Journal of Magnetism and Magnetic Materials 198}199 (1999) 525}527 First-principles calculation for spin-density wave in Fe/Cr multilayers Kunitomo Hirai* Department of Physics, Nara Medical University, Kashihara, Nara 634-8521, Japan Abstract A "rst-principles electronic structure calculation for Fe/Cr multilayers is presented, where a spin-density-wave order in the Cr layer is taken into account in addition to an antiferromagnetic one. The interlayer magnetic coupling between ferromagnetic Fe layers is investigated, and oscillation of the coupling 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 is furthermore demonstrated. 1999 Elsevier Science B.V. All rights reserved. Keywords: First-principles calculation; Multilayers; Spin-density wave As an archetype of spin-density wave (SDW) in itiner- magnetic (AF) and the SDW order has hardly been ant electron systems, Cr has been extensively investigated considered [3]. up to the present. An interest has recently focused on We here present a "rst-principles electronic structure SDW in Fe/Cr multilayers, in connection with the inter- calculation for Fe/Cr multilayers, which is performed by layer magnetic coupling between ferromagnetic Fe means of the Korringa}Kohn}Rostoker (KKR) Green layers. The coupling of magnetizations of the two function method within the framework of the local spin successive Fe layers oscillates between parallel and density (LSD) functional formalism. The calculation is antiparallel with a two-monolayer period of the spacer carried out for periodic multilayers which consist of thickness of the Cr layer, and for thicker Cr layers the ferromagnetic Fe layers and AF or SDW Cr layers, with oscillation is followed by periodic phase slips, which are the magnetizations of the two successive Fe layers being considered to be due to incommensurability of an SDW aligned parallel or antiparallel, in other words, with the order in the Cr layer [1]. One of the points at issue is interlayer magnetic coupling being ferromagnetic or anti- whether the SDW in Fe/Cr multilayers is attributed to ferromagnetic. The purpose of this work is to survey the Fe proximity layers or to the Fermi-surface nesting, variation of the interlayer magnetic coupling with respect which is intrinsic in the bulk Cr and is closely related to to the thickness of the Cr layer, together with an invest- its electronic structure [2]. The electronic structure igation for the possibility of the appearance of an SDW calculation is therefore expected to supply signi"cant order in the Cr layer. It is to be added that the present knowledge for the discussion about the SDW in Fe/Cr author performed a "rst-principles calculation for bulk multilayers; there have been some electronic structure SDW Cr with results in good agreement with experi- calculations for Fe/Cr multilayers, but in these calcu- ments; for example, the wave vector of SDW of the lowest lations Cr layers have been assumed to be antiferro- total energy per atom is found to be aH(19/20, 0, 0), which is close to the observed one aH(0.952, 0, 0), where aH"2 /a and a denotes a lattice constant of the chem- * Corresponding author. Tel.: #81-744-22-3051 ext 2270; ical bcc lattice [4]. fax: #81-744-25-7657. In the present calculation, we consider perfect multi- E-mail address: khirai@nmu-gw.naramed-u.ac.jp (K. Hirai) layers without interfacial roughness, which usually leads 0304-8853/99/$ } see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 9 8 ) 0 1 2 3 6 - 0 526 K. Hirai / Journal of Magnetism and Magnetic Materials 198}199 (1999) 525}527 the coupling of the local magnetic moments between Fe and Cr atoms across the interface to a strongly antiparal- lel one, in comparison with that between Cr atoms [3]. This antiparallel interfacial coupling between Fe and Cr layers may in#uence the magnetic order of the inner Cr layer to some extent, especially when Fe layers exist both sides of the Cr layer, as is the case of periodic Fe/Cr multilayers or Fe/Cr/Fe sandwiches. When the magneti- zations of two successive Fe layers are aligned parallel, an AF order is commensurate with a Cr layer of an odd N! and needs one defect layer for that of an even N!, where N! (N$ ) denotes the number of monolayers of the Cr (Fe) layer. We do not consider such unfounded defect layers but instead consider an SDW order such that a Cr layer contains a half period of SDW (half SDW order), Fig. 1. The variation of the interlayer magnetic coupling be- which is really commensurate with a Cr layer of an even tween Fe layers J N $ with respect to the spacer thickness of the Cr !. On the contrary, when the magnetizations of two layer. successive Fe layers are aligned antiparallel, an AF order is commensurate with a Cr layer of an even N! and a half SDW order is commensurate with that of an odd N!. Thus the magnetic order of the Cr layer may be governed a"5.45 a.u. illustrates the oscillation of J$ with a two- by the parity of N!, and the oscillation of the interlayer monolayer period but does not indicate the appearance magnetic coupling with a two-monolayer period seems of the SDW order in the Cr layer. quite natural. It is to be emphasized that the wave vector The calculation is furthermore carried out for two of the SDW is not determined by the Fermi-surface cases of N!"9 and N!"19 with varying the lattice nesting but by N!; the half SDW order corresponds to constant a to determine an equilibrium lattice constant an SDW order of a wave vector aH(q, 0, 0) with a at which the total energy becomes minimum. In Fig. 2, we show E N , E , and their di!erence J$ as a function of q" !!2 (1) the lattice constant a. It is found that E N  and E  become !!1 minimum at around 5.32 a.u., that is, aK5.32 a.u. for both cases of N and accordingly to that of bulk Cr when N !; this value is about 98% of the experi- ! is around mental value, which is common with the use of the LSD 20. When N! is not so large, it is not likely that a Cr formalism. The curves of E layer of the multilayers admits a part of the SDW with  and E  do not cross for N a speci"c wave vector, which is determined by the Fermi- !"9, but they cross each other for N!"19, with a reversal of the sign of J surface nesting and is usually incommensurate with the $ , as can be seen in Fig. 2b. For the case of a"a chemical lattice. , J$ is positive for N!"9 and negative for N Let us discuss results of the calculation, which is car- !"19, which means that in the Cr layer the AF order is favourable for N ried out for the multilayers with N !"9 whereas the half SDW !)21 and with order is favourable for N N !"19. This is di!erent from $ "3 for an odd N! and N$ "4 for an even N!. We the case of a"5.45 a.u., where the AF order is favour- adopt the experimental lattice constant of bulk Cr (that able even for N is, a"5.45 a.u.) to view variation of the interlayer mag- !"19. The di!erence between these two cases of a can be ascribed to the magnitude of the local netic coupling J$ with respect to N!; J$ is de"ned as magnetic moment of the Cr atom. For the case of J a"5.45 a.u., we have the larger magnitude of the local $ "E !E, (2) magnetic moment, which usually makes the AF order where E  and E are the total energy per one atom when favourable. When the magnitude of the local magnetic the magnetizations of the two successive Fe layers are moment is not so large and N! is around 20, where the aligned antiparallel and parallel, respectively. In Fig. 1, half SDW order approaches the SDW order of bulk Cr, we show J$ for the case of a"5.45 a.u., and it is found an energy gain due to the nesting mechanism may that J$ is positive (parallel coupling is favored) for an become dominant to make the half SDW order favour- odd N! and negative (antiparallel coupling is favored) able. Thus the calculation for the case of the equilibrium for an even N!. The result is consistent with the fact that lattice constant surely indicates the appearance of the the interlayer magnetic coupling oscillates with a two- SDW order in the Cr layer. monolayer period, and it means that in the Cr layer the Here we brie#y discuss the interlayer magnetic coup- AF order is more favourable than the half SDW one, in ling J$ for the case of a"a. We expect that the oscilla- so far as N!)21. The calculation for the case of tion of J$ with a two-monolayer period basically does K. Hirai / Journal of Magnetism and Magnetic Materials 198}199 (1999) 525}527 527 AF and half SDW orders is reversed between 9 and 19. At this critical N!, correspondence between the sign of J$ and the parity of N! is also reversed, and this reversal of the correspondence gives rise to a phase change by in the oscillation of J$ , which is nothing but a phase slip of the oscillation. In conclusion, we investigate the interlayer magnetic coupling between ferromagnetic Fe layers with respect to the spacer thickness of the Cr layer on the basis of the "rst-principles KKR-LSD calculation. It is shown that the interlayer magnetic coupling oscillates with a two- monolayer period of the Cr layer thickness. It is also demonstrated that an SDW order in the Cr layer can appear when N! becomes large, and the relation be- tween the appearance of the SDW order and the phase slip in the oscillation of the interlayer magnetic coupling Fig. 2. Dependence of (a) the total energy per atom E, E  and is discussed. (b) their di!erence J$ upon the lattice constant a for the cases of N!"9 and N!"19. The reference energy E is E for the case of a"5.33 a.u. References [1] J. Ungris, R.J. Celotta, D.T. Pierce, Phys. Rev. Lett. 69 (1992) 1125. not change, since the oscillation may be in consequence [2] Z.P. Shi, R.S. Fishman, Phys. Rev. Lett. 78 (1997) 1351. of the commensurability within the Cr layer of the AF or [3] D. Stoe%er, F. Gautier, J. Magn. Magn. Mater. 121 (1993) half SDW order. We furthermore expect that there exists 259. a critical N! across which the relative stability between [4] K. Hirai, J. Phys. Soc. Japan 66 (1997) 560.