Disorder-induced cubic phase in Fe 2 -based Heusler alloys
Janos Kiss, Stanislav Chadov, Gerhard H. Fecher, Claudia Felser
aa r X i v : . [ c ond - m a t . m t r l - s c i ] F e b Disorder-induced cubic phase in Fe -based Heusler alloys. Janos Kiss, Stanislav Chadov, Gerhard H. Fecher, ∗ and Claudia Felser Max-Planck-Institut f¨ur Chemische Physik fester Stoffe,N¨othnitzer Strasse 40, 01187 Dresden, Germany.
Based on first-principles electronic structure calculations, we analyze the chemical and magneticmechanisms stabilizing the cubic phase in Fe -based Heusler materials, which were previously pre-dicted to be tetragonal when being chemically ordered. In agreement with recent experimental data,we found that these compounds crystallize within the so-called “inverted” cubic Heusler structureperturbed by a certain portion of the intrinsic chemical disorder. Understanding these mechanismsis a necessary step to guide towards the successful future synthesis of the stable Fe -based tetrag-onal phases, which are very promising candidates for the fabrication of rare-earth-free permanentmagnets. Keywords: rare-earth-free, hard magnets, Heusler alloys, chemical disorder
One of the oldest problems within the field of ma-terials science is the search for the inexpensive hardmagnets, i. e., for materials retaining their magnetisa-tion after being once magnetized. Their role in thedaily life can be hardly overestimated: hard magnets arewidely used in automotive applications, telecommunica-tions, data processing, consumer electronics, instrumen-tation, aerospace and bio-surgical applications. In par-ticular, they play a unique role in renewable energy tech-nologies based on electric generators (e. g., rotors in wind-turbines, small hydroelectric systems etc.). However, ma-terials exhibiting outstanding hard-magnetic propertiestogether with high magnetization and high Curie tem-perature are rather expensive as being based on combina-tions involving rare-earth elements (e. g., Sm-Co, Nd-Fe-B) [1, 2]. Thus, the development of new inexpensive com-pounds with hard-magnetic properties (i. e., rare-earth-free hard magnets) which can be industrially mass pro-duced is important and highly relevant (for the reviewsee e. g., Ref. [2]). The recent explosion of attention forthe tetragonally-distorted magnetic Heusler systems orig-inates at a large extent from this prospective as well [3].Indeed, apart of being promising candidates for tun-neling magneto-resistance and spin-torque-transfer ap-plications [3, 4], this family may also provide materialscombining the tetragonal distortion with a large mag-netic moment and high Curie temperature, suitable ashard magnets. The group of Fe YZ-based Heusler com-pounds (with Y and Z being the transition and the main-group element, respectively), theoretically predicted tobe tetragonal with a large magnetization (4–5 µ B /f.u.,f.u.=formula unit) would be one of such promising ma-terials sources [5]. In contrary, the subsequent synthesis,XRD and M¨ossbauer characterization have shown thatall these compounds crystallize in the cubic phase [6]. Tounderstand which ingredients can lead to their tetrago-nal distortion obviously implies an important preliminarystep – a detailed understanding of the mechanisms sta-bilizing their cubic phase. This is the main point of the ∗ [email protected] FIG. 1. (color online) The cubic unit cell of the point-symmetry group No. 216 together with its schematic graph-ical diagram and the corresponding written notation. Thehigh-symmetric Wyckoff positions 4 a , 4 b , 4 c and 4 d are dis-tinguished by different colors. present study.Before proceeding to the results, first we would liketo introduce the notations extensively used throughoutin the text (see Fig. 1). In the most general case anycubic Heusler system corresponds at least to the point-symmetry group No. 216. In order to distinguish betweendifferent chemical configurations, we will use the specialwritten notation according to the occupations of the fourhigh-symmetric Wyckoff positions: first we will writedown the occupants of the 4 a and 4 b sites followed bythe slash sign, then - of the 4 c and 4 d , i. e., “4 a b /4 c d ”.Thus, e. g. the so-called “regular” and “inverted” vari-ants (terms introduced in Ref. [5]) of Fe CuGa can bewritten as CuGa/FeFe and FeGa/CuFe, respectively. Incase if certain Wyckoff position is occupied by severalatomic sorts randomly, e. g. by A and B with probabil-ities x and 1 − x , it is noted using the square brackets:[A x B − x ]. In case of the tetragonal distortion ( c = a ) thesymmetry reduces at least to the point group No. 119.The sequence for the written notation in this case doesnot change, it implies only the usage of the differentWyckoff positions: 2 a b /2 c d .To clarify the discrepancy between the theoretical pre-dictions from Ref. [5] and the experimental data [6], inthe following we will study the relative stability of thecubic and tetragonal phases of the Fe -based Heuslersystems by optimizing both structural, magnetic andchemical degrees of freedom based on ab-initio density-functional calculations. As a suitable numerical toolwhich accounts for these factors simultaneously, we usethe fully-relativistic Green’s function formalism imple-mented within the SPR-KKR (Spin-Polarized fully Rel-ativistic Korringa-Kohn-Rostoker) method [7]. The ran-dom occupation is described in terms of the CPA (Coher-ent Potential Approximation) [8]. Despite its mean-fieldnature (the effective averaging of the short-range ordereffects) CPA remains the most practical technique whichincludes the essential features of randomness. In orderto ensure that the CPA result is not an artifact of thesingle-site approximation, we performed additional su-percell calculations. It was also found, that the usageof the full potential (i. e., the non-spherical potential) ismuch more essential for the adequate description thana particular choice of the exchange-correlation potential.For this reason, the presented calculations correspondsto the fully-relativistic and full-potential results employ-ing the local density approximation for the exchange-correlation functional [9]. The calculations for different c/a ratios are performed for the fixed volume, which wastaken from the available experimental data [6].In order to explain the mechanism which keeps theFe -based systems cubic, throughout the discussion wewill focus on the Fe CuGa system, because as we found,all basic conclusions valid for this system can be trans-ferred without restrictions onto other compounds in thisseries (i. e., Fe CuAl, Fe NiGa, Fe NiGe and Fe CoGe)synthesized experimentally [6].The main outcome of the present study is summarizedin Fig. 2, which represents the dependency of the totalenergy on c/a ratio for various alloy configurations. Theordered “regular” Heusler structure in cubic phase turnsout to be unstable (indicated by the corresponding en-ergy curve maximum at c/a = 1, black line in Fig. 2 (a)),whereas at about c/a = 1 .
54 the system falls into therelatively deep energy minimum. For the fixed chem-ical order (i. e., CuGa/FeFe) the tetragonal distortionis the only mechanism which can relax the instabilityof the cubic phase, since the magnetic degrees of free-dom are already in use (for more detailed description oftetragonal distortion mechanisms, see e. g. [3, 5]). Asit follows, the gradual transition towards the “inverted”Heusler structure, realized by random chemical Cu-Feinter-layer exchange starts gradually to develop an en-ergy minimum for the cubic phase. Although the config-urations with intermediate Cu-Fe site occupations (e. g.,[Cu . Fe . ]Ga/[Cu . Fe . ]Fe) exhibit an energy minimafor both tetragonal and cubic phases, the limiting orderedsystem ( x = 1, i. e. the fully “inverted” FeGa/CuFe) isstable only within the cubic phase (Fig. 2 (a) or (b),red curve). Despite the large energy difference (about −
170 meV/f.u.) between the cubic “regular” and “in-verted” phases, the deepest absolute energy minimumis found for the tetragonally-distorted “regular” config-uration (see Fig. 2 (a)), which is by about 20 meV/f.u. c/a ratio T o t a l e n e r gy [ e V / f . u . ] (a) c/a ratio -0.050.000.050.100.15 T o t a l e n e r gy [ e V / f . u . ] (b) FIG. 2. (color online) Total energy of Fe CuGa Heusler al-loy calculated as a function of c/a ratio for various distri-butions of Fe and Cu (indicated by the box-like diagram;green, yellow and white colored areas correspond to the Fe, Cuand Ga occupations, respectively). (a) Thick black and redcurves correspond to CuGa/FeFe (“regular”) and FeGa/CuFe(“inverted”) configurations, respectively. Thinner violet lineshows the intermediate [Cu . Fe . ]Ga/[Cu . Fe . ]Fe case be-tween “regular” and “inverted”. The absolute energy min-imum of the “regular” (CuGa/FeFe) tetragonal phase istaken as a reference. (b) Solid red curve represents thesame “inverted” configuration as on the left panel, whereasthe solid blue line shows the most stable configuration:FeGa/[Cu . Fe . ][Cu . Fe . ], obtained from the ordered “in-verted” by mixing Fe and Cu randomly in-plane. The dashedred and blue lines correspond to the distributions derived fromthe previous two configurations via additional in-plane ran-dom spread of Ga and Fe: [Fe . Ga . ][Fe . Ga . ]/CuFe (reddashed) and [Fe . Ga . ][Fe . Ga . ]/[Cu . Fe . ][Cu . Fe . ](blue dashed). The absolute energy minimum of the “in-verted” (FeGa/CuFe) phase is taken as a reference. more stable compared to the “inverted” cubic configura-tion. Thus, for the ordered systems our results agreeswith the former calculations [5]. This means, that themechanisms stabilizing the cubic phase involve degreesof freedom which where neglected so far, e. g. the chem-ical disorder [6].It is important to note, that the huge energy decrease(about −
170 meV/f.u.) gained by going from the “reg-ular” cubic to the “inverted” cubic configuration (thelargest energy scale in the diagram on Fig. 2) is mostlikely of the magnetic origin. The latter is due to the op-timization of the magnetic exchange coupling within theFe sublattice, since the nearest magnetic neighbors (i. e.,the Fe atoms from the adjacent layers within “inverted”cubic FeGa/CuFe) are sitting closer to one another com-pared to the “regular” cubic CuGa/FeFe setup, in whichthey are in-plane. For this reason, by searching for themore stable configurations, we start from the “inverse”cubic system and perturb it by in-plane chemical disor-der (i. e., by conserving the total amount of Cu and Gawithin adjacent layers). Hence, there are two importantin-plane disorder scenarios: random in-plane mixtures ofFe-Ga, and that of Fe-Cu.We found, that the random in-plane spread of Ga andFe (case [Fe . Ga . ][Fe . Ga . ]/CuFe, red dashed linein Fig. 2 (b)) leads to the increase of the total energy(compared to the “inverted” configuration, FeGa/CuFe)by about 150 meV. In contrast, the random in-planespread of Cu and Fe (case FeGa/[Cu . Fe . ][Cu . Fe . ],solid blue line in Fig. 2 (b)) leads to an energy gain ofabout −
40 meV (again, compared to the “inverted” case,FeGa/CuFe). The key observation is, that this −
40 meVenergy gain is enough to stabilize the cubic structure(in FeGa/[Cu . Fe . ][Cu . Fe . configuration), which fi-nally becomes more stable than the tetragonal “regular”ordered CuGa/FeFe by about 40 −
20 = 20 meV/f.u., inagreement with experiment.Our results show, that these two effects (Fe-Ga and Fe-Cu in-plane random mixtures) are rather independent onone another: i. e. disregarding the particular arrangementof atoms within the adjacent layer, the energy changed by150, 40 or 150 ±
40 meV while going from one distribu-tion to another within all four cases. The large increasein energy by 150 meV in the first case is mainly due tothe distinct nature of Fe and Ga. So, within the fixedsquare lattice it is unfavorable to form separate clustersof Fe and Ga, since each atomic sort would prefer to cre-ate its own lattice within a cluster which will be ratherdifferent from another. For this reason, any perturbationof the perfect chemical order in Fe-Ga layers will increasethe total energy. This issue, however, is not critical forthe second case: the separation of Fe and Cu within thegiven lattice does not cost so much energy, since bothatom types are much more similar. To ensure that the −
40 meV energy gain in this case is not just an artifact ofthe single-site nature of the CPA, we have performed su-percell calculations by systematically increasing the num-ber of Fe-Cu in-plane swaps, mimiquing Fe-Cu disorder.This has shown that by increasing the degree of Fe-Cuseparation the total energy is indeed reduced by around −
40 meV.The subsequent calculations of the magnetic exchangecoupling constants J ij (Fig. 3) of the classical Heisenbergmodel ( H = − P i>j J ij ˆe i ˆe j , where ˆe i,j are the unity vec-tors along the magnetization directions on local sites i and j ) revealed the magnetic origin of both stabilizationmechanisms responsible for the atomic rearrangementfrom the “regular” into the “inverted” phase and for thechemical disorder within the Fe-Cu layers. Namely, thestrong Fe-Fe inter-layer coupling ( J inter-layer ≈
25 meVbetween the adjacent Fe-Ga and Fe-Cu layers) keeps thewhole system ferromagnetic. This is in agreement withthe high Curie temperature (798 K) measured in Ref. [6].Although the in-plane couplings appear to be an or-der of magnitude weaker, still, as we mentioned, theiroptimization plays a crucial role in the stabilization ofthe cubic phase. In the ordered “inverted” configura-tion (FeGa/FeCu) the nearest in-plane Fe atoms tendto couple antiferromagnetically ( J in-plane = − . FIG. 3. (color online) Comparison of the magnetic exchangecoupling in the “inverted” FeGa/FeCu (a) and in the moststable FeGa/[Fe . Cu . ][Fe . Cu . ] (b) configurations. Theatoms are arranged within Fe-Ga and Fe-Cu layers marked bylight- and dark-blue horizontal planes, respectively. Fe, Cuand Ga atoms are shown as green, yellow and white spheres,respectively. Magnetic moments are shown by arrows. Thebond thickness reflects the strength of the exchange inter-action. The inter-layer Fe-Fe interactions are dominating: J inter-layer ≈
25 meV (thick green bonds). The in-plane in-teractions are negligibly small, except those in Fe-Cu planes.Case (b) illustrates the typical distinction from the ordered“inverted” structure: the random in-plane swap of one Fe andone Cu atoms which brings two Fe atoms closer to one anotherwithin the Cu-Fe plane. This alters the nearest in-plane Fe-Feexchange from antiferromagnetic ( J in-plane = − . J in-plane = 5 . coupling. Thus, the magnetic energy can be furtherreduced by bringing the Fe atoms closer together asshown in Fig. 3 (b), so that they couple ferromagnetically( J in-plane = 5 . . Cu . ][Fe . Cu . ]configuration.The considerations presented above can be futher sup-ported by comparing the electronic structures as shownon Fig. 4. The instability of the electronic subsystemis typically related to the strength of the DOS peaks inthe vicinity of the Fermi energy. In case of the “regular”CuGa/FeFe cubic system, a huge instability peak at E F (total DOS( E F ) ≈ . E - E F [ e V ] E - E F [ e V ] E - E F [ e V ] (a)(b)(c)Γ ΓWX L FIG. 4. (color online) Comparison of the spin-resolvedelectronic band structures and related densities of statesfor the (a) “regular” cubic CuGa/FeFe, (b) “inverse” cubicFeGa/CuFe and (c) FeGa/[Fe . Cu . ][Fe . Cu . ] configura-tions. The majority- and minority-spin states are distin-guished by red and blue, respectively. the 4 a (or 4 b ) site will split the minority-spin states atW-point far away from the Fermi energy, noticeably re-ducing the DOS peaks (see Fig. 4 (b) and (c)). This stepis related to the largest ( −
170 meV/f.u.) energy gain at-tributed to the inter-layer magnetic exchange optimiza-tion discussed above. Now by going from the ordered“inverted” Heusler variant (FeGa/CuFe) to the most sta-ble case of FeGa/[Fe . Cu . ][Fe . Cu . ], the DOS at E F is further reduced (from 3.2 to 2.9 sts./eV). Indeed,as we have shown above, the Fe-Cu disordered configu- ration benefit from the in-plane magnetic optimization,and therefore it is by 40 meV/f.u. lower in energy com-pared to the “inverted” FeGa/CuFe system. We wouldlike to point out, that very similar stabilization mech-anisms characterized by comparable energy scales cantake place in other Fe -based cubic Heusler compounds.For example, for Fe CuAl and Fe NiGe the inter-planeexchange energy optimization (i. e. by going from the“regular” to the “inverted” cubic phase) gains −
367 and −
168 meV/f.u., whereas the in-plane optimization (dueto Fe-Y in-plane disorder) contributes with −
27 and −
40 meV/f.u., respectively to the energy. Thus the sta-bilization mechanisms presented in this letter are rathergeneral within the group of Fe YZ materials.To conclude, we emphasize that the presented analysisexplains the stability of the cubic phase in Fe YZ Heuslercompounds, and provides a clear explantion for the dis-crepancy between experimental results and theory. Theactual stabilizing mechanism appears to be the chemi-cal disorder, which optimizes the magnetic exchange cou-pling within Fe-Y layers of the initially ordered FeZ/FeYcubic phase. At the same time, the FeZ layers remainchemically ordered due to a large difference (i. e. atomicradius, valency, electronegativity etc.) between Fe andthe main-group element Z. Thus, the most stable config-urations can be written as FeZ/[Fe . Y . ][Fe . Y . ]. Ingeneral, the important prerogative enabling the chemi-cal disorder is the “inverted” ordered structure: as wehave seen, the effect of the rearrangement from YZ/FeFeinto FeZ/YFe within the cubic phase is comparably ef-ficient to the tetragonal distortion in YZ/FeFe. On theother hand, the first scenario allows to further optimizethe system by chemical disorder, whereas the second onedoes not. ACKNOWLEDGMENTS
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