Magnetic anisotropy in Permalloy: hidden quantum mechanical features
Debora C M Rodrigues, Angela B Klautau, Alexander Edström, Jan Rusz, Lars Nordström, Manuel Pereiro, Björgvin Hjörvarsson, Olle Eriksson
MMagnetic anisotropy in Permalloy: hidden quantum mechanical features
Debora C M Rodrigues,
1, 2
Angela B Klautau, Alexander Edstr¨om, Jan Rusz, Lars Nordstr¨om, ManuelPereiro, Bj¨orgvin Hj¨orvarsson, and Olle Eriksson Department of Physics and Astronomy, Uppsala University, Box 516, SE-751 20 Uppsala,Sweden Faculdade de F´ısica, Universidade Federal do Par´a, CEP 66075-110, Bel´em-PA,Brazil (Dated: 11 October 2018)
By means of relativistic, first principles calculations, we investigate the microscopic origin of the vanishinglylow magnetic anisotropy of Permalloy, here proposed to be intrinsically related to the local symmetries ofthe alloy. It is shown that the local magnetic anisotropy of individual atoms in Permalloy can be severalorders of magnitude larger than that of the bulk sample, and 5-10 times larger than that of elemental Feor Ni. We, furthermore, show that locally there are several easy axis directions that are favored, dependingon local composition. The results are discussed in the context of perturbation theory, applying the relationbetween magnetic anisotropy and orbital moment. Permalloy keeps its strong ferromagnetic nature due to theexchange energy to be larger than the magnetocrystalline anisotropy. Our results shine light on the magneticanisotropy of permalloy and of magnetic materials in general, and in addition enhance the understanding ofpump-probe measurements and ultrafast magnetization dynamics.PACS numbers: 75.30.Gw,71.15.Mb,71.70.Ej,71.20.BeKeywords: Permalloy; magnetocrystalline anisotropy; orbital moment anisotropy
Introduction – Random alloys can be viewed as a dis-tribution of clusters of different composition, that havean underlying crystal structure in common. The config-urational space is enormous for these systems and anymacroscopic property is the result of averaging of a im-mense amount of local clusters with different configura-tion and composition . Random alloys often have prop-erties that stand out from the pure elements they arebuild up from, i.e. the mixing of elements may produceproperties that are completely unexpected.One of the most prominent examples is Permalloy(Py), the common name for Fe x Ni − x alloys with x ∼ . . These attributes elevate Py to a stan-dard material in magnetism and advantageous soft mag-net for technological applications.One might ask what the mechanism of the vanishingMAE of Py really is. One attempt to explain it is theresultant MAE picture , which reinforces that an appro-priate mixture of two elements with distinct easy axis(as bcc Fe and fcc Ni with (cid:104) (cid:105) and (cid:104) (cid:105) direction,respectively) would result in a material without strongpreferential easy magnetization axis, i.e. a low MAE.Nevertheless, recent experiments suggest that the Fe-Nihybridization in the alloy environment is the major causeof low MAE in Py, rather than the easy axis of its sepa-rated constituents. Other experiments show the existence of orbital mo-ments at the individual chemical species in Py . Addi-tionally, it is known that the magnetic anisotropy is pro-portional to the anisotropy of orbital moment for transi-tion metals . Thus, the anisotropy at atomic level mayexist, although diminished at the bulk. Considering this, as a random alloy, Py may be viewed as a huge ensem-ble of interconnected clusters of Fe-Ni atoms distributedon an fcc lattice, in which the macroscopic properties re-flect the configurational average of different such clusters.Then different parts of the alloy would have competinglocal anisotropies, that effectively average out, leading toa fairly isotropic state. This microscopic scenario has, tothe best of our knowledge, not been considered so far asa possible mechanism for low MAE in Py.For that, in this Letter, we present first principles cal-culations to investigate this local MAE. However it is noteasy to directly study the MAE, particularly not fromfirst principles theory as it is hard to uniquely and ac-curately decompose the energy into local contributionsand the numerical challenges are countless. Instead wewill study another related quantity induced by the spin-orbit coupling (SOC), namely the local anisotropy of theorbital moments, since it is known to reveal also infor-mation about the MAE . With this information we in-vestigate the role of these local competing anisotropiesand how they reveal information about the soft magneticbehavior of Py. Method – The study was designed as the following:first, we performed ab-initio calculations of a fcc ma-trix ( ∼ . The fcc unit cell was considered to have the samenumber of valence electrons as Py (9.6 e − ). After theself-consistent procedure, clusters composed by Fe andNi, with different configurations, were embedded in thePy-VCA matrix. The cluster region was self-consistentlyupdated while the potential parameters of the Py-VCAmatrix were kept fixed. Finally, the magnetic spin andorbital moments were computed for every site in the clus-ter. a r X i v : . [ c ond - m a t . m t r l - s c i ] F e b As the configuration space for the clusters is vast, ourinvestigation does not cover all possible configurations.We have, however, investigated a large number of geome-tries (84 excluding configurations that are symmetricalto these), and we illustrate as an example a few typi-cal geometries in Fig. 1c. These clusters present distinctconfigurations due to the Fe distribution. Note that, lo-cally in a cluster, the number of Fe and Ni atoms canvary, although a configurational average over all clustersof the material would naturally result in a concentrationof Fe and Ni that reflected the alloy concentration, i.e.20 % Fe and 80 % Ni (see dashed lines in Fig. 1b). x yz
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18 19 c)a)b) site
Fe Ni p r obab ili t y p r obab ili t y FIG. 1. (Color online) Distributions of Fe (red) and Ni (grey)atoms at cluster sites (1: central site, 2-13: first neigbours and14-19: second neigbours), considering (a) 8 or (b) 84 configu-rations. Three examples of configurations that may be foundin Permalloy are illustrated in (c). The dashed (dot-dashed)line represent the average Fe (Ni) concentration.
In our investigations we choose to study clusters with19 atoms, of which 4 or 5 are Fe atoms and the restbeing Ni atoms. The atoms are sorted from a centralsite (labeled 1), followed by its first (labeled 2-13 ) andsecond neighbors (labeled 14-19 ).The electronic structure and magnetism of VCA-Py and the clusters were evaluated using the first-principles real-space linear muffin-tin orbital methodwithin the atomic sphere approximation (RS-LMTO-ASA) . This method follows the steps of the LMTO-ASA formalism but uses the recursion method tosolve the eigenvalue problem directly in real space. Thecalculations presented here are fully self-consistent, theexchange and correlation terms were treated within the local spin density approximation (LSDA) , and the SOCterm was included at each variational step . The RS-LMTO-ASA method is particularly designed to treat lowsymmetry systems as the embedded clusters presentedhere, without the need of periodic boundary conditions.Regarding the calculations for the matrix of VCA-Py,the resulting spin moment ( m s ) is 1 . µ B per atom,which is in acceptable agreement with the experimentalvalue of approximately 1 . µ B per atom and previouscalculations . Therefore, we conclude that the ef-fective medium that is considered to host the differentclusters reproduces the main features of Py.For the different clusters in this investigation we haveestimated the local anisotropy from a well defined quan-tity – the orbital moment anisotropy, which is the differ-ence of the orbital moment projection L for two differentglobal quantization axes ∆ L = L ˆn − L ˆn . Since ∆ L isdefined as a local quantity, it is numerically easy to evalu-ate from first principles theory in contrast to the tiny en-ergy difference needed for the MAE. It is established thatthe energy difference between two states with the magne-tization direction along two different global directions is E MAE = − ξ µ B ∆ L , where ξ is the SOC constant. Oneof the key assumptions in deriving this relation is thatspin diagonal matrix elements of the spin orbit couplingshould dominate the contribution to the MAE . SincePy is a strong ferromagnet (the majority spin band isessentially filled) only minority spin states contribute tothe density of states at the Fermi energy, and this cri-terion is expected to be fulfilled. When minority spinstates dominate the MAE, the easy axis is parallel to thedirection of maximum orbital magnetic moment. To ex-emplify the numerical advantage of the approach adoptedhere, we note that values of 1 µ Ry for the MAE are re-lated to orbital anisotropies of 10 − µ B , which are valueswell defined by the method’s precision. Thus, it serveswell as the relevant quantity to evaluate and to quantifythe local anisotropies in alloys. Results – Before we discuss the results of the MAE, wenote that for all configurations investigated here, the cal-culated individual moments were close to m F es = 2 . µ B and m Nis = 0 . µ B , for spin, and L F e = 0 . µ B and L Ni = 0 . µ B for orbital ones. These values are inagreement with previous theoretical and experimen-tal studies.From each cluster of our investigation, we estimatedthe ∆ L , between two magnetization directions, for allatoms. Therefore, the global direction ˆn = [001]was considered as reference and the orbital momentanisotropy computed as ∆ L = L [001] − L ˆn , with ˆn =[110] and [111] directions. Comparing the ∆ L values onecan obtain the direction of maximum L, i.e. the local easyaxis. This information is summarized in Fig. 2, whichshows the distributions of easy axis directions per site.In Fig. 2a we show results formed from an average over 8different clusters and in Fig.2b the average is made over84 different configurations. We note that the number ofconfigurations favoring the [100] easy axis is larger thanthe [110] and [111] easy axis directions. We will returnto this fact below. a)b)
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18 19 site p r obab ili t y p r obab ili t y FIG. 2. (Color online) Likelyhood of different easy axis di-rections for each of the 19 atomic sites considered for eachcluster. Averages are formed from 8 configurations (a) and84 configurations (b). The easy axis direction are representedby the color bars.
In addition to these values of ∆ L we also estimated thesite resolved E MAE , for each type of cluster. For that, weused the calculated SOC constants for ξ Fe = 4 . ξ Ni = 6 . MAE and the direc-tion of the easy axis for each atom are shown in Fig. 3(for sake of simplicity only 8 configurations are shown).Note from the figure that we show local easy axis di-rections that in general is different for each atom in acluster, and sometimes even have different local easy axisdirections. Furthermore, the 8 clusters considered in thisfigure all show rather different behaviors when it comesto the MAE. For some of them, e.g., site 3 in one clustercan have the [100] easy axis direction, but other config-urations could for this site favor the [110] or the [111]easy axis direction. As is clear from the figure we findvalues that are typically 5-10 times larger compared tothe values of bcc Fe ( ∼ . µ Ry) or fcc Ni ( ∼ . µ Ry). Further, these local MAE values are, remarkably, ordersof magnitude larger compared to the almost vanishingvalue of the MAE of bulk Py.A key point in the Fig. 3 is that the symmetry of eachcluster is not cubic. Hence, spin-orbit effects enter as alocal uniaxial anisotropy and it depends of second-orderanisotropy terms instead of fourth-order as cubic envi-ronments have. This is the primary reason why the localMAE values of Py are bigger than those of bcc Fe andfcc Ni.It is interesting to compare the results of Fig. 2 and -6.0-5.0-4.0-3.0-2.0-1.00.01.02.03.04.05.06.0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 M A E ( µ R y ) site [001] [110] [111] FIG. 3. (Color online) MAE per site with the easy axis di-rection represented by squares (for [100]), circles (for [110])and triangles (for [111] direction). Negative values of MAEsymbolize an [001] easy axis. The 8 different configurationsconsidered here are represented by different colors. Note thatsome data points are superposed, since the same MAE valueis found for an atom placed in a given site of different config-urations.
Fig. 3 to recent supercell calculations for the MAE ofFeCo based alloys . There it is also found that thelocal anisotropy of various atomic configurations variesstrongly, and even changes sign, while the alloy MAEis described by the average over many configurations.Those systems are, however, very different in that theyhave a large MAE, meaning that one direction of magne-tization should be over-represented among different clus-ter configurations. Py differs in that the MAE is vanish-ingly small, meaning that there must be a balance fromdifferent local anisotropy contributions.
Discussion and Conclusion –
We consider here themagnetic anisotropy of a macroscopic sample as a con-figurational average of local anisotropies, for a diversedistribution of clusters like the ones shown in Fig. 1c.Each cluster may have several atoms with large localanisotropies directed in any of the common crystallo-graphic axes ( (cid:104) (cid:105) , (cid:104) (cid:105) and (cid:104) (cid:105) ), but since theinter-atomic exchange interaction of Py (not shown here)is much stronger and ferromagnetic, the resulting mag-netic configuration is a collinear ferromagnet, where, af-ter a proper configurational average is made, the result-ing MAE is expected to be vanishingly small.In the presented study, were investigated only clus-ters with approximately the same concentration of Py(Fe . Ni . ). In a real sample such constrain does notexist, and configurations involving, e.g., 1 Ni and 18 Featoms and vice-verse must also be considered. Once aproper configurational average of a huge set of clustersis considered, the proper macroscopic MAE can be ob-tained, and we suggest this leads to a vanishingly smallMAE for Py.The scenario proposed here is principally different thansimply making a linear interpolation of anisotropy con-stants of bcc Fe and fcc Ni and adopting an interpolatedvalue for all atoms of the alloy. We have shown that thelocal anisotropy is orders of magnitude larger than theobserved anisotropy in Py. We therefore argue that thevanishing anisotropy in bulk Py arises from the cancella-tion of these local anisotropies. It is likely that the sce-nario put forward here also applies to other magnetic pa-rameters, like the damping parameter or potentially theasymmetric exchange (like a local Dzyaloshinskii-Moriyainteraction). We also note that experiments showedthat amorphous materials present orbital induced mag-netic anisotropy explained by the random anisotropymodel. Note that in amorphous materials the lack ofsymmetry (chemical and crystalline) allows the emer-gence of orbital anisotropy.As a final comment, we note that the local anisotropyeffects discussed here might affect the magnetization dy-namics in thin films of Py . For that, adopting a sce-nario of locally unique information, as proposed here,would be relevant for the interpretation of pump-probemeasurements and crucial to simulations involving an ef-fective spin-Hamiltonian. Acknowledgments –
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