MagNet machine learning enhanced three-dimensional magnetic reconstruction Boyao Lyu1 2Shihua Zhao3 4 5Yibo Zhang3 6Weiwei Wang7Haifeng Du1and Jiadong Zang3 8 1Anhui Province Key Laboratory of Condensed Matter Physics at Extreme Conditions

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MagNet: machine learning enhanced three-dimensional magnetic reconstruction

Boyao Lyu,1, 2, ∗Shihua Zhao,3, 4, 5, ∗Yibo Zhang,3, 6 Weiwei Wang,7Haifeng Du,1and Jiadong Zang3, 8, †

1Anhui Province Key Laboratory of Condensed Matter Physics at Extreme Conditions,

High Magnetic Field Laboratory of Chinese Academy of Sciences, Hefei, 230031, China

2University of Science and Technology of China, Hefei, 230031, China

3Department of Physics and Astronomy, University of New Hampshire, Durham, New Hampshire 03824, USA

4Department of Physics, The City College of New York, New York, NY 10031, USA

5Physics Program, Graduate Center of the City University of New York, New York, NY 10016, USA

6Department of Chemistry, University of New Hampshire, Durham, New Hampshire 03824, USA

7Institutes of Physical Science and Information Technology, Anhui University, Hefei 230601, China

8Materials Science Program, University of New Hampshire, Durham, New Hampshire 03824, USA

Three-dimensional (3D) magnetic reconstruction is vital to the study of novel magnetic materials

for 3D spintronics. Vector ﬁeld electron tomography (VFET) is a major in house tool to achieve that.

However, conventional VFET reconstruction exhibits signiﬁcant artefacts due to the unavoidable

presence of missing wedges. In this article, we propose a deep-learning enhanced VFET method to

address this issue. A magnetic textures library is built by micromagnetic simulations. MagNet, an

U-shaped convolutional neural network, is trained and tested with dataset generated from the library.

We demonstrate that MagNet outperforms conventional VFET under missing wedge. Quality of

reconstructed magnetic induction ﬁelds is signiﬁcantly improved.

Keywords: Vector ﬁeld electron tomography, deep learning, micromagnetism

I. INTRODUCTION

Recent studies of novel magnetic materials with topo-

logical textures, such as skyrmionic families1–4, become

an important driving force in spintronics to develop next

generation nano-electronic devices5,6. In addition to

two-dimensional (2D) topological textures, their three-

dimensional (3D) counterparts are emergent, such as the

skyrmion bundle and magnetic hopﬁon7,8. 3D magnetic

textures are prominent due to their potentially larger

volume-density and novel dynamics. However, imaging a

3D magnetic conﬁguration is a major obstacle. Most ex-

isting magnetic imaging tools such as Kerr microscopy9,

magnetic force microscopy10, and spin-polarized scan-

ning tunneling microscopy11 can only resolve magnetic

conﬁgurations on the 2D surface of a sample. Recent

advances in 3D magnetic imaging have been made. Neu-

tron scattering12–14, magnetic X-ray dichroism15–18 and

Lorentz transmission electron microscopy (LTEM)19 can

probe the internal magnetic structure of a sample. Com-

pared to neutron scattering and X-ray dichroism, LTEM

and its derivatives can achieve sub-Angstrom20 resolu-

tion without accelerating particles with a synchrotron.

It is thus attractive to enable LTEM-based 3D magnetic

reconstructions.

3D vector ﬁeld electron tomography (VFET), i.e. 3D

magnetic reconstruction from electron phase shifts re-

trieved from electron holography (EH)21 or transport of

intensity (TIE) equation22,is a relatively new but fast de-

veloping 3D magnetic imaging technique. Compared to

∗These authors contributed equally to this work

†Jiadong.Zang@unh.edu

LTEM, phase retrieval in EH signiﬁcantly elevates the

spatial resolution of the imaging. Since its earliest pro-

posal by Lai et al. in 199423, the theoretical foundation

of VFET has been established24–26. Once clean electron

phase shifts of two orthogonal and complete tilt series

are collected, two components of the magnetic induction

ﬁeld Bcan be reconstructed separately by the central

slicing theorem in scalar tomography. The third com-

ponent of Bcan then be calculated by the constraint

∇ ⋅ B=0. Thus conventional analytical algorithms, such

as weighted backprojection method (WBP) and regrid-

ding reconstruction method (Gridrec) can be directly ex-

tended to VFET27. However, in real experiments, there

are many sources of inevitable errors during electron

phase shifts collection, such as noise, sparsity, misalign-

ment, and missing wedge. Those errors thus lead to sig-

niﬁcant inevitable artefacts. Iterative algorithms such

as algebraic reconstruction technique (ART) and simul-

taneous iterative reconstruction technique(SIRT), as the

second generation of reconstruction algorithms, show the

capability of working with data with missing-wedge prob-

lem and sparse sampling problem27. Recent advances of

iterative algorithms, such as model based iterative recon-

struction (MBIR)28,29, incorporate with physical knowl-

edge and geometrical information of the sample as prior

knowledge and can reconstruct the three components si-

multaneously. But iteratively minimizing a cost function

has to pay a price of eight times longer run-time com-

pared to conventional analytical methods29.

With the development of machine learning techniques,

deep learning tomography (DLT) is emergent as the third

generation reconstruction algorithm. Model with Unet30

architecture has already shown its capability in remov-

ing artefacts in limited-angle tomography31. Instead of

building an end-to-end DLT algorithm, combining con-

arXiv:2210.03066v1 [cond-mat.mtrl-sci] 6 Oct 2022

FIG. 1: Workﬂow of MagNet.

(a) Limited-angle phase shifts φof xand ytilt series. Gray images indicate phase shifts within missing wedge. (b)

Conventional VFET reconstructed Bin. (c) MagNet enhanced Bout. (d) The neural network architecture of MagNet. Every

convolutional layer is labeled with it’s output channel number.

ventional reconstruction with deep learning is an alter-

native approach to improve the reconstruction results32.

In this article, we are focusing on solving the missing-

wedge problem in VFET by a DLT algorithm. By at-

taching an Unet architecture machine learning model to

conventional VFET, we build a data-driven DLT algo-

rithm that can work end-to-end from phase shifts to B.

We will start section II with the theoretical background

and dataﬂow of our reconstruction model followed by a

description of our MagNet architecture. Details about

training and testing samples generation as well as the

creation of 3D magnetic textures library will also be dis-

cussed in this section. In section III, reconstruction re-

sults will be shown with comparison to the conventional

method. Model performance at diﬀerent missing-wedge

conditions are also discussed there. Conclusions and out-

look will be discussed in section IV.

II. THEORETICAL BACKGROUND,

NETWORK ARCHITECTURE, DATA LIBRARY

AND MODEL TRAINING

Our MagNet framework is shown in Figure 1. Limited

angle phase shifts ﬁrst enter a VFET module. At the

presence of missing wedge, the VFET module gives a

defective reconstructed magnetic induction ﬁeld noted as

Bin.Bin is then fed to the Unet module. Unet module

outputs an enhanced reconstruction result noted as Bout.

In the VFET module, the projected components of the

magnetic induction are the gradient of the phase along

orthogonal directions26,

∂yφ(x, y)=−e

hBx(x, y, z)dz

∂xφ(x, y)=e

hBy(x, y, z)dz

(1)

where φ(x, y)=−e

h∫Az(x, y, z)dz is the phase shift, eis

electron charge, and his the Planck constant. Because

Bxcomponent is invariant under the rotation about x-

axis and Bycomponent is invariant under the rotation

about y-axis, the reconstruction of Bxand Bycan be

simpliﬁed as two scalar tomography. In this paper, this

tomography is achieved by a simple k-space bilinear in-

terpolation33. And the third component is calculated by

the constraint ∇ ⋅ B=0. The Unet module inherits the

3D-Unet skeleton34. Convolution blocks are used to ex-

tract features. The unpool layer is concatenated with

skip layer to return the same dimension as the input.

Details of our Unet architecture are shown in Figure 1

(e).

The geometry of magnetic textures are set as cylin-

ders with radius of 40 pixels and thickness varying from

10 to 80 pixels. The majority of magnetic structures in

our library are generated from micromagnetic software

JuMag.jl35 by setting various simulating parameters and

initial states. Magnetic textures are then manually se-

lected from simulation results to make sure that sample

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MagNet:machinelearningenhancedthree-dimensionalmagneticreconstructionBoyaoLyu,1,2,˜ShihuaZhao,3,4,5,˜YiboZhang,3,6WeiweiWang,7HaifengDu,1andJiadongZang3,8,1AnhuiProvinceKeyLaboratoryofCondensedMatterPhysicsatExtremeConditions,HighMagneticFieldLaboratoryofChineseAcademyofSciences,Hefei,230031,China2...

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MagNet machine learning enhanced three-dimensional magnetic reconstruction Boyao Lyu1 2Shihua Zhao3 4 5Yibo Zhang3 6Weiwei Wang7Haifeng Du1and Jiadong Zang3 8 1Anhui Province Key Laboratory of Condensed Matter Physics at Extreme Conditions.pdf

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MagNet machine learning enhanced three-dimensional magnetic reconstruction Boyao Lyu1 2Shihua Zhao3 4 5Yibo Zhang3 6Weiwei Wang7Haifeng Du1and Jiadong Zang3 8 1Anhui Province Key Laboratory of Condensed Matter Physics at Extreme Conditions

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