RAPID COMMUNICATIONS PHYSICAL REVIEW B VOLUME 59, NUMBER 10 1 MARCH 1999-II Spin-density wave in Fe/Cr superlattices: A first-principles study Kunitomo Hirai Department of Physics, Nara Medical University, Kashihara, Nara 634-8521, Japan Received 17 November 1998 A first-principles electronic-structure calculation for Fe/Cr superlattices is presented, where a spin-density- wave order in the Cr layer is considered in addition to an antiferromagnetic one. The interlayer magnetic coupling between ferromagnetic Fe layers is investigated, and the oscillation of the interlayer magnetic cou- pling with a two-monolayer period of the spacer thickness of the Cr layer is illustrated. The appearance of the spin-density-wave order in the Cr layer, which gives rise to a phase slip of the oscillation, is furthermore demonstrated. S0163-1829 99 51010-2 As an archetype of spin-density wave SDW in itinerant therefore expected to supply significant knowledge for the electron systems, Cr and its alloys have been extensively quantitative discussion on the SDW in Fe/Cr superlattices. In investigated up to the present.1 An interest has recently fo- most of the electronic structure calculations for Fe/Cr super- cused on SDW in a Cr spacer of Fe/Cr superlattices, in con- lattices, however, the AF order has been considered as the nection with the interlayer magnetic coupling between ferro- magnetic order in the Cr layer. The SDW order has never magnetic Fe layers.2 The coupling of magnetizations of two been considered properly, since the calculation for the SDW successive Fe layers oscillates between parallel and antipar- order is rather time consuming on account of the large num- allel with a two-monolayer period of the spacer thickness of ber of atoms in the unit cell;11 a calculation for the SDW a Cr layer, and for thicker Cr layers the oscillation is fol- order was once done without fulfillment of a self-consistent lowed by periodic phase slips, as is revealed by scanning procedure,12 but it is questionable whether such a non-self- electron microscopy with polarization analysis SEMPA consistent procedure is competent to discuss a delicate sta- studies.3 These phase slips are supposed to be due to incom- bility between the AF and SDW orders. mensurability of an SDW order in the Cr layer, and the ap- In this paper, we present a first-principles electronic struc- pearance of an SDW order is ascertained by neutron- ture calculation for Fe/Cr superlattices, which is performed diffraction studies for the superlattices with a relatively thick by means of the Korringa-Kohn-Rostoker KKR Green's- Cr layer of more than 36 monolayers.4 The SDW order in function method within the framework of the local spin den- Fe/Cr superlattices is thus substantial when the spacer thick- sity LSD functional formalism. The calculation is a self- ness of a Cr layer is large enough. On the other hand, it is consistent one without adjustable parameters and is carried generally accepted, at least in a theoretical aspect, that the Cr out for periodic superlattices which consist of ferromagnetic layer has a nonmagnetic or antiferromagnetic AF order Fe layers and AF or SDW Cr layers, with the magnetizations when the spacer thickness of a Cr layer is small;5 either of of two successive Fe layers being aligned parallel or antipar- the orders roughly accounts for the oscillation of the inter- allel. The purpose of this work is to supply reliable knowl- layer magnetic coupling.6 The nonmagnetic order, which edge for the quantitative discussion on the SDW in Fe/Cr may be justified by the existence of interfacial roughness,7 superlattices. We survey variation of the interlayer magnetic has a magnetism induced by the proximity Fe layer.8 The AF coupling with respect to the spacer thickness of a Cr layer, order, to the contrary, is inherent in the Cr layer itself and is together with an investigation for the possibility of the ap- expected to appear when the spacer thickness is small com- pearance of an SDW order in the Cr layer. We add that the pared with the period of the SDW, as is suggested in elec- present author already performed a first-principles calcula- tronic structure calculations for Fe/Cr superlattices.7 The tion for bulk SDW Cr with results in good agreement with question to be discussed here is what the magnetic order in experiments concerning its magnetism; for example, the the Cr layer is, specifically, whether the order is AF or of wave vector of SDW of the lowest total energy per atom is SDW, when the spacer thickness of a Cr layer is found to be a*(19/20,0,0), which is close to the observed intermediate.9 one at low temperature a*(0.952,0,0), where a* 2 /a and There are two points at issue; one is a critical thickness of a denotes a lattice constant of the chemical bcc lattice.13 a Cr layer above which the SDW order appears, and the other Through self-consistent calculations with different initial is whether the SDW in Fe/Cr superlattices is attributed to the magnetic orders in the Cr layer, we can confirm that the proximity Fe layer or to the Fermi-surface nesting respon- coupling of the local magnetic moments between Fe and Cr sible mostly to the Cr layer, which is intrinsic in the bulk Cr. atoms across the interface is a strongly antiparallel one in These points have so far been discussed qualitatively by comparison with that between Cr atoms, which may be means of a phenomenological model,10 but they have hardly readily understood in the case of the superlattices without been discussed quantitatively, although the Fermi-surface interfacial roughness.7 The strong interfacial coupling be- nesting, which is closely related to its electronic structure, is tween Fe and Cr layers influences the magnetic order in the a seriously delicate factor. The electronic structure calcula- inner Cr layer, especially when Fe layers exist on both sides tion of first principles, which is at present most reliable, is of the Cr layer, as is the case of periodic Fe/Cr superlattices 0163-1829/99/59 10 /6612 4 /$15.00 PRB 59 R6612 ©1999 The American Physical Society RAPID COMMUNICATIONS PRB 59 SPIN-DENSITY WAVE IN Fe/Cr SUPERLATTICES: . . . R6613 difference between the AF and half-SDW orders in the cor- respondence of the parity of NCr with the coupling of the magnetizations of the Fe layers. In addition to the half-SDW order, we have an SDW order such that the spacer-Cr layer contains one period of SDW one-SDW order , and this one- SDW order is commensurate within the spacer-Cr layer un- der the same situation as the AF order see Fig. 1 . Because of the strong interfacial coupling, the antinode of SDW is generally fixed on the interface, and SDW orders commen- surate within the spacer-Cr layer may be restricted to half- SDW, one-SDW, 32 -SDW, and so on, when the spacer thick- ness of a Cr layer, that is, NCr is not so large.14 It is to be emphasized that such commensurability within the spacer Cr layer is not an assumption but a consequence of the self- consistent calculations. The wave vector of the SDW orders is thus not determined by the Fermi-surface nesting but by NCr , so long as NCr is not so large; for example, the half- SDW order corresponds to an SDW order of a wave vector a*(q,0,0) with q (NCr 2)/(NCr 1) and accordingly to that of bulk Cr when NCr is 20 24. When NCr is not so large, it is not likely that the Cr layer in superlattices admits an incomplete portion of SDW with a specific wave vector, which is usually determined by the Fermi-surface nesting and is in principle incommensurate with the underlying lat- tice.Let us discuss results of the calculation, which is carried out for the superlattices with NCr 21 and with NFe 3 for an FIG. 1. Magnetic order in a Fe/Cr superlattices and in the odd NCr and NFe 4 for an even NCr , where NFe is the num- spacer-Cr layer for the cases of b odd NCr and c even NCr . An ber of monolayers of an Fe layer. We adopt an experimental arrow indicates the direction of the magnetization of an Fe layer, lattice constant of bulk Cr, that is, a 5.45 a.u. to view varia- and a bar indicates the local magnetic moment of a Cr atom. The tion of the interlayer magnetic coupling between Fe layers local magnetic moment of a Cr atom at the interface is antiparallel JFe with respect to NCr ; we assume that the superlattice re- to the magnetization of the neighboring Fe layer. tains cubic lattice spacing. Here JFe is defined as or Fe/Cr/Fe sandwiches. This can be seen in Fig. 1 of the magnetic order, and NCr proves to play a key role, where NCr JFe Eap Ep , 1 denotes the number of monolayers of the spacer Cr layer. When the magnetizations of two successive Fe layers are where Eap and Ep are the total energy per one atom when the aligned parallel, an AF order is commensurate within the magnetizations of the two successive Fe layers are aligned spacer-Cr layer of an odd N antiparallel and parallel, respectively, and the AF and half- Cr . For the spacer-Cr layer of an even N SDW orders in the Cr layer are considered.15 Figure 2 shows Cr , however, an AF order needs a defect in a mag- netic sense, and we can hardly obtain a self-consistent solu- JFe , and it is found that JFe is positive parallel coupling is tion of the calculation if such a defect exists. Instead of an favored for an odd NCr and negative antiparallel coupling is AF order with a defect, we can obtain a self-consistent solu- favored for an even NCr . The result is consistent with the tion of an SDW order such that the spacer-Cr layer contains fact that the interlayer magnetic coupling oscillates with a a half period of SDW with its antinodes located at interfaces two-monolayer period, and it means that in the Cr layer the half-SDW order ; the strong interfacial coupling favors a AF order is more favorable than the half-SDW one, in so far larger magnitude of the local magnetic moment of a Cr atom as NCr 21, though the energy difference between the AF at the interface. This half-SDW order is really commensurate and half-SDW orders becomes smaller as NCr becomes large. within the spacer-Cr layer of an even NCr . To the contrary, The calculation for the case of a 5.45 a.u. thus illustrates when the magnetizations of two successive Fe layers are the oscillation of JFe with a two-monolayer period but does aligned antiparallel, an AF order is commensurate within the not indicate the appearance of the SDW order in the Cr layer. spacer-Cr layer of an even NCr and a half-SDW order is The conclusion mentioned above, however, is drawn from commensurate within that of an odd NCr . The difference in the result of the calculation for the case of the experimental the half-SDW order between two cases of odd and even NCr lattice constant. For a crucial discussion concerning the consists in the presence of a nodal monolayer; there is really stable magnetic order, we have to discuss the case of an a nodal monolayer at the center of the Cr layer for an odd equilibrium lattice constant a0 at which the total energy be- NCr while not for an even NCr . comes minimum. We therefore carry out the calculation with The magnetic order in the Cr layer is thus governed by varying lattice constant a to determine a0 , for two cases of whether NCr is even or odd, that is, the parity of NCr , and the NCr 9 and NCr 19. In Fig. 3, we show Ep , Eap , JFe , and oscillation of the interlayer magnetic coupling with a two- the magnitude of the local magnetic moment of a Cr atom at monolayer period seems quite natural. There is however a the interface mifCr , as a function of a. It is found that Ep and RAPID COMMUNICATIONS R6614 KUNITOMO HIRAI PRB 59 FIG. 2. Variation of the interlayer magnetic coupling between Fe layers JFe with respect to the spacer thickness of the Cr layer. Here JFe for the case of a 5.45 a.u. is shown as a function of NCr . Eap both become minimum at around a 5.32 a.u., that is, a0 5.32 a.u. for both cases of NCr 9 and NCr 19; this value is about 98% of the experimental value, which is com- mon with the use of the LSD formalism.16 The curves of Ep and E FIG. 3. Dependence of the total energy and the magnitude of the ap do not cross as a varies for the case of NCr 9, but they cross each other for the case of N local magnetic moment of a Cr atom upon the lattice constant a for Cr 19, with a reversal of the sign of J the cases of N Fe . At the equilibrium lattice constant a0 , JFe Cr 9 and 19. Here a Ep and Eap , b their differ- if is positive for N ence JFe , and c mCr are shown, where the reference energy E0 is Cr 9 while negative for NCr 19, which means that in the Cr layer the AF order is favorable for Ep of a 5.33 a.u. A vertical arrow indicates the position of the N equilibrium lattice constant a0 . Cr 9 whereas the half-SDW order is favorable for NCr 19. This is different from the case of a 5.45 a.u., where the AF order is always favorable. The difference between We may furthermore expect that there is a critical NCr , c c these two cases of a can be ascribed to the magnitude of the which is denoted by NCr , between 9 and 19; below NCr , the local magnetic moment of a Cr atom. As can be seen in Fig. AF order is more favorable than the half-SDW one, and c 3 c , for the case of a 5.45 a.u., we have a larger magnitude above NCr , the half-SDW order is more favorable than the of the local magnetic moment of a Cr atom in comparison AF one. At NcCr , the relative stability between the AF and with that for the case of a a half-SDW orders is reversed, and accordingly correspon- 0 , which usually makes the AF order favorable.17 When the magnitude of the local magnetic dence between the parity of NCr and the sign of JFe is also moment of a Cr atom is not so large and N reversed; that is, when N c , J Cr is around 20, Cr NCr Fe 0 for odd NCr and where the half-SDW order approaches the SDW order of J c Fe 0 for even NCr , whereas when NCr NCr , JFe 0 for bulk Cr, an energy gain due to the nesting mechanism may odd NCr and JFe 0 for even NCr . This reversal of the cor- become dominant to make the half-SDW order favorable. respondence gives rise to a phase change by in the oscil- Thus the calculation for the case of the equilibrium lattice lation of JFe , which is nothing but a phase slip of the oscil- constant surely indicates the appearance of the SDW order in lation. Such a phase slip is also expected to appear at another the Cr layer. critical NCr for the stability between the half-SDW and one- Although the calculation to determine the equilibrium lat- SDW orders or such, since the stable magnetic order in the tice constant a0 is only performed for NCr 9 and 19, we can Cr layer will change as AF, half-SDW, one-SDW, sufficiently discuss the variation of J 3 Fe with respect to NCr 2 - SDW, . . . , when N Cr increases. The phase slips are thus for the case of a a0 , with particular attention to the phase irrelevant to incommensurability of the SDW order itself, as slips of the oscillation of the interlayer magnetic coupling is shown in the discussion based on a phenomenological observed in SEMPA. We expect that the oscillation of JFe model.10 Here we will not discuss values of NCr of the phase with a two-monolayer period basically does not change, slips, the reported values of which are 24, 44, and 64,3 since since the two-monolayer period is originated from the parity the present calculation has not been completed yet; the dis- of NCr , in other words, in consequence of the commensura- cussion will be presented in a future publication, with respect bility within the spacer-Cr layer of the AF and SDW orders. to effects of temperature or interfacial roughness. RAPID COMMUNICATIONS PRB 59 SPIN-DENSITY WAVE IN Fe/Cr SUPERLATTICES: . . . R6615 Finally, we briefly mention a helical or spiral SDW or- 34 , 54 , . . . ) period will have an energy gain due to the nesting der in the Cr layer, which is a noncollinear magnetic order, mechanism and might be stabilized when NCr is around 11 unlike the sinusoidal one that we have so far considered. In or 33,55,... . bulk Cr, the helical SDW order never appears, since the en- In conclusion, we present a first-principles electronic- ergy gain due to the nesting mechanism of the helical SDW structure calculation for the SDW in Fe/Cr superlattices for order is always smaller than that of the sinusoidal one with the first time. We investigate the interlayer magnetic cou- the same wave vector.1,17 Similarly, in the Fe/Cr superlat- pling between ferromagnetic Fe layers in the Fe/Cr superlat- tices, it is not likely that the helical SDW order becomes tices with respect to the spacer thickness of the Cr layer. It is more stable than the sinusoidal one when the magnetizations shown that the interlayer magnetic coupling oscillates with a of the Fe layers are aligned parallel or antiparallel, that is, two-monolayer period of the spacer thickness, which may be collinear. However, when the magnetizations of the Fe layers due to the commensurability within the spacer-Cr layer of are not collinear,18 there seems a possibility of the appear- the AF and SDW orders. It is also demonstrated that an ance of the helical SDW order, in which the local magnetic SDW order can appear in the Cr layer when NCr is large at moments of the Fe and Cr atoms across the interface can be least, when NCr 19), and the phase slip in the oscillation of aligned antiparallel; the helical SDW order possibly does not the interlayer magnetic coupling is discussed in connection suffer so large an energy loss caused by compulsory twisting with the appearance of the SDW order. of the local magnetic moments as the sinusoidal one suffers. We hope that the helical SDW order might be stabilized in the Fe/Cr superlattices under an optimum condition for the The author would like to express his sincere thanks to angle between the magnetizations of the Fe layers and the Professor H. Akai for helpful discussion concerning the spacer thickness of the Cr layer;19 for example, in the case KKR-LSD calculation. The author thanks the Supercomputer where the magnetizations of two successive Fe layers are Center, Institute for Solid State Physics, the University of perpendicular, a helical SDW order with a quarter or Tokyo, for the use of the FACOM VPP500. 1 E. Fawcett, Rev. Mod. Phys. 60, 209 1988 ; E. Fawcett, H. L. S. Fishman, and Z. P. Shi, J. Phys.: Condens. Matter 10, L277 Alberts, V. Yu. Galkin, D. R. Noakes, and J. V. Yakhmi, ibid. 1998 . 66, 25 1994 . 11 A real-space method of electronic structure calculation adopted in 2 A. Fert, P. Grušnberg, A. BartheŽleŽmy, F. Petroff, and W. Zinn, J. Ref. 7 is not an appropriate one for the purpose of discussing the Magn. Magn. Mater. 140-144, 1 1995 . SDW order, since the method cannot appreciate the Fermi- 3 J. Unguris, R. J. Celotta, and D. T. Pierce, Phys. Rev. Lett. 69, surface nesting. 1125 1992 . 12 M. van Schilfgaarde and F. Herman, Phys. Rev. Lett. 71, 1923 4 E. E. Fullerton, S. D. Bader, and J. L. Robertson, Phys. Rev. Lett. 1993 . 77, 1382 1996 . 13 K. Hirai, J. Phys. Soc. Jpn. 66, 560 1997 . 5 The nonmagnetic order in the Cr layer is still controversial, 14 When N though it is addressed in some experiments, for example, J. Cr becomes large, the antinode of these SDW orders may sometimes move away from the interfacer, with the period of the Meersschaut, J. Dekoster, R. Schad, P. Beliešn, and M. Rots, SDW orders deviating from a multiple of a half. Phys. Rev. Lett. 75, 1638 1995 . 15 6 S. Mirbt, A. M. N. Niklasson, B. Johansson, and H. L. Skriver, The one-SDW order will be included in the calculation if we Phys. Rev. B 54, 6382 1996 . consider a case of NCr 21. 16 7 D. Stoeffler and F. Gautier, J. Magn. Magn. Mater. 121, 259 K. Hirai, J. Phys. Soc. Jpn. 67, 1776 1998 . 17 1993 ; 147, 260 1995 . K. Hirai, J. Phys. Soc. Jpn. 62, 690 1993 ; 65, 586 1996 . 8 18 J. C. Slonczewski, Phys. Rev. Lett. 67, 3172 1991 . The noncollinear magnetic coupling in Fe/Cr superlattices is ac- 9 Terminology of AF and SDW is sometimes confused. The present tively investigated now, for example, A. Schreyer et al., Phys. way of discrimination of these two terms is the following; there Rev. Lett. 79, 4914 1997 . is no node in the AF order, while there is at least one node in the 19 This is one mechanism of the noncollinear magnetic coupling SDW order. peculiar to Cr superlattices, which has not been discussed in J. 10 Z. P. Shi and R. S. Fishman, Phys. Rev. Lett. 78, 1351 1997 ; R. C. Slonczewski, J. Magn. Magn. Mater. 150, 13 1995 .