PHYSICAL REVIEW B VOLUME 57, NUMBER 13 1 APRIL 1998-I Superlattice symmetry in magnetic multilayer systems J. Zabloudil Center for Computational Materials Science, Vienna, Austria C. Uiberacker and C. Blaas Institute for Technical Electrochemistry, Technical University of Vienna, Vienna, Austria U. Pustogowa Center for Computational Materials Science, Vienna, Austria L. Szunyogh Center for Computational Materials Science, Vienna, Austria and Department of Theoretical Physics, Technical University of Budapest, Budapest, Hungary C. Sommers Laboratoire de Physique des Solides, Campus d'Orsay, Orsay, France P. Weinberger Center for Computational Materials Science, Vienna, Austria and Institute for Technical Electrochemistry, Technical University of Vienna, Vienna, Austria Received 20 October 1997 The problem of superlattice symmetry, i.e., the question of periodicity along the growth direction surface normal in magnetic multilayer systems, is discussed using discrete Fourier transformations for the anisotropy energy, as well as, for the antiparallel and perpendicular interface exchange coupling. We analyze the system Cu 100 / Cu3Ni3) n , where n is the number of repetitions, for the case of free surfaces and surfaces capped semi-infinitely by Cu 100 . It will be shown that for some magnetic properties, and only in certain situations, almost periodic behavior with respect to n applies, while for other properties an oscillatory behavior is characteristic. Also discussed are implications with respect to typical experimental situations and with respect to traditional supercell approaches. S0163-1829 98 00313-0 I. INTRODUCTION ergies are chosen as characteristic examples. Frequently when discussing physical properties of mag- netic multilayer systems, periodicity along the surface nor- II. THEORETICAL ASPECTS AND COMPUTATIONAL mal is assumed in most theoretical approaches, but also in DETAILS analyzing experimental data. Theoretically very often super- The fully relativistic spin-polarized version10 of the cell calculations are performed, which of course, indepen- screened Korringa-Kohn-Rostoker method11 for layered dent of the number of atoms per unit cell, use implicitly systems12 is applied to calculate selfconsistently the elec- cyclic boundary conditions along the growth direction.1­4 In tronic structure and the magnetic properties of a free sur- the same manner, there is a tendency to interpret and explain faces of Cu 3Ni3) n on Cu 100 , denoted in the following as results of experimental investigations in terms of superlattice Cu 100 / Cu 3Ni3) n/Vac, and b semi-infinitely capped sur- effects.5­9 Therefore, it seems that there is a definite need for faces, denoted by Cu 100 / Cu3Ni3) n/Cu 100 , whereby all investigating the applicability of such approaches. For interlayer distances refer to a fcc parent lattice13 correspond- this reason in the present paper magnetic properties of ing to the experimental lattice spacing of Cu no surface or Cu 3Ni3) n on Cu 100 , where n is the number of repeti- interface relaxations . For each system, i.e., for each n, first tions, are determined by considering free surfaces and sur- the electronic and magnetic structure of the magnetic con- faces capped semi-infinitely by Cu 100 . Quite clearly one figuration corresponding to a uniform in-plane orientation of such unit Cu3Ni3) serves as a building block and-by as- the magnetization in the Ni layers magnetic reference con- suming periodicity along 100 -has to be viewed as the unit figuration C0, see also Table I is calculated self-consistently cell. By employing an approach that makes use only of two- using 45 k points in the irreducible part of the surface Bril- dimensional translational symmetry, namely, within the louin zone ISBZ and the local density functional form of planes of atoms, and that allows one to vary n, periodicity Ref. 14. The obtained self-consistent layer-resolved effective with respect to n can manifest itself if it really exists for potentials and layer-resolved effective magnetization fields various magnetic properties. The magnetic anisotropy energy in the spin-polarized Kohn-Sham-Dirac Hamiltonian see, and specific forms of multi-interface exchange coupling en- e.g., Ref. 15 are then used to evaluate the following differ- 0163-1829/98/57 13 /7804 10 /$15.00 57 7804 © 1998 The American Physical Society 57 SUPERLATTICE SYMMETRY IN MAGNETIC . . . 7805 TABLE I. Magnetic configurations. Conf. MNi MCu MNi MCu MNi MCu MNi MCu MNi orientation of magnetization C0: *** *** *** *** uniform in-plane C1: *** *** *** *** uniform perpendicular to plane C2: *** *** *** *** antiparallel in-plane C3: *** *** *** *** alternating in-plane and perpendicular to plane ences in the band energies with respect to the magnetic con- 1/2 figurations given in Table I: A F q;Ci dq 1, 5 0 E Ci E Ci E C0 . 1 comparison can made between different cases such as free It should be noted that in Table I configuration C and capped surfaces, and different numbers n of repetitions. 1 refers to the case that uniformly in all Ni layers the orientation of the Quite clearly any other layer-resolved quantity such as the magnetization is perpendicular to the planes of atoms. Con- magnetic moments or the Madelung potentials12 can be figuration C transformed in the same manner. 2 comprises again a case of in-plane orientations, arranged, however, alternatively antiparallel, while in con- If the physical property investigated is periodic with re- figuration C spect to the building block a period of six, i.e., a pronounced 3 in-plane and perpendicular-to-plane orienta- tions alternate. E(C maximum in F(q;C 1) refers to the band energy contribu- i) at q 1/6 has to show up, since one tion to the magnetic anisotropy energy,10,16 while E(C unit Cu 2) 3Ni 3) consists of six layers. A period of twelve and E(C (q 1/12) is characteristic if the quantity is periodic with 3) reflect multi-interface exchange coupling. In principle the magnetic anisotropy energy and the multi- respect to twice a building block. Of course the interesting interface exchange coupling energy also contains a contribu- and physically relevant question of using such discrete FT's tion arising from the magnetic dipole-dipole interaction,10 is how large n has to grow in order to trace a periodic be- which, however, only grows more or less linear with the havior. In the present paper the number of repetitions is re- number of magnetic layers.10,17­19 In the case of the mag- stricted to n 11, which in turn implies a maximum netic anisotropy energy, e.g., the magnetic dipole-dipole en- multilayer thickness of about 225 a.u. ergy predominantly determines the so-called volume By relating the band energy differences E(Ci) in Eq. 1 anisotropy.20,19 In the present investigations the magnetic to the number of repetitions band energy difference per unit dipole-dipole interaction is not included. cell , one further can examine whether for increasing n, If P denotes the total number of atomic layers in the E(Ci)/n approaches a constant or oscillates. The next sec- intermediate regime,12 i.e., the total number of atomic layers tion will show examples for both kinds of behavior. between the nonmagnetic semi-infinite systems, then It should be noted that both, discrete FT's and quantities E(C per repetition unit cell , are essential in describing what is i) can be partitioned into layerwise contributions E sometimes called colloquially superlattice symmetry or col- p(Ci), loquial lattice,13 namely, in defining periodic behavior with P P respect to the surface normal. E Ci Ep Ci Ep Ci Ep C0 , 2 p 1 p 1 III. RESULTS AND DISCUSSION which in turn can be Fourier transformed using the following discrete Fourier transformation FT A. Layer-resolved band energies In the following, in all figures showing layer-resolved P quantities, the indexing of atomic layers starts at the Cu 100 F q;Ci eiqp Ep Ci 3 p 1 substrate, i.e., the Cu substrate is to the left of the interme- diate region and vacuum or the cap is to the right. For n 5 or, in relation to their mean value E(Ci)/P, as the layer-resolved band energy contributions Ep(C1) to the magnetic anisotropy energy are displayed in Fig. 1 for free P E C and capped surfaces. For the free surface case one can easily F q;C i i eiqp Ep Ci , 4 see the strong perturbation caused by the surface, however, p 1 P for n 4 a period of six emerges since with increasing n the where q is given in units of 2 /d with d being the interlayer number of nearly identical peaks in the interior of the film is distance.21,19 increasing. For the capped case the effect of the interfaces All band energy differences presented in this paper were with the semi-infinite substrate on both sides is fairly mar- evaluated within the force theorem approximation see in ginal: for each n the corresponding entries are practically particular Ref. 16 by using 990 k points in the ISBZ and by characterized by n identical peaks in Ep(C1). applying the group theoretical methods described in Ref. 10. Completely different in shape are the layer-resolved band By normalizing the absolute value of the Fourier transform energy contributions Ep(C2) to the antiparallel interface in a suitable manner, e.g., to a unit area A, coupling energy shown in Fig. 2 a in the case of capped 7806 J. ZABLOUDIL et al. 57 FIG. 1. Layer-resolved band energy contribution to the magnetic anisotropy energy Ep(C1) in Cu3Ni3) n multilayers on Cu 100 . Left: free surfaces Cu 100 / Cu3Ni3) n/Vac, right: semi-infinitely capped surfaces Cu 100 / Cu3Ni3) n/Cu 100 . The number of repetitions n is marked explicitly. systems. Obviously, odd and even number of repetitions dis- right-hand-side boundary. Note that for even n's the shapes play a different kind of boundary conditions: for odd n pe- of Ep(C2) in Fig. 2 a will be reversed with respect to the riods of 12 are terminated symmetrically due to an inversion labeling of layers if the orientation of magnetic moments is with respect to the geometrical center of the multilayer, reversed simultaneously in each layer. It is easy to guess that whereas for even n this is not possible. Therefore, in Fig. for n 9 the pattern to be seen follows the one shown in the 2 a the entries corresponding to odd n's have a symmetric right column of Fig. 2 a . shape, while those corresponding to an even number of rep- As can be seen from Fig. 2 b , for free surfaces a similar etitions are asymmetric with a peak of positive sign at the pattern as illustrated in Fig. 2 a applies: for odd n the con- 57 SUPERLATTICE SYMMETRY IN MAGNETIC . . . 7807 FIG. 2. Layer-resolved band energy differences for antiparallel interface exchange coupling Ep(C2) in the case of a Cu 100 / Cu3Ni3) n/Cu 100 and b for n 7,8 for free surfaces Cu 100 / Cu3Ni3) n/Vac left side and for semi-infinitely capped surfaces Cu 100 / Cu3Ni3) n/Cu 100 right side . The number of repetitions n is marked explicitly. 7808 J. ZABLOUDIL et al. 57 FIG. 3. Magnetic moments in Cu3Ni3) 8 multilayers on Cu 100 . Top: free surface, middle: capped surface, bottom: abso- lute value of the discrete Fourier transformation A 1 F(q;C1) of the magnetic moments for the free surface dashed line and the capped surface solid curve . FIG. 4. Layer-dependent Madelung constants in Cu3Ni3) 8 tributions from the two boundaries of the multilayer remain multilayers on Cu 100 . Top: free surface, middle: capped surface, approximately equal in size and identical in sign, whereas for bottom: absolute value of the discrete Fourier transformation A 1 F(q;C even n the surface induces a much bigger perturbation than 1) of the layer-dependent Madelung constants for the free surface dashed line and the capped surface solid curve . for an odd n. The above features of multi-interface exchange coupling are also characteristic for the pattern of E heights being only marginal. As to be expected with increas- p(C3) not shown , i.e., for the case of layer-resolved band energy con- ing n the width of this peak shrinks and its height increases, tributions to the perpendicular interface coupling, whereby while simultaneously-due to the increasing number of the peaks are about half as big as those in Figs. 2 a and 2 b . terms summed over-the background reduces. A completely different pattern arises when considering the distribution of Madelung potentials corresponding to the B. Magnetic moments and Madelung potentials reference configuration C0 Fig. 4 . Clearly enough in terms In Figs. 3 and 4 one particular case, namely, n 8, is of electrostatics a remarkable perturbation at a surface has to examined in some detail. Figure 3 shows the distribution of be encountered, however, it is somewhat surprising to see magnetic moments for free and capped surfaces correspond- that for the free surface in the discrete FT of the layer- ing to the reference configuration C0 and their discrete FT's. resolved Madelung potentials the peak at q 1/6, which As to be expected near the surface the Ni moment is slightly characterizes the capped system, is completely whipped out. enhanced, however, in terms of the distribution of magnetic It should be noted that in the latter case also a kind of ``alias- moments in the multilayer system, the free surface case dif- ing'' at q 1/3, i.e., a period of 3 can be seen. In particular fers only very little from that of the capped surface. This is Fig. 4 illustrates and explains quite convincingly the differ- directly mapped in the corresponding discrete FT's: in both ent behavior of free and capped surfaces in Fig. 1 and Fig. cases a strong peak at q 1/6 is seen, the difference in peak 2 b . 57 SUPERLATTICE SYMMETRY IN MAGNETIC . . . 7809 FIG. 5. Absolute value of the discrete Fourier transformation A 1 F(q;C1) of the band energy contribution to the magnetic anisotropy energy for free surfaces dashed lines and capped surfaces solid curves . The number of repetitions n is marked explicitly. C. Discrete Fourier transformations of layer-resolved band the magnetization, since in every second building block the energy differences same orientation of the magnetic field applies. Somewhat For n 10 the discrete FT's of E surprising is that also a period of about 2.5 (q 0.41) can be p(C1) are displayed in Fig. 5 for free and capped surfaces. In this figure one can see seen, which quite likely refers to the so-called short period that for capped surfaces with increasing n the peak at q 1/6 recorded in magnetic systems with a Cu spacer,22,23 and gets substantially sharper, while in the case of free which in the asymptotic limit frequently is related to a par- surfaces-even after 10 repetitions-this peak is still rather ticular vector of the Fermi surface of fcc Cu.22,24 At first weak and quite some intensity remains for q 0.3. For rea- glance for n 8 the difference between free and capped sur- sonably large n the capped surfaces obviously show a well- faces seems to become unimportant. However, a closer in- defined period of 6, whereas for free surfaces the presence of spection of the peak at q 1/2 shows that a for odd num- the interface to the vacuum almost prevents such a periodic- bers of repetitions the peak height is considerably larger than ity see also Figs. 1 and 4 . for even numbers and b for odd n there is almost no dif- In the discrete FT's of Ep(C2) Fig. 6 several rather ference between free and capped surfaces, while for even n a well-developed peaks emerge with increasing n, namely, at clear difference exists. Going back to Figs. 2 a and 2 b one q 1/12, 1/6, 1/3, 0.41, and 1/2. The main periods of 6 and can correlate this particular peak to the peaks in the layer- 12 obviously reflect the number of atomic layers per building resolved quantities near the interfaces, which with alternat- block and the geometrical arrangement of the orientations of ing n alternate in sign. In particular from Fig. 2 b it is clear 7810 J. ZABLOUDIL et al. 57 FIG. 6. Absolute value of the discrete Fourier transformation A 1 F(q;C2) of the band energy contribution to the antiparallel interface exchange coupling energy for free surfaces dashed lines and capped surfaces solid curves . The number of repetitions n is marked explicitly. that for odd n the difference between free and capped sur- for the interface coupling situation. Quite obviously, if a pe- faces is much smaller than for an even number of repetitions. riod of 12 occurs aliasing signals corresponding to periods In Fig. 7 the results for the discrete FT's of the layer- of 6 and 3 show up. resolved band energy differences due to perpendicular multi- interface exchange interactions are compiled. As one can see D. Energetic contributions per repetition there is one prominent peak at q 1/12 evolving with in- In Fig. 8 the total band energy contribution to the anisot- creasing n. As compared to Fig. 6 the peaks at q 1/6 and ropy energy, and to the two types of multi-interface ex- 0.41 are less pronounced, while those at q 1/3 and 1/2 are change coupling are shown together with the corresponding of about the same peak heights. Unlike in the case of anti- quantity per repetition. In the case of capped surfaces parallel interface exchange coupling the height of the peak at E(C1)/n is already a constant for n 8, while for free sur- q 1/2 does not oscillate with respect to even and odd num- faces the values for E(C1)/n still go slightly up with in- bers of repetitions. creasing n. Since for an increasing number of repetitions In comparing Figs. 5­7 the discrete FT's reveal periods of E(C1)/n asymptotically becomes a constant, this particular 6 building block in the case of the anisotropy energy, and quantity, namely, E(C1)/n, then has to be called periodic in of 12 arrangement of the orientations of the magnetization n. Stated differently, this implies that even for capped 57 SUPERLATTICE SYMMETRY IN MAGNETIC . . . 7811 FIG. 7. Absolute value of the discrete Fourier transformation A 1 F(q;C3) of the band energy contribution to the perpendicular interface exchange coupling energy for free surfaces dashed lines and capped surfaces solid curves . The number of repetitions n is marked explicitly. surfaces such repeated multilayer systems have to be at least representation no longer results in an oscillating quantity. as thick as about 150 a.u. in order to justify a ``periodic'' However, as shown in Fig. 9 the convergence of E(C2)/n approach. and E(C3)/n with respect to n is far from convincing. Even For antiparallel and perpendicular multi-interface ex- for a multilayer thickness of about 200 a.u., these two quan- change coupling the total band energy per repetition shows tities have not reached a constant value. oscillatory behavior with respect to n. In both cases a period of two can be read off from the corresponding entries in Fig. 8. Quite clearly E(C E. Comparison to experiment 2)/n and E(C3)/n are not periodic in n, since their values oscillates with n. These oscillations Experimentally mostly free surfaces of Ni on Cu 100 seem to fit very well the theoretical predictions made by were investigated,26­29 including in some cases surfaces Aristov25 in discussing indirect Ruderman­Kittel­Kasua­ capped by Cu.27 For less than 7 monolayers ML of Ni the Yosida RKKY interactions. orientation of the magnetization is in-plane,29 above 8 ML a In view of the period of 12 occurring in the FT's in Figs. reorientation transition to a perpendicular orientation occurs. 6 and 7, it is intriguing to consider multilayers of the type For thin films a tetragonal distortion of the parent substrate Cu 3Ni3) 2n , i.e., to double the building block unit cell and fcc lattice was suggested,1 while for thick Ni films the mag- then relate the total band energy to n. Clearly enough such a netic moments vary with film thickness, having a maximum 7812 J. ZABLOUDIL et al. 57 FIG. 8. Band energy contribution to the mag- netic anisotropy energy top , the antiparallel in- terface exchange coupling energy and the perpen- dicular interface exchange coupling energy for free surfaces circles , and capped surfaces squares for Cu3Ni3) n multilayers on Cu 100 . The right column shows the corresponding quan- tity per repetition. at a nominal thickness of about 100 Å.30 There seem to be no experimental results for repeated multilayer systems like the one investigated in this paper. IV. CONCLUSION In the present paper we tried to address two important questions connected with physical properties of multilayer systems, namely, 1 is there a characteristic volume unit cell such that when repeated a particular quantity stays con- stant and 2 are there pronounced peaks in the discrete FT of the corresponding layer-resolved quantity with respect to the interlayer distance that suggest an almost Bloch periodic be- havior in the direction of the surface normal. Quite clearly in the presence of three-dimensional translational symmetry unit cells and three-dimensional Bloch periodicity are au- tomatically provided. At least for the systems and properties chosen here, no straightforward answer can be given. Free surfaces differ considerably from surfaces capped semi- infinitely with Cu 100 . As was shown recently31 different cap materials can induce, e.g., large effects in the anisotropy energy, just as a variation of the cap thickness can cause oscillations of the interface exchange coupling energy.32 In view of these facts it has to be stated that only two, very FIG. 9. Band energy contribution to the antiparallel top and specific types of systems have been considered. For the band perpendicular bottom interface exchange energy per repetition for energy part of the anisotropy energy it was found that re- Cu3Ni3) 2n multilayers on Cu 100 . Free surfaces: circles; capped peated multilayer systems have to be at least 150­200 a.u. surfaces: squares. 57 SUPERLATTICE SYMMETRY IN MAGNETIC . . . 7813 thick to permit the use of a unit cell in practical terms. For however, even a colloquial use of periodicity along the multi-interface exchange interactions this thickness has to be growth direction can obscure considerably the physics to be almost twice as big. seen. The discrete FT's prove that with a few repetitions of Cu 3Ni3), namely, n 6, periods can be traced mapping the number of layers per repetition 6 or of the characteristic ACKNOWLEDGMENTS sequence of the orientation of the magnetization 12 : the peak positions stay constant, whereas the peak heights and This paper resulted from a collaboration partially funded widths change when increasing n. by the TMR network on ``Ab-initio calculations of magnetic The present paper also shows that any interpretation of properties of surfaces, interfaces, and multilayers'' Contract experimental results in terms of fitting models based on pe- No. EMRX-CT96-0089 . Financial support was provided riodicity along the surface normal have to be used with ex- also by the Center of Computational Materials Science Con- treme care. It might very well turn out that such models are tract No. GZ 308.941 , the Austrian Science Foundation only helpful for reasonably thick multilayers, namely those Contract No.'s P11626 and P12352 , and the Hungarian Na- with a thickness of several hundred a.u. This in turn is ex- tional Science Foundation Contract No. OTKA T021228 . actly the regime where supercell calculations, i.e., computa- We also wish to thank the computing center IDRIS at Orsay tional schemes using three-dimensional translational symme- as part of the calculations was performed on their Cray T3D try, will be very useful. For thin multilayer systems, and T3E machines. 1 O. Hjortstam, K. Baberschke, J. M. Wills, B. Johansson, and O. 17 B. U´jfalussy, L. Szunyogh, and P. Weinberger, Phys. Rev. B 54, Eriksson, Phys. Rev. B 55, 15 026 1997 . 9883 1996 . 2 E. Y. Tsymbal and D. J. Pettifor, Phys. Rev. B 54, 15 314 1996 . 18 L. Szunyogh, B. U´jfalussy, and P. Weinberger, Phys. Rev. B 55, 3 K. Kyuno, R. Yamamoto, and S. 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