89 Average=2.88 virtual bound state is lower in energy, show that this Average=2.90 linear behaviour holds to a lesser extent, and much less so for Ni, where the intra-atomic exchange field is even 7A 3.04 7B 8 2.91 2.91 weaker [16]. A similar difference among Fe, Co, and Ni 2.91 2.91 2.89 2.54 3.04 2.91 2.45 2.91 clusters on Ag(001) has been shown in Ref. [11] and [9]. 2.91 2.48 2.91 On the other hand, Mn and Cr should have a strong 2.75 2.73 2.75 2.75 2.91 2.91 intra-atomic exchange field, but they present tendency 3.04 3.03 for antiferromagnetism or even non-collinear magnetic Average=2.85 Average=2.86 Average=2.83 structures [16]. In this respect, Fe is expected to be perhaps unique among the transition elements in showing 3.4 such a clear cut linear trend. Similar is the case of Fe clusters on the more compact 3.2 (B) Fe on Ni(111) Ni(111) substrate. The considered clusters, sized up to )B 8 atoms, are shown together with the spin moments in µnt ( 3 Figure 2A. Since the Fe adatomon the (111)surface has e m 2.8 only 3 Ni neighbours instead of 4 on the (001) surface, o m its spin moment is higher than on the (001) surface. On n 2.6 the other hand, the number of Fe neighbours (in-plane) pi S can increase up to Nc = 6, which makes the Fe moment 2.4 decreaseup to 2.45µB; the collectedstatistics (spin mo- 2.2 ments vs. Nc) are presented in Figure 2B. Again we see 0 1 2 3 4 5 6 a linear behaviour of the moment vs. Nc. Number of Fe neighbours Next we turn to Fe clusters on Cu(001) and Cu(111) surfaces,for whichwe consideredthe same types of clus- FIG. 2: A: Spin moments (in µB) of atoms in 15 different ters as in the case of Ni. Since the Cu lattice parameter Feclustersrangingfrom1to8atomsontheNi(111) surface, andaverage(peratom)momentsoftheclusters(theclusters is slightly larger than the one of Ni, one might expect are viewed from the top, i.e., all atoms lie on the surface). a lesser degree of hybridisation and therefore increased B:LineartrendfortheatomicFespinmomentasfunctionof spin moments at the Fe atoms. Nevertheless, the mo- thecoordination numberof Fe neighbours. ments are slightly lower,presumably because in the case of Ni the magnetisation of the substrate assists the spin polarisation of the cluster. The results for clusters on Cu(001)andonCu(111)areshowninFigure3Aand 3B of zero. This regime is not accessible by a mean field respectively. The result for the single adatom, showing theory as the LSDA. Thus the result shown here for the again the highest moment, should not be taken literally: adatom should be considered either as the trend of the it is well known that magnetic impurity atoms in the LSDAcalculation,orwhatoneexpectsabovethe Kondo bulk and on the surface of nonmagnetic hosts enter the temperature. For magnetic clusters (dimers and larger) Kondoregimeatlowtemperatures,whence the spinmo- the quantum fluctuations of the moment are strongly ment shows quantum fluctuations and has a net average suppressedandthe Kondotemperaturedecreasesdrasti- 4 Surf. a(µB) b(µB) 3.2 (A) Fe on Cu(001) Ni(001) 0.17 3.29 Ni(111) 0.15 3.33 )B µnt ( 3.0 CCuu((010111)) 00..1152 33..2213 e m o 2.8 m TABLE I: Parameters a and b from the linear fit of the spin n moment according to Eq. (1). pi S 2.6 2.4 WewouldliketothankP.H.Dederichsforfruitfuldis- 0 1 2 3 4 cussions. Financial support by the Priority Programme Number of Fe neighbours “Clusters in Contact with Surfaces” (SPP1153) of the Deutsche Forschungsgemeinschaft is gratefully acknowl- 3.2 (B) Fe on Cu(111) edged. )B µnt ( 3.0 e m o 2.8 m [1] J.T. Lau, A. Föhlisch, R. Nietuby`c, M. Reif, and pin W. Wurth,Phys. Rev.Lett. 89, 057201 (2002). S 2.6 [2] P. Gambardella, S. Rusponi, M. Veronese, S.S. Dhesi, C.Grazioli,A.Dallmeyer,I.Cabria,R.Zeller,P.H.Ded- erichs,K.Kern,C.Carbone,andH.Brune,Science300, 2.4 0 1 2 3 4 5 6 1130 (2003). [3] R. Röhlsberger, J. Bansmann, V. Senz, K. L. Jonas, Number of Fe neighbours A. Bettac, O. Leupold, R. Ru¨ffer, E. Burkel, and K. FIG.3: Lineartrendfor theatomic Fespin momentas func- H. Meiwes-Broer, Phys. Rev. Lett. 86, 5597 (2001); R. tion of the coordination in Fe clusters on Cu surfaces. A: on Röhlsberger,J.Bansmann,V.Senz,K.L.Jonas,A.Bet- Cu(001), and B: on Cu(111). tac, K. H. Meiwes-Broer, and O. Leupold, Phys. Rev. B 67, 245412 (2003). [4] K.Potzger,A.Weber,H.H.Bertschat,W.-D.Zeitz,and M. Dietrich, Phys.Rev.Lett. 88, 247201 (2002). cally with cluster size; thus the results givenhere for the [5] O. Pietzsch, A. Kubetzka, M. Bode, and R. Wiesendan- ger, Phys. Rev.Lett.92, 057202 (2004). clusters are realistic. [6] D.SpiˇsákandJ.Hafner,Phys.Rev.B65,235405(2002); The linear dependence of the spin moments is again Claude Ederer, Matej Komelj, and Manfred Fähnle, evident, except perhaps the result for the single adatom Phys. Rev. B 68, 052402 (2003); A. B. Shick, F. Máca, (Nc = 0). A fit of the calculated data, using a linear and P. M. Oppeneer,Phys. Rev.B 69, 212410 (2004). equation of the form [7] K. Wildberger, V.S. Stepanyuk, P. Lang, R. Zeller, and P.H. Dederichs, Phys. Rev. Lett. 75, 509 (1995); M =−aNc+b, (1) V.S.Stepanyuk,W.Hergert,K.Wildberger,S.K.Nayak, and P. Jena, Surf. Sci. 384, L892 (1997); V. S. Stepa- where M is the local spin moment, results in values for nyuk, W. Hergert, P. Rennert, B. Nonas, R. Zeller, and a and b given in Table I. We note that calculations on P. H. Dederichs, Phys.Rev.B 61, 2356 (2000); Fe nanostructures on Ag(001), reported for instance in [8] N. Papanikolaou, R. Zeller, P. H. Dederichs, J. Phys.: Refs. 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