3b2v7 MAGMA : 8364 Prod:Type:com ED:JollySamson pp:123ðcol:fig::NILÞ PAGN: mohance SCAN: Shobha ARTICLE IN PRESS 1 3 Journal of Magnetism and Magnetic Materials ] (]]]]) ]]]­]]] 5 7 Calculated angular dependence of the interlayer couplings in 9 Fe/Cr superlattices having imperfect interfaces 11 Daniel Stoeffler*,Clara Cornea 13 Institut de Physique et de Chimie des Mat!eriaux de Strasbourg, UMR 7504 du CNRS, Groupe d'Etude des Mat!eriaux M!etalliques, 23 rue du Loess F-67037, Strasbourg, France 15 17 Abstract 19 We present calculations of the non-collinear magnetic structure in Fe/Cr superlattices having imperfect interfaces 21 modeled by considering atomic steps in the Cr layers and Fe/Cr interfacial ordered compounds. The interlayer couplings are obtained directly from self-consistent tight binding band structure calculations. We show that the 23 bilinear­biquadratic expression for the coupling energy fits nicely the calculated interlayer couplings curves. r 2001 Published by Elsevier Science B.V. 25 Keywords: Electronic structure; Exchange couplingFbilinear­biquadratic; Interface roughness; MultilayersFmetallic 27 29 This last decade,considerable effort has been devoted and limiting the determination of the interlayer coupling 57 31 to enhance the theoretical description of the magnetic to the relative energy between parallel and antiparallel configuration at the atomic scale in more and more magnetic states [1,2]. 59 33 complex nanostructured systems combining angular and The study of Fe/Cr multilayers needs to include magnitude degrees of freedom of the magnetic moment. interfacial imperfections being at the origin of non- 61 35 When non-collinear magnetic structures are considered collinear magnetic features [3,4] as well as strong in metallic systems,the itinerant character of the reductions of the Cr local magnetic moments [5]. This 63 37 transition metals requires a band structure description. can only be done with simplified electronic structure Such an approach is needed to describe the interlayer descriptions (like the real space tight binding approach) 65 39 exchange coupling in metallic multilayers in particular allowing to build large unit cells containing a few tens of when the spacer layer,separating two ferromagnetic non-equivalent atoms. Our method is completely 67 41 layers,is itself magnetically ordered,like in the Fe/Cr equivalent to ab initio ones for the determination of case,where magnetic frustrations play a major role on total energy differences in Fe/Cr systems as shown by 69 43 the overall properties. The method we use describes the comparison between our results (Fig. 2 in Ref. [6]) to correctly the Cr magnetism in Fe/Cr multilayers for the the recently ab initio one published by the group of 71 45 range of Cr thickness we consider here even if it does not Dresden (Figs. 2 and 3 in Ref. [7]) obtained with spiral include the SDW. Frustrations due to imperfections are magnetic states in the Cr spacer. This comparison 73 47 present whatever the thickness is since they originate shows a very good quantitative agreement between both from the interfaces and can extend over a significant results : they obtain a total energy difference of 40.8 meV 75 49 fraction of 77 51 79 53 *Correspo UNCORRECTED PROOF the spacer layer [1,2]. Up to now, only self- per unit cell for 6Fe/6Cr superlattices whereas we consistent collinear solutions have been considered obtained 41.2 meV for 5Fe/5Cr (a more recent calcula- resulting in an overestimation of the frustration energy tion for 5Fe/6Cr gives 37.8 meV per unit cell). However, since they do not allow the direction of the magnetic moments to relax,they obtain only a bilinear contribu- nding author: Tel.: +33-88-107065; fax: +33-88- 107249. tion to the interlayer coupling energy. Details on the 81 55 E-mail address: daniel.stoeffler@ipcms.u-strasbg.fr calculation method can be found in previous papers (D. Stoeffler). [8,9]. In order to reduce the computer time, we assume 0304-8853/01/$ - see front matter r 2001 Published by Elsevier Science B.V. PII: S 0 3 0 4 - 8 8 5 3 ( 0 1 ) 0 0 7 7 3 - 9 MAGMA : 8364 ARTICLE IN PRESS 2 D. Stoeffler, C. Cornea / Journal of Magnetism and Magnetic Materials ] (]]]]) ]]]­]]] 1 that all magnetic moments remain in the same plane and minimum. This shows clearly that,for perfect interfaces, 57 we build the smallest possible unit cells by considering nearly spiral magnetic states occur in the Cr spacer for n 3 straight line [0 1 0] steps and ordered compounds in a large enough. 59 doubled in plane cell. Our calculations remain never- Up to now we have only considered the coupling 5 theless at the limit of what can be done with the most energies for perfect interfaces due mainly to computer 61 recent generation of massively parallel computers. time limitation. The speed of the convergence and the 7 As previously reported for perfect interfaces [6],the access to recent powerful massively parallel computers 63 calculated interlayer coupling energy ECðDyMÞ between allow to determine directly the interlayer coupling for 9 the magnetisation of successive Fe layers follow the large cells taking into account interfacial imperfections. 65 bilinear­biquadratic expression As usual in such kind of studies,we increase the in plane 11 E cell and include Fe and Cr atoms in the same atomic 67 CðDyMÞ ¼ J1 cosðDyMÞ þ J2 cos2ðDyMÞ; ð1Þ layer. Mixed interfaces are modeled by c2 2 and 2 2 13 where J1 and J2 are,respectively,the bilinear and in plane chemical cells [1,2]. The c2 2 cell corresponds 69 biquadratic coupling constants and DyM is the angle to a Fe0:5Cr0:5 mixed interfacial atomic layer and the 15 between the magnetisation of the two successive Fe 2 2 one corresponds to two Fe0:75Cr0:25=Fe0:25Cr0:75 71 layers. For small Cr thickness n (smaller than 11 atomic mixed interfacial atomic layers. Atomic steps,separating 17 layers (AL)),collinear magnetic states are obtained for Cr and Fe terraces corresponding to a periodic repe- 73 DyM ¼ 01 and 1801: This is related to the vanishing of tition of l Cr and l Fe atomic rows,at one interface are 19 the inner Cr magnetic moments when frustration built with a 2l 1 in-plane cell; these cases are denoted 75 increases. On the contrary,for n larger than 12 AL, EC by the terraces size TS ¼ l þ l: Finally,with a 2l 2 in 21 ðDyMÞ follows the parabolic expression first evidenced by plane cell,we can consider simultaneously both inter- 77 Slonczewski [4] faces; these cases are denoted by TS ¼ ðl þ lÞ 2: 23 Dy 2 The coupling energy for a given value of Dy is 79 E M Dymin CðDyMÞ ¼ C ; ð2Þ calculated by maintaining fixed the angle for all Fe p 25 atoms of the atomic layer in the middle of the Fe layers 81 where C and Dymin are,respectively,the coupling (see Fig. 1 of Ref. [6]). We define DyM by the angle 27 constant and the angle corresponding to the energy between the two successive magnetisation resulting from 83 29 85 (x 10) (x 100) 31 87 40 1x1 : n = 4 4 TS = 2+2 0.4 TS = (3+3)x2 33 1x1 : n = 5 TS = 3+3 89 2x2 : n = 4 TS = 5+5 30 2x2 : n = 5 3 0.3 35 c2x2 91 20 TS = 1+1 2 0.2 37 93 10 1 0.1 39 95 0 0 0.0 41 97 _10 _1 _0.1 43 99 ) (meV/in plane atom) M _20 _2 _0.2 45 ( 101 E C _30 _3 _0.3 47 103 _40 _4 _0.4 49 105 _50 _5 _0.5 51 107 0 30 60 90 120 150 180 0 30 60 90 120 150 180 0 30 60 90 120 150 180 53 Fig. 1. UNCORRECTED PROOF (o) (o) (o) M M M 109 Calculated (symbols) and fitted (solid lines) coupling energies EC as a function of DyM for all Fe/Cr multilayers considered in 55 this work having a Cr thickness between 4 and 5 atomic layers. The two graphs at right correspond to a zoom of the graph at left for 111 small energies. MAGMA : 8364 ARTICLE IN PRESS D. Stoeffler, C. Cornea / Journal of Magnetism and Magnetic Materials ] (]]]]) ]]]­]]] 3 1 Table 1 57 Coupling constants J1; J2; C (meV/in plane atom) obtained by fitting the calculated interlayer magnetic couplings curves displayed in 3 Fig. 1. DyMðminÞ corresponds to the position of the energy minimum 59 1 1 n ¼ 4 n ¼ 5 n ¼ 10 n ¼ 11 n ¼ 20; 21 5 61 J1 26.9 20.6 15.4 15.2 C20 ¼ 29:0 7 J2 2.23 1.15 5.40 4.58 C21 ¼ 26:8 63 DyMðminÞð1Þ 180 0 180 0 180/0 9 65 2 2 : n ¼ 4 n ¼ 5 c2 2 TS ¼ 1 þ 1 J 11 1 5.15 3.55 11.2 4.40 67 J2 0.35 0.65 1.76 1.67 DyMðminÞð1Þ 180 0 0 0 13 69 TS 2 þ 2 3 þ 3 5 þ 5 ð3 þ 3Þ 2 15 J1 1.94 0.17 1.17 0.09 71 J2 1.99 2.77 4.03 0.16 17 DyMðminÞð1Þ 60 88 99 107 73 19 75 21 the calculation at fixed DyM: The substitution of Dy by realistic simulations. To our knowledge,it is the first 77 DyM introduces a slight distortion in the ECðDyMÞ curves time that a direct band structure calculation in Fe/Cr 23 varying mainly their curvature and affecting directly the has been done taking into account both imperfections 79 coupling constants listed in Table 1 obtained by fitting evidenced experimentally [10,12] and giving coupling 25 the calculated values (Fig. 1) with expressions 1 and 2. strengths values similar to experiment. The coupling 81 Fig. 1 shows ECðDyMÞ for all cases considered for n strength experimentally found by Heinrich et al. [10] are 27 varying between 4 and 5 AL. The most remarkable jJ1j ¼ 0:35 and J2 ¼ 0:13 meV/in plane atom. The 83 result is that for a Cr thickness smaller than 12 atomic coupling strength experimentally found by Schreyer 29 layers,all calculated coupling curves fit nicely expres- et al. [11] are Jþ ¼ 2:5 and J ¼ 1:0 meV/in plane atom 85 sion 1. For perfect interfaces (first line of Table 1),the (+ and corresponding,respectively,to an odd and 31 coupling constants fluctuate significantly for small Cr even number of Cr atomic layers). 87 thickness with a surprising negative J2 for n ¼ 4 whereas The authors acknowledge the Institut du D!eve- 33 for the largest thickness they become similar in loppement et des Ressources en Informatique Scientifi- 89 magnitude. In agreement with the experiments [10],we que of the CNRS,the Centre d'Informatique National 35 found J2 nearly equal to jJ1j=3 but the coupling strength de l'Enseignement Sup!erieur and the Institut du Calcul 91 is overestimated by at least one order of magnitude. The Parall"ele de Strasbourg of the University Louis Pasteur 37 introduction of interfacial mixing at one interface for having allowed the access to their massively parallel 93 (second line of Table 1) reduces significantly the computers. 39 coupling strength by a factor 3 to 4 and the energy 95 minimum still corresponds to a collinear magnetic 41 solution since J2ojJ1j=2: For imperfections having a References 97 larger periodicity (third line of Table 1) than these last 43 cases,the energy minimum occurs for non collinear [1] D. Stoeffler,F. Gautier,Phys. Rev. B 44 (1991) R10389. 99 magnetic states with even J2 larger than jJ1j: When the [2] J. Magn. Magn. Mater. 156,(1996) 114. 45 terrace size l increases for TS ¼ l þ l; (i) J1 diminishes [3] A. Schreyer,et al.,Phys. Rev. Lett. 79 (1997) 4914. 101 rapidly and becomes negative for l larger than 3 atomic [4] J.C. Slonczewski,J. Magn. Magn. Mater. 150 (1995) 13. 47 rows and (ii) J2 increases. This shows clearly that the [5] K. Mibu,et al.,Phys. Rev. Lett. 84 (2000) 2243. 103 coupling strength which was determined directly differs [6] C. Cornea,D. Stoeffler,Europhy. Lett. 49 (2000) 217. 49 completely [7] B. Yu. Yavorsky,et al.,Phys. Rev. B 62 (2000) 9586. 105 51 107 53 UNCORRECTED PROOF from the value deduced by averaging over values calculated for perfect interfaces. However,with [8] C. Cornea,D. Stoeffler,Comput. Mater. Sci. 10 (1998) these stepped interfaces,the coupling strength remains 245­ 249. significantly higher than the experimental one. It is only [9] C. Cornea,Thesis of the University Louis Pasteur, Strasbourg,April 1999,unpublished. for the last system we considered,with TS ¼ ð3 þ 3Þ2; [10] B. Heinrich,J.F. Cochran,T. Monchesky,R. Urban,Phys. 109 that we obtain extremely small coupling constants Rev. B 59 (1999) 14520. 55 (Table 1) around 0.1 meV/in plane atom demonstrating [11] A. Schreyer,et al.,Phys. Rev. B 52 (1995) 16066. 111 that both kind of imperfections are required for a [12] A. Davies,et al.,Phys. Rev. Lett. 76 (1996) 4175.