Probing the timescale of the exchange interaction
in a ferromagnetic alloy
Stefan Mathiasa,b,1,2, Chan La-O-Vorakiata,1, Patrik Grychtola,c, Patrick Granitzkaa,b, Emrah Turguta, Justin M. Shawd,
Roman Adamc, Hans T. Nembachd, Mark E. Siemensa,e, Steffen Eichb, Claus M. Schneiderc, Thomas J. Silvad,
Martin Aeschlimannb, Margaret M. Murnanea, and Henry C. Kapteyna
aDepartment of Physics and JILA, University of Colorado, Boulder, CO 80309-0440;
of Standards and Technology, Boulder, CO 80305-3328; and
bUniversity of Kaiserslautern and Research Center OPTIMAS, 67663,
dElectromagnetics Division, National Institute
eDepartment of Physics and Astronomy, University of Denver, Denver, CO 80208-6900
cPeter Grünberg Institute, PGI-6, Research Center Jülich, 52425, Jülich, Germany;
Contributed by Margaret M. Murnane, January 27, 2012 (sent for review December 22, 2011)
The underlying physics of all ferromagnetic behavior is the coop-
erative interaction between individual atomic magnetic moments
that results in a macroscopic magnetization. In this work, we use
extreme ultraviolet pulses from high-harmonic generation as an
element-specific probe of ultrafast, optically driven, demagnetiza-
tion in a ferromagnetic Fe-Ni alloy (permalloy). We show that for
times shorter than the characteristic timescale for exchange cou-
pling, the magnetization of Fe quenches more strongly than that
of Ni. Then as the Fe moments start to randomize, the strong fer-
romagnetic exchange interaction induces further demagnetization
in Ni, with a characteristic delay determined by the strength of
the exchange interaction. We can further enhance this delay by
lowering the exchange energy by diluting the permalloy with Cu.
This measurement probes how the fundamental quantum mechan-
ical exchange coupling between Fe and Ni in magnetic materials
influencesmagneticswitchingdynamics in ferromagneticmaterials
relevant to next-generation data storage technologies.
magnetism ∣ quantum ∣ ultrafast
are engineered at the nanometer scale. Heat-assisted magnetic
recording (1), bit-patterned data storage media (2), all-optical
magnetization reversal (3), and giant tunneling magnetoresistive
disk drive read sensors are examples of such technologies (4).
Next-generation devices will require that the magnetic state of
materials be manipulated on fast timescales and at the nanometer
level. However, a complete microscopic understanding of magne-
tization dynamics that involves the correlated interactions of
spins, electrons, photons, and phonons on femtosecond time-
scales has yet to be developed. Two reasons for this lack of
fundamental understanding of ultrafast magnetism at the micro-
scopic scale are the complexity of the problem itself, as well as the
experimental challenge of accessing ultrafast and element-speci-
fic magnetization dynamics. One approach for addressing the ex-
perimental challenge is to use X-ray magnetic circular dichroism
(XMCD) employing X-rays generated by a synchrotron light
source. XMCD has the inherent advantage of element-specific
detection, and “sliced” synchrotron pulses are already used for
ultrafast studies (5–9). In an alternative approach, we recently
demonstrated that coherent extreme ultraviolet (XUV) beams
from a tabletop high-harmonic source (10, 11) can also be used
to probe ultrafast element-specific magnetization dynamics in
permalloy (Ni0.8Fe0.2) (12). For that demonstration, we took
advantage of magnetic birefringence at the M-edge in transition
metals to independently follow dynamics for Ni and Fe. However,
the time resolution available in that initial experiment was insuf-
ficient to observe any differences in the response of the consti-
tuent elements on very short timescales.
In this work, we experimentally answer the fundamental
question of whether the magnetization dynamics of individual
elements in a ferromagnetic alloy can differ on ultrafast time-
rogress in magnetic information storage and processing tech-
nology is intimately associated with complex materials that
scales. This is a very important fundamental question that has
not been addressed either theoretically or experimentally to date,
the answer to which reveals how the exchange interaction can
control the ultrafast dynamics of elemental spin subsystems
in complex materials. To answer this question, we rapidly excite
permalloy with an ultrashort (≈25 fs) laser pulse and probe the
element-specific demagnetization dynamics using <10 fs high-
harmonic pulses. The superior time resolution of our experiment
allows us to observe that the magnetization dynamics of Fe
and Ni are transiently delayed with respect to each other—by
about 18 fs in permalloy and 76 fs in Cu-diluted permalloy
(ðNi0.8Fe0.2Þ1-xCux). We ascribe this transient decoupling in the
magnetic behavior to the finite strength of the fundamental quan-
Specifically, for times shorter than the characteristic timescale for
exchange coupling, the magnetization of Fe quenches more
strongly than that of Ni. Then, as the Fe moments start to rando-
mize, the strong ferromagnetic interatomic exchange interaction
between Fe and Ni induces further demagnetization in Ni, with
a characteristic delay determined by the strength of the Fe-Ni
exchange interaction. Interatomic exchange energies of transition
metal alloys are in the 10–100 meV range, yielding characteristic
exchange times in the femtosecond range which corresponds to
finite spin-flip scattering times of 10–100 fs (9). Our findings pro-
vide crucial information for open questions in femtosecond mag-
In our experiment, sub-10 fs XUV light pulses from high-harmo-
nic generation (HHG) are produced by focusing 2 mJ femtose-
cond laser pulses into a Ne-filled waveguide. The harmonic
photon energy range of 35 to 72 eV spans the M absorption edges
of Fe and Ni at ≈54 eV and ≈67 eV, respectively (see Fig. 2B).
In the transverse magneto-optical Kerr-effect (T-MOKE) geome-
try used for these measurements, the intensity of the reflected
HHG light is proportional to the magnetization transverse to
the plane of incidence (12). We probe the magnetization by re-
flecting the XUV beam from a magnetic diffraction grating struc-
ture, as shown in Fig. 1A. We used gratings with 1 μm lines and
a 2 μm period patterned in three different ways: (i) alternating
Author contributions: S.M., C.L., P. Grychtol, J.M.S., H.T.N., M.E.S., C.M.S., T.J.S., M.A.,
M.M.M., and H.C.K. designedresearch; S.M., C.L., P. Grychtol, P. Granitzka, E.T., R.A., M.E.S.,
and S.E. performed research; J.M.S., H.T.N., C.M.S., T.J.S., M.A., M.M.M., and H.C.K.
contributed new reagents/analytic tools; S.M., C.L., P. Grychtol, P. Granitzka, E.T., T.J.S.,
M.M.M., and H.C.K. analyzed data; and S.M., C.L., P. Grychtol, E.T., J.M.S., R.A., H.T.N.,
M.E.S., C.M.S., T.J.S., M.A., M.M.M., and H.C.K. wrote the paper.
The authors declare no conflict of interest.
Freely available online through the PNAS open access option.
1S.M. and C.L. contributed equally to this work.
2To whom correspondence should be addressed. E-mail: firstname.lastname@example.org.
This article contains supporting information online at www.pnas.org/lookup/suppl/
www.pnas.org/cgi/doi/10.1073/pnas.1201371109PNAS Early Edition
1 of 6
elemental Fe and Ni stripes to probe the behavior of the pure
materials; (ii) permalloy (Ni0.8Fe0.2); and (iii) permalloy-Cu
(ðNi0.8Fe0.2Þ0.6Cu0.4). The Curie temperature Tcfor permalloy is
850 K, while for permalloy-Cu, Tc≈ 400 K. The HHG spectrum
diffracted from the grating sample is focused onto an X-ray CCD
camera.In order to determine theT-MOKE asymmetry, the change
in reflected HHG intensity at the M-absorption edges is monitored
while the magnetization direction of the sample is switched. The
T-MOKE asymmetry parameter A is calculated from the experi-
mental data as
A ¼Iþ− I−
where Iþand I−denote the reflected XUV intensities for the two
magnetization directions. More details of the measurement method
can be found in Refs. (12–14).
The asymmetry for the permalloy sample was measured using
XUV radiation from both the HHG source and a synchrotron
source. Fig. 2A shows the dependence of the magnetic asymmetry
onthe angle ofincidence and photonenergyinthe form ofa color-
coded contour plot. Fe and Ni are easily distinguished by strong,
citation of the localized M-shell electrons into unoccupied states
above the Fermi energy. XUV T-MOKE is therefore similar to
XMCD, providing a localized probe of magnetic moments. More-
over, the magnetic dynamics in pure Ni measured by XUV T-
MOKE are in excellent agreement with visible MOKE probes
(14, 15). Note that both peaks for the two elements have widths
of several eV. The magnetic asymmetry signal shows bipolar con-
tributions over an extended energy range for Fe and Ni, below and
above anenergyofabout 60 eV(white linein Fig. 2A), respectively
(13). The detailed peak structure is made complicated by the con-
volution of the finite lifetime of the p-orbital holes and the weak
splitting of the shallow M2and M3levels. The splitting is largest
The largest magnetic asymmetry occurs at an angle of inci-
dence of 45° (black dashed line in Fig. 2A, which corresponds to
the geometry used in the HHG setup). Fig. 2B shows the mea-
sured magnetic asymmetries using synchrotron and HHG light
at a 45° angle of incidence. The spectra are in good agreement
with each other. We attribute the minor discrepancies to the qua-
litatively different spectra for HHG and synchrotron radiation,
which is composed of discrete harmonic lines for HHG and is
a quasicontinuum for synchrotron radiation. The good agreement
in the asymmetry spectra between the HHG and synchrotron
sources validates our approach for measuring ultrafast demagne-
tization dynamics using HHG radiation.
For these measurements, the sample is transiently demagne-
tized using a focused ultrafast laser pump pulse (25 fs duration,
780 nm wavelength) that rapidly excites the electronic system.
After the excitation of the electron system in the material, various
scattering processes between electrons and phonons (with and
without spin-flips) determine the dynamical response of the sys-
tem on femtosecond to nanosecond timescales (see Fig. 1B). In
our experiment, the demagnetization is captured by measuring
(Top) Ultrafast XUV pulses (A) are reflected from a
permalloy grating sample, which spatially separates
the harmonics to form a spectrum on a CCD camera.
The reflected HHG intensity at the Fe and Ni M-shell
absorption edges (red and blue) depends on the
magnetization transverse to the optical plane of in-
cidence that is periodically reversed by transverse-
mounted Helmholtz coils. Exciting the sample with
an infrared laser pulse (red) causes the material to
demagnetize on femtosecond timescales. (B) After
rapid excitation of the electron system by a femtose-
cond laser pulse, various scattering processes be-
tween electrons and phonons (with and without
spin-flips) determine the dynamical response of the
system. First, the strongly excited electron gas ther-
malizes by predominantly electron-electron scatter-
ing to a Fermi-Dirac distribution. The ferromagnet
starts to demagnetize because of spin-flip scattering
events during this thermalization process. Electron-
phonon scattering processes transfer energy from
the excited electron gas to the lattice, and thermal
equilibrium is typically reached on picosecond time-
scales. Finally, on nanosecond timescales, the materi-
al cools by thermal diffusion. The red and blue
arrows in the lower boxes show the observed distinct
demagnetization dynamics of Fe and Ni in permalloy.
Schematic of the physics and experiment.
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www.pnas.org/cgi/doi/10.1073/pnas.1201371109Mathias et al.
the asymmetry A as a function of time delay between the infrared
pump and the XUV probe pulses (see Movie S1). We start with
a simultaneous measurement of the demagnetization dynamics
of elemental Fe and Ni using a sample with interleaved stripes
of Fe and Ni (Fig. 3A). After excitation at a laser fluence of
≈2 mJ∕cm2, the magnetization
quenched by about 19% for Fe and 45% for Ni. Using a double
Δm½1 − expð−t∕τmÞ?expð−t∕τrÞ, we measure demagnetization
times of τm¼ 98 ? 26 fs for Fe and 157 ± 9 fs for Ni (with re-
covery time constants τr¼ 11 ? 7 ps for Fe, and τr¼ 9 ? 1 ps for
Ni, respectively), in agreement with earlier studies (17, 18).
Now, moving from single-species metals to the more complex
binary alloy permalloy, where the constituents Fe and Ni are mis-
cible and strongly exchange coupled—one might expect identical
demagnetization dynamics for the two elements if one assumes a
completely delocalized, itinerant spin-polarized band structure
i.e. if the Fe and Ni contributions to the magnetization are indis-
tinguishable. If this were the case, even though T-MOKE probes
the local magnetic signal in the vicinity of the Fe and Ni atoms,
one would expect identical demagnetization timescales at the
two different sites. Note here the inherent difference between
our measurements in a strongly coupled 3d ferromagnetic system
and a recent study by Radu et al. of demagnetization dynamics
in the 3d-4f ferrimagnet GdFeCo (9). In that work, distinctly
different dynamics of the weakly exchange-coupled elements
arranged in sublattices were observed, a natural consequence of
the different temperature-dependent properties of the localized
4f Gd moment and less localized 3d Fe moment when in thermo-
dynamic equilibrium (a property that gives rise to a magnetic
compensation point whereby the rare earth and transition metal
sublattices are of equal but opposite magnetic moment).
Fig. 3B shows the measured element-specific demagnetization
of Fe and Ni in permalloy following excitation by a pump pulse
with fluence of ≈2 mJ∕cm2. As expected in a strongly exchange
coupled 3d alloy, the magnetization decreases rapidly for both
elements to a common minimum of about 70% of the total mag-
netization. Somewhat surprisingly, however, an inspection of the
data on short timescales clearly shows that the demagnetization
of Fe precedes that of Ni by approximately 10–20 fs (Inset,
Fig. 3B). This relative difference between Fe and Ni in permalloy
was not previously observed in Ref. (12), because the temporal
resolution in that experiment was insufficient to resolve such a
small shift in the onset of demagnetization. We stress that the
demagnetization data for Fe and Ni are collected at the same time
in this measurement, precluding any mismatch between the two
mðtÞ ¼ 1−
elements in the determination of the arrival time for the pump
and probe pulses.
The experimental results of Fig. 3B directly demonstrate that
the spin-dependent part of the electronic wave functions in the
itinerant 3d bands must also exhibit a local character. Differing
dynamics in the vicinity of the Fe and Ni atoms shows that con-
tributions of Fe and Ni to the total magnetic moment can be
clearly distinguished. This is a very surprising result, and since
we focus in the following discussion on the origin of these distin-
guishable parts of the Fe and Ni magnetic contributions, we for
simplicity denote them as demagnetization dynamics of Fe and
The degreeto which demagnetization dynamicscan be different
for Fe and Ni in permalloy necessarily depends on the strength of
the Fe-Ni interatomic exchange coupling between neighboring
magnetic moments: the weaker the Fe-Ni exchange coupling, the
more the dynamics can differ without incurring too large of an
energy cost. In the particular case of permalloy, the interatomic
exchange coupling is substantial, as indicated by the Curie tem-
perature TCof 850 K. Motivated by this line of reasoning, we
repeated our measurements with the tertiary alloys of permalloy
diluted by Cu (permalloy-Cu). Fe, Ni, and Cu are all miscible at
room temperature when one dilutes permalloy with Cu (19, 20).
of the volume-averaged exchange parameter through the reduc-
tion of the number of ferromagnetic nearest-neighbor atoms.
Such alloys also retain the high permeability associated with pure
permalloy and avoid any discontinuous crystallographic phase
transitions with varying Cu content. This, in turn, provides us with
the ability to tune TC(see Supporting Information) over a broad
temperature range. For fixed temperature measurements, the ex-
change coupling is further reduced by the concomitant renorma-
lization of the effective exchange integral near TC(21, 22).
We prepared a sample of ðNi0.8Fe0.2Þ0.6Cu0.4by cosputtering
from permalloy and Cu targets. X-ray diffraction verified that
our samples are a solid solution (i.e., random placement of the
Fe, Ni, and Cu atoms in the crystal lattice) fcc phase (see Methods
and Materials and Supporting Information). Fig. 3C shows a plot
of the element-selective, time-resolved T-MOKE signal for a
permalloy-Cu sample with TC¼ 406 ? 3 K. We unambiguously
observe a significant demagnetization delay for Ni of approxi-
mately 76 fs relative to Fe, as indicated by the arrows. Interest-
ingly, after accounting for the delay in the demagnetization, the
exponential decay of the magnetization for each of the elements
is identical within our error bars, yielding fitted values for the
effective demagnetization time τEffof 242 ? 12 fs for Fe and
measured using synchrotron radiation. The asymmetry signal of Fe (≈54 eV) is clearly separated from Ni (≈67 eV). (B) HHG XUV spectra reflected from
the permalloy grating sample at an angle of incidence of 45°, shown as green solid and dotted lines for the two different magnetization directions. The
blue line is the calculated asymmetry from the HHG spectra, and the black line the asymmetry from synchrotron data that corresponds to the spectral
cut shown as a black dashed line in (A).
XUV spectra and magnetic asymmetry. (A) Magnitude of the asymmetry, coded in color, as a function of photon energy and angle of incidence,
Mathias et al.PNAS Early Edition
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236 ? 13 fs for Ni. (Note the difference between the effective
demagnetization time of the respective element in the alloy and
the intrinsic elemental demagnetization times used in the model
The dynamics of ultrafast demagnetization are complex. A pro-
ven theory that completely describes all the interactions between
photons, electrons, spins, and phonons at a microscopic level
does not yet exist. However, it is known that femtosecond infra-
red pulses coherently interact with the electric charges and spins
in the material within ≈0–50 fs (23). Subsequently, the highly ex-
cited electrons relax to a thermalized population, accompanied
by spin-flip scattering processes that lead to ultrafast demagne-
tization on timescales of ≈100–1; 000 fs (Fig. 1B) (18, 24–27).
These details of the scattering processes remain the subject of
intense debate in ultrafast magnetism (7, 24, 25, 28–33). More-
over, nonadiabatic heating processes of the electron, spin, and
lattice subsystems on such ultrafast timescales, together with
strongly nonequilibrium transient phase states, necessarily com-
plicate our understanding of the underlying physics for ultrafast
demagnetization. It is therefore important to include the laser-
induced hot electrons in a discussion of magnetic dynamics on
such ultrashort <100 fs timescales.
Hot electrons can induce demagnetization by superdiffusive
spin transport (33), and also by screening the Coulomb potentials
on femtosecond timescales (34, 35). While superdiffusive spin
transport leads to a direct demagnetization process, screening
might indirectly act on the magnetization of the material by tran-
siently modifying the exchange interaction in ferromagnetic con-
ductors (36) during the 100–500 fs needed for the relaxation of
the pump-induced highly excited electrons. Such a modification
of the exchange interaction then has been shown to directly
influence the ultrafast magnetization dynamics (37). Note that
the screening process itself evolves on attosecond timescales in
metals, but is active until the highly excited electrons relax their
energy. If superdiffusive spin transport or any hot-electron in-
duced modification of the exchange coupling contributed signif-
icantly to the observed delay of the demagnetization dynamics of
Fe and Ni in permalloy, then we would expect a strong depen-
dence of the delay times on the pump fluence, since the pump
fluence controls the number of excited hot electrons. However,
the demagnetization delays for permalloy-Cu do not change when
the pump fluence is varied between 250 and 360 mW, which cor-
responds to a variation in the relative change in magnetization,
ΔM∕M, between about 50% and 80%, respectively (data shown
in Supporting Information). We therefore conclude that neither
superdiffusive spin transport nor a transient modification of
decay fits yield the demagnetization constants of (A) elemental Fe and Ni, and “effective” demagnetization constants τEfffor Fe and Ni in (B) permalloy, and
(C) permalloy-Cu, data set (see text). Fits to the model (solid lines) are used to extract the intrinsic demagnetization times for Fe and Ni in the alloys, τFeand τNi,
as well as the exchange time τEx, after which the Fe and Ni spin baths return to equilibrium with respect to each other with an effective demagnetization time
constant of τEff. The data for permalloy-Cu (C) is also shown in log-scale as a function of the normalized asymmetry changes ΔA ¼ ðA-AminÞ∕ðA0-AminÞ, where A0
the total asymmetry and Aminthe minimum asymmetry reached in the demagnetization process. We stress that the demagnetization data for Fe and Ni are
collected at the same time in this measurement, precluding any mismatch between the two elements in the determination of time-zero between pump and
probe laser pulses.
Ultrafast demagnetization of Fe (red dots) and Ni (blue dots) for elemental Fe and Ni (A), in permalloy (B), and in permalloy-Cu (C). Simple exponential
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www.pnas.org/cgi/doi/10.1073/pnas.1201371109Mathias et al.
the exchange interaction (e.g. due to hot electrons) are the domi-
nant processes causing the demagnetization delay of Ni. Rather,
the demagnetization delay between Ni and Fe in permalloy and
permalloy-Cu is an intrinsic property that depends upon the
strength of the interatomic Fe–Ni exchange interaction, since the
demagnetization delay is increased when the exchange interac-
tion is reduced in permalloy-Cu.
To gain more physical insight into demagnetization dynamics
in ferromagnetic systems, we need to take the interatomic Fe–Ni
exchange coupling into account. Our data clearly shows that using
a double exponential fitting function for elemental Fe and Ni
is not sufficient to describe the coupled dynamics in the alloyed
systems. Therefore, to extract quantitative timescales for the de-
magnetization process, we modeled our experimental data using
the following first-order coupled rate equations:
where mFeand mNiare the normalized Fe and Ni magnetizations,
τFeand τNiare the intrinsic decay times for Fe and Ni in the
absence of exchange coupling between them, and τExthe charac-
teristic “exchange time” that describes the thermodynamic cou-
pling of spins in the Fe and Ni systems. Solving the equations for
the limit where τFe≪ τNi, τExemerges as the delay time between
Fe and Ni, where both species have the same effective time con-
stant τEff(See Fig. 3B); i.e., the exchange time and the measured
delay time are equivalent quantities. The solution for the initial
condition mFeðt ¼ 0Þ ¼ mNiðt ¼ 0Þ ¼ 1 is:
?1 − η−
?1 − η−
?1 − ηþ
?1 − ηþ
mNiðtÞ ¼ ηþ
Fitting our permalloy-Cu data to this model reproduces the
distinct demagnetization dynamics of Fe and Ni on timescales
shorter than the exchange time τEx, and also the observed delay of
Ni with respect to Fe at times larger than τEx. It additionally yields
a smaller intrinsicdemagnetizationconstant(i.e.,the“virtual”con-
stant in the absence of Fe–Ni interatomic exchange coupling)
for Fe in comparison to Ni; i.e., τNi> τFe, qualitatively consistent
with our observations for elemental Fe and Ni (see Fig. 3). For
permalloy-Cu, a reasonable fit requires τNi> 500 fs, an indication
that the Ni itself is barely affected by the pump pulse (see
Supporting Information, where as expected, the demagnetization
times are somewhat different in the alloy and the pure material).
The fit to our model thus uncovers a picture validated by measure-
ments: after a characteristic exchange time τEx, the Ni magnetization
the rapidly demagnetizing Fe moments. At this point both moments
decay at the same effective time scale τEff∼ 2τFe. We note that the
Ni magnetization, when alloyed with Fe, is only weakly affected by
the pump pulse immediately after excitation—leading to the very
large intrinsic demagnetization constants of τNiwhen the data are
fitted to the model. For the case of permalloy with stronger intera-
tomic Fe–Ni exchange coupling, a smaller exchange time τExis ex-
pected. Indeed, using the same rate equations, we can reproduce
the demagnetization dynamics of permalloy, which is not possible
with the usual double exponential decay function. Fitting all our
data yields mean values of τEx¼ 18 ? 10 fs for permalloy, and
76 ? 9 fs for permalloy-Cu (see Supporting Information).
Additional support for our interpretation can be found by
considering the Landau–Lifshitz equation for magnetization
dynamics, where spin relaxation in ferromagnets proceeds at a
rate proportional to the gyromagnetic precession frequency. In
the present case of disproportionate demagnetization between
the Ni and Fe components, we expect local gyromagnetic dy-
namics to be dominated by interatomic exchange coupling. Based
on the values of Tc, the average exchange energy for permalloy-
Cu is approximately 3.3 times less than that of pure permalloy
(see Supporting Information). The ratio of τExextracted from
our data for permalloy-Cu relative to permalloy is 4.2 ? 2.8 (cor-
responding to characteristic exchange times of τEx¼ 18 ? 10 fs
and 76 ? 9 fs, respectively). Thus, the scaling of exchange energy
and τExbetween permalloy and permalloy-Cu are comparable,
supporting our interpretation.
The significantly higher intrinsic demagnetization times ex-
tracted for Ni, τNi> 500 fs, compared to Fe, (τFe≈ 89 ? 8 fs in
permalloy and τFe≈ 126 ? 9 fs in permalloy-Cu) indicate that
the uniformity of the Ni spins in the alloy are most strongly
influenced by the exchange coupling to the Fe, and much less
influenced by the laser excitation in comparison to the pure
material (τNi;elemental≈ 157 fs). In contrast, our data indicate that
the laser excitation induces demagnetization for Fe on a compar-
able timescale to that for an elementally pure material
(τFe;pure≈ 98 ? 26 fs). Because of interatomic Fe–Ni exchange
coupling, the Ni spins eventually demagnetize with the same
time-constant as Fe in the alloys via the thermodynamic contact
to the Fe spin bath—but with an apparent delay that is given by
τEx. Our data indicate that this delay is larger in permalloy-Cu
than in permalloy due to the reduced exchange energy. To our
knowledge, such a delayed behavior of magnetization dynamics
in metallic alloys has not been previously predicted or observed.
Current macroscopic and microscopic models that explain de-
magnetization dynamics for pure materials need to be extended
to alloyed magnetic materials. The absence of such microscopic
models for multicomponent systems prevents us from addressing
why Ni intrinsically reacts slower in the specific alloys in compar-
ison to the pure Ni material. However, our experiment provides a
clear observation of how the strength of the exchange coupling
between the constitutive atomic components can influence mag-
netization dynamics in alloys on ultrafast timescales. As such, our
data help elucidate the microscopic role of the fundamental
quantum mechanical exchange interaction in the ultrafast demag-
In summary, we explore the consequences of the fundamental
quantum exchange interaction in strongly coupled ferromagnetic
systems, showing that quantitatively different magnetization dy-
namics of the individual elements can be observed on timescales
shorter than the characteristic exchange timescale. On longer
timescales, the dynamics are dominated by the faster of the two
species. Analysis of our data indicates that the observed differ-
ences in demagnetization rate are primarily determined by intrin-
sic properties of the material rather than the result of photo-
induced ultrafast transient changes in the material, e.g., hot-elec-
Mathias et al. PNAS Early Edition
5 of 6
tron-gas screening or nonequilibrium phases. This fact has signif-
icant impact for fundamental models of ultrafast magnetism, and
for the dynamical magnetic behavior for all types of exchange-
coupled materials, including both the alloys and multilayer struc-
tures that are widely used for data storage.
Materials and Methods
Experimental Setup.We generate coherent high-harmonics (HHG) by focusing
25 fs laser pulses (780 nm central wavelength) into a neon-filled hollow wa-
veguide. The laser operates at 2 kHz repetition rate with the pulse energy of
approximately 2.2 mJ∕pulse. Ninety percent of the laser power is used for
HHG, while the remaining light is used to excite the sample. The waveguide
is filled with neon gas that is pressure tuned to approximately 400 torr in
order to phase-match a broad range of harmonics in the range of extreme
ultraviolet (XUV) from 35 to 70 eV (21st–43rd harmonic), covering the region
of the spectrum where the M-edge resonances of 3d ferromagnetic metals
are located. A 200 nm thick Al filter blocks the fundamental laser light. The Al
filter limits our highest energy HHG photons to 72 eV as a result of strong
absorption above the Al L2;3edges. The HHG beam is refocused onto the
grating sample using a grazing incident toroidal mirror. The HHG spot size
on the sample is less than 500 μm, which is smaller than the pump laser spot
size of approximately 1–2 mm to ensure good spatial overlap and uniform
demagnetization. Water-cooling stabilizes the sample temperature at 293 K.
Sample Fabrication. A 10 nm thick permalloy-Cu (ðNi0.8Fe0.2Þ0.6Cu0.4) alloy thin
film was grown by cosputter deposition with permalloy (Ni0.8Fe0.2) and pure
Cu targets. The rates from a permalloy target and a pure Cu target were ca-
librated using a quartz crystal monitor and a profilometer. A thin 3 nm Ta
seed layer was first sputter deposited onto a thermally oxidized Si(100) wafer
to provide a strong (111)-texture and good adhesion prior to depositing the
permalloy-Cu alloy. Diffraction gratings were patterned from the film via
optical lithography and a subsequent Ar ion milling at 300 eV. The grating
consisted of an array of 1 μm wide stripes with a center-to-center spacing of
2 μm. The 10 nm thick permalloy (Ni0.8Fe0.2) diffraction grating was fabri-
cated by a direct liftoff process from a film grown by ion beam deposition
with a target made from the same source material that was used for sputter-
ing of the permalloy-Cu film. A 3 nm Ta seed layer was also used for adhesion
to promote strong (111)-texture prior to depositing the permalloy layer.
X-ray diffraction data, magnetometry measurements, ferromagnetic reso-
nance measurements, and static element-specific magnetization measure-
ments presented in the SI verify a random placement of the Fe, Ni, and
Cu atoms in single fcc-phase crystal lattice.
ACKNOWLEDGMENTS. Contribution of the National Institute of Standards and
Technology, an agency of the U.S. government, not subject to U.S. copyright.
S.M. and M.A. thank Daniel Steil, Tobias Roth, and Mirko Cinchetti for helpful
discussion. This work was supported by U.S. Department of Energy Office
of Basic Energy Sciences and used facilities from the National Science
Foundation Engineering Research Center for Extreme Ultraviolet Science
and Technology. S.M. was supported by the European Community’s FP7
under Marie Curie International Outgoing Fellowship GA 253316, P.Grychtol
by BMBF Project No. 05KS7UK1 and the German Academic Exchange
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www.pnas.org/cgi/doi/10.1073/pnas.1201371109 Mathias et al.
Mathias et al. 10.1073/pnas.1201371109
SI Materials and Methods
A) Sample characterization. We would like to present measure-
ments to disprove any possible phase segregation in the alloys
used in the experiment, which would lead to distinct differences
in magnetization dynamics between Fe and Ni. The results pre-
sented below clearly show that phase segregation is not present in
The sample used for the measurement of elemental Fe and Ni
were similar to the grating structures used for alloys, except that
the grating consisted ofalternating stripes of pure Fe and pure Ni.
In this case, 1 μm stripes of Ni with a 4 μm center-to-center
spacing were first fabricated by a direct liftoff process. A second
lithography step was used to pattern and liftoff 1 μm wide Fe
stripes in-between the previously fabricated Ni stripes, yielding
alternating stripes of Fe and Ni with a center-to-center spacing
of 2 μm. In both cases, the thickness of the individual Ni and
Fe layers was 10 nm and a 2 nm Ta seed layer was initially depos-
ited for adhesion. The Fe stripes are capped with a Ta layer
(2.5 nm) to prevent oxidation. After removal from the deposition
chamber, these samples were quickly transferred to a vacuum sto-
rage chamber to minimize oxidation of the surface.
A.1. X-ray diffraction. X-ray diffraction (XRD) measurements
were performed using a parallel beam configuration where a Cu
Kαsource was conditioned with a wavelength-specific X-ray
mirror. The diffracted beam optics consisted of a parallel plate
collimator, Soller slit, and graphite monochromator prior to de-
tection by a proportional counter. The sample was mounted on a
4-circle goniometer with an instrumental resolution of 0.0001° in
2θ and ω. After a direct-beam alignment of 2θ ¼ 0”, the sample
height was adjusted until it cut the beam in half. The sample tilt
angles were then rocked and iterated with the sample height po-
sition to ensure that the sample was centered with the surface
parallel with respect to the incident X-ray beam.
Fig. S1 shows 2θ- ω scans for the permalloy and permalloy-Cu
thin films as well as the permalloy-Cu grating sample used in the
experiment. Both the (111) and (222) peaks are present in all the
spectra, consistent with a well (111)-textured face centered cubic
(fcc) structure. Rocking curves peak widths of ≈3–4° at FWHM
on the (111) peaks further indicate the high quality of (111)-tex-
ture. No additional peaks corresponding to additional phases or
segregation of species are present. The measured values of the
lattice parameters are 0.3547 nm, 0.3573 nm, and 0.3572 nm
for the permalloy thin film, permalloy-Cu thin film, and permal-
loy-Cu grating, respectively. Furthermore, if a linear relationship
of the lattice constant is assumed, then the ideal lattice constant
of a solid solution of 60% permalloy and 40% Cu is calculated to
be 0.3574 nm (using the measured lattice constant for permalloy
above and the bulk value of 0.3615 nm for Cu). The good agree-
ment between the measured and calculated lattice parameters,
combined with the lack of any additional peaks in the XRD spec-
tra, confirm that the sample consists of a single fcc phase, solid
solution of Ni, Fe, and Cu, as expected for this system.
A.2. SQUID magnetometry. Magnetometry measurements were
performed using a superconducting quantum interference device
(SQUID) magnetometer. The saturation magnetization was mea-
sured as function of temperature from 10 K to 400 K, from which
TCwas determined for the permalloy-Cu alloy via extrapolation
from the power law dependence of M on T (Fig.S2A). TCis iden-
tical for Fe and Ni, as expected. In addition, we measure a smooth
and continuous change of TCas a function of Cu doping for a full
series of samples where the Cu content was varied from 60%
to 40% (Fig. S2B). This trend further confirms that our samples
consist of a complete intermixture of Fe, Ni, and Cu, without
A.3. Ratio of exchange energies permalloy/permalloy-Cu. Based on
the SQUID data for 60∶40 permalloy-Cu, the ratio of the Curie
temperature for permalloy/permalloy-Cu is 2.1. The ratio of the
exchange energy is therefore 2.1:1 at 0 K. At room-temperature,
we need to account for the renormalization of exchange, which
scales in proportion to MðTÞ∕MðT ¼ 0Þ. The ratio of this quan-
tity is 1.56, according to the SQUID data. Therewith, the ratio of
exchange energies between permalloy and permalloy-Cu at 300 K
is about 1.56 × 2.1 ¼ 3.3.
A.4. Ferromagnetic Resonance (FMR).We measured the ferromag-
netic resonance (FMR) of the permalloy-Cu grating sample with
a broadband FMR spectrometer in a perpendicular applied field
(P-FMR) geometry. Such measurements provide accurate deter-
mination of magnetic homogeneity in the material and the intrin-
sic damping parameter, α. Details of the experimental technique
can be found in Refs. (1–3). The real and the imaginary parts of
the magnetic contribution to the transmission parameter S21are
fit simultaneously to the complex susceptibility χðHÞ.
Fig. S3shows the FMRdata taken on the permalloy-Cu grating
sample. Most importantly, we observe a single FMR peak, as ex-
pected for an alloy without segregation. Furthermore, we fit the
measured resonant fields (blue circles) with the Kittel equation
(red line), as shown in Fig. S3.
The Kittel equation in this perpendicular geometry is
where Meffis the effective magnetization, γ ¼ gμB∕h is the gyro-
magnetic ratio, μ0is the permeability of free space, f is the ap-
plied microwave frequency, μBis the Bohr magneton, and g the
spectroscopic splitting factor. The fit to the data yields μ0Meff¼
0.293 ? 0.001 T and g ¼ 2.026 ? 0.004. The value for μ0Meff
is lower than the value for the saturation magnetization μ0Msob-
tained by SQUID magnetometry. This is likely due to edge effects
for a finite width grating structure and anisotropy, which will low-
er μ0Meffcompared with μ0Ms. The measured line width ΔHðfÞ
(yellow diamonds) is fit with the phenomenological equation
(green line) (4):
ΔHðfÞ ¼ ΔH0þ4πα
where ΔH0is the inhomogenous line width broadening, generally
attributed to locally varying magnetic properties of the sample.
The linear fit yields μ0ΔH0¼ 5.6 ? 0.2 mT and α ¼ 0.0158 ?
0.0002. Both of these values are elevated compared to permalloy
films without alloying Cu. The value of α is higher than that of
0.005 for pure Ni0.8Fe0.2(5), which is expected since TCof the
permalloy-Cu sample is relatively close to room temperature (6).
The inhomogeneity of the perpendicular anisotropy μ0ΔH0¼
5.6 ? 0.2 mT is indicative of a high quality thin film with rela-
tively small variation of magnetic properties within the material.
Mathias et al. www.pnas.org/cgi/doi/10.1073/pnas.12013711091 of 8
A.5. Static T-MOKE asymmetry measured by HHG. We also
measured the static asymmetry parameter for permalloy-Cu as
a function of sample temperature up to 425 K, which exceeds
the Curie-temperature. Fig. S4 shows the measurement results.
The Fe and Ni M-edge asymmetry signals have the same tem-
perature dependence and both asymmetries gradually reduce
to zero near the Curie temperature. The power law fit gives
the Curie temperature of 407.5 ? 3.7 K for Ni and 403.5 ?
1.1 K with critical exponent β of little less than 0.5—the value
from mean-field theory. The fact that both the Fe and Ni signals
show the same Tc to within error bars (and not 1043 K and 631 K
as expected for bulk Fe and Ni, respectively) implies that we have
a single-phase alloy without any segregation between the Fe and
Ni. The value of Curie temperature is 5% less than the value mea-
sured by SQUID method (Fig. S2) due to expected variations in
the sputter deposition rate from different sample to sample.
B) Experimental methods.
B.1. T-MOKE asymmetry. By matching the electromagnetic boundary
conditions at the interface, the reflection coefficient in T-MOKE
geometry for p- and s-polarized light can be written as (7, 8)
where n0and n are refractive indices of the incident nonmagnetic
medium (vacuum, in our case) and the reflecting magnetic sample,
respectively. θtis the refracted angle and θiis the angle of inci-
dence onto the magnetized sample. The first terms in each equa-
tion are from the optical response (Fresnel coefficients) while the
Voigt parameter Q describes the magneto-optic effect, which is
related to the magnetization vector through the off-diagonal ele-
ment of the dielectric tensor. Only the p-polarized reflection de-
pends on the magnetization (9). The sign of rppdepends on the
orientation of the magnetization aligned by the external magnetic
field. Finally, the asymmetry parameter A described in the text is
related to the rppcoefficient as:
A ¼Iþ− I−
The shape of the T-MOKE asymmetry near an absorption edge
is understood (9–14). The function depends sensitively on θi, the
photon energies through n, and Q. The asymmetry A is usually
maximized near an absorption edge. We operated at an angle of
incidence near the Brewster’s angle of θi≈ 45°to maximize A.
B.2. Time zero determination. The absolute time zero of the dy-
namics is determined experimentally via autocorrelation with a
BBO crystal positioned between the pump laser and the funda-
mental laser beam, where the second harmonic propagates col-
linearly with the high-harmonics. The accuracy is within ?10 fs.
Finally, the zero delays for Fe and Ni signals reported in the main
text are identical since we record all data in parallel.
B.3. Photon energy calibration. To calibrate the photon energy E
of the high-order harmonics, we use the diffraction formula for
gratings generalized for any angle of incidence α and diffracted
angle β (15)
dðsinα − sinβÞ ¼ mλ;
where d is the grating period (2 μm for permalloy and permalloy-
Cu, 4 μm for Fe-Ni stripes), m is the diffraction order and λ is the
wavelength. The photon energies of the harmonics are odd multi-
ples (N) of the fundamental energy (E ¼hc
angle of incidence is α ¼ 45°.
From geometry, the diffracted angle β can be related to the
angle of incidence α, sample-to-CCD distance z, and the dif-
fracted distance along the CCD x by the following relation
λ¼ NE0Þ, and the
β ¼ α∓x
where the minus (plus) sign is used in the case of positive (nega-
tive) diffraction orders.
After expanding the diffracted formula around α ¼ 45°, the
equation reduces to
This equation allows us to relate the measurable distance x to
the known order of harmonics N by performing a fit between the
two parameters and setting E0as a fitting parameter.
C) Data Analysis.
C.1. Influence of sample grating structure and different photon at-
tenuation lengths. In the main text, we extract the time-resolved
data at the locations of the high harmonics that give the largest T-
MOKE asymmetry and have the highest photon flux. We aver-
aged over 25 time-resolved traces to get sufficient signal-to-noise
ratios for the fitting functions. The best signal of Ni (Fe) can be
extracted from the harmonics at 67 eV (54 eV) for the permalloy
The high harmonic at 67 eV (Ni) had sufficient intensity to give
good signal/noise in the first and second order diffraction pattern,
see Fig. S5A, Inset (blue and green bar, respectively). When we
extract the demagnetization dynamics from both orders, we find
very good agreement of our fitting results (See Fig. S5A). This
measurement confirms that the grating structure of the sample
does not artificially contribute spurious signal to our time-
Different attenuation lengths of the pump pulse and the har-
monics used to extract the demagnetization dynamics at the
Fe and Ni M-edges yield a different probing depth of the sample,
with different effective fluences (excitation densities) probed.
However, for a sample thickness of 10 nm and attenuation
lengths varying between about 9 nm and 15 nm in permalloy and
permalloy-Cu in the relevant XUV range [from The Center for
X-ray Optics, http://www.cxro.lbl.gov] (see Fig. S5B), effective
fluence differences are in the 1–2% range. For example, the lar-
gest attenuation length is found for photons just below the Fe
absorption edge in permalloy and is about 15 nm. Using Beer-
Lambert law, the sample depth at which the mean of the Beer-
Lambert function for a sample thickness of 10 nm and photons
travelling in and out of the sample is found is 4.45 nm from the
surface. With an attenuation length of 17 nm for the infrared
pump pulse (16), this corresponds to an effectively measured flu-
ence of 77% of the total impinging fluence. On the other hand,
the lowest attenuation length is found for photons with energies
just above the Ni edge and is about 9.1 nm, yielding a mean of the
Beer-Lambert exponential decay function at 4.12 nm sample
depth. This corresponds to a fluence of 78.5% of the total
fluence. As can be seen from Ref. (17) of the main text, fluence
differences in the 1–2% region influence the exponential decay
constant only on a sub-2 fs timescale for Ni (where typical demag-
netization times in the alloys are ≈160 fs). Please note, however,
that such probing-depth-induced fluence-dependence can affect
the demagnetization constants, but cannot produce the observed
delayed behavior as discussed in the paper.
Mathias et al. www.pnas.org/cgi/doi/10.1073/pnas.12013711092 of 8
In the case of elemental Fe and Ni, the attenuation lengths in
the XUV vary between 6 and 25 nm, see Fig. S5B. The attenua-
tion lengths of the pump pulse for elemental Fe and Ni also need
to be considered and are in the region of 35 nm and 24 nm,
respectively, calculated using Fresnel equations and refractive in-
dexes (17). The same calculation with Beer-Lambert exponential
decay function for the according attenuation lengths in Fe and Ni
yields effective fluence differences in the 2–4% region, so that the
difference of the exponential demagnetization decay constant is
on a sub-5 fs timescale.
Finally, the different harmonics take the same physical path in
the beamline, using grazing-incidence reflective optics, and a thin
filter to reject the laser light. The relative delay through such fil-
ters have been measured in the attosecond range (18). Thus, the
harmonics all arrive at the sample at the same time (well within
our error bars).
C.2. Fitting results When we fit our data to the rate-equation
model (Eq. 2), and we allow all parameters to vary, the intrinsic
demagnetization time for Ni τNidiverges [the intrinsic demagne-
tization times are the artificial time constants for Fe and Ni given
by the rate-equation model, which describes Fe and Ni being
(i) alloyed to permalloy and (ii) in the absence of exchange cou-
pling]. This indicates that the Ni spins are only weakly influenced
by the pump pulse such that we are insensitive to the value of τNi
in the context of fitting the data with Eq. 2. To determine a rea-
sonable lower bound for τNi, we investigated the fractional uncer-
tainty in the exchange time τExusing fixed values of τNi. This
procedure establishes a reasonable lower bound for τNi. Using
the criterion that the fractional error in τExdoes not exceed twice
the minimum possible, we find τNi> 504 fs for permalloy-Cu. If
τNiis constrained to be smaller than this amount, the error for τEx
is unacceptably large. The fitting results presented in the manu-
script are extracted when all parameters are allowed to vary.
Fluence-dependent data of the ultrafast demagnetization in
permalloy-Cu yield identical dynamics to within our error bars.
In particular, we do not see any fluence dependence in the de-
magnetization delay between Fe and Ni, which we would have
expected if highly excited hot electrons were responsible for
the observed dynamics. Fig. S6 shows the data for pump fluences
of 300 mW (q ¼ 0.6) and 250 mW (q ¼ 0.5) with the fit to the
model (with q the maximum quenching of the magnetization
q ¼ Amin∕A0). [Data for pump fluence of 360 mW (q ¼ 0.77)
is shown in the main paper]. The fluence-dependent data has
been collected in the same measurement period in order to
achieve maximum comparability. All fitting results are summar-
ized in Tables S1 and S2. The mean values for the exchange times
τExare 18±10 fs for permalloy and 76±9 fs for permalloy-Cu.
We note once again that the Ni magnetization, when alloyed
with Fe, is only weakly affected by the pump pulse immediately
after excitation—leading to the very large intrinsic demagnetiza-
tion constants of τNiwhen the data are fitted to the model.
The elemental data for Fe and Ni was fitted using a simple
Δm½1 − expð−t∕τmÞ?expð−t∕τrÞ. The fitting results are, as given
in the paper, τm¼ 98 ? 26 fs for elemental Fe and 157±9 fs
for elemental Ni (with recovery time constants τr¼ 11 ? 7 ps
for Fe, and τr¼ 9 ? 1 ps for Ni, respectively). Here, we note that
one usually compares demagnetization constants for identical
quenching parameters q, in which case one would expect the op-
posite behavior to a first approximation: Ni should demagnetize
faster than Fe due to the much lower Curie temperature of Ni
(19). Here, however, the measurement is carried out in a parallel
detection scheme with identical laser fluence, so that the magne-
tization quenching q of Fe is considerably lower than that of
Ni (19% vs. 45%, respectively, see Fig. 3A in main text). As a
consequence, the demagnetization constant for Fe is smaller
than the demagnetization constant for Ni under these excitation
mðtÞ ¼ 1−
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20 40 60
NiFe or NiFeCu
Intensity (arb. units)
NiFe or NiFeCu
Fig. S1.XRD 2θ- ω scans for the permalloy thin film (NiFe film), permalloy-Cu thin film (NiFeCu film), and permalloy-Cu grating (NiFeCu grating) sample.
temperature is TC¼ 423 K and the critical exponent is β ¼ 0.53. (B) Measured Curie temperature (red square dot, right axis) and saturation magnetization
(blue circle, left axis) as a function (x) of permalloy (NiFe) content (or 1—x as copper content).
(A) SQUID measurement of permalloy-Cu thin film that shows the saturation magnetization as a function of temperature. The fitted Curie
sample. The red and green lines through the data are fits used to determine the magnetic parameters.
Ferromagnetic resonance field (blue circles) and line width (orange diamonds) as a function of microwave frequency for the ðNi0.8Fe0.2Þ0.6Cu0.4grating
Mathias et al. www.pnas.org/cgi/doi/10.1073/pnas.12013711094 of 8
HHG spectrum (green) and asymmetry (blue) recorded at room temperature are shown in the inset. The harmonics used to extract the asymmetry around Fe
and Ni M-edges are shaded in red and blue, respectively. The sample is magnetized to saturation by a magnetic field of ≈ ? 40 Oe.
Static asymmetry parameter of the permalloy-Cu grating sample as a function of temperature. Fe (Ni) M-edge signal is shown as red (blue) dots. The
Mathias et al. www.pnas.org/cgi/doi/10.1073/pnas.12013711095 of 8
diffraction pattern. Clearly, the grating does not artificially contribute signal to our time-resolved measurements. (B) Attenuation lengths in Fe, Ni, permalloy
and permalloy-Cu in the XUV energy region [from The Center for X-ray Optics, http://www.cxro.lbl.gov].
(A) Element-selective laser induced demagnetization dynamics of Ni extracted from first (blue) and second order (green) of the 67 eV harmonic in the
Mathias et al. www.pnas.org/cgi/doi/10.1073/pnas.1201371109 6 of 8
achieve maximum comparability. The corresponding data for a pump intensity of 360 mW and q ¼ 0.77 is shown in the main paper, Fig. 3C.
Pump-fluence dependent data of the permalloy-Cu sample. The data series has been collected in a single experimental measurement period to
around 66 eV. The movie shows the evolution of the asymmetry signal as a function time delay between pump and probe pulses, as indicated by “Time Delay”
at the bottom of the graph. Right top and bottom: according extracted demagnetization signal for Fe (bottom) and Ni (top). The circles indicate the
momentary extracted data points of the magnetization of Fe and Ni, which correspond to the current “Time Delay” in the movie.
Element-specific ultrafast demagnetization dynamics of Fe and Ni in permalloy-Cu. Left: Magnetic asymmetry signal of Fe around 54 eV and Ni
Mathias et al. www.pnas.org/cgi/doi/10.1073/pnas.12013711097 of 8
Table S1. Fitting results for all collected datasets: permalloy Download full-text
5.70 × 1005
1.00 × 1005
8.02 × 1004
6.43 × 1004
3.1 × 1004
3.1 × 1003
4.7 × 1003
3.7 × 1003
q is the maximum quenching of the magnetization; i.e., the minimum asymmetry value reached in the time-
resolved experiment normalized by the total asymmetry. τReis the exponential time constant for the return of the
asymmetry signal to its thermal equilibrium value
Table S2. Fitting results for all collected datasets: permalloy-Cu
1.07 × 1008
1.45 × 1006
1.86 × 1006
7.36 × 1005
1.39 × 1005
4.9 × 1006
3.2 × 1004
4.9 × 1004
1.9 × 1004
4.3 × 1003
2.28 × 1009
q is the maximum quenching of the magnetization; i.e., the minimum asymmetry value reached in the time-resolved
experiment normalized by the total asymmetry. τReis the exponential time constant for the return of the asymmetry
signal to its thermal equilibrium value
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