# Bridging atomistic spin dynamics methods and phenomenological models of
single pulse ultrafast switching in ferrimagnets

Florian Jakobs  Dahlem Center for Complex Quantum Systems and Fachbereich
Physik, Freie Universität Berlin, 14195 Berlin, Germany  Unai Atxitia  Dahlem
Center for Complex Quantum Systems and Fachbereich Physik, Freie Universität
Berlin, 14195 Berlin, Germany

###### Abstract

We bridge an essential knowledge gap on the understanding of all-optical
ultrafast switching in ferrimagnets; namely, the connection between atomistic
spin dynamics methods and macroscopic phenomenological models. All-optical
switching of the magnetization occurs after the application of a single
femtosecond laser pulse to specific ferrimagnetic compounds. This strong
excitation puts the involved degrees of freedom, electrons, lattice and spins
out-of-equilibrium between each other. Atomistic spin models have
quantitatively described all-optical switching in a wide range of experimental
conditions, while having failed to provide a simple picture of the switching
process. Phenomenological models are able to qualitatively describe the
dynamics of the switching process. However, a unified theoretical framework is
missing that describes the element-specific spin dynamics as atomistic spin
models with the simplicity of phenomenology. Here, we bridge this gap and
present an element-specific macrospin dynamical model which fully agrees with
atomistic spin dynamics simulations and symmetry considerations of the
phenomenological models.

##  I Introduction

Since its experimental discovery [1], the theoretical description of laser
induced all-optical switching (AOS) of the magnetization in GdFeCo
ferrimagnetic alloys has remained a challenge. Despite intense experimental
and theoretical research in the field [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12],
an established and unified picture of the process is still missing.
Experimental findings are mostly compared or interpreted in terms of atomistic
spin dynamics simulations [13, 14, 15, 16, 17], multisublattice spin dynamics
based on symmetry arguments [5, 18, 19], and based on the Landau-Lifshitz-
Bloch equation [20, 21, 22]. The main goal of the present work is the
revision, extension and merging of these approaches into a unified model.

Atomistic spin dynamics (ASD) models have been used before to quantitatively
describe ultrafast dynamics in $3d$ transition metals [23, 24] and $4f$ rare-
earth ferromagnets [25, 26]. They have also been used in GdFeCo, to describe
the equilibrium thermal properties [13], the thermal character of AOS [4], the
so-called transient ferromagnetic-like state [3], the demonstration of spin-
current-mediated rapid magnon localisation and coalescence [27] and the
possibility of AOS using picosecond-long laser pulses [16]. Results from
atomistic spin models also compare qualitatively well to an analytical theory
based on the excitation of spin-wave exchange modes [8], provide insights for
optimal electron, phonon and magnetic characteristics for low energy switching
[28] and predict maximum repetition rate using two consecutive laser pulses
[29]. More sophisticated, orbital-resolved atomistic models provide insights
on the role of the intra-exchange coupling between $4f$ and $5d$ electrons in
the dynamics of GdFeCo alloys[14]. Atomistic models can naturally describe
switching in Gd/Fe multilayers composed of very thin layers [30, 31]. Recent
observations [32, 33] of single pulse switching in Mn2RuxGa alloys are also
well-described by ASD methods [34]. Despite the demonstrated success in
modeling AOS, ASD simulation results are cumbersome to interpret without an
analytical model that unveils the role of the different processes and
interactions during the switching process. This potential semi-analytical
model has to capture most of the features of the ASD simulations.

Semi-phenomenological models describing switching already exist. A macroscopic
theory for the description of the dynamics and relaxation of the macroscopic
(sublattice) magnetization of ferromagnets and antiferromagnets was developed
originally by Baryakhtar [35, 9]. An extension of such phenomenology to
ferrimagnets in the context of ultrafast spin dynamics was introduced in Ref.
[5]. At the ultrafast scale, magnetization dynamics are dominated by atomic
scale spin excitations, these spin dynamics are driven by dissipative
processes which in ferrimagnets are two-fold, relativistic and exchange
driven. Relativistic processes allow for exchange of angular momentum between
the spins and lattice degree of freedom due to the presence of spin-orbit
interaction connecting them. Exchange processes can arise due to transport of
spin angular momentum – spin and magnon transport – which is the only mean to
exchange angular momentum in ferromagnets. In multisublattice magnets another,
different pathway opens, namely, local exchange of angular momentum. To
account for such local exchange processes in ferrimagnets, the equation of
motion for the magnetization dynamics proposed by Landau and Lifshitz [36] is
enhanced by an exchange relaxation term [5, 19, 9, 37]. Within this
macroscopic model, the exchange relaxation dominates the dynamics when the
magnetic sublattices are driven into mutual non-equilibrium. Qualitative
agreement to experiments in two-sublattice magnets has been demonstrated [19],
such as AOS in ferrimagnetic GdFeCo using fs laser pulses [5] and ps laser
pulses [38], AOS in Heusler semimetals Mn2RuxGa [39], or element-specific
demagnetization of ferromagnetic NiFe alloys [18]. Quantitative comparison of
this model to neither experiments nor ASD simulations have been conducted so
far. While the arguments behind such phenomenology are robust, the range of
applicability and the validity of the model parameters could be questioned.
For instance, the parameters defining the relativistic and exchange relaxation
are assumed to be constant and of the same order. The magnetic free energy
functional is calculated for near thermal equilibrium states. This implies a
relatively strong coupling to the heat-bath, while switching conditions are
supposedly fulfilled when exchange relaxation between sublattices dominates
over the relaxation to the heat-bath.

An alternative macroscopic model directly derived from an atomistic spin model
has also been proposed. This model is based in the Landau-Lifshitz-Bloch (LLB)
equation of motion [40, 41, 20, 42, 43]. The LLB model for two-sublattice
magnets [20, 42] has been used in the context of AOS in GdFeCo, e.g. the
element-specific demagnetization rates compare well to experiment, and it
predicts that near the magnetic phase transition the otherwise slower Gd
sublattice becomes faster than Fe [22], as recently observed [44]. The LLB
model has been demonstrated to provide accurate analytical expressions for the
temperature dependence of the relativistic relaxation parameter as well as for
the non-equilibrium effective fields below and above the critical temperature
[42]. Moreover, the LLB model also describes the transverse motion of the
magnetization. This makes it the preferred model for computer simulations of
heat-assisted magnetic recording [45] and realistic description of all-optical
switching [46], and ultrafast spintronics, such as domain wall motion [47, 48]
or skyrmion creation by ultrafast laser pulses [49]. So far the LLB model and
Baryakhtar-like models have been considered as complementary approaches. Here,
we merge them into one unified approach.

In this work we address the issues discussed above by directly comparing both
phenomenological models to ASD simulations. We do so since ASD simulations
have been already quantitatively compared to experiments in literature. We
find that quantitative comparison between ASD and both phenomenological models
is partially possible for laser excitation producing small deviation from
equilibrium. However, those models hardly reproduce magnetic switching using
the same parameter values describing the relaxation of small perturbations.
Here, based upon those phenomenological models, we propose a macroscopic model
that compares precisely to the magnetization dynamics calculated using ASD
simulations, including element-specific magnetization relaxation and
switching. This model bridges atomistic spin dynamics based models and
previously proposed phenomenological models. Notably, it provides a deeper
understanding to the parameters entering the phenomenological models and sheds
some light into the process of ultrafast switching in ferrimagnets.

The work is broken down in the following way: in Sec. II, we present the
atomistic spin model for the calculation of the magnetic equilibrium
properties and non-equilibrium dynamics. The equilibrium properties are
compared to a mean field model. We then provide atomistic calculations of the
ultrafast magnetization dynamics with input from the two temperature model.
These results are the basis for the comparison to the phenomenological models
presented in Sec. III. Firstly, we present the Baryakhtar model and the
Landau-Lifshitz-Bloch model. Secondly, we compare the ultrafast magnetization
dynamics calculated with those models to the atomistic spin dynamics results.
Finally, in Sec. III.3 we present the unified phenomenological model, a hybrid
model combining Baryakhtar and LLB models, and its comparison to atomistic
spin dynamics.

##  II Atomistic Spin Model

Ferrimagnetic materials characterise by spontaneous magnetization as a
resultant of two or more components of non-parallel magnetic moments [50].
Atomistic spin models based on the Heisenberg Hamiltonian can be considered
one of the simplest microscopic models able to reproduce the equilibrium
properties of ferrimagnets. The spin system energy due to only the exchange
interactions can be described by an effective Heisenberg model:

$\displaystyle\mathcal{H}=-\sum_{i\neq
j}J_{a}\mathbf{S}_{a,i}\cdot\mathbf{S}_{a,j}-\sum_{i\neq
j}J_{b}\mathbf{S}_{b,i}\cdot\mathbf{S}_{b,j}-\sum_{i\neq
j}J_{ab}\mathbf{S}_{a,i}\cdot\mathbf{S}_{b,j}$ (1)

where $J_{a(b)(ab)}$ is the exchange constant between neighboring sites
represented by two classical spin vectors $\mathbf{S}_{i}$ and
$\mathbf{S}_{j}$ ($|\mathbf{S}|=1$). Further, one can include magnetic
anisotropy terms to Eq. (1) to set a preferential axis for the magnetization.
However, since the anisotropy energy is relatively low it plays a marginal
role in the switching process. This makes for a simpler Hamiltonian and a more
direct comparison to the phenomenological models. To model a ferrimagnet, one
needs to consider two alternating sublattices of unequal and antiparallel
moments, with three exchange coupling constants: ferromagnetic for each
sublattice ($J_{a}$ and $J_{b}$) and a third for the antiferromagnetic
interaction between them, $J_{ab}$. For instance, GdFeCo alloys are composed
of a transition metal FeCo and a Gd rare-earth sublattices. We model the Fe
and Co spins as only one magnetic sublattice, and we assume a common atomic
magnetic moment of $\mu_{\rm{FeCo}}=1.94\mu_{\textrm{B}}$. In these alloys the
rare-earth impurities add localised $4f$ spins to the system assumed to be,
$\mu_{\rm{Gd}}=7.6\mu_{\textrm{B}}$. The amorphous nature of GdFeCo is
modelled by using a simple cubic lattice model but with random placements of
Gd moments within the lattice to the desired concentration. The applicability
of the Heisenberg approximation relies on the stability of local moments under
rotation and at high temperature where Stoner excitations are generally weak
[51]. It is assumed that the electronic properties are temperature-independent
in the range where the system is magnetically ordered.

###  II.1 Atomistic spin dynamics

Equilibrium and non-equilibrium element specific magnetic properties of a
ferrimagnet are calculated using atomistic spin dynamics simulations which are
based in the stochastic-Landau-Lifshitz-Gilbert equation (s-LLG) [52]

$\displaystyle(1+\lambda_{i}^{2})\mu_{s,i}\dot{\mathbf{S}}_{i}=-\gamma\mathbf{S}_{i}\times\left[\mathbf{H}_{i}-\lambda_{i}\;\left(\mathbf{S}_{i}\times\mathbf{H}_{i}\right)\right],$
(2)

where $\gamma$ is the gyromagnetic ratio, and $\lambda_{i}$ is the so-called
phenomenological sublattice specific damping parameter. By including a
Langevin thermostat the spin dynamics including statistical – equilibrium and
non-equilibrium thermodynamic properties can be obtained. An effective field-
like stochastic term $\boldsymbol{\zeta}_{i}$ is added to the effective field
$\mathbf{H}_{i}=\boldsymbol{\zeta}_{i}(t)-\frac{\partial\mathcal{H}}{\partial\mathbf{S}_{i}}$,
with white noise properties [53]:
$\langle\boldsymbol{\zeta}_{i}(t)\rangle=0\quad\text{and}\quad\langle\boldsymbol{\zeta}_{i}(0)\boldsymbol{\zeta}_{j}(t)\rangle=2\lambda_{i}k_{\text{B}}T\mu_{s,i}\delta_{ij}\delta(t)/\gamma.$
The variance of the Langevin noise is chosen such that the fluctuation-
dissipation theorem is full filled.

Figure 1: Equilibrium magnetization of a GdFeCo alloy for Gd concentration,
$x_{\rm{Gd}}=25\%$. Element-specific normalized equilibrium magnetization and
net equilibrium magnetization,
$M(T)=x_{\rm{Gd}}\mu_{\rm{Gd}}m_{\rm{Gd}}-x_{\rm{Fe}}\mu_{\rm{Fe}}m_{\rm{Fe}}$,
where $\mu_{\rm{Gd(Fe)}}$ is the atomic magnetic moment of Gd(Fe). Lines
correspond to the mean-field approximation with renormalized exchange
parameters. Symbols correspond to atomistic spin dynamics simulations.

###  II.2 Mean-field approximation

Exact analytical expressions for the $M(T)$ curve are cumbersome to derive due
to the many body character of the problem. Here we resort the mean field
approximation (MFA), already used in previous works [13, 8, 54]. We note that
to be able to apply the MFA for the GdFeCo impurity model, and thus
translation non-symmetric with respect to spin variables $\mathbf{S}_{i}$, we
need to transform the Heisenberg Hamiltonian to a symmetric one. We use the
spin analogy of the virtual crystal approximation (VCA) to transform the
disordered lattice Hamiltonian $\mathcal{H}$ to a symmetric VCA Hamiltonian
$\mathcal{H}_{\mathrm{VCA}}$. Within the VCA we evaluate the effective
sublattice exchange parameters, given by the sum of the exchange interactions
of a given spin at a site $\mathbf{r}_{i}$ of sublattice $i$ with all other
atoms of this sublattice. This involves weighting the exchange parameters by
the relative composition, $x_{i}\equiv\mathrm{concentration\ species}\ i$ [8],

$J_{i}=\sum_{\mathbf{r}_{i},\mathbf{r}^{\prime}_{i}}J(\mathbf{r}_{i},\mathbf{r}^{\prime}_{i})\underbrace{\equiv}_{\mathrm{VCA}}\
\ x_{i}J(\mathbf{r}_{i},\mathbf{r}^{\prime}_{i})\ \ \
\mathrm{intrasublattice}$ (3)

whereas the intersublattice effective exchange reads

$J_{ij}=\sum_{\mathbf{r}_{i},\mathbf{r}^{\prime}_{j}\notin
A_{i}}J(\mathbf{r}_{i},\mathbf{r}^{\prime}_{j})\underbrace{\equiv}_{\mathrm{VCA}}\
\ x_{i}J(\mathbf{r}_{i},\mathbf{r}^{\prime}_{j})\ \ \
\mathrm{intersublattice}$ (4)

Thus the VCA Hamiltonian reads

$\mathcal{H}_{\mathrm{VCA}}=\sum_{j\in
A_{i}}J_{i}\mathbf{S}_{i}\cdot\mathbf{S}_{j}+\sum_{j\notin
A_{i}}J_{ij}\mathbf{S}_{i}\cdot\mathbf{S}_{j}$ (5)

where $A_{i}$ represent the magnetic sublattice of the spin $\mathbf{S}_{i}$.
In the exchange approximation we define the MFA field as

$\mu_{a}H_{a}^{\rm{MFA}}=z_{a}J_{aa}m_{a}+z_{ab}J_{ab}m_{b}$ (6)

The element-specific equilibrium magnetization is calculated via the self-
consistent solution of $m_{a}=L(\beta\mu_{a}H^{\rm{MFA}}_{a})$ and
$m_{b}=L(\beta\mu_{b}H^{\rm{MFA}}_{b})$. $z_{a}$ and $z_{ab}$ correspond to
the number of first nearest neighbours of type $a$ and $b$, respectively. It
is well-known that the MFA overestimates the value of the critical temperature
$T_{C}$. However, a very good agreement between ASD and MFA can be obtained by
using a reduced value for the exchange parameters, even for multilattice
magnets [54]. Figure 1 shows element-specific $M_{a}=x_{a}\mu_{a}m_{a}(T)$
using ASD simulations and renornalized MFA for $x_{\rm{Gd}}=25\%$. Net
magnetization is also shown in Fig. 1, which is defined as
$M(T)=x_{\rm{Gd}}\mu_{\rm{Gd}}m_{\rm{Gd}}-x_{\rm{Fe}}\mu_{\rm{Fe}}m_{\rm{Fe}}$.
The agreement between ASD and MFA is good enough for all the temperature
regions. We observe the presence of compensation temperature $T_{\rm{M}}$ at
room temperature for $x_{\rm{Gd}}=25\%$ at which the thermally average
magnetization of both sublattices are equal but opposite, so that the
magnetization of the system is equal to zero $M(T_{\rm{M}})=0$. The mapping of
the atomistic spin model and the corresponding mean-field approximation turns
out to be necessary for a quantitative comparison to the phenomenological
models, and thereby paramount for the unification of both pictures.

###  II.3 Two Temperature Model

Single pulse all-optical switching has been demonstrated to be a thermal
process in ferrimagnetic GdFeCo alloys [4] and in Mn2RuxGa Heusler semi-metals
[32]. Ultrafast heating by optical or electric means are sufficient to achieve
switching in specific GdFeCo alloys [55]. Although the minimum achievable
duration of the electric pulses are limited to picoseconds, those are better
suited for potential integration into applications. Laser pulses can be as
short as only a few femtoseconds, which permits to excite the electron system
in timescales of the order of the exchange interaction allowing for the
investigation of fundamental physics governing switching. In this work, we
center in excitation of the ferrimagnetic GdFeCo using femtosecond laser
pulses. When a metallic ferrimagnetic thin film is subjected to a near
infrared laser pulse, only the electrons are accessible by the photon electric
field. Initially, the absorbed energy is barely transferred to the lattice and
consequently the electron system heats up. The electron and phonon
temperatures are decoupled for up to several picoseconds until the electron-
phonon interaction equilibrates the two heat-baths. This phenomenology is well
captured by the so-called two-temperature model (2TM) [56, 57] which can be
written as two coupled differential equations:

$\displaystyle C_{\rm{el}}\frac{\partial T_{\rm{el}}}{\partial t}$
$\displaystyle=-g_{\rm{ep}}\left(T_{\rm{el}}-T_{\rm{ph}}\right)+P_{l}(t)$ (7)
$\displaystyle C_{\rm{ph}}\frac{\partial T_{\rm{ph}}}{\partial t}$
$\displaystyle=+g_{\rm{ep}}\left(T_{\rm{el}}-T_{\rm{ph}}\right).$ (8)

$C_{\rm{el}}=\gamma_{\rm{el}}T_{\rm{el}}$ where $\gamma_{\rm{el}}=6\times
10^{2}$ J/m3K2, and $C_{\text{ph}}=3.8\times 10^{6}$ J/m3K represent the
specific heat of the electron- and phonon system. The electron-phonon coupling
is taken temperature independent, $G_{\rm{ep}}=7\times 10^{17}$ J/m3K. Here,
$P(t)$ is a Gaussian shaped pulse with a duration of 55 fs. The exact values
of the parameters entering the TTM in GdFeCo are still unknown. The values we
use here are close to the commonly used, e.g. Refs. [8, 4, 34].

###  II.4 Ultrafast magnetization dynamics using ASD

Element-specific magnetization dynamics induced by a femtosecond laser pulse
are calculated by combining the atomistic s-LLG equation for the spin dynamics
(Eq. (2)) and the 2TM for the electron temperature (Eq. (7)). The electron
system acts as heat-bath for the atomic spins. We consider a lattice with
$N=50\times 50\times 50$ spins, and damping parameters,
$\lambda_{\rm{Gd}}=0.01=\lambda_{\rm{Fe}}$. Figure (2) shows, for $t<0$, the
dynamics of the element-specific magnetization from an initial saturated state
($T=0$ K), towards thermal equilibrium with the heat-bath which is set to
$T=300$ K. The relaxation dynamics of Fe sublattice is faster than those of
the Gd sublattice. This comes out naturally as the element-specific
dissipation of angular momentum scales as
$\dot{m}_{z}\sim\gamma\lambda/\mu_{s}$, in Gd
($\mu_{\rm{Gd}}=7.6\mu_{\rm{B}}$) is slower than in Fe sublattice
($\mu_{\rm{Gd}}=1.94\mu_{\rm{B}}$). Once the magnetic system is in thermal
equilibrium with the heat-bath, we apply the laser pulse, $t>0$, which
introduces energy into the electron system and induces ultrafast magnetization
dynamics. To illustrate the switching and no switching dynamics we consider
two limiting cases, dynamics induced by low laser power, $P_{0}$, and large
laser power, $2P_{0}$. The electron temperature increases up and above the
Curie temperature in time scales of a few hundreds of femtoseconds Fig. (2)
(a). This reflects in the magnetic system as a fast demagnetization of both Fe
and Gd sublattices. For relatively low laser power, $P_{0}$, the magnetization
of both sublattices reduces while the electron temperature remains relatively
high. Once the electron temperature reduces and equalizes to the lattice
temperature, the magnetization recovers to the thermal state given by the
heat-bath temperature, which is higher than initially ($T=300$ K). This is why
the final magnetization value is smaller than the initial one. For higher
laser powers, $2P_{0}$, the magnetization of both sublattices reduces quickly.
The Fe sublattice faster than the Gd one. Once the magnetization of the Fe
sublattice hits zero, instead of remaining demagnetized, the magnetization
starts to develop toward the opposite direction, while the magnetization of
the Gd sublattice is still in the process of demagnetization. During a couple
of picoseconds, both sublattice magnetization are aligned along the same
direction, similar to a ferromagnet. Consequently, this non-equilibrium state
has been named the transient ferromagnetic-like state [3]. One can observe in
Fig. (2) (b) that the demagnetization rates of both sublattices slow down when
the Fe magnetization crosses zero. This change reveals the set in of a process
driving the magnetization dynamics different to the one driving the initial
demagnetization. It has been argued that at this point direct exchange of
angular momentum between sublattices dominates over processes of relativistic
origin, which in turn dissipate angular momentum into the heat-bath.
Interestingly, soon after switching, both sublattice magnetization rapidly
relax to equilibrium indicating that relaxation into the heat-bath dominates
the dynamics.

Figure 2:  (a) Electron and lattice temperature dynamics for two laser pulse
power values, $P_{0}$ and $2P_{0}$. Both electron and lattice temperature are
kept constant, $T=300$K, for $t<0$. At $t=0$ a laser pulse is applied and the
dynamics of the electron and lattice temperature heat up. The dynamics of
those temperatures are theoretically described by the two-temperature model.
(b) Element-specific magnetization dynamics induced by the heat profile at
(a). The dynamics are calculated using atomistic spin dynamics methods. For
lower laser powers $P_{0}$, the magnetization of both sublattices demagnetize
rapidly and remagnetize towards the new equilibrium. For laser power $2P_{0}$,
the magnetization of both sublattices demagnetizes and switches. After
switching they relax towards the thermal equilibrium state. GdFeCo alloys with
$x_{\rm{Gd}}=25\%$ are calculated.

##  III Phenomenological models

Differently to ASD simulations, phenomenological models describe the element-
specific magnetization dynamics by solving two coupled equations of motion,
one for each sublattice. In this work we aim at finding a phenomenological
model that describes the same element-specific magnetization dynamics as those
coming out from the ASD simulations (Fig. 2). The starting point is the
comparison of the ASD simulations to well-known phenomenological models. We
show that those models are unable to describe in a satisfactory way the
different element-specific magnetization dynamics studied in the previous
section and summarized in Fig. 2.

###  III.1 Baryakhtar model

The simplest model to describe element-specific magnetization dynamics and
switching in ferrimagnets was proposed by Mentink and co-workers [5].
Longitudinal spin dynamics was derived from Onsager’s relations

$\displaystyle\frac{\mu_{a}}{\gamma_{a}}\frac{dm_{a}}{dt}$ $\displaystyle=$
$\displaystyle\alpha^{\rm{B}}_{a}\mu_{a}H_{a}+\alpha^{\rm{B}}_{e}(\mu_{a}H_{a}-\mu_{b}H_{b})$
(9) $\displaystyle\frac{\mu_{b}}{\gamma_{b}}\frac{dm_{b}}{dt}$
$\displaystyle=$
$\displaystyle\alpha^{\rm{B}}_{b}\mu_{b}H_{b}+\alpha^{\rm{B}}_{e}(\mu_{b}H_{b}-\mu_{a}H_{a})$
(10)

here, $\alpha^{\rm{B}}_{a,b}$ stands for the relaxation parameter of
relativistic origin, which dissipates angular momentum out of the spin system,
and $\alpha^{\rm{B}}_{e}$ stands for the exchange relaxation parameter and
describes the rate of dissipation of angular momentum between sublattices. By
construction exchange relaxation conserves the total angular momentum. We
emphasize here the difference in the notation between the atomic relaxation
parameter, $\lambda$, describing the dissipation of the atomic spins in ASD
simulations and the macrospin relaxation parameter, $\alpha$, describing the
dissipation of the whole magnetic sample. Within this model, the values for
$\alpha^{\rm{B}}_{a,b}$ and $\alpha^{\rm{B}}_{e}$ are unknown but used as
fitting parameters when compared to experiments. The internal effective field
$H_{a(b)}$, acting on sublattice $a(b)$ are derived from a non-equilibrium
mean-field approximation,

$\displaystyle\mu_{a}H_{a}=-\beta^{-1}L^{-1}(m_{a})+\mu_{a}H_{a}^{\rm{MFA}}$
(11)

where, $L^{-1}(x)$ is the inverse Langevin function, $\beta=1/k_{B}T$, where
$T$ represents the temperature of the heat-bath to which the spin system is
coupled to. At equilibrium, the effective field is $H_{a}=0$, as
$m_{a}=L(\beta\mu_{a}H_{a}^{\rm{MFA}})$. The same arguments apply for
sublattice $b$. It turns out that by solving Eqs. (9) and (10) together with
the 2TM, described in Eqs. (7) and (8), one obtains similar ultrafast
magnetization dynamics as those using ASD simulations (Fig. (2)). Element-
specific demagnetization [18] and switching dynamics [19] based on this
approach have been discussed thoughtfully before. On those works, the values
for the relaxation parameters, relativistic and exchange, are taken constant
and of the same order,
$\alpha^{\rm{B}}_{\rm{Fe}}\approx\alpha^{\rm{B}}_{\rm{Gd}}\approx\alpha^{\rm{B}}_{e}$.
We note that here $\alpha^{\rm{B}}_{a}$ defines the rate of change of angular
momentum ($m\mu/\gamma$). It differs from the definition of intrinsic damping
parameters in ASD, which are related to the rate of change of the
magnetization ($m$). Similarly to ASD methods though, within the Baryakhtar
model the observed fast dynamics of the Fe sublattice is related to a smaller
value of atomic magnetic moment.

The switching process within the Baryakhtar-like model is explained in the
following manner. Since the Fe sublattice reacts faster than Gd to heating it
is expected to remain closer to thermal equilibrium with the heat-bath. This
translates into a smaller non-equilibrium effective field acting on Fe than in
Gd, $H_{\rm{Fe}}\ll H_{\rm{Gd}}$, during the action of the laser pulse. For
strong enough pulses, the Fe magnetization rapidly reduces,
$m_{\rm{Fe}}\approx 0$, still $H_{\rm{Fe}}$ is small in comparison to
$H_{\rm{Gd}}$, in a way that the dynamics of Fe can be fairly approximated by
$\dot{m}_{\rm{Fe}}\approx\alpha^{\rm{B}}_{e}H_{\rm{Gd}}$. This drives the
magnetization of Fe towards the opposite direction. The field, $H_{\rm{Gd}}$
is defined by the energy of the system, $H^{\rm{MFA}}_{\rm{Gd}}$ (Eq. (6)) and
$\alpha^{\rm{B}}_{e}$ from the coupling between the Gd and the Fe sublattices.
After switching, $H_{\rm{Fe}}\approx H_{\rm{Gd}}$ and relativistic relaxation
processes dominate the dynamics and drive magnetization to complete the
switching. The question here is to what extent the non-equilibrium fields as
given by Eq. (11) are accurate, and how are the relaxation parameters related
to atomic damping parameters in ASD.

Figure 3: Element-specific magnetization dynamics of GdFeCo calculated using
atomistic spin dynamics (symbols) and macroscopic Baryakhtar-like equation
(solid lines) for two laser pulse power values, (a) $P_{0}$ and (b) $2P_{0}$.
Both electron and lattice temperature are kept constant, $T=300$ K, for $t<0$.
At $t=0$ a laser pulse is applied. In the Baryakhtar-like model the
relativistic relaxation parameters $\alpha^{\rm{B}}_{a}$ have a value
different to the Gilbert damping in ASD simulations,
$(\gamma/\mu_{\rm{Fe}})\alpha^{\rm{B}}_{\rm{Fe}}=0.005$ and
$(\gamma/\mu_{\rm{Gd}})\alpha^{\rm{B}}_{\rm{Gd}}=0.01$. The exchange
relaxation parameter is varied,
$\alpha^{\rm{B}}_{e}/\alpha^{\rm{B}}_{\rm{Fe}}=0,0.3$ and 3. The relaxation to
thermal state ($t<0$) is only well described for the Fe sublattice. (a) For
$P_{0}$, the laser induced dynamics is well described by
$\alpha^{\rm{B}}_{e}/\alpha^{\rm{B}}_{\rm{Fe}}=0.1$. (b) For $2P_{0}$ the
demagnetization phase of both sublattices is relatively well described in
comparison to ASD simulations. Switching is also possible, here one instance,
for a value $\alpha^{\rm{B}}_{e}/\alpha^{\rm{B}}_{\rm{Fe}}=3$.

So far the connection between the relaxation parameters in the ASD and
Baryakhtar-like model is unknown. In ASD simulations shown in Fig. 2 we have
used $\lambda_{\rm{Fe}}=\lambda_{\rm{Gd}}=0.01$ as atomistic relaxation
parameter. One would expect that the relaxation parameters in the atomistic
and macroscopic models are related as
$\lambda_{a}\approx\alpha^{\rm{B}}_{a}(\gamma_{a}/\mu_{a})$. In an attempt to
find this correspondence, we directly compare results from ASD simulations and
Baryakhtar-like models for different values of $\alpha^{\rm{B}}_{a}$ and
$\alpha^{\rm{B}}_{e}$ in Eqs. (9) and (10). We numerically solve Eqs.
(9),(10), and (11) coupled to the 2TM with exactly the same parameters as for
the ASD simulations. After exploring the results of the Baryakhtar model for a
range of values for $\alpha^{\rm{B}}_{a}$ and $\alpha_{e}$, we find that for
some values the agreement is good, as one observes in Fig. 3, however, it is
not possible to find a good match for all scenarios.

In order to illustrate this, we first focus on the dynamics induced by the
laser pulse with power $P_{0}$ (Fig. (3)(a)). We find a good match for the
laser induced magnetization dynamics ($t>0$ for
$(\gamma/\mu_{\rm{Fe}})\alpha_{\rm{Fe}}=0.005$ and
$(\gamma/\mu_{\rm{Gd}})\alpha_{\rm{Gd}}=0.01$, and for values of exchange
relaxation of up to $\alpha^{\rm{B}}_{e}/\alpha^{\rm{B}}_{\rm{Fe}}=0.3$. For
values $\alpha^{\rm{B}}_{e}/\alpha^{\rm{B}}_{\rm{Fe}}<0.3$, thermal relaxation
($t<0$) of the Fe is also well described, however the relaxation of the Gd
sublattice is significantly faster. For larger values of the exchange
relaxation $\alpha^{\rm{B}}_{e}/\alpha^{\rm{B}}_{\rm{Fe}}=3$, the dynamics of
both sublatttices are substantially speed up and strongly disagree with ASD
simulations.

For larger laser pulse power $2P_{0}$ the magnetization switches using ASD
simulations. We keep the same values for the relaxation parameters in
Baryakhtar-like model as for $P_{0}$, and compare to the ASD simulations. For
small values of $\alpha^{\rm{B}}_{e}$ (Fig. (3)(b)), differently to the
$P_{0}$ case (Fig. (3)(a)), the dynamics described by the Baryakhtar-like
model is not only slower than those of ASD simulations but it hardly
reproduces magnetization switching. In order to reproduce switching, we need
to use larger values of the exchange relaxation parameter,
$\alpha^{\rm{B}}_{e}/\alpha^{\rm{B}}_{\rm{Fe}}=3$. These findings are in
agreement with previous works using Baryakhtar-like model where switching was
reproduced for comparable values of $\alpha^{\rm{B}}_{e}$. However, as we have
discussed before, for those values of $\alpha^{\rm{B}}_{e}$, thermal
relaxation dynamics ($t<0$) is much faster than in ASD simulations. This
brings us to the question of how much understanding about switching can we
gain by using this bare Baryakhtar-like model, are we missing something?

###  III.2 The Landau-Lifshitz-Bloch model

Since the Baryakhtar-like model is based on symmetry arguments, the
macroscopic magnetization dynamics coming out from ASD simulations should also
be described by that model with adequate expression for the relaxation
parameters and non-equilibrium effective fields. The magnetization dynamics
coming out from ASD simulations is well described by the LLB equation of
motion.

$\frac{dm_{a}}{dt}=\Gamma_{\|,a}\left(m_{a}-m_{0,a}\right),$ (12)

where

$\Gamma_{\|,a}=2\lambda_{a}\frac{\gamma}{\mu_{a}}k_{B}T\frac{1}{\xi_{a}}\frac{L(\xi_{a})}{L^{\prime}(\xi_{a})},$
(13)

with $\xi_{a}=\beta\mu_{a}H_{a}^{\rm{MFA}}$, where $H_{a}^{\rm{MFA}}$ is given
in Eq. (6), and $m_{0,a}=L(\xi_{a})$. The same equation applies to the second
sublattice $b$. Here, the relaxation rate $\Gamma_{\|,a}$ depends non-linearly
on the non-equilibrium sublattice magnetization, $m_{a(b)}$, through the
parameter $\xi_{a}$. We note that Eq. (12) can be expanded around equilibrium
for small perturbations of the magnetization. By doing so, the relaxation
rates and effective fields are expressed in terms of equilibrium properties
such as equilibrium magnetization and zero-field susceptibilities [20]. In the
present work, however, we use the version in Eq. (12).

Figure 4: Element-specific magnetization dynamics of GdFeCo calculated using
atomistic spin dynamics (symbols) and macroscopic LLB equation (solid lines)
for two laser pulse power values, (a) $P_{0}$ and (b) $2P_{0}$. For $t<0$,
electron and lattice temperature are $T=300$K, and at $t=0$ a laser pulse is
applied. The exchange relaxation parameter is varied,
$\alpha_{e}/\alpha_{a}=0,0.1$ and 1, where $\alpha_{a}=0.01$, and $a=$FeCo or
Gd. The initial relaxation dynamics is well described by
$\alpha_{e}/\alpha_{a}=0$. (a) For laser power $P_{0}$, the element-specific
dynamics is well-described for $\alpha_{e}/\alpha_{a}=0.1$. (a) For
$\alpha_{e}/\alpha_{a}=1$, exchange relaxation dominates and the element-
specific dynamics are similar. (b) For laser power $2P_{0}$, the switching
dynamics is not described by the LLB model.

Direct comparison between ASD simulations and the LLB model of element-
specific magnetization dynamics is possible and with relatively good
agreement. Importantly, since the LLB model is derived directly from the ASD
microscopic model, the damping parameters, $\lambda_{a(b)}$ in Eqs. (13) and
(2) stand for the same physics, the rate of angular momentum dissipation of
the atomic spins. Differently to the Baraykhtar model where
$\alpha_{a(b)}^{\rm{B}}$ is taken as a fitting parameter, within the LLB model
the value of $\lambda_{a(b)}$ in Eq. (13) is the same as in the ASD
simulations. A key difference between the Baryakhtar-like model and the LLB
model is that in the latter an exchange relaxation term is missing. In order
to find a meeting point between these phenomenological models, we rewrite Eq.
(12) in terms of a damping term multiplied by an effective field,

$\frac{dm_{a}}{dt}=\frac{2\lambda_{a}L(\xi_{a})}{\xi_{a}}\frac{\gamma}{\mu_{a}}\frac{m_{a}-m_{0,a}}{\beta
L^{\prime}(\xi_{a})}=\gamma\alpha_{a}H_{a},$ (14)

where

$\alpha_{a}=2\lambda_{a}\frac{L(\xi_{a})}{\xi_{a}}.$ (15)

Differently to Baryakhtar-like model, in the LLB model, the relaxation
parameter strongly depends on temperature and non-equilibrium sublattice
magnetization through the thermal field,
$\xi_{a}=\beta\mu_{a}H_{a}^{\rm{MFA}}$. At the same time, the non-equilibrium
fields $\mu_{a}H_{a}$ within the LLB and Baryakhtar-like models differ. The
effective field in the LLB model is defined as

$\mu_{a}H_{a}=\frac{(m_{a}-m_{0,a})}{\beta L^{\prime}(\xi_{a})}.$ (16)

Equation (16) provides a microscopic description of the effective field
driving the magnetization dynamics in ferrimagnets, based on the Heisenberg
spin model (Eq. (1)). Under the assumption of small perturbations around the
equilibrium both, LLB and Baryakhtar-like effective fields, simplify to
Landau-like expressions [19]. Equation (14) describes with a very good degree
of accuracy the relaxation of the angular momentum via dissipation to the
heat-bath, which corresponds to the relativistic term in Eqs. (9) and (10).
Previously, it has been found that ASD simulations compare well to Eq. (14)
for coupling parameters of around $\lambda_{a}\approx 0.1-1$ [20, 42]. These
values can be considered to correspond to the intermediate-to-high coupling
regime. Direct comparison between ASD simulations and experiments of single
pulse switching in GdFeCo has suggested values of $\lambda_{\rm{Fe}}\approx
0.06$ and $\lambda_{\rm{Gd}}\approx 0.01$ [16]. In the context of the present
work we find that Eq. (14) describes relatively well the thermal relaxation
dynamics in direct comparison to ASD simulations (Fig. (4)).

In order to account for the exchange relaxation in the LLB model, we follow
the Baryakhtar-like model ((9) and (10)), and add an exchange relaxation term
to Eq. (14),

$\frac{\mathrm{d}m_{a}}{\mathrm{d}t}=\gamma\alpha_{a}H_{a}+\gamma\frac{\alpha_{e}}{\mu_{a}}(\mu_{a}H_{a}-\mu_{b}H_{b})$
(17)

where $\alpha_{e}$ is a phenomenological exchange relaxation parameter to be
determined by comparison to ASD dynamics. The inclusion of the exchange
relaxation (second term in r.h.s) in the LLB improves the agreement to ASD
simulations. With this addition, the LLB model describes well thermal
relaxation for small values of the ratio $\alpha_{e}/\alpha_{a}$ as
demonstrated in Fig. 4. For large values of $\alpha_{e}$ the LLB model is
unable to describe thermal relaxation dynamics ($t<0$ in Fig. 4(a) and (b)).
For laser power $P_{0}$ (Fig. 4(a) ($t>0$)) the magnetization dynamics is
slightly slower using the LLB model than those gained by ASD simulations for
$\alpha_{e}/\alpha_{a}=0$. For $\alpha_{e}/\alpha_{a}=0.1$, the agreement is
even better than without exchange relaxation. The agreement vanishes when the
exchange relaxation is increased to $\alpha_{e}/\alpha_{a}=1$. Critically,
when the laser power is increased from $P_{0}$ to $2P_{0}$, for which ASD
simulations show ultrafast switching, the LLB model only shows
demagnetization-remagnetization of both sublattices. We find some agreement on
the demagnetization time scales when a quite large exchange relaxation is
used, $\alpha_{e}/\alpha_{a}=1$. These dynamics are similar to those observed
using the Baryakhtar-like model for intermediate values of the exchange
relaxation parameter (Fig. (3)). It has been demonstrated previously that by
including the transverse components of the equation of motion, switching is
possible via a precessional path when a canting between the magnetization of
each sublattice exists [21]. Here, we restrict to purely longitudinal
switching within the LLB model.

###  III.3 Unified phenomenological model

Figure 5: Element-specific magnetization dynamics of GdFeCo calculated using
atomistic spin dynamics (symbols) and the unified phenomenological model
derived here, following Eq. (17) (solid lines) for two laser pulse power
values, (a) $P_{0}$ and (b) $2P_{0}$. Both electron and lattice temperature
are kept constant, $T=300$ K, for $t<0$. At $t=0$ a laser pulse is applied,
the same as in Figure (2). GdFeCo alloys with $x_{\rm{Gd}}=25\%$ are
calculated.

So far we have constructed a phenomenological model based on the LLB and
Baryakhtar-like models, the dynamics is given by Eq. (17), the effective field
by Eq. (16) and the relativistic relaxation parameter Eq. (15). We still need
an expression for the exchange relaxation parameter. We construct this
expression starting with single species ferromagnets, where sublattices $a$
and $b$ represent the same spin lattice, hence exchange of angular momentum is
non-local. Therefore,
$\mu_{a}H_{a}-\mu_{b}H_{b}=\mu_{a}H_{\rm{ex}}a_{0}^{2}\Delta m_{a}$, with
$a_{0}$ representing the lattice constant. Hence, the rate of non-local
angular momentum transfer reads $\Gamma_{\rm{ex}}^{\rm{non-
loc.}}=\alpha_{\rm{ex}}(\mu_{a}H_{a}-\mu_{b}H_{b})=\alpha_{a}(A/M_{a}(T))\Delta
m_{a}$, where $A$ is the so-called micromagnetic exchange stiffness [58].
$M_{a}(T)=(\mu_{a}/\upsilon_{a})m_{a}$ is the magnetization density at
temperature $T$, where $\upsilon_{a}$ is the unit cell volume. Therefore, we
find that $\alpha_{\rm{ex}}=\alpha_{a}/(zm_{a})$. By considering that the
exchange relaxation rate should conserve the symmetry under the exchange of
lattice index, $\alpha_{ex}(M_{1},M_{2})=\alpha_{ex}(M_{2},M_{1})$, we find
that

$\alpha_{\rm{ex}}=\frac{1}{2}\left(\frac{\alpha_{a}}{z_{ab}m_{a}}+\frac{\alpha_{b}}{z_{ba}m_{b}}\right).$
(18)

This expression is the extension of the non-local exchange relaxation in
ferromagnets to local exchange relaxation in ferrimagnets. This explicit
expression for the exchange relaxation parameter in Eq. (18) completes our
unified model, which bridges the atomistic spin dynamics model and the
Baryakhtar and LLB macroscopic models.

We find that the agreement between our unified phenomenological model and ASD
simulations is excellent, see Fig. (5)(a) and (b). Figure 5(a) shows that for
$t<0$, the sublattice magnetization relaxation towards thermal equilibrium
value is described with a high level of accuracy by our model. For $t>0$ and a
relatively low laser power $P_{0}$, the agreement is also excellent for the
demagnetization and remagnetization dynamics. Figure 5(b) shows the comparison
between the unified model and ASD simulations of the switching dynamics. We
conclude that Eq. (17) for the sublattice magnetization dynamics together with
the Eq. (16) for the effective field and Eqs. (15) and (18) for the relaxation
parameters, unify the Barayakhtar and the LLB phenomenological models for
single-pulse all-optical switching in ferrimagnets.

##  IV Discussion and conclusion

The macroscopic model presented in this work solves some open questions in the
field of ultrafast magnetization dynamics in ferrimagnets. For example, it
answers the question of the range of applicability and the validity of the
parameters of the Barayakhtar and LLB phenomenological models. In the one
hand, within our model, the relativistic relaxation parameters ($\alpha_{a}$)
are element-specific and strongly depend on both the temperature and the non-
equilibrium sublattice magnetization. The temperature and magnetization
dependence of the relativistic relaxation parameters are well described by the
LLB model. In the other hand, the exchange relaxation parameter
($\alpha_{\rm{ex}}$) is cast in terms of the element specific relativistic
relaxation parameters and sublattice magnetization. We have demonstrated that
in order to reproduce the ASD simulations results, the relaxation parameters
in the Barayakhtar model have to be both temperature and magnetization
dependent. The explicit expression of the exchange relaxation parameter is the
main result of the present work since it allows us to unify the Barayakhtar
and LLB models. While for the Barayakhtar model $\alpha_{e}$ is unconnected to
$\alpha_{a}$, within our proposed model they are proportional to each other,
$\alpha_{e}\sim\alpha_{a}/m_{a}$. This relation is the key to bridge both ASD
simulations and Barayakhtar and LLB models together. Additionally, we have
also demonstrated the validity of the non-equilibrium effective fields given
in Eq. (16) as derived in the LLB model instead of the Barayakhtar model.

Single-pulse switching in ferrimagnets has been described before by the
Baryakhtar model. A necessary condition for switching is that the system
transits from the relativistic relaxation regime to the so-called exchange-
dominated relaxation regime. Although details of switching in such a regime
have been already discussed in detail [5, 19], so far it has remained unknown
how this transition could be described theoretically. Our model resolves this
question. When the system is at equilibrium or weakly excited, the exchange-
relaxation parameter fulfills, $\alpha_{e}\ll\alpha_{a}$. For strong
excitation, such that the magnetic order of one sublattice reduces
significantly, close to zero $m_{a}\rightarrow 0$, the exchange relaxation
will dominate the dynamics since
$\alpha_{e}\sim\alpha_{a}/m_{a}\gg\alpha_{a}$. From our model, one can derive
universal criteria for switching in ferrimagnets, including GdFeCo and
Mn2RuxGa [59].

The provided understanding is paramount for further research on material
engineering, for example, to find alternative material classes showing all-
optical switching. Notably, our model predicts that the exchange relaxation
term is enhanced as the number of neighbours reduces. This dependence suggests
that magnetic systems of lower dimension, e.g. 2D magnets [60], could show a
faster, more efficient switching than bulk materials. Further, the extension
of our model to the micromagnetic level will allow to optimize switching
conditions. The use of micromagnetic computational solvers permits for a
realistic description of ultrafast AOS processes in ferrimagnetic alloys, such
as helicity-independent and helicity-dependent AOS, where multidomain states
and thermal gradients play an important role in the process [46].

To summarize, in the present work we have presented a unified model for
single-pulse all-optical switching in ferrimagnets. Our model merges and
improves previous semi-phenomenological models, the Landau-Lifshitz-Bloch
model and Barayakhtar-like models. To verify the accuracy of the proposed
model, we directly compare the laser induced magnetization dynamics to
atomistic spin dynamics computer simulations. Differently to previous models,
our model has the advantage that it can be directly compared to ASD
simulations. Further, we have established the connection between ASD and
macroscopic equations of motion. Importantly, we provide here the stepping
stone for the construction of a micromagnetic model valid for ferrimagnets
including exchange relaxation between sublattices. This is paramount for a
robust construction of a multiscale scheme of the switching process in which
not only local magnetization dynamics is described but also magnetic domain
nucleation and motion under strong non-equilibrium. Multiscale-based
micromagnetic models will allow for the description of realistic sample sizes
and describe recent spintronics phenomena using laser pulses, e.g. magnetic
skyrmion creation/deletion with fs laser pulses, or domain-wall motion under
dynamics thermal gradients.

###### Acknowledgements.

The authors acknowledge support from the Deutsche Forschungsgemeinschaft
through SFB/TRR 227 ”Ultrafast Spin Dynamics”, Project A08.

## References

  * Stanciu  _et al._ [2007] C. Stanciu, F. Hansteen, A. Kimel, A. Kirilyuk, A. Tsukamoto, A. Itoh, and T. Rasing, All-Optical Magnetic Recording with Circularly Polarized Light, Physical Review Letters 99, 047601 (2007).
  * Vahaplar  _et al._ [2009] K. Vahaplar, A. Kalashnikova, A. V. Kimel, D. Hinzke, U. Nowak, R. Chantrell, A. Tsukamoto, A. Itoh, A. Kirilyuk, and T. Rasing, Ultrafast Path for Optical Magnetization Reversal via a Strongly Nonequilibrium State, Physical Review Letters 103, 117201 (2009).
  * Radu  _et al._ [2011] I. Radu, K. Vahaplar, C. Stamm, T. Kachel, N. Pontius, H. A. Dürr, T. A. Ostler, J. Barker, R. F. L. Evans, R. W. Chantrell, A. Tsukamoto, A. Itoh, A. Kirilyuk, T. Rasing, A. V. Kimel, and H. A. Dürr, Transient ferromagnetic-like state mediating ultrafast reversal of antiferromagnetically coupled spins., Nature 472, 205 (2011).
  * Ostler  _et al._ [2012] T. Ostler, J. Barker, R. Evans, R. Chantrell, U. Atxitia, O. Chubykalo-Fesenko, S. El Moussaoui, L. Le Guyader, E. Mengotti, L. Heyderman, F. Nolting, A. Tsukamoto, A. Itoh, D. Afanasiev, B. Ivanov, A. Kalashnikova, K. Vahaplar, J. Mentink, A. Kirilyuk, T. Rasing, and A. Kimel, Ultrafast heating as a sufficient stimulus for magnetization reversal in a ferrimagnet, Nature Communications 3, 666 (2012).
  * Mentink  _et al._ [2012] J. H. Mentink, J. Hellsvik, D. V. Afanasiev, B. A. Ivanov, A. Kirilyuk, A. V. Kimel, O. Eriksson, M. I. Katsnelson, and T. Rasing, Ultrafast Spin Dynamics in Multisublattice Magnets, Physical Review Letters 108, 057202 (2012).
  * Le Guyader  _et al._ [2012] L. Le Guyader, S. El Moussaoui, M. Buzzi, R. V. Chopdekar, L. J. Heyderman, a. Tsukamoto, A. Itoh, A. Kirilyuk, T. Rasing, A. V. Kimel, and F. Nolting, Demonstration of laser induced magnetization reversal in GdFeCo nanostructures, Applied Physics Letters 101, 022410 (2012).
  * Graves  _et al._ [2013] C. E. Graves, A. H. Reid, T. Wang, B. Wu, S. de Jong, K. Vahaplar, I. Radu, D. P. Bernstein, M. Messerschmidt, L. Müller, R. Coffee, M. Bionta, S. W. Epp, R. Hartmann, N. Kimmel, G. Hauser, A. Hartmann, P. Holl, H. Gorke, J. H. Mentink, A. Tsukamoto, A. Fognini, J. J. Turner, W. F. Schlotter, D. Rolles, H. Soltau, L. Strüder, Y. Acremann, A. V. Kimel, A. Kirilyuk, T. Rasing, J. Stöhr, A. O. Scherz, and H. A. Dürr, Nanoscale spin reversal by non-local angular momentum transfer following ultrafast laser excitation in ferrimagnetic GdFeCo., Nature Materials 12, 293 (2013).
  * Barker  _et al._ [2013] J. Barker, U. Atxitia, T. A. Ostler, O. Hovorka, R. W. Chantrell, O. Chubykalo-Fesenko, and R. W. Chantrell, Two-magnon bound state causes ultrafast thermally induced magnetisation switching., Scientific Reports 3, 3262 (2013).
  * Baryakhtar  _et al._ [2013] V. G. Baryakhtar, V. I. Butrim, and B. A. Ivanov, Exchange relaxation as a mechanism of the ultrafast reorientation of spins in a two-sublattice ferrimagnet, JETP Letters 98, 289 (2013).
  * Gridnev [2016] V. N. Gridnev, Ultrafast heating-induced magnetization switching in ferrimagnets, Journal of Physics: Condensed Matter 28, 476007 (2016).
  * Schellekens and Koopmans [2013] A. J. Schellekens and B. Koopmans, Microscopic model for ultrafast magnetization dynamics of multisublattice magnets, Phys. Rev. B 87, 020407(R) (2013).
  * Mangin  _et al._ [2014] S. Mangin, M. Gottwald, C.-H. Lambert, D. Steil, V. Uhlír, L. Pang, M. Hehn, S. Alebrand, M. Cinchetti, G. Malinowski, Y. Fainman, M. Aeschlimann, and E. E. Fullerton, Engineered materials for all-optical helicity-dependent magnetic switching., Nature materials 13, 286 (2014).
  * Ostler  _et al._ [2011] T. A. Ostler, R. F. L. Evans, R. W. Chantrell, U. Atxitia, O. Chubykalo-Fesenko, I. Radu, R. Abrudan, F. Radu, A. Tsukamoto, A. Itoh, A. Kirilyuk, T. Rasing, and A. V. Kimel, Crystallographically amorphous ferrimagnetic alloys: Comparing a localized atomistic spin model with experiments, Physical Review B 84, 024407 (2011).
  * Wienholdt  _et al._ [2013] S. Wienholdt, D. Hinzke, K. Carva, P. M. Oppeneer, and U. Nowak, Orbital-resolved spin model for thermal magnetization switching in rare-earth-based ferrimagnets, Physical Review B 88, 020406(R) (2013).
  * Chimata  _et al._ [2015] R. Chimata, L. Isaeva, K. Kádas, A. Bergman, B. Sanyal, J. H. Mentink, M. I. Katsnelson, T. Rasing, A. Kirilyuk, A. Kimel, O. Eriksson, and M. Pereiro, All-thermal switching of amorphous Gd-Fe alloys: Analysis of structural properties and magnetization dynamics, Physical Review B 92, 094411 (2015).
  * Jakobs  _et al._ [2021] F. Jakobs, T. A. Ostler, C.-H. Lambert, Y. Yang, S. Salahuddin, R. B. Wilson, J. Gorchon, J. Bokor, and U. Atxitia, Unifying femtosecond and picosecond single-pulse magnetic switching in Gd-Fe-Co, Physical Review B 103, 104422 (2021).
  * Ceballos  _et al._ [2021] A. Ceballos, A. Pattabi, A. El-Ghazaly, S. Ruta, C. P. Simon, R. F. L. Evans, T. Ostler, R. W. Chantrell, E. Kennedy, M. Scott, J. Bokor, and F. Hellman, Role of element-specific damping in ultrafast, helicity-independent, all-optical switching dynamics in amorphous (Gd,Tb)Co thin films, Physical Review B 103, 024438 (2021).
  * Radu  _et al._ [2015] I. Radu, C. Stamm, A. Eschenlohr, F. Radu, R. Abrudan, K. Vahaplar, T. Kachel, N. Pontius, R. Mitzner, K. Holldack, A. Föhlisch, T. A. Ostler, J. H. Mentink, R. F. L. Evans, R. W. Chantrell, A. Tsukamoto, A. Itoh, A. Kirilyuk, A. V. Kimel, and T. Rasing, Ultrafast and Distinct Spin Dynamics in Magnetic Alloys, SPIN 5, 1550004 (2015).
  * Mentink [2017] J. H. Mentink, Manipulating magnetism by ultrafast control of the exchange interaction, Journal of Physics: Condensed Matter 29, 453001 (2017).
  * Atxitia  _et al._ [2012] U. Atxitia, P. Nieves, and O. Chubykalo-Fesenko, Landau-Lifshitz-Bloch equation for ferrimagnetic materials, Physical Review B 86, 104414 (2012).
  * Atxitia  _et al._ [2013] U. Atxitia, T. A. Ostler, J. Barker, R. F. L. Evans, R. W. Chantrell, and O. Chubykalo-Fesenko, Ultrafast dynamical path for the switching of a ferrimagnet after femtosecond heating, Physical Review B 87, 224417 (2013).
  * Atxitia  _et al._ [2014] U. Atxitia, J. Barker, R. W. Chantrell, and O. Chubykalo-Fesenko, Controlling the polarity of the transient ferromagneticlike state in ferrimagnets, Phys. Rev. B 89, 224421 (2014).
  * Zahn  _et al._ [2021] D. Zahn, F. Jakobs, Y. W. Windsor, H. Seiler, T. Vasileiadis, T. A. Butcher, Y. Qi, D. Engel, U. Atxitia, J. Vorberger, and R. Ernstorfer, Lattice dynamics and ultrafast energy flow between electrons, spins, and phonons in a 3d ferromagnet, Physical Review Research 3, 023032 (2021).
  * Zahn  _et al._ [2022] D. Zahn, F. Jakobs, H. Seiler, T. A. Butcher, D. Engel, J. Vorberger, U. Atxitia, Y. W. Windsor, and R. Ernstorfer, Intrinsic energy flow in laser-excited $3d$ ferromagnets, Phys. Rev. Research 4, 013104 (2022).
  * Frietsch  _et al._ [2015] B. Frietsch, J. Bowlan, R. Carley, M. Teichmann, S. Wienholdt, D. Hinzke, U. Nowak, K. Carva, P. M. Oppeneer, and M. Weinelt, Disparate ultrafast dynamics of itinerant and localized magnetic moments in gadolinium metal, Nature Communications 6, 8262 (2015).
  * Frietsch  _et al._ [2021] B. Frietsch, A. Donges, R. Carley, M. Teichmann, J. Bowlan, K. Döbrich, K. Carva, D. Legut, P. M. Oppeneer, U. Nowak, and M. Weinelt, The role of ultrafast magnon generation in the magnetization dynamics of rare-earth metals, Science Advances 6, eabb1601 (2021).
  * Iacocca  _et al._ [2019] E. Iacocca, T.-M. M. Liu, A. H. Reid, Z. Fu, S. Ruta, P. W. Granitzka, E. Jal, S. Bonetti, A. X. Gray, C. E. Graves, R. Kukreja, Z. Chen, D. J. Higley, T. Chase, L. Le Guyader, K. Hirsch, H. Ohldag, W. F. Schlotter, G. L. Dakovski, G. Coslovich, M. C. Hoffmann, S. Carron, A. Tsukamoto, A. Kirilyuk, A. V. Kimel, T. Rasing, J. Stöhr, R. F. L. Evans, T. Ostler, R. W. Chantrell, M. A. Hoefer, T. J. Silva, and H. A. Dürr, Spin-current-mediated rapid magnon localisation and coalescence after ultrafast optical pumping of ferrimagnetic alloys., Nat Commun 10, 1756 (2019).
  * Atxitia  _et al._ [2015] U. Atxitia, T. A. Ostler, R. W. Chantrell, and O. Chubykalo-Fesenko, Optimal electron, phonon, and magnetic characteristics for low energy thermally induced magnetization switching, Applied Physics Letters 107, 192402 (2015).
  * Atxitia and Ostler [2018] U. Atxitia and T. A. Ostler, Ultrafast double magnetization switching in GdFeCo with two picosecond-delayed femtosecond pump pulses, Applied Physics Letters 113, 62402 (2018).
  * Xu  _et al._ [2016] C. Xu, T. A. Ostler, and R. W. Chantrell, Thermally induced magnetization switching in Gd/Fe multilayers, Physical Review B 93, 054302 (2016).
  * Gerlach  _et al._ [2017] S. Gerlach, L. Oroszlany, D. Hinzke, S. Sievering, S. Wienholdt, L. Szunyogh, and U. Nowak, Modeling ultrafast all-optical switching in synthetic ferrimagnets, Physical Review B 95, 224435 (2017).
  * Banerjee  _et al._ [2020] C. Banerjee, N. Teichert, K. E. Siewierska, Z. Gercsi, G. Y. P. Atcheson, P. Stamenov, K. Rode, J. M. D. Coey, and J. Besbas, Single pulse all-optical toggle switching of magnetization without gadolinium in the ferrimagnet Mn2RuxGa, Nature Communications 11, 4444 (2020).
  * Banerjee  _et al._ [2021] C. Banerjee, K. Rode, G. Atcheson, S. Lenne, P. Stamenov, J. M. D. Coey, and J. Besbas, Ultrafast double pulse all-optical reswitching of a ferrimagnet, Phys. Rev. Lett. 126, 177202 (2021).
  * Jakobs and Atxitia [2022a] F. Jakobs and U. Atxitia, Atomistic spin model of single pulse toggle switching in Mn2RuxGa Heusler alloys, Applied Physics Letters 120, 172401 (2022a).
  * Baryakhtar and Danilevich [2013] V. G. Baryakhtar and A. G. Danilevich, The phenomenological theory of magnetization relaxation (review article), Low Temperature Physics 39, 993 (2013).
  * Lifshits and Landau [1935] E. Lifshits and L. Landau, On the theory of dispersion of magnetic permeabilty in ferromagnetic bodies, Phys. Zeitsch. der Sow. 8 (1935).
  * Kamra  _et al._ [2018] A. Kamra, R. E. Troncoso, W. Belzig, and A. Brataas, Gilbert damping phenomenology for two-sublattice magnets, Phys. Rev. B 98, 184402 (2018).
  * Davies  _et al._ [2020a] C. S. Davies, T. Janssen, J. H. Mentink, A. Tsukamoto, A. V. Kimel, A. van der Meer, A. Stupakiewicz, and A. Kirilyuk, Pathways for single-shot all-optical switching of magnetization in ferrimagnets, Phys. Rev. Applied 13, 024064 (2020a).
  * Davies  _et al._ [2020b] C. S. Davies, G. Bonfiglio, K. Rode, J. Besbas, C. Banerjee, P. Stamenov, J. M. D. Coey, A. V. Kimel, and A. Kirilyuk, Exchange-driven all-optical magnetic switching in compensated $3d$ ferrimagnets, Phys. Rev. Research 2, 032044(R) (2020b).
  * Garanin [1997] D. A. Garanin, Fokker-Planck and Landau-Lifshitz-Bloch equations for classical ferromagnets, Physical Review B 55, 3050 (1997).
  * Chubykalo-Fesenko  _et al._ [2006] O. Chubykalo-Fesenko, U. Nowak, R. W. Chantrell, and D. Garanin, Dynamic approach for micromagnetics close to the Curie temperature, Physical Review B 74, 094436 (2006).
  * Nieves  _et al._ [2014] P. Nieves, D. Serantes, U. Atxitia, and O. Chubykalo-Fesenko, Quantum Landau-Lifshitz-Bloch equation and its comparison with the classical case, Physical Review B 90, 104428 (2014).
  * Atxitia  _et al._ [2016] U. Atxitia, D. Hinzke, and U. Nowak, Fundamentals and applications of the Landau–Lifshitz–Bloch equation, Journal of Physics D: Applied Physics 50, 33003 (2016).
  * Hennecke  _et al._ [2019] M. Hennecke, I. Radu, R. Abrudan, T. Kachel, K. Holldack, R. Mitzner, A. Tsukamoto, and S. Eisebitt, Angular Momentum Flow During Ultrafast Demagnetization of a Ferrimagnet, Physical Review Letters 122, 157202 (2019).
  * Vogler  _et al._ [2019] C. Vogler, C. Abert, F. Bruckner, and D. Suess, Stochastic ferrimagnetic Landau-Lifshitz-Bloch equation for finite magnetic structures, Physical Review B 100, 054401 (2019).
  * Raposo  _et al._ [2022] V. Raposo, F. García-Sánchez, U. Atxitia, and E. Martínez, Realistic micromagnetic description of all-optical ultrafast switching processes in ferrimagnetic alloys, Phys. Rev. B 105, 104432 (2022).
  * Schlickeiser  _et al._ [2014] F. Schlickeiser, U. Ritzmann, D. Hinzke, and U. Nowak, Role of Entropy in Domain Wall Motion in Thermal Gradients, Physical Review Letters 113, 097201 (2014).
  * Moretti  _et al._ [2017] S. Moretti, V. Raposo, E. Martinez, and L. Lopez-Diaz, Domain wall motion by localized temperature gradients, Physical Review B 95, 064419 (2017).
  * Lepadatu [2020] S. Lepadatu, Emergence of transient domain wall skyrmions after ultrafast demagnetization, Physical Review B 102, 94402 (2020).
  * Barker and Atxitia [2021] J. Barker and U. Atxitia, A Review of Modelling in Ferrimagnetic Spintronics, Journal of the Physical Society of Japan 90, 81001 (2021).
  * Chimata  _et al._ [2012] R. Chimata, A. Bergman, L. Bergqvist, B. Sanyal, and O. Eriksson, Microscopic Model for Ultrafast Remagnetization Dynamics, Physical Review Letters 109, 157201 (2012).
  * Nowak [2007] U. Nowak, Classical Spin Models in Handbook of Magnetism and Advanced Magnetic Materials (John Wiley and Sons, Ltd 2007) (2007).
  * Atxitia  _et al._ [2009] U. Atxitia, O. Chubykalo-Fesenko, R. W. Chantrell, U. Nowak, and A. Rebei, Ultrafast Spin Dynamics: The Effect of Colored Noise, Physical Review Letters 102, 057203 (2009).
  * Hinzke  _et al._ [2015] D. Hinzke, U. Atxitia, K. Carva, P. Nieves, O. Chubykalo-Fesenko, P. M. Oppeneer, and U. Nowak, Multiscale modeling of ultrafast element-specific magnetization dynamics of ferromagnetic alloys, Physical Review B 92, 054412 (2015).
  * Yang  _et al._ [2017] Y. Yang, R. B. Wilson, J. Gorchon, C.-H. Lambert, S. Salahuddin, and J. Bokor, Ultrafast magnetization reversal by picosecond electrical pulses, Science Advances 3, e1603117 (2017).
  * Kaganov  _et al._ [1957] M. I. Kaganov, I. M. Lifshitz, and L. V. Tanatarov, Relaxation between electrons and crystalline lattices, JETP 173 (1957).
  * Chen  _et al._ [2006] J. Chen, D. Tzou, and J. Beraun, A semiclassical two-temperature model for ultrafast laser heating, International Journal of Heat and Mass Transfer 49, 307 (2006).
  * Atxitia  _et al._ [2010] U. Atxitia, O. Chubykalo-Fesenko, J. Walowski, A. Mann, and M. Münzenberg, Evidence for thermal mechanisms in laser-induced femtosecond spin dynamics, Physical Review B 81, 174401 (2010).
  * Jakobs and Atxitia [2022b] F. Jakobs and U. Atxitia, Universal criteria for single femtosecond pulse ultrafast magnetization switching in ferrimagnets, arXiv:2201.09067 (2022b).
  * Wang  _et al._ [2022] Q. H. Wang, A. Bedoya-Pinto, M. Blei, A. H. Dismukes, A. Hamo, S. Jenkins, M. Koperski, Y. Liu, Q.-C. Sun, E. J. Telford, H. H. Kim, M. Augustin, U. Vool, J.-X. Yin, L. H. Li, A. Falin, C. R. Dean, F. Casanova, R. F. L. Evans, M. Chshiev, A. Mishchenko, C. Petrovic, R. He, L. Zhao, A. W. Tsen, B. D. Gerardot, M. Brotons-Gisbert, Z. Guguchia, X. Roy, S. Tongay, Z. Wang, M. Z. Hasan, J. Wrachtrup, A. Yacoby, A. Fert, S. Parkin, K. S. Novoselov, P. Dai, L. Balicas, and E. J. G. Santos, The magnetic genome of two-dimensional van der waals materials, ACS Nano 10.1021/acsnano.1c09150 (2022), pMID: 35442017, https://doi.org/10.1021/acsnano.1c09150 .