Effect of stacking faults on the magnetocrystalline anisotropy of hcp Co: a first-principles study
EEffect of stacking faults on the magnetocrystallineanisotropy of hcp Co: a first-principles study
C.J. Aas , L. Szunyogh , R.F.L. Evans , R.W. Chantrell Department of Physics, University of York, York YO10 5DD, United Kingdom Department of Theoretical Physics and Condensed Matter Research Group ofHungarian Academy of Sciences, Budapest University of Technology and Economics,Budafoki ´ut 8., H1111 Budapest, Hungary
Abstract.
In terms of the fully relativistic screened Korringa-Kohn-Rostoker methodwe investigate the effect of stacking faults on the magnetic properties of hexagonalclose-packed cobalt. In particular, we consider the formation energy and the effecton the magnetocrystalline anisotropy energy (MAE) of four different stacking faultsin hcp cobalt – an intrinsic growth fault, an intrinsic deformation fault, an extrinsicfault and a twin-like fault. We find that the intrinsic growth fault has the lowestformation energy, in good agreement with previous first-principles calculations. Withthe exception of the intrinsic deformation fault which has a positive impact on theMAE, we find that the presence of a stacking fault generally reduces the MAE of bulkCo. Finally, we consider a pair of intrinsic growth faults and find that their effect onthe MAE is not additive, but synergic. a r X i v : . [ c ond - m a t . m t r l - s c i ] A p r ffect of stacking faults on the magnetocrystalline anisotropy of hcp Co: a first-principles study
1. Introduction
Within the magnetic recording industry, cobalt alloys such as CoPt and CoPd are ofgreat interest due to their large magnetocrystalline anisotropy energies (MAE) [1]. Forthe purpose of magnetic recording, a large MAE of the recording medium is crucial inorder to maintain stability of the written information as larger areal information stor-age densities require smaller grain sizes [2]. In close-packed metals and alloys, stackingfaults are known to form relatively easily [3]. This is one of the contributing factors tothe relatively large ductility and malleability that are observed in many such materi-als [3]. For a magnetic recording medium, the presence of stacking faults is generallyconsidered detrimental, as disturbances in the microstructure will generally worsen thesignal-to-noise ratio of the medium [4]. Stacking faults may also break the local latticesymmetry and, therefore, drastically influence the MAE.Experimentally, the effects of stacking faults are generally measured in terms of thestacking fault density, which can be determined from X-ray diffraction spectra (seee.g. [1, 4, 5]). There are a large number of experimental studies into stacking faultformation energies and the effect of the stacking fault density on the magnetocrystallineanisotropy for various magnetic recording alloys [1, 6]. However, in experiment, thereal effect of a stacking fault might be obscured by other phenomena, such as migrationof impurities along the stacking fault, synergies of closely spaced stacking faults, etc.Consequently, a number of theoretical methods have been developed for determiningthe properties and effects of stacking faults, see e.g. [7]. In particular, there is a largenumber of first-principles studies of the formation energies of given types of stackingfaults in metals [3, 8]. It has been suggested that stacking fault formation energiesdetermined from first-principles may be more accurate than experimental measurements[3] as theoretical calculations separate the formation energy from any other correlatedeffects on the total energy. The effect on the MAE of a particular stacking fault is,however, less commonly explored. In this work, we aim to determine from first principlesthe effect on the MAE of four different types of stacking faults in hcp cobalt.
2. The stacking faults
Hexagonal planes can be packed either in an ...ABAB... sequence, yielding a hexagonalclose-packed lattice structure, or in an ...ABCABC... sequence, yielding a face-centredcubic lattice structure [9]. In the hexagonal close-packed lattice structure, the stackingdirection corresponds to the (0001) axis of the lattice, whereas for the face-centred cubiclattice structure, the stacking direction is parallel to the (111) axis of the lattice. In ahcp lattice, a stacking fault is defined as an interruption in the ...ABAB... stacking of thehexagonal planes. While there are of course any number of conceivable stacking faults,their varying degrees of formation energies and formation mechanisms mean they havedifferent probabilities of occurrence [8]. In line with previous work [3, 10], we consider ffect of stacking faults on the magnetocrystalline anisotropy of hcp Co: a first-principles study , I , E andT [11, 12]: • I (intrinsic): · · · B A B A B C B C B · · ·• I (intrinsic): · · · A B A B | C A C A · · ·•
E (extrinsic): · · ·
A B A B C A B A B · · ·• T (twin-like): · · · A B A B C B A B A · · ·
Here the bold face letters or vertical line denote the plane of reflection symmetry of thestacking fault. In an intrinsic stacking fault (I and I ), the stacking fault is simply ashift of one lattice parameter and the stacking on either side is correct all the way upto the very fault [9]. The stacking fault I is a growth fault while the stacking fault I is a deformation fault [3]. In the extrinsic stacking fault (E), a plane has been insertedso that it is incorrectly stacked with respect to the planes on either side of it [9, 13].In a twin -like fault (T ), the stacking sequence is reflected in the fault layer [3]. In thefollowing, we refer to the centre of reflection symmetry as the zeroth layer. The twolayers adjacent to the centre of reflection symmetry are then indexed ±
1, and so on.Note that in the case of a stacking fault of type I , the plane of reflection symmetry liesin between two atomic layers. Therefore, in the following, for type I the atomic layerswill be labelled by ± , ± , . . . , rather than by 0 , ± , ± , . . . as for the other types ofstacking faults.
3. Computational details
For our theoretical study we employed the fully relativistic Screened Korringa-Kohn-Rostoker (SKKR) method, in which the Kohn-Sham scheme is performed in terms ofthe Green’s function of the system (rather than the wavefunctions) and the treatmentof extended layered systems is particularly efficient [14, 15, 16]. We used the local spindensity approximation (LSDA) of density functional theory (DFT) as parametrised byVosko and co-workers [17]. The effective potentials and fields were treated within theatomic sphere approximation (ASA) and an angular momentum cut-off of (cid:96) max = 2 wasused. The magnetocrystalline anisotropy energy (MAE) is calculated using the mag-netic force theorem [18], within which the MAE is defined as the difference in the bandenergy of the system when magnetised along the easy axis (0001) and perpendicular tothe easy axis. Alternatively, the uniaxial MAE can be calculated from the derivativeof the band energy, for more details see Ref. [19]. Only when calculating the MAE, weused an angular momentum cut-off of (cid:96) max = 3.The LSDA+ASA fails in describing the orbital moment and the MAE of Co accurately.Similar to our previous work [19] we, therefore, employed the orbital polarisation (OP)correction [20, 21, 22], as implemented within the KKR method by Ebert and Battocletti[23]. Note that the OP correction was applied only for the (cid:96) = 2 orbitals. Excludingthe OP correction we obtained an easy-plane magnetisation and a MAE of 6.7 µ eV per ffect of stacking faults on the magnetocrystalline anisotropy of hcp Co: a first-principles study µ eV per cobalt atom. This is in goodagreement with the experimental value of 65.5 µ eV [24] and with the experimental easyaxis being parallel to the (0001) direction. Our result also compares well with that ofTrygg et al. [25], who calculated K = 110 µ eV for hcp cobalt using a full-potentialLMTO method including the OP correction.In this study we consider an infinite cobalt system, consisting of two semi-infinite bulkcobalt systems and an internal region (region I ). Region I contains the stacking faultand is positioned in between the semi-infinite regions. The combined system is periodicand infinite in the plane normal to the (0001) direction. Due to the long-ranged natureof the stacking fault effects on the MAE (see section 4.3), the region I in this studyhad to be kept at a size of around 80 atomic layers. More specifically, for stackingfaults I and I , systems of 80 atomic layers were required, while for stacking faultsE and T , 74 atomic layers were required. In order to keep the calculations tractablewe limited the self-consistent calculations only for a number of atomic layers near thestacking fault, and then appended the bulk potentials for the atomic layers further awayfrom the stacking fault. We found that it was sufficient to treat only the 20 centremostlayers self-consistently. In line with previous first-principles studies of stacking faults inclose-packed metals, we ignored any atomic and volume relaxations (see e.g. [8]). Theeffects of such relaxations are normally negligible because atoms in the faulted part ofthe system tend to retain their close-packed coordination numbers despite the presenceof the fault [3, 13, 26, 27, 28, 29]. Throughout, therefore, we have used the experimentallattice parameter for cobalt, a = 2 .
507 ˚A.
4. Results
Before exploring how the stacking faults influence the MAE of bulk Co, we would like togain an idea of their formation energy. Within the SKKR-ASA scheme, the LSDA totalenergy can be cast into contributions related to individual atomic cells, E i , comprisingthe kinetic energy, the intracell Hartree energy and the exchange-correlation energy,and into the two-cell Madelung (or intercell Hartree) energy, E Mad [14]. For a simplebulk metal, like hcp Co, E Mad is, in practice, negligible, while in the presence of stack-ing faults it gives a non-negligible contribution due to charge redistributions. However,from our self-consistent calculations we found that E Mad is in the order of 0 . − . E i for i = −
10 (layer with effective potential from a self-consistent stacking fault calculation) ffect of stacking faults on the magnetocrystalline anisotropy of hcp Co: a first-principles study E i for i = 10 (layer with appended bulk potential), since due to the mirror symme-try, these two contributions should be identical. Reassuringly enough, they agreed towithin a relative error of 10 − , which is well within intrinsic and numerical error of ourcomputational method.The layer-resolved contributions to the total energy across the systems containing thestacking faults I , I , E and T is shown in Fig. 1. Herein we observe the expectedmirror symmetry and that the layer-resolved total energy contributions approach thebulk total energy, E Co = − .
459 eV, towards the edges of each system. From thisfigure it is obvious that the deviation of E i from E Co is significant up to about 15 layersaway from the centre of stacking fault. Figure 1.
The layer-resolved contributions to the total energy in four hcp cobaltsystems, each exhibiting one of the four different types of stacking fault. The label 0refers to the plane of mirror symmetry. Solid lines serve as a guide for the eyes. ffect of stacking faults on the magnetocrystalline anisotropy of hcp Co: a first-principles study E (I , E , T ) ( N ) = N (cid:88) − N E i − (2 N + 1) E Co , (1)and ∆ E I ( N ) = N − (cid:88) − N + E i − N E Co . (2)The formation energy of a given stacking fault X = I , I , E , T , E ( X ) form , is then definedas E ( X ) form = lim N →∞ (∆ E X ( N )) . (3) Figure 2.
The cell-like part of the formation energy, ∆ E X ( N ), see Eqs. (1) and (2),of stacking faults I , I , E and T in hcp cobalt, displayed as a function of the numberof layers, N , considered in the system on either side of the stacking fault. Solid linesserve as guides for the eyes. The calculated values of ∆ E X ( N ) are shown in Fig. 2. Quite obviously, for all types ofstacking faults, nearly 30 atomic layers (i.e., 15 layers on either side of the stacking fault)are required in order to obtain well-converged stacking fault formation energies. Thefact that the effect of the stacking fault is relatively long-ranged could have significantimpact on nano-sized systems as the formation energy and, consequently, the likelihoodof occurrence of a stacking fault could be different depending on its location in relationto, e.g., other imperfections as well as surfaces or interfaces in the sample. We obtainthe following formation energies, with a possible error of ∼ . − . ffect of stacking faults on the magnetocrystalline anisotropy of hcp Co: a first-principles study E ( I ) form ≈
16 meV ≈
40 mJ · m − E ( I ) form ≈
48 meV ≈
122 mJ · m − E ( E ) form ≈
62 meV ≈
160 mJ · m − E ( T ) form ≈
39 meV ≈
100 mJ · m − . As expected, all stacking faults incur a positive change in the total energy. Of the fourtypes of stacking faults considered here, the intrinsic stacking fault I has the lowestformation energy and the stacking fault E exhibits the highest one. While there is noavailable experiment in literature, the overall results agree well with e.g. Ref. [8]: theextrinsic stacking fault formation energy for the close-packed fcc metals in this study isgenerally significantly larger than that of the intrinsic and twin faults. Moreover, ourcalculated values for the hcp Co growth stacking fault I and the hcp Co extrinsic faultE are close to those obtained by Crampin and co-workers for Ni (which is next to Co inthe periodic table) [8]: 28 mJ · m − for the intrinsic stacking fault and 180 mJ · m − forthe extrinsic fault. Because it is calculated directly from the band energy, the MAE can naturally be re-solved into layer-dependent contributions, D i , see Ref. [19]. These layer-resolved con-tributions are depicted in Fig. 3 for the different types of stacking faults. Note thatthe mirror symmetry is well reproduced in the layer-resolved MAE contributions forall stacking faults. Moreover, the MAE approaches the bulk MAE, K Co = 84 . µ eV,towards the edges of all four systems. For stacking faults of type I , I and T , the MAEcontributions become negative at the centre of the fault, favoring thus an in-plane easyaxis in these layers. This could indicate that these types of stacking faults may act aspinning sites. For the type E stacking fault, the layer-resolved MAE contributions nearthe centre are also reduced, retaining, however, very small positive values.Furthermore, we note that all stacking faults induce long-ranged oscillations in theMAE. For layers of about | i | >
15, the four stacking faults exhibit very similar trendsin the layer-resolved MAE contributions. In other words, at about 15 layers away fromthe stacking fault, the presence of a stacking fault still influences the MAE, while theparticular type of the stacking fault is less significant. This will, however, obviouslydepend on the size of the sample.
The long-ranged oscillations in the MAE could cause significant finite-size effects in theexperimental determination of the MAE of nano-sized samples. We therefore consider ffect of stacking faults on the magnetocrystalline anisotropy of hcp Co: a first-principles study Figure 3.
Calculated layer-resolved contributions to the MAE for stacking faults I and I (upper panel) and E and T (lower panel). The horizontal line refers to thebulk MAE, 84.4 µ eV/atom. Solid lines serve as guide for the eyes. the following cumulative sums, K (I , E , T ) ( N ) = N (cid:88) − N D i − (2 N + 1) K Co , (4)and K I ( N ) = N − (cid:88) − N + D i − N K Co , (5)where the MAE of the stacking fault systems of finite width is related to the MAE ofhcp Co. ffect of stacking faults on the magnetocrystalline anisotropy of hcp Co: a first-principles study Figure 4.
Cumulative sums of layer-resolved contributions to the MAE, K X ( N ) ( X =I , I , E and T ), see Eqs. (4) and (5). Solid lines serve as guide for the eyes. Fig. 4 shows K X ( N ) for the four different stacking faults as a function of N . Surprisingly,for N ≥ type stacking fault appears to increase the MAE, i.e., to strengthen theeasy axis (0001) ( positive effect ). As seen from Fig. 3, this is due to the positive MAEcontributions induced by the stacking fault on neighbouring layers. These apparentlyoutweigh the strongly negative MAE contributions induced in the centre of stackingfault type I . This is an unexpected result as stacking faults are typically reported tolower the MAE (see e.g. [4]). It should be noted, however, that, of the stacking faultsstudied here, type I has the next highest formation energy and it is therefore less likelyto occur in an equilibrated sample. For stacking faults of types I , E and T , the overallchange in the MAE with respect to hcp Co is negative ( negative effect ). As noted earlier,in the vicinity of these stacking faults, the easy dirction is rotated normal to the (0001)axis. This is consistent with the reduction in the total MAE observed experimentallyby Sokalski et al. in[4].It is quite a remarkable feature that, as seen from Fig. 4, the layer-resolved MAEcontributions do not settle until at about approximately 35 layers on either side ofthe stacking fault. This long-ranged behaviour could give rise to significant finite-sizeeffects in nano-sized samples. Moreover, this might have consequences for theoreticalinvestigations into the formation and effects of stacking faults on magnetic properties.Typically, in Monte Carlo simulations of stacking faults, interactions between stackingfaults is kept to around three neighbouring planes [4]. In light of our results, this appearsto be an uncertain assumption. ffect of stacking faults on the magnetocrystalline anisotropy of hcp Co: a first-principles study Experimentally, the presence of stacking faults is normally quantified in terms of thestacking fault density, which is partly a measure of how close the stacking faults arelocated. As the simplest assumption, the change in the MAE due to the presence ofa number stacking faults in a sample is approximated by the sum of the changes inthe MAE due to each individual stacking fault. If this were the case, the effect of anisolated stacking fault on the MAE could quite straightforwardly be transformed intothe change in MAE as a function of the stacking fault density. However, the long-rangedoscillations in the MAE caused by the presence of each stacking fault indicates that thesituation is far more complex.In particular, we considered two stacking faults of type I , separated by three atomiclayers. In other words, the system exhibits the composite stacking fault: · · · A B A B C B C B A B A · · ·
Note that by removing one of the two C-B pairs of atomic layers, a twin-like stackingfault T is obtained. We have chosen three atomic layers between the centres of thetwo stacking faults, since in dynamical models it is often used as the distance beyondwhich the interaction between stacking faults is neglected (see e.g. Ref. [4]). Moreover,we deal with a pair of I type stacking faults because this type of stacking fault has thelowest formation energy and is, therefore, expected to occur more commonly than theother three types of stacking faults.The difference between the layer-resolved MAE contributions and the MAE of bulk hcpCo, ∆ D (I I ) i = D (I I ) i − K Co , (6)is shown in Fig. 5 for the composite stacking fault. As a comparison, we also show theaverage deviations in the layer-resolved MAE contributions from the bulk MAE of twoindependent type I stacking faults,∆ D (I +I ) i = 12 (cid:16) D (I ) i +2 + D (I ) i − (cid:17) − K Co . (7)If ∆ D (I I ) i and ∆ D (I +I ) i were equal for each atomic layer i , the change of the MAEdue to the presence of the composite stacking fault would be exactly twice that of asingle I stacking fault. However, as shown in Fig. 5, ∆ D (I I ) i and ∆ D (I +I ) i deviatesignificantly, particularly in the layers | i | ≤
2, i.e., in the layers between the two stackingfaults. Beyond | i | >
3, the magnitudes of the MAE contributions are similar for ∆ D (I I ) i and ∆ D (I +I ) i , but with a phase shift of approximately one layer. ffect of stacking faults on the magnetocrystalline anisotropy of hcp Co: a first-principles study Figure 5. • : Calculated deviations in the layer-resolved MAE contributions, ∆ D (I I ) i ,of the composite stacking fault from the bulk Co MAE, see Eq. (6), and + : thecorresponding average deviations, ∆ D (I +I ) i , of two superposed I type stacking faultscentred on atomic layers i ±
2, see Eq. (7). Solid lines serve as guide for the eyes.
Similar to the case of single stacking faults, we calculate the cumulative sum of theMAE contributions of the composite stacking fault, K I I ( N ) = N (cid:88) i = − N D (I I ) i − (2 N + 1) K Co , (8)and plot it in Fig. 6. Apparently, K I I ( N ) converges to approximately − .
18 meVfor large N , which is almost three times the change of the MAE of the single typeI stacking fault ( ∼ − .
40 meV, see Fig. 4). Also shown in Fig. 6 is the difference K I I ( N ) − K I ( N ), which appears to settle at approximately − .
38 meV. In otherwords, the two stacking faults interact to yield a stronger negative effect on the totalMAE as compared to two isolated type I stacking faults. This appears to be mainlydue to MAE contributions from the atomic layers located in between the two type I stacking faults. This could have significant consequences for predicting the resultingMAE in dynamical models used to explain experimental data. To draw any definiteconclusions, a systematic study of the stacking fault types and separations would berequired. We expect that such a study would be computationally extremely intensive asinterlayers (or supercells) of up to approximately 160 atomic layers would be requiredin order to reach the limit in which the two stacking faults are far enough apart not tointeract.
5. Summary and Conclusions
Using the fully relativistic screened Korringa-Kohn-Rostoker method, we have studiedthe MAE of bulk hcp cobalt in the vicinity of four different types of stacking faults.We find that, in accordance with experiment, most stacking faults have a detrimen- ffect of stacking faults on the magnetocrystalline anisotropy of hcp Co: a first-principles study Figure 6. (cid:63) : Change in the MAE of hcp Co due the composite stacking fault, K I I ( N ), as defined in Eq. 8). • : Interaction term of the two stacking faults in theMAE, K I I ( N ) − K I ( N ), see. Eqs. (4) and (8) Solid lines serve as guide for the eyes.. tal overall effect on the MAE. The one exception to this overall conclusion is the type I intrinsic stacking fault, which, however, exhibits a relatively high formation energyand which may, consequently, occur relatively infrequently under standard experimen-tal conditions. The effect of a stacking fault on the layer-resolved contributions to theMAE is long-ranged, in the order of 15 atomic layers on either side of each stackingfault. Motivated by this observation, we investigated a particular composite stackingfault and concluded that the MAE of the composite stacking fault is not identical tothe sum of the MAE of the two isolated stacking faults. A further challenging studyis proposed regarding the dependence of the ’interaction’ of two stacking faults on theseparation between them.CJA is grateful to EPSRC and to Seagate Technology for the provision of a research stu-dentship. Support of the EU under FP7 contract NMP3-SL-2012-281043 FEMTOSPINis gratefully acknowledged. Financial support was in part provided by the New Sz´echenyiPlan of Hungary (T ´AMOP-4.2.2.B-10/1–2010-0009) and the Hungarian Scientific Re-search Fund (OTKA K77771). [1] B. Lu, T. Klemmer, K. Wierman, G. Ju, D. Weller, A. G. Roy, D. E. Laughlin, C. Chang, andR. Ranjan, J. Appl. Phys. (2002) 8025[2] D. Weller, A. Moser, L. Folks, M. Best, W. Lee, M. Toney, M. Schwickert, J.-U. Thiele, andM. Doerner, IEEE Trans. Mag. (2000) 10[3] N. Chetty and M. Weinert, Phys. Rev. B (1997) 10844[4] V. Sokalski, D. E. Laughlin, and J.-G. Zhu, J. Appl. Phys. (2011) 093919[5] G. B. Mitra and N. C. Hadler, Acta Crystallographica , vol. 17, pp. 817–822, July 1964.[6] S. Saito, A. Hashimoto, D. Hasegawa, and M. Takahashi,
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