Magnetic anisotropy of FePt: effect of lattice distortion and chemical disorder
MMagnetic anisotropy of FePt: effect of lattice distortion and chemical disorder
C.J. Aas , L. Szunyogh , J.S. Chen , and R.W. Chantrell Department of Physics, University of York, York YO10 5DD, United Kingdom Department of Theoretical Physics, Budapest University of Technologyand Economics, Budafoki ´ut 8. H1111 Budapest, Hungary and Department of Materials Science and Engineering,National University of Singapore, 117576, Singapore (Dated: November 3, 2018)We perform first principles calculations of the magnetocrystalline anisotropy energy in the veL1 FePt samples studied experimentally by Ding et al. [J. App. Phys. 97, 10H303 (2005)]. Theeffect of temperature-induced spin fluctuations is estimated by scaling the MAE down accordingto previous Langevin dynamics simulations. Including chemical disorder as given in experiment,the experimental correlation between MAE and lattice mismatch is qualitatively well reproduced.Moreover we determine the chemical order parameters that reproduce exactly the experimentalMAE of each sample. We conclude that the MAE is determined by the chemical disorder ratherthan by lattice distortion.
PACS numbers: 75.30.Gw 75.50.Ss 71.15.Mb 71.15.Rf
Due to its extraordinarily high magnetocrystallineanisotropy energy (MAE), L1 FePt is of considerable in-terest to the development of ultrahigh density magneticrecording applications and spintronics devices. From atheoretical point of view, there is an obvious need fora complete first principles model of FePt to be used ingenerating effective spin Hamiltonians for the purposeof atomistic and multiscale modelling. Amongst manyother issues, this requires an understanding of the role ofinterfacial effects and chemical disorder. The large effectof chemical disorder on the MAE of FePt has alreadybeen outlined both experimentally and theoretically .Recently, the experiments were extended to thin filmsof FePt deposited on different substrates . A strong cor-relation was revealed between the MAE of the FePt sam-ple and the lattice mismatch of the FePt films with re-spect to the substrate . The experimental data are sum-marised in Table I. The chemical order parameter, s , isdefined as the probability of finding an Fe atom on a nom-inal Fe site or, equivalently, as the probability of findinga Pt atom on a nominal Pt site. In the experiment, thechemical order parameters were derived from the X-raydiffraction intensities I (001) and I (002), (( xyz ) denot-ing the plane of diffraction), through the relationship s ∼ (cid:112) I (001) /I (002) and normalizing s to unity for sam-ple no. 3. We refer to the experimentally obtained chemi-cal order parameters as s e for distinction from the chemi-cal order parameters s obtained later by fitting calculatedMAE-values to experiment.The aim of the present work is to investigate in de-tail the effect of lattice distortion and chemical order onthe MAE of FePt by means of the relativistic Korringa-Kohn-Rostoker method as combined with the coherentpotential approximation (KKR-CPA). In order to dif-ferentiate between the two main properties characterizingthe samples, namely, the lattice distortion and the chem-ical disorder, we perform calculations with and withoutthe inclusion of chemical disorder. We then fit the calcu- TABLE I: Summary of experimental results by Ding et al. ;lattice parameters a and c , chemical order parameter, s , mag-netocrystalline anisotropy energy per formula unit, K , anddiffraction intensity ratio, I (001) /I (002).Sample a (˚A) c (˚A) s e K (meV) I (001) /I (002)1 3.88673 3.69977 0.709 0.493 12 3.88279 3.69387 0.978 0.696 1.93 3.89752 3.68964 1.000 0.841 1.9854 3.89646 3.69175 0.965 0.788 1.855 3.86954 3.71378 0.615 0.271 0.7536 lated MAE to the experimental values using the chemicalorder parameter, s , as a fitting parameter and draw con-clusions from the results of our calculations. We find thatchemical disorder of each sample is the more importantfactor in determining the experimental MAE.As the relativistic KKR method is well documented inthe literature (see e.g. 7), here we merely describe somedetails of our calculations. We used Density FunctionalTheory within the Local Spin-Density Approximation(LSDA) as parametrised by Vosko et al. . The effec-tive potentials and fields were treated within the atomicsphere approximation (ASA). As the thin-film samples inthe experiment had a thickness of approximately 20 nm(60 formula units), surface contributions to the MAEshould be negligibly small. We therefore modelled theFePt samples as face-centered-tetragonal (fct) bulk lat-tices with lattice constants as displayed in Table I. Theself-consistent calculations were performed by using thescalar-relativistic approximation, i.e., by neglecting spin-orbit coupling and solving the Kohn-Sham-Dirac equa-tion using a spherical wave expansion up to an angularmomentum quantum number of (cid:96) = 3. As in earlier the-oretical work, we used the coherent potential approx- a r X i v : . [ c ond - m a t . m t r l - s c i ] A ug imation (CPA) to elucidate long-range chemical disor-der effects in FePt. In combination with KKR, the CPAhas proved particularly useful in calculating the physicalproperties of chemically disordered alloys . The partiallydisordered FePt alloy is modelled by a stack of alternat-ing layers with the chemical compositions of Fe s Pt − s and Pt s Fe − s .The MAE is then evaluated using the magnetic forcetheorem , which states that the difference in a system’stotal energy for two different directions of magnetiza-tion can be approximated by the corresponding differenceof the band energies, neglecting further self-consistency,i.e., keeping the effective potentials and fields fixed. Fromprevious experience we know that for transition metalsystems these potentials and fields can safely be takenfrom self-consistent scalar-relativistic calculations . Inorder to achieve a relative accuracy within 5 % for theMAE, the associated energy integration was performedby sampling 20 energy points along a semi-circular con-tour in the upper complex half-plane. At the energy pointclosest to the real axis the k -integration was calculatedusing 5050 k -points in the irreducible segment of the two-dimensional Brillouin zone.As the MAE should vanish at the Curie temperature,it is a rapidly decreasing function of temperature. Whilstthe temperature dependence of the MAE of ordered FePthas been previously calculated in terms of different the-oretical methods , in the present work we do notmake an attempt to carry out a similar process, sincesite-resolved information is currently not available for achemically disordered system. Instead, for an approx-imate comparison with experiments at room tempera-ture, we use the scaling obtained for perfectly orderedL1 FePt in terms of Langevin dynamics simulations ,namely, K T =293 K ∼ . K T =0 K .Using the methods described above we performed sys-tematic calculations of the MAE of each of the FePt sam-ples in Table I. In order to separate the effects of thelattice distortion and the chemical disorder, we split ourstudy into three stages. In our first set of calculations, theFePt samples were modelled as perfectly ordered alloyswith lattice parameters according to Table I. As can beinferred from Fig. 1, our calculated values spread around3 meV/f.u. and show a very minor dependence on thevariation of the lattice parameters. Moreover, this mod-erate variation between the samples is contrary to theexperimentally observed trend.Although high in comparison to experiment, our cal-culated MAE values are in good agreement with othertheoretical results based on the LSDA or the LSDA+Uapproach . One obvious reason for the discrepancybetween the theoretical and experimental values is thestrong temperature dependence of the MAE. We estimatethis contribution by scaling the calculated MAE downby an approximate factor of 0.6, as described above. Thecorresponding MAE-values (also shown in Fig. 1) are stilltoo high as compared to experiment. Thus we concludethat, even when taking temperature-induced spin fluctu- M AE ( m e V / f. u . ) Sample NoKKR (ordered) T=0 KKKR (ordered) T=298 K (est)experimental
FIG. 1: Crosses (solid line): Calculated MAE per formula unitfor each of the FePt samples in Table I modelled as perfectlyordered alloys. Circles (dashed line): The same values scaleddown by a factor of 0.6 in order to account for temperatureinduced effects. Stars (dotted line): The experimental values. ations into account, lattice distortion alone can explainneither the size nor the trend of the MAE obtained inthe experiment.Subsequently, the chemical disorder of each sample asgiven in Table I was taken into account using the co-herent potential approximation. The corresponding re-sults are shown in Fig. 2. In accordance with earlierwork , long-range chemical disorder drastically reducedthe MAE; for sample no. 1 ( s e = 0 . s e = 0 . s to 0.5 caneven cause a change of sign of the MAE. In contrast, forsamples no. 2 and 4 with a high degree of chemical orderthe MAE was reduced by less than 10 %, and for sampleno. 3 ( s e = 1) the MAE remained unchanged with respectto our previous calculations. Taking into account again areduction by a factor of 0.6 due to temperature effects, itis obvious that the inclusion of chemical disorder has sig-nificantly improved the agreement between experimentand theory: the trend of the MAE between the differentsamples is now correct and the magnitudes of the MAEare closer to the range reported by the experiment.As mentioned above, the chemical order parametersin Table I were derived from measured diffraction inten-sity ratios . However, due to an incomplete rockingcurve , the measured diffraction intensities, and therebythe experimentally obtained chemical disorder parame-ters, can only be considered approximate values. Fur-thermore, we note the assumption that the sample withhighest MAE, sample no. 3, refers to perfect chemicalorder, s e = 1. This seems a reasonable working hypoth-esis, but one worth investigating theoretically since it iscentral to the interpretation.The above uncertainties motivated us to perform athird set of calculations, in which the theoretical MAEwas fitted to the experimental MAE using the chemical M AE ( m e V / f. u . ) Sample NoKKR (disordered) T=0 KKKR (disordered) T=293 K (est)experimental
FIG. 2: Crosses (solid line): Calculated MAE per formula unitfor each of the FePt samples in Table I modelled as partiallydisordered alloys with the degree of disorder given by theexperiment. Circles (dashed line): The same values scaleddown by a factor of 0.6 in order to account for temperatureinduced effects. The experimental values are also displayedby stars (dotted line). M AE ( m e V / f. u . ) Chemical order parameter, s1 2 345
FIG. 3: Magnetic anisotropy energy (MAE) calculated as afunction of chemical order parameter, s , for the FePt samples:1 +, 2 ∗ , 3 (cid:4) , 4 × , 5 • . Solid lines serve as a guide for theeye. Open circles are placed at the best-fit chemical orderparameter for each of the samples. Dashed line: Linear fit. order parameter, s , as a fitting parameter. In Fig. 3, foreach of the samples we present the calculated MAE for an appropriate set of chemical order parameters. Firstly,for a given sample, i.e. for fixed lattice parameters, thetheoretical MAE shows a non-linear dependence on s . InFig. 3 the circles indicate the intersection of the calcu-lations with the experimental values for each sample asindicated. This determines the best-fit order parameterthat corresponds to the experimental MAE value. Ascan be clearly inferred from Fig. 2, for samples no. 2, 3and 4, a smaller degree of chemical order was fitted thanpredicted by the experiment, namely, s (cid:39) . , . s (cid:39) . s , there is a nearly perfect linear correlation between theexperimental MAE and the best-fit chemical order pa-rameters as indicated by the dashed line in Fig. 3. Ob-viously, this remarkable linear behavior is the result ofa subtle interplay of the dependence of the MAE on thelattice distortion and the chemical disorder. This is prob-ably specific to the data set investigated here rather thanbeing a general property.In summary, our first principles calculations imply thatlattice distortion in the FePt samples has only a minoreffect on the MAE, even opposite to the experimentaltrend. Calculating the MAE using the highly approx-imate experimental chemical order parameters signifi-cantly improves the agreement between theory and ex-periment, in particular with regards to the relative dif-ferences in the MAE between the samples. This indicatesthat the substrate-sample lattice mismatch effect on theMAE reported by Ding et al. is mainly due to the varia-tion in chemical disorder. To circumvent the uncertaintyof the experimental determination of chemical disorder,we, furthermore, determined theoretical chemical orderparameters that reproduced the experimental MAE val-ues. Interestingly, a linear correlation between the MAEand the best-fit chemical order parameters is found. Itshould be mentioned that work is underway to performconstrained Monte-Carlo simulations of K ( T ) for chem-ically disordered FePt, since this is clearly an importantfactor in relation to experimental data.Financial support was provided by the Hungarian Re-search Foundation (contract no. OTKA K77771) andby the New Sz´echenyi Plan of Hungary (Project ID:T ´AMOP-4.2.1/B-09/1/KMR-2010-0002). CJA is grate-ful to EPSRC for the provision of a research studentship. S. Okamoto, N. Kikuchi, O. Kitakami, T. Miyazaki, Y. Shi-mada and K. Fukamichi, Phys. Rev. B , 024413 (2002). J.B. Staunton, S. Ostanin, S.S.A. Razee, B. Gy-orffy, L. Szunyogh, B. Ginatempo and E. Bruno,J. Phys.: Cond. Mat. , S5623 (2004). Y.F. Ding, J.S. Chen, E. Liu, C.J. Sun and G.M. Chow,J. App. Phys. , 10H303 (2005) J.A. Christodoulides, P. Farber, M. Daniil, H. Okumura, G.C. Hadjipanayis, V. Skumryev, A. Simopoulos andD. Weller, IEEE Trans. Mag. , 1292 (2001). J. Korringa, Physica , 392 (1947). W. Kohn and N. Rostoker, Phys. Rev. , 1111 (1954). J. Zabloudil, R. Hammerling, L. Szunyogh and P. Wein-berger,
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