PHYSICAL REVIEW B VOLUME 57, NUMBER 10 1 MARCH 1998-II Magnetic superlattices, classical spin chains, and the Frenkel-Kontorova model Leonardo Trallori Dipartimento di Fisica dell'Universita di Firenze e Sezione INFM di Firenze, Largo Enrico Fermi 2, I-50125 Firenze, Italy Received 25 July 1997 An equivalence between a simple magnetic model subject to an intrinsically nonconvex Heisenberg inter- action and a Frenkel-Kontorova-type model is formulated. The magnetic model is able to reproduce the experimental results obtained on Fe/Cr 211 superlattices and for this reason has already attracted theoretical attention. The problem is formulated via a two-dimensional area-preserving map and the effect of the surfaces in the magnetic model, which are introduced by appropriate boundary conditions, is shown to be equivalent to the introduction of a discommensuration in a Frenkel-Kontorova-type chain. Further analogies between the two models are presented. The analysis is used to reconsider and clarify some of the features of the magnetic model and to add some contributions to its phase diagram in the space of parameters. S0163-1829 98 02009-8 I. INTRODUCTION effective potential methods produced the phase diagram of In the last decade great attention has been devoted to the same model in the space of parameters.9 magnetic films and superlattices;1 this interest has been One of the essential characteristics of the Fe/Cr 211 su- driven by the enormous experimental advances in the growth perlattices is the competition between the antiferromagnetic and characterization techniques of these materials, as well as exchange interaction and the Zeeman interaction. The first in the investigation ones, which have led to the realization of one favors an antiparallel orientation of neighboring spins, i.e., a difference of in their orientation, while the second high quality samples and the discovery of many new and one tends to align all the spins in the field direction, with a interesting phenomena, such as the oscillatory exchange cou- zero or equivalently 2 ) difference in the orientation of pling between ferromagnetic films separated by nonmagnetic neighboring spins. spacers,2 the giant magnetoresistence effect,3 and the biqua- On a general grounds, competitive interactions can make dratic exchange coupling.4 the determination of the ground state far from trivial, as the Additional interest in magnetic superlattices comes from internal degrees of freedom are frustrated by two or more their lack of translational invariance in the direction normal interactions, which favor different equilibrium to the surfaces and by the high-ratio surface to volume of configurations.10 A clear example of such a difficulty is these systems, so that they are ideal candidates to investigate given by the ground-state properties and the spectacular the effect of surfaces and/or interfaces.5 phase diagram of the Frenkel-Kontorova11,12 model and of In this framework, Fe/Cr 211 superlattices have recently the axial next-nearest-neighbor Ising ANNNI model,13 both received consistent attention. These superlattices, grown by a of these models being characterized by two natural length sputtering technique as single-crystal samples with a uniaxial scales, just as the uniaxial antiferromagnet in the presence of anisotropy, show an antiferromagnetic interlayer coupling an applied field. for a suitable choice of the Cr thickness 11 Å .6 Magnetic As a consequence of this simple observation, it is obvi- measurements7 performed via magneto-optical Kerr effect ously tempting to describe the magnetic system in the wider MOKE and superconducting quantum interference device framework of frustrated models,12 and this is exactly what is SQUID techniques showed two main peaks in the magnetic done in this paper. susceptibility at the so-called surface and bulk spin-flop tran- In fact, I will show in the following how it is possible to sitions and other minor peaks in the intermediate region, in establish an equivalence between a Frenkel-Kontorova-type the case of an even number of Fe layers. In the considered model and the uniaxial antiferromagnet with an external situation, the system is isomorphic to a classical two- magnetic field in the direction of the easy magnetization sublattice antiferromagnet, with ferromagnetic planes anti- axis. The equivalence I am going to formulate connects an ferromagnetically coupled, and it is suitably described by a intrinsically nonconvex interaction-such as the Heisenberg simple uniaxial antiferromagnetic model. Starting from this interaction in the magnetic model considered-with the para- model, the behavior of the system with an even number of digmatic convex one given by the interparticle interaction in planes was explained, using a numerical self-consistent ap- the Frenkel-Kontorova model. In particular, I will show that proach, as due to jumps of a Bloch wall.7 The occurrence of the effect of the surfaces in the magnetic model is equivalent this series of experimentally detected phase transitions to the introduction of a discommensuration in a Frenkel- stimulated further theoretical work: a determination of the Kontorova chain. The importance of the concept of discom- ground state of the system formulated as a two-dimensional mensuration in the analysis of the considered magnetic sys- area preserving map emphasized the importance of the dis- tem was foreseen by Micheletti et al.9 My approach goes in creteness of the magnetic lattice and of the chaotic nature of the well-estabilished path in statistical mechanics of reducing the map,8 due to the high ratio between the uniaxial anisot- a wide series of observations to a restricted number of mod- ropy and the exchange interaction; a very recent study using els, and the analysis performed is also used to reconsider and 0163-1829/98/57 10 /5923 10 /$15.00 57 5923 © 1998 The American Physical Society 5924 LEONARDO TRALLORI 57 clarify some of the many interesting features of the magnetic jectory returns to any neighborhood of any point of the tra- system and to add some contributions to its phase diagram in jectory. The reason for this distinction is that under certain the space of parameters. boundary conditions, for example, The paper is then organized as follows: In Sec. 2, for the sake of completeness, some of the basic properties of the uN uN Frenkel-Kontorova model are recalled; in Sec. III the mag- lim N N 2a, 5 netic model is introduced, and the determination of its N N ground state is formulated by means of a two-dimensional the equilibrium configuration which has the lowest energy is area-preserving map. The equivalence between the magnetic a discommensuration, which is not associated with a recur- model and a Frenkel-Kontorova-type model is then shown. rent trajectory and is not considered as a ground state, since The interpretation of the magnetic field dependence of the this kind of solution is a zero-measure set in the map repre- ground-state configuration and of the magnetic susceptibility sentation. Despite their zero measure, discommensurations of the Fe/Cr 211 superlattices given the established equiva- play an essential role in understanding the generation of sto- lence and using the concept of discommensuration is pre- chasticity in the map phase space. For our purposes, they sented in Sec. IV, whereas the low anisotropy case for the have even a greater importance, since they are the configu- film and the semi-infinite system is considered in Sec. V. rations providing the ground state of the magnetic system in Conclusions are drawn in Sec. VI. the presence of surfaces, as I will show in the following sections. A rigorous definition of discommensuration will be II. FRENKEL-KONTOROVA MODEL given in a moment. Any ground state is specified by the winding number l/2a In this section I will outline some of the basic properties of the corresponding trajectory in the standard map,11,18,19 of the ground state of the Frenkel-Kontorova model;15 as the where subject is extensively treated in numerous papers and re- views see, for example, Refs. 11, 12, and 16­23 , my de- uN uN scription will be no more than schematic. l lim N N . 6 The Frenkel-Kontorova model describes a chain of elasti- N N cally coupled atoms submitted to a periodic potential at zero The winding number represents the mean number of revolu- temperature . The energy of the system can be written as tions around the cylinder per iteration of the map, and it can be either rational or irrational. In the first case the ground 1 u U i state is commensurate with the lattice: 2 ui 1 ui 2 1 cos , 1 i 2 i a ui il i , 7 where ui is the abscissa of the ith particle. The goal is to determine the ground state of this system where i is an arbitrary phase factor; the corresponding tra- and its properties, which turns out to be a far from trivial task jectory is associated with fixed points27 i.e., periodic cycles as the harmonic interaction and the sinusoidal potential favor in the standard map. Moreover, the set of commensurate different equilibrium positions of the particles. All the equi- ground states can be either continuous or discontinuous. In librium configurations of the system are given by the former situation it can be parametrized by a set of func- tions ui( ) , where is a continuous factor, varying from U to . In the latter situation this is not possible and u 0 i. 2 there exist two ground state configurations i Introducing the new variable p i ui ui 1, Eq. 2 can be vi il i , vi il i , 8 written as a two-dimensional recursive mapping: such that no other ground state exists in between them. This u is a sufficient and necessary condition for the existence of a p i i 1 pi 2a sin a , 3 discommensuration, that is, a minimum energy configuration ui , such that ui 1 ui pi 1 , 4 v i ui vi , 9 which is known as the standard map.24­26 The standard map is area preserving, since its Jacobian and for an advanced discommensuration see Fig. 4 in Ref. J 1, and setting 19 i ui(mod2a) the mapping can be folded onto a torus 0,2a 0,2a . As the map is derived lim u 0. 10 from Eq. 2 , any trajectory in its phase space is associated i vi i with an equilibrium configuration of the system. By the analysis of the standard map Aubry was able to The continuous case is an exceptional situation, which determine the properties of the ground state of the system. occurs only in integrable maps; it corresponds to the absence First of all he introduced a distinction between the concept of of a lattice locking of the commensurate ground state. On the minimum energy configuration and the concept of ground contrary, in the discontinuous case, the lattice does apply a state.11,18,19 The concept of a ground state is restricted to locking effect, and this means that none of the particles can minimum energy configurations which can be represented by be at a maximum of the sinusoidal potential in physically recurrent trajectories in the associated map. A recurrent tra- stable solutions.11,22 It was proved by Aubry11,18,19 that a 57 MAGNETIC SUPERLATTICES, CLASSICAL SPIN . . . 5925 discommensuration corresponds with the set of homoclinic tive integers. The two phases, labeled A and B, are separated intersections associated to the corresponding fixed points. by horizontal lines corresponding to first-order transitions. In In the case with l/2a irrational, Aubry proved11,17­19 that states of type A there are particles at the minima of the a transition exists at a well-defined value c of the constant potential V if Q is odd, while if Q is even the phase is depending on the value of l), which he called transition characterized by the absence of particles at the minima of V; by breaking of analyticity. This transition can be related to in the phase B, on the contrary, there are particles at the the discrete nature of the structure and the existence of a maxima of V, regardless of the parity of Q. nonvanishing Peierls-Nabarro barrier,28 i.e., the smallest en- ergy barrier that has to be overcome to move a domain wall III. MAGNETIC MODEL along a chain. The transition by breaking of analyticity is associated with the breaking of a Kolmogorov-Arnold-Roser Consider now a magnetic superlattice composed by ferro- KAM curve of the standard map, and precisely of a KAM magnetic films separated by nonmagnetic spacers, in the con- curve which encircles the torus; a curve with such a charac- dition of antiferromagnetic coupling between the ferromag- teristic is called nonhomotopic to zero, since it cannot be netic films. Suppose that the only effect of the intrafilm reduced to a point on the toroidal surface by a continuous interaction, which is usually much stronger than the inter- deformation. films one, is to keep all the spins belonging to the same film The number of results proven by Aubry applies not only parallel to each other and that each film can be represented to the Frenkel-Kontorova model, but also to any chain of by a single layer. Under these assumptions, the determina- particles with a nearest-neighbor interaction, such that its tion of the ground state of the three-dimensional superlattice energy can be written as reduces to a one-dimensional problem in the direction nor- mal to the films surfaces. E L In the case of Fe/Cr 211 superlattices,7,6 the system is ui ,ui 1 , 11 i isomorphic to a classical two-sublattice, uniaxial, antiferro- magnet, so that, denoting by provided that the function L satisfies some conditions;11,18 i the angle formed by the magnetization of the ith layer with the direction of the ap- the most restrictive of these conditions is that L(x,y) is twice plied field, the energy of the system at T 0 can be written differentiable and satisfies for all x and y as 2L x,y E x y c 0, 12 N HE cos i i 1 HA cos2 i 2H cos i , S i which means that the interparticle interaction must be strictly 15 convex. This condition ensures that the corresponding map is unambiguously determined. which is equivalent to the energy of a classical spin chain. On the other hand, the work by Aubry was not devoted to HE and HA are the exchange interaction and the uniaxial the determination of the actual ground state of the system for anisotropy constants, and H is the external magnetic field. different values of the parameters. This determination is due For an infinite system with i 0, 1, 2,...), a first-order to Griffiths and Chou,20,21 which developed and employed an phase transition occurs when H exceeds a critical value, effective potential method which works for both convex and driving the system in the so-called bulk spin-flop phase.29 In nonconvex interactions W, given the energy of the system in this phase, the spins belonging to the two sublattices form an the form angle /2 with the direction of the applied field. The antiferromagnetic AF phase and the bulk spin-flop BSF phase have equal energy for H H 2 . The E V u B 2HEHA HA i W ui ui 1 . 13 i upper and lower boundaries of the metastability region associated with the first-order nature of the transi- Using the effective potential method, Griffiths and Chou tion are H 2 and H 2 )/ were able to produce the phase diagram of the model in the BSF 2HEHA HA B (2HEHA HA 2 space of parameters.20,21 2HAHE HA, respectively.30 For H Hsat 2HE HA the An interesting variation of the Frenkel-Kontorova model saturation regime, with all the spins aligned with the mag- is the one in which the potential V is a Fourier series:20,21 netic field, is reached. All the equilibrium configurations are obtained by K V u E i 2 2 k 1 cos 2 kui , 14 k 1 0 i, 16 i where the coefficients k are assumed to be sufficiently small that no other minima or maxima are introduced in the poten- which leads to tial V with respect to the Frenkel-Kontorova case which is obtained for HE sin i 1 i sin i 1 i 2H sin i k 0 with k 2). The phase diagram of the above model with 1 1, 2 0.1, and k 0 for k 3 was H again obtained numerically by Griffiths and Chou.20,21 Their A sin 2 i 0. 17 analysis shows that in this situation one can distinguish two Introducing H/HE , HA /HE , and si sin( i i 1), different types of commensurate ground state with the same Eq. 17 can be rewritten as a two-dimensional recursive winding number P/Q, where P and Q are irreducible posi- mapping 5926 LEONARDO TRALLORI 57 si 1 si 2 sin i sin 2 i , 18a turn by nearly /2 near the surface and asymptotically reach the antiferromagnetic configuration in the bulk. It was also i 1 i sin 1 si 1 , 18b suggested34 that the extension of the region of turned spins The map is area preserving, since its Jacobian should increase continuously with increasing H, until the on- J 1, and it is invariant with respect to the transformation ( ,s) set, for H HBSF , of a uniform bulk spin-flop state, with all ( , s). Its domain is the spins rotated by nearly /2. 0,2 ) 1,1 , once the vari- For a film with an even number of planes, one finds an able is defined as mod(2 ). Trajectories in the ( ,s) analogous behavior of the excitations. There are two surface space are associated with equilibrium configurations. The AF and BSF phases are reproduced by second-order fixed points, modes, and for H HSSF the one localized at the surface with the spins antiparallel to H shows a complete softening.36 The such that ( AF n 2 ,sn 2) ( n ,sn) . They are P (0,0), surface transition is again a first-order phase transition. HSSF PAF BSF BSF ( ,0), P ( ¯ , sin2 ¯), and P ( ¯,sin 2 ¯), is therefore the upper boundary of the associated metastabil- where cos ¯ /(2 ). A linear stability analysis shows that ity region, whose lower boundary will be HS , below which the AF fixed points PAF AF and P are hyperbolic for no nonuniform configuration can be minima for the system. H HBSF and elliptic for higher fields. On the other hand, the In between these two values, HS is the field of energetic BSF fixed points PBSF BSF equivalence between nonuniform configurations and the an- and P are hyperbolic for H HB and elliptic for lower fields. In the metastability region, both tiferromagnetic one. Both HS and HS have to be determined the AF and the BSF fixed points are hyperbolic. In this re- numerically. From the analysis of the map phase portrait at gion we can still distinguish two regimes; for H HB the different values of the field, one can find the ground-state stable and unstable manifolds associated with the BSF fixed configurations for the semi-infinite system and for the film, points are enclosed by the stable and unstable ones associ- as will be shown in Secs. IV and V. ated to the AF fixed points, and the opposite happens for H HB .31 Let us now turn to the semi-infinite and the film cases. In Equivalence with the Frenkel-Kontorova model the semi-infinite situation Eq. 17 for i 1 is modified in Consider for the moment the infinite system. In the map- ping, Eq. 18 the presence of the sin 1 function is due to the HE sin 2 1 2H sin 1 HA sin 2 1 0, 19 Heisenberg interaction between neighboring spins, and it which introduces a boundary condition for the map of the makes the map multivalued, as a direct consequence of the infinite system; this condition is taken into account32,14 with nonconvexity of the Heisenberg interaction. At each step two the introduction of a fictitious plane for i 0 such that choices are possible: s1 sin 1 0 0. 20 n 1 n 1 , 0,1, 23 For an N-planes film, two fictitious planes must be introduced,14 for i 0 and i N 1, respectively, so that in where sin 1(s this case the boundary conditions are n 1) is the principal value of the trigono- metric function sin 1 and is the branch index. Each pair of s initial conditions generates 2N trajectories after N iterations 1 sin 1 0 0, 21 of the map. The choice of the branch index is trivial only for s bulk systems characterized by low periodic configurations. N 1 sin N 1 N 0. 22 Anyway, in the case of small parameters ( , 1), i.e., when Only those trajectories of the map Eq. 18 which satisfy the the exchange interaction is largely dominant with respect to boundary conditions represent therefore equilibrium configu- the Zeeman and the anisotropy ones, there is a natural choice rations of the system in the presence of surfaces. For the of the branch index to obtain the ground state. Neighboring semi-infinite system, it is possible to obtain a nonuniform spins tend to align in a nearly antiparallel way, and this situ- ground state if the inflowing orbit to a hyperbolic fixed point ation is reproduced by the choice 1 at each step. The intersects the boundary condition line (s 0); in fact, far same reasoning does not hold for comparable interactions, from the surface the system must have the same configura- which turns out to be the case pertinent to the experimental tion as the bulk. For the film, the trajectories associated with situation. Several years ago, Belobrov et al.37 proposed a equilibrium configurations must have two intersections with local minimization criterion to select the branch index. The the s 0 line, exactly separated by N steps. criterion consisted in two conditions: a 2E/ 2n 0 for all It is interesting to consider the effect of an applied mag- n, and b when condition a is satisfied by both the netic field on the semi-infinite system and the film. The semi- branches, one has to choose the branch which gives the low- infinite system, in the limit of small external field and small est energy at that step. Unfortunately, we verified that this anisotropy with respect to the exchange interaction, was ana- criterion, and more generally any local criterion, does not lyzed several years ago.33,34 In a uniaxial semi-infinite anti- apply in our situation: we followed all the trajectories for ferromagnet with the surface spins antiparallel to the mag- which condition a was satisfied, and it usually happened netic field (AF ), a surface transition occurs at a field that the configuration with the lowest energy for n n2 did H 2 SSF HAHE HA HBSF / 2, as pointed out by the soft- not originate from the lowest-energy one for n n1, with ening at H HSSF of the surface mode.33,35 This instability n1 n2. was predicted to drive the system in the so-called surface Myself and co-workers proposed38 a different criterion: spin-flop state.33 In this phase, the spins were predicted33 to the branch index is always a constant of the mapping, once 57 MAGNETIC SUPERLATTICES, CLASSICAL SPIN . . . 5927 the parameters are fixed, and the choice to adopt is the one sinusoidal potential Eq. 14 . The second-order fixed points which reproduces the correct ground state for the associated P AF and PBSF are the periodic configurations with winding infinite system. number w 1/2 which give the ground state for 1/2. The In the infinite system there exists a threshold value A phase is equivalent to the bulk spin-flop one, and the B Hth (2HE HA)cos( /4) for which the angle formed by phase to the antiferromagnetic configuration. The horizontal two neighboring spins is exactly /2. So, for H Hth the line separating the A and B phases is equivalent to the bulk choice one has to fulfill to reproduce the uniform bulk spin- spin-flop transition. Identified 2 with K 1 and with 2K 2 flop configuration is 1, while for H Hth it is 0. For see Eqs. 14 and 15 , this transition, for 1/2, is located H Hth , the two choices are equivalent. We believe the con- at stancy of the branch to hold for finite and semi-infinite sys- tems, too, and we have numerical evidence for all the situa- 16 2 tions we considered, although we were not able to give an K 2 16 2 . 28 analytical explanation. The criterion is expected to hold as 1 2 far as the uniaxial anisotropy becomes so large that the sys- The condition which ensures no additional minima or tem is more adequately described by an Ising model rather maxima in the potential V becomes in our case H HA , than a Heisenberg one. which is again the interesting region of our system.9 The invariance of the branch index under map iteration Although in the following only the 1 choice will be has important consequences: as is apparent considering the considered, it is interesting to take briefly into account the form of the Heisenberg interaction and Eq. 23 , as far as regime in which the particles are subject to the nonconvex 1 the coordinates i are subject only to the convex part part of the Heisenberg potential. First of all, it must be of the Heisenberg interaction, while for 0 they are sub- stressed that this region is ``explored'' only due to the pres- ject only to the nonconvex part of the interparticle interac- ence of the external magnetic field. Actually, Griffiths and tion. This is equivalent to say that, when 1, the map, Eq. Chou20 showed that the ground state of a system with an 18 , satisfies the twist condition39 which, given the defini- energy given by Eq. 13 , with a Heisenberg interaction form tion of the variable si , is for W and the following one for V, i 1 V i h cos p i , 29 s 0. 24 i never experiences the nonconvex part of W for any p 2, i putting on a rigorous ground an earlier hypothesis by Baner- Since the coordinates are subject only either to the convex jea and Taylor40 for the p 2 case. The presence of a p 1 part of the Heisenberg interaction or to the nonconvex one, it contribution makes the nonconvex part accessible, as actu- is possible to introduce a harmonic approximation, which is ally happens for the chiral model41 or other types of expected to be correct for values of the magnetic field suffi- models.42,43 In our model, too, for very high values of the ciently far from Hth . The energy of the system in this ap- magnetic field, i.e., near the saturation value, all the spins are proximation becomes canted in the field direction and the ground state is provided by a trajectory obtained with the 0 choice.38 To summarize, the constant branch index criterion we H H 1 th , E i 2 i i 1 2 2 cos i have introduced has two main consequences: the equiva- lence, for a wide range of the applied field H, of the mag- netic model with a Frenkel-Kontorova-type model with a cos2 i, 25 misfit d 1/2 and the possibility to tackle a series of models with nonconvex interparticle interactions, like the Josephson 1 junction arrays44,45 which seemed precluded, up to now, H H th, E from a map approach analysis. i 2 i i 1 2 2 cos i IV. Fe/Cr 211... SUPERLATTICES cos2 i. 26 Consider for the moment the situation in which , 1. Consider the first case, which is the one defined in the inter- The phase portrait see Fig. 1 is extremely regular and this esting range of H, H means that we are in a quasi-integrable limit. th being much higher than HBSF and close to the saturation value H On the contary, Fe/Cr 211 superlattices are characterized sat . For 0 our system is now equivalent to the Frenkel-Kontorova model 1 , with by a relatively high ratio between the uniaxial anisotropy and a or, equivalently, with d ( 2a)/2a 1/2] and the exchange interaction. A reasonable estimate is HE 2.0 kG and HA 0.5 kG, and the corresponding phase portrait ui a2 for a magnetic field in the range of interest is shown in Fig. a i , 4 2 , 27 2. Strong nonintegrability effects and, in particular, the ho- moclinic intersections between the inflowing and the out- or, in the notation of Griffiths and Chou,20 with flowing orbits from the antiferromagnetic fixed points are (1 d) 1/2 and K 2 . The presence of the uniaxial apparent. Moreover, these manifolds oscillate and intersect anisotropy makes the system equivalent to a Frenkel- the boundary condition line s 0. This means that several Kontorova model with a second-harmonic contribution to the metastable configurations coexist and the ground state must 5928 LEONARDO TRALLORI 57 FIG. 1. Phase portrait obtained from mapping Eq. 18 for small anisotropy 0.01 and an applied field such that HSSF H HB . The various labels denote a hyperbolic fixed points PAF , b el- liptic fixed points PBSF , and c nonhomotopic to zero trajectories. FIG. 3. Magnetic susceptibility as a function of the applied filed Arrows denote inflowing and outflowing orbits associated with the for a N 22 film, with HE 2.0 kG and HA 0.5 kG. hyperbolic fixed points. method, the problem is more complex than the actual be determined by a comparison of their energy. The presence ground-state one, as each trajectory in the map phase dia- of metastable states is intimately connected with the strong gram is associated to an equilibrium configuration, regard- nonintegrability of the map and the pinning effect produced less of its energetic stability or of its energy. The determina- by the lattice.22 It is interesting to note that the stable and tion of the ground state is possible only once all the stable unstable manifolds ``cover'' all the domain of the variable s, equilibrium configurations have been determined, by a com- so that nonhomotopic to zero curves cannot exist. This is the parison of their energies. Moreover, energetic stable configu- characteristic of the stocasticity regime defined by Greene25 rations correspond to topologically unstable in his analysis of the standard map. trajectories.11,16,46,47 The advantage is that a map is an exact Nonetheless, the determination in an accurate way of the representation of the equilibrium configuration space and equilibrium configurations of a film is still possible, given a displays in a graphical way the various kinds of solution the limited number of planes, and it was performed for N 22 system can have, the presence of possible metastable con- and, for a restricted range of H, for N 16, too. At the grow- figurations, the importance of effects due to the discreteness ing of the magnetic field some intersections of the manifolds and to the nonintegrability of the system, and the occurrence with the s 0 line disappear, or a previously metastable state of strongly chaotic regimes, which is exactly the present situ- becomes the ground-state; this causes abrupt changes in the ation. ground-state configuration and consequently in the magneti- The behavior of the system becomes comprehensible in a zation M(H) of the system. Correspondingly to the jumps of more direct way if one considers the connection existing be- M(H), one has spikes in the magnetic susceptibility tween the mestastable configurations and the homoclinic in- dM/dH see Fig. 3 . tersections. As it is shown in Figs. 4 and 5, the ground state The first peak for H HS at H 1.045 kG was not is provided almost exactly by a subset of the homoclinic present in our earlier version8,38 of Fig. 3, as the true ground- intersections between the manifolds relative to the AF fixed state for this and lower values of H is embedded in a very points. The only coordinates which slightly deviate from this chaotic region of the map phase portrait, and we missed it. subset are the ones on the first and last planes. This is a clear example of the difficulties of a map approach If we take for a moment i as a coordinate along a to the ground-state determination. In fact, using the map chain rather than on a circumference, it is clear that the ground state is represented by a discommensuration see Fig. 6 . The explanation is straightforward: the spins on the sur- FIG. 2. Phase portrait obtained from mapping Eq. 18 for HE 2.0 kG, HA 0.5 kG, and H 1.073 kG. The inflowing and FIG. 4. Inflowing orbit in ( ,0), outflowing one from (0,0), outflowing orbits associated with the AF fixed points are shown, and ground-state configuration solid circle for the odd planes of a and their homoclinic intersections are apparent. N 22 film, with HE 2.0 kG, HA 0.5 kG, and H 1.113 kG. 57 MAGNETIC SUPERLATTICES, CLASSICAL SPIN . . . 5929 FIG. 5. The same as in the previous figure, with H 1.120 kG. FIG. 7. Inflowing orbit in ( ,0), outflowing one from (0,0), The inset shows the outflowing orbit and the ground-state configu- and ground-state configuration solid circle for the odd planes of a ration in the proximity of (0,0). N 22 film, with HE 2.0 kG, HA 0.5 kG, and H 1.147 kG. faces have a reduced number of neighbors, so that they are increasing of H the discommensuration proceeds discontinu- the most influenced by the magnetic field, which tends to ously into the film, until a symmetric configuration is make them parallel to its direction; these spins belongs to reached for H HSSF .7,9 This description is confirmed by the opposite sublattices (N is even and this means that a rota- analysis of the N 16 situation. In this case a reduced num- tion of nearly 2 between the first and last spins of the chain ber of discontinuities are detected, since the ground state must intervene. This is exactly the definition of discommen- reaches the symmetric configuration with fewer jumps. For a suration in the Frenkel-Kontorova model see Eq. 5 , given film with N 22 a similar behavior is expected, with the the substitution u discontinuities accumulating at H i( /a) i see Eq. 27 . The effect of the SSF .9 surfaces is then to introduce a discommensuration. For H HSSF the effect of the magnetic field starts to Micheletti et al.9 were the first to use the concept of dis- overcome the other interactions and the spin configuration in commensuration in the analysis of this system. Using the the middle of the sample resembles the bulk spin-flop phase, effective potential method, they studied the phase diagram of which is close to the top of the energy barrier produced by a film with an even number of planes and of a semi-infinite the uniaxial anisotropy see Fig. 7 . The transitions in this system in the parameter space ( , ). Their results for range of the magnetic field increase the number of spins in 0.25 are in excellent agreement with ours. this position see Fig. 8 . The change from a configuration Together with the presence of a discommensuration, the with neighboring spins almost mutually antiparallel to a con- other key ingredient is the high value of the anisotropy. For figuration with an increasing number of spins near the bulk H H spin-flop phase is responsible for the steep increase of the SSF 1.118 kG, the magnetic field fixed, all the meta- stable configurations and the ground state are characterized background magnetization,7,8 to which the jumps are super- by a ``forbidden zone'' around the maximum energetic cost imposed. The transitions occurring for H HSSF were for the anisotropy: the homoclinic intersection around /2 interpreted9 as due to the enlargement of the core of what is always avoided see Figs. 4 and 5 . This means that the was called a bulk discommensuration, positioned between ``particles'' are pinned by the lattice. The high value of the two tails reaching the AF phase. In this description the dis- anisotropy determines a Peierls-Nabarro barrier which for- continuities cease when the two AF-BSF interfaces reach the bids a continuous evolution of the ground state at the in- surfaces. Our description is substantially equivalent, even if creasing of the magnetic field, and the discontinuities occur a growing of the core is still observed after that the last jump when H forces the particles to overcome the barrier. So, for takes place. From a mapping perspective it is instead worth- H while to stress that the transitions cease when the trajectory S H HSSF , the ground-state evolution is the following: at H providing the ground state is distinct from a subset of the S a discommensuration48 nucleates at the surface with spins antiparallel to the applied field in the AF phase; at the homoclinic points not only in the proximity of the surfaces, but for all layers, and reduces to a regular nonhomotopic to zero curve.49 FIG. 6. Ground-state configuration for a N 22 film, with HE 2.0 kG, HA 0.5 kG, and H 1.118 kG. FIG. 8. The same as in the previous figure, with H 1.294 kG. 5930 LEONARDO TRALLORI 57 FIG. 10. Inflowing orbits in ( ,0), for 0.01 and different FIG. 9. Ground-state configuration for a N 22 film, with values of the magnetic field in the metastability region of the sur- HE 2.0 kG, HA 0.02 kG, and H 0.201 kG; the lines in the inset face transition. are the best fit of the coordinates in the first and second halves of the film. length of the BSF core joining the AF tails exist all the way down to 0. Actually, these transitions may exist if one V. LOW-ANISOTROPY CASE considers the homoclinic intersections, or subsets of them, which exist as soon as 0 being associated with the non- Let us now turn to the low-anisotropy case. As we already integrability of the map, but no transition is indeed possible noted the map phase portrait is extremely regular, reflecting if one considers the actual ground state. the quasi-integrability of the system. Due to the low value of In other words, the form of the ground state as a discom- the anisotropy, the dimensions of both the surface and the mensuration and the impossibility to have particles at the top bulk spin-flop metastability regions are greatly reduced. For of the sinusoidal potential mirror the finite size effect,22 but HE 2.0 kG and HA 0.02 kG we have, for example, this does not necessarily mean that the ground-state evolu- HSSF 0.201 kG and we found HS 0.200 kG. tion is discontinuous and we numerically verified that it is Consider initially the effect of the magnetic field on the certainly not for N up to 100, for 0.01). semi-infinite system AF , for H HSSF . This is the regime Let us now turn to magnetic field values lower than H in which the existence of a surface spin-flop configuration SSF . In this regime there is a range of magnetic field in which the was originally proposed. The analysis of the map showed inflowing orbits intersect the s 0 line see Fig. 10 . that this is impossible as the inflowing orbit does not inter- This is the metastability region for the semi-infinite sys- sect the s 0 line,14 which is the necessary condition to have tem. Its lower bound is given by the condition that the in- a surface localized nonuniform configuration. It was then flowing orbit be tangent to the s 0 axis; this happens for inferred by energetic arguments that the system becomes un- 2 stable with respect to the nucleation of a domain wall which H HS HEHA HA. Surface-localized, nonuniform, con- produces an interchange of the two sublattices, making the figurations, i.e., a true surface spin-flop phase see Fig. 11 , spins on the surface layer parallel to the applied field.14 This of lower energy than the AF configuration are obtained9,38 result was recently confirmed.9 for HS H HSSF . Consider now the film case in the same regime of the The surface spin-flop configuration has the form of a dis- magnetic field. A nonuniform ground state is provided by commensuration, even if the discontinuity is too small to be nonhomotopic to zero curves. The effect of the surface is still detected in Fig. 11; the distance of the discommensuration to introduce a discommensuration see Fig. 9 , but in this from the surface grows with the magnetic field, until it is case the notion itself of discommensuration for the finite pushed to infinity, accomplishing the interchange of the two system must be reconsidered. In fact, at variance with the sublattices, at H HSSF . high-anisotropy case, even for large values of N, the ground The question addressing the continuity or the discontinu- state is now provided by a nonhomotopic to zero curve ity of the motion of the discommensuration from the surface which is clearly distinct from the set of homoclinic points. Actually, the considered trajectory is likely to be a KAM curve so that there would be no discommensuration, and on the contrary there would be particles arbitrarily near the maximum of the sinusoidal potential, if this curve were con- sidered for the infinite system. Moreover, at variance again with the high-anisotropy case and confirming the connection between metastability and pinning effects, the ground state is the only stable equilibrium configuration given our boundary conditions; following Aubry's notation,18 the ground state is then said to be undefectible. As a consequence its evolution with the magnetic field is continuous. Some attention is then necessary considering the phase FIG. 11. Stable equilibrium configuration on the odd planes of a diagram of the system. In this range of fields it was argued9 semi-infinite system, for 0.01 and a magnetic field in between that the transitions between configurations with a different HS and HSSF . 57 MAGNETIC SUPERLATTICES, CLASSICAL SPIN . . . 5931 for H due to their zero measure in the equilibrium configurations S H HSSF may be answered considering the film case. The surface localized configuration for the semi-infinite set, are therefore of the greatest importance for the consid- system has a counterpart in nonsymmetric ground-state con- ered magnetic model. figurations for films with a sufficiently high value of N for Moreover, the results for the low-anisotropy case seem to example, the N 22 ground state is always symmetric . indicate the existence of a threshold line ( ) in the These nonsymmetric ground states are characterized by the model parameter space, separating an unpinned and undefec- same distance of the discommensuration from the surface as tible ground-state region from one in which a nonvanishing the semi-infinite system. The evolution of these configura- Peierls-Nabarro barrier forbids the continuous evolution of tions to the symmetric one at the increasing of the magnetic the ground state. In the ``noncontinuous'' regime, the ground field is continuous. This is inferred by a numerical analysis state is provided almost exactly by a subset of homoclinic of the corresponding magnetization, which shows no jumps, intersections, and the discontinuities cease when the trajec- but above all by the fact that the ground state is always tory associated with the ground state becomes a regular non- undefectible; no pinning effects and consequent discontinui- homotopic to zero curve, as in the low-anisotropy case. This ties are therefore expected. We then argue that the evolution could suggest a possible connection with the transition by of the spin-flop configuration for the semi-infinite system is breaking of analyticity in the Frenkel-Kontorova model, continuos too; i.e., no Peierls-Nabarro barrier exists either in whose investigation is beyond the scope of this paper. Actu- this region or in the H H ally, it must be stressed that the transition by breaking of SSF range for 0.01. analitycity occurs at a fixed, and irrational, winding number; VI. CONCLUSIONS the analysis I performed is different, as I focused on the ground-state problem of the system once the surfaces are I showed in this paper how it is possible to establish an properly taken into account, rather than on a peculiar wind- equivalence between a simple and realistic magnetic model ing number. and a Frenkel-Kontorova model with an additional second- My analysis obviously applies also to finite Frenkel- harmonic contribution to the sinusoidal potential and a fixed Kontorova chains, with an even number of sites. To my misfit d 1/2. The determination of the ground state of the knowledge, this problem was treated only by Braiman magnetic model-a uniaxial antiferromagnet in the presence et al.,50 for a pure Frenkel-Kontorova model i.e., without of an applied field H-was formulated as a two-dimensional the second-harmonic contribution . A comparison with their area-preserving map; the results are consistent with the ex- results emphasized the importance of the uniaxial anisotropy perimental data on Fe/Cr 211 superlattices7 and other theo- in our model; in fact, for an even number of sites they always retical works.7,9 The effect of the surfaces, introduced by found only symmetric configurations. appropriate boundary conditions, is shown to be equivalent to the introduction of a discommensuration, whose meaning ACKNOWLEDGMENTS both in the high- and low-anisotropy cases is discussed, to- gether with its connection with metastability and pinning ef- Many interesting discussions with M.G. Pini, P. Politi, fects. 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