††thanks: These authors contributed equally.††thanks: These authors
contributed equally.

# Probing Laser-Induced Spin-Current Generation in Synthetic Ferrimagnets
Using Spin Waves

Tom Lichtenberg  Youri L.W. van Hees  Maarten Beens  Caspar J. Levels  Reinoud
Lavrijsen  Department of Applied Physics, Eindhoven University of Technology,
P.O. Box 513, 5600 MB Eindhoven, The Netherlands  Rembert A. Duine  Department
of Applied Physics, Eindhoven University of Technology, P.O. Box 513, 5600 MB
Eindhoven, The Netherlands  Institute for Theoretical Physics, Utrecht
University, Leuvenlaan 4, 3584 CE Utrecht, The Netherlands  Bert Koopmans
Department of Applied Physics, Eindhoven University of Technology, P.O. Box
513, 5600 MB Eindhoven, The Netherlands

###### Abstract

Several rare-earth transition-metal ferrimagnetic systems exhibit all-optical
magnetization switching upon excitation with a femtosecond laser pulse.
Although this phenomenon is very promising for future opto-magnetic data
storage applications, the role of non-local spin transport in these systems is
scarcely understood. Using Co/Gd and Co/Tb bilayers we isolate the
contribution of the rare-earth materials to the generated spin currents by
using the precessional dynamics they excite in an adjacent ferromagnetic layer
as a probe. By measuring THz standing spin-waves as well as GHz homogeneous
precessional modes, we probe both the high- and low-frequency components of
these spin currents. The low-frequency homogeneous mode indicates a
significant contribution of Gd to the spin current, but not from Tb,
consistent with the difficulty in achieving all-optical switching in Tb-
containing systems. Measurements on the THz frequency spin waves reveal the
inability of the rare-earth generated spin currents to excite dynamics at the
sub-ps timescale. We present modelling efforts using the $s$-$d$ model, which
effectively reproduce our results and allow us to explain the behavior in
terms of the temporal profile of the spin current.

##  I Introduction

Over the past few decades, femtosecond (fs) laser excitation of magnetic
materials has led to the discovery of a rich collection of physical phenomena.
Among these, single fs laser pulse all-optical switching (AOS) of the
magnetization in rare-earth transition-metal ferrimagnetic alloys [1, 2] and
multilayers [3, 4] appears to be especially promising for future memory
applications [5]. This phenomenon was shown to arise from the transfer of
angular momentum between two magnetic sublattices [6]. In the same period the
fs laser excitation of spin currents, mobile electrons carrying spin angular
momentum, has been gaining significant interest from a fundamental perspective
[7, 8].

A particularly relevant use for these optically excited spin currents exploits
their ability to transfer angular momentum between two ferromagnetic layers.
This was first demonstrated in an experiment investigating the ultrafast laser
induced magnetization dynamics of two ferromagnetic layers separated by a
conductive spacer layer [9]. A diverse body of research into optically
generated spin currents has since arisen [7, 10, 11, 12, 13, 14, 15]. In
recent years it has been shown that novel device functionality can be achieved
at the intersection of AOS and optically excited spin currents. These works
focus on systems where an all-optically switchable ferrimagnetic layer is
separated by a spacer layer from a ferromagnetic layer. Depending on the
precise composition of the layers, either the spin current coming from the
ferromagnet can influence the AOS process [16], or the ferromagnet can be
switched by the spin current coming from the switchable layer [17, 18, 19].
This last case demonstrates the strength of the spin current generated upon
excitation of an all-optically switchable system, and begs the question to
what extent this spin current plays a role in the switching mechanism.

Although a great deal of AOS research has been performed on Gd(Fe)Co and Co/Gd
systems, recent research indicates that Tb can be used as rare-earth (RE)
component instead of Gd [4, 20]. However, in these works AOS has been found
only for very specific layer thicknesses, a requirement which is not present
in layered ferrimagnets containing Gd [21]. Work by Choi et al. has shown that
spin currents generation in Tb is significantly weaker than in Gd [22],
hinting at an explanation for the increased difficulty in attaining AOS. Here
the accumulation of spins at the far end of a thick conductive spacer layer
was measured, which can lead to a distorted spin current profile due to
diffusive electron transport [8]. Moreover, the use of alloys makes it
difficult to disentangle the spin current contributions originating from
different elements.

In this work, we systematically study spin current generation in synthetic
ferrimagnets using the collective spin precession they excite in a neighboring
layer as a probe. The basic experimental concept and the noncollinear magnetic
bilayers used in this work are sketched in Fig. 1(a). The generation layers
are synthetic ferrimagnetic bilayers, consisting of Co which couples
antiferromagnetically to either Gd or Tb. Although these latter materials are
paramagnetic at room temperature in bulk, a ferromagnetic phase can be
stabilized when interfaced with a ferromagnetic material [23], which decays
exponentially away from the interface. Upon laser excitation of the out-of-
plane magnetized generation layer, a spin current which is spin-polarized in
the direction of the local magnetization is excited and injected into a
ferromagnetic absorption layer with in-plane magnetization. There the spins
exert a torque on the local magnetic moments and excite both the homogeneous
precessional mode [24, 8] as well as higher-order inhomogeneous spin waves
[25, 26]. By extracting the phase, amplitude and frequency of the FMR mode and
THz spin waves, we indirectly study the absorbed spin current [27], which is
commonly expected to scale with the time derivative of the magnetization
($\text{d}m/\text{d}t$), fulfilling conservation of angular momentum [8, 27].
In this framework, these spin-wave parameters are directly proportional to the
corresponding parameters of the Fourier component of the spin-torque pulse at
the spin-wave frequency. Specifically, we probe the phase of the precessional
dynamics to investigate the temporal profile of the spin current, and the
amplitude to study the spin-current strength and attenuation. Changing the
thickness of the RE layer changes both spin-current properties, which leads to
significant variations in the measured parameters of the precessions. By
measuring both the homogeneous FMR mode ($\sim\\!10$ GHz) and the first-order
inhomogeneous mode ($\sim\\!0.5$ THz), the spin current can be studied on two
distinct timescales, giving access to both the fast as well as the slow
components of the spin current separately. We corroborate our experiments with
an $s$-$d$ model which treats local and non-local spin dynamics in a joint
description [28, 29, 30, 31].

Our results on the FMR mode show a strong contribution to the excited spin
current from Gd, which is in line with previous work [17, 22] and confirmed by
our modelling efforts. In contrast, the THz mode cannot be efficiently excited
by Gd, hinting at the relatively slow nature of the spin current contribution
from Gd. In the Co/Tb system, both the FMR and THz modes are found to vanish
with only a slight addition of Tb, indicating weak spin current generation in
Tb but strong spin absorption, consistent with the high spin-orbit coupling
(SOC) in this material. These results shed new light on non-local spin
dynamics in highly technologically relevant synthetic ferrimagnets, and help
elucidate the role of these processes in all-optical magnetic data recording.

##  II Experimental Details

The noncollinear magnetic bilayer central to this work, as sketched in Fig.
1(a), consists of an out-of-plane magnetized Co/RE generation layer (where RE
is either Gd or Tb) and an in-plane magnetized Co absorption layer. Spin
transport between the two magnetic layers is facilitated by a thin Cu spacer
layer, which also serves to magnetically decouple the layers. The samples are
fabricated using DC magnetron sputtering, where we vary the RE layer thickness
along the sample length to ensure consistency between measurements. The full
sample stack is given by
Si:B(substrate)/Ta(4)/Pt(4)/Co(1)/RE($X$)/Cu(2.5)/Co(5)/Pt(2.5) (numbers in
brackets indicate the thickness in nm, with $X$ being the variable RE
thickness).

We employ pump-probe spectroscopy to measure the magnetization dynamics upon
laser-pulse excitation, using the magneto-optical Kerr effect (MOKE) in the
polar configuration to probe the magnetization. The sample is placed in a
magnetic field with a variable angle of up to 20° with the sample plane. A
mode-locked Ti:Sapphire laser is used to generate pulses at a repetition rate
of 80 MHz and with a wavelength of 780 nm. The pulse length at sample position
is approximately 150 fs. Both pump and probe pulses are focused onto the
sample with a spot size of approximately 16 and 8
$\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{2}$ respectively, and the pump
fluence is about $1\text{\,}\mathrm{mJ}\text{\,}{\mathrm{cm}}^{-2}$. We use
_Complex MOKE_ to separate the magnetic contrast of both magnetic layers [32].
To accurately determine the absolute spin-wave phase in our experiments, the
so-called coherence peak is used, which arises at temporal and spatial pump-
probe overlap due to interference effects [33]. For more details on the
experimental setup, see Sec. I of the Supplementary Information [34].

Figure 1:  (a) Schematic overview of the experiments, where the effect of the
generation layer composition on the response of the absorption layer is
sketched for each of the three studied configurations (Co, Co/Gd and Co/Tb).
(b) Example magnetization response calculated using the _s_ -_d_ model for Co
and for Gd in a Co/Gd bilayer. (c, d) Response of the absorption layer for
each studied configuration. Both (c) the homogeneous ($H_{\text{ext}}=100$ mT)
and (d) the first order inhomogeneous mode ($H_{\text{ext}}=0$ mT) are
plotted, where the latter is measured using _complex MOKE_.  Figure 2:  (a)
Measured FMR mode ($H_{\text{ext}}=100$ mT) for several Gd thicknesses, fitted
with a damped cosine function. The orange dashed line indicates the FMR phase
shift and the thickness calibration is discussed in Sec. 1 of the
Supplementary Information [34]. (b,c) FMR phase as a function of Gd (b) and Tb
(c) thickness. (d,e) FMR amplitude as a function of Gd (d) and Tb (e)
thickness, normalized to the amplitude without RE layer for both datasets. The
orange solid line in (b) and (d) represents the modeled behavior using eq. 1,
and the black solid line in (d) is a fit with an exponential function.

##  III Experimental results

As a first experiment, we investigate the general behavior of the precessional
dynamics for the three different generation layer configurations (Fig. 1(a)).
In order to isolate the contribution of the RE layers, in which the
magnetization decays exponentially away from the Co interface, their
thicknesses need to be chosen carefully. Previous work gives a typical
lengthscale of the magnetization decay in Gd of 0.45 nm [3], with a similar
value to be expected for Tb. In Fig. 1(c) and (d) we present typical
measurements of the THz spin-wave and FMR mode respectively for the three
different generation layer configurations, namely Co(1), Co(1)/Gd(0.53), and
Co(1)/Tb(0.45). Our results show that the composition of the generation layer
significantly influences the spin-wave behavior, specifically the amplitude
and phase. To better understand this observation, we show representative
modelled traces (See Sec V of the Supplementary Information [34]) of the
magnetization response of Co and Gd upon fs laser excitation, in Fig. 1(b).
The precessional behavior can then generally be understood with a simple
picture where the generated spin current is assumed to be proportional to
$\text{d}m/\text{d}t$ [8, 27]. Considering the antiferromagnetic coupling
between Co and Gd (or Tb), the spin currents generated by the RE material and
by Co should also have the opposite sign. Therefore, we expect that the
polarization of the total spin current changes sign (Fig. 1(a)) when the RE
material becomes the dominant contributor, which could explain the
experimental observation of the large phase shift of the FMR mode. However, no
such sign change is observed in the THz measurements, hinting at a less
significant contribution of the RE material to the sub-ps spin current
dynamics. This is consistent with the expected magnetization dynamics plotted
in Fig. 1(b), which for Gd take place on the timescale of multiple ps, leading
to a relatively slow spin current profile. Additionally, we find a strongly
reduced amplitude for precession excited with a Co/Tb generation layer. This
can be attributed to strong absorption of the spin current generated in Co,
consistent with the large SOC in Tb [35], as well as weak spin current
generation in Tb.

###  III.1 FMR mode

To get a better understanding of the observed behavior, we systematically
measure both the FMR and THz mode as a function of the Gd and Tb thickness in
the generation layer. First we discuss the results on the FMR mode, of which
we show a selection of measurements for a Co(1)/Gd(X) generation layer in Fig.
2(a). To extract the phase and amplitude of the FMR mode, these measurements
are fitted with a damped cosine function, indicated by the solid black lines,
where the frequency is shared between all datasets.

We first discuss the FMR mode phase, of which we already observe a shift as a
function of Gd thickness in Fig. 2(a), as indicated by the dotted orange line
connecting the shifted maxima of the fits. The phase we extract from the fits
is plotted in Fig. 2(b) and (c) for Co/Gd and Co/Tb respectively. With the
addition of only 1 nm of Gd, a phase shift of more than 130° is observed,
whereas no consistent phase shift is measured when adding Tb. This again
indicates a strong contribution to the spin current from Gd, but a small to
non-existent contribution from Tb. For large thicknesses of Gd the phase
remains constant, which can be explained by the paramagnetic state further
away from the Co/Gd interface.

Next, we focus on the amplitude of the FMR mode as a function of RE thickness,
which is normalized to the amplitude for a pure Co generation layer and
plotted in Fig. 2(d) and (e) for Co/Gd and Co/Tb respectively. This amplitude
has been corrected for changes in light absorption using transfer matrix
calculations [36], as described in Sec. II of the Supplementary Information
[34]. For Gd, an initial dip and succesive rise of the amplitude is observed,
followed by a gradual decrease. This behavior, in combination with the
observed change of the FMR phase can be captured in a simple toy model where
we assume the spin current contributions from Co and Gd excite two spin waves
with different amplitudes and phases, but with the same frequency. As
discussed previously, the Gd is magnetized due to exchange coupling with the
Co, decaying at a characteristic length scale $\lambda_{\text{mag}}$ which
determines its contribution to the total spin current. Furthermore, the
addition of Gd introduces a characteristic length scale
$\lambda_{\text{diff}}$ over which spin information is lost due to spin flip
scattering events, known as the spin diffusion length. For now we assume that
spin diffusion is independent of the magnetic state of Gd. Writing the phase
difference $\delta$ between the two excited precessions using Euler’s formula,
the FMR mode can then be described as

$\displaystyle{A}_{\text{FMR}}~{}e^{i\phi}=\left(A_{\text{Co}}-A_{\text{Gd}}~{}e^{i\delta}\left(1-e^{-\frac{t_{\text{Gd}}}{\lambda_{\text{mag}}}}\right)\right)e^{-\frac{t_{\text{Gd}}}{\lambda_{\text{diff}}}}\,,$
(1)

where $A_{\text{Co}}$ and $A_{\text{Gd}}$ are the dimensionless amplitudes of
the precessions excited by the magnetic volume of Co and Gd respectively.
These then result in a combined precession with amplitude ${A}_{\text{FMR}}$
and phase $\phi$. Expressions for these two parameters can be derived, as is
shown in Sec. III of the Supplementary Information [34]. Now $A_{\text{Co}}=1$
can be fixed by normalization, and $A_{\text{Gd}}=3.2$ and
$\delta=-140^{\circ}$ are chosen to match the maximum of the amplitude and
saturation of the phase, respectively. Within these constraints, valid values
for the remaining parameters are found by manual adjustment, and are found to
be $\lambda_{\text{mag}}=0.4\pm 0.1$ nm and $\lambda_{\text{diff}}=2.0\pm 0.2$
nm. The amplitude of the Gd-excited precession $A_{\text{Gd}}$, when corrected
for the expected equivalent magnetic thickness of Gd (0.45 nm), gives a Gd
contribution that is approximately 7 times larger per nm than that of Co. A
complete understanding of this difference is outside of the scope of this
work, but some factors of relevance are the differences in magnetization, the
amount of magnetic moment lost during demagnetization, and the spin-wave
excitation efficiency, which will be discussed later. The value for
$\lambda_{\text{mag}}$ closely matches the experimentally determined length
scale for the loss of magnetization in Gd of $\sim 0.45$nm [3]. We note that
this length scale could also affect the rate of spin flip scattering, such
that the parameter $\lambda_{\text{diff}}$ might not provide a full
description. Fitting the data for the FMR amplitude for thicknesses where Gd
is expected to be paramagnetic (from 1.5 nm onwards) with an exponential
function, indicated by the solid black line in Fig. 2(d) results in a
$\lambda_{\text{diff,Gd}}$ of $1.9\pm 0.2$ nm, which has not been measured
before to the best of our knowledge. The close agreement between the two
descriptions indicates that the magnetic state is not very relevant for spin
flip scattering in these weakly magnetic systems.

The reasonable agreement between experiments and calculations indicate that
the most important elements of this complex system are captured by this simple
model. However, the model can not capture the behavior we observe for a Co/Tb
generation layer, as plotted in Fig. 2(c) and (e). Here we instead find only a
rapid decrease of the amplitude with the addition of Tb. This could indicate a
very short diffusion length $\lambda_{\text{diff}}$ for Tb, which precludes
any statements about the spin-current generation strength in Tb. We attribute
this discrepancy to a larger degree of scattering of the mobile electrons in
Tb due to the high SOC compared to Gd [35], leading to a loss of the spin
information over shorter length scales.

Figure 3:  (a) Measured THz mode for several Gd thicknesses, fitted with a
damped cosine function. The orange dashed line indicates the spin-wave phase
shift. (b) Spin-wave amplitude, orange line indicates a solution of Eq. 1
using $A_{\text{Gd}}=0$ and $\lambda_{\text{diff,Gd}}=1.9$ nm and (c)
frequency as a function of Gd thickness.

###  III.2 THz mode

We measure the THz spin-waves as a function of RE thickness to investigate the
spin-current generation on the ps timescale, and show a selection of the
measurements for a Co/Gd injection layer in Fig. 3(a). Because THz mode
excitation takes place on the same timescale as laser-induced demagnetization
and spin-current generation, we disregard the first 2 ps of the data, in
accordance with Ref. [27]. We again used a damped cosine function to fit the
data, indicated by the solid black lines, in order to extract the spin-wave
amplitude, phase, and frequency. At around 1 nm, the signal-to-noise ratio is
too low to extract spin-wave parameters reliably. The amplitude of the spin
waves, which is plotted in Fig. 3(b), drops significantly with Gd thickness.
Contrary to the behavior for the FMR mode, no initial rise of the amplitude
due to a Gd contribution is observed. We therefore show the expression for the
amplitude derived from Eq. 1 using $A_{\text{Gd}}=0$ and
$\lambda_{\text{diff,Gd}}=1.9$ nm as the solid orange line in the figure,
which is equivalent to an exponential decay describing only the spin diffusion
due to Gd. The good agreement indicates that Gd is not actively contributing
to the excitation of the THz mode, which shows that this mode is excited by
the Co-dominated fast component of the generated spin current [37]. This is
further confirmed by the observation that the spin-wave phase is independent
of the Gd thickness, as indicated by the orange dashed line in Fig. 3(a). The
same measurements are repeated for Tb and presented in Sec. IV of the
Supplementary Information [34]. No significant difference between the THz
spin-wave frequency and phase for the two materials is observed, again
confirming the dominant role of Co in exciting these spin waves. However, a
full analysis of the data is complicated by a more rapid decrease of the spin-
wave amplitude with increasing Tb thickness, which could again be attributed
to high spin-flip scattering due to SOC.

Contrary to our measurements of the FMR mode we observe a significant decrease
of the THz spin-wave frequency for increasing Gd thickness, plotted in Fig.
3(c), which is not predicted by our simulations (see Sec. VI of the
Supplementary Information [34]). We believe there are several mechanisms that
could explain this observation, the foremost being a growth related change of
the exchange stiffness [38, 26] and coupling between spin waves in the
generation and absorption layers. The mechanisms are discussed in detail in
Sec. VII of the Supplemental Material [34] and predict the same direction of
the shift of the THz spin-wave frequency. However, a full understanding of
this effect is beyond the scope of this work and requires further research.

##  IV $s$-$d$ modelling

To better understand the mechanisms governing our observations, as well as the
underlying physics in general, we modeled the generated spin current in the
synthetic ferrimagnetic generation layer using an extension of the $s$-$d$
model [28, 39, 40, 8, 29, 30, 31]. This model describes the coupling of local
spins, in this case the $3d$ and $4f$ electrons in the RE-TM ferrimagnet, to a
system of itinerant spins ($s$ electrons). The latter system includes
diffusive spin transport, similar to Refs. [8, 30, 31]. To model the
experiments, we define a discretized material system consisting of a
ferrimagnetic (Co/Gd) region and a nonmagnetic spacer layer (Cu). The Co/Gd
bilayer is modeled in a layered manner, where the local Co and Gd
concentration is sampled from a function that represents an intermixed
transition from pure Co to Gd, similar to Ref. [41]. This process is discussed
in Sec. V of the Supplementary Material [34]. Furthermore, the absorption
layer is implemented as an ideal spin sink connected to the spacer layer,
which is a valid assumption considering the experimental absorption layer
thickness of 5 nm and the transverse spin diffusion length in Co of
approximately 1 nm [26]. This allows us to calculate the absorbed spin current
for varying composition of the Co/Gd layer. Using a linearized Landau-
Lifshitz-Gilbert equation, including the anti-damping spin-transfer torque
exerted by the absorbed spin current, we calculate the absolute phase shift of
the excited homogeneous precession. Although the used model includes multiple
assumptions that disqualify any precise quantitative statements [34], it gives
a complete description of the qualitative characteristics of the spin current
and excited precession. Further details on the modeling, including all the
used material parameters, are presented in Sec. V of the Supplemental Material
[34].

Figure 4:  (a) Calculated FMR mode excited by the spin current from a Co(1)
(black) and Co(1)/Gd(2) (blue) generation layer. (b) Absolute phase of the
excited FMR mode as a function of the effective thickness $t_{\text{eff,Gd}}$
of the (generation) Gd layer. (c) Absorbed spin current as function of time
generated by a Co (black) and a Co/Gd (blue) generation layer.

In Fig. 4(a) we present traces of the modeled $z$-component of the FMR mode
excited by a Co(1) and a Co(1)/Gd(2) generation layer, in black and blue,
respectively. Note that the amplitude of the FMR mode excited by pure Co has
been multiplied by 10 for clarity, as it is significantly weaker than that
excited by Co/Gd. This is likely a result of the assumptions made in the
model, such as the simplistic description of optical absorption and the
parameters chosen for Gd, as the exact shape of the spin current pulse
strongly affects the excitation efficiency. Similar to the experimental
results, our model shows a shift of the phase of the FMR mode when Gd is added
to the generation layer. In Fig. 4(b) the absolute phase is plotted as a
function of the effective Gd thickness $t_{\text{eff,Gd}}$, which for our
purposes is analogous to the Gd layer thickness, and is further described in
Sec. V of the Supplemental Material [34]. The figure clearly shows a
qualitative agreement with the experiments (Fig. 2(d)), with a shift of
approximately 130° upon adding a few monolayers of Gd. The exact value of the
phase shift can be explained by a combination of factors. Naively, one would
expect a 180° phase shift if only the Gd spin-current would excite dynamics
once it starts to dominate the total spin current. The actual shift however is
lowered by the contribution of Co, as well as the relatively slow (Type II
[37]) magnetization dynamics of the Gd, as will be demonstrated in the
following.

To clarify the origin of the observed phase shift, we plot the absorbed spin
currents for a pure Co(1) injection layer and a Co(1)/Gd(2) bilayer calculated
using the $s$-$d$ model in Fig. 4(c). The addition of Gd leads to changes in
both the amplitude of the first, fast peak of the spin current, as well as a
much longer negative tail. Both of these can be understood by the
demagnetization behavior of Gd, as plotted in Fig. 1(b). The reduction of the
amplitude of the first peak of the spin current is caused by the initial,
relatively rapid demagnetization of Gd. At longer timescales, when the Co
generated spin current is negligible, a long tail in the Co/Gd generated spin
current is observed. This is in turn caused by the secondary, slower
demagnetization of Gd, and results in a spin current that is mostly polarized
in the opposite direction. This explains the abrupt phase jump in Fig. 2(d)
and 4(b) for increasing Gd thickness. Moreover, the long duration of this
component of the spin current gives rise to a deviation of the phase shift
from 180°. This is to be expected, as the Gd demagnetization takes place over
tens of ps, which is comparable to the precessional period ($\sim$ 80 ps). The
small increase of the phase from -130° to -110° that is observed in Fig. 4(b)
for a further increase of the Gd thickness is attributed to an additional
slowing down of the Gd magnetization dynamics. More specifically, we find that
transfer of angular momentum from the Co, which speeds up the Gd magnetization
dynamics near the interface between Co and Gd, has a smaller total effect when
the Gd thickness is increased.

In Sec. VI of the supplementary information [34] we show calculated traces of
the THz mode for increasing Gd thickness. Our calculations indicate a phase
shift of up to 10° when adding a 2 nm thick Gd layer to a 1 nm thick Co layer.
Contrary to previous work [27] we were unable to resolve this potential phase
shift, because the resolution is determined by the strength of the spin
current and thus the total magnetic moment of the generation layer. In future
research, Co/Gd multilayers could be used to study THz spin-wave generation in
rare-earth transition-metal ferrimagnetic system in more detail.

##  V Conclusion

We have shown that examining the parameters of precessional modes excited by
ultrafast optically generated spin currents can be a powerful tool to
elucidate the behavior of these spin currents. Synthetic ferrimagnets offer a
novel platform to systematically investigate these phenomena. The large
discrepancy in spin current generation between Gd and Tb could shed new light
on the relative difficulty of achieving AOS in systems containing only the
latter as RE material. Using spin-wave modes with THz frequencies has
additionally allowed us to probe the high-frequency component of the generated
spin currents. Here we again find a link between the intrinsic speed of the
magnetization dynamics and the behavior of the excited spin current. This
gives additional weight to the notion that angular momentum which is lost
during demagnetization can be transferred to mobile spins [8]. This notion
also underlies the $s$-$d$ model, which we have successfully used to describe
our experimental results. Our experiments give new insight in the
magnetization dynamics of rare-earth materials, which could prove critical in
future spintronic memory devices.

## Supplementary Material

##  I Additional experimental details

In this work, ultrafast laser-pulse induced dynamics are measured using pump-
probe spectroscopy. A photo-elastic modulator is used to modulate the
polarization of the probe, aligned such that the polarization oscillates
between linearly and circularly polarized light at 50 kHz. To measure the
pump-induced change of the magnetization after excitation, a mechanical
chopper at around 69 Hz is used to modulate the pump. A dual lock-in scheme,
locked to the two aforementioned frequencies, is used to measure the Kerr
rotation induced in the sample. A more detailed description of the technique
can be found in ref. [42].

In our samples, the Gd and Tb layers were deposited while moving a shutter
over the sample, leading to a thickness gradient along the sample. In order to
determine the absolute thickness on this wedge an additional marker layer of
Pt was deposited while keeping the shutter at the starting point of the path
followed in the previous step. The reflectivity of the sample as a function of
position then exhibits a step where this marker layer starts, giving a
calibration point for the gradient. Due to shadowing effects the width of the
step in this measurement is $\sim$0.3 mm, which corresponds to 0.3 nm of RE
thickness. Also due to imperfections in the positioning of the shutter, the
exact starting point of the gradient can not be determined with greater
accuracy than this 0.3 nm. Within this range, we define 0 nm as the final
measurement point along the sample before the behavior of the FMR or THz mode
begins to change. Note that the length of the gradient can be accurately
determined, and the relative variation in thickness is accurate to within a
few percent.

To accurately determine the exact moment of excitation, the so-called
coherence peak is used [33], which is centered around the temporal pump-probe
overlap. This is a non-magnetic effect, so it can be measured by measuring the
demagnetization after saturation in opposite directions. Adding the two curves
leaves only the non-magnetic effects that are independent of the magnetization
direction, which in our case is only the coherence peak. Using this technique,
the moment of excitation can be determined with an uncertainty in the order of
10 fs, necessary for phase measurements of the THz spin waves. A more detailed
description of the measurements, as well as an example, can be found in ref.
[27].

##  II Transfer matrix calculations for optical absorption

In Fig. 2b of the main paper we show the amplitude of the FMR mode for Gd
thicknesses in the generation layer of up to 10 nm. At these thicknesses the
optical absorption starts to play a significant role, and needs to be taken
into account when determining the actual amplitude. As the film thickness
increases, the absorbed laser energy also increases. Therefore heating and
thus demagnetization is greater for thicker Gd layers, potentially generating
stronger spin currents. To calculate the size of this effect, we perform
transfer matrix calculations [36] using the _tmm_ Python package [43]. Values
for the refractive indices at the central laser wavelength (780 nm) are taken
from literature [44, 45].

Figure S1:  (a) Instantaneous light absorption profile calculated with
transfer matrix method for a
Si:B(substrate)/Ta(4)/Pt(4)/Co(1)/Gd(3)/Cu(2.5)/Co(5)/Pt(2.5) stack. (b) Total
absorption in the metallic layers as a function of the Gd thickness.

We calculate the instantaneous absorption profile for the metallic stack used
in the experiments (with $t_{\text{Gd}}$ = 3 nm), which we plot in Fig. S1(a).
In metallic layers with thicknesses of only a few tens of nm, thermal
equilibration between the layers proceeds on the timescale of mere hundreds of
fs [46]. As this is comparable to the demagnetization timescale in Co, and
much shorter than this timescale in Gd [37], the absorption in the entire
metallic stack is the relevant parameter to determine the ultimate
magnetization quenching. Integrating the absorption profile over the thickness
of the metallic layers then gives this total absorption. Repeating this for
the range of Gd thicknesses used in the main work gives the curve shown in
Fig. S1(b). Here we indeed find an increase of the absorption with the Gd
thickness. To finally arrive at the data shown in Fig. 2b of the main paper,
the measured amplitude of the FMR spin-waves is divided by the total
absorption for each value of the Gd thickness.

##  III Derivation of the phase and amplitude of toy model FMR mode

In the main text, we explain the Gd thickness dependence of the FMR amplitude
and phase by decomposing the total FMR signal into a separate Co and Gd
contribution, each with its own respective amplitude (${A}_{\text{Co}}$ and
${A}_{\text{Gd}}^{\dagger}={A}_{\text{Gd}}(1-\text{exp}(-t/\lambda_{\text{mag}}$))
and phase ($0$ and $\delta$). Using Euler’s formula, this can be written as:

$\displaystyle{A}_{\text{FMR}}e^{i(\omega t+\phi)}={A}_{\text{Co}}e^{i\omega
t}+{A}_{\text{Gd}}^{\dagger}e^{i(\omega t+\delta)}\,.$ (2)

As the frequency $\omega$ is assumed to be the same for all contributions, we
can multiply Eq, 2 with $e^{-i\omega t}$, yielding:

$\displaystyle{A}_{\text{FMR}}e^{i\phi}={A}_{\text{Co}}+{A}_{\text{Gd}}^{\dagger}e^{i\delta}\,.$
(3)

First, we find an expression for ${A}_{\text{FMR}}$ by taking the complex
absolute value of Eq. 3.

$\displaystyle{A}_{\text{FMR}}$
$\displaystyle=\sqrt{|{A}_{\text{Co}}+{A}_{\text{Gd}}^{\dagger}e^{i\delta}|^{2}}$
$\displaystyle=\sqrt{{{A}_{\text{Co}}}^{2}+{{A}_{\text{Gd}}}^{2}\left(1-e^{-\frac{t}{\lambda_{\text{mag}}}}\right)^{2}+2{A}_{\text{Co}}{A}_{\text{Gd}}\cos(\delta)\left(1-e^{-\frac{t}{\lambda_{\text{mag}}}}\right)}\,.$
(4)

Next, an equation for the phase of the FMR mode is derived by taking the real
and imaginary part of Eq. 3:

$\left.\begin{aligned}
{A}_{\text{FMR}}&=\frac{{A}_{\text{Co}}+{A}_{\text{Gd}}^{\dagger}\cos\delta}{\cos\phi}\\\
{A}_{\text{FMR}}&=\frac{{A}_{\text{Gd}}^{\dagger}\sin\delta}{\sin\phi},\end{aligned}\right\\}\qquad\phi=\tan^{-1}\left(\frac{\sin(\delta){A}_{\text{Gd}}\left(1-e^{-\frac{t}{\lambda_{\text{mag}}}}\right)}{{A}_{\text{Co}}+\cos(\delta){A}_{\text{Gd}}\left(1-e^{-\frac{t}{\lambda_{\text{mag}}}}\right)}\right)\,.$
(5)

Eq. III and 5 are used to describe the data presented in Fig. 2b and d of the
main text.

##  IV THz spin-waves excited by Co/Tb

THz spin-wave measurements are performed for the Co/Tb generation layers as
well, as presented in Fig. S2(a). Due to strong attenuation of the measured
signal for added Tb, only Tb thicknesses of up to 0.4 nm can be reliably
considered. A decrease of the spin-wave frequency for increasing Tb thickness
is observed (Fig. S2(b)), similar to our observation for Gd, presented in the
main text. We should note that these measurements were performed under
slightly different growth conditions, leading to a slightly different
absorption-layer thickness and thus a different spin-wave frequency. To be
able to compare Gd and Tb, we performed the same experiments for a sample
containing Gd, grown under the same conditions. As can be seen in Fig. S2(b),
the frequency does not depend on the RE material, indicating that THz spin-
wave generation is dominated by the Co. Furthermore, the observed frequency
differences between identical samples gives us a strong indication that the
observed anomalous frequency behavior reported in the main text is related to
growth.

Figure S2:  (a) Spin wave measurements for the Co/Tb generation layer. (b)
Extracted frequency as a function of Tb and Gd thickness. Data for Gd on a
similar sample are inc;luded for comparison.

##  V _s_ -_d_ modelling

###  i The description of the magnetic stack

As mentioned in the main text, the generation layer is described as a system
of layered CoGd alloys with a varying local Co concentration describing an
intermixed Co/Gd bilayer [21]. Furthermore, this Co/Gd bilayer is connected to
a Cu spacer layer of $d=2.5\mbox{nm}$. We define the interface between the
generation layer and the spacer layer as $z=0$, where $z$ represents the out-
of-plane spatial coordinate. Now the local concentration of Co at position $z$
is modeled by the function

$x_{\mathrm{Co}}=\dfrac{1}{2}\bigg{[}1-\mbox{erf}\Big{(}(z-z_{\mathrm{int}})/w\Big{)}\bigg{]},$
(6)

where $z_{\mathrm{int}}$ is the position of the the (intermixed) Co/Gd
interface. We set $w=0.5\mbox{ nm}$. This yields an intermixing region with a
width in the order of nanometers, an order of magnitude larger than the
discretization step size (lattice spacing $a$).

Due to the intermixing region the layer thicknesses are not well defined. We
define $\tilde{d}_{\mathrm{Gd}}=z_{\mathrm{int}}+a$, where $a$ is the lattice
constant. $\tilde{d}_{\mathrm{Gd}}$ determines the total Gd in the system and
plays the role analogue to a layer thickness. The total thickness of the
system is then given $d=d_{\mathrm{Co}}+\tilde{d}_{\mathrm{Gd}}$, where
$\tilde{d}_{\mathrm{Co}}=1\mbox{ nm}$ is set constant and plays a the role of
a thickness for the Co. To be clear, using this approach $d_{\mathrm{Co}}$ and
$\tilde{d}_{\mathrm{Gd}}$ do not precisely define a layer thickness, since we
assume an intermixed interface. Nevertheless, for increasing value of
$\tilde{d}_{\mathrm{Gd}}$ the relative amount of Gd increases in an
monotonuous way.

###  ii The calculation of the phase

To calculate the phase of the excited homogeneous precession in the absorption
layer, we write down the Landau-Lifshitz-Gilbert-Slonzcewski equation for a
homogeneous magnetization in the form [47]

$\dfrac{d\mathbf{m}}{dt}=-\omega_{0}\mathbf{m}\times\hat{z}+\alpha\mathbf{m}\times\dfrac{d\mathbf{m}}{dt}+\mbox{\boldmath$\tau$}_{\mathrm{STT}},$
(7)

where $\omega_{0}$ is the angular FMR frequency and $\alpha$ the effective
Gilbert damping. $\mbox{\boldmath$\tau$}_{\mathrm{STT}}$ is the (anti-damping)
spin-transfer torque exerted by the absorbed spin current. The latter is
proportional to [47]

$\mbox{\boldmath$\tau$}_{\mathbf{STT}}\propto\mathbf{m}\times(j_{s}(t)\hat{x}\times\mathbf{m}).$
(8)

We focus on small perturbations around the ground state $\mathbf{m}=\hat{z}$,
assuming small transverse components in the $x$ (OOP direction) and $y$
direction $\delta m_{x},\delta m_{y}\ll 1$. Furthermore, we define
$\psi=\delta m_{x}+i\delta m_{y}$. Eqs. 7 and 8 then reduce to

$-i(1-i\alpha)\dfrac{d\psi}{dt}=\omega_{0}\psi-i\tau_{\mathrm{STT},x}(t),$ (9)

which can easily be solved using the Green’s function method. Assuming
$|\alpha|^{2}\ll 1$, we have the Green’s function

$G_{0}(t-t^{\prime})\propto\theta(t-t^{\prime})\exp(i\omega_{0}(1+i\alpha)(t-t^{\prime})).$
(10)

The transverse magnetization is expressed in terms of a convolution

$\delta m_{x}(t)\propto\mbox{Re}\Big{\\{}G_{0}\ast j_{s}\Big{\\}}.$ (11)

Eq. 11 is used to determine the phase of the homogeneous precession excited by
$j_{s}(t)$ as calculated by the method in the previous section.

Extending the model to include the excited inhomogeneous modes can be
accomplished by taking the following steps. The first step is to make the
normalized magnetization $\mathbf{m}$ spatial dependent. The second step is to
implement the exchange field in Eq. 7, which is of the form
$\mathbf{H}_{\mathrm{ex}}\propto A\nabla^{2}\mathbf{m}$ with $A$ the spin-wave
stiffness [26]. In analogy with Eq. 9, the resulting equation of motion can be
expressed in terms of the complex function $\psi(x,t)$, which is now dependent
on spatial coordinate $x$. Importantly, it is assumed that all incoming
(perpendicular) spins are absorbed at the interface. Now, the spin-transfer
torque will appear as a boundary condition for the function $\psi(x,t)$.
Finally, the function $\psi(x,t)$ is expanded in cosines $\cos(n\pi x/L)$ that
represent the standing spin-wave solutions. In that way, an equation for every
separate mode can be derived. It is straightforward to show that the
contribution $\delta m_{x,n}$ of mode $n$ to the amplitude of the transverse
magnetization is equivalently determined by

$\delta m_{x,n}(t)\propto\mbox{Re}\Big{\\{}G_{n}\ast j_{s}\Big{\\}},$ (12)

with the response function $G_{n}(t)$ for mode $n$

$G_{n}(t-t^{\prime})\propto\theta(t-t^{\prime})\exp(i\omega_{n}(1+i\alpha_{n})(t-t^{\prime})),$
(13)

where $\omega_{n}$ is the frequency of mode $n$ and given by
$\omega_{n}=\omega_{0}+(A/\hbar)(n\pi/L)^{2}$ with $L$ the thickness of the
absorption layer. Furthermore, $\alpha_{n}$ is the mode-dependent damping
parameter. For simplicity, the latter is assumed to be independent of the mode
and set equal to the Gilbert damping.

###  iii System parameters

Table S1:  The typical values used for the material parameters in the calculation of section .. . symbol | meaning | value
---|---|---
$T_{\mathrm{amb}}$ | ambient temperature | $295\mbox{ K}$
$\gamma$ | electronic heat capacity parameter | $2000\mbox{ J\,m}^{-3}\mbox{\,K}^{-2}$
$C_{p}$ | phonon heat capacity | $4\cdot 10^{6}\mbox{ J\,m}^{-3}\mbox{\,K}^{-1}$
$P_{0}$ | absorbed pulse energy | $2\cdot 10^{8}\mbox{ J\,m}^{-3}$
$\sigma$ | pulse duration | $0.15\mbox{ ps}$
$\tau_{D}$ 11footnotemark: 1 | heat dissipation time scale | $100\mbox{ ps}$
$a$ | lattice spacing | $0.25\mbox{ nm}$
$T_{C,\mathrm{TM}}$ | TM Curie temperature | $1000\mbox{ K}$
$T_{C,\mathrm{RE}}$ | RE Curie temperature | $292\mbox{ K}$
$S_{\mathrm{TM}}$ | spin quantum number | $1/2$
$S_{\mathrm{RE}}$ | spin quantum number | $7/2$
$\mu_{\mathrm{at,TM}}$ | TM atomic magnetic moment | $2.0\,\,\mu_{B}$
$\mu_{\mathrm{at,RE}}$ | RE atomic magnetic moment | $7.0\,\,\mu_{B}$
$\rho_{sd}$ |  $s$-$d$ coefficient Ref. [31] | $1.0\mbox{ eV}$
$\tau_{sd,\mathrm{TM}}$ | TM $s$-$d$ scattering time | $0.1\mbox{ ps}$
$\tau_{sd,\mathrm{RE}}$ | RE $s$-$d$ scattering time | $20.0\mbox{ ps}$
$\tau_{s}$ | spin-flip scattering time (magnetic region) | $0.2\mbox{ ps}$
$\tau_{s,\mathrm{N}}$ | spin-flip scattering time (nonmagnetic region)[30] | $17\mbox{ ps}$
$\sigma$ | conductivity (magnetic region) [30] | $6.7\cdot 10^{6}\mbox{ S\,m}^{-1}$
$\sigma_{N}$ | conductivity (spacer layer)[30] | $39\cdot 10^{6}\mbox{ S\,m}^{-1}$
$D$ | spin diffusion coefficient (magnetic region)[30] | $250\mbox{ nm}^{2}\mbox{\,ps}^{-1}$
$D_{N}$ | spin diffusion coefficient (spacer layer)[30] | $9500\mbox{ nm}^{2}\mbox{\,ps}^{-1}$
$g$ | interfacial conductance parameter | $0.4\cdot 10^{6}\mbox{ m}^{-2}$
$\omega_{0}/(2\pi)$ | FMR frequency | $12\mbox{ GHz}$
$\alpha$ | effective damping | 0.05
$j_{\mathrm{Co-Gd}}$ | interatomic exchange constant | $2.0\mbox{ meV}$
00footnotetext: The demagnetization traces imply that this time is extremely slow for the used stack, thereby set to an extremely long time. |  |

The values of the material parameters as used in the calculations presented in
the main text are given in Table S1. We stress that although we use the
terminology of Co/Gd components in the text, the material parameters presented
here correspond to the values for a general TM-RE system. The latter is
motivated by that the modeling only justifies qualitative statements, since
the implementation required a large number of assumptions and approximations.

##  VI Modelling THz spin waves using the _s_ -_d_ model

Using the $s$-$d$ framework discussed in Sec. IV of this, the THz response of
the absorption layer is calculated. The results are plotted in Fig. S3 for
Co(1) (black line) and Co(1)/Gd(2) (Blue line).

Our calculations indicate a slight decrease of the amplitude, which in
experiments we expect to be more pronounced due to spin diffusion in the Gd
layer. Furthermore, a slight phase shift is expected. The calculated shift is
in the order of 60 fs, which corresponds to a phase shift of about 10°. This
is too small to pick up in our measurements.

Figure S3:  Calculated THz mode excited by the spin cur- rent from a Co(1)
(black) and Co(1)/Gd(2) (blue) generation layer.

##  VII Potential explanations for the THz spin-wave frequency shift

###  i Growth Related frequency shift

A growth-related change in the exchange stiffness scales linearly with the
spin-wave frequency [48]. Eyrich _et al._ demonstrated that the effective
exchange stiffness of ultrathin ($<$ 10 nm) Co can drop significantly when
alloyed with certain non-magnetic materials [38], and similar effects were
later reported in Cu/Co bilayers due to intermixing at the interface [26]. We
postulate that due to the addition of a RE dusting below the Cu, subsequent
growth becomes more disordered, leading to more intermixing and thus a lower
exchange stiffness. Although we expect the $\sim\\!45\%$ reduction of the
exchange stiffness required to explain the observed frequency shift completely
is improbably large [38, 26], the order of magnitude of this effect is unknown
for our system specifically and requires further research.

###  ii Interlayer coupling between two eigenmodes

We postulate that the frequency is significantly affected by the interlayer
coupling of the simultaneous transverse magnetization dynamics in the
generation layer and absorption layer. This coupling arises from the
combination of spin pumping, interlayer spin transport, and the resulting
spin-transfer torques. It will be maximized in case the eigenmodes of the
absorption layer (the standing spin waves) and the eigenmodes of the
generation layer (e.g., the exchange mode of the synthetic-ferrimagnetic
multilayer) have a similar frequency.

In order to reach a mathematical description of this concept, we formulate an
expression of the form of Eq. (7) for both layers separately. The noncollinear
nature of the full magnetic stack is captured in the definitions of the
effective fields within the distinct layers. In analogy with the previous
subsection, the effective field within the absorption is pointing along the
$z$ axis, and its magnitude is expressed in terms of the eigenfrequency
$\omega_{A}$. For the generation layer, we define the effective field to be
directed along the positive $x$ direction, with a magnitude determined by
$\omega_{G}$. Additionally, we include the interlayer coupling terms.
Specifically, the equation for the absorption layer includes the torque [47]

$\mbox{\boldmath$\tau$}_{\mathbf{\mathrm{interlayer-
STT}}}\propto\,\,g_{A}\,\mathbf{m}_{A}\times\bigg{(}\bigg{(}\mathbf{m}_{G}\times\dfrac{d\mathbf{m}_{G}}{dt}\bigg{)}\times\mathbf{m}_{A}\bigg{)},$
(14)

where $\mathbf{m}_{A}$ describes the normalized magnetization of the
absorption layer and $\mathbf{m}_{G}$ the normalized magnetization of the
generation layer. Furthermore, the dimensionless factor $g_{A}$ is determined
by the efficiency of spin pumping and the interlayer spin transfer, and mainly
depends on the spin-mixing conductances. The expression describes an anti-
damping spin-transfer torque mediated by the pumped spin current from the
generation layer. A similar torque will be present in the equation for the
magnetization dynamics in the generation layer, where the subscripts are
interchanged $A\leftrightarrow G$. We linearize the set of equations using
$\mathbf{m}_{A}=(\delta m_{A,x},\delta m_{A,y},1)$ and
$\mathbf{m}_{G}=(1,\delta m_{G,y},\delta m_{G,z})$. By collecting the linear
terms, and switching to the frequency domain, we find the following equation
for the eigenfrequency $\omega$

$((\omega_{A}+i\alpha_{A}\omega)^{2}-\omega^{2})((\omega_{G}+i\alpha_{G}\omega)^{2}-\omega^{2})-g_{A}g_{G}\omega^{2}(\omega_{A}+i\alpha_{A}\,\omega)(\omega_{G}+i\alpha_{G}\,\omega)=0,$
(15)

where $\alpha_{A}$ and $\alpha_{G}$ are the effective damping for the layer-
specific eigenmodes. To purely extract the role of the effect discussed in
this subsection, we remove the terms dependent on the damping
$\alpha_{A}=\alpha_{G}\rightarrow 0$. By solving Eq. (15), it can be shown
that the maximal shift of the frequency (compared to the layer-specific
eigenfrequencies $\omega_{A}$ and $\omega_{G}$) is reached in case the
eigenfrequencies are equal $\omega_{A}=\omega_{G}\equiv\omega_{0}$. Then, the
frequency is given by

$\omega=\omega_{0}\sqrt{1+\dfrac{g_{A}g_{G}}{2}\pm\sqrt{g_{A}g_{G}}\sqrt{1+\dfrac{g_{A}g_{G}}{4}}}.$
(16)

For $\sqrt{g_{A}g_{G}}\ll 1$ this expression is equivalent to

$\omega=\omega_{0}(1\pm\sqrt{g_{A}g_{G}}/2),$ (17)

indicating that the change is linear in $\sqrt{g_{A}g_{G}}$.

Considering that the factors $g_{A}$ and $g_{G}$ are related to the effective
damping (which for ultrathin films is strongly enhanced by spin pumping),
using an effective damping of $\alpha=0.1$ we estimate that the reduction of
the frequency resulting from this effect is approximately $5\%$. Considering
that the (not investigated) eigenmode of the ferrimagnetic generation layer
may have a stronger damping, higher values for the frequency shift may be
possible. Although this mechanism can not explain the large frequency
reduction as observed in the experiments, the order of magnitude of this
results proofs that this contribution is nonnegligible.

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