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Modeling and Characterization of Materials with Unique Magnetic, Electric and Mechanical Properties (Second Volume)

A special issue of Materials (ISSN 1996-1944). This special issue belongs to the section "Electronic Materials".

Deadline for manuscript submissions: 20 July 2025 | Viewed by 7334

Special Issue Editor


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Guest Editor
Department of Solid State Physics, Institute of Physics, University of Silesia, 75-Pułku Piechoty 1A, 40-500 Chorzów, Poland
Interests: magnetic and related properties of amorphous, nanocrystalline, and composite materials; solid state physics
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Modern materials are a source of technology progress in many areas. There are well known applications in electronics, energetics (including the green technologies), motor and car industry.  For example, modern soft magnetic materials as a core of transformers or electric motors cause a significant increase of their energetic efficiency, saving energy in a global meaning. On the other hand, the hard magnetic materials are widely used in computer and energetic technologies as data storage media and high-efficiency electric generators, respectively. Between this two groups, one can observe a family of magnetic materials with excellent properties for broad application spectra (sensors, actuators, energy harvesting devices, magnetic refrigerators etc.). Other interesting property is electric transport in solids giving new conducting and semiconducting materials for modern electronic and computing machines. In this aspect, it is worth to mention topological insulators which sims to be a future for high efficiency electronic devices.

The progress in magnetic materials, not restricted to the mention above groups, would not be possible without basic science including technology, characterization and modeling in the atomic as well as large scale level. Such researches are useful for designing new systems with unique properties required for different applications.

The special issue entitled “Modeling and Characterization of Materials with Unique Magnetic, Electric and Mechanical Properties (Second Volume)” refers to reviews and/or original research papers in the broad area of materials characterized by some unique features, leading to new application spectra.

It is my pleasure to invite you to submit a manuscript for this Special Issue. Full papers, communications, and reviews are welcome.

Prof. Dr. Artur Chrobak
Guest Editor

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Keywords

  • magnetism
  • magnetic materials
  • electric transport
  • electronic structure
  • intermetallic compounds
  • mechanical properties
  • materials simulations

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Published Papers (5 papers)

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Research

14 pages, 7106 KiB  
Article
Numerical Investigation and Device Architecture Optimization of Sb2Se3 Thin-Film Solar Cells Using SCAPS-1D
by Chung-Kuan Lai and Yi-Cheng Lin
Materials 2024, 17(24), 6203; https://doi.org/10.3390/ma17246203 - 19 Dec 2024
Viewed by 213
Abstract
Antimony selenide (Sb2Se3) shows promise for photovoltaics due to its favorable properties and low toxicity. However, current Sb2Se3 solar cells exhibit efficiencies significantly below their theoretical limits, primarily due to interface recombination and non-optimal device architectures. [...] Read more.
Antimony selenide (Sb2Se3) shows promise for photovoltaics due to its favorable properties and low toxicity. However, current Sb2Se3 solar cells exhibit efficiencies significantly below their theoretical limits, primarily due to interface recombination and non-optimal device architectures. This study presents a comprehensive numerical investigation of Sb2Se3 thin-film solar cells using SCAPS-1D simulation software, focusing on device architecture optimization and interface engineering. We systematically analyzed device configurations (substrate and superstrate), hole-transport layer (HTL) materials (including NiOx, CZTS, Cu2O, CuO, CuI, CuSCN, CZ-TA, and Spiro-OMeTAD), layer thicknesses, carrier densities, and resistance effects. The substrate configuration with molybdenum back contact demonstrated superior performance compared with the superstrate design, primarily due to favorable energy band alignment at the Mo/Sb2Se3 interface. Among the investigated HTL materials, Cu2O exhibited optimal performance with minimal valence-band offset, achieving maximum efficiency at 0.06 μm thickness. Device optimization revealed critical parameters: series resistance should be minimized to 0–5 Ω-cm2 while maintaining shunt resistance above 2000 Ω-cm2. The optimized Mo/Cu2O(0.06 μm)/Sb2Se3/CdS/i-ZnO/ITO/Al structure achieved a remarkable power conversion efficiency (PCE) of 21.68%, representing a significant improvement from 14.23% in conventional cells without HTL. This study provides crucial insights for the practical development of high-efficiency Sb2Se3 solar cells, demonstrating the significant impact of device architecture optimization and interface engineering on overall performance. Full article
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Figure 1
<p>Schematic diagram of the proposed solar-cell structure: (<b>a</b>) p-n substrate configuration, (<b>b</b>) p-n superstrate configuration, (<b>c</b>) n-p-p<sup>+</sup> substrate configuration.</p>
Full article ">Figure 2
<p>Energy band diagrams of different device configurations: (<b>a</b>) substrate and (<b>b</b>) superstrate structures.</p>
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<p>Performance characteristics of different device configurations: (<b>a</b>) current–voltage curves and (<b>b</b>) external quantum efficiency spectra.</p>
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<p>Energy band diagrams of various HTL materials in the device structure.</p>
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<p>PCE comparison of different HTL materials.</p>
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<p>Relationship between Cu<sub>2</sub>O HTL thickness and device performance.</p>
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<p>Relationship between Cu<sub>2</sub>O HTL shallow acceptor density and device performance.</p>
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<p>Effect of shallow acceptor density on Sb<sub>2</sub>Se<sub>3</sub> solar-cell efficiency at different Cu<sub>2</sub>O HTL thicknesses.</p>
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<p>Numerical analysis of series and parallel resistance on device performance. (<b>a</b>) Open-circuit voltage (Voc) variation, (<b>b</b>) Short-circuit current density (Jsc) response, (<b>c</b>) Fill Factor (FF) dependence, and (<b>d</b>) Device efficiency changes with respect to series and parallel resistance.</p>
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13 pages, 572 KiB  
Article
High Spin Magnetic Moments in All-3d-Metallic Co-Based Full Heusler Compounds
by Murat Tas, Kemal Özdoğan, Ersoy Şaşıoğlu and Iosif Galanakis
Materials 2023, 16(24), 7543; https://doi.org/10.3390/ma16247543 - 7 Dec 2023
Cited by 2 | Viewed by 1321
Abstract
We conduct ab-initio electronic structure calculations to explore a novel category of magnetic Heusler compounds, comprising solely 3d transition metal atoms and characterized by high spin magnetic moments. Specifically, we focus on Co2YZ Heusler compounds, where Y and Z [...] Read more.
We conduct ab-initio electronic structure calculations to explore a novel category of magnetic Heusler compounds, comprising solely 3d transition metal atoms and characterized by high spin magnetic moments. Specifically, we focus on Co2YZ Heusler compounds, where Y and Z represent transition metal atoms such that the order of the valence is Co > Y > Z. We show that these compounds exhibit a distinctive region of very low density of minority-spin states at the Fermi level when crystallizing in the L21 lattice structure. The existence of this pseudogap leads most of the studied compounds to a Slater–Pauling-type behavior of their total spin magnetic moment. Co2FeMn is the compound that presents the largest total spin magnetic moment in the unit cell reaching a very large value of 9 μB. Our findings suggest that these compounds are exceptionally promising materials for applications in the realms of spintronics and magnetoelectronics. Full article
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Figure 1
<p>Schematic representation of the <math display="inline"><semantics> <mrow> <mi>L</mi> <msub> <mn>2</mn> <mn>1</mn> </msub> </mrow> </semantics></math> structure adopted by the full-Heusler compounds and the <math display="inline"><semantics> <mrow> <mi>X</mi> <mi>A</mi> </mrow> </semantics></math> structure adopted by the inverse Heusler compounds, The black spheres, pink spheres, yellow squares, and green spheres are widely called A, B, C, and D sites, respectively. The large cube in the figure contains exactly four primitive unit cells. On the right, the nearest and next-nearest neighbors are depicted.</p>
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<p>Total density of states (DOS) for all studied compounds for both the <math display="inline"><semantics> <mrow> <mi>L</mi> <msub> <mn>2</mn> <mn>1</mn> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>X</mi> <mi>A</mi> </mrow> </semantics></math> lattice structures. Positive (negative) DOS values correspond to the majority (minority)-spin electronic band structure.</p>
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<p>Atom-resolved density of states (DOS) for the Co<math display="inline"><semantics> <msub> <mrow/> <mn>2</mn> </msub> </semantics></math>MnZ compounds. Details as in <a href="#materials-16-07543-f002" class="html-fig">Figure 2</a>.</p>
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<p>Atom-resolved DOS for the Co<math display="inline"><semantics> <mrow> <msub> <mrow/> <mn>2</mn> </msub> <mi>Y</mi> </mrow> </semantics></math>Sc compounds. Details as in <a href="#materials-16-07543-f002" class="html-fig">Figure 2</a>.</p>
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<p>Total DOS for the Co<math display="inline"><semantics> <msub> <mrow/> <mn>2</mn> </msub> </semantics></math>Fe(Ti, V, Cr) compounds calculated at the equilibrium lattice constant and for a lattice constant expanded by 10%. Details as in <a href="#materials-16-07543-f002" class="html-fig">Figure 2</a>.</p>
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<p>Ab-initio calculated total spin magnetic moments, <math display="inline"><semantics> <msub> <mi>M</mi> <mi>t</mi> </msub> </semantics></math>, in <math display="inline"><semantics> <msub> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">B</mi> </msub> </semantics></math> as a function of the total number of valence electrons, <math display="inline"><semantics> <msub> <mi>Z</mi> <mi>t</mi> </msub> </semantics></math>, in the primitive unit cell for the studied compounds at their equilibrium lattice constant assuming the <math display="inline"><semantics> <mrow> <mi>L</mi> <msub> <mn>2</mn> <mn>1</mn> </msub> </mrow> </semantics></math> lattice structure (red spheres). With blue empty spheres, we present the results for the three compounds studied at a lattice constant expanded by 10% (see <a href="#materials-16-07543-t001" class="html-table">Table 1</a>). The dashed line represents the <math display="inline"><semantics> <mrow> <msub> <mi>M</mi> <mi>t</mi> </msub> <mo>=</mo> <msub> <mi>Z</mi> <mi>t</mi> </msub> <mo>−</mo> <mn>24</mn> </mrow> </semantics></math> Slater–Pauling rule.</p>
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<p>Schematic representation of the character of the bands at the <math display="inline"><semantics> <mi mathvariant="sans-serif">Γ</mi> </semantics></math> point in the minority-spin band structure of Co<math display="inline"><semantics> <msub> <mrow/> <mn>2</mn> </msub> </semantics></math>YZ compounds when crystallizing in the <math display="inline"><semantics> <mrow> <mi>L</mi> <msub> <mn>2</mn> <mn>1</mn> </msub> </mrow> </semantics></math> lattice structure (see text for details). Note that below the Fermi level, there are 12 orbitals.</p>
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11 pages, 5754 KiB  
Article
Modeling of Severe Plastic Deformation by HSHPT of As-Cast Ti-Nb-Zr-Ta-Fe-O Gum Alloy for Orthopedic Implant
by Dan Cătălin Bîrsan, Carmela Gurău, Florin-Bogdan Marin, Cristian Stefănescu and Gheorghe Gurău
Materials 2023, 16(8), 3188; https://doi.org/10.3390/ma16083188 - 18 Apr 2023
Viewed by 1247
Abstract
The High Speed High Pressure Torsion (HSHPT) is the severe plastic deformation method (SPD) designed for the grain refinement of hard-to-deform alloys, and it is able to produce large, rotationally complex shells. In this paper, the new bulk nanostructured Ti-Nb-Zr-Ta-Fe-O Gum metal was [...] Read more.
The High Speed High Pressure Torsion (HSHPT) is the severe plastic deformation method (SPD) designed for the grain refinement of hard-to-deform alloys, and it is able to produce large, rotationally complex shells. In this paper, the new bulk nanostructured Ti-Nb-Zr-Ta-Fe-O Gum metal was investigated using HSHPT. The biomaterial in the as-cast state was simultaneously compressed up to 1 GPa and torsion was applied with friction at a temperature that rose as a pulse in less than 15 s. The interaction between the compression, the torsion, and the intense friction that generates heat requires accurate 3D finite element simulation. Simufact Forming was employed to simulate severe plastic deformation of a shell blank for orthopedic implants using the advancing Patran Tetra elements and adaptable global meshing. The simulation was conducted by applying to the lower anvil a displacement of 4.2 mm in the z-direction and applying a rotational speed of 900 rpm to the upper anvil. The calculations show that the HSHPT accumulated a large plastic deformation strain in a very short time, leading to the desired shape and grain refinement. Full article
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Figure 1
<p>Processing of the blanks by High Speed High Pressure Torsion. (<b>a</b>,<b>b</b>) HSHPT scheme, (<b>c</b>,<b>d</b>) compression staples blank, (<b>e</b>) HSHPT tools, (<b>f</b>) HSHPT experimental setup.</p>
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<p>Simulation bodies: (<b>a</b>) upper punch geometry, (<b>b</b>) 3D upper punch, (<b>c</b>) active part of upper punch, (<b>d</b>) sample, (<b>e</b>) lower punch.</p>
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<p>Flow stress for different temperatures.</p>
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<p>Force variation in the HSHPT process. (a) FEA, (b) experimental for disk-shaped workpiece.</p>
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<p>HSHPT experimental parameters.</p>
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<p>Pressure field: (<b>a</b>) at 2.6 s from the SPD beginning and (<b>b</b>) at 8 s from the SPD beginning.</p>
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<p>Temperature profile at two moments in time, 6 s (<b>a</b>) and 8 s (<b>b</b>) from the SPD beginning.</p>
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<p>Von Mises stress distribution: (<b>a</b>) on the upper surface of the workpiece at the time interval between 0 and 1 s, (<b>b</b>) on the upper surface of the workpiece between 1 s and 8 s of the HSHPT process, (<b>c</b>) Von Mises stress versus time in the points located on the upper surface of the workpiece, (<b>d</b>) Von Mises stresses at two moments of time, 6 s and 8 s.</p>
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<p>HSHPT strain.</p>
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10 pages, 6071 KiB  
Article
Magnetic Anisotropies and Exchange Bias of Co/CoO Multilayers with Intermediate Ultrathin Pt Layers
by Dimitrios I. Anyfantis, Camillo Ballani, Nikos Kanistras, Alexandros Barnasas, Ioannis Tsiaoussis, Georg Schmidt, Evangelos Th. Papaioannou and Panagiotis Poulopoulos
Materials 2023, 16(4), 1378; https://doi.org/10.3390/ma16041378 - 7 Feb 2023
Cited by 2 | Viewed by 1933
Abstract
Co/CoO multilayers are fabricated by means of radio-frequency magnetron sputtering. For the formation of each multilayer period, a Co layer is initially produced followed by natural oxidation. Platinum is used not only as buffer and capping layers, but also in the form of [...] Read more.
Co/CoO multilayers are fabricated by means of radio-frequency magnetron sputtering. For the formation of each multilayer period, a Co layer is initially produced followed by natural oxidation. Platinum is used not only as buffer and capping layers, but also in the form of intermediate ultrathin layers to enhance perpendicular magnetic anisotropy. Three samples are compared with respect to the magnetic anisotropies and exchange bias between 4–300 K based on superconducting quantum interference device magnetometry measurements. Two of the multilayers are identical Co/CoO/Pt ones; one of them, however, is grown on a Co/Pt “magnetic substrate” to induce perpendicular magnetic anisotropy via exchange coupling through an ultrathin Pt intermediate layer. The third multilayer is of the form Co/CoO/Co/Pt. The use of a “magnetic substrate” results in the observation of loops with large remanence when the field applies perpendicular to the film plane. The CoO/Co interfaces lead to a significant exchange bias at low temperatures after field cooling. The largest exchange bias was observed in the film with double Co/CoO/Co interfaces. Consequently, significant perpendicular anisotropy coexists with large exchange bias, especially at low temperatures. Such samples can be potentially useful for applications related to spintronics and magnetic storage. Full article
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Figure 1

Figure 1
<p>XRR patterns (up) and film drawings (down) for three Co/CoO/Pt multilayers with different formation and repetitions. The fitted patterns using the GenX code are also shown (continuous thin lines). The patterns of samples B and C have been vertically shifted for clarity of presentation.</p>
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<p>(<b>a</b>) XTEM image of sample A. One may see the Si substrate, native oxide, Pt buffer layer, “magnetic substrate”, Co/CoO/Pt multilayer, and Pt capping layer. (<b>b</b>) A magnified part of (<b>a</b>). (<b>c</b>) Electron diffraction pattern.</p>
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<p>Hysteresis curves of sample A for both in-plane and out-of-plane geometries recorded by SQUID at (<b>a</b>) 300 K, (<b>b</b>) 200 K, (<b>c</b>) 100 K, and (<b>d</b>) 4 K.</p>
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<p>Temperature-dependent magnetization hysteresis loops with the external field applied perpendicular (red circles) and parallel (black squares) to the film plane for sample B.</p>
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<p>Temperature-dependent magnetization hysteresis loops with the external field applied perpendicular (red circles) and parallel (black circles) to the film plane for sample C.</p>
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<p>Temperature-dependent coercivity and exchange bias field for all samples measured with the external field parallel and perpendicular to the film plane. Lines serve as guides to the eye.</p>
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<p>Temperature-dependent normalized saturation magnetizations Ms and remanences Mr along the in-plane (i) or out-of-plane (o) directions. Lines serve as guides to the eye.</p>
Full article ">
18 pages, 4408 KiB  
Article
Solid-State Dewetting as a Driving Force for Structural Transformation and Magnetization Reversal Mechanism in FePd Thin Films
by Arkadiusz Zarzycki, Marcin Perzanowski, Michal Krupinski and Marta Marszalek
Materials 2023, 16(1), 92; https://doi.org/10.3390/ma16010092 - 22 Dec 2022
Cited by 4 | Viewed by 1962
Abstract
In this work, the process of solid-state dewetting in FePd thin films and its influence on structural transformation and magnetic properties is presented. The morphology, structure and magnetic properties of the FePd system subjected to annealing at 600 °C for different times were [...] Read more.
In this work, the process of solid-state dewetting in FePd thin films and its influence on structural transformation and magnetic properties is presented. The morphology, structure and magnetic properties of the FePd system subjected to annealing at 600 °C for different times were studied. The analysis showed a strong correlation between the dewetting process and various physical phenomena. In particular, the transition between the A1 phase and L10 phase is strongly influenced by and inextricably connected with solid-state dewetting. Major changes were observed when the film lost its continuity, including a fast growth of the L10 phase, changes in the magnetization reversal behavior or the induction of magnetic spring-like behavior. Full article
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Figure 1
<p>(<bold>a</bold>) Diffractograms of FePd thin alloy films after different times of annealing. The blue and red colors correspond to the A1 phase and the L1<sub>0</sub> phase, respectively. The parameters extracted from the XRD patterns are presented below: (<bold>b</bold>) cell parameters, (<bold>c</bold>) cell volume, and (<bold>d</bold>) coherence length. The inset of figure (<bold>c</bold>) shows the tetragonal distortion ratio c/a.</p>
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<p>SEM studies of morphology evolution as a function of annealing time for the FePd alloy thin film.</p>
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<p>(<bold>a</bold>) Histogram of particle sizes from <inline-formula><mml:math id="mm200"><mml:semantics><mml:mrow><mml:mi>SEM</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> microscopy fitted with gamma distribution, different colors correspond to different times of annealing; (<bold>b</bold>) particle size mean values; (<bold>c</bold>) <inline-formula><mml:math id="mm39"><mml:semantics><mml:mrow><mml:mi>SEM</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> image at the void edge of the sample annealed for 4 min showing successive changes in film morphology. The orange line on the right side of the image shows the variation in the grayscale of the <inline-formula><mml:math id="mm40"><mml:semantics><mml:mrow><mml:mi>SEM</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> signal, where <inline-formula><mml:math id="mm41"><mml:semantics><mml:mn>0</mml:mn></mml:semantics></mml:math></inline-formula> corresponds to the darkest regions, i.e., uncovered silica substrate, and <inline-formula><mml:math id="mm42"><mml:semantics><mml:mn>1</mml:mn></mml:semantics></mml:math></inline-formula> corresponds to the regions of unchanged FePd thin film.</p>
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<p>Magnetic hysteresis loops for annealed samples measured with the magnetic field perpendicular (black curves) and parallel (red curves) to the sample surface. Graphs (<bold>a</bold>–<bold>f</bold>) refer to different times of annealing indicated in the left top corner.</p>
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<p>(<bold>a</bold>) Coercive field <inline-formula><mml:math id="mm201"><mml:semantics><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mi>C</mml:mi></mml:msub></mml:mrow></mml:semantics></mml:math></inline-formula> vs. annealing time for in-plane and out-of-plane geometries and (<bold>b</bold>) angular dependence of the <inline-formula><mml:math id="mm68"><mml:semantics><mml:mrow><mml:msub><mml:mi>M</mml:mi><mml:mi>R</mml:mi></mml:msub><mml:mo>/</mml:mo><mml:msub><mml:mi>M</mml:mi><mml:mi>S</mml:mi></mml:msub></mml:mrow></mml:semantics></mml:math></inline-formula> ratio where different colors are for samples annealed with different time.</p>
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<p>Switching field (top) and <inline-formula><mml:math id="mm202"><mml:semantics><mml:mrow><mml:mi>M</mml:mi><mml:mfenced><mml:mi>H</mml:mi></mml:mfenced></mml:mrow></mml:semantics></mml:math></inline-formula> curves (bottom) for the sample annealed for <inline-formula><mml:math id="mm104"><mml:semantics><mml:mrow><mml:mn>60</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> min with the magnetic field applied at an angle of <inline-formula><mml:math id="mm105"><mml:semantics><mml:mrow><mml:mn>40</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> degrees. The red line is a fit with asymmetric Lorentz function.</p>
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<p>Angular dependence of switching field for samples annealed for different times with solid lines corresponding to fitted functions of (<bold>a</bold>) the modified Kondorsky model and (<bold>b</bold>) the M-type model of the multidomain ferromagnet. (<bold>c</bold>) Critical angle of reversal magnetization mechanism <inline-formula><mml:math id="mm203"><mml:semantics><mml:mrow><mml:msubsup><mml:mi>φ</mml:mi><mml:mi>C</mml:mi><mml:mrow><mml:mi>M</mml:mi><mml:mo>−</mml:mo><mml:mi>t</mml:mi><mml:mi>y</mml:mi><mml:mi>p</mml:mi><mml:mi>e</mml:mi></mml:mrow></mml:msubsup></mml:mrow></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm114"><mml:semantics><mml:mrow><mml:msubsup><mml:mi>φ</mml:mi><mml:mi>C</mml:mi><mml:mrow><mml:mi>m</mml:mi><mml:mi>o</mml:mi><mml:mi>d</mml:mi><mml:mi>K</mml:mi></mml:mrow></mml:msubsup></mml:mrow></mml:semantics></mml:math></inline-formula>.</p>
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<p>(<bold>a</bold>) Ratio between the nucleation field of the inverse domain wall and the rotation field of the magnetization vector, <inline-formula><mml:math id="mm204"><mml:semantics><mml:mrow><mml:msubsup><mml:mi>H</mml:mi><mml:mn>0</mml:mn><mml:mrow><mml:mi>D</mml:mi><mml:mi>W</mml:mi><mml:mi>M</mml:mi></mml:mrow></mml:msubsup><mml:mo>/</mml:mo><mml:msubsup><mml:mi>H</mml:mi><mml:mn>0</mml:mn><mml:mrow><mml:mi>C</mml:mi><mml:mi>R</mml:mi></mml:mrow></mml:msubsup></mml:mrow></mml:semantics></mml:math></inline-formula>, in the modified Kondorsky model and ratio of demagnetization factors <inline-formula><mml:math id="mm138"><mml:semantics><mml:mi>N</mml:mi></mml:semantics></mml:math></inline-formula> in the M-type model as a function of annealing time. (<bold>b</bold>) Correlation of the <inline-formula><mml:math id="mm139"><mml:semantics><mml:mrow><mml:msubsup><mml:mi>H</mml:mi><mml:mn>0</mml:mn><mml:mrow><mml:mi>D</mml:mi><mml:mi>W</mml:mi><mml:mi>M</mml:mi></mml:mrow></mml:msubsup><mml:mo>/</mml:mo><mml:msubsup><mml:mi>H</mml:mi><mml:mn>0</mml:mn><mml:mrow><mml:mi>C</mml:mi><mml:mi>R</mml:mi></mml:mrow></mml:msubsup></mml:mrow></mml:semantics></mml:math></inline-formula> parameter with the critical angle <inline-formula><mml:math id="mm140"><mml:semantics><mml:mrow><mml:msubsup><mml:mi>φ</mml:mi><mml:mi>C</mml:mi><mml:mrow><mml:mi>m</mml:mi><mml:mi>o</mml:mi><mml:mi>d</mml:mi><mml:mi>K</mml:mi></mml:mrow></mml:msubsup></mml:mrow></mml:semantics></mml:math></inline-formula> in the modified Kondorsky model and the ratio of demagnetization factors <inline-formula><mml:math id="mm141"><mml:semantics><mml:mi>N</mml:mi></mml:semantics></mml:math></inline-formula> with <inline-formula><mml:math id="mm142"><mml:semantics><mml:mrow><mml:msubsup><mml:mi>φ</mml:mi><mml:mi>C</mml:mi><mml:mrow><mml:mi>M</mml:mi><mml:mo>−</mml:mo><mml:mi>t</mml:mi><mml:mi>y</mml:mi><mml:mi>p</mml:mi><mml:mi>e</mml:mi></mml:mrow></mml:msubsup></mml:mrow></mml:semantics></mml:math></inline-formula> in the M-type model.</p>
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<p>(<bold>a</bold>) Switching field distribution for samples treated with different times of annealing. The <inline-formula><mml:math id="mm205"><mml:semantics><mml:mrow><mml:msub><mml:mi>ω</mml:mi><mml:mi>L</mml:mi></mml:msub></mml:mrow></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm178"><mml:semantics><mml:mrow><mml:msub><mml:mi>ω</mml:mi><mml:mi>R</mml:mi></mml:msub></mml:mrow></mml:semantics></mml:math></inline-formula> are the widths of the left and right sides of the switching field distribution. (<bold>b</bold>) Difference <inline-formula><mml:math id="mm179"><mml:semantics><mml:mrow><mml:mi mathvariant="sans-serif">Δ</mml:mi><mml:mi>ω</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> between the right <inline-formula><mml:math id="mm180"><mml:semantics><mml:mrow><mml:msub><mml:mi>ω</mml:mi><mml:mi>R</mml:mi></mml:msub></mml:mrow></mml:semantics></mml:math></inline-formula> and left <inline-formula><mml:math id="mm181"><mml:semantics><mml:mrow><mml:msub><mml:mi>ω</mml:mi><mml:mi>L</mml:mi></mml:msub></mml:mrow></mml:semantics></mml:math></inline-formula> width of the switching field distribution at different stages of annealing. (<bold>c</bold>) Map of <inline-formula><mml:math id="mm182"><mml:semantics><mml:mrow><mml:mi mathvariant="sans-serif">Δ</mml:mi><mml:mi>ω</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> data with colors corresponding to the renormalized <inline-formula><mml:math id="mm183"><mml:semantics><mml:mrow><mml:mi mathvariant="sans-serif">Δ</mml:mi><mml:mi>ω</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> value. The dashed line marks angles where the maximum of the <inline-formula><mml:math id="mm184"><mml:semantics><mml:mrow><mml:mi mathvariant="sans-serif">Δ</mml:mi><mml:mi>ω</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> is observed. (<bold>d</bold>) Correlation of the critical angle of the reversal mechanism and the angle of maximal asymmetry in the <inline-formula><mml:math id="mm185"><mml:semantics><mml:mrow><mml:mi>SFD</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>. The colors on (<bold>a</bold>,<bold>b</bold>) refer to samples annealed for different times.</p>
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