# Non-equilibrium cluster-cluster aggregation in the presence of anchoring
sites

Renaud Baillou  current address: PMMH, ESPCI, 75005 Paris, France  Institut de
Biologie de l’ENS, Ecole Normale Supérieure, CNRS, Inserm, Université PSL, 46
rue d’Ulm, 75005 Paris, France  Jonas Ranft<EMAIL_ADDRESS>Institut
de Biologie de l’ENS, Ecole Normale Supérieure, CNRS, Inserm, Université PSL,
46 rue d’Ulm, 75005 Paris, France

(August 31, 2024)

###### Abstract

Non-equilibrium cluster-cluster aggregation of particles diffusing in or at
the cell membrane has been hypothesized to lead to domains of finite size in
different biological contexts such as lipid rafts, cell adhesion complexes, or
postsynaptic domains in neurons. In this scenario, the desorption of particles
balances a continuous flux to the membrane, imposing a cut-off on possible
aggregate sizes and giving rise to a stationary size distribution. Here, we
investigate the case of non-equilibrium cluster-cluster aggregation in two
dimensions where diffusing particles and/or clusters remain fixed in space at
specific anchoring sites, which should be particularly relevant for synapses
but may also be present in other biological or physical systems. Using an
effective mean-field description of the concentration field around anchored
clusters, we derive an expression for their average size as a function of
parameters such as the anchoring site density. We furthermore propose and
solve appropriate rate equations that allow us to predict the size
distributions of both diffusing and fixed clusters. We confirm our results
with particle-based simulations, and discuss potential implications for
biological and physical systems.

##  I Introduction

The aggregation of particles into larger structures is observed not only in
inactive physical [1, *smoluchowski1918versuch, 2] but also in biological
systems, where it plays an important role e.g. in the formation of larger
protein assemblies in cells. Such protein clusters typically form in or at
cellular membranes; examples range from the clustering of chemotactic
receptors in bacteria [3, 4] to the aggregation of cell-cell adhesion proteins
in epithelia into dense clusters [5]. More generally, aggregation processes
have been hypothesized to govern the spatio-temporal organization of domains
in cellular membranes e.g. in the context of lipid raft formation [6, 7] or
the maturation of the Golgi apparatus [8]. Such instances of ‘healthy’
aggregation are not to be confounded with the abnormal protein aggregation
often linked to neurodegenerative diseases [9], which has also received
theoretical attention [10, 11].

In the context of diffusion-limited growth, aggregates (domains) grow by the
addition of individual particles or by fusion with other clusters (domain
coalescence). Whereas the former typically occurs at the molecular scale in
supersaturated solutions and is described by Lifshitz-Slyozov-Wagner theory
[12], the latter case of cluster-cluster aggregation or domain coalescence is
more relevant for (typically larger) strongly interacting particles with small
or vanishing nucleation barrier and has been described by Smoluchowski
coagulation equations [1, *smoluchowski1918versuch, 5, 6, 7, 11]. Submonolayer
molecular beam epitaxy, during which monolayer islands form due to the
diffusion and aggregation of deposited adatoms on the crystal surface, is
another growth scenario that has been described using rate equations for the
concentrations of islands of a given size [13, 2]. In the absence of a
specific mechanism for growth arrest, eventually only one large component will
remain and grow until the depletion of available particles. On the contrary,
biological systems are generally out of equilibrium and can use energy to
maintain a stationary state with finite aggregate or domain sizes. For
example, arrested liquid-liquid phase separation (LLPS) has become a very
active field of research since the discovery of its importance in the
formation of P-granules in the _C. elegans_ oocyte [14, 15, 16]. In the
context of LLPS, liquid droplets of the minority phase are limited in size by
chemical reactions that are driven out of equilibrium by externally imposed
gradients. In the context of particle aggregation or domain coalescence in
cellular membranes, the recycling of individual particles, clusters or
membrane domains, i.e. their removal from the surface, can serve to maintain a
stationary distribution of sizes of diffusing clusters or domains despite
continuous fusion and aggregation [5, 6, 7].

Recently, a similar mechanism has been hypothesized to underly the formation
of postsynaptic domains (PSDs) at inhibitory synapses [17] (but see [18]). In
neurons, synaptic transmission depends on the concentration of transmembrane
neurotransmitter receptors at PSDs due to transient interactions with scaffold
proteins that form larger domains by homotypic interactions; both excitatory
and inhibitory PSDs have been shown to be highly dynamic structures subject to
continuous turnover of their constituent particles [19, 20], raising the
question of how domains of a well-defined size may be maintained. In ref.
[17], the authors characterized the stationary size distributions that arise
from the interplay of the (possibly size-dependent) diffusion of particles and
clusters, aggregation, and the turnover of individual particles. Here, we will
follow up on this earlier work by extending the analysis to include the
existence of anchoring sites that can locally pin clusters at specific
positions at the surface and hinder their diffusion. At synapses,
transsynaptic adhesion complexes link the presynaptic bouton to the PSD [21]
and can thus be expected to act as such anchoring sites in the postsynaptic
membrane. Beyond the specific example of PSDs, our theory generalizes earlier
approaches to describe two-dimensional diffusion-limited aggregation with
turnover to the case where surface impurities or interactions with cytoplasmic
or extracellular structures locally impedes the diffusion of protein clusters
or membrane domains, with the aim to predict the stationary size distribution
of fixed and diffusing domains as a function of parameters.

This article is organized as follows. In Sec. II, we first derive a self-
consistent equation for the radius of the fixed domains based on the solution
of the diffusion equation for particles outside the domains. While this
approach provides insight into the scaling of the domain size with anchoring
site density, it does not account for the distribution of sizes of domains. In
Sec. III, we then present the results for those stationary size distributions
obtained by rate and master equations for diffusing and fixed domains,
respectively. In Sec. IV, we finally compare our results to particle-based
Brownian dynamics simulations and discuss the role of spatial order in the
distribution of anchoring sites, before we conclude with a short summary
discussion of our results in Sec. V.

##  II Mean-field description based on the diffusion equation

Figure 1:  Sketch of diffusion-aggregation dynamics with turnover in the
presence of anchoring sites. (a) Particles and clusters that are not attached
to anchoring sites can diffuse freely as indicated by the thin black arrows.
Clusters at anchoring sites do not diffuse. Individual particles are subject
to turnover (“monomer recycling scheme”) as indicated by the blue arrows. (b)
Corresponding mean-field picture with a smooth concentration field
$c(\mathbf{r},t)$ (gray shading) outside of fixed domains (red).

In a first approach, we describe the system as follows, see sketch of Fig. 1.
Diffusing particles outside of anchored domains are captured by a
concentration field $c(\mathbf{r},t)$, $\mathbf{r}\in\mathbb{R}^{2}$, which
obeys the diffusion equation

$\displaystyle\partial_{t}c(\mathbf{r},t)=D\Delta
c(\mathbf{r},t)-kc(\mathbf{r},t)+J$ (1)

between fixed domains. Here, $D$ is the diffusion constant, $k$ is the
recycling (or turnover) rate of individual particles, and $J$ is a flux that
re-injects recycled particles into the system. Diffusion and turnover define a
characteristic length scale $\lambda=\sqrt{D/k}$, and the recycling constants
$k$ and $J$ define an equilibrium (average) concentration $c_{0}=J/k$.

Upon encounter with a fixed domain, diffusing particles are adsorbed and added
to that domain. Each domain $i$ thus acts as a local sink for $c(\mathbf{r})$
with a flux $B_{i}$ that depends on its radius $R_{i}$. In principle, this is
described by absorbing boundary conditions for $c(\mathbf{r},t)$ at each
domain boundary. However, for simplicity we consider point-like sinks located
at anchoring sites $\mathbf{r_{j}}$, $j\neq i$, when describing the
concentration field around a specific domain $i$, assuming that the typical
extension of the domains is small compared to the distance between domains. In
this limit, the concentration field obeys the equation

$\displaystyle\partial_{t}c=D\Delta c-kc+J-\sum_{j\neq
i}B_{j}\delta(\mathbf{r}-\mathbf{r_{j}})\ .$ (2)

The size $N_{i}$ of each domain $i$ expressed as the number of constituent
particles obeys

$\displaystyle\frac{d}{dt}N_{i}=B_{i}-kN_{i}\ ,$ (3)

where $B_{i}$ is the flux of adsorbed particles and $kN_{i}$ is the particle
loss due to turnover within the domain.

We can now try and coarse-grain Eq. (2) with respect to the (random) positions
of the individual domains when all anchored domains are considered to be at
steady state and of the same size $N$, i.e. $B_{i}=kN_{i}=kN$. Stationarity
furthermore implies a constant concentration field
$c(\mathbf{r},t)=c(\mathbf{r})$. Introducing the density $n$ of anchoring
sites and assuming radial symmetry around the specific domain that we
consider, Eq. (2) then becomes

$\displaystyle 0=D\Delta c(r)-k(c(r)-c_{0})-kNn\ ,$ (4)

which needs to be solved with the absorbing boundary condition $c(R)=0$ at the
domain edge $r=R$. One then readily obtains

$\displaystyle
c(r)=(c_{0}-Nn)\left(1-\frac{K_{0}(r/\lambda)}{K_{0}(R/\lambda)}\right)\ ,$
(5)

where $K_{\alpha}$ are the modified Bessel functions of the second kind.

Figure 2:  Mean-field theory for a single domain based on the diffusion
equation. (a) Variation of the size of anchored domains with anchoring site
density $n/c_{0}$. The asymptote $N=c_{0}/n$ when all mass is concentrated in
fixed clusters is also shown, see text. (b) Concentration field $c(r)$ outside
of the anchored domains for different values of $n\lambda^{2}$. (c) Mass
fraction of anchored domains as a function of anchoring site density
$n/c_{0}$. (d) Size of anchored domains as a function of $c_{0}\lambda^{2}$
for fixed values of $n/c_{0}$. For all plots, $\rho/c_{0}=7000$. Cluster radii
$R$ (and equivalently cluster sizes $N$) were obtained from numerically
solving Eq. (7).

The radius $R=\sqrt{N/(\pi\rho)}$ is linked to the domain size via the domain
surface density $\rho$. The domain size $N$ itself follows self-consistently
from the flux equilibrium condition $dN/dt=0$ (Eq. (3)), namely

$\displaystyle kN=2\pi RD\partial_{r}c(r)|_{R}\ .$ (6)

Here, the lateral influx is given by the diffusive particle flux
$\mathbf{j}=-D\bm{\nabla}c(r)$ integrated along the domain circumference. With
Eq. (5), this condition leads to the following implicit equation for $R$:

$\displaystyle\frac{R}{\lambda}=2\left[\frac{c_{0}}{\rho}-\pi\left(\frac{R}{\lambda}\right)^{2}n\lambda^{2}\right]\frac{K_{1}(R/\lambda)}{K_{0}(R/\lambda)}\
.$ (7)

The resulting domain size $N=\pi R^{2}\rho$ can thus be numerically determined
as a function of the parameters of the system. For fixed $c_{0}$, $\rho$, and
$\lambda$, we show the predicted variation of $N$ with the density of
anchoring sites $n$ in Fig. 2a. The concentration field around anchored
domains for different values of $n$ is shown in Fig. 2b.

When the density $n$ of anchoring sites increases, an increasing fraction of
available particles remains trapped in the anchored clusters. We can obtain
the mass fraction of anchored particles directly from the size $N$ of fixed
domains via $Nn/c_{0}$, which we plot in Fig. 2c as a function of $n/c_{0}$.
Note that _a priori_ $N$ changes with $n$ in a non-trivial fashion (cf. Eq.
(7), Fig. 2a); however, when almost all particles are trapped at anchoring
sites, $N\approx c_{0}/n$ which we also plot on Fig. 2a for comparison.

Finally, we can ask how the domain size $N$ changes with particle turnover
rate $k$ for a fixed value of $n$, or more generally with $c_{0}$, $D$, and
$k$ at a fixed relative value $n/c_{0}$. This is shown in Fig. 2d. Note that
while in the dilute limit ($n=0$) the domains can become arbitrarily large,
their size can become at most $c_{0}/n$ (in the mean-field regime) when $n>0$,
which corresponds to limiting case when all particles are trapped at anchoring
sites.

So far, we have only considered aggregation at the fixed anchoring sites but
completely ignored possible effects due to aggregation and cluster formation
of diffusing particles outside of the anchored domains. Assuming a vanishing
nucleation barrier for cluster formation of particles coming into contact, one
however cannot exclude that particles diffuse as entire clusters in the free
surface. In order to take into account the potentially non-negligible size of
diffusing clusters, we introduce a typical radius $R_{\rm typ}$ and a typical
diffusion constant $D_{\rm typ}$ of diffusing clusters and modify the above
Eqs. (4) and (6) accordingly, i.e.,

$\displaystyle 0$ $\displaystyle=D_{\rm typ}\Delta c(r)-k(c(r)-c_{0})-kNn\ ,$
(8) $\displaystyle kR^{2}\rho$ $\displaystyle=2R_{\rm eff}D_{\rm
typ}\partial_{r}c(r)|_{R_{\rm eff}}\ ,$ (9)

where $R_{\rm eff}=R+R_{\rm typ}$ is the effective radius at which anchored
and diffusing clusters fuse. With $\bar{\lambda}=\sqrt{D_{\rm typ}/k}$, the
implicit equation for $R$ (Eq. (7)) then becomes

$\displaystyle\frac{R^{2}}{\bar{\lambda}R_{\rm
eff}}=2\left[\frac{c_{0}}{\rho}-\pi\left(\frac{R}{\bar{\lambda}}\right)^{2}n\bar{\lambda}^{2}\right]\frac{K_{1}(R_{\rm
eff}/\bar{\lambda})}{K_{0}(R_{\rm eff}/\bar{\lambda})}\ .$ (10)

To be able to make an educated guess about $R_{\rm typ}$ and $D_{\rm typ}$ if
not to outright calculate them, we need a theory that allows to predict the
size-distribution of diffusing clusters, which will be the subject of the next
section.

##  III Cluster size distributions obtained from rate equations

In an earlier work, some of us have shown that in the dilute limit $n=0$,
Smoluchowski rate equations capture extremely well the size distribution of
diffusing clusters that one obtains in full particle-based simulations of
Brownian dynamics with particle aggregation and turnover, in which a
stationary size distribution is observed after a characteristic time $1/k$
[17]. In the same spirit, we propose here rate equations that describe the
evolution of the (spatially averaged) concentrations $c_{m}(t)$ of diffusing
clusters of size $m$, where we introduce an additional term that accounts for
the fusion with anchored domains. In particular, this approach allows us to
take into account a possible size dependence of the diffusion constant of
diffusing clusters, which we assume to be

$D_{m}=D_{0}m^{-\sigma}$ (11)

in a scale-free way such that no additional length scale is introduced in the
system. Once we have obtained the size distribution of diffusing clusters, we
can use a master equation approach to calculate the size distribution of fixed
clusters, providing us with a full description of the system that we set out
to study.

In the presence of anchoring sites at a density $n$, the rate equations now
read

$\frac{d}{dt}c_{m}=-kmc_{m}+k(m+1)c_{m+1}+J\delta_{m1}\\\
-K\sum_{j}(D_{j}+D_{m})c_{j}c_{m}+K\sum_{j<m}(D_{j}+D_{m-j})c_{j}c_{m-j}\\\
-KD_{m}c_{m}n\ ,$ (12)

where $K$ is a kinetic coefficient (see e.g. ref. [22] for a discussion). In
the dilute limit $n=0$ and for $\sigma=0$, an analytical solution for the
stationary solution $c^{*}_{m}$ can be found [17]. For $n>0$ and/or $\sigma>0$
one can determine the $c^{*}_{m}$ numerically by integration of Eq. (12) until
a stationary state is reached. It follows from the above equation that the
normalized cluster concentrations $c^{*}_{m}/c_{0}$ only depend on three
dimensionless parameters,

$\displaystyle c^{*}_{m}=c_{0}f_{n}(c_{0}\lambda^{\prime
2},\sigma,n/c_{0})\,,$ (13)

where we introduced the characteristic length scale
$\lambda^{\prime}=\sqrt{KD_{0}/k}$ reminiscent of $\lambda$ introduced in the
previous section.

While the stationary cluster size distribution depends on $n$ via the last
term of Eq. (12), it does not depend on the size $N$ of the fixed domains. To
obtain $N$, we can proceed along two lines. Let us first invoke average
particle number balance on the surface. The total concentration of particles
$c_{\rm tot}$ is given by the weighted sum of concentrations $c_{m}$ plus the
surface density of anchored particles which is given by $Nn$. At the same
time, the balance of desorption and adsorption to the surface requires
$kc_{\rm tot}=J$, or $c_{\rm tot}=c_{0}$. One thus obtains an average size for
the anchored domains

$\displaystyle N=(c_{0}-\sum_{m=0}^{\infty}mc^{*}_{m})/n$ (14)

in the stationary state. Note that with Eq. (13) it follows that $N$ only
depends on the same three dimensionless parameters as $c^{*}_{m}/c_{0}$.

Second, we can use a previously developed master equation approach to compute
the distribution of sizes for fixed clusters that fluctuate in size due to the
stochastic aggregation with diffusing clusters and recycling [22]. For the
sake of completeness we repeat here the governing equation for probabilities
$p_{l}(t)$ for a domain to be of size $l$,

$\frac{d}{dt}p_{l}=-klp_{l}+k(l+1)p_{l+1}-p_{l}K\sum_{j}D_{j}c_{j}\\\
+K\sum_{j<l}D_{l-j}c_{l-j}p_{j}$ (15)

for $l>0$, where the first two terms describe size changes due to the
desorption of individual particles and the last two terms size changes due to
fusion with diffusing aggregates. Note that strictly speaking the $p_{l}$ for
$l\geq 1$ do not define a probability distribution, as $(d/dt)\sum_{l\geq
1}p_{l}=-kp_{1}$ and $l=0$ is an absorbing state. However, one can define a
quasi-stationary solution $\hat{p}_{l}$ such that $p_{l}(t)=\hat{p}_{l}e^{-\nu
t}$ on long times, with $\nu\ll k$ and $\sum_{l\geq 1}\hat{p}_{l}=1$. The
$\hat{p}_{l}$ can be obtained in a semi-analytical way from a recursion
relation [22] or via the relaxation of an initial $p_{l}(0)$ towards the
(quasi-)stationary state by integrating Eq. (15) numerically. Before solving
for the $\hat{p}_{l}$, we can again make a scaling argument using Eq. (13) and
find they also depend only on $c_{0}\lambda^{\prime 2}$, $\sigma$, and
$n/c_{0}$.

Assuming now that each anchoring site indeed harbors a fixed domain, the
absolute concentrations of fixed domains of size $l$ are then simply given by

$\displaystyle\hat{c}_{l}=n\hat{p}_{l}\ .$ (16)

The total density of clusters of size $m$ follows as $c^{\rm
tot}_{m}=c^{*}_{m}+\hat{c}_{m}$. Finally, following this approach the average
size of anchored domains is found to be

$\displaystyle N=\sum_{l}l\hat{p}_{l}\ ,$ (17)

which happens to give basically the same result as Eq. (14) when evaluated.

Figure 3:  Characterization of diffusing and anchored domains. (a,b) Size
distributions for different values of $n/c_{0}$, for (a) $\sigma=0$ and (b)
$\sigma=1$. Solid lines: $\hat{c}_{m}$, dotted lines: $c^{*}_{m}$, dashed-
dotted lines: $c^{\rm tot}_{m}$; colors indicate the value of $n/c_{0}$, see
legend in panel (b). (c) Average (typical) size of the anchored (diffusing)
clusters as a function of $n/c_{0}$ for different values of $\sigma$. (d)
Relative mass fraction of anchored domains as a function of $n/c_{0}$ for
different values of $\sigma$. (c,d) The mean-field theory of Sec. II is shown
for comparison with dots (“naïve” theory, Eq. (7)) and crosses (“effective”
theory, Eq. (10)). For the rate equations, $K=2$ was used.

The results for the size distributions of diffusing and anchored clusters for
different values of the anchoring site density $n$ and size-depedence exponent
$\sigma$ of the diffusion constant are shown in Fig. 3a,b. In the dilute limit
$n=0$, a strong dependence of the size distribution $c^{*}_{m}$ of diffusing
clusters on $\sigma$ has been described [17]. For $\sigma=0$, one can
furthermore theoretically predict in this dilute limit that the size
distribution follows a power law according to $c^{*}_{m}\sim m^{-1}$ until a
typical size

$\displaystyle M=\frac{\sum_{m}m^{2}c_{m}^{*}}{\sum_{m}mc_{m}^{*}}$ (18)

that scales as $M^{\rm dil}_{0}\approx c_{0}\lambda^{\prime 2}$. In the more
general case of $\sigma>0$ in the dilute limit, the power law no longer holds,
while the typical size of diffusing clusters follows the more general scaling
$M^{\rm dil}_{\sigma}\approx(c_{0}\lambda^{\prime 2})^{\frac{1}{1+\sigma}}$.
This picture changes in the presence of a finite $n>0$. Notably, the total
concentration $c^{\rm tot}_{m}$ starts to deviate from a power law for
$\sigma=0$ (Fig. 3a), and differences between the size distributions of
diffusing clusters $c^{*}_{m}$ for different $\sigma$ become less pronounced
with increasing $n$ (compare Fig. 3a,b). For increasing $n$, the size
distributions of diffusing and anchored clusters overlap less and less, which
manifests itself by an increasingly prominent trough in the total
concentrations $c^{\rm tot}_{m}$ that separates both types of clusters.

The average size $N$ of the fixed domains as a function of $n$ as predicted by
the rate-equation approach is shown in Fig. 3c. We first observe that in line
with the results of the previous section, $N$ decreases with the density of
anchoring sites $n$. Moreover, the difference between the average size for
different values of $\sigma$ does indeed decrease with $n$, as one can expect
from the fact that it does only depend on the concentrations of diffusing
clusters $c^{*}_{n}$, which become increasingly similar. With increasing $n$,
the mass fraction $nN/c_{0}$ of anchored domains does increase despite
decreasing $N$, as shown in Fig. 3d. Consequentially, as more and more mass is
concentrated in the anchored domains, their average size must decrease as
$N\approx c_{0}/n$ for large $n$, see Eq. (14) (Fig. 3c).

Before we turn to particle-based simulations in the next section, we can
already ask how well the “effective” theory of the previous section (Eq. (10))
accounts for the average size $N$ of anchored domains when taking into account
a typical radius $R_{\rm typ}=\langle{\langle{r_{m}}\rangle}\rangle$ and
diffusion constant $D_{\rm typ}=\langle{\langle{D_{m}}\rangle}\rangle$ of
diffusing clusters, where we defined
$\langle{\langle{a_{m}}\rangle}\rangle={\sum_{m}ma_{m}c_{m}^{*}}/{\sum_{m}mc_{m}^{*}}$
to be the mass-weighted average over diffusing clusters and
$r_{m}=\sqrt{m/(\pi\rho)}$. The comparison between the “naïve” theory (Eq.
(7)) with $D=D_{0}$, the “effective” theory with $R_{\rm typ}$ and $D_{\rm
typ}$ informed from the rate-equation approach, and the result for $N$ from
the rate-equation approach itself is also shown in Fig. 3. Perhaps not
surprisingly, the corrections based on $R_{\rm typ}$ and $D_{\rm typ}$ almost
entirely account for the discrepancy between the “naïve” theory and the result
from the rate-equation approach.

##  IV Particle-based simulations

Figure 4:  Particle-based simulations. (a) Snapshot of a simulation with 144
randomly distributed anchoring sites in a square box of side length $L=2000$.
Lengths are in units of single particle diameter $a$, see refs. [17, 22];
cluster radii are not to scale but increased five-fold for better visibility.
Clusters at anchoring sites are colored in red. (b) Size distributions of
anchored and diffusing clusters for different values of $n/c_{0}$, comparing
simulations with randomly distributed anchoring sites to the rate-equation-
based theory. Note that the theory has no free parameter. (c,d) Same as (a,b)
but for simulations with regularly distributed anchoring sites. (e) Average
size $N$ of anchored clusters (solid lines) and typical size $M$ of diffusing
clusters (dotted lines) as a function of $n/c_{0}$ as predicted by theory
(blue, large dots) and obtained by simulations with randomly (green, crosses)
or regularly (brown, small dots) distributed anchoring sites. For all
simulations, parameters were identical to case II of ref. [22]
($c_{0}/\rho=9\cdot 10^{-4}$, $c_{0}D_{0}/k=45$, $\sigma=0.5$), with a
simulation box size $L=2000$ and for a duration $T=5\cdot 10^{7}$.

To validate our theory, we implemented particle-based simulations following
the model of refs. [17, 22]. In brief, $c_{0}L^{2}$ particles were distributed
on a square simulation box of lateral dimension $L$ with periodic boundary
conditions. Upon diffusive encounter, particles fuse and form larger clusters,
which can also diffuse. Here, we assumed that particles within clusters
rearrange quickly and considered disc-like clusters the radius of which is
related to the number of constituent particles according to
$R=\sqrt{N/(\pi\rho)}$, where $\rho$ is the particle density within clusters,
but note that we have shown in earlier work that the resulting cluster size
statistics is almost identical in the absence of any particle rearrangement
[17]. Individual particles (monomers or constituent particles of larger
clusters) are removed from the surface with rate $k$ and randomly injected at
the surface to keep the total particle number constant. In addition, $nL^{2}$
anchoring sites are distributed on the surface either randomly or forming a
square lattice. While particles and clusters in principle diffuse with a size-
dependent diffusion constant $D_{m}$, clusters that are in contact with these
anchoring sites do not diffuse. Example snapshots of the simulations are shown
in Fig. 4(a,c). We compare the simulation results for the observed size
distributions of diffusing and anchored domains with the corresponding
predictions from the rate-equation approach of Sec. III in Fig. 4(b,d). (Note
that the predictions are obtained without any free adjustable parameter as we
chose the kinetic coefficient $K=1.81$ according to the calibration in the
absence of anchoring sites of ref. [22].) The comparison of the average
(typical) size of fixed and diffusing clusters as a function of anchoring site
density for both random and ordered cases with the theoretical predictions is
shown in Fig. 4(e).

In the disordered case, the excellent agreement between the size distributions
for specific parameter choices [Fig. 4(b)] and the dependence of average sizes
on anchoring site density [Fig. 4(e)] allows us to conclude that our theory
provides a quantitatively precise account of the aggregation dynamics and the
ensuing non-equilibrium stationary state. As biological systems often display
disorder, as can be seen for example in the random distribution of inhibitory
PSDs on the dendritic membrane, our theory can thus be considered as
quantitatively relevant beyond the general qualitative agreement between
simulation and theory independent of spatial order. Interestingly, the
agreement becomes less exact in the case of orderly distributed anchoring
sites [Fig. 4(d,e)], in which case fixed clusters tend to be slightly larger
then predicted by our theory, and diffusing clusters conversely slightly
smaller. It seems to make intuitively sense that the disordered case is better
described by the mean-field rate equations for the concentrations of domains
of a given size, assuming that potential spatial correlations between
neighboring clusters effectively self-average and cancel each other in this
case. While it is not entirely clear why spatial correlations in the ordered
case should give rise to an increased average size of the anchored domains,
one may speculate that the regular tiling of space tends to suppress the
growth of larger diffusing clusters and thus leads to the accumulation of a
larger mass fraction of particles in fixed clusters.

##  V Conclusion

To summarize, we developed a framework to predict the size distribution of
diffusing and fixed domains that form by irreversible aggregation of particles
combined with particle turnover in the presence of anchoring sites. In doing
so, we followed three separate approaches that provided increasing insight
into the statistics of the resulting non-equilibrium stationary state. First,
by considering the concentration field of diffusing particles around a
stationary domain, we derived an implicit, self-consistent equation for the
domain radius. The calculation predicts a decrease of the domain size with the
density of anchoring sites and a concomitant increase in the fraction of
particles at fixed domains (Sec. II, Fig. 2). We then combined Smoluchowski
rate equations for the freely diffusing clusters with a Master equation for
the stochastic size dynamics of fixed clusters to predict more precisely the
shapes of the distribution of sizes of both types of cluster (Sec. III, Fig.
3). Notably, the results obtained using the diffusion-equation approach
capture rather well the average size of fixed domains predicted by the rate-
and Master equation approach [Fig. 3(e,f)]. Given that both approaches are
quite different insofar as the first relies on the spatial gradient in the
concentration field whereas the second discards spatial variations completely
and focuses on the kinetics of aggregation instead, this is in and of itself
indicative of the validity of our theory. Finally, we used particle-based
simulations to assess the quantitative agreement between our mean-field theory
and the aggregation dynamics in space (Sec. IV). Here, we found that a random
distribution of anchoring sites leads to a non-equilibrium stationary state
the size statistics of which are very well captured by our theory [Fig.
4(b,e)]. In the (biologically less relevant) case of ordered anchoring sites,
the agreement becomes less precise, although our theory still captures
qualitatively the dependence of the size distributions and average sizes on
anchoring site density [Fig. 4(d,e)].

Our results indicate that the presence of anchoring sites modifies the cluster
size statistics in several ways: (i) Diffusing and fixed clusters show
distinct characteristic size distributions, the typical (average) sizes of
which increasingly differ with increasing cluster site density; (ii) fixed
clusters have a well-defined average size for arbitrary anchoring site density
$n$ whereas (depending on the diffusion size-dependence exponent $\sigma$)
diffusing clusters may show a power-law dependence until a cut-off size
imposed by turnover at vanishing $n$, with large sizes becoming increasingly
suppressed with increasing $n$. As a corollary, even if the diffusion constant
does not depend on cluster size ($\sigma=0$), the size distribution of
diffusing and fixed clusters combined would show a deviation from a power-law
scaling at finite anchoring site density $n$. Such a deviation is therefore
not alone sufficient to indicate that diffusion is indeed size-dependent, as
would be the case in the absence of any anchoring.

We note that the results we have obtained here for the case of a finite
density of anchoring sites are somewhat similar to the finite-size effects
discussed in the context of membrane nanodomains in vesicles [23], where the
finite amount of aggregating particles leads to the separation of clusters
into one “giant” cluster and small clusters when the system size becomes
smaller or comparable to the typical cluster size in the infinite system.
While the authors considered two distinct recycling schemes (monomer vs.
cluster recycling), they did not investigate the effects of a possible size-
dependence of diffusion and used a simplified Master equation for the size
evolution of the single large cluster; their study can thus be considered
complementary to our work. In addition, our particle-based simulations reveal
the subtle importance of spatial correlations in the resulting size
statistics, which is a property of the extended system, i.e. containing much
more particles than the typical cluster size. It would be interesting to see
whether analytical approaches beyond pure mean-field description may shed
additional light onto these kinds of effects (see e.g. [24, 25] for a
discussion of the interaction between domains).

Another interesting consequence of the clear distinction between diffusing and
fixed clusters is that the conceptual “boundary” between (actively suppressed)
Ostwald ripening on the one hand [26, 24] and domain or droplet coalescence on
the other becomes somewhat blurred, as mass becomes increasingly concentrated
in the larger fixed clusters which are replenished by diffusing small
clusters.

Here, we restricted ourselves to the statistics of the non-equilibrium
stationary state that can arise in systems where aggregation is combined with
turnover [6, 23]. Recently, the dynamics of the cluster size distribution for
size-dependent recycling has been analytically studied using a continuum
version of Smoluchowski coagulation equations, including results for the
transient dynamics of the average cluster size for given initial conditions
[7]. It would be worthwhile to see whether these approaches may be extended to
the case with size-dependent diffusion coefficient and a finite density of
anchoring sites. Given the quantitative agreement between our stationary
solution and the simulations, we would expect that the time evolution of the
cluster size distribution from non-stationary initial conditions would equally
well be captured using the time-dependent (numerical) solutions of the rate
equations and Master equation, although admittedly this does not amount to an
analytical theory.

This work has been inspired by the observation that inhibitory
neurotransmitter receptors and scaffold proteins can spontaneously organize
into surface domains of a well-defined size even in non-neuronal cells [27,
28], which lead to the hypothesis that these domains form according to the
aggregation process with turnover described here [17]. A recent study assessed
whether the proposed non-equilibrium recycling scheme could explain the
observed receptor and scaffold kinetics at synapses upon the suppression of
lateral receptor fluxes at the extrasynaptic membrane, and concluded that
direct recruitment of scaffold proteins to synaptic sites cannot be neglected
[18]. Given the ubiquity of protein diffusion and aggregation in membranes,
and the potential sites for protein anchoring at the submembrane structures
such as the cytoskeleton, transmembrane or extracellular structures, we are
confident that the results presented here remain nevertheless relevant for a
large number of biological systems.

Acknowledgments. The authors would like to thank Vincent Hakim and Benjamin
Lindner for comments on an earlier version of this manuscript.

## References

  * Smoluchowski [1916] M. v. Smoluchowski, Zeitschrift für Physik 17, 557 (1916); M. v. Smoluchowski, Zeitschrift für physikalische Chemie 92, 129 (1918).
  * Einax et al. [2013] M. Einax, W. Dieterich, and P. Maass, Reviews of Modern Physics 85, 921 (2013).
  * Sourjik [2004] V. Sourjik, Trends in Microbiology 12, 569 (2004).
  * Greenfield et al. [2009] D. Greenfield, A. L. McEvoy, H. Shroff, G. E. Crooks, N. S. Wingreen, E. Betzig, and J. Liphardt, PLoS Biology 7, e1000137 (2009).
  * Quang et al. [2013] B.-A. T. Quang, M. Mani, O. Markova, T. Lecuit, and P.-F. Lenne, Current Biology 23, 2197 (2013).
  * Turner et al. [2005] M. S. Turner, P. Sens, and N. D. Socci, Physical Review Letters 95, 168301 (2005).
  * Rautu et al. [2018] S. A. Rautu, G. Rowlands, and M. S. Turner, Europhysics Letters 121, 58004 (2018).
  * Vagne et al. [2020] Q. Vagne, J.-P. Vrel, and P. Sens, Elife 9, e47318 (2020).
  * Tyedmers et al. [2010] J. Tyedmers, A. Mogk, and B. Bukau, Nature Reviews Molecular Cell Biology 11, 777 (2010).
  * Lenz and Witten [2017] M. Lenz and T. A. Witten, Nature Physics 13, 1100 (2017).
  * Miangolarra et al. [2021] A. M. Miangolarra, A. Duperray-Susini, M. Coppey, and M. Castellana, Biophysical Journal 120, 2394 (2021).
  * Bray [2002] A. J. Bray, Advances in Physics 51, 481 (2002).
  * Bales and Zangwill [1997] G. Bales and A. Zangwill, Physical Review B 55, R1973 (1997).
  * Brangwynne et al. [2009] C. P. Brangwynne, C. R. Eckmann, D. S. Courson, A. Rybarska, C. Hoege, J. Gharakhani, F. Jülicher, and A. A. Hyman, Science 324, 1729 (2009).
  * Hyman et al. [2014] A. A. Hyman, C. A. Weber, and F. Jülicher, Annual Review of Cell and Developmental Biology 30, 39 (2014).
  * Weber et al. [2019] C. A. Weber, D. Zwicker, F. Jülicher, and C. F. Lee, Reports on Progress in Physics 82, 064601 (2019).
  * Ranft et al. [2017] J. Ranft, L. G. Almeida, P. C. Rodriguez, A. Triller, and V. Hakim, PLoS Computational Biology 13, e1005516 (2017).
  * Chapdelaine et al. [2021] T. Chapdelaine, V. Hakim, A. Triller, J. Ranft, and C. G. Specht, Biophysical Journal 120, 805 (2021).
  * Choquet and Triller [2013] D. Choquet and A. Triller, Neuron 80, 691 (2013).
  * Maynard et al. [2023] S. A. Maynard, J. Ranft, and A. Triller, Nature Reviews Neuroscience 24, 4 (2023).
  * Varoqueaux et al. [2006] F. Varoqueaux, G. Aramuni, R. L. Rawson, R. Mohrmann, M. Missler, K. Gottmann, W. Zhang, T. C. Südhof, and N. Brose, Neuron 51, 741 (2006).
  * Hakim and Ranft [2020] V. Hakim and J. Ranft, Physical Review E 101, 012411 (2020).
  * Vagne et al. [2015] Q. Vagne, M. S. Turner, and P. Sens, PLoS One 10, e0143470 (2015).
  * Bressloff [2020] P. C. Bressloff, Physical Review E 101, 042804 (2020).
  * Bressloff [2023] P. C. Bressloff, Proceedings of the Royal Society A 479, 20220831 (2023).
  * Zwicker et al. [2015] D. Zwicker, A. A. Hyman, and F. Jülicher, Physical Review E 92, 012317 (2015).
  * Calamai et al. [2009] M. Calamai, C. G. Specht, J. Heller, D. Alcor, P. Machado, C. Vannier, and A. Triller, Journal of Neuroscience 29, 7639 (2009).
  * Haselwandter et al. [2011] C. A. Haselwandter, M. Calamai, M. Kardar, A. Triller, and R. A. da Silveira, Physical Review Letters 106, 238104 (2011).


# Speed limits of the laser-induced phase transition in FeRh

M. Mattern  Institut für Physik und Astronomie, Universität Potsdam, 14476
Potsdam, Germany  J. Jarecki  Institut für Physik und Astronomie, Universität
Potsdam, 14476 Potsdam, Germany  J. A. Arregi  CEITEC BUT, Brno University of
Technology, 61200 Brno, Czech Republic  V. Uhlíř  CEITEC BUT, Brno University
of Technology, 61200 Brno, Czech Republic  Institute of Physical Engineering,
Brno University of Technology , 61669 Brno, Czech Republic  M. Rössle
Helmholtz-Zentrum Berlin für Materialien und Energie GmbH, Wilhelm-Conrad-
Röntgen Campus, BESSY II, 12489 Berlin, Germany  M. Bargheer  Institut für
Physik und Astronomie, Universität Potsdam, 14476 Potsdam, Germany  Helmholtz-
Zentrum Berlin für Materialien und Energie GmbH, Wilhelm-Conrad-Röntgen
Campus, BESSY II, 12489 Berlin, Germany<EMAIL_ADDRESS>
###### Abstract

We use ultrafast x-ray diffraction (UXRD) and the polar time-resolved magneto-
optical Kerr effect (tr-MOKE) to study the laser-induced metamagnetic phase
transition in two FeRh films with thicknesses below and above the optical
penetration depth. In the thin film, we identify an intrinsic timescale for
the light-induced nucleation of ferromagnetic (FM) domains in the
antiferromagnetic material of $8\,\text{ps}$ that is substantially slower than
the speed of sound. For the inhomogeneously excited thicker film, only the
optically excited near-surface part transforms within $8\,\text{ps}$. For
strong excitations we observe an additional slow rise of the FM phase, which
we experimentally relate to a growth of the FM phase into the depth of the
layer by comparing the transient magnetization in front- and backside
excitation geometry. In the lower lying parts of the film, which are only
excited via near-equilibrium heat transport, the FM phase emerges
significantly slower than $8\,\text{ps}$ after heating above the transition
temperature.

First-order phase transitions are characterized by an abrupt change of
structural, electronic or/and magnetic properties and a co-existence of
multiple phases that introduces nucleation and domain growth to the kinetics
of the phase transition [1, 2, 3, 4, 5, 6, 7, 8, 9, 10].

The abrupt change of properties accompanying the emerging phase as a
consequence of a fine interplay of spin, charge, orbital and lattice degrees
of freedom [6, 11] predestine materials featuring first-order phase
transitions for ultrafast laser control of functionalities [12]. In this
context, the first-order antiferromagnetic to ferromagnetic (AFM-FM) phase
transition of FeRh at $370\,\text{K}$ attracted considerable attention in
terms of ultrafast generation of ferromagnetic order [13, 14] that extends the
more extensively studied ultrafast demagnetization [15, 16] and magnetization
reversal [17, 18, 19].

The metamagnetic phase transition in FeRh is parameterized by the expansion of
the unit cell [20], the change in the electronic band structure [21, 22] and
the arising magnetization [23, 24], which each serve as order parameters for
different aspects of the ultrafast laser-induced phase transition. Time-
resolved photoelectron spectroscopy experiments reveal the formation of an
electronic signature of the ferromagnetic state by a photo-induced change of
the band structure on a sub-picosecond timescale [25]. X-ray magnetic circular
dichroism [26, 27], magneto-optical Kerr effect (MOKE) [28] and double-pulse
THz emission spectroscopy [29] report a subsequent formation of an in-plane
magnetization on a $100\,\text{ps}$ timescale. While the rise of a macroscopic
magnetization is dominated by the slow coalescence and alignment of the
nucleated domains in an external magnetic field [14, 29], the large FM lattice
constant as structural order parameter is independent of the orientation of
the arising magnetization [28]. Hence, ultrafast x-ray diffraction (UXRD)
directly accesses the nucleation and growth of FM domains that also determines
the rise of the laser-induced magnetization within the first picoseconds [28,
29]. Previous UXRD studies report nucleation timescales ranging from $15$ to
$90\,\text{ps}$ depending on the probing depth and fluence [28, 30]. Thus, the
kinetics of the nucleation and growth of the laser-induced FM phase remains
unclear and controversial.

Here, we use UXRD experiments on a homogeneously optically excited
$12\,\text{nm}$ thin FeRh film to identify an intrinsic fluence-, temperature-
and field-independent $8\,\text{ps}$ rise time of the structural order
parameter. This nucleation of the FM phase is not limited by the speed of
sound, which would require only $2.5\,\text{ps}$. For the inhomogeneously
excited $44\,\text{nm}$ thick film, we observe the same intrinsic
$8\,\text{ps}$ nucleation timescale and an additional delayed slow rise for
high fluences when the deposited energy is sufficient to heat the lower lying
parts beyond the transition temperature $T_{\text{T}}$ by heat diffusion. This
unlocks a growth of the FM phase into the depth of the film. Modeling the UXRD
results shows that this growth driven by near-equilibrium heat transport is
considerably slower than the $8\,\text{ps}$ nucleation timescale after heating
above $T_{\text{T}}$, indicating the crucial role of optically induced non-
thermal states for the kinetics of the phase transition. The heat transport
timescale is cross-checked by a buried tungsten detection layer, which
measures the energy transmitted through FeRh. To complement the insights from
the structural order parameter, we probe the subsequent formation of a
macroscopic out-of-plane magnetization within $180\,\text{ps}$ by polar tr-
MOKE. When we excite the thick FeRh film from the backside, we find a
considerable slower and fluence-dependent rise compared to frontside
excitation, which experimentally verifies the slow out-of-plane growth of the
FM phase.

The two samples are sketched in Fig. 1(a) and (b) and consist of a
$12.6\,\text{nm}$ thick FeRh film on MgO(001), and a $43.8\,\text{nm}$ thick
FeRh film embedded in a metallic heterostructure on MgO(001) consisting of a
$2\,\text{nm}$ Pt capping and a $8\,\text{nm}$ W buffer layer. The thickness
of the layers has been characterized via x-ray reflectivity (XRR) and the FeRh
films were grown using magnetron sputtering from an equiatomic FeRh target
[20]. Both FeRh films exhibit the first-order metamagnetic phase transition
characterized by the temperature-dependent magnetization (solid lines) and
average out-of-plane lattice constant $d$ (symbols) displayed in Fig. 1(c) and
(d). They were measured in thermal equilibrium via vibrating sample
magnetometry (VSM) using a QuantumDesign VersaLab magnetometer and XRD
performed at the KMC-3 XPP endstation at BESSY II [31], respectively.

The thin film exhibits a reduced mean transition temperature of
$T_{\mathrm{T}}=365\,\text{K}$ in comparison to the thick film
($375\,\text{K}$) and a residual FM phase fraction of around $20\,\%$
originating from interface effects [32, 33, 34]. This reduces the relative
expansion associated with the AFM-FM phase transition from $0.6\,\%$ in the
thick film to $0.48\,\%$ in the thin film. While the magnetization and lattice
constant as order parameters of the temperature-induced phase transition
nicely agree for the thick film, the inhomogeneity of the thin film results in
a narrower hysteresis for the locally probed lattice constant in contrast to
the global magnetization determined by VSM.

In the combined UXRD and tr-MOKE experiment, the FeRh layers are excited by
$p$-polarized pump pulses with a central wavelength of $800\,\text{nm}$ and
$100\,\text{fs}$ pulse duration that are incident under $50^{\circ}$ with
respect to the sample surface. Utilizing UXRD, we probe the transient out-of-
plane strain response of the FeRh layers via reciprocal space mapping [35] of
the FeRh(002) Bragg peak at a table-top laser-driven plasma x-ray source [36]
providing $200\,\text{fs}$ hard x-ray pulses with a photon energy of $\approx
8\,\text{keV}$. The Bragg peak position in reciprocal space is given by the
average out-of-plane lattice constant $d$ of the FeRh films via
$q_{\text{z}}=4\pi/d$. Therefore, the laser-induced peak shift accesses its
change $\Delta d$ determining the lattice strain $\eta_{\text{FeRh}}=\Delta
d/d_{0}$ as the relative change with respect to its value $d_{0}$ before
excitation.

Figure 1:  Characterization of FeRh films: (a–b) Sketch of the sample
structures containing a $12\,\text{nm}$ and a $44\,\text{nm}$ FeRh film, the
excitation profile (red) and the probing profile of UXRD (black) and MOKE
(blue). The combined UXRD and MOKE experiment is sketched in (a). Panels (c)
and (d) compare the temperature-dependent magnetization (solid lines) with the
out-of-plane lattice constant (symbols) parametrizing the AFM-FM phase
transition in the thin and thick FeRh film, respectively.

Figure 2 displays the laser-induced strain response of both FeRh films and the
buried W layer to a weak sub-trehsold excitation that is not able to drive the
AFM-FM phase transition [28, 27]. Thus, the strain response is the
superposition of only two contributions: A quasi-static expansion due to
heating and coherently driven propagating strain pulses (partially) reflected
at the surface and interfaces. In the homogeneously excited $12\,\text{nm}$
FeRh film (optical penetration depth $\lambda=13\,\text{nm}$), the laser
excitation launches a strain pulse that is reflected at the surface and
partially transmitted into the substrate. This results in a decaying
oscillation with a period of $2L_{\text{FeRh}}/v_{\text{s}}$ determined by the
layer thickness $L_{\text{FeRh}}$ and the sound velocity $v_{\text{s}}$ [38]
that is superimposed with a decreasing quasi-static expansion due to heat
transport into the substrate (see Fig. 2(a)). For the thick film sample, the
bipolar strain pulse launched at the optically excited surface is partially
reflected at the FeRh-W interface, which leads to a more complex shape of the
oscillations in the FeRh strain response (see Fig. 2(b)). The compression of
the W layer is caused by the leading part of the bipolar strain pulse
generated in FeRh, and hence essentially senses the profile of optical energy
deposition [38]. The W strain turns positive, when the expansive part of the
strain pulse enters and the compressive part again exits the layer. The slowly
rising expansion of W accesses the heat transport from FeRh into W on tens of
picoseconds. These thermoelastic strain contributions scale linearly with the
deposited energy [38].

In the following, we utilize this calibration of the thermoelastic strain
response in the absence of a laser-induced AFM-FM phase transition to extract
the transient FM volume fraction $V_{\text{FM}}$ from the strain response to
above-threshold excitations that additionally contains a strain signature from
the forming FM phase. For this approach and to access the inhomogeneous
spatio-temporal temperature profile within the $44\,\text{nm}$-thick FeRh
film, we modeled the strain response by the modular Python library udkm1Dsim
[39] utilizing literature values for the thermo-elastic properties stated in
Tab. S1. We calculate the absorption profile and solve the one-dimensional
heat diffusion equation determining the spatio-temporal temperature that
determines the stress on the lattice [38]. By solving the linear one-
dimensional elastic wave equation for this stress we finally calculate the
strain response [38] (see supplementary material S1 for a detailed description
of the strain modeling). Figures 2(c) and (d) display the spatio-temporal
strain $\eta$ and temperature increase $\Delta T$ for
$F_{\text{st}}=1.6\,\text{mJ/cm}^{2}$, respectively. Averaging the strain
$\eta(z,t)$ over a respective layer yields the layer-specific strain response
measured in our UXRD experiment. We find excellent agreement of the modeled
strain response (solid lines) with the measurements in both samples. The
excellent agreement with both the $44\,\text{nm}$ FeRh and the buried W layer
for a single set of parameters pinpoints the modeled spatio-temporal
temperature shown in Fig. 2(c). The shape of the initial compression of W
quantifies the absorption profile and its slowly rising expansion probes the
heat propagated through the inhomogeneously excited FeRh layer [38].

Figure 2: Thermoelastic strain response calibrated by UXRD: (a) Transient
average strain of the $12\,\text{nm}$ FeRh film (symbols) upon an excitation
of $F_{\text{st}}=0.5\,\text{mJ/cm}^{2}$. (b) Transient average strain
response of the $44\,\text{nm}$ FeRh and the W layer (symbols) for
$F_{\text{st}}=1.5\,\text{mJ/cm}^{2}$. The solid lines denote our strain model
described in the text. The perfect agreement shown in (b) is a strong evidence
for the correct modeling of the corresponding the spatio-temporal temperature
increase $\Delta T(z,t)$ (c) and strain $\eta(z,t)$ (d).

Figure 3(a) displays the strain response of the $12\,\text{nm}$-thin FeRh film
for various fluences below and above the threshold of the phase transition.
The dashed line denotes the modeled thermoelastic strain scaled to
$1.8\,\text{mJ/cm}^{2}$, i.e. the hypothetical strain response without phase
transition. The very large difference to the actual measurement (coloured
area) highlights the signature of the phase transition in the strain response.
The strain measured in an external magnetic field of $1\,\text{T}$ (squares)
is identical to the response without an applied magnetic field (dots). This
shows that the signature of the phase transition in the structural response is
independent of the external magnetic field. We relate the additional strain
contribution to $V_{\text{FM}}(t)$ by considering the expansion of $0.48\,\%$
for a complete phase transition in thermal equilibrium (see Fig. 1(c)). In
addition, we considered the thermal expansion coefficient of the FM phase (see
Tab. S1) and that the energy consumed by the latent heat $4.2\,\text{J/gK}$
[40] at the transition temperature does not contribute to thermoelastic
expansion. Figure 3(b) displays the extracted FM volume fraction
$V_{\text{FM}}(t)$ that parameterizes the laser-driven phase transition. We
observe that $V_{\text{FM}}(t)$ rises as a single exponential according to:

$\displaystyle V_{\text{FM}}(t)$
$\displaystyle=V_{\text{FM}}^{*}\cdot\left(1-e^{-t/\tau}\right)\;,$ (1)

where $V_{\text{FM}}^{*}$ denotes the final fraction of the film in the FM
phase and $\tau$ is the rise time, independent of the fluence and the applied
magnetic field. This model of $V_{\text{FM}}$ has been discussed and
successfully applied in a previous UXRD study of FeRh [28] and assumes the
nucleation of FM domains at independent sites according to Avrami [1]. We
observe an intrinsic $\tau=7.8\pm 0.6\,\text{ps}$ timescale from the fit of
all measurements in Fig. 3(b). This is clearly slower than the relaxation of
the lattice: The propagation of strain pulses through the thin film at the
speed of sound only takes the time $L_{\text{FeRh}}/v_{s}=2.5\,\text{ps}$ (cf.
Fig. 3(a)). We can therefore definitely exclude the widely accepted hypothesis
that the sound velocity $v_{s}$ sets the speed limit for the AFM-FM phase
transition in FeRh [28, 29, 41]. Instead, the structural order parameter
intrinsically responds on an $8\,\text{ps}$ timescale to direct optical
excitation. The transformed volume fraction $V_{\text{FM}}^{*}$ increases with
fluence and saturates at $F_{\text{sat}}=2.1\,\text{mJ/cm}^{2}$, when the
complete film is driven into the FM phase. Figure 3(b) shows that
$F=3.4\,\text{mJ/cm}^{2}$ yields the same $V_{\text{FM}}^{*}$, although the
thermoelastic strain contributions grow (Fig. 3(a)).

We complement the insights of the UXRD measurements by measuring the transient
out-of-plane magnetization by polar MOKE in the very same experimental setup
under identical excitation conditions. We applied a maximum out-of-plane
magnetic field of $1\,\text{T}$ provided by an electromagnet. The transient
magnetization is extracted by the difference of the transient MOKE signal for
opposite field-polarities. The $100\,\text{fs}$-long $p$-polarized probe pulse
with a central wavelength of $400\,\text{nm}$ is focussed through the pole of
the magnet and is incident under less than $2^{\circ}$ with respect to the
sample normal.

Figure 3:  Intrinsic timescale of nucleation: (a) Fluence-dependent strain
response of the $12\,\text{nm}$ thick FeRh film. The strain response for a
field of $1\,\text{T}$ (squares) matches the one without external field
(dots). The dashed line is the modeled strain for $0.5\,\text{mJ/cm}^{2}$
scaled to $1.8\,\text{mJ/cm}^{2}$, i.e. the hypothetical response without
phase transition.(b) Transient FM volume fraction $V_{\text{FM}}$ extracted
from the difference (coloured area) between measurement and modeled
thermoelastic strain. (c) Fluence-dependent transient out-of-plane
magnetization from polar MOKE. Field-dependent MOKE data are presented in Fig.
S4 in the supplementary material. Panels (d–h) sketch the kinetics of the
phase transition in regard of the structural order parameter (pink color) and
the magnetization (arrows) in line with previous results [28, 14, 29, 43].
After the nucleation of FM domains the local magnetization precessionally
tilts out-of-plane plane (g,h).

The transient out-of-plane magnetization in Fig. 3(c) verifies this fluence
dependence of $V_{\text{FM}}^{*}$. Up to the threshold fluence of
$F_{\text{th}}=0.6\,\text{mJ/cm}^{2}$ we observe no macroscopic magnetization
at all and a fluence of $2.1\,\text{mJ/cm}^{2}$ fully saturates the laser-
induced magnetization. We observe a fluence-independent rise time with the
maximum signal at $180\,\text{ps}$ for $1\,\text{T}$, i.e. much slower than
the nucleation of the FM domains. This slow formation of a macroscopic
magnetization by the alignment of the local magnetization of the nucleated
domains was reported previously [28, 27, 29]: The local magnetization
initially lies in the sample plane along four equal directions determined by a
cubic anisotropy of around $100\,\text{mT}$ that represent the magnetic easy
axes due to the shape anisotropy field of $1.38\,\text{T}$ for thin FeRh films
[28, 42]. For an in-plane magnetic field, a macroscopic in-plane magnetization
is formed by field-driven coalescence of the FM domains via domain wall motion
in agreement with the observed linear increase of the growth rate of the
macroscopic magnetization with the field [29]. For the out-of-plane
magnetization, in contrast, we observe the maximum to be established faster
for smaller out-of-plane fields (Fig. S4). This is consistent with
precessionally tilting [14] the magnetization of the nucleated domains out of
the sample plane. Figures 3(d–h) sketch the series of events.

Figure 4:  Kinetics of out-of-plane growth of the FM phase: (a) Fluence-
dependent field-free strain response of the $44\,\text{nm}$ thick FeRh film.
(b) Transient FM volume fraction derived analogously to the thin film by
comparing the measured strain to a scaled sub-threshold strain model. (c,d)
Fluence-dependent magnetization rise measured by polar MOKE with a magnetic
field of $1\,\text{T}$ in front- and backside excitation geometry,
respectively. (e–h) Sketch of the out-of-plane growth of the FM phase
indicated by the green area for $5.4$ (left) and $11.9\,\text{mJ/cm}^{2}$
(right) that is induced by a fluence-dependent heating above $T_{\text{T}}$ at
the backside of the film via near-equilibrium heat transport.

These MOKE results upon direct optical excitation of the thin film serve as
reference for the inhomogeneously excited thick film where the redistribution
of energy within the layer by heat transport extends the dynamics of the phase
transition. We utilize the finite probing depth of MOKE to gain insights into
the out-of-plane evolution of the FM phase by comparing the rising
magnetization at the sample surface for front- and backside excitation in
Figs. 4(c and d). For frontside excitation, we observe the same fluence-
independent rise time as for the thin film in Fig. 3(c). In contrast, exciting
through the substrate leads to a delayed and slower rise. Additionally, the
rise time strongly depends on the fluence (see Fig. S3 for a comparison
normalized to the maximum). The delayed and much slower rise for backside
excitation is fluence-dependent, indicating a slow out-of-plane growth of
$V_{\text{FM}}$ via near-equilibrium heat transport that brings the upper part
of the film above the transition temperature. However, the tr-MOKE
measurements cannot quantify whether it is the heat transport or different
kinetics of the phase transition in near-equilibrium that yield the increased
timescale.

In order to determine the speed at which the FM phase nucleates and grows in
the thick film, we again subtract the fluence-scaled strain response to a sub-
threshold excitation $F_{\text{st}}$ in Fig. 4(a) measured by UXRD from the
strain measured above the threshold. The resulting transient FM volume
fraction in Fig. 4(b) displays the change from a single exponential rise for
low fluences to a two-step rise for high fluences ($>6\,\text{mJ/cm}^{2}$).
For all fluences $V_{\text{FM}}$ rises according to Eq. 1 with the intrinsic
$8\,\text{ps}$ timescale already observed in the thin film during the first
$30\,\text{ps}$ (see solid colored lines in Fig. 4(b)). The amplitude of the
additional contribution in the high fluence regime increases with the fluence.
The maximum $V_{\text{FM}}$ is reached at increasing delays up to
$250\,\text{ps}$ for the highest fluence, where the complete FeRh film is
driven across the AFM-FM phase transition. Reaching the full transformation to
the FM phase in Fig. 4(b) shows that for high fluences the temperature at the
backside finally exceeds the transition temperature $T_{\text{T}}$, thus
enabling the phase transition in near-equilibrium. This results in the two-
step rise of $V_{\text{FM}}$ for high fluences in contrast to a phase
transition on the $8\,\text{ps}$ timescale limited to the optically excited
near-surface region for low fluences as sketched in Fig. 4(e–h). Note, that
the laterally homogeneous growth of the FM phase into the depth, which follows
the inhomogeneous optical excitation, differs from the dynamics observed for
homogeneous equilibrium heating [2], where columns through the FeRh film are
formed after the nucleation of FM domains at both interfaces before the in-
plane domain growth.

To crosscheck our interpretation, we repeated the UXRD varying the initial
sample temperature (cf. Fig. S2). At temperatures only slightly below
$T_{\text{T}}$ the laser excitation is sufficient to drive the phase
transition in the not directly excited bottom part of FeRh, which results in a
two-step rise of $V_{\text{FM}}$. At lower temperatures, the heating is
insufficient and we observe only a rise on the $8\,\text{ps}$ timescale. In
this temperature-dependent experiment, we directly accessed $V_{\text{FM}}$ by
the transient relative amplitudes of the structural Bragg peaks of the AFM and
FM phase that are separated in reciprocal space due to the excellent
collimation of the x-ray beam at the KMC-3 XPP endstation at BESSY II [31].
This parameter-free analysis also applied by Mariager and co-workers [28]
serves as a crosscheck of our findings in Fig. 3 and 4. The slow growth of the
FM phase into the depth unlocked by strong excitations explains the different
timescales for different fluences and probing depth reported in previous UXRD
experiments on inhomogeneously excited FeRh films [28, 30]. The exponential
probing profile in grazing incidence geometry in previous experiments [28] may
have masked the clear two-step behaviour in our experiments.

We obtain additional insights into the growth of the FM phase within the
inhomogeneously excited FeRh layer from the modeled spatio-temporal
temperature profile $T(z,t)$ in Fig. 2(c). From this analysis, we obtain the
gray solid line in Fig. 4(b), which denotes the fraction of the film
transiently heated above $T_{\text{T}}=375\,\text{K}$ for an excitation of
$11.9\,\text{mJ/cm}^{2}$. This analysis shows that already within the first
$30\,\text{ps}$ a considerably large fraction and within $80\,\text{ps}$ the
complete film is heated above the transition temperature. This is much faster
than the observed rise of $V_{\text{FM}}$ within $250\,\text{ps}$ and
indicates that the growth of the FM phase does not simply follow the heating
of the backside above $T_{\text{T}}$ but exhibits intrinsic kinetics. If we
assume the FM phase to locally rise with an $8\,\text{ps}$ timescale as soon
as the local temperature exceeds $T_{\text{T}}$, the modeled rise of
$V_{\text{FM}}$ (gray dashed line) is still significantly faster than in the
measurement. Therefore, establishing the FM phase by only near-equilibrium
heating must be considerably slower than the nucleation in the optically
excited near-surface part of the film. The gray dotted line in Fig. 4(b)
represents the combination of an $8\,\text{ps}$ nucleation timescale for the
optically heated unit cells and a second $50\,\text{ps}$ timescale for the
formation of the FM phase for unit cells heated above $T_{\text{T}}$ by heat
transport. This approach matches the delay when $V_{\text{FM}}^{*}$ is reached
and provides an estimation of the intrinsic timescale related to the phase
transition driven by near-equilibrium heating via thermal electrons and
phonons.

In summary, we discovered an intrinsic fluence-, temperature- and field-
independent $8\,\text{ps}$ timescale for locally establishing macroscopic
properties of the FM phase in FeRh via nucleation of domains upon direct
optical excitation. This timescale is not limited by the relaxation of the
lattice with sound velocity as stated previously [28, 41] but represents
intrinsic kinetics of the first-order phase transition. In the high fluence
regime and for sample temperatures near the transition temperature, we
additionally observe a delayed and slow growth of the FM phase into the depth
of an inhomogenously excited FeRh layer driven by near-equilibrium heat
transport. This observation explains the different rise times of the
structural order parameter in previous UXRD experiments [28, 30]. Our modeling
reveals that the rise of the FM phase after heating above the transition
temperature by near-equilibrium heat transport is substantially slower than
nucleation driven by direct optical excitation. This hints to the crucial role
of modifying the electronic band structure within the first picosecond via
photoexcited electrons[25] for the kinetics of the subsequent formation of the
equilibrium FM phase.

We acknowledge the DFG for financial support via No. BA 2281/11-1 and Project-
No. 328545488 – TRR 227, project A10. V. U. acknowledges the project
CZ.02.01.01/00/22_008/0004594. Access to the CzechNanoLab Research
Infrastructure was supported by the MEYS CR (LM2023051). Beamtimes at the
KMC-3 XPP endstation of the synchrotron radiation facility BESSY II at the
Helmholtz Zentrum Berlin were required for thorough sample characterization
and parameter-free crosscheck measurements.

## References

  * Avrami [1939] M. Avrami, Kinetics of phase change. i general theory, The Journal of chemical physics 7, 1103 (1939).
  * Gatel  _et al._ [2017] C. Gatel, B. Warot-Fonrose, N. Biziere, L. Rodríguez, D. Reyes, R. Cours, M. Castiella, and M.-J. Casanove, Inhomogeneous spatial distribution of the magnetic transition in an iron-rhodium thin film, Nature Communications 8, 15703 (2017).
  * Uhlíř  _et al._ [2016] V. Uhlíř, J. A. Arregi, and E. E. Fullerton, Colossal magnetic phase transition asymmetry in mesoscale ferh stripes, Nature communications 7, 13113 (2016).
  * Arregi  _et al._ [2023] J. A. Arregi, F. Ringe, J. Hajduček, O. Gomonay, T. Molnár, J. Jaskowiec, and V. Uhlíř, Magnetic-field-controlled growth of magnetoelastic phase domains in ferh, Journal of Physics: Materials 6, 034003 (2023).
  * Roy  _et al._ [2004] S. Roy, G. Perkins, M. Chattopadhyay, A. Nigam, K. Sokhey, P. Chaddah, A. Caplin, and L. Cohen, First order magnetic transition in doped cefe 2 alloys: Phase coexistence and metastability, Physical review letters 92, 147203 (2004).
  * De Jong  _et al._ [2013] S. De Jong, R. Kukreja, C. Trabant, N. Pontius, C. Chang, T. Kachel, M. Beye, F. Sorgenfrei, C. Back, B. Bräuer, _et al._ , Speed limit of the insulator–metal transition in magnetite, Nature materials 12, 882 (2013).
  * Randi  _et al._ [2016] F. Randi, I. Vergara, F. Novelli, M. Esposito, M. Dell’Angela, V. Brabers, P. Metcalf, R. Kukreja, H. A. Dürr, D. Fausti, _et al._ , Phase separation in the nonequilibrium verwey transition in magnetite, Physical Review B 93, 054305 (2016).
  * Qazilbash  _et al._ [2007] M. M. Qazilbash, M. Brehm, B.-G. Chae, P.-C. Ho, G. O. Andreev, B.-J. Kim, S. J. Yun, A. Balatsky, M. Maple, F. Keilmann, _et al._ , Mott transition in vo2 revealed by infrared spectroscopy and nano-imaging, Science 318, 1750 (2007).
  * Baldasseroni  _et al._ [2012] C. Baldasseroni, C. Bordel, A. Gray, A. Kaiser, F. Kronast, J. Herrero-Albillos, C. Schneider, C. Fadley, and F. Hellman, Temperature-driven nucleation of ferromagnetic domains in ferh thin films, Applied Physics Letters 100, 262401 (2012).
  * Keavney  _et al._ [2018] D. J. Keavney, Y. Choi, M. V. Holt, V. Uhlíř, D. Arena, E. E. Fullerton, P. J. Ryan, and J.-W. Kim, Phase coexistence and kinetic arrest in the magnetostructural transition of the ordered alloy ferh, Scientific reports 8, 1778 (2018).
  * Polesya  _et al._ [2016] S. Polesya, S. Mankovsky, D. Ködderitzsch, J. Minár, and H. Ebert, Finite-temperature magnetism of ferh compounds, Physical Review B 93, 024423 (2016).
  * Wegkamp  _et al._ [2014] D. Wegkamp, M. Herzog, L. Xian, M. Gatti, P. Cudazzo, C. L. McGahan, R. E. Marvel, R. F. Haglund Jr, A. Rubio, M. Wolf, _et al._ , Instantaneous band gap collapse in photoexcited monoclinic vo 2 due to photocarrier doping, Physical review letters 113, 216401 (2014).
  * Ju  _et al._ [2004] G. Ju, J. Hohlfeld, B. Bergman, R. J. van de Veerdonk, O. N. Mryasov, J.-Y. Kim, X. Wu, D. Weller, and B. Koopmans, Ultrafast generation of ferromagnetic order via a laser-induced phase transformation in ferh thin films, Physical review letters 93, 197403 (2004).
  * Bergman  _et al._ [2006] B. Bergman, G. Ju, J. Hohlfeld, R. J. van de Veerdonk, J.-Y. Kim, X. Wu, D. Weller, and B. Koopmans, Identifying growth mechanisms for laser-induced magnetization in ferh, Physical Review B 73, 060407 (2006).
  * Beaurepaire  _et al._ [1996] E. Beaurepaire, J.-C. Merle, A. Daunois, and J.-Y. Bigot, Ultrafast spin dynamics in ferromagnetic nickel, Physical review letters 76, 4250 (1996).
  * Kirilyuk  _et al._ [2010] A. Kirilyuk, A. V. Kimel, and T. Rasing, Ultrafast optical manipulation of magnetic order, Reviews of Modern Physics 82, 2731 (2010).
  * Stanciu  _et al._ [2007] C. D. Stanciu, F. Hansteen, A. V. Kimel, A. Kirilyuk, A. Tsukamoto, A. Itoh, and T. Rasing, All-optical magnetic recording with circularly polarized light, Physical review letters 99, 047601 (2007).
  * Radu  _et al._ [2011] I. Radu, K. Vahaplar, C. Stamm, T. Kachel, N. Pontius, H. Dürr, T. Ostler, J. Barker, R. Evans, R. Chantrell, _et al._ , Transient ferromagnetic-like state mediating ultrafast reversal of antiferromagnetically coupled spins, Nature 472, 205 (2011).
  * Remy  _et al._ [2023] Q. Remy, J. Hohlfeld, M. Vergès, Y. Le Guen, J. Gorchon, G. Malinowski, S. Mangin, and M. Hehn, Accelerating ultrafast magnetization reversal by non-local spin transfer, Nature Communications 14, 445 (2023).
  * Arregi  _et al._ [2020] J. A. Arregi, O. Caha, and V. Uhlíř, Evolution of strain across the magnetostructural phase transition in epitaxial ferh films on different substrates, Physical Review B 101, 174413 (2020).
  * Gray  _et al._ [2012] A. Gray, D. Cooke, P. Krüger, C. Bordel, A. Kaiser, S. Moyerman, E. Fullerton, S. Ueda, Y. Yamashita, A. Gloskovskii, _et al._ , Electronic structure changes across the metamagnetic transition in ferh via hard x-ray photoemission, Physical review letters 108, 257208 (2012).
  * Pressacco  _et al._ [2018] F. Pressacco, V. Uhlíř, M. Gatti, A. Nicolaou, A. Bendounan, J. A. Arregi, S. K. Patel, E. E. Fullerton, D. Krizmancic, and F. Sirotti, Laser induced phase transition in epitaxial ferh layers studied by pump-probe valence band photoemission, Structural Dynamics 5, 034501 (2018).
  * Maat  _et al._ [2005] S. Maat, J.-U. Thiele, and E. E. Fullerton, Temperature and field hysteresis of the antiferromagnetic-to-ferromagnetic phase transition in epitaxial ferh films, Physical Review B 72, 214432 (2005).
  * Stamm  _et al._ [2008] C. Stamm, J.-U. Thiele, T. Kachel, I. Radu, P. Ramm, M. Kosuth, J. Minár, H. Ebert, H. Dürr, W. Eberhardt, _et al._ , Antiferromagnetic-ferromagnetic phase transition in ferh probed by x-ray magnetic circular dichroism, Physical Review B 77, 184401 (2008).
  * Pressacco  _et al._ [2021] F. Pressacco, D. Sangalli, V. Uhlíř, D. Kutnyakhov, J. A. Arregi, S. Y. Agustsson, G. Brenner, H. Redlin, M. Heber, D. Vasilyev, _et al._ , Subpicosecond metamagnetic phase transition in ferh driven by non-equilibrium electron dynamics, Nature Communications 12, 5088 (2021).
  * Ünal  _et al._ [2017] A. Ünal, A. Parabas, A. Arora, J. Ehrler, C. Barton, S. Valencia, R. Bali, T. Thomson, F. Yildiz, and F. Kronast, Laser-driven formation of transient local ferromagnetism in ferh thin films, Ultramicroscopy 183, 104 (2017).
  * Radu  _et al._ [2010] I. Radu, C. Stamm, N. Pontius, T. Kachel, P. Ramm, J.-U. Thiele, H. Dürr, and C. Back, Laser-induced generation and quenching of magnetization on ferh studied with time-resolved x-ray magnetic circular dichroism, Physical Review B 81, 104415 (2010).
  * Mariager  _et al._ [2012] S. O. Mariager, F. Pressacco, G. Ingold, A. Caviezel, E. Möhr-Vorobeva, P. Beaud, S. Johnson, C. Milne, E. Mancini, S. Moyerman, _et al._ , Structural and magnetic dynamics of a laser induced phase transition in ferh, Physical Review Letters 108, 087201 (2012).
  * Li  _et al._ [2022] G. Li, R. Medapalli, J. Mentink, R. Mikhaylovskiy, T. Blank, S. Patel, A. Zvezdin, T. Rasing, E. Fullerton, and A. Kimel, Ultrafast kinetics of the antiferromagnetic-ferromagnetic phase transition in ferh, Nature Communications 13, 2998 (2022).
  * Quirin  _et al._ [2012] F. Quirin, M. Vattilana, U. Shymanovich, A.-E. El-Kamhawy, A. Tarasevitch, J. Hohlfeld, D. von der Linde, and K. Sokolowski-Tinten, Structural dynamics in ferh during a laser-induced metamagnetic phase transition, Physical Review B 85, 020103 (2012).
  * Rössle  _et al._ [2021] M. Rössle, W. Leitenberger, M. Reinhardt, A. Koç, J. Pudell, C. Kwamen, and M. Bargheer, The time-resolved hard x-ray diffraction endstation kmc-3 xpp at bessy ii, Journal of Synchrotron Radiation 28, 948 (2021).
  * Pressacco  _et al._ [2016] F. Pressacco, V. Uhlíř, M. Gatti, A. Bendounan, E. E. Fullerton, and F. Sirotti, Stable room-temperature ferromagnetic phase at the ferh (100) surface, Scientific reports 6, 22383 (2016).
  * Fan  _et al._ [2010] R. Fan, C. J. Kinane, T. Charlton, R. Dorner, M. Ali, M. De Vries, R. M. Brydson, C. H. Marrows, B. J. Hickey, D. A. Arena, _et al._ , Ferromagnetism at the interfaces of antiferromagnetic ferh epilayers, Physical Review B 82, 184418 (2010).
  * Chen  _et al._ [2017] X. Chen, J. Feng, Z. Wang, J. Zhang, X. Zhong, C. Song, L. Jin, B. Zhang, F. Li, M. Jiang, _et al._ , Tunneling anisotropic magnetoresistance driven by magnetic phase transition, Nature Communications 8, 449 (2017).
  * Schick  _et al._ [2013] D. Schick, R. Shayduk, A. Bojahr, M. Herzog, C. v. Korff Schmising, P. Gaal, and M. Bargheer, Ultrafast reciprocal-space mapping with a convergent beam, Journal of Applied Crystallography 46, 1372 (2013).
  * Schick  _et al._ [2012] D. Schick, A. Bojahr, M. Herzog, C. v. K. Schmising, R. Shayduk, W. Leitenberger, P. Gaal, and M. Bargheer, Normalization schemes for ultrafast x-ray diffraction using a table-top laser-driven plasma source, Review of Scientific Instruments 83, 025104 (2012).
  * Willig  _et al._ [2019] L. Willig, A. von Reppert, M. Deb, F. Ganss, O. Hellwig, and M. Bargheer, Finite-size effects in ultrafast remagnetization dynamics of fept, Physical Review B 100, 224408 (2019).
  * Mattern  _et al._ [2023] M. Mattern, A. von Reppert, S. P. Zeuschner, M. Herzog, J.-E. Pudell, and M. Bargheer, Concepts and use cases for picosecond ultrasonics with x-rays, Photoacoustics 31, 100503 (2023).
  * Schick [2021] D. Schick, udkm1dsim–a python toolbox for simulating 1d ultrafast dynamics in condensed matter, Computer Physics Communications 266, 108031 (2021).
  * Richardson  _et al._ [1973] M. Richardson, D. Melville, and J. Ricodeau, Specific heat measurements on an fe rh alloy, Physics Letters A 46, 153 (1973).
  * Kang  _et al._ [2023] K. Kang, H. Omura, S. Inoue, D. Yesudas, author O. Lee, author K. J. Lee,author H. W. Lee,T. Taniyama, and G. M. Choi, Spin current driven by ultrafast magnetization of FeRh, Nature communications 14, 3619 (2023).
  * Cao  _et al._ [2008] J. Cao, N. T. Nam, S. Inoue, H. Y. Y. Ko, N. N. Phuoc, and T. Suzuki, Magnetization behaviors for ferh single crystal thin films, Journal of Applied Physics 103, 07F501 (2008).
  * Agarwal [2021] N. Agarwal, Ultrafast dynamics of electronic structure and domain nucleation during photo-induced phase transition in FeRh, Ph.D. thesis, Staats-und Universitätsbibliothek Hamburg Carl von Ossietzky (2021).