Reduced order model for hard magnetic films

H. Moustafa, A. Kovacs, J. Fischbacher, M. Gusenbauer, Q. Ali, L. Breth, Y.
Hong, W. Rigaut, T. Devillers, N. Dempsey, et al.

<figure><img src="image_1.png" /></figure>

To cite this version:

H. Moustafa, A. Kovacs, J. Fischbacher, M. Gusenbauer, Q. Ali, et al. .. Reduced order model for hard magnetic films. AIP Advances, 2024, 14 (2), pp.025001. 10.1063/9.0000816 . hal-04729290

HAL Id: hal-04729290

https://hal.science/hal-04729290v1

Submitted on 16 Oct 2024

**HAL** is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_2.png" /></figure>
# Reduced order model for hard magnetic films

Cite as: AIP Advances 14, 025001 (2024); doi: 10.1063/9.0000816

Submitted: 4 October 2023 • Accepted: 30 November 2023 •

Published Online: 1 February 2024

<figure><img src="image_3.png" /></figure>

View Online

<figure><img src="image_4.png" /></figure>

Export Citation

<figure><img src="image_5.png" /></figure>

CrossMark

H. Moustafa,<fnref n="1" />,ª ID A. Kovacs,<fnref n="1" /> ID J. Fischbacher,<fnref n="1" /> ID M. Cusenbauer,<fnref n="1" /> ID Q. Ali,<fnref n="1" />,<fnref n="2" /> ID L. Breth,<fnref n="1" /> ID Y. Hong,<fnref n="3" /> ID
W. Rigaut,<fnref n="3" /> ID T. Devillers,<fnref n="3" /> ID N. M. Dempsey,<fnref n="3" /> ID T. Schrefl,<fnref n="1" />,<fnref n="2" /> ID and H. Oezelt<fnref n="1" />

## AFFILIATIONS

Note: This paper was presented at the 68th Annual Conference on Magnetism and Magnetic Materials.

ªAuthor to whom correspondence should be addressed: heisam.moustafa@donau-uni.ac.at

## ABSTRACT

In the pursuit of rare earth-lean permanent magnets for green technologies, microstructural optimisation offers a promising strategy to enhance coercivity while minimising critical element content. For this approach, the combination of experimental work on hard magnetic films and numerical investigations is necessary. However, computational limitations restrict micromagnetics to small systems, motivating the development of a reduced order model for investigating large multigrain systems. The model is based on the embedded Stoner-Wohlfarth method and is used to investigate the influence of a nonmagnetic grain boundary phase thickness and the aspect ratio of the magnetic grains on the overall coercivity. It is possible to simulate large NdFeB multigrain structures which can be compared to hard magnetic films. We derive design recommendations to increase coercivity by increasing the grain boundary phase thickness and the aspect ratio of the grains in hard magnetic materials.

© 2024 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/9.0000816

## I. INTRODUCTION

Permanent magnets can be found in a large variety of applications, e.g., in electronic and medical devices. Especially due to the transition to green technologies, high-performance permanent magnets are needed for windmills and (hybrid) electric cars. The permanent magnets must withstand high opposing magnetic fields at elevated temperatures. The required high coercive fields are generally achieved in rare-earth based magnets like NdFeB-type magnets. However, the energy transition increases the demand for rare-earth elements like Nd drastically, making it a critical material in regards of supply and prices. Therefore, to ensure the success of these new technologies, the content of rare-earth elements in permanent magnets needs to be reduced while maintaining their high performance.

To improve magnet performance, optimizing microstructural properties, like grain size distribution or shape, is crucial. Introducing Dy along grain boundaries reduces heavy rare-earth content in NdFeB grains, while maintaining high coercivity.<fnref n="1" /> Another approach is grain size refinement or substituting Nd by

less critical elements like La or Ce while maintaining high coercive fields at elevated temperatures.<fnref n="2" /> Tang et al. showed experimentally and by numerical approaches that thermal stability of coercivity can be tuned by changing the aspect ratio of grains or by engineering a nonmagnetic intergranular phase.<fnref n="3" /> Vial et al. used an optimised heat treatment to create a nonmagnetic rare-earth-rich grain boundary phase displaying increased coercivity.⁴ The influence of the grain boundary phase on the coercivity was investigated by Chen et al.⁵ using micromagnetic simulations with a system of about a micrometer in size.

Only recently, data-driven approaches have been used in the workflow of magnetic materials design.⁶ Reliable and quantitative models for coercivity of permanent magnets require well prepared training data, ideally from experimental and fast simulation results.

Hard magnetic films pose as beneficial model systems since they offer the advantage of controlling microstructural properties by adjusting growth parameters<fnref n="2" /> diverse material compositions.⁸

NdFeB-based films typically show grain sizes of up to several micrometers. Therefore, a computational examination of structural

<footnote n="1">¹Department for Integrated Sensor Systems, University for Continuing Education Krems, Wr. Neustadt, Austria</footnote>

<footnote n="2">²Christian Doppler Laboratory for Magnet Design Through Physics Informed Machine Learning, Department for Integrated Sensor Systems, University of Continuing Education Krems, Wr. Neustadt, Austria</footnote>

<footnote n="3">³Université Grenoble Alpes, CNRS, Grenoble INP, Institute Néel, Grenoble, France</footnote>
properties like grain aspect ratio distribution need geometrical models of at least 1 µm in size. This would result in hundreds of millions of finite elements which requires supercomputing systems and huge amounts of computing resources. To generate a significant amount of data through micromagnetic simulations for large multigrain structures observed in hard magnetic films, it is necessary to reduce the complexity of the simulation models. A reduced order approach based on the embedded Stoner-Wohlfarth model has been established⁹ which allows to investigate granular structures of films at their actual size.

In this work we will give a brief overview of the reduced order model and demonstrate its capabilities by investigating the impact of nonmagnetic grain boundary phase thickness and grain aspect ratio of hard magnetic films. The magnetic systems we investigate consist of realistically shaped grains with a mean grain size of 375 nm and on average, were computed within 6 min.

## II. REDUCED ORDER MODEL FOR HARD MAGNETIC SYSTEMS

The reduced order model (ROM) is based on four assumptions: i) the grains are homogeneously magnetised before and after switching. The switching process itself happens by nucleation of a domain wall and immediate propagation through the whole grain.¹⁰ Hence this is not reflected in the ROM and ii) the entire grain is switched in a single step. iii) Magnetisation reversal always starts at the grains' surface. This has been observed in both micromagnetic simulations and experimental studies.¹¹ In our first implementation of the ROM iv) the grains are decoupled by a nonmagnetic boundary phase.

In classical finite element micromagnetism, the magnetisation of the system is defined by the magnetic moments on the nodes of the mesh. Because of the first assumption the magnetic moments can be reduced to just one magnetic moment per grain. For each grain we have to detect if magnetic reversal happens at a certain applied field. Since reversal starts at the border of the grains, we only need a 2D-mesh close to the surface. This means a substantial reduction of required computation resources compared to full micromagnetic simulations. In Fig. 1 the difference between the two meshes is shown by the sketches of the grains on the left. Manually testing 2D meshes of various sizes for the ROM showed that using 25 evaluation points per grain is sufficient to replicate the reversal curve calculated by the full micromagnetic model.

Each node of this mesh acts like a field sensor. This is realised by calculating the total magnetic field

$$H_{\text{total}} = H_{\text{ext}} + H_{\text{dmag}} + H_{\text{exch}} \quad (1)$$

acting on this node. Here, $H_{\text{ext}}$ is the external field acting on the whole system, $H_{\text{dmag}}$ is the demagnetisation field exerted by the neighbouring grains, and $H_{\text{exch}}$ is the exchange field in each grain, as pictured in Fig. 1 on the right side. We compute $H_{\text{dmag}}$ from the surface charge density by following the work of Guptasarma and Singh and use analytical formulas for polyhedral geometries.¹³ To reduce computation time and memory requirement, we use hierarchical matrices as implemented with h2tools¹⁴ where the demagnetisation field of nearby grains is computed exactly, while the fields from grains far away from the respective node are approximated. The exchange field is calculated for each grain by $H_{\text{exch}} = M/l^2$ with M

<figure><img src="image_6.png" /><figcaption>FIG. 1. Comparison of volumetric finite element mesh for classical micromagnetic simulations and the 2D-surface mesh of the reduced order model depicted in a 2D sketch. The mesh is moved inwards by a certain distance which was found in our previous work.¹² Each node represents an embedded Stoner-Wohlfarth particle used to decide if a grain gets reversed. On the right hand side, the three field contributions, acting on the particles are shown.</figcaption></figure>

being the magnetisation of the respective grain and l being a phenomenological distance parameter, also tuned by comparison to full micromagnetic simulations in our previous work.¹² The direction of the magnetisation M is calculated in the centre of each grain analytically by the Stoner-Wohlfarth model according to the work of Wood.¹⁵ Using this approach, we can track the reversible part of the demagnetisation curve of each grain, and therefore of the entire multigrain system.

The local Stoner-Wohlfarth switching field, which covers the anisotropy field, defined as

$$H_{\text{SW}} = \frac{2K_1}{\mu_0 M_s} (\sin^{2/3} \psi + \cos^{2/3} \psi)^{-3/2} \quad (2)$$

is computed analytically at each node and compared to the total field. Here $\psi$ is the angle between the total field acting on the respective node and the anisotropy axis, and $2K_1/(\mu_0 M_s)$ is the nucleation field. $K_1$ is the magneto-crystalline anisotropy constant, $\mu_0$ the magnetic permeability constant in vacuum, and $M_s$ the saturation magnetisation. Given our third assumption of single step switching, once $H_{\text{SW}} < H_{\text{total}}$, the entire grain will be reversed. The nodes of the 2D mesh act like SW particles, hence this approach is also called embedded Stoner-Wohlfarth model.¹⁶

The demagnetisation curve of a multigrain system is computed by applying an external field $H_{\text{ext}}$ in the same direction as the system is magnetically saturated. $H_{\text{ext}}$ is reversed step by step in the opposite direction until the system reverses as well. For each field step, $H_{\text{total}}$ is evaluated in all embedded SW particles of all grains. If a grain includes a particle which shows $H_{\text{SW}} < H_{\text{total}}$, it gets reversed. If multiple grains include particles which meet this requirement, the one with the smallest difference between $H_{\text{SW}}$ and $H_{\text{total}}$ is switched. Sequentially, $H_{\text{total}}$ of all particles need to be recomputed. For each field step, the reversal of grains and recomputation of the entire system is done iteratively until a magnetic equilibrium state is reached. After that the external field is further reduced by one step.

In full micromagnetics for permanent magnets we are interested in the local configuration of the magnetization as a response to an externally applied field. For this purpose time-dependent magnetization dynamics is irrelevant and it is sufficient to compute
<figure><img src="image_7.png" /><figcaption>**FIG. 2.** Demagnetisation curves computed by a full micromagnetic model and by the reduced order model. For each model three snapshots of the reversed grains at different applied fields are shown.</figcaption></figure>

stable or meta-stable equilibrium states by minimizing the total
Gibb's free energy. For micromagnetic minimization, the magni-
tude of the magnetization has to be constant over the entire system
and depends on the temperature, which is constant in time and
space. To validate our ROM we use a micromagnetic finite-element
code for energy minimization which takes advantage of massively
parallized graphical processing units.¹⁷ We compute the demagneti-
sation curve of a Nd₂Fe₁₄B cube with 150 nm edge length with
both models. Typical intrinsic properties for this materials are used
with μ₀M_s = 1.61 T, K₁=4.9 × 10⁶ J m⁻³ and A_exch=8 × 10⁻¹² J m⁻¹.
The magnetocrystalline anisotropic easy axes of the 36 grains are
uniformly distributed with a maximal deviation angle of 30° with
respect to the applied field. The grains, column-shaped as observed
in thin hard magnetic films, are assumed to be separated by a non-
magnetic boundary phase. The microstructure, intrinsic properties,
and simulation setup are the same for both the micromagnetic and
the ROM model. We begin by saturating the cube magnetically
in the z-direction and then apply an increasing external field in
the opposite direction in 5 mT increments from 0 to -6 T. The
reversible part of the curve, up to -3.5 T, is tracked perfectly by
the ROM (see Fig. 2). Each small step in the curve represents a
switch of a single grain. The overall conformity of both curves is
good and only small deviations can be observed (see third snap-
shot). The computation time is drastically reduced and depends
on the number of grains and the embedded Stoner-Wohlfarth
particles per grain. For the full micromagnetic model the com-
putation time is 202 minutes (GPU). For the ROM an average
of 25 evaluation points per grain were calculated, resulting in a
computation time of 6 minutes (CPU). This is a nearly 34-fold
speed up.

### III. FAST SIMULATIONS OF LARGE MULTICRAIN SYSTEMS

After describing the model and demonstrating its accuracy for
the calculation of coercivity, we increase the grains to an average
size of 375 nm and investigate two microstructural features: i) the
impact of the thickness of a nonmagnetic grain boundary phase
between the grains, and ii) the influence of the aspect ratio of the
grains on the coercivity of a magnetic system. For both investigations

we start by generating cuboid models with one micron edge-length,
tesselated into 36 grains by the software Neper.¹⁸ The geometric
models are then modified to introduce the grain boundary phase and
meshed by Salome.¹⁹ The demagnetization curves are computed by
the ROM and the coercive field, defined as the field which is needed
to reduce the magnetisation from saturation to zero in the applied
field's direction, is recorded for each variation and parameter set.

### A. The effect of grain boundary phase thickness

At first we investigate the impact of the grain boundary phase thickness on the coercivity. Experimentally, heat treatments have been proposed to introduce non-magnetic grain boundaries by creating Nd-rich phases.⁴ The ROM enables us to quickly explore the possibilities of this approach for large multigrain systems.

Figure 3 shows the coercivity as a function of the grain boundary thickness. Each dot represents the mean coercivity of ten simulated random structure variations. The software Neper offers the possibility to create random granular structures by Voronoi tesselation. While the structures vary in the size, shape, and position of the individual grains, the quantity, average grain size, and aspect ratio of the grains are maintained. This is done to mitigate the finite size effect. The grain boundary phase thickness is increased in 1 nm-steps from 2 nm to 20 nm. The coercivity is rising from approximately 3.65 T to 4 T in this range. The dependency is not linear, it is starting with a steep gradient and getting more and more flat as the grain boundary phase gets thicker. This trend was also observed by Chen et al.⁵ A possible explanation is the reduced influence of the stray field with increasing distance between grains. It should be noted, that an increasing grain boundary phase thickness comes with two implications: i) the aspect ratio of the grains is increased, ii) the magnetic volume of the grains is reduced. Both parameters are contributing to the increased coercivity. These effects have also been seen in experiments, where an increasing Nd-content in permanent

<figure><img src="image_8.png" /><figcaption>**FIG. 3.** Coercivity as a function of thickness of a nonmagnetic grain boundary phase. Ten random variations of granular structures with an average grain aspect ratio of 3.38 are simulated for each thickness value. Hence, for each value a distribution is shown. The dots mark the mean; the black lines show the distance between minimum and first quartile (Q1) and maximum and third quartile (Q3), respectively. The inset shows an example of the structure. The grain boundary phase is transparent, the surface of the grains is coloured blue, and the embedded SW particles in the cutout are shown in red.</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>FIG. 4. Coercivity as a function of the aspect ratio. For each aspect ratio we show the distribution of five random variations of granular structures.</figcaption></figure>

magnets increases the coercivity but decreases the magnetic rema-
nence due to the loss of magnetic volume.²⁰

B. Impact of aspect ratio of grains

Another important factor influencing the behaviour of a per-
manent magnet can be the shape of the magnetic grains. Here, we
examine the aspect ratio of the grains in relation to the direction of
the applied field and its impact on coercivity. In our previous work²¹
it has been shown, that by varying the deposition temperature of a
permanent magnetic film the shape of the grains can be tuned. In
the simulation we can easily change the aspect ratio of grains to look
at the impact on the coercive field. Figure 4 shows the coercivity as
a function of a changing aspect ratio. The aspect ratio was altered
from 1.0 to 4.0 in steps of 0.1. Here it is defined as the length of the
grain in z-direction, i.e., the direction of the applied field, divided
by the width in x-direction. The coercivity is continuously rising
with increasing aspect ratio from approximately 3.15 T at equiaxed
grains to 3.8 T at an ratio of 3.2. The gradient is slowly decreasing
and above the aspect ratio of 3.2 up to 4 no coercivity enhancement
was observed. Tang et al.<fnref n="3" /> could show the same behaviour when
increasing the aspect ratio.

IV. CONCLUSION

Large hard magnetic multi-grain-models, beyond the limits of
conventional micromagnetism, can be calculated using the reduced
order model based on the embedded Stoner-Wohlfarth method. The
presented reduced order model drastically reduces the computa-
tional effort in terms of memory and processing time by a) requiring
only a 2D surface mesh instead of a volumetric finite element mesh,
b) approximate the grains' reversal by analytic computation of the
local switching field, and c) compute the demagnetization field ana-
lytically employing hierarchical matrices. We used this model to
investigate the influence of the thickness of a nonmagnetic grain
boundary phase and the aspect ratio of grains on the magnet's coer-
civity. An increased thickness of the boundary phase improves the
coercivity of the magnet but the effect weakens with increasing
thickness. Compared to equiaxed grains, columnar grains with an

aspect ratio of 3 exhibit an increase by 0.65 T (21%) in the direction
of the longer axis. These results can be taken as guiding princi-
ples to increase coercivity in permanent magnets. The thickness of
an ideally nearly non-magnetic grain boundary phase in NdFeB-
type magnets can be influenced to some extend by methods such as
increasing the Nd content²² or through grain boundary diffusion.²³
The aspect ratio of grains could be modified using manufacturing
processes like hot-pressing or annealing. Nevertheless, a satisfactory
level of control over this parameter has not been reached yet.²⁴

ACKNOWLEDGMENTS

This research was funded in whole or in part by the Austrian
Science Fund (FWF) I 6159-N and the French National Research
Agency ANR-22-CE91-0008.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

**H. Moustafa:** Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal). **A. Kovacs:** Methodology (equal); Software (equal); Supervision (equal); Visualization (equal); Writing – review & editing (equal). **J. Fischbacher:** Data curation (equal); Methodology (equal); Software (equal). **M. Gusenbauer:** Methodology (equal); Software (equal). **Q. Ali:** Software (equal). **L. Breth:** Methodology (equal); Software (equal). **Y. Hong:** Data curation (equal); Investigation (equal); Resources (equal). **W. Rigaut:** Data curation (equal); Investigation (equal); Resources (equal). **T. Devillers:** Investigation (equal); Resources (equal). **N. M. Dempsey:** Project administration (equal); Resources (equal). **T. Schrefl:** Conceptualization (equal); Methodology (equal); Software (equal). **H. Oezelt:** Conceptualization (equal); Funding acquisition (equal); Project administration (equal); Software (equal); Supervision (equal); Writing – review & editing (equal).

DATA AVAILABILITY

The data that support the findings of this study are available
from the corresponding author upon reasonable request.

REFERENCES

<footnote n="1">¹H. Nakamura, *Scripta Materialia* **154**, 273 (2018).</footnote>

<footnote n="2">²C. V. Colin, M. Ito, M. Yano, N. M. Dempsey, E. Suard, and D. Givord, *Applied Physics Letters* **108**, 242415 (2016).</footnote>

<footnote n="3">³X. Tang, J. Li, Y. Miyazaki, H. Sepehri-Amin, T. Ohkubo, T. Schrefl, and K. Hono, *Acta Materialia* **183**, 408 (2020).</footnote>

<footnote n="4">⁴F. Vial, F. Joly, E. Nevalainen, M. Sagawa, K. Hiraga, and K. Park, *Journal of Magnetism and Magnetic Materials* **242–245**, 1329 (2002).</footnote>

<footnote n="5">⁵F. Chen, S. Lu, Y. Zhao, W. Zhao, and Z. Xu, *Journal of Magnetism and Magnetic Materials* **582**, 171017 (2023).</footnote>
<footnote n="6">⁶H.-K. Park, J.-H. Lee, J. Lee, and S.-K. Kim, *Sci. Rep.* **11**, 3792 (2021).</footnote>

<footnote n="7">⁷K. Hono and H. Sepehri-Amin, *Scripta Materialia* **67**, 530 (2012).</footnote>

<footnote n="8">⁸D. Le Roy, O. Akdogan, G. Ciuta, N. Dempsey, and D. Givord, *Journal of Applied Physics* **115**, 17A738 (2014).</footnote>

<footnote n="9">⁹A. Kovacs, J. Fischbacher, H. Oezelt, A. Kornell, Q. Ali, M. Gusenbauer, M. Yano, N. Sakuma, A. Kinoshita, T. Shoji, A. Kato, Y. Hong, S. Grenier, T. Devillers, N. M. Dempsey, T. Fukushima, H. Akai, N. Kawashima, T. Miyake, and T. Schrefl, *Frontiers in Materials* **9**, 1094055 (2023).</footnote>

<footnote n="10">¹⁰M. E. Schabes, *J. Magn. Magn. Mater.* **95**, 249 (1991).</footnote>

<footnote n="11">¹¹T. Helbig, K. Loewe, S. Sawatzki, M. Yi, B.-X. Xu, and O. Gutfleisch, *Acta Materialia* **127**, 498 (2017).</footnote>

<footnote n="12">¹²A. Kovacs, J. Fischbacher, H. Oezelt, T. Schrefl, A. Kaidatzis, R. Salikhov, M. Farle, G. Giannopoulos, and D. Niarchos, *IEEE Trans. Magn.* **53**, 1 (2017).</footnote>

<footnote n="13">¹³D. Guptasarma and B. Singh, *GEOPHYSICS* **64**, 70 (1999).</footnote>

<footnote n="14">¹⁴A. Y. Mikhalev and I. V. Oseledets, *Numer Linear Algebra Appl* **23**, 230 (2016).</footnote>

<footnote n="15">¹⁵R. Wood, *IEEE Trans. Magn.* **45**, 100 (2009).</footnote>

<footnote n="16">¹⁶L. Exl, J. Fischbacher, A. Kovacs, H. Özelt, M. Gusenbauer, K. Yokota, T. Shoji, G. Hrkac, and T. Schrefl, *Journal of Physics: Materials* (2018).</footnote>

<footnote n="17">¹⁷T. Schrefl, G. Hrkac, S. Bance, D. Guess, O. Ertl, and J. Fidler, *Handbook of Magnetism and Advanced Magnetic Materials*, 1st ed (Wiley, 2007).</footnote>

<footnote n="18">¹⁸R. Quey, P. Dawson, and F. Barbe, *Computer Methods in Applied Mechanics and Engineering* **200**, 1729 (2011).</footnote>

<footnote n="19">¹⁹Salomé, www.salome-platform.org (Accessed: August 2023) (2023).</footnote>

<footnote n="20">²⁰J. Fidler, W. Winkler, and J. Bernardi, in *9th International Symposium on Magnetic Anisotropy and Coercivity in Rare Earth-Transition Metal Alloys*, Sao Paulo, 1996, pp. 419–428.</footnote>

<footnote n="21">²¹N. M. Dempsey, A. Walther, F. May, D. Givord, K. Khlopkov, and O. Gutfleisch, *Applied Physics Letters* **90**, 092509 (2007).</footnote>

<footnote n="22">²²J. Liu, H. Sepehri-Amin, T. Ohkubo, K. Hioki, A. Hattori, T. Schrefl, and K. Hono, *Acta Materialia* **61**, 5387 (2013).</footnote>

<footnote n="23">²³D. Salazar, A. Martín-Cid, R. Madugundo, J. M. Barandiaran, and G. C. Hadjipanayis, *Applied Physics Letters* **113**, 152402 (2018).</footnote>

<footnote n="24">²⁴J. Li, H. Sepehri-Amin, T. Sasaki, T. Ohkubo, and K. Hono, *Science and Technology of Advanced Materials* **22**, 386 (2021).</footnote>