Also at ]Institute of Physics of the NAS of Ukraine, 46 Nauky Ave., Kyiv
03028, Ukraine Also at ]Institute of Magnetism of the NAS of Ukraine and MES
of Ukraine, 03142 Kyiv, Ukraine

# Magnetocaloric effect in multilayers studied by membrane-based calorimetry

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

###### Abstract

We study magnetic multilayers, incorporating dilute ferromagnetic spacers
between strongly-ferromagnetic layers exhibiting a proximity-enhanced
magnetocaloric effect. Using magnetometry and direct measurements of the
adiabatic temperature change based on a nanomembrane-calorimetry, we find that
the magnetocaloric effect in the studied multilayer is indeed enhanced
compared to that in the bulk spacer material. We develop a phenomenological
numerical model of the studied trilayer and find that a long-range exchange
interaction through the weakly-ferromagnetic spacer is required to adequately
describe the magnetic and magnetocaloric properties of the system.

††preprint: APS/123-QED

##  I Introduction

Since the discovery of the giant magnetocaloric effect (MCE) in Gd5(Si2Ge2)
[1], the literature reveals a steadily growing interest in the area of MCE and
magnetic cooling, in particular. The related technology of adiabatic
demagnetization refrigeration (ADR) is expected to benefit from the new
materials and offer cheaper, robust, more compact and long-lasting solutions
than the commonly used gas-based heat pumps, with the ultimate goal of
reducing the energy loss and the greenhouse gas emissions [2, 3]. Even though
some of the currently available MCE materials can be relatively eco-friendly
in the long run, they often need expensive, difficult to mine rear-earth
elements, which can neutralize their benefits when compared to the gas-based
systems.

The majority of the MCE-related literature is on developing new complex alloys
and compounds in the bulk form; only a small fraction of the MCE-related
articles are on MCE in films, multilayers, and other low-dimensional
structures. In contrast to bulk materials, nanostructures exhibit finite-size
effects (due to surfaces and interfaces), which can significantly alter their
MCE with respect to that intrinsic to the incorporated materials [4, 5, 6, 7,
8, 9]. This can yield novel properties and offer benefits that are unavailable
with bulk MCE materials. For example, for a spin valve-like structure with a
weakly ferromagnetic spacer [10, 11, 12], the field as small as a hundred
Oersted could be enough to induce significant MCE [13] due to the magnetic
proximity effect (_direct_ interlayer exchange across the interfaces). Recent
works[14, 15, 16, 17] have investigated such multilayered structures using
magnetometry and concluded that the MCE is indeed enhanced, however, no direct
magneto-caloric measurements are available. In spin valve-type structures with
_indirect_ interlayer exchange (RKKY), arranged such that its constructive
versus destructive interference (on parallel to antiparallel switching of the
spin-valve’s magnetization) yields a change in the effective field acting on
the dilute ferromagnetic spacer layer of the order of tens of Tesla [18]. Such
field is strong enough to de/magnetize the spacer layer and induce a giant
MCE. Another mechanism for enhancing MCE was demonstrated [8] using
CoFe2O4/La0.7Sr0.3MnO3 heterostructures and relies on the interfacial strain-
induced magnetostructural coupling in the system. The possibility of tuning
between positive and negative MCE in Co/Cr superlattices has been reported
[6]. The above mentioned results illustrate how an enhanced MCE can be
obtained from tailoring the interlayer interactions in a nanostructure, rather
than by changing the intrinsic properties of the constituent materials. Often,
the task is reduced to enhancing the effective de/magnetization field via
interfacial spin-spin or spin-orbit interactions and focusing it on the MCE-
active region of the nanostructure, designed to have high magnetic
susceptibility.

One of the promising nanostructured materials expected to have enhanced MCE is
a trilayer of two strongly ferromagnetic films separated by a thin weakly
ferromagnetic spacer, F${}_{\text{p}}$/f/F [10, 11, 12, 13]. Here,
F${}_{\text{p}}$ and F denote a pinned and a free strongly ferromagnetic
layers, respectively, with their Curie temperature much higher than the
operating temperature range of the MCE material. f is a weakly ferromagnetic
spacer of, typically, a transition metal alloy, with the Curie temperature
($T_{\text{C}}^{\text{s}}$) much lower than that of the F-layers, designed to
lie within the desired operating temperature range by tailoring the magnetic
dilution of the alloy. The pinning of the F${}_{\text{p}}$ layer can be
achieved by exchange bias, high coercive field, or any other effect that is
capable to strongly pin the magnetization of the layer. Exchange enhanced MCE
in this design is due to the interplay of the magnetic proximity effects from
the two F/f interfaces within the nearly paramagnetic spacer when the F-layers
are switched between the parallel (P) and antiparallel (AP) magnetization
orientations. Such magnetic multilayer systems were investigated previously in
view of the influence of the proximity effect on, e.g., the induced magnetic
properties of a Pt spacer [19] and the transport properties of a dilute
magnetic semiconductor [20].

Previous work [14, 15, 16, 17] on experimentally determining the magnitude of
the magnetocaloric effect in the F${}_{\text{p}}$/f/F system was conducted
using an indirect method relying on macroscopic magnetization measurements,
with the assumption that the analysis of the $M-H$ curves based on the Maxwell
relation applies to the orientational transition of the strongly ferromagnetic
outer layers (P-to-AP), namely, that the P-AP switching is affected by the
state of the weakly paramagnetic spacer (f). The model took the magnetization
throughout all layers to be along one axis (z-axis, collinear with P/AP
magnetization states) as well as only nearest neighbor interatomic exchange in
the _paramagnetic_ spacer. Others [21, 22, 23] have modeled the system and
obtained a twisted spin state in the spacer in the AP configuration, similar
to a domain wall, highly sensitive to temperature due to the low Curie point
of f. Additionally, it was shown [24] that a more accurate description of the
switching behavior in a metallic F/f/F trilayer requires a long-range
interaction term in the total energy of the system, especially within the
dilute ferromagnetic spacer.

The applicability of the thermodynamic Maxwell relation to modeling the
magnetization reversal loops of a system of type F${}_{\text{p}}$/f/F must be
carefully examined as $M-H$ loops obtained using regular magnetometry (such as
VSM) represent an orientational change of the field-projection of the total
magnetization dominated by the F-layers, whereas the MCE is related to the
changes in the magnetization of the spacer, which are most often
indistinguishable by magnetometry. Compounding the difficulties with the
Maxwell-relation approach, the presence of typically significant anisotropy
and coercivity in the magnetization reversal behavior of the strongly
ferromagnetic layers must be carefully analyzed and understood as their
contribution to the overall $H-M$ response of the system can be predominant.
These considerations make the indirect (via magnetometry) methods of
evaluating the MCE in the discussed F/f/F system uncertain and prompted us to
implement a nanomembrane-based measurement technique for directly measuring
the adiabatic temperature change of the multilayer material undergoing P-AP
magnetization reversal in an applied magnetic field.

##  II Samples and Methods

###  II.1 Magnetic Multilayers

The multilayer samples under investigation were fabricated using magnetron
sputtering from multiple targets in a UHV chamber (ATC Orion system by AJA
Inc.). The base pressure in the chamber was lower than $1\mathrm{E}{-8}$ Torr.
The Ar pressure during the deposition was $3\mathrm{E}{-3}$ Torr. The
deposition rates of Py (Fe20Ni80), Fe, FeMn (Fe50Mn50), and Cr were calculated
from a nanoprofiler-based calibration of the thickness of the films deposited
for each target separately. The Fe${}_{\text{x}}$Cr${}_{\text{100-x}}$ (x=30)
alloy was deposited by co-sputtering of the Fe and Cr targets. The atomic
ratio x in the Fe${}_{\text{x}}$Cr${}_{\text{100-x}}$ alloy was controlled by
the ratio of the deposition rates of Fe and Cr. The obtained thickness of the
deposited stacks was in good agreement with the calculated one. To select a
direction for exchange bias of the pinned layer, a DC magnetic field of 400 Oe
was applied to the substrate during the deposition process. During the film
growth, to achieve a more uniform deposition, the substrate (with DC magnets)
was rotated with a frequency of 32 rpm.

The spacer was chosen to be FeCr because of a straightforward correlation of
the proportion of Fe in Cr with the Curie temperature of the alloy. The value
of the MCE in bulk Fe30Cr70 is relatively low [25], nevertheless, it was found
to be sufficient in strength and optimal methodologically to act as a
calibration for the direct MCE-measurement technique used.

The deposited F${}_{\text{p}}$/f/F multilayers have a well-known structure of
a spin valve. In a conventional spin valve, the spacer between the pinned and
free layers is nonmagnetic, whereas in the system under investigation the
spacer is a dilute ferromagnetic alloy with the bulk Curie temperature
designed to be close to 160 K. The schematic of the stack is shown in figure
1. To provide better adhesion and improve the growth conditions a 5 nm Cr seed
layer was used. Also, a 5 nm Cr layer was used to protect the top of the
multilayered film. For a higher signal-to-noise ratio in both vibrating sample
magnetometry (VSM) and direct MCE measurements, the Py/Fe30Cr70/Py/FeMn/Cr
structure was sequentially deposited 3 times and the Cr layers served as
magnetic isolation between the top FeMn layer and the bottom free Permalloy
layer of the next stack.

Figure 1:  Schematic of Py/Fe${}_{\text{x}}$Cr${}_{\text{100-x}}$/Py/FeMn
stack.

For this study, we deposited 6 samples, 3 of which onto thermally oxidized Si
substrates (“s” series) and 3 onto home-fabricated sensors based on 100 nm SiN
thin membranes for direct MCE measurements (“m” series). The naming and
description of the samples are shown in table 1.

Table 1:  Multilayer composition and sample notations. Si/SiO substare | SiN membrane | Deposited structure
---|---|---
sA | mA | Cr/Py/Fe30Cr70/Py/FeMn/Cr
sB | mB | Cr/Py/Cr/Py/FeMn/Cr
sC | mC | Cr/Fe30Cr70/Cr

The magnetic properties of the multilayers deposited onto oxidized Si
substrates should be essentially same as those deposited on membrane-sensors,
which are capped with a 15 nm thick insulating layer of SiO2 (sensor details
below). To verify this two identical magnetic stacks were deposited onto a
bare oxidized-Si substrate and onto a Ti-O layer (sensor material) prepared on
the same substrate, where the Ti-O was subsequently capped with a 15 nm of
SiO2. VSM $M-H$ measurements for the two samples showed only minor differences
in the behavior of the magnetization and, therefore, justified a comparative
analysis magnetometry-vs-MCE presented below.

Samples sA and mA were the focus of the study, expected to show an enhanced
MCE compared to that in sample mC with the reference value of MCE of the bulk
spacer material. The thickness of the Fe30Cr70 alloy film in samples sC and mC
was 50 nm and the $T_{\text{C}}$ close to that of the bulk material [26]. We,
therefore, in what follows refer to samples sC and mC as _bulk_ samples.

Samples sB and mB have the same structure as sA and mA but a nonmagnetic
spacer, which at 6 nm thickness should not mediate any significant magnetic
exchange (direct or indirect) between the free and pinned strongly
ferromagnetic layers. This sample is used as an additional control sample in
the direct measurements of the adiabatic temperature change of the
multilayers, as it should not show any signal related to the MCE in the spacer
(Cr spacer is non-magnetic), while it should reveal any spurious contributions
that may be present in a membrane-based MCE measurement.

###  II.2 Sensor Fabrication

To detect the adiabatic temperature change of the magnetic multilayer a micro-
sensor was fabricated consisting of a 50 nm thick Ti-O planar thermistor with
high temperature coefficient of resistance (TCR=1-3%) deposited on a 100 nm
SiN membrane (Silson Ltd, UK). The thermistors were deposited using reactive
magnetron sputtering from a Ti target in a mixed atmosphere of Ar and O2. The
thickness control of the Ti and SiO2 was done in the same way as for the
magnetic multilayer. To electrically isolate the thermistor from the
conductive magnetic film, a 15 nm layer of SiO2 was deposited on top of the
Ti-O layer. The key properties of nm-thin SiN membranes are their low heat
conductivity and heat capacity, which make the MCE measurement conditions
close to adiabatic [27, 28]. The micrograph and schematic of the membrane with
the thermistor and the magnetic multilayer are shown in figure 2.

Figure 2: Microphotograph (a) and schematic (b) of sample on Ti-O sensor, on
SiN membrane, with following bottom-to-top layer sequence: (1) Si-SiN
substrate (7.5 mm $\times$ 7.5 mm $\times$ 381 µm); (2) SiN membrane window
(1.5 mm $\times$ 1.5 mm $\times$ 100 nm); (3) contact pads of thermistor with
thickness 60 nm (50 nm of Ti, 10 nm of Au); (4) Ti-O planar thermistor (880 µm
$\times$ 840 µm $\times$ 50 nm) capped with SiO2 insulation layer (15 nm); (5)
magnetic stack (700 µm $times$ 800 µm). Dark lines on contact pads in (a) are
bonded wires.

###  II.3 VSM Measurements

The VSM measurements were carried out using a Lake Shore 7300 vibrating sample
magnetometer equipped with a nitrogen gas-flow cryostat. The isothermal
magnetization reversal loops were measured with the field applied in the plane
of the films, with the exchange bias direction oriented along the positive
direction of the field. The measured magnetization was normalized by the
magnetic volume of the samples.

###  II.4 Membrane-based Calorimetry

The schematic of the measurement setup is shown in figure 3. R${}_{\text{x}}$
denotes the Ti-O planar thermistor, deposited directly onto the SiN membrane,
acts as the substrate for deposition of the magnetic multilayer to be studied.
DC voltage U${}_{\text{drive}}$ is applied to voltage divider
R${}_{\text{b}}$-R${}_{\text{x}}$, where R${}_{\text{b}}$ is a current
limiting resistor. The voltage drop across R${}_{\text{x}}$ is fed into a
lock-in amplifier (LA). The membrane with the sensor and magnetic stack is
placed in a vacuum inside a cryostat (Oxford OptistatDN, modified to operate
without gas in the sample space to prevent parasitic sample-gas-chamber heat
exchange). The cryostat is placed inside an electromagnet (M), which is
powered by a power supply unit (PSU) capable of supplying an AC+DC current
(Kepco BOP).

Figure 3:  Schematic of electrical circuit for measuring resistance of
membrane thermistor (R${}_{\text{x}}$, see figure 2), calibrated to yield
temperature change of membrane with sample.

When a low frequency (11.7 Hz) AC magnetic field is applied to a magnetic
sample, sufficient to induce MCE (strong enough to produce P-AP switching in
spin-valve samples), the temperature of the sample placed in direct contact
with the sensor/membrane changes synchronously with the external field. Due to
the low heat conductivity of the membrane, a momentary change in its
temperature equilibrates over a relatively long period of time, $\tau>$0.1 s
in our case (time constant of our membrane sensor). Thus, for the applied
field frequency that exceeds 1/$\tau$ the condition for the temperature change
of the sample to be adiabatic is fulfilled (it changes/oscillates faster than
it can dissipate to the bigger substrate via the membrane). The magnetic
sample is in direct contact with the thermistor, whose resistance
R${}_{\text{x}}$ oscillates producing a variation in the voltage drop across
it, which is then detected using a lock-in amplifier synced to the frequency
of the applied AC magnetic field. Generally, depending on the value of the DC
field offset and the properties of the magnetic system, the MCE signal can be
detected at the first or the second harmonic. In what follows, we limit the
discussion to the most straightforward case of a low-amplitude AC field
applied at various fixed biasing fields, such that the signal is found
predominantly at the 1${}^{\textrm{st}}$ harmonic (details below).

The size of the planar thermistor must fulfill two main criteria: the volume
of the active MCE material must be relatively large to provide a significant
change in temperature of the membrane/sensor/trilayer stack; the
sensor/trilayer must be relatively small to avoid the unwanted heat exchange
with the bulk Si substrate. We have modelled these aspects numerically, that
for a given membrane geometry, AC-field frequency, and material parameters,
and arrived at the optimal size of the sensor – the one used in this article.

As the MCE-related signal is detected at the first harmonic of the AC magnetic
field, the inductive pick-up is detected together with the MCE signal. It is
unwanted and is removed using the phasor approach to the signal processing.
Two measurements are carried out consecutively - with positive and negative
U${}_{\text{drive}}$ applied to the R${}_{\text{b}}$-R${}_{\text{x}}$ voltage
divider. Denoting the vector of the parasitic inductive pick-up as P, the
vectors of the MCE signal as $\textbf{S}_{1}$ and $\textbf{S}_{2}$ (for
positive and negative U${}_{\text{drive}}$), and the vectors measured by lock-
in signals as $\textbf{L}_{1}$ and $\textbf{L}_{2}$, we have

$\left\\{\begin{matrix}{\textbf{S}}_{1}=-{\textbf{S}}_{2}={\textbf{S}}_{\text{MCE}}\\\
{\textbf{P}}+{\textbf{S}}_{1}={\textbf{L}}_{1}\\\
{\textbf{P}}+{\textbf{S}}_{2}={\textbf{L}}_{2}\par\end{matrix}\right.$ (1)

The solution of this system gives us

$\begin{split}S_{\text{MCE}}&=|\textbf{S}_{1}|=\frac{1}{2}\sqrt{R_{1}^{2}+R_{2}^{2}-2R_{1}R_{2}\cos(\theta_{1}-\theta_{2})}\
,\\\
\theta_{\text{MCE}}&=\text{atan2}\left(R_{1}\cos\theta_{1}-R_{2}\cos\theta_{2},R_{1}\sin\theta_{1}-R_{2}\sin\theta_{2}\right)\
,\end{split}$ (2)

where $R_{1}$, $R_{2}$, $S_{\text{MCE}}$ and $\theta_{1}$, $\theta_{2}$,
$\theta_{\text{MCE}}$ are $R$ and $\theta$ components of the measured and the
MCE signal phase vectors, $\textbf{L}_{1}$ and $\textbf{L}_{2}$,
$\textbf{S}_{\text{MCE}}$, respectively.

The AC component of the resistance of the sensor ($R_{\text{AC}}$) is equal to

$\begin{split}R_{\text{AC}}&=\frac{\big{(}R_{\text{DC}}R_{\text{I}}+R_{\text{b}}(R_{\text{DC}}+R_{\text{I}})\big{)}^{2}S_{\text{MCE}}}{-R_{\text{DC}}R_{\text{I}}^{2}U_{\text{drive}}+(R_{\text{b}}+R_{\text{I}})\big{(}R_{\text{DC}}R_{\text{I}}+R_{\text{b}}(R_{\text{DC}}+R_{\text{I}})\big{)}S_{\text{MCE}}}.\end{split}$
(3)

Here $R_{\text{DC}}$ and $R_{\text{I}}$ are the DC resistance of the sensor at
a given temperature and the input impedance of the lock-in amplifier (or
preamplifier, if used).

For small changes in the thermistor resistance due to the AC-field induced MCE
(our case), the temperature coefficient of resistance ($k_{\text{TCR}}$) of
the sensor at a given temperature can be considered constant, hence the
adiabatic temperature change becomes

$\Delta
T_{\text{ad}}^{\text{measured}}=\frac{-R_{\text{AC}}100\%}{k_{\text{TCR}}R_{\text{DC}}}\
.$ (4)

The heat capacity of the active MCE material is often comparable to or less
than that of the sensor-membrane material, a scaling coefficient for $\Delta
T_{\text{ad}}$ needs to be introduced to reflect the ratio of the MCE-active
thermal mass to the total thermal mass of the device. Straightforward general
considerations show that this thermal-mass correction coefficient
($k_{\text{A}}$) and the resulting $\Delta T_{\text{ad}}^{\text{MCE}}$ of the
magnetic sample only should be equal to

$k_{\text{A}}=\frac{C_{\text{tot}}}{C_{\text{MCE}}},\quad\Delta
T_{\text{ad}}^{\text{MCE}}=\Delta
T_{\text{ad}}^{\text{measured}}k_{\text{A}},$ (5)

where $C_{\text{tot}}$ is the total heat capacity of the device including the
membrane, the sensor with the SiO2 insulation, the MCE material, and the other
auxiliary materials not contributing to the exchange-enhanced MCE in the
weakly ferromagnetic spacer (Cr underlayer, biasing antiferromagnet, etc);
$C_{\text{MCE}}$ is the heat capacity of the active MCE material only
(Fe30Cr70 in our case); $\Delta T_{\text{ad}}^{\text{measured}}$ is the raw
value of the adiabatic temperature change measured directly; $\Delta
T_{\text{ad}}^{\text{MCE}}$ is the normalized value of the adiabatic
temperature change in the MCE material. All values of $\Delta T_{\text{ad}}$
discussed below are scaled by $k_{\text{A}}$ to show the characteristics of
the studied magnetic material rather than a combined property skewed by the
unimportant details of the measurement device (its ‘dead’ thermal mass).

##  III Results

###  III.1 VSM Measurements

The magnetization loops for samples sA and sB measured at different
temperatures are shown in figure 4.

Figure 4: Magnetization reversal loops of samples sB (a), sA (b,c), and single
repetition of trilayer of type sA (d).

Sample sB is a control sample, a spin valve with a nonmagnetic spacer, for
which the magnetization reversal loops at $T$=180 and $T$=290 K are shown in
figure 4(a). The behavior is classical behavior for a spin valve [29] – the
hysteresis is composed of two characteristic minor loops. One of the minor
loops has low coercivity and is centered at zero field, easily identifiable as
switching of the free Py layer. The second minor loop has a higher coercivity
and is offset in field, which corresponds to the exchange-biased transition of
the FeMn-pinned Py layer. As expected, cooling from 290 to 180 K leaves the
minor loop of the free Permalloy layer almost unchanged, while the offset of
the pinned loop significantly increases due to a stronger exchange-bias from
the antiferromagnetic FeMn at low temperature. The amplitudes of the free and
pinned layers’ minor loops have a ratio of 6:4, which corresponds to the
thickness ratio of the respective layers.

The magnetic properties of the sA sample are similar to those observed in [10,
11, 16, 14]. The $M-H$ loops for three representative temperatures are shown
in figure 4(b). At high temperatures, when the spacer is in its nominally
paramagnetic state, the free and pinned layers are almost fully decoupled and
their minor loops are detached from each other. Approaching the effective
$T_{\text{C}}^{\text{s}}$ from above, the exchange coupling through the weakly
ferromagnetic spacer increases, which mediates unidirectional anisotropy from
the pinned layer and results in a visibly offset loop of the free layer
(partially free at 210 K).

The temperature dependence of the bias field for the free and pinned layer
loops for sample sA are shown in figure 5. The strength of the biasing field
of the free layer increases from essentially zero at high temperature to about
170 Oe at 130 K. The bias-field strength of the pinned layer shows an increase
from room temperature (RT) down to about 210 K, after which it weakens on
further lowering the temperatures. This non-monotonic behavior is due to an
interplay of two effects: first, the well-known increase and eventual
saturation of the strength of the antiferromagnetic pinning at low temperature
[30]; the second is related to a higher Zeeman torque, proportional to the
effective thickness of the now strongly coupled Py layers at low-$T$, which
counteracts the exchange bias and lowers the loop offset [30, 31]. The
resulting functional form of $H_{\text{bp}}^{\text{VSM}}$ versus $T$ is non-
monotonic, as shown in figure 5 (red curve).

Figure 5:  Temperature dependence of bias field of free layer
($\text{H}_{\text{bf}}^{\text{VSM}}$) obtained from $M$-$H$ loops of sample sA
(blue circles); bias field of free layer ($\text{H}_{\text{bf}}^{\text{MCE}}$)
obtained from direct measurements of adiabatic temperature change for sample
mA (purple triangles); bias field of pinned layer
($\text{H}_{\text{bp}}^{\text{VSM}}$) obtained from $M$-$H$ loops of sample sA
(red diamonds).

From the low-temperature (150 K) magnetization reversal curve in figure 4(b)
it may appear that the switching of the free layer is a viscous process
accompanied by significant coercivity. This conclusion would however not be
correct as the partial reversal of only the free layer’s magnetization plotted
in figure 4(c) show very low coercivity. The relatively high coercivity of the
full $M-H$ loop in the field region of the switching of the free layer is due
to the overlap of the narrow switching loop of the free layer and the wide
switching loop of the pinned layer.

Comparing the partial magnetization reversal loops of the free layer at room
temperature for the samples with the triple and single base spin-valve
structure (Figs. 4(c) and 4(d), respectively) suggests that, in addition to
the magnetic state/dilution of the spacer, the strength of the interlayer
coupling depends on the roughness of the interfaces, which increases somewhat
for thicker films. It is visible that for the second and third repetitions of
the deposited spin-valve structure, the RT exchange bias of the free layer
gradually increases (figure 4(c)).

###  III.2  Direct Measurements of MCE

Using the membrane-based MCE device described above we have measured the
magnitude of the temperature variation caused by the AC modulation of the
field applied to the sample. Typical 2D maps of the MCE signal versus peak-to-
peak AC field and DC offset field for samples mA, mB, and mC are shown in
figure 6.

Figure 6: Typical $\Delta T_{ad}$ maps for samples mA (e, f), mB (c, d), and
mC (a, b) at following temperatures: (a) – 160 K; (b) – 295 K; (c) – 150 K;
(d) – 295 K; (e) – 170 K; (f) – 295 K. Horizontal axis represents DC bias
field in Oe; vertical axis – peak-to-peak value of AC field modulation.

The data for the 50 nm thick film of Fe30Cr70 alloy (mC sample) are shown in
figure 6(a) and (b) for 150 and 295 K, respectively. At the high temperature,
no signal is visible above the background noise. As the temperature is
lowered, a clear signal appears and increases, reaching a maximum at $\sim$160
K. The typical appearance of the MCE signal shown in figure 6(a) remains the
same at all temperatures where the signal is detected, changing only in
intensity. The gap near zero DC field can be explained by the fact that at
$|H_{\text{DC}}|<\frac{1}{2}H_{\text{AC}}^{\text{pp}}$ the value of the field-
dependent entropy change
$S(H_{\text{DC}}+\frac{1}{2}H_{\text{AC}}^{\text{pp}})-S(H_{\text{DC}}-\frac{1}{2}H_{\text{AC}}^{\text{pp}})$
is lower than for the case of
$|H_{\text{DC}}|>\frac{1}{2}H_{\text{AC}}^{\text{pp}}$, as $S(H)$ is an even
function and becomes zero at $H_{\text{DC}}=0$.

The sample with a spin-valve structure and a nonmagnetic spacer (mB) shows a
relatively weak peak at 295 K (figure 6(d), marked with arrow), localized at
the bias field of the pinned layer, determined independently from the
respective magnetization reversal loop in figure 4(a), for the same
temperature. At lower temperatures, this peak broadens and shifts to a higher
negative DC field (not shown), eventually (toward 170 K) moving out of the
experimentally available field range (both AC and DC). This well defined peak
clearly visible at 295 K is heating/cooling due to the reversal of the pinned
Py layer, which is likely demagnetizing/frustrating a portion of the spins at
the F/AF interface, some of which are uncompensated [32]. This MCE of the F/AF
interface is outside the focus of the present study and will not be discussed
further, however, it is an interesting topic in itself and deserves a separate
investigation.

A background signal linearly proportional to the strength of the AC field is
detected at lower temperatures. It is DC-field unspecific and, therefore, does
not complicate the interpretation of MCE peaks related to the F-layer
magnetization switching, which is found at specific DC fields.

For sample mA, two well-distinguishable peaks are observed in the temperature
range 150-290 K. At room temperature the first peak (marked with arrow 1 in
figure 6(f)) is located near zero field and has a somewhat lower strength
(less bright) than the second peak (arrow 2). Additionally, the signal for
peak 1 appears and saturates at a lower AC modulation field than that for peak
2. At low temperatures, both peaks shift to higher negative DC fields and
increase in both magnitude and width. The data for $T=170$ K corresponding to
the maximum MCE we have observed is shown in figure 6(e). At temperatures
lower than 170 K both signals undergo a rapid change, both in magnitude and
shape - they merge and become hardly distinguishable.

We identify peaks 1 and 2 as corresponding to the P-to-AP and AP-to-P
switching of the free and pinned layers in the sample (mA – trilayer with
dilute ferromagnetic spacer). Peak 1 corresponds to the transition where the
magnetization of the pinned layer stays unchanged while that of the free layer
reverses. Peak 2, then, is where the switching occurs in the pinned layer
while the free layer is fixed by the relatively high DC field offset. This
interpretation is supported by the independently measured $M-H$ switching
fields (see magnetometry data above), and naturally explains why the AC field
required to induce peak 2 is higher than that for peak 1 – the width of the
partial $M-H$ loop of the pinned layer is much larger than that of the free
layer.

It should be mentioned that peak 2 in figure 6(f) at high temperatures
possibly contains two magnetocaloric contributions – one from the proximity-
enhanced MCE in the paramagnetic spacer, described in the previous paragraph,
and one from the MCE that can be related to the F/AF interface and is observed
in figure 6(d). As the amplitude of the latter is much lower and is observed
only at high temperatures, we can neglect its contribution to the most
prominent MCE regime depicted in figure 6(e), clearly identifiable as due to
the relatively narrow P-to-AP transition on switching of the free layer at
about 50-100 Oe in DC offset.

The dependence of $\Delta T_{\text{ad}}$ on $H_{\text{AC}}$ for samples mA and
mC at the temperature of the maximum MCE signal are shown in figure 7. Since
the $\Delta T_{\text{ad}}$ signal for sample mC is delocalized in DC offset,
the profile presented in figure 7 for this sample is the average value of
$\Delta T_{\text{ad}}$ for
$|H_{\text{DC}}|>\frac{1}{2}H_{\text{AC}}^{\text{pp}}$. For sample mA, the
data presented in figure 7 is the profile of the map in figure 6(e)
corresponding to the center of peak 1. The dependence for sample mC is linear
with a positive slope, precisely as expected for a bulk-like alloy. At the
same time for sample mA, the behavior of $\Delta T_{\text{ad}}$ vs AC-field is
clearly nonlinear, with a region of rapid growth followed by saturation. We
denote the value of $\Delta T_{\text{ad}}$ at saturation for sample mA as
$\Delta T_{\text{ad}}^{\text{s}}$.

Figure 7:  Profiles of $\Delta T_{\text{ad}}$ vs AC field peak-to-peak
amplitude, experimental points with best fits. Blue circles and green
diamonds: $\Delta T_{\text{ad}}$ for sample mA at 170 and 295 K at DC bias of
maximum signal; 295 K data scaled up tenfold are shown as green dotted curve
to demonstrate same functional form seen in 170 K data. Red triangles: average
$\Delta T_{\text{ad}}$ for sample mC at 160 K at DC bias fields
$|H_{\text{DC}}|>\frac{1}{2}H_{\text{AC}}^{\text{pp}}$.

The temperature dependence of the bias field of the directly measured MCE in
sample mA (figure 5 \- purple curve) and the bias field of the partial M-H
loop of the free layer in sample sA measured by VSM (figure 5 \- blue curve)
are essentially identical. This makes us conclude that the strongest MCE in
the F${}_{\text{p}}$/f/F structure is manifest on switching of the free layer,
producing the P-to-AP transition in the system, with the weakly ferromagnetic
spacer more frustrated in terms of its spatial spin orientation (higher
magnetic disorder/entropy) between the oppositely magnetized interfaces of the
outer F-layers in the AP state.

Figure 8 shows the data for the adiabatic temperature change measured directly
on the membrane-fabricated samples, as well as that reported in the literature
for the same composition bulk-like Fe-Cr alloy and extracted from magnetometry
using the Maxwell relation [25] at the temperature of maximum MCE near 160 K.
For samples mA and mC, MCE maps similar to those shown in figure 6 were
recorded for the temperature range of 130-295 K. Using the data in figure 4(a)
of Ref. [25] for $\Delta S$ for a 88 nm Fe30Cr70 film, the value of $\Delta
S=0.024$ J/kg K is obtained for the field changing from 0 to 250 Oe. Assuming
a constant value of the heat capacity, C=462 J/kg K, the maximum adiabatic
temperature change in this case of the bulk-like alloy is $|\Delta
T_{ad}|\approx 0.008$ K at $\sim 160$ K (brown dash-dotted line in figure 8).
The value of the peak adiabatic temperature change we measure directly for the
same alloy (our 50 nm thick film of Fe-Cr; sample mC) and the one from the
literature obtained indirectly are in good agreement. The Curie temperature
for the Fe30Cr70 sample estimated using the Arrot plots for the VSM M-H data
taken at different temperatures for sample sC, is close to 160 K, which is in
good agreement with the position of the maximum entropy change in our direct
measurement as well as the literature [25, 26].

Figure 8:  Adiabatic temperature change obtained from direct MCE measurements:
blue circles for spin valve with Fe30Cr70 spacer, red squares for 50 nm film
of Fe30Cr70, both for $\Delta H=$250 Oe; brown dash-dotted line was obtained
using Maxwell’s equation for 88 nm of Fe30Cr70 [25].

The $\Delta T_{\text{ad}}^{\text{s}}$ vs T dependence for sample mA (figure 8
blue circles) shows a wide asymmetric peak with a maximum at $\sim$170 K which
is $\sim$10 K higher than the Curie temperature of the Fe30Cr70 alloy. The
maximum measured MCE in the spin valve with the dilute ferromagnetic spacer is
$\sim$2.5 greater than the intrinsic MCE of the spacer material (taken here
for the same external AC field amplitude). Further, the respective half-widths
of the MCE signal are $\sim$50 K versus 25 K, which means that, overall, the
relative cooling power per unit volume is approximately 5 times higher for the
F${}_{\text{p}}$/f/F trilayer 111We note that using the indirect Maxwell-
relation approach with the magnetization data for the trilayer sample yielded
MCE, which did not correlate with the direct measurements either in magnitude
or temperature dependence, and was discarded from further discussion and
analysis..

Even though, at room temperature, the 6 nm Fe30Cr70 spacer is in a nearly
paramagnetic state and the coupling between the two outer F${}_{\text{p}}$ and
F layers should be weak-to-absent [11], both the exchange bias of the F layer
(nominally free) and the MCE signal of the trilayer are nonzero. These
observations are rather fascinating since the correlation length of the
ferromagnetic order parameter is nominally less than 1 nm for Fe30Cr70 (at
RT), suggesting that the direct interatomic exchange should not be expected to
provide the observed finite exchange coupling through the 6 nm thick spacer at
a temperature that exceed the spacer’s $T_{\text{C}}^{\text{s}}$ by nearly a
factor of two.

An important characteristic of the investigated effect conclusively following
from the data of figure 7 is the saturation of the MCE signal with the applied
AC field amplitude for sample mA. This means that the magnitude of the MCE in
the structure (almost entirely within the weakly magnetic spacer; see
Discussion below) predominantly depends on the mutual orientations of the
outer strongly ferromagnetic layers and not on the strength of the applied
magnetic field, as is the case for the bulk-like Fe-Cr alloy film with a
linear MCE-vs-Field dependence.

##  IV Discussion

The switching of the free layer and the corresponding $\Delta T_{\text{ad}}$
signal in the studied system (sA and mA samples) are characteristically offset
in field even at room temperature, which indicates a non-zero interlayer
exchange coupling through the spacer, nominally too thick to mediate direct
exchange as the spacer should be in a paramagnetic state, much above its Curie
point. The presence of such exchange coupling at temperatures much higher than
$T_{\text{C}}^{\text{s}}$ in similar F/f/F structures was observed by several
groups previously [10, 19, 11, 24, 14]. It has been suggested [24] that a
long-range magnetic interaction (much longer than the usual inter-atomic
exchange length scale) should be added to the energetics of the system in
order to properly account for the observed interlayer coupling between the
outer ferromagnetic layers at high temperatures and for thick (vs atomic
scale) spacers.

A possible origin of such long-range coupling through a metallic spacer is
exchange mediated by the polarized conduction electrons [34, 35] that traverse
the stack and can transfer spin over longer distances. The spin flip length
($l_{\text{sf}}$) in magnetic metals, which is the key characteristic in this
process, varies from a few to a few tens of nanometers [36]. It is then
reasonable to expect that in the dilute ferromagnetic metal-alloy spacer
electrons polarized by the strongly ferromagnetic outer layers can mediate a
significant amount of angular momentum over distances much exceeding the
atomic spacing. The thickness of the spacer in our case is 6 nm, so no
significant direct spin-transfer between the F-layers due to the conduction
electrons is expected (RKKY essentially vanishes beyond 3 nm of Cr [37, 38,
39]). Therefore, in the model below, which uses a long-range interaction term,
we restrict the action to the spacer spins only (40 monolayers for 6 nm thick
spacer). The spins in the spacer are affected by this long-range exchange from
both F and F${}_{\text{p}}$ and, effectively, form a spin-chain mediating a
relatively small but finite F${}_{\text{p}}$-F exchange over several
nanometers, at temperatures much exceeding the intrinsic Curie point of the
spacer alloy where the material is nominally paramagnetic.

The need for a long-range spin-spin interaction term in quantitative modeling
of the studied system can be illustrated with the the following simplified
argument based on our MCE vs $T$ data of figure 8. For the spacer in a
paramagnetic state (at $T>T_{\text{C}}$), its magnetic moment would be

$\mathbf{M}_{\text{sp}}(z)=\chi\mathbf{H}_{\text{eff}}(z)\ ,$ (6)

where $\chi$ is the magnetic susceptibility of the paramagnetic spacer and
$\mathbf{H}_{\text{eff}}$ is the effective field within the spacer, which
depends on the $z$ coordinate normal to the plane of the
$\text{F}_{\text{p}}/$f$/\text{F}$ stack (figure 9):

$\mathbf{H}_{\text{eff}}(z)=\mathbf{H}_{\text{ext}}+\mathbf{H}_{\text{f}}(z)+\mathbf{H}_{\text{p}}(z)+\mathbf{H}_{\text{fd}}(z,T)+\mathbf{H}_{\text{pd}}(z,T)\
.$ (7)

Here $\mathbf{H}_{\text{ext}}$ is external magnetic field,
$\mathbf{H}_{\text{f}}(z)$ and $\mathbf{H}_{\text{p}}(z)$ are long-range
effective field terms for the F/f and $\text{F}_{\text{p}}$/f interfaces,
respectively, $\mathbf{H}_{\text{fd}}(z)$ and $\mathbf{H}_{\text{pd}}(z)$ are
short-range effective exchange fields (direct-exchange proximity effect),
which generally depend on both the temperature and coordinate, for the
respective interfaces. At a temperature much above $T_{\text{C}}^{\text{s}}$,
the magnetization induced in the paramagnetic spacer at its f/F interfaces by
the direct exchange fields ($\mathbf{H}_{\text{fd}}$,$\mathbf{H}_{\text{pd}}$)
is significant, however, the respective penetration depth (due to direct
inter-atomic exchange) is much smaller than the total thickness of the spacer.
As a result, the induced magnetization vanishes off the two interfaces toward
the center of the spacer; their overlap is essentially zero and, therefore,
the difference in magnetization between the parallel and antiparallel
configurations of the F${}_{\text{p}}$/f/F system expected from the short-
range, direct exchange can be neglected. At temperatures comparable or lower
than $T_{\text{C}}^{\text{s}}$, this short-range term does produce an overlap
in magnetization (_different_ for P and AP, so $\Delta M\neq 0$ on P-to-AP
switching) and must be taken into the account. A schematic of the multilayer
with the spatial profile of the above effective fields is shown in figure 9.
For the sake of this illustrative argument, we take the long-range exchange
field (mediated by polarized electrons) as temperature independent, as it is
expected to vary much weaker vs $T$ than the direct-exchange term, scaled by
roughly the Fermi vs the Curie temperature.

Spin polarization of the electrons decays exponentially off the F/f interfaces
into the spacer over a characteristic spin-flip length, $l_{\text{sf}}$, so
the spatial dependence of the respective effective magnetic fields,
$\textbf{H}_{\text{f}}(z)$ and $\textbf{H}_{\text{p}}(z)$, has the form

$\mathbf{H}_{\text{f}}(z)=\mathbf{M}_{\text{f}}J_{\text{f}}e^{-z/{l_{\text{sf}}}},\quad\mathbf{H}_{\text{p}}(z)=\mathbf{M}_{\text{p}}J_{\text{p}}e^{(z-t)/{l_{\text{sf}}}},\
$ (8)

where $t$ is the thickness of the spacer, $J_{\text{f}}$ and $J_{\text{p}}$ –
properly normalized exchange constants between the pairs of materials
$\text{F}/$f and $\text{F}_{\text{p}}/$f, $\mathbf{M}_{\text{f}}$ and
$\mathbf{M}_{\text{p}}$ – magnetization vectors of the free and pinned layers,
respectively. In figure 9 the fields for the long-range exchange are shown out
of proportion for visual clarity – estimated to be about two orders of
magnitude smaller than the corresponding short-range exchange fields (2% range
used in the numerical simulations below).

Figure 9:  Schematic of $\text{F}_{\text{p}}/$f$/\text{F}$ multilayer and
profiles of effective exchange fields. Arrows within F and F${}_{\text{p}}$
layers indicate up and down direction of layer’s magnetization.

With the magnetization of the spacer denoted as $\textbf{M}_{\text{sp}}(z)$,
the energy areal density is

$\varepsilon(z)=\mathbf{M}_{\text{sp}}(z)\mathbf{H}_{\text{eff}}(z)\ .$ (9)

Using (6) and integrating (9) over $z$, the magnetic energy per volume of the
paramagnetic spacer becomes

$E=\chi\int_{0}^{t}\mathbf{H}_{\text{eff}}^{2}(z)\text{d}z\ .$ (10)

The energy difference, $\Delta E$, between the P ($\uparrow\uparrow$) and AP
($\uparrow\downarrow$) states of the trilayer is then

$\Delta E=\chi\left[\int_{0}^{t}\mathbf{H}_{\text{eff}}^{\uparrow\uparrow
2}(z)\text{d}z-\int_{0}^{t}\mathbf{H}_{\text{eff}}^{\uparrow\downarrow
2}(z)\text{d}z\right].$ (11)

Taking $H_{\text{DC}}$ as the field-offset and $H_{\text{AC}}$ as the width of
the P-AP transition, the effective fields for the two states projected on to
the P/AP axis ($x$) are

$\begin{split}H_{\text{eff}}^{\uparrow\uparrow}(z)=&H_{\text{DC}}+H_{\text{AC}}+M_{\text{f}}J_{\text{f}}e^{{-z}/{l_{\text{sf}}}}+M_{\text{p}}J_{\text{p}}e^{{(z-t)}/{l_{\text{sf}}}},\\\
H_{\text{eff}}^{\uparrow\downarrow}(z)=&H_{\text{DC}}-H_{\text{AC}}-M_{\text{f}}J_{\text{f}}e^{{-z}/{l_{\text{sf}}}}+M_{\text{p}}J_{\text{p}}e^{{(z-t)}/{l_{\text{sf}}}}.\end{split}$
(12)

According to the Landau theory of phase transitions, above the Curie point,
$\chi\propto C\tau^{-\gamma}$, where $\tau=T-T_{\text{C}}$ and $\gamma$ is the
critical exponent for the Curie transition. With constant heat capacity $C$,
this gives for the entropy change on P-AP switching versus temperature

$\Delta S=\frac{\partial\Delta
E}{\partial\tau}=-C\gamma\tau^{-(\gamma+1)}\int_{0}^{t}\left[H_{\text{eff}}^{\uparrow\uparrow
2}(z)-H_{\text{eff}}^{\uparrow\downarrow 2}(z)\right]\text{d}z\,$ (13)

or $\Delta S(T)\propto\tau^{-k}$, with $k=\gamma+1$. figure 10 shows $\Delta
S$ converted from the experimentally measured $\Delta T_{\text{ad}}$ data of
figure 8, alongside $C(T-T_{C})^{-k}$ fitting at high temperatures, with
$T_{\text{C}}=162$ K and $k=2.28\pm 0.25$ and $2.23\pm 0.1$ for the mA and mC
samples, respectively. The fitting yields the critical index, $\gamma$, close
to 1.3 _for both samples_ , which is rather interesting as this value is
characteristic for paramagnetic $\chi(T)$ and, at first glance, not expected
to apply to the F/f/F structure. The explanation of the high-temperature
critical-exponent behavior discussed below sheds additional light on the MCE
mechanism at play in the studied system.

For the bulk-like sample (mC), obviously, both short- and long-range effective
exchange fields are absent as there are no interfaces, so only the external
fields affect the material. In this case, at a given temperature (12) and (13)
reduce to $\Delta S(H)\propto H_{\text{DC}}H_{\text{AC}}$. With
$H_{\text{DC}}=\text{const}$, $\Delta T_{\text{ad}}$ (or, equivalently,
$\Delta S(H)$) should be linearly proportional to $H_{\text{AC}}$, which is
indeed observed experimentally; see figure 7, red triangles for sample mC.

In contrast, for the F${}_{\text{p}}$/f/F trilayer, the P-AP switching of the
magnetization results in a rather sharp, step-like behavior, with a
characteristic saturation at a field where the magnetization of the free layer
is fully rotated; figure 7, blue circles for sample mA. The effect is
normalized to the volume of the MCE active layer (spacer or thick film), so
the fact that the initial $\Delta T_{\text{ad}}$ vs $H_{\text{AC}}$ of the
trilayer is much steeper than that of the thick film indicates that the
effective (exchange) field producing the response is much stronger than the
externally applied field. Naturally, as the magnetization vector of the free
layer rotates, its proximity field within the spacer (mostly long-range
component at higher temperatures and mostly direct exchange near
$T_{\text{C}}$) follows the rotation and partially cancels out that from the
pinned layer when the rotation is complete (AP state). The resulting
relatively large (as it is of exchange origin) effective-field differential as
given by (13) is expected to scale with the angle of rotation, produce a steep
slope, and saturate when in the AP state. This picture applies above as well
as near the Curie point and describes well the experimentally observed
behavior, shown in figure 7.

Interestingly, the fact that the 295 K data has the same functional form of
step-saturation as the 170 K data (figure 7, green diamonds), justifies the
assumption that a weakly temperature dependent, long-range exchange is present
in the structure at temperatures much higher than the transition point.
Otherwise, with the direct-exchange proximity fields (very short-range at
high-$T$) fast-decaying and not overlapping in this regime, the external field
would be expected to produce a linear slope of the type seen in the mC sample
data. In the Simulations section below, we will discuss in detail domain-wall
like spin distributions within the spacer in the AP state responsible for this
MCE behavior and how they vary with temperature and exchange characteristics.

Figure 10:  Entropy change obtained from $\Delta T_{ad}$ data of figure 8 for
samples mA (trilayer; blue circles) and mC (bulk alloy; red squares); $\Delta
H=250$ Oe. Half filled symbols mark data used for fitting $\Delta
S(T)\propto\tau^{-k}$ at $T>T_{\text{C}}$ (dotted lines), with parameters
found in main text.

The above qualitative argument offer a compelling illustration of the
mechanisms behind the studied phenomenon. However, the presence of the
internal exchange inside the spacer material, including the range above its
Curie temperature, should be expected to contribute and likely modify, at
least quantitatively and potentially significantly, the MCE in the F/f/F
system near and below $T_{\text{C}}^{\text{s}}$. We therefore have conducted
in-depth numerical simulations of the system in a wide parameter space,
including direct-proximity as well as long-range exchange interactions, for
which a computationally efficient phenomenological model has been developed
for large phase-space mapping of our system, which is difficult to do using
other approaches, such as the well-established but more resource intensive
atomistic spin dynamics (discussed in detail elsewhere [40]; showing good
qualitative agreement with the conclusion herein in regions of overlapping
parameter space, e.g., only short-range exchange). The results of the
numerical simulations are presented below and show that the AP state of the
trilayer near and above $T_{\text{C}}$ has a nontrivial spin configuration
similar to a Bloch domain wall, and it is this configuration left out by the
paramagnetic-spacer model that actually determines the strength of the MCE in
the system and its variation with temperature and composition.

###  IV.1 Numerical Simulations

Figure 11: Magnetization reversal loops of $\text{F}_{\text{p}}$/f/F system
simulated to fit measured $M-H$ loops of figure 4: (a) without long-range
exchange, with short-range exchange only; (b) with long-range exchange as well
as short-range exchange.

Atomistic spin simulations of our system [40], which would incorporate long-
range spin-spin interactions (several nanometers, full width of spacer, upto
40 nearest neighbors), would be extremely time and resource consuming. We,
therefore, develop here a phenomenological computational model, simulating the
F/f/F system as a stack of magnetization vectors representing atomic
monolayers throughout the trilayer. The magnitude of the magnetization of a
monolayer is defined by the Brillouin function of temperature and the local
effective field, consisting of a Zeeman term and the intra- and inter-
monolayer exchange contributions. The direction of the magnetization vector is
the same as the direction of the local effective field acting on it. The
strong out-of-plane demagnetization in our thin film geometry restricts the
spins orientationally to the film plane. One of the ferromagnetic layers is
‘pinned’ by a fixed static field of 450 Oe, the other has no in-plane
anisotropy and would be free to rotate in a variable external field in case
the spacer was fully nonmagnetic.

The overall minimization algorithm consists of separate consecutive steps of
recursive calculations of the magnetization ($M$) and a minimization of its
orientation ($\varphi$) by the gradient descent method – this sequence
constitutes a single ‘iteration’. The exit from the iteration loop is possible
on two conditions - either $\Delta M<10^{-7}$ $\text{emu cm}^{-3}$ and
$\Delta\varphi<10^{-6}$ deg are reached, or the maximum number of iterations
is reached. The final state of the system for various parameter combinations
showed only minor changes after about 90k iterations, which was taken to be
the maximum simulation length, sufficient to achieve good accuracy.

The values of the saturation magnetization and Curie temperature used in the
simulations were obtained from the literature [41]. The Curie temperature of
the spacer was chosen to be 160 K as this value was observed on the
experiment. The Curie temperature of the free and pinned ferromagnetic layers
was chosen to be 2000 K to minimize the variation of the F-layers’
magnetization with temperature. This approximation speeds up the simulations
and essentially limits the variable part of the system to only the spacer, as
we specifically aim to investigate the effect of the properties of the spacer
on the overall behaviour of the trilayer. The strength of the intrinsic
exchange in the spacer was chosen so as to reproduce the experimentally
observed Curie temperature (160 K). The strength of the short-range proximity
exchange between the FeCr and Py layers (at F/f interfaces) generally depends
on the quality of the interfaces and, for our samples, was numerically
estimated to be about 150% of the intrinsic exchange in the spacer.

To qualitatively reproduce the magneto-statics observed on the experiment, two
simulations of the F/f/F system were performed using the aforementioned
algorithm. We have compared the magnetostatic behaviour of the system for the
case where the only exchange interaction available is direct exchange with the
nearest and the next nearest neighbours, and the case where the direct short-
range exchange is supplemented by long-range exchange interactions between the
interface spins of the F layers and the spins in all of the spacer (with
spatially decaying strength). In what follows, we refer to these two types of
simulations as “without long-range exchange” and “with long-range exchange”.
The long-range exchange constant is taken to have the same $z$-dependence as
the corresponding exchange field given by (8):

$J_{\text{f}}(z)=J_{\text{0f}}e^{{-z}/{l_{\text{sf}}}},\quad
J_{\text{p}}(z)=J_{\text{0p}}e^{{(z-t)}/{l_{\text{sf}}}},\ $ (14)

where $J_{\text{f}}(z)$ and $J_{\text{p}}(z)$ are the $z$-dependent exchange
constants proximity-induced within the spacer by the free and pinned F/f
interfaces, $J_{\text{0f}}$ and $J_{\text{0p}}$ – their interface values (at
$z=0$), $z$ – the coordinate normal to the film plane, $l_{\text{sf}}$ \- the
characteristic decay length of the long-range exchange interaction (the spin-
flip length in the spacer, assuming the effect is due to polarized electrons).

The simulated $M-H$ loops for the two model cases, namely, without and with
long-range exchange, are shown in figure 11. Parameters $l_{\text{sf}}=4$ nm
and $J_{\text{0f}}=J_{\text{0p}}=$ 2% of the intrinsic exchange in the spacer
were found to optimally fit the measured data of figure 4 (although likely not
the only one possible, this parameter combination is representative of the
optimal range and is quite reasonable based on the general considerations the
energetics and scales of spin relaxation and spin transfer in dilute
ferromagnetic alloys). It is clearly visible that with only the direct
exchange, the coupling between the $\text{F}_{\text{p}}$ and F layers through
a 6-nm spacer fully vanishes at $T>T_{\text{C}}^{\text{s}}$, as the minor loop
of the F layer is centered at zero field. In contrast, for the case with both
the direct and long-range exchange, the inter-layer coupling survives up to
room temperature, yielding a nonzero exchange bias of the free layer. This
specific feature of the simulated magnetization reversal loop, which we find
to be critically dependent on the presence of long-range exchange, reproduces
the behavior observed experimentally in (figure 4(b-d)) and previously in [14,
19, 10, 11, 24].

Figure 12: Simulated magnetization magnitude and orientation through thickness
($z$) of F${}_{\text{p}}$/f/F system: (a) without and (b) with long-range
exchange; $z=0\div 4$ nm is occupied by pinned layer, $4\div 10$ nm – spacer,
$10\div 16$ nm – free layer. Solid lines indicate magnitude of magnetization
vector (left axis scale) and dashed lines indicate its orientation (right axis
scale). Insets: magnetization reversal loops with characteristic points
indicated by numerals, $n=1,2,3$, where point $n$ corresponds to magnetization
profile $M_{n}(z)$ and orientation profile $\theta_{n}(z)$. $T=174$ K.

A completed iteration sequence yields the equilibrium magnitude and
orientation of the magnetization of each monolayer, for a given parameter set.
As an example, the evolution of the simulated magnetization profile during the
switching of the free layer, with the long-range exchange disabled and
enabled, are shown in figure 12(a) and (b), respectively, for $T$=174 K.
Analyzing and comparing the data in figure 12 (a) and (b), several important
observations can be made. In the parallel state and with only short-range
exchange enabled (point 1 in (a)), the proximity-induced magnetization decays
exponentially from the F/f interfaces toward the center of the spacer, which
in fact is expected for penetration of the magnetic order parameter in the
Landau theory (for individual interfaces). Despite the fact that the direct
exchange may induce substantial magnetization inside the spacer in the direct
proximity of the F layers, this magnetization decays rapidly and vanishes
toward the center of the spacer, and therefore can not provide either a
significant change in magnetization on P-to-AP switching or any appreciable
coupling between the outer F layers. With both direct and long-range exchange
present (figure 12(b)), the P-state magnetization deviates from the above
behavior and is an order of magnitude higher in the middle of the spacer
compared to that for only direct exchange. A similar tendency is observed for
the AP-state (point 2 in (a) and (b)) - the magnetization profile functionally
deviates from an exponential decay and this deviation is much stronger in the
case where the long-range exchange is enabled. Due to a cancellation of the
opposing effective exchange fields (AP-state), overlapping in the center of
the spacer, the magnetization at $z=7$ nm is drastically decreased, with the
difference on P-to-AP switching being much bigger when the long-range exchange
is present. The intermediate-state orientational profile (figure 12(b), point
3, dashed line for $\theta_{3}(z)$) reveals a scissor-like state in the
system, with a $\sim 90$-degree ‘exchange spring’ in the spacer, which must be
the reason for the absence of any significant cancellation of the effective
exchange at $z=7$ nm and hence no significant additional magnetic disorder. In
contrast, the orientational transition is sharp in the AP-state
($\theta_{2}(z)$) with a well defined 180-wall, a single nanometer in
thickness, producing strong local magnetic disorder.

The switching of the free layer in the model with only direct exchange (inset
to figure 12(a)) is step-like and no intermediate state is observed between
the parallel and antiparallel configurations at all temperatures higher than
the Curie temperature. Enabling long-range exchange results in non-vanishing
interlayer coupling in a wide temperature range above
$T_{\text{C}}^{\text{s}}$ and, consequently, in rotational states intermediate
between P and AP. Due to this coupling, both magnetization vectors of the
pinned and free layers tilt to an equilibrium orientation that is generally
different from the direction of the external field (e.g., point 3 in inset to
figure 12(b)). The magnetization magnitude in the spacer decreases only
slightly from its value in the P state for intermediate angles. Only in the
vicinity of the AP state the spacer demagnetizes significantly in the very
center, showing a narrow orientational spin transition similar to a Bloch
domain wall.

The numerical simulations have a high illustrative and interpretational value
as they yield the spatial spin profiles in the structure, which are difficult,
if at all possible, to obtain on the experiment – VSM magnetometry yields only
a projection of the net magnetic moment of the structure as a whole. Having
the microscopic spin distribution sheds light on the mechanism behind the
system’s magneto-caloric response and its strength. Thus, extracting the
change in the magnetization magnitude on P-to-AP switching yields the change
in magnetic disorder (or entropy of the magnon sub-system; between points 1
and 2 in the inset to figure 12(a) and (b)). This normalized magnetization
difference is shown in figure 13, simulated with (blue) and without (green)
long-range exchange. For comparison, the magnetization change in the bulk
spacer material in response to 250 Oe of external field is shown in red. To
convert the magnetization change to the magnetic entropy change, different
approaches can be used, but the general relation is that $\Delta
S\propto\Delta m$.

The key result presented in figure 13 is that for the trilayer system modelled
with the long-range exchange enabled (blue) the strong magnetic disorder (MCE)
induced by the P-AP transition persists to temperatures significantly higher
than $T_{\text{C}}^{s}$, with the maximum $\Delta m$ shifted to slightly
higher than $T_{\text{C}}^{\text{s}}$. This is in excellent agreement with the
results of our direct MCE measurements shown in Figs. 8,10. At the same time,
$\Delta m$ for the case of the trilayer with no long-range exchange as well as
for the bulk system (simulated with direct exchange only) peak at
$T_{\text{C}}^{\text{s}}$ and vanish directly above it. It is clear that the
strength of the field-induced demagnetization is enhanced in the case of the
F/f/F trilayer system, both with and without long-range exchange taken into
account, compared to the case of the bulk alloy (thick film). Estimating
potential MCE efficiency of the proximity-enhanced material versus that of the
bulk spacer alloy, we note that the ratio of areas under $\Delta m$ of the
respective materials is proportional to the ratio of their relative cooling
power (RCP) [42], which means that the RCP of the proximity-enhanced spacer
(simulated with long-range exchange) can be up to two order of magnitude
higher than that of its constituent material. The simulated case with only
direct exchange (green in figure 13) does show enhanced MCE and RCP, however
it clearly disagrees with the experiment as to the peak position and,
especially, the high temperature range where the effect is pronounced.

Each magnon corresponds to one Bohr magneton of thermal demagnetization (one
$\hbar$ in angular momentum), distributed in energy up to
$k_{B}T_{\text{C}}^{\text{s}}$. Taking as an estimate the mean magnon energy
for our spacer as 10 meV and the simulated peak value of $\Delta m=0.6$
emu/cm-3 for the trilayer, we arrive at the peak adiabatic temperature change
of approximately 30 mK, which agrees well with 25 mK obtained on the
experiment (figure 8).

Comparing the simulated and the experimental results (figure 13 versus figure
10) we conclude that long-range exchange is necessary to adequately explain
and quantify the observed MCE effect. The long-range exchange affects the
magnetization profile within the spacer in a significant way, magnetizing the
otherwise paramagnetic material into a domain-like spin distribution of
spatially variable magnitude, which shows a strong change on P-to-AP switching
in the structure, in a comparatively very low external field ($\sim 100$ Oe,
$2\div 3$ orders of magnitude lower than for conventional ADR systems). Long-
range exchange is the key to have MCE peak noticeably above
$T_{\text{C}}^{\text{s}}$ and extend in a strong fashion to temperatures
significantly higher than $T_{\text{C}}^{\text{s}}$, which should be useful
for extending the material’s operating range in device applications. Our
experiment and simulations confirm that the concept of enhanced MCE in F/f/F
systems is valid both in terms of higher peak entropy-change values as well as
much higher RCP.

Figure 13:  Simulated temperature dependence of magnetic disorder $\Delta m$
induced by P-to-AP switching for trilayer with _both_ direct and long-range
exchange (blue) and _only_ direct exchange (green), in units of former.
Simulated $\Delta m$ for bulk Fe-Cr alloy (thick film) in applied field of 250
Oe (corresponds to switching field of trilayer), likewise normalized (red).

##  V Conclusion

We have performed, using a membrane-based nano-calorimetry method, direct
measurements of the adiabatic temperature change in magnetic multilayers
designed for enhanced MCE. The measurements show that the volume and field
normalized MCE induced by the magnetic proximity effect in a specially
designed trilayer can be significantly higher than the conventional MCE in the
corresponding bulk material. The respective relative cooling power is vastly
stronger, one to two orders of magnitude, due in particular to a much broader
operating temperature range of the exchange-enhanced material. We therefore
conclude that the approach of exchange-enhancing MCE in magnetic
nanostructures is valid and effective in reducing the requirements on the
externally applied magnetic field. The experimental MCE measurement technique
was the key to these findings as the commonly used indirect magnetometry-based
methods were found to be inaccurate when applied to the orientational magnetic
transition in the studied multilayered system.

The obvious challenge with thin F/f/F type multilayered structures is the
relatively small volume of the active material with enhanced MCE properties.
This issue may be overcome by using spacer materials having intrinsically
higher MCE as well as by vertical stacking of the core trilayer structure. The
presence of the assisting F-layers roughly doubles the volume of the stack at
no direct benefit to MCE, however, the fact that they can switch in low fields
($\sim$10 mT) at very high speed (sub-$\mu$s) make feasible high-frequency
cooling cycles, orders of magnitude faster than the typical paramagnetic ADR
cycles where ramping up/down $\sim 5$ T field over minutes is required.
Particularly attractive may be cooling of micro/nano-devices (compact sensors,
sources, gates, etc.) based on the very membrane platform we demonstrate,
provided a mechanical arrangement for heat outflow can be integrated and
synced with the AC field cycle.

We find that a long-range exchange interaction, alongside the direct exchange,
must be taken into account in order to adequately describe the observed
properties of the studied metallic trilayer system. This interaction can be
due to the conduction electrons polarized by the strongly ferromagnetic outer
layers, traversing the weakly ferromagnetic spacer, thereby mediating spin-
spin exchange over several nanometers. Our phenomenological model, which
includes such long-range exchange, has allowed efficient, wide parameter-space
numerical simulations, which explain the experimentally observed behavior in
an illustrative as well as quantitatively accurate way. The enhanced MCE
effect is shown to be due to an exchange-spring in the spacer, pinched-off in
the center on rotation into the antiparallel state of the device.

###### Acknowledgements.

The authors are thankful to V.M. Kalita and Yu.I. Dzhezherya for fruitful
discussions. Support by Olle Engkvist Foundation (207-0460) and Swedish
Research Council (VR 2018-03526) are gratefully acknowledged.

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