# Picosecond all-optical switching of Co/Gd based synthetic ferrimagnets

Pingzhi Li∗<EMAIL_ADDRESS>Department of Applied Physics, Eindhoven University
of Technology, P. O. Box 513, 5600 MB Eindhoven, The Netherlands  Thomas J.
Kools  Department of Applied Physics, Eindhoven University of Technology, P.
O. Box 513, 5600 MB Eindhoven, The Netherlands  Hamed Pezeshki  Department of
Applied Physics, Eindhoven University of Technology, P. O. Box 513, 5600 MB
Eindhoven, The Netherlands  Joao M. B. E. Joosten  Department of Applied
Physics, Eindhoven University of Technology, P. O. Box 513, 5600 MB Eindhoven,
The Netherlands  Jianing Li  Department of Applied Physics, Eindhoven
University of Technology, P. O. Box 513, 5600 MB Eindhoven, The Netherlands
Junta Igarashi  Institut Jean Lamour, UMR CNRS 7198 – Université de Lorraine –
Boulevard des Aiguillettes, BP 70239, Vandoeuvre cedex F-54506, France  Julius
Hohlfeld  Institut Jean Lamour, UMR CNRS 7198 – Université de Lorraine –
Boulevard des Aiguillettes, BP 70239, Vandoeuvre cedex F-54506, France
Reinoud Lavrijsen  Department of Applied Physics, Eindhoven University of
Technology, P. O. Box 513, 5600 MB Eindhoven, The Netherlands  Stephane Mangin
Institut Jean Lamour, UMR CNRS 7198 – Université de Lorraine – Boulevard des
Aiguillettes, BP 70239, Vandoeuvre cedex F-54506, France  Gregory Malinowski
Institut Jean Lamour, UMR CNRS 7198 – Université de Lorraine – Boulevard des
Aiguillettes, BP 70239, Vandoeuvre cedex F-54506, France  Bert Koopmans
Department of Applied Physics, Eindhoven University of Technology, P. O. Box
513, 5600 MB Eindhoven, The Netherlands

###### Abstract

Single pulse all-optical switching of magnetization (AOS) in Co/Gd based
synthetic ferrimagnets carries promises for hybrid spintronic-photonic
integration. A crucial next step progressing towards this vision is to gain
insight into AOS and multi-domain state (MDS) behavior using longer pulses,
which is compatible with state-of-the-art integrated photonics. In this work,
we present our studies on the AOS and MDS of [Co/Gd]n ($n$ = 1, 2) using ps
optical pulses across a large composition range. We theoretically and
experimentally show that a large Gd layer thickness can enhance the AOS energy
efficiency and maximum pulse duration. We have identified two augmenting roles
of Gd in extending the maximum pulse duration. On the inter-atomic level, we
found that more Gd offers a prolonged angular momentum supply to Co. On the
micromagnetic level, a higher Gd content brings the system to be closer to
magnetic compensation in the equilibrized hot state, thereby reducing the
driving force for thermally assisted nucleation of domain walls, combating the
formation of a MDS. Our study presents a composition overview of AOS in
[Co/Gd]n and offers useful physical insights regarding AOS fundamentals as
well as the projected photonic integration.

##  I Introduction

Single-pulse all-optical switching of magnetization (AOS) in Gd-based metallic
ferrimagnetsStanciu  _et al._ (2007); Ostler  _et al._ (2012); Radu  _et al._
(2011), has attracted significant interests within the community for its
application potential in memory and storage, owing to its ultrafast switching
speed and very low energy dissipationKimel and Li (2019); Kim  _et al._
(2022). Initially discovered in GdFeCo alloysOstler  _et al._ (2012), AOS was
later also found in multi-layer synthetic ferrimagnets, such as Co/GdLalieu
_et al._ (2017) and [Co/Tb]nAvilés-Félix  _et al._ (2019). A unique merit of
Co/Gd based synthetic ferrimagnets is their large composition (layer
thickness) range for low threshold energy AOSLalieu  _et al._ (2017); Beens
_et al._ (2019); Li  _et al._ (2022a, 2021) in comparison with CoGd based
alloysXu  _et al._ (2017); Beens  _et al._ (2019); Wei  _et al._ (2021) and
[Co/Tb]nPeng  _et al._ (2023); Avilés-Félix  _et al._ (2019). This difference
has been attributed to a different switching mechanismBeens  _et al._ (2019);
Gerlach  _et al._ (2017), related to the notion that in [Co/Gd]n, AOS is
suggested to be driven by interfacial exchange scattering. Moreover, the
switching speed of AOS in Co/GdPeeters, van Ballegooie, and Koopmans (2022)
was found to be similar to that of CoGd alloysRadu  _et al._ (2011); Xu  _et
al._ (2017) , which displays a zero crossing of the magnetization within 1 ps.
[Co/Gd]n is also wafer-scale production compatible and allows for additional
interface engineering, such as combining with a possibility of adding Tb
(Gd/Co/Tb) for improved anisotropy without losing the fast switching speed of
AOSHintermayr  _et al._ (2023).

In recent years, interests involving integrating AOS in spintronics has
naturally emergedAlexey V.Kimel (2019); Sobolewska  _et al._ (2020); Becker
_et al._ (2017). One approach is to create AOS compatible magnetic tunnel
junctions (AOS-MTJ), in which the free-layer can be all-optically switched.
Synthetic ferrimagnets-based AOS-MTJs, such as those using Co/GdWang  _et al._
(2022) and [Co/Tb]nSalomoni  _et al._ (2023) as free layer, have yielded
significantly better tunneling magneto-resistance than the GdFeCo alloy
counterpartChen  _et al._ (2017). Meanwhile, another typical attempt is to
interface integrated photonics monolithically with spintronic racetrack
memoryLalieu, Lavrijsen, and Koopmans (2019); Li  _et al._ (2022a); Pezeshki
_et al._ (2022, 2023) for direct transport of information from the photonic
domain to the magnetic domain. Recently, progress demonstrating AOS in the
integrated photonics circuits has been reportedLi (2024).

As for compatibility of AOS with integrated photonic platforms, fs pulses (as
typically used in research) impose significant challenges. In particular, the
large transient electric field induces strong non-linear absorptionPezeshki
_et al._ (2023); Sobolewska  _et al._ (2020); Bente  _et al._ (2021); Pezeshki
_et al._ (2024). Moreover, the short pulse duration is accompanied by large
dispersion, jitter and modal competitions, which impede the use of fs pulses
for data transport in integrated photonics. In contrast, ps pulses can be
handled by passive waveguides with high pulse energy as well as generated with
high energy efficiency using on-chip laser sourcesVan Gasse  _et al._ (2019).

Another important consideration regards the undesired occurrence of a multi-
domain state (MDS) that typically occurs at fluences well above the AOS
thresholdStanciu  _et al._ (2007); Ostler  _et al._ (2012); Radu  _et al._
(2011); Lalieu  _et al._ (2017). To establish deterministic performance of
devices, the occurrence of MDS formation should be avoided. In practice, this
provides an incentive to optimize the fluence window, defined as the
difference between threshold fluences of MDS and that of AOS at a certain
pulse duration (see a conceptual illustration in FIG. 1). In work on alloys,
it has been reported that the fluence window strongly decreases upon
increasing the pulse durationWei  _et al._ (2021); Zhang  _et al._ (2022);
Verges  _et al._ (2024) (see FIG. 1). This behavior can be explained easily by
the notion that AOS intrinsically requires a strong non-equilibrium between
electron and phonon temperatures to occurBeens  _et al._ (2019); Mentink  _et
al._ (2012); Jakobs  _et al._ (2021). Using longer pulses, this non-
equilibrium reduces, therefore leading to a higher AOS threshold energy. In
contrast, in the case of MDS, the threshold energy is theoreticallyJakobs  _et
al._ (2021); Wei  _et al._ (2021); Zhang  _et al._ (2022) and
experimentallyVerges  _et al._ (2024); Wei  _et al._ (2021); Zhang  _et al._
(2022) shown to be bound by the pulse energy at which the phonon temperature
transiently exceeds the Curie temperature, a process called thermal
demagnetization. This MDS threshold remains almost flat for a low heat
conductive substrate. Consequently, a right triangular region of AOS is
resulting in a phase diagram, in which the fluence window linearly decreases
as a function of pulse duration. A narrow fluence window can be problematic
for device operation, particularly in cases of nano objects in which light
absorption can be strongly inhomogeneousPezeshki  _et al._ (2022, 2023). Thus,
when aiming at ps pulse operation, a combination of large fluence window with
homogenous light absorption in a spintronic nano-object lying within the
fluence window is desired during the integrated photonic design. Few studies
have been carried out so far on above mentioned synthetic Co/Gd using ps
pulses.

Figure 1: A conceptual illustration of the phase diagram of all-optical
switching (AOS), thermal demagnetization or multi-domain states (MDS) and
delayed MDS as a function of pulse duration and energy. The trend of the
threshold fluence of AOS and MDS (and delayed-MDS) is indicated by a solid
line and a dashed line respectively. The intersection between the solid and
dashed lines indicate the maximum pulse duration for AOS.

In this paper, we present our theoretical and experimental studies on AOS and
MDS in synthetic ferrimagnets of [Co/Gd]n as a function of composition and
pulse duration. In order to build up a physical understanding of the effect of
pulse duration on the microscopic level, we conducted theoretical studies
based on the microscopic-three-temperature model (M3TM)Koopmans  _et al._
(2010); Schellekens (2014); Beens  _et al._ (2019). We simulated the AOS and
the thermal demagnetization using longer pulses in Co/Gd multilayers in a
layer-resolved fashion and constructed its phase diagram, including a special
region in which MDS is occurring in a delayed fashion without entire thermal
demagnetization of the material (similar to FIG. 1). Guided by these
simulations, we further conducted experiments using ps laser pulses with
varying pulse duration. After confirming that the AOS can be achieved with ps
pulses, we investigated the composition dependence. We found that a higher
relative Gd content can both enhance the ps-AOS energy efficiency and prolong
the maximum pulse duration for AOS, which is in a qualitative agreement with
our theoretical studies. Furthermore, we found that a high Gd content in our
[Co/Gd]n suppresses the energy efficiency for AOS using short pulses but
significantly enhances the energy efficiency at longer pulses. In contrast,
the maximum pulse duration for a Co dominated samples is significantly lower
as a result of a delayed behavior of MDS compared to that of Gd dominated
samples. The phase diagram of Co dominated samples deviates from a right
triangular shape, which indicates extra mechanisms beyond microscopic level
also contribute to their MDS formation. Our studies reveal the composition
dependence of the AOS in [Co/Gd]n using ps pulses, which is an essential piece
of information to establish compatibility of [Co/Gd]n with integrated
photonics.

##  II Theoretical Investigation

Figure 2: The (absorbed) threshold energy density of AOS and thermal
demagnetization (MDS) as a function of pulse duration for various composition
(constituent layer thickness) (a) Co(4L)/Gd(2, 3, 4, 8 L), (b) Co(2, 3, 4,
5L)/Gd(4L), (c) Gd(1, 2, 3, 4L)/Co(4L)/Gd(1, 2, 3, 4L), (d) Gd(4L)/Co(2, 3, 4,
5L)/Gd(4L), as obtained from our layer-resolved M3TM-based simulation. The
solid symbols indicate the threshold energy for AOS. The open symbols indicate
the threshold fluence for thermal demagnetization. The maximum pulse durations
(within our computation resolution) for each composition were obtained from
the intersection between the extrapolated trends of the two curves. Their
values are marked by vertical dashed lines.

In order to gain microscopic insight on the influence of composition on AOS
and MDS, we first present our theoretical investigations on AOS and thermal
demagnetization in Co/Gd multilayers. We base our theoretical approach on the
M3TM with an addition of angular momentum exchange scattering between Co and
Gd sublatticesKoopmans  _et al._ (2000); Schellekens (2014). In our model, the
magnetization of each atomic layer is assumed to be a macrospin, which is
governed by electron and phonon temperature dynamics. The magnetization of
each layer is exchange-coupled and obtained following Weiss’s mean field
theoryKoopmans  _et al._ (2010) (see Sup. A.1). Within our model, during the
transient non-adiabatic heating process induced by a short laser pulse, the
angular momentum can locally be lost following the Elliot-Yafet spin-flip
scattering and can be also changed via exchange scattering with the
neighboring layers . We assume that the electron and phonon heat conductivity
are high enough to have the electron and phonon temperature being uniform
throughout our metallic stack. We treat the heat sinking to the substrate
phenomenologically with a characteristic timescale of 20 ps (see Sup. A.2). We
keep the parameters in our simulation (see the table in Sup. A.2) similar to
our previously presented works Koopmans  _et al._ (2010); Schellekens (2014);
Beens  _et al._ (2019); Li  _et al._ (2022b) but have adjusted the exchange
scattering and Elliot-Yafet spin-flip scattering strength to match the
simulated results with experimental ones within an order of magnitude. In this
way, we simulate the AOS of our designated system in the most idealistic case
with a computationally viable approach.

In our investigations, we simulated the switching dynamics of the Co/Gd
multilayer structure at different pulse energy. Scanning the pulse energy
upwards, at low energies, the simulated time traces only show a fast
demagnetization followed by a remagnetization in the original direction. Above
a certain threshold energy, a zero crossing of the magnetization in all atomic
layers is found, followed by a remagnetization in the opposite direction (see
FIG. 7 in Sup. A.2), thus AOS takes place. We term such a threshold value as
the threshold energy for AOS.

At even higher pulse energies, the magnetization stays close to zero for very
long time (see FIG. 7 in Sup. A.3). This observation is related to the fact
that the phonon temperature remains above or close enough to the Curie
temperature for a significant amount of time (see FIG. 8 Sup. A.3). In
reality, it is well-known that a too high pulse energy above a certain
threshold value brings the system into a MDSLalieu  _et al._ (2017); Gorchon
_et al._ (2016); Wei  _et al._ (2021); Zhang  _et al._ (2022). We thus
associate our observation of this full thermal demagnetization to MDS. In our
simulation, we define the threshold energy for the thermal demagnetization (or
MDS), as the pulse energy above which, the averaged normalized magnetization
of all Co layer remains smaller (less negative) than $-$10$\%$ in the reversed
direction at 6 ps after arrival of the peak of pump pulse. This choice was
motivated by the fact that, as a real system remains demagnetized for longer
than 5 ps before switching, the thermal fluctuations would already eliminate a
possible switching scenario at a later time or already disrupt a slightly
switched process.

We start by discussing the generic behavior of the effect of the pulse
duration on the threshold energy. FIG. 2a and 2b show the extracted threshold
energies for AOS and MDS, i.e. a phase diagram, for a bilayer Co/Gd sample.
Using that of Co(4L)/Gd(2L) as a typical example (number in parantheses
indicates the number of atomic layers), the threshold energy for AOS increases
as a function of the pulse duration. We attribute this behavior to the fact
that prolonged pulses reduce the non-adiabatic temperature difference between
electrons and phonons, thus, the peak electron temperature reduces, which is
commonly considered a key criterion for AOSMentink  _et al._ (2012); Beens
_et al._ (2019). We also see that the threshold energy for MDS remains almost
flat, which is consistent with the earlier experimental observations in GdCo-
based alloysWei  _et al._ (2021); Zhang  _et al._ (2022) and earlier
atomistic-LLG simulationsRaposo  _et al._ (2022); Jakobs  _et al._ (2021); Wei
_et al._ (2021); Zhang  _et al._ (2022). As the pulse duration increases, it
can be seen that the switching window gradually closes till the two curves
intersect at the maximum pulse duration. Such a behavior is consistently
observed in all compositions we have simulated.

After having verified the generic behavior, we continue our discussion on the
influence of the composition on the threshold energy and maximum pulse
duration. We will first address the dependence on the number of Gd layers.
FIG. 2a displays results from different Gd thickness, while the effect of
changing the Co thickness is shown in FIG. 2b. As compared with the bilayer
composition of Co(4L)/Gd(2L), FIG. 2a shows that adding one additional or more
Gd layers almost does not change the threshold energy for both AOS and thermal
demagnetization. Since the Curie temperature of Gd is below room temperature,
we understand this by the notion that the Curie temperature is dominated by
the Co layer (see Sup. A.1). It can be also observed that the maximum pulse
duration increases adding one additional Gd layer. Next we consider the
influence of the amount of Co layers (see FIG. 2b). We can see that the
threshold energy for both AOS and thermal demagnetization increases upon
adding more Co layers from 2 to 7, as a consequence of the associated increase
of Curie temperature with added Co layers, which enhance the exchange
stabilization (see Sup. A.1). The maximum pulse duration displays an opposite
trend, i.e. it decreases from approximately 7 to 3.5 ps with increasing number
of Co layers, which will be explained in a later discussion.

We next shift our focus to the effect of an added Gd interface (see FIG.
2c-d). Here, we address the differences between a Co/Gd bilayer and a Gd/Co/Gd
trilayer. In FIG. 2c, we show the threshold energy for both AOS and thermal
demagnetization as a function of Gd layers sandwiching Co (4L). We see that
adding Gd can both reduce the threshold energy for AOS and extend maximum
pulse duration in a similar fashion to the bilayer case, however, in a more
pronounced way. Comparing Co(4L)/Gd(2L) with Gd(2L)/Co(4L)/Gd(2L) (see FIG. 2a
and c), it can be seen that, despite Gd(2L)/Co(4L)/Gd(2L) exhibiting a
significant higher threshold energy for thermal demagnetization, its AOS
threshold energy is lower than Co(4L)/Gd(2L). Moreover, the maximum pulse
duration of Gd(2L)/Co(4L)/Gd(2L) is 50$\%$ higher. Now comparing between
Gd(1L)/Co(4L)/Gd(1L) with Gd(2L)/Co(4L)/Gd(2L), it can be seen that
Gd(2L)/Co(4L)/Gd(2L) displays a lower AOS threshold energy and a higher
thermal demagnetization threshold due to a higher Curie temperature as well as
a significantly longer max pulse duration. Adding more Gd away from the two
interfaces continues this trend, which is similar to the bilayer case of
Co(4L)/Gd(2-8L) (see FIG. 2a), but with a significantly larger impact (compare
between FIG. 2 a and c). It is also worth noting that for short pulses, the
AOS threshold energies of the compositions in FIG. 2c are similar, while for
longer pulses, those in Gd(2-4L)/Co(4L)/Gd(2-4L) are much lower.

Lastly, we discuss the behavior of Gd(4L)/Co(X)/Gd(4L), in which X is the
number of Co layers (see FIG. 2b and d). FIG. 2d shows the dependence of
threshold energy for both AOS and thermal demagnetization as a function of
number of Co layers. We see that by reducing the number of Co layers, the
threshold energy for both AOS and thermal demagnetization significantly
reduce, while the maximum pulse duration significantly increases. This trend
is similar to the behavior of corresponding bilayer Co(X)/Gd(4L) as discussed
above in FIG. 2b. However, with an added Co/Gd interface, the AOS threshold
energy is much lower than the bilayer while with similar thermal
demagnetization threshold consequently the maximum pulse duration is
significantly longer for the trilayer case.

Thus, based on our theoretical investigations, we can briefly summarize at
this point that an added Gd content by either increasing the amount of Gd
layers, reducing the amount of Co layers or adding a Co/Gd interface can
increase the energy efficiency of AOS and extend the maximum pulse duration.
The changes of threshold energy for AOS induced by Co are mainly due to the
modification of the Curie temperature, while that induced by Gd are mainly a
result of the larger angular momentum transfer from added Gd contents. For
instance, the bilayer and trilayer share similar threshold energy for thermal
demagnetization, but the trilayer displays a much lower threshold energy for
AOS. As for the maximum pulse duration, we understand this behavior based on
the fact that an increased relative Gd content effectively increases the time
of the demagnetization process and the angular momentum transfer via the
exchange scattering with the Gd sub-system Jakobs  _et al._ (2021); Davies
_et al._ (2020); Kim  _et al._ (2022). During these processes, the Gd content
can continuously supply reversed angular momentum to already demagnetized Co
assisting it to switch sign at a lower pulse energy. Moreover, using longer
pulses leads to prolonged AOS dynamics, which is more susceptible to the
thermal agitation. More reversed angular momentum from Gd also helps to
stabilize switching, which facilitates the AOS at longer pulses.

##  III Experimental Setup

Figure 3: Kerr images of switched domains of various composition, pulse energy
and duration. All the images share the same scale. The identification as
either AOS or MDS is of given in some of the images. A whole image is marked
as AOS if not the whole area is MDS. (a)-(c): Kerr image after the first and
second shot of the samples with various composition, pulse energy and pulse
duration. (a) and (b) are for composition Co(0.7)/Gd(1.75)/Co(0.5)/Gd(1.5),
(c) is for Co(0.7)/Gd(1.0)/Co(0.5)/Gd(1.5). (d) Kerr images of the first shot
for composition (shown left of the corresponding row of image) after
illumination with a laser pulse of various duration at a pulse energy of 1000
nJ.

We now proceed with experimental investigations on AOS and MDS in [Co/Gd]n. We
used DC magnetron sputtering to create our magnetic thin film sample, in which
a moving wedge shutter is incorporated during deposition to create a spatial
gradient of film thickness. For this study, we have wedge deposited (thickness
in nm in parentheses) Ta(4)/Pt(4)/Co(0.7)/Gd(0-2.25)/Co(0-1.5)/Gd(1.5)/TaN(5)
on a Si/SiO2(100) substrate at a base pressure of less than $1\times 10^{-8}$
mBar. Ta/Pt serves as a seed layer and induces a large perpendicular magnetic
anisotropy (PMA)Lalieu  _et al._ (2017); Li  _et al._ (2022b), while TaN is
used as a capping layer to preserve the magnetic stability of GdKools  _et
al._ (2023); Li  _et al._ (2022a); Kools  _et al._ (2022) at ambient
condition. From here on, we will not mention the seed layer and capping layer
when referring to the exact composition. Using a double wedgeKools  _et al._
(2022) allows us to study both the case of bilayer [Co/Gd] and quadlayer
[Co/Gd]2 in a single deposition run.

In our experiments, samples were illuminated by linearly polarized laser
pulses with Gaussian spatial energy profile with center wavelength of 800 nm
at about 45 degrees from perpendicular incidence with known pulse energy and
pulse duration. As a result, AOS takes place at a certain fluence range and
leaves with a switched domain. AOS was verified to take place at a ps
timescale. Some examples of switching dynamics of [Co/Gd]n illuminated using a
0.05 ps pulse are given in FIG. 9g-h from Sup. A.4. A chirped-pulse amplifier
(CPA) setup was used to stretch the pulse duration while keeping the same
pulse energy. An auto-correlator was used to determine the pulse duration
right after adjusting the configuration of the CPA. We exploited polar Kerr
microscopy to image the switched domains at given pulse durations and
energies.

Some example Kerr images of AOS and MDS following the laser illumination with
the same spot size are presented in FIG. 3. We determine the threshold fluence
following the method commonly usedLiu (1982); Lalieu  _et al._ (2017); Li  _et
al._ (2021); Xu  _et al._ (2017); Wei  _et al._ (2021), which is based on
fitting the dependence of switched domain area on pulse energy to our Gaussian
laser profile. We note that we identify a given case as switching only when we
see a 100$\%$ switching after a single shot and 100$\%$ switching-back using a
second shot with the same laser pulse energy at the same location as shown in
FIG. 3a-c. Here, we specifically note that the fringes at the edge after a
second shot are due to the vibration of the sample during the measurement and
pulse energy fluctuations, such that the second shot cannot fully overlap with
the first shot. A MDS is identified by the appearance of more than two domains
or a single domain deviating significantly from an elliptical shape in the
region of illumination at a given laser fluence. Sometimes, the multi-domain
region may contain more than 50% of switched domains. In this case, we still
count these cases as MDS. For example, as shown in FIG. 3d, for the MDS cases
of both Co(1.0)/Gd(1.5) and Co(0.85)/Gd(1.5), with a single shot using a 5 ps
pulse, the domains created are not uniform, and with a second shot, the
domains cannot be switched back (not shown).

##  IV Experimental Results

Figure 4:  (a)-(d) Threshold fluence for AOS (closed symbol) and multi-domain
state (open symbol) for bilayer Co(X)/Gd(Y) (a)-(b) and quadlayer
Co(0.7)/Gd(X)/Co(0.5)/Gd(1.5) (c)-(d). (a),(c) The switching phase diagram
(threshold fluence as a function of pulse duration) of bilayer Co(X)/Gd(Y) (a)
and quadlayer Co(0.7)/Gd(X)/Co(0.5)/Gd(1.5) (c). (b),(d) Composition
dependence of threshold fluence for AOS and multi-domain state for bilayer
Co(X)/Gd(Y) (b) and [Co/Gd]2 Co(0.7)/Gd(X)/Co(0.5)/Gd(1.5) (d). In (a), a
color plot was made based on the phase diagram of a bilayer of
Co(1.2)/Gd(1.5), indicating the region of switching (orange), non-switching
(purple) and multi-domain state (green). In (b), the thickness of the Co and
Gd is indicated below the figure. In (c), the phase diagram of Co dominated
samples are presented in the top part and Gd dominated samples are presented
in the bottom part. In (b) and (d), the Co and Gd dominated regime are color
coded by red and blue, respectively. Their bordering vertical line indicate
the composition of magnetic compensation. A guide to the eye (in the form of
colored arrow) is added in panel (b) and (d) connecting AOS and MDS threshold
curves with the pulse duration.

We begin our discussion with the switching experiments on Co/Gd bilayers. We
plot the threshold fluence of AOS and MDS of Co-dominated Co/Gd as a function
of pulse duration in a phase diagram as shown in FIG. 4a. A selection of
results for several Co- and Gd- dominated samples are collected in FIG. 4b,
ordered from low to high Gd content, where Co- and Gd- dominated regions are
accentuated by a red and blue background respectively. We first focus on the
Co-dominated samples in FIG. 4a. It can be seen that as the pulse duration
increases, the AOS threshold fluence increases almost linearly, corresponding
to a reduction of domain size in FIG. 3d, whereas the MDS threshold almost
stays constant. This makes a right triangular phase diagram. This behavior is
relatively well in line with our simulation shown in FIG. 2 as well as
experiments and atomistic LLG simulation results reported in alloysWei  _et
al._ (2021); Zhang  _et al._ (2022); Verges  _et al._ (2024); Jakobs  _et al._
(2021). The linear increase of AOS threshold fluence can be explained by the
reduced non-adiabatic difference between electron and phonon temperature. In
addition, in FIG. 4b left data points, it can be observed that the threshold
fluence for AOS increases with Co thickness, which is attributed to the
increase of Curie temperature as our simulation predicted (see FIG. 2). In
Sup. A.4, we provide more detailed composition dependence of threshold fluence
using 0.05 ps for AOS of Co/Gd bilayers.

Despite these qualitative agreements, the slight decrease of the threshold
fluence for MDS as a function of pulse duration as observed in FIG. 4a, makes
the shape of the resulting phase diagram deviating from the right-triangular
phase diagram. Additionally, the relative increase of MDS threshold with Co
thickness is much slower than for the AOS threshold fluence, i.e. the relative
fluence gap at 0.05 ps reduces from $\sim$ 6 to $\sim$ 3.5 varying Co layer
thickness from 0.7 to 1.2 nm. In our simulation (see FIG. 2), the threshold
energies for AOS and MDS almost scale proportionally, which means that the
relative fluence gap stays unaltered. Moreover, we see the negative slope for
MDS threshold fluence with respect to the pulse duration is largest for the
largest Co layer thickness of 1.2 nm, which leads to an acute triangular phase
diagram and a small value of the maximum pulse duration of 2.5 ps. Our
observations suggest that some other processes apart from local thermal
demagnetization also play their roles in forming a MDS, as will be elaborated
in the later discussion.

We have also measured Gd-dominated bilayer samples Co(0.7)/Gd(2.1, 3.1) Kools
_et al._ (2023) (see FIG. 4b, right most region). We see that the AOS
threshold fluence reduces with Gd thickness. This is in qualitative agreement
with our theoretical investigation (see FIG. 2a and c). Nevertheless, the
maximum pulse duration for these samples is less than our Co dominated samples
(except for the thickness of 1.2 nm). The reason behind this observation is
not clear. As for all bilayer samples (including Co-dominated samples), we
found that the maximum pulse duration for AOS is below 5 ps.

Now we shift gear to the discussion on quadlayer samples of [Co/Gd]2 at
composition Co(0.7)/Gd(X)/Co(0.5)/Gd(1.5). FIG. 4c and d show the threshold
fluence of both AOS and MDS of these compositions as a function of pulse
duration. As for the dataset for 0.05 ps laser pulses shown by black symbols
in FIG. 4d, it can be seen that the threshold fluence reduces as Gd increases,
approaching a minimum close to magnetization compensation (t${}_{\text{Gd}}$ =
1.09 nm)Li  _et al._ (2022a) (see also a zoomed-in view in Sup. A.4 in FIG.
9f), and staying almost flat at the Gd dominated regime unlike the monotonic
trend in bilayers as shown in FIG. 4b. On the other hand, the multi-domain
threshold fluence decreases, which we attribute to the reduction of Curie
temperature as Gd dilutes the exchange coupling strength between the two Co
layers. The relative fluence gap between MDS and AOS decreases from close to 8
to 5 going deeper from the compensation to Gd dominated regime. Our
observation suggests that, within the Gd dominated regime in quadlayer
[Co/Gd]2, the added Gd content suppresses the energy efficiency of AOS using
short pulses. As the pulse duration increases, interestingly, the threshold
fluence increases less drastically for samples with more Gd content. Moreover,
the minimum of AOS threshold fluence at compensation vanishes at longer
pulses. We conclude that added Gd content is beneficial for energy efficient
AOS with longer pulses.

After having explored the threshold fluence dependence in bilayer as a
function of composition, we focus on the phase diagram of [Co/Gd]2
Co(0.7)/Gd(X)/Co(0.5)/Gd(1.5) quad-layers. For a Gd layer thickness of 0.25 nm
and below, we found that the MDS threshold fluence decreases rapidly with
pulse duration, forming an “acute triangular” phase diagram (see FIG. 4c and
FIG. 1) similar to that of the bilayer with large Co content Co(1.2)/Gd(1.5)
(see FIG. 4a). At higher Gd layer thickness, the phase diagram becomes a
familiar “right triangle”. Interestingly, as the Gd thickness is above 0.5 nm,
the maximum pulse duration for AOS in this case can be extended to more than 5
ps, which is longer than the maximum pulse duration for all our Co/Gd
bilayers. Moreover, at the Gd dominated regime (see FIG. 4c bottom panel), the
maximum pulse duration can be extended beyond 7.5 ps. We attribute such an
enhancement to the added Gd content induced by an extra interface and more Gd
content away from the interface, qualitatively in line with our theoretical
investigation. Likewise, various other theoretical studiesDavies  _et al._
(2022); Jakobs  _et al._ (2021) also show that a higher Gd content in CoGd
alloys will facilitate a longer pulse AOS. Thus, we argue that a higher Gd
content is also crucial for extending the maximum pulse duration for our
synthetic [Co/Gd]2 structure. It is worth mentioning that the AOS thresholds
and fluence windows of Gd dominated samples at 5 ps (around $50\&$ of its AOS
threshold fluence) can satisfy the requirement of AOS imposed by our
integrated photonic designPezeshki  _et al._ (2023, 2024) without suffering
from significant non-linear absorption losses.

##  V Discussion on MDS formation

It is widely accepted that the primary factor leading to the formation of a
MDS is the elevation of the phonon temperature beyond the Curie temperature, a
viewpoint supported by referencesBeens  _et al._ (2019); Jakobs  _et al._
(2021); Mentink  _et al._ (2012); Davies  _et al._ (2020); Kim  _et al._
(2022); Wei  _et al._ (2021); Gorchon  _et al._ (2016). This aligns with our
M3TM-based model as well. However, based on our experimental phase diagrams
(see FIG. 4a and c), we have observed regions within samples with high Co
contents, in which MDS appears at fluences well below that required to
overheat phonons above the Curie temperature. This is clearly visible by the
normalized phase diagram shown in FIG. 5, which shows the data from two
characteristic samples with Co domination ($t_{\text{Gd}}=0.25$ nm) and Gd
domination ($t_{\text{Gd}}=1.25$ nm) after scaling the respective fluences
such that the AOS threshold fluences overlap. This rescaling accounts for the
fact that the Co dominated sample has a much higher Curie-temperature. since
AOS and MDS threshold fluences are expected to scale in the same way with the
Curie temperature (see our simulation results in FIG. 2), it would have been
anticipated that after rescaling also the MDS threshold for the two data sets
would overlap. However, the observed trend is strongly in contrast with the
prediction. More specifically, the normalized MDS threshold for the Co
dominated sample is significantly smaller than expected, leading to a
switching window that is reduced by a factor of almost 2 and leaving behind an
extra region of MDS, as compared with the Gd dominated sample. Recent studies
by Verges et al.Verges  _et al._ (2024), Linet al.Lin  _et al._ (2023) and Wei
et al.Zhang  _et al._ (2024) have suggested that the MDS formation in
perpendicularly magnetized alloys traces its origin also in the influence of
dipolar field. Hereby, we conjecture that this region corresponds to a delayed
formation of a MDS, well after the initial occurrence of successful AOS. This
transition is driven by thermally assisted processesLin  _et al._ (2023);
Zhang  _et al._ (2024) while still being in the hot state – albeit well below
the Curie temperature.

The reason for such a delayed MDS formation at fluences where the lattice
temperature remains well below the Curie temperature can be understood as
follows. The completion of the AOS process results in a uniform switched
domain. For this uniform state to remain stable at a longer timescale, it is
required that the domain wall energy is higher than the dipolar energy of the
switched region. However, as for Co dominated samples, the dipolar energy is
high and effective anisotropy is low, which leads to a low domain wall energy
as well as a high imbalance between the dipolar energy and domain wall energy.
Right after the AOS is completed, the system still remains in a hot state, in
which the relative Gd content becomes even less due to the steeper decay of Gd
magnetization as a function of temperature as compared to Co. This further
exacerbates this imbalance, making the system to reduce its dipolar energy by
forming an MDS, nucleated by thermally-assisted processes at the elevated
temperature. Moreover, with longer pulses, the degree of switching extending
to the opposite side becomes lessZhang  _et al._ (2022). This creates an
initial condition for thermalization closer to MDS, which might explain the
negative slope of MDS threshold fluence (see FIG. 5 for example). While
conditions for this delayed MDS formation are optimally met for Co dominated
samples, they are not for Gd dominated samples. Right after AOS, Gd dominated
system may transiently become Co dominated, again due to the stronger decay of
Gd magnetization with temperature. Upon cooling down, the system will remain
for a long time near the compensation temperature while returning from the Co
to the Gd dominated state. During this phase, the dipolar energy is much lower
compared to the domain wall energy, which can much easier stabilize a uniform
domain. Thus, we conclude that the delayed transition to a MDS should
preferentially occur in Co-dominated samples, in line with our experimental
findings. Here, it is noteworthy to mention that a full understanding of the
domain formation process requires a thorough micromagnetic study including
thermally assisted nucleation events, which is beyond the scope of this paper.

Figure 5:  The phase diagram for the quadlayer sample [Co/Gd]2
Co(0.7)/Gd(0.25)/Co(0.5)/Gd(1.5) ($t_{\text{Gd}}=0.25$ nm) and
Co(0.7)/Gd(1.25)/Co(0.5)/Gd(1.5) ($t_{\text{Gd}}=1.25$ nm). The threshold
fluence of each is normalized to their respective AOS threshold fluence using
0.05 ps pulses.

##  VI Conclusion

In conclusion, in this study, we theoretically and experimentally investigated
the AOS and MDS formation behavior of [Co/Gd]n in a large composition range
using ps pulses. Based on our M3TM studies, we have confirmed the augmenting
role of the relative Gd content on the inter-atomic level to enhance the
energy efficiency and prolong the maximum pulse duration for AOS. Our
experiments on [Co/Gd]2 show that an added Gd content can improve the energy
efficiency of AOS at ps pulse duration, despite suppressing the energy
efficiency of AOS at short pulses. It stabilizes a right triangular shaped
phase diagram and extends the maximum pulse duration for AOS beyond 7 ps. In
contrast, deep Co dominated samples exhibit an acute triangular shaped phase
diagram, in which the MDS threshold energy is much lower than required for the
phonon temperature to exceed the Curie temperature. We argue that the origin
lies in the competition between dipolar and domain wall energy after the AOS
dynamics. This, on the other hand, suggests another augmenting role of Gd on
the micromagnetic level. Our study offers fresh insights in ps-AOS and
likewise provides with guidance on designing the Co/Gd based out-of-plane
magnetic stack for energy efficient all-optically switchable devices in hybrid
integration between spintronics and integrated photonics.

###### Acknowledgements.

This project has received funding from the European Union’s Horizon 2020
research and innovation programme under the Marie Skłodowska-Curie grant
agreement No.860060. This work was part of the research program Foundation for
Fundamental Research on Matter (FOM) and Gravitation program “Research Center
for Integrated Nanophotonics,” which are financed by the Dutch Research
Council (NWO). This work was supported by the Eindhoven Hendrik Casimir
Institute (EHCI). This work is supported by the ANR-20-CE09-0013 UFO,
ANR-20-CE24-0003 SPOTZ, and ANR-23-CE30-0047 SLAM, the Institute Carnot ICEEL,
the Région Grand Est, the Metropole Grand Nancy for the project ‘‘OPTIMAG’’
and FASTNESS, the interdisciplinary project LUE ‘‘MAT-PULSE’’, part of the
French PIA project ‘‘Lorraine Université d’Excellence’’ reference
ANR-15-IDEX-04-LUE. This work was supported by the ANR through the France 2030
government grants EMCOM (ANR-22-PEEL-0009) and PEPR SPIN (ANR-22-EXSP-0002).
J.I. acknowledges support from JSPS Overseas Research Fellowships.

## Author Contribution

Pingzhi Li: Conceptualization (lead); Data Curation (lead); Formal Analysis
(lead); Investigation (equal); Validation (lead); Methodology (equal);
Visualization (lead); Supervision (equal); Writing - original draft (lead);
Writing - review editing (equal);

Thomas Kools: Methodology (equal); Investigation (Supporting); Writing -
original draft (supporting); Writing - review editing (supporting);

Hamed Pezeshki: Conceptualization(supporting); Methodology (supporting);
Writing - original draft (supporting); Writing - review editing (supporting);

Joao Joosten: Methodology (supporting); Writing - original draft (supporting);
Writing - review editing (supporting);

Jianing Li: Methodology (supporting); Writing - original draft (supporting);
Writing - review editing (supporting);

Julius Hohlfeld: Investigation (Supporting); Writing - review editing
(supporting);

Junta Igarashi: Methodology (supporting); Investigation (Supporting); Writing
- review editing (supporting);

Reinoud Lavrijsen: Funding Acquisition (equal); Writing - original draft
(supporting); Writing - review editing (supporting);

Stephane Mangin: Methodology(supporting); Investigation (supporting);
Validation (supporting); Supervision (supporting); Funding Acquisition
(supporting); Writing - review editing (supporting);

Gregory Malinowski: Methodology(equal);Investigation (equal); Validation
(supporting); Supervision (equal); Funding Acquisition (equal); Writing -
original draft (supporting); Writing - review editing (supporting);

Bert Koopmans: Investigation (equal); Supervision (equal); Funding Acquisition
(equal); Validation (supporting); Writing - original draft (supporting);
Writing - review editing (equal);

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##  Appendix A Supplementary information

###  A.1 Sup: Magneto-static estimate of Co/Gd using Weiss’s mean field model

In this section, we provide a simplified theoretical insight into the magneto-
statics of Co/Gd. As demonstrated in Figure 4b (and Figure 9b), we observe a
phenomenon where the magnetism induced in Gd can potentially surpass that of
Co in specific compositions of the Co/Gd system. We aim to theoretically
elucidate how the magnetism in Gd, initially induced from a pristine Co/Gd
interface, evolves in a static context with varying layer numbers. To quantify
this evolution, we employ exchange splitting defined by applying Boltzmann
statistics, based on an fcc oriented magnetic structure, following:

$\Delta_{ex,i}=\frac{J^{-}_{i}m_{i-1}}{4}+\frac{2J_{i}m_{i}}{4}+\frac{J^{+}_{i}m_{i+1}}{4}.$
(1)

The spin angular momentum (normalized to $\mu_{B}$) at each layer is expressed
as following in equation 2, in which $m_{i}$ and $S_{i}$ are normalized
magnetizaiton (relative to its saturation magnetization) and spin quantum
number at i-th layer, and $J^{-}_{i}$, $J_{i}$ and $J^{+}_{i+1}$ are exchange
coupling energy per atom with the previous, current and next neighboring
layer.

$m_{i}=\frac{1}{S_{i}}\frac{\sum_{s=-S_{i}}^{s=S_{i}}\text{exp}(\frac{s\,\Delta_{ex,i}}{k_{B}T})}{\sum_{s=-S_{i}}^{s=S_{i}}\text{exp}(\frac{\,\Delta_{ex,i}}{k_{B}T})}\\\
=\frac{1}{S_{i}}\frac{\sum_{s=-S_{i}}^{s=S_{i}}s\,\text{exp}(\frac{s(\frac{J^{-}_{i}m_{i-1}}{4}+\frac{2J_{i}m_{i}}{4}+\frac{J^{+}_{i}m_{i+1}}{4})}{k_{B}T})}{\sum_{s=-S_{i}}^{s=S_{i}}\text{exp}(\frac{s(\frac{J^{-}_{i}m_{i-1}}{4}+\frac{2J_{i}m_{i}}{4}+\frac{J^{+}_{i}m_{i+1}}{4})}{k_{B}T})}$
(2)

Exchange coupling at i-th layer $J_{i}$is defined as

$J_{i}=\frac{3k_{B}T_{C}^{i}}{S_{i}+1},$ (3)

whereas, the interlayer exchange between Co and Gd interface was set to be
$J_{Co/Gd}/k_{B}=1000K$Coey (2010); Kittel (2004). Here we take $T_{C}$ for Co
and Gd as 1388 K and 292 K respectivelyCoey (2010); Kittel (2004). Considering
the size of the Co and Gd atom, we approximate the ratio of Gd magnetic moment
per layer over that of Co to be a factor of two. We plot the normalized
magnetic moment (to the magnetization of a single Co layer) of the Co/Gd
bilayer as a function of underlying layer thickness in Figure 6. We found that
the magnetic compensation can be obtained up to 3 Co monolayers for less than
10 layers of Gd. Our study represent the most ideal case of induced magnetism
between Co and Gd.

Figure 6: The net normalized magnetization of Co/Gd bilayer (normalized to the
saturated magnetization of single Co layer) as a function of Gd layer number.
The positive sign is defined as the case in which the magnetism is Co
dominated. An orange dashed line marked the compensation.)

###  A.2 Sup: M3TM-based theoretical modeling of AOS with stretched pulse

To generate the simulation results presented in Figure 2, illustrating the
dependence of threshold energy of AOS and the thermal demagnetization on pulse
duration and composition, we employed a model that simulates AOS dynamics
based on M3TM. This model is also complemented by incorporating
phenomenological exchange scattering in accordance with prior
publicationsKoopmans  _et al._ (2010); Schellekens (2014); Beens  _et al._
(2019); Li  _et al._ (2022b).

In our simulation, temperatures are considered homogeneous over the full stack
and thermal parameters of Co and Gd were taken the same. The temperature
dynamics of electronic and lattice sub-systems are modeled as:

$\gamma
T_{e}\frac{dT_{e}}{dt}=g_{ep}(T_{P}-T_{e})+\frac{P_{0}}{\sigma\sqrt{2\pi}}e^{\frac{-t^{2}}{2\sigma^{2}}}$
(4)

$C_{p}\frac{dT_{p}}{dt}=g_{ep}(T_{e}-T_{p})-C_{p}\frac{T_{p}-T_{\text{amb}}}{\tau_{D}},$
(5)

in which, $T_{e}$ and $T_{p}$ are the electron and phonon (lattice)
temperature, respectively, $\gamma T_{e}$ and $C_{p}$ are the electron and
phonon heat capacity, $g_{ep}$ is the electron-phonon temperature coupling,
$\tau_{D}$ is the characteristic timescale of heat dissipation ($\tau_{D}=$ 20
ps) and $T_{amb}$ is the ambient (room) temperature (295 K). Although full
thermal relaxation can be many hundreds of ps, fast equilibration process
between overheated electron reservoir with the cooler substrate can already
have taken place within a few ps. Mobile electrons can pick up the energy of
the fast oscillating electric field from a laser pulse. To reflect this, the
laser energy dynamics is coupled as a second term in equation 4, in which
$P_{0}$ is the pulse energy in units of $\text{J/m}^{3}$ and $\sigma$ is the
standard deviation of the Gaussian pulse duration. The FWHM quoted in this
paper follows FWHM = $2\sqrt{2\text{ln}2}\sigma$. In our study, we assume all
layer are in thermal equilibrium with respect to each other, i.e. $T_{e}$ and
$T_{p}$ are uniform in all layers.

The magnetization (in each individual layer $m_{i}$) in our system is modeled
following a Weiss mean-free field approach (see Sup. A.1) following equation
2. The electron temperature dynamics is directly responsible for the
magnetization dynamics. A first contribution to the magnetization dynamics is
Elliot-Yafet spin-flip scatteringKoopmans  _et al._ (2010), which follows

$\frac{dm_{i}}{dt}|_{\text{EY}}=R_{i}\frac{\Delta_{ex,i}T_{p}}{2k_{B}T_{C,i}^{2}}[1-m_{i}\text{coth}(\frac{\Delta_{ex,i}}{2k_{B}T_{e}})]$
(6)

in which, $\Delta_{ex,i}$ is the exchange splitting of the i-th layer and
$R_{i}$ is the demagnetization rate of the i-th layer. A second contribution
is the exchange scattering of the angular momentumKittel (2004); Beens  _et
al._ (2019), the angular momentum exchange follows

$\frac{dm_{i}}{dt}|_{ex,i,i\pm 1}=\frac{\eta_{i,i\pm
1}\mu_{B}}{\mu_{at,i}}T_{e}^{3}\\\ [-\text{SI}(\frac{-\Delta_{ex,i,i\pm
1}}{k_{B}T_{e}})(\frac{1+m_{i}}{2})(\frac{1-m_{i\pm 1}}{2})\\\
+\text{SI}(\frac{\Delta_{ex,i,i\pm
1}}{k_{B}T_{e}})(\frac{1-m_{i}}{2})(\frac{1+m_{i\pm 1}}{2})]$ (7)

where, SI is a functionBeens  _et al._ (2019) integrating all the occupation
level of the density of states over energy assuming a Fermi-Dirac
distribution, $\mu_{at,i}$ is the atom size of the i-th layer, and
$\eta_{i,i+1}$ is the exchange scattering rate between i-th layer and i+1-th
layer.

The magnetization dynamics of the i-th layer involving the mentioned
contribution can be summarized as follows

$\frac{dm_{i}}{dt}=\frac{dm_{i}}{dt}|_{\text{EY}}+\frac{dm_{i}}{dt}|_{ex,i,i+1}+\frac{dm_{i}}{dt}|_{ex,i,i-1}$
(8)

Previous work following the above-mentioned identical approach, predicts an
AOS dynamic with a dramatic "plateau", i.e. the magnetization stuck at 0 for
significant long time, such dynamics is however not experimentally
observed(see ref. Peeters, van Ballegooie, and Koopmans (2022) and results in
Figure 9g-h). We understand this behavior as the strength in exchange
scattered angular momentum is not sufficient while the electron reservoir is
sufficiently heated above the Curie temperature. Moreover, keeping the
original parametersSchellekens and Koopmans (2013); Beens  _et al._ (2019);
Koopmans  _et al._ (2010), the resulting threshold energy for AOS is much
larger than experimentally quantified value and the energy window (between AOS
energy and Curie energy) is very little compared with our experiment of Co/Gd
multilayer (see Figure 4). Given that the exact report on the value of
exchange scattering efficiency is lacking and the mechanism is highly
phenomenological, we increased the exchange scattering strength between Co and
Gd ($\eta_{\text{Co,Gd}}$) by a factor 10 and reduced the spin-flip
probability of Gd ($R_{\text{Gd}}$) by a factor of two (while keeping the rest
of the parameters as identical as in ref. Koopmans  _et al._ (2010);
Schellekens (2014); Beens  _et al._ (2019)), in this way, we can significantly
reduce the "plateau"Beens  _et al._ (2019) (see Figure 7) to match the dynamic
behaviour similar to our experimental results presented in Figure 9g-h
qualitatively, and also create a much larger energy windowBeens  _et al._
(2019) as similar to our experiment.

The parameters mentioned in above discussion are presented in the table below

Parameter | Value | Unit
---|---|---
$C_{\mathrm{p}}$ | 4$\times$106 | Jm${}^{-3}K^{-1}$
$g_{\mathrm{ep}}$ | 4.05$\times$106 | Jm-3K-1ps-1
$\gamma$ | 2.0$\times$103 | Jm-3K-2
$\tau_{\mathrm{D}}$ | 20 | ps
$R_{\mathrm{Co}}$ | 9.550$\times$105 | Am-1ps-1
$R_{\mathrm{Gd}}$ | 0.196$\times$105 | Am-1ps-1
$T_{\mathrm{C,Co}}$ | 1388 | K
$T_{\mathrm{C,Gd}}$ | 292 | K
$\mu_{\mathrm{at,Co}}$ | 1.72 | $\mu_{B}$
$\mu_{\mathrm{at,Gd}}$ | 7.55 | $\mu_{B}$
$S_{\mathrm{Co}}$ | 1/2 |
$S_{\mathrm{Gd}}$ | 7/2 |
$\eta_{\mathrm{Co-Co}}$ | 4.615 | ns-1
$\eta_{\mathrm{Gd-Gd}}$ | 0.012 | ns-1
$\eta_{\mathrm{Co-Gd}}$ | 0.554 | ns -1
Table 1: Summary of the parameters in equations shown in Sup. A.1 and A.2.

We now present some characteristic results of the dynamics of AOS using pulses
with different pulse durations (see Figure 7). In this case, we used pulses
identical in pulse energy ($P_{0}$) but differ in pulse duration
($2\sqrt{2ln2}\sigma$). We see, in Figure 7, as $\sigma$ increases from 0.1 ps
to 1 ps, the dynamics gradually transits from AOS switching dynamics towards
demagnetization dynamics. In the meantime, the transient difference between
electron and phonon temperature reduces. We interpret this behavior as the
strength of exchange scattering and spin-flip scattering are monotonically
related to the non-adiabatic temperature difference and the peak electron
temperature. As the pulse is prolonged, especially closer to the
characteristic timescale of electron-phonon couplingKoopmans  _et al._ (2010),
the phonon temperature can more easily follow the electron temperature,
resulting a low temperature difference and low peak temperature. Our results
here also qualitatively matches that of atomistic-LLBJakobs  _et al._ (2021);
Raposo  _et al._ (2022); Wei  _et al._ (2021).

Figure 7: The simulated AOS dynamics (up) and two-temperature evolution
(electron and phonon) (down) of Co(2L)/Gd(4L) structure excited by laser
pulses of standard deviation $\sigma$ = 100 fs, 300 fs and 1000 fs
respectively (note that FWHM = $2\sqrt{2\text{ln}(2)}\sigma$) as a function of
pulse energy. The electron temeprature and phonon temperature is represented
in the same color, but can be differentiated by the underlying behavior as
electron temperature can temporarily rise up a lot higher than phonon
temperature. The pulse energy (in in unit of $10^{8}$ J/m3) is presented in
the color bar.

###  A.3 Sup: Determination of threshold energy for AOS and thermal
demagnetization from M3TM-based theoretical modeling

Now we proceed to extract the threshold energy for AOS and the maximum pulse
duration from the M3TM-based simulations mentioned earlier. To illustrate such
a process, we focus on the composition of Co(4L)/Gd(4L) as a characteristic
example. Figure 8 presents a plot of the magnetization at 6 ps (note that the
pulse peak occurs at 0 ps) for different pulse energies and durations.

When using a pulse with $\sigma$ = 0.05 ps and low energy, we observe a
transient quenching of magnetization followed by a remagnetization process.
This demagnetization process is also evident with a longer pulse ($\sigma$ = 1
ps in Figure 8), but at a higher pulse energy. As the pulse energy surpasses a
certain threshold, AOS in magnetization occurs. We define this threshold as
the threshold energy for AOS.

With further increases in energy, AOS continues to occur, but with an extended
plateau (see Figure 8) and a less pronounced end state on the opposite side.
Our model attributes this behavior to the fact that overheating reduces
magnetization, increases spin-flip probability, and decreases exchange
scattering rate, as described by equations 6 and 7.

Figure 8:  (Center) A plot of the normalized magnetization of Co component
(averaged over all Co layer) from Co(4L)/Gd(4L) obtained at the 6 ps of its
dynamics after the pulse illumination (peak reached at time 0 ps) as a
function of standard deviation $\sigma$ of the Gaussian pulse and pulse energy
(in unit of $10^{8}$J/m3). -10$\%$ of the end state after 6 ps is also marked
using a horizontal black line, which is used to differentiate cases between
switching and thermal demagnetization. The threshold pulse energy for thermal
demagnetization, as the intersection between the black line and the plots with
different pulse duration, is also marked up as a vertical colorbar, which also
indicates the threshold energy for different pulse duration (expressed in
$\sigma$). Note that the pulse duration (FWHM) of the gaussian pulse, as
quoted in our paper, is expressed as FWHM = $2\sqrt{2ln2}\sigma$. Some
examples of magnetization dynamics of pulses with $\sigma$ = 0.05 ps (dark
blue), $\sigma$ = 1.0 ps (light blue) and $\sigma$ = 2.4 ps (red) are
presented along side the plot. The pulse energy and pulse duration of these
selected values for pulse duration and energy are also marked up in the center
plot. The threshold pulse energy for different pulse duration (expressed in
$\sigma$), at which the peak phonon temperature during its dynamics exceeds
Curie temperature, is marked up as a vertical color bar.

As the pulse energy is raised even higher, the final magnetization state
approaches zero or very close to it (see Figure 8). In these cases,
magnetization remains zero after losing its angular momentum to the lattice
without crossing zero, or crossing 0 after staying at 0 for a prolonged time
(see Figure 8). It is also worth noting that, in these cases, the peak phonon
temperature is already very close to Curie temperature (see Figure 8).
Specifically, with short pulses, the peak phonon temperature is already higher
than the Curie temperature, whereas with long pulses, the peak phonon
temperature can be slightly lower. Additionally, it’s evident that as pulse
duration increases, zero crossing becomes less likely until it eventually
disappears. This aligns with the established behaviors of AOS processes at
longer pulse durations, both experimentally and theoretically, as demonstrated
in atomistic-LLG Jakobs  _et al._ (2021); Raposo  _et al._ (2022). We
attribute this behavior to the well-documented thermal demagnetization
processes commonly associated with multi-domain states, as observed in
numerous experimental studiesStanciu  _et al._ (2007); Ostler  _et al._
(2012); Lalieu  _et al._ (2017); Wei  _et al._ (2021); Zhang  _et al._ (2022).

In our studies, we chose to define the threshold energy for thermal
demagnetization from our model based on the criteria that, the end state at 6
ps obtained from our simulation should be at least lower (more negative) than
-10$\%$ to be able to qualify as AOS. This choice is motivated by the fact
that, thermal fluctuation field (which is not implemented in our model) can be
expected to easily interrupt a long dwelling at zero. As such a fluctuation
introduces randomistic direction of small angular momentum, which is enough to
break the degeneracy at zero developing to a undeterministic state in reality.
We, thus, consider the switching processes with dwelling time longer than 6 ps
or a slight switching more positive than -10$\%$ as also thermal
demagnetization.

###  A.4 Sup: AOS in [Co/Gd]n by a 50 fs laser pulse

In our study, additional samples as represented in Figure 9a were created to
give further information on the generic behaviour of magnetostatics of Co/Gd
bilayers, and also AOS statics and dynamics of [Co/Gd]n using a 50 fs laser
pulse. The samples were created following the method presented in our main
paper.

Figure 9: (a) The schematic illustration of the wedge sample stacks including
Co/Gd bilayer Co(X)/Gd(4), Co(0.7)/Gd(4) and [Co/Gd]2. The substrate Si/SiO100
of the all the sample is not present in the illustration. (b) - (e) include
the wedge scan study of magneto-statics characterized by MOKE and threshold
fluence of AOS using a 50 fs laser pulse. Here, we note that, due to the large
spot size of our probe laser, inaccuracy of 5$\%$ as the absolute position
error bar should be expected (not present in the figure). (b) MOKE signal of
Bi:Co, where as the signal of the sample saturated at +H (red) and -H (blue),
and Coercivity (black) as a function of Co thickness. (c) MOKE signal of
Bi:Gd, where as the signal of the sample saturated at +H (red) and Coercivity
(black) as a function of Gd thickness. (d)-(e)-(f) Threshold fluence (by 50 fs
laser pulse) of Bi:Co, Bi:Gd and Quad:Gd as a funciton of wedge layer
thickness. The magnetic compensation is marked in the figure as a vertical
line. (g)-(h) The switching dynamics of various composition (marked in the
legend) of Bi:Co and Quad:Gd. The incident fluence is marked in the figure.
Magnetization characterized using TRMOKE is normalized, the M>0 corresponds to
the direction of the Co magnetic moment parrallel to the externally applied
magnetic field.

First, we present the magnetic statics of Co/Gd bilayer, whose ps-AOS behavior
are presented in Figure 4 in the main text. We characterized the magnetic
statics of the presented samples using polar-MOKE ellipticity at wavelength of
658 nm. In particular, we measured the coercivity and the MOKE contrast as a
function position (composition) by sweeping magnetic field along the out of
plane direction at the same rate. The positive sign of MOKE signal corresponds
to the direction of the Co magnetic moment aligned with the direction of the
external magnetic field. We present our results on bilayer wedges of
Co(0.6-1.2)/Gd(4) and Co(0.7)/Gd(0-4). In Figure 9b, we plot the coercivity
and MOKE signal of Co(0.6-1.2)/Gd(4). At the Co thickness of 0.88 nm, the
bilayer is at magnetization compensation, which is evidenced by a divergent
coercivity and flipping sign of MOKE signal. Below this compensation
thickness, the system is in Gd dominated regime. Similarly, for
Co(0.7)/Gd(0-4) (see Figure 9c), the Gd dominated regime is reached at a Gd
thickness above 2.2 nm. The existence of Gd dominated regime at room
temperature, to our knowledge have not been reported in Co/Gd bilayer
systemPham  _et al._ (2016); Bläsing  _et al._ (2018); Lalieu  _et al._
(2017); Lalieu, Lavrijsen, and Koopmans (2019); Tan  _et al._ (2019). It is
worth noting that the Co(0.7)/Gd(2.1) presented in our main paper (see Figure
4) is Gd dominated. The underlying difference, i.e. the difference in
compensation thickness, we believe, lies in the sample to sample thickness
variation, uncertainties in the thickness calibrations of both Co and Gd.

Our theoretical calculation, based on Weiss mean-field model, shows that
compensation could not be obtained with more than 2 monolayers of Co from a
pristine Co/Gd interface (see Sup. 6). Thus, we also consider the interfacial
intermixing between Co and Gd, which can potentially induce more magnetization
of Gd, as another contributing mechanism leading to the room temperature
compensation.

Next, we present our AOS threshold fluence using 50 fs linearly polarized
laser pulses as a function of composition. We first present the AOS threshold
fluence of Co(0.6-1.2)/Gd(4), Co(0.7)/Gd(0-4) and
Co(0.7)/Gd(0-2.2)/Co(0.5)/Gd(1.5), which is shown in Figure 9d-f. As for
Co(0.6-1.2)/Gd(4) (see 9d), it can be observed that the threshold fluence
decreases with Co thickness, which is consistent with our theoretical
calculation (see Figure 2) and our experimental result (see Figure 4) due to
the reduction of Curie temperatureBeens  _et al._ (2019); Jakobs  _et al._
(2021); Lalieu  _et al._ (2017); Li  _et al._ (2021). Interestingly, crossing
compensation to the Gd domination (at thinner Co), it can be seen that the
trend of decrease is flattened and eventually tilt up. As for
Co(0.7)/Gd(0-4)(see 9e), as Gd thickness increases from the lowest thickness
of Gd (0.2 nm) enabling the existence of AOS, the AOS threshold fluence
decreases by a factor of four at the Co dominated regime and levels up in in
the Gd dominated regime. The behavior in Gd dominated regime is similar to the
experimental results of that for [Co/Gd]2 for Gd thickness beyond compensation
(see Figure 4d and a zoomed in view in Figure 9f). As for qualdyaer,
consistently with previous reportLi  _et al._ (2022a), the threshold fluence
decreases as Gd thickness increases, reaching a minimum at magnetization
compensation and remains almost unchanged at Gd domination at even thicker Gd.
A common feature can be briefly summarized here, using a 50 fs laser pulse,
higher Gd content decreases AOS threshold fluence at Co domination, while at
Gd domination, the AOS energy efficiency is suppressed.

Having demonstrated the static AOS, we now switch gear to the AOS dynamics of
[Co/Gd]2. The dynamics measurement follows identical protocol as in previous
worksXu  _et al._ (2017); Zhang  _et al._ (2022). Here we denote the positive
sign of MOKE signal as the direction of Co magnetic moment is parallel with
the external magnetic field applied during the measurement. As for bilayer
Co/Gd (see Figure 9g), it can be seen that the magnetization of Co crosses 0
at less than one ps as a result of a strong exchange scattering. While at
longer timescale (50-400 ps), the Co dominated sample was able to reset its
magnetization. On the other hand, for the Gd dominated sample, the switching
continues and was able to reset at a much longer timescale. We attribute this
effect to the presence of an external magnetic field. As cobalt switches its
sign, its magnetization becomes aligned with the external magnetic field. The
fact that the Co magnetization in both sample follows external magnetic field,
indicates both systems are in Co dominated regime at the timescale of 50-400
ps. Similar behaviors are observed in Co(0.7)/Gd(X)/Co(0.5)/Gd(1.5) (see
Figure 9h). Nevertheless, as the composition is getting closer to the
compensition (Gd thickness of 1.1 nm), the magnetization dynamics at longer
timescale becomes less susceptible to the magnetic field compared with bilayer
case. This means the system is closer to magnetization compensation.