Journal of Magnetism and Magnetic Materials 198}199 (1999) 455}457 The observation of non-homogeneous FMR modes in multilayer Fe/Cr structures A.B. Drovosekov , D.I. Kholin , A.N. Kolmogorov , N.M. Kreines *, V.F. Mescheriakov , M.A. Miliayev , L.N. Romashev , V.V. Ustinov P. Kapitza Institute for Physical Problems, Russian Academy of Sciences, Kosygina Str. 2, 117973 Moscow, GSP-1 Russia MIREA, Vernadskogo 78, 117464 Moscow, Russia Institute of Metal Physics, Ural Division of Russian Academy of Sciences, S. Kovalevskaja Str. 18, 620219 Ekaterinburg, Russia Abstract The spectrum of excitations in [Fe/Cr]L structures with a non-collinear magnetic ordering was studied by means of the FMR technique. The measurements were carried out at room temperature in the frequency range of 9.5}37 GHz with both static and microwave magnetic "elds lying in the "lm plane. Along with the acoustic FMR mode several additional modes were observed in the longitudinal pumping con"guration. The resonance spectrum of an in"nite magnetic superlattice was calculated analytically on the basis of the biquadratic exchange coupling model. Both cases were considered: external magnetic "eld H parallel and perpendicular to the "lm plane. It was shown that the observed additional FMR modes correspond to the excitation of standing spin waves with wave vectors perpendicular to the superlattice plane. 1999 Elsevier Science B.V. All rights reserved. Keywords: Multilayers; Ferromagnetic resonance; Biquadratic exchange coupling The ferromagnetic, antiferromagnetic and non-col- FMR technique was widely used by several experimental linear magnetic ordering were experimentally observed groups (see overview in Ref. [6]) for investigation of in magnetic superlattices [1}5]. Along with the usual magnetic superlattices, but only a few works included the Heisenberg-form energy term, the so-called biquad- biquadratic coupling into the analysis of the obtained ratica one J results [7}9]. (MM) should be taken into account to explain this non-collinear ordering [2]. In spite of nu- We used two samples: S9 } [Cr(10.4 As)/Fe(21.2 As)] merous experimental evidences and theoretical consider- and S14 } [Cr(7.7 As)/Fe(33.2 As)] epitaxially grown on ations, the origin of J the MgO substrate. The crystallographic plane (1 0 0) of  is not clear yet. In this work we use the ferromagnetic resonance Fe layers was parallel to the sample plane. (FMR) technique and static magnetization measure- The magnetization measurements were performed on ments to investigate the [Fe/Cr] a vibrating sample magnetometer in two orientations of L multilayers with a non-collinear magnetic ordering. The results of static the applied magnetic "eld: H parallel (up to 17 kOe) and and resonance experiments are discussed in the frame- H perpendicular to the "lm plane (up to 9 kOe). The work of the biquadratic exchange coupling model. The in-plane magnetization curves demonstrated a large value of residual magnetic moment typical for systems with non-collinear magnetic ordering. The correspond- * Corresponding author. Tel.: 7-095-9382029; fax: 7-095- ing angles of neighboring magnetic layers ordering were 9382030. 1453 for sample S9 and 1103 for sample S14 (see details in E-mail address: kreines@kapitza.ras.ru (N.M. Kreines) Ref. [10]). 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 1 6 3 - 9 456 A.B. Drovosekov et al. / Journal of Magnetism and Magnetic Materials 198}199 (1999) 455}457 The FMR measurements were performed in the where M is the projection of the iron layers magneti- 9.5}37 GHz frequency range with both static H and zation on the "eld direction, which can be de"ned as an microwave h magnetic "elds lying in the "lm plane. The implicit function of H: setup con"guration allowed to employ both the trans- verse (hNH ) and the longitudinal (h#H ) FMR excita- H"AM#BM for H)H1 tion. In the case of transverse pumping we observed one very strong absorption line which corresponded to the M"M for H'H1. (3) homogeneous precession of all iron magnetic moments Here A"(4J (acoustic mode). In the case of longitudinal pumping !8J)/dM1, B"16J/dM1, H1"(4J# 8J several other less intensive modes were observed. The )/dM1 is the saturation "eld in the parallel con"gura- tion. The usual Landau}Lifshitz equations without the results of FMR experiments for two samples are present- dissipation term are used to "nd the resonance spectrum ed in Fig. 1 by open (acoustic branch) and solid points. of the system: To explain the origin of several FMR modes observed in the longitudinal pumping con"guration, we carried \M out an analytical calculation of in"nite superlattice res- G"![MG H G ], onance spectra in the framework of the biquadratic ex- where H G "!(RE/RMG)d\. (4) change coupling model. Supposing that the magnetic moment of every iron layer oscillates as a single vector, After linearization of Eq. (4) near the equilibrium posi- we can write the magnetic energy per unit surface area as tion (2, 3), the solution can be found in the form of a spin follows: wave propagating along the "lm plane normal. The oscil- lation frequency O is determined by the constant pre- L J L\ cession phase shift q between the neighboring magnetic E"!d (M  G H )# (MG MG>) layers. Omitting the intermediate calculations, we are G M1 G giving only the "nal expressions for O: J L\ K L #  (M (M M G MG>)#d G z), (1) 1 G 2 G O "[C(1#cosq)#K ][CM(1#cosq) where J and J are the bilinear and biquadratic inter- layer coupling constants. M #(C#BM) (M G is the magnetization of the 1!M) (1!cos q)] ith iron layer, d the iron layers thickness, K the e!ective for H)H surface anisotropy factor and z the unit normal vector to 1, the "lm plane. We ignored the fourfold in-plane aniso- 1!cos q 1!cos q tropy in our calculations as its usual value (about 500 Oe O " H!H1 2 ; H!H1 2 [8,9]) is much less than the interlayer exchange "elds (about 10 KOe) in our samples. The following calcu- lations will be performed separately for two cases: H par- #K M1 allel and H perpendicular to the "lm plane. (a) H parallel to the xlm plane. The minimization of for H'H energy (1) gives the equilibrium angles between the ex- 1. (5) ternal magnetic "eld direction and M where C"(A#BM)/2. G: (b) H perpendicular to the ,lm plane. A similar proce- G"(!1)Garccos(M/M1), (2) dure for the perpendicular case gives the following Fig. 1. Experimental and theoretical FMR spectra for two samples. H is parallel to the "lm plane. A.B. Drovosekov et al. / Journal of Magnetism and Magnetic Materials 198}199 (1999) 455}457 457 expressions for magnetization in the "eld direction: M"H/K for H)K M, H"(A#K )M#BM for K M(H(H1#K M1, M"M1 for H*H1#K M. (6) The frequencies O for this orientation of magnetic "eld are given by the formula O H "K B (M1!M); M!K (1!cos q) Fig. 2. Calculated spectra for sample S14. H is perpendicular to for H)K the "lm plane. M, O "C(1#cosq)[CM(1#cosq) for S14. It allows us to a$rm that a series of standing spin #[K #(C#BM) (1!cos q)] (M1!M)] waves with wave vectors perpendicular to the "lm plane was excited in our experiments. for K M(H(H1#K M1, O 1!cos q Acknowledgements "H!K M1!H1 2 for H*H The research was supported by Grant Nos. 96-02- 1#K M1. (7) 16687, 98-02-17517 and 98-02-16797 of the Russian For q"0, Eqs. (5) and (7) give the frequency of the Foundation of Basic research and Grant No. RP1-207 of acoustic, and for q" } of the optical branches of FMR the US Civilian Research & Development Foundation spectra, which coincide with the result, obtained in [7]. for the Independent States of the Former Soviet Union Supposing H"0 in Eqs. (5) and (7) we come to the (CRDF). spectrum obtained in Ref. [11] in the absence of the external "eld. The values of J, J, M1 and K , derived from the References approximation of experimental magnetization curves with expressions (3) and (6) are the following: [1] P. GruKnberg, R. Schreiber, Y. Pang et al., Phys. Rev. Lett. 57 (1986) 2442. J"0.40 erg/cm, J"0.23 erg/cm, [2] P. GruKnberg, S.O. Demokritov, A. Fuss et al., J. Appl. M Phys. 69 Part 2A (1991) 4789. 1"1.62;10 emu/cm, [3] V.V. Ustinov, L.N. Romashev, V.I. Minin et al., Fiz. Met. K Metalloved. 80 (1995) 71 (in Russian). "11!for sample S9; [4] A. Schreyer, J.F. Ankner, Th. Zeidler et al., Phys. Rev B 52 J"0.22 erg/cm, J"0.39 erg/cm, (1995) 16 066. [5] V.V. Ustinov, N.G. Bebenin, L.N. Romashev et al., Phys. M1"1.59;10 emu/cm, Rev. B 54 (1996) 15 958. K [6] P.E. Wigen, Z. Zhang, Brazilian J. Phys. 22 (1992) 267. "13!for sample S14. [7] N.M. Kreines, A.N. Kolmogorov, V.F. Mescheriakov, These values of magnetic constants were used to calcu- J. Magn. Magn. Mater. 177}181 (1998) 1189. late the theoretical spectra according to formulas (5) and [8] S.M. Rezende, C. Chesman, M.A. Lucena et al., J. Magn. (7) (lines in Figs. 1 and 2). The "gures show the acoustic Magn. Mater. 177}181 (1998) 1213. (q"0), the optical (q" ) and several modes with inter- [9] A. Azevedo, C. Chesman, M. Lucena et al., J. Magn. Magn. Mater. 177}181 (1998) 1177. mediate q values. The results of the calculation, in which [10] A.B. Drovosekov, N.M. Kreines, D.I. Kholin et al., JETP only the static data were used, exhibit a reasonable agree- Lett. 67 (9) (1998) 727. ment with available experimental resonance data, [11] N.G. Bebenin, V.V. Ustinov, Fiz. Met. Metalloved. 84 a qualitative one for sample S9, and a quantitative one (1997) 29 (in Russian).