Dynamic and rotatable exchange anisotropy in Fe/KNiF3/FeF2 trilayers

Results from ferromagnetic resonance experiments carried out on epitaxially grown Fe/KNiF3/FeF2 trilayers are presented. Exchange coupling between the KNiF3, a weak anisotropy antiferromagnet, and the Fe leads to shifts in the resonance field of the ferromagnet. The field shifts can be described by a temperature-dependent exchange anisotropy . depends on the orientation direction of the applied field relative to the magnetic anisotropy axis, and a non-monotonic dependence on KNiF3 thickness. Three thickness regimes appear that correspond to different values of exchange bias in each region. A qualitative understanding of the basis for these three thickness regimes due to spin canting at the interfaces is presented. Our results illustrate a method to tune the value of exchange anisotropy using a combination of different antiferromagnets.


Introduction
Exchange bias with natural antiferromagnets is typically investigated for systems with only a single magnetic interface [1][2][3][4][5][6][7]. Of these, one of the best understood in terms of atomic level spin configurations is the epitaxial Fe/FeF 2 film system. The present work explores exchange bias in a structure in which an epitaxially grown KNiF 3 film is used to separate the Fe and FeF 2 . The uniaxial magnetic anisotropy of the KNiF 3 is much weaker than that of the FeF 2 , and the corresponding exchange bias and exchange anisotropies are also different.
By changing the thickness of the KNiF 3 , we show that it is possible to change the exchange anisotropy and bias from that of the FeF 2 to that of the KNiF 3 . In the limit of very thick KNiF 3 , we observe unidirectional, uniaxial and rotatable exchange anisotropies. Competing effects from the FeF 2 are observed as the KNiF 3 thickness is reduced, allowing us to probe magnetic lengthscales associated with spin ordering near the KNiF 3 interfaces.
We used ferromagnetic resonance (FMR) to obtain values for exchange anisotropies and bias. The FMR response is provided by the Fe in our Fe/KNiF 3 /FeF 2 systems, and magnetic anisotropies and other parameters are obtained by fitting the raw data to well known resonance conditions for thin films. In this paper we show that the fitted values reveal an exchange anisotropy that appears to be dependent on applied field orientation relative to the magnetic anisotropy symmetry axes. Throughout the rest of the paper we label this anisotropy H AFM dyn . The onset of H AFM dyn with temperature is similar to that of a rotatable anisotropy observed previously by us for KNiF 3 systems [8,9], and we suggest that H AFM dyn in this paper consists of an isotropic 'rotatable' component H rot in addition to an orientation dependent, unidirectional component H E .
Results of magnetometry and FMR studies of bilayered thin films of single-crystal Fe(0 0 1) and either single crystal or polycrystalline KNiF 3 are given in [8,9]. The smallest lattice mismatch to the Fe is 1.2%, assuming that the Fe/KNiF 3 interface coincides at the (1 0 0) face of both materials. This good lattice match between the ferromagnetic and antiferromagnetic layers preserves the cubic structure of both.
The (1 0 0) plane of KNiF 3 should be compensated with both sublattices present in equal numbers. This should also be true for polycrystalline KNiF 3 on Fe since all possible growth planes are reasonably well lattice matched with Fe. The most striking feature observed was a blocking temperature in the range 50-80 K . This temperature is much lower than the 250 K Néel temperature expected for bulk KNiF 3 . Particularly relevant for this work was the observation (by FMR) of a rotatable anisotropy that was an order of magnitude larger than the exchange bias.
As noted earlier, the Fe/FeF 2 system has been particularly well studied. Nogués and Schuller [1] have shown that the exchange bias depends strongly on the spin structure at the interface and in particular, on the angle between the FM and AFM spins. Work by Fitzsimmons et al [18][19][20][21][22] using polarized neutron reflectometry and magnetometry shows that spin configurations at the FeF 2 interface differ from the bulk, and can be linked to exchange anisotropies and bias phenomena.
We have used MOKE microscopy to observe domains in a Fe/FeF 2 system to identify interface regions in which spin arrangements are distinct from either AFM or FM magnetic spin arrangements [23]. Results indicate that the crystallographic arrangement affects the value of the exchange bias but not the temperature dependence. Measurements of the temperature dependence of domain density in conjunction with coercivity field enhancement and exchange bias suggest that exchange bias and coercivity are different in origin in the sense that the unpinned magnetic moments at the AFM/FM interface are responsible for the enhancement of the coercivity field while pinned moments shift the hysteresis loop.
In what follows we first discuss the sample structure and characterization in section 2. This is followed in section 3 by a discussion of reference FM/AFM1 and FM/AFM2 bilayers, and FM/AFM1/AFM2 trilayer results. In section 4 we present our micromagnetic model and discuss the possibility of different thickness regimes. Results and discussion are summarized in section 5.

Sample preparation and structural characterization
Samples were grown on GaAs(0 0 1) substrates using molecular beam epitaxy. Wafers were annealed at elevated temperatures (∼450 • C) and sputtered with Ar ions until 4 × 6 reconstruction on the surface of GaAs was clearly visible. The Fe was deposited using K-cells and the fluorides were grown using e-beam evaporation at room temperature. A 0.6 nm thick Fe seed layer was followed by a 75 nm thick Ag buffer layer. At this point the structure was annealed at 350 • C for 12 h. From RHEED we confirmed that annealing produced an average terrace size of ∼13 nm. Next an Fe(0 0 1) layer was grown at room temperature followed by a KNiF 3 layer and topped with a 50 nm FeF 2 layer. Series with different KNiF 3 thicknesses between 0 and 90 nm were prepared. The structures were capped with a 2.5 nm thick Au layer to protect samples during measurements under ambient conditions.
In figure 1(a), example RHEED intensity oscillations measured at the specular spot during the growth are shown, confirming layer by layer growth. Also, from RHEED we determined that that the Fe layer is monocrystalline, and the fluoride layers are polycrystalline. Note that in order to reduce variations under growth conditions within the series, two thicknesses of KNiF 3 were grown on each Fe film, thereby forming two samples. The geometry is sketched in figure 1(b) for the 2.8 and 3.6 nm samples from the series. For each pair of samples the same roughness at the Fe/KNiF 3 interface is expected.

Experimental details and results
FMR was carried out at 24 GHz at temperatures between 24 and 300 K, using the TE 011 mode of a cylindrical cavity. The temperature dependence measurements were carried out in a dewar equipped with a closed-cycle helium refrigerator. The temperature of the sample cavity was monitored with two E-type thermocouples. The dc signal was measured on a diode, and the first derivative of the power absorption signal with respect to the applied field was detected. A lock-in amplifier technique was used employing a weak, 0.5 Oe, 155 Hz ac modulation field superimposed on the applied dc magnetic field. The FMR absorption spectra were fit with standard Lorentzian function, providing directly the resonance field, H res , and FMR linewidth, H .
A sketch of the experimental geometry is shown in figure 1(c). In-plane angles for magnetization and applied field (ϕ F and ϕ H , respectively) are defined relative to the Fe[1 1 0] direction. Measurements were carried out by varying the angle ϕ H . An applied magnetic field was swept between 1 and 6 kOe. These fields were large enough to saturate the samples to within 2 • for all angles of applied field orientation. The samples were first measured at a room temperature, then were field cooled to 24 K in the cooling field H cf = 0.87 kOe and H cf = 2.07 kOe for Fe easy axis (denoted as EA; ϕ H = 45 • ) and Fe hard axis (denoted as HA; ϕ H = 0 • ), respectively. These values of the cooling fields are significantly larger than field needed to saturate FM (∼600 Oe along HA).
The effective magnetization 4πM eff , and a fourfold anisotropy field, H = 2K /M s can be determined with FMR by measuring H res for different orientations of the static applied field H a relative to a magnetocrystalline anisotropy axis. An example for Fe thin film (2.59 nm) is shown in figure 1(d). All FMR measurements were fit to the following resonance condition equation [8,9]:  where the first term represents a constant shift in the resonance field baseline. This shift is due to the fact that 4πM eff and (ω/γ ) are typically an order of magnitude larger than Fe fourfold anisotropy. The second term consists of an induced shift in the position of the resonance line due to exchange coupling which has been attributed to rotatable anisotropy. The third and fourth terms contain the angular variation of H res caused by the Fe fourfold and coupling-induced unidirectional anisotropies, respectively. The easy direction of the unidirectional anisotropy is given by ϕ E . Both terms H rot and H E are temperature dependent. As noted above, equation (1) is valid only for H res large enough to saturate the magnetization.
A summary of results for effective magnetization and fourfold anisotropy fields determined from fits of H res to equation (1) are summarized in table 1. The fourfold anisotropy field of the 18 ML Fe films increased from ∼440 Oe (average value for all measured samples) at room temperature to ∼680 Oe at 24 K. For 36 ML thick Fe layers, a similar increase in the fourfold anisotropy field was observed (∼540 Oe to ∼710 Oe from room temperature to 24 K ).
The effective demagnetizing field (4πM eff ) value is less than 21.4 kOe at room temperature. A possible explanation for this is the existence of surface anisotropy and uniaxial anisotropy fields, H s and H u , respectively, such that 4πM eff = 4πM S − H u − H S . It is interesting to note that the 4πM eff for the thinner Fe layers decreases slightly with decreasing temperature from ∼15.1 to ∼14.1 kG between room temperature and 24 K. The thicker Fe layers show 4πM eff nearly constant (∼19.0 kG) between room temperature and 24 K. This is suggestive of an out of plane anisotropy induced by the KNiF 3 interface.

Fe/FeF 2 and Fe/KNiF 3 reference structures
The resonance field for the reference bilayers is less than that of the single Fe film data. This reduction is attributed to the AFM-induced dynamic anisotropy field H AFM dyn . The temperature dependence of the resonance field for EA and HA field orientation for bilayer Fe/FeF 2 is given in figure 2(a) and the inset shows the experimental results for the single Fe film. In figures 2(b) and (c) the temperature dependence of H AFM dyn for EA and HA field orientation for reference bilayers Fe/FeF 2 and Fe/KNiF 3 is presented. The values of H AFM dyn were obtained as the difference between the resonance field data of bilayers and the linear fit extrapolated to low temperature. For each sample the linear fit comes from the high-temperature (i.e. greater than blocking temperature) resonance field dependence.
The values for the exchange bias H E were obtained from scans with positive and negative applied magnetic fields. According to equation (1) the asymmetry between resonance fields at each 180 • interval is 2H E .
For Fe(2.60 nm)/FeF 2 (50.1 nm) structure value of H E at 24 K was found to be 185 Oe and is significantly smaller (by a factor of four) than H AFM dyn determined for this sample (see figure 2(b)). At the same time, the FMR results show a low value of H AFM dyn for Fe(2.62 nm)/KNiF 3 (16.0 nm) bilayer (see figure 2(c)). An exchange bias field of H E = 19 Oe was found, in agreement with the value reported by Wee et al [8,9]. Values extracted for H AFM dyn display several interesting features. The first is that H AFM dyn is a non-monotonic function of field orientation. Figure 3 shows the temperature dependence of the H AFM dyn for Fe(2.6 nm)/KNiF 3 (0.8 nm)/FeF 2 (50 nm) structure for three different orientations of the applied field: along an EA, a HA and midway between (IA; ϕ H = 22.  20  We also mention results from field cooling. It has repeatedly been shown that the value of the unidirectional anisotropy H E depends upon the cooling field strength. A stronger cooling field leads to more pinned spins and larger values for the exchange bias. We have tested the dependence of H AFM dyn on cooling field strength and determined that there is   Doubling the thickness of the Fe reduces H AFM dyn by approximately a factor of two for each KNiF 3 thickness, as one would expect for an interface effect. We therefore conclude that the results displayed in figure 4 may be related to spin structure within the KNiF 3 .

FMR linewidth
FMR linewidth is the primary parameter, outside resonance field, considered in the experiment. Figures 5(a), (b)   The FMR linewidth for the single Fe layer displays a linear dependence on temperature. In contrast, the observed temperature changes in linewidth of the exchange coupled systems are strong and non-linear. For example, H of Fe(2.60 nm)/KNiF 3 (0.8 nm)/FeF 2 (50.1 nm) increased 12 times with decreasing temperature to 24 K when the magnetic field was applied along an EA. When the FMR measurements were carried out with the field applied along HA, the observed increment was eighfold.
We proposed an explanation for FMR linewidth behaviour that depends on inhomogeneity existing in the local fields acting on the ferromagnet. Some evidence for this may come from our MOKE magnetometer studies on Fe/FeF 2 structure [23] and the observations carried out by some of us for Fe/KNiF 3 system [8,9]. For these reference bilayers a close correspondence between linewidth and coercive field H C (see figure 5(a)) was noted. Similarly to FMR linewidth theory, the spatial inhomogeneities in the local internal fields can have a large effect on the coercive fields and magnetization loop widths.

Discussion
The temperature dependence of H AFM dyn described above behaves generally like that observed previously for Fe/KNiF 3 bilayers. A rotatable anisotropy was able to describe well FMR results in the previous work. Possible mechanisms underlying the formation of the rotatable anisotropy were discussed in [8,9], and argued to be consistent with measured linewidths. The essential idea is that the Fe exchange couples to different regions of the KNiF 3 interface, and coupling between these regions can produce a rotatable anisotropy and coercivity as observed experimentally.
Similar arguments should apply in the present case. In the case of trilayers additional measurements were carried out in relation to reference bilayers, namely for Fe/KNiF 3 (0.8 nm)/FeF 2 sample temperature dependence of the H AFM dyn for three field orientations (EA, HA, IA) was studied. It turned out that the H AFM dyn is a non-monotonic function of the field orientation. We have at present no evidence to suggest a mechanism for this anisotropy, but can speculate that the presence of the second interface with high anisotropy FeF 2 introduces an additional competition that may affect alignment of regions in the KNiF 3 .
It seems that a great deal of information to verify the introduced model of H AFM dyn could be obtained from studies of Fe/AFM1/AFM2 with single crystalline AFMs. However, the studies of Fe/AFM bilayers, where AFM was deposited onto single crystal of Fe either in the polycrystalline or singlecrystal form, do not give a clear picture. The systematic study of the influence of in-plane crystalline quality of the antiferromagnet on anisotropies in Fe/FeF 2 structure was examined by Fitzsimmons et al [22]. In this study three types of samples were investigated with polycrystalline ferromagnetic Fe thin films and antiferromagnetic FeF 2 as (i) untwinned single crystal; (ii) twinned single crystal and (iii) textured polycrystal. The results obtained suggest that the value of exchange bias depends on many conditions, such as the orientation between the spins in AFM and FM during field cooling; the choice of cooling field orientation relative to the AFM; engineering the AFM microstructure and so on. The bilayer Fe/AFM structures consisting of single-crystal Fe and KNiF 3 , or KCoF 3 , or KFeF 3 films were investigated by some of us [8,9,24,25]. All antiferromagnets in the samples had single crystalline structure or a polycrystalline one with a high degree of texture. The crystalline quality of the antiferromagnets significantly affects the size of training effects, the magnitude of the parameters such as FMR linewidth, the blocking temperature, the exchange bias.
H s of the samples with the single-crystal fluorides were reduced compared with the polycrystalline structures. At the same time, the observed changes of the blocking temperature, the exchange bias, the coercivity versus the crystallinity of the antiferromagnet film were different for different fluorides. It therefore seems indisputable that only the FMR linewidth is a parameter that uniquely altered depending on poly-or single crystalline nature of AFMs.
Perhaps the most curious feature observed was the complex dependence of H AFM dyn on KNiF 3 thickness as illustrated in figure 4. Three different behaviours were observed, corresponding to three thickness regions: 0 to 2 nm, 2 to 6 nm and others greater than 6 nm. The first two regions involve only a few monolayers of KNiF 3 . Some insight into reasons underlying the existence of these three regions can be obtained by considering possible spin configurations in a model trilayer. The reason is that KNiF 3 is a weak antiferromagnet, and so we might expect that strong interlayer coupling to the Fe and FeF 2 might result in large modifications of the spin ordering near the interfaces. Characteristic length scales would correspond to a few nanometres. We explore this idea in some detail with a model described in the appendix. The model makes use of an iterative energy minimization scheme and was used to examine static equilibrium configurations  According to our calculations, for example for trilayer structure Fe/KNiF 3 (3 ML)/FeF 2 , we find that for angles θ H (the direction of the applied field is characterized by the angle θ H ) from 0 • to −95 • the net moments in the first layers of both AFMs oppose the direction of the applied field. Between -95 • and −96 • a sudden change in spin orientation appears in the first layer of the FeF 2 . In order to understand how these results may illuminate our measured results for H AFM dyn , we note first that the canted spin configuration calculated for all KNiF 3 thicknesses is suggestive of Koon's model [26] for exchange bias. In this model, a net moment at the interface is created by spin canting at a compensated interface of the antiferromagnet in contact with the ferromagnet. It was later pointed out by Schulthess and Butler [27] and discussed by Stiles and McMichael [28] that this canting was not itself sufficient to produce bias due to instabilities in the canted configuration. The instabilities allow the canted moment to reverse into a configuration whose energy is the same as the original field cooled orientation energy.
The same principle applies in our case for the KNiF 3 films with the largest thickness. However for small KNiF 3 thicknesses, the energy of the reversed state is not the same as for the field cooled orientation. The difference in energies for the θ H = 0 and θ H = 180 • configurations can be sizable, and is shown in figure 6 as a function of KNiF 3 thickness where 1 ML = a(KNiF 3 ) = 0.4013 nm (Muller et al [10]).
This asymmetry in energies disappears when the KNiF 3 is thicker than 4 ML. This thickness is characteristic of the depth over which a twist can develop in the KNiF 3 . Above 4 ML, the twist can extend to 180 • , and behave like the 'partial' wall in the Stiles and McMichael picture of exchange bias [28,29].
On the basis of this, we suggest that H AFM dyn is associated with the gradient of the energy as a function of θ H , and that this energy is different for small and large θ H for KNiF 3 thicknesses less than a domain wall width. KNiF 3 thicknesses sufficiently larger than this support formation of a partial wall that effectively isolates the Fe from the KNiF 3 interface. This results in an energy whose gradient is the same for 0 • and 180 • orientation of the field. As such, the H AFM dyn arises in this model as a kind of susceptibility of the spin configuration to the orientation of the Fe magnetization (which is controlled by the external applied field).
In this model for H AFM dyn , the smallest KNiF 3 thickness region creates asymmetry in energies at 0 • and 180 • because the KNiF 3 is not wide enough to support a complete partial wall. In this region the effect of FeF 2 is apparent. The thickest KNiF 3 films support a complete partial wall, and the resulting H AFM dyn are associated with energy gradients that are symmetric with respect to reversal of the applied field. The intermediate thickness region allows spin configurations that are strongly thickness dependent with strongly distorted partial walls. The H AFM dyn in this region vary strongly with KNiF 3 thickness, in a manner determined by details of configurational energies of the distorted partial wall. Lastly, we note that this picture can also explain the dependence of H AFM dyn on field orientation with respect to easy and hard directions. This follows because the magnitude of H AFM dyn will depend on orientation of the Fe with respect to anisotropy symmetry axes.

Conclusions
In summary, we show that a ferromagnet exchange coupled to an antiferromagnetic bilayer can allow the character and strength of exchange anisotropy to be modified. We have studied using FMR exchange bias and exchange anisotropy for Fe exchange coupled to KNiF 3 /FeF 2 . The trilayer was grown by MBE. The temperature dependence of the ferromagnetic resonance peak position shows a characteristic negative shift of the resonance from that of a single Fe layer. This negative shift is a direct result of the exchange coupling between ferromagnetic and antiferromagnetic layers and results in a dynamically induced field H AFM dyn . The value of H AFM dyn is different for measurements performed along the ferromagnets' easy and hard axes for all the bilayer and trilayer samples. The largest values were found for the field along Fe hard axis. The dependence of the H AFM dyn on the thickness of the KNiF 3 is non-monotonic, reaching a minimum for 2.0 nm thick KNiF 3 . For structures with thinner KNiF 3 , the magnitude of the H AFM dyn increases with decreasing thickness, reaching the maximum expected for the bilayer Fe/FeF 2 .
Results from a calculational model for the equilibrium configuration of spins in a ferromagnet/antiferromagnet1/ antiferromagnet2 trilayer suggest that a form of spin canting may occur at the antiferromagnetic interfaces. For sufficiently thin KNiF 3 , significant spin canting at the Fe and FeF 2 interfaces occurs due to exchange coupling. Effects of the FeF 2 are effectively isolated by KNiF 3 thick enough to support a partial magnetic domain wall. As a general result, we suggest that H AFM dyn is a measure of the susceptibility of the interface spin ordering to interface coupling in the KNiF 3 .
A numerical mean field model for equilibrium spin orientations in a ferromagnet/antiferromagnet/antiferromagnet is described in this appendix. The spins are treated as classical vectors on a lattice.
The energy of a spin configuration is determined in the following way. An atomistic approach is employed in a mean field approximation. A cubic lattice of vector spins is considered consisting of N layers representing the first antiferromagnet (called AFM1 and representing the KNiF 3 ) and M layers representing a second antiferromagnet (called AFM2 and representing the FeF 2 ). The outermost layer of AFM1 is in contact with a block spin representing the FM. Each layer is described by a unit cell of spins representing the two antiferromagnetic sublattices. Periodic boundary conditions are assumed in the plane of each layer.
The ground state spin configuration is found as follows. The applied magnetic field is set, and a spin site in the FM/AFM1/AFM2 trilayer is chosen. The algorithm is begun by picking a set of values for the initial configuration of spins in each layer corresponding to a field cooled orientation. Exchange coupling between spins is taken into account in addition to anisotropy energies, so that the energy of the spins in any given layer depends on the orientation and magnitude of the spins in the nearby layers. The orientation of a spin in layer 'n' is characterized by the angle θ n carried out with respect to the applied field. A low energy magnetic configuration is obtained when the energy is minimized [30], i.e. the angle θ n is determined by rotating the spin into the direction of a local effective field calculated from the gradient of the energy at the spin site. The procedure determines the configuration in the following manner: (i) a given layer is randomly chosen and the spins in that layer are rotated to be parallel to the local effective field; (ii) the process is continued until one has a self-consistent, stable state where all the spins are aligned with the local effective fields produced by the neighbouring spins.
In this model, the multilayer is treated as an effective one-dimensional system with one spin representing an entire sublattice in a given layer. Only nearest-neighbour exchange coupling is considered. The energy is defined by where H AFM is the Hamiltonian representing a given antiferromagnet with its both sublattices, and H int is the Hamiltonian describing the interface contribution.
The geometry is defined in figure A1. The direction normal to the layer planes is y. Each xz plane, for a given value of y, depicts one layer of an AFM.
Both AFMs, KNiF 3 and FeF 2 , exhibit uniaxial anisotropy with easy axes in the +z and −z directions. The first AFM is KNiF 3 and is known G-type meaning that the nearest-neighbour coupling is antiferromagnetic. The nearestneighbour exchange constants for each AFM1 layer (J NNP ) and between AFM1 layers (J NN ) are defined as negative values. Each layer of the KNiF 3 is assumed to be compensated such that both AFM1 sublattices are present within a given layer. The second AFM is FeF 2 and is regarded as completely uncompensated within a given layer. In this case, the J NNP2 exchange coupling constant along x is defined as positive.  The exchange coupling constants are defined as • J NN , J NN2 are the exchange interaction constants between spins at 'n' and 'n + m' layers of AFM1 and AFM2, respectively, • J NNP , J NNP2 are the exchange interaction constants within nth layer of AFM1 and AFM2, respectively, • J FAF is the interface exchange coupling constant between the FM layer and the first antiferromagnet, • J 12 is the interface exchange coupling constant between the first and the second AFM1/AFM2 layer.
Note that the sign of these exchange interactions has been introduced through the numerical values of the parameters.
All the values of anisotropy and exchange coupling are given in field units and are summarized in table 3. All calculations were carried out for 50 atomic layers in AFM2. An example arrangement of spins in the AFM1 and AFM2 is depicted in figure A1. Bulk values of the anisotropy and the exchange coupling constants for KNiF 3 and FeF 2 have been used in all calculations. The interface exchange coupling constant between AFMs (J 12 ) have been assumed to be a geometrical mean of the exchange coupling constants of the adjacent layers. The interface exchange coupling between FM and AFM1 is defined as ZJ NN , i.e. the exchange coupling within the first AFM1 multiplied by the number of nearest neighbours of each Fe spin site.
After converging to a stable configuration, the total energy per spin site was calculated as a function of the angle of the applied magnetic field, θ H . Example results are shown in figure A2(a) for different thicknesses of the first AFM1.
Note that the horizontal axis (θ H ( • )) is the same for all calculated structures, but the total calculated energy is not to scale in order to visibly depict the calculated results for each thickness of AFM1. A discontinuity in the total energy per spin site appears for thicknesses of AFM1 ranging from 1 ML to 9 ML. The values of angle θ H for which the discontinuity appears (θ Hdis ) is presented in figure A2(b). Three regions can be distinguished in these results: 1 ML to 4 ML, 5 ML to 9 ML and >9 ML.
The different regions correspond to different spin arrangements within each AFM layer. An example for the 3 ML thick AFM1 is shown in figure A3. The spin arrangements are presented in terms of the angle that each spin vector makes with the z-direction. Here, the blue arrows represent the applied field, and red arrows are used for the spins in the three layers of AFM1 and green arrows represent the first few spin sites of AFM2.
The comparison between a spin arrangement for θ H = −95 • (top diagrams) and for θ H = −96 • (bottom diagrams) is shown. The central diagram in figure A3 shows the spin arrangement when the field is applied along the +z direction. The first layer of AFM1 experiences spin canting. An energy minimum for the fully compensated interface has been obtained for perpendicular interfacial coupling between the FM and AFM spins. The spin canting and the resulting net moment induced within the first layer of the AFM1 always opposes the direction of the applied field. Moreover, spin canting appears also in the second antiferromagnet AFM2 for  Figure A2. (a) The total energy per spin site as a function of the angle of the applied magnetic field θ H for different thicknesses of the first AFM1. The horizontal axis (θ H ( • )) is the same for all calculated structures, but the total calculated energy is not to scale in order to visibly depict the calculated results for each thickness of AFM1, (b) the values of angle θ H for which the discontinuity appears (θ Hdis ) for thicknesses of AFM1 ranging from 1 ML to 9 ML. The lines are a guide to the eye. Figure A3. The spin arrangements are presented in terms of the angle that each spin vector makes with the z-direction for the 3 ML-thick AFM1. Blue represents the applied field H a , red arrows are used for the spins in the three layers of AFM1 and green represents the first few spin sites of AFM2. the first two layers. It is important to note that the direction of the net moment induced in the first layer of AFM2 does not correlate directly with the direction of the applied magnetic field.