PHYSICAL REVIEW B VOLUME 59, NUMBER 2 1 JANUARY 1999-II Surface-induced low-field instability of antiferromagnetic multilayers A. L. Dantas and A. S. Carric¸o* Departamento de Fi´sica Teo´rica e Experimental, Universidade Federal do Rio Grande do Norte, CEP 59.072-970­Natal, RN, Brazil Received 3 August 1998 We discuss the surface-induced low-field instability of the antiferromagnetic phase of magnetic multilayers. The threshold field is calculated analytically for multilayers of arbitrary thickness containing an even number of layers. We show that the threshold is given by H 2 SSF HeHa Ha, where He and Ha are the effective exchange and anisotropy fields. The effective anisotropy field Ha may include both uniaxial and fourfold crystalline anisotropy. Numerical simulations of the equilibrium phases, based on a self-consistent effective field method, are used to obtain the magnetization pattern. We find that thick uniaxial multilayers display a three-stage transition from the antiferromagnetic to the field-aligned phases, whereas in thin multilayers the transition is from the antiferromagnetic to a nearly spin-flop structure, which gradually aligns with the applied field. If the films composing the multilayer have uniaxial and crystalline anisotropy, the magnetization profile in the multilayer and the nature of the transition depend on the relative values of the uniaxial and crystalline anisotropies. S0163-1829 99 11301-8 I. INTRODUCTION plored and terms proportional to the anisotropy field were neglected. In the bulk the AF phase becomes unstable at an The properties of magnetic multilayers have recently at- applied field strength of H 2 SF 2HeHa Ha. He and Ha are tracted a great deal of research interest. This is largely due to the exchange and anisotropy fields and HSF is the bulk spin- the technological potential of the measured giant flop field. Therefore the surface-mediated instability occurs magnetoresistance1 of transition metal multilayers. Also the at a field much lower than the bulk spin-flop field for low- possibility of tailoring a wide class of magnetic multilayers, anisotropy antiferromagnetic materials. with films down to a few atomic planes thicknesses, has Recently it has been shown that a low-field surface- motivated basic research. New phases not encountered in the induced spin-flop transition also occurs in uniaxial antiferro- parent materials have been found. The magnetic properties of magnetic films.5 These results were for two-sublattice these structures depend significantly on the layering pattern uniaxial antiferromagnetic fluorides (FeF2 , CoF2 , MnF2). and have been the subject of a great deal of experimental and At a field strength lower than the bulk spin-flop field a phase theoretical work in the last decade.2 transition is nucleated at the surface where spins point oppo- A large fraction of phenomena in magnetic multilayers site to the applied field. The same threshold field was found has been so far studied by assuming invariance in the direc- for films containing an even number of planes, no matter tions parallel to the interfaces, since the main features in- how large is the number of planes. duced by surfaces originate in the magnetization variations For antiferromagnetic films with an odd number of layers in the direction perpendicular to the surface. Therefore these size effects are relevant. In this case the surface spins are systems have most commonly been modeled by a stacking of softer and more easily kept parallel to the applied field. inequivalent moments, each representing a layer parallel to Therefore the threshold field for AF instability is larger than the surface, coupled through an effective exchange field and the bulk value. We have shown5 that the threshold field for subjected to anisotropy fields. This covers aspects of major the surface-induced spin-flop transition is thickness depen- interest, like surface and interface effects, size effects, and dent. For thin films the transition requires large applied mutual stabilization between components of magnetic fields. For thick films the threshold field was shown to be superlattices.3 equal to the bulk spin-flop field. Surface effects are particularly relevant to the stability of To our knowledge, there is no experimental report on the the antiferromagnetic phase of antiferromagnetic AF mul- surface-induced instability of antiferromagnets. However, a tilayers. Surface spins are softer and can more easily be stacking of thin uniaxial ferromagnetic films coupled through turned in the direction of the applied field. In fact a surface- nonmagnetic spacers can be regarded isomorphic to a two- nucleated field-induced phase transition has been predicted sublattice antiferromagnet, if the interfilm coupling favors an for two-sublattice uniaxial antiferromagnets4 and shown to antiferromagnetic alignment of neighboring films. The first occur at a lower field strength than the bulk spin-flop field. In experimental verification of the surface spin-flop transition this work the equilibrium equations were solved for a semi- was reported on transition metal multilayers. The simulta- infinite system of spins coupled antiferromagnetically, neous analysis of Magneto-optic Kerr effect MOKE and through an exchange field He , and subjected to an external superconducting quantum interference device SQUID mag- field along the easy axis of the uniaxial anisotropy. It was netization curves of a Fe/Cr multilayer demonstrated clearly shown that the phase transition occurs when an external field that the transition is nucleated at the surface where spins of magnitude given by Hext HeHa is applied antiparallel point opposite to the applied field.6 to surface spins. In this work the limit of Ha He was ex- Brillouin light scattering as well as ferromagnetic reso- 0163-1829/99/59 2 /1223 9 /$15.00 PRB 59 1223 ©1999 The American Physical Society 1224 A. L. DANTAS AND A. S. CARRIC¸O PRB 59 nance FMR provide useful information regarding the ex- zero applied field. However, if the number of ferromagnetic change and anisotropy fields.7 Typical values of the ex- films is even, then the net magnetization of the multilayer is change and anisotropy fields in transition metal multilayers zero. We show presently that the magnetization jump at the are found in recent experimental data8­13 and review field-induced transition is controlled by the nature of the an- articles.14 In a number of metallic multilayers of current in- isotropy of the ferromagnetic films composing the terest the magnetic symmetry is controlled by a combination multilayer. of crystalline anisotropy and surface- or strain- induced We restrict our present analysis to multilayers with an uniaxial anisotropy. The relative strength of these contribu- even number of layers. For these multilayers, in the antifer- tions to the effective anisotropy of thin films depends on a romagnetic state, the spins of one of the surfaces are opposite number of factors, including the growth process itself, the to the spins of the other surface of the multilayer. Therefore substrate, and the crystallographic orientation of the stacking one of the surfaces of the multilayer has spins opposite to the of films. The surface contribution typically varies as the in- external field. The instability of the antiferromagnetic state is verse of the magnetic film thickness, leading in some cases nucleated at this surface. This is a genuine surface effect and to a crossover, at thicknesses of the order of a few ang- requires a lower value of the external field strength, com- stroms, between the uniaxial-anisotropy- and the crystalline- pared to the bulk spin-flop field. Therefore it may have in- anisotropy-dominated regimes. The orientation of the surface-induced anisotropy easy terest for the study of multilayers designed for devices that axis with respect to the crystalline anisotropy easy directions should respond at low field values. Furthermore, the value of is of particular interest. In a recent report the magnetic prop- the external field strength which produces the instability is erties of Fe/Cr 211 superlattices grown on MgO 110 sub- independent of the multilayer thickness.5 Thus, it is possible strates were studied.11 It was shown that Fe films exhibit a to calculate analytically the threshold field as a function of strong uniaxial anisotropy along the 0,1¯,1 direction in the the characteristic fields exchange and anisotropy of the Fe 211 plane, with the easy axis making an angle of ap- multilayer. proximately 40° with the easy axis of the crystalline anisot- If the multilayer contains an odd number of layers, then ropy. Therefore the uniaxial axis is nearly in the hard direc- the nature of the process that leads to the instability of the tion of the crystalline anisotropy. In this work the authors antiferromagnetic state is distinct. In this case it is not pos- found an appreciable change in the uniaxial anisotropy if the sible to single out a particular layer where the nucleation of Fe thickness is varied from 14 to 90 Å , while the reported the instability occurs. Instead, the multilayer as a whole re- values of the crystalline anisotropy are practically equal to sponds to the external field, and the value of the threshold the bulk value. The crossover inverse thickness, for which field is dependent upon the surface to volume ratio of the the uniaxial and crystalline anisotropy energies are equal, is multilayer. For a multilayer with N magnetic layers, the sur- around 0.04 Å 1. This corresponds to an Fe film thickness face to volume ratio is 2/(N 2). For large values of N the of tFe 25 Å see Fig. 3 of Ref. 11 . contribution of the surface region to the magnetic energy is The field dependence of the giant magnetoresistance of negligible. In this limit the threshold field is equal to the bulk transition metal antiferromagnetic multilayers is associated spin-flop field. However, for small values of N surface ef- with the changes in the relative orientation of the magneti- fects are relevant. The surface spins, with lower coordina- zations of the ferromagnetic layers.1 Therefore the threshold tion, are more easily kept parallel to the external field. Thus field for instability of the antiferromagnetic phase of these the effect of surfaces in odd-numbered thin multilayers is to multilayers is a key parameter. The effective exchange of increase the threshold field for instability of the antiferro- transition metal multilayers varies with spacer thickness. magnetic state. We have found that in this case the threshold Therefore the value of the anisotropy energy is not necessar- field for instability of the antiferromagnetic state may be ily small compared to the effective exchange energy. The much larger than the bulk spin-flop field.5 We have shown anisotropy to exchange ratio may vary significantly accord- that as the surface to volume ratio is decreased, the threshold ing to the stacking pattern. Thus, it is useful to obtain a field drops and reaches the value of the bulk spin-flop field general expression for the threshold field, valid for any val- for sufficiently thick multilayers. Therefore thin multilayers ues of the exchange and anisotropy fields. have large values of the threshold field. This case is less We presently investigate the low-field surface-induced in- attractive for current applications, since modern devices stability of the AF phase of an antiferromagnetic multilayer. based on magnetic multilayers are designed to respond at The results apply to two-sublattice uniaxial antiferromag- low values of the external field strength. Furthermore, it is netic thin films as well as to antiferromagnetic multilayers not practical to calculate analytically the threshold field, constructed with thin transition metal films. We show that, since it depends on the multilayer thickness. Thus, we pres- for AF multilayers with an even number of ferromagnetic ently concentrate on a study of multilayers with an even layers, the threshold field for instability of the AF phase is number of layers. not dependent on the multilayer thickness. Therefore it pro- The basic structure of the calculation is initially set for an vides one more function of the exchange and anisotropy antiferromagnetically coupled stacking of ferromagnetic lay- fields for the interpretation of the magnetic properties of an- ers with uniaxial anisotropy in Sec. II. In Sec. III the theory tiferromagnetic multilayers. is extended to include a contribution from fourfold symmetry Furthermore, the large variation of magnetization in the crystalline anisotropy. In the last two sections we discuss the field-induced instability of the AF state might be of practical nature of the transition according to the relative strengths of relevance. Antiferromagnetic multilayers with an odd num- the uniaxial and crystalline anisotropies and present our con- ber of ferromagnetic films exhibit finite magnetization at clusions. PRB 59 SURFACE-INDUCED LOW-FIELD INSTABILITY OF . . . 1225 II. FIELD-INDUCED INSTABILITY 1 m We consider each layer of the multilayer represented by a ii h cos i cos i 2 sin2 i 2 cos i i 1 single spin variable S n , with components only in the plane cos of the layer. We consider only nearest neighbor interactions i i 1 , and call z the uniaxial axis and y the axis normal to the surface. The external field is along the uniaxial axis and only 1 multilayers with an even number of layers N are consid- mij 2 cos i j j,i 1 j,i 1 , 3 ered. We use the principle of induction to examine the stability of the antiferromagnetic configuration. First a system con- where nm is the Kronecker delta function. sisting of a pair of layers is considered. Then a multilayer In the AF phase the matrix elements mij are constructed with four layers is studied and finally we show that the from Eqs. 3 , with i i 1 . For surface spins the ex- threshold field for a multilayer with N 2 layers is the same change part in the diagonal elements, mii , contains only one as that of a multilayer with N layers. Thus, the threshold field of the cosine terms. If the applied field is smaller than the is an intrinsic value. It is defined by the exchange and an- threshold for instability of the AF phase, all the eigenvalues isotropy fields of the multilayer, and is not dependent upon of the matrix M are positive. Instability occurs when one the number of layers, N. eigenvalue becomes zero. In this case the matrix M becomes The internal energy, written in units of g singular. The elements of the matrix M depend on the value BHES, where H of the magnetic field, and the critical field for instability of E is the exchange field coupling neighboring layers, is given by the AF phase is calculated by finding the lowest field value for which the determinant of M is zero. N For a multilayer with N ferromagnetic layers, M is an E 1 . 1 N Nmatrixand,exceptforthematrixelementsoftheprin- n 1 2 cos n n 1 2 cos2 n h cos n cipal diagonal and the two secondary diagonals, all the ma- The first term corresponds to the exchange coupling between trix elements are zero. The elements of the principal diagonal adjacent layers, the second is the uniaxial anisotropy energy, alternate between those corresponding to positive and nega- and the third term is the Zeeman energy. Ha /He and h tive values of the Zeeman energy. Also the matrix elements H/He are the uniaxial anisotropy field, and applied field, in m11 and mNN are distinct by having only half coordination units of exchange field. The exchange term is not included and, therefore, half the exchange of the others. for the Nth layer. For a pair of layers M is given by The magnetic phases are described by the angles i ,i 1, . . . ,N . A given profile corresponds to an extremum of the energy if E/ i 0 for i 1,2, . . . ,N. The relevant M , 4 equations are 2 a b b c E 1 where we have used a h 1/2, c h 1/2, and h cos 1 sin 1 1 2 sin 1 2 0, b 1/2. These matrix elements correspond to choosing E 1 1 and 2 0. The AF instability occurs when ac b2. This corresponds to an applied field HSSF given by h cos 2 sin 2 2 2 sin 2 1 sin 2 2 3 0, 2 HSSF HeHa Ha. 5 E 1 h cos 3 sin 3 Notice that the decrease in the threshold field, compared 3 2 sin 3 2 to the bulk value 2H 2 eHa Ha, corresponding to an infinite sin 3 4 0, stacking of layers, results from the reduced exchange of both spins. ] For a multilayer with four layers we have E 1 h cos N sin N N 2 sin N N 1 0. a b 0 0 b c b b 0 The N conditions imposed by Eqs. 2 are automatically M4 . 6 0 b a b b satisfied in the AF phase, where the angles ( i) alternate between and 0. However, in order to satisfy the condition 0 0 b c for a minimum of the energy it is necessary that all the ei- genvalues of the matrix M, formed with elements given by We use elementary matrix algebra to write the determinant mij 2E/ i j , be positive.15 The matrix elements mij are of M as the product of the diagonal elements dn of the upper given by triangular matrix 1226 A. L. DANTAS AND A. S. CARRIC¸O PRB 59 b2 b 0 a b2 0 0 a b b2 b D4 a b 0 0 0 c b c b a . 7 b2 c 0 0 0 b2 a b b2 c b a Therefore b2 b2 b2 Det M 4 a c b a a b b2 c . 8 c b b2 a a b b2 c b a In this case it easy to show that the lowest field value for The recurrence relation between the diagonal elements of which the determinant of M4 is zero is the field for which the matrix M Eqs. 10 is obtained from a composition of ac b2, as in the case of a pair of layers Eq. 5 . By in- two continued fractions. The structure of the equations incor- spection of Eq. 8 we find that the factors in Det(M4) are porates the fact that in the AF phase the spins directions given by a, b,a, and 0 if ac b2. The odd-numbered di- alternate along the field and opposite to it. For the discussion agonal elements are given by hSSF 0.5 with hSSF that follows it is convenient to group the diagonal elements HSSF /He), while the d2 0.5 and d4 0. dn(H) into two sequences, one corresponding to odd val- The threshold field for a multilayer with N layers is also ues of n and another corresponding to even values of n. found by putting MN into an upper triangular form. The de- Therefore the sequences correspond to spins pointing in the terminant of MN is then simply the product of diagonal ele- direction of the applied field even sequence and opposite to ments (dn) of the transformed upper triangular matrix DN . it odd sequence . The recurrence relations for the even and We have odd sequences are Det MN d1d2d3***dN 1dN* . 9 b2 for 4 n N n even , The diagonal elements (d b2 n) are the following functions of a b h: d d n 2 n c b d b2 1 a, a b b2 for 3 n N 1 n odd , b2 c b d c b n 2 d for 2 n N n even , d n 1 n b2 b2 a b d . 11 d for 3 n N 1 n odd , N * c b2 n 1 10 a b dN 2 b2 The elements d d 1 and d2 are given by Eqs. 10 . N * c d . If h 0, then c a and both sequences are described by N 1 the same continued fraction. In this case we verified that all We use dN* for the Nth diagonal element of DN for a d's are positive and that d1 d2 d3 *** dN 1 dN* , multilayer with N layers. The first and last elements d1 and with d1 1/2 see Fig. 1 . dN* correspond to surface spins, the first pointing opposite to We have found that for h 0 each element of the even the field direction and the last aligned along the field, in the sequence initially increases with field and then decreases to AF state. The absence of the b term in the expression for dN* converge to 1/2 for any value of n N) at the threshold results from the reduced coordination. The expression for d1 field. The elements of the odd sequence for any value of n) comes from the same fact. decrease and converge to h 1/2 at the critical field PRB 59 SURFACE-INDUCED LOW-FIELD INSTABILITY OF . . . 1227 0.05. The elements of the even sequence (n 8) converge to 0.5 at the threshold field while the elements of the odd sequence converge to hSSF 0.5. Notice that for any value of H, d8* is smaller then all the other elements. The choice of 0.05 and N 8 is a matter of convenience. The basic features displayed in the field dependence of the diag- onal elements dn (n 1,2, . . . ,8), in Fig. 1, are also found for any other value of N even and . III. CRYSTALLINE ANISOTROPY Metallic multilayered structures comprising alternating ferromagnetic and nonferromagnetic layers are nearly iso- morphic to an antiferromagnetic film. There are, however, FIG. 1. Field dependence of the diagonal elements dn for a special features of the metallic multilayers that may lead to stacking of eight layers. The applied field is shown in units of the significantly different magnetic behavior. In a metallic surface spin-flop field. The numbers by the curves indicate the val- multilayer the effective antiferromagnetic exchange, cou- ues of n. Open symbols are used for n odd and solid symbols are pling neighboring ferromagnetic layers, can be varied by or- used for n even. The lines are just a guide to the eyes. ders of magnitude by choosing the spacer thickness appro- priately. Furthermore, the anisotropy of the individual which makes ac b2. Furthermore, dN* 0 for h hSSF . ferromagnetic layers may have a complex structure. Most Therefore hSSF is the lowest value of the applied field for commonly there are two major contributions to the effective which the determinant of MN vanishes. This is the threshold anisotropy: a uniaxial part induced by strains during the film field for instability of the AF state. growth or due to surface effects and a crystalline part which At the threshold field h hSSF , the limits of the even and is intrinsic to the material.14 odd sequences can be derived from Eqs. 10 and 11 . Start- We consider multilayers with the magnetic moments in ing with d1 a we obtain, from Eqs. 10 , d2 b if ac the plane of the films. The actual form of the crystalline b2. For the even sequence, from Eqs. 11 , for any value anisotropy depends on the crystallographic orientation of the of n N, we have dn c b b2/a if dn 2 b. Since d2 stacking. We presently study a particular case when the crys- b when ac b2, any element of the even sequence con- talline anisotropy has fourfold symmetry in the plane of the verges to the same value (dn b). It is also clear from layers. In this case the crystalline anisotropy contributes to Eqs. 11 that, for this value of the applied field and any even the magnetic energy with a term, for each layer, of the fol- value of N, d lowing form: N * 0. A similar analysis of the limit of any element (dn) of the odd sequence, as h approaches h 1 SSF , can easily be made. All E 2 S2 . 13 the elements of the odd sequence are finite for 0 h c 2 KcSnx nz hSSF . We now show, using the principle of induction, that h We made this particular choice of the symmetry of the SSF is independent of N. In other words, the threshold for insta- crystalline anisotropy term to allow a simple discussion of bility of the AF phase is the same for multilayers with an relevant features of the magnetic pattern for applied fields even number of layers. We start by proving that if d just above the threshold field for instability of the antiferro- N * is zero for a multilayer with N layers, then d magnetic phase of the multilayer. The results can be ex- N * 2 is also zero. From Eqs. 11 we find that for a multilayer with N 2 tended to other symmetries without much effort. layers the last element of the diagonal of M, d For the present discussion we assume that the uniaxial N * 2, is related anisotropy determines the orientation of the magnetization of to the corresponding element of a multilayer with N layers the ferromagnetic layers in the absence of applied fields. We according to also assume that the equilibrium configuration, in the ab- sence of an external field, consists in an antiferromagnetic b2 d arrangement of the ferromagnetic layers. Notice that if Kc N * 2 c . 12 b2 0, then the easy directions of the crystalline anisotropy are a b 0, /2, , while if K b d c 0, then the easy directions of N * the crystalline anisotropy are /4, 3 /4. In order to extend the results of Sec. II, we write the By inspection of Eq. 12 we find that if dN* 0 at the field crystalline anisotropy per layer as value for which ac b2, then dN * 2 is also zero at the same applied field value. From Eq. 4 we have that d2* c KcS4 KcS4 b2/a. Thus d E 2* 0 for ac b2. Then it follows by induc- c 2 cos n sin n 2 8 sin 2 n 2. 14 tion that the result is valid for any even value of N. In Fig. 1 we show the field dependence of the two se- The contributions of this new term to the equilibrium quences of diagonal elements of D8 . We selected a equations Eqs. 2 as well as the contributions to the matrix multilayer with eight layers and an anisotropy ratio M Eqs. 3 are easily calculated. We note that 1228 A. L. DANTAS AND A. S. CARRIC¸O PRB 59 2Ec K 2 cS4cos 4 n . 15 n Therefore, in the AF phase ( n 0 for n even and n for n odd) the above term Eq. 15 just adds a con- stant to the diagonal elements of the second derivative matrix obtained from the exchange, Zeeman, and uniaxial anisot- ropy energies Eqs. 3 . This corresponds to defining an ef- fective anisotropy parameter which includes the anisotropy field amplitudes for the uniaxial and crystalline contribu- tions. We define c KcS2/2J where J is the AF exchange cou- pling, and obtain the threshold field for instability of the AF phase. In units of the exchange field, the critical field is given by h 2 SSF e f f e f f , with an effective anisotropy constant FIG. 2. The magnetization curves for multilayers of various defined by eff c . thicknesses and a single value of the uniaxial anisotropy to ex- The effect of adding a contribution from the crystalline change ratio ( 0.15). The numbers by the curves indicate the anisotropy is to introduce a shift in the calculated threshold number of layers. The applied field is shown in units of the bulk field given by Eq. 5 . The remaining results of Sec. II are spin-flop field (HSF), and the magnetization is shown in units of the not modified. However, compared to the system with only saturation magnetization. uniaxial anisotropy, the actual spin profile after the threshold field may differ considerably if the crystalline energy is com- anisotropy of the ferromagnetic layers is composed of parable to the uniaxial energy. As we shall discuss below, uniaxial and fourfold crystalline anisotropy contributions, this aspect is relevant for the interpretation of MOKE mag- then the nature of the phase transition for h hSSF depends netization measurements in metallic multilayers. on the orientation of the uniaxial anisotropy field relative to Our discussion centers on the existence of a given the crystalline anisotropy easy directions. uniaxial anisotropy; then the possible effects of crystalline The spin profiles are calculated for arbitrary field values anisotropy are introduced. This corresponds to the situation by a numerical self-consistent algorithm, which consists of that might be found in thin film transition metal multilayers. finding the equilibrium configuration by allowing the spins We consider two distinct cases according to the orientation to align with the local effective field. This method has been of the crystalline anisotropy easy axis with respect to the used to study antiferromagnetic films, and the reader is re- uniaxial axis. ferred to Ref. 3 for details. We note that if Kc 0, then the uniaxial axis is an easy For purely uniaxial multilayers we have found that al- axis of the crystalline anisotropy. In this case the AF state is though the critical field is given by Eq. 5 for any value of further stabilized by the crystalline anisotropy and the the anisotropy to exchange fields ratio ( Ha /He), the spin threshold field for instability increases. If the uniaxial axis is profile just after the threshold field is dependent upon the a hard axis of the crystalline anisotropy (Kc 0), the effec- values of N and . tive anisotropy parameter eff decreases and the instability In Fig. 2 we show the magnetization curves for various occurs at a lower field compared to the purely uniaxial values of N and a fixed value of 0.15. The magnetization case . If the magnitudes of and c are comparable, the is shown in units of the saturation magnetization, and the threshold field may turn out to be weak compared to the applied field is shown in units of the bulk spins flop field exchange field. (HSF). For the chosen value of the surface spin-flop field We have found that the order of the phase transition, as is given by HSSF 0.73HSF . As seen in Fig. 2 the instability well as the width of the surface modified region, containing of the antiferromagnetic state occurs at the same value of the the layers directly affected at the transition, depend on the applied field (H/HSF 0.73) for all the multilayers chosen. magnitudes of the anisotropies and the relative orientation of For the multilayers with N 10 and N 20, there is a single the easy axis. These two aspects are relevant to the interpre- magnetization jump. For these two cases when H HSSF the tation of MOKE magnetization measurements and will be magnetic pattern consists of a surface-modified spin-flop explored in examples in the next section. phase SMSF in which, except for a few layers near the surfaces, the structure resembles a spin-flop pattern. IV. SPIN PROFILES AND THE NATURE For the multilayers with N 20 there are two jumps in the OF THE TRANSITION magnetization. In these cases for H HSSF the magnetic phase consists of an almost antiferromagnetic pattern near In this section we discuss the magnetic structure when the the surface with spin, in the AF state, parallel to the applied external field strength is equal or larger than the surface spin field, and a region with spins canted towards the field direc- flop field (hSSF). We divide the discussion into two parts. In tion, near the other surface. When the field is increased, be- the first part we consider purely uniaxial multilayers. In this yond the threshold value, the canted region moves without case the anisotropy of each ferromagnetic layer is uniaxial appreciable change in width to the center of the multilayer. and for h hSSF we find a first-order transition with a jump By further increasing the applied field the canted region wid- in the magnetization. In the second part we show that if the ens up, initially very slowly until another critical field value PRB 59 SURFACE-INDUCED LOW-FIELD INSTABILITY OF . . . 1229 is reached when the canted region spreads over the whole multilayer, leading to a surface-modified spin-flop phase. In this phase, except for the surface region, the spins are in the spin-flop state. However the angles of orientation with the applied field differ from those for a bulk spin-flop state. These results conform with those reported for Fe/Cr multilayers.6 The second jump of magnetization in Fig. 2, at H* 0.92HSF , corresponds to the canting of spins largely con- centrated in the center of the multilayer spreading over the whole multilayer, leaving the system in a SMSF phase. The major fraction of the spins of thick multilayers is in an al- most AF state for lower fields above the critical value HSSF); therefore there is a considerable increase in magneti- zation in passing to the SMSF state. The threshold field for this transition is size dependent. This is not clearly seen in the picture due to small value of anisotropy ( 0.15) used. We also note that H* is not the bulk spin-flop transition field. H* is smaller than the bulk spin-flop field. For thick multilayers, when the transition must show unambiguously, the magnetic state before the transition occurs (HSSF H H*) consists of a mixed phase with most of the multilayer in an almost AF state but a fraction of spins in the center of the multilayer is in a canted state similar to a spin-flop phase. The existence of this region of canted spins lowers the sta- bility of the phase with respect to further increase in the applied field, if compared to a purely AF state. Therefore the second transition occurs for a field strength lower then the bulk spin-flop field (H* HSF). We have found that for a given value of there is a critical number of layers (Nc) below which the above three- FIG. 3. a The magnetization of a 30-layer AF multilayer for stage process turns into a two-stage process. For N Nc the positive and negative values of c see text for details . The applied surface-modified spin-flop state sets in just above the thresh- field is shown in units of the surface spin-flop field (HSSF) and the old field. The critical value N magnetization is shown in units of the saturation magnetization. c decreases with increasing uniaxial anisotropy. The uniaxial anisotropy is 0.2 and two values of the effective We now discuss the nature of the field-induced phase anisotropy parameter eff , corresponding to c 0.1 and c transition, for H H 0.1, are indicated by the numbers by the curves. b Profile of SSF , when the anisotropy of the ferro- magnetic layers is the sum of two contributions, one with the angles n , shown in degrees, wih the z axis. Open symbols for uniaxial symmetry and another with fourfold symmetry. In eff 0.3 and solid symbols for eff 0.1. The lines through the Fig. 3 we show the total magnetization, for field applied points are just a guide to the eyes. along the uniaxial axis, and profiles of the layer magnetiza- are almost perpendicular to the applied field. The middle of tions for an AF multilayer with 30 layers. We have chosen the multilayer displays a spin-flop pattern and there are small 0.2 and c 0.1. The chosen values of the crystalline modifications near the surfaces. The magnetic state consists anisotropy ( c 0.1) are of the same order of magnitude of a SMSF state. The formation of this state is responsible as that of the uniaxial anisotropy. As a result there are rel- for the jump in the magnetization seen in Fig. 3 a for evant differences in the magnetization profile, near the e f f 0.3. threshold field, if the sign of c is changed. For In Fig. 3 a the applied field is shown in units of H c 0.1 an almost AF state is seen. The majority of SSF for spins is in an AF phase. Only near the surface where spins convenience. Notice that c 0.1 corresponds to eff 0.3 point opposite to the applied field is there a small field effect. and c 0.1 corresponds to eff 0.1. For c 0.1 the Therefore the total magnetization is rather small, as seen in transition is of first order and the magnetization jump at the Fig. 3 a for threshold field corresponds to a transition to a surface- e f f 0.1. For this value of e f f the magneti- zation increases continuously near the threshold field. Upon modified spin-flop state which covers all the multilayer. For further increase in the applied field the almost AF state c 0.1 the transition is to an almost AF state with weak evolves continuously to a pattern in which a region of canted modifications near the surface with spins opposite to the ap- spins is formed near the surface. This turns into a SMSF plied field. In this case the transition is of second order with state for larger field values. a continuous increase in the magnetization. In Fig. 3 b we display the angles with the z axis at the threshold field. This picture is complementary to Fig. 3 a . V. CONCLUSIONS The magnetic profiles at the threshold field (H HSSF) help We presented a calculation of the surface-induced low- to identify the nature of the transition. For c 0.1 the spins field instability of the AF state of a multilayer with negative 1230 A. L. DANTAS AND A. S. CARRIC¸O PRB 59 bilinear coupling between the layers and in-plane anisotro- We have also examined the influence of fourfold symme- pies. The results are valid for any value of the effective an- try crystalline anisotropies. For magnetization in the plane of isotropy parameter ( eff), provided the equilibrium configu- the layers, we considered a particular kind of symmetry ration for H 0 is an antiferromagnetic arrangement of which amounts to adding to the energy of each layer a term spins, aligned along the uniaxial easy axis, and for any value of the form KcS2cos2( n)sin2( n). This term affects the en- of the multilayer thickness, if the number of spins is even. ergy of the spin-flop-like phase, which forms for applied We started by setting the basic structure of the calculation fields beyond the critical value HSSF , and also the actual for uniaxial antiferromagnetic multilayers. These results spin pattern beyond the critical field. However, we have were extended to include contributions from crystalline an- shown that for multilayers containing an even number of isotropy, so as to apply to the metallic multilayers of current ferromagnetic layers the threshold field for instability of the interest. AF state is not thickness dependent. The critical field is In the case of a purely uniaxial antiferromagnetic given by Eq. 5 with an effective anisotropy parameter eff multilayer, the presently calculated threshold field (HSSF) is incorporating the uniaxial and crystalline anisotropy energy the field that makes surface spin waves soft, as shown by contributions. Saslow and Mills for a semi-infinite uniaxial We have shown that either a first-order or a second-order antiferromagnet.16 We have presently shown that the critical phase transition may occur for a field-induced instability of field is thickness independent for finite multilayers, provided the AF phase of finite antiferromagnetic multilayers. For there is a surface layer with spins opposite to the applied multilayers comprising an even number of thin ferromag- field. netic films with uniaxial anisotropy, the transition is of first The first reports on surface-nucleated field-induced phase order with a jump in the magnetization. In this case the width transitions of AF systems referred to two-sublattice uniaxial of the surface-modified region depends both on the antiferromagnets.4 These works dealt with the phase transi- multilayer thickness and on the strength of the uniaxial an- tions of a semi-infinite antiferromagnet. Under the assump- isotropy field. tion of a small anisotropy to exchange field ratio, the ener- If the anisotropy energy of the ferromagnetic films have gies of the antiferromagnetic state as well as the energy of both uniaxial and fourfold symmetry contributions, then the the surface-modified spin-flop state were calculated and the nature of the phase transition depends on the relative orien- threshold field was found as the value of applied field tation of the easy axis as well as on the relative strength of strength which makes these energies equal. Wang et al. re- the anisotropy fields. We have found that if the magnitude of ported experimental verification of the surface-induced insta- the anisotropy constants ( and bility of the AF phase of finite multilayers.6 By simultaneous c) are comparable, as might be expected for thin films, and the easy axis of the MOKE and SQUID measurements of the magnetization of uniaxial anisotropy is along a hard direction of the fourfold Fe/Cr multilayers with even number of Fe films, it was pos- anisotropy, then the transition is of second order, with a con- sible to identify in a clear manner the surface-induced insta- tinuous variation of the magnetization. Furthermore, in this bility. The narrow width of the surface modified region was case the critical field may turn out to be rather small, since explored to identify the surface-induced instability. the critical field is a function of the effective anisotropy pa- Our results are consistent with these previous reports. rameter Furthermore, we have shown that H e f f c . SSF is determined by The relevant changes of the giant magnetoresistance the values of exchange and anisotropy energies per layer and of transition metal multilayers set in at a field value is independent of the multilayer thickness. In some cases which makes the AF state unstable and saturates at a field the experimental results indicate that the transition strength comparable to the exchange field. We hope occurs when the applied field strength is such that the ener- our present results will be helpful in studying the low-field gies of the antiferromagnetic and spin-flop-like phases are limit, since H equal, as pointed out by Wang et al.6 The magnetic energy of SSF is an upper bound for the stability of the AF state. finite AF multilayers is thickness dependent. Thus it is valu- able to have an upper bound for the threshold field. HSSF is a fixed upper bound for the stability of the AF state and ACKNOWLEDGMENT should help to examine the magnetic properties of AF multilayers. This research was partially supported by the CNPq. *Electronic address: acarrico@dfte.ufrn.br 5 A. S. Carric¸o, R. E. Camley, and R. L. Stamps, Phys. Rev. B 50, 1 M. N. Baibich, J. M. Broto, A. Fert, F. Nguyen Van Dau, P. 13 453 1994 . Pretoff, Etiene, G. Creuzet, A. Friedich, and J. Chazelas, Phys. 6 R. W. Wang, D. L. Mills, E. E. Fullerton, J. E. Mattson, and S. D. Rev. Lett. 61, 2472 1988 . Bader, Phys. Rev. Lett. 72, 920 1994 . 2 R. E. Camley and R. L. Stamps, J. Phys.: Condens. Matter 5, 7 P. E. Wigen and Z. Zhang, Braz. J. Phys. 22, 267 1992 . 3727 1993 . 8 M. Gester, C. Daboo, R. J. Hicken, S. J. Gray, and J. A. C. Bland, 3 F. C. Nortemann, R. L. Stamps, A. S. Carric¸o, and R. E. Camley, Thin Solid Films 275, 91 1996 . Phys. Rev. B 46, 10 847 1992 ; A. S. Carric¸o and R. E. Cam- 9 K. Inomata, Y. Saito, and R. J. Highmore, J. Magn. Magn. Mater. ley, ibid. 45, 13 117 1992 . 137, 257 1994 . 4 D. L. Mills, Phys. Rev. Lett. 20, 18 1968 ; F. Keffer and H. 10 P. Kabos, C. E. 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