# Subpicosecond metamagnetic phase transition driven by non-equilibrium
electron dynamics

Federico Pressacco  The Hamburg Centre for Ultrafast Imaging, Hamburg
University, Luruper Chaussee 149, 22761, Hamburg, Germany  DESY Photon
Science, Hamburg Germany  e-mail<EMAIL_ADDRESS>Davide Sangalli
Istituto di Struttura della Materia—Consiglio Nazionale delle Ricerche (CNR-
ISM), Division of Ultrafast Processes in Materials (FLASHit), Via Salaria Km
29.5, CP 10, I-00016 Monterotondo Stazione, Italy  European Theoretical
Spectroscopy Facility (ETSF)  Vojtěch Uhlíř  CEITEC BUT, Brno University of
Technology, Purkyňova 123, 612 00 Brno, Czech Republic  Institute of Physical
Engineering, Brno University of Technology, Technická 2, 616 69 Brno, Czech
Republic  Dmytro Kutnyakhov  DESY Photon Science, Hamburg Germany  Jon Ander
Arregi  CEITEC BUT, Brno University of Technology, Purkyňova 123, 612 00 Brno,
Czech Republic  Steinn Ymir Agustsson  Johannes Gutenberg-Universität,
Institute of Physics, Staudingerweg 7, 55128 Mainz, Germany  Günter Brenner
DESY Photon Science, Hamburg Germany  Harald Redlin  DESY Photon Science,
Hamburg Germany  Michael Heber  DESY Photon Science, Hamburg Germany  Dmitry
Vasilyev  Johannes Gutenberg-Universität, Institute of Physics, Staudingerweg
7, 55128 Mainz, Germany  Jure Demsar  Johannes Gutenberg-Universität,
Institute of Physics, Staudingerweg 7, 55128 Mainz, Germany  Gerd Schönhense
Johannes Gutenberg-Universität, Institute of Physics, Staudingerweg 7, 55128
Mainz, Germany  Matteo Gatti  Laboratoire des Solides Irradiés, École
Polytechnique, CNRS, CEA/DRF/IRAMIS, Institut Polytechnique de Paris, F-91128
Palaiseau, France  European Theoretical Spectroscopy Facility (ETSF)
Synchrotron SOLEIL, L’Orme des Merisiers, Saint-Aubin, BP 48, F-91192 Gif-sur-
Yvette, France  Andrea Marini  Istituto di Struttura della Materia—Consiglio
Nazionale delle Ricerche (CNR-ISM), Division of Ultrafast Processes in
Materials (FLASHit), Via Salaria Km 29.5, CP 10, I-00016 Monterotondo
Stazione, Italy  European Theoretical Spectroscopy Facility (ETSF)  Wilfried
Wurth  The Hamburg Centre for Ultrafast Imaging, Hamburg University, Luruper
Chaussee 149, 22761, Hamburg, Germany  DESY Photon Science, Hamburg Germany
Fausto Sirotti  Synchrotron SOLEIL, L’Orme des Merisiers, Saint-Aubin, BP 48,
F-91192 Gif-sur-Yvette, France  Physique de la Matiére Condensée, CNRS and
École Polytechnique, IP Paris, F-91128 Palaiseau, France

###### Abstract

Femtosecond light-induced phase transitions between different macroscopic
orders provide the possibility to tune the functional properties of condensed
matter on ultrafast timescales. In first-order phase transitions, transient
non-equilibrium phases and inherent phase coexistence often preclude non-
ambiguous detection of transition precursors and their temporal onset. Here,
we present a study combining time-resolved photoelectron spectroscopy and ab-
initio electron dynamics calculations elucidating the transient subpicosecond
processes governing the photoinduced generation of ferromagnetic order in
antiferromagnetic FeRh. The transient photoemission spectra are accounted for
by assuming that not only the occupation of electronic states is modified
during the photoexcitation process. Instead, the photo-generated non-thermal
distribution of electrons modifies the electronic band structure. The
ferromagnetic phase of FeRh, characterized by a minority band near the Fermi
energy, is established $350\pm 30$ fs after the laser excitation. Ab-initio
calculations indicate that the phase transition is initiated by a photoinduced
Rh-to-Fe charge transfer.

### Introduction

Emergence of long-range ordered states in condensed matter is typically a
consequence of a fine interplay between the coupled spin, charge, orbital, and
lattice degrees of freedom [1, 2, 3, 4]. The mechanisms vary between different
correlated oxides and metallic systems leading to specific dynamical behavior.
Excitation with ultrashort electromagnetic pulses offers the most efficient
means to control the physical properties of condensed matter systems on a
femtosecond time scale [5, 6, 7]. Materials featuring first-order phase
transitions (FOPTs) with abrupt changes in their order parameters are
especially appealing for ultrafast devices based on a functionality switch. In
this regard, prominent examples are the insulator-metal transition in VO2 [8,
9] or 1T–TaS2 [10, 11].

In order to obtain a good understanding of the relevant mechanisms triggering
the transition, it is necessary to explore the fundamental timescale of FOPTs.
However, this is often challenging with multiple coupled degrees of freedom
displaying complex dynamics upon laser-induced excitation. Moreover,
macroscopic phase coexistence at the FOPT, specifically, the processes of
nucleation and domain growth, complicate the disentanglement of dynamic
changes in order parameters. The fundamental question, whether the
modification of electronic structure drives the transition, is extensively
debated in the literature, giving key arguments on the role of photoexcited
states in double-exchange interactions [12], electronic precursors closing the
insulating gap [13], or the existence of intermediate transient phases [14, 2,
15, 9].

In the case of ferromagnetic materials, magnetization dynamics triggered by a
laser pulse leads to ultrafast demagnetization [16], associated with changes
in the spin polarization [17, 18, 19, 20], which is impacted by the spin-
dependent mobility of electrons [21, 22, 23]. In contrast, disentangling the
ultrafast response of coupled order parameters of magnetic FOPTs has been far
less investigated. Ultrafast generation of ferromagnetic (FM) order has been
observed so far in a relatively small group of materials such as manganites
[24, 1] or CuB2O4 [25]. Understanding the subpicosecond generation of FM order
across a FOPT is still a major challenge in femtomagnetism [26].

Figure 1:  Electronic properties of FeRh across the metamagnetic phase
transition. (a) Sketch of the isostructural metamagnetic phase transition in
FeRh. At room temperature (top) the system is AFM showing atomic magnetic
moments only at the Fe atom sites ($m_{\text{Fe}}=\pm 3.3$ $\mu_{\text{B}}$).
Above 360 K (bottom) the system has ferromagnetically coupled magnetic moments
at the Fe ($m_{\text{Fe}}=$ 3.1 $\mu_{\text{B}}$) and Rh ($m_{\text{Rh}}=$ 0.9
$\mu_{\text{B}}$) sites. The whole unit cell expands isotropically by about 1%
in volume [27]. (b) Schematic representation of the two possible paths in the
AFM to FM phase transition: direct thermally-driven transition to the FM phase
(green arrow), and two-step transition going through a transient electronic
state reached during photoexcitation (red arrow) followed by relaxation to the
equilibrium FM state (blue and black arrows). (c) Calculated spin-resolved
electronic density of states in the AFM and FM phases. The filled areas
represent the electronic occupation at thermal equilibrium, with the green
area highlighting the position of the Fe minority band (see manuscript text).
(d) Measured x-ray photoelectron spectra of FeRh in the AFM (black dots) and
FM (red dots) phase. The solid curve at the bottom of the graph is the
difference between the two spectra, which allows to appreciate the relative
electron density change across the transition. The data correspond to quasi-
static thermal cycling experiments prior to the time-resolved measurements.

In this work we focus on FeRh, a metallic material that undergoes a
metamagnetic FOPT from antiferromagnetic (AFM) to FM order at $T_{M}\sim$ 360
K and exhibits coupled structural, magnetic and electronic order
parameters[28, 29] (see Fig. 1a). The thermally induced, quasi-static phase
transition in FeRh (depicted in Fig. 1b by the green arrow) has been
extensively studied by following the sample magnetization, lattice parameter
or resistivity [30, 31, 32, 33]. Moreover, numerous works have studied the
AFM-FM phase transition by means of time-resolved techniques, where
photoexcitation above a threshold intensity results in a nonzero net
magnetization. Seminal pump-probe magneto-optical studies of FeRh films
suggested subpicosecond generation of FM order [34, 35]. These results were
discarded by subsequent works, which implied a much slower transition on the
order of several picoseconds [36, 37, 38, 39]. Time-resolved x-ray diffraction
indicated that the speed of the transition might be set by the time scale of
the structural changes and thus limited by the speed of sound ($\sim 5$
nm/ps), such that magnetic and structural order emerge concurrently [38]. The
establishment of long range magnetic order is naturally slower, since it is
mediated by phase boundary expansion, domain coalescence, and magnetic moment
alignment. Thus, detection of FM phase via techniques susceptible to the
magnetization direction results in a perceived delay in the emergence of FM
order [36, 37, 38].

However, FM order can also be traced by directly exploiting the specifics of
the electronic structure, naturally manifested in terms of spin unbalance and
the appearance of majority and minority spin bands [40, 41, 42]. This is
independent of spin alignment along a particular direction, and thus allows
inspecting FM order via x-ray photoelectron spectroscopy (XPS) [43, 44, 45].
Similar to the electronic signature of the insulator-metal transition [10, 8],
it was demonstrated that the modification of electronic bands might prove
equally useful to investigate laser-induced generation of FM order across the
magnetic FOPT in FeRh [46].

Here, utilizing time-resolved photoelectron momentum microscopy and supported
by first principle calculations, we demonstrate that it is the light-induced
modification of the electronic band structure that triggers the phase
transition in FeRh. In particular, we show that ultrafast laser excitation
induces a charge transfer between the Rh and Fe atoms, serving as a non-
equilibrium precursor for the formation of the FM band structure on the
subpicosecond timescale.

### Results

The establishment of the FM phase in FeRh is accompanied by the appearance of
a narrow peak located about $150$ meV below the Fermi energy $E_{\text{F}}$
due to the occupation of a spin-polarized Fe band (see green highlighted area
in Fig. 1c). The photoelectron spectroscopy data shown in Fig. 1d are the
momentum integrated energy distribution curves measured at room temperature
for the AFM phase (black dots) and at $420$ K for the FM phase (red dots). We
use this spectral feature to follow the emergence of the FM phase after laser
excitation. Pump-probe experiments were performed at the FLASH Free Electron
Laser (FEL) in Hamburg using near-infrared ($800$ nm, $1.55$ eV) pulses of
$90$ fs coupled with $130$ fs soft x-ray pulses with a photon energy of
$\hslash\omega=123.5$ eV (see Methods for details). The laser fluence was
$5.6$ mJ/cm2, which is above the threshold value to induce the FOPT [34].

Figure 2: Time-resolved x-ray photoelectron spectroscopy at room temperature.
(a) Energy- and delay-dependent matrix of the measured spectra. The vertical
dashed line marks the position of the Fermi level $E_{\text{F}}$, while the
horizontal solid line designates the time zero, $t_{0}$. The appearance of
electronic density in the unoccupied states is a fingerprint of the laser
excitation. We used this spectral feature to identify the temporal overlap
between the optical pump and the x-ray probe pulses. (b) Differential
photoelectron spectra at selected delays. To enhance the signal to noise
ratio, we average the unpumped spectra (between $-1$ ps and $-0.5$ ps) and
subtract the average spectrum from each row of the matrix. This allows
evaluating the statistical noise at negative time delays ($-500$ fs) as well
as accentuating the temporal evolution of the photoelectron spectra. Red and
blue shaded areas indicate an increase and a reduction of the electron density
with respect to the spectra at negative delays, respectively.

Fig. 2a presents the time-resolved, $k$-integrated photoelectron spectra
measured as a function of the delay between the optical pump and x-ray probe
pulses, focusing on a 4 ps window around time zero $t_{0}$. One can clearly
identify distinct regions in the time-dependent spectra: the temporal overlap
between the optical and x-ray pulses (about 100 fs around $t_{0}$), the
relaxation of electrons towards the Fermi energy on the 100 fs timescale, and
the subsequent changes in the density of states near the Fermi level,
associated with the formation of the Fe minority band. Differential energy-
dependent profiles, reported in Fig. 2b, provide a clearer picture. These are
retrieved by averaging the measured photoelectron spectra within a $\pm$ 50 fs
temporal region for each of the indicated time delays, and subtracting the
average photoelectron spectra at negative time delays.

The photoexcited electrons relax via electron-electron and electron-phonon
scattering, leading to the onset of a Fermi-like distribution (see the red
shaded areas for $E-E_{\text{F}}>0$ in Fig. 2b). A reduction of the electron
density below the Fermi level is also observed around $t_{0}$ (blue shaded
areas for $E-E_{\text{F}}<0$ in Fig. 2b). At the same delay, the spectrum
shows excitation of electrons up to $3.1$ eV above $E_{\text{F}}$ (see also
the inset in Fig. 3). Note that the electron density above 1.5 eV is almost
two orders of magnitude lower. The changes in the population above the Fermi
level can be explained by one- and two-photon absorption processes, such that
laser excitation ($\hslash\omega_{p}=1.55$ eV) promotes electrons into states
within the energy range [$E_{\text{F}},E_{\text{F}}+2\hslash\omega_{p}$].
Then, electrons start to relax towards the Fermi level and accumulate in an
energy region of a few hundred meV above $E_{\text{F}}$. On the other hand,
the transient depletion of electronic density below the Fermi level is
concentrated between $-3.1$ eV and $-1.55$ eV
[$E_{\text{F}}-2\hslash\omega_{p},E_{\text{F}}-\hslash\omega_{p}$]. Assuming
the photo-induced depletion, one would expect to observe measurable changes in
the photoelectron yield only for energies $1.55$ eV below the $E_{\text{F}}$
(two photon contribution should be negligible). This suggests that the laser
induced changes in the electronic distribution close to the Fermi level cause
severe modifications of the deeper lying bands. This implies that the in-fieri
FOPT involves changes in the overall electronic structure of the system.

Figure 3: Comparison of experimental and computed photoelectron spectra of
FeRh. Spectra at equilibrium ($t=-0.5$ ps) and at the maximum of the pump
pulse ($t=0$ ps) are shown in linear (top panel) and logarithmic scales
(inset). The bottom panel exhibits the difference between the two spectra in
the top panel. The green (red) region represents the energy range to (from)
which electrons are excited with one-photon (1.55 eV) processes. The light
green region in the top panel inset identifies the energy range that can be
reached by two-photon (3.1 eV) processes.

To elucidate the electronic dynamics near the zero time delay, we compare the
experimental results with time-dependent density functional theory (TD-DFT)
calculations of the electronic structure performed on FeRh (see Fig. 3). Here,
we select a laser fluence which gives a good quantitative agreement with the
measured photoelectron spectra. The calculated one- and two-photon absorption
intensity above the Fermi level presented in the inset of Fig. 3 corresponds
to an excitation of $0.25$ electrons per FM unit cell of FeRh (see Fig. 1a),
i.e. $\sim 10^{22}\leavevmode\nobreak\ \hbox{cm}^{-3}$. This is close to the
experimental estimate of $0.15$ electrons per unit cell for a $5.6$ mJ/cm2
fluence. In Fig. 3, the calculated XPS spectrum and its time evolution are
obtained considering (i) the time evolution of the electronic distribution
function $f_{n\mathbf{k}}(t)$, and (ii) the time evolution of the electronic
band structure $\epsilon_{n\mathbf{k}}(t)$ (see Methods). The changes in
$f_{n\mathbf{k}}(t)$ give rise to a significant signal only in the red and
green shaded regions between $-1.55$ eV and $+1.55$ eV, while the effect of
changes in $\epsilon_{n\mathbf{k}}(t)$ is most prominent well below the Fermi
level. Thus, the signal below $-1.55$ eV is fully due to changes in
$\epsilon_{n\mathbf{k}}(t)$. The fact that it is negative implies a reduction
in the density of states (see the relative height difference of the red and
blue solid curves in Fig. 3).

On the other hand, the changes in the region between $-1.55$ eV and 0 eV
result from two effects which sum to a negligible signal: $f_{n\mathbf{k}}(t)$
gives a negative contribution due to the promotion of electrons into the
originally unoccupied states while the change in $\epsilon_{n\mathbf{k}}(t)$
must result in an increase of the DOS. The region above the Fermi level seams
to be mainly governed by $f_{n\mathbf{k}}(t)$. The overall agreement between
theory and experiment is very good. The observed slight differences may be due
to relaxation processes in the experiment already active during the pumping
phase, making the electron distribution more peaked towards the Fermi level.

The relaxation of excited electrons proceeds with the formation of a peak
above the Fermi level between $100$ fs and $300$ fs (see Fig. 2b). At a time
delay of around $400$ fs, the peak crosses the Fermi level and after $500$ fs,
its position stabilizes at the energy value which is characteristic for the Fe
minority band of the FM phase in FeRh [46].

Figure 4: Subpicosecond generation of the electronic FM order in FeRh. (a)
Energy- and delay-dependent differential matrix of the measured photoelectron
spectra. We subtracted the average of unpumped spectra (between $-$1 ps and
$-0.5$ ps) from each measured spectrum. The effect of laser excitation is
evident around time zero, indicating depletion of the occupied states and the
corresponding population of the empty states. In addition, an increase in the
electronic density close to the Fermi level is observed at positive time
delays. The red, blue, and black bars on the right hand side mark the
representative integration regions for tracking the electronic dynamics in the
unoccupied states, the formation of the Fe minority peak, and the modification
of the deeper bands, respectively. (b) Temporal evolution of the electronic
density in the three characteristic energy regions marked in (a). The
population of states above the Fermi level shows a fast rise and a consecutive
decay around $t_{0}$ (empty red circles). The deeper bands (empty black
diamonds) show a corresponding depletion and recovery which reflects the
dynamics of the unoccupied states. However, their occupation level stabilizes
$300$ fs after $t_{0}$ and remains constant thereafter. The electronic density
slightly below $E_{\text{F}}$ (filled blue circles) shows a moderate increase
during the laser excitation up to $300$ fs delay, followed by a pronounced
increase due to the shift of Fe minority band below $E_{\text{F}}$ at a delay
$t_{D}$ of 350 fs. This value was extracted by fitting the error function to
the experimental results (black solid line). Subsequently, the minority Fe
band peak intensity remains constant throughout the investigated delay range.

Further insights into the FOPT dynamics are obtained from the complete
differential matrix, presented in Fig. 4a, monitoring the modification of the
electronic density. We selected three energy ranges marked by the red, blue,
and black bars placed on the right hand side of Fig. 4a. The first (red)
accounts for the electronic density above $200$ meV and identifies the total
number of electrons injected into the unoccupied states upon photoexcitation.
The second (blue) goes from $-240$ meV to $0$ meV and is used to monitor the
formation of the Fe minority peak across the phase transition. The third
(black) includes the region from $-1.8$ eV to $-2.8$ eV and is used to follow
the modification of deeper bands.

Fig. 4b shows the time evolution of the integrated signal in each range. The
injection of electrons into the unoccupied states takes place near zero delay
(we recall that our system’s response function is $150$ fs) and then rapidly
decays (red empty circles), the process being finished by about $500$ fs. The
intensity at the position of the Fe minority peak starts to grow already
during the laser excitation (filled blue circles), due to both the
modification of the electronic structure and the Fermi level smearing.
However, it is only about $300$ fs after the excitation that it displays a
fast density increase. The characteristic time delay $t_{D}$ and subsequent
rise time $\tau$ of the transition are obtained by fitting the integrated
electron density at the Fe minority peak region with an error function (see
Methods) and yields values of $t_{D}=350\pm 30$ fs and $\tau=220\pm 110$ fs.
In addition, the data represented by black diamonds in Fig. 4b, which indicate
the time-dependent population of bands in a region below $E_{\text{F}}$,
nearly mirror the population of the unoccupied levels up to $300$ fs, with a
fast depletion and recovery. The curve stabilizes thereafter at a finite
value, implying permanent modifications of the deeper bands already after
$300$ fs, during which the Fe minority peak is still shifting towards its
final position at 150 meV below $E_{\text{F}}$. Similar differences in the
behaviour of the electronic density at and below the Fermi level have been
observed during ultrafast demagnetization process in Fe [17] and Co [18]. In
the present case, this behavior shows that the FOPT in FeRh is mediated by a
transient electronic phase in which the electronic structure is different from
both the AFM and FM phases. The transient phase exists in a delay range from
up to $500$ fs after laser excitation.

### Discussion

The unexpected reduction of the photoelectron yield below the one-photon
absorption range ($\sim 1.55$ eV), an energy region where the electronic
populations $f_{n\mathbf{k}}(t)$ cannot be strongly affected by the laser
excitation, is explained by theoretical calculations. The effect can only be
accounted for by considering the photoinduced change in the electronic band
structure (i.e. density of states). Time-resolved XPS spectra are usually
described and interpreted in terms of changes in the population
$f_{n\mathbf{k}}(t)$ only. This is clearly insufficient for the ultrafast
dynamics in FeRh, since changes in $\epsilon_{n\mathbf{k}}(t)$ must be
considered on the same level even before the phase transition is complete.

Figure 5: Computed photoinduced changes in charge and spin density in FeRh.
(a, b) Changes in the charge density ${n({\bf r},t_{f})-n^{eq}({\bf r})}$ and
(c) spin density ${S_{z}({\bf r},t_{f})-S_{z}^{eq}({\bf r})}$ at the end of
the laser pulse ($t=t_{f}$) with respect to the equilibrium configuration.
Densities are represented in atomic units on two different FeRh{111}-planes:
the plane just below the $Fe(\uparrow)$ atoms (panel a) and the plane
containing the Rh atoms (panels b, c). Fe and Rh atom positions are indicated
by red and grey segments of the lattice, respectively. Charge is transferred
from the occupied (bonding) orbitals of Rh atoms (panel b), to the unoccupied
(anti-bonding) orbitals of Rh and Fe (panel a). Since the Fe anti-bonding
levels are filled in this process, the local Fe spin density is reduced. As a
result, there is a redistribution of spin density around the Rh atoms and a
reduction of the local spin moment at the Fe atoms (panel c). The movies in
the Supplementary Information show the simulated time evolution of the charge
density variations at the Fe and Rh sites as represented in panels (a) and (b)
at the beginning and the end of the laser pulse.

The changes in $\epsilon_{n\mathbf{k}}(t)$ during and after photoexcitation
provide insights into why FeRh would relax towards the FM phase. The
photoexcitation process in the AFM state depletes the valence states, which
are characterised by hybridized Fe-Rh bonding states, and fills the unoccupied
states, where empty Fe “local minority-spin” anti-bonding states are mainly
available. As a result two processes occur: a charge (and spin) transfer from
Rh to Fe (${Rh\rightarrow Fe}$), and a symmetric spin transfer from Fe “local
majority” $\rightarrow$ Fe “local minority” (${Fe(\uparrow)\,\leftrightarrow
Fe(\downarrow)}$). Simulated charge and spin density changes upon
photoexcitation clearly show this, Fig. 5. Here, the blue regions around the
Rh atoms indicate a charge depletion, which can only partially be explained by
a local redistribution. Most of the charge is transferred to the Fe atoms (see
Fig. 5a), while there is a strong charge depletion around Rh atoms (see Fig.
5b). The ${Rh\rightarrow Fe}$ process also increases the spin density around
the Rh atoms (see Fig. 5c). On the other hand, the
${Fe(\uparrow)\,\leftrightarrow Fe(\downarrow)}$ is a pure spin transfer
process, with a strong reduction ($\sim 10\%$) of the local momentum of the Fe
atoms, which is transferred into the vicinity of the Rh atoms (see Fig. 5c).

In the AFM phase, the zero magnetic moment around the Rh atoms is a result of
a non-vanishing spin density that integrates to zero because of the
hybridization with surrounding Fe atoms with opposite moments [47, 3, 42]. The
photoexcitation alters this delicate balance. As shown for the magnetic
disorder associated with the temperature increase, the decrease of the Fe-Fe
first-neighbor AFM couplings favors the FM order of the Fe subsystem, while
inducing magnetic fluctuations on the Rh sites [3]. In turn, the induced Rh
magnetic moments stabilize the FM over the AFM state [48, 34, 49, 3]. Our
simulations therefore suggest that the change of Fe-Rh hybridization
(${Rh\rightarrow Fe}$ process) plays a critical role in the photoinduced
transition. The ${Fe(\uparrow)\,\leftrightarrow Fe(\downarrow)}$ process
corresponds to the intersite spin transfer which has been recently proposed,
on the basis of TD-DFT simulations, as a key mechanism also in other
multicomponent magnetic materials [50, 51, 52]. Here, it causes to weaken the
AFM ordering but is not sufficient to trigger the magnetic transition alone
(just after the photoexcitation, the system is still in the AFM phase).

The theoretical simulation, not including dissipating effects, cannot describe
the dynamics after the photoexcitation when electron-electron, electron-phonon
and electron-magnon interactions are at play, and the actual phase transition
takes place. Taking into account dissipating effects, one would expect further
dynamics of both $f_{n\mathbf{k}}(t)$ and $\epsilon_{n\mathbf{k}}(t)$: the
formation of a Fermi distribution and its subsequent cooling (for
$f_{n\mathbf{k}}(t)$), and the formation of the FM band structure (for
$\epsilon_{n\mathbf{k}}(t)$). Instead, our experimental time resolution is
fast enough to allow the identification of a bottleneck time in this process,
i.e., the metamagnetic transformation exhibiting a 350 fs delay. It is
associated with a change in the band structure, with the spin-minority Fe band
slightly pushed below the Fermi level and getting filled by the electrons that
progressively cool down. This transformation occurs on a subpicosecond
timescale that is faster than what was determined by previous experiments on
FeRh with a lower time resolution [46]. Most importantly, our results set a
new timescale that is faster than the lattice expansion and the establishment
of the macroscopic, long-range magnetic order [36, 37, 38].

The process is schematically depicted in Fig. 1b. Immediately after the action
of the pulse, a significant number of electrons is excited to unoccupied
states, with a non-thermal distribution of electrons $f_{n\mathbf{k}}(t)$ (red
arrow in Fig. 1b) decaying towards the Fermi level on a time scale of about
200 fs. This is accompanied by the slower dynamics of the band structure
$\epsilon_{n\mathbf{k}}$ with the formation of the peak of the Fe minority
band. The peak crosses the Fermi level at about 350 fs delay, and stabilizes
at $-150$ meV binding energy after 400 fs. During this step, the system
undergoes a purely electronic transition through a transient phase, where the
electronic band configuration evolves from $\epsilon_{n\mathbf{k}}(t)$ to an
intermediate electronic FM phase $\epsilon_{n\mathbf{k}}^{e-FM}$ (blue arrow
in Fig. 1b). Once the electronic distribution reaches the configuration of the
FM phase (after 400 fs), the relaxation of the lattice parameter towards the
equilibrium value of the FM phase then follows on a longer time scale (black
arrow in Fig. 1b).

### Conclusion

In conclusion, we determine the existence of a transient electronic phase
needed to induce the AFM to FM phase transition of FeRh using pump-probe
photoelectron spectroscopy at the FLASH FEL facility. The results are
supported by electronic structure calculations, which explain the details of
the dynamics following the laser excitation. The time-resolved photoemission
experiment at a laser fluence of $5.6$ mJ/cm2 is well reproduced assuming the
excitation of $0.25$ electrons per unit cell of FeRh. At these fluences, the
photon absorption cannot be described simply as promotion of electrons from
filled to empty states of the calculated band structure, but the modified
electron population induces a modification of the band structure as well,
which is confirmed by the good agreement between theory and experiments. The
laser excitation results in the transfer of electrons from the occupied $d$
orbitals below the Fermi level to the unoccupied $d$ orbitals above the Fermi
level, with a partial transfer of electrons from the Rh to the Fe sites. The
transient electronic phase exists up to $500$ fs after the laser excitation.
The emergence of the FM phase can be followed by the appearance and position
of the Fe minority band near the Fermi level with characteristic time of
$\tau=220\pm 110$. Photoexcited electrons relax across the Fermi level and
establish the FM electronic band structure 400 fs after the laser excitation.
We thus conclude that metamagnetism in FeRh is triggered on a subpicosecond
time scale. Further exploration of the laser-induced dynamics in ultrathin and
nanoscale confined FeRh [32] could lead to ultrafast devices based on magnetic
order-order phase transitions at room temperature.

## Methods

Sample and surface preparation. The sample consists of an epitaxial 80-nm-
thick FeRh(001) film grown onto a MgO(001) substrate by dc magnetron
sputtering using an equiatomic target. The films were grown at $725$ K and
post-annealed in situ at $1070$ K for $45$ minutes in order to achieve CsCl-
type chemical ordering. Upon cooling down the samples in the ultra high vacuum
chamber, single-layer graphene is formed on top of the FeRh surface by
segregating the carbon from the film [53]. This provides oxidation protection
in air and avoids the need for further capping layers of the FeRh layer to be
transported to the FEL facility without degradation. The good quality of the
crystallographic texture and the existence of the magnetic phase transition
were confirmed using x-ray diffraction and vibrating sample magnetometry,
respectively [54]. The sample surface was prepared via annealing only, in
order to preserve the graphene layer, and tested with XPS prior to the time-
resolved experiments as described in earlier works [42, 46]. Examination with
low-energy electron diffraction revealed the expected reconstruction pattern
of the FeRh(001) surface.

Experiment. The experiments are performed at the plane grating monochromator
beamline [55, 56] at FLASH [57, 58], using the HEXTOF end-station [59]. The
pump-probe scheme is established by a near-infrared pulse of $90$ fs coupled
with a FEL pulse of about $130$ fs (both values are the full width half
maximum, FWHM), which provide an estimated system response function [59],
i.e., the effective pump-probe correlation, of $\sim 150$ fs FWHM. The optical
pump and x-ray probe energies were set to $1.55$ eV and $123.5$ eV,
respectively. The energy resolution of $\sim 150$ meV is extracted from the
Fermi level fit.

Photoexcited electrons near normal emission were detected using a momentum
microscope, which has an acceptance angle of $2\pi$ above the sample surface
and can image the full Brillouin zone (BZ) with up to $7\,{\text{\AA}}^{-1}$
diameter [60, 59]. We used a negative extractor voltage ($\sim 40$ V with
respect to the sample potential). This retarding field between the sample and
extractor effectively removes the slow secondary electrons originating from
the x-ray photons and pump-laser-induced slow electrons. All background
electrons with energies less than $\sim 4$ eV are thus repelled within the
first $400$ $\upmu$m above the sample surface. This removal of space charge
comes at the expenses of $k$-resolution and causes a reduction of the
$k$-field-of-view to $1.3\,\text{\AA}^{-1}$. Integrating over this $k$-field
represents the integral of 60$\%$ of the BZ of FeRh, which was sufficient to
well identify the peak associated with the FM phase. We characterize the
sample surface by measuring the spectra of the system in the AFM and FM phases
at fixed temperatures, and obtained line-shapes equivalent to those reported
in ref. 46 (see Fig. 1d). The presence of the peak at about $\sim 150$ meV
below the Fermi level is the signature of the Fe minority band characteristic
of the FM phase. To fit the experimental data in Fig. 4b, we used an error
function of the following form:

$f(t)=y_{0}+A\left[1+\text{erf}\left(\frac{t-t_{D}}{\tau}\right)\right]$

where $y_{0}$ is a vertical offset, $A$ is the amplitude, $t_{D}$ is the
temporal onset of the transition (with respect to $t_{0}$), and $\tau$ is the
characteristic rise time.

Theory. We calculate from first principles the equilibrium and non–equilibrium
properties of FeRh using the pw.x and yambo codes [61, 62, 63, 64] within
Density Functional Theory (DFT) and its Time Dependent (TD–DFT) extension. At
equilibrium both the FM and AFM phases are computed within the local density
approximation (LDA) fully including spin–orbit coupling (SOC). An energy cut-
off of $65$ Ry is used for the wave–functions with a $5\times 5\times 5$
sampling of the BZ for the self–consistent calculation. The experimental
lattice parameter, 5.966 Å, is chosen for the AFM structure: an FCC unit cell
containing 4 atoms (there is a factor 2 compared to the parameter of the BCC
unit cell with 2 atoms used in ref. 65). The FM ground state is then computed
for the same unit cell and for a unit cell with a $1$% lattice expansion. LDA
gives a low negative stress using the experimental value of the lattice
parameters in both the FM and the AFM phase. We use the experimental values
and we verified that changes in the lattice parameters very weakly affect the
electronic density of states.

We also verified that the Generalized Gradient Approximation (GGA) gives small
improvements in comparison with experimental values, which however are not
relevant to the present work. For this reason we used LDA which is more easily
handled in the non equilibrium TD–DFT simulations with SOC. Finally we
verified that the AFM structure displays a phonon instability as reported in
the literature [65, 66, 67].

Subsequently, a non self–consistent calculation (NSCF) on a $8\times 8\times
8$ sampling of the BZ is performed. The electronic density of states (DOS) of
the two structures reported in Fig. 1c is computed starting from such NSCF
calculation. We then construct the XPS spectrum from the projection of the DOS
on the atomic orbitals of Fe and Rh. The projected DOS are weighted using
tabulated photoionization cross sections [68] for a probe of $125$ eV (in
practice the signal is dictated by Fe(3d) orbitals). Moreover, we use energy
dependent lifetimes of the form
$\gamma_{n\mathbf{k}}=A+B\,d(\epsilon_{n\mathbf{k}})+C(\epsilon_{F}-\epsilon_{n\mathbf{k}})^{2}$,
where the first constant contribution $A=60$ meV mimics the experimental
resolution, the second term $B$, proportional to the electronic DOS
$d(\epsilon)$, mimics the electron–phonon lifetimes and, finally, the term
which grows quadratically away from the Fermi level mimics the
elecron–electron lifetimes. Finally, the effect of temperature is included in
the Fermi distribution used for the electronic occupations. The resulting
spectrum is shown in Fig. 1c.

The TD–DFT simulations, as implemented in the yambo code [69], are then
performed propagating the Kohn–Sham density matrix in the basis–set of the
equilibrium wave–functions under the action of the same pump pulse used in the
experiment. The NSCF DFT calculation is used as a starting point for TD–DFT.
The laser pulse parameters are equivalent to the experimental conditions. In
particular, the fluence is chosen considering: (i) the experimental fluence,
(ii) the fact that part of the pulse is reflected by the sample, and (iii) the
fact that the external field is renormalized by the induced field. The effect
of both (ii) and (iii) is estimated taking into account the dielectric
function of bulk FeRh. In particular, point (iii) needs to be considered since
we adopt the so called transverse gauge, where the macroscopic (or $G=0$)
component of the Hartree field is subtracted from the microscopic TD-DFT
equations. The density matrix is calculated on a $8\times 8\times 8$ grid of
$\mathbf{k}$ points in the BZ including all states from $-3.5$ up to $5.5$ eV.
The Kohn–Sham field felt by the electrons is updated at each time step during
the simulation. The non-equilibrium DOS is then computed by diagonalizing the
Hamiltonian evaluated from the photoexcited electron density after the action
of the pump pulse. The adiabatic non-equilibrium XPS spectrum is finally
computed using the same smearing used for the equilibrium spectra. The
electronic occupation is obtained from the diagonal elements of the density
matrix in the basis updated during the course of the simulation.

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## Acknowledgements

This work is dedicated to Wilfried Wurth, who passed away on May 8, 2019. We
acknowledge support by the scientific and technical staff of FLASH, as well as
Holger Meyer and Sven Gieschen form University of Hamburg. This work was
supported by the excellence cluster “The Hamburg Centre for Ultrafast Imaging
- Structure, Dynamics and Control of Matter at the Atomic Scale” of the
Deutsche Forschungsgemeinschaft (DFG EXC 1074) and through the SFB 925
"Lichtinduzierte Dynamik und Kontrolle korrelierter Quantensysteme" (project
B2). It received funding from the EU-H2020 research and innovation program
under European Union projects “MaX” Materials design at the eXascale
H2020-EINFRA-2015-1 (Grant Agreement No. 824143) and “NFFA” Nanoscience
Foundries and Fine Analysis-Europe H2020-INFRAIA-2014-2015 (Grant Agreement
No. 654360) having benefited from the access provided by the ISM node (CNR,
Italy). We acknowlege the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) – TRR 173 – 268565370 (projects A02 and A05). Access to
the CEITEC Nano Research Infrastructure was supported by the Ministry of
Education, Youth and Sports (MEYS) of the Czech Republic under the projects
CEITEC 2020 (LQ1601) and CzechNanoLab (LM2018110). We acknowledge funding from
the Italian project MIUR PRIN Grant No. 20173B72NB. This work has received
funding from the European Union’s Horizon 2020 research and innovation program
under the Marie Skłodowska-Curie and it is co-financed by the South Moravian
Region under grant agreement No. 665860.

## Author contributions

F.P., V.U., J.A.A., M.G., D.S. and F.S. designed the project. F.P., D.K.,
M.H., S.Y.A., G.B., H.R., D.V., V.U., J.A.A. and F.S. performed the time-
resolved XPS experiments and analyzed the data. M.G., D.S. and A.M. designed
the theoretical approach to the problem. J.A.A. and V.U. prepared and
characterized the samples. All authors discussed the results. F.P., V.U.,
J.A.A., D.S., M.G. and F.S. wrote the paper with contributions from all
authors and critical revision from J.D., G.S. and W.W.

## Competing financial interests

The authors declare no competing interests.

## Supplementary Information

We provide four animations showing the simulated dynamics of charge density
variations $n({\bf r},t)-n^{eq}({\bf r})$ as represented in panels a and b of
Fig.5 during two time-windows. The first time window
($-65\,\text{fs}<t<-55\,\text{fs}$) corresponds to the beginning of the
photoexcitation process and shows the laser-induced charge oscillations for
about four periods of the main laser frequency (for $\lambda=800\,\text{nm}$,
$T=\,2.67\,\text{fs}$). During this time window, $n({\bf r},t)-n^{eq}({\bf
r})\sim 10^{-7}\,\text{a.u.}$ (see Supplementary Videos 1 & 2). The second
time window ($135\,\text{fs}<t<137.5\,\text{fs}$) is close to the end of the
optical pulse where a persistent change of the charge density is observed.
This variation is due to the update of the mean field felt by the electrons,
which redistribute accordingly in the unit cell. Laser-induced charge
oscillations are still present, but only constitute a correction on top of the
much larger persistent variation. At the end of the laser pulse, $n({\bf
r},t)-n^{eq}({\bf r})\sim 10^{-5}\,\text{a.u.}$ (see Supplementary Videos 3 &
4.)