Magnetic anisotropy energy of disordered tetragonal Fe-Co systems from ab initio alloy theory
aa r X i v : . [ c ond - m a t . m t r l - s c i ] N ov Magnetic anisotropy energy of disordered tetragonal Fe-Co systems from ab initioalloy theory
I. Turek ∗ Institute of Physics of Materials, Academy of Sciences of the Czech Republic, ˇZiˇzkova 22, CZ-616 62 Brno, Czech Republic
J. Kudrnovsk´y † Institute of Physics, Academy of Sciences of the Czech Republic,Na Slovance 2, CZ-182 21 Praha 8, Czech Republic
K. Carva ‡ Charles University, Faculty of Mathematics and Physics,Department of Condensed Matter Physics, Ke Karlovu 5, CZ-121 16 Praha 2, Czech Republic (Dated: August 30, 2018)We present results of systematic fully relativistic first-principles calculations of the uniaxial mag-netic anisotropy energy (MAE) of a disordered and partially ordered tetragonal Fe-Co alloy usingthe coherent potential approximation (CPA). This alloy has recently become a promising system forthin ferromagnetic films with a perpendicular magnetic anisotropy. We find that existing theoreticalapproaches to homogeneous random bulk Fe-Co alloys, based on a simple virtual crystal approxima-tion (VCA), overestimate the maximum MAE values obtained in the CPA by a factor of four. Thispronounced difference is ascribed to the strong disorder in the minority spin channel of real alloys,which is neglected in the VCA and which leads to a broadening of the d -like eigenstates at the Fermienergy and to the reduction of the MAE. The ordered Fe-Co alloys with a maximum L -like atomiclong-range order can exhibit high values of the MAE, which, however, get dramatically reduced bysmall perturbations of the perfect order. PACS numbers: 71.15.Rf, 71.23.-k, 75.30.Gw, 75.50.Bb
I. INTRODUCTION
The binary ferromagnetic Fe-Co alloy has been knownfor a long time as a system with the maximumspontaneous magnetization among the transition-metalsystems. Its basic magnetic properties, such as the con-centration dependence of the alloy magnetization (theSlater-Pauling curve) in the ground-state body-centeredcubic (bcc) structure, have been reproduced successfullyby ab initio electronic structure calculations in a num-ber of studies.
This system attracted renewed interestseveral years ago, after a theoretical prediction of a giantuniaxial magnetic anisotropy energy (MAE) of the body-centered tetragonal (bct) Fe − x Co x alloys. The uniaxialMAE is defined quantitatively as the difference of totalenergies for the magnetization direction parallel to thetetragonal a and c axis, K u = E (100) − E (001) . The cal-culated MAE reached high values, K u ≈ µ eV/atom,obtained however only in a narrow range of the Co con-centration, 0 . ≤ x ≤ .
65, and of the tetragonal strain,1 . ≤ c/a ≤ .
25. The combination of the high MAEwith the high magnetization makes the bct Fe-Co systema promising material for fabrication of ferromagnetic thinfilms with a perpendicular magnetic anisotropy (the mag-netic easy axis perpendicular to the film plane), whichmight be relevant as high-density magnetic recording me-dia.Most of experimental realizations of this tetrago-nal system employed the possibility to grow epitaxiallystrained Fe-Co alloy films of a varying composition on different non-magnetic transition-metal substrates withthe face-centered cubic (fcc) structure, such as Pd(001),Rh(001), and Ir(001).
The different lattice parame-ters of the substrates enable one to scan a relativelywide interval of the c/a -ratio of the Fe-Co films, namely1 . ≤ c/a ≤ .
24 (note that c/a = 1 and c/a = √ in situ magneto-optical Kerr effect (MOKE) confirmed the theoretical prediction of Ref. 5 ona qualitative level; in particular, a strong perpendicularmagnetic anisotropy of the films was observed for compo-sitions and tetragonal distortions in a rough agreementwith the calculated trends of the bulk K u . However, sys-tematic quantitative experimental studies of the MAE ofthe films have not been performed yet; available valuesof the K u for a few selected systems are appreciablysmaller than the calculated ones. The missing informa-tion on the MAE has partly been compensated by mea-surements of X-ray magnetic circular dichroism (XMCD)spectra, which provide orbital magnetic moments of bothalloy constituents. The latter measurements, performedon the Rh(001) substrate corresponding to c/a ≈ . − x Co x filmsaround x ≈ .
6, in a close relation to the predicted max-imum of the bulk uniaxial MAE.Nevertheless, the existing agreement between the the-ory and the experiment should be taken with caution.The original approach of Ref. 5 employed the so-calledvirtual crystal approximation (VCA), in which the truespecies of a random binary alloy are replaced by a sin-gle element with an effective atomic number given by thealloy composition. The VCA has been implemented invarious ab initio methods and used for a number of al-loys of neighboring elements in the periodic table.
Its application to the bcc Fe-Co and fcc Co-Ni systems yields concentration trends of the spin magnetic momentin a very good agreement with experiment; the orbitalmagnetic moments seem to be described reliably in theVCA as well. However, a recent first-principles studyof the random bct Fe − x Co x alloys, based on a super-cell technique applied to a few special Co concentrations( x = 0 .
5, 0.625, and 0.75), has shown that a realistictreatment of the chemical disorder can reduce the MAEby a factor of 1.5 − ab initio calculation of theMAE for the disordered bulk tetragonal Fe-Co systemsobtained by using the coherent potential approximation(CPA) as a basic tool of the theory of metallic alloys. For homogeneous bct alloys, we compare results of theVCA and the CPA, show the big difference between them,and identify the underlying physical mechanism in termsof the electronic structure. Moreover, we investigate theeffect of a complete and an incomplete atomic long-rangeorder on the MAE; this study is motivated by the exist-ing prediction of a large uniaxial MAE in stoichiometricperfectly ordered tetragonal FeCo and FeCo systems. Similar theoretical studies have so far been done mainlyfor tetragonal FePt alloys and very recently also forthe FeCo alloy. These works are confined to stoichio-metric equiconcentration alloys; in general, their resultsreveal a decrease of the MAE due to imperfect chemicalordering.
II. MODELS AND COMPUTATIONAL DETAILS
All calculations of this study employed a bct latticewith the Wigner-Seitz s radius equal to that of pure bcciron, s = 2 . a where a = 5 . × − m denotesthe Bohr radius. The neglect of volume relaxations onthe MAE in a broad range of lattice parameters and of al-loy concentrations is justified by a more general study ofRef. 14 within the VCA. This structure model—the vol-ume conserving tetragonal distortion (the Bain path)—coincides with that used in the original study and in therecent study of the partially ordered FePt alloy. The effect of atomic long-range ordering has been stud-ied for the case of L (CuAu) order relevant for com-positions not far from the equiconcentration Fe . Co . system. The bct structure is partitioned in two simpletetragonal sublattices (alternating atomic planes perpen-dicular to the tetragonal c axis). For the Fe − x Co x sys-tem, an additional concentration variable y is introduced,such that 0 ≤ y ≤ min { x, − x } , which defines the chemi-cal composition of both sublattices: Fe − x + y Co x − y in theFe-enriched planes and Fe − x − y Co x + y in the Co-enrichedplanes. Note that the homogeneous solid solution corre- sponds to y = 0, while the other end of the y -intervaldescribes the maximum L order compatible with thegiven total Co concentration x . This model is a naturalgeneralization of the model used for the stoichiometricFePt systems. The eletronic structure calculations were done bymeans of the tight-binding linear muffin-tin orbital (TB-LMTO) method in the atomic sphere approximation(ASA) with a full inclusion of relativistic effects bysolving the one-electron Dirac equation with the spd -basis of the valence orbitals. The effective spin-polarizedpotential was constructed in the local spin-density ap-proximation (LSDA) using the parametrization of theexchange-correlation term according to Ref. 25. Theeffects of alloying were treated in the CPA for allsystems; for the homogeneous random alloys ( y = 0),the simple VCA was used as well.The reliable evaluation of the MAE in metallic systemsrepresents a difficult task, especially for cubic 3 d transi-tion metals owing to the weakness of the spin-orbit in-teraction and the high symmetry of cubic structures. The situation is more favorable for uniaxial systems(tetragonal, trigonal, hexagonal) and for systemswith f -electrons, see Ref. 30 for a review. In a number ofexisting studies, including the very recent ones, themagnetic force theorem is used and the total energy dif-ference K u is approximated by the change in the sum ofoccupied valence one-particle eigenvalues. In this study,we go beyond this approximation and evaluate the MAEdirectly from the total energies of the fully self-consistentsolutions for both magnetization directions, whichrequires high accuracy in total-energy calculations. Wehave used uniform sampling meshes of about 10 k -pointsfor averages over the full Brillouin zone (BZ) of the bctlattices, while about 5 × k -points have been used forthe BZ of the simple tetragonal Bravais lattices of the L ordered systems. The total energies were convergedto 0.1 µ eV/atom for all systems; this accuracy is suffi-cient in view of the resulting values of K u (see below). III. RESULTS AND DISCUSSIONA. Homogeneous random alloys
1. Magnetic anisotropy energy
The dependence of the uniaxial MAE of the randomFe − x Co x alloys on the chemical composition and the c/a -ratio, calculated in the VCA and in the CPA, isshown in Fig. 1; for better transparency, only the pos-itive values of the K u are displayed in the plots, whilethe cases with negative K u have been omitted (marked bywhite color). The VCA results (Fig. 1a) agree nicely withboth previous studies; in particular, a sharp maxi-mum of the MAE, K u ≈ µ eV/atom, obtained for x = 0 . c/a = 1 .
24, is clearly visible. The maxi-mum MAE of K u ≈ µ eV/atom for x = 0 . c/a between 1.22 and 1.24 was reported in Ref. 14 (with aslight sensitivity to the particular model of the volume re-laxation employed) and practically the same values wereobtained in the original VCA study. Minor quantitativedifferences might be ascribed to different computationalschemes employed. The MAE in the CPA (Fig. 1b) ex-hibits a similar trend as in the VCA; however, the CPAmaximum is significantly smaller, K u ≈ µ eV/atom.The latter value is obtained for x = 0 . c/a = 1 . K u values can be found in a broad region of 0 . ≤ x ≤ . . ≤ c/a ≤ .
28, see Fig. 1b. The maximum K u in the CPA is reduced by a factor of four as comparedto that in the VCA, which is an even stronger reductionthan that reported in Ref. 14.The small MAE values in the CPA might be used to ex-plain the difference between the large K u values predictedin the VCA and the much smaller values inferred frommeasurements on thin Fe-Co films. Undoubtedly, thebetter description of the chemical disorder in the CPAas compared to the VCA contributes to the reduction ofthe MAE of prepared alloy thin films. However, a thor-ough quantitative analysis of this point is impossible atpresent and it could even be misleading because of thewell-known uncertainty of the LSDA to yield quantita-tively correct MAEs in 3 d transition-metal systems. Forthis reason, an analysis of the trends seems to be moreappropriate, see, e.g., Ref. 29 and references therein.In general, the LSDA underestimates both the MAEsand the orbital magnetism. A counterexampleto this rule is represented by the MAE of the orderedFePt alloy, which is overestimated in the LSDA prob-ably owing to an interplay of intraatomic Coulomb cor-relations, weak exchange splitting and strong spin-orbitinteraction of Pt atoms. The present results for the bctFe − x Co x alloys seem to confirm the general rule as canbe documented by two facts. First, the effective MAEof strained Fe-Co thin films, given by the difference ofthe bulk K u and the magnetostatic shape anisotropy en-ergy K sh , comes out positive for alloys in the vicinity of x ≈ . c/a ≈ .
25 (near the maximum of K u in theCPA). This situation would thus lead to an out-of-planeorientation of the thin-film magnetization, in a qualita-tive agreement with existing experiments. Note that K sh = ( µ / M /V , where µ is the permeability of thevacuum and where M and V denote, respectively, the al-loy magnetic moment per atom and the atomic volume.In the present case, M ≈ . µ B , where µ B is the Bohrmagneton, one obtains K sh ≈ µ eV/atom, which liesbelow the bulk MAE ( K u ≈ µ eV/atom). However,for the case of x = 0 . c/a = 1 .
13, correspondingroughly to the Fe . Co . films grown on the Pd(001)substrate, our calculated values are K u = 124 µ eV/atom, M ≈ . µ B , and K sh ≈ µ eV/atom, which indi-cates an in-plane orientation of the magnetization, incontrast to the out-of-plane orientation observed at lowtemperatures. Second, the calculated orbital magneticmoments in the bulk Fe − x Co x alloys, plotted in Fig. 2 c / a (a) 0200400600800 0 0.5 1Co concentration 1.15 1.2 1.25 1.3 1.35 c / a (b) 04590135180 FIG. 1. (Color online) The uniaxial MAE of random bctFe − x Co x alloys as a function of the Co concentration andof the tetragonal strain c/a : calculated in the VCA (a) andin the CPA (b). Only positive values of the K u are dis-played in both plots; the corresponding coloured scales arein µ eV/atom. M o r b ( µ B ) c/aVCA Fe (CPA)Co (CPA) FIG. 2. (Color online) The orbital magnetic moments in therandom bct Fe . Co . alloy with magnetization along z axisas functions of the c/a -ratio: the alloy orbital moment in theVCA and the species-resolved orbital moments in the CPA. for x = 0 .
6, are appreciably smaller than the values ob-tained from the measured XMCD spectra by employ-ing the sum rules. In particular, the measured val-ues for films with x = 0 . c/a = 1 .
24, namely, M Feorb = 0 . ± . µ B and M Coorb = 0 . ± . µ B , exceedtheir bulk theoretical counterparts by a factor of three. Asimilar relation of the experiment and the LSDA theory(quantified roughly by a factor between 1.5 and 2) wasobtained for bcc Fe-Co alloys. Moreover, the datain Fig. 2 prove that the VCA and the CPA yield quali-tatively different trends of the orbital magnetism versusthe c/a -ratio: the pronounced maximum at c/a = 1 . In view of the above mentioned problems to evaluatereliably the MAEs and the orbital magnetic moments, wehave not attempted to recalculate our results by tech-niques going beyond the LSDA, but have fo-cused on the strong difference between the VCA andthe CPA and on the role of disorder on the MAE in thetetragonal Fe-Co systems.
2. Electronic structure
The giant MAE obtained in the VCA was ascribedoriginally to the changes in the band structure accom-panying the tetragonal distortion of the effective FeCocrystal. The explanation rests on a coincidence of twoparticular eigenvalues at the Γ point, which occurs inthe minority spin (spin-down) channel. For a special al-loy composition and a special c/a -ratio, this coincidencetakes place just at the Fermi energy corresponding to theparticular filling of the spin-down band. The large con-tribution of this eigenvalue pair to the MAE can be un-derstood in terms of the second-order perturbation the-ory, in which the effect of the weak spin-orbit interactioncan be safely included. In this approach, the enhancedvalue of the K u of the bulk Fe-Co alloys can be explaineddue to the coupling between d x − y and d xy states me-diated by the orbital momentum operator L z , see Ref. 5for details. The physical origin of the perpendicular mag-netic anisotropy of the strained Fe-Co films is thus simi-lar to that of Au covered Co monolayers on an Au(111)substrate. In the latter case, however, the resulting pos-itive MAE is caused by the L z coupling between d xz and d yz states.The usual band structures are not relevant for theelectronic structure of random alloys, for which moreappropriate quantities, such as the densities of states(DOS) and the Bloch spectral functions (BSF), have tobe studied. Motivated by the above explanation ofthe giant MAE, we present these quantities for the ran-dom bct Fe − x Co x alloys in the CPA, evaluated withoutthe spin-orbit interaction, i.e., in the scalar-relativistic
20 10 0 10 20 30-0.8 -0.4 0 D O S ( s t a t e s / R y ) E - E F (Ry) FeCo FIG. 3. (Color online) The spin-polarized local densities ofstates of the random bct Fe . Co . alloy with c/a = 1 .
25 asfunctions of the energy. The dotted vertical line denotes theposition of the Fermi level. approximation.
The spin-polarized local DOSs are shown in Fig. 3 forthe system with x = 0 . c/a = 1 .
25. The shapesof the individual DOSs prove a very weak disorder inthe majority spin (spin-up) channel, whereas the spin-down band exhibits a regime of a stronger scattering.This type of spin-dependent disorder was found in thebcc Fe-Co system a long time ago; it is responsible,e.g., for the observed concentration trend of the residualresistivity. Note that the strong disorder is present inthe spin-down d -band, which is only partially occupiedand which thus contributes a lot to the alloy total energyand, consequently, to the MAE.The spin-polarized BSFs at the Γ point for theFe . Co . alloy with three tetragonal distortions areplotted in Fig. 4. The BSFs were resolved according tothe irreducible representations of the point group D of the bct lattice. The odd (ungerade) representations,belonging to p orbitals, have been omitted in the plotssince their contribution is negligible in the displayed en-ergy interval. The relevant even (gerade) representationsare: A (orbitals s and d z ), B ( d x − y ), B ( d xy ),and E g ( d xz and d yz ). One can see that all spin-up BSF’s,consisting of narrow Lorentzian peaks, can be interpretedas Bloch-like eigenstates with finite lifetimes due to aweak disorder. The spin-down BSF’s above the Fermienergy (mainly A -like) reflect effects of strong disorder(non-quasiparticle behavior); the profiles at and belowthe Fermi energy are Lorentzian peaks again. However,their widths are clearly bigger than the spin-up widths, inagreement with the spin-dependent disorder manifestedin the DOS.
100 50 0 50 100 150 BS F ( s t a t e s / R y ) c/a = 1.15A B B E g
100 50 0 50 100 150 BS F ( s t a t e s / R y ) c/a = 1.25A B B E g
100 50 0 50 100 150-0.3 -0.2 -0.1 0 0.1 0.2 BS F ( s t a t e s / R y ) E - E F (Ry) c/a = 1.35A B B E g FIG. 4. (Color online) The spin-polarized symmetry-resolvedBloch spectral functions of the random bct Fe . Co . alloyat k = Γ as functions of the energy for three values of the c/a -ratio. The dotted vertical lines denote positions of theFermi levels. For details, see the text. The maximum of the MAE in the VCA is found for analloy with x = 0 . c/a = 1 .
25; its spin-down BSF in the CPA (Fig. 4,middle panel) shows an analogy of the coincidence of the B ( d x − y ) and B ( d xy ) levels at the Fermi energy.However, the disorder-induced smearing of both peaks isat least as big as their separation, which suppresses thecontribution of this eigenvalue pair to the MAE in theframework of the second-order perturbation theory. Thisfeature proves that the minority-spin Fe-Co disorder isstrong enough to reduce significantly the MAE, whichexplains the very big difference between the VCA andthe CPA results, see Section III A 1. B. Partially ordered alloys
The adopted model of the partially ordered tetrag-onal Fe − x Co x alloys, namely, the model of two sub-lattices with chemical compositions Fe − x + y Co x − y andFe − x − y Co x + y (Section II) and the use of the CPA (incontrast to supercell techniques) enable one to study theMAE as a function of three continuous variables: x , y ,and c/a . Such a full three-dimensional scan is, however,computationally very demanding; we have thus confinedour study to Co concentrations close to x = 0 .
6, for whichthe maximum MAE was found in the homogeneous bctalloys both in the VCA and in the CPA. The depen-dence of the calculated K u on the tetragonal distortionfor x = 0 . L order. One can see that the very flat low maximum K u ≈ µ eV/atom for the completely disordered alloy( y = 0) treated in the CPA is replaced by a significantlyenhanced pronounced maximum K u ≈ µ eV/atom ob-tained for the alloy with the maximum order ( y = 0 . x = 0 . K u ≈ µ eV/atom), which can beascribed to the fact that the system with x = 0 . y = 0 . L order ( y = 0 .
2) exhibits themaximum and the trend very close to the fully disorderedalloy.The detailed dependence of the K u on the concentra-tion variable y is presented in Fig. 6 for two cases: the sto-ichiometric alloy ( x = 0 . c/a = 1 .
24) and the previousoff-stoichiometric alloy ( x = 0 . c/a = 1 . c/a -ratio chosen in both cases corre-spond to the maximum K u value obtained for alloys withthe maximum order, i.e., for x = 0 . y = 0 . x = y = 0 .
5. One can see that the perfectly ordered( y = 0 .
5) stoichiometric alloy leads to a very high MAE of K u ≈ µ eV/atom. This value agrees well with Ref. 14,where the maximum MAE for the same L ordered sys-tem was obtained as K u ≈ µ eV/atom. The convexshape of both dependences in Fig. 6 proves that the very K u ( µ e V / a t o m ) c/a VCAy=0.4y=0.2y=0 FIG. 5. (Color online) The uniaxial MAE of the bctFe . Co . alloy as a function of the tetragonal strain c/a fordifferent degrees of the L -like atomic order: the maximumorder ( y = 0 .
4, diamonds), an intermediate order ( y = 0 . y = 0, crosses),all treated in the CPA. For a comparison, the MAE of thecompletely random alloy in the VCA is displayed as well (tri-angles). high MAEs can be obtained only for systems with themaximum order; even a small amount of additional dis-order is detrimental to the MAEs and reduces them tomuch lower values of the completely random alloys. Strong sensitivity of the MAE to the degree of the L atomic order has recently been reported for the stoicho-metric FePt alloy on an fcc lattice with small tetrago-nal distortions. The results of Ref. 19 indicate that thechemical ordering is a more important factor for highMAE values than the tetragonal distortion. Our resultsfor the Fe-Co system witness that both factors are ofequal importance, see Fig. 5. We believe that validity ofthis type of conclusions depends also on the range of rele-vant variables: the tetragonal distortion was varied over anarrower interval (0 . ≤ c/a ≤ .
06) in Ref. 19, whereasa wider interval (1 . ≤ c/a ≤ .
35) has been coveredin our study. Another difference between the two sys-tems lies in the dependence of the MAE on the c/a -ratio:the MAE is an ever increasing function of c/a in FePt, in contrast to the maxima found for the Fe-Co systemsaround c/a ≈ .
25, see Fig. 1 and Fig. 5. These trendsreflect probably the different origins of the high MAEin these systems. In the FePt alloys, the iron sites are
100 300 500 0 0.1 0.2 0.3 0.4 0.5 K u ( µ e V / a t o m ) concentration yx=0.5, c/a=1.24x=0.6, c/a=1.27 FIG. 6. The uniaxial MAE of two bct Fe − x Co x alloys as afunction of the degree of the L -like atomic order: for x = 0 . c/a = 1 .
24 (full circles), and for x = 0 . c/a = 1 . responsible for strong exchange fields and the platinumsites provide strong spin-orbit interaction, both effectsbeing only little dependent on the tetragonal distortionof the lattice. In the Fe-Co alloys, both species are fea-tured by strong exchange splittings and weak spin-orbitcouplings; the high MAEs are heavily based on collectiveproperties (band structure) of the tetragonal systems. IV. CONCLUSIONS
We have shown by means of first-principles LSDA cal-culations that chemical disorder has a strong influenceon the uniaxial MAE of the bulk tetragonal Fe-Co sys-tems. First, the complete neglect of the disorder, in-herent to the simple VCA, overestimates the MAE by alarge factor, whereas the more sophisticated CPA leadsto a drastic reduction of the MAE. The latter low MAEsare quite close to the magnetic shape anisotropy energyand the resulting estimated stability of the perpendicularmagnetic anisotropy of thin strained Fe-Co films comesout probably smaller than the measured one. Second,the disorder-induced reduction of the MAE is due tothe strong scattering regime in the minority-spin chan-nel, which smears the Bloch-like eigenstates at the Fermienergy. Third, an analysis of the calculated orbital mag-netic moments points to a non-negligible underestima-tion of the orbital magnetism and, most probably, of theMAEs by the LSDA. Fourth, the L -like atomic long-range order leads to an enhancement of the MAE values.The most pronounced enhancement is obtained for themaximum degree of the order, while imperfect orderingreduces the MAE very rapidly as compared to the per-fectly ordered systems. Similar correlations between theatomic order and the MAE have recently been found inthe FePt alloy and observed in artificially synthe-sized FeNi films. We believe that this interplay should be taken into account in a future search of advanced ma-terials for high-density magnetic recording.
ACKNOWLEDGMENTS
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