The influence of anti-site defects and stacking faults on the magneto crystalline anisotropy of FePt
TThe influence of anti-site defects and stacking faults on the magneto crystallineanisotropy of FePt
M. Wolloch, ∗ D. Suess, and P. Mohn Institute of Applied Physics, Vienna University of Technology,Wiedner Hauptstr. 8-10/134, 1040 Vienna, Austria Institute of Solid State Physics, Vienna University of Technology,Wiedner Hauptstr. 8-10/134, 1040 Vienna, Austria
We present density functional theory (DFT) calculations of the magnetic anisotropy energy (MAE)of FePt, which is of great interest for magnetic recording applications. Our data, and the major-ity of previously calculated results for perfectly ordered crystals, predict an MAE of ∼ . PACS numbers: 71.15.Mb, 75.50.Bb, 75.30.Gw
I. INTRODUCTION
Storage density of hard disc drives (HDDs) have in-creased over 8 orders of magnitude since their first intro-duction in the 1950s, peaking in more than 100% increaseper year in the late 1990s and reaching 100 Gb/in in 2002and, after a period of slower growth, finally 500 Gb/in This tremendous achievement was mainly realizedthrough minimization of the read-write head and thin-ner recording media and reduced grain size. However,to keep storage density growing, grain sizes need to befurther reduced and then can be effected by the super-paramagnetic limit. Here the magnetic energy stored in asingle grain, the product of grain volume V and magneticanisotropy constant K u , approaches the size of thermalenergy k B T . The thermal stability requirements havethus shifted the focus to materials with very high K u ,especially FePt alloys. While the anisotropy constant ofordered FePt is large enough to allow storage densities ofup to 4 Tb/in , a further problem arises with the limitedwrite fields employed by conventional read-write heads. Two promising solution to this problem have been pro-posed, giving a thermal write assist using a laserpuls con-centrated by near field laser optics, or exchange springcoupled multilayer media, which reduce the switchingfield while maintaining good thermal stability. A com-bination of both approaches is especially promising, forexample combining the first order magnetic phase tran-sition of FeRh (see Ref. 8 and Refs. therein) with a smallthermal assist to write on extremely hard magnetic FePtalloys. To date the highest demonstrated recording den-sity of 1.402 Tb/in was reported in 2015 by using FePtwith heat assisted magnetic recording (HAMR). Stoichiometric FePt exists both in the disordered fcc A L phase, whereFe and Pt layers are alternating along the c direction. The extremely large magnetic anisotropy energy (MAE)is only found for the ordered phase, which is stable be-low ∼ ◦ C. For magnetic recording, thin films ofthe material are mainly fabricated by co-sputtering fromelemental or alloyed targets, and by electron beamevaporation, but molecular beam epitaxial deposi-tion and other methods are also feasible. The substrateis mainly MgO(001) and deposition temperature as wellas sputtering gas pressure have a large effect on thedegree of ordering. Growing extremely highly orderedFePt films with small grains is a challenging endeavor, but extensive research is performed in order to improvethe growth of grains with the easy axis perpendicular tothe film plane using special buffer- and seed layers. Thishelps with the reduction of the in-plane components ofthe magnetization, which are a serious noise sources inHAMR.Computationally it is of course much easier to inves-tigate fully ordered FePt using periodic boundary con-ditions then simulating disordered structures. However,even though FePt has an extremely large MAE it is stillonly in the meV range per formular unit (f.u.) and calcu-lated by subtracting comparatively large numbers fromeach other. Additionally, the MAE is a Fermi surfaceeffect and thus very sensitive to the k-point sampling ofthe Brillouin zone. These effects make accurate calcula-tion challenging and one should not be surprised to findlarge variations in the results of ab-initio calculations inthe literature published in the last decades. In Fig. 1 wesort 29 previously calculated values for the MAE of FePt,published in 19 different papers, in 0.25 meV widebins and fit the data with a Gaussian distribution. Allresults have been calculated ab-initio with density func-tional theory (DFT), but involve multiple codes, meth-ods, lattice parameters, and exchange-correlation poten-tials. Nearly half of the results fall into the bin between a r X i v : . [ c ond - m a t . o t h e r] M a r µ = 2 .
88 meV / f . u . but quite alarge standard deviation of σ = 0 .
64 meV / f . u . . FIG. 1:
Histogram of 29 previously calculated MAE val-ues of FePt with different DFT codes, computational pa-rameters and exchange and correlation approximations.The black curve is a Gaussian fitted to the data resultingin µ = 2 .
88 meV / f . u . and σ = 0 .
64 meV / f . u . . The dataare collected from Refs 19–37.These values are often compared with the bulk experi-ments of Ivanov et al. from 1973, who used the ballisticthrow method and also a vibration magnetometer to mea-sure the magnetic properties of annealed FePt powders,reporting an anisotropy constant of K = 7 . / m ,corresponding to ∼ . etal. , and Staunton et al. are close to this value, at1.30 meV/f.u. and 1.70 meV/f.u., respectively. Ivanov etal. argue that their sample is fully ordered, because itexhibits especially high magnetic anisotropy, but take nofurther measures to actually quantify the degree of order.From more recent experiments on thin films and powders,however (see Sec. III B), we know that full order is notnecessary to measure anisotropies of 7.0 MJ/m or higherin FePt, leading us to believe that the sample of Ivanov et al. was indeed highly, but not fully ordered .In a very recent paper Khan et al. publishedbenchmark calculations for the MAE of fully or-dered FePt. They employ both full-potential linearaugmented plane wave (FLAPW) and full-potentialKorringa-Kohn-Rostoker (KKR) Green function meth-ods to calculate the MAE within the local density ap-proximation (LDA). The authors of Ref. 28 took utter-most care in converging their computational parametersand the results of both different DFT methods are in verygood agreement with each other. We consider those cal-culation the most accurate ones published to date. How-ever, since the calculated MEA of ∼ . II. METHODOLOGICAL DETAILS
We have performed spin polarized DFT compu-tations employing the Vienna Ab-Initio SimulationPackage
VASP version 5.4.1, using the ProjectorAugmented-Wave (PAW) method.
The plane waveenergy cutoff was chosen to be 900 eV, which is morethan 230% (300%) higher than the recommended valuefor the Fe (Pt) PAW potentials (set of 2003) which treatthe 3 s , 3 p , 3 d , and 4 s (5 s , 5 p , 5 d , and 6 s ) electrons (32per FePt pair) as valence. We sample the Brillouine zonewith generalized Monkhorst-Pack grids as described byWisesa et al., finding a significantly quicker convergencewith their server generated grids than for those generatedby the VASP routines. Unless otherwise noted their pa-rameter r min , which describes the distance between lat-tice points on the real-space superlattice and increasesk-mesh density if increased, was set to 65 ˚A. The cho-sen energy cutoff might seem large, but only with thiscutoff we can achieve a total energy convergence of lessthan 0.1 meV, and thus properly quantify the MCA. Toapproximate the effects of exchange and correlation thegeneralized gradient correction (GGA) as parametrizedby Perdew, Burke, and Ernzerhof (PBE) has been used. To ensure accurate forces during relaxations we use anadditional superfine fast Fourier transform (FFT) grid forthe evaluation of the augmentation charges and a smear-ing of ≤ . (first order). For total energy calculations the tetrahe-dron method with Bl¨ochl corrections has been used. Inall total energy GGA calculations we explicitly accountfor non spherical contributions of the gradient correc-tions inside the PAW spheres. Electronic relaxations areconverged to 10 − meV, while forces in ionic relaxationswhere converged to ≤ µ eV range. We chose the [110] direction of the L unit cell as the hard axis (corresponding to the [001]direction if distorted fcc unit cell is used). III. RESULTS AND DISCUSSIONA. Pristine FePt
The high MAE of FePt in the L1 phase is mainly dueto the large spin orbit coupling in the Pt atoms. Theyshow magnetic moments induced by the Fe 3 d orbitals,and the d orbital of both species hybridize with eachother. Detailed discussions about the origin and natureof the large MAE can be found in the literature and will not be discussed here further.In contrast to Ref. 28 we use the PBE functional, whichbelongs to the class of GGAs, instead of the LDA. Thischoice was made due to the better equilibrium volumeobtained by PBE of 28.02 ˚A , which at +2%, is muchcloser to the experimental value of 27.5 ˚A than LDAat 24.55 ˚A ( − L FePt with PBE yields lattice parameters a = 2 . c = 3 . c / a ratioof 1.38, about 1.5% larger than the experimental valueof 1.36. While this difference will influence the MAEsomewhat, the effect of the c / a ratio is considered to besmall compared to disorder in the sample. For PBE we calculate an MAE of 2.74 meV/f.u., cor-responding to an anisotropy constant K u of 15.7 MJ/m .Increasing the total number of k-points in the Bril-louin zone by ∼
55% from 7317 to 11340 (via adjust-ing r min to 75 ˚A) does not change this value. This re-sult compares well to the mean value of previously pub-lished results of Fig. 1, and is in excellent agreementwith the benchmark calculations of Ref. 28, which re-ports a value of 2.73 meV/f.u. for PBE. The angular de-pendence of the anisotropy Energy E can be fitted to E ( θ ) − E (0) = K sin ( θ )+ K sin ( θ ), where we find that K with 2.67 meV is more than one order of magnitudelarger than K with 0.13 meV. Employing the LDA func-tional (for the same PBE-relaxed unit cell) yields a some-what larger value of 3.11 meV/f.u. (which is virtually un-changed if we calculate the MAE for the slightly smallerlattice parameters used in Ref. 28), again in good agree-ment with the benchmark calculations (reporting valuesfrom 2.85 to 3.12 meV/f.u., depending on method andcode). The local magnetic moments as well as the or-bital moments calculated with PBE, LDA, and LDA+Ucan be found in Tab. I.
1. Analyzing correlation effects
Although we are able to reproduce the reference val-ues of Khan et al. very well with less computational ef-fort, we fail to reproduce the LDA+U results publishedpreviously by Shick and Myrasov. Using their lattice
TABLE I:
Local spin moments ( m loc ) and orbital mag-netic moments ( m orb ) for FePt. (cid:107) and ⊥ mark orien-tation of the spin moments parallel and normal to thez-axis, respectively. m loc [ µ B ] m orb [ µ B ]Fe Pt Fe (cid:107) Pt (cid:107) Fe ⊥ Pt ⊥ PBE 2.83 0.39 0.056 0.044 0.052 0.057LDA 2.69 0.37 0.058 0.043 0.052 0.055LDA+U 2.83 0.36 0.056 0.043 0.054 0.055 parameters and values for both U and J, we calculatea MAE of 2.79 meV/f.u., which is about 0.32 meV lessthan our result with LDA for their lattice parameters,but still about twice as large as their published result of1.3 meV/f.u.. Switching from the rotationally invariantLDA+U flavor of Liechtenstein et al., to the simplifiedversion by Dudarev et al., did not change our resultsignificantly.To investigate correlation effects further we employthe adiabatic connection fluctuation-dissipation theoremin the random phase approximation (ACFDT-RPA), asimplemented in the VASP package. Here the cor-relation energy is computed via the plasmon fluctuationequation by calculating the independent particle responsefunctions using occupied and unoccupied states. Theexchange energy is calculated by Hartree-Fock theory.Both contributions are calculated non self-consistentlyusing DFT orbitals and are added to Hartree, kinetic,and Ewald energies to obtain the total ACFDT-RPA en-ergy. As prudent for metallic systems, we neglected longwavelength contributions. Due to the huge computational effort needed for suchcalculations, we where not able to perform them with thesame accuracy as our other MAE calculations. Further-more the integral over ω in the plasmon equation hasto be solved numerically using a fixed number of sam-pling points N ω . Fortunately convergence with respectto N ω (which can be troublesome for metals) was quickfor FePt, and the necessary accuracy ( ∼ . N ω = 10. Calculating the RPA en-ergy for significantly more than ∼ r min parameter to maximally 34 ˚A, resultingin 1088 k-points in the full Brillouin zone. The planewave cutoff was also reduced to 600 eV, which then leadsto an MAE of 3.1 meV/f.u. on the PBE level, about 10%higher than for our converged computational parameters.As can be seen from Fig. 2, the ACFDT-RPA calcula-tions are not at all converged at 1088 k-points, much incontrast to standard DFT, which is qualitatively correcteven for comparatively low k-mesh densities and for anenergy cutoff of 600 eV. For example 420 k-points areenough to approach the converged value of the MAEwithin ∼
10% for PBE, while on the RPA level not eventhe easy axis is correctly predicted. At least we are ableto show a clear trend in our data, where the MAE in-creases monotonically with the number of k-points forACFDT-RPA calculations. Of course it is not possibleto predict at what value of the MAE the ACFDT-RPAresults will converge from the trend at low k-mesh densi-ties, but we are fairly confident that the data is sufficientto predict a higher value than 2.0 meV/f.u..
FIG. 2:
MAE calculated with on the ACFDT-RPAlevel with respect to the number of k-points in the fullBrillouin zone. The PBE results for an energy cutoff of600 eV are also plotted for comparison. The MAE in-creases monotonically with increased number of k-pointsand should be higher than 2.0 meV/f.u..Although we could not converge our ACFDT-RPA cal-culations, the trend we observe make us confident thatcorrelation effects alone are not able to reduce the MAEof FePt by a factor of two. In the following section wewill show that disorder is able to reconcile experimentand calculations much more satisfyingly than a high leveltreatment of exchange and correlation.
B. Defects in FePt
As discussed in section I, experimental measurementsof the MAE of FePt are always performed for a somewhatdisordered crystal. Disorder in a crystal can be quantifiedby the long range order parameter S . In the case of astoichiometric FePt crystal, the fractions of Fe and Ptatoms sitting on their correct respective lattice sites mustbe equal ( r Fe = r Pt = r ), thus the equation for S reducesto S = r Fe + r Pt − r − . (1)For a totally disordered crystal S = 0, as each atom has50% probability to sit on its preferred lattice site, while S = 1 is achieved for perfect order. Experimentallythe order parameter usually is estimated by the relativestrength of integrated X-ray diffraction peaks I and I according to the formula S = ( I /I ) meas ( I /I ) S =1calc , (2) where the numerator consists of the measured values andthe denominator uses calculated intensities for perfectorder, assuming atomic scattering factors, Debye-Wallercorrection, Lorentz polarizations factors and structurefactors. However, in a recent investigation of a multi-grain FePt nanoparticle by 3D atomic electron tomogra-phy, it was observed that L1 order might be wronglyattributed in standard 2D methods due to overlappingL1 grains, although this seems unlikely in highly sto-ichiometric samples. In Fig. 3 we have plotted sev-eral experimentally determined values for the magneticanisotropy constant K u = K + K . Values are givenin MJ/m and have also been converted to meV/f.u.,for easier comparison to calculations. Most measure-ments have been performed at room temperature (RTshown with red symbols) but Okamoto et al. and Lyu-bina et al. have also provided low temperature mea-surements at 10 and 5 K, respectively (LT shown withblue symbols). From their data we see that the MAEis reduced by ∼
20% to 30% at RT compared to LT.More generally, the temperature dependence of the firstorder anisotropy constant K is coupled to the temper-ature dependence of the magnetization M S to approxi-mately second power, K ( T ) /K (0) = ( M S ( T ) /M S (0)) ,as measured by Refs. 12 and 13, and calculated byRefs. 24, 37, 57, and 58. From Fig. 3 we also see thatthe spread of values for high order parameters is quitelarge, an effect which can be explained in part by the dif-ferent compositions of the samples (see legend of Fig. 3),but also indicates the difficulties in accurately measuringsuch a large anisotropy with usual laboratory fields. Forexample Thiele et al. give two values for the MAE ofthe same sample, once measured by torque magnetome-try (3.96 MJ/m , (cid:79) in Fig 3), and once deduced from sat-uration magnetization and dipolar length measurements(10 MJ/m , (cid:77) in Fig 3), which differ by more than 100%.Computational studies investigating the MAE of dis-ordered FePt in the L1 structure have been conductedby Staunton, Burkert, Kota and their respectivecoworkers . Several studies also deal with the electronicstructure and magnetic properties of the fully disorderedalloy in the face centered cubic structure (e.g. Ref. 61 andthe references therein). Generally the coherent potentialapproximation (CPA) was used in these papers to modelthe disorder effects on a mean field basis. While the re-sults calculated by Ref. 59 and Ref. 29 fit the experimen-tal data very well (see Fig. 4), they arrive at considerablylower MAE values for the fully ordered system than themajority of other calculations and the new benchmarkstudy by Khan and coworkers. Furthermore, at cer-tain order parameters some experimental data points liehigher than the CPA calculations, which seems unlikelygiven that surface effects, grain boundaries and varyinggrain orientations in experimental samples will likely de-crease the MAE compared to the infinite crystal size ofthe calculations. Burkert et al. report data that agreewith the benchmarks for full order and approach the ex-perimental data nicely for lower values of S (see Fig. 4). FIG. 3:
Experimental anisotropy constant K u inMJ/m and plotted over the long range order parame-ter S . The right axis is a conversion to the MAE inmeV. Data is taken from Refs. as indicated in thelegend. Red symbols are measurements at room temper-ature, while blue symbols stand for low temperature.However, the mean field approach is unable to predictwhich types of defects are responsible for the significantdrop of K u with decreasing order and the divergence be-tween the studies by Staunton, Kota, and Burkert, allusing very similar methods, is a little unsatisfactory. FIG. 4:
MAE in meV/f.u. plotted over the long rangeorder parameter S . Purple crosses are the ab-inito resultsfor the fully ordered system as presented also in Fig.1.Black symbols represent experiments at room tempera-ture (circles) and around 10 K (squares). The red circles,blue squares, and green diamonds represent the calcula-tions by Kota, Burkert, and Staunton. We, on the other hand, are more interested in the in-fluence of single localized defects in the FePt crystal, es-pecially anti-site defects (ASD) and stacking faults (SF). An ASD consists of one Fe and one Pt atom exchangingtheir place in the lattice, while a SF can occur duringgrowth of FePt thin films if instead of perfect alternatingstacking of Fe and Pt planes, two planes of the same ma-terial follow each other. We distinguish between localizeddefects, where two neighboring atoms are exchanged forASDs and two layers of one type are followed by two lay-ers of the other in SF, and dispersed defects, where theexchanged atoms and the double-planes are far away fromeach other. These basic defects are depicted in Fig. 5.
FIG. 5:
Depiction of a localized (a) and a dispersed (b)anti-site defect, as well as a localized (c) and a dispersed(d) stacking fault in stoichiometric FePt alloy. ASDs aremodeled in a 54 atom and SFS in a 16 atom supercell,with Fe shown in gold and Pt in silver. Arrows mark thedefect positions.
1. Defect formation energies
While we do not consider a change in cell volume, theatomic positions in all of our supercells have been relaxedcarefully and separate static calculations are used to de-termine the defect formation energies. As we only con-sider defects where two (or more) atoms exchange theirpositions and keep the alloy fully stoichiometric, the de-fect formation energy (DFE) is simply the total energy ofthe supercell containing the defect minus n times the to-tal energy of a fully relaxed FePt unit cell, where n is thenumber of FePt pairs in the supercell, E df = E sc − nE uc .Defect formation energies for different super cell sizes (de-scribed as multiples of the unit cell in a , b , and c direc-tion) are given in table II for both ASDs and SFs. If 2defects are considered in a cell the DFEs are averagedover several configurations. For example if 2 local ASDsare put into a 2 × × a (equivalent to a shift by b ), a lattice vec-tor c , by both a and c (equivalent to b and c ), a and b ,or by a , b , and c together. We thus arrive at 5 differentpossibilities, of which two have to be counted twice sincethey have less symmetry. The DFEs are actually quitedifferent, ranging from 435 meV for stacking along c , to690 meV for stacking along a or b . TABLE II:
Defect formation energies E df per defect forASDs and SFs with corresponding order parameters S indifferent supercells and configurations. N is equal to thenumber of defects per cell, while L or C denote a local ora dispersed defect. ASDs N L/D
S E df [meV]4 × × × × × × × × × × × × N L/D
S E df [meV]1 × ×
12 1 L 0.83 449.41 × ×
10 1 L 0.80 451.61 × ×
10 1 D 0.80 455.81 × × × × × × × × × × × × × × We immediately notice that SFs have a lower DFEthan ASDs, and that they are very well decoupled fromeach other, since the energy stays nearly constant at ∼
450 meV. If the stacking fault is localized, with 2Pt layers followed immediately by two Fe layers, or dis-persed, where the double layers are far away from eachother does not matter much from an energetic point ofview. On the other hand, a single local ASD shows quitedifferent E df depending on supercell size. For the 3 × × × × E df . This indicates that ASDs are not decoupled, andinteract attractively in close proximity, as can be seenfrom the averaged DFE for two ASDs in a 2 × × ∼
566 meV per defect, is considerablylower than an isolated ASD ( ∼
736 meV).From the data of table II we see that the DFEs arequite sizable at ∼ . S would lower thefree energy enormously, even considering the decreasedentropy. The difficulties in producing highly ordered films of the material experimentally (see section I), leadsthus to the conclusion that the barriers for defect healingmust be comparably large.
2. Magnetic anisotropy energy
The magnetic anisotropy energy was calculated for thesupercells of table II analogous to the method used forthe perfect crystal with the same energy cutoff and k-griddensity. Our results for both SFs and ASDs are plottedin Fig. 6 alongside the mean field data from Burkert etal. and the experimental data detailed in Fig. 3. FIG. 6:
MAE in meV/f.u. plotted over the long rangeorder parameter S . Black symbols represent experimentsat room temperature (circles) and around 10 K (squares).The red circles are for SFs, green diamonds are for ASDs,and blue squares are the KKR-CPA results of Ref. 22. SFdata are fitted with a straight line, all other lines serveonly as guides to the eye.It is obvious that although the DFE of ASDs is higherthan for SFs, the former are responsible for the strong de-crease of the MAE at reduced order. For stacking faultsa decrease in S leads to a linear reduction of the MAE.This means that the defects are not only well isolatedfrom each other regarding the DFE, but also regardingthe MAE. As the fraction of correctly ordered unit cellsin the supercell decreases, the MAE decreases propor-tionally, right down S = 0 .
5. If the cell size is further re-duced to six atoms ( S = 0 .
33) or 4 atoms ( S = 0), whichare not shown in Fig. 6, it is not really appropriate tospeak of a SF, as 2/3 or more of the material is layeredin the wrong way. For S = 0 .
33 we calculate an MAEof 1.40 meV, slightly higher than the linear trend wouldpredict, while for S = 0 the MAE drops to 0.18 meV. Ananti site defect on the other hand, has much larger effect,which can also be reasoned intuitively, as there are sig-nificantly more unit cells directly influenced by a singlelocalized ASD (8) than by a SF (2), disregarding lowerorder effects like atoms sitting on wrong lattice sites in aneighboring unit cell or relaxations. Furthermore ASDsperturb the surrounding of Pt atoms (which are mainlyresponsible for the large MAE) in 3D, while SFs onlychange the surrounding in 2D, having a diminished ef-fect on the MAE. While our calculations for ASDs arein reasonable agreement with the experimental data overthe whole range, the agreement is certainly a lot betterfor lower values of S . This might indicate that ASDs doindeed cluster together in FePt, as our data for S = 0 . et al. lies be-tween our SF and ASD data, although closer to the ASDpoints. This is to be expected from a mean field ap-proach for random disorder, which should encounter SFlike regions less than ASDs. Delocalized defects wherenot included in Fig. 6, due to generally higher DFE, butthe cases that we tested showed MAE differing less than5% from the localized defects in the same super cell.In the single-atom resolution images from Ref. 56 ASDsare also commonly observed and their density is still ∼
3% in the highest ordered grain centers of the nanopar-ticle . This is a strong indication that ASDs are alsocommon in stoichiometric FePt, although they have arather large DFE. Although we can not completely ruleout that the MAE might also be lowered slightly by cor- relation effects (see section III A 1), we believe that theinclusion of ASDs is sufficient to explain the experimentalMAE data on the basis of standard DFT calculations. IV. CONCLUSIONS
We have shown that ASDs are responsible for the largereduction of the MAE in FePt with decreasing long rangeorder parameter S . Experimental measurements and ab-initio DFT calculations are thus also comparable withoutincluding many body effects beyond the LDA or GGAlevel. Qualitative calculations using the ACDFT-RPAshow that the effect of more accurate treatment of cor-relations is probably smaller than that of disorder. Thiswill allow future DFT calculations to accurately modelthin FePt films and layered systems useful for magneticrecording with reasonable effort. ACKNOWLEDGMENTS
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