Interfacial exchange interactions and magnetism of Ni2MnAl/Fe bilayers
Rocio Yanes, Eszter Simon, Sebastian Keller, Balazs Nagyfalusi, Sergii Khmelevsky, Laszlo Szunyogh, Ulrich Nowak
IInterfacial exchange interactions and magnetism of Ni MnAl/Fe bilayers
R. Yanes , E. Simon , S. Keller , B. Nagyfalusi , S. Khmelevsky , L. Szunyogh , and U. Nowak Department of Physics, University of Konstanz, Germany Department of Theoretical Physics, Budapest University ofTechnology and Economics, Budafoki ´ut 8., H-111 Budapest, Hungary Institute for Solid State Physics and Optics, Wigner Research Centre for Physics,Hungarian Academy of Sciences, P.O. Box 49, H-1525 Budapest, Hungary Center for Computational Materials Science, Institute for Applied Physics,Vienna University of Technology, Wiedner Hauptstrasse 8, A-1060, Vienna, Austria MTA-BME Condensed Matter Research Group, Budafoki ´ut 8., H-111 Budapest, Hungary
Based on a multi-scale calculations, combining ab-initio methods with spin dynamics simulations,we perform a detailed study of the magnetic behavior of Ni MnAl/Fe bilayers. Our simulationsshow that such a bilayer exhibits a small exchange bias effect when the Ni MnAl Heusler alloy is ina disordered B2 phase. Additionally, we present an effective way to control the magnetic structureof the Ni MnAl antiferromagnet, in the pseudo-ordered B2-I as well as the disordered B2 phases,via a spin-flop coupling to the Fe layer.
PACS numbers: 75.50.Ss, 75.60.Jk, 75.70.Cn, 75.30.Gw
I. INTRODUCTION
Antiferromagnets build a class of materials which isused in magnetic multilayer devices, such as GMR sen-sors or magnetic tunnel junctions, to stabilize and controlthe magnetization of a ferromagnetic compound. Thisfact has increased the demand of antiferromagnets andit has led to an increasing interest in novel antiferromag-netic materials, with Heusler alloys as promising candi-dates for that .Heusler alloys are ternary inter-metallic compoundswith the general formula X YZ, in which X and Y aretypically transition metals and Z is an main group el-ement. This kind of alloys has been in the center ofintensive studies in the last decades, mainly due to thewide range of their multifunctional properties. These in-clude magnetic shape memory effects, magneto-caloricand spintronic effects, as well as thermoelectric proper-ties amongst others. Heusler alloys can be categorized in two distinct groupsby their crystalline structures: half Heusler alloys withthe form of XYZ in the C b structure and full Heusleralloys with the form of X YZ in the L2 structure . Theunit cell of the L2 structure consists of four interpen-etrating face-centered cubic (fcc) lattices, while that ofthe C b structure is formed by removing one of the Xsites. The L2 structure transforms into the so-calleddisordered B2 phase when the Y and Z atoms are mixed,replacing each other at random.The majority of magnetic Heusler alloys are ferromag-netic though it has been reported that some of themare ferrimagnets or even antiferromagnets. In particu-lar, those compounds with 3d elements where only theMn atoms carry magnetic moments at Y site are anti-ferromagnets in the disordered B2 phase. In this con-text Ni MnAl is especially interesting since it has beenreported to exhibit an antiferromagnetic behavior in thedisordered B2 phase as well as in the pseudo-order phase B2-I . The latter is certain limit of the disordered B2phase, where all the Mn atoms are located in the same(001) plane. Ni MnAl in the disordered B2 phase hasa perfectly compensated antiferromagnetic ground state,where the Al and Ni atoms posses no net magnetic mo-ment and the site and anti-site Mn atoms are equivalent .Ni MnAl has been in the center of former studies re-garding shape memory applications because of its ca-pability to change its magnetic order along with its chem-ical order. The existence of exchange bias (EB) wasreported for Heusler alloys which undergo martensiticphase transitions , for Ru MnGe/Fe bilayers and, re-cently, Tsuchiya et al. published an experimental studyof EB in Ni MnAl/Fe bilayers . In general, EB is re-lated to the coupling between a ferromagnet (FM) andan antiferromagnet (AF) and its strength depends on theexchange interaction across the interface and the stabil-ity provided by the AF. This calls for a detailed studyof the interfacial exchange interactions in Ni MnAl/Febilayers.This paper is organized as follows: first we introducea spin model which is based on first-principles calcula-tions. Then, we analyze the exchange interactions inthe bulk and across the interface for the Ni MnAl(B2-I;B2)/Fe interfaces. In the next section, we present spin-dynamics simulations and analyze the possibility to con-trol the magnetic state of the Ni MnAl layer via the Felayer. We finish with a discussion of the origin of a smallin-plane EB found in the Ni MnAl(B2)/Fe system.
II. MODEL AND NUMERICAL APPROACH
In the following we study the magnetic propertiesof Ni MnAl(B2-I)/Fe and Ni MnAl(B2)/Fe interfaces inthe spirit of a multi-scale model, linking ab initio calcu-lations with dynamical spin model simulations. In termsof the fully relativistic Screened Korringa-Kohn-Rostoker a r X i v : . [ c ond - m a t . m e s - h a ll ] J u l (SKKR) Green’s function method we perform self-consistent calculations of the Ni MnAl/Fe bilayers in thedisordered local magnetic moment approach . We usedthe general gradient approximation (GGA) in connec-tion with the atomic sphere approximation and an an-gular momentum cut-off of l max = 3. We derive theexchange interactions between the magnetic momentsby using the spin-cluster expansion (SCE) technique that provides a systematic parametrization of the adi-abatic energy of an itinerant magnetic system. Com-bining this method with the relativistic disordered lo-cal moment (RDLM) scheme , the parameters of thespin-Hamiltonian below can be determined on a quitegeneral level . It is important to note that, due to therelativistic spin-orbit coupling, the exchange interactionsbetween two spins form a 3 × H = − (cid:88) i,j (cid:126)s i J ij (cid:126)s j − (cid:88) i (cid:126)s i K i (cid:126)s i − (cid:88) i µ i (cid:126)H A (cid:126)s i , (1)where the (cid:126)s i represent classical spins, i. e. unit vectorsalong the direction of each magnetic moment at sites i .The first term stands for the exchange contribution to theenergy, with J ij denoting the tensorial exchange interac-tion between moment i and j . The second term comprisesthe on-site anisotropy as well as the magneto-static en-ergy, where K i is called the anisotropy matrix. In thepresence of an external magnetic field, (cid:126)H A , the last termadds a Zeeman contribution to the Hamiltonian, where µ i is the magnetic moment of the atom i .The exchange tenors J ij can be further decomposedinto three parts, J ij = J isoij I + J Sij + J Aij , with theisotropic exchange interaction J isoij = Tr (cid:2) J ij (cid:3) , thetraceless symmetric (anisotropic) part J Sij = ( J ij + J Tij ) − J isoij I , and the antisymmetric part J Aij = ( J ij − J Tij ). The latter one is clearly related to theDzyaloshinskii-Moriya (DM) interaction, (cid:126)s i J Aij (cid:126)s j = (cid:126)D ij · ( (cid:126)s i × (cid:126)s j ), with the DM vector (cid:126)D ij . The DM interactionarises due to spin-orbit coupling and favors a perpendic-ular alignment of the spins (cid:126)s i and (cid:126)s j .Our first principle calculations show that the Nickelas well as the Alumina atoms have negligible magneticmoments in both phases of the Ni MnAl compound, thepseudo-ordered B2-I phase as well as the disordered B2phase. Therefore we restrict our spin dynamics analysisto the evolution of Fe and Mn moments only.To study ground state properties along with spin dy-namics at zero and finite temperatures we solve the stochastic Landau-Lifshitz-Gilbert (SLLG) equation, ∂(cid:126)s i ∂t = − γ (1 + α ) µ s (cid:126)s i × (cid:126)H i − γα (1 + α ) µ s (cid:126)s i × (cid:16) (cid:126)s i × (cid:126)H i (cid:17) , (2)by means of Langevin dynamics, using a Heunalgorithm . The SLLG equation includes the gyro-magnetic ratio γ , a phenomenological damping parame-ter, α , and the effective field (cid:126)H i = (cid:126)ζ i ( t ) − ∂ H ∂(cid:126)s i , (3)which considers also the influence of a temperature T byadding a stochastic noise term (cid:126)ζ i ( t ), obeying the proper-ties of white noise with (cid:104) (cid:126)ζ i ( t ) (cid:105) = 0 , (4) (cid:104) ζ ηi ( t ) ζ θj ( t (cid:48) ) (cid:105) = 2 k B T αµ s γ δ ij δ ηθ δ ( t − t (cid:48) ) . (5)Here i, j denote lattice sites and η and θ Cartesian com-ponents of the stochastic noise.
III. RESULTS AND DISCUSSIONSA. Ab initio results
For the two cases investigated in this work, theNi MnAl(B2-I)/Fe and Ni MnAl(B2)/Fe bilayer, we firstcalculated the exchange interactions, the magnetic mo-ment and the on-site anisotropy layered resolved with themethods described above. In Fig. 1 the isotropic contri-bution of the exchange interaction between Mn-Mn andMn-Fe neighbors are presented as a function of the dis-tance between spin pairs. For the isotropic Mn-Mn ex-change interactions our results indicate a similar behav-ior for the pseudo-ordered B2-I and the disordered B2phase. The dominant nearest neighbor Mn-Mn exchangeinteraction, J , Mn − Mn ≈ −
15 meV, supports antiferro-magnetic order while the magnitude of the exchange in-teractions between Mn atoms in successive shells decayrapidly (Fig. 1(a)).The exchange interactions between Mn and Fe atomsacross the interface are plotted in Fig. 1(b). The dom-inant Mn-Fe exchange interaction is again the nearestneighbor one, favoring antiferromagnetic alignment. Re-markably, this interaction is even larger in magnitudethan the nearest neighbor Mn-Mn interaction in thebulk. It should also be mentioned that the magnitudeof the nearest neighbor exchange interaction in bulk Fe, J , Fe − Fe ≈
50 meV, is again much larger in magnitudethan the above interactions. A summary of the mostrelevant isotropic exchange parameters is given in TableI. Another important parameter, which influences themagnetic behavior of a magnetic bilayer and which can (a) J iso Mn − Mn (B2) J iso Mn − Mn (B2-I)Distance [ a ] I s o t r o p i c E x c h a n g e I n t e r a c t i o n [ m e V ] (b) J iso Mn − Fe (B2) J iso Mn − Fe (B2-I)Distance [ a ] I s o t r o p i c E x c h a n g e I n t e r a c t i o n [ m e V ] FIG. 1. (Color online). Isotropic exchange interaction as afunction of pair distance, (a) between Mn-Mn atoms in B2 andB2-I phases of Ni MnAl bulk, (b) across the interface betweenMn-Fe atoms in Ni MnAl(B2-I)/Fe and Ni MnAl(B2)/Fe bi-layers.TABLE I. Calculated maximum isotropic exchange interac-tions between nearest neighbors, J iso ij (in meV) and magni-tudes of the magnetic moment.Material J iso , Mn − Mn J iso , (Mn − Fe) J iso , Fe − Fe µ Mn µ Fe Ni MnAl(B2-I)/Fe -13.71 -16.45 52.62 3.32 2.4 Ni MnAl(B2)/Fe -15.1 -18.61 52.67 3.35 2.45 lead to the existence of exchange bias is the magneticanisotropy energy (MAE). It has been reported thatbulk Ni MnAl(B2-I) has a small in-plane anisotropy witha magnitude of 0 .
19 meV per spin, while in the case ofthe perfectly disordered B2 phase the MAE is negligible.Close to the AF/FM interface, however, the magneticanisotropy is modified. In case of the Ni MnAl(B2-I)/Feinterface the preferred magnetic orientation is in-planewith an energy of 0 .
03 meV in the interface Mn layerand 0 .
06 meV in the Fe layer. Similarly, an easy planeanisotropy was determined for the Ni MnAl(B2)/Fe in-terface, with a MAE of 0 .
05 meV and 0 .
10 meV in theinterface Mn and Fe layers, respectively.
B. Spin dynamics simulations
For our spin-dynamics simulations we use the modelparameters as determined above from first principles.We suppose the Ni and Al atoms to be nonmagneticand only consider the dynamics of the Mn and Fe mo-ments. The antiferromagnet is hence modeled by theMn sub-lattice, forming in total 30 × × t AF unitcells and the ferromagnet by 30 × × t AF denoting the number of Ni MnAl atomic monolayersperpendicular to the interface (in the following labeled[Ni MnAl(B2I;B2)] t AF /[Fe] ). We consider open bound-ary conditions.For the case of the disordered B2 phase, the Mn atomsare statistically distributed. The magnitudes of the mag-netic moments of Mn and Fe atoms were taken uniformlyin the sample using the values given in Table I. Addition-ally we approximate the effects of the magneto-static in-teraction in the FM layer as an uniaxial shape anisotropywith K Fe = − .
134 meV and the magnetic hard axis per-pendicular to the FM/AFM interface.In the following sections we will analyze the magneticproperties of the two types of bilayers described above.We evaluate the in-plane hysteresis loops and explore theexistence of EB and the switching of the magnetic struc-ture of the Ni MnAl layer.
1. Hysteresis in the pseudo-ordered Ni MnAl(B2-I)/Febilayer
To study the magnetic behavior of this bilayer wecalculate hysteresis loops as a succession of quasi-equilibrium states determined by the numerical integra-tion of the SLLG equation applied to the spin model de-scribed above. The tensorial exchange interactions areconsidered up to 11th neighbor. Initially we preparethe system similar to experiments by simulating a field-cooling process. This process starts from a random spinconfiguration in the AFM at an initial temperature T above the N´eel temperature of the AFM but below theCurie temperature of the FM, and proceeds to a finaltemperature under the influence of an in-plane magnetic H FC .After the field cooling process Mn as well as Femagnetic moments are oriented in-plane which is incorrespondence with the calculated in-plane magneticanisotropy. Importantly, the direction of the Mn mo-ments is nearly perpendicular to that of the Fe moments,a consequence of the so-called spin-flop coupling . Nearthe interface, the Mn moments are slightly tilted fromthis perpendicular ( x -) direction, leading to a very smallnet magnetic moment, anti-parallel to the Fe moments.This configuration follows from the strong antiferromag-netic exchange interaction between Mn-Fe moments andthe fact that the interface between Ni MnAl(B2-I)/Feis compensated (equal number of Mn moments in bothmagnetic sub-lattice). This spin configuration is shownin Fig. 2. (a)(b)FIG. 2. (Color online).(a) Sketch of the magnetic state of theNi MnAl(B2-I)/Fe interface after the field cooling process.(b) Spin configurations of the first two Mn layers starting atthe interface to the Fe (interface Ni layer has layer index 0).
We investigate the switching mechanism and the pos-sible existence of EB for different values of the thickness t AF of the Ni MnAl(B2-I) layer. Our findings indicatethat for perfect bilayers there is no EB within our nu-merical error of ± t AF is increased. As an exam-ple in Fig. 3 we show hysteresis loops for two differentthicknesses of the AF, focusing on the evolution of themagnetization of the FM along the direction of the ap-plied field, M x (FM), the total magnetization of the sys-tem along the direction of the applied field, m h (Tot), aswell as the in-plane antiferromagnetic order parameter, M st y , perpendicular to the applied field.We observe that during hysteresis the Fe moments ro-tate coherently, staying mostly in-plane. For the smallerthickness, due to the strong exchange interactions be-tween Mn-Fe moments, the small net magnetic momentof the AF close to the interface also rotates, maintainingthe antiferromagnetic order. Finally the AF switches fol-lowing the FM (see Fig. 3 (a)). When the thickness ofthe AF increases, and concomitantly the relevance of theon-site MAE of the AF, the AF cannot switch anymoreand the antiferromagnetic order parameter, M st y , remainsclose to unity (see Fig. 3 (b)). Nevertheless, the smallcanting of the Mn moments at the interface switches withthe FM so that the magnetic moment of the AF main-tains its direction antiparallel to the Fe moments.These results indicate that for sufficiently thin lay-ers it is possible to manipulate the magnetic order ofthe antiferromagnetic Ni MnAl layer through the mag-netization of the Fe layer. A similar control of the AFmagnetization by the FM layer has been reported for (a) M x (FM) M st y m h (Tot) H A [T] m h ( T o t ) , M s t y a nd M x ( F M ) (b) M x (FM) M st y m h (Tot) H A [T] m h ( T o t ) , M s t y a nd M x ( F M ) FIG. 3. (Color online). In-plane hysteresis loops for a[Ni MnAl(B2-I)] t AF /[Fe] bilayer. (a) thickness of the AF t AF = 20 a z and (b) t AF = 30 a z . Shown are the normalizedmagnetization of the FM along the applied field direction, M x (FM), and the total magnetization of the system alongthe direction of the applied field, m h (Tot), as well as thein-plane normalized antiferromagnetic order parameter M st y perpendicular to the applied field. NiFe/IrMn/MgO/Pt heterostructure as a key point tothe use of that system in an AF-based tunnel junction.Our finding opens hence the door for new Heusler-alloy-based antiferromagnetic spintronic devices.
2. Hysteresis in the disordered Ni MnAl(B2)/Fe bilayer
Calculations similar to the ones described above wereperformed for the disordered Ni MnAl(B2)/Fe system.First of all, it is important to note that, as a result of thechemical disorder in the B2 phase and its low effectiveanisotropy, much more complex spin structures appear inthe AF after the field cooling process (see Fig. 4). As be-fore, the Fe moments are aligned along the x − direction,the direction of the field during cooling. Again, we ob-serve a kind of spin-flop coupling with the AF orderedmostly perpendicular to the FM and in-plane. However,the canting of the Mn moments at the interface is muchmore pronounced as compared to the Ni MnAl(B2-I)/Fesystem (see the red and yellow Mn moments in the firstlayer of Fig. 4 b)). The reason for this much strongercanting is the structural disorder in Mn moment posi- (a)(b)FIG. 4. (Color online).(a) Sketch of the magnetic state of the[Ni MnAl(B2)] /[Fe] interface after the field cooling pro-cess. (b) Spin configurations of the first two Mn layers start-ing at the interface to the Fe (interface Ni layer has layerindex 0). tions. Due to the statistical distribution of the Mn mo-ments with some probability clusters of moments withinthe same sub-lattice appear. In these clusters the mo-ments have a smaller connectivity to Mn moments ofthe other Mn sub-lattice where the antiferromagnetic ex-change would counteract the canting. As a consequence,larger tilting angles and with that a larger net magnetiza-tion antiparallel to the Fe magnetization appears. How-ever, the effective coupling between Mn and Fe layersis still smaller since only 50 % of the sites of the layerwhich is closest to the Fe are occupied. For comparison,in the pseudo-ordered Ni MnAl(B2-I)/Fe interface 100%of these sites are occupied by Mn atoms.In Fig. 5 (a) hysteresis curves are presented. Thesehysteresis loops are shifted horizontally, correspondingto an exchange bias field of H EB = 200 ± x, y and z components of the Fe magnetization areplotted. However a small out-of-plane component of themagnetization appears as well during the switching inboth branches of the hysteresis loops. The coercive fieldis much smaller than in the previous case of the pseudo-ordered bilayer. This smaller coercive field is due to thefact that the effective interface coupling is smaller be-cause of the smaller occupancy with magnetic Mn atoms (a) M x (FM) M st y m h (Tot) H A [T] m h ( T o t ) , M s t y a nd M x ( F M ) (b) m z m y m x H A [T] ( F M ) m x , m y a nd m z State (II)State (I) . . − . − . . . − . − . (c)FIG. 5. (Color online). (a) In-plane hysteresis loops fora disordered [Ni MnAl(B2)] /[Fe] bilayer. Shown are thenormalized magnetization of the FM, M (FM), and the to-tal magnetization of the system along the direction of theapplied field, m h (Tot) as well as the in-plane normalized an-tiferromagnetic order parameter, M st (AF), perpendicular tothe applied field. (b) Components of the Fe magnetization.(c) Sketches of the magnetic state of the Ni MnAl(B2)/Febilayer before (I) and during (II) the switching process. at the interface. Furthermore, the anisotropy of the AFis smaller which leads to a smaller stability of the AFagainst switching.For an investigation of the thermal stability of the EBeffect mean hysteresis loops where calculated as an av-erage over 5 hysteresis loops performed using the samespacial distribution of Mn-Al atoms. The EB we findis not only rather small and but also unstable againstthermal fluctuations (see Fig. 6). Our results suggest ablocking temperature below 100K.Over all our simulations indicate that the EB is relatedto the disorder — the lack of perfect compensation dueto the random distributions of the Mn and Al atoms into T = 100 K T = 20 K T = 0 K H A [T] m h ( T o t ) . . − . − . . − . − FIG. 6. (Color online). In-plane hysteresis loops for a dis-ordered [Ni MnAl(B2)] /[Fe] bilayer at different tempera-tures. the Y-Z positions in the Heusler alloy — in combinationwith the anisotropy in the AF. As a consequence a smallpart of the interface magnetization of the AF becomesfrozen and does not switch with the FM which leads tothe EB. This conclusion is supported by the fact that theEB vanishes for increasing lateral size. IV. CONCLUSION
In summary, by means of a multi-scale modeling we in-vestigate the interfacial magnetic interactions, the mag-netic state and the hysteresis loops of Ni MnAl(B2-I;B2)/Fe bilayers. Based on first principles calculationswe find a strong negative Mn-Fe interface interaction,exceeding the antiferromagnetic interactions within theNi MnAl. For the disordered Ni MnAl(B2)/Fe bilayerwe find a small EB at low temperatures in agreementwith recent measurements . The existence of such anexchange bias is related to the disorder in the AF andwith that to a lack of perfect compensation at the inter-face. More importantly, we have shown that it is possibleto switch the magnetic structure of the antiferromagneticNi MnAl layer in both, the pseudo-ordered B2-I and dis-ordered B2 phase, via a spin-flop coupling to the ferro-magnetic Fe capping layer. This open the doors for thecontrol of antiferromagnetic Heusler alloys in spintronicdevices with antiferromagnetic components.
ACKNOWLEDGMENTS
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