# Spin wave study of magnetic perpendicular surface anisotropy
in single crystalline MgO/Fe/MgO films

J. Solano  Institut de Physique et Chimie des Matériaux de Strasbourg, UMR
7504 CNRS, Université de Strasbourg, 23 rue du Loess, BP 43, 67034 Strasbourg
Cedex 2, France  O. Gladii  Helmholtz-Zentrum Dresden - Rossendorf, Institute
of Ion Beam Physics and Materials Research, 01328 Dresden, Germany  P. Kuntz
Entroview, 11 rue des mathématiques, 38402 Saint Martin d’Hères, France  Y.
Henry  D. Halley  M. Bailleul<EMAIL_ADDRESS>Institut de
Physique et Chimie des Matériaux de Strasbourg, UMR 7504 CNRS, Université de
Strasbourg, 23 rue du Loess, BP 43, 67034 Strasbourg Cedex 2, France

###### Abstract

Broadband ferromagnetic resonance is measured in single crystalline Fe films
of varying thickness sandwiched between MgO layers. An exhaustive magnetic
characterization of the films (exchange constant, cubic, uniaxial and surface
anisotropies) is enabled by the study of the uniform and the first
perpendicular standing spin wave modes as a function of applied magnetic field
and film thickness. Additional measurements of non-reciprocal spin wave
propagation allow us to separate each of the two interface contributions to
the total surface anisotropy. The results are consistent with the model of a
quasi-bulk film interior and two magnetically different top and bottom
interfaces, a difference ascribed to different oxidation states.

††preprint: APS/123-QED

##  I Introduction

In recent years, there has been an increased interest in magnetic films with
large perpendicular magnetic anisotropy due to their potential to improve the
efficiency and non-volatility of spin transfer torque magnetoresistive random
access memories [1, 2, 3, 4, 5]. In this search for new materials, Fe films
interfaced with MgO are of particular interest due to their favourable
properties: a small magnetic damping, a high tunneling magnetoresistance and a
large perpendicular surface anisotropy (PSA) [6, 7, 2, 8, 3].

There is a good general agreement between experimental and theoretical
investigations on the nature and order of magnitude of this PSA, with values
ranging between 0.86 and 3.15 $\text{mJ/m}^{2}$[3, 8, 9, 10, 11, 12, 13, 14,
15, 16, 17]. These are up to two times larger than usual PSA found at the
interfaces between transition metals and heavy metals [18, 6, 7] despite weak
spin orbit coupling[15]. Theoretical works have attributed this large PSA to
the hybridization between interfacial oxygen and iron atoms [13, 14].

Experimentally, it is very challenging to access the internal magnetic
environment of ultrathin films and separate the contributions of the top and
bottom interfaces to the total PSA of a film. So far, experimental
characterizations [3, 19, 20, 21, 8, 9, 10, 11, 12] have relied on
magnetometry measurements to estimate the surface magnetic anisotropy. This
means that additional hypothesis were needed to extract the individual surface
anisotropies, including comparison with a reference interface or assumptions
regarding possible bulk magnetoelastic contributions.

In this work, we separate the top and bottom perpendicular surface
anisotropies of single crystalline MgO/Fe/MgO films resorting exclusively to
spectroscopic measurements of inhomogeous magnetization dynamics. To achieve
this, we perform a broadband ferromagnetic resonance characterization of a
thickness series of films MgO/Fe$\text{(}t\text{)/}$MgO
($t\\!=\\!10\text{-}30$nm) and combine it with a careful study of non-
reciprocal spin-wave propagation. Our results show that the films of the
entire series behave as a quasi-bulk film interior with two Fe/MgO interfaces
that are not magnetically equivalent. We deduce then different top and bottom
surfaces anisotropies that are in good agreement with theoretical calculations
for ultra-thin films[13, 14, 15, 16, 17]. This work not only presents a new
characterization methodology, but also provides evidence that the large PSA of
the technologically relevant ultra-thin films also exists in the thicker films
traditionally used in material science.

##  II Broadband ferromagnetic resonance

###  II.1 Film growth

The studied films were grown by molecular beam epitaxy on commercial MgO(001)
substrates and consist of the following stacks:
substrate/MgO(20)/Fe($t$)/MgO(8)/Ti(4.5) (thicknesses in nm). The MgO buffer
film was deposited on top of a polished MgO surface at 550 °C. The Fe film,
with thickness $t$ = 10, 15, 20, 25, 30 nm, was subsequently grown at 100 °C
(stair step structure obtained with a movable shutter). Finally, the sample
was annealed at 480 °C and capped with the MgO and Ti layers, both grown at
room temperature. The epitaxial relationship between Fe and MgO is such that
the [010] and [100] in-plane directions (magnetic easy axes of the Fe film)
are rotated by 45° with respect to those of the MgO films (aligned with the
edges of the substrate).

The crystalline quality of the samples was confirmed in-situ by RHEED. After
growth, an X-ray diffraction study revealed a slight tetragonal distortion of
the Fe lattice with respect to the bulk, more precisely a 0.5$\%$ out-plane
compression accompanied by a 0.7$\%$ in-plane expansion [22].

###  II.2 Vector Network Analyzer-Ferromagnetic Resonance

The dynamic magnetic properties of the films are characterized by Vector
Network Analyzer - Ferromagnetic Resonance. The sample
($1.8\\!\times\\!1.8\text{mm}^{2}$ piece cut from a film) lies on a 50
$\Omega$ channelized coplanar waveguide (CPW) [23] with a 300 $\mu$m center
line separated from the lateral ground planes by 100 $\mu$m gaps (see inset in
Fig. 1). The 50 $\mu$m thick copper/gold top metallization rests on a 127 µm
thick PTFE/glass Rogers RT5880 substrate backed with a very thick copper
layer. The CPW’s top and bottom grounds are connected through rows of vias
parallel to the center line in order to ensure a single-mode propagation in
the entire 0-50 GHz frequency range. The part of the waveguide on which the
sample is placed has a tapered center line (width 200 $\mu$m, thickness 30
$\mu$m) that compensates for the impedance change caused by the presence of
the conductive film on top of the CPW [24].

Figure 1:  Ferromagnetic resonance spectrum measured at 40 GHz for the t=20 nm
film. The main panel shows the real and imaginary components of the change of
effective permeability as function of the in-plane applied magnetic field
(along an the [100] direction of Fe). One distinguishes both a main peak
(uniform resonance, n=0) and a satellite one (perpendicular standing spin
wave, n=1). The inset contains a photograph of the CPW loaded with a sample at
its center (top) and a cross-section sketch showing the tapered CPW, the
magnetic film, and the thickness profiles of the dynamic magnetization for the
two resonance modes (bottom, not to scale).

To perform the magnetic measurements, the CPW and sample are inserted in the
gap of an electromagnet and connected to a 2-port vector network analyzer via
2.4 mm connectors and coaxial microwave cables. The analyzer can excite and
measure the microwave response of the CPW: microwave reflection on each port
and transmission between the two ports. The excitation of the ferromagnetic
sample results in a modification of the waveguide’s impedance. After a
suitable calibration and deembedding procedure [25], we extract the magnetic
field-induced change of the effective permeability $\Delta\mu_{\text{eff}}$ of
the waveguide loaded with the ferromagnetic sample. This magnetic response
exhibits clear resonances when the microwave frequency matches the field-
dependent magnetization precession frequency.

Figure 1 shows the ferromagnetic resonance spectrum measured on a 20 nm thick
Fe film at a microwave frequency of 40 GHz with an external field $\mathbf{H}$
applied in-plane, along the [100] magnetic easy axis (see Fig. 1 inset). One
recognizes an intense peak centered at 634 mT, which we attribute to the
uniform resonance mode ($n$=0 in the inset of Fig.1). A satellite peak
centered at 112 mT is also observed: we attribute this to the first
perpendicular standing spin wave mode, corresponding to an inhomogeneous
precession across the film thickness, with opposite phases at the two surfaces
and zero amplitude at the center ($n$=1 in the inset of Fig. 1). The
observation of this satellite peak might be surprising at first glance since
the microwave magnetic field produced by the coplanar waveguide is expected to
be homogeneous over the thickness of the magnetic film, thus preventing the
excitation of inhomogeneous modes. However, the fact that the ferromagnetic
film is conductive leads to the occurrence of electromagnetic shielding. This
effect is characterized by the creation of electrical currents in the metallic
film which tend to confine the electromagnetic field within the space between
the waveguide and the sample [24]. This results in a very inhomogeneous
microwave magnetic field distribution across the ferromagnetic film thickness
leading to the excitation of non-uniform magnetization precession modes [26].

Similarly, ferromagnetic resonance spectra are recorded for the various
thicknesses of Fe with frequency in the range 1.4-50 GHz and external field
(up to 2.7 T) applied either in-plane, along the [100] direction of Fe, or
out-of-plane (along [001]). Each resonance spectrum is fitted with a complex
lorentzian function. From these fits we extract the resonance fields of the
two modes, in the two field configurations (Fig. 2). From the fit of the
resonance peak of mode $n$=0 in the out-of-plane configuration, we also
extract the linewidth and from its frequency dependence we finally estimate
the damping factor $\alpha$=2.6$\times 10^{-3}$. It must be noted that mode
$n$=1 could not be observed for $t$=10 nm and 15 nm with in-plane applied
field and for $t$=10 nm with out-of-plane field because the corresponding
resonance frequencies lie beyond the 50 GHz experimental limit.

Figure 2:  Measured frequency vs. resonance magnetic field of both the uniform
mode (n=0) and first perpendicular standing spin wave mode (n=1) for the
entire thickness series. In the case of in-plane applied field, squares
(triangles) correspond to mode n=0 (n=1). For the case of out-of-plane applied
field, circles (stars) correspond to mode n=0 (n=1). The lines are the
corresponding Kittel fits [see Eqs. (5)].

###  II.3 Theoretical model

To interpret the resonance data of Figure 2, we use the so-called Kittel
formulas [27]. These simple expressions are known to be exact in the case of a
uniform resonance mode in a high symmetry configuration (equilibrium
magnetization parallel or perpendicular to the anisotropy axes). This section
shows how they can be extended also to the case of inhomogeneous dynamics
($n\\!=\\!1$) in films with moderate thickness and sizable surface
anisotropies. Our starting point is the linearized Landau-Lifshitz equation
for plane spin-waves:

$i\omega\textbf{m}=\gamma\mu_{o}(\textbf{H}_{\text{eq}}\times\textbf{m}-\textbf{M}_{\text{eq}}\times\textbf{h}).$
(1)

Here, $\omega$ is the angular frequency, $\gamma$ is the gyromagnetic ratio
and $\mu_{0}$ is the permeability of vacuum. $\textbf{M}_{\text{eq}}$ and m
are the static and dynamic components of the magnetization, respectively.
Similarly, $\textbf{H}_{\text{eq}}$ and h are the static and dynamic parts of
the effective magnetic field, respectively. The effective field derives from
the total magnetic energy [28], which in the present case contains five
contributions: i) the exchange and ii) demagnetizing contributions present in
any ferromagnet, iii) the cubic volume anisotropy known to exist in iron, iv)
surface perpendicular anisotropies at the two Fe/MgO interfaces, and v) an
additional volume anisotropy with uniaxial symmetry and perpendicular-to-plane
axis which, we argue, is created by strain through a magnetoelastic coupling
(see discussion section).

When the external magnetic field $\mathbf{H}$ is applied along the easy axes
of the Fe films, the static effective field writes

$H_{\text{eq}}(\xi)=\begin{cases}H\quad&\text{if}\,${H}$\>\parallel[100]_{\text{Fe}}\\\
H+H_{\text{U}}+h_{\text{S}}(\xi)-M_{\text{s}}\quad&\text{if}\,${H}$\>\parallel[001]_{\text{Fe}},\\\
\end{cases}$ (2)

while the dynamic effective field writes

$\textbf{h}=\frac{1}{M_{\text{s}}}\bigg{\\{}\left[\frac{2A}{\mu_{0}M_{\text{s}}}\nabla^{2}+H_{\text{K}}\right]\textbf{m}+\left[H_{\text{U}}+h_{\text{S}}(\xi)-M_{\text{s}}\right]m_{\xi}\bm{\hat{\xi}}\bigg{\\}}.$
(3)

Here, $M_{\text{s}}$ is the saturation magnetization, $A$ is the exchange
stiffness constant, $\bm{\hat{\xi}}$ is a unit vector along the direction
perpendicular to the film and $m_{\xi}$ is the dynamic magnetization component
along this direction. Additionally,
$H_{\text{K}}\\!=\\!\frac{2K_{1}}{\mu_{0}M_{\text{s}}}$, where $K_{1}$ is the
volume cubic anisotropy constant, and
$H_{\text{U}}\\!=\\!\frac{2K_{\text{U}}}{\mu_{0}M_{\text{s}}}$, where
$K_{\text{U}}$ is the volume uniaxial magnetoelastic anisotropy constant.
Finally, the field
$h_{\text{s}}(\xi)=\frac{2}{\mu_{0}M_{\text{s}}}[K^{\text{bot}}_{\text{S}}\delta(\xi)+K^{\text{top}}_{\text{S}}\delta(\xi-t)]$
models the perpendicular surface anisotropies with constants
$K_{\text{S}}^{\text{top}}$ and $K_{\text{S}}^{\text{bot}}$ at the top and
bottom interfaces, respectively [29]. Note that $m_{\xi}$=0 when a saturating
magnetic field is applied perpendicular to the plane of the film
($\textbf{M}_{\text{eq}}\parallel\bm{\hat{\xi}}$), making Eq. (3) valid for
the two experimental configurations considered in Eq. (2).

The system of equations (1-3) is effectively a fourth order differential
equation for the dynamic magnetization with mixed-type boundary conditions
(so-called surface pinning), which does not have an exact analytical solution
[30]. However, it is possible to obtain approximate solutions in some limiting
cases. In particular, if the exchange energy $A/t$ is much larger than the
surface anisotropy $K_{\text{S}}$, we can expand the dynamic magnetization in
a Fourier series of cosine thickness modes (unpinned standing spin wave modes)
and, in the spirit of the Kalinikos-Slavin theory of dipole-exchange spin-
waves [31, 32], limit ourselves to the first two terms of the series. Then we
may write the complex amplitude of the dynamic magnetization
$\textbf{m}^{*}(\xi)$ as

$\begin{split}\textbf{m}^{*}(\xi)=&\quad\>[m^{0}_{x}\Phi_{0}+m^{1}_{x}\Phi_{1}(\xi)]\>\bm{\hat{x}}\\\
&+[m^{0}_{y}\Phi_{0}+m^{1}_{y}\Phi_{1}(\xi)]\>\bm{\hat{y}},\end{split}$ (4)

with $x$ and $y$ denoting two directions orthogonal to the static
magnetization $\textbf{M}_{\text{eq}}$. $\Phi_{0}\\!=\\!1/\sqrt{t}$ is the
lowest order term in the Fourier series corresponding to a normalized uniform
distribution, and
$\Phi_{1}(\xi)\text{=}\sqrt{2}\cos(\frac{\pi}{t}\xi)/\sqrt{t}$ is the second
term corresponding to a normalized non-uniform distribution with a thickness-
profile which is antisymmetric with respect to the center of the film (see
their sketch in the inset of Fig. 1). In the basis of the four orthogonal
vector modes
$\left[\Phi_{0}\mathbf{\hat{x}},\;\Phi_{1}(\xi)\mathbf{\hat{x}},\;\Phi_{0}\mathbf{\hat{y}},\;\Phi_{1}(\xi)\mathbf{\hat{y}}\right]$,
the complex amplitude of the dynamic magnetization can be conveniently
expressed as $\textbf{m}^{*}$=$(m^{0}_{x},m^{1}_{x},m^{0}_{y},m^{1}_{y})$.

After substituting Eqs. (3) and (4) in Eq. (1), one can project the system of
equations (1-3) on this new four-mode basis (see Ref. 33 for a detailed
treatment of the projection) and obtain a simplified eigenvalue equation of
the form $i\omega\mathbf{m}^{*}=C\mathbf{m}^{*}$, where $C$ is the 4$\times$4
dynamic matrix [29]. The eigenvalues of this matrix are the resonance
frequencies. By replacing the eigenvectors of matrix $C$ back in Eq. (4) one
may recover the actual oscillation modes of the magnetization.

The dynamic matrix in the presence of surface anisotropies is given in
Appendix A for the case of a spin wave with wavevector k parallel to the
$[010]_{\text{Fe}}$ axis. It is important to note that this matrix depends
explicitly not only on the total surface anisotropy
$K_{\text{S}}\\!=\\!K_{\text{S}}^{\text{bot}}\\!+\\!K_{\text{S}}^{\text{top}}$
but also on the difference of surface anisotropies at the two interfaces
$\Delta
K_{\text{S}}\\!=\\!K_{\text{S}}^{\text{bot}}\\!-\\!K_{\text{S}}^{\text{top}}$
[29].

Let us now concentrate on the case of ferromagnetic resonance ($k\\!=\\!0$).
Up to first order in $\Delta K_{S}$, our approach produces Kittel-like[27]
expressions for the resonance frequencies of the first two standing spin wave
modes:

$\displaystyle f_{\parallel
n}=\frac{\gamma\mu_{0}}{2\pi}[(H+H_{Xn})(H+H_{Yn})]^{1/2},$ (5a) $f_{\perp
n}=\frac{\gamma\mu_{0}}{2\pi}(H-H_{Zn}).$ (5b)

Here $n\\!=\\!0,1$ is the mode index, and $\parallel$ and $\perp$ refer to the
configurations with $\mathbf{H}\parallel[100]_{\text{Fe}}$ and
$\mathbf{H}\parallel[001]_{\text{Fe}}$, respectively. $H_{\alpha n}$ are
orientation and mode-dependent stiffness fields whose expressions are given in
Table 1,

Table 1: Explicit expressions of stiffness fields in the resonance frequencies of Eqs.(5).  Field | $n=0$ | $n=1$
---|---|---
$H_{Xn}$ | $H_{\text{K}}$ | $H_{\text{K}}+H_{\text{E}}$
$H_{Yn}$ | $M_{\text{s}}+H_{\text{K}}-H_{\text{U}}-H_{\text{S}}$ | $M_{\text{s}}+H_{\text{K}}-H_{\text{U}}-2H_{\text{S}}+H_{\text{E}}$
$H_{Zn}$ | $M_{\text{s}}-H_{\text{K}}-H_{\text{U}}-H_{\text{S}}$ | $M_{\text{s}}-H_{\text{K}}-H_{\text{U}}-2H_{\text{S}}-H_{\text{E}}$

where the following thickness-dependent exchange and surface anisotropy fields
have been used:

$\displaystyle H_{\text{E}}=\frac{2\pi^{2}A}{\mu_{0}M_{\text{s}}t^{2}},$ (6a)
$H_{\text{S}}=\frac{2K_{\text{S}}}{\mu_{0}M_{\text{s}}t}.$ (6b)

At the chosen level of approximation, which is valid only for small
differences in surface anisotropies ($\Delta K_{\text{S}}\>t/A\ll 1$), the
resonance frequencies (Eqs. 5) depend on $K_{S}$ but not on $\Delta K_{S}$,
although the $C$ matrix depends explicitly on it. This may be explained as
follows. With $k$=0, the mutual demagnetization factor $Q$ vanishes (see
Appendix A) and so does the largest source of hybridization between the
uniform and antisymmetric basis modes. Only terms in $\Delta K_{\text{S}}$
remain non zero in the off diagonal blocks of the $C$ matrix (Eq. 9), meaning
that the difference in surface anisotropies becomes the sole source of
hybridization. It happens that the corresponding coupling is proportional to
$\Delta K_{\text{S}}^{2}$ and is thus neglected in the above approximation. We
note also that the contribution of surface anisotropies to the stiffness
fields is doubled in the case of mode $n$=1 as compared to mode $n$=0. This is
a direct consequence of mode $n$=0 being uniform at this level of
approximation while mode $n$=1 is fully asymmetric with large amplitude at the
surfaces, which makes it more sensitive to PSA.

###  II.4 Results

Fitting the experimental data in Fig. 2 to the corresponding Eqs. (5) yields
the values of the stiffness fields $H_{Xn}$, $H_{Yn}$, and $H_{Zn}$
$(n\\!=\\!0,1)$ presented in Fig. 3 as open circles. For this extraction, we
assume a unique value for $\gamma$ (see Table 2),

Table 2: Magnetic parameters obtained for MgO/Fe($t$)/MgO films ($t\\!=\\!10\\!-\\!30\text{nm}$). $\mu_{0}M_{\text{s}}$ [T] |  $\gamma/2\pi$ [GHz/T] |  $A$ [pJ/m] |  $K_{1}$ [kJ/$\text{m}^{3}$]
---|---|---|---
2.15111Tabulated value for bulk iron at room temperature | $29.1\pm 0.6$ | $19.4\pm 0.1$ | $52\pm 1$
|  |  |
$K_{\text{U}}$ [kJ/$\text{m}^{3}$] |  $K_{\text{S}}$ [mJ/$\text{m}^{2}$] |  ${\Delta K_{\text{S}}}$ [mJ/$\text{m}^{2}$]
$-45\pm 28$ | $2.3\pm 0.3$ |  $0.8\pm 0.1$222Value obtained from non-reciprocal spin wave measurements (See Section III)

which is the average over all film thicknesses of the individual $\gamma$
values obtained by fitting $f_{\perp 0}(H)$ experimental data to Eq. (5b).

Figure 3:  a) Stiffness fields as a function of Fe thickness obtained from
Kittel-like fits of the experimental resonance frequencies (open circles)
compared with the analytical model of Table 1 (solid lines) and the values
obtained from Kittel-like fits of SWIIM simulations (dotted lines). b)
Stiffness field $H_{X1}$ as a function of $1/t^{2}$. c) Stiffness fields
$H_{Y0}$ and $H_{Z0}$ as a function of $1/t$.

Values of the stiffness fields associated with the two oscillation modes
$(n\\!=\\!0,1)$ can be readily treated and combined sequentially so as to
extract most of the magnetic parameters of the iron films. As a starting point
and in agreement with SQUID characterization, the saturation magnetization
value is set to that of bulk iron $\mu_{0}M_{\text{s}}\\!=\\!2.15$ T. Next, we
observe that $H_{X0}$ is thickness independent and we extract the cubic
anisotropy constant $K_{1}$ from $H_{X0}\\!=\\!H_{\text{K}}$. Then, since
$H_{X1}\\!-\\!H_{X0}\\!=\\!H_{\text{E}}$ varies as $t^{-2}$ [Fig. 3(b)], we
confidently extract a thickness independent exchange constant $A$ (Eq. 6a). We
subsequently notice that the thickness dependent parts of $H_{Y0}$ and
$H_{Z0}$ both vary as $t^{-1}$, with similar slopes [Fig. 3(c)], from which we
determine the average total surface anisotropy constant $K_{\text{S}}$ (Eq.
6b). Finally, we determine the unixial anisotropy constant $K_{\text{U}}$
using
$H_{Y0}\\!+\\!H_{Z0}\\!-\\!2(M_{\text{s}}\\!-\\!H_{\text{S}})\\!=\\!2H_{\text{U}}$.

The obtained magnetic parameters are summarized in Table 2. In the chosen
parametrization, the negative sign of $K_{\text{U}}$ indicates an easy-plane
parallel to the film’s plane and the positive sign of $K_{\text{S}}$ an easy-
axis along the film normal.

The lines in Fig. 3 are the theoretical stiffness fields calculated by
injecting the parameters just determined (Table 2) back into our analytical
model. The good agreement obtained illustrates that a single set of thickness
independent magnetic parameters is indeed enough to capture most features of
the magnetization dynamics in the studied MgO/Fe/MgO films. Furthermore, the
fact that the agreement also applies to those stiffness fields which have not
been considered in the above analysis ($H_{Y1}$ and $H_{Z1}$) can be
considered as a validation of the model. There are however points of slight
disagreement between the experimentally determined values and the
corresponding analytical predictions. In particular, predicted values of the
difference $H_{Y0}\\!-\\!H_{Z0}$ are significantly larger than the ones
determined experimentally. As may be inferred from Appendix B, modifying the
analytical model (Eqs. 5) to include terms up to second order in $\Delta
K_{\text{S}}$ allows one partially reducing the disagreement. This second
order approximation however produces cumbersome expressions which are
unpractical and, even more importantly, unable to provide information
regarding the sign of $\Delta K_{\text{S}}$. This points at the need for an
accurate determination of this additional parameter through an experimental
technique which is sensitive to it at first order, namely spin wave frequency
non-reciprocity.

##  III Non-reciprocal spin wave propagation

Propagating spin wave spectroscopy has been shown to be very sensitive to
magnetic asymmetries across the thickness of thin films [29], including
differences in surface anistropies at the two interfaces ($\Delta
K_{\text{S}}\\!\neq\\!0$). The principle of such measurement is sketched in
the inset of Fig. 4. A spin-wave propagating in the so-called Damon-Eshbach
geometry (i.e. with its wave-vector $\mathbf{k}$ oriented perpendicular to the
in-plane applied magnetic field $\mathbf{H}$) is known to exhibit a mode
profile non-reciprocity. This means that the wave has an asymmetric
distribution across the film thickness, with more amplitude on one side of the
film than on the other. This asymmetric profile is reversed when changing the
sign of $\mathbf{k}$, i.e. for spin waves propagating in the opposite
direction [34] (see inset of Fig. 4). Consequently, an inhomogeneous magnetic
environment across the thickness will have different effects on the dynamics
of counter-propagating spin waves. This can be measured experimentally as a
difference between their resonance frequencies which becomes a spectroscopic
signature of the film’s asymmetric magnetic environment [29].

Figure 4:  Measured imaginary component of the mutual inductance between
antennas due to counter-propagating spin waves (with wave numbers: $k_{1}$=3.9
$\text{rad/}\mu\text{m}$ and $k_{2}$=1.57 $\text{rad/}\mu\text{m}$) in a strip
of $t$=20 nm Fe film under a 120 mT magnetic field (Damon-Eshbach geometry).
The inset shows an electronic microscope image of the experimental device
(top) and a sketch of the asymmetric modal profiles of counter-propagating
waves across the thickness of the film and its interplay with asymmetric
magnetic surface anisotropies (bottom).

To measure this frequency non-reciprocity, suitable devices are fabricated
from the Fe samples. This is achieved by patterning the film into a strip
geometry and fabricating a pair of conductors on top. These conductors with a
meander geometry serve as antennas for exciting and detecting spin-waves of
controlled wavelength. With the design chosen in the present work (see inset
in Fig. 4) the most important excitation occurs around two particular wave
vectors $k_{1}$=3.9 $\text{rad/}\mu\text{m}$ and $k_{2}$=1.57
$\text{rad/}\mu\text{m}$. By using two separate antennas, and interchanging
their role, it is possible to measure the changes in mutual inductance
corresponding to spin waves propagating with positive and negative wave
vectors $\Delta L_{k>0}$, $\Delta L_{k<0}$ [35]. More details on the
fabrication process of the devices and experimental procedure can be found
elsewhere [36].

Fig. 4 shows the measured change in mutual inductance corresponding to spin
wave propagation in a $t$=20 nm device under a 120 mT field applied along
[100]. One can observe directly a frequency difference between
counterpropagating waves, both for the main spin-wave excitation
[$f_{\text{NR}}(k_{1})$] and for the secondary one [$f_{\text{NR}}(k_{2})$].
The value of this frequency non-reciprocity is followed as a function of the
applied magnetic field in the range 30-200 mT. The different symbols in Fig. 5
show the frequency non-reciprocity measured for three different samples:
namely devices from the FMR series with $t\\!=\\!10$ nm and 20 nm, and a third
device with $t\\!=\\!20$ nm but without Ti capping, labelled thereafter 20
nm*. We observe that $f_{\text{NR}}$ is roughly field independent.

Figure 5:  Frequency non-reciprocity measurements as a function of applied
magnetic field for devices $t$=10 nm, 20 nm and 20 nm* and each wave numbers
$k_{1}$=3.9 $\text{rad/}\mu\text{m}$, $k_{2}$=1.57 $\text{rad/}\mu\text{m}$.
The lines correspond to the average value of $f_{\text{NR}}$ over the
experimental field range.

To analyze quantitatively the data in Fig. 5 and estimate the difference in
magnetic surface anisotropy $\Delta K_{\text{S}}$, we use the theory developed
in the previous section with some modifications. To account for the
propagating character of the spin waves, the space-dependent part of the
dynamic component of the magnetization [Eq. (4)] becomes
$\textbf{m}^{*}(\xi)e^{-ik\eta}$ where $\hat{\eta}$ is a direction vector
along the ferromagnetic strip. Now, the dynamic magnetic field writes

$\begin{split}\textbf{h}=&\frac{1}{M_{\text{s}}}\bigg{\\{}\left[\frac{2A}{\mu_{0}M_{\text{s}}}\nabla^{2}+H_{\text{K}}\right]\textbf{m}+\left[H_{\text{U}}+h_{\text{s}}(\xi)\right]m_{\xi}\bm{\hat{\xi}}\bigg{\\}}\\\
&+\int_{0}^{t}G_{k}(\xi-\xi^{\prime})\textbf{m}\,d\xi^{\prime},\end{split}$
(7)

where the last term is the dipolar contribution and $G_{k}$ the magnetostatic
Green’s tensor [37]. As in the previous section, the system of Eqs. (1,7) can
be projected onto the spin wave modes basis which allows one to obtain an
eigenvalue equation with a dynamical matrix $C$ given explicitly in Appendix A
(note that in this case $k\\!\neq\\!0$). In the Damon-Eshbach configuration,
the non-uniform character of magnetization along $\hat{\eta}$ gives rise to
dipolar coupling between Fourier components $n\\!=\\!0$ and $n\\!=\\!1$
[through the factor $Q\\!\neq\\!0$ in the non-diagonal blocks of the dynamic
matrix $C$, see Eq. (9a)], which is non-reciprocal with respect to the wave
number. This, combined with the asymmetry produced by $\Delta K_{\text{S}}$,
is at the very origin of the frequency non-reciprocity.

Once again we can calculate the resonance frequencies from the dynamic matrix
C, and in this case, obtain the frequency difference between
counterpropagating waves. Assuming $\Delta K_{\text{S}}t/A\\!\ll\\!1$, an
approximate expression can be derived in which the frequency non-reciprocity
of mode $n\\!=\\!0$ is a linear function of both the wave number $k$ and the
difference in surface anisotropy $\Delta K_{\text{S}}$ [29]:

$\begin{split}f_{\text{NR}}&=f_{\parallel 0}(k<0)-f_{\parallel 0}(k>0)\\\
&\approx\frac{8\gamma\>\Delta
K_{\text{S}}}{\pi^{3}M_{\text{s}}(1+\frac{\Lambda^{2}\pi^{2}}{t^{2}})}\>k\end{split}$
(8)

Since $f_{\text{NR}}$ shows no systematic variation with the external magnetic
field (Fig. 5), we consider below its average value over the 30-200 mT range
and plot it in Fig. 6 as a function of $k$ for each of the studied films.
Applying Eq. (8) and using values of $\gamma$, $M_{\text{s}}$, and $A$ found
in Table 2, we finally extract estimates for $\Delta K_{\text{S}}$ from the
slopes of linear fits: 0.8, 1.1, 1.6 $\text{mJ/m}^{2}$ for the 10 nm, 20 nm
and 20nm* films, respectively.

Figure 6:  Frequency non-reciprocity as a function of wave number for the
three films under study. The points represent the average experimental values
in the applied field range 30-200 mT (see Figure 5). The solid lines are the
corresponding linear fits to Eq. (8). The dashed lines show the mean value as
calculated with SWIIM while using values for $\Delta K_{S}$ that best adjust
the experimental points (see text for details).

These results confirm the asymmetry of the two film interfaces suggested by
FMR measurements. They show that $\Delta K_{\text{S}}$ is undoubtedly
positive, which means that, in all films, the bottom interface has a stronger
PSA than the top one. Since the above characterization is based on several
approximations [Eqs. (4, 8)], the magnitude of $\Delta K_{\text{S}}$ should
however be refined before comparing and contrasting the films and their
respective interfaces. To this aim, we finally turn to numerical simulations.

We resort to the code SWIIM [38] which provides a finite-difference numerical
solution of Eqs. (1,7) to calculate the difference between the frequencies of
counterpropagating waves as a function of wave number $k$ in the Damon-Eshbach
configuration. For each sample, we adjust $\Delta K_{\text{S}}$ (the remaining
magnetic parameters are taken from Table 2), so as to best reproduce the
experimental $k_{\text{NR}}(k)$ data in Fig. 6 (dashed lines). The values of
$\Delta K_{\text{S}}$ obtained in this way are 0.7, 0.8, and 1.2
$\text{mJ/m}^{2}$ for the 10 nm, 20 nm and 20 nm* films, respectively.
Comparing them with values obtained from Eq. (8), we observe that the
analytical approach systematically underestimates the effect of $\Delta
K_{\text{S}}\\!\neq\\!0$. Noticeably, using numerical simulations allows us
reducing the difference between the values of $\Delta K_{\text{S}}$ for the 10
nm and 20 nm samples to almost nothing, which is of course expected for films
of similar composition. We then choose the value $\Delta
K_{\text{S}}\\!=\\!0.8$ mJ/m2 as a reference for our MgO/Fe/MgO system.

Having refined the magnitude of $\Delta K_{\text{S}}$, we finally go back to
the ferromagnetic resonance case and we use SWIIM to calculate the resonance
frequencies of the two lowest FMR modes as a function of magnetic field using
the now completed set of magnetic parameters (Table 2). Then we fit the
frequencies determined numerically to Eqs. (5) and extract the corresponding
stiffness fields. As expected, accounting for the difference in surface
anisotropy evidenced through propagating spin wave spectroscopy allows one
improving slightly the agreement between experimentally and numerically
determined $H_{Y0}$, $H_{Z0}$, $H_{Y1}$, and $H_{Z1}$ stiffness fields (Fig.
3). Note that despite the introduction of $\Delta K_{\text{S}}$ and the exact
treatment of hybridization effects by SWIIM, numerical data remain however
rather close to predictions of our analytical model. This proves the
suitability of our choice of a limited four-vector mode basis (Sec. II.3).

##  IV Discussion

Starting from a simplified analytical model, we have described above a method
for extracting the magnetic parameters of ferromagnetic films with moderate
thickness from broadband ferromagnetic resonance and propagating spin wave
spectroscopy measurements. The positions of the ferromagnetic resonance peaks
measured over large field and frequency ranges are first fitted to Kittel
formulas (Fig. 2). Then, the extracted stiffness fields are confronted to
explicit expressions (Fig. 1) allowing one to extract successively five
magnetic parameters. The deviations between the model and the experiments do
not exceed 3$\%$ (Fig. 3), which we find very satisfactory given the wide
range of field, frequency and film thickness investigated, and the limited
number of parameters involved. Moreover, the extension of the model to
propagating spin waves accounts for the measured frequency non-reciprocity, a
quantity from which we extract a sixth magnetic parameter. The value of the
latter is eventually refined by confronting frequency non reciprocity data to
full micromagnetic calculations.

The above ferromagnetic resonance study suggests that the entire thickness
series can be described quite accurately with a thickness independent set of
magnetic parameters. The exchange stiffness constant we determined lies within
the range of values reported in literature for Fe at room temperature
$A\\!=\\!19\text{-}23~{}\text{pJ/m}$ [39, 40, 41] and agrees particularly well
with the recent determinations by Niitsu [40] and Kuz’min et al. [41]. The
measured cubic anisotropy constant, on the other hand, is slightly larger than
the values measured on bulk iron and thin iron films
($K_{1}\\!=\\!44\text{-}49~{}\text{kJ/m}^{3}$) [42, 43, 44, 45], but it agrees
well with results from first principles calculations
($K_{1}\\!=\\!52~{}\text{kJ/m}^{3}$) [45].

The third volume parameter, namely the uniaxial anisotropy $K_{\text{U}}$,
accounts for the difference between the saturation magnetization
$M_{\text{s}}$ determined from SQUID magnetometry and the so-called effective
magnetization $M_{\text{eff}}\\!=\\!(M_{\text{s}}\\!-\\!H_{\text{U}})$
measured from FMR. Now, we argue that this anisotropy originates from a
distortion of the iron lattice. Indeed, the -4$\%$ mismatch between Fe and the
MgO substrate is known to relax only partly through a dense array of
dislocations formed in the first Fe atomic layers, thus leaving a small
residual strain in nanometer thick films [46]. Accordingly, a tetragonal
distortion is measured in the samples under study consisting of a 0.7$\%$ mean
in-plane expansion and a 0.5$\%$ out-of-plane compression [22]. Such vertical
lattice compression is expected to enhance the spin-orbit mediated
interactions between electronic states which favor an in-plane orientation of
the magnetization [47]. To relate phenomenologically this extra magnetic
anisotropy to the measured lattice distortion one may use the magnetoelastic
coupling constants of bulk iron [48]. The obtained uniaxial anisotropy
constant $K_{\text{U}}\\!=\\!-41$ kJ/m3 (see Sander [49] for calculation
details) is in very good agreement with our experimental observations
regarding both its sign and its magnitude.

In terms of total perpendicular surface anisotropy, our results (see Table 2)
agree well with what is expected for an Fe ultra thin film sandwiched between
two MgO layers [13, 14, 15, 16, 17]. From the joint results of broadband FMR
and spin wave propagation, we can extract the values for the two individual
PSA constants: $K^{\text{bot}}_{\text{S}}\\!=\\!(K_{\text{S}}+\Delta
K_{\text{S}})/2\\!=\\!1.55$ mJ/m2 for the bottom interface (MgO buffer/Fe) and
$K^{\text{top}}_{\text{S}}\\!=\\!(K_{\text{S}}-\Delta
K_{\text{S}})/2\\!=\\!0.75$ mJ/m2 for the top one (Fe/MgO capping). These two
values are within the range for the PSA obtained by ab-initio calculations
[13, 14, 15, 16, 17] and measurements on ultrathin films with a single MgO/Fe
interface [10, 11, 12, 50, 51]. However, in these previous experimental works,
the extracted values always included the contributions from two interfaces,
and some hypothesis based on reference interfaces (e.g. V/Fe) needed to be
included to extract individual values. In the present study we provide
individual $K_{\text{S}}$ values for both interfaces which allows us to
compare them directly with results from ab-initio simulations and evidence
that the ultra-thin interface physics can be extrapolated to thicker films
[13, 14, 15, 16, 17, 10, 11, 12, 50, 51].

It has been shown theoretically that over/under oxidation at the Fe/MgO
interface reduce drastically its surface anisotropy [15, 16]. Moreover, using
Mossbaüer spectroscopy, it has been shown that the Fe/MgO and MgO/Fe
interfaces of a film can exhibit different amount of interfacial Fe oxidation
[52]. Therefore, we attribute the difference in PSA at the two interfaces to a
difference in their oxidation states. We assume that the distinct temperature
treatments to which the bottom and top interfaces are subjected during growth
is the reason for that: the 480°C annealing, performed just after iron
deposition, promotes a better epitaxy and higher value of surface anisotropy
for the bottom MgO/Fe interface [12]. On the other hand, the top interface is
not annealed, which likely leads to an over oxidation of the interfacial Fe
atoms and a lower value of the PSA [15, 16]. This behaviour is corroborated by
the larger value of $\Delta K_{\text{S}}$ observed in the 20 nm* sample
without Ti protection (see blue dots in Fig. 6). For this sample, indeed, we
argue that a further oxidation of the top interface may take place after the
unprotected 8 nm thick MgO capping layer is exposed to water [53, 54] both
during the fabrication of this specific device and later under ambient
conditions.

##  V Conclusion

The magnetization dynamics of a thickness series of MgO/Fe($t$)/MgO epitaxial
films ($t\\!\text{=}\\!10\text{-}30$ nm) was characterized using a combination
of ferromagnetic resonance and non-reciprocal spin wave propagation
measurements. Our rather versatile Kittel model accounts consistently for the
frequencies of the uniform mode of magnetization precession and also for the
inhomogeneous first standing spin-wave mode. Noticeably, the ability to probe
both of these modes over a wide range of film thicknesses allowed us to
determine the exchange stiffness constant and the perpendicular surface
anisotropy, two quantities which are inaccessible through the sole study of
homogeneous dynamics.

With our detailed ferromagnetic resonance characterization, we evidenced that
the entire thickness series can be described with a single set of magnetic
parameters. The magnetic volume parameters, cubic anisotropy and exchange
stiffness, agree very well with what is expected for bulk iron. Also, an
additional uniaxial perpendicular anisotropy was identified and attributed to
a slight tetragonal distortion of the Fe lattice. Finally, it was possible to
separate contributions of individual film interfaces to the perpendicular
surface anisotropy with the help of complementary propagating spin wave
spectroscopy measurements. The sizeable asymmetry between the top and bottom
interfaces was attributed to the different oxidation states of each interface.
Our characterization suggest that 10-30 nm thick single crystalline Fe films
have a well defined quasi-bulk magnetic interior, while the interfaces with
MgO retain the large perpendicular surface anisotropy found in ultra-thin
film.

Our work provides new light into the technologically-relevant ferromagnet/MgO
interfaces and their effect on spin waves, while it also validates a new
method for characterizing magnetic interfaces. Note that our methodology could
be extended to alloys or multilayer systems, for which it could provide key
information about possible inhomogeneities / asymmetries of the magnetic
properties across the film thickness.

###### Acknowledgements.

We thank Arnaud Boulard, Benoît Leconte, Daniel Spor, Jérémy Thoraval and
Fares Abiza for assembling and testing the broadband FMR setup; Jérôme Robert
for SQUID magnetometry measurements; Romain Bernard, Sabine Siegwald and
Hicham Majjad for technical support during nanofabrication work in the STnano
platform; and Matías Grassi for useful discussion. We acknowledge financial
support by the Interdisciplinary Thematic Institute QMat, as part of the ITI
2021-2028 program of the University of Strasbourg, CNRS and Inserm, IdEx
Unistra (ANR 10 IDEX 0002), SFRI STRAT’US project (ANR 20 SFRI 0012) and
ANR-17-EURE-0024 under the framework of the French Investments for the Future
Program. We also acknowledge financial support from Region Grand Est through
its FRCR call (NanoTeraHertz and RaNGE projects) and from Agence Nationale de
la Recherche (France) under Contract No. ANR-20-CE24-0012 (MARIN).

##  Appendix A Dynamic matrices

To find the resonance frequencies of the magnetization modes one has to solve
the Landau-Lifshitz (LL) equation. For this, we substitute Eq. (2) for the
static field, Eq. (7) for the dynamic field, and Eq. (4) for the dynamic
magnetization into the linearized LL Eq. (1) and project the latter onto the
space of cosine thickness modes, as explained in Section II [31]. To be able
to derive useful analytical solutions, it is convenient to restrict this
projection to the first two modes, yielding a total of four basis vectors
$\left[\Phi_{0}\mathbf{\hat{x}},\;\Phi_{1}(\xi)\mathbf{\hat{x}},\;\Phi_{0}\mathbf{\hat{y}},\;\Phi_{1}(\xi)\mathbf{\hat{x}}\right]$
(two modes per spacial coordinate of the dynamic magnetization). After
projection, the linearized LL equation [Eq. (1)] takes the form of an
eigenvalue equation: $i\omega\bar{\textbf{m}}^{*}=C\bar{\textbf{m}}^{*}$,
where $C$ is the so called dynamic matrix, which in the present case, is
4$\times$4 [29]. The eigenvalues of this matrix are the resonance frequencies
and the eigenvectors describe the corresponding dynamic magnetization mode
amplitudes. In Eq. 9, we provide expressions for the dynamic matrices in cases
where the external magnetic field is applied in-plane ($C_{\parallel}$) and
out-of-plane ($C_{\perp}$).

${C_{\parallel}\text{=}\gamma\mu_{\text{o}}\begin{pmatrix}0&H_{\parallel}(k)\text{+}M_{\text{s}}P_{00}(k)&-iM_{\text{s}}Q(k)&0\\\
\text{-}[H_{\parallel}(k)\text{-}H_{\text{U}}\text{-}H_{\text{S}}\text{+}M_{\text{s}}[1\text{-}P_{00}(k)]]&0&\text{-}\frac{2\sqrt{2}\Delta
K_{\text{S}}}{\mu_{\text{o}}tM_{\text{s}}}&iM_{\text{s}}Q(k)\\\
iM_{\text{s}}Q(k)&0&0&H_{\parallel}(k)\text{+}H_{\text{E}}\text{+}M_{\text{s}}P_{11}(k)\\\
\text{-}\frac{2\sqrt{2}\Delta
K_{\text{S}}}{\mu_{\text{o}}tM_{\text{s}}}&\text{-}iM_{\text{s}}Q(k)&\text{-}[H_{\parallel}(k)\text{+}H_{\text{E}}\text{-}H_{\text{U}}\text{-}2H_{\text{S}}\text{+}M_{\text{s}}[1\text{-}P_{11}(k)]]&0\\\
\end{pmatrix}}$ (9a)
${C_{\perp}\text{=}\gamma\mu_{\text{o}}\begin{pmatrix}0&H_{\perp}(k)\text{+}M_{\text{s}}P_{00}(k)&0&\text{-}\frac{2\sqrt{2}\Delta
K_{\text{S}}}{\mu_{\text{o}}tM_{\text{s}}}\\\
\text{-}H_{\perp}(k)&0&\frac{2\sqrt{2}\Delta
K_{\text{S}}}{\mu_{\text{o}}tM_{\text{s}}}&0\\\
0&\text{-}\frac{2\sqrt{2}\Delta
K_{\text{S}}}{\mu_{\text{o}}tM_{\text{s}}}&0&H_{\perp}(k)\text{+}H_{\text{E}}\text{+}H_{\text{S}}\text{+}M_{\text{s}}P_{11}(k)\\\
\frac{2\sqrt{2}\Delta
K_{\text{S}}}{\mu_{\text{o}}tM_{\text{s}}}&0&-[H_{\perp}(k)\text{+}H_{\text{E}}\text{+}H_{\text{S}}]&0\\\
\end{pmatrix}.}$ (9b)

The fields
$H_{\parallel}(k)\text{\\!}=\\!H\\!+\\!H_{\text{K}}\\!+\\!M_{\text{s}}\Lambda^{2}k^{2}$
and
$H_{\perp}(k)\\!=\\!H+H_{\text{K}}\\!+\\!H_{\text{U}}\\!+\\!H_{\text{S}}+M_{\text{s}}(\Lambda^{2}k^{2}$-1)
are intermediate parameters introduced to simplify those expressions.
$P_{00}(k)\\!=\\!1\\!-\\!\frac{1-e^{-|k|t}}{|k|t}$ and
$P_{11}(k)\\!=\\!\frac{(kt)^{2}}{\pi^{2}+(kt)^{2}}\left(1\\!-\\!\frac{2(kt)^{2}}{\pi^{2}+(kt)^{2}}\frac{1+e^{-|k|t}}{|k|t}\right)$
are self demagnetizing factors for the cosine thickness modes $n\\!=\\!0$ and
$n\\!=\\!1$, and $Q\\!=\\!\frac{\sqrt{2}kt}{\pi^{2}+(kt)^{2}}(1+e^{-|k|t})$ is
a mutual demagnetizing factor responsible for hybridization between those
modes; all three factors being part of the magnetostatic Green’s function that
describes the dipolar interaction [37].

##  Appendix B Effect of a non-zero $\Delta K_{\text{S}}$ on the stiffness
fields in the case of ferromagnetic resonance

In Section II we have derived Eq. (5) for the resonance frequencies of the
modes $n\\!=\\!0,1$ by considering the effect of $\Delta K_{\text{S}}$ only up
to first order. In the infinite wavelength limit ($k\\!=\\!0$), those
frequencies become fully independent of $\Delta K_{\text{S}}$. Now, we proceed
to consider the approximation up to second order in this parameter and study
its effect on Eq. (5).

When keeping terms proportional to $\Delta K_{\text{S}}^{2}$ in the model, the
stiffness fields need to be modified as follows: $H_{\text{X}n}\to
H_{\text{X}n}$, $H_{\text{Y}n}\to H_{\text{Y}n}+g_{\text{Y}n}$,
$H_{\text{Z}0}\to H_{\text{Z}0}+g_{\text{Z}}$, and $H_{\text{Z}1}\to
H_{\text{Z}1}-g_{\text{Z}}$, where the field corrections $g_{\text{Y}n}$
($n\\!=\\!0,1$) and $g_{\text{Z}}$ are given by

${g_{Y0}(H,t)=\frac{-8\frac{H_{\text{E}}+H_{\text{K}}+H}{({\mu_{0}M_{\text{s}}t)}^{2}}\Delta
K_{\text{S}}^{2}}{\scriptstyle{H_{\text{S}}(H_{\text{K}}+H)+H_{\text{E}}[H_{\text{U}}-M_{\text{s}}-H_{\text{E}}+2(H_{\text{S}}-H_{\text{K}}-H)]}}}$
(10a)
${g_{Y1}(H,t)=\frac{8\frac{H_{\text{K}}+H}{({\mu_{0}M_{\text{s}}t)}^{2}}\Delta
K_{\text{S}}^{2}}{\scriptstyle{H_{\text{S}}(H_{\text{K}}+H)+H_{\text{E}}[H_{\text{U}}-M_{\text{s}}-H_{\text{E}}+2(H_{\text{S}}-H_{\text{K}}-H)]}}}$
(10b) ${g_{Z}(t)=\frac{8\Delta
K_{\text{S}}^{2}}{({\mu_{0}M_{\text{s}}t)}^{2}(H_{\text{E}}+H_{\text{S}})}\text{.}}$
(10c)

$g_{\text{Z}}$ is completely independent of the external field $H$ and
$g_{\text{Y}n}$ depends only very weakly on it in the range of interest (Fig.
7) so that it can also be considered as constant. Therefore, under this second
order approximation, the resonance frequencies $f_{\parallel n}$ and $f_{\perp
n}$ retain approximately the Kittel-like form of Eq. 5, with the external
field dependence contained entirely in the explicit $H$ term. This justifies
our fitting of the ferromagnetic resonance frequencies in Fig. 2 to Eq. 5 even
in the presence of a sizable $\Delta K_{\text{S}}$. We note in passing that,
given their smallness (Fig. 7), the correction fields will have a negligible
effect on the determination of the main parameters $K_{\text{S}}$ and
$K_{\text{U}}$.

Figure 7:  Second order correction functions for the stiffness fields (Eq. 10)
as calculated with $\Delta K_{\text{S}}=1.65$ mJ/m2 to obtain a sizable
effect. Functions $g_{Yn}$ and $g_{\text{Z}}$ are the stiffness field
corrections for in-plane applied field, while $g_{\text{Z}}$ applies to the
out-of-plane case. The shaded zones around the lines account for the
variations of $g_{\alpha n}$ with applied magnetic field in the range 0-1.3 T.

As mentioned in the body of the paper (Sec. II.4), moving to second order
approximation allows one to improve qualitatively the agreement between
theoretical and experimental values of the stiffness fields (Fig. 8). However,
when assuming the value $\Delta K_{\text{S}}\\!=\\!0.8$ mJ/m2 determined from
spin-wave spectroscopy (Sec. III) the improvement remains marginal, especially
regarding $H_{Y0}$, and only a much bigger value of $\Delta K_{\text{S}}$
allows one reaching a reasonably good matching (note the different vertical
scales in Figs. 3 and 8). Here again, we evidence the tendency of our
analytical approach to underestimate the effect of a difference in surface
anisotropies. We attribute this quantitative discrepancy partially to the
hypothesis made in writing Eq. 4, which is to neglect higher order terms in
the Fourier series. We overcome this limitation with our numerical analysis
(SWIMM code), which allows us to obtain a better overall agreement with
broadband FMR and PSWS (see dotted and dashed lines in Figs. 3 and 8).

Figure 8: Stiffness fields $H_{\text{Y}0}$ (red) and $H_{\text{Z}0}$ (blue) as
functions of the Fe film thickness $t$. The open circles are the
experimentally determined values and the lines are predictions of our
analytical approach. Continuous, dashed, and dotted lines correspond,
respectively, to first order approximation, second order approximation with
$\Delta K_{\text{S}}\\!=\\!0.8$ mJ/m2, and second order approximation with
$\Delta K_{\text{S}}\\!=\\!2.3$ mJ/m2.

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# Potrika: Raw and Balanced Newspaper Datasets in the Bangla Language with
Eight Topics and Five Attributes

Istiak Ahmad  Department of Computer Science, Faculty of Computing and
Information Technology, King Abdulaziz University, Jeddah 21589, Saudi Arabia
0000-0001-9914-4116 Fahad AlQurashi  Department of Computer Science, Faculty
of Computing and Information Technology, King Abdulaziz University, Jeddah
21589, Saudi Arabia  0000-0002-7919-747X Rashid Mehmood  High Performance
Computing Center, King Abdulaziz University, Jeddah 21589, Saudi Arabia
Corresponding author<EMAIL_ADDRESS>0000-0002-4997-5322

###### Abstract

Knowledge is central to human and scientific developments. Natural Language
Processing (NLP) allows automated analysis and creation of knowledge. Data is
a crucial NLP and machine learning ingredient. The scarcity of open datasets
is a well-known problem in machine and deep learning research. This is very
much the case for textual NLP datasets in English and other major world
languages. For the Bangla language, the situation is even more challenging and
the number of large datasets for NLP research is practically nil. We hereby
present Potrika, a large single-label Bangla news article textual dataset
curated for NLP research from six popular online news portals in Bangladesh
(Jugantor, Jaijaidin, Ittefaq, Kaler Kontho, Inqilab, and Somoyer Alo) for the
period 2014-2020. The articles are classified into eight distinct categories
(National, Sports, International, Entertainment, Economy, Education, Politics,
and Science & Technology) providing five attributes (News Article, Category,
Headline, Publication Date, and Newspaper Source). The raw dataset contains
185.51 million words and 12.57 million sentences contained in 664,880 news
articles. Moreover, using NLP augmentation techniques, we create from the raw
(unbalanced) dataset another (balanced) dataset comprising 320,000 news
articles with 40,000 articles in each of the eight news categories. Potrika
contains both the datasets (raw and balanced) to suit a wide range of NLP
research. By far, to the best of our knowledge, Potrika is the largest and the
most extensive dataset for news classification.

_K_ eywords Natural Language Processing $\cdot$ Single-Label Text Dataset
$\cdot$ Bangla Text Analysis $\cdot$ News Article Analysis $\cdot$ Machine
Learning $\cdot$ Deep Learning

##  1 Introduction

Knowledge is central to human and scientific developments. Natural Language
Processing (NLP) allows automated analysis and creation of knowledge. Data is
a crucial NLP and machine learning ingredient. Textual datasets are critically
needed for NLP and machine learning research and development, in all human
languages including English, more so in languages such as Bangla where NLP
research is limited. The existing datasets for newspaper classification in the
Bangla language are limited in their size (the number of news article,
sentences, and words) and attributes (see Section 2 where we provide a summary
of the existing datasets and establish the research gap).

We present here Potrika, the largest and the most extensive dataset to date
for news classification containing 185.51 million words and 12.57 million
sentences in 664,880 news articles, providing five attributes, classified into
eight distinct categories. The data contains news articles for the period
2014-2020. It is a single-label dataset that can be used for text
classification, text summarization, text generation, and other NLP-related
research. It is made available publicly and freely.

Potrika comprises two datasets, a raw dataset, and a balanced dataset created
using NLP augmentation techniques. This balanced dataset contains 320,000 news
articles with 40,000 articles in each of the eight news categories. Potrika
can be used as a benchmark dataset. It may, for example, be used to create a
diversity of word embedding models. Aside from that, Potrika comprises eight
distinct news categories that have been carefully picked to reduce ambiguity
and, as a consequence, render it robust for text classification. It may also
be used to summarize content by referring to the news article and its
associated headline. Furthermore, it can be used to investigate NLP techniques
focussed on temporal aspects, such as identifying which category has the most
influence at a particular time or which keywords are utilized more frequently
over time.

This extensive dataset, Potrika, provides an opportunity to investigate and
improve the efficacy of applying machine learning and deep learning algorithms
for natural language processing on textual data in the Bangla language. The
Potrika datasets (the raw dataset and the balanced dataset) are described in
detail along with various dataset statistics, examples of news article with
associated categories, and data collection method, architecture, and
algorithms.

The rest of the paper is organised as follows. Section 2 provides a
description of the Potrika dataset and its comparison with the existing
datasets. Section 3 provides the design and methodology of data collection and
Section 4 concludes the paper.

Table 1: Potrika: Comparison with Existing Bangla Newspaper Datasets Dataset |  | Articles
---
(K)
| Sentences
---
(M)
| Words
---
(M)
| Categories
---
(Number)
Categories | Balanced
| BARD [1]
---
2018
376 | – | 88 | 5 |  | Economy, State
---
Entertainment,
International,
Sports,
$\times$
| Hossan [2]
---
2018
40 | – | – | 12 |  | Economy,
---
Opinion,
Entertainment,
International,
Sports,
Bangladesh,
Technology,
Education,
Durporobash
Art-and-literature,
North America,
Lifestyle
$\times$
| Shahin [3]
---
2020
10 | – | – | 8 |  | Economy,
---
International,
Entertainment,
Sports,
Science &
Technology,
Opinion, Politics
Country
$\times$
|  Potrika [4]
---
2021
665 | 125 | 185 | 8 |  | Economy,
---
Entertainment,
International,
Sports,
National, Politics
Science &
Technology,
Education
✓

##  2 Data Description

Data is a crucial NLP and machine learning ingredient. NLP allows automated
analysis and creation of knowledge – knowledge that is central to human and
scientific developments. This paper proposes the Potrika dataset that is a
vast collection of news articles in the Bangla language collected from six
prominent news portals in Bangladesh, including Jugantor [5], Jaijaidin [6],
Ittefaq [7], Kaler Kontho [8], Inqilab [9], and Somoyer Alo [10].

The scarcity of open datasets is a well-known problem in machine and deep
learning research. This is very much the case for textual NLP datasets in
English and other major world languages. For the Bangla language, the
situation is even worse and the number of large datasets for NLP research is
practically nil. There is little research undertaken on the Bangla text
classification challenge applied to small datasets and limited article
categories. The situation is evident from Table 1 that compares Potrika with
the existing related datasets and shows that (apart from Potrika) the existing
datasets for newspaper classification in the Bangla language are limited in
their size and other attributes.

The largest dataset after the Potrika dataset (see Table 1) is the BARD [1]
dataset that provides news articles alone without giving the headings,
publication dates, and the newspaper sources of the articles. Also, the BARD
[1] dataset is unbalanced. It contains around 240K articles related to the
State or National category, whereas the categories Economy, International,
Entertainment, and Sports have 19K, 32K, 31K, and 50K news articles,
respectively. An unbalanced dataset is not suitable for news article
classification. The other datasets mentioned in the table, Hossan [2] and
Shahin [3], have a much smaller number of news articles and these are not
suitable for deep learning algorithms. Moreover, the BARD and Hossan datasets
were provided in 2018 and therefore these are relatively old. The comparison
given in the table makes it evident that there is a scarcity of balanced and
comprehensive datasets and NLP research for Bangla news articles.

Potrika comprises two datasets, a raw dataset, which is described in Section
2.1 and a balanced dataset described in Section 2.2.

Figure 1: Potrika: News Article Examples for Each News Category

###  2.1 Potrika: The Raw Dataset

The dataset contains 665K articles divided into eight distinct categories
(National, Sports, International, Entertainment, Economy, Education, Politics,
and Science & Technology). We have collected five attributes for each document
that are News Article, Category, Headline, Publication Date, and Newspaper
Source. The dataset can be used in a variety of natural language processing
tasks such as single-label text classification, text summarization, named
entity recognition, word embedding model generation, news analysis task,
machine translation, and so on. Figure 1 shows one example of news article for
each of the eight categories with the respective four other attributes
Category, Headline, Publication Date, and Newspaper Source.

In the raw dataset folder, the news articles are organised in eight different
directories respective to each of the eight news categories. Each directory
contains eight CSV files for the six newspapers (one CSV file for each
newspaper except Jugantor and Ittefaq, which have two files each due to their
website structure). Each CSV file has five columns, one for each of the five
attributes that we have collected for each news article. These five attributes
have already been mentioned. After web scraping, the data is maintained in its
raw form with no cleaning, stemming, or other pre-processing. Duplicate
articles were removed from the dataset. Some English symbols, punctuations,
arithmetic, and special characters can be found throughout the articles.

Figure 2 depicts the number and percentage of the news articles for each
newspaper. Most of the news articles (around 85%) belong to the top three
newspapers: Inqilab, Jugantor, and Ittefaq. The newspapers Kaler Kontho,
Somoyer Alo, and Jaijaidin have contributed to 11%, 3% and 1% of the news
articles, respectively. Additional information about the distribution of news
articles across the six newspapers and eight news categories is shown in Table
2. The table shows that most of the news articles came from the Inqilab
newspaper. Note that the National category has the highest number of articles
among all the eight categories. The Kaler Kontho newspaper archive has zero
articles in the Entertainment category.

Figure 2: Potrika: News Article Distribution by Newspaper Table 2: Distribution of News Articles across Six Newspapers and Eight News Categories | Inqilab | Jugantor | Ittefaq |  | Kaler
---
Kontho
| Somoyer
---
Alo
Jaijaidin | Total
National | 103602 | 82322 | 80388 | 18003 | 6365 | 4424 | 295104
International | 47026 | 30352 | 32618 | 7 | 4026 | 545 | 114574
Sports | 26197 | 38550 | 18683 | 17618 | 3492 | 499 | 105039
Entertainment | 14898 | 8134 | 11414 | 0 | 2906 | 333 | 37685
Economy | 11156 | 6084 | 3718 | 15306 | 2475 | 47 | 38786
Education | 663 | 4838 | 3225 | 12704 | 426 | 171 | 22027
ScienceTech | 544 | 5993 | 5179 | 11476 | 396 | 53 | 23641
Politics | 468 | 15733 | 9248 | 1628 | 646 | 301 | 28024
Total | 204,554 | 192,006 | 164,473 | 76,742 | 20,732 | 6,373 | 664,880

Table 3 lists the number of news articles, sentences, and words for each of
the eights categories in the Potrika raw dataset. There are a total of about
665K news articles that contain 12.57 million sentences and 185.51 million
words. The data is presented in the table the descending order of the number
of news article in each category. The National category has the most articles
(295K) in the dataset, followed by the Sports and International categories,
105K and 114K articles, respectively. Education and Science & Technology
categories have the lowest number of articles compared to the other
categories.

Table 3: Potrika Raw Dataset: Number of News Articles, Sentence, and Words per Category Category |  Articles (K) |  Sentence (M) |  Words (M)
---|---|---|---
National | 295 | 5.46 | 87.34
International | 114 | 1.80 | 26.34
Sports | 105 | 2.21 | 29.81
Economy | 39 | 0.71 | 11.65
Entertainment | 38 | 0.62 | 8.01
Politics | 28 | 0.53 | 8.4
Science & Technology | 24 | 0.32 | 4.8
Education | 22 | 0.92 | 9.16
Total | 665 | 12.57 | 185.51

Figure 3 plots the number of news articles (the blue bars) for an increasing
number of words per document (news article). The majority of news articles
contain between 100 to 250 words (see the graph peaks). The number of news
articles that contain over 800 words is relatively small. The maximum number
of articles in any document in the data is approximately 8000 (7939 to be
precise). The maximum number of articles or documents for any given number of
words is around 3000. The zoomed plot inside the figure is provided for the
reader’s convenience. The figure also plots the density against an increasing
number of words per document. See the y-axis on the right of the figure and
the black plot around the blue bars. The maximum density is around 0.004.

Figure 3: Number of News Articles against Word Count Per News Article

Figure 4 plots density and the number of news articles against the word count
per document for each category. The zoomed plots inside the figures are
provided for the reader’s convenience. Note that the National news category
holds the highest number (around 1200) of news articles comprising around 300
words per article (for the peak). The International category has just over 600
news articles with this peak (the maximum number of articles) of around 250
words per article. The majority of news articles on entertainment, sports,
politics, and the economy are around 500 words in length (look at the peaks in
the zoomed figures). Note also that the Entertainment news category has the
highest density of about 0.007 whereas Education has the least density of just
over 0.002. The reason for the low density for the Education category is the
fact that the maximum number of news articles is relatively low (just over 50)
and are spread across the wider spectrum of the number of words per document
(up to 1500 or more).

Figure 4: Density and Number of News Articles against Word Count for Each
Category

###  2.2 Potrika: The Balanced Dataset

An unbalanced dataset where the number of instances in the training dataset
per class label is not balanced can produce low accuracy for a predictive
classification model. In the Potrika dataset, we added a folder called
“BalancedDataset” that contains a balanced dataset where each news category
has 40,000 articles and the total number of articles for all the categories is
320,000. The balanced dataset is created using NLP augmentation techniques
that will be described in Section 3.3.

##  3 Design and Methodology

Figure 5 depicts a high-level architecture of our data collection process. The
web scraping and Python tools including BeautifulSoup, Requests, Pandas, and
others were used to collect the articles. Our data gathering procedure was
separated into two parts. First, in algorithm 1, article links (URL) are
gathered from each newspaper archive, and then algorithm 2 retrieves the news
articles, Category, Heading, Publish Date, and Source from the link using the
HTML and JavaScript tags. Finally, we export the data into a CSV file.

Figure 5: Dataset Collection Architecture

###  3.1 Link Collection for News Articles

Initially, we assembled the links to the news from six Bangla newspaper
archives using web scraping and python libraries including BeautifulSoup,
Requests, Pandas, etc. There are two different versions of online archives for
the newspapers Jugantor and Ittefaq; for other newspapers, there are single
online archives. There are many additional links in the archive, but we
deleted all the additional links and only saved the targeted links.

Algorithm 1 News Articles Link Collection

procedure articlelink($archive$) $\triangleright$ Newspaper archive link

for <startYear to endYear> do

for <startMonth to endMonth> do

for <startDay to endDay> do

$url\leftarrow archive/year/month/day$

$content\leftarrow\textit{fetch the content from the URL}$

$links\leftarrow\textit{save the link from the content}$ $\triangleright$ News
articles link

end for

end for

end for

end procedure

###  3.2 Data Collection

Retrieving the data from the website automatically is most challenging because
an enormous amount of additional data is frequently available on the website
link. Initially, we analyzed the pattern of the URL (Uniform Resource Locator)
and found that all the newspapers have almost the same pattern in their online
archive URLs. For example, the pattern is like
“newspaperlink/archive/year/month/day” or “newspaperlink/category/articleID”,
etc. Based on the pattern, we separately fetch the content of the URL from
different newspapers and retrieved the targeted data.

Algorithm 2 Data Collection

procedure datacollection($links$)

for <i = 0 to total number of links> do

$link\leftarrow links[i]$

$content\leftarrow\textit{fetch the content from the link}$

$article\leftarrow\textit{fetch article from content}$

$headline\leftarrow\textit{fetch headline from content}$

$category\leftarrow\textit{fetch category from content}$

$time\leftarrow\textit{fetch publication date from content}$

end for

save the dataset as CSV file

end procedure

###  3.3 Balancing the Dataset using NLP Data Augmentation

NLP data augmentation techniques are most commonly used on tiny and unbalanced
datasets. There are three well-known augmentation strategies. %1) Back
Translation: this approach involves translating the text material into another
language and then back into the original language. This technique preserves
the context of the textual data. 2) Easy Data Augmentation (EDA) consists of
four methods: synonym replacement, random insertion, deletion, and text
swapping. 3) NLP Albumentation: For duplicate value, sentences are shuffled or
excluded.

  1. 1.

Back Translation: this approach involves translating the text material into
another language and then back into the original language. This technique
preserves the context of the textual data.

  2. 2.

Easy Data Augmentation (EDA) consists of four methods: synonym replacement,
random insertion, deletion, and text swapping.

  3. 3.

NLP Albumentation: for duplicate value, sentences are shuffled or excluded.

The Potrika dataset comprises a relatively small number of documents for five
categories (Education, Economy, Science & Technology, Politics, and
Entertainment) compared to the other three categories, National,
International, and Sports, which have more than 100k articles for each
category. We used back translation methods on the datasets from these five
categories to create a balanced dataset. We use the Google Translator API to
convert Bangla text to English and then back to Bangla. The Google Translate
API only accepts legitimate text with a limit of 5000 characters; otherwise,
it cannot be translated. Each of the news category in the balanced dataset has
40,000 articles after performing the data augmentation approach. The balanced
dataset has 320,000 news articles in total (each of the eight categories have
40,000 articles).

##  4 Conclusion, Utilization, and Outlook

The Potrika dataset introduced in this paper is the largest and the most
extensive dataset for news classification. Textual datasets are in critical
demand for NLP and machine learning research and development, in all human
languages including English, more so in languages such as Bangla where NLP
research is limited. Apart from our dataset Potrika, the existing datasets for
newspaper classification in the Bangla language are limited in their size,
attributes, and other properties.

We introduced the Potrika dataset in this paper, a large single-label Bangla
news article textual dataset curated for NLP research from six popular online
news portals in Bangladesh. The articles were classified into eight distinct
categories providing five attributes for each news article. The raw dataset
contains over 185 million words and 12 million sentences contained in nearly
665 thousand news articles. Moreover, we created and included in Potrika an
additional dataset that is balanced to facilitate those who intend to research
in NLP involving a balanced dataset. The Potrika raw and balanced datasets can
be used for text classification, text summarization, text generation, and
other NLP-related research. Potrika is made available publicly and freely.

We are working on providing applications of NLP research over the Potrika
datasets, particularly on investigating performance of machine and deep
learning algorithms for news classification and on the use of clustering for
automatic labelling. The research is completed, is in the write-up stage and
will soon be submitted for publication. Our earlier research on NLP has
focussed in multiple languages on a range of problems including transportation
[11, 12, 13], logistics [14, 15], healthcare [16, 17, 18], and urban
development and governance [19, 20, 21]. These works fall under the broader
umbrella of smart cities and societies [22, 23] using big data and artificial
intelligence technologies [24, 25] where we have developed significant
research outputs. We plan to extend our research on the Bangla language for
these problems in the future. This will involve curating additional open
datasets in the Bangla language and developing NLP research over Potrika and
additional datasets.

## Ethics Statement

The reuse of article text from Bangladesh’s online newspaper complies with the
terms of use. All the articles were acquired with the consent of the people,
groups or organizations.

## Acknowledgments

The work reported in this paper is supported by the High Performance Computing
Centre (HPC Center) at King Abdulaziz University, Saudi Arabia. The computing
tasks reported in this paper were performed on the Aziz supercomputer at the
HPC Center, King Abdulaziz University. We would also like to express our
gratitude to the authors of Jugantor, Inqilab, Jaijaidin, Kaler-kontho,
Ittefaq, and Somoyer Alo, for making their news article archives publicly
available.

## Declaration of Competing Interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

## References

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  * [2] Md. Zakir Hossan.  40k bangla newspaper article. september 2018.  https://www.kaggle.com/zshujon/40k-bangla-newspaper-article, 2018\.
  * [3] Md Mahmudul Hasan Shahin, Tanvir Ahmmed, Shahriar Hasan Piyal, and Md Shopon.  Classification of bangla news articles using bidirectional long short term memory.  In 2020 IEEE Region 10 Symposium (TENSYMP), pages 1547–1551. IEEE, 2020.
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