# The effect of iron layer thickness on the interlayer exchange coupling in
Fe/MgO (001) superlattices

Anna L. Ravensburg  Department of Physics and Astronomy, Uppsala University,
Box 516, 75120 Uppsala, Sweden  Matías P. Grassi  Department of Physics and
Astronomy, Uppsala University, Box 516, 75120 Uppsala, Sweden  Björgvin
Hjövarsson  Department of Physics and Astronomy, Uppsala University, Box 516,
75120 Uppsala, Sweden  Vassilios Kapaklis  Department of Physics and
Astronomy, Uppsala University, Box 516, 75120 Uppsala, Sweden

###### Abstract

We describe the effect of the Fe layer thickness on the antiferromagnetic
interlayer exchange coupling in [Fe/MgO]N superlattices. An increase in
coupling strength with increasing Fe layer thickness is observed, which
highlights the need of including the extension of both the layers when
discussing the interlayer exchange coupling in superlattices.

††preprint: APS/123-QED

##  I Introduction

Epitaxial Fe/MgO heterostructures have sparked an intense research effort due
to their large tunneling magnetoresistance [1, 2]. High crystal quality
trilayer Fe/MgO/Fe heterostructures exhibit antiferromagnetic interlayer
exchange coupling (IEC) [3, 4, 5, 6] mediated by spin-polarized tunneling
through the MgO barrier [3, 7, 8]. The coupling strength is exponentially
decaying with increasing MgO layer thickness $t_{\text{MgO}}$ [9, 10, 11]. In
[Fe/MgO]N superlattice structures with $N$ Fe/MgO bilayers, the interlayer
exchange coupling has also been observed to exponentially decay with
$t_{\text{MgO}}$ between 16.3 and 22.1 Å MgO [5]. Furthermore, in these
epitaxial [Fe/MgO]N superlattices, discrete layer-by-layer magnetic switching
is observed, resulting from the competition between the interlayer coupling
and magnetocrystalline anisotropy [5]. However, understanding and tuning the
sequential magnetic switching in these heterostructures and the precise
mechanism governing it has been proven challenging [6, 12]. The MgO layer
thickness, impurities or defects in the MgO layer, temperature, and growth
conditions are important parameters affecting the interlayer coupling [3, 13,
9, 14, 15, 5, 11, 6]. Oxygen vacancies [13, 9], along with magnetic impurities
which might be present in the MgO layers [15] have also been shown to be
relevant. Finally, a non-trivial dependence of the coupling strength in
[Fe/MgO]N superlattices on the number of bilayer repetitions, i.e., the
extension of the whole stack has been reported [12].

Systematic variations of the MgO layer thickness in previous works [9, 10, 11,
5] relate to the modification of the tunneling barrier. On the other hand, the
study of the dependence on $N$ in superlattices, highlights potential
collective effects between quantum well states in the Fe layers making up the
[Fe/MgO]N superlattices [12]. A detailed investigation of the Fe layer
thickness dependence, would thus shed light on the importance of the extension
of the quantum wells formed in the Fe layers for the interlayer coupling
strength. To this end, we investigate the influence of the Fe layer thickness
on the IEC and sequential switching in [Fe/MgO]N superlattices. The Fe layer
thickness was varied while the MgO layer thickness was kept constant. We
present a detailed characterization of the layering and crystal structure of
the superlattices, employing x-ray scattering techniques, and report on the
increase of the IEC with increasing Fe layer thickness.

##  II Methods

Superlattices of [Fe/MgO]N with $N=$ 8 and 10 bilayer repetitions were
deposited on single crystalline MgO ($001$) substrates by direct current (dc)
and radio frequency (rf) magnetron sputtering. Within one deposition process,
identical copies were grown on a 10$\times$10 mm2 and a 20$\times$20 mm2 sized
substrate, eliminating this way all uncertainties concerning differences in
growth conditions. The thicknesses of the MgO layers were kept constant at
$t_{\text{MgO}}$ = 17(1) Å, while the thickness of the Fe layers was chosen to
vary in the range of $t_{\text{Fe}}$ = 11 to 23 Å. Prior to the deposition,
the substrates were annealed in vacuum at 1273(2) K for 600 s. The base
pressure of the growth chamber was below 5$\times$10-7 Pa. The target-to-
substrate distance in the deposition chamber was approximately 20 cm. The
depositions were carried out in an Ar atmosphere (gas purity $\geq$ 99.999 %,
and a secondary getter based purification) from an elemental Fe (50 W, dc,
diameter: 5.08 cm) and an MgO compound target (50 W, rf, diameter: 5.08 cm) at
438(2) K and 0.67 Pa. Target power and deposition temperature were optimized
previously with respect to well-defined layering in combination with the
highest crystal quality. In order to prevent surface oxidation of the films,
the samples were capped at ambient temperature ($<$ 313(2) K) and 1.07 Pa with
50 Å Pt (50 W, dc, diameter: 5.08 cm). The targets were cleaned by sputtering
against closed shutters for at least 60 s prior to the deposition of each
layer. The deposition rates (Fe: 0.1 Å/s, MgO: 0.01 Å/s, Pt: 0.8 Å/s) were
calibrated prior to the growth of the samples. In order to ensure thickness
uniformity, the substrate holder was rotated at 30 rpm during the deposition.

Figure 1: a) X-ray scattering pattern of a [Fe/MgO]10 superlattice with 20(1)
Å Fe layer thickness. In b) x-ray reflectivity is displayed separately
including a fit. The scattering length density profile based on the fit is
displayed in the inset. c) X-ray diffraction of [Fe/MgO]10 superlattice
samples with different Fe layer thicknesses.

X-ray reflectometry (XRR) and diffraction (XRD) were carried out using a Bede
D1 diffractometer equipped with a Cu K${}_{\alpha_{1}}$ x-ray source operated
at 35 mA and 50 kV. For monochromatizing the beam by reducing the Cu Kβ
radiation, the setup included a Göbel mirror and a Ni filter on the incidence
and detector side, respectively. A circular beam mask (diameter: 0.005 m),
incidence and detector slits (both 0.0005 m) were also used. The x-rays were
detected with a Bede EDRc x-ray detector. The instrument angles for the
coupled $2\theta$-$\theta$ scans were aligned to the sample surface for XRR
and to the [Fe/MgO] crystal planes for XRD measurements. The measured XRR data
was fitted using GenX [16, 17] enabling the determination of the scattering
length density (SLD) profile, which includes information on layer thickness
and roughness. Rocking curve measurements in XRD were fitted with a Pseudo-
Voigt function to mimic the effect of mosaic spread.

Magnetization measurements were performed at ambient temperature using a
longitudinal magneto-optical Kerr effect (L-MOKE) setup with $s$-polarized
light. The magnetic response was measured parallel to an in-plane applied
magnetic field along the Fe magnetic easy and hard axes, i.e. Fe [$100$] and
Fe [$110$], respectively. The data was averaged over 10 full loop recordings.
Additional measurements were conducted in an Evico Magnetics KerrLab Kerr
microscope in longitudinal and transverse mode. The determined saturation
fields are corrected for the coercivity. Displayed error bars of the
saturation fields and the remanence magnetization correspond to the standard
deviation of the extracted parameters from the two symmetrical axes
(90-degrees sample rotation), two field-scan directions, and two field
polarities.

##  III Results and Discussion

###  III.1 Layering and crystal structure

Figure 2: Magnetic hysteresis loops of [Fe/MgO]10 superlattices measured with
L-MOKE on 10$\times$10 mm2 samples with varying Fe layer thickness with an
applied field along the a) Fe magnetic easy axis, i.e., Fe [$100$] and b) Fe
magnetic hard axis, i.e., Fe [$110$].

A full x-ray scattering pattern of an [Fe/MgO]10 superlattice with 20(1) Å Fe
and 17(1) Å MgO layer thicknesses is displayed in Fig. 1a. Below 10 degrees in
2$\theta$, pronounced Kiessig fringes [18] are observed, consistent with the
presence of flat and distinct layering with low layer roughness (below 5(1)
Å), which is confirmed in the scattering length density (SLD) profile based on
a fit displayed in Fig. 1b. All nominal Fe and MgO layer thicknesses within
this study were determined using XRR. Analysis of the results confirmed that
the MgO layer thickness was the same, 17(1) Å, while the thicknesses of the Fe
layers were determined to be 11(1) Å, 15(1) Å, 20(1) Å, and 23(1) Å.
Henceforth, all presented thicknesses correspond to the experimentally
determined thicknesses of the Fe and MgO layers.

Diffraction peaks at around 39 and 43 degrees in 2$\theta$ are observed,
arising from the Pt capping and the MgO ($001$) substrate, respectively. Laue
oscillations around the Pt Bragg peak are obtained for all the samples, see
Fig. 1c, providing conclusive evidence for well-defined texture and thickness
of the Pt layer [19]. Fe/MgO superlattice peaks are visible between 52 and 70
degrees in 2$\theta$. The results of the analysis of the diffraction data are
consistent with the ones obtained from the XRR analysis. No Laue oscillations
are visible between the superlattice Bragg peaks for [Fe/MgO]8 and [Fe/MgO]10.
This is attributed to the mismatch between the layers, giving rise to defects
causing incoherent scattering from the Fe atomic planes [20, 21]. The out-of-
plane atomic distances in Fe and MgO are 2.866 Å [22] and 4.212 Å [23],
respectively. Hence, the atomic step heights in the Fe and MgO lattices are
incommensurate, yielding finite terrace widths in the superlattice, which can
suppress both, diffraction peaks as well as Laue oscillations [24, 19]. This
is in stark contrast to, e.g., the growth of [Fe/V]N (001) superlattices,
within which the layers have similar lattice parameters [25]. Substrate
twinning is common in commercially available MgO substrates [26]. More twins
and a larger mosaic spreads were observed for 20$\times$20 mm2 compared to
10$\times$10 mm2 substrates. In specular scans for samples on 20$\times$20 mm2
substrates, multiple reflections attributed to asymmetric substrate peaks were
observed, indicating defects in the single crystalline structure. This
difference in crystal quality between the substrates is found to alter the
magnetic properties in [Fe/MgO]N superlattices (see Supplemental Material
(SM)). Therefore, a comparison between samples, in particular for different
(substrate) batches, has to be done with care, as observations in magnetic
properties need to be related to the crystal structure.

###  III.2 Magnetic characterization

Magnetic hysteresis loops of [Fe/MgO]10 superlattices, measured with an
applied magnetic field along the Fe easy axis, are displayed in Fig. 2a.
Sequential switching of the layers in the superlattices is apparent from the
presence of discrete steps in the field response. Coming from positive
saturation, the first switching occurs at a positive applied field, in
agreement with an antiferromagnetic IEC of the Fe layers in all samples [5,
6]. Kerr microscopy revealed domain wall nucleation and motion during the
magnetization reversal, as expected in sequential switching of the magnetic
layers. The hysteresis loops obtained for the samples having 8 bilayers, are
shown in the SM.

Figure 3: Remanent magnetization $M_{R}$ normalized to the saturation
magnetization $M_{S}$ displayed over Fe layer thickness $t_{\text{Fe}}$ for
[Fe/MgO]N superlattices with 8 and 10 bilayer repetitions $N$ on 10$\times$10
mm2 substrates for an applied magnetic field along the Fe easy axis.

The remanent configuration changes with Fe layer thickness. For the sample
with the thinnest Fe layers, 11(1) Å, the remanent state corresponds to
0.48(6)$\times$MS, consistent with every other layer pointing along the
sensitivity axis and the others oriented perpendicular to that [6, 12]. The
inferred 90 degrees configuration could either be enabled by the presence of
an activation barrier for the switching or by the presence of a biquadratic
coupling [15]. A 90 degrees remanent configuration has been observed for
[Fe/MgO]10 superlattices with thicker MgO barriers of 19.6 Å, which was
attributed to the coupling strength being smaller than the nucleation field of
the domain reversal in the samples [6]. In contrast, the remanent
magnetization in the samples with larger Fe layer thickness is closer to being
compensated, i.e., 0.151(3)$\times$MS, 0.039(4)$\times$MS, and
0.031(5)$\times$MS for the samples with 15(1), 20(1), and 23(1) Å Fe layer
thickness, respectively. A compensated remanent state corresponds to a fully
antiferromagnetic alignment and was reported for [Fe/MgO]10 with thinner MgO
barriers of 16.4 Å [6], having larger coupling strength as compared to the
thicker layers [5]. In these samples, the coupling between the Fe layers seems
however sufficiently strong to drive the order to an antiferromagnetic
remanent state. The normalized remanent magnetization $M_{R}/M_{S}$ is plotted
as a function of Fe layer thickness in Fig. 3 for [Fe/MgO]8 and [Fe/MgO]10.
Independent of whether a superlattice consists of 8 or 10 bilayer repetitions,
the remanent magnetization decreases with increasing Fe layer thickness.

The same relative change in remanent magnetization $M_{R}/M_{S}$ with Fe layer
thickness is observed in loops measured with an applied field along the Fe
magnetic hard axis, which are displayed in Fig. 2b for [Fe/MgO]10
superlattices. Here we notice that more than four times larger fields are
required to overcome the crystalline anisotropy as compared to the saturation
field along the easy axis of the samples. Differences in the switching are
also noticed for the two magnetic field directions, where discrete steps are
observed up until 2 to 5 mT, when the field is applied along the hard axis of
the samples. The magnetization at these fields is approximately
0.63$\times$MS, indicating a mixture of switching and coherent rotation of
layers. Hence, when the applied field is along the easy axis, the individual
layers do switch, one or more at the time, which is not observed when the
field is applied along the hard axis. Above the threshold, the change in the
alignment of magnetization of the Fe layers appears to be dominated by
coherent rotation for fields along the hard axis. The saturation fields along
the easy and hard axes, $H_{S}^{\text{EA}}$ and $H_{S}^{\text{HA}}$, were
extracted from the hysteresis loops. The results are shown in Fig. 4. As seen
in the figure, the saturation field increases with increasing Fe layer
thickness in both cases.

Figure 4: Saturation field $H_{S}$ displayed over Fe layer thickness
$t_{\text{Fe}}$ for [Fe/MgO]N superlattices with 8 and 10 bilayer repetitions
$N$ on 10$\times$10 mm2 substrates for an applied magnetic field along a) the
Fe easy axis, b) the Fe hard axis.

To obtain quantitative determination of $J$ and $K$, we define the energy
density $E_{i}$ of the layer $i$ as:

$\begin{split}E_{i}=&-\mu_{0}M_{S}H\,\mathrm{cos}(\theta_{i})+\frac{K}{4}\,\mathrm{sin}^{2}(2\theta_{i})\\\
&-J\,\mathrm{cos}(\theta_{i}-\theta_{i+1})-J\,\mathrm{cos}(\theta_{i}-\theta_{i-1}),\end{split}$
(1)

where $M_{S}=1.71$ kA/m is the saturation magnetization [27], $\mu_{0}$ is the
vacuum permeability, $H$ is the applied field, and $\theta_{i}$ is the angle
of the magnetization of the layer $i$ measured from the easy axis. For the
outermost layers, with only one neighboring Fe layer present, one of the last
two terms vanishes. With this definition, $J$ is negative for an
antiferromagnetic IEC.

Figure 5: a) Fe cubic anisotropy $K$ and b) volumetric coupling strength $J$
displayed over Fe layer thickness $t_{\text{Fe}}$ for [Fe/MgO]N superlattices
with 8 and 10 bilayer repetitions $N$ on 10$\times$10 mm2 substrates
determined from hysteresis loops measured with an applied magnetic field along
the Fe easy and hard axes.

From this expression, it is possible to calculate the saturation field along
the hard axis $\ H_{S}^{\text{HA}}$ (see SM) as:

$\,H_{S}^{\text{HA}}=\frac{2\,K-4\,J}{\mu_{0}\,M_{S}}.$ (2)

The dependence of the saturation field along the easy axis $H_{S}^{\text{EA}}$
as a function of $K$ and $J$ is not accessible in an analytical form for a
multilayer structure [28, 29] and has therefore to be extracted from
simulations of the data. The functional dependence of $H_{S}^{\text{EA}}(K,J)$
was calculated, and it was found that it decays with $K$, converging to
$H_{S}^{\text{EA}}=-2J/M_{S}$ in the limit of large $K/|J|$ (for details see
SM). With both experimental values for $H_{S}^{\text{HA}}$,
$H_{S}^{\text{EA}}$ and Eq. 2 as well as $H_{S}^{\text{EA}}(K,J)$ (see SM), it
is possible to calculate the values of the coupling strength $J$ and the Fe
cubic anisotropy $K$, which are shown in Fig. 5. The anisotropy is determined
to be weakly changing with Fe thickness (60(4) kJ/m3) in the range studied
here. For 20 Å Fe on MgO, hence in a comparable thickness range, a $K=56$
kJ/m3 was reported [30]. Moreover, in line with the findings about the
remanent state being affected by the Fe layer thickness, the antiferromagnetic
coupling strength is found to increase with Fe layer thickness, independent of
the number of bilayer repetitions.

##  IV Summary

The antiferromagnetic coupling strength [Fe/MgO]N superlattices with 17 Å
thick MgO layers is found to linearly increase with Fe layer thickness, in the
range between 11 and 23 Å. The Fe cubic anisotropy was determined to be weakly
increasing in the same thickness range, being close to the value reported for
Fe single layers. These findings indicate that the coupling in [Fe/MgO]N
superlattices is not solely defined by the thickness of the MgO barriers, but
along with the number of bilayer repetitions $N$ [12], the thickness of the
individual Fe layers is also an essential parameter. This calls for a closer
look on the coupling mechanism and effects in such superlattices, highlighting
the importance of all length scales present in these ($t_{\text{MgO}}$,
$t_{\text{Fe}}$, and $N$($t_{\text{MgO}}+t_{\text{Fe}}$)). These observations
challenge our current understanding of electronic and spin effects in magnetic
quantum well structures comprised of metal-oxide heterostructures, with
potential impact on emergent technologies employing their magnetic properties.

###### Acknowledgements.

VK would like to acknowledge financial support from the Swedish Research
Council (Project No. 2019-03581). MPG and VK also acknowledge support from the
Carl Trygger Foundation (Project No. CTS21:1219).

## Data availability

The data that support the findings of this study are available from the
authors upon reasonable request.

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Supplemental Material: The effect of iron layer thickness on the interlayer
exchange coupling in Fe/MgO (001) superlattices

##  S-I Magnetic hysteresis loops of superlattices with 8 bilayer repetitions

The magnetic hysteresis loops of [Fe/MgO]8 superlattices with varying Fe layer
thickness, measured with an applied field along the Fe magnetic easy and hard
axis, are shown in Supplementary Fig. 1a and b, respectively. Qualitatively,
the loops exhibit the same shape as the loops recorded for superlattices with
10 bilayer repetitions. All extracted values for relative remanent
magnetization $M_{R}/M_{S}$ and saturation fields $\mu_{0}H_{S}^{EA}$ and
$\mu_{0}H_{S}^{HA}$ are displayed in the main text.

Supplementary FIG. 1: Magnetic hysteresis loops of [Fe/MgO]8 superlattices
measured with L-MOKE on 10$\times$10 mm2 samples with varying Fe layer
thickness with an applied field along the a) Fe magnetic easy axis, i.e., Fe
[$100$] and b) Fe magnetic hard axis, i.e., Fe [$110$].

##  S-II Dependence of the magnetic properties on the substrate size

All analysis presented in the main text was conducted on [Fe/MgO]N
superlattices grown on 10$\times$10 mm2 MgO ($001$) substrates. However,
copies of the deposited [Fe/MgO]10 superlattices on 20$\times$20 mm2 MgO
($001$) substrates show differences in the magnetization reversals, displayed
in Supplementary Fig. 2, despite an identical layering and a similar crystal
structure. All 20$\times$20 mm2 samples exhibit a remanent state around
0.5(0)$\times$MS corresponding to a 90 degrees orientation of the
magnetization in the Fe layers at remanence. The sample with 15(1) Å Fe layer
thickness shows a slightly lower remanent magnetization but exhibits
0.5(0)$\times$MS already at applied fields below 1 mT. A strong dependence of
the shape of the magnetic hysteresis loop on the measured position on the
sample was found. Furthermore, the saturation field of the [Fe/MgO]10
superlattices on 20$\times$20 mm2 MgO ($001$) substrates is significantly
lower and in the order of 10 to 15(1) mT, compared to their copies on
10$\times$10 mm2 substrates. Based on the previous analysis of remanent state
and saturation field, the coupling strength in the samples on 20$\times$20 mm2
is smaller compared to their 10$\times$10 mm2 counterparts. Nevertheless, the
trends observed for the saturation fields with Fe layer thickness were found
to be valid for both, 20$\times$20 mm2 and 10$\times$10 mm2 samples in
relative comparison.

Supplementary FIG. 2: Magnetic hysteresis loops of [Fe/MgO]10 superlattices
measured with a Kerr microscope on 20$\times$20 mm2 samples with varying Fe
layer thickness with an applied field along the Fe magnetic easy axis, i.e.,
Fe [$100$].

The change in magnetic properties is attributed to differences in the
substrate quality. As the switching mechanism in [Fe/MgO]N superlattices is
found to rely on domain wall movements, crystallographic defects may inhibit
this motion and lower the coupling strength between the layers. More surface
area of an MgO ($001$) substrate increases the probability for the presence of
crystallographic twins and an increase in mosaic spread, i.e., crystal
misorientation, in line with the analysis of the crystal structure of the
superlattices on these substrates, presented above. Therefore, the coupling
strength in [Fe/MgO]N superlattices is found to be highly dependent on
substrate quality and crystal structure/quality.

##  S-III Dependence of the easy-axis saturation field on $K$ and $J$

The dependence of $H_{S}^{EA}$ as a function of the antiferromagnetic IEC $J$
and the cubic anisotropy $K$ only becomes trivial in the limit of large $K$
[1, 2], where:

$H_{S}^{\text{EA}}=\frac{2\,J}{\mu_{0}\,M_{S}}.$ (S1)

In this limit, the switching process is modelled by a first-order transition
where the magnetization of one layer is exactly 90 or 180 degrees from the
direction defined by the applied field. This process is known as spin-flip.
When the cubic anisotropy is weaker, the magnetization of the different layers
can explore a broader range of configurations in order to minimize the energy.

In the case where the magnetization of one layer rotates to $\theta=\pi/2$,
the system can further minimize its energy if the magnetization of the two
neighboring layers drives away from saturation. This could be understood from
the large slope of the term $J\,\mathrm{cos}(\theta_{i}-\theta_{\pm 1})$ in
Eq. 1 in the main text, when $\theta_{i}-\theta_{\pm 1}=\pi/2$, and the small
slope of $\mu_{0}HM_{S}\,\mathrm{cos}(\theta_{\pm 1})$ when $\theta_{i}=0$.
Therefore, it is expected that the value of $H_{S}^{\text{EA}}$ depends on
both $K$ and $J$ as $H_{S}^{EA}=F(J,K)$. This dependence can be written as
$h=\frac{f(k)}{\mu_{0}M_{S}}$, where $h=H_{S}^{\text{EA}}/|J|$ and $k=K/|J|$
are adimensional parameters. In order to explore this dependence, Eq. 1 in the
main text was used to determine the angle $\theta_{\pm 1}$, which minimizes
the energy, and the field at which the spin-flip becomes energetically
favorable. The minimal model to study the deviation from Eq. S1 requires
considering 5 layers. To calculate the reduced saturation field $h$, a
switched state is proposed, and its energy is compared with the saturated
state. In the switched configuration, the angles $\theta_{i}$ of the
magnetization of the layer $i$ are:

$\begin{split}\theta_{1}=\theta_{5}=0,\\\ \theta_{2}=\theta_{4}=\delta,\\\
\theta_{3}=\pi/2.\end{split}$ (S2)

It must be noted that if $\theta_{3}$ is unconstrained, it always minimizes
the energy at the exact angle of $\pi/2$. By taking a step in $\delta$ of
0.009 rad, and 0.0011 in $h$ it was possible to obtain $f(k)$. A similar
procedure was followed to calculate the reduced field related to a spin-flop
transition (second-order transition), where the system transitions from
saturation to a configuration with
$\theta_{\text{odd}}=-\theta_{\text{even}}$. In this case, only two layers
have to be considered. However, to take the boundary conditions imposed by the
outermost layers into account, the IEC term in Eq. 1 in the main text has to
be multiplied by a factor $(N-1)/N$, where $N$ is the amount of layers of the
superlattice. The results for both spin-flip and spin-flop are shown in
Supplementary Fig. 3. For values under $k=1$, spin-flop dominates at higher
fields.

Supplementary FIG. 3: Calculated values of $f$ as a function of the reduce
parameter $k=K/|J|$. For each $k$, the highest value dominates the transition
from saturation.

It can be observed that in the limit of large $k$ the factor $f(k)$ for spin-
flip converges to 2 in agreement with Eq. S1. From Postava  _et al._ [3] and
Eq. S1 is it possible to estimate that $k$ should lie close to 3. Therefore,
$f(k)$ can be approximated in the range $k\in[2.8;\,3.9]$ by a linear function
$f_{l}(k)=A\,k+B$. With this simplified expression, the saturation field can
be calculated as:

$H_{S}^{\text{EA}}=\frac{A\,K+B\,J}{\mu_{0}\,M_{S}},$ (S3)

where $A=-$0.04 and $B=-$2.26.

##  S-IV Dependence of the hard-axis saturation field on $K$ and $J$

For extracting values for the coupling strength from hard axis hysteresis
loops, a ferromagnetic multilayer with a fourfold magnetocrystalline
anisotropy is considered. The interlayer exchange coupling is
antiferromagnetic ($J<0$). When the external magnetic field $\bm{H}$ is
reduced from saturation, each layer system goes through a second-order
transition at the critical field $H_{S}^{HA}$, as shown in Supplementary Fig.
1b or in the main text. At lower fields, the angle $\theta_{i}$ between the
magnetization of each layer $\bm{M}_{i}$ and the magnetic field takes a non-
zero value. Therefore, the total magnetization along the field direction can
be calculated as $\frac{M_{S}}{N}\sum^{N}_{i}\mathrm{cos}(\theta_{i})$, where
$N$ is the number of layers.

However, if $N>2$, the layers with two neighbors present a higher saturation
field $H_{S}^{HA}$ which can be extracted from the measured hysteresis loops.
For each of these "middle" layers, if even and odd layers angles satisfy
$\theta=-\theta$, the energy density per layer is reduced to:

$E=-\mu_{0}M_{S}H\,\mathrm{cos}(\theta)-\frac{K}{4}\,\mathrm{sin}^{2}(2\theta)-J\,\mathrm{cos}(2\,\theta).$
(S4)

From here, the minima can be found from:

$\frac{\partial
E}{\partial\theta}=\mu_{0}M_{S}H\,\mathrm{sin}(\theta)-K\,\mathrm{sin}(2\theta)\,\mathrm{cos}(2\theta)+2J\,\mathrm{sin}(2\,\theta)=0.$
(S5)

Using $\mathrm{sin}(2\theta)=2\,\mathrm{sin}(\theta)\mathrm{cos}(\theta)$,
this equation can be written as:

$\mu_{0}M_{S}H\,\mathrm{sin}(\theta)-4K\,\mathrm{sin}(\theta)\mathrm{cos}(\theta)\,\mathrm{cos}(2\theta)+4J\,\mathrm{sin}(\theta)\mathrm{cos}(\theta)=0.$
(S6)

One trivial solution is $\theta=0$ and it represents the ground state at
$H>H_{S}^{HA}$. We know that below the saturation field, this minimum will no
longer be the ground state and the system will transition to a solution with
$\theta\neq 0$. However, at the critical field, this second solution should
present $\theta=0$. Therefore, by discarding the trivial solution and imposing
$\theta=0$ we obtain $\mu_{0}M_{S}H-8K+4J=0$, and the critical field is
calculated as:

$H_{S}^{\text{HA}}=\frac{2\,K-4\,J}{\mu_{0}M_{S}}.$ (S7)

At this field, all the layers are expected to go through a soft transition.
From this expression and Eq. S3, it is possible to calculate the values of $J$
and $K$ from $H_{S}^{\text{HA}}$ and $H_{S}^{\text{EA}}$.

## References

  * Dieny  _et al._ [1990] B. Dieny, J. Gavigan, and J. Rebouillat, J. Phys. Condens. Matter. 2, 159 (1990).
  * Folkerts [1991] W. Folkerts, J. Magn. Magn. Mater. 94, 302 (1991).
  * Postava  _et al._ [1997] K. Postava, H. Jaffres, A. Schuhl, F. N. Van Dau, M. Goiran, and A. Fert, J. Magn. Magn. Mater. 172, 199 (1997).