# Ultrafast magnetization reversal in ferromagnetic spin-valves: an $s$-$d$
model perspective

Quentin Remy<EMAIL_ADDRESS>Cavendish Laboratory, University of
Cambridge, Cambridge, United Kingdom  Université de Lorraine, CNRS, IJL,
F-54000 Nancy, France

###### Abstract

We present an extension to simple $s$-$d$ models, aiming at simulating
ultrafast magnetization dynamics and spin transport in metallic
heterostructures. In particular, we consider an alternative spin dissipation
channel due to a finite exchange splitting of the $s$ band. From this theory,
we show three different mechanisms governing the dynamics of spin
accumulation. On top of the already widely discussed
“$\text{-}\mathrm{d}M/\mathrm{d}t$” electron-magnon mechanism, we study the
role of a dynamic change of exchange splitting (of conduction electrons) as
well as the rotation of spins reflected at an interface with a ferromagnet.
Finally, we use the presented theory to explain the recent observation of
subpicosecond reversal of a ferromagnet in rare-earth free spin-valves. Our
conclusion agrees with the one of reference [1] favoring magnetization
reversal due to the rotation of the spin polarization of a reflected spin
current.

##  I Introduction

11footnotetext: Certain works can be assigned to several categories, for
instance extensions of the 3TM which are derived from a Boltzmann equation.

Ever since the first investigations of ultrafast magnetization dynamics of
metals [2, 3, 4, 5, 6, 7], numerous and diversified theoretical approaches
have been attempted to understand its origin [8]. The proposed contributing
microscopic mechanisms are typically classified as spin transport [9, 10, 11,
12, 13, 14, 15], spin-flip scattering [16, 17, 18, 13, 19] and magnon
generation [20, 21, 22, 23, 24] processes. A sharp separation of the latter
two mechanisms is however not always clear, mostly because the magnon
generation mechanism is generally discussed in the framework of the electron-
magnon interaction [25, 22] where the creation of a magnon always comes with a
spin-flip, and also because both magnons and spin flips (Stoner excitations)
are two specific cases of magnetic excitation [26, 27]. Assuming a clear
separation between both types of excitation is possible, recent experimental
results seem to confirm that both magnon generation and spin-flips are
happening during ultrafast demagnetization (UDM) [28, 29] and the final
destination of angular momentum during magnetization quenching is the crystal
lattice [30]. The generation of circularly polarized phonons was also observed
for nickel [31]. A transfer to the electromagnetic field has been estimated
negligible [32, 33] and a possible transient role of the orbital degree of
freedom [34] via, for instance, an increase of the orbital angular momentum
[20, 35], still has not been observed to the best of our knowledge.
Theoretical works however indicate that such an increase cannot be observed
because orbital angular momentum in metals is transferred to the lattice with
a characteristic time around one femtosecond [36, 37]. The computational
frameworks that can incorporate some or all of these processes include the
real-time time dependent density functional theory (rt-TDDFT) [38, 15, 35, 39,
40, 41, 42, 43], a direct propagation of the system wave function with a
parametrized many-body Hamiltonian [44, 36, 37, 45], the semiclassical
Boltzmann equation [46, 47, 9, 48, 10, 49, 18, 50, 22, 23, 51, 52, 24, 53,
54], quantum kinetics [55, 19, 56, 57], molecular dynamics [58, 59], the
stochastic atomistic Landau-Lifshitz-Gilbert equation [60, 61, 62, 59], the
Landau-Lifshitz-Bloch equations [63, 64] as well as other more
phenomenological parametrized models [65] such as the so-called three
temperature model (3TM) [4] and its various extensions [16, 21, 66, 67, 68,
69]. We note that in parametrized approaches, parameters may be obtained from
ab initio calculations [70, 17, 71, 72, 73]. In particular, recent works seem
to validate the use of temperature based models for a description of
magnetization dynamics [61, 62, 68, 59] provided that one properly accounts
for energy conservation and the temperature dependence of all parameters.
Because there is so far no method that is able to completely solve the problem
of UDM of metals for all relevant spatial and time scales, such simplified
bath models prove to be quite useful.

Moreover, the field of ultrafast magnetization dynamics in metals is not
limited to UDM. A plethora of consequent phenomena have been observed such as
all-optical magnetization switching [74, 75, 76], ultrafast spin injection
[77, 78, 79, 80, 81, 13, 82, 83, 84, 85, 86, 65], terahertz electromagnetic
pulse emission [79, 87, 65, 88], ultrafast spin transfer torque [89, 90, 85]
and magnetization reversal of ferromagnets using non-local transfer of spin
[91, 92, 93, 94, 95] or spin-orbit torque [96]. We also note the apparent
thermal/incoherent nature of these phenomena as experimentally highlighted by
the fact that they do not intrinsically depend on the external energy stimulus
used to trigger the dynamics [97, 98, 99, 100]. Thus, the simplified
parametrized models can be applied to study or predict more complex
magnetization dynamics subsequent to UDM. It is also easier to enforce angular
momentum conservation (although in a more phenomenological way), which is
believed to not always be satisfied in more complicated approaches such as rt-
TDDFT [101].

In this work, we use a simple reservoir approach for the dynamics and
transport of both energy and angular momentum, presented in Sec. II, which is
suitable for the study of ultrafast thermal effects in metallic multilayers
with possibly several magnetic layers. It is based on a $s$-$d$ model which
can incorporate both transverse ($\sim$ magnons) and longitudinal (spin flips)
magnetic excitations. In particular, this framework allows us to simulate a
more complex spin accumulation dynamics which could explain recent experiments
[93, 1]. Simulations of UDM and spin accumulation generation in systems with a
single magnetic layer are briefly shown in Sec. III. Then, magnetization
reversal in ferromagnetic spin-valves is discussed in Sec. IV. We conclude and
discuss possible improvements in Sec. V.

##  II Model

In order to provide some general context, we start the description of our
model with a general model Hamiltonian $\mathcal{H}$ suitable for a $s$-$d$
model, in the absence of any external field and for a homogeneous material,
where itinerant ($s$) electrons, phonons and localized ($d$) electrons are
distinct quantities defined by their respective free (quasi)particle
Hamiltonian terms $\mathcal{H}_{e}$, $\mathcal{H}_{p}$ and $\mathcal{H}_{d}$:

$\mathcal{H}=\mathcal{H}_{e}+\mathcal{H}_{p}+\mathcal{H}_{d}+\mathcal{H}_{ep}+\mathcal{H}_{ed}$
(1)

together with two interaction terms $\mathcal{H}_{ep}$ and $\mathcal{H}_{ed}$
for the electron-phonon and $s$-$d$ interactions respectively. Two important
approximations of this model are (i) the lack of hybridization between the $s$
and $d$ states and (ii) $d$ states do not contribute to (heat and charge)
transport. The real impact of these approximations on the simulated dynamics
is unclear since the final equations rely on parameters whose input values are
often taken from experiments or ab initio calculations that do not make these
approximations. We write the $s$-$d$ interaction term as [102, 103, 100]

$\mathcal{H}_{ed}=-J\sum_{i}\left[\dfrac{1}{2}\left(\hat{S}_{i}^{+}\hat{s}_{i}^{-}+\hat{S}_{i}^{-}\hat{s}_{i}^{+}\right)+\hat{S}_{i}^{z}\hat{s}_{i}^{z}\right]$
(2)

where $J>0$ is the $s$-$d$ interaction constant for ferromagnetic coupling
between $s$ and $d$ electrons,
$\hat{\bm{S}}_{i}=(\hat{S}_{i}^{x},\hat{S}_{i}^{y},\hat{S}_{i}^{z})$ are the
spin operators for $d$ electrons localized on lattice site $i$,
$\hat{\bm{s}}_{i}=\frac{1}{2}\hat{c}_{i}^{\dagger}\bm{\sigma}\hat{c}_{i}$ are
the spin operators for $s$ electrons with $\hat{c}_{i\sigma}^{\dagger}$ and
$\hat{c}_{i\sigma}$ the creation and annihilation operators for a Wannier
state at lattice site $i$ and spin $\sigma$ and
$\bm{\sigma}=(\sigma^{x},\sigma^{y},\sigma^{z})$ is the vector of Pauli
matrices. Superscripts $+$ and $-$ denote ladder operators as usual, for
instance $\hat{S}_{i}^{\pm}=\hat{S}_{i}^{x}\pm\hat{S}_{i}^{y}$. The term with
the $z$ components is treated in the mean field approximation [102, 103] and
we write the average (over all quantum states and lattice sites) of the $z$
components of the itinerant and localized spins operators as $s^{z}$ and
$S^{z}$ respectively. These two quantities contain all the angular momentum,
per atom in units of $\hbar$, possessed by electrons within the mean field
approximation. The Hamiltonian then becomes:

$\displaystyle\mathcal{H}$
$\displaystyle=\mathcal{H}_{e}^{\prime}+\mathcal{H}_{p}+\mathcal{H}_{d}^{\prime}+\mathcal{H}_{ep}+\mathcal{H}_{ed}^{\pm}$
(3a) $\displaystyle\mathcal{H}_{e}^{\prime}$
$\displaystyle=\mathcal{H}_{e}-JS^{z}\sum_{\bm{k}}\dfrac{1}{2}\left(\hat{c}_{\bm{k}\uparrow}^{\dagger}\hat{c}_{\bm{k}\uparrow}-\hat{c}_{\bm{k}\downarrow}^{\dagger}\hat{c}_{\bm{k}\downarrow}\right)$
(3b) $\displaystyle\mathcal{H}_{d}^{\prime}$
$\displaystyle=\mathcal{H}_{d}-Js^{z}\sum_{i}\hat{S}_{i}^{z}$ (3c)
$\displaystyle\mathcal{H}_{ed}^{\pm}$
$\displaystyle=-J\sum_{i}\dfrac{1}{2}\left(\hat{S}_{i}^{+}\hat{s}_{i}^{-}+\hat{S}_{i}^{-}\hat{s}_{i}^{+}\right)$
(3d)

Where we wrote the mean field term felt by the $s$ electrons in the Bloch
representation (with states indexed by a wavevector $\bm{k}$).

From this Hamiltonian, the system dynamics is obtained in the framework of
perturbation theory: the interaction terms $\mathcal{H}_{ep}$ and
$\mathcal{H}_{ed}^{\pm}$ are assumed to be much smaller than the free particle
terms. The interaction terms are then used to determine exchange of energy (as
well as momentum and angular momentum) while the free particle terms are used
for conservation of energy (as well as momentum and angular momentum).
Notably, the interaction energies are neglected in the conservation of energy,
which is not true in general [100], but is valid whenever perturbation theory
can be applied.

We first discuss how the free particle terms are treated. The itinerant free
electron term $\mathcal{H}_{e}^{\prime}$ describes renormalized (for instance
due to the electron-phonon and electron-electron interaction) Bloch states
which are assumed to be characterized by a thermal distribution for each
(pure) spin state. The electronic temperature $T_{e}$ is assumed to be the
same for both spin species $\sigma=\pm 1/2$ while the chemical potential
$\mu_{\sigma}$ and the density of states $D_{\sigma}$ are different for
different spins. The mean field term in Eq.(3b) adds an exchange splitting to
the $s$ band. Noting $E_{\sigma}^{0}$ the lowest energy of the $s$ band, the
exchange splitting is seen to be
$E_{\uparrow}^{0}-E_{\downarrow}^{0}=-JS^{z}$. The free phonon term
$\mathcal{H}_{p}$ represents renormalized phonons [104] described as Debye
phonons in equilibrium at a temperature $T_{p}$. The localized electrons term
$\mathcal{H}_{d}$ can contain contributions due to $d$-$d$ exchange
interaction, magnetocrytalline anisotropy and dipolar interaction [23]. It is
also treated within the mean field approximation [52]. We consider for
simplicity the case where the spin quantum number $S$ of the $d$ electrons is
1/2. The total mean field, including the one generated by the $s$-$d$
interaction (second term of Eq.(3c)), induces an energy splitting
$\Delta=2mk_{B}T_{C}$ between the $d$ electrons energy levels where
$m=-2S^{z}$ is the magnetization of the $d$ electrons normalized to its zero
temperature value and $T_{C}$ is the Curie temperature of the ferromagnetic
layer. We note that the $d$ electrons are not assumed to be in internal
equilibrium (in the $d$ electrons bath itself) for a general value of $S$, but
for $S=1/2$, in this mean field approximation, there is no difference between
internal equilibrium and out-of-equilibrium [95].

The electron-phonon term $\mathcal{H}_{ep}$ is treated in perturbation theory
via Fermi’s Golden rule [105, 104, 106]. We consider the usual high
temperature case where the energy transfer between electrons and phonons is
found to be $g_{ep}(T_{e}-T_{p})$ with $g_{ep}$ the electron-phonon coupling
considered as temperature independent. We also consider that this term induces
angular momentum transfer between the $s$ electrons and the lattice and which
is phenomenologically given by, following references [102, 103],
$(s^{z}-s^{z}_{\text{ie}})/\tau_{s}$ with $\tau_{s}$ a spin relaxation time.
The instantaneous equilibrium $s$ electrons spin polarization
$s^{z}_{\text{ie}}$ is defined such that $s^{z}-s^{z}_{\text{ie}}$ is the
excess of spin due to the out-of-equilibrium state of the itinerant electrons
[23]. It is discussed bellow. The exchange of energy due to the $s$-$d$
interaction term $\mathcal{H}_{ed}^{\pm}$ is obtained via a slightly
generalized version of Fermi’s Golden rule [107, 102, 52] which leads to a
typical two-level dynamics [16, 108, 52]:

$\dfrac{dm}{dt}=\dfrac{1}{\tau_{m}}\left(m-\dfrac{\Delta\mu}{2k_{B}T_{C}}\right)\left[1-m\coth\left(\dfrac{2mk_{B}T_{C}-\Delta\mu}{2k_{B}T_{e}}\right)\right]$
(4)

Where $\tau_{m}$ is the characteristic time for angular momentum transfer from
$d$ to $s$ electrons, which needs to be taken as an additional parameter [52,
100], and $\Delta\mu=\mu_{\uparrow}-\mu_{\downarrow}$ is the spin accumulation
of the $s$ electrons.

The equations governing the dynamics of the system come from the fact that
conservation equations must be fulfilled while transfers happen as given by
interactions, which we just discussed. We only focus on energy and angular
momentum conservation equations, and discard effects appearing when one also
considers charge [67] and momentum [109] conservation, as we wish to discuss
new effects arising from angular momentum conservation driven by energy
transfer. The energy conservation equation for the total Hamiltonian Eq.(1)
reads:

$\dfrac{\partial}{\partial t}\left(\dfrac{\gamma}{2}T_{e}^{2}+C_{p}T_{p}-\rho
m\dfrac{\Delta}{2}\right)+\bm{\nabla}\cdot\left(\bm{Q}_{e}+\bm{Q}_{p}\right)=0$
(5)

Where we chose the middle of both $d$ electrons levels as the reference energy
for $d$ electrons, $\rho$ is the number of atoms per unit volume, $\gamma
T_{e}$ is the standard expression for a free electron gas volumetric heat
capacity, $C_{p}$ is the phonon volumetric heat capacity and $\bm{Q}_{e}$ and
$\bm{Q}_{p}$ are the electronic and phononic heat current densities
respectively. The phononic heat current is given by the standard Fourier’s law
$\bm{Q}_{p}=-\kappa_{p}\bm{\nabla}T_{p}$, with $\kappa_{p}$ the phonon heat
conductivity, while the electronic heat current is given by
$\bm{Q}_{e}=-\kappa_{e}(T_{e}/T_{p})\bm{\nabla}T_{e}$ with $\kappa_{e}$ the
equilibrium (when the electronic plus phononic system is in equilibrium)
electronic heat conductivity. Such a description of the energy flow within
electrons and phonons was recently used to successfully describe the ultrafast
strain dynamics of heterostructures similar to the ones considered in our work
[110]. The resulting heat equations for the system, modeled as one dimensional
in the thin film limit, are:

$\displaystyle\begin{split}\gamma T_{e}\dfrac{\partial T_{e}}{\partial
t}&=\dfrac{\partial}{\partial
z}\left(\kappa_{e}\dfrac{T_{e}}{T_{p}}\dfrac{\partial T_{e}}{\partial
z}\right)-g_{ep}\left(T_{e}-T_{p}\right)\\\ &+2\rho
mk_{B}T_{C}\dfrac{dm}{dt}+S(z,t)\end{split}$ (6a)
$\displaystyle\begin{split}C_{p}\dfrac{\partial T_{p}}{\partial
t}&=\kappa_{p}\dfrac{\partial^{2}T_{p}}{\partial
z^{2}}+g_{ep}\left(T_{e}-T_{p}\right)\end{split}$ (6b)

The energy dynamics for the $d$ electrons is given by Eq.(4) since there is a
one-to-one relationship between the energy density $-\rho m\Delta/2$ and the
absolute value of magnetization $|m|$ in this model. The magnetization
dependent term in Eq.(6a) [111, 68] comes from the requirement that Eq.(5)
must be fulfilled. The last term in Eq.(6a) is due to energy transfer from an
external laser pulse as computed in reference [112] where it is argued that
energy conservation in the total system, including the electromagnetic field,
is significantly broken. Satisfying conservation of energy when the
electromagnetic field is included is however irrelevant to the phenomena
discussed in this work and beyond the scope of this work [113].

The angular momentum conservation equation is:

$\dfrac{\partial}{\partial
t}\left(s^{z}+S^{z}+S_{p}^{z}\right)+\bm{\nabla}\cdot\bm{J}_{s}=0.$ (7)

Similarly to energy conservation, quantities appearing after the time
derivative operator are intensive quantities. In this case we take them as
angular momentum (or spin polarization for the electrons) per atom in units of
$\hbar$ to have notations consistent with the previously introduced averaged
spins. $S_{p}^{z}$ refers to the angular momentum dissipated in the lattice
which, according to our previous discussion, satisfies $\partial
S_{p}^{z}/\partial t=(s^{z}-s^{z}_{\text{ie}})/\tau_{s}$. $\bm{J}_{s}$ is the
spin current density and we neglect a contribution to this current density due
to angular momentum transport in the localized $d$ electrons [114, 24] or in
phonons [115]. Moreover, we assume that spin transport only happens via
conduction electrons close to the Fermi level [54] and so depends on the spin
accumulation only [52, 85, 65]. We will detail the spin current term when we
use it in section IV.

To close our system of equations, and because spin transport depends on the
spin accumulation, we need to rewrite Eq.(7) in terms of $\Delta\mu$. The
relation we need is [50, 23]

$s^{z}-s^{z}_{\text{ie}}=\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu\left(\Delta\mu-\delta\right),$ (8)

Which is valid to first order in both the spin accumulation $\Delta\mu$ and
the change of exchange splitting $\delta=-J(S^{z}-S^{z}_{\text{ie}})$ with
$S^{z}_{\text{ie}}$ the instantaneous equilibrium value of $S^{z}$. $N\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu=D_{\uparrow}(\varepsilon_{F})D_{\downarrow}(\varepsilon_{F})/(D_{\uparrow}(\varepsilon_{F})+D_{\downarrow}(\varepsilon_{F}))$,
with $\varepsilon_{F}$ the equilibrium Fermi level and $N$ the total number of
$s$ electrons, and it is taken as a parameter [23, 52]. A derivation of Eq.(8)
is provided in Appendix A. This equation also allows us to consider the change
of exchange splitting in the conduction electrons which was argued to be
fundamental to describe ultrafast magnetization dynamics of itinerant
ferromagnets [116, 50] and later considered in both itinerant and localized
electrons [23]. We also need to calculate the instantaneous equilibrium values
of $s^{z}$ and $S^{z}$. For $s^{z}_{\text{ie}}$, we follow the argumentation
of Gridnev [103], simplified to the case of ferromagnets:

$s^{z}_{\text{ie}}(t)=\chi S^{z}(t)$ (9)

With $\chi$ a spin susceptibility. When ferromagnetic order only arises due to
the $s$-$d$ interaction, $\chi=4k_{B}T_{C}/J$. The choice of
$S^{z}_{\text{ie}}(t)$ is more complicated. The naive case where
$S^{z}_{\text{ie}}(t)$ is given by its equilibrium value for a temperature
given by the electronic temperature at the instant of interest $T_{e}(t)$
would be unphysical. In particular, it would lead to a fast change (even a
discontinuity in the mean field approximation) of the slope of the spin
accumulation dynamics curve when the electronic temperature crosses the Curie
temperature. Rather, we follow the physics of out-of-equilibrium spin
relaxation, where the dynamics of localized spins for $S=1/2$ is governed by
an equation with the following form [16, 108, 52, 95]:

$\displaystyle\dfrac{dm}{dt}_{|\text{relaxation}}$
$\displaystyle=-\dfrac{m(t)-m_{\text{ie}}(t)}{\tau(t)}$ (10a) $\displaystyle
m_{\text{ie}}(t)$
$\displaystyle=-2S^{z}_{\text{ie}}(t)\equiv\tanh\left(\dfrac{2k_{B}T_{C}m(t)-\Delta\mu(t)}{2k_{B}T_{e}(t)}\right)$
(10b)

Where $\tau(t)$ is a characteristic time that depends on time and, in general,
$\Delta\mu$ represents an energy splitting due to an external (to the $d$
electrons subsystem) source of angular momentum. For instance, for the $s$-$d$
model [52] fundamentally describing, at each instant $t$, spin relaxation of
$d$ electrons in the thermal bath of $s$ electrons

$\tau(t)=\dfrac{m_{\text{ie}}(t)\tau_{m}}{m(t)-\Delta\mu(t)/(2k_{B}T_{C})},$
(11)

While for Elliott-Yafet scattering as computed by Koopmans et al. [16]

$\tau(t)=\dfrac{m_{\text{ie}}(t)T_{C}}{m(t)RT_{p}(t)}$ (12)

With $R$ the demagnetization rate in the Elliot-Yafet model [16] and an
external source of angular momentum can also be considered [117]. Eq.(10a) is
more general than a Bloch equation, and even more general than (the
longitudinal term of) the self-consistent Bloch equation [118, 119] because
$\tau$ depends on time in a complicated way. Here we keep $\tau(t)=\tau_{s}$
which is consistent with the naive description of spin dissipation Eq.(7)
[102, 103] and the self-consistent Bloch equation [118, 119].

Using equations (8) and (9), Eq.(7) becomes [100]:

$\dfrac{d\Delta\mu}{dt}=\left(\dfrac{S}{\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu}(1+\chi)+JS\right)\dfrac{dm}{dt}-JS\dfrac{dm_{\text{ie}}}{dt}-\dfrac{\bm{\nabla}\cdot\bm{J}_{s}}{\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu}-\dfrac{\Delta\mu}{\tau_{s}}+JS\dfrac{m-m_{\text{ie}}}{\tau_{s}}$ (13)

Which together with equations (4), (6) and (10b) form the set of equations we
wish to solve to obtain the dynamics of $T_{e}$, $T_{p}$, $m$ and $\Delta\mu$.
Other quantities such as the exchange splitting or the total spin polarization
in the electronic subsystem can be obtained from the latter four quantities.
Eq.(13) is valid for any value of $S$ but more equations are then needed to
calculate the dynamics of $m$ and $m_{\text{ie}}$. Eq.(13) generalizes
previous approaches [103, 52] mainly because it includes a dynamic exchange
splitting of the $s$ electrons (terms proportional to $JS$). It also includes
an equilibrium spin polarization of these electrons which was not in the model
of Beens et al. [52]. This dynamic exchange splitting was considered before by
Tveten et al. [23] and we note a similarity between the form of our equations
(4) and (13) and equations (5) and (4) of reference [23] if one replaces $m$
and $m_{\text{ie}}$ by the out-of-equilibrium and equilibrium (Bose-Einstein)
magnon distribution respectively. The system of equations of Tveten et al. is
however much more complicated than ours to solve and it is also not clear
whether the magnonic description of the magnetization dynamics via the
Holstein-Primakoff expansion is valid especially since we wish to model
situations where magnetization can be fully quenched or even reversed [120].
Finally, a new mechanism appearing in our approach is a spin dissipation in
the lattice due to a non-zero value of the out-of-equilibrium magnetization
$m-m_{\text{ie}}$ of $d$ electrons. All previous works so far (in this
framework) have, as far as we know, only been considering a spin dissipation
due to the presence of a spin accumulation as defined above. Within the
context of the derivation of Eq.(8), this means that we consider spin
relaxation in the lattice due to a spin non-equilibrium in $s$ electrons close
to the Fermi level (the spin accumulation dissipation term
$-\Delta\mu/\tau_{s}$) as well as all the other ones (the dynamic exchange
splitting dissipation term $JS(m-m_{\text{ie}})/\tau_{s}$). This additional
contribution comes from the fact that all electronic states are considered to
contribute to the spin dissipation when the phenomenological term $\partial
S_{p}^{z}/\partial t=(s^{z}-s^{z}_{\text{ie}})/\tau_{s}$ [102, 103] is
assumed. It was argued, in a different framework, that a relaxation-time
approximation can be used to simulate UDM, with a relaxation-time identical
for all electronic states [53]. One potential interesting consequence of this
additional term is that it can change the sign of the spin accumulation as
compared to what can be expected from the usual
“$\text{-}\mathrm{d}M/\mathrm{d}t$” law for the spin generation rate (first
term on the right hand side of Eq.(13)) since when $dm/dt$ is negative,
$m-m_{\text{ie}}$ is usually positive (this depends on the dynamics of the
electronic temperature and the spin accumulation). Also note that, even though
equation (4) can be written as equation (10a), $dm/dt$ is not proportional (as
a function of time) to $m-m_{\text{ie}}$ due to the complex time dependence of
$\tau$.

In order to facilitate the numerical implementation of this model and to
explain its limitations, it is instructive to expand Eq.(13) using Eq.(10b):

$\displaystyle\begin{split}\dfrac{d\Delta\mu}{dt}=\dfrac{1}{1-\dfrac{JS\eta(t)}{2k_{B}T_{e}(t)}}\times\left[\left(\dfrac{S}{\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu}\left(1+\chi\right)+JS\left(1-\eta(t)\dfrac{T_{C}}{T_{e}(t)}\right)\right)\dfrac{dm}{dt}\right.\\\\[8.61108pt]
+\left.\dfrac{JS\eta(t)\xi(t)}{T_{e}(t)}\dfrac{dT_{e}}{dt}-\dfrac{\bm{\nabla}\cdot\bm{J}_{s}}{\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu}-\dfrac{\Delta\mu(t)}{\tau_{s}}+\alpha
JS\dfrac{m(t)-m_{\text{ie}}(t)}{\tau_{s}}\right]\end{split}$ (14a)
$\displaystyle\begin{split}\eta(t)\equiv\operatorname{sech}^{2}\left(\xi(t)\right)\quad\text{;}\quad\xi(t)\equiv\dfrac{2k_{B}T_{C}m(t)-\Delta\mu(t)}{2k_{B}T_{e}(t)}\end{split}$
(14b)

Additional terms appears due to the dependence of $m_{\text{ie}}$ on $T_{e}$,
$m$ and $\Delta\mu$. These terms are proportional to the quantity $\eta$ which
becomes sizable when $\xi$ is small i.e. at large electronic temperature and
for values of magnetization where effects such as spin cooling and spin
heating [95] are expected to become significant (when
$2k_{B}T_{C}m(t)\sim\Delta\mu(t)$). Moreover, we notice that the change of
spin accumulation diverges when $(JS\eta(t))/2k_{B}T_{e}(t))\sim 1$, which may
happen even when $\xi$ is large depending on the complexity of the dynamics of
$T_{e}$, $m$ and $\Delta\mu$. This divergence happens when the first order
expansions of the density of states of $s$ electrons leading to equation (8)
are no longer valid approximations. These approximations are valid as long as
$J\ll 1/\mkern 1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu$ (see
Appendix A). This is consistent with the fact [121, 52] that the width of the
conduction band is usually larger than $JS$. The terms proportional to $J$
should then be treated as a correction to the model of Beens et al. [52]
however we will show that they can significantly change the magnetization
dynamics when $dm/dt\sim 0$ in the presence of an external source of spin
accumulation. Equation (14a) not only shows that the spin accumulation and the
spin current are not always proportional to $-dm/dt$ [24] but the spin
generation rate itself [80] is also not always proportional to $-dm/dt$ due to
the dynamic exchange splitting. We introduced a parameter $\alpha$ which is
equal to 1 in our model but we will also set it to 0 in the next section to
see the effect of the corresponding term in equation (14a).

Because both $s$ and $d$ electrons carry angular momentum in this model, even
at equilibrium, the question arises as to which quantity is measured in
experiments. Indeed, it is not clear whether an optical probe measuring
magnetization via magneto-optical effects would be as sensitive to both kinds
of electrons or not. Here we will assume that the experimentally measured
quantity is proportional to the total spin angular momentum of the system as
we are only interested in qualitative modeling. Then a magnetic signal will be
proportional to

$\begin{split}-S^{z}_{\text{tot}}&\equiv-\left(S^{z}+s^{z}\right)\\\
&=S\left(m\left(1+\chi\right)+J\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu\left(m-m_{\text{ie}}\right)\right)-\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu\Delta\mu\end{split}$
(15)

Showing that even under our simple approximation Eq.(9), the signal will not
be proportional to the $d$ electrons normalized magnetization $m$ due to a non
zero spin accumulation [52] and dynamic exchange splitting. All the data we
plot is normalized by the equilibrium spin angular momentum $-Sm(1+\chi)$ and
we note the corresponding normalized magnetization $m_{\text{tot}}$.

The systems we wish to simulate are actually multilayers and are thus not
homogeneous. We thus assume as usual that all the previous equations are valid
within each layer of the multilayer. We solve the conservation of energy
equation (5) by discretizing each layer and we use appropriate boundary
conditions for each interface [67, 100]. For the conservation of angular
momentum, we assume that magnetization is constant along the thickness of each
magnetic layer and use the average of the electronic temperature in the
corresponding layer as the input temperature appearing in equations (4) and
(13). Thus the term $2mk_{B}T_{C}dm/dt$ in Eq.(5) is identical for all depths
in a given magnetic layer. Going beyond this approximation would require a
generalization of this $s$-$d$ model to include either a direct $d$-$d$
coupling [122] or indirect $s$-$d$ coupling between neighboring atomic layer.
Such approach is beyond the scope of this work.

##  III Ultrafast demagnetization of a single layer

We first present results for a multilayer structure with a single
ferromagnetic layer, namely Sapphire(Substrate) / Ta(5) / Pt(4) / [Co/Pt](3.2)
/ Ta(5) similarly to reference [1] where numbers between brackets are
thicknesses in nm and [Co/Pt] is a ferromagnetic multilayer which is simulated
as an effectively homogeneous medium. We neglect spin transport in this case.
We use a 50 fs (gaussian) laser pulse, with normal incidence (coming from the
sample side i.e. the air/Ta interface) and 800 nm central wavelength to bring
the system out of equilibrium. We study the dynamics of various quantities as
a function of time delay with respect to the time instant where the position
of the center of the laser pulse is at the air/Ta interface. The sample is
initially at equilibrium with room temperature chosen to be 300 K. The
parameters entering the energy conservation equation are taken from our
previous works [93, 100, 95] and we choose $\rho=7.5\times
10^{28}~{}\text{m}^{-3}$ which lies between the values of pure Co ($\sim
9\times 10^{28}~{}\text{m}^{-3}$) and Pt ($\sim 6.5\times
10^{28}~{}\text{m}^{-3}$). For the angular momentum parameters we choose
$\tau_{m}=$ 100 fs, $\tau_{s}=$ 20 fs and $1/\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu=$ 1 eV as in reference
[52]. We choose to study $\chi=$ 1 such that equation (13) reduces to equation
(6) of reference [52] when $J=$ 0 eV, since we have $S=1/2$. These parameters
will be kept in the rest of this work. In this section, we choose $J=$ 0 or
0.1 eV and $T_{C}=$ 700K. We will also study the effect of the newly proposed
spin dissipation channel by using $\alpha=$ 0 or 1. All the reported fluences
are external fluences i.e. not the absorbed ones.

Figure 1: (a) Normalized magnetization $m_{\text{tot}}$ of the [Co/Pt]
multilayer as a function of time delay for various fluences as noted on the
right of the plot (in $\text{mJ/cm}^{\text{2}}$). (b) Comparison of the
magnetization dynamics for a fluence of 4 $\text{mJ/cm}^{\text{2}}$ and
different models; “Full” corresponds to the full equation (14a) with $J=$ 0.1
eV and $\alpha=$ 1; “$J=$ 0” is obtained by setting $J=$ 0 eV;
“$\dot{E}_{\text{es}}=$ 0” is obtained by neglecting the magnetization
dependent term in equation (6a); “$\alpha=$ 0” is obtained by setting $\alpha$
to zero. (c) Same comparison as in (b) but for the electron and phonon
temperature as indicated. (d) Same comparison as (b) but for the spin
accumulation as indicated.

Fig. 1(a) shows the magnetization dynamics of the [Co/Pt] multilayer for
various fluences. We recover the standard behavior, with an UDM followed by a
“fast” recovery (where spins and electrons do not form an equilibrated sub-
bath of the system) and a “slow” recovery (where spins and electrons are
equilibrated and the dynamics is driven by the phonon temperature dynamics via
heat dissipation in the substrate). At higher fluences, we also recover the
so-called Critical Slowing Down (CSD) of magnetization dynamics [60, 16, 63,
21, 118, 95]. In Fig. 1(b), (c) and (d), we plot the normalized magnetization,
temperatures and spin accumulation respectively, for four different models.
All four models lead to qualitatively identical dynamics. A quantitative
difference is observed for fluences such that there is a significant quenching
of magnetization and yet at the same time a significant remagnetization. All
four models lead to almost identical dynamics when there is a significant CSD
(not shown). Neglecting the dynamic exchange splitting can change the value of
the normalized magnetization by a few percent during the UDM and “fast”
recovery phases. By only turning off the spin dissipation channel due to the
dynamic exchange splitting, we recover almost the same dynamics as when we
completely turn off the dynamic exchange splitting, indicating that the
dissipation part dominates the dynamics induced by the dynamic exchange
splitting. A bigger effect on the magnetization dynamics is obtained by
turning off transfer of energy from $d$ to $s$ electrons. This is because it
modifies the temperature dynamics as shown in Fig. 1(c).

Overall, we do not observe any drastic difference between the model of Beens
et al. [52] and ours, even in the high fluence limit. The reason is that, when
$dm/dt\sim 0$ and so when terms proportional to $J$ could dominate, the
temperature dynamics is already much slower and the self-consistency of our
system of equations forces $d\Delta\mu/dt\sim 0$. The situation will be
completely different in the next section where an external source of angular
momentum (due to an additional ferromagnetic layer) can also drive the spin
accumulation dynamics.

##  IV Subpicosecond magnetization switching in ferromagnetic spin-valves

The aim of this section is to study potential mechanisms that can lead to the
subpicosecond magnetization switching of ferromagnets observed in reference
[1]. In this case, the system is Sapphire(Substrate) / Ta(5) / Pt(4) /
[Co/Pt](7) / Cu(10) / [Co/Pt](3.2) / Ta(5) where the first [Co/Pt] multilayer
(7 nm thick) is referred to as the “Reference” layer and the second [Co/Pt]
multilayer (3.2 nm thick) is referred to as the “Free” layer. A magnetic
configuration of the system where the magnetizations of each ferromagnetic
multilayer are parallel is noted “P” and if they are antiparallel, we note it
“AP”. The main result of reference [1] was to show that upon a single
femtosecond laser pulse irradiation of the sample, the free layer can reverse
its magnetization. For such a thickness of Cu, this can happen whether the
sample is initially prepared in either a P or an AP configuration. However,
less fluence is systematically required to reverse the magnetization of the
free layer from a P configuration compared to the AP configuration. Also, the
dynamics of the reversal was measured for an initial P configuration and the
free layer magnetization was seen to cross zero before the reference layer
starts to remagnetize. It was shown in reference [1] that the model of Beens
et al. [52] can reproduce the magnetization reversal from the P configuration,
but magnetization crosses zero during the remagnetization of the reference
layer and it can therefore not explain the experimental measurements of
reference [1]. Igarashi et al. [1] therefore concluded that another mechanism
has to come into play and suggest a mechanism where a spin current generated
by the demagnetization of the free layer is reflected at the Cu/Reference
layer interface, and upon this reflection, the polarization of the spin
current can be rotated. This phenomenon is already well-known in the context
of spin transfer torque in non-colinear spin configurations [121]. We propose
here to make a simplified model of this mechanism to show that it can explain
the qualitative behavior observed in reference [1].

The parameters that we use are the same as in the previous section. The only
exception is that we take a Curie temperature of 500 Kelvin and $J=$ 0.05 eV
for the free layer. We now need to include spin transport and so we no longer
neglect the spin current term of Eq.(13) unless it is explicitly stated. We
make use of the now well-established result that spin currents are
proportional to the spin accumulation when a ferromagnetic layer is in contact
with a good spin sink [85, 24, 54, 65]. In our case, considering a given
ferromagnetic layer, we assume ballistic spin transport in the Cu spacer layer
and this spin sink is the other ferromagnetic layer. For a single
ferromagnet/Cu interface, we then assume that the spin current exiting the
ferromagnet obeys the following equation [52, 85]:

$\dfrac{\bm{\nabla}\cdot\bm{J}_{s}}{\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu}=\dfrac{\Delta\mu}{\tau_{B}}$ (16)

Where, following Beens et al. [52] we take $\tau_{B}=$ 10 fs for our 10 nm of
Cu. Our approach of the simulation of ballistic spin transport is similar to
Ref. [52]. We have a single (i.e. depth independent) $d$ electrons
magnetization $m^{\text{free}}$ and $m^{\text{ref}}$ in the free and reference
layers respectively, as well as depth independent spin accumulations
$\Delta\mu^{\text{free}}$ and $\Delta\mu^{\text{ref}}$. The main difference is
that we consider that the Cu/Reference layer interface has a separate spin
accumulation $\Delta\mu^{\text{int}}$. The situation is summarized in Fig.2.

Figure 2: Schematic description of the ballistic spin transport model. Black
arrows represent the various spin currents considered at each interface. The
corresponding terms in the spin transport equations (17) and (18) are shown
next to each arrow. “Spacer” refers to the Cu spacer layer and “Int” to the
Cu/Reference layer interface.

Because we want to simulate reflection at the Cu/Reference layer interface, we
introduce a parameter $r\in[0,1]$ that quantifies the amount of spin that is
reflected from the interface. $r$ times the spin current coming from the free
layer is then transferred to the interface while a fraction $(1-r)$ is
transmitted to the reference layer spin accumulation. To model the rotation of
the spin polarization of the spin current, we then assume that
$-\Delta\mu^{\text{int}}/\tau_{B}$ is transferred from the interface to the
free layer (see Fig.2). This corresponds to a full rotation of the spin
polarization. Considering a partial rotation would require, in our simple
approach, to introduce an additional parameter which we wish to avoid for this
qualitative modeling. In order to conserve angular momentum, however, we need
to have a dissipation of the extra $2\Delta\mu^{\text{int}}/\tau_{B}$ that is
generated. Realistically, this dissipation should be independent of the
thickness of the spacer layer (which is not the case here since $\tau_{B}$ is
given by the spacer thickness divided by the Fermi velocity in the spacer).
But this relationship is enforced by conservation of angular momentum and our
assumption of full spin rotation upon reflection. Finally, all the spin
current generated by the reference layer is assumed to be completely
transmitted to the free layer without transiently stopping by the interface.
The complete situation is summarized in Fig.2 and the corresponding spin
current terms entering the spin accumulation dynamics equation (13) of each of
the considered ferromagnetic layers are:

$\displaystyle\dfrac{\bm{\nabla}\cdot\bm{J}_{s}^{\text{free}}}{\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu}=\dfrac{\Delta\mu^{\text{free}}}{\tau_{B}}-\left(-\dfrac{\Delta\mu^{\text{int}}}{\tau_{B}}\right)-\dfrac{\Delta\mu^{\text{ref}}}{\tau_{B}}$
(17a) $\displaystyle\dfrac{\bm{\nabla}\cdot\bm{J}_{s}^{\text{ref}}}{\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu}=\dfrac{\Delta\mu^{\text{ref}}}{\tau_{B}}-(1-r)\left(\dfrac{\Delta\mu^{\text{free}}}{\tau_{B}}\right)$
(17b)

While the dynamics of the interfacial spin accumulation is:

$\dfrac{d\Delta\mu^{\text{int}}}{dt}=r\dfrac{\Delta\mu^{\text{free}}}{\tau_{B}}-\left(-\dfrac{\Delta\mu^{\text{int}}}{\tau_{B}}\right)-2\dfrac{\Delta\mu^{\text{int}}}{\tau_{B}}$
(18)

Where we wrote all equations in such a way that each term appears in Fig.2. By
setting $r$ to zero, one retrieves ballistic spin transport as it is modeled
by Beens et al. [52]. Overall, this model generalizes the model of reference
[95] and so it should also be suitable to reproduce ultrafast magnetization
reversal provided that the calculated spin accumulation has the right dynamics
and amplitude. For the simulations shown below, we use $r=$ 0.1 unless it is
stated otherwise.

Figure 3: Normalized magnetization dynamics of the spin-valve sample where the
dynamics of the free layer is shown in (a) and the one of the reference layer
is shown in (b), for various fluences as indicated in (b). The inset of (a)
shows a zoom between 2 and 5 ps where the normalized magnetization has been
scaled by a factor of 100. (c) shows the same data as (a) and (b) for a
fluence of 7 $\text{mJ/cm}^{\text{2}}$ together with the normalized
instantaneous equilibrium $d$ electrons magnetization of each layer; the
vertical dotted line indicates the time instant where the magnetization of the
reference layer starts to recover. (d) shows the calculated spin accumulation
dynamics in each layer as well as at the Cu/Reference layer interface.

###  IV.1 P configuration

In Fig.3, we show the results of the computed angular momentum dynamics in the
spin-valve sample with our model. Fig.3(a) shows the normalized magnetization
in the free layer while Fig.3(b) shows the normalized magnetization of the
reference layer. The dynamics is computed for several fluences as shown in
Fig.3(b). We see that the reference layer exhibits the standard UDM plus
recovery behavior without any special feature. Consistently with Fig.1, there
is no CSD observable as the normalized magnetization never even reaches 0.2
for these fluences. We note however that the spin current coming from the
demagnetization of the free layer hinders the demagnetization of the reference
layer. This behavior is known to happen in real systems [123]. For the free
layer, we observe only UDM plus recovery at low fluence, magnetization
reversal above $F1\sim$ 4 $\text{mJ/cm}^{\text{2}}$ and only a transient
switching for fluences above $F2\sim$ 10 $\text{mJ/cm}^{\text{2}}$. The
transient nature of the latter switching is best observed in the inset of
Fig.3(a). This behavior is consistent with the experimental observation of
Igarashi et al. [1] that the free layer is only permanently reversed for a
bounded range of fluences $[F1,F2]$ when starting from the P configuration.
For fluences which are greater than $F2$ (yet still below the threshold
fluence to generate a multidomain state due to a complete quenching of
magnetization in the sample), the system remains in the P configuration on a
long timescale. A key characteristic of the dynamics observed in reference [1]
is that the free layer magnetization crosses zero before the reference layer
starts to remagnetize. The $s$-$d$ model of Beens et al. cannot reproduce this
feature [1]. In Fig.3(c), we plot the magnetization dynamics of both layers
for a fluence greater than $F1$ and lesser than $F2$. We also plot the
instantaneous equilibrium magnetization of the $d$ electrons, for reference.
We can see that with our model, the normalized magnetization of the free layer
does cross zero before the reference layer starts to remagnetize (this instant
is indicated by the vertical dotted line). The time delay between the free
layer magnetization zero crossing and the beginning of the remagnetization of
the reference layer increases with fluence. To provide more insight regarding
the dynamics of angular momentum and spin transport, we plot the spin
accumulations in each layer and at the Cu/Reference layer interface in
Fig.3(d). We first note that the spin accumulation inside the free layer has
the bipolar shape used in Ref. [95], together with the same order of
magnitude, to obtain magnetization reversal of a [Co/Pt] multilayer subjected
to a spin current coming from a ferrimagnetic GdFeCo. It is known that the
spin accumulation needs to be positive to induce the reversal of
magnetization. This happens naturally at lower fluences (see Fig.1(d)) due to
remagnetization. For fluences above $F1$, the electronic temperature overcomes
the critical temperature of the free layer (see Fig.4(a)) and, without an
external source of angular momentum, CSD will appear [95]. From Eq.(13), it
follows that the spin accumulation will be negative at all times (the terms
containing the instantaneous equilibrium magnetization can lead to a positive
spin accumulation when the light source term is not zero, but this effect is
negligible). This ensures that a ferromagnetic layer cannot reverse its
magnetization because of the spin accumulation it generates, in normal
circumstances. The positive spin accumulation peak of the free layer in
Fig.3(d) is due to the spin current reflection mechanism. One can see that
after a certain delay due to the ballistic spin transport, a spin accumulation
starts reaching the Cu/Reference layer interface. Because the corresponding
angular momentum is reversed upon being reflected back to the free layer, this
leads to an increase of the spin accumulation of the free layer which
eventually becomes positive. Once the free layer spin accumulation becomes
positive, the spin accumulation at the interface starts to decrease. The spin
accumulation of the reference layer also has this bipolar structure because
(i) the reference layer remagnetizes and (ii) the ballistic spin transport
Eq.(17b) tends to bring the spin accumulation curve of the reference layer
closer to the one of the free layer (and vice versa).

We now look at some other effects predicted by our model in this spin-valve.
First, we look at the electron and phonon temperatures dynamics in each
ferromagnetic layer. This is shown in Fig.4(a) and (b) for the free and
reference layers respectively.

Figure 4: Temperature dynamics of electrons (solid lines) and phonons (dashed
lines) for the (a) free and (b) reference layers. The dynamics is calculated
for various fluences as indicated in (b). The Curie temperature of each layer
is indicated in each case by the horizontal dotted line.

A standard dynamics, as obtained from the two temperature model, is obtained
at first glance. Upon closer inspection, we can see however some bumps or
additional peaks in the electron temperature dynamics for certain fluences. We
adjusted the scale so as to make this effect obvious for the free layer,
therefore cutting off the first peak of the dynamics which does not show any
interesting feature. The deviation from the standard two temperature model
dynamics is especially large when no reversal of magnetization is obtained.
These extra features, such has the peak around 2,5 ps in the free layer for 4
$\text{mJ/cm}^{\text{2}}$ is due to the increase of magnetization (in absolute
value) via the magnetization dependent term in Eq.(6a). Although these peaks
may be overestimated due to the fact that the mean field model tends to
overestimate the speed of remagnetization around $m=0$, we still observe a
sizable effect of the transfer of energy from $d$ to $s$ electrons in the
reference layer while such effect did not exist for a single layer with
parameters identical to that reference layer (Section III). We therefore
conclude that an external source of angular momentum, via a spin current, can
heat up electrons in a way that should be observable experimentally.

Now, we look at the role of the dynamic exchange splitting, spin currents and
$d$ to $s$ energy transfer on the magnetization dynamics of spin-valves. To do
so, we plotted in Fig.5 the magnetization dynamics of the free layer for two
fluences and different models.

Figure 5: Normalized magnetization dynamics of the free layer computed for (a)
4 and (b) 7 $\text{mJ/cm}^{\text{2}}$ for different models. “Full” corresponds
to the full equation (14a) with $J=$ 0.1 (reference layer) and 0.05 (free
layer) eV and $\alpha=$ 1; “$J=$ 0” is obtained by setting $J=$ 0 eV for both
ferromagnetic layers; “$\dot{E}_{\text{es}}=$ 0” is obtained by neglecting the
magnetization dependent term in equation (6a); “No spin current” is obtained
by neglecting all spin transport; “Beens et al.” is obtained by setting $r=$ 0
and $J=$ 0.

First, we can neglect the role of dynamic exchange splitting (“$J=$ 0”). For
the highest fluence, where magnetization does not stay around zero, a very
little difference is obtained. On the other hand, for the lower fluence, we
can see a significant difference which results either in a reversal or not.
Including the dynamic exchange splitting will not qualitatively change the
magnetization dynamics but it can have sizable quantitative effects and so
should also be considered in more realistic models. Then, we can also neglect
the $d$ to $s$ electrons energy transfer (“$\dot{E}_{\text{es}}=$ 0”). This
can lead to large effects but the reason is, just as for the single layer
case, that it will change the maximum electron temperature. We can also block
all spin transport which prevents any magnetization reversal even for high
fluences (not shown). We can see that blocking this spin current also has a
large effect on the magnetization dynamics. Even though our value of the
reflection parameter $r$ is quite arbitrary, this should not come as a
surprise as it was observed experimentally that spin heating (i.e. a decrease
of magnetization solely due to an external source of angular momentum) can
lead to a change of magnetization of up to 50$\%$ [95]. Finally, we compare
these results with the calculations obtained with the model of Beens et al.
[52]. This model can also lead to magnetization reversal of the free layer but
at higher fluence and the zero crossing happens for larger time delays [1].
Because temperatures are higher for higher fluences, the maximum reachable
normalized magnetization is smaller and the effect of the spin current in
Beens et al. model is smaller, at least for our choice of parameters. Overall,
Fig.3 and 5 illustrate the rich varieties of dynamics which can be expected in
spin-valves compared to single layers where there is only demagnetization
followed by remagnetization. This diversity and its strong fluence/parameters
dependence should be kept in mind when studying the ultrafast magnetization
dynamics of spin-valve heterostructures. Such diversity of behaviors has
already been observed experimentally [93, 100].

###  IV.2 AP configuration

To finish this section, we calculate the dynamics of the spin-valve when it is
initially prepared in an AP configuration. In practice, we run exactly the
same simulations except that out of the two stable equilibrium magnetizations
of the reference layer, we select the solution with a negative sign instead of
the positive one. We always keep the positive solution for the free layer. We
assume that the reflection mechanism is identical to the P case. In
particular, the spin polarization is rotated the same way independently of the
magnetic configuration of the reference layer, consistently with the model
introduced above. The results are shown in Fig.6(a) and (b).

Figure 6: Normalized magnetization dynamics for the (a), (c) free and (b), (d)
reference layers. In (a) and (b), we use $r=$ 0.1 while in (c) and (d) we use
$r=$ 0\. For (a) and (b), the considered fluences (in
$\text{mJ/cm}^{\text{2}}$) are indicated on the right of plot (a) while for
(c) and (d) they are indicated on the right of plot (c).

Apart from the obvious sign change for the reference layer, the results are
almost identical to the simulations of the P case. A difference of interest,
consistent with experiments [1] is that it is harder to switch the free layer
from the AP configuration compared to the P configuration, although the effect
is much smaller than in the experiments (check the slightly reduced quenching
of magnetization for a fluence of 4 $\text{mJ/cm}^{\text{2}}$ in both cases).
However, in the experiments, the transient reversal of the free layer from the
P configuration starts happening for fluences $F2$ almost identical to the
threshold fluence $F1^{\prime}$ required to observe a reversal of the free
layer from the AP configuration i.e. $F1^{\prime}\sim F2$. This is not the
case in our simulations where $F1^{\prime}\gtrsim$ 4 $\text{mJ/cm}^{\text{2}}$
while $F2\sim$ 10 $\text{mJ/cm}^{\text{2}}$. We note that Igarashi et al. [1]
do not consider this reflection mechanism for the AP configuration. We thus
perform simulations for $r=$ 0 in Fig.6(c) and (d). However, no permanent
reversal of the free layer magnetization is observed. Although a transient
reversal is observed starting from fluences around 12
$\text{mJ/cm}^{\text{2}}$, the remagnetization of the reference layer prevents
the reversal from being permanent. Even at much higher fluences, when the
remagnetization of the reference layer is greatly hindered due to CSD, a
permanent reversal is still not possible. This is because the magnetization of
the free layer is also further reduced which makes it even more sensitive to
spin currents. In real systems, a multidomain state would be generated at such
high fluences. The transient nature of the reversal at these higher fluences
could also be due to the overestimation of remagnetization by mean field
models.

It could also be possible that a reflection mechanism is still present in the
AP configuration, even though it would be different, for reasons we do not
explain, from the reflection mechanism when one starts from the P
configuration. This is supported by some experimental measurements of Igarashi
et al. [1] where a reversal of the free layer from an AP configuration is
still observed for a copper spacer with a thickness of 40 nm. In this case,
only a small amount of light can reach the reference layer which also supports
a contribution of the spin current reflection mechanism. In Fig.7, we compute
the magnetization dynamics of both layers for a fluence around $F2$ and for
different values of $r$.

Figure 7: Normalized magnetization dynamics, of the (a) free and (b) reference
layers, in the AP configuration and for different values of $r$ as indicated.

The magnetization dynamics of the reference layer remains qualitatively the
same for the considered reflection parameters. The one of the free layer is
however significantly modified. As a general trend, we see that lowering the
value of the $r$ parameter slows down the reversal dynamics i.e. the
normalized magnetization crosses zero for larger time delays. It is also
interesting to note that in all cases, the temperature dynamics are almost
identical (plus or minus two Kelvins) even though the $d$ to $s$ electrons
energy transfer is different for different values of $r$. This means that one
cannot in general conclude that a magnetic subsystem is at equilibrium from
the fact that its magnetization almost no longer changes (as it is observed
for instance in Fig.7(a) for $r=$ 0, 0.01 and 0.1). It is possible that an out
of equilibrium situation is sustained due to a spin current emitted by another
layer where the dynamics is still not over (here the reference layer; see
Fig.7(b)).

We conclude this section by highlighting that our spin current reflection
mechanism is probably oversimplified. We note however that our approach, based
on parameters with reasonable values, shows that this mechanism, if it exists,
generates spin accumulations with a reasonable order of magnitude and triggers
a magnetization dynamics with a realistic speed.

##  V Conclusion

In this work, we presented an extension of the $s$-$d$ model based on
references [102, 23, 103, 52] which includes a dynamic exchange splitting (and
equilibrium spin polarization of the $s$ band), energy transfer from $d$ to
$s$ electrons [111, 68] as well as a newly proposed spin current reflection
mechanism [1]. This model leads to qualitatively (and also mostly
quantitatively) identical results in single ferromagnetic layer systems
compared to the case where all these additional effects are neglected. In the
case where there is an external/non-local source of angular momentum such as
in spin-valves, we showed that these effect can drastically change the
observed magnetization dynamics. In particular, we could reproduce the
magnetization reversal of the free layer of a ferromagnetic spin-valve which
cannot be qualitatively reproduced by previous models. We also predict that
the electronic temperature dynamics may be strongly affected, for instance in
such spin-valves, when a ferromagnetic material is subjected to an external
source of angular momentum.

The advantage of our model is that it is not computationally expensive but as
pointed out in this paper, it lacks quantitative predictive power. In
particular, the mechanism for the reflection of the spin current is still
unclear. It is also not clear whether the $s$-$d$ model is well suited for
such simulations as it does not contain magnons which are believed to play a
fundamental role at the ferromagnetic/paramagnetic transition [124].
Transverse excitations however are included in this mean-field model, but the
fast recovery of magnetization (we mean even in the absence of an external
source of angular momentum), once it has almost been quenched, compared to
experiments [95] or atomistic simulations [60], seem to indicate that at least
the mean-field approximation needs to be lifted in order to make more
quantitative predictions. Still, our model is attractive for its simplicity
and its capability to explain the diversity of ultrafast magnetization
dynamics behaviors.

A quantitative agreement could also perhaps be obtained by following the
approach of reference [68], by using for instance temperature dependent
parameters, but considering the number of parameters required in such
simulations, it is not yet clear whether these calculations can have any
quantitative predicting power for such complex systems. An alternative route,
largely unexplored in the field of ultrafast magnetization dynamics, would be
to treat the electron-magnon problem within the framework of the Fermi-liquid
theory [125].

##  Appendix A Derivation of equation (8)

In this appendix, we provide a derivation of Eq.(8). This equation was first
given in reference [23], to the best of our knowledge, but its derivation and
validity was not discussed. This derivation is useful to establish the
limitations of the equations given in this work as well as to show the need
for a more realistic description of the materials densities of state. We need
to calculate the $s$ electrons spin polarization

$s^{z}=\dfrac{N_{\uparrow}-N_{\downarrow}}{2N}$ (19)

With $N=N_{\uparrow}+N_{\downarrow}$ the total number of $s$ electrons and
$N_{\sigma}$ is the number of electrons with spin $\sigma$. Note that when all
$s$ electrons have an up spin, $s^{z}=1/2$ and so this definition of the spin
polarization is consistent with the equations of the main text. The spin
dependent electronic numbers are obtained from

$N_{\sigma}=\int D_{\sigma}(E)f(E;T_{e},\mu_{\sigma})dE$ (20)

With $f$ the Fermi-Dirac distribution function. However, Eq.(8) does not
dependent on the electronic temperature explicitly. Thus, we first need to
approximate the Fermi-Dirac functions by step functions. This means that the
thermal energy should be much smaller than the width of the $s$ band:
$k_{B}T_{e}\ll\mu_{\sigma}-E^{0}_{\sigma}$. Then

$N_{\sigma}=\int_{E^{0}_{\sigma}}^{\mu_{\sigma}}D_{\sigma}(E-E^{0}_{\sigma})dE=\mathcal{D}_{\sigma}(\mu_{\sigma}-E^{0}_{\sigma})-\mathcal{D}_{\sigma}(0)$
(21)

Where we shifted the functions representing the densities of state for
convenience and we also assumed that these functions are continuous such that
they all have an antiderivative $\mathcal{D}_{\sigma}$. We now perform three
consecutive first order Taylor expansions of
$\mathcal{D}_{\sigma}(\mu_{\sigma}-E^{0}_{\sigma})$, assuming that
$\Delta\mu$, $\mu-\mu_{\text{ie}}$ and
$E^{0}_{\sigma}-E^{0}_{\sigma,\text{ie}}$ are small compared to
$\mu_{\sigma}-E^{0}_{\sigma}$. $\mu_{\sigma}=\mu+\sigma~{}\Delta\mu/2$ and the
“ie” subscript refers to instantaneous equilibrium value as for the spin
polarization $s^{z}_{\text{ie}}$. We obtain:

$\begin{split}N_{\sigma}&\simeq\mathcal{D}_{\sigma}(\mu_{\sigma,\text{ie}}-E^{0}_{\sigma,\text{ie}})-\mathcal{D}_{\sigma}(0)\\\
&+\sigma\dfrac{\Delta\mu}{2}D_{\sigma}(\mu-E^{0}_{\sigma})\\\
&+(\mu-\mu_{\text{ie}})D_{\sigma}(\mu_{\text{ie}}-E^{0}_{\sigma})\\\
&-(E^{0}_{\sigma}-E^{0}_{\sigma,\text{ie}})D_{\sigma}(\mu_{\text{ie}}-E^{0}_{\sigma,\text{ie}})\end{split}$
(22)

To the same level of approximation, all the density of states factors should
be taken equal, and after shifting back the function representing the
densities of state, one has

$N_{\sigma}\simeq
N_{\sigma,\text{ie}}+D_{\sigma}(\varepsilon_{F})\left[(\mu-\mu_{\text{ie}})+\sigma\dfrac{\Delta\mu}{2}-(E^{0}_{\sigma}-E^{0}_{\sigma,\text{ie}})\right]$
(23)

Using the fact that the total number of $s$ electrons does not change
$N=N_{\text{ie}}=N_{\uparrow,\text{ie}}+N_{\downarrow,\text{ie}}$, one
obtains:

$\begin{split}\mu-\mu_{\text{ie}}&=-\dfrac{D_{\uparrow}(\varepsilon_{F})-D_{\downarrow}(\varepsilon_{F})}{D_{\uparrow}(\varepsilon_{F})+D_{\downarrow}(\varepsilon_{F})}\dfrac{\Delta\mu}{2}\\\
&+\dfrac{D_{\uparrow}(\varepsilon_{F})}{D_{\uparrow}(\varepsilon_{F})+D_{\downarrow}(\varepsilon_{F})}(E^{0}_{\uparrow}-E^{0}_{\uparrow,\text{ie}})\\\
&+\dfrac{D_{\downarrow}(\varepsilon_{F})}{D_{\uparrow}(\varepsilon_{F})+D_{\downarrow}(\varepsilon_{F})}(E^{0}_{\downarrow}-E^{0}_{\downarrow,\text{ie}})\end{split}$
(24)

Equation (8) is readily obtained by combining equations (19), (23) and (24).

Now $(E^{0}_{\sigma}-E^{0}_{\sigma,\text{ie}})\sim J$ and Eq.(21) implies that
$\mu_{\sigma}-E^{0}_{\sigma}\sim 1/\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu$. So to be consistent,
our theory should be such that $\Delta\mu$, $J$ and $k_{B}T_{e}$ are much
smaller than $1/\mkern 1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu$.
We have used $J=0.1/\mkern 1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu$ and for the highest studied fluences, we have $k_{B}T_{e}\simeq
0.2/\mkern 1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu$ and
$\Delta\mu\simeq 0.1/\mkern 1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu$. However, this does not invalidate the qualitative nature of our
results since we obtain similar results for $J=0$ and the magnetization
switching in the spinvalve appears at relatively low fluences. Nevertheless,
we note that using larger values of $J$ (in combination with fluences
comparable to the ones used in this work) leads to large divergences in the
spin accumulation because of the factor $1/(1-JS\eta(t)/(2k_{B}T_{e}(t)))$ in
Eq.(14a). No such behavior was observed for the results presented in this
work.

###### Acknowledgements.

The author thank Jon Gorchon, Grégory Malinowski, Junta Igarashi, Guillermo
Nava Antonio, Chiara Ciccarelli, Philippe Scheid and Stéphane Mangin for
valuable discussions. This work is supported by the ANR-20-CE09-0013 UFO, by
the Institute Carnot ICEEL for the project “CAPMAT” and FASTNESS, by the
Région Grand Est, by the Metropole Grand Nancy, for the Chaire PLUS by the
impact project LUE-N4S, part of the French PIA project “Lorraine Université
d’Excellence” reference ANR-15-IDEX-04-LUE, by the “FEDERFSE Lorraine et
Massif Vosges 2014-2020”, a European Union Program, by the European Union’s
Horizon 2020 research and innovation program COMRAD under the Marie
Skłodowska-Curie grant agreement No 861300. This article is based upon work
from COST Action CA17123 MAGNETOFON, supported by COST (European Cooperation
in Science and Technology). We also acknowledge funding by the German Research
Foundation (DFG) through the collaborative research center SFB TRR 227
“Ultrafast spin dynamics” (Project ID 328545488, project B02) and the European
Union H2020 program through the FET project SKYTOP/Grant No. 824123.

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