VOLUME 80, NUMBER 10 P H Y S I C A L R E V I E W L E T T E R S 9 MARCH 1998 Theory of the Magnetization Reversal of Ultrathin Fe Films on a Cr Substrate A. F. Khapikov Institute of Solid State Physics, 142432 Chernogolovka, Moscow District, Russia (Received 5 August 1997) A micromagnetic theory based on the thickness-averaged Landau-Lifshitz equation is proposed to describe the magnetization behavior of ultrathin Fe films on Cr. The calculations predict the appearance of an effective uniaxial anisotropy of Fe below the Cr spin-flip transition temperature when its magnetic state is characterized by a longitudinal spin density wave. This anisotropy results in a perpendicular coupling of Fe and Cr spins, suggesting the modification of the coercive behavior. The calculations provide a possible explanation for recently discovered anomalous magnetic properties of Fe films grown on Cr(100). [S0031-9007(97)05178-8] PACS numbers: 75.70.Ak, 75.50.Bb, 75.50.Ee, 75.70.Cn The behavior of mixed ferro/antiferromagnetic systems First we offer an approach to average the Landau- is a subject of interest since the discovery forty years ago Lifshitz equation over the thickness of a thin magnetic of the exchange anisotropy phenomenon [1] and more re- sandwich, taking into account interface/surface boundary cently of a variety of intriguing properties of layered mag- conditions. Second, for Fe on Cr, a mechanism of the netic sandwiches, including oscillatory exchange coupling appearance of an effective uniaxial anisotropy below through a nonmagnetic spacer [2­5], biquadratic coupling TSF is proposed, similar to a change of the effective [6­8], and giant magnetoresistance [9]. It is well estab- potential of a pendulum in a rapidly oscillating field. The lished that magnetic and transport properties of ultrathin calculations give the energy minimum when the wave layered systems are in a great extent governed by inter- vector of the longitudinal SDW is perpendicular to the face phenomena. However, an outstanding problem in the Fe magnetization. The change of the ground state of thin-film magnetism is to understand the mechanism of Fe Cr system below TSF alters the magnetization reversal the influence of microscopic interfacial interactions on the mechanism and results in a drastic increasing in the macroscopic magnetic properties as exchange shift in the coercivity. Third, to be specific, we calculate nonuniform hysteresis loop, coercive force, and remanent magnetiza- nucleation modes localized near the film edge and at tion. Theoretical models for exchange coupling in ferro/ Fe Cr interface steps to find the coercive field below and antiferromagnetic systems so far have concentrated on at- above TSF. The thickness dependence of nucleation fields tempts to account for exchange shift and paying no atten- allows a detailed comparison with the experiment [12]. tion to the coercivity and remanent magnetization behavior To derive the thickness-average Landau-Lifshitz equa- [10,11]. tion, we write the three-dimensional micromagnetic en- Recently, an anomalous temperature behavior of the co- ergy functional as the sum of volume and surface parts ercivity and of the remanent magnetization of thin Fe lay- Z ers grown on Cr(100) substrate has been reported [12]. E A f 2 1 K sin2 f cos2 f dV Among layered materials Fe Cr systems demonstrate a Z unique set of unusual physical properties such as extreme 1 J cos f 2 c dS, (1) magnetoresistance [9,13], two periods in the bilinear in- terlayer coupling [2­5], and non-Heisenberg biquadratic where the Fe and Cr in-plane magnetization is character- coupling [6,8,14]. The most intriguing problem for Fe Cr ized by f and c angles with respect to the x axis in the systems is to understand the correlation in the mag- film plane [Fig. 1(a)], A and K are the Fe exchange and netic behavior of Fe and Cr. At the Néel tempera- the cubic anisotropy constants, and J . 0 is the Fe Cr in- ture TN 311 K, bulk Cr is magnetically ordered as a terface exchange coupling per unit area which is expected transverse spin density wave (SDW), while below the to be antiferromagnetic. The spin distribution of Cr is as- spin-flip transition temperature TSF 123 K a phase sumed to be "frozen". Below we discuss this restriction. transition to a longitudinal SDW is observed [15]. Tem- Variational procedure for Eq. (1) leads to the volume perature anomalies of macroscopic magnetic properties re- and surface static equilibrium conditions ported in Ref. [12] have been observed around TSF and 22d2 2f 1 sin 2f cos 2f 0 , (2a) qualitatively have been associated with the change of the Cr magnetic ordering. In this Letter we propose a quan- J fzjz 0 sin f 2 c , fzjz d 0 . (2b) titative micromagnetic theory of the magnetization behav- 2A p ior of Fe films on Cr substrate. Some conclusions of the where d A K, and d is the Fe thickness. We theory can be applied not only to the Fe Cr system, but seek solutions of Eq. (2a) in the form f x, y, z also to other layered magnetic materials. tan21 f j, h g z , where j x d, h y d, and 0031-9007 98 80(10) 2209(4)$15.00 © 1998 The American Physical Society 2209 VOLUME 80, NUMBER 10 P H Y S I C A L R E V I E W L E T T E R S 9 MARCH 1998 antiferromagnet play the role of an effective in-plane mag- netic field He J 2Msd. In principle, this particular result might be obtained from a qualitative consideration of the equilibrium condition of a fictitious domain wall in the ferromagnet. In our case the effective magnetic field results from a ferromagnet spin variation along the z axis due to different boundary conditions at free surface and at ferro/antiferromagnetic interface. Although we concentrate here on the case of a ferro/antiferromagnetic bilayer with "frozen" spins in the antiferromagnet, this procedure can be extended for other magnetic sandwiches with flexible spin structure in all layers. Further, we use Eq. (4) to analyze the magnetic be- havior of Fe on Cr. For simplicity, we assume the Cr magnetic surface structure is given by the bulk terminated magnetic configuration. Below TSF the surface Cr spins in the longitudinal SDW vary as S S0 cos kx or S S0 cos ky [Fig. 1(a)] where the wave vector k 2p l with l 50 Å [15]. The spin variations result in a peri- odic spatial dependence of the exchange coupling magni- tude J J0 cos ky. Because l ø d further simplifying can be achieved by averaging Eq. (4) over rapid spatial spin oscillations, similar to a pendulum in a rapidly oscil- FIG. 1. (a) The ground state of Fe Cr system below the Cr lating field [17]. Consider a single-domain Cr state with spin-flip transition temperature. 90± coupling between Fe and the longitudinal SDW along the y axis c p 2 . Let Cr spins results in an effective uniaxial anisotropy. (b) The magnetostatic field at the edge of a thin magnetic film. (c) The F x, y F x, y 1 x y , where F x, y describes av- nucleation mode near the film edge. eraged over the rapid oscillations magnetization "motion", and x y represents small oscillations of Fe magnetic mo- ment around F x, y . Averaging Eq. (4) yields z z d. Then Eqs. (2) are rewritten as K f 2A Fxx 1 Fyy 2 sin 4 F 2 Keff u sin 2 F 0 , (5) jj 1 fhh g 1 fgzz 1 1 f2g2 2 2 2fg f2j 1 f2h g2 1 f2g2z 2 fg 1 2 f2g2 0 , where Keff u J20 32Ak2d2 is the additional effective uniaxial anisotropy constant, and J (3a) 0 is the amplitude of the interface exchange interaction between Fe and Cr. Jd The density of the effective anisotropy energy may be fgz jz 0 fg cos c 2 1 2 f2g2 sin c , 2A written as the sum of cubic and uniaxial terms g w z jz d d 0 . (3b) a K sin2 F cos2 F 1 Keff u sin2 F . (6) Because of Lamb's remark [16] we imply g2z bg2 2 a Thus, below TSF the fourfold symmetry of Fe is bro- where a and b are arbitrary constants. The solu- ken by interactions at the Fe Cr interface and the energy tion of the equation satisfying the boundary condi- minimum is achieved by 90± coupling between Fe and Cr tion at the free ferromagnet surface is given by g p p spins. Note that the same result has been found by num- a b cosh b z 2 d d . For thin films d d ø 1 erical calculations for the case of the ordinary antiferro- p p g 0 a b, g magnet with fully compensated interface spin structure zjz 0 2 ab d d. Integrating Eq. (3a) over the ferromagnet thickness d taking into ac- [18]. It has been pointed out that the 90± coupling is count the boundary condition at ferro/antiferromagnet in- similar to the spin-flop state of an antiferromagnet in an terface and introducing the variable F tan21 f ? g 0 , external magnetic field. In the case of Fe on Cr we are the following equation is obtained: able to give an analytic treatment of this phenomenon. K J Further, we calculate the coercive force below TSF. The 2A Fxx 1 Fyy 2 sin 4F 1 sin F 2 c 0 . reversal behavior of ferromagnets is often discussed in 2 2d terms of the coherent spin rotation. In the case of ultrathin (4) extended layers that mechanism seems to be extremely This equation describes the thickness-averaged in-plane unlikely. Because of the presence of such defects as film magnetization behavior of the cubic ferromagnet in edges and interface steps which serve as nucleation centers, contact with an antiferromagnet. Interface spins of the we consider incoherent reversal mechanisms. It is very 2210 VOLUME 80, NUMBER 10 P H Y S I C A L R E V I E W L E T T E R S 9 MARCH 1998 important to note that the interfacial roughness plays no tween magnetic and exchange energy. Spin deviations in essential role in the reversal at T , TSF when Fe and Cr the nucleation mode (9b) reach a maximum at the distance spins are perpendicular in the magnetized state [Fig. 1(a)]. from the film edge of order of 4A M2sd [Fig. 1(c)]. Hence, Below we suggest that the coercive force below TSF is for a particular case of Fe Ms 1700 emu cm3, A determined by the instability of a nonuniform mode that 2 3 1026 erg cm our representation of the film edge as in the absence of great volume defects is assumed to a "charged" line remains valid for the film thickness of p be localized at the film edge. The magnetostatic field d & 2 A Ms 160 Å. The second term in the expres- due to Fe edge for thin films can be determined as sion (9a) is comparable with the first term even for the being caused by a charged line with the linear "charge" thinnest films and is increased crucially with the film density 2Msd cos F 0 , where Ms is the Fe saturation thickness (see Fig. 3, curve 1), we assume for Fe K magnetization [Fig. 1(b)]. The integration of the Maxwell 4.6 3 105 erg cm3. In contrast, the effective uniaxial equation yields the radial component of the magnetostatic anisotropy constant Keff u is small as compared to the Fe field H m r 2pr 24pMsd cos F 0 r ¿ d . The field cubic anisotropy and does not substantially contribute to component along x direction when jxj ¿ d may be written the coercivity. Actually, assuming J0 1.5 erg cm2 (see as H m below) we obtain Keff x 22Msd cos F 0 jxj. For the stability analysis u 104 erg cm3 for d 20 Å. Al- we consider the equation of motion of the Fe magnetization together, this anisotropy establishes the perpendicular ori- in the limit of a large damping a entation of Fe and Cr spins below TSF, making the role of interface steps in the reversal insignificant [Fig. 1(a)]. aM K 2 s F sin 4 F 1 Keff The situation is radically changed above TSF when Fe g t 2 2A Fxx 1 2 u sin 2 F and Cr spin suggest 0± or 180± coupling [Fig. 2(a)]. In M2 this case we should take into account a possibility of 2 s d cos F 0 sin F 1 HM x s sin F , (7) the nucleation at interface steps. In the simplest version of our model, the exchange coupling J x per unit area where g is the gyromagnetic ratio, and H is the external is presented by the step function with period 2L. The magnetic field along x axis. Linearization of Eq. (7) fluctuations in coupling due to monoatomic terraces at F F0 1 c x, t around the reference state F0 0 and Fe Cr interface are shown in Fig. 2(b). Further, we substitution c 1 C x exp nt [19] yields the equation approximate the step function as J x J0 cos px L. 2pk C In this case the instability of the reference state F 0 xx 1 2p2 1 C 0 , (8) x is determined by the existing of periodic solutions of Mathieu's equation where p2 2K 1 2Keff u 1 HMs 1 aMsn g 2A, µ k M2 px s d 4pA. This is well known "radial Schrödinger equation" for the hydrogen atom with zero or- Cxx 1 a 2 2q cos C 0 , (10) L bital moment. The localized solutions are given by where a 2 2K 1 HM C x exp 2px s 2A, q J0 4Ad. The bot- 1F1 1 2 k, 2; 2px , where 1F1 is the tom of zero zone of periodic solutions of Mathieu's equa- degenerated hypergeometric function, and k 1, 2, . . . . tion can be approximated for q & 1.5 by a 2q2 2 [20] The reference state F0 0 becomes unstable when n 0 [19]. The largest (the smallest in absolute value) magnetic field of the instability is determined by the condition k 1. This yields the nucleation field and the associated nucleation mode 2 K 1 Keff M3 HW u s d2 n HW c 2 1 , (9a) Ms 8A µ M2 C x exp 2 s d x . (9b) 4A Evidently, this result is valid not only for Fe Cr system, but for any thin magnetic film. It is clear that the linear nucleation mode (9b) shown schematically in Fig. 1(c) de- velops eventually into a domain wall. The first term of the expression (9a) would give the coercive force for the case of the coherent magnetization rotation. In contrast, the appearance of the second term is entirely due to an in- coherent spin rotation near the film edge. In accordance FIG. 2. (a) The arrangement of Fe and Cr spins at the interface with atomic steps above the Cr spin-flip transition with an intuitive understanding of the nonuniform reversal temperature. (b) The variations of the exchange coupling picture, its magnitude is determined by the competition be- between Fe and Cr in the presence of the interface roughness. 2211 VOLUME 80, NUMBER 10 P H Y S I C A L R E V I E W L E T T E R S 9 MARCH 1998 For the particular case of Fe on Cr such a crossover occurs with the temperature variation due to the spin-flip transition in Cr. We believe that this crossover leads to the abrupt change of the coercive field reported in Ref. [12]. Strictly speaking, we find the expressions for the coer- civity of Fe on Cr above and below TSF in the frames of a micromagnetic theory assuming "frozen" Cr spin distri- bution. A more realistic picture would have the frustrated magnetic order located near the Fe-Cr interface but within the Cr layer. Microscopic calculations of stepped Fe Cr interface demonstrate that nodes in the Cr spin density wave could be moved toward the interior of the Cr layer during the Fe magnetization. Electronic structure calcula- tions [23] are not in contradiction with the micromagnetic FIG. 3. The thickness dependence of the nucleation field (in description while we study the instability condition of the absolute value) of both the domain wall near the Fe film edge magnetized Fe state. The exchange coupling constant in (curve 1) and of the ripple structure of Fe Cr interface steps (11) can simply be thought as a phenomenological param- (curve 2). Exchange coupling per unit area J0 1.5 erg cm2; eter which should be determined from the experiments. the lateral size of monoatomic terraces L 300 Å. This work was supported by the Russian Foundation for Basic Research, under Grant No. 97-02-16879. which yields the nucleation field above TSF 2K 1 J2 L2 HR 0 [1] W. H. Meiklejohn and C. P. Bean, Phys. Rev. B 105, 904 n HR c 2 1 . (11) Ms 16p2 MsA d2 (1957). In contrast to the case T , T [2] P. Grünberg et al., Phys. Rev. Lett. 57, 2442 (1986). SF when the reversal is [3] S. S. P. Parkin, N. More, and K. P. Roche, Phys. Rev. Lett. governed by the nucleation of the domain wall near the 64, 2304 (1990). film edge, here we expect the occurrence of a specific [4] J. Unguris, R. J. Celotta, and D. T. Pierce, Phys. Rev. Lett. ripple structure associated with the interface roughness. 67, 140 (1991). The typical thickness dependence of the ripple nucleation [5] S. T. Purcell et al., Phys. Rev. Lett. 67, 903 (1991). field is shown in Fig. 3 (curve 2) (we assume J0 [6] M. Rührig et al., Phys. Status Solidi (a) 125, 635 (1991). 1.5 erg cm2, L 300 Å). The nucleation field alters [7] B. Heinrich et al., Phys. Rev. B 44, 9348 (1991). the sign below the film thickness of about d 25 Å, [8] J. C. Slonczewski, Phys. Rev. Lett. 67, 3172 (1991). resulting in decreasing the remanent magnetization. This [9] M. N. Baibich et al., Phys. Rev. Lett. 61, 2472 (1988). fact is in good agreement with the experiment [12]. Note [10] D. Mauri, H. C. Siegmann, P. S. Bagus, and E. Key, that no decrease in the remanent magnetization of the J. Appl. Phys. 62, 3047 (1987). thinnest Fe film has been observed below T [11] A. P. Malozemoff, Phys. Rev. B 35, 3679 (1987). SF. [12] A. Berger and H. Hopster, Phys. Rev. Lett. 73, 193 (1994). It is clear that the expression (11) can be applied not [13] R. Schad et al., Appl. Phys. Lett. 64, 3500 (1994). only for Fe on Cr but also for other magnetic layered [14] E. E. Fullerton et al., Phys. Rev. Lett. 75, 330 (1995). structures. A ferromagnet grown onto a conventional an- [15] E. Fawcett, Rev. Mod. Phys. 60, 209 (1988). tiferromagnet NiFe NiO , two ferromagnetic films sepa- [16] G. L. Lamb, Jr., Rev. Mod. Phys. 43, 99 (1971). rated by a nonmagnetic layer Co Cu Co and spin valves [17] L. D. Landau and E. M. Lifshitz, Mechanics (Pergamon provide some examples. For such structures, we would Press, Oxford, 1960). predict a crossover in the hysteresis behavior at a criti- [18] N. Koon, J. Appl. Phys. 81, 4982 (1997). cal film thickness d [19] W. F. Brown, Jr., Micromagnetics (Wiley Interscience, c using calculated thickness depen- dence of the nucleation field for the domain wall and for New York, 1963). the ripple structure (Fig. 3). Namely, the reversal pro- [20] Handbook of Mathematical Functions, edited by M. ceeds by nucleation and motion of the domain wall above Abramowitz and I. A. Stegun (National Bureau of Stan- dards, Washington, DC, 1964). d . dc 60 Å, whereas below dc the occurrence of [21] L. H. Bennett et al., Appl. Phys. Lett. 66, 888 (1995). ripple structure should be observed. The magneto-optical [22] A. F. Khapikov et al., Phys. Rev. B (to be published). indicator film imaging technique (see, for example, [23] D. Stoeffler and F. Gautier, Phys. Rev. B 44, 10 389 [21,22]) study of that phenomenon is in progress. (1991). 2212