Long-range magnetic interactions and proximity effects in an amorphous exchange-spring magnet

Abstract

Low-dimensional magnetic heterostructures are a key element of spintronics, where magnetic interactions between different materials often define the functionality of devices. Although some interlayer exchange coupling mechanisms are by now well established, the possibility of direct exchange coupling via proximity-induced magnetization through non-magnetic layers is typically ignored due to the presumed short range of such proximity effects. Here we show that magnetic order can be induced throughout a 40-nm-thick amorphous paramagnetic layer through proximity to ferromagnets, mediating both exchange-spring magnet behaviour and exchange bias. Furthermore, Monte Carlo simulations show that nearest-neighbour magnetic interactions fall short in describing the observed effects and long-range magnetic interactions are needed to capture the extent of the induced magnetization. The results highlight the importance of considering the range of interactions in low-dimensional heterostructures and how magnetic proximity effects can be used to obtain new functionality.

ARTICLE

Received 28 Aug 2015 | Accepted 13 May 2016 | Published 13 Jun 2016

F. Magnus1,2, M.E. Brooks-Bartlett3, R. Moubah1,4, R.A. Procter5, G. Andersson1, T.P.A Hase5, S.T. Banks3,w

& B. Hjrvarsson1

Low-dimensional magnetic heterostructures are a key element of spintronics, where magnetic interactions between different materials often dene the functionality of devices. Although some interlayer exchange coupling mechanisms are by now well established, the possibility of direct exchange coupling via proximity-induced magnetization through non-magnetic layers is typically ignored due to the presumed short range of such proximity effects. Here we show that magnetic order can be induced throughout a 40-nm-thick amorphous paramagnetic layer through proximity to ferromagnets, mediating both exchange-spring magnet behaviour and exchange bias. Furthermore, Monte Carlo simulations show that nearest-neighbour magnetic interactions fall short in describing the observed effects and long-range magnetic interactions are needed to capture the extent of the induced magnetization. The results highlight the importance of considering the range of interactions in low-dimensional heterostructures and how magnetic proximity effects can be used to obtain new functionality.

DOI: 10.1038/ncomms11931 OPEN

Long-range magnetic interactions and proximity effects in an amorphous exchange-spring magnet

1 Department of Physics and Astronomy, Uppsala University, Box 516, Uppsala 751 20, Sweden. 2 Science Institute, University of Iceland, Dunhaga 3, Reykjavik IS-107, Iceland. 3 Department of Chemistry, University College London, 20 Gordon Street, London WC1H 0AJ, UK. 4 LPMMAT, Universit Hassan II de Casablanca, Facult des Sciences Ain Chock, Marif B.P. 5366, Morocco. 5 Department of Physics, University of Warwick, Coventry CV4 7AL, UK. w Present address: Faculty of Engineering Sciences, University College London, Gower Street, London WC1E 6BT, UK. Correspondence and requests for materials should be addressed to F.M. (email: mailto:[email protected]

Web End [email protected] ).

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Magnetism is an intrinsically quantum mechanical phenomenon but classical approaches can nonetheless be very useful for describing certain magnetic proper

ties1. Many features of ferromagnetic ordering such as the asymptotic changes in magnetization with temperature as well as the ordering temperature itself can be described well using atomistic models with only nearest-neighbour spinspin interactions. However, as the spatial dimensions of the ferromagnet are reduced, the surfaces and interfaces, where the atomistic interactions are truncated, start to have a dening effect on the magnetic properties. Modelling such nite size effects has typically required the assumption of weakened or enhanced magnetic interactions at the boundaries, which approach the bulk state exponentially2. The need for such assumptions is removed if longer-range (beyond nearest-neighbour) magnetic interactions are allowed37. Although this approach is rarely used due to its computational complexity, it has been employed successfully to capture nite size effects on the magnetic ordering and describe the spatial variation of the magnetization in very thin, free-standing lms8.

Extending these ideas to model magnetic heterostructures comprising multiple magnetic or non-magnetic layers can give a fresh insight into the interface phenomena, which are central to many current and emerging magnetic technologies. One of the most important, but often overlooked, interface phenomena in magnetic heterostructures is the magnetic proximity effect9. In general, this refers to the inuence that an ordered magnetic state in one layer has on an adjacent layer10. When two magnetic materials are in direct contact, the inuence of the proximity effects is mutual, leading to a variation in the coupling strength across the interface, which has been modelled previously using mean eld theory11. This can for example give rise to changes in the ordering temperature of the two materials12 or even result in a single singularity in the susceptibility11. In another type of proximity effect between two ferromagnets with different directions of anisotropy, a mutual imprinting of domain structures has been observed13. Alternatively, at a ferro-magnetic/non-magnetic interface a magnetization can be induced within the non-magnetic material. The extent of this proximity-induced magnetic region is typically quite short, of the order of only a few atomic distances, but in materials which are close to satisfying the Stoner criterion, such as Pd and Pt, it is known to extend up to a few nanometres1416.

When two ferromagnetic layers are separated by a non-magnetic layer, proximity effects arise at both interfaces, which can give rise to long-range interlayer exchange coupling16, changes in ordering temperature17 and/or non-oscillatory

alignment of the magnetic layers15,18. Here we provide experimental evidence that such a proximity effect can result in direct exchange coupling across a 40-nm-thick paramagnetic layer, which is more than an order of magnitude longer than previously demonstrated. The proximity-induced magnetization can give rise to both spring magnet behaviour and exchange bias, with the two having a different temperature dependence and extension. The extent of the proximity-induced magnetization and the richness of the resulting magnetic interlayer coupling effects are of importance for any magnetic metallic multilayered system. Furthermore, we show how such a proximity effect can be rationalized using an atomistic spin model with long-range magnetic interactions and thus move beyond the common assumption of only nearest-neighbour magnetic interactions.

ResultsMagnetization measurements. Our experimental system is an amorphous exchange-spring magnet19, composed of a magnetic trilayer as illustrated in Fig. 1a. Such an amorphous hetero-structure is ideal for examining magnetic interface effects, as amorphous materials form highly homogeneous and at layers20, which are free of step edges and grain boundaries. A strong uniaxial anisotropy is imprinted21 in the bottom layer (C) and the ferromagnetic ordering temperature of the three layers TXc (X A, B or C) is, through the choice of composition, as

illustrated in Fig. 1b. The anticipated effect of the ferromagnetic proximity in the paramagnetic layer B can be seen in Fig. 1a. At low temperature ToTBc

a normal spring-magnet behaviour is expected, whereas at high temperature T TBc

the top layer(A) is expected to act independently of the bottom layer (C). At intermediate temperatures T\TBc

a ferromagnetic coupling may exist between layers A and C, arising from the proximity-induced ferromagnetism in layer B. The trilayer architecture and material choice are crucial, as they allow us to determine the range of any proximity-induced magnetization in layer B by determining the temperature dependence of the coupling between layers A and C for different thicknesses of layer B. However, as the ordering temperature of layer B and the coercivity of layer C can be tuned through composition, the temperature ranges and interaction strengths could be changed as desired.

Representative easy axis magnetic hysteresis loops of a trilayer structure with a middle layer thickness dB 10 nm are shown in

Fig. 2a, measured by magneto-optical Kerr effect (MOKE) magnetometry. Layer C (Sm10Co90) has a large imprinted in-

plane uniaxial anisotropy resulting in a square magnetization loop with substantial coercivity along the easy magnetic axis22.

Co85(AIZr)15

Co60(AIZr)40

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Figure 1 | Design of the experiment. (a) A simplied schematic of the layer structure of the sample, showing the three different magnetic layers, A, B andC. Layer C has a large imprinted uniaxial anisotropy (parallel to the large grey arrow), whereas layer A has a small imprinted anisotropy (in the same direction) and layer B is isotropic. The magnetization prole during magnetization reversal, in different temperature regimes, is shown by the round coloured arrows, demonstrating the exchange-spring magnet behaviour and the magnetic proximity effect. The large grey arrow shows the direction of the applied magnetic eld. (b) An illustration of the temperature dependence of the magnetization in the three layers, showing the three different ordering temperatures TXc (X A, B or C) of the layers.

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Figure 2 | Spring-magnet behaviour and enhanced coercivity. (a) The magnetization along the easy axis for three different temperatures, showing the exchange coupling between the top and bottom layers. The middle layer thicknesses dB is 10 nm. (b) The lower coercive eld HAc as a function of temperature (absolute and normalized by TBc), for three different middle layer thicknesses dB. The dashed vertical line indicates the intrinsic transition temperature of Co60AlZr40. A coupling between the top and bottom layers is seen in a region well above the intrinsic transition temperature of the middle layer when dB 10 nm (highlighted by the blue

shaded area).

Comparatively, layer A (Co85AlZr15) has a small in-plane anisotropy and layer B (Co60AlZr40) is isotropic, although the magnetization is constrained to lie in the plane of the lms by the shape anisotropy. As the growth temperature is above the ordering temperature of layer B, no anisotropy is imposed during the growth of that layer. At room temperature (290 K), layer B should be paramagnetic (see Supplementary Fig. 1) and the A and C layers act independently and display switching elds HAc and HCc, respectively. Consequently, the magnetic contributions of the two layers simply add, resulting in a two-step, square hysteresis curve as seen in Fig. 2(a). When the temperature is lowered below TBc the magnetic coupling between adjacent layers causes the magnetization of the soft A and B layers to be pinned by the strongly anisotropic layer C. Therefore, the coercivity of layer A is strongly enhanced as compared with its value determined from an isolated lm. A two-step switching is again observed where the lower switching eld HAc can be associated with the ipping of the top part of the trilayer (mostly layer A), whereas the high-eld switch HCc corresponds to the ipping of the magnetization of the bottom part of the trilayer (mostly layer

C). In between the two switching elds, the characteristic exchange-spring magnet behaviour is observed with a gradual

increase in the magnetization19. This gradual increase in the magnetization arises from the continuous in-plane rotation of the magnetization between the ipped top layer and the pinned bottom layer (analogous to a torsion spring), shown schematically in Fig. 1a.

For the B layer thicknesses dB 40 nm and dB 20 nm, the rise

in coercivity occurs at the intrinsic transition temperature of layer B (103 K), as seen in Fig. 2b. Strikingly, when the thickness of the middle layer is reduced to dB 10 nm, the coupling between the

top and bottom layers persists to above 150 K, which is 450% above the intrinsic ordering temperature of layer B (Co60AlZr40). The coupling can also be seen in the susceptibility when the eld is applied parallel to the hard axis (see Supplementary Fig. 2). This apparent change in TBc can not be attributed to nite size effects as the smallest thickness of dB 10 nm could only account

for changes in Tc of 12% compared with the bulk23,24. Therefore, it is clear that the proximity of the ferromagnetic A and C layers to the intrinsically paramagnetic B layer induces some ferromagnetic ordering and an associated exchange stiffness within layer B, far above its intrinsic transition temperature. At 150 K, the proximity-induced ferromagnetic state extends at least 5 nm into the layer from both interfaces. With an interatomic distance of 0.15 nm, this corresponds to well above 30 atomic distances.

The two coercive elds HAc and HCc can be related to the energy barrier between the spiral and collinear magnetic states. The coercivity of layer C is at least an order of magnitude smaller than its saturation eld along the hard axis (see ref. 22) and the reverse is true for the top layer (at least at low temperature). Therefore, the switching energy of both layers is associated with the domain wall forming within the trilayer. As the precise position of the domain wall and the magnetization prole within layer B is not known, we cannot calculate these energies precisely. However, using the coercive elds of layer A and layer C, and the associated magnetizations (MAs and MCs), we can estimate the energy barrier using JA;Cc m0HA;CcMA;Cs. This gives an energy barrier of

JAc 1:3 104 Jm 3 for the switching of the A layer and

JCc 2:7 104 Jm 3 for the C layer at 5 K. These energies are

highly temperature dependent as Fig. 2 shows.

Element-specic X-ray resonant magnetic scattering has been carried out to further examine the inuence of proximity on the magnetic response in the thick layer limit (dB 40 nm).

Measurements of the Co and Sm magnetization loops were performed at 100 K (at TBc) and 300 K. The results showed

two-step switching of the Co and rounding at low temperature, in good agreement with the MOKE results. Minor loops were obtained by successively increasing the maximum positive eld, as shown in Fig. 3a for the Co. There is no signature of exchange-spring behaviour or enhanced coercivity at high temperature, which is consistent with the MOKE data. However, the minor loops are clearly shifted to positive elds, demonstrating that layer A is in fact exchange biased by the high anisotropy layer C, even though there are no exchange-spring effects observed in hysteresis loops taken to complete saturation. We note that even the switching elds of layer C are shifted, implying that the Co in the bottom layer is also exchange biased. Comparing the Sm and Co magnetization loops reveals that the Co in the SmCo layer switches in smaller elds than the Sm, as described in detail elsewhere25. This decoupling of the Sm and Co sub-networks within the SmCo layer therefore results in an exchange bias from the Sm sub-network acting on the Co. The Co in the C layer then in turn results in an exchange bias in layer A.

The exchange eld Hex decreases as the maximum applied eld increases as seen in Fig. 3b and is equal in size at both 100 and 300 K. This decrease in Hex with applied eld is a signature of the gradual switching of the Sm, which is weakly temperature

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Figure 3 | Exchange bias. (a) Room-temperature element-specic minor magnetization loops of Co, measured by X-ray resonant magnetic scattering, for successively higher maximum applied eld in the positive direction Hmax. The sample is a trilayer with dB 40 nm. The loops are

shifted towards positive eld (as highlighted by the dashed line at zero eld), showing the presence of exchange bias. (b) The exchange bias Hex as

a function of Hmax for both 100 and 300 K. The exchange bias is the same at both temperatures and tends to zero (dashed line) as Hmax increases.

Figure 4 | Simulations of Tc in a trilayer. (a) The magnetization versus temperature in the middle of a single B-layer and in the middle of a B-layer sandwiched between an A and C layer. The magnetization is calculated by including up to eighth nearest-neighbour interactions and the temperature is normalized by the intrinsic ordering temperature of the B layer found in the simulations Tr T=TBc

. (b)

The susceptibility of the middle atomic layer of a single B-layer and in the middle of a B-layer, sandwiched between an A and C layer. The shift in the ordering temperature of the sandwiched layer due to the proximity to the A and C layers can be seen clearly. Only a small subset of symbols is shown.

dependent. Owing to this complex magnetization switching mechanism and the unknown magnetization prole within layer B, we can only estimate the coupling strength. Using the magnetization of layer A only MAs

and the maximum obtained exchange bias eld (from Fig. 3b) we nd the coupling strength at 300 K to be approximately Jex 4 10 5 J m 2 using the

relation Jex m0HexMAsdA. In any case, these data show that

layer A is still coupled to layer C at 300 K, or three times the intrinsic ordering temperature of the 40-nm-thick B layer.

Monte Carlo simulations. To examine the root cause of the observed proximity effects we have carried out Monte Carlo simulations of a model trilayer structure, resembling the sample shown in Fig. 1. A simple cubic model is used for simplicity and we adopt a terminology where a layer comprises a number (one or more) of identical monolayers. The simulations are based on a single crystal approach, which at rst glance appears inconsistent with the experimental conditions. However, the modelling can be viewed as a rationalization of an arbitrary sample, where each atom represents a given volume fraction. Thus, the results can be scaled and generalized to capture the experimental ndings, but are also applicable to other materials systems. A range of coupling schemes were used, from simple nearest neighbour up to eighth

nearest neighbour, as described in the Methods section. In each case, the temperature was varied from well below to well above the ordering temperature of layer B.

When using a nearest-neighbour interaction scheme, the induced magnetization in the paramagnetic layer was restricted to the near-interface region, which is clearly not consistent with the experimental results. Consequently, we will emphasize the inuence of the range of interactions on the extent of the proximity effects. The temperature dependence of the magnetization and susceptibility of the eighth nearest-neighbour model is shown in Fig. 4. Two cases are considered: (i) a monolayer in the centre of a free-standing thin lm of B (40 monolayers thick) and(ii) a monolayer in the centre of a B layer sandwiched by ferromagnetic A and C layers (10, 8 and 10 monolayers thick, respectively). Central monolayers are chosen as they are least affected by surface effects and for similar reasons, the properties of the monolayer in case (i) is considered to represent the bulk properties of the B layer. The temperature dependence of the magnetization for case (i) is as expected for the model as seen in Fig. 4a and the temperature dependence of the susceptibility (Fig. 4b) shows a sharp peak at the ordering temperature. The magnetization of the B-layer in the trilayer structure (case (ii))

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1.2

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Figure 5 | Magnetization prole in a trilayer. The simulated magnetization prole throughout the trilayer for a few temperatures above and below the TBc of a single B-layer. The temperature is normalized by the intrinsic ordering temperature of the B layer found in the simulations Tr T=TBc

.

The magnetization decays into layer B away from the interfaces but a signicant magnetization extends through the layer well above TBc.

1.2

20 25 30

Figure 6 | Range of interaction. The magnetization throughout the trilayer for different interaction ranges: rst, fourth and eighth nearest neighbours (n.n.), at T TBc DT, where TBc corresponds to the bulk ordering

temperature for the respective range and DT is arbitarily chosen as 0.20. Monolayer 1 and 28 are the top and bottom surfaces of the sample, respectively. The background colour is a guide to the eye, showing the trilayer structure. A longer-range interaction is needed to capture the proximity-induced magnetization in the middle layer.

shows a slower approach to the high-temperature behaviour, resembling the response of a ferromagnet in an external eld (see Supplementary Fig. 1). The behaviour of the susceptibility suggests that the Tc of the middle monolayer of B is increased by B10% when the B-layer is sandwiched between layers A andC. More extensive nite size analysis would be required to determine this shift more precisely. The peak in the susceptibility is also signicantly broadened, similar to the effect of applying an external magnetic eld.

Figure 5 shows the magnetization prole for the trilayer for a selection of temperatures. As the intrinsic ordering temperature of layer B (Tr 1) is reached, a rapid reduction in the layer-

resolved magnetization is seen, decaying from the interfaces into the centre of layer B. Despite this, layer B exhibits an observable magnetization at and somewhat above its intrinsic ordering temperature. Only for temperatures above Tr 1.5 does the

magnetization reach 0 in the centre of layer B. We also note that signicant ferromagnetic proximity effects are present at the interfaces at temperatures that are as high as twice the ordering temperature of layer B.

The extent of the proximity-induced magnetization and the ability to sustain this induction for T4TBc is greatly inuenced by the range of the direct exchange interactions. This effect is illustrated in Fig. 6, which shows the magnetization prole in the trilayer structure for three different interaction schemes: nearest neighbour (corresponding to the rst coordination sphere of each spin), up to fourth nearest neighbour (corresponding to the second coordination sphere) and up to eighth nearest neighbour (corresponding to the third coordination sphere). The proles for each range are plotted at T TBc DT, where TBc corresponds to

the bulk ordering temperature for the respective range and DT is arbitarily chosen as 0.20. It is clear that a nearest-neighbour model does not capture the extent of the proximity-induced magnetism in the paramagnetic layer (B) that is determined experimentally when T4TBc. We also note that the proximity effect increases with increasing range of interaction in the simulations. From the regions near the interfaces, where there is little overlap between the proximity tails from either side of the B layer, we can estimate that the extent of the proximity region increases approximately linearly by a factor of 0.65 monolayers with the coordination sphere number. From this, one can estimate that the simulations would need to include up to the

45th coordination sphere, to obtain an extension of the induced magnetization of 30 atomic distances from the interfaces, but this is not feasible with the current computational methods.

DiscussionThe simulations show that an induced magnetization can be obtained well above the intrinsic ordering temperature of the B layer through proximity-induced magnetization. However, despite allowing interactions up to eighth nearest neighbours, our simulations underestimate the extent of the regions with induced magnetization. Increasing the range of interactions increases the extent of the proximity-induced magnetization, suggesting that even longer-range interactions are at play. However, another possible contribution to the observed range of the induced magnetization are atomic correlations within the amorphous layers. The disordered atomic structure can contain a modulation in the Co density, with interconnected regions of higher Co composition than the average. The presence of a modulation in the atomic specic density2628 will result in regions of larger exchange coupling than the average coupling JBB

assumed within the B layer and are therefore more susceptible to the proximity of the adjacent ferromagnetic layers. Apparent structural disorder has indeed been shown to result in enhanced magnetic correlations above the ordering temperature in amorphous magnetic thin lms with an associated high magnetic polarizability29,30. Therefore, it is possible that amorphous materials can exhibit larger proximity effects than their crystalline counterparts.

Independent of the root cause of the proximity effect, there are clearly two distinct coupling regimes, which have a different temperature dependence and extension. In the temperature regime 100oTt150 K, there are B5-nm-thick regions in the

B layer at its interfaces with an induced magnetization and an exchange stiffness, which exerts a torque on the A layer, enhancing its coercivity and resulting in spring-magnet behaviour. This can be considered as the proximity-induced ferromagnetic region. At greater distances from the interfaces and at temperatures up to at least room temperature, the B layer is still strongly polarized, although the spin stiffness is small or

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zero. This can be considered as the proximity-induced super-paramagnetic region, where there is no torque on the A layer but the internal eld results in an exchange bias effect. This state can be thought of as a magnetic liquid state in layer B. It is important to note that the temperature regions of these magnetic phases will be composition dependent and can thus be chosen for specic applications.

The results raise a number of fundamental questions about the range of magnetic interactions and the effect of structural disorder on magnetic properties. For example, from the results of the Monte Carlo simulations it appears that long-range interactions are the key to obtaining a substantial extension of a proximity-induced magnetization at interfaces. This has far reaching consequences concerning simulations of conned magnetic systems, not least the temperature dependence of the magnetization. Furthermore, the large extent of the proximity effect and its importance for exchange-spring behaviour and exchange bias imply that it needs to be considered in a range of structures showing interlayer exchange coupling. Finally, the two distinctly different regions of observed magnetic coupling, with different temperature dependence and extent, hint at the existence of a rich magnetic phase diagram for amorphous materials and an extensive scope for tailoring of their properties. This tuneability through temperature or composition can, for example, allow the increase of operating frequencies in microwave devices31, increase the performance of exchange-spring layer recording media32,33 or even add new functionality in areas such as magnetic sensors or logic, where a controllable interlayer coupling is desired. As a result, amorphous magnetic lms may have an important role to play in future spintronic devices.

Methods

Sample growth and characterization. The samples were grown by dc magnetron sputtering in a sputtering chamber with a base pressure below 5 10 10 Torr. The

sputtering gas was Ar of 99.9999% purity and the growth pressure was 2.0 mTorr. Si(100) substrates with the native oxide layer were used, 0.5 mm thick and with an area of 10 10 mm2. To remove surface impurities, the substrates were annealed in

vacuum at 550 C for 30 min before growth. First, a 2-nm-thick buffer layer of AlZr was deposited on the substrate from an Al70Zr30 alloy target of purity 99.9%. The buffer layer promotes the at amorphous growth of the following layers. Subsequently, a 20-nm-thick Sm10Co90 alloy lm was grown by co-sputtering from 2

elemental targets of Co (99.9% purity) and Sm (99.9% purity), after which a Co60(AlZr)40 layer in the thickness range 1040 nm and a Co85(AlZr)15 layer of 15 nm were grown by co-sputtering from the Co and AlZr targets. Finally, a 3-nm-thick capping layer of AlZr was grown to protect the magnetic trilayer from oxidation. The simplied sample structure can be seen in Fig. 1a. All lms were grown at room temperature, without any substrate cooling. The sample holder is equipped with two permanent magnets, which give a magnetic eld of approximately Him 0.1 T parallel to the plane of the lms. This magnetic eld induces a

uniaxial in-plane anisotropy in the layers, which are magnetic at room temperature. The sample holder design is described in detail in ref. 21. Structural characterization, attesting to the amorphicity of all layers and layer perfection, has been performed by grazing incidence X-ray diffraction, X-ray reection and transmission electron microscopy (see ref. 22 for more details).

The magnetic characteristics of the samples were determined by magneto-optical Kerr effect (MOKE) measurements in the longitudinal geometry with s polarized light. The sample was rotated around the azimuthal angle f (around the sample normal) and the full hysteresis loop recorded at 5 intervals, to determine the in-plane magnetic anisotropy. Full hysteresis loops were also recorded over the temperature range 5380 K, for f 0 and f 90 with respect to the magnetic

easy axis.

X-ray magnetic reectivity was performed on the X13A beamline at the National Synchrotron Light Source34, using circular polarized X-rays tuned to the Co L3 edge. Keissig fringes with a period corresponding to the total trilayer thickness were observed, conrming that we are probing the entire trilayer thickness. Hysteresis loops were recorded by measuring the eld dependence of the asymmetry ratio (I I )/(I I ), with I

(

The spins, s, are three-dimensional vectors where one or more components may be xed as zero to constrain the spin dimensionality (or type). In all cases the spins are of length 1. The system consists of N 28,672 spins, 1,024 per 32 32 layer

with 8 layers of A spins and 10 layers each of B and C spins. The exchange parameter Jab is a member of the set (JAA, JBB, JCC, JAB and JBC) according to the layers to which spins i and j belong.

The reduced energy scale of the exchange couplings is normalized to JCC 1.0.

Other inter- and intra-layer couplings were then chosen as JAA 1.0, JBB 0.40

and JAB JBC 0.90. These choices reect the relative magnetic content (mole

fraction of cobalt) of the layers, together with the fact that experimental results indicate that coupling within the B layer is signicantly weaker than both of the inter-species couplings (Fig. 2).

The algebraic decay of the exchange couplings varies as rd sij after Fisheret al.36, where d is the dimension of the system and s is positive (and generally small). Although our slab is nite in the z direction, we assume that it is sufciently large to take d 3. We initially simulated a range of s between 0.1 and 2.0.

Qualitatively, the results showed no variation and here we report the case s 0.5.

A fast decay (s42) in the interaction37 implies an approach to the nearest-neighbour model (the short-range limit). Our choice avoids this limiting behaviour, although allowing the further neighbour interactions to decay sufciently quickly such that nite size effects are minimized (relative to the critical system at s 0)37.

In our simulations we used a single spin-ip Metropolis algorithm with 104 Monte Carlo steps per spin for equilibration and 105 Monte Carlo steps per spin for observation at each temperature. We believe the spatial extent in the xy plane,32 32 spins, is a reasonable compromise between accuracy and computational

expense. Systems of this size allowed for accurate modelling of the critical phase of the two-dimensional XY model38 and it is known that nite size effects are smaller in three-dimensional systems39.

Data availability. The data that support the ndings of this study are available from the corresponding author upon request.

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the scattered intensity for X-rays of opposite helicity. In this geometry, the signal measured is element specic and sensitive to the in-plane ferromagnetic moment35. All loops were tted to a modied Langevin function to quantify the coercivity and exchange bias.

Model. The aim of the Monte Carlo simulations is to investigate the extent to which a simple, exchange-only, classical spin model can capture the essential

physics of the experimental sample. Our model consists of a simple cubic lattice trilayer with periodic boundaries in the (x, y) plane and free boundaries in thez direction. The spin dimensionalities within each layer have been chosen as realistic representations of the magnetic moments in the layers. Using the labelling introduced in Fig. 1, layers A and B consist of only XY spins, constrained by the shape anisotropy to lie in the (x, y) plane of the lattice. Layer C contains Ising spins, constrained to point in only the y direction due to the strong uniaxial anisotropy.

The Hamiltonian governing our model is dened by the equation

H Xi;jJijsi sj 1

where the sum is over all pairs of spins with iaj and the exchange couplings decay algebraically up to a hard cutoff at r rc,

Jij

a r

d

sJab; if rij rc0; otherwise rij4rc:

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NATURE COMMUNICATIONS | DOI: 10.1038/ncomms11931 ARTICLE

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Acknowledgements

This work was funded by the Swedish research council (VR), STINT, the Knut and Alice Wallenberg foundation (KAW) and the Carl Trygger Foundation. Work undertaken at the National Synchrotron Light Source was supported by the U.S. DoE, Ofce of Science, Ofce of Basic Energy Sciences, under contract DE-AC02-98CH10886.

Author contributions

F.M. and B.H. designed the experiment. F.M. and R.M. carried out the sample preparation as well as the structural and magnetic characterization. F.M., G.A. and B.H. carried out experimental data analysis and interpretation. M.E.B.-B., S.T.B. and B.H. designed and carried out the Monte Carlo simulations. R.A.P. and T.H. carried out and analysed the X-ray resonant magnetic scattering measurements. All authors participated in the writing of the manuscript.

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How to cite this article: Magnus, F. et al. Long-range magnetic interactions and proximity effects in an amorphous exchange-spring magnet. Nat. Commun. 7:11931 doi: 10.1038/ncomms11931 (2016).

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