PHYSICAL REVIEW B VOLUME 56, NUMBER 5 1 AUGUST 1997-I Surface and interface phase transitions in thin magnetic films with frustrated exchange interactions D. Spis aŽk* and J. Hafner Institut fušr Theoretische Physik and Center for Computational Materials Science, Technische Universitašt Wien, Wiedner Hauptstraße 8-10/136, A-1040 Wien, Austria Received 4 Feburary 1997 Detailed studies of magnetic phase transitions in thin magnetic films with frustrated exchange interactions on nonmagnetic surfaces are presented. In the first part of the work we use a self-consistent real-space tight- binding linear-muffin-tin orbital approach to determine the magnetic structure of face-centered cubic Fe films on Cu 100 substrates and a Green's-function technique to calculate the exchange pair interactions. The results demonstrate a ferromagnetic coupling at the free surface and antiferromagnetic coupling in the interior of the films. The competition between ferro- and antiferromagnetism leads to a pronounced enhancement of the exchange coupling at the surface and at the interface with the nonmagnetic substrate and a strong reduction frustration in the inner layers. In the second part we use these results to formulate an Ising model for magnetic films with frustrated exchange interactions and to perform extended Monte Carlo simulations of magnetic phase transitions. The results demonstrate a rich scenario of two-dimensional surface and interface phase transitions, coupled through weak magnetic fluctuations in the interior of the film. In addition, spin- reorientation transitions reversible and irreversible between high- and low-moment states are observed. S0163-1829 97 05529-X I. INTRODUCTION Thouless type.7 This also implies that the transition at the multiciritical point i.e., the ``special'' transition has a dif- The nature of the phase transitions at the surface of mag- ferent character. netic materials and in thin magnetic films has been studied Parallel to the progress in statistical-mechanical studies of repeatedly in recent years.1­5 The main aim of these studies idealized model systems, advanced experimental techniques was to characterize the critical behavior at the surface of a allowed for an investigation of the magnetic properties of magnetic material or at the interface of a magnetic film with surfaces and ultrathin films with unprecedented accuracy.8 In a nonmagnetic surface and to find out whether it can be general one finds that, due to the reduced coordination num- related to the properties of the bulk materials. The earliest ber, the critical temperature is lower at the surface and in thin studies in this direction were performed by Binder and films and decreases with decreasing film thickness.9­11 How- Hohenberg6 who demonstrated that in Ising and Heisenberg ever, this does not represent a universal behavior as predicted systems with different nearest-neighbor coupling in the bulk by a scaling hypothesis.12 An enhanced Curie temperature at and at the surface distinct phase transitions can occur at the the surface and an ordered surface coexisting with a disor- surface and in the bulk. Later extensive Monte Carlo calcu- dered bulk has been experimentally observed for Gd.13 For lations for semi-infinite three-dimensional Ising models have face-centred-cubic Fe films on Cu 100 substrates the Curie been used to establish the phase diagram for a material where temperature increases strongly from one to two monolayers, the exchange interaction Js at the surface differs from that in but decreases in thicker layers.14,15 In addition, different be- the bulk.4 It has been shown that a surface phase transition haviors have been reported for the variation of the magneti- decoupled from the phase transition in the bulk can occur in zation as a function of temperature, ranging from a linear two different regimes: a J 0 and Js 0. Here the bulk is temperature dependence16,17 to a behavior similar to that in ferromagnetically ordered below a bulk critical temperature bulk ferromagnets with a slow variation at low temperatures Tcb , and the interface orders antiferromagnetically at a tem- and a sharp drop of the magnetization as the critical tempera- perature Tcs . The phase boundaries for the surface and bulk ture is approached.18,19 Again fcc Fe/Cu 100 films are a phase transitions cross at a decoupled tetracritical point. b very interesting case for study, because the magnetization J 0 and Js Jsc J. If the exchange coupling at the surface curves have been shown to change from a shape character- is much stronger than in the bulk, the surface remains ferro- istic for an almost ideal anisotropic Heisenberg ferromagnet magnetically ordered above Tcb and the surface phase tran- for the thinnest films 2 monolayers ML to a linear sition shows two-dimensional critical behavior. At temperature dependence in the thicker ( 7 ML films.15 Js Jsc 1.52 J bulk and surface become simultaneously Several attempts have been made to relate the variation of critical and the phase boundaries meet at a different multi- the critical temperature at the surface or in a thin film not critical point sometimes also referred to as the ``special only to the reduced coordination number, but also to surface transition'' . Later these investigations were extended to magnetic moments s s and exchange coupling constants Ji j continuous models like the classical XY model5 and it was deviating from their bulk values in the region of the surface shown that the surface transition is of the Kosterlitz- or interface.20­22 For the free surfaces of both bcc Fe and fcc 0163-1829/97/56 5 /2646 15 /$10.00 56 2646 © 1997 The American Physical Society 56 SURFACE AND INTERFACE PHASE TRANSITIONS IN . . . 2647 Ni the reduction of the coordination number leads to a nar- netic and low-moment predominantly antiferromagnetic rowing of the d band at the surface and further to an en- solutions coexist in films with more than three ML as ener- hancement of s by 20­30 % relative to the bulk value getically almost degenerate stable and metastable solutions, for rough films even noncollinear spin structures have been b .21,23 At the interface with a Cu substrate, the Fe-interface moments show a similar enhancement,24 whereas the Ni mo- predicted.50 The only certain feature is that the moment at ments are reduced by a comparable amount.25 Jensen et al.20 the free surface is always strongly enhanced even compared have used a generalized mean-field theory to show that the to the bulk value in bcc ferromagnetic Fe and ferromagneti- observed thickness dependence of T cally coupled to the moments in the first subsurface layer. c in fcc Cu/Ni/Cu 100 sandwiches may be explained in terms of the variation of the These results show that in fcc Fe/Cu 100 the ferromag- magnetic i.e., Ni-Ni coordination number and the magnetic netic coupling is strongly enhanced at the free surface and moment at the Ni/Cu interface, but assuming a surface ex- eventually also at the interface with the nonmagnetic sub- change coupling equal to the value in the bulk. On the other strate , but frustrated in the interior of the film due to com- hand, in Fe/Cu 100 films a ratio of Fe surface and bulk peting antiferromagnetic interactions. In such a case one moments as large as would expect that the phase transitions at the surface or in- s / b 1.8 has to be assumed to re- produce the maximum in the T terface are effectively decoupled from the magnetic transi- c/thickness dependence ob- served experimentally.14,15 This disagreement is indicative of tions in the interior of the film. In the present work this the special nature of the magnetic interactions in the fcc Fe conjecture has been further explored. films that could also lead to a peculiar nature of the magnetic Our approach consists of two distinct steps. In the first we phase transitions. These transitions form the central subject use local-spin-density theory to calculate the magnetic struc- of this study. ture of Fe/Cu 100 films with up to 6 ML Fe and a real-space Extensive experimental work on the magnetic properties Green's-function approach to calculate the exchange cou- of Fe films grown epitaxially on Cu 100 substrates has es- pling between pairs of magnetic moments. This confirms tablished a complex phase diagram, with the physical prop- both the existence of stable/metastable ferro- and antiferro- erties depending on the film thickness t and other experimen- magnetic configurations, and the enhancement of the ex- tal parameters. It is possible to distinguish three different change interactions at the surfaces, as well as their strong regions. reduction frustration in the interior of the film. In the sec- a In region I with t 5 6 ML the easy axis of magne- ond step we map the magnetic interactions on an effective tization is perpendicular to the surface and a competition of Ising-Hamiltonian and we perform extensive Monte-Carlo high-moment ferromagnetic26­34 and low moment studies of the magnetic phase transitions. A detailed analysis antiferromagnetic35,36 states has been reported. The structure of the data demonstrates the existence of surface and inter- of the films has been described as tetragonally distorted face phase transitions and shows in addition that spin- (c/a 1 or c/a 1) fcc,37,38 and complex (4 1) and reorientation transitions can lead to temperature-dependent (5 1) reconstructions have been proposed on the basis of transformations between high- and low-moment phases. low-energy electron-diffraction data.39,40 b Films in region II (6 7 ML t 10 12 ML) re- II. MAGNETIC STRUCTURE AND EXCHANGE-PAIR main fcc with eventually a small tetragonal distortion37,41,40 INTERACTIONS IN FE FILMS ON CU 100... SUBSTRATES and show paramagnetism or low-moment antiferromagnetism.42 In this region the magnetic anisotropy A. Theory switches from perpendicular to in-plane. The thickness Our investigations of the magnetic properties of thin films where the crossover occurs depends on the preparation con- are based on self-consistent spin-polarized electronic- ditions: film prepared at low temperature and hence prob- structure calculations performed with a local-spin-density ably rougher than films prepared at room temperature34 LSD Hamiltonian51 in a scalar-relativistic approximation, show in-plane anisotropy already for t 5 6 ML, whereas using a real-space tight-binding linear-muffin-tin-orbital room-temperature prepared films acquire in-plane anisotropy TB-LMTO technique.52­54 Given the initial charge densi- only at 10 to 12 ML. Low-temperature prepared films with ties and potential parameters, the partial local spin-polarized 5­6 ML show a reversible spin-reorientation transition as a densities of states DOS's were computed using the real- function of temperature: the direction of the magnetic mo- space recursion method.55 From the moments of the DOS's ment switches from perpendicular to in-plane and back as the integrated up to the Fermi level, the updated charge and spin temperature is increased and decreased.27 densities, local magnetic moments, and potential parameters c Films with t 10 12 ML are bcc and ferromagnetic were calculated. The self-consistency iterations were stopped with in-plane anisotropy. Early local-spin-density calcula- after the difference of all the magnetic moments in two suc- tions of the magnetic properties of free-standing Fe ceeding iterations became smaller than 10 4 B . films43­45 agree on a ferromagnetic coupling between the The recursion calculations were performed for large clus- moments in the surface and subsurface layers and an antifer- ters of atoms with periodic boundary conditions in lateral romagnetic coupling between the deeper layers. In contrast directions and free boundary conditions in the direction of to bulk fcc and fct iron,46 only a weak dependence of the the surface normal. Each cluster consists of three layers of magnetic moments on a tetragonal distortion of the film has empty spheres to account for the spilling out of charge into been predicted.45 The most recent calculations of the mag- the vacuum, one to six Fe layers, three Cu interface layers in netic structure and anisotropy of Fe/Cu 100 films in regions these layers charge and spin densities are calculated self- I and II show that the scenario is in reality more consistently plus up to six Cu layers with the potential pa- complex:24,47­49 high-moment predominantly ferromag- rameters fixed at the values characteristic for bulk Cu. The 2648 D. SPIS AŽK AND J. HAFNER 56 interatomic distance is equal to that in bulk Cu, no relaxation renz and Hafner24,47 and the results of Ujfallussy et al.48 and in the Fe overlayer has been allowed. Each fcc 001 layer Szunyogh et al.49 based on a relativistic screened Kohn- contains 288 atoms in a (12 2 12 2) cell. Twenty exact Korringa-Rostocker KKR technique. Coexisting low- and recursion levels in the continued fraction were used for s and high-spin solutions were first found for 5 and 7 ML films p states, 40 recursion levels were used for d states. The using a technique allowing for a continuous rotation of the Beer-Pettifor terminator56 was used to get a smooth DOS. local spin quantization axes which makes it easier to relax Recently we have shown21,22 that the real-space tight- the spin-configuration to the ground state. These calculations binding approach may be used for an efficient calculation of have been performed using a Hubbard-Stoner-type exchange the exchange-pair interactions Jij between the local magnetic Hamiltonian59 - this explains the slightly different values moments i , following the torque-force approach pioneered for the moments obtained here with the full LSD Hamil- by Small and Heine.57 The exchange coupling between a pair tonian. The KKR calculations are based on a fully relativistic of magnetic moments at sites i and j can be expressed as Hamiltonian and have been performed for ferro- as well as antiferromagnetic configurations at all thicknesses. The appearance of both ferromagnetic high-spin and par- EF J i j tially antiferromagnetic AFM low-spin solutions in fcc Fe i j 2 Im TrGij Gji dE, 1 films is, of course, closely related to the frustrated exchange interactions in bulk fcc Fe. Depending on the atomic volume, where a constrained collinear calculation results in stable/unstable i stands for the local exchange splitting and where Gss low/high-spin AFM configurations. However, at least for a i j is an intersite Green's function of the system in the ground state. In principle this is equivalent to a mapping of wide range of densities, the magnetic ground state is defi- the LSD Hamiltonian of an itinerant 3d magnet on a classi- nitely a noncollinear state, probably a spin spiral.58,60,61 In cal Heisenberg spin Hamiltonian describing a localized mag- ideally flat Fe films, the calculations of Lorenz and net. Independently, a closely related approach to the Hafner24,47 have so far not found any indications for a pos- exchange-pair interactions has recently been proposed by sible noncollinear solution. Mryasov et al.58 For films with one and two monolayers all calculations We have recently used this approach and its generaliza- agree on a ferromagnetic ground state with strongly en- tion to biquadratic exchange coupling to calculate the ex- hanced moments. For a three-layer film we find three meta- change interactions in bulk bcc ferromagnetic Fe, at Fe stable solutions: the low-spin solution with an antiferromag- surfaces21 and at the interface of ferromagnetic Fe substrates netic coupling close to the substrate we adopt the notation with antiferromagnetic Mn overlayers.22 For the present con- , starting from the free surface and proceeding towards text, the main results of these studies besides the confirma- the substrate is marginally lower in energy than the ferro- tion that good agreement with experimentally measured spin- magnetic high-spin solution . The symmetrical low-spin wave stiffness constants and related properties can be solution leads to a strongly reduced moment in the cen- achieved are i The exchange-pair interactions are long tral layer. This is a consequence of the frustration of the ranged. Quantitatively converged values for Curie tempera- interactions between the surface and subsurface layer. Such a tures, spin-wave parameters, etc., can be achieved only after solution is stabilized in an Fe-film sandwiched between thick taking the sum over 12­15 shells of neighbors. ii Like the Cu layers.49 magnetic moments, both the nearest- and next-nearest- For a four-layer film both TB-LMTO calculations with neighbor exchange interactions are strongly enhanced at the Hubbard-Stoner and LSD exchange, respectively converge surface the nearest-neighbor interaction in the 001 surface to the configuration with ferromagnetic coupling at the increases from J 16.27 meV in the bulk to Js 44.30 meV surface and antiferromagnetic coupling in the deeper layers. in the surface and Js 1 18.14 meV in the subsurface layer, However, both calculations and the KKR result disagree on the moment from b 2.21 B to s 3.04 B and the degree of quenching of the antiferromagnetic moments. s 1 2.09 B) . iii The reduced coordination numbers and the enhanced moments and exchange couplings have We have verified that both sets of calculations are well con- opposite effects on the local Curie temperature. Within a verged with respect to the local minimum, hence these re- mean-field approximation, the highest transition temperature sults indicate a very flat distribution of the total energy inde- is calculated not for the surface but for the first subsurface pendence on the magnetic moments. layer. iv These results, together with the fact that the For the 5 ML film the ground state is the low-spin con- nearest-neighbor coupling is enhanced not only in, but also figuration that was also found in the KKR calcula- perpendicular to the surface layer indicates that in a realistic tions, but according to Lorenz and Hafner24,47 a high-spin surface the decoupling of surface and bulk-phase transitions solution with only a single antiferromagnetic layer is might be more difficult to achieve than in a model where almost equal in energy. For the 6 ML film all calculations only the in-plane coupling is enhanced. agree on a ground-state configuration with antifer- romagnetically coupled double layers, enhanced moments in B. Coexisting high- and low-spin solutions the outer layers and quenched moments in the interior of the film. However, here again slight differences in the setup Table I summarizes our results for the layer-resolved scalar-relativistic fully relativistic, LSD-exchange magnetic moments in the Fe/Cu 100 films, together with the Hubbard-Stoner exchange result in relatively large differ- earlier k -space LMTO and r -space TB-LMTO results of Lo- ences in the magnitude of the quenched moments. 56 SURFACE AND INTERFACE PHASE TRANSITIONS IN . . . 2649 TABLE I. Average magnetic moment m and layer-resolved C. Effective exchange coupling magnetic moments mi in B) Table II summarizes our results for the nearest- and next- a LMTO calculation after Ref. 24 nearest exchange interactions in the Fe/Cu 100 films. The l 1 2 3 4 51 52 6 exchange coupling has been calculated for the ground state. As in bulk Fe and at Fe surfaces, the exchange interactions m 2.72 2.70 2.65 1.40 1.59 0.56 0.87 m are quite long ranged, but for clarity we report only the 1 2.72 2.86 2.87 2.86 2.92 2.79 2.81 m2 2.53 2.50 2.40 2.49 2.24 2.28 dominant short-range interactions where the enhancement m3 2.58 2.01 2.42 1.68 2.38 due to the reduced coordination close to the surface or inter- m4 2.35 2.24 1.69 2.35 face is most pronounced. m5 2.35 2.26 2.32 In the monolayer limit we note a huge enhancement of the m6 2.54 nearest-neighbor NN exchange coupling: JNN 42.2 meV, compared to JNN 16.3 meV in bulk ferromagnetic FM Fe, b TB-LMTO calculation after Ref. 24 l 1 2 3 4 5 6 comparable to JNN 44.3 meV at the surface of FM bcc Fe see Ref. 21 . Even the next nearest-neighbor NNN inter- m 2.71 2.49 2.45 1.41 0.58 0.82 action in the monolayer is stronger than the nearest-neighbor m1 2.71 2.76 2.82 2.81 2.75 2.77 coupling in the bulk. The surface-related enhancement of the m2 2.22 2.34 2.38 2.27 2.20 m NN coupling within the layers is only slightly weaker in a 2 3 2.19 1.95 1.70 2.36 m4 2.41 1.86 2.31 ML slab, but the interlayer coupling is almost the same as in m5 2.27 2.11 the bulk. At the interface with the Cu substrate we note a m6 2.51 relatively weak NNN coupling. The effect of competing ferro- and antiferromagnetic po- c KKR calculation ­ ferromagnetic solution after Ref. 48 larizations appears first in the 3 ML film. The intralayer in- l 1 2 3 4 5 6 teractions in the surface and interface layers are both en- m 2.78 2.69 2.62 2.56 2.54 2.53 hanced over the bulk values, more strongly at the free m1 2.78 2.79 2.82 2.83 2.84 2.85 surface than at the interface. The asymmetry can be attrib- m2 2.59 2.49 2.47 2.49 2.50 uted mainly to the different behavior of the s,p electrons at m3 2.56 2.38 2.39 2.42 the surface and at the interface. For the bulk our analysis21 m4 2.54 2.43 2.41 has shown that the strong FM d-d coupling is partly reduced m5 2.55 2.45 by negative s,p-d contributions. At the surface, the s,p elec- m6 2.56 trons partly relax into the vacuum and the enhancement of the d-d coupling due to a narrower d band is fully effective. d KKR calculation ­ antiferromagnetic solution after Ref. 49 No s,p relaxation occurs at the interface and the Fe-d ­Cu- l 1 2 3 4 5 6 d hybridization limits the narrowing of the Fe-d band. m 2.78 0.01 1.00 1.47 -0.55 0.92 Within the central Fe layer the exchange coupling is compa- m1 2.78 2.36 2.82 2.79 -2.79 2.79 rable to that in the bcc Fe bulk the different atomic structure m2 2.34 2.30 2.26 -2.25 2.16 of the film seems to be of minor importance . The most m3 2.12 1.43 1.51 2.02 m interesting effect, however, is the competition between the 4 2.26 1.45 2.03 m5 2.22 2.09 strong AFM-NNN coupling between the surface and inter- m6 2.52 face layer (JNNN S (S 2) 22.0 meV and the FM-NN cou- pling between the central and the interface layer e TB-LMTO calculation present results , the index 1 indicates the (JNN (S 1) (S 2) 10.8 meV . While the former interaction fa- ground state vors the low-spin configuration, the latter favors the l 1 2 31 32 33 4 5 6 high-spin FM configuration. In either configuration, one m 2.76 2.60 0.93 1.13 2.44 1.37 0.49 0.92 of these interactions is necessarily frustrated. m1 2.76 2.77 2.80 2.50 2.81 2.81 2.79 2.74 In the 4 ML film the intralayer couplings in the surface m2 2.43 2.03 -1.22 2.09 1.92 1.85 1.78 and interface layers are about the same as in the 3 ML film, m3 2.05 2.10 2.42 1.36 1.40 1.64 m the coupling in the subsurface layer is bulklike. The most 4 2.10 1.30 1.68 m striking effect is the extremely weak exchange coupling 5 2.08 1.85 m6 2.33 within the antiferromagnetically polarized third layer. The antiferromagnetic moments are stabilized by a strong AFM- NNN coupling to the surface-layer (JNNN S (S 2) 12.5 meV , f MC simulation present results , the index 1 indicates the ground and a weaker AFM-NN coupling to the interface layer. In the state configuration the FM-NN coupling between the sec- l 1 2 3 4 51 52 61 62 ond and third layers is necessarily frustrated. m1 A similar situation is established for the 5 ML film in the m2 ground state: enhanced coupling within surface and m3 interface layers, bulklike interactions in the subsurface layer, m4 very weak exchange interactions in the third and fourth lay- m5 m ers in the interior of the film. This concerns not only the 6 intralayer coupling, but the interlayer coupling as well. The 2650 D. SPIS AŽK AND J. HAFNER 56 TABLE II. Effective exchange parameters JNN(l), JNN(l 1) and JNNN(l), JNNN(l 1) in meV . NN stands for nearest-neighbor, NNN for next-nearest-neighbor coupling, l for coupling within the same layer, l 1 for coupling to an atom in the neighboring layer. No. of Layers Layer Orientation JNN(l) JNNN(l) JNN(l 1) JNNN(l 2) 1 1 42.20 19.19 1 37.67 14.11 18.26 2 2 37.65 9.15 1 29.02 9.86 19.37 22.04 3 2 15.80 7.77 10.79 3 20.66 9.62 1 31.26 8.59 13.61 12.54 2 11.76 5.94 2.62 7.25 4 3 0.22 3.59 4.51 4 19.71 10.08 1 29.16 7.58 12.56 12.88 2 11.64 5.48 3.92 2.05 5 3 2.12 3.91 4.26 6.11 4 1.75 3.00 3.32 5 19.36 10.86 1 23.63 6.99 12.45 14.64 2 12.91 7.16 9.40 5.28 3 0.28 7.75 5.66 7.47 6 4 1.28 7.50 9.22 10.29 5 10.04 8.97 5.64 6 23.41 11.44 spin configuration is stabilized mainly by a strong ferromagnetic NN interlayer coupling indicate the possibility AFM-NNN coupling between the surface and the third layer of even more complex transitions. In the following these (JNNN S (S 2) 12.9 meV , a modest FM-NNN coupling of magnetic transitions will be studied using Monte Carlo simu- this layer to the interface layer (JNNN (S 2) (S 4) 6.1 meV , as lations. well as weaker AFM-NN interlayer couplings in the deeper However, this is not an easy task: the exchange-pair in- layers. The FM-NN coupling between layers S-1 and S-2 teractions we have calculated are, in principle, Heisenberg- is frustrated in this configuration. type and long ranged. Monte Carlo simulations for all 1­6 A different scenario appears for the AFM coupled double ML films, on ensembles that are sufficiently large to allow layers forming the ground state of the 6 ML film. In for a characterization of the magnetic phase transitions, this almost symmetric configuration we find enhanced intra- would hence be prohibitively expensive. We therefore de- layer interactions in surface and interface layers, bulklike cided to simplify the task by a restricting the interactions to interactions in the adjacent layers, and quenched interactions nearest- and next-nearest-neighbor sites as given in Table in the paired central layers but note the relatively strong II , and b performing the simulations for an Ising instead of FM-NNN intralayer coupling . All NN-interlayer interac- a Heisenberg model. This leads to a tractable model. The tions are ferromagnetic, all NNN-interlayer interactions are cutoff operated on the range of these interactions could, in antiferromagnetic and stronger than the FM-NN interactions principle, be compensated in part by a renormalization of the if a surface or interface layer is involved. This leads naturally short-range interactions such that the critical temperatures to the double-layer configuration where only every are correctly described. Here we do not proceed to a renor- fourth NN or NNN interlayer interaction is frustrated. malization, but the necessity should be kept in mind before Altogether our results demonstrate that due to the exis- making any comparisons with experiment. To use an Ising tence of frustrated ferro- and antiferromagnetic exchange in- instead of a Heisenberg model appears to be a more serious teractions in fcc Fe, the magnetic interactions in fcc limitation, especially as the exchange interactions have been Fe n/Cu 100 films are much more complex than assumed in calculated for an infinitesimal rotation of the moments. How- the simple scenario underlying previous studies of magnetic ever, previous Ising as well as Heisenberg MC simulations surface phase transitions. Strongly enhanced interactions with a fixed set of exchange interactions for an AFM Mn ML near the surface or interface, quenched coupling in the inte- on a Fe 100 substrate22,62 show that both simulations lead to rior of the films suggest the occurrence of surface and even- equivalent scenarios for the magnetic transition, although the tually also interface phase transitions. Antiferromagnetic critical temperature scales by about a factor of 2. Hence NNN interlayer interactions that are even stronger than the Ising-MC simulations should be sufficient to characterize the 56 SURFACE AND INTERFACE PHASE TRANSITIONS IN . . . 2651 FIG. 1. Variation of the internal energy E , the average magnetization per atom m , the specific heat C, the susceptibility , and the fourth-order cumulant UL with temperature, calculated for a monolayer of Fe on top of a nonmagnetic Cu substrate. The size of the model has been varied from 8 8 to 64 64. possible phase transitions in a qualitative manner, but the III. ISING MONTE CARLO STUDIES results should be taken as a representative model of magnetic OF PHASE TRANSITIONS IN THIN FILMS films with frustrated exchange interactions, and not immedi- A. Background ately for Fe films on Cu substrates. Our simulations are based on a three-dimensional Ising Hamiltonian with the moments listed in Table I and the NN and NNN exchange couplings listed in Table II. The simu- lations were carried out for (L L) cells in the film plane (L 8,16,32,64) and 1­6 layers thick. Periodic boundary conditions were applied parallel to the surfaces, free bound- ary conditions to both top and bottom surfaces. A standard single spin-flip Monte Carlo method with a Metropolis algo- rithm for the flipping probability63 was used. Data sampling was extended over 8 104 (L 8) up to 12 105 (L 64) MC steps. For each film we report the variation of the average en- ergy E , the average magnetic moment m , the specific heat C, and of the susceptibility with temperature. The specific heat and susceptibility were calculated according to the fluctuation-dissipation theorem as FIG. 2. Variation of the critical temperature Tc with the nmber E2 E 2 C of monolayers in the Fe film, calculated using Monte Carlo simu- kBT2 2 lations and truncated nearest- and next-nearest-neighbor interac- tions as given in Table II , and calculated using mean-field theory and long-range exchange coupling. Cf. text. and 2652 D. SPIS AŽK AND J. HAFNER 56 FIG. 3. Summary of the MC simulations for the 3 ML-Fe/Cu 100 film, cf. Fig. 1. m2 m 2 tions, decoupled from the global magnetic properties of the k film, can occur is one of the central objectives of this study. BT . 3 Another important quantity for determining the transition B. Results temperature is the fourth-order cumulant UL defined as64 1. Mono- and bilayer films m4 The Fe monolayer is expected to behave as a classical U L L 1 3 m2 2 . 4 two-dimensional 2D Ising ferromagnet, and this is con- L firmed by our results compiled in Fig. 1. From the cumulant analysis we determine a critical temperature A convenient method for locating the phase transition is T to follow the variation of U c(1ML) 1784 K, the analysis of the magnetization in the L with temperature for various vicinity of the critical point leads to a critical exponent in L and to look where these curves intersect. good agreement with the 2D Ising exponent 1/8. Essen- In addition to the quantities characterizing the global tially identical results are obtained for a 2 ML film. The properties of the films, we monitor also the layer-resolved critical temperature is even higher, T average moments m c(2ML) 2214 K. i and susceptibilities i again as de- Again this is as expected, because the intralayer coupling is termined from the fluctuation of the moments in the respec- almost the same in both layers, and the interlayer coupling is tive layer. The layer moments and susceptibilities display reduced only by a factor of 2 compare Table II . The high critical behavior with characteristic exponents and ampli- values of the critical temperatures are due to the cutoff of the tudes, exchange interactions. If we calculate Tc from the total long- range exchange interactions according to mean-field theory, m i Biti 5 1 1 i Cit T J J i , 6 c 3k i i j , 7 B i 3kB i,j j i where t 1 T/Tci . The local critical temperature Tci agrees with the critical temperature Tcf of the entire film we obtain critical temperatures that are lower by more than a only if the coupling between the layers is comparable to the factor of 2. Figure 2 shows that this scaling factor holds not intralayer coupling. Whether two-dimensional phase transi- only for the mono- and bilayer case, but for all film thick- 56 SURFACE AND INTERFACE PHASE TRANSITIONS IN . . . 2653 increase and the antiferromagnetic orientation is induced by the strong AFM coupling between surface and interface lay- ers, the approach to saturation is slowed down because of the frustrated FM coupling between interface and subsurface layer see Table II . Hence a true phase transition occurs only in the two top layers, whereas the frustration suppresses a phase transition in the interface layer. 3. 4 ML films The scenario for the magnetic ordering in the 4 ML films is even more complex: from the cumulant analysis see Fig. 5 we determine a critical temperature of Tc surface 4 ML 1441 K, and specific heat and susceptibility show critical behavior at this temperature. The average magnetization, however, approaches saturation only very slowly, and the specific heat shows two side maxima at lower temperatures. From the analysis of the layer-resolved moments and suscep- tibilities Fig. 6 we learn that again the critical point corre- sponds to the PM-FM transition in the strongly coupled sur- face and subsurface layers. The interface layer orders ferromagnetically at a lower temperature of Tc(interface) 1200 K estimated from the peak in the layer susceptibility 4. At this point it is necessary to point out that the total susceptibility is not simply the sum of the layer susceptibilities. Interlayer correlations here mainly be- tween surface and subsurface layer can make quite impor- tant contributions. The ratio of the two transition tempera- tures corresponds roughly to the strengths in the exchange coupling in the surface bilayer and in the interface layer. The PM-FM transition in the interface layer is somewhat slug- gish. The approach to the saturation of the magnetic mo- ments in the interface layer does not represent a genuine FIG. 4. Layer-resolved average moments mi a and suscep- tibilities two-dimensional PM-FM phase transition. This is indicated i b in a 3 ML-Fe/Cu 100 film (32 32 3 lattice . The layer moments m by the absence of a corresponding peak in the specific heat i have been multiplied with the sign corre- sponding to the global magnetization in each layer. and the smearing of the peak in the suceptibility. The reason is that the coupling to the surface bilayer across the mag- nesses covered in our study. Hence, in principle, the interac- netically essentially ``dead'' interior of the film ordered al- tions could be renormalized by this factor if one wants to ready at higher temperatures creates a magnetic field acting proceed to a more direct comparison with experimental re- on the interface moments. It is well known that an Ising sults. Note that the trend reflects the initial increase of T ferromagnet in a magnetic field does not show critical behav- c up to two layers, and the decrease for thicker layers observed in ior, because there is a nonzero magnetization above Fe/Cu 100 . However, for the correct interpretation of T T c , c(interface). A magnetic moment in the S-2 layer devel- see below. ops only slowly because of competing couplings to the neighboring layers. Note also that the MC simulation con- 2. 3 ML films verges to a ferromagnetic configuration and not to the configuration determined as the ground state in the In the three-monolayer films with the ground-state LSD calculations. Altogether our results raise interesting configuration, we observe a more complex behavior: at the questions concerning the character of the phase transitions in critical temperature of Tc(3ML) 1680 K, the magnetization two weakly coupled two-dimensional ferromagnets which increases first rapidly but goes through a maximum about certainly deserve further investigations. 250 K below the critical point and saturates at a lower level. Specific heat and susceptibility show that characteristic criti- cal divergences see Fig. 3 occur at T 4. 5 ML films c , the specific heat shows in addition a broad shoulder at the low-T side of the Fluctuation effects are found to have a large effect on the critical peak. The analysis of the layer-resolved moments and ordering transitions in the 5 ML films. Simulations for a susceptibilities Fig. 4 demonstrates that the critical point (8 8 5) ensemble show an onset of a magnetic ordering corresponds to the paramagnetic PM to ferromagnetic tran- transition at Tc 1350 K, but then strong fluctuations of the sition in the ferromagnetically coupled surface and subsur- total magnetization in the temperature range between 750 face layers. The magnetic moments of the interface layer and 500 K before the moments converge to a high-spin so- show no critical behavior, but only a linear increase below lution where all layers are ferromagnetically aligned Fig. the critical temperature of the two overlayers. This linear 7 a . Simulations performed on a (16 16 5) ensemble 2654 D. SPIS AŽK AND J. HAFNER 56 FIG. 5. Summary of the MC simulations for the 4 ML-Fe/Cu 100 film, cf. Fig. 1 and text. follow first the same pattern, but at the temperature where high-spin state. We find that, as in the 4 ML film, two dis- the fluctuations in the smaller ensemble begin, the average tinct PM-FM transitions occur: first at Tc(surface) 1350 K magnetization breaks down and the simulation converges to in the surface bilayer, than at Tc(interface) 900 K in the a configuration with almost zero total moment. interface layer. The second transition is now much sharper Hence the breakdown of the total magnetization is the signa- than in the 4 ML film because the two ``magnetically dead'' ture of a spin-reorientation transition in part of the film. layers in the interior of the film decouple the surface and Simulations for a still larger (32 32 5) ensemble con- interface more effectively. The moments in the third and verge to the ferromagnetic high-spin solution, but the varia- fourth layers do not show critical behavior, they are induced tions of the shape of the magnetization vs temperature curve, by the weak coupling to the magnetic layers. the specific heat, and the susceptibility indicate that there are For the PM-FM surface phase transition the susceptibility at least two, if not three distinct transitions. The ferromag- shows a size-dependent singularity coincident with the sin- netic and the layered antiferromagnetic con- gularity in the specific heat see Figs. 7 a,b . For the figurations differ in energy by only about 1 meV/atom with PM-FM interface phase transition the singularities in the sus- the low-spin solution being slightly lower in energy , but ceptibility and in the specific heat coincide only for those none of the two agrees with one of the stable or metastable runs for the (32 32 5) ensemble bypassing the spin- and configurations found in the LSD calcu- reorientation transition see Fig. 7 a . If a spin reorientation lations cf. Table I . takes place, it is signaled by a dominant peak in the total The system can also be driven reversibly through the re- susceptibility at temperatures that are lower than the critical orientation transition see Fig. 7 b , an analysis of the spe- temperatures T cific heat, susceptibility, and the cumulants indicates a se- c(interface) 900 K for the interface phase transition T 750 K for the (16 16 5) ensemble . In this quence of three transitions ( three peaks in the specified case the broad susceptibility peak of the reorientation transi- heat . At the two high-temperature transitions, the singularity tion covers the singularity associated with the PM-FM inter- in the specific heat shows the dependence on the size of the face transition. This singularity is, however, clearly resolved MC ensemble expected for a second-order phase transition, in the layer-decomposed susceptibilities see Fig. 8 c where whereas the low-temperature transition is almost size inde- the reorientation transition is not manifest because it in- pendent and hence does not correspond to a genuine phase volves mainly a change in the interlayer coupling. Note that transition. Figure 8 a shows the temperature dependence of the reorientation transition is not signaled by a singularity in the layer-resolved magnetic moments for the transition to the the specific heat. 56 SURFACE AND INTERFACE PHASE TRANSITIONS IN . . . 2655 ture Tc 6 ML 1315 K estimated from the cumulant analy- sis and converges to a low-spin configuration with an average moment of only 0.06 B/atom Fig. 9 . This con- figuration is about 4 meV/atom higher in energy than the symmetric high-spin ground-state with an average moment of 0.92 B/atom resulting from the LSD calculations cf. Table I . On heating the stable ground-state configura- tion, the lower half of the film undergoes a spin-reorientation transition at about 1000 K to the low-spin configu- ration see Fig. 10 a and goes reversibly through the maxi- mum in the magnetization before the film becomes paramag- netic. The origin of the magnetization maximum becomes clear when we analyze the layer-resolved magnetic moments mi and susceptibilities i see Figs. 10 a ­10 c ; on cool- ing, a ferromagnetic ordering transition occurs first in the surface bilayer at Tc surface 1315 K, while the bilayer close to the interface orders only at Tc interface 1220 K in an orientation aligned antiferromagnetically relative to the surface layer, resulting in a decrease of the total moment note that the specific heat and total susceptibility show only a single broad peak . The small difference in the critical temperatures scales rather well with the exchange coupling within the bilayers see Table II . The magnetic moments in the two central layers do not show critical behavior, but only a slow, almost linear increase. At higher temperature where the average moments in the central layers are still rather small the low-spin configuration with an antiparallel orientation of the ferromagnetically or- dered surface and interface bilayers is apparently entropy stabilized, because it allows for a wider range of spin fluc- tuations in the interior of the film. As the temperature is lowered, the high-spin configuration becames energetically FIG. 6. Layer-resolved average moments mi a and suscep- favored, but the reorientation of the spin in the entire lower tibilities i b for a 4 ML-Fe/Cu 100 film (32 32 4 lattice . Cf. half of the film would be possible only by overcoming a text. substantial barrier. Hence the metastable low-spin configura- tion is quenched. The transition to the low-spin state occurs through a sud- den flipping of the moments in the fifth to third layers after reaching a value close to saturation see Fig. 8 b . Such a IV. DISCUSSION AND CONCLUSIONS spin flip would be favored by the AFM-NNN coupling of the surface to the central layer and the FM-NNN coupling of the In the first part of this work we have presented real-space central layer, while frustrating the weaker NN interlayer cou- TB-LMTO calculations of the magnetic exchange-pair inter- pling cf. Table II . These results suggest that the reorienta- actions for fcc Fe films on Cu 100 substrates. Our calcula- tion transition is not a phase transition in the thermodynamic tions show that competition between ferro- and antiferro- sense, but merely a transition between two different meta- magnetic exchange interactions characteristic for the fcc stable relative orientations of two ferromagnetic layers phase of iron leads to the existence of a variety of metastable coupled via weak magnetic fluctuation in the interlayer. low- and high-spin configurations in films with more than 2 The layer-resolved susceptibilities Fig. 8 c show that ML, confirming earlier results24,47­49 obtained using different the surface and interface phase transitions are neatly decou- techniques. The calculations of the exchange-pair interac- pled. The critical exponents of the magnetization close to the tions using a torque-force method leads to rather surprising critical points are analyzed in Fig. 8 d . While for the inter- results: a The exchange coupling is strongly enhanced in face phase transition the critical exponent ( 0.150) is the boundary layers of the film, both at the free surface and close to the value expected for a 2D-Ising critical point, the at the interface with the nonmagnetic substrate. The enhance- effective critical exponents determined for the surface and ment decreases with increasing thickness of the films. b subsurface layers are distinctly larger. This indicates that in The exchange interactions are drastically reduced in the in- the surface bilayer the coupling is no longer strictly two- terior of the films where the moments in neighboring layers dimensional. are antiferromagnetically aligned. The reduction is strongest for the nearest-neighbor intralayer coupling which almost vanishes in films with 4­6 ML, while the next-nearest- 5. 6 ML films neighbor intralayer and the interlayer coupling remain rela- On cooling, the magnetization of the 6 ML-Fe/Cu 100 tively strong. However, in all cases at least some of the in- goes through a maximum slightly below the critical tempera- terlayer couplings are frustrated - the ground-state 2656 D. SPIS AŽK AND J. HAFNER 56 FIG. 7. a Temperature dependence of the internal energy E, magnetization M, specific heat C, and susceptibility as recorded in three MC cooling runs for different ensembles representing a 5 ML-Fe/Cu 100 film. b Average energy, magnetic moment, specific heat, susceptibility, and cumulant for a (16 16 5) ensemble going reversibly through the spin-reorientation transition. Cf. text. 56 SURFACE AND INTERFACE PHASE TRANSITIONS IN . . . 2657 FIG. 8. a Temperature dependence of the layer-resolved average moments mi for the high-spin PM-FM transition in the 5 ML-Fe/ Cu 100 film on cooling. b Same for the high-spin­low-spin reorientation transition on heating the configuration. Note that here we plot mi and not mi -in this case the transition is more abrupt. c Temperature dependence of the layer-resolved susceptibilities i for the high-spin transition in the 5 ML-Fe/Cu 100 film. Cf. text. d Variation of the magnetic moments in the surface and subsurface layer (m1 ,m2) and in the interface layer (m5) as a function of the reduced temperature 1 T/Tc . The straight lines show the linear interpolations used to estimate the critical exponents. configuration is just the one with the minimal frustrations. next-nearest interactions taken from the TB-LMTO calcula- c Long-range interactions exist at about the same level as in tions for Fe/Cu 100 films. In principle, the effects of the the bulk and close to the surface of a semiinfinite crystal. truncation could be reduced by a renormalization of the Hence the exchange interactions in the films are considerably strength of the interactions. However, this would essentially more complex than a simple picture based on a surface- reduce the critical temperatures without changing the sce- induced enhancement would suggest. nario of the phase transition. We also have to emphasize that The second part of this study was devoted to a Monte the exchange coupling has been determined from infinitesi- Carlo study of magnetic phase transitions in films with frus- mal rotation of the moments from a ground-state configura- trated exchange interactions. However, because of the high tion and depend on that configuration. Hence the results of computational effort that simulations with long-range the simulations should be taken as representative for the tran- Heisenberg-type interactions would require, the simulations sitions in a model with frustrated interactions and not be were performed for an Ising Hamiltonian with nearest- and considered as quantitative predictions for Fe/Cu 100 . 2658 D. SPIS AŽK AND J. HAFNER 56 FIG. 9. Variation of the average energy E , the average magnetic moment m , specific heat C, susceptibility , and fourth-order cumulant UL in a 6 ML-Fe/Cu 100 film. Open symbols: cooling from high temperature, ending in a metastable low-spin configuration. Full symbols: heating from the high-spin ground-state-a spin-reorientation transition in one-half of the film occurs close to T 1000 K. Cf. text. The MC simulations reveal a very complex scenario for two-dimensional ferromagnets that are weakly coupled via the magnetic phase transitions: a films with one or two the fluctuating moments in the interlayers, and not a genuine monolayers show a simple paramagnetic-ferromagnetic tran- phase transition. e Surface and interface phase transitions, sition. b Films with three monolayers show a followed by a fluctuation-induced bifurcation towards high- paramagnetic-ferromagnetic transition in the surface bilayer, and low-spin solutions have been found also in the 6 ML but the antiferromagnetic moment in the interface layer in- films. creases only slowly below Tc - this leads to a maximum in Our MC simulations have been performed with an Ising the total magnetization just below Tc . c In a four- Hamiltonian. Hence they cannot give immediate information monolayer film we find a surface phase transition in the on the magnetic anisotropy of the films and its possible strongly coupled surface bilayer, followed by a somewhat variation with temperature. However, the low-spin/high-spin sluggish transition in the interface layer. The moment in the reorientation transitions observed in the 5 and 6 ML films are interior of the film increases only very slowly. Altogether clearly related to the in-plane/perpendicular reorientation this leads to a slow increase of the total moment, as observed transitions observed in fcc Fe/Cu 100 films with 5­6 ML experimentally in Fe/Cu 100 films of similar thickness.15 d Ref. 27 : The change of the relative orientations of the fer- A similar scenario with two distinct surface and interface romagnetic polarization of surface and interface layers influ- phase transitions is also found in 5 ML films. Even after the ences both the spin-orbit and dipolar contributions to the second phase transition the interior of the film remains es- anisotropy energies see Ref. 24 , but it is not a priori clear sentially nonmagnetic. Once the developing magnetic mo- in which direction note that both low- and high-moment ments in the interior of the films lead to a coupling of surface solutions can appear as low-temperature phases . This point bilayer and interface layer, the film can either develop a fer- deserves further investigation. Altogether our results show romagnetic high-spin state, or fluctuation effects can lead to that very large changes in the exchange interactions relative a spin-reorientation transition of the lower part of the film to to their values in the bulk can be found in films where a an energetically almost degenerate antiferromagnetic low- strong ferromagnetic coupling at the surfaces competes with spin state. Note that the reorientation transition describes a antiferromagnetic interactions in the interior of the films. transition between two different relative orientations of two Such a situation leads to a strong frustration of the pair in- 56 SURFACE AND INTERFACE PHASE TRANSITIONS IN . . . 2659 FIG. 10. a Variation of the layer-resolved moments mi through the spin-reorientation transition on heating. b,c Variation of the layer-resolved moments mi and mi during a cooling run. The comparison of these two diagrams shows that the ferromagnetic ordering in the surface bilayer occurs first in small domains with opposite local moments. After the interface bilayer orders ferromagnetically, the fluctuations are effectively suppressed. The critical temperatures for surface and interface phase transitions estimated from the layer-resolved susceptibilities are marked Tc(s) and Tc(i). d Layer-resolved susceptibilities i , cf. text. teractions in the interior of the magnetic film, resulting in the ACKNOWLEDGMENTS existence of energetically nearly degenerate metastable mag- netic configurations and to a decoupling of the magnetic or- This work has been supported by the Austrian Ministery dering transitions at the surface and at the interface with the for Science, Research and Art through the Center for Com- substrate from the phase transitions in the inner layers. Our putational Materials Science within the project ``Magnetism MC simulations show a very rich scenario of surface, inter- on the Nanometer-Scale'' GZ 45.378/2-IV/6/94 . D.S. face, and spin-reorientation transitions driven by the frus- gratefully acknowledges the Department of Experimental trated exchange coupling. 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