Document [original]

PHYSICAL REVIEW B 109, 184440 (2024)

Editors’ Suggestion

Relationship between magnetic asymmetry and magnetization in ultrafast transverse

magneto-optical Kerr effect spectroscopy in the extreme ultraviolet spectral range

Johanna Richter ,1Somnath Jana ,1Martin Hennecke ,1Daniel Schick ,1Clemens von Korff Schmising ,1,*

and Stefan Eisebitt 1,2

1Max-Born-Institut für Nichtlineare Optik und Kurzzeitspektroskopie, Max-Born-Straße 2A, 12489 Berlin, Germany

2Institut für Optik und Atomare Physik, Technische Universität Berlin, 10623 Berlin, Germany

(Received 9 February 2024; accepted 7 May 2024; published 30 May 2024)

Ultrafast transverse magneto-optical Kerr effect (T-MOKE) spectroscopy in the extreme ultraviolet spectral

range provides element-specific information about the magnetization dynamics of complex magnetic structures.

However, the relationship between the T-MOKE observable, denoted magnetic asymmetry, and the magnetization

of a sample can exhibit significant nonlinearities, even in the case of magnetization changes that are homoge-

neous along the depth of the sample and without considering any nonequilibrium spin dynamics. Here, we

combine static and time-resolved experimental data with simulations based on a wave propagation algorithm for

a prototypical magnetic heterostructure that exhibits pronounced deviations from a linear relationship between

magnetic asymmetry and magnetization, including increasing values of asymmetry in spite of a reducing

magnetization. As an outlook, we describe sample structures and experimental geometries for which a linear

response of the T-MOKE observable remains a valid approximation.

DOI: 10.1103/PhysRevB.109.184440

I. INTRODUCTION

Measurements using transverse magneto-optical Kerr ef-

fect (T-MOKE) spectroscopy in the XUV spectral range

have become an established method for the investigation of

ultrafast magnetization dynamics in multicomponent mate-

rials and multilayer systems. T-MOKE is characterized by

an asymmetry in the reflectance for two opposite magnetiza-

tion directions aligned perpendicular to the reflection plane

of the incident p-polarized radiation. This effect originates

from the magnetic medium’s finite off-diagonal elements

in the dielectric tensor [1] and is strongly enhanced at an

atomic resonance, thus providing element selectivity. The

experimental geometry is shown schematically in Fig. 1:

broadband XUV radiation, incident at the normal angle θi,is

reflected off a magnetic sample and subsequently spectrally

dispersed and detected by a camera. Ultrafast dynamics are

recorded employing a pump-probe technique in which an

optical laser pulse excites the sample at varying time delays

before the XUV pulse probes the evolving magnetization of

the sample.

In contrast to magnetic circular dichroism (MCD) ex-

periments in transmission geometry [2,3], T-MOKE does

not require circularly polarized light. This avoids addi-

tional experimental difficulties in controlling the polariza-

tion of the XUV radiation, either intrinsically within the

high-harmonic generation (HHG) process itself [4–6], ex-

trinsically via specialized optics [7,8], or by performing

a polarization analysis [9–12]. Measurements in reflection

geometry also allow investigating a wider range of sam-

ples, including systems grown on crystalline substrates, and

*korf[email protected]

facilitate easier heat management in high-repetition rate,

time-resolved pump-probe experiments. Furthermore, analy-

sis of the time-dependent reflectance as a function of either

the angle of incidence or photon energy allows obtain-

ing detailed information on transient magnetization depth

profiles [13,14].

Time-resolved T-MOKE measurements using radiation

from high-harmonic generation sources were first success-

fully conducted by La-O-Vorakiat et al. [15] to investigate

the ultrafast response of a permalloy sample after optical

excitation. Further development of T-MOKE spectroscopy al-

lowed tackling important questions regarding the microscopic

origin of ultrafast magnetization dynamics. Attempts were

made to disentangle and quantify contributions stemming

from Stoner (reduction of exchange splitting) and Heisenberg

(generation of magnons) excitations [16–18] and to track ul-

trafast superdiffusive spin transport in multilayer structures

[19–21]. Recently, MCD [3,22]aswellasT-MOKE[23–25]

experiments in the XUV spectral range provided the first ex-

perimental evidence of optical intersite spin transfer, a process

theoretically predicted by Dewhurst et al. [26] in 2018. Here,

spins are transferred between two sublattices by the optical

excitation itself, while the efficiency and the direction of the

transfer are dictated by the availability of empty states above

the Fermi energy. In the T-MOKE experiments an increase

of magnetic asymmetry at certain photon energies was inter-

preted as an increase of magnetization at the corresponding

sublattices upon laser excitation.

However, from the very beginning T-MOKE experiments

have attracted a considerable amount of criticism because the

reflectance of a laser-excited magnetic film can, in principle,

be influenced by distinct changes in the real and imagi-

nary parts of the nonmagnetic [27–29] as well as magnetic

[30] index of refraction. The controversy was further fueled

JOHANNA RICHTER et al. PHYSICAL REVIEW B 109, 184440 (2024)

by conflicting demagnetization time constants of Fe and Ni

obtained from L-edge MCD transmission experiments [31]

and XUV T-MOKE measurements [32], as well as by the

difficulty in reproducing T-MOKE results, showing an en-

hancement of the Fe magnetization in interlayer coupled

Fe/Ni systems [16], in L-edge MCD transmission experi-

ments [33]. The work by Jana et al. [34] is important in this

respect since it provides the first systematic comparison be-

tween L-edge MCD and M-edge T-MOKE. Here, a delayed

onset of the demagnetization of Ni in FeNi alloys, previously

observed in a number of T-MOKE experiments [32,35], was

confirmed employing both techniques.

Finally, the T-MOKE observable is generally assumed to

be directly proportional to the magnetization, an approxi-

mation, however, which is derived for the case of a small

magnetic contribution to the total reflectance and for a single

vacuum/magnetic layer interface. In this work, we scruti-

nize this assumption by investigating the relationship between

magnetic asymmetry and magnetization for realistic sample

structures and common experimental geometries. To achieve

this objective, we compare static and time-resolved measure-

ments with detailed simulations using a wave propagation

matrix formalism taking into account reflection and refrac-

tion at each interface. We demonstrate that particularly for

measurements performed in the vicinity of the Brewster angle

around 45◦, where the T-MOKE observable is maximized but

at the same time the nonmagnetic reflectance is minimized,

pronounced deviations from a direct proportionality between

asymmetry and magnetization arise. We emphasize that this is

already the case for the simplest scenario, where the magne-

tization decreases homogeneously within the probed sample

volume and without considering any specific excited state

or nonequilibrium demagnetization dynamics. Finally, we lay

out strategies to retain a linear relationship between T-MOKE

measurements and magnetization.

II. T-MOKE ASYMMETRY

In the following, we present a brief derivation of the

commonly used approximation for the T-MOKE asymmetry

at a single vacuum/magnetic sample interface. We start by

separating the nonmagnetic rn

pand magnetic rm

preflection

coefficients:

p=ncos θi−cos θt

ncos θi+cos θt

p=2inQcos θisin θt

(ncos θi+cos θt)2,(1)

where nis the refractive index of the magnetic layer, θi,tare

the normal incidence and refracted angles, respectively, and Q

describes the magneto-optical constant, which is proportional

to the magnetization M[1]. The magnetic asymmetry is de-

fined as the difference between the reflectances for the two

magnetization directions R±

p, normalized to their sum:

A=R+

p−R−

R+

p+R−

=rn

p+rm

p2−rn

p−rm

p2

rn

p+rm

p2+rn

p−rm

p2.(2)

Rewriting Ausing the identity z+z∗=2Re(z), where Re(·)

denotes the real part of the complex quantity and ∗represents

FIG. 1. Schematic of the transverse magneto-optical Kerr effect

(T-MOKE) geometry. A p-polarized, ultrashort, and broadband XUV

pulse impinges on the sample at the normal angle θi, and the reflected

spectrum is detected by a spectrometer, consisting of a grating and a

camera. The magnetization Mof the sample is set perpendicular to

the plane of incidence.

its complex conjugate, yields

A=2Rern

p·rm∗

p

rn

p2+rm

p2=2Rerm

p⎛

⎜

⎝

1+rm

p

rn

p

⎞

⎟

⎠

 

.(3)

We now approximate the expression by a Taylor expansion of

the second term T2 around |rm

p|2/|rn

p|2=0 and retain only the

zero-order term, yielding the following expression:

Aapp ≈2Rerm

p1−|rm

p|2

|rn

p|2≈2Rerm

p.(4)

Plugging in the reflection coefficients rn,m

p, we reach the

approximation that is generally used to relate measured mag-

netic asymmetries to the magnetization:

Aapp ≈2Rexy sin 2θi

n4cos2θi−n2+sin2θi∝M,(5)

where xy =iQn2is the off-diagonal elements of the dielectric

function. As n≈1 in the XUV spectral range, the asymmetry

is maximized for θi≈45◦, i.e., at the Brewster angle θB.We

can readily test the validity of the approximation by calculat-

ing the second term T2 of Eq. (3). As examples, we choose

a vacuum/Fe interface and a vacuum/Ni interface, calculate

xy =2in(δ −iβ), and use tabulated values for n[36]as

well as for δ and β [37,38]. In Fig. 2, we display the results

as a function of both the incidence angle θiand photon energy

Eph. In the vicinity of the Brewster angle, the prerequisite of

the approximation |rm

p||rn

p|is no longer valid, leading to

values T2 1. Within the θi-Eph map, shown in Fig. 2,we

find minimum values of T2 =0.06 and 0.60 for Fe and Ni,

respectively. We also note that for experimental conditions

where T2 is significantly less than 1, the approximated asym-

metry Aapp reaches values exceeding ±1, which is inconsistent

with a normalized quantity. Obviously, care must be taken

when using expression (5) to characterize magnetic asymme-

tries in T-MOKE experiments. While these limitations were

discussed in some detail in the Ph.D. thesis of Turgut [39], we

emphasize that, with the majority of T-MOKE experiments

having been performed in the vicinity of the Brewster angle,

this important aspect has not been given sufficient attention.

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FIG. 2. Calculated term T2 as a function of the incidence angle

θiand photon energy Eph, describing the deviation between Aand

Aapp for (a) a vacuum/Fe interface and (b) a vacuum/Ni interface.

In the vicinity of the Brewster angle, T2 reaches values significantly

less than 1. Note that the color maps are scaled differently in the two

panels.

The second approximation in Eq. (5) limits the analysis of

Ato a single vacuum/magnetic interface. This is generally

not fulfilled because even experimental studies of magnetic

monolayers are effectively complex heterostructures, as thin

film samples often rely on seed layers on a substrate to tune

their magnetic properties, and are also often covered by a cap-

ping layer to prevent oxidation. Additionally, in time-resolved

experiments, where the magnetic layer is optically excited, the

layer thickness of the magnetic film is deliberately limited to

the optical penetration depth of the pump pulse of approxi-

mately 10–15 nm in order to avoid depth-dependent excitation

profiles. In this common situation, the reflectance of the XUV

probe pulse is also influenced by the interfaces between the

magnetic film and the seed layer and between the seed layer

and the substrate. In these cases, the T-MOKE asymmetry can

be calculated by simulating the reflectance of the investigated

sample heterostructure using the well-established matrix for-

malism first introduced by Parratt [40] and later extended to

describe magnetic systems [41–43]. Here, one solves the wave

field while taking into account the Fresnel equations and using

a recursive algorithm to describe multibeam interference from

complex structures with multiple interfaces or density and

magnetization gradients. In this work, we use an implemen-

tation provided by the UDKM1DSIM toolbox [44]. We note

that the main additional challenge to achieve reliable results

in such calculations is connected to accurate knowledge of

the sample structure, i.e., the layer thicknesses, densities, in-

terface roughnesses, and surface oxidation, as well as of the

exact values of the atomic and magnetic form factors at the

investigated resonances.

III. EXAMPLE STRUCTURE

A. Static characterization

In the following, we present experimental T-MOKE mea-

surements in which both assumptions leading to the linear

relationship between magnetic asymmetry and magnetization

are violated; i.e., the magnetic reflectance dominates for cer-

tain photon energies and incidence angles, and the reflectance

is not given by a single vacuum/magnetic film interface. We

present calculations based on the above-mentioned matrix for-

malism that demonstrate pronounced nonlinearities between

changes in the magnetization of a magnetic heterostruc-

ture and corresponding changes in the magnetic asymmetry,

i.e., A∝ M.

The investigated sample structure Al(30nm)/Fe50

Ni50(5nm)/Ta(2 nm)/glass is grown via electron beam

evaporation on a glass substrate. The Ta layer acts as

a seed layer for the growth process, while the Al layer

prevents oxidation. We emphasize that due to the small

absorption cross section of Al for energies below its L

edge, the relatively large thickness of the Al capping layer

has an almost negligible influence on the reflectance in the

XUV spectral range. The FeNi alloy exhibits an in-plane

magnetization with a square hysteresis loop, characterized by

a small coercive field less than 10mT.

The measurements are performed using XUV radiation

with a photon energy between 45 and 72 eV, generated by

a HHG source. We focus laser pulses from a Ti:sapphire

based amplifier system with a pulse duration of 25 fs, a

repetition rate of 3kHz, a pulse energy of ≈2.5mJ, and a

center wavelength of λ=800nm into a gas cell filled with

helium. Depending on the setting of the chirp and energy

of the laser pulses, either the resulting XUV spectrum is

characterized by discrete narrow-bandwidth emission peaks,

separated by ≈3.1eV, or it is quasicontinuous. To determine

the magnetic asymmetry, an electromagnet toggles the mag-

netization direction of the sample between the two opposite

directions perpendicular to the plane of incidence, yielding the

reflectances R±

p. For more details on the experimental setup

and time-resolved techniques, we refer to Refs. [45,46].

Figure 3(a) displays the reflectance measured by eight

discrete harmonic emission peaks for the two magnetization

directions of the FeNi alloy, as well as the magnetic asymme-

try measured using a continuous XUV spectrum (solid green

line) in the energy range of the Fe and Ni M3,2resonances. We

observe large asymmetry values with a bipolar shape almost

approaching ±100% for Fe around 54 eV and up to 50%

for Ni around 67eV. The dots indicate photon energies of

harmonics H33, H35, and H39, for which we will later discuss

time-resolved data (see Sec. IIIB).

The dashed violet line in Fig. 3(a) indicates the simulated

magnetic asymmetry obtained from the wave propagation

algorithm implemented by the UDKM1DSIM toolbox. The com-

plex atomic and magnetic form factors are retrieved from the

tabulated values provided by Henke et al. [36], complemented

by MCD and Faraday measurements for the M-edge reso-

nance of Fe and Ni [37,38]. The thicknesses of the different

layers are determined during the growth process by a quartz

crystal microbalance, calibrated by atomic force microscopy.

To yield good agreement between simulated and experimental

asymmetry, we allow small variations of the absolute values

of the elemental magnetic moments. We also use a reduced

density of Ta, in very close agreement with tabulated density

values of Ta(V) oxide, suggesting a partial oxidation of the

evaporation target. This is consistent with previous work in

the literature [47,48] and by us, in which we confirmed the

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JOHANNA RICHTER et al. PHYSICAL REVIEW B 109, 184440 (2024)

50 55 60 65 70

Energy (eV)

-1

-0.5

0.5

Asymmetry, A (~45°)

Reflectance (norm.)

(a)

simulation

data

tr data

(b)

Energy, E

(eV)

(deg)

Energy, E

(eV)

-7

-6

-5

-4

-3

log

(Reflectance)

(d)

30 35 40 45 50 55 60

(deg)

Energy, E

(eV)

-1

-0.5

0.5

Asymmetry

54 54.5 55

Energy, E

(eV)

0.5

Asymmetry, A (~45°)

(f)

m = 1

m = 0.8

m = 0.5

m = 0

52 53 54 55 56

Energy, E

(eV)

-7

-6

-5

-4

Reflectance (~45°)

(e)

FIG. 3. (a) Measured (solid green line) and simulated (dashed

purple line) continuous spectra of the magnetic asymmetry, as well

as measured reflectance of the discrete harmonic emission peaks for

the two magnetization directions of the FeNi alloy in the photon

energy range of the Fe and Ni M3,2resonance. The dots indicate

the photon energies of HHG peaks EH33 =51.3eV,EH35 =54.4eV,

and EH39 =66.8eV, for which we later discuss the ultrafast mag-

netization dynamics. (b) and (c) Simulations of the reflectance as a

function of the photon energy Eph and angle of incidence θifor both

transverse magnetization directions. (d) Corresponding simulation of

the magnetic asymmetry. (e) Reflectance at 45◦around the Fe M3,2

resonance for the m↑(dashed lines) and m↓(solid lines) magne-

tization directions, calculated for decreasing values of the magnetic

moment. (f) Corresponding asymmetries, clearly showing a strongly

nonlinear behavior at H35, where a decrease in magnetization leads

to an increase in A.

reduced Ta density by independent hard x-ray reflectance

measurements [14]. Finally, we assume that the top Al layer

is oxidized and model this by a 2 nm thin Al2O3layer [47].

We note that XUV reflectances and magnetic asymmetries are

especially sensitive to the oxidation of heavy elements since

their densities are significantly reduced when oxidized.

The determined magnetic and structural properties of the

FeNi system now allow us to calculate the reflectances for the

two transverse magnetization directions as a function of both

the incidence angle θiand photon energy Eph. The results are

shown on a logarithmic scale in Figs. 3(b) and 3(c). Around

the Brewster angle, i.e., in the vicinity of 45◦, the reflectance

is strongly reduced and reaches values down to 1 ×10−7.In

this region we observe large differences in reflectance for the

two opposite directions of magnetization, caused by changes

in the dichroic index of refraction. This behavior is partic-

ularly pronounced for H35 at the Fe M3,2resonance and a

clear indication that the magnetic contribution to the total

reflectance starts to dominate. This is also reflected in the

corresponding asymmetry map shown in Fig. 3(d), revealing

the largest asymmetry amplitudes varying between negative

and positive polarity around the Brewster angle and the M3,2

resonance.

In order to examine the reflectance around the Brewster

angle in more detail, we show line plots of the simulated

spectra at θi=45◦from52to56eVinFig.3(e). We observe

pronounced minima of the reflectance, shifted for the two

magnetization directions m↑(dashed lines) and m↓(solid

lines) by approximately 1.2 eV. As we reduce the normalized

magnetization m=M/M0of the FeNi alloy, the difference in

the minimum reflectances decreases and, as expected, con-

verges to a single value at ≈54 eV for m=0 (violet dotted

line). In this photon energy range, the linear approximation (5)

evidently does not hold: the reflectance is strongly influenced

by the magnetic part rm

p.InFig.3(f), we turn again to the

asymmetry A, showing its response at θi=45◦and zoom in

on the photon energy range between 54 and 55 eV, where the

asymmetry is characterized by a very steep slope, increasing

from 0 to almost 1. Strikingly, as we again examine the effect

of a reduction of the magnetization, we calculate an increase

in the magnetic asymmetry; compare the response of Aas m

is reduced from 1 (green line) to 0.8 (red line) and 0.5 (blue

line). We again mark the photon energy of H35 at 53.4eV,for

which we measured time-resolved data. We note that for the

case of a single vacuum/FeNi interface, we do not observe

that the asymmetries for different values of mcross each

other, suggesting that the complicated response is a direct

consequence of the multilayer structure.

B. Time-resolved response

In the following, we present ultrafast demagnetization

measurements which confirm our simulations regarding a

nonlinear relationship between magnetic asymmetry and mag-

netization. We excite the sample with laser pulses with a

wavelength of λ=800nm and a pulse duration below 30 fs at

the sample position. The pump fluence is set to ≈39 mJ/cm2,

defined as the incident pulse energy divided by the effective

footprint of the laser on the sample surface, calculated using

the full width at half maximum of the Gaussian beam shape

as the diameter. The light is almost exclusively absorbed

within the Al layer, generating hot electrons which are

injected into the FeNi alloy, causing its ultrafast loss of mag-

netization. This type of indirect excitation has been shown to

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0.25

0.5

0.75

time (ps)

(b)

= 54.4 eV

0.25

0.5

0.75

time (ps)

0.8

0.9

1.1

norm. Asymmetry, A/A

(a)

= 51.3 eV

0.25

0.5

0.75

time (ps)

0.8

0.9

1.1

norm. Magnetization, m

(c)

= 66.8 eV

A data

A model

FIG. 4. Time-resolved magnetic asymmetry A/A0for the photon

energies (a) EH33 =51.3eV, (b) EH35 =54.4 eV, and (c) EH39 =

66.8 eV. The dashed pink line shows a single-exponential, spatially

homogeneous decrease of the magnetization, while the solid lines

displays the corresponding calculated magnetic asymmetry based on

the wave propagation algorithm. For 54.4 eV, the magnetic asymme-

try exhibits a strongly nonlinear relationship to magnetization.

lead to efficient and ultrafast magnetization dynamics [49,50].

Also see our very recent work on a direct comparison between

direct and indirect excitation of an FeNi alloy [51].

The experimental results are summarized in Fig. 4for the

three photon energies marked in the static asymmetry spectra

in Fig. 3(a). We show the measured normalized asymmetry

A(t)/A(t<0) =A/A0(dots) as a function of pump-probe

delay up to 1ps. To describe the data, we calculate the

asymmetry for a normalized magnetization that decays expo-

nentially with time but homogeneously along the depth of the

sample according to

m=[1 −C(1 −e−t/τ )](t)∗G(t).(6)

Here, we additionally take into account the temporal res-

olution of the experiment of approximately 35fs [45]by

convolution with a Gaussian function G(t). is the Heaviside

function at time delay zero. We then vary the amplitude C

and time constant τuntil the calculated asymmetry matches

the measured data. We show the calculated asymmetry as

a solid line and the corresponding magnetization as a pink

dashed line. At Eph =51.3eV, we find τ≈250 fs and C=

0.22. Importantly, at this energy A∝Mprovides a very good

approximation. The situation changes dramatically when we

examine the response at 54.4eV, shown in Fig. 4(b). Here,

for the same evolution of m(pink dashed line), we observe an

ultrafast increase of the measured asymmetry by about 7%, in

agreement with the results shown in Fig. 3(f). We examine one

further photon energy, now in resonance with Ni at 66.8eV.

Here, we again observe an exponential decay of the measured

asymmetry. The calculated asymmetry, which describes the

data [solid yellow line in Fig. 4(c)], corresponds to a very

similar demagnetization amplitude compared to Fe and is

shown again as a pink dashed line. We find C=0.21 and

τ≈180 fs. Importantly, we find that the measured asymmetry

and magnetization show a non-negligible absolute deviation

of about 3%. Assuming direct proportionality between Aand

0.8

0.6

0.4

0.2

norm. Magnetization, m

0.2

0.4

0.6

0.8

1.2

norm. Asymmetry, A/A

= 45

NL =

0.63

(a)

51.3 eV

54.4 eV

66.8 eV

(b)

vacuum/Al(30nm)

/FeNi(5nm)/Ta(2nm)/glass

(deg)

Energy, E

(eV)

0.04

0.16

0.36

0.64

FIG. 5. (a) Normalized magnetic asymmetry A/A0for the sample

Al(30 nm)/Fe50Ni50(5 nm)/Ta(2 nm)/glass as a function of magne-

tization calculated for an angle in the vicinity of the Brewster angle

at θ=45◦. (b) The nonlinear response (NL) for the same sample as a

function of normal incidence angles θiand photon energies Eph.Note

that the color map is scaled quadratically.

m, one would underestimate the relative demagnetization am-

plitude by more than 15%.

C. Quantifying the nonlinear response of the magnetic

asymmetry

We now introduce a simple metric for the nonlinear re-

sponse of the magnetic asymmetry as the magnetization is

reduced for every point in space spanned by the incidence

angles θiand photon energies Eph. In order to do so, we first

calculate the normalized asymmetry A/A0as a function of

the normalized magnetization mfor the experimental geom-

etry discussed above, i.e., θi=45◦. We again consider only

a reduction of the magnetization that is homogeneous along

the depth of the FeNi alloy and, for simplicity, identical for

both elements Fe and Ni. This allows us to compare the

relationship between A/A0and mfor the same three photon

energies discussed above. Figure 5(a) summarizes the results.

While we find a nearly perfect linear dependence for EH33 =

51.3eV and only moderate deviations for EH39 =66.8eV,at

EH35 =54.4 eV the response strongly differs from the approx-

imation A/A0∝m. First, the asymmetry increases by about

15% for a magnetization amplitude of 0.6 before it crosses

A/A0=1form=0.37 and finally drops to zero. It becomes

apparent that this highly nonmonotonic and nonlinear de-

pendence of the asymmetry on the magnetization not only

influences the amplitude but also the perceived dynamics of

the pump-induced change. As the slope of the asymmetry as

a function of magnetization depends sensitively on the struc-

tural and magnetic parameters of the sample, this may offer

a potential explanation for the observed discrepancy in the

onset of measured and calculated time-resolved asymmetries

at EH35 =54.4eVinFig.4(b). Furthermore, we note that the

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JOHANNA RICHTER et al. PHYSICAL REVIEW B 109, 184440 (2024)

very early dynamics may be influenced by more complicated

nonequilibrium magnetization dynamics [18,30,45,52], which

are not taken into account in our simulation.

To quantify the nonlinear response of the asymmetry for

varying magnetization amplitudes, we introduce the metric

NL, defined as the maximum difference between the lin-

ear function A/A0=mand the actual, calculated values of

A(m)/A(m=1) (see Fig. 5). For EH35 =54.4 eV, we find

NL =0.63; for EH39 =66.8eV we find a much smaller value

of NL =0.04. The values of NL as a function of both exper-

imental parameters, i.e., the angle of incidence θiand photon

energy Eph, are summarized in Fig. 5(b) for the investigated

sample structure Al(30 nm)/Fe50Ni50(5 nm)/Ta(2 nm)/glass.

We find very large values around the Brewster angle, reaching

values of up to NL =1.2 around the Fe M3,2resonance. Note

that in regions where Aexhibits a zero crossing, NL has been

set to zero.

IV. OPTIMIZED SAMPLE

AND EXPERIMENTAL GEOMETRIES

In the final part of our study, we discuss strategies to

avoid or minimize nonlinearities in optical-pump–XUV-probe

T-MOKE experiments. It is important to note that although

correcting for a nonlinear relationship between A/A0and m

is possible in theory, reflectance simulations require precise

knowledge of both the structural parameters and the magnetic

and nonmagnetic indices of refraction. We find that particu-

larly for very large values of NL found close to the Brewster

angle, even small uncertainties in the input parameters can

result in significant effects on the asymmetry, making numer-

ical corrections difficult. To obtain a precise measure of the

magnetization and its dynamics, it is thus advantageous to

avoid such experimental geometries. Furthermore, a suitable

choice of the material for the antioxidation cap layer can

significantly change the situation.

In the design of an optimized sample structure, we are

guided by the condition for which the approximation in Eq. (5)

holds, namely, that the total reflectance has to be dominated by

the nonmagnetic reflectivity, i.e., |rm

p||rn

p|. This suggests

choosing a high-Zmaterial as a capping layer to increase

the nonmagnetic reflectivity, albeit at the cost of reduced

magnetic asymmetry. As an example, we chose Ta instead

of Al. To facilitate a direct comparison with the investigated

Al-capped magnetic FeNi layer, we adopt the same sample

parameters and replace only the capping layer by 2 nm of Ta.

We assume a thin Ta oxide layer of 1.5nm[47,48] and model

it with tabulated values for Ta(V) oxide. Since oxidation of

the top layer reduces its density and thus the nonmagnetic

reflectance of the structure, the calculated values of NL can

be considered an upper limit.

The magnetic asymmetry and NL landscape of the cor-

responding Ta(2 nm)/Fe50Ni50(5nm)/Ta(2 nm)/glass sample

are shown in Figs. 6(a) and 6(b), respectively. We still observe

sizable values of the magnetic asymmetry, but now with neg-

ligible values of NL <2×10−3.

To gain an overview of how capping layers (CLs) grown

with different materials influence the asymmetry and NL val-

ues, we repeat the above simulations for the same magnetic

heterostructure: CL(2 nm)/Fe50Ni50(5nm) /Ta(2 nm)/glass.

(a)

(deg)

Energy, E

(eV)

-0.1

0.1

Asymmetry

(b)

vacuum/Ta(2nm)/

FeNi(5nm)/Ta(2nm)/glass

(deg)

10-3

FIG. 6. (a) Magnetic asymmetry and (b) NL map as a function of

the incidence angle θiand photon energy Eph of the sample structure

Ta(2 nm)/Fe50Ni50(5nm)/Ta(2 nm)/glass. Note that changing the

capping layer to a high-Zelement significantly reduces the maximum

values of the nonlinear response (NL).

For low-Zmaterials such as Al, SiN, MgO, and native ox-

ide layers such as NiO, we find large maximum asymmetry

amplitudes approaching 1, but accompanied by very large

maximum values of NL >2. These maxima are found in

the vicinity of the Brewster angle and atomic resonances,

where, evidently, the assumption of a linear relationship be-

tween asymmetry and magnetization completely breaks down.

On the other hand, choosing a high-Zmaterial as a capping

layer, for example Pt, Ru, or Ta, allows recovering the linear

relationship between magnetic asymmetry and magnetiza-

tion. While we find significantly lower asymmetry amplitudes

on the order of A=0.1, the highest NL values remain

below 0.01.

As the magnetic properties of the magnetic film can be

strongly influenced by an adjacent layer, the choice of ma-

terials can be a constraint depending on the nature of the

investigation. In particular, interfaces with a heavy metal can

lead to perpendicular magnetic anisotropy [53] and spin-orbit

torques [54] in the presence of currents in the sample. In-

terdiffusion can lead to magnetically dead layers near the

interface, which is the case for Ta and Ru, for example [55,56].

Although platinum is a good element for oxidation protection,

it forms a strong interfacial magnetization leading to over-

lapping magnetic asymmetries of its O3,2resonance in the

spectral regions of the Fe and Co M3,2edges [46]. In these

cases, as well as for low-Zcapping materials and capping lay-

ers with an unknown oxidation state, it may be preferable to

perform experiments away from the Brewster angle to ensure

direct proportionality between asymmetry and magnetization.

V. CONCLUSION

We have discussed the limitations of the commonly used

approximation of the T-MOKE observable, which predicts

a direct proportionality between magnetic asymmetry and

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magnetization. In the vicinity of the Brewster angle, devi-

ations between the exact and approximated representations

of Acan exceed 100%, leading to potentially large errors in

determining the demagnetization amplitude in time-resolved

measurements. Furthermore, we showed that in magnetic het-

erostructures wave propagation and interference effects can

significantly alter the reflectance and corresponding asym-

metry, leading to a complex, nonmonotonic and nonlinear

relationship between A/A0and m. We provided experimental

results—both static and time resolved—in conjunction with

simulations using a wave propagation matrix formalism for a

prototypical magnetic heterostructure. Our most striking ex-

perimental observation is that for photon energies tuned to the

Fe M3,2resonance and for measurements in the vicinity of the

Brewster angle, the magnetic asymmetry increases in spite

of a decreasing magnetization, indicating a clear breakdown

of A∝M. Our investigation of the relationship between

magnetic asymmetry and magnetization as a function of both

the angle of incidence and photon energy suggests two strate-

gies to avoid a nonlinear T-MOKE response, albeit at the

cost of a lower magnetic asymmetry: One should perform

the measurement away from the Brewster angle and/or cap

the magnetic films with a high-Zmaterial, such as Ta, Pt,

or Ru. Both approaches increase the nonmagnetic part of

the total reflectance and allow retaining a linear relationship

between magnetic asymmetry and magnetization.

ACKNOWLEDGMENTS

C.v.K.S., J.R., and S.E. acknowledge financial support

from the Deutsche Forschungsgemeinschaft (DFG, German

Research Foundation), Project ID No. 328545488, TRR 227,

Project No. A02.

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