PHYSICAL REVIEW B VOLUME 56, NUMBER 10 1 SEPTEMBER 1997-II Interfacial alloying and interfacial coupling in Cr/Fe 001... M. Freyss, D. Stoeffler, and H. Dreysse´ Institut de Physique et de Chimie des Mate´riaux de Strasbourg, 23, rue du Loess, 67037 Strasbourg Cedex, France Received 27 March 1997; revised manuscript received 22 May 1997 The magnetic order of Cr layers on Fe 001 is studied taking into account interfacial alloying and possible interdiffusion. The interfacial alloy is modeled by either a one-monolayer or a two-monolayer ordered com- pound whose concentration is varied. The spin-polarized electronic structure is determined self-consistently by solving a d-band tight-binding Hamiltonian. We determine the concentration for which the phase of the layer-by-layer antiferromagnetic structure of Cr changes in the Cr film on Fe 001 . We find that in the case of two interfacial mixed layers, a phase shift occurs at a Cr concentration between 33% and 50% when three monolayers ML of Cr are deposited. The occurrence of this phase shift changes to a concentration between 11% and 25% for a more important coverage of Cr namely 11 ML . When only one mixed layer is considered, the phase of the antiferromagnetic stacking of Cr changes at a concentration between 25% and 33% for 3 ML of Cr and between 33% and 50% for 11 ML of Cr. A simulation of the variation of the magnetization during Cr growth shows that the more Cr and Fe are interdiffused at the interface, the more important is the decrease of the magnetization. We compare our results to the many experimental data available. S0163-1829 97 10234-X I. INTRODUCTION More recently, other evidence for alloying at the Cr/ Fe 001 interface was found by Davies et al.9 and Pfand- The exchange coupling between Fe layers through a Cr zelter et al.10 By means of scanning electron microscopy spacer has been widely studied since the discovery of its STM , Davies et al. measured the concentration of Cr on the oscillating behavior as a function of the Cr thickness. Even if surface layer during its growth. They showed that after the the origin of the oscillation is now well understood, and ex- deposition of 1 ML of Cr, there only remains 10% of Cr on plained in connection with the Fermi surface topology,1,2 the surface layer, indicating that most of the Cr atoms have some discrepancies remain between experimental observa- penetrated into the Fe substrate. Furthermore, it is only after tions and theoretical results. For instance, even though the the deposition of 2­3 ML of Cr that the surface layer con- two-monolayer ML period of the oscillation observed tains more Cr than Fe. The same trend was found by Pfand- experimentally3,4 is in accordance with numerous calcula- zelter et al.10 by means of proton- and electron-induced tions, its phase is exactly opposite to that theoretically Auger-electron spectroscopy. The Cr concentration profile predicted:5,6 for an odd number of Cr layers, an antiferro- they measured is, from the surface layer to the bulk: 45%, magnetic AFM coupling is observed whereas calculations 55%, and 0% for 1 ML of Cr deposited, and 70%, 100%, and predict a ferromagnetic FM coupling, and inversely for an 30% for 2 ML of Cr deposited. These results are different even number of Cr layers. The same phase shift is observed from those of Venus et al.7 who show a gradual decrease of by Unguris et al.3 in the layered antiferromagnetic structure the Cr concentration from the surface to the bulk. of Cr on Fe 001 : due to the strong antiferromagnetic inter- More quantitative studies were made by Bayreuther and facial coupling between Fe and Cr and the layered antiferro- co-workers: they measured with an alternating gradient mag- magnetic structure of Cr, one expects for an even resp. odd netometer AGM the variation of the total magnetization number of Cr layers deposited on Fe 001 the magnetic mo- during the growth of Cr on Fe 001 substrates with different ments of the Cr overlayer to be positive resp. negative . The degrees of roughness. With a flat Fe surface, they observed a experimental results show just the opposite. This discrepancy strong decrease in the magnetization of approximately 5 B can be accounted for by the abrupt interface considered in per interface atom.11 However, such a big decrease has not theoretical studies. In experiments, such perfect interfaces do been reproduced until now. The more important the rough- not exist due to roughness or interdiffusion. In particular, ness is, the smaller the decrease in the magnetization is:12,13 Venus and Heinrich7 have recently shown by angular re- on a highly stepped Fe 001 substrate, no variation at all of solved Auger electron studies ARAES the existence of a the total magnetization occurs during the Cr growth. strong interdiffusion of Cr atoms at the Cr/Fe 001 interface: In order to understand the experimental results, we study at a growth temperature of 246 °C the Cr atoms penetrate the magnetism of thin Cr layers deposited on a semi-infinite into the second atomic layer counting from the surface and Fe 001 substrate with either a two-monolayer or a one- the concentration of Cr is found to decrease from the surface monolayer ordered alloy at the interface to simulate interdif- to the substrate. On the other hand, no interdiffusion is pre- fusion. Interface alloying is expected to have an important dicted at the Fe/Cr 001 interface.8 Heinrich et al. thus sug- influence on the magnetic structure of the deposited film, as gest that the phase shift observed in the oscillating exchange has already been shown theoretically for Rh and Ru on coupling in Fe/Cr/Fe 001 systems is the result of interface Ag 001 by Turek et al.14 In their study, they considered a alloying at the Cr/Fe 001 interface. completly disordered alloy. Here, we calculate the magnetic 0163-1829/97/56 10 /6047 9 /$10.00 56 6047 © 1997 The American Physical Society 6048 M. FREYSS, D. STOEFFLER, AND H. DREYSSE´ 56 such an interface could reproduce very well the strong de- crease observed by Turtur and Bayreuther.11 Here, we con- sider a more general situation by introducing mixed Fe-Cr layers at the interface with different concentration and per- forming a systematical study. We also make a simulation of the variation of the total magnetization during Cr growth for various concentrations in case I. In the following of the paper, we present in Sec. II the formalism used to compute the magnetic order. Then, in Sec. III we report the results obtained with two interfacial mixed layers case I and with one interfacial mixed layer case II . II. FORMALISM The magnetic moments are calculated self-consistently by solving a d-band tight-binding Hamiltonian, and by means of the real-space recursive method. Only collinear magnetic or- ders are assumed. The advantage of the recursive method is that it does not require a symmetry in the direction perpen- dicular to the layers and enables thus to study systems with a low degree of symmetry. In the basis of atomic orbitals im (i stands for the atomic site, m for the orbital symmetry, and 1 for the spin , the Hamiltonian is given by FIG. 1. Unit cells within the first interface mixed layer counting from the Fe substrate . The white circles represent Cr atoms, the black ones Fe atoms. The unit cells of the second mixed layer is H im i m o ii mm i m ii obtained by exchanging Fe and Cr atoms. mm 1 structure as a function of the concentration of an interfacial tim i 1 ii ordered alloy in the following situations: m im 2 IiMi im im . 1 o Cr The first term of H is the band term. i m is the center of the n /Cr1 xFex /CrxFe1 x /Fe 001 case I , m band on the site i and tim i m the intersite spin-independent Cr hopping integrals. The second term of H is the exchange n /CrxFe1 x /Fe 001 case II , term which accounts for magnetism. Ii is the effective ex- change integral and M with x 0, 1 i the local magnetic moment on the 9 , 14 , 13 , 12 , 23 , 34 , 89 ,1. site i. The parameters tim i m and Ii were chosen to reproduce The coverage of Cr is varied from n 0 to 2 ML. In some ab initio calculations with the FLAPW full potential aug- cases, the calculation is made up to n 10 ML. The symme- mented plane waves method. They have already been used try of the unit cell within the mixed interface layers is shown satisfactorily in previous studies.5,16 on Fig. 1 for the different concentrations considered. This Self-consistency is obtained by requiring that the band unit cell is repeated by translation within the layer. occupation on each site is equal to the bulk value Noi local The aim of our study is to discuss the magnetic structure neutrality approximation , and that the exchange splitting of the Cr film as a function of the alloy concentration x, and i i is equal to IiM i : to determine for which concentration the experimentally ob- served phase in the layer-by-layer AFM structure of Cr can N o i Ni Ni , 2 be obtained. The case of two interfacial mixed layers case I corresponds to an integer number of Cr layers deposited, i i IiM i Ii Ni Ni . 3 namely (n 1), and can thus easily be compared to experi- mental results. The situation is different with one interfacial The energy levels i and i are thus determined so that mixed layer case II : as x increases, the quantity of Cr de- both equations are verified: Equations 2 and 3 are solved posited varies from n monolayers for x 0) up to (n 1) for a given site i, keeping the energy levels of all other sites monolayers for x 1). fixed, and this process is repeated for all sites until the self- D. Stoeffler et al.15 already have considered the border- consistency relations are satisfied. The band occupation is line case x 1 in situation I. Such a case corresponds to the obtained by integration of the density of states up to the exchange of a complete monolayer of Cr and Fe at the inter- Fermi level. The density of states is calculated by means of face. It was shown that such an exchange induces a phase the real-space recursive method with eight levels of the con- shift in the AFM stacking of Cr: For an even number of Cr tinuous fraction and using the Beer-Pettifor terminator. layers deposited the surface layer displays positive magnetic The calculation of the magnetic moments is self- moments, as experimentally observed. The simulation of the consistently made on all inequivalent atoms of the Cr film, variation of the total magnetization during Cr growth with on all inequivalent atoms of the mixed layers, and on the Fe 56 INTERFACIAL ALLOYING AND INTERFACIAL . . . 6049 atoms on 10 layers below the interface. The moments of the In the two metastable solutions, the Cr overlayer displays a rest of the semi-infinite Fe substrate are frozen to the bulk c(2 2) magnetic order, which induces frustations with the value. Fe atoms of the layer right below. Again, the origin of the The results are then discussed in term of the relative en- unstability of these solutions can be attributed to the local- ergy between the different magnetic solutions obtained for a ization at the surface of the frustrations. given atomic configuration. The absence of repulsive term in With two pure Cr layers n 2, Fig. 2 c , we find the two the Hamiltonian can be justified by considering it indepen- possible layer-by-layer antiferromagnetic structure of the dent of the magnetism. Its contribution to the total energy pure Cr film. In the ground state, no Cr pair is frustated as thus disappears when calculating the relative energy between the Cr atoms are antiferromagnetically coupled from one the different solutions. Energy calculations with such a plane to the other. The only frustrations are between the Fe model has already proven to be satisfactory in a previous and the Cr atoms of the two mixed layers. The surface layer study of the Fe/Cr interlayer magnetic coupling.17 displays positive magnetic moments. On the other hand, in It is also to be noted that our study will only give a quali- the metastable solution, there are more frustrations: between tative understanding of the effect of interdiffusion since the the Cr atoms of the first pure Cr layer and their Fe neighbors, and between the Cr atoms of the lower and the upper mixed interfacial alloy is approximated by an ordered compound. A layers. In this case, the surface layer displays negative mag- more precise treatment of the alloy would require the use of netic moments, meaning that the phase of the AFM stacking the coherent potential approximation CPA in the model. of the pure Cr film is opposed to that of the ground state situation. III. RESULTS Figure 3 shows the results concerning the phase of the For a given value of x, the number of magnetic configu- antiferromagnetic stacking of Cr as a function of the alloy rations obtained by our self-consistent calculation can be concentration with 3 ML and 11 ML of Cr deposited (n 2 very important when one or no pure Cr layer is considered and n 10). As previously said, the phase of the stacking can (n 0 or n 1). These different magnetic configurations are be charaterized by the sign of the magnetic moments on the obtained by choosing a different initial magnetic configura- surface layer. Accordingly, Fig. 3 shows the average value of tion in the self-consistent process. On the other hand, the the surface layer magnetic moments as a function of the in- number of configurations obtained with two pure Cr layers terfacial alloy concentration x. A clear trend can be noticed: (n 2) or more reduces to one or two, whatever the value of With 3 ML of Cr, negative surface layer moments are ob- x is. In that case, the solutions always present one of the two tained when x is smaller than 1/2. The same sign is also possible antiferromagnetic layer-by-layer structure for the obtained with a perfect interface i.e., no interdiffusion . On pure Cr film. In the following, first the case with two mixed the other hand, positive moments are obtained when x is layers is discussed, then the case with one mixed layer. larger or equal to 1/2, that is to say when there are more Cr than Fe atoms on the first mixed layer counting from the substrate , and more Fe than Cr atoms on the second mixed A. Crn/Cr1 xFex/CrxFe1 x/Fe 001... layer. This latter situation displays the phase experimentally As a example of multiple solutions obtained for a given observed in the AFM stacking of Cr. The magnetic structure value of x, Fig. 2 shows the solutions for x 1/2 and n 0, 1, of Cr on Fe 001 can thus be accounted for by a partial and 2. In the presence of the mixed layers, the antiferromag- exchange of a monolayer of Cr and Fe at the interface. We netic coupling between Cr and Fe first neighbors on the one can also see on Fig. 3 that the value of the surface layer hand, and Cr first neighbors pairs on the other hand cannot magnetization is almost constant as a function of the alloy always be satisfied and magnetic frustrations are induced. It concentration x. is probably due to such frustrations that a very large number It is to be noted that when x 1/9, that is when the alloy- of magnetic configurations can be obtained for a given value ing is very weak and gradual from the Fe substrate to the Cr of x. film, only one solution could be found. When other configu- With n 0 Fig. 2 a , in the ground state situation the Cr rations with the other phase in the AFM stacking of Cr were atoms of the lower mixed layer are frustrated with Fe and introduced as the initial configuration in the self-consistent have an almost vanishing magnitude which stabilizes this process, the phase would shift during the calculation, and the configuration relatively to the second solution obtained. In- calculation would converge to the unique solution. This ob- deed, in the second solution, the Cr atoms of the surface servation can be related to a result obtained by Stoeffler18,16 layer are frustrated with the Fe atoms of the plane below. which showed that with a perfect interface (x 0), no And as the frustrated Cr atoms are at the surface, they have frustated configuration could be obtained in the AFM stack- an enhanced magnetic moment. Thus the magnetic frustra- ing of Cr when the thickness of the film was smaller than 6 tion cost more energy than in the case of small frustrated Cr ML. This is apparently also the case when the alloying is atoms below the surface. The magnetic structure at the inter- weak and the Cr thickness is small. face differs a lot when more Cr is added on top as shown in The phase shift in the magnetic structure of Cr between Figs. 2 b and 2 c . x 1/3 and x 1/2 occurs in the following way: For x 1/3 With one pure Cr monolayer n 1, Fig. 2 b , it is the the Fe atoms in the first mixed layer, in larger amount than metastable interfacial configuration obtained with n 0 that Cr, impose negative moments on Cr in the second mixed is displayed in the ground state and the Cr overlayer is fer- layer due to the strong AFM coupling between Fe and Cr. romagnetic with negative moments. In this situation, the Then, those negative Cr moments impose positive moments frustrations occur between the Fe atoms and the Cr atoms of on the pure Cr layer above, inducing thus no phase change in the lower and upper mixed layers, that is below the surface. the magnetic structure of the Cr film compared to the case 6050 M. FREYSS, D. STOEFFLER, AND H. DREYSSE´ 56 FIG. 2. Ground state and metastable solutions for Crn /Cr1/2Fe1/2 /Cr1/2Fe1/2 /Fe(001) with a n 0, b n 1, and c n 2. The energy indicated is the interfacial energy relative to the ground state. Only the unit cell of each layer is shown, as well as only two layers of the semi-infinite Fe substrate. 56 INTERFACIAL ALLOYING AND INTERFACIAL . . . 6051 FIG. 3. Value of the surface layer average magnetic moment as a function of the concentration x when 3 Cr layers filled circles and 11 Cr layers squares are deposited and with two interfacial mixed layers. The lines are guides for the eyes. The sign of the surface layer magnetic moments characterizes the phase of the an- tiferromagnetic layer-by-layer structure of Cr. with a perfect interface. For x 1/2 see Fig. 2 c , the Fe atoms of the second mixed layer impose negative moments on the first pure Cr layer, inducing thus the phase shift in the layer-by-layer AFM structure of Cr. Calculations have also been perfomed with a bigger thick- ness of Cr, 11 ML, to verify if the Cr magnetic stacking is not changing as the thickness increases. Surprisingly, we have obtained that the phase shift occurs for a smaller con- centration x compared to the situation with 3 ML of Cr. The phase shift now occurs between x 1/9 and x 1/4 as can be seen on Fig. 3. In this case, a small interdiffusion is enough to reverse the magnetization of the layers of the Cr film. As a consequence, in Fe/Cr/Fe 001 trilayers, if interdif- fusion is assumed on both interface in the range of concen- tration x 1/4, we find that the coupling between Fe layers is the same as in the case of perfect interfaces ferromagnetic for an odd number of Cr layers, and antiferromagnetic for an even one , as shown on Fig. 4. And if only one interface is interdiffused, namely the Cr/Fe 001 interface as it is as- sumed by experimentalists, the coupling between the Fe lay- ers is reversed, and the coupling experimentally observed is obtained: ferromagnetic for an even number of Cr layers, and antiferromagnetic for an odd one. The fact that the phase shift occurs for a smaller concen- tration with 11 ML of Cr than with 3 ML also means that for the concentrations x 1/4 and x 1/3 a change occurs in the magnetic structure of the alloy layers as the Cr coverage increases. This change for x 1/4 occurs at a coverage of 2 to 5 ML of Cr. In this range of coverage, the surface Cr moments are always negative. The alternating sign of the surface layer moments as a function of the coverage starts only after 5 ML of Cr deposited. As a consequence, as Fe and Cr couple antiferromagnetically, if the surface of Cr has FIG. 4. Ground state for a Fe5 /Cr12 /Fe(001) trilayer with two negative moments, when covering it with Fe, Fe would then interdiffused interfaces. Only the unit cell of each layer is shown, as be ferromagnetically coupled to the lower Fe layer. Such a well as only two layers of the semi-infinite Fe substrate. 6052 M. FREYSS, D. STOEFFLER, AND H. DREYSSE´ 56 centration x larger than 3/4. As the Fe surface gets rougher, they find a decrease of the magnetization less important:12,13 only 1.4 B with an intermediate step density, and even zero B with a highly stepped surface. Even if the structure of the interface of their samples is not precisely known, those latter results would be consistent with alloy concentrations x of around 2/3­3/4 and 1/4­1/2, respectively. By our electronic structure calculations we can thus give an estimation of the chemical structure at the interface of the samples. B. Crn/CrxFe1 x/Fe 001... We have also calculated the magnetic order of Cr films on Fe 001 with only one interfacial mixed layer: Crn /CrxFe1 x /Fe(001) case II with n equal to 0, 1, 2 and in some cases 10 ML. With n 0, that is with a mixed layer on top of the pure Fe substrate, all Cr moments are antiferromagnetically coupled to the Fe moments of the substrate and of the mixed FIG. 5. Simulation of the variation of the total magnetization layer. There is no magnetic frustation in the ground state. during Cr growth for different interfacial alloy concentrations x. The values of the surface layer moments are almost constant as a function of x, especially the Cr ones. The values of the surface Cr moments are of the order of 3.33 0.05 B and the result could explain the ferromagnetic interlayer coupling be- value of the surface Fe moments are of the order of tween Fe layers observed up to 4­5 ML of Cr in the trilayers 2.65 0.11 B . Fe/Cr/Fe 001 .3 With one pure Cr layer added on top of the mixed layer Figure 5 shows the simulation of the variation of the total (n 1), the ground state changes according to the value of x. magnetization during Cr growth with various concentrations. First, for x in the range of 8/9 to 2/3 included, the Cr mo- The model used is presented in the Appendix. Let us recall ments of the mixed layer are all always antiferromagnetically that we have supposed an important non-layer-by-layer coupled to the Fe atoms, and the Cr moments of the pure growth mode. A trend can be noticed on the figure: in the surface layer are all positive, that is antiferromagnetically first stage of the growth coverage smaller than 3 ML , the coupled to the Cr moments of the mixed layer below. The magnetization decreases whatever the value of x is. For number of Fe-Cr first-neighbor bindings being for these con- larger coverage, the magnetization continues to decrease up centrations smaller than the number of Cr-Cr first-neighbor to an asymptotic value when x is larger than 1/2. And the bindings, the layer-by-layer antiferromagnetic structure of Cr more x is large the more the decrease in the magnetization is prevails on the Fe-Cr antiferromagnetic coupling. As a con- large: almost 5 B when x equals 8/9 and 1. The decrease of sequence of the magnetic frustrations between Fe and Cr, the the magnetization for these concentrations is related to the Fe magnetic moments on the mixed layer are reduced and strong reduction of the Fe moments of the upper mixed layer. smaller than 1 B . When x is larger or equal to 1/2, the Indeed, in this range of concentrations, those Fe moments are number of Fe-Cr first-neighbor bindings is no longer smaller of the order of 1 B or much less. On the other hand, when x than the Cr-Cr ones. A different configuration thus arises for is smaller or equal to 1/2, the magnetization increases for the ground state. Except for x 1/9, the ground state displays coverage larger than 3 ML and then stabilizes to an asym- the following features: in the mixed layer, the Cr moments potic value. The increase is especially important for x equal are antiferromagnetically coupled to Fe, as in the previous to 1/4. This can be explained by the fact that x 1/4 is the cases. The difference is in the pure Cr layer, which does not concentration for which changes occur in the magnetic struc- display an in-plane ferromagnetic order anymore: there are ture within the mixed layers with increasing Cr thickness, as Cr moments with different signs within the pure layer, form- previously mentionned. The consequence is that the magne- ing complex magnetic configurations. The ground state for tization increases when the coverage of Cr is larger than 3 x 1/9 is particular as the Cr moment of the mixed layer is ML. It can also be noticed that the curves for x 2/3 and positive, that is ferromagnetically coupled to the Fe sub- x 3/4 on the one hand, and the curves for x 8/9 and x 1 strate, contrary to the configuration found for all other con- on the other hand tend to the same asympotic values. This centrations. The pure Cr layer also displays a complex mag- can be attributed to the fact that for these concentrations, the netic structure with moments of different signs. values of the average Fe moments of the upper mixed layer With n 2 two pure Cr layers , the situation is more are almost the same, namely about 0.75 B for both x 2/3 simple: only solutions corresponding to a layer-by-layer an- and 3/4, and about 0.5 B for both x 8/9 and x 1. tiferromagnetic structure of the pure Cr film could be found. This simulation can be compared to the experimental re- The phase of the Cr magnetic stacking in the ground state, sults of Bayreuther and co-workers. During the deposition of given in Fig. 6, shows the variation of the surface layer mag- Cr on a flat Fe surface, they show a very important decrease netization as a function of x with 2 and 10 pure Cr layers of the magnetization of approximately 5 B .11 These data are deposited. With 2 pure Cr layers, we see that the antiferro- in accordance with our results in the case of an alloy con- magnetic stacking of the Cr film changes when x is larger 56 INTERFACIAL ALLOYING AND INTERFACIAL . . . 6053 than 1/4. This is quite surprising because it would have been expected that the transition occurs rather around x 1/2, that is to say that the first pure Cr layer would have been ex- pected to couple antiferromagnetically to the Fe atoms of the mixed layer as long as the Fe atoms are in majority in the mixed layer. This is actually not the case as shown in Fig. 7, which represents the ground state obtained for x 1/4 and 1/3, i.e., the concentrations between which the transition oc- curs. We see that for x 1/4, the first pure Cr layer is anti- ferromagnetically coupled to the Fe substrate. On the other hand, for x 1/3 the first pure Cr layer is coupled ferromag- netically to Fe, leading to slightly smaller Cr moments on this layer. On Fig. 7, one can also see that the value of the magnetic moments of both Fe and Cr of the mixed layer vary with x: For Fe, they are of order of 1.5­1.7 B when x 1/3 and 1.9 B when x 1/4. For Cr, they are of order of 1.2 B when x 1/3 and 0.9 B when x 1/4. This result is in contradic- tion with the results obtained by Coehoorn:19 by means of the augmented spherical waves ASW method, he computed FIG. 6. Value of the surface layer average magnetic moment as the magnetic moments in superlattices of the type 5 ML Fe/1 a function of the concentration x when n 2 ML filled circles and ML FexCr1 x/5 ML Cr/1 ML Fe1 xCrx with x 0, 1/4, 1/2, n 10 ML squares , and with one interfacial mixed layer. The lines 3/4, and 7/8, the alloy layer being modeled as an ordered are guides for the eyes. The sign of the surface layer magnetic compound as well. The value of the moments were found moments characterizes the phase of the antiferromagnetic layer-by- approximately constant whatever the value of x, and of order layer structure of Cr. of 2.0 B for Fe and 0.5 B for Cr. We have also considered the cases x 1/3, 1/4, and 1/2 with 10 layers of pure Cr (n 10) in order to see if, as in the FIG. 7. Ground state for Cr2 /Cr1/3Fe2/3 /Fe(001) and Cr2 /Cr1/4Fe3/4 /Fe(001). The white circles represent Cr atoms, the grey ones Fe atoms. A distinction is made between positive and negative Cr magnetic moments by means of the linewidth of the circles. In the case x 1/3, the vertical scale has been expanded. Only the unit cell of each layer is shown, as well as only two layers of the semi-infinite Fe substrate. 6054 M. FREYSS, D. STOEFFLER, AND H. DREYSSE´ 56 previous case with two interfacial mixed layers, the transi- tion occurs for a different concentration when the coverage of Cr increases. It is indeed the case as can be seen on Fig. 6: The antiferromagnetic structure of Cr changes when x is larger than 1/3, that is for a slightly larger value compared to the case with three monolayers of Cr deposited. Our results concerning the reversal of the sign of the magnetic moments in the pure Cr film are in good agreement with those of Coehoorn:19 he found the reversal in the sign of the Cr mag- netic moments in his 5 ML thick pure film for x between 0 and 1/4, which is close to our value of x obtained with a 2 ML thick pure Cr film (x between 1/4 and 1/3 . IV. CONCLUSION We have calculated the magnetic moments of Cr films on a Fe 001 substrate with mixed layers at the interface and shown how the concentration of these mixed layers affects the magnetic order of the Cr film. We have shown that in- terdiffusion could account for experimental results: with two interfacial mixed layers, the exchange of one quarter of a monolayer of Fe and Cr is enough to reverse the layer-by- layer antiferromagnetic structure of a 10 ML thick Cr film. Which concentration profile accurately describes the Cr growth on Fe 001 is highly dependent on the growth condi- tions. This is reflected by the various concentration profiles FIG. 8. a Macroscopic growth model of Cr on the Fe 001 experimentally observed by the different groups7,9,10 and by substrate. n is the Cr coverage on the nth layer. b Cr coverage the various results obtained by Bayreuther et al.11­13 Our n(t) on the surface as a function of the amount of Cr deposited simulation of the variation of the magnetization during Cr n n(t) in monolayers . The different curves correspond to the growth can give an estimation of the degree of interfacial filling of the successive layers. interdiffusion. In our calculations, the magnetic order is restricted to a collinear order. It is highly probable that a noncollinear order n is more stable, at least near the interface, in order to mini- t n Mi Crn 1 /Cr1 xFex /CrxFe1 x /Fe 001 ... mize the effect of the magnetic frustrations induced by the i 9 interdiffusion. Unfortunately, calculations of the magnetic n order with noncollinear moments in nonperfect systems still Mti Fe 001 ..., require too large a computing time.20 i 9 t ACKNOWLEDGMENTS where Mi is the average magnetic moment of the ith layer. Nine layers of Fe below the interface are taken into account The Institut de Physique et de Chimie des Mate´riaux de in the calculation of the difference of magnetization. Strasbourg is Unite´ Mixte 46 du CNRS-Universite´ Louis The most important thing is the simulation of n(t), that Pasteur de Strasbourg. We would like to thank G. Bayreuther is the growth mode of Cr. Here, we model n(t) in the fol- and S. Miethaner for fruitful discussions. lowing way: APPENDIX: SIMULATION OF THE VARIATION OF MAGNETIZATION DURING GROWTH 0 fort n 1 The variation of magnetization during Cr growth is simu- n t 1 cos t n 1 ... otherwise. lated using a simple macroscopic model of growth. At time t, 2 the variation of magnetization M(t) is given by When n(t) is equal to 0, no Cr is present on layer n, and M when t n(t) is equal to 1, the nth layer is complete. In our n n t n 1 t ..., n 1 theoretical study, the notion of time is arbitrary. A more relevant quantity which is equivalent to the time is the where n is the index of the Cr layers deposited. n(t) is the amount of matter deposited. It is defined as n n(t). In the Cr coverage on the nth layer at time t see Fig. 8 a . n is model, the only parameters to choose are and . The pa- the difference of magnetization between n Cr layers depos- rameter is the ``time'' between the beginning of two suc- ited on Fe 001 with two mixed layers at the interface and cessive layers. When is 1, a layer starts once the previous the pure Fe 001 surface: one is complete, corresponding thus to a perfect layer-by- 56 INTERFACIAL ALLOYING AND INTERFACIAL . . . 6055 layer growth. When is inferior to 1, the growth mode cor- the second layer starts. We choose 1( ) 0.06 and responds to the formation of islands. In our simulation, in- 0.06. Figure 8 b shows the behavior of n(t) in the stead of setting , we set an equivalent quantity, 1( ), simulation. With the parameters chosen, the seventh layer which is the amount of Cr deposited on the first layer when starts when the first one is completed. 1 P. Bruno and C. Chappert, Phys. Rev. B 46, 261 1992 . 11 C. Turtur and G. Bayreuther, Phys. Rev. Lett. 72, 1557 1994 . 2 S. Mirbt, A. M. N. Niklasson, B. Johansson, and H. L. 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