Correlation effects and orbital magnetism of Co clusters
L. Peters. I. Di Marco, O. Grånäs, E. Şaşıoğlu, A. Altun, S. Rossen, C. Friedrich, S. Blügel, M. I. Katsnelson, A. Kirilyuk, O. Eriksson
aa r X i v : . [ phy s i c s . a t m - c l u s ] M a y Correlation effects and orbital magnetism of Co clusters
L. Peters, ∗ I. Di Marco, O. Gr˚an¨as, E. S¸a¸sıo˘glu, A. Altun, S. Rossen,
1, 3
C. Friedrich, S. Bl¨ugel, M. I. Katsnelson, A. Kirilyuk, and O. Eriksson Institute for Molecules and Materials, Radboud University Nijmegen, NL-6525 AJ Nijmegen, The Netherlands Department of Physics and Astronomy, Uppsala University, Box 516, SE-75120, Uppsala, Sweden Peter Gr¨unberg Institut and Institute for Advanced Simulation,Forschungszentrum J¨ulich and JARA, 52425 J¨ulich, Germany Department of Physics, Fatih University, 34500 Istanbul, Turkey (Dated: June 1, 2016)Recent experiments on isolated Co clusters have shown huge orbital magnetic moments in com-parison with their bulk and surface counterparts. These clusters hence provide the unique possibilityto study the evolution of the orbital magnetic moment with respect to the cluster size and how com-peting interactions contribute to the quenching of orbital magnetism. We investigate here differenttheoretical methods to calculate the spin and orbital moments of Co clusters, and assess the perfor-mances of the methods in comparison with experiments. It is shown that density functional theory inconventional local density or generalized gradient approximations, or even with a hybrid functional,severely underestimates the orbital moment. As natural extensions/corrections we considered theorbital polarization correction, the LDA+U approximation as well as the LDA+DMFT method.Our theory shows that of the considered methods, only the LDA+DMFT method provides orbitalmoments in agreement with experiment, thus emphasizing the importance of dynamic correlationseffects for determining fundamental magnetic properties of magnets in the nano-size regime.
I. INTRODUCTION
The orbital magnetic moments of transition-metal bulkmagnets are largely quenched , while transition-metalsurfaces have extremely large orbital moments . Un-derstanding the nature of the orbital moment is a prob-lem of fundamental interest, and has over the years at-tracted much experimental attention through techniquessuch as XMCD and ferromagnetic resonance . The or-bital moment is important for several reasons. First, itcontributes to the total magnetic moment of a system,and second it is together with the spin magnetic momenta measure of the extent of spin-orbit coupling in gen-eral and magnetic anisotropy in particular . The latterproperty is also known to couple to the anisotropy of theorbital moment . As a result, a detailed knowledge oforbital magnetism is crucial in designing new materialswith desired hardness and saturation moments. Most in-terestingly, recent XMCD experiments on transition-metal clusters showed huge orbital moments in compar-ison with their bulk counterparts. Hence these systemsmay possess a large magnetic anisotropy energy, whichmakes them potentially interesting for several technolog-ical applications. In this letter we investigate this possi-bility by means of several computational approaches.Since the orbital magnetic moments of transition-metalclusters lie between those of the quenched values of thebulk systems and the large values of the isolated atoms,clusters provide a unique opportunity for studying themechanisms that affect the orbital magnetic momentssystematically. Unfortunately, there is up to now notheory available that can reproduce the experimentallyobserved large orbital moments for the transition-metalclusters. We formulate here, using several computationalmethods, a theory of orbital and spin magnetism for these clusters. In particular, we take Co clusters as a test case,because Co atoms possess the largest orbital momentsamong the transition-metals in all their forms, as clus-ters , surfaces and bulk .Previous theoretical studies on the magnetic structureof transition-metal clusters focus primarily on thespin moment, due to the difficulties in estimating the or-bital moment. The only theoretical study that addressesthe orbital magnetic moment of pure clusters is focusedon Co , which is technically more treatable. In Ref. 10it is shown that the calculated spin moment in general isin reasonable agreement with experiment. Although thecalculated spin and orbital magnetic moments of Co clusters capped with Pt are available, there are no ex-perimental data to compare with these calculations. Thelatter are also performed within GGA, which is in thepresent work shown to be inadequate for these systems,e.g. by severely underestimating the orbital magneticmoments with respect to experiment . Therefore, itis expected that also the previous study on the orbitalmagnetism of Pt capped Co clusters suffers from theinadequacy of GGA functionals.The problem to face when calculating the orbital mo-ment from conventional density functionals is the ab-sence of Hund’s second rule, which is the primary rea-son for the orbital moment and is driven by intra-shellelectrostatic interaction. The crystal field effect com-petes against this interaction, and results in a quenchingof the orbital magnetism. It will be shown here thatfor complex systems like clusters, a high level theoryis required to properly describe the subtle competitionamong these effects. More precisely, it will be shown thatplain density-functional theory (DFT) in its conven-tional LDA/GGA or hybrid forms severely under-estimates the orbital moment of Co clusters. Also anapproximate consideration of Hund’s second rule withinextensions of plain DFT like the orbital polarization cor-rection or the LDA+ U approach results in asevere underestimation. Among the theories exploredin this study, the only one that is consistent with themeasured orbital moments is the combination of DFTand dynamical mean-field theory (usually addressed asLDA+DMFT ), which treats onsite correlations andthus Hund’s second rule exactly. This demonstrates theimportance of dynamical correlations on orbital mag-netism of magnetic transition-metal clusters as well asthe fact that DMFT can give reasonable results for clus-ters too.For Co impurities in gold it has already been demon-strated that LDA+DMFT is required to produce thelarge orbital moments observed in experiments . In thisstudy, the authors used the spin-polarized T-matrix fluc-tuation exchange solver, because the correlation effectsare not very strong and can be treated perturbatively.As a matter of fact, we did try to use this solver for thepresently studied systems but found that it was inappro-priate to describe the formation of orbital moments whichare close to their atomic values. Therefore, we exploitedthe exact diagonalization routine for the impurity part ofDMFT. II. THEORYA. Theoretical methods
The focus of this work is on the calculation of thespin and orbital moments of pure Co clusters. For thispurpose several codes based on density-functional theory(DFT), and extensions, have been used. The extensionsare used to incorporate step by step a more sophisticatedtreatment of the onsite Coulomb repulsion. Namely it isthis onsite Coulomb interaction that leads to the forma-tion of local spin and orbital moments, which in atoms issummarized by the first and second Hund’s rules. On theother hand the crystal field effects compete against thismechanism quenching both moments. A proper theoryshould take these two competing effects accurately intoaccount, but at the moment such a theory exists onlyfor pure atoms or dimers at most. Therefore we haveexplored several suitable techniques for our purposes.The first method considered here is plain Kohn-ShamDFT, with an exchange-correlation functional in the localdensity approximation (LDA) as formulated by Perdewand Wang (PW) , in the generalized-gradient approx-imation (GGA) as formulated by Perdew, Burke andErnzerhof (PBE) and in the B3LYP hybrid approxi-mation . In all these approximations the second Hund’srule is completely neglected, and orbital moments areinduced by the spin moment through the spin-orbit cou-pling. Further, LDA and GGA are derived in the limitof a (nearly) uniform electron gas, while the hybrid func-tional treats the electron exchange of the inhomogeneous system partially exactly. Therefore, DFT in these formsonly describes onsite Coulomb effects in a very rough ap-proximation as far as orbital magnetism is concerned.The situation improves when an explicit onsiteCoulomb repulsion term is considered, leading to a gen-eralized Hubbard model . The idea behind this is tocombine DFT and the Hubbard model. Here, we ex-ploit the fact that DFT works well for the (weakly cor-related) delocalized electrons in the system, while theonsite Coulomb repulsion term is crucial for the descrip-tion of (strongly correlated) localized electrons as knownfrom studies of the Hubbard model. There are basicallytwo methods available to approximately solve this gen-eralized Hubbard model. The first is the static meanfield approximation, i.e. the LDA+U method . Thisshould describe to a certain extent the effects due to thesecond Hund’s rule, although at the price of a forcedbroken symmetry, which is not a problem in the presentcase . The second approach to this problem is based onthe dynamical mean-field approximation , which leadsto the LDA+DMFT approach . The LDA+DMFTapproach becomes exact in the atomic limit or equiv-alently when hybridization effects can be neglected, inthe non-interacting limit, and in the limit of an infinitenumber of nearest neighbors. Within this respect theregime of small clusters is rather far from the limit ofinfinite neighbors, although it has been shown that inpractical terms this limit is reached very fast, even for asmall number of nearest neighbors . In order to evalu-ate the influence of hybridization effects we perform twotypes of LDA+DMFT calculations. The first is a sim-plified version of the LDA+DMFT method in the limitof zero hybridization, i.e. Hubbard-I approximation .The second is a more accurate version, where the hy-bridization is considered within the exact diagonalizationroutine.Due to the inclusion of the onsite Coulomb interac-tion term, the Hubbard U and Hund exchange J pa-rameters of Co clusters are required as an input for theLDA+U and LDA+DMFT calculations. It is not clearfrom the beginning what the Hubbard U value in thecluster regime is but it is reasonable to assume that it isintermediate between the bulk value of about 3 eV andthe atomic value of about 14 eV . To obtain theHubbard U and Hund exchange J of Co clusters we per-formed calculations using the constrained random phaseapproximation (cRPA) . The results are reported anddiscussed below.In the LDA+U calculations there is a great risk toobtain a solution that corresponds to a local minimuminstead of the global one. To avoid this problem we haveused the method of Ref. 43, which consists in startingfrom a converged DFT calculation and then increasing U and J step-by-step. For completeness this type of cal-culation is compared with a LDA+U calculation startingfrom a converged DFT calculation, but without a stepwise increase of the Hubbard U and Hund exchange J value.Finally, most codes evaluate the orbital moments onlywithin certain spheres around the atomic sites . How-ever, here we have also evaluated the contribution to theorbital moment given by the interstitial region in betweenthese spheres via the Modern Theory of Orbital Polar-ization .Unfortunately, the plethora of these calculations couldnot be made by means of a single code. Therefore, differ-ent codes have been used for different purposes. Cal-culations based on LSDA and GGA were made using RSPt , VASP and Quantum ESPRESSO (with or-bital moment obtained from Modern Theory of OrbitalPolarization ). The cRPA calculations of Hubbard U and Hund exchange J were made using FLEUR and SPEX . The LSDA+U calculations were made with VASP , whereas the LSDA+DMFT calculations weremade with
RSPt . All the computational details are con-tained in the Appendix.
B. Hubbard U and Hund exchange J parameters The Hubbard U and Hund exchange J parameters arerequired as an input for LDA+U and LDA+DMFT cal-culations. The cRPA method was used to calculate theseparameters, which are reported in Table 1 for Co to Co .We find that U and J are slightly different for inequiva-lent atomic sites in a given cluster. Therefore, the valuesshown in Table 1 are average values.From Table 1 one can observe that the Hubbard U value decreases with increasing size, which indicates thatthe screening becomes more effective with increasing clus-ter size. Comparing these results with the U and J valuespredicted for Co bulk, U = 2 − J = 0 . − . U value is significantly larger, but J is about the same .Note that it is well known that the Hund exchange J isan atomic like quantity which is practically system in-dependent. Therefore, it is not unexpected to find theHund exchange J to be independent of cluster size andalmost equal to the bulk value.We note here that in the calculation of the Hubbard U and Hund exchange J parameters within the cRPAmethod the d - d screening channel is excluded. There-fore, the Hubbard U and Hund exchange J values of Ta-ble 1 can be directly used for a LDA+DMFT calculation,where the d - d screening is taken into account explicitly.As explained in the main text, to allow a better compar-ison among the LDA+DMFT calculations for different cRPA Co Co Co Co Co Co U (eV) 9.7 8.8 8.3 7.7 7.3 7.2 J (eV) 0.8 0.7 0.7 0.7 0.7 0.7TABLE 1. The Hubbard U and Hund exchange J parame-ters in eV obtained from cRPA calculations for Co to Co clusters. cluster sizes, the same value of 8 eV was used for U . III. RESULTSA. Spin and orbital moments from GGA andLDA+U
We start by reporting on the spin and orbital momentsfrom GGA (PBE) and LDA+U in Table 2. These cal-culations were made using the VASP code , and werefocused on clusters of different size, from Co to Co . Toallow a better comparison, also for the LDA+U methodthe GGA (PBE) functional was used. Thus, note thatthe nomenclature LDA+U in this work should be inter-preted as GGA+U. Further, in order to analyze the U dependence of the spin and orbital moment, two sets ofLDA+U calculations were performed. One set, labeledas LDA+U(1), is for a Hubbard U corresponding to thebulk value of 3 eV. The other one, labeled as LDA+U(2),is instead for a U calculated appropriately for the clusterregime (see Table 1), which is of 7 eV. For both calcula-tions the same J value of 0.9 eV is used, as in Ref. 35. Method Co Co Co Co Co Co Co Co GGA 2.08 2.33 2.50 2.60 2.33 2.14 2.0 2.110.34 0.21 0.14 0.15 0.12 0.13 0.12 0.12LDA+U(1) 1.99 2.32 2.49 2.59 2.32 2.13 1.99 2.100.28 0.30 0.31 0.29 0.32 0.30 0.30 0.26LDA+U(2) 1.98 2.32 1.97* 2.00* 2.32 2.13 1.94* 1.98*0.27 0.29 0.28 0.27 0.27 0.26 0.27 0.25Experiment – – – – – – 2.6 2.1– – – – – – 0.7 0.65TABLE 2. The spin (upper line) and orbital (lower line) mo-ments in µ B /atom obtained from GGA, LDA+U and experi-ment . Here LDA+U(1) and LDA+U(2) correspond respec-tively to a LDA+U calculation with U = 3 eV and J = 0 . U = 7 eV and J = 0 . d configuration)and 0.13 µ B /atom . The comparison of the computational results withthe available experimental values reveals that bothGGA and LDA+U severely underestimate the orbitalmoments, while the computed spin moments are quiteclose to the experimental values (Table 2). Further, itcan be observed that LDA+U calculations in general re-sult in an orbital moment which is roughly a factor 2larger than the value of GGA. This increase can be un-derstood from the fact that the GGA calculations do notgive any account of the orbital polarization induced bythe second Hund’s rule, while this contribution is par-tially described in LDA+U . The comparison betweenLDA+U(1) and LDA+U(2) emphasizes that spin and or-bital moments do not depend strongly on the Hubbard U parameter nor the cluster size. However, it is importantto mention that for the LDA+U(2) setup for some clustersizes (i.e. Co , Co , Co and Co ) an antiferromagneticmagnetic structure was favoured with respect to the fer-romagnetic structure. More precisely, for Co a magneticground state with two moments pointing up and two mo-ments pointing down was found. For Co there were fourmoments pointing up and one down, for Co six momentswere pointing up and two down, and for Co eight mo-ments were pointing up and one moment was pointingdown. For these antiferromagnetic structures the valuesin Table 2 correspond to the site averaged absolute valueof the spin and orbital moment. The experimental dataof Refs. 9, 10, and 52 indicate a ferromagnetic alignmentof the Co moments for Co , Co , Co and Co , hence theresults of LDA+U(2) in Table 2 are inconsistent withexperiments.It is interesting to analyze the GGA calculations forCo more in detail. Namely this calculation can be com-pared with the work of Refs. 18 and 19. For the groundstate of Co a theoretical orbital moment of 0.39 µ B /atomand a spin moment of 1.95 µ B /atom were reported inRef. 18, while in Ref. 19 the values of orbital and spinmoments were respectively 1 µ B /atom and 2.05 µ B /atom.By using different starting densities for our self-consistentcalculations, we managed to obtain a state with an or-bital moment of 0.34 µ B /atom and a spin moment of2.08 µ B /atom (see GGA result in Table 2) and anotherstate with an orbital moment of 0.94 µ B /atom and a spinmoment of 2.11 µ B /atom. The former state was foundto be 31 meV lower in energy with respect to the lat-ter. This shows that it is possible to stabilize differentstable and meta-stable configurations, and may explainthe different results of Refs. 18 and 19. The small dis-crepancies between our values for the magnetic momentsand the values of Refs. 18 and 19 are reasonable in termsof slight changes in the computational strategies and inthe exchange-correlation functionals. Therefore, the twostates with different orbital moments that were reportedearlier should be interpreted as two different energy min-ima, and the state with low orbital moment is the groundstate of the GGA functional. B. Co as a test case None of the GGA or LDA+U results in Table 2 repro-duce the experimental orbital moment, and there couldbe several reasons for this discrepancy. For examplethe XMCD experiment is performed on charged clusters,while the theoretical calculations are for neutral clusters.Another reason could be the consideration of an erro-neous geometry. For Co it is for example known fromindirect vibrational spectroscopy experiments that thegeometry is a planar rhombus, while theory not always finds this to be lowest in energy . Further, the em-ployment of an inappropiate functional could also lead toa discrepancy. In order to test the influence of the ion-ization of the cluster (i.e. charge), geometry and func-tional, Co is used as a test case. We selected Co as atest case, since each atom has already a three-fold coordi-nation while the computational effort is still manageablefor exploring different methods. In Table 3 the spin andorbital moments of Co are reported, as obtained via var-ious approaches. Method Spin moment Orbital moment( µ B /atom) ( µ B /atom)LDA (planar) RSPt 2.44 0.10LDA (tetra) RSPt 2.44 0.12GGA (planar) VASP 2.50 0.17GGA (tetra) VASP 2.50 0.14B3LYP (planar) VASP 2.50 0.25B3LYP (tetra) VASP 2.50 0.20GGA+OPC (planar) RSPt 2.48 0.33GGA+OPC (tetra) RSPt 2.48 0.21LDA+U(1) (planar) VASP 2.49 0.31LDA+U(1) (tetra) VASP 2.49 0.31LDA+U(2) (planar) VASP* 1.96 0.26LDA+U(2) (tetra) VASP* 1.97 0.28GGA charged (planar) VASP 2.25 0.18GGA charged (tetra) VASP 1.75 0.13TABLE 3. The spin and orbital moments (in µ B /atom) of Co as obtained from different methods are given. The geometryis indicated within the round brackets, where ’planar’ refers tothe planar rhombus and ’tetra’ to the (distorted) tetrahedron.Further, OPC refers to the orbital polarization correction .The asterisks indicate that an antiferromagnetic ground stateis obtained instead of a ferromagnetic one. The analysis of the results of Table 3 is much simplifiedby first noticing that the theoretical data reported in Ta-ble 2 show that the orbital moment does not change muchin the range of cluster sizes considered here. This is con-sistent with the experiments of Refs. 9 and 10, where theorbital moment is found to exhibit only a weak depen-dence on the cluster size. Taking theory and experimen-tal results together, an orbital moment of at least about0.7 µ B /atom is naively expected for Co . Later it willbe shown that this seems to be correct. One can imme-diately notice that none of the orbital moments reportedin Table 3 is close to a value of 0.7 µ B /atom, reflecting aproblem with the theoretical description.Further, the results in Table 3 show that the geometryhardly influences the values of the spin and orbital mo-ments. Very small changes are also found when changingthe exchange-correlation functional from LDA to GGA,while the hybrid (B3LYP) functional leads to a some-what larger increase in orbital magnetism. Consideringa charged cluster leads instead to an interesting depen-dence of the magnetic moments on the assumed geom-etry. For the planar arrangement spin and orbital mo-ments are similar to those of a non-charged cluster, whilefor the tetrahedron geometry the charge has a large in-fluence on the spin moment. The orbital polarizationcorrection increases the orbital moments obtained withLDA and GGA slightly and makes the results very sim-ilar to those obtained with a hybrid functional. Thelargest values for the orbital moment are obtained fromthe LDA+U calculations, albeit with values far from theexpected experimental results. However, for the large U setup, i.e. LDA+U(2), an unexpected antiferromag-netic ground state is obtained, which consists for bothgeometries in a configuration where the magnetic mo-ments point up at two atomic sites and down at the othertwo.Another possible source of error not considered so farcould be the contribution of the interstitial region tothe orbital moment. Namely in RSPt and VASP onlythe contribution to the orbital moment within a certainsphere around the atomic sites is considered. Therefore,the Quantum Espresso code was used in order to evalu-ate the interstitial region contribution to the orbital mo-ment. For Co , Co and Co respectively the interstitialcontribution to the total orbital moment was found to be1 %, 4 % and 15 %. Taking 15 % of the largest value forthe orbital moment found so far, i.e. 0.3 µ B /atom, givesroughly an 0.05 µ B /atom orbital moment contribution ofthe interstitial region. This is obviously much too smallto cover the difference between experiment and theory. C. LDA+DMFT
From the results obtained for Co one can concludethat for all the approaches tried, the orbital moment isunderestimated with respect to our extrapolation of theexperiments. Thus, replacing the exchange correlationfunctional with one of the most common formulations,adjusting the geometry and charge of the clusters, orincluding the interstitial contributions does not lead toa substantial increase in the orbital moments. There-fore, we resort here to a more sophisticated method,the LDA+DMFT approach, where atomic-like effects aretreated via a multi-configurational solution of the many-body problem . While in LDA+U it is a commonpractice to perform calculations for different values ofthe Hubbard U , in LDA+DMFT one can use directlythe values calculated through constrained random-phaseapproximation (cRPA) , which removes a parameterfrom the calculations and makes them fully ab-initio.These values are usually not used for LDA+U due tothat this approach does not account for the dynamicalscreening due to 3d electrons themselves, which reducesthe effective value of U by an unknown amount . Ourcalculated values of the Hubbard parameter U and Hundexchange parameter J are reported in Table 1. For sim-plicity, and for offering a better comparison among clus-ters of different size, we used U = 8 eV for all calcu-lations. We also checked the effect of a larger U = 9eV for Co , in agreement with Table 1, and we found no major changes (see below). We consider two differentapproximations of the local impurity problem to investi-gate separately the influence of the static crystal field andhybridization (kinematic effect) on the orbital moment.First, we evaluate the performance of the LDA+DMFTmethod without including any effect of the hybridization,which corresponds to the Hubbard-I approximation .Then, a more accurate solution is obtained by consideringthe hybridization effects through the exact diagonaliza-tion solver .The Hubbard-I approximation calculations were per-formed for clusters ranging from Co to Co , while ex-perimental data are available for only Co and Co (seeTable 1) . For all cluster sizes the same geometries asthose in Table 1 are considered. The only exception isCo for which a planar rhombus is considered, because itis experimentally known to be the ground state insteadof the (distorted) tetrahedron . Further, we analyse dif-ferent directions for the magnetization for the Hubbard-I approximation calculations of Co , Co and Co , asshown in Fig. 1. Since from Table 4 one can infer that thedirection of the magnetization axis is not so crucial forthe magnetic properties, for clusters of larger size onlyone direction is reported. Co is a trigonal bipyramidwith the spin axis orthogonal to the common base ofboth pyramids. Co is an octahedron, where the spin axis’connects’ the most distant atoms. Co is a capped octa-hedron for which the spin axis is chosen to ’connect’ themost distant atoms of the octahedron-part of the struc-ture. Co and Co are respectively a bicapped and adistorted tricapped octahedron for which the spin axis ischosen equivalently to Co .In Table 4 the spin and orbital moments obtainedwithin the Hubbard-I approximation are shown. SinceHund’s second rule effects might be sensitive to a changein the Hund’s rule J parameter, we performed calcula-tions for J = 0 . J = 0 . J . It is also clear thatalready within this simplified version of the LDA+DMFTmethod, the orbital moments of Co and Co are in verygood agreement with the experimental values (see Table1). Further, Table 4 clearly shows that the orbital mo-ment does almost not depend on the cluster size, whichwas also observed from Table 1 and experiments . Anexception here is Co , which has a substantially larger or-bital moment than observed for larger clusters. As men-tioned above for Co , judging the quality of the resultsfor clusters smaller than Co requires an extrapolation ofthe experimental data obtained for Co and Co . Thisextrapolation is based on the experimental value for Co and on the fact that both experiments and calculations(via DFT and LDA+U) show a very weak depedence ofthe orbital moment on the cluster size (see Table 2 andRefs. 9 and 10). Thus, from this extrapolation orbitalmoments of approximately 0.7 µ B /atom are expected forclusters from Co to Co . The results of Table 4 clearlyconfirm this expectation, with the notable exception ofCo , which will be discussed in more detail below. System Spin moment Orbital moment( µ B /atom) ( µ B /atom)Co saxis1 2.97 2.99 0.75 0.75Co saxis2 2.98 - 0.73 -Co saxis1 2.49 - 0.64 -Co saxis2 2.49 2.49 0.74 0.74Co µ B /atom cal-culated with the LDA+DMFT method within the limit ofzero hybridization, i.e. Hubbard-I approximation, for Co to Co clusters. Note that Co and Co are measured ex-perimentally with an orbital moment of respectively 0.7 and0.65 µ B /atom . Here, saxis refers to spin axis direction. ForCo saxis1 is in the triangular plane and saxis2 is orthogonalto the triangular plane (see Fig. 1). For Co both spin axesare in plane (see Fig. 1). Further, the first column of the ’Spinmoment’ and ’Orbital moment’ column refers to a J = 0 . J = 0 . Furthermore, it can be observed from Table 4 that thespin moment of Co is in good agreement with experi-ment, while for Co it is a bit off. On the other hand thespin moment of Co is in good agreement with that ofCo and smaller clusters, which is the trend one wouldexpect for a ferromagnetically coupled system. Regard-ing such a trend, the spin moment of Co is in very goodagreement with what one would expect from the experi-mental data of Ref. 10. FIG. 1. The geometry and spin axes are indicated for (a) Co ,(b) Co and (c) planar rhombus Co . For Co the second spinaxis (saxis2) is orthogonal to the triangular plane. In the following we will report on how a more accurateversion of LDA+DMFT, i.e. also including hybridizationeffects within the exact diagonalization solver, changesthe orbital moments. From the very good match of theHubbard-I results with experiment for Co and Co , one would expect hybridization effects to be small. Due tocomputational reasons, and also in light of the observedsize independence of the orbital moment, we have con-sidered only Co , Co and Co clusters for these moreaccurate LDA+DMFT calculations. For these clustersthe same geometries as for the Hubbard-I approxima-tion calculations are used. In Fig. 1 the used geometriestogether with the directions of the magnetization axisunder consideration are depicted. System Spin moment Orbital moment( µ B /atom) ( µ B /atom)Co U = 9 eV 2.97 0.71Co IAD = 2 . saxis1 2.98 0.86Co saxis2 2.98 0.76Co saxis1 2.47 0.73Co saxis2 2.48 0.80TABLE 5. The spin and orbital moments in µ B /atom calcu-lated with the LDA+DMFT method are printed for Co , Co and Co clusters. Here IAD stands for interatomic distance,which is 2.2 ˚A for the Co calculations without IAD specifi-cation. Further, saxis refers to spin axis direction. For Co saxis1 is in the triangular plane and saxis2 is orthogonal tothe triangular plane (see Fig. 1). For Co both spin axes arein plane (see Fig. 1). In Table 5 the spin and orbital moments obtainedwithin the more accurate execution of the LDA+DMFTmethod are shown. From this table it is observed thatthe effect of the hybridization on the spin and orbital mo-ments is indeed small for Co and Co , while it is largefor Co . This large influence on the orbital moment forCo can be traced back to the energy difference betweenthe many body eigenstates obtained in the Hubbard-Iapproximation. Namely, for Co the energy differencebetween the ground state and the first two higher ly-ing states is at least an order of magnitude smaller thanwhat is observed for Co and Co . Since for clustersfrom Co to Co this energy difference is of the sameorder of what is found for Co and Co , hybridizationeffects should be small also for these clusters. This dis-cussion leads us to conclude that our calculated value ofthe orbital moment for clusters from Co to Co is indeedapproximately 0.7 µ B /atom, which is exactly what is ex-pected from extrapolations from experimental data. Fur-thermore, one can conclude that LDA+DMFT already inits most simplified form (the Hubbard-I approximation)provides very accurate orbital moments except for Co ,i.e. when hybridization effects are expected to be large.In this case, and in general for all systems with largehybridization effects, the more accurate exact diagonal-ization version of the LDA+DMFT method should beemployed.From the discussion above it is clear that the calculatedorbital moment for Co and Co within the Hubbard-Iapproximation is in good agreement with experiment.Then, from a comparison between LDA+DMFT calcu-lations with and without hybridization effects included,we could show that the orbital moment for Co to Co is also approximately 0.7 µ B /atom. It is important tostress that this in principle only holds for neutral clus-ters. Future (XMCD) experiments should show whetherthis also holds for charged clusters. Very recently XMCDexperiments have been performed on Co +2 , for which theground state is found to be of 2 S + 1 = 6 and L = 1type . This result is obtained from a discussion, whichis entirely based on the ratio of the spin and orbital mo-ment. In this way difficulties due to the unknown numberof d-holes, ion temperatue and degree of circular polariza-tion due to the incident photon beam are circumvented.However, in this work also an estimation of the orbitaland spin moment is made, i.e. respectively 0.29 µ B /atomand 1.18 µ B /atom. Both orbital and spin moment arethus found to be about a factor 2 smaller than whatwe obtain from our best LDA+DMFT calculations (Ta-ble 5). Here we should note that the ratio of orbitaland spin moment found by us, i.e. 0.24, is exactly thesame as what was observed experimentally. Further, itis difficult to reconcile an orbital and spin moment of0.29 µ B /atom and 1.18 µ B /atom for a ground state of2 S + 1 = 6 and L = 1. Therefore, for completenesswe also performed a LDA+DMFT calculation with hy-bridization effects included for Co +2 (with the same interatomic distance as used for Co ). We find an orbital mo-ment of 0.69 µ B /atom, which again shows the very smallinfluence of the charge on the orbital moment. Further-more, the ratio of orbital and spin moment is found tobe 0.20, which is within their error bars, i.e. 0.24 ± U from 8 to 9 eV has little influence on the spin and or-bital moments of Co (Table 5). The same holds for anincrease of the interatomic distance from 2.2 to 2.4 ˚A.Thus, although DMFT is supposed to work better forincreasing cluster size due to the increasing number ofnearest neighbors, the orbital moment of Co is alreadyin agreement with our expectation of 0.7 µ B /atom. How-ever, note that according to an exact (many body) con-sideration, the sum of the spin and orbital angular mo-ment along the dimer axis should be integer or half in-teger. From an inspection of Table 5 it is clear that thisis not the case. This could be due to an overestimationof the spin, since it is subtantially larger than the valuesobtained by GGA and LDA+U (see Table 2). Further,it is well known that approximate methods like GGA,LDA+U and LDA+DMFT can violate rigorous symme-try considerations.For Co the LDA+DMFT calculations have been per-formed for two spin axes: one with a spin axis in thetriangular plane and another with a spin axis orthogo-nal to the triangular plane, see Fig. 1. From Table 5 itcan be seen that for the in-plane spin axis the orbitalmoment is 0.1 µ B /atom larger than for the out of planespin axis. For both spin axes the orbital moment is ingood agreement with the roughly expected orbital mo- -6 -4 -2 0 2 4 - do s E (eV) m l =2m l =1m l =0m l =-1m l =-2 LDA -10 -8 -6 -4 -2 0 2 4m l =2m l =1m l =0m l =-1m l =-2 - do s E (eV)LDA+U -15 -12 -9 -6 -3 0 3 6m l =2m l =1m l =0m l =-1m l =-2 - do s E (eV)LDA+DMFT
FIG. 2. The m l -projected 3 d density of states for a planarCo cluster with LDA (top), LDA+U (middle) LDA+DMFTwith hybridization (bottom). ment of 0.7 µ B /atom. The spin moment is a bit largerthan obtained from GGA and LDA+U in Table 2.For Co two different spin axes in the plane of therhombus are considered (Fig. 1). As can be observedfrom Table 5, the orbital moment is very similar for bothspin axes. Further, both orbital moments are in goodagreement with the 0.7 µ B /atom orbital moment, whichis roughly expected. The spin moment is very similar tothat obtained for GGA and LDA+U in Table 2.In order to visualize the difference in orbital momentbetween the LDA, LDA+U and LDA+DMFT (with hy-bridization effects included) methods, we took the pla-nar structure of Co as a typical example to plot theprojected 3 d density of states for (see Fig. 2). Fromthis figure it can be observed that the density of stateschanges drastically between the methods. Furthermore,by a detailed inspection one may observe that the dif-ference between the m l = 1 and m l = −
1, as wel as m l = 2 and m l = − d density of states of m l and − m l , and the dashed linescorrespond to the integrals of the these differences. Fromthese dashed lines it is clear where and how the differencein orbital moment between the different methods occurs.In fact the enhanced orbital moment of the LDA+DMFTmethod, compared to the other two methods, is not theresult of a m l projection or states in a narrow energyinterval. Instead Fig. 3 shows that the LDA+DMFTcalculations result in large contributions of the orbitalmagnetism over the entire occupied energy interval andfor all m l projections (except m l = 0).Finally, we would like to come back to Table 4.It would be interesting to see for what cluster sizesthe orbital moment reaches the (hcp) bulk value of0.13 µ B /atom . We speculate that a central atom in acluster with nearest and next-nearest neighboring atomswill have a bulk like orbital moment. This speculation isbased on the observation that for surfaces in general thethird layer already behaves bulk like . Thus, we expectthat in order to obtain a bulk like total orbital momentthe major part of atoms in a cluster should have nearestand next-nearest neighboring atoms. IV. CONCLUSION
The size and direction of spin and orbital momentsare determined by several interactions of a material,e.g. kinematic effects, crystal field interaction and on-site Coulomb repulsion between electrons. All this canamount to a complex dynamical interaction which crit-ically influences all properties, in particular magnetism.The present investigation is mainly focused on spin andorbital magnetism of clusters, where we find that the or-bital magnetism behaves differently from the large valueof the atomic limit as well as the reduced value of bulkand thin films. We here focus on clusters since they areexcellent model systems that allow for an investigation -15 -10 -5 0 5LDALDA+ULDA+DMFT do s ( m l = ) − do s ( m l = - ) E (eV) -15 -10 -5 0 5 do s ( m l = ) − do s ( m l = - ) E (eV)
LDALDA+ULDA+DMFT
FIG. 3. For a planar Co cluster the difference betweenthe m l = 1 and m l = − m l = 2 and m l = − where different interactions can have different relative im-portance, and hence allow for a means to elucidate theimportance of different contributions. We consider sev-eral levels of theory to undertake this investigation, withincreasing level of accuracy, e.g. GGA, orbital polar-ization correction, LDA+U and hybrid functionals. Wefound that none of these approximations resulted in cal-culated orbital moments that are in agreement with ex-periments. Only when one considers a description basedon multiple Slater determinants, as in the LDA+DMFTmethod, the theory predicts the orbital moments is inaccordance with experiment. Thus, for a proper treat-ment of the orbital moment it is absolutely crucial to takethe onsite Coulomb correlations accurately into accountin a dynamical fashion. Furthermore, from comparingLDA+DMFT calculations with and without hybridiza-tion effects, we can conclude that the static crystal fieldpotential is the dominant quenching mechanism for theorbital moment except for Co , where hybridization ef-fects are also very important. Since LDA+DMFT be-comes exact in the limit of negligible hybridization, it isnot surprising that it already works for small cluster sizes,Co to Co . Our findings in this work are relevant notonly for Co clusters, but have bearing also for isolated Coatoms on substrates, e.g. as reported in Refs. 57 and 58or as impurities . These studies can be summarized asall showing large orbital moments (in the range of ∼ ∼ µ B /atom) in experiment, which was not repro-duced by first principles theory (on GGA or LDA level).In these works, as well as in previous investigations, the effects of reduced symmetry, correlation effects asso-ciated with narrow bands, and spin-orbit effects of ligandorbitals were discussed. However, a clear understandingof which effect dominates for specific systems was notobtained. The present investigation clearly points to theimportance of electron correlation as a general cause oflarge orbital magnetism of narrow band systems. Acknowledgements
We acknowledge support from the Swedish ResearchCouncil (VR), eSSENCE, STANDUPP, and the SwedishNational Allocations Committee (SNIC/SNAC). TheNederlandse Organisatie voor Wetenschappelijk Onder-zoek (NWO) and SURFsara are acknowledged for theusage of the LISA supercomputer and their support.The calculations were also performed on resources pro-vided by the Swedish National Infrastructure for Com-puting (SNIC) at the National Supercomputer Center(NSC), the High Performance Computing Center North(HPC2N) and the Uppsala Multidisciplinary Centerfor Advanced Computational Science (UPPMAX). O.E.also acknowledges support from the KAW foundation(projects 2013.0020 and 2012.0031). M.I.K. acknowl-edges a support by European ResearchCouncil (ERC)Grant No. 338957. L.P. also acknowledges Dr. Da-vide Ceresoli for his support in the usage of the Quan-tum Espresso code. Further, L.P. also acknowledges Dr.Soumendu Datta for discussions on the VASP calcula-tions, and Dr. Gustav Bihlmayer and Dr. Timo Schenafor discussions on FLEUR calculations.
Appendix
Below the computational details are given for each ofthe used methods. Since all codes are k -space codes, asupercell approach was used, with a large empty spacebetween clusters that were repeated in a periodic lattice.In practice a large unit cell of at least 14 ˚A dimensionswas used to prevent the interaction between clusters ofdifferent unit cells. The only k -point considered was the Γ point, and all calculations included the spin-orbit cou-pling.Before providing all the computational details it is im-portant to say something about the geometry of the clus-ters. Every theoretical consideration about clusters re-quires the (ground state) geometry. Although the spinand orbital moments can be obtained experimentally, itis a real challenge to probe the geometry of the cluster.Bulk-like experimental techniques, i.e. those based onx-ray diffraction, cannot be employed for obtaining thegeometry of isolated clusters in the gas phase due the di-luteness of the gas. The geometry of the cluster with thelowest total energy is considered as the cluster geometry.To obtain this structure properly, the geometries are cal-culated with DFT for all possible spin and orbital mag-netic moments, in other words, spin states and electronicconfigurations . Another method in selecting theproper cluster geometry is to compare experimental vi-brational spectra with those obtained theoretically forthe different geometries . The second method is espe-cially useful in case of doubt about the computationaltotal energies, for example, when the total energies oftwo or more structures are very close.
1. RSPt
RSPt software (http://fplmto-rspt.org/) is a full-potential linearized muffin-tin orbital method (FP-LMTO) developed by Wills et al. . In the calculationspresented here the space was divided in muffin-tin sphereswhose radius was of 1.95 a.u., and an interstitial region.The main valence basis functions included 4 s , 4 p and 3 d states, while 3 s and 3 p states were treated as pseudocorein a second energy set . Three kinetic energy tails wereused for the 4 s and 4 p states, with values -0.3, -2.8 and-1.6 Ry. For the plain DFT calculations the LDA (PW)functional was used.RSPt includes an implementation of the orbital polar-ization correction (OPC) as described in Refs. 29 and30. The main idea of this correction is to include an ap-proximate description of the second Hund’s rule into theDFT problem. From a multipolar decomposition it canbe shown that the orbital polarization correction termis contained in the LDA+U method . For the orbitalpolarization calculations the GGA (PBE) functional wasused.The RSPt code was also used to perform theLDA+DMFT calculations both with and without hy-bridization effects included, where for both problems theexact diagonalization solver is used. For details on theimplementation of this routine see Refs. 34, 54, and 63.The local orbitals used in LDA+DMFT were constructedby considering only the so-called “head” of the LMTOs,which correspond to the MT orbitals of Refs. 29 and 36.In the case where hybridization effects are included thenumber of auxiliary bath states per 3 d orbital (used in theexact diagonalization) is one, i.e. there are ten 3 d states0and ten auxiliary bath states to consider in the many-body problem. The fully localized limit (FLL) was usedas the double counting correction. For the LDA+DMFTcalculations the LDA (PW) functional was used.Since this code is a collinear spin code with fixed spinquantization axis, different spin quantization axes wereconsidered. Furthermore, the calculations performedwith RSPt are for fixed geometry.
2. VASP
The Vienna ab-initio simulation package (VASP) isa DFT implementation based on a pseudopotentialaugmented-plane-wave method . As a cut-off of theplane wave basis set a kinetic energy of 400 eV wasused. The calculations were considered converged forchanges of the total energy smaller than 10 − eV betweentwo consecutive iterations. The geometry was consideredconverged, when the forces on all atoms were smaller than5 meV/˚A. For the LDA+U calculations we employed therotationally invariant formulation of Lichtenstein et al. and the GGA (PBE) functional. For the plain DFT cal-culations the GGA (PBE) and hybrid (B3LYP) function-als were considered. Since the geometry of Co clustershas been extensively investigated in Ref. 11, we usedthese ground state geometries and magnetic structuresas starting points. Further, the calculations were spinpolarized with non-collinearity, and the spin-orbit cou-pling was also included. In order to avoid to get trappedin a local minimum of the magnetic structure, differentstarting directions of the spin quantization axis were con-sidered.
3. FLEUR and SPEX
The FLEUR code is based on DFT and is an implemen-tation of the full-potential linearized augmented planewave (FLAPW) method . As a cutoff for the planewaves 3.6 Bohr − was taken, while l cut = 8 was usedfor the angular momentum. Moreover, the GGA (PBE)functional was employed. Based on the DFT calcula-tions, the SPEX code was used in combination withthe WANNIER90 code to perform cRPA calcula-tions of the Hubbard U and Hund exchange J parame-ters. The WANNIER90 code is used for the constructionof the maximally localized Wannier functions (MLWF).In the construction of the MLWF’s six states per Coatom are included, i.e. five d states and the valence s state. Finally, the geometry was fixed in these calcula-tions, corresponding to the optimized geometry obtainedwith VASP in a collinear spin-polarized scalar relativis-tic (without spin-orbit coupling) approximation in GGA(PBE), which were also obtained in Ref. 11.
4. Quantum ESPRESSO
Quantum ESPRESSO is a DFT implementation basedon a pseudopotential plane wave method . This codewas used to evaluate the interstitial region contributionto the total orbital moment . The interstitial regionis defined as the region outside the spheres around theatomic sites. These spheres were constructed with a ra-dius of 2.0 a.u. For the plane wave basis a kinetic energycut-off of 90 Ry was used. Furthermore, the GGA (PBE)functional was used. These calculations were performedfor fixed geometries, which were also obtained from thescalar relativistic GGA VASP calculations, i.e. the samegeometries as for the FLEUR/SPEX calculations wereused. ∗ [email protected] H. C. Siegmann and J. St¨ohr,
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