# Magnetic moment orientation and in depth distribution of dysprosium near the
surface of DyCo4.6 thin films determined by the analysis of their X ray
circularly polarized absorption Dy $M_{5}$ spectra

J. Díaz,1,2<EMAIL_ADDRESS>C. Blanco-Roldán,1,2 1Universidad de Oviedo,
Calle Federico García Lorca, 18, Oviedo 33007, Spain  2CINN (CSIC-Universidad
de Oviedo), 33940 El Entrego, Spain

###### Abstract

We have investigated the RE atomic distribution and its magnetic moment
orientation at the region near the surface of DyCo4.4&4.6 ferrimagnetic
amorphous films with perpendicular magnetic anisotropy (PMA). XMCD
spectroscopy of the films at the Dy $M_{4,5}$ and Co $L_{2,3}$ edges using
total electron yield (TEY) detection was performed at 2 K and 300 K
temperatures, and at sample orientations ranged from 0∘ to 70∘ with respect to
the magnetic easy axis. The measurements showed an apparent partial decoupling
between the cobalt and dysprosium magnetic sublattices. At RT, the magnetic
moment per atom of dysprosium was below the minimum value expected if all
dysprosium moments were AF coupled to cobalt. At 2 K, the cobalt sublattice
presented a stronger magnetic anisotropic behavior than the dysprosium
sublattice. A detailed analysis of the circularly polarized spectra of the Dy
$M_{5}$ edge, based on the deconvolution of the spectra in their related
parallel, antiparallel and transverse to $J_{z}$ spectral components,
demonstrates that the spectra are composed by dysprosium with different
magnetic moment distributions. The fit of the Dy $M_{5}$ spectra using the
$J_{z}$ spectral components evidenced a gradation of dysprosium concentration
due to segregation at the region probed by TEY. The topmost layer was
magnetically uncoupled from cobalt. At RT, 25$\%$ of the dysprosium magnetic
moments in the under layer were found averaged oriented in the same direction
as cobalt. The expected weak magnetic coupling of these dysprosium atoms to
cobalt should explain the surprisingly lower magnetic anisotropy of the
dysprosium sublattice compared to that of cobalt probed by TEY at 2 K.

###### pacs:

75.30.Gw, 75.50.Kj,75.70.Ak, 78.70.Dm,81.05.Bx

##  I Introduction

RE-TM alloys are well known magnetic materials since decades ago [1, 2],
constituting a key component in a variety of industrial and technological
applications. A renewed interest in these materials has grown because their
richness in magnetic behaviors and their relative flexibility for tailoring
their properties to fit in specific magnetic applications and devices. RE-TM
alloys are present in spring magnets[3, 4, 5], magnetic topological formations
[6, 7, 8], spin wave functional devices [9] and all optical magnetic switching
[10, 11, 12]. Their use is also favored by its simple thin film preparation
process which can be done at RT.

The extraordinary magnetic properties of these alloys are based on the
strength of the TM-TM exchange coupling, the high magnetic moment of the RE,
and their RE-TM indirect exchange coupling interaction. The high orbital
moment of its unquenched $4f$ orbital and the high energy of its spin-orbit
coupling, makes the RE acts as a strong and localized magnetic moment whose
orientation depends on the interatomic exchange and the crystal field at its
particular local atomic environment. These two interactions are mainly
provided by their TM neighbors whose RE-TM interaction energies overwhelms
those of RE-RE. Since the RE-TM magnetic exchange coupling is
antiferromagnetic in spin, RE-TM alloys are ferrimagnetic for heavy REs like
dysprosium. This interaction is usually treated as a molecular field whose
intensity is calculated to be of the order of 200 T for crystalline DyCo5
[13]. The energy of this interaction is, at least, one order of magnitude
smaller than the TM-TM exchange. These range of values are comparable to
$k_{B}T$ energies, giving rise to different magnetic configurations as a
function of temperature and RE concentration. A characteristic parameter that
define these ferrimagnetic alloys is their compensation temperature,
$T_{Comp}$, the temperature where the magnetic moment of the RE and TM
sublattices cancel to each other.

Giving the radical different magnetic behavior of the RE and the TM, the
understanding of the magnetism of these alloys requires a precise
characterization of the magnetic moment and anisotropy of the RE and TM
sublattices separately. The perfect technique to do so is XMCD spectroscopy.
The magnetic properties of the RE and TM sublattices can be studied separately
by tuning the incident circularly polarized X rays to the corresponding
absorption edges of the RE ($M_{4,5}$, probing its $4f$ orbital) and the TM
($L_{2,3}$, probing its 3d orbital)[14, 15, 16]. This is actually the
technique of choice for the study of RE-TM alloys in the form of thin films
because it allows ex-situ sample preparation and it can be sensitive to
regions at different depths of the sample by changing the way the X ray
absorption spectra is detected. Surface sensitivity is attained using TEY
detection, with probed depths of the order of 2 to 3 nm, whereas fluorescence
yield or X ray transmission are more bulk sensitive techniques. The
simultaneous use of these two detection modes has been proved to be important
for a correct understanding of these alloys. The different $T_{Comp}$ measured
in these alloys using TEY and bulk detection[5], together with the reported RE
segregation at their surface[17, 18], has been used as an argument to explain
interesting phenomena that occurs at temperatures near their $T_{Comp}$ when
the applied magnetic field is intense enough. The hysteresis loops of some
DyCo alloys of similar concentration than the samples used in this study
presented side wing loops at high applied fields as the presented in some
spring magnet-like bilayer structures [19, 3, 4]. The effect was explained as
derived from the effective different RE concentration at the bulk and the
region near the surface which will yield two different $T_{Comp}$ for each
region [5, 20].

In these studies, the $T_{Comp}$ at the region near the surface was estimated
from the magnetic moments of cobalt and dysprosium deduced by XMCD
measurements done using TEY detection. In all the cases, the analysis have
always assumed the presence of a single dysprosium magnetic phase. However,
due to the structural disorder of these alloys, it is expected different
grades of intensity in the magnetic exchange interaction between dysprosium
and cobalt. Such effects should be stronger at the surface whose assumed
larger RE concentration is driven by RE-RE bonding preference. Previous XMCD
studies in NdCo alloys conducted by us using TEY detection [21] reported a
substantial proportion of RE atoms that behaved as a paramagnetic, indicating
that not all the RE atoms probed might have the same exchange interaction
strength with the TM and, therefore, the same magnetic orientation
distribution. Although those experiments were sensitive to the surface, their
exact distribution was not demonstrated. Actually, EXAFS experiments performed
by us in similar samples deduced the possible presence of RE segregation in
the bulk as well[22]. The possible existence of paramagnetic RE atoms non-
exchange coupled to the TM due to disorder or/and RE segregated would
contribute to overestimate the $T_{Comp}$ at the surface because its average
magnetic moment orientation would be in the same direction of the appied
field.

This effective bilayer interpretation of the side wing loops in DyCo
monolayers has been contested by others who consider that the observed effect
is due to a spin flop phase transition[23]. The range of temperatures and
fields under which this transition is produced can be predicted by the $H-T$
(applied field $H$-Temperature) phase diagram of the alloy which is built by
considering all the interactions present in the alloy. These studies shows
that the relative magnetic anisotropy of the RE and TM sublattices [23], and
the interaction of the alloy at the interface with other metals [23] are key
to understand these spin flop transitions.

It is clear, then, the importance of having a good characterization of the
magnetism of these alloys at the region near the surface or at their interface
with other materials whose magnetic structure could be complex. The purpose of
the experiment presented in this work is to improve the spectroscopical tools
to understand the magnetic behavior of the TM and the RE sublattices, their
mutual interaction and their magnetic anisotropy using the information
extracted from the X ray absorption spectra of the RE atoms, and considering
their possible inhomogeneous distribution in depth at the region probed by
TEY.

For this experiment, we prepared DyCo thin films anisotropically uniaxial with
high PMA energies. Their anisotropy fields at 2 K, $H_{K}$, were well above
the range of available field intensity in the experiment. Measurements were
done at 2 K and RT. These two temperatures were distant enough from the
$T_{Comp}$ of the alloys to avoid possible spin flip (and flop) effects. Also,
it permitted to measure in the regions where each of the sublattices were
magnetically dominant (cobalt at RT, and dysprosium at 2 K). At 2 K, thermal
disorder was reduced to a minimum.

The ferrimagnetic character of DyCo alloys permits to detect dysprosium atoms
magnetically uncoupled from cobalt since they will be oriented in the opposite
direction than their counterpart exchange coupled dysprosium atoms at
temperatures above $T_{Comp}$ (cobalt sublattice magnetically dominant). Their
detection requires a deconvolution of the RE $M_{4,5}$ spectra in their
parallel, antiparallel and transverse components of $J_{z}$ which are
specially well defined in dysprosium. This deconvolution has been presented
before in DyCo films [24], but it was not linked to the moment orientation of
the dysprosium magnetic moments as it is done and explained in detail in this
work. Also, we use the XMCD spectra to deconvolve these components from the
circularly polarized spectra, instead of relying completely on their
theoretically calculated shapes, as the mentioned work tried. Actually, this
is the first time that this kind of analysis is applied in a certainly complex
system like the studied in this work.

We show the potential of this novel technique in the study of the interaction
between RE and cobalt atoms at the region probed by TEY, by measuring at
different orientation angles under a strong PMA. This technique is able to
detect the presence of a RE segregated layer that affects in a significant way
the measured moment of the RE. The presence of this layer seems intrinsic to
the thin film growth in RE-TM alloys. Even removing the effect of this layer,
the proportion of dysprosium atoms directly engaged in the PMA anisotropy of
the alloy probed by TEY was not majority, possibly due to the extended
thickness of the cobalt depleted layer caused by the dysprosium segregation.

The paper is organized as follows: First, sample preparation and experimental
details for XMCD data acquisition are described. Next, after showing the VSM
magnetometry characterization of the measured films, the XMCD experimental
results for each elemental sublattice are presented and discussed separately.
The following section is dedicated to explain how the deconvolution of the Dy
$M_{5}$ spectra is made, with the calculation details shown in an Appendix
section. After this explanation, a model to fit the Dy $M_{5}$ of the measured
samples using the deconvolved spectral components is proposed, which consisted
in two layer with different dysprosium moment distributions. The results of
the fits are presented and discussed. The final section are the conclusions of
the work.

##  II Experiment

The two studied DyCo thin films were prepared at RT by magnetron sputtering at
a base pressure of $10^{-8}$ mbar and $10^{-3}$ mbar Ar pressure. They were
grown onto silicon wafer substrates using two separate magnetron guns set at
normal (cobalt gun) and at 30∘ angle incidence (dysprosium) with respect to
the normal to the sample. Dysprosium and cobalt concentrations were calibrated
using a quartz balance. The deposition method was different for each sample.
Sample called DCC was prepared by codeposition of cobalt and dysprosium.
Sample called DCM was grown by the alternate deposition of cobalt and
dysprosium layers. The nominal thickness of the cobalt and dysprosium layers
was 4.9 Å and 2.8 Å respectively. The topmost deposited layer was cobalt. This
second preparation method was intended as a way to estimate the importance of
the interdiffusion between the elements forming the alloy in their structure
and magnetic properties. All the samples were protected with a 20 Å thick
aluminum capping layer. The mean atomic concentration in the alloys was
determined by electron induced fluorescence spectroscopy. The difference in
concentration between both films was relatively small. Sample DCC contained
more dysprosium (DyCo4.4±0.05) than the DCM film (DyCo4.6±0.05). Their nominal
thickness was 35 nm.

X ray circularly polarized absorption spectra were obtained at the HECTOR
endstation [25] of the BOREAS BL-29 beamline at the ALBA synchrotron using
total electron yield detection. HECTOR has a cryomagnet that can apply up to
$\pm$6 T along the x-ray beam direction at different sample orientations,
which in our case ranged from normal incidence (0∘) to near grazing incidence
(70∘). The cryomagnet works at ultra-high vacuum conditions (pressure within
the 10-10 mbar range). A liquid He cryostat permits to fix the sample
temperature between 2 K and 350 K. Circularly polarized light was produced by
an APPLE II elliptical undulator. Each XMCD spectra was the result of 4
spectra taken at opposite circular polarization helicities and magnetic field
orientations.

##  III EXPERIMENTAL RESULTS and DISCUSSION

###  III.1 Magnetometry

Both samples presented a PMA even at RT. Table 1 shows the magnetization in
remanence of samples DCC and DCM measured using a Vibrating Sample
Magnetometer (VSM). The values were obtained after magnetic saturation of the
films with a field of 9 T at 5 K applied normal to their plane [26]. The same
table gives the compensation temperatures, $T_{Comp}$, of the alloys and their
coercive fields at 2 K and RT. $T_{Comp}$ temperature was smaller in sample
DCM (90 K) than in sample DCC (120 K), in agreement with their different RE
concentration, and with the expected by comparing with the reported by others
in DyCo alloys of similar concentrations [5, 20], assuming a linear
relationship between $T_{Comp}$ and the atomic concentration of the alloy
[18]. The higher cobalt concentration in sample DCM causes a marked lower
magnetic remanence at 10 K than in sample DCC. This explains the large
difference between the coecive fields measured at 10 K, higher in sample DCM
(3.5 T) than in sample DCC (1.9 T). However, sample DCM has a higher $H_{C}$
at RT, when its $M_{S}$ is higher. X ray reflectometry shows rougher surfaces
in DCM than in DCC thin films, suggesting that the larger $H_{C}$ of the DCM
thin film is probably caused by its higher density of domain wall pinning
defects. The coercive fields of both samples were notoriously higher than the
reported in DyCo4 by others [5, 20], with $H_{C}$ values lower than 1 T at a
similar decrement ($\Delta$T=150 K) in temperature from their $T_{Comp}$ (250
K). This might account for a higher PMA energy and/or domain wall pinning
defects density in our samples. None of the samples presented triple
hysteresis loops at fields below 6 T at temperatures close to their $T_{Comp}$
as the reported in samples of similar concentration and thickness [5, 20].

Table 1: Compensation temperature, $T_{Comp}$, Magnetic remanence, $M_{R}$, and coercive field, $H_{C}$, of samples DCC and DCM at 2 K and RT[26]. Sample | $T_{Comp}$ | Temperature |  $M_{R}$ (emu$\cdot$cm-3) |  $H_{C}$ (T)
---|---|---|---|---
DCC | 90 K | 2 K | 125 | 1.9
|  | 300 K | 250 | 0.2
DCM | 125 K | 2 K | 75 | 3.5
|  | 300 K | 275 | 0.4

###  III.2 XMCD hysteresis loops and magnetic moments

Figure 1 shows the hysteresis loops of sample DCC at RT and 2K for the cobalt
and dysprosium sublattices obtained by measuring the intensity of the Co
$L_{3}$ peak and the Dy $M_{5}$ peak respectively. Their change in shape with
field orientation at RT (figures 1 (a) and (b)), from squared at 0∘
orientation to ”S” shaped at 70∘ field orientation, shows that the samples had
a PMA, as observed by VSM magnetometry. The loops of the cobalt and dysprosium
sublattices are nearly identical. A small decoupling between both sublattices
is only noticeable at high fields. The magnetization of cobalt remains
constant at high fields up to 6 T whereas that of dysprosium seems to
decreases steadily with the increasing field from zero field.

Figure 1: Hysteresis loops taken at the Dy $M_{5}$ (pink) and Co $L_{3}$
(blue) edges at RT: (a) normal orientation, (b) grazing incidence (70∘). The
hysteresis loop of the dysprosium sublattice multiplied by -1 has been drawn
in red behind the cobalt loop for comparison. (c) Dysprosium hysteresis loop
taken at 2 K at grazing incidence (70∘).

A similar effect, although much more pronounced than in the present case, has
been observed in NdCo films [21]. The effect was demonstrated to be caused by
a portion of paramagnetic Nd. A similar explanation would justify the
observation in the present case since, at this temperature, any dysprosium
magnetically uncoupled to cobalt would be oriented towards the field direction
as cobalt, reducing its total magnetization.

At 2 K, only the hysteresis loop of the dysprosium sublattice in sample DCM
was taken at near the plane field orientation (70∘), shown in figure 1(c). The
shape of the loop shows that at 6 T the films were far from being magnetically
saturated, indicating that the PMA energy of the samples was strongly
increased at this temperature. The lack of coercive field in the loop shows
that the measured magnetic moment experience a progressive rotation with the
applied field intensity. The loop shows a shape asymmetry between the positive
and negative branches.

###  III.3 XMCD Cobalt

The magnetic moments of cobalt were deduced from their $L_{2,3}$ spectra. The
method used to extract the cobalt absorption coefficient to correctly apply
the XMCD sum rules [27, 28], which considers saturation effects, is fully
described in [21]. The number of holes was calculated comparing their
unpolarized absorption spectra with that of a pure cobalt reference sample
deposited in similar conditions than the rest of the films. The number of
holes for this reference sample was set to the tabulated for pure cobalt,
2.49, yielding a magnetic moment at 2 K (1.79$\pm$0.02$\mu_{B}$) similar to
the measured by others (1.77 $\mu_{B}$) [29]. As it can be observed from
figures 2(a) and 2(b), the shape of the spectra was almost identical to the
cobalt reference except in their intensity which was lower and almost
independent of the sample orientation angle or temperature. Their number of
holes reduced to approximately 91$\%$ those of pure cobalt. The total magnetic
moment of the samples together with the related orbital and spin components
and orbital to spin ratios are shown in table 2. The total magnetic moment of
the two alloys at 2 K and normal incidence (easy magnetic axis) was similar
within the error, of the order of 1.33$\pm 0.03\mu_{B}$. This value is in
agreement with the expected from those observed in NdCo alloys as a function
of the number of holes [21], and they would correspond to a RE concentration
higher than the nominal, of the order of DyCo3.5. The magnetic moment obtained
at RT (only measured at normal incidence in DCC) was practically the same than
at 2 K, 1.34$\pm 0.03\mu_{B}$.

Table 2: Total magnetic moment (mtot), efective spin (m${}^{*}_{s}$) moment, orbital (mo) moment, and its ratio mo/m${}^{*}_{s}$, of cobalt obtained by XMCD in DyCo alloys (samples DCC and DCM) and YCo alloys (YCC and YCM). All moments are given in $\mu_{B}$ units. m${}^{*}_{s}$ and therefore, mtot have not been corrected by the mTz term, the dipole moment of spin. Error bars are of the order of 2$\%$ for $m_{s}$ and $m_{o}$. As a reference, pure cobalt (thin film) at 2 K: m${}_{s}=$1.58 $\mu_{B}$; m${}_{o}=$0.21 $\mu_{B}$; m${}_{tot}=$1.79 $\mu_{B}$ Sample | T |  | Field Orientation Angle
---|---|---|---
|  |  | 0∘ | 20∘ | 45∘ | 70∘
DCC | RT | m${}^{*}_{s}$ | 1.22 | - | - | 1.11
|  | mo | 0.12 | - | - | 0.16
|  | mtot | 1.34 | - | - | 1.25
|  | m${}_{o}/$m${}^{*}_{s}$ | 0.09 | - | - | 0.12
| 2 K | m${}^{*}_{s}$ | 1.17 | 1.15 | 0.91 | 0.41
|  | mo | 0.16 | 0.10 | 0.08 | 0.06
|  | mtot | 1.33 | 1.25 | 0.99 | 0.50
|  | m${}_{o}/$m${}^{*}_{s}$ | 0.14 | 0.09 | 0.09 | 0.10
DCM | RT | m${}^{*}_{s}$ | - | - | - | 1.08
|  | mo | - | - | - | 0.16
|  | mtot | - | - | - | 1.24
|  | m${}_{o}/$m${}^{*}_{s}$ | - | - | - | 0.15
| 2 K | m${}^{*}_{s}$ | 1.15 | 1.10 | 0.79 | 0.54
|  | mo | 0.15 | 0.10 | 0.06 | 0.04
|  | mtot | 1.30 | 1.20 | 0.85 | 0.58
|  | m${}_{o}/$m${}^{*}_{s}$ | 0.13 | 0.09 | 0.08 | 0.09
YCC | 2 K | m${}^{*}_{s}$ | 1.28 | - | 1.24 | 1.28
|  | mo | 0.14 | - | 0.19 | 0.16
|  | mtot | 1.42 | - | 1.43 | 1.44
|  | m${}_{o}/$m${}^{*}_{s}$ | 0.11 | - | 0.16 | 0.13
YCM | 2 K | m${}^{*}_{s}$ | 1.26 | - | 1.18 | 1.22
|  | mo | 0.07 | - | 0.18 | 0.17
|  | mtot | 1.33 | - | 1.36 | 1.39
|  | m${}_{o}/$m${}^{*}_{s}$ | 0.06 | - | 0.15 | 0.14
Figure 2: X ray absorption (XAS) spectra obtained at the Co $L_{2,3}$ edge of
a pure cobalt thin film compared with the spectra of cobalt in (a) DCC and (b)
DCM thin films taken at 2K and 0∘ (blue), 20∘ (green),45∘ (orange), and 70∘
(red) beam incidence angles (magnetic field orientation angle) with respect to
the normal to the sample. The most intense spectrum (black line) is from pure
cobalt. In the inset, the XMCD spectra taken at different orientation field
angles, using the same code color.

The effective spin and orbital magnetic moments of cobalt changed with the
field orientation angle at RT. The effective spin moment meff obtained from
the sum rules is the sum of the spin moment $m_{s}$ and the dipolar moment
$m_{T}$ which is angle dependent:

$m^{\theta}_{eff}=m^{\theta}_{s}-7m^{\theta}_{T}$ (1)

The dipolar moments at normal and in plane field orientation are -0.013
$\mu_{B}$ and 0.004 $\mu_{B}$, respectively, assuming a zero value for the
dipolar moment when it is angle averaged[30, 14]. The orbital anisotropy is
also oriented in the plane,
$\Delta$m${}_{L}=$m${}_{0^{\circ}}$-m${}_{70^{\circ}}=$-0.02. The anisotropy
of this term is usually linked to the anisotropy in the orbital moment [14,
31, 32] as observed in other 5d-3d systems like Au/Co/Au [33]. This means that
the cobalt sublattice of the alloys have an in plane magnetic anisotropy at
RT, in coincidence with what we observed in NdCo alloys [21]. Actually, the in
plane anisotropy of the cobalt sublattice is the expected for the oblate shape
of the Nd and Dy $4f$ orbital, which is the one that generates the expected
crystal field orientation for the RE out of plane anisotropy. Nevertheless,
the anisotropy of the observed values is small, with values below the
experimental error. This would discard cobalt as the origin of the PMA of the
alloy at RT.

The total magnetic moment measured at 2 K is displayed in figure 3(a) as a
function of the field orientation angle. It decreases in both samples with
increasing field angle orientations because of the uncompleted magnetic
saturation of the alloy at the 6 T applied field. Due to the higher strength
of the exchange coupling between cobalt atoms compared to the indirect
exchange interaction with the dysprosium moments, the orientation angle
dispersion of their magnetic moments is expected to be much smaller than those
of the dysprosium atoms. The dashed curve displayed in figure 3(a) is the fit
of the averaged magnetic moments of the two samples using a cosine function
plus a constant. The constant could be interpreted as the portion of the
cobalt magnetic moment that it is oriented parallel to the field. This
constant is only 10 $\%$ of the total intensity, indicating that most of the
cobalt magnetic moments were fixed at the easy axis direction.

Figure 3:  Total magnetic moment of cobalt (solid dots) and dysprosium (empty
dots) in samples DCC (red dots) and DCM (blue dots) measured at 2 K. The black
dashed line in (a) is the fit to a cosine function plus a constant. Black dots
an triangles are cobalt moments measured at RT. The estimated error of the
measurement is the width of the dots.

The orbital magnetic moment at 0∘ field orientation was bigger than at RT,
increased from 0.12 $\mu_{B}$ to 0.15 $\mu_{B}$. Both samples showed similar
values. For the rest of the field orientation angles, only the orbital to
effective spin moments ratio can be compared because the uncompleted magnetic
moment saturation. This ratio is the highest at 0∘ and it decreases
significantly in both samples, from 0.14 at normal orientation, to 0.09 (0.10
at 70 ∘ in sample DCC). This indicates that the perpendicular magnetic
anisotropy of the alloy involves either an anisotropy in the orbital moment of
cobalt in the same direction of the easy axis or a larger spin component at
angles far from the easy axis.

In order to check that if this effect is caused by the magnetic interaction at
cobalt with the RE, YCo films deposited in similar conditions of
concentration, film thickness and deposition process were measured by XMCD.
The chemical interaction of Yttrium is equivalent to that of RE, with a nearly
empty $4d$ band, but without an occupied $4f$ orbital, i.e., with no $4f$
magnetic moment. Actually, their RE-Co phase diagram is very similar [34, 35].
Therefore, the atomic structure of the YCo alloy should be similar to that of
DyCo. The magnetic moments of cobalt in YCo measured at RT (normal orientation
only) and 2 K are displayed in table 2. The values are about 10 $\%$ smaller
in the multilayer than in the continuous film and they increase in both films
when temperature goes from RT to 2 K, in contrast to the observed in the DyCo
alloys where there was almost no difference. The YCo alloys were magnetically
soft and their magnetic anisotropy was in the plane. Their orbital to
effective spin ratios were similar to the found at normal orientation in the
DyCo films. But they did not show the strong decrease in their value with the
increasing orientation angle as the observed in DyCo. This fact, together with
the cosine like variation in the total magnetic moment of cobalt in the DyCo
films, indicates that their probed cobalt atoms are mostly located at
magnetically anisotropic environments. By comparison with the YCo alloy, the
magnetic anisotropy found in the cobalt sublattice of the DyCo films must be
induced by the dysprosium atoms. It would be expected, therefore, a similar
angle variation in the magnetic moment of the dysprosium sublattice as the
found in cobalt.

###  III.4 XMCD Dy

In dysprosium, the most relevant changes in the spectra occur in the Dy
$M_{5}$ edge. Figure 4 shows the Dy $M_{5}$ taken in sample DCM at RT and 2 K,
for the two circular polarizations, at different angles. Sample DCC showed
similar changes. The Dy $M_{5}$ spectral shape is defined by three intense
peaks, marked in the figure as peaks AP, TR and PP, which are located at
photon energies 1294.5 eV, 1296.7 eV, and 1298.6 eV, respectively. Their
relative intensity changes with the polarization sign and with the field
orientation angle. At RT, the changes with the angle are small, but they are
very evident at 2 K. At this temperature, peak PP is the most intense at
$C^{+}$ polarization and normal incidence. Within the same polarization, peak
PP decreases in intensity as the field orientation angle (incident X ray beam
angle) increases. The opposite is observed using $C^{-}$ polarization. In this
case, peak marked AP is the most intense at normal incidence. For both
polarizations, peak TR increases in intensity when the incident angle
increases. As it will be shown, peaks PP, AP and TR are associated to
electronic transitions to states where the magnetic moment of the related $4f$
orbital is parallel, antiparallel or transverse to the circularly polarized x
ray beam wave vector, respectively. It is important to notice that, at RT,
peak PP is the most intense also in $C^{-}$ polarization. This is opposite to
what happen at 2 K, specially at 70∘, where the measured magnetic moment of
dysprosium is the lowest and comparable to the measured at RT.

Figure 4:  Dy $M_{5}$ spectra of sample DCM obtained at RT (spectra at (a) and
(b)) and 2 K (spectra (c) and (d)). The spectra taken with right handed
circular polarization are (a) and (c). (b) and (d) are the spectra taken using
left handed circular polarization. RT spectra were taken at 0∘ (dark blue
line) and 70∘ (red line) incident angles. 2 K spectra were taken at 0∘ (dark
blue line), 20∘ (light blue line), 45∘ (orange line) and 70∘ (red line)
incident angles.

The XMCD spectra of Dy $M_{4,5}$ edge were analyzed following the process
explained in [21]. Table 3 shows the magnetic moments of dysprosium in each of
the samples at RT and 2 K. The relative weakness of the AF interaction with
cobalt and the structural disorder of the alloy makes the magnetic moment of
the dysprosium atoms to be dispersed in its orientation with respect to the
cobalt magnetic moment forming what is called an asperomagnet [36]. The moment
orientation of the RE atoms when the alloy is magnetized in the easy axis is
usually taken as uniform and symmetrically distributed around this axis within
a cone with a half opening angle $\theta_{C}$ ($\theta_{C}=\pi/2$ represents
the distribution of moments in a semisphere). For field orientations away from
the easy axis, a dispersion in the shape and opening angle of the cone is
expected. In this case, we assume that the resulted distribution of the
dysprosium can be approximated by a cone that has its symmetry axis deviated
from the field direction an angle $\varphi_{C}$. The magnetic moment measured
is the averaged sum within the cone of the magnetic moment component along the
field orientation axis. At the easy axis, this sum is
$M_{Dy}\cos^{2}\theta_{C}/2$, where $M_{Dy}$ is 9.8 $\mu_{B}$, the dysprosium
magnetic moment of its $4f$ orbital. The magnetic anisotropy would then change
the value of the measured magnetic moment by opening the cone angle and
changing its symmetry axis orientation.

Table 3: Efective spin moment,m${}^{*}_{s}$, orbital moment, mo, dipole moment of spin, mTz, and total magnetic moment, mtot, of dysprosium obtained by XMCD in samples DCC and DCM. All moments are given in $\mu_{B}$ units. The estimated error bars is of 2$\%$ in m${}^{*}_{s}$, mo and mTz. mTz=$\frac{m_{o}}{6}\left[\frac{m^{*}_{s}}{m_{o}}-\frac{m_{s}}{m_{o}}\right]$ [15], using the theoretical value of $\frac{m_{s}}{m_{o}}=0.475$ [16]. mtot=mo+(m${}^{*}_{s}$-6mTz) Sample | T |  | Field Orientation Angle
---|---|---|---
|  |  | 0∘ | 20∘ | 45∘ | 70∘
DCC | RT | m${}^{*}_{s}$ | -1.92 | - | - | -1.92
|  | mo | -1.42 | - | - | -1.41
|  | mTz | -0.09 | - | - | -0.10
|  | mtot | -2.78 | - | - | -2.74
| 2 K | m${}^{*}_{s}$ | 2.78 | 2.51 | 2.19 | 1.46
|  | mo | 4.19 | 3.81 | 3.42 | 2.23
|  | mTz | 0.26 | 0.23 | 0.19 | 0.13
|  | mtot | 8.18 | 7.43 | 6.67 | 4.35
DCM | RT | m${}^{*}_{s}$ | - | - | - | -0.91
|  | mo | - | - | - | -1.44
|  | mTz | - | - | - | -0.07
|  | mtot | - | - | - | -2.81
| 2 K | m${}^{*}_{s}$ | 2.84 | 2.64 | 2.14 | 1.61
|  | mo | 4.21 | 3.93 | 3.17 | 1.61
|  | mTz | 0.28 | 0.26 | 0.21 | 0.15
|  | mtot | 8.2 | 7.66 | 6.19 | 4.73

At RT, the dysprosium magnetic moment has almost no difference at both
orientations, as expected from the hysteresis loops. The PMA of dysprosium has
not enough energy to distort its magnetic moment distribution cone at RT.
There is not a significant variation between the values obtained in both
samples, which are about -2.8 $\mu_{B}$ (sample DCM only measured at 70∘ field
orientation). This quantity is small. The lowest magnetic moment that can be
measured assuming that all dysprosium atoms are AF coupled to Co, i.e.,
uniformly distributed within a cone of half opening angle $\theta_{C}=\pi/2$
(semisphere), is $M_{Dy}/2=$ -4.9 $\mu_{B}$, i.e., half the value of its total
magnetic moment. This means that there should be dysprosium atoms with their
moment orientation opposite to that of those AF coupled to cobalt. Theoretical
calculations estimated that the averaged magnetic moment of Dy decays to half
its value at about 300 K in crystalline DyCo5 [37]. The lower values observed
in the present case should have to do with the intrinsic disorder of the
measured alloys, with dysprosium atoms distributed in different atomic
environments where, in some of them, the exchange coupling to cobalt should be
weaker than in crystalline DyCo5 or possibly nonexistent. The analysis of the
Dy $M_{5}$ spectra exposed in a later section searches to understand how these
dysprosium atoms are distributed through the depth proved by TEY and how this
can affect the TEY measurements including the possible presence of segregated
dysprosium at the interface with the aluminum capping layer.

Figure 3 (right side scale) displays the variation of the dysprosium magnetic
moment as a function of the field orientation angle at 2 K compared to that of
cobalt. Both samples have similar values which decay with the angle at a
slower pace than in the cobalt sublattice. This magnetically less anisotropic
behavior of dysprosium is somehow contrary to what expected: the magnetic
moment of the RE should be, compared to cobalt, the one fixed with the highest
energy to the easy axis since the PMA in these alloys must stems from it, as
deduced from the analysis done in the previous section (III.3). The total
magnetic anisotropy of the alloy should depend on the distribution of the
crystal field orientation at the RE sites[2], and its influence on the
magnetic moment of the bonding TMs at each RE site. Following the single ion
anisotropy model [2], both crystal field and RE-TM indirect exchange are
expected to be tightly related. Therefore, the RE environments that most
contribute to the magnetic anisotropy of the alloys should be strongly
exchange coupled to the TM as well.

The magnetically less anisotropic character of the measured dysprosium
sublattice indicates that only a portion of the dysprosium atoms probed by TEY
should be active magnetic anisotropy generators. The magnetic field felt by
dysprosium is the sum of the external applied field and the molecular field,
which is mainly provided by the interatomic exchange interaction with cobalt
atoms, whose strength can be estimated in, at least, more than 150 T [21]. As
it has been observed, the rotation of the cobalt moment at grazing
orientations is small. Therefore, the effective molecular field felt by the
dysprosium should be closer to that of the applied field at those sample
orientation angles, meaning that a portion of the probed dysprosium should be
somehow decoupled or poorly coupled to cobalt, as stemmed from the analysis
done at RT. One of the challenges in the analysis of the Dy $M_{5}$ spectra,
shown in the next sections, is to determine the location of the dysprosium
which are apparently weakly interacting to cobalt.

A way to compare the magnetization measured in the bulk with the magnetization
measure near the surface (XMCD using TEY) is to evaluate how these magnetic
moments found by XMCD fit with the deduced from VSM, shown in table 1. When
the magnetic moment of the cobalt used to calculate the total magnetization of
the alloy is the deduced by XMCD, 1.33 $\mu_{B}$, the magnetic moment of
dysprosium that it is needed to match the magnetization measured by VSM at RT
is 3.8 $\mu_{B}$, 1 $\mu_{B}$ above the measured by XMCD. This value is still
below the expected if all dysprosium magnetic moments were AF coupled to
cobalt. When the same estimation is done at 2 K, the magnetic moment expected
for dysprosium is 7 $\mu_{B}$, 1 $\mu_{B}$ smaller than the found by XMCD.
These differences between bulk (VSM) and surface (XMCD) are similar to the
reported in [5, 20]. These were justify there because it was assumed a higher
dysprosium concentration at the region near surface than in the bulk due to RE
segregation. If that was happening in our samples, their cobalt and dysprosium
moments in the bulk should be higher than the measured by XMCD. For instance,
if the cobalt moment in the bulk raised to 1.50 $\mu_{B}$, the related
dysprosium moment should be 4.7 $\mu_{B}$, which is almost 2 $\mu_{B}$ higher
than the measured by XMCD. This would set the surface at a higher $T_{Comp}$
than in the bulk. But none of our samples showed wing side hysteresis loops at
7 T at temperatures near the $T_{Comp}$, as the reported in [5, 20] in alloys
of similar thickness and concentration.

The $T_{Comp}$ of a ferrimagnetic RE-TM alloy essentially depends on the
magnetic exchange strength between the RE and the TM atoms. If the segregated
RE atoms at the region near the surface do not bond tight to cobalt, their
exchange interaction will be weaken. The effect, in terms of magnetic moments,
will be similar to an increase of the $T_{Comp}$ because, above this
temperature, there will be dysprosium moments that will not be AF coupled to
cobalt, reducing the total moment of the dysprosium sublattice. Below
$T_{Comp}$, the effect will be the opposite, i.e., they will add up in moment
to the moment of the AF coupled-to-cobalt dysprosium. This is not an unlikely
situation regarding the magnetic moments of dysprosium measured by XMCD, which
are well below the expected if all RE atoms were AF coupled to cobalt.
Moreover, if RE segregation occurs is because it favors RE-RE metallic
bonding. To demonstrate the presence of these two kinds of RE atoms requires a
technique able to detect them. The next section shows a way to do it in a
quantitative way by analyzing the circularly polarized Dy $M_{5}$ spectra.
Actually, dysprosium is specially well adapted for this kind of analysis,
although it should be applicable to most of the REs, with the exception of
those with $J=0$ like $Gd^{+3}$.

##  IV Spectral analysis Method for Dy $M_{4,5}$ spectra

Like in most of the RE, the electrons in the $4f$ orbital of dysprosium are
well screened by the valence band orbitals and behave as in an isolated atom.
Actually, the Dy $M_{4,5}$ edge spectrum, which involves electronic
transitions form the 3d to the $4f$ orbital, is practically insensitive to the
dysprosium chemical environment and it is well fitted by calculating it using
only intratomic interactions[38, 39]. The relative intensity of these
transitions can be notably influenced by the bonding with other magnetic atoms
through its indirect exchange magnetic interaction and the resulting crystal
field by breaking its degeneracy in the $J_{z}$ momentum. Their relative
weakness permits to treat them as perturbations whose most important effect is
to orient the magnetic moment of $4f$ orbital along a specific direction,
i.e., to break the degeneracy of the quantum number $M$ of the $4f$ angular
moment component $J_{z}$.

Therefore, to a good approximation, only the intensity, but not the shape, of
the Dy $M_{4,5}$ lines are modulated by the orientation of the total angular
moment of the $4f$ orbital with respect to the direction of the polarized X
ray. The allowed transitions in the dipole approximation are those in which
$\Delta J=0,\pm 1$. If light is circularly polarized, only $\Delta m=\pm 1$
transitions are allowed. Therefore, if the beam is perfectly oriented parallel
(antiparallel) to the magnetic moment of the $4f$ orbital, the resulting
$M_{4,5}$ spectrum will be built with only those excitations where $\Delta
J=0,\pm 1$ and $\Delta m=1(-1)$. We will call to this spectral components PP
if $\Delta m=1$ and AP when $\Delta m=-1$. The circular dichroism spectrum,
XMCD, is the difference between these two spectrum. The sum rules related to
these transitions gives rise to the XMCD sum rules that permits to determine
both the magnetic spin and orbital moment of the $4f$ orbital. $\Delta m=0$
transitions occurs when the light is linearly polarized along the magnetic
moment direction. This spectral component is not present in the XMCD spectrum
because depends on $\left\langle M^{2}\right\rangle$ [39]. We will call to
this component TR.

The excitations of all the three spectral components PP, AP and TR occurs when
the $4f$ magnetic moment and the incident circularly polarized light are not
parallel. This is because the electric field felt by the $4f$ electrons is
equivalent to the sum of two circularly polarized fields with opposite
helicities, and a linear polarized field aligned with the direction of the
magnetic moment. The detailed calculation of the amplitude of these fields is
explained in appendix A. The dependency of the amplitude of these polarized
field with the magnetic moment orientation angle $\theta$ is:

$P=\frac{e^{\pm i\varphi}}{\sqrt{2}}\left[\frac{(\cos\theta\pm
1)}{\sqrt{2}}P^{-1}_{z_{m}}+\frac{(\cos\theta\mp
1)}{\sqrt{2}}P^{1}_{z_{m}}+P^{0}_{z_{m}}\sin\theta\right]$ (2)

$P^{-1}_{z_{m}}$, $P^{1}_{z_{m}}$ and $P^{0}_{z_{m}}$ are the dipole operators
for right and left circular polarizations, and linear polarization along the
$z$ axis, respectively. Taking into account the previous expressions, the
modulation of the intensity of each of the spectral components (PP, AP and TR)
with the orientation angle $\theta$ is given by the following expression (see
appendix A):

$PP+AP+TR=\left[\frac{(\cos\theta\pm
1)^{2}}{4}A^{-1}_{JJ^{\prime}}+\frac{(\cos\theta\mp
1)^{2}}{4}A^{1}_{JJ^{\prime}}+\frac{\sin^{2}\theta}{2}A^{0}_{JJ^{\prime}}\right]$
(3)

where $A^{q}_{JJ^{\prime}}$ are the reduced angular integers. They are
directly related to the transitions $\Delta J=0,\pm 1$ and $q=\Delta M$. The
distribution in energy and intensity of these transitions are well tabulated,
both theoretically and experimentally [39, 40, 38]. Figure 5(a) displays the
coefficients that multiply to the angular integers $A^{q}_{JJ^{\prime}}$
showed in 3 as a function of the angle $\theta$. As expected, the coefficients
that multiply to the TR ($A^{0}_{JJ^{\prime}}$) and AP ($A^{1}_{JJ^{\prime}}$)
components increase their value only at angles closer to ${\pi/2}$.

Figure 5:  Angle coefficients for PP, AP and TR as a function of (a) magnetic
moment orientation angle with respect to the circularly polarized incident
beam, $\theta$, and (b) cone half opening angle $\theta_{C}$.

The most common situation found in the studied alloys, which have no defined
crystal structure, is the one in which the orientation of the magnetic moment
of dysprosium is not well defined but it is distributed over a range of angles
due to structural and thermal disorder. This distribution is assumed to be
uniform and symmetrical with respect to the axis of a cone with a half opening
angle $\theta_{C}$. This distribution modifies the coefficients of the angular
integers. The new coefficients in this situation are obtained by averaging the
angle coefficients in 3 for each component within this angle, resulting in the
following expression as a function of $\theta_{C}$:

$PP+AP+TR=\frac{1}{3}\frac{\left(1-\cos^{6}\frac{\theta_{C}}{2}\right)}{\sin^{2}\frac{\theta_{C}}{2}}A^{-1}_{JJ^{\prime}}+\frac{1}{3}\sin^{4}\frac{\theta_{C}}{2}A^{1}_{JJ^{\prime}}+\frac{1}{3}\frac{\left(\cos^{3}\theta_{C}-3\cos\theta_{C}+2\right)}{1-\cos\theta_{C}}A^{0}_{JJ^{\prime}}$
(4)

Figure 5(b) shows the coefficients of each excitation component as a function
of the cone half opening angle $\theta_{C}$. The coefficients of the TR and AP
components increment its value at large angles, but not as fast as in the case
of a defined angle orientation. This is better observed when the ratio between
the TR and AP components, shown in both figures, are compared. These ratios
are substantially smaller in the cone angle coefficients (figure 5(b)).

When the cone has an inclination angle $\varphi_{C}$ with respect to the beam
direction, the components PP’ AP’ and TR’ of the magnetic moment in the cone
are projected in the beam axis following expression 3, substituting the angle
$\theta$ by $\varphi_{C}$. The PP’ component is calculated using equation 4,
the AP’ component is the same as the equation 3, but interchanging the factors
$A^{-1}_{JJ^{\prime}}$ by $A^{1}_{JJ^{\prime}}$, and the TR’ component is
calculated by multiplying TR by $\sin^{2}\theta_{C}/2$.

Since the cross section values of the angular integers are tabulated [38, 39]
and the angle dependent modulation of the PP, AP and TR components are known
(equations 3 and 4)), it is possible to determine the orientation of the
magnetic moment of any RE in any chemical environment just by identifying
their PP, AP and TR components and checking their relative intensities. This
is something that can be inferred also by comparing the magnetic moment
deduced from the application of the XMCD sum rules and compare it with the
total magnetic moment of the RE. But this is only valid if there is a single
magnetic form of dysprosium. Things become more complicated if the analyzed
material contains RE in different magnetic states, i.e., RE with different
magnetic moment orientation distributions. The deconvolution of the spectra in
their PP, AP and TR components is then required. This could be done by their
direct calculation using numerical methods as the employed by Thole et al.
[38]. This is the approach used in [24]. But it is important to notice that
the build of the RE $M_{4,5}$ spectrum only requires, to a good approximation,
the shape of the spectral component, which can be isolated by spectral
methods. The Dy $M_{5}$ spectrum is specially suited for this kind of
analysis.

Due to the large value of the orbital moment, $L$, of the $4f$ orbital in
dysprosium, the splitting between the PP and AP components in the $M_{5}$ edge
is the largest among the RE. Dy is the RE that has the lowest overlap between
these two sets of excitations, which is estimated in less than $5\%$ [39].
This eases the extraction of both spectral components from any XMCD spectra at
the Dy $M_{5}$ edge. Figure 6 shows the XMCD spectrum for Dy $M_{5}$. Only the
PP and AP components are in the spectra, which are well distinguished because
they have opposite sign. These two spectral shapes can be taken, as a first
approximation, to the PP and AP true components of the spectra. The TR
spectral component is extracted by subtraction of these two approximated
components to the related circularly polarized spectra. For the subtraction,
each of the components must be multiplied by a coefficient which is related to
its relative spectral intensity, which depends on the sum of all the RE
magnetic moment orientation distributions probed. These coefficients must be
the same in both circularly polarized spectra, bearing in mind that the PP and
AP components are interchanged in the two circular polarization spectra taken
at the same magnetic field direction. The determination of the value of these
coefficients is not precise, since it depends on how well the shape of the TR
component is known. In our approach, we used the spectral lineshape
theoretically calculated by Thole et al. [38]. This so extracted TR component
contains the overlap region between the PP and AP components. This produces
some shape differences between the TR obtained at different beam orientations
angles, because the proportion of the TR component respect to the PP and AP
components changes. In our case, the spectra taken at 70∘ has, clearly, the
lowest overlap proportion in its TR component. Then, the PP-AP overlapping
region can be extracted by subtracting this TR component to the TR component
withdrawn at 0∘, being the former conveniently normalized to the intensity of
the 0∘ TR component. This overlapping component is added to the PP and AP
components directly extracted from the XMCD spectra obtained at any angle and
temperature. The process is then repeated for the extraction of the related TR
components, reducing in this way its overlapping portion, and approaching to
the true TR component.

The result of applying this process in our samples is shown in figure 7.
Although the overall shape of each of the components is very similar for any
spectra, they were not exactly the same. For instance, we observed a reduction
in the energy distance between the AP and PP peaks position with increasing
field orientation angle, which is visible in their XAS spectrum shown in
figure 7 (a) for sample DCC. The same observation was made in sample DCM and
at RT. The reason of this is beyond the scope of this work. More specific
experiments are required to determine if these differences, which are small in
any case, are caused by experimental factors or because the spectra are
actually sensitive to the chemical and/or crystal field environments [41, 21,
42]. However, such differences imposed that the deconvolution of each spectrum
and, later on, its model fitting, had to be done using its own PP, AP and TR
components. We noticed that the obtained coefficients for each of the
components used in the deconvolution of the spectra have little variations
when using the corrected or the uncorrected spectral components.

Figure 6:  (a) XMCD spectrum of the DCC sample at RT and normal incidence, and
its related PP and AP components; (b) $C^{-}$ polarized spectrum of DCC sample
taken at RT and normal incidence with their related PP, AP and TR components;
(c) $C^{+}$ polarized spectrum of DCC sample taken at RT and normal incidence
with their related PP, AP and TR components  Figure 7:  (a) XAS spectrum
($C^{+}+C^{-}$) of sample DCC taken at 2 K and at different orientation
angles: 0∘ (dark blue line), 20∘ (light blue line), 45∘ (orange line) and 70∘
(red line). (b) PP and AP components, and (c) TR components extracted from the
spectra of the DCC sample at 2 K. The colors code is the same as in (a).

Figure 8 shows the normalized coefficients of the three spectral components
obtained from the deconvolution of the spectra of the two samples DCC and DCM
at RT and 2 K, as a function of the beam and magnetic field orientation angle.
The plot was done to compare it with the angle dependent coefficients plotted
in figures 5(a) and 5(b), which are cross section independent. The
normalization of the coefficients was done by dividing all the components by
the intensity of the PP component measured at normal incidence, which is the
one with the highest intensity, and corrected by their relative cross
sections. Cross sections were obtained from [39]. The used cross section ratio
between the PP and the AP components were 3.4, and 1.85 between PP and TR
components. The TR component is divided by 2, since there are two possible
traversal directions to $J_{z}$. The coefficients obtained at 2 K had the TR
component smaller than the AP component at any angle. There is not any
dysprosium moment orientation with such a coefficient configuration in figure
5. The same occurs at RT. In this case, the AP experimental coefficient is
overwhelm higher than the PP coefficient. Moreover, the PP coefficient reaches
such a value only when the dysprosium magnetic moment orientation is in the
plane. All these means that the dysprosium spectra of the DCC and DCM samples
can not be understood using a single distribution of dysprosium magnetic
moments.

Figure 8:  Normalized PP (blue dots), AP (red dots) and TR (yellow dots)
components in Dy $M_{5}$ spectra of samples (a) DCC and (b) DCM taken at 2 K
as a function of the circularly polarized incident beam. Empty squares are
related to the components of the Dy $M_{5}$ spectra taken at RT.

This finding supports the conclusion derived from the low magnetic moment
value of dysprosium at RT in section III.4: the observed spectra must be the
result of the combination of the spectra from, at least, two different
distribution of dysprosium magnetic moments. One of them must have the
dysprosium poorly magnetically coupled or even uncoupled from cobalt. For this
to happen, a possibility is to have this dysprosium physically separated from
the DyCo alloy by segregation at the interface with the aluminum cap layer. RE
segregation is a common effect in RE-TM alloys, with the RE always located at
the free surface region [17, 22, 21, 18]. Then, the total spectra can not be
built by the simple summation of two spectra. That would be the case if the
two types of dysprosium atoms were uniformly distributed all through the depth
of the film. When TEY detection is used, the measured spectral intensity is
build from the secondary electrons current which depends on the depth from
where they are emitted. Therefore, the excitations produced at the surface are
more heavily weighted than the originated in the layers underneath the
segregated dysprosium. For instance, a possible way to explain the larger than
expected intensity of component AP found in the spectra is by the presence of
dysprosium with low magnetic moment (large cone opening angle) at the topmost
surface of the alloy.

The intensity of the secondary electrons collected by the detector depends on
the depth from where they are generated by two factors: their scape depth in
the excited layers ($\lambda_{e}$), and the X ray intensity which is absorbed
as it propagates through them. This causes distortions in the measured spectra
with respect to the true absorption coefficient of the analyzed sample. These
saturation effects becomes more important as the angle of incidence is
increased, because the X ray penetration depth becomes similar to the
secondary electrons scape depth [43]. Such effects are taken into account when
the spectra are analyzed to recover the absorption coefficients that were
originated using a method which is fully described in [21]. Note that this
analysis process assumes the presence of a single layer.

Then, two steps are required to untangle the experimental spectra in their
contributions from the different regions at different depths from the films
surface. The first step is to model the experimental spectra assuming the
presence of, at least, two layers with different thickness, dysprosium
concentrations, magnetic moment distributions and secondary electron scape
lengths. The modeling of the spectra is done using the absorption cross
section of the PP, AP and TR components (i.e., the related peak shapes
$A^{-1}_{JJ^{\prime}}$, $A^{1}_{JJ^{\prime}}$ and $A^{0}_{JJ^{\prime}}$),
multiplied by their corresponding angle dependent coefficients shown before.
The sum of the contribution of each layer is weighted depending on their
specific thickness and depth. The resulting spectral intensity follows the
expression:

$I(E)\approx\frac{I_{0n}}{\cos\theta_{i}}\left(\mu_{1}(E)\frac{(1-e^{-\frac{z_{1}}{\lambda_{e1}}})}{\frac{\lambda_{e1}\mu_{1}(E)}{\cos\theta_{i}}+1}+\mu_{2}(E)\frac{\lambda_{e2}}{\lambda_{e1}}\frac{e^{-\frac{z_{1}}{\lambda_{e2}}}}{\frac{\lambda_{e2}\mu_{2}(E)}{\cos\theta_{i}}+1}\right)$
(5)

This equation has been calculated assuming that the first layer is thinner
than the absorption coefficient ($z<<\mu_{1,2}(E)$), which is applicable in
our case.

In the second step, the modeled spectrum passes through the same process as
the actual experimental spectrum, described in [21] to extract what it could
be called the effective absorption coefficient. This is a necessary step to
compare the saturation corrected experimental effective absorption
coefficient, where only a single layer is assumed in the spectrum saturation
correction process, with the modeled one.

The parameters to enter in the modeling of the spectra are the thickness of
the first layer, $\tau_{1}$, the dysprosium concentration $\rho_{i}$ of the
top layer and the underlayer, the secondary electron scape length
$\lambda_{ei}$ for each of the layers, the overall escape length used to
correct saturation effects $\lambda_{ei}$, and the opening and inclination
angles of the cone of each layer, $\theta_{Ci}$ and $\varphi_{Ci}$. The model
that significantly best fitted the spectra required an additional parameter
for each layer which represented the portion of dysprosium atoms with moments
in the opposite orientation.

Some assumptions were done to chose the value of some of the parameters of the
model. The cobalt concentration of the first layer was set to zero in the
first layer in base of the expected RE segregation. The secondary electron
scape length was set to 25 Å for both layers. This is the length used in the
saturation correction of the experimental spectrum, which was the one that
worked the best in these alloys [21]. The validity of this parameter to
correct the saturation effects was tested by measuring the experimental ratio
between the AP and PP component extracted from the XMCD spectra. This ratio
has to be constant for any field orientation and probed sample. Its value was
identical to the theoretical one [39] within the experimental error. The
goodness of the fits were evaluated by minimizing a $\chi^{2}$ function.
Different values of the above mentioned fixed parameters were tried to check
the incidence in the results of the fits. Their variation over a relatively
wide range of values did not change in a substantial way the conclusions of
the spectral fits here presented.

The accuracy of the fitted parameters and the discrimination accuracy of the
possible configurations was increased by fitting at the same time the spectra
taken at opposite circular polarizations. Both spectra are symmetrical in
their spectral components and, therefore, they must use the same fitting
parameters: the TR component does not change with circular polarization light
and the PP and AP components becomes AP and PP components in the opposite
helicity. Obviously, the condition for the fits of yielding the same magnetic
moment value as the experimental one was imposed. The inadequacy of the
spectra to be fitted with a single component was tested. The quality of the
fits was very low in this case, with $\chi^{2}$ values two orders of magnitude
higher than the resulted from the proposed models, confirming the previous
conclusion extracted from the deconvolution of the spectra in their PP, AP and
TR components.

The spectral fittings were done using the PP and AP components with
(corrected) and without (uncorrected) the overlapping component, using the
related TR component in each case. We will only show the results corresponding
to the corrected components. The results were essentially the same using the
uncorrected components.

The described analysis of the spectra used two layers for the spectra
modeling. Include a larger number of layers might compromise the accuracy of
the analysis due to the consequent increase in the number of variables. Also,
the thickness of the effective probed layer, which is estimated to be of the
order of 2 to 3 nm, leaves little room for a better in depth accuracy for the
method, because the intensity is exponentially reduced with depth. The use of
two layers is also justified by posterior measurements on the same samples by
X ray photoemission which showed a significantly larger dysprosium
concentration at the surface than the nominally expected. The next section
discusses the results of the fits.

###  IV.1 Results and discussion of the fits

Table 4 shows the result of the fits with the lowest $\chi^{2}$ of the spectra
taken at RT and 2 K. Figure 9 shows the fitting of the circularly polarized
spectra of sample DCC at RT an 0∘ incident angle as an example of the quality
of the fit. In the model used for these fits, the top layer had its average
moment direction opposite to that of the dysprosium in the under-layer. The
underlayer had two dysprosium moment components that shared the same moment
distribution, but having their average moment in opposite directions. Figure
10 shows the spectra of each of the layers used to adjust to the spectra taken
at RT and at 0∘ and 70∘ incident angles. The proportion of the component not
AF oriented to cobalt was estimated in 25$\%$ of the total. The cone opening
angle of these two components was of about 50∘. In the top layer, the moment
orientation was distributed uniformly in a semi-sphere. These parameters did
not change too much with field angle orientation or sample. The only observed
changes occurred in the effective atomic cobalt concentration, which increased
from 3.6 to 4.9 atoms of cobalt per 1 of dysprosium from normal to grazing
incidence. This change has the effect of increasing the spectral weight of the
top layer with increasing field orientation angle in a slight higher
proportion than the predicted by equation 5. This effect was even stronger in
the fits of the spectra taken at 2 K. The possible explanation to this will be
discussed later on.

Figure 9:  Fit of the circularly polarized spectra (a) C+ and (b) C- of Dy
$M_{5}$ of sample DCC taken at RT. The dark blue line is the experimental
spectrum, the dashed light blue line is the fitted curve, and the red line is
the residual. The light red line is the spectral component for the first layer
and the blue line is the spectral component for the under layer.

The total moment of dysprosium in the underlayer was 3.8$\mu_{B}$. It matched
the expected from the VSM data (see table 1) using the cobalt moment deduced
from XMCD. It is important to notice that this value was obtained considering
the presence of a significant portion of dysprosium atoms whose average
magnetic moment orientation was the same as cobalt. There is no other way to
obtain such a low magnetic moment in dysprosium. This is the reason why the AP
component is so intense at this temperature. The presence of this phase in the
underlayer indicates that it should also exists in the bulk, although in a
lower proportion in base of the estimated value of its dysprosium moment in
that region. The magnetic interaction of this kind of dysprosium with cobalt
must be either too weak or null and it might be caused by a segregation
process and/or by disorder. The relatively narrow moment distribution of this
dysprosium compared to the fitted on the top layer could be explained by being
the former in closer interaction with other magnetic atoms increasing the mean
field felt by them, whereas the latter might be mixed with those of the
capping layer at the interface. At low temperature, this type of dysprosium in
the under-layer should contribute to increase the total magnetic moment of the
dysprosium sublattice. The presence of these weakly coupled dysprosium atoms
at the region probed by TEY would explain the apparent increase in the
$T_{Comp}$ at the surface with respect to the bulk measured by XMCD using TEY
detection, since its average magnetic moment is oriented in the same direction
as the applied field. It should be also the cause,of the observed small
decoupling between the cobalt and dysprosium hysteresis loops (see section
III.1).

Figure 10:  Espectral components of the Dy $M_{5}$ circularly polarized
spectra of sample DCC taken at RT and at (a) 0∘ and (b) 70∘ orientation field
angles. For C+ polarization, blight blue line for the top layer and dark blue
line for the under layer. For C- polarization, blight orange line for the top
layer and red line for the under layer.  Figure 11:  Espectral components of
the circularly polarized spectra Dy $M_{5}$ taken at 2 K and at (a) 0∘ and (b)
70∘ orientation field angles for sample DCC, and at (c) 0∘ and (d) 70∘ for
sample DCM. For C+ polarization, blight blue line for the top layer and dark
blue line for the under layer. For C- polarization, blight orange line for the
top layer and red line for the under layer.

At 2 K, the underlayer has a single component, but the top layer required two
types of dysprosium with opposite averaged moment directions. The moment
distribution for this layer caused a reduction of its total moment to almost
zero, becoming close to a magnetically dead layer. This change in the magnetic
behavior of the top layer could be attributed to the presence of some oxidized
dysprosium which is AF below 4 K [44]. The oxidation state does not cause any
change in the Dy $M_{4,5}$ spectra [45] but it does in the Co $L_{2,3}$
spectra. Cobalt spectra did not show any feature related to oxidized cobalt,
what it could be explained by the expected location of the possibly oxidized
segregated dysprosium at the interface with the aluminum cap layer. Figure 11
shows the fitted components for both layers at the two circular polarizations
for 0∘ and 70∘ incidence angles at 2 K, and for both samples.

Table 4: Fitting parameters of Dy $M_{5}$ spectra of samples DCC and DCM taken at RT and 2K, and at different sample orientations with respect to the applied field: $\tau_{1}$: thickness of the top layer; $\theta_{C1}$ and $\theta_{C2}$: half opening cone angle for the fanning distribution of the magnetic moment of dysprosium in the top and under the top probed layers, respectively; $\varphi_{C2}$: cone inclination angle with respect to the beam (field) orientation in the underlayer; $m_{C2}$ estimated magnetic moment of the underlayer; $\rho_{2}$: effective atomic concentration of dysprosium at the under layer; AP1: proportion of antiparallel dysprosium in the first layer (only at 2 K); $\chi^{2}$: statistical goodness of the fit. Sample | T |  | Field Orientation Angle
---|---|---|---
|  |  | 0∘ | 20∘ | 45∘ | 70∘
DCC | RT | $\tau_{1}$ | 1.6(1) Å | - | - | 1.5(1) Å
|  | $\theta_{C1}$ | 90∘ | - | - | 90∘
|  | $\theta_{C2}$ | 53.5(1)∘ | - | - | 54.1(1)∘
|  | $AF_{C2}$ | 0.26(1) | - | - | 0.26(1)
|  | $m_{C2}$ | -3.8(1) | - | - | -3.8(1)
|  | $\rho_{2}$ | DyCo3.6(2) | - | - | DyCo4.9(3)
|  | $\chi^{2}$ | 0.6 | - | - | 0.8
| 2 K | $\tau_{1}$ | 2.1(1) Å | 2.0(1) Å | 2.8(1) Å | 5.6(1) Å
|  | $\theta_{C1}$ | 90∘ | 90∘ | 90∘ | 90∘
|  | $\theta_{C2}$ | 35.9(1)∘ | 45.7(2)∘ | 47.5(2)∘ | 40.2(1)∘
|  | $\varphi_{C2}$ | 0 | 10∘ | 9∘ | 47∘
|  | $m_{C2}$ | 8.9 | 8.2 | 8.1 | 5.8
|  | $\rho_{2}$ | DyCo5.1 | DyCo5.0 | DyCo5.2 | DyCo5.8
|  | AP1 | 56$\%$ | 64$\%$ | 56$\%$ | 39$\%$
|  | $\chi^{2}$ | 1.74 | 3.23 | 2.0 | 0.9
DCM | RT | $\tau_{1}$ | - | - | - | 1.5 (1) Å
|  | $\theta_{C1}$ | - | - | - | 90∘
|  | $\theta_{C2}$ | - | - | - | -56.7(1)∘
|  | $AF_{C2}$ | - | - | - | 0.26(1)
|  | $m_{C2}$ | - | - | - | -3.7(1)
|  | $\rho_{2}$ | - | - | - | DyCo5.3
|  | $\chi^{2}$ | - | - | - | 0.4
| 2 K | $\tau_{1}$ | 2.6(2) Å | 2.4(2) Å | 3.3(2) Å | 3.7(2) Å
|  | $\theta_{C1}$ | 90∘ | 90∘ | 90∘ | 90∘
|  | $\theta_{C2}$ | 27.0(1)∘ | 32(1)∘ | 44(1)∘ | 48(1)∘
|  | $\varphi_{C2}$ | 0(1)∘ | 6(1)∘ | 25(1)∘ | 46(1)∘
|  | $m_{C2}$ | 9.3(2) | 9.0(2) | 7.8(2) | 6.1(2)
|  | $\rho_{2}$ | DyCo4.9(1) | DyCo5.3(1) | DyCo5.4(1) | DyCo5.4(1)
|  | AP1 | 61$\%$ | 61$\%$ | 58$\%$ | 52$\%$
|  | $\chi^{2}$ | 3.3 | 3.3 | 3 | 2.8
Figure 12: (a) Comparison between the magnetic moment of dysprosium in the
underlayer resulted from the ”‘two layers model” fits (red dots, sample DCC;
blue dots, sample DCM), and the magnetic moments measured considering a single
dysprosium magnetic state, as a function of the field orientation angle. (b)
Comparison between the spectra of the Dy $M_{5}$ circularly polarized spectra
of samples DCC and DCM taken at 2 K and at 70∘. Dark (C+) and light (C-) blue
lines for sample DCM and red (C+) and orange (C-) lines for sample DCC.

Due to the low magnetization of the top layer, the values of the dysprosium
moments in the under-layer deduced from the fits were larger than the
extracted from the XMCD measurements (see table 3). They are compared in
figure 12 (a). The values are similar in both samples. An important
consequence of the values of the cobalt and dysprosium moments found by the
fits at 2 K is the expected enhancement of the magnetization at the region
probed by TEY. At normal incidence, and keeping the nominal dysprosium to
cobalt concentration, both samples yielded the same magnetization, 345
$emu/cm^{3}$. This is almost 5 times for sample DCM, and 3 times for DCC,
their magnetization in the bulk measured by VSM at 10 K. The magnetization
enhancement would be even higher if the dysprosium concentration was higher.
This high magnetization of the surface region has to do mainly with the
increased momentum of dysprosium with respect to the expected in the bulk.
Attending to the results of the fits obtained at RT, the increment in the
dysprosium total magnetic moment can be explained by the portion of the weakly
magnetically exchange-coupled-to-cobalt dysprosium atoms which, at this
temperature, are ferromagnetic. The magnetic anisotropy of these dysprosium
atoms is expected to be lower than those exchange coupled to cobalt, what
would explain the apparent lower anisotropy of the dysprosium sublattice.
Also, the anisotropy field $H_{k}$ is expected to decrease due to the increase
in the magnetization. In reference to this, the deviations of the dysprosium
cone distribution with respect to the beam (angle $\varphi_{C2}$) and the
increasing opening of the cone angle (angle $\theta_{C2}$) with the
increassing field orientation angle $\theta_{i}$ with respect to the easy axis
is the qualitatively expected behavior in the presence of an uniaxial PMA
where dysprosium atoms are subject to different local anisotropy energies. The
variation in the dysprosium magnetic distribution were similar in both samples
(see table 4). A higher insight of the magnetic interactions and anisotropy of
the dysprosium and cobalt atoms could be gain by calculating them in base to
the observed magnetic moment distributions resulted from the fits [46].

The required use of dysprosium with opposite average moment directions in the
fits of the spectra at RT and at 2 K is consequent with the observations done
in the previous section, where it was shown that the intensity of the
deconvolved AP component was too high compared to the TR component. The fits
showed, however, some apparent inconsistencies in the values of the thickness
of the first layer and the cobalt concentration of the underlayer that were
pointed out at the beginning of this section. These consisted in an increased
value of both parameters with the incident angle, indicating that the spectral
weight given to the top layer in equation 5 was below the required in the
fits. The second inconsistency was an increase in the values of these
parameters when the temperature decreased to 2 K. These two effects could be
attributed, at least partially, to the simplicity of the model, which does not
consider any gradation in the moment distribution from one layer to the other,
and to the fact that the spectra represents moment distributions, which might
change with the temperature, and not chemical changes.

The thickness values for the top layer used in the fits are small, of the
order of one dysprosium monolayer, and its highest value is not higher than
two monolayers. These values might not be the actual ones. For instance, they
are below the estimated roughness of the film surfaces. Moreover, it is
expected some diffusion of dysprosium within the aluminum capping layer that
should extend the thickness of the top layer dysprosium, decreasing its
density and concentration. These details are not contemplated in the fit,
where the concentration of the top layer was fixed to pure dysprosium.
Therefore, thickness and concentrations would be acting as effective
parameters. This would explain the apparent increase in the thickness of the
top layer and/or in cobalt concentration in the underlayer at higher incident
angles. In this situation, the signal from the underlayer will be taken from
deeper regions, which have lower secondary electrons emitted and a larger path
for them to scape up to the surface of the sample. The final effect is an
increase in the spectral weight of the top layer in the spectra.

However, this can not fully explain the larger spectral weight of the top
layer in the spectra taken at 2 K and 70∘ angle incidence with respect to the
spectra taken at RT. This can be checked by comparing the spectral components
used to fit the spectra taken at RT in figure 10 with those taken at 2 k shown
in figure 11. Moreover, the spectral weight of the top layer is different
depending on the sample, being specially large in sample DCC, as it can be
observed in figure 11. The difference of the fitted parameters in these two
samples at 70∘ incidence angle is related to their different intensity in
their AP component, as it is shown in figure 12 (b).

At 2 K, a forced reduction in the thickness of the first layer in the fits of
the spectra taken at high angle of incidence to match those at low incidence
angles was possible but the fits yielded a reduction in the magnetization of
the under layer. The related magnetic configurations had to be discard because
the yielded dysprosium moments for the underlayer were below the deduced from
XMCD (table 3). Moreover, such a low magnetic moment would be inconsistent
with the values obtained at lower angles. If these changes were related to an
irregular distribution of segregated dysprosium across the film, their lateral
size should be in the range of the size of the beam which could be in this
case of the order of 300 $\mu$m.

A plausible explanation for this effect might have to do with the sensitivity
of the dysprosium spectrum to the moment distribution across the thickness of
the film and not to its chemistry. The magnetization of the probed dysprosium
increases from RT to 2 K. Pure dysprosium becomes ferromagnetic at
temperatures below 80 K and oxidized dysprosium is AF at temperatures below 4
K. This means that the changes in the orientation of the dysprosium moment
should be more gradual from the top layer to the under-layer at 2 K than at
RT, as the fits suggests. As the orientation of the applied magnetic field
deviates from the easy axis, the top layer with its nearly zero average
magnetic momentum would penetrate into the film driven by the increasing
energy of the anisotropy field that reduced the magnetization in the direction
of the applied field [47]. More measurements done at different temperatures
are needed to fully test this explanation, what will definitively prove the
high sensitivity of the proposed analysis method to the variations in the
magnetic moment of the REs at the region probed by TEY.

##  V Conclusions

To summarize, the magnetism of the dysprosium and the cobalt sublattices in
ferromagnetic DyCo thin films with PMA has been investigated by XMCD
spectroscopy using TEY detection. Some unexpected results were observed in the
magnetic behavior of the dysprosium sublattice. The measurements done at the
Co $L_{2,3}$ showed that the PMA of the films must stem from the RE
sublattice. However, at 2 K, when the PMA energy of the films was the highest,
the probed cobalt sublattice resulted to be magnetically more anisotropic than
the dysprosium sublattice. Additionally, the magnetic moment of the dysprosium
sublattice measured at RT was lower than expected if all moments were AF
coupled to cobalt. To understand the cause of these apparent anomalies, a
method to analyze the circularly polarized Dy M5 spectra obtained by TEY has
been presented, which is based in the deconvolution of their parallel,
antiparallel and transverse to $J_{z}$ spectral components. This is the first
time that this kind of analysis is applied in this kind of systems. This
spectral analysis reveals the presence of a relatively large proportion of
dysprosium with its average moment oriented in the same direction than cobalt,
and a thin layer of segregated dysprosium at the top of the alloy which should
be not exchanged coupled to cobalt. Some of the dysprosium in this layer is AF
coupled at 2 K, suggesting that either it is partially oxidized or there exist
a proportion of dysprosium which is magnetically dead at this temperature. The
apparently lower magnetic anisotropy of dysprosium with the field orientation
angle is explained by the thickness of the segregated dysprosium layer at the
region near the surface. A similar kind of measurements should be conducted
using bulk detection sensitivity to determine the precise distribution of RE
sites that contribute to the PMA of the alloy. This seems mandatory to be able
to predict with accuracy the possible magnetic configurations resulting from
the interaction between RE and TM sublattices.

The presented results show that, if the RE is segregated at the surface, care
must be taken to remove the effect that it causes in the value of the RE
magnetic moment measured using TEY detection. This affects, for instance, to
the $T_{Comp}$ values of RE-TM alloys measured at the surface in this way,
which could be overestimated.

The proposed analysis method can serve to detect the presence of RE in
different magnetic states, either in the surface or in the bulk. The method
can be extended to the circularly polarized spectra of other RE at the
$M_{4,5}$ edge, whenever the angular moment $J$ of its $4f$ orbital is
different of zero (i.e.,it can not be applied to Gd+3). In those other RE, the
identification of the PP, AP and TR spectral components requires a different
methodology than the presented here for dysprosium since their overlap is much
important. This might not be a problem because these components are well
approximated by theory. They can be also isolated by spectroscopic methods
using circular but also linear polarized spectroscopy at the related RE
$M_{4,5}$ edges. Nevertheless, more additional work is needed to improve and
test the accuracy of the method by better characterizing the region probed
with other surface sensitive techniques.

The experiment confirms the presence of segregated RE reported in other RE-TM
systems. The preparation method of the alloy did not produced substantial
differences in the magnetism of the cobalt and dysprosium sublattices at the
surface of the alloy, indicating that the origin of their different
macroscopic behaviors has to be found in the bulk or they are caused by their
different microstructure. The no-dependence of the RE segregation with the
deposition method (co-sputtering and alternate deposition) remarks that the
segregated RE is an immediate post deposition process, which can proceed
during time as it has been observed in GdCo [17]. This is also demonstrated in
the DCM film whose last deposited layer, before depositing the aluminum
capping layer, was cobalt. This means that RE atoms could diffuse also in the
bulk at defects like voids, cracks and grain borders within the bulk. This
might affect magnetic properties like, for instance, the coercive field of the
thin film since it would contribute to the formation of domain wall pinning
defects.

##  Appendix A Orientation dependence of light absorption

Here, it is shown the details of the calculation of the intensity absorbed by
circularly polarized light when the orientation of the angular moment of the
$4f$ orbital forms an angle $\theta$ with the direction of the wavevector
$\vec{k}$ which is chosen parallel to the $z$ axis. The components of the
electric field of the circularly polarized light in the frame of the $4f$
angular moment, with its $z_{m}$ axis along its quantization axis, are
calculated by making a first rotation of an angle $\varphi$ in the $xy$
polarization plane ($z$ axis as the rotation axis) followed by a rotation of
an angle $\theta$ around the resulted $y$ axis after the first rotation.

$\left(\begin{array}[]{c}\epsilon_{x_{m}}\\\ \epsilon_{y_{m}}\\\
\epsilon_{z_{m}}\\\
\end{array}\right)=\left(\begin{array}[]{ccc}\cos\varphi&-\sin\varphi&0\\\
\sin\varphi&\cos\varphi&0\\\ 0&0&1\\\
\end{array}\right)\left(\begin{array}[]{ccc}\cos\theta&0&-\sin\theta\\\
0&1&0\\\ \sin\theta&0&\cos\theta\\\
\end{array}\right)\left(\begin{array}[]{c}\epsilon_{x}\\\ \epsilon_{y}\\\
\epsilon_{z}\\\ \end{array}\right)$ (6)

Therefore, for right ($-$) and left ($+$) circular polarization:

$\left(\begin{array}[]{c}\epsilon_{x_{m}}\\\ \epsilon_{y_{m}}\\\
\epsilon_{z_{m}}\\\ \end{array}\right)=\frac{\epsilon e^{\pm
i\varphi}}{\sqrt{2}}\left(\begin{array}[]{c}\cos\theta\\\ \mp i\\\
\sin\theta\\\ \end{array}\right)$ (7)

By using this result, the electric dipole operator $P$ written in the frame of
the 4$f$ orbital reads:

$P=\frac{e^{\pm i\varphi}}{\sqrt{2}}\left[P^{0}_{x_{m}}\cos\theta\mp
iP^{0}_{y_{m}}+P^{0}_{z_{m}}\sin\theta\right]$ (8)

The superscript in the dipole operator indicates the polarization state of the
electric field in the frame of the 4$f$ orbital: $0$ is lineal, $1$ circular
left handed and $+1$ circular right handed. The electric dipole operator can
be expressed in the components parallel to the quantization axis $z_{m}$ of
the 4$f$ orbital using the relations:

$P^{0}_{x_{m}}=\frac{1}{\sqrt{2}}\left[P^{-1}_{z_{m}}+P^{1}_{z_{m}}\right]$
(9)

and

$P^{0}_{y_{m}}=\frac{i}{\sqrt{2}}\left[P^{-1}_{z_{m}}-P^{1}_{z_{m}}\right]$
(10)

Then

$P=\frac{e^{\pm i\varphi}}{\sqrt{2}}\left[\frac{(\cos\theta\pm
1)}{\sqrt{2}}P^{-1}_{z_{m}}+\frac{(\cos\theta\mp
1)}{\sqrt{2}}P^{1}_{z_{m}}+P^{0}_{z_{m}}\sin\theta\right]$ (11)

The absorption cross section for the excitation from a state $\left|\alpha
mJ\right\rangle$ to a state
$\left|\alpha^{\prime}m^{\prime}J^{\prime}\right\rangle$ is, in the dipole
approximation:

$\sigma_{\alpha
mJ\longrightarrow\alpha^{\prime}m^{\prime}J^{\prime}}=4\pi^{2}\alpha_{0}\hbar\omega\left|\left\langle\alpha
mJ\right|P\left|\alpha^{\prime}m^{\prime}J^{\prime}\right\rangle\right|^{2}$
(12)

The cross section for circular polarized light is obtained substituting
equation 2 in equation 3:

$\sigma_{\alpha
mJ\longrightarrow\alpha^{\prime}m^{\prime}J^{\prime}}=4\pi^{2}\alpha_{0}\hbar\omega
S_{\alpha J\alpha^{\prime}J^{\prime}}\left[\frac{(\cos\theta\pm
1)^{2}}{4}A^{-1}_{JJ^{\prime}}+\frac{(\cos\theta\mp
1)^{2}}{4}A^{1}_{JJ^{\prime}}+\frac{\sin\theta^{2}}{2}A^{0}_{JJ^{\prime}}\right]$
(13)

where $S_{\alpha J\alpha^{\prime}J^{\prime}}$ is the radial integer and
$A^{q}_{JJ^{\prime}}$ are the angular integers. Under the assumption of a weak
crystal field, as it happens in dysprosium, the values of the radial and
angular integers do not change. Only the relative orientation of the moment
with respect to incident beam changes. We redefine the angular integers as the
absorption cross sections for the corresponding situation in which the
polarized beam and the magnetization are parallel($q=0,\pm 1$). Then, the
$\theta$ dependent functions that multiply to each of the related $q=0,\pm 1$
cross sections are the coefficients for the TR ($q=0$), PP ($q=-1$) and AP
($q=+1$) components:

$PP+AP+TR=\left[\frac{(\cos\theta\pm
1)^{2}}{4}A^{-1}_{JJ^{\prime}}+\frac{(\cos\theta\mp
1)^{2}}{4}A^{1}_{JJ^{\prime}}+\frac{\sin^{2}\theta}{2}A^{0}_{JJ^{\prime}}\right]$
(14)

###### Acknowledgements.

J.D. acknowledges M. Valvidares for BOREAS beam line set up and his help in
the beamline during data acquisition, and C. Quirós, and J. M. Alameda for
providing the samples and the VSM measurements.This project has been supported
by Spanish MINECO under grant FIS2016-76058 (AEI/FEDER, EU) and grant
PID2019-104604RB/AEI/10.13039/501100011033.

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