PHYSICAL REVIEW B VOLUME 59, NUMBER 9 1 MARCH 1999-I Surface spin-flop and discommensuration transitions in antiferromagnets C. Micheletti International School for Advanced Studies (SISSA) and INFM, Via Beirut 2-4, 34014 Trieste, Italy and The Abdus Salam Centre for Theoretical Physics, Trieste, Italy R. B. Griffiths Physics Department, Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213 J. M. Yeomans Theoretical Physics, Oxford University, 1 Keble Road, Oxford OX1 3NP, United Kingdom Received 11 September 1998 Phase diagrams as a function of anisotropy D and magnetic field H are obtained for discommensurations and surface states for an antiferromagnet in which H is parallel to the easy axis, by modeling it using the ground states of a one-dimensional chain of classical XY spins. A surface spin-flop phase exists for all D, but the interval in H over which it is stable becomes extremely small as D goes to zero. First-order transitions, separating different surface states and ending in critical points, exist inside the surface spin-flop region. They accumulate at a field H depending on D) significantly less than the value HSF for a bulk spin-flop transition. For H H HSF there is no surface spin-flop phase in the strict sense; instead, the surface restructures by, in effect, producing a discommensuration infinitely far away in the bulk. The results are used to explain in detail the phase transitions occurring in systems consisting of a finite, even number of layers. S0163-1829 99 02309-7 I. INTRODUCTION experimentally depend upon whether the number of Fe lay- ers is even or odd. The experimental work has motivated a It has been known for a long time that if an antiferromag- number of theoretical and numerical studies of finite and net with suitable anisotropy is placed in an external magnetic semi-infinite systems.4­9 Most of these have found evidence field H parallel to the easy axis the axis along which the for the existence of SSF states. spins are aligned, in opposite directions on different sublat- In the present paper we address the issue of the existence tices, in zero magnetic field and the field strength is in- of SSF phases and some related topics by studying the prop- creased, a first-order transition will occur1 in which the spins erties of the ground states of chains of antiferromagnetically are realigned in directions approximately perpendicular to coupled classical XY spins, each spin variable represented by the applied field, but with a component along the field direc- an angle between 0 and 2 , subject to a uniaxial anisot- tion. The transition to this spin-flop phase occurs when H is ropy D as well as to an external magnetic field H, as a func- equal to a spin-flop field HSF , whose value depends on the tion of D and H. One can think of as the direction of the exchange energy and the anisotropy. As H continues to in- magnetization in an Fe layer in a superlattice, or of the av- crease beyond HSF , the spins on the two sublattices rotate erage magnetization in a layer of an antiferromagnet contain- towards the field direction till eventually, if the field is suf- ing spins belonging to one type of sublattice. Minimizing the ficiently large, they are parallel to each other in a ferromag- energy of a one-dimensional model then corresponds to net structure. minimizing the free energy of a three-dimensional layered In 1968 Mills2 proposed that in an antiferromagnet with a system, provided fluctuations inside the layers do not have a free surface, spins near the surface could rotate into a flopped drastic effect. This means that the model we consider here is, state at a field HSF significantly less than HSF . This surface in its essentials, equivalent to those used in previous studies. spin-flop SSF problem was later studied by Keffer and It allows us to come to some fairly definite conclusions about Chow,3 who found a transition at HSF , but to a state having SSF phases in semi-infinite systems, and about the behavior a character rather different than that proposed by Mills. In- of systems containing a finite number of layers. Our princi- terest in this problem was recently rekindled through experi- pal conclusions were published previously in a short report;10 mental work on layered structures consisting of Fe/Cr 211 the present paper contains the complete argument, and sup- superlattices.4,5 If the thickness of the Cr layers is chosen plies a number of additional details. appropriately, adjacent Fe blocks are coupled antiferromag- In order to understand the properties of finite and semi- netically, and thus in zero magnetic field they exhibit an infinite chains, it is helpful to begin with an infinite chain antiferromagnetic structure in which the magnetization of and a defect structure known as a ``discommensuration'' or each layer is opposite to that of the adjoining layers. Apply- ``soliton'' or ``kink'' , which can occur in both the antifer- ing an external magnetic field parallel to the layers can give romagnetic and the spin-flop phases. In Sec. II we work out rise to phase transitions in which the magnetization in certain the properties of the discommensurations of minimum en- layers rotates or reverses its direction, and the results found ergy in the antiferromagnetic ground state of the XY chain. 0163-1829/99/59 9 /6239 11 /$15.00 PRB 59 6239 ©1999 The American Physical Society 6240 C. MICHELETTI, R. B. GRIFFITHS, AND J. M. YEOMANS PRB 59 spins parallel to the field, from the antiferromagnetic AF one with the spins alternating between 0 and , parallel and antiparallel to the field. Along the AF:F boundary the ground state is infinitely degenerate since it is possible to flip any number of nonadjacent spins in the F chain with no change in energy. For D 2 and intermediate values of H, the ground state no longer corresponds to spins in the Ising positions, i equal 0 or , but is a spin-flop SF phase in which the spins alternate between and , where cos H/ 4 D . 2 The spin-flop region extends between the boundaries H D(4 D) and H 4 D which are first and second or- der, respectively.11 We now consider the case when an infinite chain is con- strained by suitable boundary conditions to include a dis- commensuration for detailed studies of discommensurations in Frenkel-Kontorova models see, for example, Refs. 12­ 15 . The study of the discommensuration phase diagram is important because it helps us to understand the minimal en- ergy configurations observed both in semi-infinite and finite FIG. 1. Phase diagram for an infinite chain. The AF, F, and SF systems. A discommensuration is a defect which can arise in regions are occupied by the antiferromagnetic, ferromagnetic, and a periodic phase whose period is two or greater. In particular, spin-flop phases respectively. the AF ground state has period two and is degenerate: for Using these results, we obtain, in Sec. III, a phase diagram one ground state i 0 for i even and for i odd; for the for surface phase transitions in a semi-infinite chain. Both other, i for i even and 0 for i odd. A discommensura- discommensurations and surface phase transitions are essen- tion results if one requires that a single configuration i tial for understanding the properties of finite chains. These approaches one of these ground states as i tends to and are discussed in Sec. IV, where we provide a comprehensive the other as i tends to ; for instance, and detailed explanation of the complicated series of transi- tions found in chains containing an even number of spins. 2n 0, 2n 1 as n , The numerical procedures we used to study the phase dia- gram are described in Secs. II and III, and a certain number 2n , 2n 1 0 as n . 3 of analytic results are derived in Sec. V. The concluding Sec. VI provides a summary, and notes some topics which still The defect energy of a discommensuration is the differ- need to be studied. ence between the energy of the configuration containing the discommensuration and the energy of the corresponding II. INFINITE CHAIN ground state. Since both of these energies are infinite for an infinite chain, a proper definition requires some care; see, We consider an infinite chain of classical XY spins de- e.g., Ref. 16. We are interested in discommensurations scribed by the Hamiltonian which, for a given D and H, minimize this energy; they constitute what we call the discommensuration phase dia- D gram. It is convenient to start by considering the limiting H cos i i 1 H cos i i 4 1 cos 2 i , case D , where the spins are constrained to lie along the 1 Ising positions. For 0 H 2 the discommensuration of minimum energy is a configuration in which two successive where the antiferromagnetic exchange coefficient has been spins someplace in the middle of the chain are parallel to the taken as the unit of energy, i is the angle between the di- field H: rection of the ith spin and the external magnetic field H, and D is a twofold spin anisotropy. Our aim is to identify the . . . ,0, ,0, ,0,0, ,0, , . . . . 4 zero-temperature phases of this system, that is, those which minimize the energy. Minimizing the energy of a one- In the following we will use the notation AF to label this dimensional system corresponds to minimizing the free en- phase. When H 2, due to the absence of further-than- ergy of a layered three-dimensional system when the fluctua- nearest-neighbor interactions, there is not a unique tions within each individual layer are not playing an minimum-energy discommensuration associated with the AF important role, as is the case for the Fe/Cr superlattices men- phase. One can have any arbitrary even number of spins tioned in Sec. I. aligned with the field, not just two, as in Eq. 4 , and other, The phase diagram of the system consists of three sepa- more complicated defects are possible. The ferromagnetic rate regions, as shown in Fig. 1. For D 2, the line H 2 ground state for H 2 has period one and is nondegenerate, separates the ferromagnetic F configuration, with all the so there are no discommensurations. PRB 59 SURFACE SPIN-FLOP AND DISCOMMENSURATION . . . 6241 As the spin anisotropy D decreases from infinity, lower energies may occur if in a discommensuration the spins are not limited to the Ising values 0 and . For these cases it is difficult to find an explicit analytic form for the minimum energy discommensuration, and one has to use numerical techniques to tackle the problem. The numerical procedure that we have adopted relies on the method of effective potentials,17,18 which is very efficient for obtaining the ground state of models with short-range interactions and dis- cretized variables. The main advantage of this method is that it yields the true ground state, rather than some metastable one. The main disadvantage for our problem is that it re- quires the spin variables to be discretized: they can take on only a finite number of values. We generally used a discreti- zation grid in which each i is an integer times 2 /1400. To overcome the effects of the discretization we first fixed the anisotropy at some intermediate value, typically D 0.6, then used the Chou-Griffiths algorithm17 to identify minimal energy states of different phases for the system of discretized spins, and, finally, employed the equilibrium equations, H FIG. 2. Discommensuration phase diagram for an infinite chain. 0, 5 i The dashed phase boundaries correspond to phase transitions in the discommensuration-free chain, the solid lines in Fig. 1. for continuous spins in order to refine the configurations ob- tained using discretized spins. The phase boundaries located spin configuration resembles that in a bulk spin-flop phase, by comparing the energies of neighboring phases, calculated located between ``tails,'' each of which rapidly reverts to the using the refined configurations, were then followed as the configuration of the corresponding AF phase with increasing value of D was changed in small steps, while the spin con- distance from the core see Fig. 3 . One can think of the figurations were updated using Eq. 5 . The location of the region where the core changes into the tail as an ``interface'' phase boundaries was then checked against those obtained between the AF phase out in the tail and the SF phase in the starting with finer discretization grids. We established that, core. From this perspective, the discommensuration consists using a discretization of 2 /1400, the error in the location of of a pair of interfaces, AF-SF and SF-AF, bounding the SF the boundary, H, was in the range of 10 8­10 9 through- core. As D decreases, these interfaces broaden, making the out the range of D values we studied. The procedure just distinction between the ``tails'' and the ``core'' less clear, described was used to find minimum energy configurations but we continue to use the same label 2m for the discom- of a ring of spins periodic boundary conditions of length L mensuration which evolves continuously from the one with a with L odd, so as to produce a configuration containing a clearly-defined core of size 2m at larger D. discommensuration. When L is large we used L 31) com- An analytic calculation, see Sec. V, shows that the equa- pared to the size of the discommensuration, this is practically tion for the second-order transition between AF and 2 in the same as studying the minimal energy discommensuration Fig. 2 is in an infinite chain. The numerical results are summarized in the discommen- D H 1 1 5/3 D H, 6 suration phase diagram in Fig. 2. There are, of course, no discommensurations in the F phase. As for the SF phase, our in good agreement with our numerical calculations, and numerical results showed a smooth variation of spin angles those in Ref. 22 when H 0. At low values of H, the dis- with D and H, and consequently no phase transitions. How- commensuration 2 has the lowest energy, but upon ap- ever, various phase transitions were identified for AF phase proaching the bulk AF:SF phase boundary, one finds a se- discommensurations. In the AF region, Fig. 2, the spins in quence of phase transitions to 4 , 6 , . . . as H increases, the minimum energy discommensuration stay locked in their D positions. The persistence of this Ising spin locking for finite values of the anisotropy is a rather common feature in models with twofold spin anisotropy.19­21 Here it has the consequence that the multiphase degeneracy encountered at the point (H 2, D ) persists throughout the locus (D 2, H 2). FIG. 3. Schematic representation of phase 4 for moderate val- For values of D lying below the lower boundary of AF , ues of the spin anisotropy. The phase can be regarded as resulting but still inside the AF region in Fig. 2, ``flopped'' discom- from merging a portion of the spin-flop phase SF with two semi- mensurations of different length have lower energies than the infinite antiferromagnetic chains AF . The spins nearest the AF-SF Ising discommensuration 4 . A flopped discommensuration and SF-AF interfaces are expected to relax from their ideal AF or of type 2m consists of a ``core'' of 2m spins in which the SF angles. 6242 C. MICHELETTI, R. B. GRIFFITHS, AND J. M. YEOMANS PRB 59 ployed in Ref. 22 for small spin anisotropy. The triple points at which the phases AF , 2m and 2m 2 meet tend to an accumulation point, Q, located at H 1.58, D 0.78. This should be the point at which the energy to create a pair of AF-SF and SF-AF interfaces infinitely far apart is equal to the energy of an Ising discommensuration. III. SEMI-INFINITE CHAINS We now consider the surface states of a semi-infinite chain. The Hamiltonian for the system is the same as Eq. 1 but with the sum extending only over non-negative values of i (i 0 denotes the surface site : D H cos i i 1 H cos i i 0 4 1 cos 2 i . 7 It is useful to think of semiinfinite chains as obtained by cutting an infinite chain in two. Removing a bond in the infinite chain without allowing the spins to move will give FIG. 4. Plot of the derivative of the energy with respect to field two semi-infinite chains that we shall term unreconstructed. in the two neighboring phases, 2 , 4 along their common bound- If the spins of the unreconstructed chains are then allowed to ary, for a ring of 17 spins. The inset shows the difference of the relax, a rearrangement of the spins near the surface may take two derivatives. place, as illustrated in Fig. 5, which lowers the energy. No- tice that even though the total energy of the semi-infinite as shown in Fig. 2. Our numerical procedures found values chain is infinite, changes in the energy when a configuration of 2m up to 14, and we were able to trace the first-order lines is modified near the surface or in a way such that the modi- separating the different 2m phases down to a value of D fications decrease sufficiently rapidly with increasing dis- between 0.1 and 0.4. For smaller values of D, the difference tance from the surface are well defined. We want to con- of the energy derivatives E/ H in two neighboring sider surface states which minimize the energy in the sense phases was no longer sufficient to allow us to distinguish the that no local modifications of the configuration near the sur- phases numerically and locate the phase boundary. See the face can decrease the energy. example in Fig. 4. However, we found no evidence that these The task of finding the reconstructed surface of minimum lines terminate in critical points. The smooth decrease of energy is, in general, not simple except when all the spins in shown in the inset of Fig. 4 contrasts with what one might the chain are subject to the Ising locking . To identify the expect at a critical point as in Fig. 8 . Therefore, it seems minimal energy surface states we used numerical algorithms plausible to assume that the 2m : 2m 2 boundaries per- based on effective potential methods that, as mentioned ear- sist all the way down to D 0. lier, require a discretization of the spin variables at each site. The sequence of transitions associated with a broadening It is important to notice that, since the of the discommensuration can be understood in the following i's are constrained to take on only discrete values, after a finite distance, or ``pen- way. The defect energy of a discommensuration can be etration depth'' l from the surface the spins will be exactly in thought of as the sum of the energy required to produce a the discretized positions corresponding to a doubly infinite pair of AF-SF and SF-AF interfaces infinitely far apart, an chain or an unreconstructed surface. Configurations for the interaction energy between the interfaces which we assume infinite chain were obtained using the Floria-Griffiths is positive and rises rapidly as they approach each other, and algorithm23 which, within the limits of the discretization, a ``bulk'' contribution proportional to the size of the core, yields the exact ground state. Next, the Chou-Griffiths arising from the fact that in the AF part of the phase diagram, algorithm17 with its successive iterations was used to gener- the SF phase is metastable. In terms of which discommensu- ate reconstructed surface configurations ration has the lowest energy, the interface repulsion obvi- 0 , 1 , . . . , l . ously favors a large core, and the metastable ``penalty'' a small core. The actual size will represent some compromise between the two. Upon approaching the AF:SF boundary, the metastable penalty goes to zero, so the discommensura- tion of minimum energy should become larger and larger. Hence, one expects the 2m : 2m 2 boundary to ap- proach the AF:SF transition line as m . This is consistent FIG. 5. Cutting an infinite chain in two a while keeping the with our numerical calculations, and in agreement with the spins ``frozen'' results in two semi-infinite chains with unrecon- predictions of Papanicolaou.22 Note, however, that these structed surfaces b . Allowing the spins to relax to positions which transitions reflect the discrete nature of the spin chain and minimize the energy typically results in reconstruction of the sur- therefore are absent in the continuum approximation em- face c , a rearrangement of the spins nearest the surface. PRB 59 SURFACE SPIN-FLOP AND DISCOMMENSURATION . . . 6243 FIG. 7. Detail of the phase diagram for a semi-infinite chain FIG. 6. Phase diagram for a semi-infinite chain with a B-type with a B-type surface. surface. More details of the AF3 region are visible in Fig. 7. This should give the exact configuration minimizing the sur- and so forth, where 2n consists of 2n spins 0, , . . . in an face energy for the discrete spins. However, in practice we antiferromagnetic arrangement, followed by two spins paral- had to limit l to a maximum value l lel to the field, and then the bulk antiferromagnetic phase. max no larger than 50; thus the method could not yield the correct configuration for One can think of this reconstructed surface as an Ising dis- a larger penetration depth. The phase boundaries were then commensuration, whose core consists of two adjacent spins identified as explained in the previous section. with i 0, located a distance 2n from the surface. Because The resulting phase diagram is shown in Fig. 6. Through- the ``tails'' of this discommensuration have zero length, it out the F region the minimum energy surface states are sim- does not interact with the surface, and its energy is indepen- ply the unreconstructed surfaces; it is easy to see that making dent of its distance from the surface. While this degeneracy any changes will increase the energy. In the SF region, since persists throughout the AF2 region, along the line D 2,H the ground state of the infinite chain has period two, there are 2 the degeneracy is even greater: the set of minimum en- two unreconstructed surfaces. Each of them undergoes a re- ergy surface states includes cases where the number of con- construction in which the spins nearest the surface tilt to- secutive spins pointing along the field is not limited to 2 but wards the magnetic-field direction, as in Fig. 5 c . However, can attain any even number, e.g., 0,0,0,0, ,0, . . . or this change in spin direction occurs continuously as a func- 0, ,0, ,0,0,0,0,0,0, ,0, . . . . Incidentally, we note that tion of H and D, and so no surface phase transitions are observed inside the SF region. these degeneracies are somewhat artificial in that they would Next consider the AF part of the phase diagram. Again be lifted by introducing weak longer-range interactions in the there exist two possible surface states, A and B, whose unre- Hamiltonian 7 . constructed versions, A In the AF3 region of Fig. 6 the B-type surface again re- u and Bu , have surface spins parallel constructs, but the spin anisotropy is sufficiently low that the ( 0 0) or opposite ( 0 ) to the field direction: spins unlock from the Ising angles. As in the AF2 region, one A can think of the surface state as consisting of a discommen- u 0, ,0, ,0, , . . . , 8 suration located a finite distance from the surface, but now B this discommensuration is of the flopped type with a core of u ,0, ,0, ,0, . . . . 9 length of 2, and tails extending out on either side of the core. A surface will be said to be of type A B if the spin con- We again employ the notation 2n for the surface state with figuration tends to that of Au(Bu) far from the surface. 2n spins to the left of the core, that is, in the tail extending to Throughout the AF region of the phase diagram, the mini- the surface. Because of this tail, the discommensuration in- mum energy surface of type A is the unreconstructed Au . teracts with the surface, and the minimum surface energy However, the B-type surface shows a number of different occurs for a specific value of 2n, depending upon D and H. structures in different parts of the AF region, as indicated in Thus, in AF3 , one finds genuine spin-flop surface states. As Figs. 6 and 7. In region AF1 the unreconstructed surface Bu H increases, the discommensuration moves further from the has the lowest energy. In region AF2 , which meets AF1 surface. It does this, at least when D is large, discontinuously along a line H 1 for D larger than the value at O, it is in steps of 2, via a series of first-order phase transitions, energetically favorable to flip the surface spin so that it some of which are shown in Fig. 7, where they extend left- points along the field direction, and there is a set of degen- wards from the point P. For smaller values of D, the edges of erate equal minimum energy reconstructed surfaces the core are not as well defined, and it is more difficult to associate the 2n 2n 2 transitions with a discontinu- 0 0,0, ,0, ,0, . . . , 2 0, ,0,0, ,0, . . . , ous jump of the discommensuration. Numerically we have 10 seen states with 2n up to 14, and our results are consistent 6244 C. MICHELETTI, R. B. GRIFFITHS, AND J. M. YEOMANS PRB 59 FIG. 9. Surface spin 0 along the left edge of the AF3 region as a function of anisotropy D. The surface layer consisted of 34 spins, and the behavior of the curve at low D dashed is affected by finite-size effects in the numerical calculations. rabola D 0.5H2 to within numerical precision, which is as- ymptotically the same as Eq. 11 . We nonetheless believe that the width of AF3 remains finite as long as D 0. Nu- FIG. 8. Plot of the derivative of the energy with respect to field merical evidence for this is shown in Fig. 9 where the value in the two neighboring phases 0 and 2 along their common of the surface spin, 0 , at the left edge of the AF3 region boundary, using 50 spins in the surface layer. The inset shows the that is for H just large enough to produce the surface spin- difference of the two derivatives. flop phase is plotted as a function of D. The results are for l with n tending to infinity at the right side of the AF max 34 spins in the surface layer see the description of the 3 region, numerical approach given above . Below D 0.05 the results which our analytic calculations Sec. V , in agreement with become unreliable because l Ref. 6, show to be the line max is too small, as we can tell by carrying out calculations for different values of lmax . However, extrapolating from larger values of D indicates D 1 H2 1. 11 that as D goes to zero, 0 tends to a value near /3 or 60°, The upper boundary of the AF showing that even for very small D the discommensuration 3 region extending from O to P is a continuous second-order transition. One can think of at the threshold field is still a finite distance from the surface. it as the limit of stability of the Ising surface phase 0 as D This situation is quite distinct from that in region AF1 , decreases inside AF where 2 . An analytic calculation, Sec. V, shows 0 , and in AF4 , discussed below, where 0 0. that the implicit equation for the boundary is Between AF3 and the AF:SF bulk phase boundary lies region AF4 , see Figs. 6 and 7, in which the flopped discom- 2 D H 1/a 1 2 D H a, mensuration is repelled by the surface, so that its minimum energy location is in the bulk infinitely far away from the aªH D 1/ 1 H D . 12 surface, as noted in Ref. 6. Thus there is no minimum-energy reconstructed B surface, or, properly speaking, a ``surface Thus the point P, where all the phases 2n come together, spin-flop phase'' in region AF4 . It seems better to identify lies at H 4/3, D 2/3, the intersection of Eqs. 11 and AF4 , thought of as part of the B-type surface phase diagram, 12 . Both Eqs. 11 and 12 agree with our numerical re- as a ``discommensuration phase,'' since the minimum en- sults. ergy surface will always be of the A-type, with the surface We find that the first-order lines extending downwards spin 0 0. and leftwards from P in Fig. 7, separating phases 2n from In Fig. 10 the discommensuration phase diagram for the 2n 2 , end in critical points as D decreases. This is clearly infinite chain Fig. 2 , represented by dashed lines, is super- visible in the example in Fig. 8, which shows the typical imposed on the B-type surface diagram for the semi-infinite behavior of the energy derivatives of two neighboring phases chain, represented by solid lines, in the vicinity of points P along their coexistence line. Near a critical point D Dc one and Q, which are common to both diagrams, as is the broken expects to vary as D Dc, in qualitative agreement with line shown dashed from P to Q. Note that the OP line of what we observed. The larger the value of n, the further the the surface diagram, Fig. 7, lies above the lower boundary of first-order line extends towards the origin of the H,D plane, the AF' region of the discommensuration phase diagram in but presumably for any finite value of n the difference be- Fig. 2. Thus to the left of P, for H 4/3, as D decreases the tween the phases 2n and 2n 2 eventually disappears at reconstructed B-type surface phase changes from Ising to a some finite value of D. Because this value decreases with flopped form before the corresponding change is energeti- increasing n, it is plausible that the corresponding critical cally favorable for the bulk discommensuration. points accumulate at the origin. In addition, Fig. 10 shows that the part of the H,D plane As is evident in Fig. 6, the region AF3 becomes extremely corresponding to 2m in the discommensuration phase dia- narrow as D decreases. The left boundary approaches a pa- gram, Fig. 2, for 2m 4 lies entirely inside the AF4 region of PRB 59 SURFACE SPIN-FLOP AND DISCOMMENSURATION . . . 6245 the behavior of the system will depend strongly on the actual length of the chain. Since we are not interested in L-dependent features of the phase diagram, apart from whether L is even or odd, we shall assume that L is suffi- ciently large to justify the use of Eq. 13 . From the discussion presented in the previous sections one can predict that a finite chain will not undergo any phase transition for values of D and H inside the SF and F regions. On the other hand, it can also be anticipated that the behavior of the chain in the AF region will be rather complicated. As noted in Refs. 4,5,7, the behavior of the chain for values of D and H in the AF region changes dramatically according to whether the length of the chain is even or odd. If L is odd, both ends of the chain have to be of the same type, A or B, unless a discommensuration is present. Having two A-type surfaces gives a lower energy than two B-type FIG. 10. Discommensuration phase diagram Fig. 2 , using surfaces, because the former results in a net magnetization in dashed lines, superimposed on the phase diagram for a semi-infinite the direction of the field, and the latter a net magnetization chain with a B-type surface Fig. 7 , using solid lines, in the vicinity opposite to the field. Similar considerations show that of the point P. The broken line connecting P with Q is part of both phase diagrams. throughout the AF region it is energetically unfavorable to insert a dislocation, thus producing one A-type and one Fig. 6 and 7 for the surface phase diagram. This is consis- B-type surface. Hence for odd L, the minimum energy cor- tent with our observation that as long as the discommensu- responds to two unreconstructed A-type surfaces at either ration is a finite distance from the surface, in the AF end of the chain, and no discommensurations. 3 region, it is always of the type 2m 2. Thus as H increases, it is only On the other hand, when L is even, the two surfaces have after the discommensuration has moved infinitely far from to be of different types, unless a discommensuration is the surface, and thus has no influence on the surface phase present. The analysis of Sec. II has shown that discommen- diagram, that its core begins to broaden. surations are not favored energetically outside region AF4 . In retrospect it seems likely that the broadening of the Thus, for D and H falling in region AF1 or AF3 , one expects SSF transition mentioned in the abstract of Ref. 3 actually one surface of type A and the other of type B. Moreover, refers to broadening of the bulk discommensuration which, from the results of Sec. III, we expect that in region AF1 the as noted above, occurs as H approaches the AF:SF boundary B surface remains unreconstructed, whereas surface spin-flop inside region AF states should be observed in AF 4 . It appears that no work prior to ours has 3 owing to the reconstruction correctly identified the stable SSF phase at small values of D, of the B-type end of the chain. The A-type end of the chain characterized when it first appears with increasing H by a remains, of course, in its unreconstructed state. Next, in re- surface spin with a value very near 60° Fig. 9 . The narrow- gion AF4 the energy is minimized using two A-type surfaces ness of the AF and a discommensuration, which lies at the center of the 3 region for small D may be why it was over- looked. finite chain because it is repelled by both surfaces. Finally, in AF2 , because of the degeneracy due to the Ising spin lock- ing, one has either a reconstructed B-type surface or a dis- IV. FINITE CHAIN commensuration, depending upon what one wants to call it, We now move on to consider the case of a chain of finite and an A-type surface at the other end of the chain. length L. Since a surface reconstruction can occur at both Consequently, if D is smaller than the value correspond- ends of the chain, and it is also possible for a discommensu- ing to point P in Fig. 7, we expect a finite system with even ration to be present in the interior of the chain, we write its L to undergo the following set of transitions with increasing total energy in the form H. At H 0, Fig. 11 a , there are unreconstructed surfaces of types A and B at opposite ends of the chain. When H reaches E L R L L E the threshold for the formation of an SSF phase, the B-type s Es Ed , 13 surface restructures discontinuously, b to form a type 2 where is the bulk energy, the ground-state energy per spin discommensuration which then, as H increases, moves to- for an infinite chain, EL R s and Es are the energies of the left wards the center of the chain in a series of discontinuous and right surfaces, respectively, and Ed is the energy of a steps, c and d , some of which may be continuous if D is discommensuration in the chain if present . Minimizing the smaller than the value for the corresponding critical point, total energy for fixed L is equivalent to finding the spin con- see Sec. III. figuration that minimizes EL R s Es Ed . The discommensuration will reach the center of the chain, In writing Eq. 13 , L was assumed to be sufficiently large Fig. 11 d , when H is close to the threshold for the AF4 or that the interaction between the two ends of the chain, and discommensuration region in Fig. 6. Further increases of H between each end and the discommensuration, if present, can will lead to a broadening of the discommensuration, with be neglected. For any given L this condition can always be 2m going through the sequence 2 , 4 , 6 , . . . of Fig. satisfied by choosing a large enough value for the spin an- 2; see Figs. 11 d to 11 g . While these transitions are likely isotropy. Outside the range of D for which Eq. 13 holds, to be discontinuous for larger values of D, it may be hard to 6246 C. MICHELETTI, R. B. GRIFFITHS, AND J. M. YEOMANS PRB 59 FIG. 11. Schematic representation of the series of different phases encountered in a chain of 10 spins for increasing values of H. FIG. 13. Plot of the susceptibility in arbitrary units for a chain see the discontinuities when D is small. The center of the of 22 spins for D 0.3. 2m discommensuration in Fig. 11 does not fall at the pre- cise center of the chain when m is even; the offset is needed so that the surface spins can both be approximately paral- the AF1 region into the surface spin-flop AF2 , phase 0 . lel, rather than antiparallel, to the field direction. For L The first series of spikes, for H between 0.9 and 1.13, is 12 the offset occurs when m is odd. associated with first-order spin-flop transitions, in agreement The AF-SF and SF-AF interfaces on either side of the with Refs. 7,9. For H between 1.13 and 1.32, one observes a core move outwards as the discommensuration expands, and second series of transitions associated with the broadening of eventually they reach the surfaces of the chain, Fig. 11 g , at the discommensuration. Figure 13 shows the susceptibility a field very close to that required to produce the bulk spin- for the same length of chain (L 22) with a smaller anisot- flop transition. At still higher fields the entire chain can be ropy, D 0.3. The spikes are smaller than in Fig. 12 due to a thought of as being in the bulk spin-flop phase, with appro- decrease in anisotropy, and some of the surface spin-flop priate reconstructed surface configurations corresponding peaks have disappeared, which is what one would expect in to this phase. Sufficiently large values of H will eventually view of the critical points along the 2n : 2n 2 phase force all of the spins into the ferromagnetic configuration boundaries noted in Sec. III. A recent study by Papanicolaou8 of the dynamics of a i 0. The scenario just described is basically consistent with model similar to Eq. 1 , but with three-dimensional classi- previous numerical studies, including two that have appeared cal spins, shows evidence for metastability and hysteresis as quite recently,8,9 and our own numerical work. Thus Fig. 12 the magnetic field H is varied, as one would expect for a shows the magnetic susceptibility M/ H,M the magne- first-order SSF transition. Additional hysteresis is seen as the tization, for a chain of L 22 spins when D 0.5. The spikes field is increased beyond the SSF transition, consistent with appearing in Fig. 12 should be Dirac delta functions. Here additional first-order transitions of the sort discussed above. they appear to have a finite height because of the finite in- Small differences in detail between these results and ours can cremental step H chosen for the numerical calculation. The probably be explained in terms of hysteresis effects, or pos- sibly as due to the fact that the models are not identical. A first spike in Fig. 12 for H 0.9) signals the transition from numerical study of Eq. 1 by Trallori,9 using an area- preserving map, is also in very good agreement with all of our results, except that certain transitions which we would expect to be first order as the discommensuration moves to the center of the chain and broadens are found to be continu- ous when D is very small. But this difference is probably not important, since the discontinuities will in any case be very small when D is small, and could be absent because L is finite. V. ANALYTICAL RESULTS In this last section we give a detailed derivation of the analytical results presented earlier in the paper. As already noted, analytical solutions to the problem of minimizing the energy are, in general, only available when the spins are in Ising position, 0 or . However, when deviations from these values are small, systematic approximations are pos- FIG. 12. Plot of the susceptibility in arbitrary units for a chain sible. Throughout this section we shall use 0i to indicate of 22 spins for D 0.5. Ising or ``locked'' spin values, i for the actual canted val- PRB 59 SURFACE SPIN-FLOP AND DISCOMMENSURATION . . . 6247 FIG. 14. Schematic representation of the canted discommensu- FIG. 15. Schematic representation of the surface phase 0 . ration phase 2 . The only set of values (H,D) for which Eqs. 16 can be ues, and 0 simultaneously satisfied under the constraint that the modu- i i i for the deviations of the latter from the locked values. lus of s1 and s2 cannot exceed 1 so that the spin deviations To obtain an analytic expression for a second-order decay to zero infinitely far from the discommensuration boundary separating locked and canted versions of a spin core has to satisfy the relation configuration, we start by expanding Eq. 5 to first order in the spin deviations, assuming that they are small, D H 1 1 5/3 D H, 18 which is the same as Eq. 6 . Equation 18 identifies the cos 0 0 0 0 i i 1 i i 1 cos i 1 i i i 1 locus of points where the spin deviations for phase 2 be- come vanishingly small, which is the second-order boundary H cos 0i D i , 14 AF : 2 . and then solving these equations self-consistently. The same method can be used to find the second-order We first apply this strategy to find the boundary separat- boundary OP between AF2 and AF3 in Fig. 6 or 7. In the 0 ing phases AF and 2 , Fig. 2, using the labels for sites in phase close to the border, with the spins labeled as in Fig. 15, the flopped discommensuration 2 given in Fig. 14. Equa- deviations from the corresponding Ising configuration, Eq. tions 14 can be written as recursion relations, in terms of 10 , will be small, and the solution to Eq. 14 takes the form ratios xi i / i 1 of successive spin deviations, in the form x x 1 2 j s2 for j 1, 2 j x2 j 1 2 D H for j 1, x2j 1 s1 for j 1, x 1 2 j 1 x2 j 2 2 D H for j 1, x1 1 H D, x 1 0 x1 D H, x 1 1 s2 H D, 19 x 1 1 x2 D H, using the same notation introduced previously, with s1 and x 1 s2 again defined by Eq. 17 . These equations yield an addi- 2 j x2 j 1 2 D H for j 1, tional relation for s2 , x 1 2 j 1 x2 j 2 2 D H for j 1, 15 s2 H D 1 H D 1, 20 with a solution which can be satisfied together with Eq. 17 only on the locus of points defined by Eq. 12 . x2j s2 for j 1, A similar analysis assuming small deviations from Ising values for the state 2 shows that the point P on , Fig. 7, x2j 1 s1 for j 1, occurs at the intersection of the curve x 1 0 x1 D H, 1 D H 1 D H 1, 21 x 1 with the boundary 18 , so that P falls at H 4/3, D 2/3, in 1 x2 D H, good agreement with our numerical results H 1.333, D x 1 0.6666. Likewise, one can show that the other 2n states 2 j 2 s2 for j 1, for n 1 meet the AF2 region at P, which is a sort of mul- x 1 ticritical point for the surface phase diagram. 2 j 1 s1 for j 1, 16 A somewhat different approach yields an equation for the obtained using techniques of continued fractions. Here s1 boundary between the AF3 and AF4 regions, that is, the left and s2 are given by edge of the AF4 region in Figs. 6 and 7. As this corresponds to an accumulation of surface spin-flop states 2n as n s1 2 2 D H 2 D H 2 D H t 1, , the distance from the surface of the chain to the core of the dislocation will become arbitrarily large, so that the spin s2 1/2 2 D H t/ 2 D H , angles in the discommensuration are essentially independent of distance from the surface,19,24 as confirmed by our nu- tª 2 D H 2 2 D H 2 4 2 D H 2 D H . merical calculations. Hence, by a route analogous to that 17 described in Refs. 19,24, it is possible to evaluate the energy 6248 C. MICHELETTI, R. B. GRIFFITHS, AND J. M. YEOMANS PRB 59 difference between two neighboring phases, En E[2n over which the SSF phase is stable when the anisotropy D is E[2n 2 , by iterating the equilibrium equations 5 on ei- small, are no doubt the reasons the two have not been dis- ther side of the discommensuration. tinguished in previous studies. Nonetheless, they are quite Using the fact that the spin deviations at the surface are different phenomena, and distinguishing them is essential for becoming vanishingly small, one obtains, to leading order at a proper understanding of phase transitions associated with large n, surfaces, both in semi-infinite and finite systems. 1 1 Our results for the discommensuration and surface phase E diagrams lead to very definite and detailed predictions, dis- n 2 1 0 2 2 2 1 2 cussed in Sec. IV, for the complicated sequence of phase 1 1 transitions occurring in a system with an even number of 2 2 2 2 layers spins as H increases at fixed D. They are in good 2 D 1 0 2 H 1 0 , 22 agreement with various numerical studies, including our where the own, if allowance is made for the uncertainties inherent in i's are the spin deviations of phase 2n 2 . The expression for E numerical work of this sort, and this gives us additional con- n can be simplified by using Eq. 14 to express fidence in the validity of our analysis. To the extent that this 1 and 2 in terms of 0, noting that when i 0, the model antiferromagnet correctly describes Fe/Cr superlat- term cos( 0 0 i i 1)( i i 1) must be omitted from Eq. 14 , tices, we can also claim to have achieved a basic understand- because i 0 represents the left edge of the finite chain, Eq. ing of the processes giving rise to the phase transitions ob- 7 . Substituting served experimentally in the latter. That does not, of course, mean that our model is adequate 1 1 D H 0 , for understanding SSF phases and other surface phase tran- sitions in more traditional antiferromagnets, such as MnF 2 . 2 2 D H 1 D H 1 1, 23 However, as noted in Sec. I, minimizing the energy of a into Eq. 22 gives one-dimensional model is the analog of minimizing the free energy of a three-dimensional layered system, whenever 1 each layer can be described, using mean-field theory or in a E 2 3 n 2 W 0 O 0 , purely phenomenological way, by means of a total magneti- zation serving as a sort of order parameter. To be sure, the Wª2D 7D2 5D3 D4 2D D2 H parameters which enter the Hamiltonian for the one- dimensional chain may not be those appropriate for three- 1 5D 2D2 H2 H3 H4. 24 dimensional system. But one can still expect qualitative simi- Note that this expression holds for all values of D as long as larities in the phase diagrams, even if certain quantitative aspects are different. 0 is small, that is, the discommensuration is very far from In that connection, it is appropriate to ask whether certain the surface. But this means that an accumulation of states features of the discommensuration and surface phase dia- 2n as n tends to infinity must lie on a locus where W in Eq. grams of the one-dimensional model depend in a sensitive 24 vanishes, because in region AF3 the discommensuration way upon the particular form of the Hamiltonian 1 . For is attracted by the surface (W 0), while it is repelled in example, it contains no spin coupling beyond nearest neigh- AF4(W 0). The relevant root of this equation takes the bors, whereas it would be physically more realistic to as- simple form sume, at the very least, some sort of exchange coupling of further neighbors, decreasing rapidly with distance. Would D 1 H2 1, 25 introducing such interactions lead to significant changes in in agreement with Ref. 6, and with our numerical calcula- the phase diagram? Could they, for example, make the SSF tions. phase disappear entirely at low values of the anisotropy? This is one of many questions which cannot be answered VI. CONCLUSIONS definitively in advance of appropriate calculations. It is worth pointing out that our physical picture of the SSF phase Our work shows that the structure of surface spin-flop as due to a discommensuration finding its minimum energy SSF states and their relationship to the behavior of finite at a finite distance from the surface does not seem to depend systems is significantly more complex than anticipated in on the absence of further-neighbor exchange or possibly previous work. In particular, the genuine SSF phase for a other types of interaction, so we can well imagine that the semi-infinite system, which we identify with region AF3 in phenomenon persists with a more realistic Hamiltonian. our surface phase diagram, Figs. 6 and 7, has previously Nonetheless, this is one respect in which our work remains been confused with what we call the ``discommensuration'' incomplete. While our numerical results, especially the ap- phase, region AF4 , in which the B-type surface has, strictly parent existence of a nonzero limit for 0 as D goes to zero, speaking, completely disappeared through a restructuring in Fig. 9, support our description in terms of a discommensu- which a discommensuration has moved infinitely far away ration, an appropriate analytic calculation in the limit of from the surface into the bulk. The fact that both the SSF and small D, of the sort which might among other things give the discommensuration phase occur at a magnetic field H the value of this limiting angle, has not been carried out. significantly below that required to produce a bulk spin-flop Such a study would probably provide insight into whether transition, together with the extremely small interval of H weak further-neighbor interactions simply change the quan- PRB 59 SURFACE SPIN-FLOP AND DISCOMMENSURATION . . . 6249 titative values of various parameters, or lead to a qualita- tions of the proper kind might result in the infinite-chain tively different result, such as the absence of the AF3 region discommensurations undergoing their broadening transitions when D is sufficiently small. at significantly smaller values of the magnetic field H. This It seems unlikely that weak further-neighbor interactions could lead to a complicated surface phase diagram in which would remove the first-order transitions between the surface the minimum energy discommensurations broaden while phases 2n and 2n 2 , or change the fact that these tran- they are still a finite distance from the surface. How this sitions terminate in critical points as D decreases. On the might effect the 2n to 2n 2 transitions and their critical other hand, such a modification of the Hamiltonian would points is hard to guess in advance of actually doing a calcu- surely remove the degeneracy of the surface states in the AF2 lation. region of Figs. 6 and 7. Thus, one would not be surprised to Hence, there is much which remains to be understood find significant modifications in the phase diagram near the about surface spin-flop transitions in antiferromagnets. multicritical point P. 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