Also at ]Institute of Physics, NASU, 03028 Kyiv, Ukraine. Also at ]Institute
of Magnetism, NASU, 03142 Kyiv, Ukraine.

# Proximity-Enhanced Magnetocaloric Effect in Ferromagnetic Trilayers

M. Persson<EMAIL_ADDRESS>M. M. Kulyk  [  A. F. Kravets  [  V. Korenivski
Nanostructure Physics, Royal Institute of Technology, 10691 Stockholm, Sweden

###### Abstract

The demagnetization and associated magnetocaloric effect in strong-weak-strong
ferromagnetic trilayers, upon a reorientation of the strong ferromagnets from
parallel to antiparallel magnetization, is simulated using atomistic spin
dynamics. The simulations yield non-trivial spin distributions in the
antiparallel state, which in turn allows entropy to be calculated directly.
Empirical functional forms are obtained for the magnetization distribution in
the spacer, differing significantly from some of the commonly used models.
Finally, we find that the magnetocaloric effect in the system can be
significantly improved by allowing the local exchange to vary through the
spacer, which in practice can be implemented by spatially tailoring the
spacer’s magnetic dilution.

††preprint: APS/123-QED

##  I Introduction

The magnetocaloric effect (MCE) relates to the energy exchange between the
phonon and magnon subsystems in a material, manifesting as a decrease in
temperature upon demagnetization and, correspondingly, an increase in
temperature upon magnetization. The effect was first observed in Nickel by P.
Weiss and A. Piccard in 1917 [1, 2] and is studied today for cooling
applications where it offers more efficiency [3, 4] and can be more
environmentally friendly than today’s vapor-compression technology, with
greater systems’ longevity and without high global warming potential (GWP)
refrigerants [5]. Many exotic materials exhibiting giant MCE have been found
and studied, but when used in their bulk form the fields required to produce
substantial values of MCE are well above 1 T [6]. With cooling of electronics
being one of the most interesting application, it is vital to reduce these
operating fields, and the search has turned to novel nanostructured materials
[7]. These may additionally reduce or eliminate the need for rare-earth
metals, which are environmentally unfriendly [8].

In a nanostructured trilayer consisting of one free and one pinned
ferromagnetic layers, separated by a weakly ferromagnetic spacer, a field of
the order of 100 Oe is sufficient to switch the outer ferromagnetic layers
from parallel to antiparallel alignment and thereby reduce the effective
exchange field in the spacer, which in turn leads to its demagnetization [9,
10]. Such trilayer systems have been rather intensively studied recently, both
in experiments and theoretically [11, 12, 13, 14, 15, 16, 17, 18, 19]. As
regards MCE, first discussed by Fraerman and Shereshevskii [13], these studies
have mostly been indirect experimentally (via magnetometry) and
phenomenological in terms of modeling [13, 18, 17, 19, 16, 15], which
introduces model-dependent uncertainties in the conclusions drawn.

In this article, the above trilayer system is simulated using the atomistic
spin dynamics in the system and the magnetic entropy (MCE) is computed
directly from the resulting atomic spin distribution at various temperatures.
The results are in good qualitative agreement with direct MCE measurements on
a prototype system [20]. In addition and quite informatively, our atomistic
simulations reveal the profile of the magnetization inside the spacer, which
in the antiparallel state is qualitatively different from the functional forms
used in the literature. Studying the temperature dependence of the spin
profiles leads to a rather intuitive understanding of what these functional
forms should be analytically in order to fit the atomistic simulations well at
all temperatures. Finally, possible nanostructure optimizations are
investigated by allowing the exchange (in practice via tailoring the
concentration of the magnetic material) to vary through the spacer, and a
significant improvement is found in terms of the entropy change per unit field
for a specific gradient-spacer design.

##  II System Description

The system under study is a trilayer F/f/F with two strongly ferromagnetic
layers, F, surrounding a weakly ferromagnetic spacer, f. One of the strongly
ferromagnetic layers is pinned, by intrinsic anisotropy or exchange, while the
other is free, so that an external magnetic field can oppose the effect of
exchange coupling through the spacer and produce an antiparallel (AP)
configuration. The idea is that the magnetization of the spacer should be
higher (ideally much higher) in the parallel (P) as compared to the AP
configuration, so that a significant change in the magnetization could be
produced when switching between the two states (P-to-AP and AP-to-P).
Furthermore, while for bulk MCE the external field competes directly with
thermal disorder and therefore has to be very large, a small field is
sufficient to switch the strongly ferromagnetic layers in F/f/F, which is then
effectively amplified by the intrinsic exchange competing with thermal
disorder within the spacer.

Figure 1:  Multilayered system under study with two 1 nm thick outer strongly
ferromagnetic layers separated by a 6 nm thick weakly ferromagnetic spacer.
Arrows show simulated spatial distribution of normalized magnetic moment
vectors of each monolayer in the antiparallel F-F magnetization state at a
temperature close to $T_{\text{C}}$. Upper black curve shows magnitude of
magnetic moments, while similar curves on the left and bottom show projections
onto $(x,y)$ and $(y,z)$ planes, respectively.

Atomistic spin dynamics simulations are carried out using the VAMPIRE [21]
software package, based on the Landau-Lifshitz-Gilbert equation with Langevin
Dynamics. The simulated system is a BCC crystal with $28\times 100\times 100$
unit cells, periodic exchange in the lateral directions, and the iron lattice
parameter of $a=2.87$ Å, which translates to roughly $8\times 29\times 29$ nm3
with 1 nm for each of the strong ferromagnetic layers and 6 nm for the weakly
ferromagnetic spacer; see Fig. 1. F. Magnus, et al. [14] found that long-range
spin-spin interactions need to be taken into account in order to properly
reproduce the experimentally observed coupling through thick paramagnetic or
weakly ferromagnetic spacers. Here, we test this premise for MCE by comparing
the results of our atomistic simulations with the 1st, 2nd, 3rd, and up to 4th
nearest neighbors included in the spin-spin exchange, to see if longer range
interactions yield a qualitative difference in interpretation. Similar to Ref.
[14] we took the longer-range exchange as decaying algebraically [22]:

$J^{i\text{th nn}}=\left(\frac{r^{\text{1st nn}}}{r^{i\text{th
nn}}}\right)^{d+\sigma}J^{\text{1st nn}},$ (1)

with $d=3$ and $\sigma=0.5$. To study the effect of only the range of the
spin-spin interactions, the values of the exchange terms were rescaled, so
that the total effective exchange field, when all spins are parallel, remains
the same. The respective scaling factor, $X_{n}$, for interactions including
up to the $n$th nearest neighbors, satisfies

$J^{\text{1st
only}}N^{\text{1st}}=X_{n}\sum_{i=1}^{n}J^{i\text{th}}N^{i\text{th}},$ (2)

where $N^{i\text{th}}$ is the $i$th coordination number. For the interface
exchange between the F and f materials, with the internal exchange values
equal to $J_{1}$ and $J_{2}$, the harmonic mean was used [23]:

$J_{1\leftrightarrow 2}=2\frac{J_{1}J_{2}}{J_{1}+J_{2}}.$ (3)

Deviations from this mean value of the order of 1% were forced by the
rescaling (2). Strong easy-plane anisotropy was used for all layers to emulate
the demagnetizing fields in the thin film geometry. The time step used is 1.0
fs which should be adequate for our system with a relatively weak effective
exchange in the spacer, even in the vicinity of the Curie temperature, since
the resulting spin precession is not exceedingly fast [21]. Furthermore, the
spin precession was slowed down and the relaxation sped up by the use of
critical damping, $\alpha=1$.

##  III Results

###  III.1 MCE versus temperature

In experiments, an external field would be required to switch and compare the
magnetization between the parallel and antiparallel states. In our
simulations, the outer ferromagnetic layers are instead fixed parallel or
antiparallel by giving them easy-axis anisotropy equivalent to 55 T
($5.65\mathrm{E}{-22}$ J/atomic-volume) – an increase of the intrinsic iron
anisotropy by three orders of magnitude [21], such that the P and AP states
are well defined. The first nearest neighbor exchange was set to
$J^{1\text{st}}=7.05\mathrm{E}{-21}$ J/link in the outer ferromagnetic layers
for iron [21], and the spacer was given 15% of this value to match the
experimentally known $T_{\text{C}}$ for Fe30Cr70 spacers [24, 25, 20]. The
atomic moments were then set to the iron value of $2.22\mu_{\text{B}}$ in the
outer layers and 30% of this, $0.666\mu_{\text{B}}$, in the spacer.

The system is relaxed at different temperatures from 0 K to well above the
Curie temperature of the spacer (200 – 214 K, depending on the interaction
range), to find the maximum magnetization difference between the P and AP
states. The temperature change is sequential with a step size of $\Delta T=2$
K and at each temperature the system is equilibrated for 20 ps, followed by 50
ps of averaging. An example of the simulated magnetization profile for the AP
state is illustrated in Fig. 1 and the spacer’s magnetization versus
temperature is shown in Fig. 2(a,b). This magnetization is not a projection as
one would obtain in magnetometry, nor is it the magnitude of the total
magnetization vector of the spacer. Instead it is the order parameter, $m$,
extracted by taking the average normalized magnetization of all monolayers in
the spacer.

Figure 2:  (a) Magnetization at the center of the spacer at different
temperatures in P and AP states, for four cases of spin-spin interaction range
from 1st through 4th nearest neighbors. (b) Total magnetization of the whole
spacer in P and AP states. (c-f) Spatial profiles of magnetization at
temperatures of maximum magnetization difference P-vs-AP (marked by vertical
lines in (a,b) panels and given as captions in (c-f) panels) for the four
cases of exchange range [1st (c) through 4th n.n. (f)].

The magnetization of the spacer in the P and AP states versus temperature
follow one another quite closely until the AP magnetization abruptly drops
approximately 10 K below the Curie transition in the P state. The longer
exchange range can at this point only be seen to have one clear effect in that
it increases the effective Curie temperature. This effect is only quantitative
and do not change the overall behavior.

The magnetization difference on P to AP switching, $\Delta m$, is shown in
Fig. 3 as a function of temperature (as $-\Delta m$), for the same four cases
of the spin-spin exchange range. In order to translate $\Delta m$ into a
difference in entropy, the magnetization in each monolayer is considered
before using the additive property of entropy, $\Delta
S=\sum_{i}^{N}S(m_{\text{ap}}^{i})-S(m_{\text{p}}^{i})$. For the entropy of a
$(2J+1)$-state ferromagnet, where $J$ is the total angular momentum, a mean
field model was presented by Tuszyński and Wierzbicki [26], in which

$\begin{split}\frac{S_{J}(m)}{Nk_{\text{B}}}=\ln\frac{\sinh\big{[}(2J+1)x\big{]}}{\sinh(x)}+2xJm,\end{split}$
(4)

where $x=g_{SLJ}\mu_{\text{B}}\nu/2$, proportional to the Lagrange coefficient
$\nu$, is found by numerically solving the equation

$2Jm=\coth(x)-(2J+1)\coth\big{[}(2J+1)x\big{]}.$ (5)

The above model is used with $S=3/2$, $L=0$, $J=S+L$, and $g_{SLJ}=1.478$ for
low temperature bulk iron [27], and when converting from the entropy per-atom
to per-mass of the active material, the spacer is considered to have 30%
magnetic (Fe) atoms, see Fig. 3. With $-\Delta m$ and $\Delta S$ plotted on
the left and right axis, respectively, a difference in the shape of the curves
is easily seen. $\Delta S$ is above $-\Delta m$ at low temperature and below
it at high temperatures, since a given change in magnetization causes a larger
entropy change near saturation compared to that close to zero magnetization.
The corresponding $m$-vs-$S$ relation is shown in the inset. A consequence is
that the peak in $\Delta S$ is at slightly lower temperature than the peak in
$-\Delta m$.

Figure 3:  Change in magnetization and entropy between P and AP states at
different temperatures. Inset shows general relation between magnetization and
entropy.

Regarding the effect of the exchange range, it may appear that the peaks in
Fig. 3 become wider (quantified via FWHM and shown in respective color) as the
$T_{\text{C}}$ increases, however the FWHM values are affected by this shift
in temperature. After rescaling the temperature axes such that the four
entropy peaks align, the widths become 11.0, 11.6, 12.1, and 10.9 K, with no
clear trend present. Vertical rescalingof ($\Delta S$) does not affect the
FWHM, but if a peak is truncated, then the FWHM is increased. This is why the
FWHM value for the $3$rd n.n. case sticks out – the 2 K increments of the
temperature sweep ‘missed’ the respective peak somewhat.

The effects of increasing the spin-spin interaction range up to 4th n.n. are
rather intuitive and found above to be relatively weak, such that the
qualitative behavior of the system, when it relates to MCE, does not change.
In order to reduce the computational complexity of the atomistic modeling of
the large phase space of the studied multilayer, in what follows the spin-spin
interaction range is restricted to the 1st nearest neighbors, which
nevertheless allows us to capture the main characteristics of interest. Long-
range exchange, going far beyond the 4th n.n., and its relevance for MCE,
calls for a somewhat less rigorous than spin-atomistics, but computationally
much more compact, phenomenological numerical model, to be discussed elsewhere
[20].

###  III.2 Magnetization Profiles

Different functional forms have been used to describe the profile of
magnetization through the spacer in a F/f/F trilayer, including linear,
hyperbolic sines and cosines, whose parameters are found by minimizing the
free energy [10, 13, 19]. The assumptions and simplifications used when
arriving at these expressions may be difficult to justify when the internal
magnetization has not been (cannot be) measured and there are no direct MCE
measurements to compare the final results to. Although the simulation model
used here is idealized on a number of aspects, such as intra-layer homogeneity
and no surface roughness, the key aspect of the spatial profile under thermal
agitation is at the ultimate microscopic precision of the atomic mesh. The
internal magnetization profiles in the exchange-proximity affected spacer we
obtain numerically are used below to support more accurate functional $m(z)$
forms, which should prove useful for future analytical modeling of the
magnetic properties in this and similar exchange-spring systems.

The hyperbolic-cosine form employed in the literature [10, 19], based on
phenomenological considerations, fits our numerical results rather well for
the P state of the system. For the AP state, however, our atomic spin profiles
data differ qualitatively from the functional forms used previously, as
illustrated in Fig. 4. Around the peak in magnetic disorder, $\Delta
S_{\text{max}}$, and at lower temperatures, $|m(z)|$ has a cusp in the center
of the spacer corresponding to a local maximum in the derivative,
$m^{\prime}(z)$, instead of a minimum in $m^{\prime}(z)$ seen with hyperbolic-
sine profiles. The form of $m(z)$ in our simulations approaches a hyperbolic
sine as the temperature is increased, but not before $-\Delta m$ (MCE) has
essentially vanished. This qualitative difference between our results and
those obtained previously [10, 19] suggests that the first-order
phenomenological assumption that the spacer is paramagnetic, while simplifying
derivations, is not fully justified. A proper analytical derivation, treating
the spacer as ferromagnetic (albeit weakly; even at $T>T_{\text{C}}$), will
not be attempted here, but better fitting expressions for $m(z)$ are presented
after the following paragraph, in which the spin-atomistically obtained
profiles are made more intuitive.

Figure 4:  Magnetization profiles in AP state of F/f/F system at temperatures
near $T_{\text{C}}$ of the spacer. High-temperature profiles resembles those
typically used in the literature, however maximum MCE is at $166$ K (top
inset), where $m(z)$ is qualitatively different. Dashed lines mark F/f and f/F
interfaces. (Bottom inset) Derivative of magnetization with respect to $z$.
Figure 5:  Projections of magnetization profiles in P and AP states. Points
are simulated data and curves are fits using Eq. (6). The symmetry of
$m_{\text{p},x}$ and $m_{\text{ap},y}$ and antisymmetry of $m_{\text{ap},x}$
was used to display all data in this compact way. Inset shows the ratio
$X=m_{\text{ap}}(z=0)/m_{\text{p}}(z=0)$, fitted domain wall length $l_{0}$,
calculated $l_{1}$ (without boundary term), and $l_{2}$ (with boundary term).

The Bloch type domain wall present in the AP state at low temperatures is
uniformly distributed across the spacer at 0 K. As the temperature increases
and the magnetization magnitude in the center decreases due to thermal
disorder, the domain wall shortens since it is less costly energetically in
regions of lower magnetization. A shorter domain wall means a larger angle
between the adjacent moments (monolayers’ magnetizations), which in turn means
a lower effective exchange field and, hence, additional reduction in the
domain wall length. This eventually leads to a ‘collapse’ of the domain wall
and a peak in $-\Delta m$. There is still a domain wall near the center of the
spacer after this peak temperature, but it is much less pronounced as its
baring magnetization essentially vanishes under high thermal agitation. These
characteristics of the simulated data suggest that the magnetization in the AP
state at a given temperature is better described by a product of the
hyperbolic cosine of the P-state $m(z)$ profile at that same temperature
multiplied by the magnetization of the domain wall, whose strength and length
decreases with increasing temperature:

$\begin{split}m_{\text{p}}(z)&=m_{\text{p},x}(z)=\alpha_{1}\cosh\big{(}\alpha_{2}z\big{)}+\alpha_{3},\\\
m_{\text{ap},x}(z)&=\begin{cases}m_{\text{p}}(z),&|z|\geq l/2,\\\
m_{\text{p}}(z)\sin\frac{\pi z}{l},&|z|<l/2,\end{cases}\\\
m_{\text{ap},y}(z)&=\begin{cases}0,&|z|\geq l/2,\\\
Xm_{\text{p}}(z)\cos\frac{\pi z}{l},&|z|<l/2,\end{cases}\end{split}$ (6)

where $X=m_{\text{ap}}(0)/m_{\text{p}}(0)$. These expressions fit the
simulated data well at all temperatures; see Fig. 5. First, parameters
$\alpha_{1}$, $\alpha_{2}$, $\alpha_{3}$ are found by fitting the parallel
state magnetization, and the resulting $m_{\text{p}}(z)$ is used for the
fitting function, $m_{\text{ap},x}(l;z)$. Fitting the AP state magnetization
gives the domain wall length, $l$, which in turn can be used for obtaining the
perpendicular magnetization component, $m_{\text{ap},y}(X,l;z)$. Factor $X$
can either be fitted for or calculated from $m_{\text{ap}}(0)$ and
$m_{\text{p}}(0)$, as was done here; see the inset to Fig. 5. The spacer has
an even number of monolayers, therefore the magnetization at $z=0$ is found by
linearly extrapolating $m(z)$ from the two closest monolayers on either side
of $z=0$, and then taking the average of these extrapolations.

For these expressions to be more than fitting functions and useful for
indirect MCE measurements (often done using magnetometry), the fitting
parameters would ideally be expressed via physical parameters that describe
the system and the constituent materials. This has already been done for the
parameters of $m_{\text{p}}(z)$ [10, 19] in such a way that they can be
estimated from hysteresis loops, although it was the case of a paramagnetic
spacer that was considered in these articles. The problem of finding $\Delta
m$ is then reduced to finding $m_{\text{ap}}(z)$, i.e. the parameters $X$ and
$l$. For fixed values of $X$ the domain wall length can be found by minimizing
the exchange energy,

$F_{\text{ex}}=\int_{0}^{d/2}\left(\frac{\text{d}\bm{m}}{\text{d}z}\right)^{2}\text{d}z,$
(7)

where no exchange parameter is included since it does not affect
$\text{d}F/\text{d}l=0$. Zeeman energy need not be considered as no external
field is applied within the spacer. Shape anisotropy is not a factor either
since the functional form $\bm{m}(z)$ already restricts the magnetic moments
to lie in the plane of the spacer. Specifics of exchange at the interfaces are
excluded as they are not important, due to the sharp growth of the hyperbolic
cosine function near the interfaces, which helps restrict the domain wall to
the spacer.

Splitting integral (7) in two, for the intervals $(0,l/2)$ and $(l/2,d/2)$,
and taking the derivative with respect to $l$ leads to, for a fixed
$X\in(0,1]$,

$\begin{split}&\frac{l}{2}m_{\text{p}}^{2}\left(\frac{l}{2}\right)=2\int\limits_{0}^{l/2}m_{\text{p}}^{2}(z)\left[\sin^{2}\Big{(}\frac{\pi
z}{l}\Big{)}+X^{2}\cos^{2}\Big{(}\frac{\pi z}{l}\Big{)}\right]\text{d}z\\\
&\quad+(1-X^{2})\int\limits_{0}^{l/2}z\sin\Big{(}2\frac{\pi
z}{l}\Big{)}\left[\frac{l}{\pi}m_{\text{p}}^{{}^{\prime}2}(z)+\frac{\pi}{l}m_{\text{p}}^{2}(z)\right]\text{d}z.\end{split}$
(8)

This equation can be solved numerically for $l$, given $X$, and the results
are displayed in the inset to Fig. 5, denoted $l_{1}$. The
$X=m_{\text{ap}}(0)/m_{\text{p}}(0)$ ratio was obtained from the simulated
spin profiles (non-trivial to obtain analytically). To achieve better
agreement at low temperatures, a boundary term of the form
$c_{b}(\bm{m}_{\text{f}}-\bm{m}_{\text{F}})^{2}|_{z=d/2}$ was added to the
free energy [13], which, with a large enough coefficient $c_{b}$, can
represent the high cost of the domain wall extending into the magnetically
hard outer F-layers. The improved results are displayed in the inset to Fig.
5, denoted $l_{2}$.

The above detailed analysis of the thermodynamics of the Bloch-type domain
wall in the AP state is motivated by the key role it plays in the F/f/F
system’s MCE, which peaks when this wall collapses in length and magnetization
near the spacer’s $T_{\text{C}}$.

Figure 6:  (a) Map over $\Delta S$ between P and AP states in F/f/F trilayers
with gradient spacer exchange. Axes show exchange of outer and inner sections
of the spacer. Profiles at maximum $\Delta S$ are shown at the top, for four
selected spacers labeled by their corresponding exchange values and marked on
the map (dots along 20-20 diagonal). Magnetization of the outer ferromagnetic
layers is fixed with strong anisotropy. (b) $\Delta S$ maps with anisotropy in
the free layer replaced by external fields (labels). (c) Ratio
$J_{\text{in}}/J_{\text{out}}$ corresponding to maximum $\Delta S$ versus
external field; ratio $\Delta S_{\text{max}}/\Delta
S_{\text{min}}^{\text{unif}}$, quantifying the improvement of using non-
uniform spacers; and $\Delta S_{\text{max}}$ per unit field.

###  III.3 MCE Optimization

When studying the profiles of magnetization and the difference between the P
and AP states, the question arises of whether it would be possible to increase
$\Delta m$ by, for example, lifting up the parallel state magnetization in the
center of the spacer. The first attempt to improve the system could therefore
be to use a spacer with a spatially varying Curie temperature, realized
experimentally by varying the concentration of the magnetic component in the
spacer alloy, similar to how it was done by A. F. Kravets et al. [9].

The spacer is divided into three equal sections, and the exchange in the inner
section $J_{\text{in}}$ is allowed to vary relative to the exchange in the
outer sections $J_{\text{out}}$. The two parameters are swept from 10%
$J_{\text{Fe}}$ to 20% $J_{\text{Fe}}$ in steps of one percentage point, such
that the ratio $J_{\text{in}}/J_{\text{out}}$ varies from 0.5 to 2. This
amount of simulations requires a large amount of time and computational
resources, even when restricting the exchange to nearest neighbors, and so the
spacer thickness was reduced from 6 nm to 3 nm. At each point
$(J_{\text{in}},J_{\text{out}})$ a temperature sweep is performed in order to
find the maximum entropy change, which represents the spacer in the resulting
map $\Delta S(J_{\text{in}},J_{\text{out}})$. For the calculation of the
entropy at any point $(J_{\text{in}},J_{\text{out}})$ one must have not just
the order parameter $m=m(z)$, but also the saturation magnetization, which
should depend on the exchange values since they represent concentration of
magnetic material. The relations between the concentration, saturation
magnetization, and Curie temperature have been well studied for alloys such as
FexCr1-x [28, 29, 30, 24, 25], and could be used to find
$M_{s}(J_{\text{in}},J_{\text{out}};z)$. However, this would make the results
less general and, for consistency and simplicity, a linear approximation is
used, with 15% exchange corresponding to 30% iron atoms, as for the system
above with the uniform 6 nm spacer, and 100% exchange corresponding to 100%
iron atoms. The saturation magnetization, for the calculation of entropy, was
then set to be proportional to the magnetic concentration, just as the atomic
moments were in the simulations.

The resulting $\Delta S(J_{\text{in}},J_{\text{out}})$ can be seen in Fig.
6(a), and the largest values lie just below the line
$J_{\text{in}}/J_{\text{out}}=1$, i.e. the best spacer is nearly uniform.
However, this is with the outer ferromagnets pinned by anisotropy fields of 55
T, meaning these $\Delta S$ values are the largest achievable for each spacer
and the external field required to saturate $\Delta S$ to these values may
vary with the exchange values. To investigate this, the 55 T anisotropy in the
free layer is replaced by a much weaker external field. Just as the anisotropy
was localized to the free layer, the external field is too.

With a field of 1 T, there is a clear change in the optimal ratio
$J_{\text{in}}/J_{\text{out}}$ in favor of the original idea of lifting up the
magnetization in the center of the spacer, i.e.
$J_{\text{in}}/J_{\text{out}}>1$; see Fig. 6(b). The largest $\Delta S$ are
comparable to those obtained with the high anisotropy, but the entropy per
unit field has increased by at least an order of magnitude. The change in the
$\Delta S$ surface suggests that 1 T is below the saturation field for
$(J_{\text{in}},J_{\text{out}})$ that are above the ridge of maximum $\Delta
S$, but still above the saturation field for points below this ridge.
Therefore, the field is further reduced by 2 and 4 times, to 0.5 T and 0.25 T;
Fig. 6(b). At 0.5 T, the largest $\Delta S$ per unit field is observed at
$(20\%,15\%)$ and is 26% greater than the minimum for a uniform spacer, while
for 0.25 T these numbers are $(19\%,13\%)$ and 98% – a significant enhancement
of a factor of two. It is important to note that this two-fold enhancement is
achieved with less Fe in the gradient spacer (31.6% Fe atoms for the case of
0.25 T reversing field) compared to the uniform spacer case (minimum found at
34.1% Fe atoms).

Units of entropy per field are not meant to imply that $\Delta S$ scales
linearly with the field, but if a smaller field can reorient the ferromagnetic
layers, then more re-orientations can be performed per unit time, and $(\Delta
S/\Delta t)/(\Delta B/\Delta t)=\Delta S/\Delta B$.

For large ratios $J_{\text{in}}/J_{\text{out}}$, the maximum $\Delta m$ occurs
when the magnetization is near zero in the outer sections of the spacer, and
the difference is concentrated in the central section. Most of the domain wall
will then be located in the outer sections, while the center is more uniform.
This is problematic as it means that an external field, which on the
experiment is not localized to the free layer, could have a significant effect
on the central magnetization in the AP state, and thereby counter-act the MCE.
Using a thicker free layer would reduce this problem and increase the MCE per
unit field, but it would also reduce the fraction of active material (spacer
vs total volume). Hence, the best way to further improve the system would be
to look for more efficient exchange (magnetic dilution) profiles $J(z)$. For
any ratio $J_{\text{in}}/J_{\text{out}}$, there should be an optimal shape of
the profile $J(z)$ with $J(0)=J_{\text{in}}$ and $J(|z|=d/2)=J_{\text{out}}$.

##  IV Conclusions

Highly parallelized computation applied to atomistic spin dynamics, in
contrast to algorithmically serial Monte-Carlo methods, makes it possible to
quickly find the spatially resolved magnetization (microscopic spin
distribution) in realistic and computationally rather large nanosystems. The
systems’ entropy can be calculated in a straightforward way, directly from the
spin distribution using a mean field model for ferromagnets with identical
spins. Extending the exchange interaction range from the 1st to the 4th
nearest neighbors results in quantitative changes of the simulated properties,
with the overall qualitative picture intact. Much longer range spin-spin
interactions, suggested by some of the experiments for the investigated
metallic F/f/F system, where the conduction band is polarized and can mediate
exchange over several nanometers, in addition to the more localized lattice-
exchange, are computationally too heavy for spin-atomistic numerics and call
for specialized phenomenological modeling.

The simulated magnetization profiles in the parallel and antiparallel states
are studied in detail. While the parallel state is well described by
hyperbolic cosines, frequently used in the literature, it is found that in
order to describe the antiparallel state the domain wall present at low
temperatures must be taken into consideration. Even at high temperatures,
after the domain wall appears to have collapsed, it is effectively present,
and a combination of a hyperbolic cosine and a domain-wall function that
varies in strength and length, fits the antiparallel state magnetization at
all temperatures. By using the hyperbolic cosine for the parallel state at the
same temperature, the problem of finding the antiparallel state magnetization
reduces to finding the domain wall length and the ratio of the parallel and
antiparallel state magnetizations at the center of the spacer. If the latter
is known, then the domain wall length can be found by numerically solving an
equation free from the material parameters, which is equivalent to minimizing
the free energy.

Finally, the effects of using a nonuniform spacer are studied, with the
exchange, or concentration of magnetic material, allowed to vary through the
spacer. With the spacer divided into three equal sections it was found that
the entropy change per unit field was significantly improved when the inner
section was given a larger exchange than the outer sections. Further
improvements should be possible with more sophisticated exchange
(concentration) profiles.

###### Acknowledgements.

Support from the Swedish Research Council (VR 2018-03526) and the Olle
Engkvist Foundation (project 2020-2022) are gratefully acknowledged. The
computations were enabled by resources provided by the Swedish National
Infrastructure for Computing (SNIC), partially funded by the Swedish Research
Council through grant agreement no. 2018-05973. We thank Lars Viklund (HPC2N)
and Tor Kjellsson (PDC) at SNIC for their assistance with installation, use
and modification of software, which was made possible through application
support provided by SNIC.

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