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Functional Inorganic Materials and Devices

Collective Spin-Wave Dynamics in Gyroid Ferromagnetic Nanostructures
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  • Mateusz Gołębiewski*
    Mateusz Gołębiewski
    Institute of Spintronics and Quantum Information, Faculty of Physics, Adam Mickiewicz University, Uniwersytetu Poznańskiego 2, 61-614 Poznań, Poland
    *Email: [email protected]
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    Orcidhttps://orcid.org/0000-0002-1948-0652
  • Riccardo Hertel
    Riccardo Hertel
    Université de Strasbourg, CNRS, Institut de Physique et Chimie des Matériaux de Strasbourg, F-67000 Strasbourg, France
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  • Massimiliano d’Aquino
    Massimiliano d’Aquino
    Department of Electrical Engineering and ICT, University of Naples Federico II, 80125 Naples, Italy
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  • Vitaliy Vasyuchka
    Vitaliy Vasyuchka
    Fachbereich Physik und Landesforschungszentrum OPTIMAS, Rheinland-Pfälzische Technische Universität Kaiserslautern-Landau, Erwin-Schrödinger-Straße 56, 67663 Kaiserslautern, Germany
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  • Mathias Weiler
    Mathias Weiler
    Fachbereich Physik und Landesforschungszentrum OPTIMAS, Rheinland-Pfälzische Technische Universität Kaiserslautern-Landau, Erwin-Schrödinger-Straße 56, 67663 Kaiserslautern, Germany
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  • Philipp Pirro
    Philipp Pirro
    Fachbereich Physik und Landesforschungszentrum OPTIMAS, Rheinland-Pfälzische Technische Universität Kaiserslautern-Landau, Erwin-Schrödinger-Straße 56, 67663 Kaiserslautern, Germany
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  • Maciej Krawczyk
    Maciej Krawczyk
    Institute of Spintronics and Quantum Information, Faculty of Physics, Adam Mickiewicz University, Uniwersytetu Poznańskiego 2, 61-614 Poznań, Poland
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    Orcidhttps://orcid.org/0000-0002-0870-717X
  • Shunsuke Fukami
    Shunsuke Fukami
    Research Institute of Electrical Communication (RIEC), Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai-shi, Miyagi 980-8577, Japan
    Center for Science and Innovation in Spintronics (CSIS), Tohoku University, 980-8577 Sendai, Japan
    Center for Innovative Integrated Electronic Systems (CIES), Tohoku University, 468-1 Aramaki Aza Aoba, Aoba-ku, 980-0845 Sendai, Japan
    WPI Advanced Institute for Materials Research, Tohoku University, 2-1-1 Katahira, Aoba-ku, 980-8577 Sendai, Japan
    Inamori Research Institute for Science, 600-8411 Kyoto, Japan
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  • Hideo Ohno
    Hideo Ohno
    Research Institute of Electrical Communication (RIEC), Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai-shi, Miyagi 980-8577, Japan
    Center for Science and Innovation in Spintronics (CSIS), Tohoku University, 980-8577 Sendai, Japan
    Center for Innovative Integrated Electronic Systems (CIES), Tohoku University, 468-1 Aramaki Aza Aoba, Aoba-ku, 980-0845 Sendai, Japan
    WPI Advanced Institute for Materials Research, Tohoku University, 2-1-1 Katahira, Aoba-ku, 980-8577 Sendai, Japan
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  • Justin Llandro
    Justin Llandro
    Research Institute of Electrical Communication (RIEC), Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai-shi, Miyagi 980-8577, Japan
    Center for Science and Innovation in Spintronics (CSIS), Tohoku University, 980-8577 Sendai, Japan
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ACS Applied Materials & Interfaces

Cite this: ACS Appl. Mater. Interfaces 2024, 16, 17, 22177–22188
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https://doi.org/10.1021/acsami.4c02366
Published April 22, 2024

Copyright © 2024 The Authors. Published by American Chemical Society. This publication is licensed under

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Abstract

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Expanding upon the burgeoning discipline of magnonics, this research elucidates the intricate dynamics of spin waves (SWs) within three-dimensional nanoenvironments. It marks a shift from traditionally used planar systems to exploration of magnetization configurations and the resulting dynamics within 3D nanostructures. This study deploys micromagnetic simulations alongside ferromagnetic resonance measurements to scrutinize magnetic gyroids, periodic chiral configurations composed of chiral triple junctions with a period in nanoscale. Our findings uncover distinctive attributes intrinsic to the gyroid network, most notably the localization of collective SW excitations and the sensitivity of the gyroid’s ferromagnetic response to the orientation of the static magnetic field, a correlation closely tied to the crystallographic alignment of the structure. Furthermore, we show that for the ferromagnetic resonance, multidomain gyroid films can be treated as a magnonic material with effective magnetization scaled by its filling factor. The implications of our research carry the potential for practical uses such as an effective, metamaterial-like substitute for ferromagnetic parts and lay the groundwork for radio frequency filters. The growing areas of 3D magnonics and spintronics present exciting opportunities to investigate and utilize gyroid nanostructures for signal processing purposes.

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1. Introduction

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Spin waves (SWs) and their intricate manipulation in magnetic materials constitute a significant part of the contemporary research. In ferromagnetic systems, the complex dynamics of SWs arises from the coexistence of magnetostatic and exchange interactions. The magnetostatic interactions, being highly anisotropic in thin structures, induce a profound dependence of SW properties on the relative orientation of magnetization and wavevector. This gives them a number of properties that are conspicuously absent in other types of waves, including negative group velocity, caustics, readily accessible nonlinearity, and dynamic reconfigurability control. (1) The potential applications are myriad, ranging from signal propagation without Joule–Lenz heat dissipation, (2,3) to the tunability of dispersion and group velocity. (4,5) The construction of magnonic systems, known for their enhanced efficiency, further emphasizes the unique properties of SWs and provides a compelling rationale for their extensive exploration. (2,6,7) Thus, the focus of current research is to harness and adapt magnetization dynamics for sophisticated industrial applications, a goal underscored by recent advances and roadmaps. (8,9)
Nanostructured 3D networks may give rise to topological and geometrical effects and emergent material properties, offering new possibilities for SW manipulation. (10−12) A fully interconnected 3D system opens up a new degree of freedom for novel phenomena, allowing interactions and collective effects in all three dimensions. (13−15) In recent years, there has been significant development of fabrication techniques such as two-photon lithography, focused electron beam deposition, and block copolymer templating, which now allow the fabrication and measurement of complex 3D structures on the nanometer scale. (16−19)
This paper analyzes a promising yet hardly explored structure in magnetism called a gyroid, which was discovered and first presented in 1970. (20) It is defined by chiral triple junctions and periodicity in all three spatial directions, classified as I4132 space group (no. 214). (21) In recent years, many studies have been published describing gyroids in the field of photonics, where they have been presented as potential chiral beamsplitters, (22) nonlinear optical metamaterials, (23,24) or photonic crystals. (25−27) It has also inspired many research groups into the fabrication of artificial systems based on this geometry. (22−24,28−34) The 3D structural unit of magnetic gyroids is in nanoscale and has both chirality and curvature, which has been shown to be highly effective in controlling noncollinear spin textures. (35) Recently, numerical and experimental visualization of the magnetic structure in a single and double Ni75Fe25 gyroid network has also been performed. (16) The effective Dzyaloshinskii–Moriya interaction and the curvature-related anisotropy are further important research topics in this context. (36−39) Curved magnetic wires and films also exhibit novel physical effects, (40) which together with chiral and topological properties, (41,42) open new avenues for future research.
SWs have been extensively researched in 2D structures; however, there are limited studies on artificial ferromagnetic systems in full 3D (43,44) and a lack of experimental research on the collective dynamics of SW in 3D nanostructures. Gyroids, due to their unique geometry and dimensions comparable to the exchange length, bear multidimensional properties of interactions with SWs through shape anisotropy, an inhomogeneous demagnetization field, and chirality. They are an intriguing candidate for the realization of artificial 3D magnonic crystals with highly coupled geometric and magnetic states. (12,45−47) Furthermore, gyroids, with a significant number of energetically equivalent stable states, may have the potential for artificial spin ice systems and active elements in unconventional computing architectures. (48)
Approaching these structures from a magnonics perspective, we employ micromagnetic simulations to delve into the collective SW dynamics. Our investigation scrutinizes the alignment of the external magnetic field in relation to the gyroid’s crystallographic axes, offering an insightful perspective on the impact of their geometry on shaping collective magnetization dynamics. To complement our simulation-based findings, we performed experimental broadband ferromagnetic resonance (BBFMR) measurements on the nickel (Ni) gyroid structure. This empirical approach enhances our numerical observations, specifically in terms of the frequencies and intensities of resonances excited at a given external magnetic field and the impact of various crystallographic domains on the half-width of the signals measured in the sample. Furthermore, the measured magnetic field dependencies suggest that the gyroid structure exhibits magnonic metamaterial-like properties. This combination of simulation and empirical experimentation fosters a comprehensive understanding of the multifaceted dynamics within these unique 3D nanostructures, highlighting their potential applications and motivating further research.

2. Geometry and Material Parameters

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The gyroidal surface was given for the first time using conjugate surface construction, (49) and in ref (50) its embedding was subsequently proved. The volume fractions of minimal and constant mean curvature gyroids have been further investigated numerically (51) with the constant mean curvature variants of the geometry.
In other fields, the gyroid is referred to as Laves’ graph of girth ten (52) and the K4 crystal. (53) It consists of cubic unit cells composed of triple bonds connected by nanorods with elliptical cross sections for the nonzero volume filling fraction ϕ, the range of which is described in ref (24). For ϕ = 0% (see Figure 1), a gyroid surface divides space into two labyrinths of paths oriented in opposite directions. The empty, unobstructed channels pass through the gyroid labyrinths in directions [100] and [111], and the paths emerge at 70.5° angles to any given channel as it passes through. Circling or gyrating down the channel in this way gives rise to the term “gyroid”. Interestingly, gyroids exist in several Schwarz surface families that preserve different symmetries of triply periodic minimal surfaces and, like many others, can be approximated by a trigonometrical equation
sin(2πx/L)cos(2πy/L)+sin(2πy/L)cos(2πz/L)+sin(2πz/L)cos(2πx/L)≤(101.5−2ϕ)/68.1
(1)
where L is the gyroid’s unit cell length.

Figure 1

Figure 1. Representation of the gyroid surface model (ϕ = 0%), highlighting the unit cell structure. The illustration accentuates the unit cell configuration along the crystallographic [111] direction in orthographic projection, unveiling the hexagonal organization inherent in the gyroid’s channels.

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The studied gyroid nanostructure was constructed by solvent vapor annealing, selective dissolution, and electrodeposition of a block copolymer template, following the protocol described in ref (16). The gyroidal structure was fabricated by applying polyisoprene-b-polystyrene-b-poly(ethylene oxide) (ISO) triblock terpolymers with block volume fractions of ϕPI = 0.30, ϕPS = 0.53, and ϕPEO = 0.17 to fluorine-doped tin oxide (FTO)-coated glass substrates. Initial cleaning with Piranha solution at 80 °C for 15 min prepared the substrates, which were then treated with octyltrichlorosilane. A 1 μm BCP film was spin-coated from a 10 wt % anhydrous anisole solution and formed into the gyroid morphology by solvent vapor annealing under nitrogen saturated with chloroform vapor at 26 °C. After annealing, the PI block was removed by UV exposure and immersion in ethanol to form gyroid polymer templates. Ni electrodeposition from a commercial solution followed in a three-electrode cell, filling the voided single-gyroid network with Ni under a constant potential at −1.05 V while monitoring the deposition charge. This method efficiently produces gyroid structures with precise Ni insertion. The material parameters used in the micromagnetic simulations represent those of Ni utilized in the fabrication of the sample, i.e., saturation magnetization Ms = 480 kA/m, exchange stiffness Aex = 8.6 pJ/m, and g-factor equal to 2.14. (54) In each relaxation simulation, the Gilbert damping was set to a large value of α = 0.5 to obtain fast convergence to the equilibrium state. Then, for all frequency-domain simulations, this parameter was set to a low value of 0.01.
The unit cell of the investigated gyroid sample measures 50 nm in each direction and has a volume fraction (ϕ) of approximately 10%, which here corresponds to a cross-sectional radius of a single gyroid node of about 4.1 nm (where it is oval in shape) and an arm length of about 19 nm. The geometric parameters of the gyroid unit cell are described in Figure 2a and are the same for all models analyzed in this work. In the experimental portion of this investigation, the gyroid sample comprises 12 unit cells of height, resulting in an overall thickness of 600 nm. Consequently, individual strut diameters are on the order of single nanometers, mirroring intrinsic magnetic length scales, including exchange length, domain wall width, and SW wavelength.

Figure 2

Figure 2. Depiction of the gyroid unit cell and geometric modeling for micromagnetic simulations. The focus of our investigation is a cubic gyroid unit cell measuring L = 50 nm, featuring a volume fraction ϕ = 10% as depicted in (a). For the purpose of micromagnetic simulations, a geometric model consisting of an aggregate 4 × 4 × 4 unit cells, or equivalently 200 × 200 × 200 [nm], was employed, as displayed in (b). Illustrated alongside are the three principal high-symmetry directions of the gyroid structure ([111]─yellow triangle, [110]─green rectangle, and [100]─red square) which are color-coordinated to match the planes intersecting the structure. Each direction reveals the unique distribution and shape of the gyroid’s channels: the [111] direction displays a hexagonal pattern and round holes, the [110] direction exhibits a square pattern and lenticular holes, while the [100] direction showcases a square pattern and round holes.

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3. Micromagnetic Simulations

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To unravel the magnetic phenomena transpiring within nanoscale gyroidal struts, we perform comprehensive micromagnetic simulations of the system, taking into account dipole and exchange interactions. In this endeavor, we harness the capabilities of our GPU-accelerated open-source finite-element (FEM) micromagnetic solver tetmag. (55) A remarkable feature of tetmag is its proficiency at resolving magnetostatic open boundary problems in large-scale micromagnetic simulations via a hybrid finite-element/boundary-element (FEM-BEM) formalism. (56) Notably, we forego the assumption of periodic boundary conditions in all of the simulations presented herein.
All of the following micromagnetic simulations are performed in two steps, where the first step is to calculate the stable (relaxed) magnetic configuration at a given field strength (see, e.g., Figure 3a,b). After the magnetization relaxation, we performed simulations of the ferromagnetic resonance using a dedicated frequency domain algorithm (57) based on a formulation proposed by d’Aquino et al. in ref (58).

Figure 3

Figure 3. Images of the postrelaxation static configurations of the reduced magnetization component mx(≡Mx/Ms) parallel to the external magnetic field, aligned in the crystallographic direction [100]. Panel (a) displays the case when Bext = 100 mT. The deviation of the magnetization vector from the x-direction is notably larger compared to (b) with Bext = 300 mT, where the magnetization distribution is nearly uniformly oriented along the direction of the external magnetic field. (c) Histogram (number of numerically calculated magnetic moments falling within specific ranges of mx) of the reduced magnetization component distribution mx in the simulated gyroid model for both magnetic field magnitudes. The number of bins indicates the number of elementary simulation elements (tetrahedrons).

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The low-amplitude alternating, and homogeneous in space, magnetic field applied in the frequency domain simulations generates a response of the magnetic system in the form of stationary magnetization oscillations with the frequency of the applied field. The magnetic susceptibility χ(ω) describes the frequency-dependent relation between the externally applied oscillating field δH and the dynamic component of magnetization δM. For a more detailed discussion of the dynamic susceptibility and its definition, see ref (57).
In the linear-response theory, the imaginary component of the susceptibility is related to dissipative processes in which the sample absorbs energy from the applied field. These absorption peaks denote resonances and generally coincide with frequencies of the maximum oscillation amplitude of the dynamic magnetization. For the case of a spatially homogeneous harmonic magnetic field applied in the simulations, we can thus analyze the spatial distribution of the magnetization oscillation amplitude at these absorption peaks to identify the modes developing at these resonances. The colors in the mode visualizations (Figures 4–6) refer to the imaginary magnetic susceptibility component of the gyroid structure, which in this case is analogous to the modulus of the dynamic components of the magnetization.

Figure 4

Figure 4. Resonance frequency spectra of 8 × 8 × 2 gyroid structures obtained from micromagnetic simulation. The graphical representations showcase the resonance derived from two different orientations of an applied magnetic field: a dashed line represents the signal from a sample subjected to a 300 mT field, directed out-of-plane (along the z-axis); the solid line, meanwhile, illustrates the response of the sample magnetized in the in-plane (x-axis) direction. The employed color scale is representative of the imaginary part of magnetic susceptibility.

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Figure 5

Figure 5. Micromagnetic simulation-derived resonance frequency spectra for 4 × 4 × 4 gyroid constructs. The spectra are derived from two scenarios: (a) in which the applied external magnetic field has a strength of 100 mT and (b) where it measures 300 mT. Within each plot, different color coding indicates the crystallographic direction in which the field is applied, with the specific points encircled on the graph signifying the ferromagnetic resonance. Visual illustrations and resonance frequency values, in sequential order of their appearance, are exhibited as insets within the plots. The coloring scheme used here corresponds directly to the imaginary component of the magnetic susceptibility.

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Figure 6

Figure 6. Spectral examination of high-intensity volume modes within a 6 × 6 × 6 gyroid structure. The lower section of the figure presents a plotted distribution of the frequency spectra, with the high-intensity volume modes depicted in the upper part distinctly marked by black circles. For the purpose of comparison, spectra corresponding to more compact structures (illustrated as dotted lines) are superimposed on the graph, thereby clearly demonstrating a marked decline in the intensity of edge modes (denoted with crosses) commensurate with the enlargement of the structure dimensions. The color gradation utilized in the visual representation of the modes is proportional to the imaginary component of the magnetic susceptibility.

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Initially, our simulation work involves the comparative analysis of the ferromagnetic response within the planar system geometry for both in-plane and out-of-plane directed fields to ascertain if a planar gyroid network exhibits macroscopic shape anisotropy. Our experimental sample, possessing a near-planar macroscopic geometry, boasts a lateral dimension spanning a few millimeters and a thickness in the submicron range (see Figure 7).

Figure 7

Figure 7. Multidomain gyroid structure employed in BBFMR measurements, illustrating the complex and varied nature of the sample. In panels (a) and (b), scanning electron microscope (SEM) topographical images offer detailed views of two distinct sample regions, characterized by different crystallographic directions and varying degrees of amorphousness. Panel (c) showcases a photograph of the entire sample, annotated with the approximate locations of several prominent domains. These domains were identified and characterized through polarized light microscopy.

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Therefore, first we analyze a quasi-planar 8 × 8 × 2 gyroid model (number of the unit cells along the x, y, and z, respectively) to verify the influence of external shape anisotropy on resonant frequencies. The model here is 400 × 400 × 100 nm in size (see Figure 4) and consists of about 366,000 tetrahedral discretization cells (∼2860 per unit cell). We consider two cases where the external magnetic field is directed out-of-plane and in-plane: [100] and [001], respectively, which are crystallographically identical. We see in Figure 4 an apparent anisotropy in the spectrum that changes in both field directions. The maximum absorption frequency shifts from about 8.25 GHz in the out-of-plane scenario to about 9.75 GHz in the in-plane scenario (Δf = 1.5 GHz). It can therefore be concluded that when simulating gyroids in a cubic simulation volume, as shown in Figure 2b, we can expect resonant frequencies with values lower than those obtained from experiments due to the significant influence of macroscopic shape anisotropy. The most accurate results of micromagnetic simulations could be obtained by simulating a much larger structure. However, due to limited computational resources and simulation time, it was necessary to limit the size of simulated structures.
To study gyroid-based effects at the nanometer scale, the model with dimensions of 4 × 4 × 4 unit cells (200 × 200 × 200 nm; see Figure 2b) was used in the second stage of micromagnetic simulations, consisting of about 171,000 tetrahedral discretization cells (∼2670 per unit cell). The cubic shape of the structure allows a more accurate analysis of the dynamic magnetization distribution and insights into the effects associated with gyroid crystallography.
In this examination, we scrutinize three crystallographic directions, namely, [100], [110], and [111], as illustrated in Figure 2b. We further engage with two distinct magnitudes of the external static magnetic field Bext, set at 100 and 300 mT, as depicted in Figure 5. The considerable computational power and time demanded by the simulations of such complex structures necessitate this restricted selection of parameters.
As shown in Figure 5, magnetic resonance simulations of gyroid structures reveal their pronounced geometric anisotropy. Upon juxtaposition of these plots with those for the planar structure depicted in Figure 4, Figure 5 exhibits a substantially richer spectrum for both external magnetic field values. This richness originates from the increased prominence of edge modes within the overall signal, a characteristic feature of cubic structures. Visual representations of these can be accessed in the Supporting Information (Figures S2 and S3). Through the identification and visualization of bulk modes, we observe that these predominantly emerge within the structure’s inner part. This observation suggests that their existence can be primarily attributed to the intrinsic gyroid geometry rather than edge features or artifacts produced in the cutting region, underscoring the distinctive and influential role that the gyroid geometry plays in modulating the magnetic response of these 3D structures. Consequently, our analysis focuses solely on the resonant modes linked to the volumetric portion of the structure, disregarding the discernible satellite peaks attributable to the edge modes.
The resonance spectra findings unequivocally assign the shifting resonance signals observed in the rotating gyroid sample to the crystallographic anisotropy, as evidenced by the plots displayed in Figure 5. A comparative assessment of the plots corresponding to 100 mT in Figure 5a and 300 mT in Figure 5b, reveals a discernible increase of frequencies. However, the order and dispersion of the resonance frequency peaks remain invariant, with the two most distal bulk modes (1.15 GHz, 4.10 GHz in Figure 5a and 6.90 GHz, 9.45 GHz in Figure 5b) separated by 2.95 and 2.55 GHz, respectively. Those separations emphasize the pivotal role played by gyroid geometry in dictating the system’s magnonic properties, demonstrating that these features are not merely a consequence of extrinsic variables.
In the final stage of our simulations, we conducted an analysis of a larger cubic gyroid structure. In the course of visualizing the modes via micromagnetic simulations (Figure 5), we uncovered a compelling localization of high-amplitude magnetization precession. In order to enhance the visibility of these modes vis-à-vis smaller structures and to incrementally amplify the signal of volume modes in relation to satellite ones, we proceeded to simulate and scrutinize a gyroid structure comprising 6 × 6 × 6 unit cells. This extensive structure was discretized into upward of 638,000 finite elements (∼2950 per unit cell). The resultant magnetization intensity distribution (Figure 6 top row) distinctly delineates planes orthogonal to the 300 mT external magnetic field axis, manifest at modes 6.95 GHz for [100], 8.15 GHz for [110], and 9.50 GHz for [111]. Furthermore, in the case of the field-aligned along [100] and [110], the modes manifest a periodicity (see the top row showing the spatial distribution of the imaginary component of the magnetic susceptibility), a characteristic not previously discernible in the 4 × 4 × 4 structure. The graph depicted at the bottom of Figure 6 showcases the FMR spectrum derived from simulations of the 6 × 6 × 6 model across the three investigated crystallographic directions. To facilitate comparison with corresponding calculations for a more diminutive structure (Figure 5b), the outcomes of these simulations were superimposed on the graph as dashed lines. It becomes evident that the influence of edge modes (denoted with a cross) in most cases diminishes in correlation with the model dimensions’ expansion, yielding a decrease of about 28% when increasing the layout from 4 × 4 × 4 to 6 × 6 × 6 unit cells, i.e., 3.375-fold. The normalized intensity of the volume modes (denoted by a circle) remains unchanged for the cases [100] and [111], but is slightly reduced for [110], most likely due to its coupling with the edge mode and/or its localization at the corners of the structure.

4. Experiment

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The main conclusion elucidated from our simulations is the intrinsic relationship between the ferromagnetic response of the gyroid network and its orientation to the axis of the static magnetic field. Figure 7 indicates that the sample embodies multiple domains, manifested as gyroid patches with varying crystallographic orientations relative to a consistent reference point. The varied orientations significantly influence the material’s magnetic response, adding complexity to its behavior in a magnetic field.
In Figure 7b, a distinct transition is observed from top to bottom, ranging from a well-defined, highly oriented region to an irregular, less-structured domain. This transition can be construed as the domain wall within the complex gyroid structure. As shown in the simulations, each distinct crystallographic orientation of the gyroid with respect to the externally applied magnetic field may lead to a nuanced alteration in the resonance response. Owing to the substantial multiplicity of domains, the anticipated FMR signal will manifest as an averaged and diluted amalgamation of contributions from individual domains.
Through the application of polarized light spectroscopy, the research facilitated the estimation of the individual domains of the gyroid structure; that is, the regions exhibiting homogeneous crystallographic alignment. As depicted in Figure 7c, the dimensions of the largest domains extend to the millimeter scale. However, it must be acknowledged that these particular measurements do not allow for the determination of the specific crystallography present within each domain. In an effort to achieve a dynamic magnetic characterization of the Ni gyroid sample, BBFMR measurements were executed in the frequency range of 0.1 to 25 GHz, following the methodology delineated by Heinrich. (59) This involved the utilization of a two-port vector network analyzer, with connections established to opposite ends of a coplanar waveguide (CPW) in accordance with the techniques described by Montoya. (60)
The gyroid probe was mounted on a CPW with a 80 μm wide center conductor facing downward to ensure maximum coupling with the microwave field (61) and the static external magnetic field applied within the sample plane, along the CPW center conductor (Figure 8a). We tested several orientations of the specimen with respect to the CPW line and discovered a signal from both gyroid-structured Ni and a uniform layer of this material (Figure 8b,c). Measurement data from more sample orientations relative to CPW are provided in the Supporting Information (Figure S1). Indeed, the precise manipulation of the sample’s position on the CPW elucidated the presence of a higher frequency signal that is uniquely associated with the homogeneous Ni layer present at one of the sample’s edge. This distinct relationship was substantiated by an excellent agreement between the observed signal and the theoretical prediction derived from the Kittel formula
f=γ2πBext(Bext+μ0Ms)
(2)
where f is the resonance frequency, μ0 is a vacuum permeability (∼4π × 10–7 T m/A), and γ is the gyromagnetic ratio. For the comparative calculations, we employed identical magnetic parameters as in the numerical simulations, specifically, the g-factor of 2.14 and saturation magnetization Ms = 480 kA/m. Contrarily, the lower frequency signal is consistently identified with the gyroid-structured region of the sample, a correlation supported by its sustained presence irrespective of orientation relative to the CPW.

Figure 8

Figure 8. BBFMR measurement conducted on the Ni gyroid structure. The sample underwent repositioning with respect to the CPW to elucidate the effect of an additional homogeneous Ni layer present within the specimen. In two distinct configurations, separate assessments were made of the energy absorption stemming from the microwave field BMW, applied perpendicular to the external static magnetic field (a). The resultant plots of dynamic magnetization amplitude as functions of static magnetic flux density and frequency for selected sample configurations are depicted in (b,c). These render a conspicuous signal attributed to the gyroid layer when the CPW is in direct alignment below it (b), and an additional, higher-frequency signal emanating from the uniform Ni layer (c) when the CPW (as delineated by the red dashed line) intersects its projected position (highlighted in purple on the sample). The dotted lines in the graphs represents the theoretical fit derived from the Kittel formula (see eq 2). For uniform Ni, parameters from micromagnetic simulations were used, while for gyroid, we implemented the calculated effective parameters, i.e., saturation magnetization Meff = 132 kA/m, and the g-factor of 2.2. Plot (d) shows a summary of the peak intensities calculations of FMR signals (blue dots for uniform Ni, and orange dots for gyroid) as a function of the external magnetic field strength. Normalized intensity values for 100, 300, and 450 mT fields are indicated. In graph (e), a cross-sectional analysis of the BBFMR signals is depicted for distinct values of the external magnetic field, where the solid blue line corresponds to Bext = 100 mT, the dashed brown line to Bext = 300 mT, and the dash-dotted green line to Bext = 450 mT. Additionally, horizontal dashed green lines mark the full width at half-maximum (FWHM) for each section, providing quantitative insights into the resonance line widths along with their respective values. The orange crosses signify peak maxima and their corresponding frequencies, pinpointing the resonant behavior within the explored frequency range. Insets furnish intensity plots from the BBFMR measurements, with the green vertical lines highlighting the specific locations of the sections for each magnetic field value. Finally, plot (f) shows the magnetic field FWHM’s as a function of frequency for BBFMR signals of gyroid (purple dots) and uniform Ni (dark red dots). Based on the experimental data and using eq 3, a linear regression was performed and the values of the determination coefficient r2, ΔH0 (from the abscissa of the lines) and the eq 3-derived damping values α (from the slope of the lines) were estimated. Parameters related to the gyroid structure are marked with a prim (′).

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Using eq 2, we not only fitted the resonance signal derived from the homogeneous Ni structure but also attempted to fit it to the experimental signal in the gyroids and estimated the effective values of the saturation magnetization and the g-factor.
As a result, a satisfactory fit of the Kittel curve to the gyroidal resonance signal for different sample orientations relative to the CPW (see Figure S4 in the Supporting Information) was obtained using Meff = 0.275Ms = 132 kA/m, and geff = 1.028g = 2.2, as shown in Figure 8b,c with a white dotted line. Such findings suggest that the given 3D structure exhibits magnonics’ metamaterial-like properties, so far considered only for the planar structures, (62−66) capable of modulating (specifically, decreasing) the effective saturation magnetization and the dynamical response of the structure in a methodical and foreseeable manner. The effective magnetization, which is higher than expected with only a 10% filling fraction, implies that the gyroid structure may have anisotropy. It can be indicative of a strong shape anisotropy inherent to the particular nanoelements of this structure, which ‘anchors’ the magnetization in place until it is forced to switch due to a change in the field direction.
To facilitate a qualitative comparison between the results of the experiment and simulation, we examined the FWHM’s. Figure 8 reveals that the signal from gyroid exhibits a broader frequency FWHM than in Ni, e.g., at 450 mT field it is 3.69 GHz for gyroid and 3.09 GHz for homogeneous Ni. This phenomenon is likely attributable to the multidomain nature of the sample. That is, the signal observed from the gyroid is the average value coming from several different domains located above the CPW and differing in crystallographic orientation. We deduce from the BBFMR measurements depicted in Figure 8e an average difference between the FWHM’s of 1.3 GHz over different values of the external magnetic field. The micromagnetic simulations for the cube shape (Figure 5) reveal a maximum peak separation of 2.55 and 2.95 GHz for the field of 300 and 100 mT, respectively (in both cases between [100] and [111] crystallographic directions). Our simulations meticulously evaluated the crystallographic directions that represent the most disparate configurations of the gyroid lattice relative to the field, most likely resulting in the largest feasible separation of resonant frequencies.
In field-swept FMR experiments conducted at a fixed frequency, the absorption line FWHM conforms to μ0ΔH = 4παf/γ. This occurs when the magnetization vector is aligned with the applied magnetic field, either in the plane or perpendicular to it. Such alignment results in a line width that scales proportionally with frequency, where the slope of this scaling is defined by the Gilbert damping parameter α. (67) Beyond this intrinsic contribution, empirical data also indicate the presence of a frequency-independent term
ΔH=ΔH0+4παμ0γf
(3)
denoted by ΔH0 that represents the inhomogeneous contributions, which add to the overall line width observed in the experiments.
Experimental investigations have enabled a linear regression analysis of FWHM across a spectrum of BMW frequencies, as depicted in Figure 8f for the designated orientations of the sample over CPW. This analytical approach is instrumental in ascertaining the damping values α and the inhomogeneous line width contribution ΔH0 for both the homogeneous Ni layer and the gyroidal structure, respectively, employing eq 3. In the first case, the derived values are α = 0.0282 ± 2.54% and ΔH0 = 0.038 ± 1.28% T, with a coefficient of determination of r2 = 0.95153. Comparatively, the gyroidal structure exhibits α′ = 0.0362 ± 6.49% and ΔH0′ = 0.101 ± 1.05% T, with r2 = 0.81639. The damping constants for Ni align with previously reported values. (68) The 95% confidence interval was used to calculate the range of estimates for the values. Larger α′ for the gyroid structure is likely attributable to the scattering of SW modes within the nanowires, which are much thinner than the bulk Ni and exhibit complex noncollinear interconnections. This difference varies with the sample’s orientation relative to the CPW, as further exemplified in the Supporting Information (Figure S4). The examples there, however, are subject to considerable uncertainty due to the nonlinearity of ΔH(f) for gyroid structures.
The disparity in the ΔH0 values between the homogeneous Ni layer and the gyroidal structure is also noteworthy. Such variations in the material’s magnetic properties, including anisotropy, manifest as a frequency-independent line width. (67) The pronounced ΔH0′ in gyroid structures corroborates the influence of crystallographic orientation on the system’s resonant frequencies, as confirmed by micromagnetic simulations and indicated in the earlier discussion. It is that, experimentally, in a multimode sample scenario depicted in Figure 7, the inhomogeneous contribution to the effective FWHM is an aggregate effect of the various crystallographic orientations present, and ΔH0′ emerges from the superposition of all resonant peaks within the spectrum bounded by the extremities observed in Figures 5 and 6 (the maximum separation between FMR peaks for the studied crystallographies in the 100 mT field is 2.95 GHz, while in the 300 mT field, 2.60 GHz).
The next step is to compare the frequencies. In the experiment, the maximum intensity at Bext = 300 mT is at 11.53 GHz (Figure 8d), while in the simulations for the gyroid of cubic shape, it is at 8.25 GHz for the [110] field orientation (among the orientations analyzed, in accordance with the above discussion about FWHM, we chose the orientation with the medium frequency, Figure 5b), giving a difference of 3.28 GHz. At 100 mT this difference decreases to 2.7 GHz (Figure 5a). The results in Figure 4 indicate that the macroscopic shape of the gyroid contributes to the results. In these simulations, considering a relatively small thin cuboid shape, the FMR frequency separation between orthogonal field orientations is Δf = 1.5 GHz. However, when considering a bulk thin film ferromagnet of Meff = 132 kA/m, the maximum influence of shape (i.e., FMR between in-plane and out-of-plane magnetic field orientations) can reach over 4 GHz. Therefore, it is reasonable to expect that the resonant frequencies in simulations will be lower than those obtained in experiments due to the significant influence of the macroscopic shape anisotropy.
A noteworthy distinction between the gyroid and homogeneous Ni signals also arises with respect to their measured intensities as a function of the external magnetic field. In the case of the gyroid structure, it shows an almost linear increase (Figure 8d), giving an almost 30% increase in intensity between 100 and 300 mT magnetic fields (and 50% between 100 and 450 mT), while for homogeneous Ni it remains fairly irregular. This intriguing behavior may reflect the intricate influence of the complex gyroid geometry on the internal demagnetization field and magnetization orientation. Such complexity may counteract the orthogonal orientation of the dynamic magnetization components with respect to the static field, thus attenuating their detectability by the microwave field, BMW. Upon increasing the field, the sample leads to a parallel orientation of the magnetization vectors, as shown in Figure 3c, thereby increasing the detectability of the magnetization precession at resonance. From this graph it can be seen that most of the bins (elementary simulation cells) are almost completely saturated for a field of 100 mT (for example, at least 94% saturation has been reached in 89% of the gyroid volume: 56,463 bins, and at 300 mT the >94% saturation is already in 100% of the structure: 63,274 bins). However, a comprehensive understanding of the FMR intensities in gyroids for larger fields would require further investigation that falls outside the scope of this paper.

5. Discussion

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Owing to the inherently multidomain architecture of the sample under investigation, the task of identifying a singular dominant crystallography and quantifying its direct influence on the resonance spectrum presents a formidable challenge. Recognizing this complexity, the study has embraced a numerical approach, rigorously testing the three selected field directions ([100], [110], and [111]) in the finite-size gyroid, to elucidate the extent to which the structure’s intrinsic complexity governs resonance frequency variations. It is imperative to note that beyond the irregular shape, size, and interfaces of the domains, which relate to the shape anisotropy, fabrication of such an intricate structure encompasses numerous factors that elude accurate prediction and, consequently, integration within the simulation framework. Such factors include, but are not limited to, irregularities and impurities in both the shape and thickness of the nanorods, compounded by a paucity of definitive information concerning the sample’s dominant crystallographic orientation in each experimental configuration. A full experimental validation of the phenomena predicted by our micromagnetic simulations is beyond the scope of this work. However, we expect that these effects will soon be observable in experiments with directed self-assembly gyroids. (69) Our experimental results provide the basis for this since the signs of the predicted phenomena have already been detected.
The research presented here highlights several emerging avenues for the application and practical use of 3D gyroidal nanostructures. By controlling ϕ, one could modulate the effective saturation magnetization, offering magnonic metamaterial of effective dynamical properties. Traditionally, the construction of artificial photonic or phononic crystals requires the use of different materials, each characterized by unique properties, e.g., dielectric constant or elastic properties. However, our results suggest that a gyroidal structure with a tunable filling factor could serve as an effective substitute for 3D magnonic crystals. This paradigm shift in fabrication methodology heralds a transition from 2D to 3D magnonic nanostructures with a wide range of novel applications and functionalities.
Due to the random distribution, shape, and size of the gyroid domains in the experiment, the sample can be approximated to a porous structure when examined collectively. (70) This simplification, however, nullifies the interpretation of the relationship between the field direction and the frequency of SWs. This may ultimately be shown in future studies e.g. by using Brillouin light scattering for individual gyroid domains.
Some of the visible SW bulk modes presented in this work show also an intriguing surface character, localizing on the sides perpendicular to the direction of the magnetic field–see, e.g. Figures 4, 6 ([110] and [100]), and in the Supporting Information Figure S2 (modes no. 7 and 8) and Figure S3 (modes no. 1 and 5). This may be the result of an additional effect arising from the strong influence of crystallography and the shape anisotropy on the localization of resonant modes in different regions and probably on the chirality of the system. However, the analysis of these effects, although interesting, is beyond the scope of this work, since it is necessary to perform micromagnetic simulations with periodic boundary conditions and to conduct experiments of a different type, where the sample containing the single-domain gyroidal structure could be selectively analyzed on a much smaller scale.
In addition, the distinct mode spectrum observed in our research has direct implications for radio frequency filtering applications. The presence of a singular bulk mode in both BBFMR experiments (as an effective response from a multidomain structure) or simulations (a strong resonance in a single domain) enables the design of selective filter systems. They could be tailored to isolate specific FMR frequencies depending on factors such as crystallographic orientation relative to the incident signal and filling factor. In addition, the three-dimensional magnonic structures are expected to exhibit superior absorption properties compared to their two-dimensional counterparts, as supported by previous studies. (44)

6. Conclusions

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In the investigation, a comprehensive ferromagnetic resonance analysis was conducted on three-dimensional gyroidal Ni nanostructures, delving into the complex interplay between magnetic properties and structural geometry. Utilizing FEM micromagnetic simulations in the time domain to determine the static magnetization structures (55) and in the frequency domain to investigate their oscillatory, magnonic properties, (57) the study embarked on a multifaceted exploration aimed not merely at interpreting experimental results but also at unraveling the intricate magnetization distribution within the gyroid nanostructure.
A major result that emerges from our simulations is the importance of the crystallographic orientation of the gyroid, relative to the magnetic field’s direction, on a frequency and spatial distribution of the collective magnetization oscillations. Complementing these simulations, experimental measurements were executed using BBFMR, where a multidomain gyroid sample was positioned on a CPW line, enabling analysis of differences and relationships between the solid, uniform Ni layer and the gyroid-structured portion of the sample. Although the measurements were not able to study individual crystallographic domains of the gyroid and thus confirm all simulation predictions, based on our findings, gyroid films can be conceptualized as homogeneous materials, i.e., magnonic metamaterials, where the effective saturation magnetization is reduced by the gyroid filling factor and the FMR signal line width encapsulates more than the inherent damping properties of the material; it is also intricately linked to the particular geometry and crystallographic orientation of the structure.
Collectively, these discoveries show a new frontier in the realm of 3D magnonics, positing the gyroid structure as a potential cornerstone in the field. The results not only augment our theoretical and experimental grasp of the magnetization dynamics in complex nanostructures but also open up promising avenues for practical applications, imbuing the gyroid configuration with the potential to become an elemental building block in emerging magnetic technologies. We propose a gyroidal structure with a tunable filling factor as a magnonic crystal and as the basis for novel 3D radio frequency filters. The study’s interdisciplinary approach, bridging numerical simulations with empirical investigation, marks a significant stride toward the comprehensive understanding and manipulation of magnetic resonance in three-dimensional architectures.

Data Availability

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The data underlying this study are openly available in Zenodo at https://doi.org/10.5281/zenodo.11004007.

Supporting Information

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The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsami.4c02366.

  • BBFMR gyroid structure analysis under in-plane field rotation in 40° increments over 180° relative to the CPW; micromagnetic results of 4 × 4 × 4 gyroid spectra at 100 mT: detailed focus on satellite peaks; micromagnetic results of 4 × 4 × 4 gyroid spectra at 300 mT: detailed focus on satellite peaks; and FWHM vs frequency: comparison of BBFMR signals of gyroids and uniform Ni, under sample rotations on CPW, revealing the variation in the resulting effective parameters of damping and inhomogeneous contributions to the line width (PDF)

Terms & Conditions

Most electronic Supporting Information files are available without a subscription to ACS Web Editions. Such files may be downloaded by article for research use (if there is a public use license linked to the relevant article, that license may permit other uses). Permission may be obtained from ACS for other uses through requests via the RightsLink permission system: http://pubs.acs.org/page/copyright/permissions.html.

Author Information

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  • Corresponding Author
  • Authors
    • Riccardo Hertel - Université de Strasbourg, CNRS, Institut de Physique et Chimie des Matériaux de Strasbourg, F-67000 Strasbourg, France
    • Massimiliano d’Aquino - Department of Electrical Engineering and ICT, University of Naples Federico II, 80125 Naples, Italy
    • Vitaliy Vasyuchka - Fachbereich Physik und Landesforschungszentrum OPTIMAS, Rheinland-Pfälzische Technische Universität Kaiserslautern-Landau, Erwin-Schrödinger-Straße 56, 67663 Kaiserslautern, Germany
    • Mathias Weiler - Fachbereich Physik und Landesforschungszentrum OPTIMAS, Rheinland-Pfälzische Technische Universität Kaiserslautern-Landau, Erwin-Schrödinger-Straße 56, 67663 Kaiserslautern, Germany
    • Philipp Pirro - Fachbereich Physik und Landesforschungszentrum OPTIMAS, Rheinland-Pfälzische Technische Universität Kaiserslautern-Landau, Erwin-Schrödinger-Straße 56, 67663 Kaiserslautern, Germany
    • Maciej Krawczyk - Institute of Spintronics and Quantum Information, Faculty of Physics, Adam Mickiewicz University, Uniwersytetu Poznańskiego 2, 61-614 Poznań, Poland;  Orcidhttps://orcid.org/0000-0002-0870-717X
    • Shunsuke Fukami - Research Institute of Electrical Communication (RIEC), Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai-shi, Miyagi 980-8577, Japan;  Center for Science and Innovation in Spintronics (CSIS), Tohoku University, 980-8577 Sendai, Japan;  Center for Innovative Integrated Electronic Systems (CIES), Tohoku University, 468-1 Aramaki Aza Aoba, Aoba-ku, 980-0845 Sendai, Japan;  WPI Advanced Institute for Materials Research, Tohoku University, 2-1-1 Katahira, Aoba-ku, 980-8577 Sendai, Japan;  Inamori Research Institute for Science, 600-8411 Kyoto, Japan
    • Hideo Ohno - Research Institute of Electrical Communication (RIEC), Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai-shi, Miyagi 980-8577, Japan;  Center for Science and Innovation in Spintronics (CSIS), Tohoku University, 980-8577 Sendai, Japan;  Center for Innovative Integrated Electronic Systems (CIES), Tohoku University, 468-1 Aramaki Aza Aoba, Aoba-ku, 980-0845 Sendai, Japan;  WPI Advanced Institute for Materials Research, Tohoku University, 2-1-1 Katahira, Aoba-ku, 980-8577 Sendai, Japan
    • Justin Llandro - Research Institute of Electrical Communication (RIEC), Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai-shi, Miyagi 980-8577, Japan;  Center for Science and Innovation in Spintronics (CSIS), Tohoku University, 980-8577 Sendai, Japan
  • Notes
    The authors declare no competing financial interest.

Acknowledgments

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The research was funded by the National Science Centre of Poland, Projects nos. UMO-2020/39/I/ST3/02413 and UMO-2023/49/N/ST3/03032. J.L. and S.F. acknowledge support from the Japan Society for the Promotion of Science (JSPS) under KAKENHI 21K04816 and 19H05622, Cooperative Research Projects of CSIS, Tohoku University, and the Graduate Program for Spintronics (GP-Spin), Tohoku University. R.H. acknowledges the High Performance Computing center of the University of Strasbourg for supporting this work by providing access to computing resources. M.W., V.V., and P.P. acknowledge funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)─TRR 173/3─268565370 Spin+X (Projects B01 and B13). The authors thank Dr. J. Dolan for sample preparation, Dr. I. Gunkel and Dr. Naëmi Leo for providing domain images, and Prof. T. Dietl and Prof. B. Hillebrands for insightful discussions.

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    A review. Traditionally, the primary field, where curvature has been at the heart of research, is the theory of general relativity. In recent studies, however, the impact of curvilinear geometry enters various disciplines, ranging from solid-state physics over soft-matter physics, chem., and biol. to mathematics, giving rise to a plethora of emerging domains such as curvilinear nematics, curvilinear studies of cell biol., curvilinear semiconductors, superfluidity, optics, 2D van der Waals materials, plasmonics, magnetism, and supercond. Here, the state of the art is summarized and prospects for future research in curvilinear solid-state systems exhibiting such fundamental cooperative phenomena as ferromagnetism, antiferromagnetism, and supercond. are outlined. Highlighting the recent developments and current challenges in theory, fabrication, and characterization of curvilinear micro- and nanostructures, special attention is paid to perspective research directions entailing new physics and to their strong application potential. Overall, the perspective is aimed at crossing the boundaries between the magnetism and supercond. communities and drawing attention to the conceptual aspects of how extension of structures into the third dimension and curvilinear geometry can modify existing and aid launching novel functionalities. In addn., the perspective should stimulate the development and dissemination of research and development oriented techniques to facilitate rapid transitions from lab. demonstrations to industry-ready prototypes and eventual products.
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  14. 14
    Cheenikundil, R.; Bauer, J.; Goharyan, M.; d’Aquino, M.; Hertel, R. High-frequency modes in a magnetic buckyball nanoarchitecture. APL Mater. 2022, 10, 81106,  DOI: 10.1063/5.0097695
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    Cheenikundil, R.; d’Aquino, M.; Hertel, R. Magnetization dynamics in a three-dimensional interconnected nanowire array. arXiv 2023, arXiv.2306.00174,  DOI: 10.48550/arXiv.2306.00174
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    preprint

  16. 16
    Llandro, J.; Love, D. M.; Kovács, A.; Caron, J.; Vyas, K. N.; Kákay, A.; Salikhov, R.; Lenz, K.; Fassbender, J.; Scherer, M. R. J. Visualizing magnetic structure in 3d nanoscale ni-fe gyroid networks. Nano Lett. 2020, 20, 3642– 3650,  DOI: 10.1021/acs.nanolett.0c00578
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    Visualizing Magnetic Structure in 3D Nanoscale Ni-Fe Gyroid Networks
    Llandro, Justin; Love, David M.; Kovacs, Andras; Caron, Jan; Vyas, Kunal N.; Kakay, Attila; Salikhov, Ruslan; Lenz, Kilian; Fassbender, Jurgen; Scherer, Maik R. J.; Cimorra, Christian; Steiner, Ullrich; Barnes, Crispin H. W.; Dunin-Borkowski, Rafal E.; Fukami, Shunsuke; Ohno, Hideo
    Nano Letters (2020), 20 (5), 3642-3650CODEN: NALEFD; ISSN:1530-6984. (American Chemical Society)
    Arrays of interacting 2D nanomagnets display unprecedented electromagnetic properties via collective effects, demonstrated in artificial spin ices and magnonic crystals. Progress toward 3D magnetic metamaterials is hampered by two challenges: fabricating 3D structures near intrinsic magnetic length scales (sub-100 nm) and visualizing their magnetic configurations. Here, we fabricate and measure nanoscale magnetic gyroids, periodic chiral networks comprising nanowire-like struts forming three-connected vertices. Via block copolymer templating, we produce Ni75Fe25 single-gyroid and double-gyroid (an inversion pair of single-gyroids) nanostructures with a 42 nm unit cell and 11 nm diam. struts, comparable to the exchange length in Ni-Fe. We visualize their magnetization distributions via off-axis electron holog. with nanometer spatial resoln. and interpret the patterns using finite-element micromagnetic simulations. Our results suggest an intricate, frustrated remanent state which is ferromagnetic but without a unique equil. configuration, opening new possibilities for collective phenomena in magnetism, including 3D magnonic crystals and unconventional computing.
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  17. 17
    Fernández-Pacheco, A.; Streubel, R.; Fruchart, O.; Hertel, R.; Fischer, P.; Cowburn, R. P. Three-dimensional nanomagnetism. Nat. Commun. 2017, 8, 15756,  DOI: 10.1038/ncomms15756
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    Three-dimensional nanomagnetism
    Fernandez-Pacheco Amalio; Cowburn Russell P; Streubel Robert; Fischer Peter; Fruchart Olivier; Hertel Riccardo; Fischer Peter
    Nature communications (2017), 8 (), 15756 ISSN:.
    There is no expanded citation for this reference.
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  18. 18
    Donnelly, C.; Hierro-Rodríguez, A.; Abert, C.; Witte, K.; Skoric, L.; Sanz-Hernández, D.; Finizio, S.; Meng, F.; McVitie, S.; Raabe, J.; Suess, D.; Cowburn, R.; Fernández-Pacheco, A. Complex free-space magnetic field textures induced by three-dimensional magnetic nanostructures. Nat. Nanotechnol. 2022, 17, 136– 142,  DOI: 10.1038/s41565-021-01027-7
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    Complex free-space magnetic field textures induced by three-dimensional magnetic nanostructures
    Donnelly, Claire; Hierro-Rodriguez, Aurelio; Abert, Claas; Witte, Katharina; Skoric, Luka; Sanz-Hernandez, Dedalo; Finizio, Simone; Meng, Fanfan; McVitie, Stephen; Raabe, Jorg; Suess, Dieter; Cowburn, Russell; Fernandez-Pacheco, Amalio
    Nature Nanotechnology (2022), 17 (2), 136-142CODEN: NNAABX; ISSN:1748-3387. (Nature Portfolio)
    Abstr.: The design of complex, competing effects in magnetic systems-be it via the introduction of nonlinear interactions1-4, or the patterning of three-dimensional geometries5,6-is an emerging route to achieve new functionalities. In particular, through the design of three-dimensional geometries and curvature, intrastructure properties such as anisotropy and chirality, both geometry-induced and intrinsic, can be directly controlled, leading to a host of new physics and functionalities, such as three-dimensional chiral spin states7, ultrafast chiral domain wall dynamics8-10 and spin textures with new spin topologies7,11. Here, we advance beyond the control of intrastructure properties in three dimensions and tailor the magnetostatic coupling of neighboring magnetic structures, an interstructure property that allows us to generate complex textures in the magnetic stray field. For this, we harness direct write nanofabrication techniques, creating intertwined nanomagnetic cobalt double helixes, where curvature, torsion, chirality and magnetic coupling are jointly exploited. By reconstructing the three-dimensional vectorial magnetic state of the double helixes with soft-X-ray magnetic laminog.12,13, we identify the presence of a regular array of highly coupled locked domain wall pairs in neighboring helixes. Micromagnetic simulations reveal that the magnetization configuration leads to the formation of an array of complex textures in the magnetic induction, consisting of vortices in the magnetization and antivortices in free space, which together form an effective B field cross-tie wall14. The design and creation of complex three-dimensional magnetic field nanotextures opens new possibilities for smart materials15, unconventional computing2,16, particle trapping17,18 and magnetic imaging19.
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  19. 19
    van den Berg, A.; Caruel, M.; Hunt, M.; Ladak, S. Combining two-photon lithography with laser ablation of sacrificial layers: A route to isolated 3D magnetic nanostructures. Nano Res. 2023, 16, 1441– 1447,  DOI: 10.1007/s12274-022-4649-z
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  20. 20
    Schoen, A. H. Infinite periodic minimal surfaces without self-intersections; National Aeronautics and Space Administration, 1970.
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  21. 21
    Lambert, C. A.; Radzilowski, L. H.; Thomas, E. L. Triply periodic level surfaces as models for cubic tricontinuous block copolymer morphologies. Philos. Trans. R. Soc., A 1996, 354, 2009– 2023,  DOI: 10.1098/rsta.1996.0089
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    Triply periodic level surfaces as models for cubic tricontinuous block copolymer morphologies
    Lambert, Charla A.; Radzilowski, Leonard H.; Thomas, Edwin L.
    Philosophical Transactions of the Royal Society of London, Series A: Mathematical, Physical and Engineering Sciences (1996), 354 (1715), 2009-2023CODEN: PTRMAD; ISSN:0962-8428. (Royal Society)
    The domains of microphase sepd. block copolymers develop interfacial surfaces of approx. const. mean curvature in response to thermodn. driving forces. Of particular recent interest are the tricontinuous triply periodic morphologies and their math. representations. Level surfaces are represented by certain real functions which satisfy the expression F(x, y, z) = t, where t is a const. In general, they are non-self-intersecting and smooth, except at special values of the parameter t. The authors construct periodic level surfaces according to the allowed reflections of a particular cubic space group; such triply periodic surfaces maintain the symmetries of the chosen space group and make attractive approxns. to certain recently computed triply periodic surfaces of const. mean curvature. This paper is a study of the accuracy of the approxns. constructed using the lowest Fourier term of the Pm3-m, Fd3-m, and I4132 space groups, and the usefulness of these approxns. in analyzing exptl. obsd. tricontinuous block copolymer morphologies at a variety of vol. fractions. The authors numerically compare surface area per unit vol. of particular level surfaces with const. mean curvature surfaces having the same vol. fraction. The authors also demonstrate the utility of level surfaces in simulating projections of tricontinuous microdomain morphologies for comparison with actual transmission electron micrographs and detn. of block copolymer microstructure.
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  22. 22
    Turner, M. D.; Saba, M.; Zhang, Q.; Cumming, B. P.; Schröder-Turk, G. E.; Gu, M. Miniature chiral beamsplitter based on gyroid photonic crystals. Nat. Photonics 2013, 7, 801– 805,  DOI: 10.1038/nphoton.2013.233
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    Miniature chiral beamsplitter based on gyroid photonic crystals
    Turner, Mark D.; Saba, Matthias; Zhang, Qiming; Cumming, Benjamin P.; Schroeder-Turk, Gerd E.; Gu, Min
    Nature Photonics (2013), 7 (10), 801-805CODEN: NPAHBY; ISSN:1749-4885. (Nature Publishing Group)
    The linearly polarizing beamsplitter is a widely used optical component in photonics. It is typically built from a linearly birefringent crystal such as calcite, which has different crit. reflection angles for s- and p-polarized light, leading to the transmission of one linear polarization and angled reflection of the other. However, the analog for splitting circularly polarized light has yet to be demonstrated due to a lack of natural materials with sufficient circular birefringence. Here, we present a nano-engineered photonic-crystal chiral beamsplitter that fulfils this task. It consists of a prism featuring a nanoscale chiral gyroid network and can sep. left- and right-handed circularly polarized light in the wavelength region around 1.615 μm. The structure is fabricated using a galvo-dithered direct laser writing method and could become a useful component for developing integrated photonic circuits that provide a new form of polarization control.
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  23. 23
    Vignolini, S.; Yufa, N. A.; Cunha, P. S.; Guldin, S.; Rushkin, I.; Stefik, M.; Hur, K.; Wiesner, U.; Baumberg, J. J.; Steiner, U. A 3D optical metamaterial made by self-assembly. Adv. Mater. 2012, 24, OP23– OP27,  DOI: 10.1002/adma.201103610
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    A 3D Optical Metamaterial Made by Self-Assembly
    Vignolini, Silvia; Yufa, Nataliya A.; Cunha, Pedro S.; Guldin, Stefan; Rushkin, Ilia; Stefik, Morgan; Hur, Kahyun; Wiesner, Ulrich; Baumberg, Jeremy J.; Steiner, Ullrich
    Advanced Materials (Weinheim, Germany) (2012), 24 (10), OP23-OP27CODEN: ADVMEW; ISSN:0935-9648. (Wiley-VCH Verlag GmbH & Co. KGaA)
    The authors demonstrate the creation of a 3-dimensional Au metamaterial based on block copolymer (BCP) self-assembly. The authors start with an isoprene-block-styrene-block-ethylene oxide (ISO) BCP that forms 2 chem. distinct, interpenetrating gyroid networks (1,0) of opposite chirality in a matrix of the 3rd block. The I gyroid network is then removed by selective UV and chem. etching and back-filled with Au by electrodeposition. The final device consists of a continuous, triply periodic network of Au. The dimension of the full unit cell is = 50 nm, which is far below optical wavelengths. This particular morphol. was chosen since it is predicted to offer a strong resonant response that depends on the relative orientation between the structure and the polarization of the incident light.
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  24. 24
    Dolan, J. A.; Wilts, B. D.; Vignolini, S.; Baumberg, J. J.; Steiner, U.; Wilkinson, T. D. Optical Properties of Gyroid Structured Materials: From Photonic Crystals to Metamaterials. Adv. Opt. Mater. 2015, 3, 12– 32,  DOI: 10.1002/adom.201400333
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    24
    Optical Properties of Gyroid Structured Materials: From Photonic Crystals to Metamaterials
    Dolan, James A.; Wilts, Bodo D.; Vignolini, Silvia; Baumberg, Jeremy J.; Steiner, Ullrich; Wilkinson, Timothy D.
    Advanced Optical Materials (2015), 3 (1), 12-32CODEN: AOMDAX; ISSN:2195-1071. (Wiley-VCH Verlag GmbH & Co. KGaA)
    The gyroid is a continuous and triply periodic cubic morphol. which possesses a const. mean curvature surface across a range of volumetric fill fractions. Found in a variety of natural and synthetic systems which form through self-assembly, from butterfly wing scales to block copolymers, the gyroid also exhibits an inherent chirality not obsd. in any other similar morphologies. These unique geometrical properties impart to gyroid structured materials a host of interesting optical properties. Depending on the length scale on which the constituent materials are organized, these properties arise from starkly different phys. mechanisms (such as a complete photonic bandgap for photonic crystals and a greatly depressed plasma frequency for optical metamaterials). This article reviews the theor. predictions and exptl. observations of the optical properties of two fundamental classes of gyroid structured materials: photonic crystals (wavelength scale) and metamaterials (sub-wavelength scale).
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  25. 25
    Michielsen, K.; Stavenga, D. G. Gyroid cuticular structures in butterfly wing scales: Biological photonic crystals. J. R. Soc., Interface 2008, 5, 85– 94,  DOI: 10.1098/rsif.2007.1065
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    25
    Gyroid cuticular structures in butterfly wing scales: biological photonic crystals
    Michielsen K; Stavenga D G
    Journal of the Royal Society, Interface (2008), 5 (18), 85-94 ISSN:1742-5689.
    We present a systematic study of the cuticular structure in the butterfly wing scales of some papilionids (Parides sesostris and Teinopalpus imperialis) and lycaenids (Callophrys rubi, Cyanophrys remus, Mitoura gryneus and Callophrys dumetorum). Using published scanning and transmission electron microscopy (TEM) images, analytical modelling and computer-generated TEM micrographs, we find that the three-dimensional cuticular structures can be modelled by gyroid structures with various filling fractions and lattice parameters. We give a brief discussion of the formation of cubic gyroid membranes from the smooth endoplasmic reticulum in the scale's cell, which dry and harden to leave the cuticular structure behind when the cell dies. The scales of C. rubi are a potentially attractive biotemplate for producing three-dimensional optical photonic crystals since for these scales the cuticle-filling fraction is nearly optimal for obtaining the largest photonic band gap in a gyroid structure.
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  26. 26
    Saranathan, V.; Osuji, C. O.; Mochrie, S. G.; Noh, H.; Narayanan, S.; Sandy, A.; Dufresne, E. R.; Prum, R. O. Structure, function, and self-assembly of single network gyroid (I4 132) photonic crystals in butterfly wing scales. Proc. Natl. Acad. Sci. U.S.A. 2010, 107, 11676– 11681,  DOI: 10.1073/pnas.0909616107
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    26
    Structure, function, and self-assembly of single network gyroid (I4132) photonic crystals in butterfly wing scales
    Saranathan, Vinodkumar; Osuji, Chinedum O.; Mochrie, Simon G. J.; Noh, Heeso; Narayanan, Suresh; Sandy, Alec; Dufresne, Eric R.; Prum, Richard O.
    Proceedings of the National Academy of Sciences of the United States of America (2010), 107 (26), 11676-11681, S11676/1-S11676/7CODEN: PNASA6; ISSN:0027-8424. (National Academy of Sciences)
    Complex three-dimensional biophotonic nanostructures produce the vivid structural colors of many butterfly wing scales, but their exact nanoscale organization is uncertain. The authors used small angle x-ray scattering (SAXS) on single scales to characterize the 3D photonic nanostructures of five butterfly species from two families (Papilionidae, Lycaenidae). The authors identify these chitin and air nanostructures as single network gyroid (I4132) photonic crystals. The authors describe their optical function from SAXS data and photonic band-gap modeling. Butterflies apparently grow these gyroid nanostructures by exploiting the self-organizing phys. dynamics of biol. lipid-bilayer membranes. These butterfly photonic nanostructures initially develop within scale cells as a core-shell double gyroid (Ia3d), as seen in block-copolymer systems, with a pentacontinuous vol. comprised of extracellular space, cell plasma membrane, cellular cytoplasm, smooth endoplasmic reticulum (SER) membrane, and intra-SER lumen. This double gyroid nanostructure is subsequently transformed into a single gyroid network through the deposition of chitin in the extracellular space and the degeneration of the rest of the cell. The butterflies develop the thermodynamically favored double gyroid precursors as a route to the optically more efficient single gyroid nanostructures. Current approaches to photonic crystal engineering also aim to produce single gyroid motifs. The biol. derived photonic nanostructures characterized here may offer a convenient template for producing optical devices based on biomimicry or direct dielec. infiltration.
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  27. 27
    Schröder-Turk, G.; Wickham, S.; Averdunk, H.; Brink, F.; Fitz Gerald, J. D.; Poladian, L.; Large, M. C.; Hyde, S. T. The chiral structure of porous chitin within the wing-scales of Callophrys rubi. J. Struct. Biol. 2011, 174, 290– 295,  DOI: 10.1016/j.jsb.2011.01.004
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    27
    The chiral structure of porous chitin within the wing-scales of Callophrys rubi
    Schroder-Turk G E; Wickham S; Averdunk H; Brink F; Fitz Gerald J D; Poladian L; Large M C J; Hyde S T
    Journal of structural biology (2011), 174 (2), 290-5 ISSN:.
    The structure of the porous three-dimensional reticulated pattern in the wing scales of the butterfly Callophrys rubi (the Green Hairstreak) is explored in detail, via scanning and transmission electron microscopy. A full 3D tomographic reconstruction of a section of this material reveals that the predominantly chitin material is assembled in the wing scale to form a structure whose geometry bears a remarkable correspondence to the srs net, well-known in solid state chemistry and soft materials science. The porous solid is bounded to an excellent approximation by a parallel surface to the Gyroid, a three-periodic minimal surface with cubic crystallographic symmetry I4132, as foreshadowed by Stavenga and Michielson. The scale of the structure is commensurate with the wavelength of visible light, with an edge of the conventional cubic unit cell of the parallel-Gyroid of approximately 310 nm. The genesis of this structure is discussed, and we suggest it affords a remarkable example of templating of a chiral material via soft matter, analogous to the formation of mesoporous silica via surfactant assemblies in solution. In the butterfly, the templating is achieved by the lipid-protein membranes within the smooth endoplasmic reticulum (while it remains in the chrysalis), that likely form cubic membranes, folded according to the form of the Gyroid. The subsequent formation of the chiral hard chitin framework is suggested to be driven by the gradual polymerisation of the chitin precursors, whose inherent chiral assembly in solution (during growth) promotes the formation of a single enantiomer.
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    Yan, C.; Hao, L.; Hussein, A.; Raymont, D. Evaluations of cellular lattice structures manufactured using selective laser melting. Int. J. Mach. Tool Manufact. 2012, 62, 32– 38,  DOI: 10.1016/j.ijmachtools.2012.06.002
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    Yánez, A.; Herrera, A.; Martel, O.; Monopoli, D.; Afonso, H. Compressive behaviour of gyroid lattice structures for human cancellous bone implant applications. Mater. Sci. Eng., C 2016, 68, 445– 448,  DOI: 10.1016/j.msec.2016.06.016
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    Compressive behavior of gyroid lattice structures for human cancellous bone implant applications
    Yanez, A.; Herrera, A.; Martel, O.; Monopoli, D.; Afonso, H.
    Materials Science & Engineering, C: Materials for Biological Applications (2016), 68 (), 445-448CODEN: MSCEEE; ISSN:0928-4931. (Elsevier B.V.)
    Electron beam melting (EBM) was used to fabricate porous titanium alloy structures. The elastic modulus of these porous structures was similar to the elastic modulus of the cancellous human bone. Two types of cellular lattice structures were manufd. and tested: gyroids and diamonds. The design of the gyroid structures was detd. by the main angle of the struts with respect to the axial direction. Thus, structures with angles of between 19 and 68.5° were manufd. The aim of the design was to reduce the amt. of material needed to fabricate a structure with the desired angles to increase the range of stiffness of the scaffolds. Compression tests were conducted to obtain the elastic modulus and the strength. Both parameters increased as the angle decreased. Finally, the specific strength of the gyroid structures was compared with that of the diamond structures and other types of structures. It is shown that, for angles lower than 35°, the gyroid structures had a high strength to wt. ratios.
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    Armatas, G. S.; Kanatzidis, M. G. Mesostructured germanium with cubic pore symmetry. Nature 2006, 441, 1122– 1125,  DOI: 10.1038/nature04833
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    Mesostructured germanium with cubic pore symmetry
    Armatas, Gerasimos S.; Kanatzidis, Mercouri G.
    Nature (London, United Kingdom) (2006), 441 (7097), 1122-1125CODEN: NATUAS; ISSN:0028-0836. (Nature Publishing Group)
    Here we describe cubic mesostructured germanium, MSU-Ge-1, with gyroidal channels contg. surfactant mols., sepd. by amorphous walls that lie on the gyroid (G) minimal surface as in the mesoporous silica MCM-48 (ref. 2). Although Ge is a high-melting, covalent semiconductor that is difficult to prep. from soln. polymn., we succeeded in assembling a continuous Ge network using a suitable precursor for Ge4- atoms. Our results indicate that elemental semiconductors from group 14 of the periodic table can be made to adopt mesostructured forms such as MSU-Ge-1, which features two three-dimensional labyrinthine tunnels obeying Ia3d space group symmetry and sepd. by a continuous germanium minimal surface that is otherwise amorphous. A consequence of this new structure for germanium, which has walls only one nanometer thick, is a wider electronic energy bandgap (1.4 eV vs. 0.66 eV) than has cryst. or amorphous Ge. Controlled oxidn. of MSU-Ge-1 creates a range of germanium suboxides with continuously varying Ge:O ratio and a smoothly increasing energy gap.
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    Hajduk, D. A.; Harper, P. E.; Gruner, S. M.; Honeker, C. C.; Kim, G.; Thomas, E. L.; Kim, G. The Gyroid: A New Equilibrium Morphology in Weakly Segregated Diblock Copolymers. Macromolecules 1994, 27, 4063– 4075,  DOI: 10.1021/ma00093a006
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    The Gyroid: A New Equilibrium Morphology in Weakly Segregated Diblock Copolymers
    Hajduk, Damian A.; Harper, Paul E.; Gruner, Sol M.; Honeker, Christian C.; Kim, Gia; Thomas, Edwin L.; Fetters, Lewis J.
    Macromolecules (1994), 27 (15), 4063-75CODEN: MAMOBX; ISSN:0024-9297.
    A new equil. microdomain morphol. was identified in an intermediate to weakly segregated diblock copolymer melt. A styrene (I)-isoprene (SI) diblock copolymer with Mw = 27,400 and 37 wt.% I thermo-reversibly transformed from the lamellar morphol. (in equil. at low annealing temps.) to a new morphol. at annealing temps. ∼50° below the order-disorder transition. SAXS and TEM study of this new morphol. revealed that the new structure had remarkable 3-dimensional long-range order, belonged to the cubic space group Ia3d, and had bicontinuous cubic microstructure. From computer simulations of model structures and comparison with microscopy results, models were proposed for the new morphol. based on the triply periodic G minimal surface (gyroid) discovered by Schoen; similar morphologies have been obsd. in a variety of microphase-sepd. surfactant-water systems. Blends of this diblock with various short-chain homopolymers were used to investigate the compositional extent of the region of Ia3d stability on the phase diagram; the results indicated that the Ia3d phase was stable over a wide range of minority component vol. fractions.
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    Kim, J. K.; Yang, S. Y.; Lee, Y.; Kim, Y. Functional nanomaterials based on block copolymer self-assembly. Prog. Polym. Sci. 2010, 35, 1325– 1349,  DOI: 10.1016/j.progpolymsci.2010.06.002
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    Functional nanomaterials based on block copolymer self-assembly
    Kim, Jin Kon; Yang, Seung Yun; Lee, Youngmin; Kim, Youngsuk
    Progress in Polymer Science (2010), 35 (11), 1325-1349CODEN: PRPSB8; ISSN:0079-6700. (Elsevier Ltd.)
    A review. Block copolymers have received considerable attention as a promising platform for the synthesis of nanomaterials and fabrication of nanostructures because of their self-assembling nature to form periodically ordered structures in the nanometer-scale range. By controlling the compn. and architecture of individual block components, a variety of nanoscale morphologies can be obtained. After a brief overview of the phase behavior of block copolymers, we highlight recent advances in the fabrication of various functional nanomaterials based on block copolymer of self-assembly and their potential applications. Future perspectives on block copolymers are briefly mentioned.
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    Bai, W.; Hannon, A. F.; Gotrik, K. W.; Choi, H. K.; Aissou, K.; Liontos, G.; Ntetsikas, K.; Alexander-Katz, A.; Avgeropoulos, A.; Ross, C. A. Thin film morphologies of bulk-gyroid polystyrene-block-polydimethylsiloxane under solvent vapor annealing. Macromolecules 2014, 47, 6000– 6008,  DOI: 10.1021/ma501293n
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    Thin Film Morphologies of Bulk-Gyroid Polystyrene-block-polydimethylsiloxane under Solvent Vapor Annealing
    Bai, Wubin; Hannon, Adam F.; Gotrik, Kevin W.; Choi, Hong Kyoon; Aissou, Karim; Liontos, George; Ntetsikas, Konstantinos; Alexander-Katz, Alfredo; Avgeropoulos, Apostolos; Ross, Caroline A.
    Macromolecules (Washington, DC, United States) (2014), 47 (17), 6000-6008CODEN: MAMOBX; ISSN:0024-9297. (American Chemical Society)
    Thin film morphologies of a 75.5 kg/mol polystyrene-block-polydimethylsiloxane (PS-b-PDMS) diblock copolymer subject to solvent vapor annealing are described. The PS-b-PDMS has a double-gyroid morphol. in bulk, but as a thin film the morphol. can form spheres, cylinders, perforated lamellae, or gyroids, depending on the film thickness, its commensurability with the microdomain period, and the ratio of toluene:heptane vapors used for the solvent annealing process. The morphologies are described by SCF theory simulations. Thin film structures with excellent long-range order were produced, which are promising for nanopatterning applications.
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    Hsueh, H. Y.; Yao, C. T.; Ho, R. M. Well-ordered nanohybrids and nanoporous materials from gyroid block copolymer templates. Chem. Soc. Rev. 2015, 44, 1974– 2018,  DOI: 10.1039/C4CS00424H
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    Well-ordered nanohybrids and nanoporous materials from gyroid block copolymer templates
    Hsueh, Han-Yu; Yao, Cheng-Thai; Ho, Rong-Ming
    Chemical Society Reviews (2015), 44 (7), 1974-2018CODEN: CSRVBR; ISSN:0306-0012. (Royal Society of Chemistry)
    A review. The design of nanostructured materials and their corresponding morphologies has attracted intense attention because of their effectiveness in tuning electronic, optical, magnetic, and catalytic properties, as well as mech. properties. Although many technologies have been explored to fabricate nanostructured materials, templated synthesis is one of the most important approaches to fabricate nanostructured materials with precisely controlled structures and morphologies from their constituent components. In this review article, we aim to highlight the use of the self-assembly of block copolymers as an emerging and powerful tool to fabricate well-defined nanomaterials with precise control over the structural dimensions and shape, as well as over the compn. and corresponding spatial arrangement. After providing a brief introduction to the synthesis of regular porous materials, including silica- and carbon-based mesoporous materials, the review focuses on the fabrication of well-ordered nanoporous polymers from the selfassembly of degradable block copolymers, in particular with gyroid-forming network morphologies, as templates for the syntheses of various materials with different entities. We highlight the principles of different templated syntheses, from the fundamentals to their practical uses in the fabrication of nanohybrids and nanoporous materials; moreover, we provide an introduction to templates, precursors, solvents, and processing. Finally, some recent examples using block copolymer structure-directed nanomaterials for applications, such as solar cells, catalysis, and drug delivery, are presented. In particular, by taking advantage of the "well-ordered" structural characteristics of the gyroid texture, the properties and applications of 3D regular nanostructures, such as the photonic behavior and optical properties of gyroid-forming nanostructures, as well as of gyroid-forming metamaterials, will be emphasized. Special attention is also given to present new developments and future perspectives in this field.
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    Lich, L. V.; Hue, D. T. H.; Giang, D. T. H.; Duc, N. H.; Shimada, T.; Kitamura, T.; Dinh, V. H. Formation and switching of chiral magnetic field textures in three-dimensional gyroid nanostructures. Acta Mater. 2023, 249, 118802,  DOI: 10.1016/j.actamat.2023.118802
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    Curvature-induced magnetochirality
    Hertel, Riccardo
    SPIN (2013), 3 (3), 1340009/1-1340009/9CODEN: SPINCC; ISSN:2010-3247. (World Scientific Publishing Co. Pte. Ltd.)
    Curved geometries like nanotubes and flexible membranes generally differ from flat films by internal strain, geodesic pathways for transport phenomena, and a break of the local inversion symmetry. In ferromagnetism, these characteristics can lead to surprising effects, esp. when the curvature radius reaches intrinsic length scales, like the domain wall width or the magnon wave length. Simulation studies demonstrate that curved ferromagnetic thin films display magnetochiral properties similar to the Dzyaloshinskii-Moriya interaction (DMI). In close analogy to the emerging field of flexoelectricity, it is suggested that the controlled bending of ferromagnetic membranes provides a new, reversible and universal method to manipulate their magnetic properties.
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    Gaididei, Y.; Kravchuk, V. P.; Sheka, D. D. Curvature Effects in Thin Magnetic Shells. Phys. Rev. Lett. 2014, 112, 257203,  DOI: 10.1103/PhysRevLett.112.257203
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    Curvature effects in thin magnetic shells
    Gaididei, Yuri; Kravchuk, Volodymyr P.; Sheka, Denis D.
    Physical Review Letters (2014), 112 (25), 257203/1-257203/5, 5 pp.CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
    A magnetic energy functional is derived for an arbitrary curved thin shell on the assumption that the magnetostatic effects can be reduced to an effective easy-surface anisotropy; it can be used for solving both static and dynamic problems. General static solns. are obtained in the limit of a strong anisotropy of both signs (easy-surface and easy-normal cases). It is shown that the effect of the curvature can be treated as the appearance of an effective magnetic field, which is aligned along the surface normal for the case of easy-surface anisotropy and is tangential to the surface for the case of easy-normal anisotropy. In general, the existence of such a field excludes the solns. that are strictly tangential or strictly normal to the surface. As an example, we consider static equil. solns. for a cone surface magnetization.
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    Streubel, R.; Fischer, P.; Kronast, F.; Kravchuk, V. P.; Sheka, D. D.; Gaididei, Y.; Schmidt, O. G.; Makarov, D. Magnetism in curved geometries. J. Phys. D: Appl. Phys. 2016, 49, 363001,  DOI: 10.1088/0022-3727/49/36/363001
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    Magnetism in curved geometries
    Streubel, Robert; Fischer, Peter; Kronast, Florian; Kravchuk, Volodymyr P.; Sheka, Denis D.; Gaididei, Yuri; Schmidt, Oliver G.; Makarov, Denys
    Journal of Physics D: Applied Physics (2016), 49 (36), 363001/1-363001/45CODEN: JPAPBE; ISSN:0022-3727. (IOP Publishing Ltd.)
    Extending planar two-dimensional structures into the three-dimensional space has become a general trend in multiple disciplines, including electronics, photonics, plasmonics and magnetics. This approach provides means to modify conventional or to launch novel functionalities by tailoring the geometry of an object, e.g. its local curvature. In a generic electronic system, curvature results in the appearance of scalar and vector geometric potentials inducing anisotropic and chiral effects. In the specific case of magnetism, even in the simplest case of a curved anisotropic Heisenberg magnet, the curvilinear geometry manifests two exchange-driven interactions, namely effective anisotropy and antisym. exchange, i.e. Dzyaloshinskii-Moriya-like interaction. As a consequence, a family of novel curvature-driven effects emerges, which includes magnetochiral effects and topol. induced magnetization patterning, resulting in theor. predicted unlimited domain wall velocities, chirality symmetry breaking and Cherenkov-like effects for magnons. The broad range of altered phys. properties makes these curved architectures appealing in view of fundamental research on e.g. skyrmionic systems, magnonic crystals or exotic spin configurations. In addn. to these rich physics, the application potential of three-dimensionally shaped objects is currently being explored as magnetic field sensorics for magnetofluidic applications, spin-wave filters, advanced magneto-encephalog. devices for diagnosis of epilepsy or for energy-efficient racetrack memory devices. These recent developments ranging from theor. predictions over fabrication of three-dimensionally curved magnetic thin films, hollow cylinders or wires, to their characterization using integral means as well as the development of advanced tomog. approaches are in the focus of this review.
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    Sander, D.; Valenzuela, S. O.; Makarov, D.; Marrows, C. H.; Fullerton, E. E.; Fischer, P.; McCord, J.; Vavassori, P.; Mangin, S.; Pirro, P. The 2017 Magnetism Roadmap. J. Phys. D: Appl. Phys. 2017, 50, 363001,  DOI: 10.1088/1361-6463/aa81a1
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    The 2017 Magnetism Roadmap
    Sander, D.; Valenzuela, S. O.; Makarov, D.; Marrows, C. H.; Fullerton, E. E.; Fischer, P.; McCord, J.; Vavassori, P.; Mangin, S.; Pirro, P.; Hillebrands, B.; Kent, A. D.; Jungwirth, T.; Gutfleisch, O.; Kim, C. G.; Berger, A.
    Journal of Physics D: Applied Physics (2017), 50 (36), 363001/1-363001/33CODEN: JPAPBE; ISSN:0022-3727. (IOP Publishing Ltd.)
    Building upon the success and relevance of the 2014 Magnetism Roadmap, this 2017 Magnetism Roadmap edition follows a similar general layout, even if its focus is naturally shifted, and a different group of experts and, thus, viewpoints are being collected and presented. More importantly, key developments have changed the research landscape in very relevant ways, so that a novel view onto some of the most crucial developments is warranted, and thus, this 2017 Magnetism Roadmap article is a timely endeavour. The change in landscape is hereby not exclusively scientific, but also reflects the magnetism related industrial application portfolio. Specifically, Hard Disk Drive technol., which still dominates digital storage and will continue to do so for many years, if not decades, has now limited its footprint in the scientific and research community, whereas significantly growing interest in magnetism and magnetic materials in relation to energy applications is noticeable, and other technol. fields are emerging as well. Also, more and more work is occurring in which complex topologies of magnetically ordered states are being explored, hereby aiming at a technol. utilization of the very theor. concepts that were recognized by the 2016 Nobel Prize in Physics. Given this somewhat shifted scenario, it seemed appropriate to select topics for this Roadmap article that represent the three core pillars of magnetism, namely magnetic materials, magnetic phenomena and assocd. characterization techniques, as well as applications of magnetism. While many of the contributions in this Roadmap have clearly overlapping relevance in all three fields, their relative focus is mostly assocd. to one of the three pillars. In this way, the interconnecting roles of having suitable magnetic materials, understanding (and being able to characterize) the underlying physics of their behavior and utilizing them for applications and devices is well illustrated, thus giving an accurate snapshot of the world of magnetism in 2017. The article consists of 14 sections, each written by an expert in the field and addressing a specific subject on two pages. Evidently, the depth at which each contribution can describe the subject matter is limited and a full review of their statuses, advances, challenges and perspectives cannot be fully accomplished. Also, magnetism, as a vibrant research field, is too diverse, so that a no. of areas will not be adequately represented here, leaving space for further Roadmap editions in the future. However, this 2017 Magnetism Roadmap article can provide a frame that will enable the reader to judge where each subject and magnetism research field stands overall today and which directions it might take in the foreseeable future. The first material focused pillar of the 2017 Magnetism Roadmap contains five articles, which address the questions of at. scale confinement, 2D, curved and topol. magnetic materials, as well as materials exhibiting unconventional magnetic phase transitions. The second pillar also has five contributions, which are devoted to advances in magnetic characterization, magneto-optics and magneto-plasmonics, ultrafast magnetization dynamics and magnonic transport. The final and application focused pillar has four contributions, which present non-volatile memory technol., antiferromagnetic spintronics, as well as magnet technol. for energy and bio-related applications. As a whole, the 2017 Magnetism Roadmap article, just as with its 2014 predecessor, is intended to act as a ref. point and guideline for emerging research directions in modern magnetism.
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    Shindou, R.; Matsumoto, R.; Murakami, S.; Ohe, J. I. Topological chiral magnonic edge mode in a magnonic crystal. Phys. Rev. B: Condens. Matter Mater. Phys. 2013, 87, 174427,  DOI: 10.1103/PhysRevB.87.174427
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    Topological chiral magnonic edge mode in a magnonic crystal
    Shindou, Ryuichi; Matsumoto, Ryo; Murakami, Shuichi; Ohe, Jun-ichiro
    Physical Review B: Condensed Matter and Materials Physics (2013), 87 (17), 174427/1-174427/11CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)
    Topol. phases have been explored in various fields in physics such as spintronics, photonics, liq. helium, correlated electron system, and cold-at. system. This leads to the recent foundation of emerging materials such as topol. band insulators, topol. photonic crystals, and topol. superconductors/superfluid. In this paper, we propose a topol. magnonic crystal which provides protected chiral edge modes for magnetostatic spin waves. Based on a linearized Landau-Lifshitz equation, we show that a magnonic crystal with the dipolar interaction acquires a spin-wave vol.-mode band with nonzero Chern integer. We argue that such magnonic systems are accompanied by the same integer nos. of chiral spin-wave edge modes within a band gap for the vol.-mode bands. In these edge modes, the spin wave propagates in a unidirectional manner without being scattered backward, which implements novel fault-tolerant spintronic devices.
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    McClarty, P. A. Topological Magnons: A Review. Annu. Rev. Condens. Matter Phys. 2022, 13, 171– 190,  DOI: 10.1146/annurev-conmatphys-031620-104715
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    Concept of artificial magnetoelectric materials via geometrically controlling curvilinear helimagnets
    Volkov, O. M.; Roessler, U. K.; Fassbender, J.; Makarov, D.
    Journal of Physics D: Applied Physics (2019), 52 (34), 345001CODEN: JPAPBE; ISSN:0022-3727. (IOP Publishing Ltd.)
    A novel type of artificial magnetoelec. material, which allows an elec. field-induced deterministic switching between magnetic states without influencing intrinsic magnetic parameters, is proposed. It refers to three dimensional curvilinear helimagnets, e.g. torsion springs, embedded in a piezoelec. matrix. In contrast to conventional strain-coupled magnetoelec. heterostructures based on piezoelec.-magnetostrictive bilayers, we exploit the geometrical coupling of the matrix to the curvilinear helimagnet with intrinsic chiral Dzyaloshinskii-Moriya interactions. Namely, the magnetic state is modified due to the change of geometrical parameters of the curved nanomagnet. Theor., the essence of the proposal is analyzed for a deformable torsional spring made of helimagnetic material. In response to the geometrical change magnetic phase transition between the homogeneous and a periodically modulated state can be driven in a wide range of geometrical parameters. Resulting transformations of the av. magnetization from non-zero to zero value can be uniquely assigned to logical '1' and '0'. As the chiral magnetic properties are easier to control by mech. distortion than effective anisotropies, our concept should lead to a robust design of novel magnetoelec. devices.
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    Beauty in the sciences: The extraordinary history of a three-periodic net and its assocd. surface, the gyroid, is recounted. These structures appear in diverse contexts in mathematics, as the topol. for crystal structures in materials, which is the basis for liq. crystal phases and derived mesoporous materials, and in insect pigments.
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    Spintronics requires the development of magnetic thin film structures having a wide range of magnetic properties. FMR is a well understood exptl. technique that has proven to be an invaluable tool to probe the static and dynamic magnetic properties of ultrathin films, multilayer nanostructures, and superlattices. In order to achieve a full characterization of thin film materials, one needs to carry out FMR measurements at a wide range of microwave frequencies. We show that one does not have to use a broadband vector network analyzer; similar performance can be achieved by a broadband microwave signal generator, a coplanar waveguide, and a broadband microwave detector. To obtain a good signal to noise ratio, one needs to employ a modulation technique to use lock-in detection;. we use low frequency external field modulation (105 Hz) and microwave power amplitude pulse modulation (10 kHz). The sensitivity and the performance of this broadband microwave system is demonstrated on 2 types of samples: MBE grown single crystal GaAs(001)/Fe/Au and sputter deposited textured Si(111)/Ta/Ru/Co/Ru superlattice structures. The samples were mounted on a coplanar waveguide, allowing one a broadband measurement, ∼0.1-50 GHz, of d.c. field swept FMR signals. The results are compared to traditional field swept, field modulated measurements in microwave cavity resonators. Despite the fact that the FMR signal can be very different from that obtained by std. microwave cavities, we show that the anal. of the FMR signal is fairly simple using an admixt. of the in-phase and out-of-phase components of r.f. susceptibility and that the resulting fitted magnetic parameters are in excellent agreement. Addnl., we demonstrate that microwave power amplitude pulse modulation can be used to greatly speed up data collection times, esp. for very weak and broad FMR signals.
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  • Abstract

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    Figure 1

    Figure 1. Representation of the gyroid surface model (ϕ = 0%), highlighting the unit cell structure. The illustration accentuates the unit cell configuration along the crystallographic [111] direction in orthographic projection, unveiling the hexagonal organization inherent in the gyroid’s channels.

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    Figure 2

    Figure 2. Depiction of the gyroid unit cell and geometric modeling for micromagnetic simulations. The focus of our investigation is a cubic gyroid unit cell measuring L = 50 nm, featuring a volume fraction ϕ = 10% as depicted in (a). For the purpose of micromagnetic simulations, a geometric model consisting of an aggregate 4 × 4 × 4 unit cells, or equivalently 200 × 200 × 200 [nm], was employed, as displayed in (b). Illustrated alongside are the three principal high-symmetry directions of the gyroid structure ([111]─yellow triangle, [110]─green rectangle, and [100]─red square) which are color-coordinated to match the planes intersecting the structure. Each direction reveals the unique distribution and shape of the gyroid’s channels: the [111] direction displays a hexagonal pattern and round holes, the [110] direction exhibits a square pattern and lenticular holes, while the [100] direction showcases a square pattern and round holes.

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    Figure 3

    Figure 3. Images of the postrelaxation static configurations of the reduced magnetization component mx(≡Mx/Ms) parallel to the external magnetic field, aligned in the crystallographic direction [100]. Panel (a) displays the case when Bext = 100 mT. The deviation of the magnetization vector from the x-direction is notably larger compared to (b) with Bext = 300 mT, where the magnetization distribution is nearly uniformly oriented along the direction of the external magnetic field. (c) Histogram (number of numerically calculated magnetic moments falling within specific ranges of mx) of the reduced magnetization component distribution mx in the simulated gyroid model for both magnetic field magnitudes. The number of bins indicates the number of elementary simulation elements (tetrahedrons).

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    Figure 4

    Figure 4. Resonance frequency spectra of 8 × 8 × 2 gyroid structures obtained from micromagnetic simulation. The graphical representations showcase the resonance derived from two different orientations of an applied magnetic field: a dashed line represents the signal from a sample subjected to a 300 mT field, directed out-of-plane (along the z-axis); the solid line, meanwhile, illustrates the response of the sample magnetized in the in-plane (x-axis) direction. The employed color scale is representative of the imaginary part of magnetic susceptibility.

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    Figure 5

    Figure 5. Micromagnetic simulation-derived resonance frequency spectra for 4 × 4 × 4 gyroid constructs. The spectra are derived from two scenarios: (a) in which the applied external magnetic field has a strength of 100 mT and (b) where it measures 300 mT. Within each plot, different color coding indicates the crystallographic direction in which the field is applied, with the specific points encircled on the graph signifying the ferromagnetic resonance. Visual illustrations and resonance frequency values, in sequential order of their appearance, are exhibited as insets within the plots. The coloring scheme used here corresponds directly to the imaginary component of the magnetic susceptibility.

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    Figure 6

    Figure 6. Spectral examination of high-intensity volume modes within a 6 × 6 × 6 gyroid structure. The lower section of the figure presents a plotted distribution of the frequency spectra, with the high-intensity volume modes depicted in the upper part distinctly marked by black circles. For the purpose of comparison, spectra corresponding to more compact structures (illustrated as dotted lines) are superimposed on the graph, thereby clearly demonstrating a marked decline in the intensity of edge modes (denoted with crosses) commensurate with the enlargement of the structure dimensions. The color gradation utilized in the visual representation of the modes is proportional to the imaginary component of the magnetic susceptibility.

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    Figure 7

    Figure 7. Multidomain gyroid structure employed in BBFMR measurements, illustrating the complex and varied nature of the sample. In panels (a) and (b), scanning electron microscope (SEM) topographical images offer detailed views of two distinct sample regions, characterized by different crystallographic directions and varying degrees of amorphousness. Panel (c) showcases a photograph of the entire sample, annotated with the approximate locations of several prominent domains. These domains were identified and characterized through polarized light microscopy.

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    Figure 8

    Figure 8. BBFMR measurement conducted on the Ni gyroid structure. The sample underwent repositioning with respect to the CPW to elucidate the effect of an additional homogeneous Ni layer present within the specimen. In two distinct configurations, separate assessments were made of the energy absorption stemming from the microwave field BMW, applied perpendicular to the external static magnetic field (a). The resultant plots of dynamic magnetization amplitude as functions of static magnetic flux density and frequency for selected sample configurations are depicted in (b,c). These render a conspicuous signal attributed to the gyroid layer when the CPW is in direct alignment below it (b), and an additional, higher-frequency signal emanating from the uniform Ni layer (c) when the CPW (as delineated by the red dashed line) intersects its projected position (highlighted in purple on the sample). The dotted lines in the graphs represents the theoretical fit derived from the Kittel formula (see eq 2). For uniform Ni, parameters from micromagnetic simulations were used, while for gyroid, we implemented the calculated effective parameters, i.e., saturation magnetization Meff = 132 kA/m, and the g-factor of 2.2. Plot (d) shows a summary of the peak intensities calculations of FMR signals (blue dots for uniform Ni, and orange dots for gyroid) as a function of the external magnetic field strength. Normalized intensity values for 100, 300, and 450 mT fields are indicated. In graph (e), a cross-sectional analysis of the BBFMR signals is depicted for distinct values of the external magnetic field, where the solid blue line corresponds to Bext = 100 mT, the dashed brown line to Bext = 300 mT, and the dash-dotted green line to Bext = 450 mT. Additionally, horizontal dashed green lines mark the full width at half-maximum (FWHM) for each section, providing quantitative insights into the resonance line widths along with their respective values. The orange crosses signify peak maxima and their corresponding frequencies, pinpointing the resonant behavior within the explored frequency range. Insets furnish intensity plots from the BBFMR measurements, with the green vertical lines highlighting the specific locations of the sections for each magnetic field value. Finally, plot (f) shows the magnetic field FWHM’s as a function of frequency for BBFMR signals of gyroid (purple dots) and uniform Ni (dark red dots). Based on the experimental data and using eq 3, a linear regression was performed and the values of the determination coefficient r2, ΔH0 (from the abscissa of the lines) and the eq 3-derived damping values α (from the slope of the lines) were estimated. Parameters related to the gyroid structure are marked with a prim (′).

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      Nano Letters (2020), 20 (5), 3642-3650CODEN: NALEFD; ISSN:1530-6984. (American Chemical Society)
      Arrays of interacting 2D nanomagnets display unprecedented electromagnetic properties via collective effects, demonstrated in artificial spin ices and magnonic crystals. Progress toward 3D magnetic metamaterials is hampered by two challenges: fabricating 3D structures near intrinsic magnetic length scales (sub-100 nm) and visualizing their magnetic configurations. Here, we fabricate and measure nanoscale magnetic gyroids, periodic chiral networks comprising nanowire-like struts forming three-connected vertices. Via block copolymer templating, we produce Ni75Fe25 single-gyroid and double-gyroid (an inversion pair of single-gyroids) nanostructures with a 42 nm unit cell and 11 nm diam. struts, comparable to the exchange length in Ni-Fe. We visualize their magnetization distributions via off-axis electron holog. with nanometer spatial resoln. and interpret the patterns using finite-element micromagnetic simulations. Our results suggest an intricate, frustrated remanent state which is ferromagnetic but without a unique equil. configuration, opening new possibilities for collective phenomena in magnetism, including 3D magnonic crystals and unconventional computing.
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    17. 17
      Fernández-Pacheco, A.; Streubel, R.; Fruchart, O.; Hertel, R.; Fischer, P.; Cowburn, R. P. Three-dimensional nanomagnetism. Nat. Commun. 2017, 8, 15756,  DOI: 10.1038/ncomms15756
      17
      Three-dimensional nanomagnetism
      Fernandez-Pacheco Amalio; Cowburn Russell P; Streubel Robert; Fischer Peter; Fruchart Olivier; Hertel Riccardo; Fischer Peter
      Nature communications (2017), 8 (), 15756 ISSN:.
      There is no expanded citation for this reference.
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    18. 18
      Donnelly, C.; Hierro-Rodríguez, A.; Abert, C.; Witte, K.; Skoric, L.; Sanz-Hernández, D.; Finizio, S.; Meng, F.; McVitie, S.; Raabe, J.; Suess, D.; Cowburn, R.; Fernández-Pacheco, A. Complex free-space magnetic field textures induced by three-dimensional magnetic nanostructures. Nat. Nanotechnol. 2022, 17, 136– 142,  DOI: 10.1038/s41565-021-01027-7
      18
      Complex free-space magnetic field textures induced by three-dimensional magnetic nanostructures
      Donnelly, Claire; Hierro-Rodriguez, Aurelio; Abert, Claas; Witte, Katharina; Skoric, Luka; Sanz-Hernandez, Dedalo; Finizio, Simone; Meng, Fanfan; McVitie, Stephen; Raabe, Jorg; Suess, Dieter; Cowburn, Russell; Fernandez-Pacheco, Amalio
      Nature Nanotechnology (2022), 17 (2), 136-142CODEN: NNAABX; ISSN:1748-3387. (Nature Portfolio)
      Abstr.: The design of complex, competing effects in magnetic systems-be it via the introduction of nonlinear interactions1-4, or the patterning of three-dimensional geometries5,6-is an emerging route to achieve new functionalities. In particular, through the design of three-dimensional geometries and curvature, intrastructure properties such as anisotropy and chirality, both geometry-induced and intrinsic, can be directly controlled, leading to a host of new physics and functionalities, such as three-dimensional chiral spin states7, ultrafast chiral domain wall dynamics8-10 and spin textures with new spin topologies7,11. Here, we advance beyond the control of intrastructure properties in three dimensions and tailor the magnetostatic coupling of neighboring magnetic structures, an interstructure property that allows us to generate complex textures in the magnetic stray field. For this, we harness direct write nanofabrication techniques, creating intertwined nanomagnetic cobalt double helixes, where curvature, torsion, chirality and magnetic coupling are jointly exploited. By reconstructing the three-dimensional vectorial magnetic state of the double helixes with soft-X-ray magnetic laminog.12,13, we identify the presence of a regular array of highly coupled locked domain wall pairs in neighboring helixes. Micromagnetic simulations reveal that the magnetization configuration leads to the formation of an array of complex textures in the magnetic induction, consisting of vortices in the magnetization and antivortices in free space, which together form an effective B field cross-tie wall14. The design and creation of complex three-dimensional magnetic field nanotextures opens new possibilities for smart materials15, unconventional computing2,16, particle trapping17,18 and magnetic imaging19.
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    19. 19
      van den Berg, A.; Caruel, M.; Hunt, M.; Ladak, S. Combining two-photon lithography with laser ablation of sacrificial layers: A route to isolated 3D magnetic nanostructures. Nano Res. 2023, 16, 1441– 1447,  DOI: 10.1007/s12274-022-4649-z
      There is no corresponding record for this reference.
    20. 20
      Schoen, A. H. Infinite periodic minimal surfaces without self-intersections; National Aeronautics and Space Administration, 1970.
      There is no corresponding record for this reference.
    21. 21
      Lambert, C. A.; Radzilowski, L. H.; Thomas, E. L. Triply periodic level surfaces as models for cubic tricontinuous block copolymer morphologies. Philos. Trans. R. Soc., A 1996, 354, 2009– 2023,  DOI: 10.1098/rsta.1996.0089
      21
      Triply periodic level surfaces as models for cubic tricontinuous block copolymer morphologies
      Lambert, Charla A.; Radzilowski, Leonard H.; Thomas, Edwin L.
      Philosophical Transactions of the Royal Society of London, Series A: Mathematical, Physical and Engineering Sciences (1996), 354 (1715), 2009-2023CODEN: PTRMAD; ISSN:0962-8428. (Royal Society)
      The domains of microphase sepd. block copolymers develop interfacial surfaces of approx. const. mean curvature in response to thermodn. driving forces. Of particular recent interest are the tricontinuous triply periodic morphologies and their math. representations. Level surfaces are represented by certain real functions which satisfy the expression F(x, y, z) = t, where t is a const. In general, they are non-self-intersecting and smooth, except at special values of the parameter t. The authors construct periodic level surfaces according to the allowed reflections of a particular cubic space group; such triply periodic surfaces maintain the symmetries of the chosen space group and make attractive approxns. to certain recently computed triply periodic surfaces of const. mean curvature. This paper is a study of the accuracy of the approxns. constructed using the lowest Fourier term of the Pm3-m, Fd3-m, and I4132 space groups, and the usefulness of these approxns. in analyzing exptl. obsd. tricontinuous block copolymer morphologies at a variety of vol. fractions. The authors numerically compare surface area per unit vol. of particular level surfaces with const. mean curvature surfaces having the same vol. fraction. The authors also demonstrate the utility of level surfaces in simulating projections of tricontinuous microdomain morphologies for comparison with actual transmission electron micrographs and detn. of block copolymer microstructure.
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    22. 22
      Turner, M. D.; Saba, M.; Zhang, Q.; Cumming, B. P.; Schröder-Turk, G. E.; Gu, M. Miniature chiral beamsplitter based on gyroid photonic crystals. Nat. Photonics 2013, 7, 801– 805,  DOI: 10.1038/nphoton.2013.233
      22
      Miniature chiral beamsplitter based on gyroid photonic crystals
      Turner, Mark D.; Saba, Matthias; Zhang, Qiming; Cumming, Benjamin P.; Schroeder-Turk, Gerd E.; Gu, Min
      Nature Photonics (2013), 7 (10), 801-805CODEN: NPAHBY; ISSN:1749-4885. (Nature Publishing Group)
      The linearly polarizing beamsplitter is a widely used optical component in photonics. It is typically built from a linearly birefringent crystal such as calcite, which has different crit. reflection angles for s- and p-polarized light, leading to the transmission of one linear polarization and angled reflection of the other. However, the analog for splitting circularly polarized light has yet to be demonstrated due to a lack of natural materials with sufficient circular birefringence. Here, we present a nano-engineered photonic-crystal chiral beamsplitter that fulfils this task. It consists of a prism featuring a nanoscale chiral gyroid network and can sep. left- and right-handed circularly polarized light in the wavelength region around 1.615 μm. The structure is fabricated using a galvo-dithered direct laser writing method and could become a useful component for developing integrated photonic circuits that provide a new form of polarization control.
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    23. 23
      Vignolini, S.; Yufa, N. A.; Cunha, P. S.; Guldin, S.; Rushkin, I.; Stefik, M.; Hur, K.; Wiesner, U.; Baumberg, J. J.; Steiner, U. A 3D optical metamaterial made by self-assembly. Adv. Mater. 2012, 24, OP23– OP27,  DOI: 10.1002/adma.201103610
      23
      A 3D Optical Metamaterial Made by Self-Assembly
      Vignolini, Silvia; Yufa, Nataliya A.; Cunha, Pedro S.; Guldin, Stefan; Rushkin, Ilia; Stefik, Morgan; Hur, Kahyun; Wiesner, Ulrich; Baumberg, Jeremy J.; Steiner, Ullrich
      Advanced Materials (Weinheim, Germany) (2012), 24 (10), OP23-OP27CODEN: ADVMEW; ISSN:0935-9648. (Wiley-VCH Verlag GmbH & Co. KGaA)
      The authors demonstrate the creation of a 3-dimensional Au metamaterial based on block copolymer (BCP) self-assembly. The authors start with an isoprene-block-styrene-block-ethylene oxide (ISO) BCP that forms 2 chem. distinct, interpenetrating gyroid networks (1,0) of opposite chirality in a matrix of the 3rd block. The I gyroid network is then removed by selective UV and chem. etching and back-filled with Au by electrodeposition. The final device consists of a continuous, triply periodic network of Au. The dimension of the full unit cell is = 50 nm, which is far below optical wavelengths. This particular morphol. was chosen since it is predicted to offer a strong resonant response that depends on the relative orientation between the structure and the polarization of the incident light.
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    24. 24
      Dolan, J. A.; Wilts, B. D.; Vignolini, S.; Baumberg, J. J.; Steiner, U.; Wilkinson, T. D. Optical Properties of Gyroid Structured Materials: From Photonic Crystals to Metamaterials. Adv. Opt. Mater. 2015, 3, 12– 32,  DOI: 10.1002/adom.201400333
      24
      Optical Properties of Gyroid Structured Materials: From Photonic Crystals to Metamaterials
      Dolan, James A.; Wilts, Bodo D.; Vignolini, Silvia; Baumberg, Jeremy J.; Steiner, Ullrich; Wilkinson, Timothy D.
      Advanced Optical Materials (2015), 3 (1), 12-32CODEN: AOMDAX; ISSN:2195-1071. (Wiley-VCH Verlag GmbH & Co. KGaA)
      The gyroid is a continuous and triply periodic cubic morphol. which possesses a const. mean curvature surface across a range of volumetric fill fractions. Found in a variety of natural and synthetic systems which form through self-assembly, from butterfly wing scales to block copolymers, the gyroid also exhibits an inherent chirality not obsd. in any other similar morphologies. These unique geometrical properties impart to gyroid structured materials a host of interesting optical properties. Depending on the length scale on which the constituent materials are organized, these properties arise from starkly different phys. mechanisms (such as a complete photonic bandgap for photonic crystals and a greatly depressed plasma frequency for optical metamaterials). This article reviews the theor. predictions and exptl. observations of the optical properties of two fundamental classes of gyroid structured materials: photonic crystals (wavelength scale) and metamaterials (sub-wavelength scale).
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    25. 25
      Michielsen, K.; Stavenga, D. G. Gyroid cuticular structures in butterfly wing scales: Biological photonic crystals. J. R. Soc., Interface 2008, 5, 85– 94,  DOI: 10.1098/rsif.2007.1065
      25
      Gyroid cuticular structures in butterfly wing scales: biological photonic crystals
      Michielsen K; Stavenga D G
      Journal of the Royal Society, Interface (2008), 5 (18), 85-94 ISSN:1742-5689.
      We present a systematic study of the cuticular structure in the butterfly wing scales of some papilionids (Parides sesostris and Teinopalpus imperialis) and lycaenids (Callophrys rubi, Cyanophrys remus, Mitoura gryneus and Callophrys dumetorum). Using published scanning and transmission electron microscopy (TEM) images, analytical modelling and computer-generated TEM micrographs, we find that the three-dimensional cuticular structures can be modelled by gyroid structures with various filling fractions and lattice parameters. We give a brief discussion of the formation of cubic gyroid membranes from the smooth endoplasmic reticulum in the scale's cell, which dry and harden to leave the cuticular structure behind when the cell dies. The scales of C. rubi are a potentially attractive biotemplate for producing three-dimensional optical photonic crystals since for these scales the cuticle-filling fraction is nearly optimal for obtaining the largest photonic band gap in a gyroid structure.
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    26. 26
      Saranathan, V.; Osuji, C. O.; Mochrie, S. G.; Noh, H.; Narayanan, S.; Sandy, A.; Dufresne, E. R.; Prum, R. O. Structure, function, and self-assembly of single network gyroid (I4 132) photonic crystals in butterfly wing scales. Proc. Natl. Acad. Sci. U.S.A. 2010, 107, 11676– 11681,  DOI: 10.1073/pnas.0909616107
      26
      Structure, function, and self-assembly of single network gyroid (I4132) photonic crystals in butterfly wing scales
      Saranathan, Vinodkumar; Osuji, Chinedum O.; Mochrie, Simon G. J.; Noh, Heeso; Narayanan, Suresh; Sandy, Alec; Dufresne, Eric R.; Prum, Richard O.
      Proceedings of the National Academy of Sciences of the United States of America (2010), 107 (26), 11676-11681, S11676/1-S11676/7CODEN: PNASA6; ISSN:0027-8424. (National Academy of Sciences)
      Complex three-dimensional biophotonic nanostructures produce the vivid structural colors of many butterfly wing scales, but their exact nanoscale organization is uncertain. The authors used small angle x-ray scattering (SAXS) on single scales to characterize the 3D photonic nanostructures of five butterfly species from two families (Papilionidae, Lycaenidae). The authors identify these chitin and air nanostructures as single network gyroid (I4132) photonic crystals. The authors describe their optical function from SAXS data and photonic band-gap modeling. Butterflies apparently grow these gyroid nanostructures by exploiting the self-organizing phys. dynamics of biol. lipid-bilayer membranes. These butterfly photonic nanostructures initially develop within scale cells as a core-shell double gyroid (Ia3d), as seen in block-copolymer systems, with a pentacontinuous vol. comprised of extracellular space, cell plasma membrane, cellular cytoplasm, smooth endoplasmic reticulum (SER) membrane, and intra-SER lumen. This double gyroid nanostructure is subsequently transformed into a single gyroid network through the deposition of chitin in the extracellular space and the degeneration of the rest of the cell. The butterflies develop the thermodynamically favored double gyroid precursors as a route to the optically more efficient single gyroid nanostructures. Current approaches to photonic crystal engineering also aim to produce single gyroid motifs. The biol. derived photonic nanostructures characterized here may offer a convenient template for producing optical devices based on biomimicry or direct dielec. infiltration.
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    27. 27
      Schröder-Turk, G.; Wickham, S.; Averdunk, H.; Brink, F.; Fitz Gerald, J. D.; Poladian, L.; Large, M. C.; Hyde, S. T. The chiral structure of porous chitin within the wing-scales of Callophrys rubi. J. Struct. Biol. 2011, 174, 290– 295,  DOI: 10.1016/j.jsb.2011.01.004
      27
      The chiral structure of porous chitin within the wing-scales of Callophrys rubi
      Schroder-Turk G E; Wickham S; Averdunk H; Brink F; Fitz Gerald J D; Poladian L; Large M C J; Hyde S T
      Journal of structural biology (2011), 174 (2), 290-5 ISSN:.
      The structure of the porous three-dimensional reticulated pattern in the wing scales of the butterfly Callophrys rubi (the Green Hairstreak) is explored in detail, via scanning and transmission electron microscopy. A full 3D tomographic reconstruction of a section of this material reveals that the predominantly chitin material is assembled in the wing scale to form a structure whose geometry bears a remarkable correspondence to the srs net, well-known in solid state chemistry and soft materials science. The porous solid is bounded to an excellent approximation by a parallel surface to the Gyroid, a three-periodic minimal surface with cubic crystallographic symmetry I4132, as foreshadowed by Stavenga and Michielson. The scale of the structure is commensurate with the wavelength of visible light, with an edge of the conventional cubic unit cell of the parallel-Gyroid of approximately 310 nm. The genesis of this structure is discussed, and we suggest it affords a remarkable example of templating of a chiral material via soft matter, analogous to the formation of mesoporous silica via surfactant assemblies in solution. In the butterfly, the templating is achieved by the lipid-protein membranes within the smooth endoplasmic reticulum (while it remains in the chrysalis), that likely form cubic membranes, folded according to the form of the Gyroid. The subsequent formation of the chiral hard chitin framework is suggested to be driven by the gradual polymerisation of the chitin precursors, whose inherent chiral assembly in solution (during growth) promotes the formation of a single enantiomer.
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    28. 28
      Yan, C.; Hao, L.; Hussein, A.; Raymont, D. Evaluations of cellular lattice structures manufactured using selective laser melting. Int. J. Mach. Tool Manufact. 2012, 62, 32– 38,  DOI: 10.1016/j.ijmachtools.2012.06.002
      There is no corresponding record for this reference.
    29. 29
      Yánez, A.; Herrera, A.; Martel, O.; Monopoli, D.; Afonso, H. Compressive behaviour of gyroid lattice structures for human cancellous bone implant applications. Mater. Sci. Eng., C 2016, 68, 445– 448,  DOI: 10.1016/j.msec.2016.06.016
      29
      Compressive behavior of gyroid lattice structures for human cancellous bone implant applications
      Yanez, A.; Herrera, A.; Martel, O.; Monopoli, D.; Afonso, H.
      Materials Science & Engineering, C: Materials for Biological Applications (2016), 68 (), 445-448CODEN: MSCEEE; ISSN:0928-4931. (Elsevier B.V.)
      Electron beam melting (EBM) was used to fabricate porous titanium alloy structures. The elastic modulus of these porous structures was similar to the elastic modulus of the cancellous human bone. Two types of cellular lattice structures were manufd. and tested: gyroids and diamonds. The design of the gyroid structures was detd. by the main angle of the struts with respect to the axial direction. Thus, structures with angles of between 19 and 68.5° were manufd. The aim of the design was to reduce the amt. of material needed to fabricate a structure with the desired angles to increase the range of stiffness of the scaffolds. Compression tests were conducted to obtain the elastic modulus and the strength. Both parameters increased as the angle decreased. Finally, the specific strength of the gyroid structures was compared with that of the diamond structures and other types of structures. It is shown that, for angles lower than 35°, the gyroid structures had a high strength to wt. ratios.
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    30. 30
      Armatas, G. S.; Kanatzidis, M. G. Mesostructured germanium with cubic pore symmetry. Nature 2006, 441, 1122– 1125,  DOI: 10.1038/nature04833
      30
      Mesostructured germanium with cubic pore symmetry
      Armatas, Gerasimos S.; Kanatzidis, Mercouri G.
      Nature (London, United Kingdom) (2006), 441 (7097), 1122-1125CODEN: NATUAS; ISSN:0028-0836. (Nature Publishing Group)
      Here we describe cubic mesostructured germanium, MSU-Ge-1, with gyroidal channels contg. surfactant mols., sepd. by amorphous walls that lie on the gyroid (G) minimal surface as in the mesoporous silica MCM-48 (ref. 2). Although Ge is a high-melting, covalent semiconductor that is difficult to prep. from soln. polymn., we succeeded in assembling a continuous Ge network using a suitable precursor for Ge4- atoms. Our results indicate that elemental semiconductors from group 14 of the periodic table can be made to adopt mesostructured forms such as MSU-Ge-1, which features two three-dimensional labyrinthine tunnels obeying Ia3d space group symmetry and sepd. by a continuous germanium minimal surface that is otherwise amorphous. A consequence of this new structure for germanium, which has walls only one nanometer thick, is a wider electronic energy bandgap (1.4 eV vs. 0.66 eV) than has cryst. or amorphous Ge. Controlled oxidn. of MSU-Ge-1 creates a range of germanium suboxides with continuously varying Ge:O ratio and a smoothly increasing energy gap.
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    31. 31
      Hajduk, D. A.; Harper, P. E.; Gruner, S. M.; Honeker, C. C.; Kim, G.; Thomas, E. L.; Kim, G. The Gyroid: A New Equilibrium Morphology in Weakly Segregated Diblock Copolymers. Macromolecules 1994, 27, 4063– 4075,  DOI: 10.1021/ma00093a006
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      The Gyroid: A New Equilibrium Morphology in Weakly Segregated Diblock Copolymers
      Hajduk, Damian A.; Harper, Paul E.; Gruner, Sol M.; Honeker, Christian C.; Kim, Gia; Thomas, Edwin L.; Fetters, Lewis J.
      Macromolecules (1994), 27 (15), 4063-75CODEN: MAMOBX; ISSN:0024-9297.
      A new equil. microdomain morphol. was identified in an intermediate to weakly segregated diblock copolymer melt. A styrene (I)-isoprene (SI) diblock copolymer with Mw = 27,400 and 37 wt.% I thermo-reversibly transformed from the lamellar morphol. (in equil. at low annealing temps.) to a new morphol. at annealing temps. ∼50° below the order-disorder transition. SAXS and TEM study of this new morphol. revealed that the new structure had remarkable 3-dimensional long-range order, belonged to the cubic space group Ia3d, and had bicontinuous cubic microstructure. From computer simulations of model structures and comparison with microscopy results, models were proposed for the new morphol. based on the triply periodic G minimal surface (gyroid) discovered by Schoen; similar morphologies have been obsd. in a variety of microphase-sepd. surfactant-water systems. Blends of this diblock with various short-chain homopolymers were used to investigate the compositional extent of the region of Ia3d stability on the phase diagram; the results indicated that the Ia3d phase was stable over a wide range of minority component vol. fractions.
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    32. 32
      Kim, J. K.; Yang, S. Y.; Lee, Y.; Kim, Y. Functional nanomaterials based on block copolymer self-assembly. Prog. Polym. Sci. 2010, 35, 1325– 1349,  DOI: 10.1016/j.progpolymsci.2010.06.002
      32
      Functional nanomaterials based on block copolymer self-assembly
      Kim, Jin Kon; Yang, Seung Yun; Lee, Youngmin; Kim, Youngsuk
      Progress in Polymer Science (2010), 35 (11), 1325-1349CODEN: PRPSB8; ISSN:0079-6700. (Elsevier Ltd.)
      A review. Block copolymers have received considerable attention as a promising platform for the synthesis of nanomaterials and fabrication of nanostructures because of their self-assembling nature to form periodically ordered structures in the nanometer-scale range. By controlling the compn. and architecture of individual block components, a variety of nanoscale morphologies can be obtained. After a brief overview of the phase behavior of block copolymers, we highlight recent advances in the fabrication of various functional nanomaterials based on block copolymer of self-assembly and their potential applications. Future perspectives on block copolymers are briefly mentioned.
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    33. 33
      Bai, W.; Hannon, A. F.; Gotrik, K. W.; Choi, H. K.; Aissou, K.; Liontos, G.; Ntetsikas, K.; Alexander-Katz, A.; Avgeropoulos, A.; Ross, C. A. Thin film morphologies of bulk-gyroid polystyrene-block-polydimethylsiloxane under solvent vapor annealing. Macromolecules 2014, 47, 6000– 6008,  DOI: 10.1021/ma501293n
      33
      Thin Film Morphologies of Bulk-Gyroid Polystyrene-block-polydimethylsiloxane under Solvent Vapor Annealing
      Bai, Wubin; Hannon, Adam F.; Gotrik, Kevin W.; Choi, Hong Kyoon; Aissou, Karim; Liontos, George; Ntetsikas, Konstantinos; Alexander-Katz, Alfredo; Avgeropoulos, Apostolos; Ross, Caroline A.
      Macromolecules (Washington, DC, United States) (2014), 47 (17), 6000-6008CODEN: MAMOBX; ISSN:0024-9297. (American Chemical Society)
      Thin film morphologies of a 75.5 kg/mol polystyrene-block-polydimethylsiloxane (PS-b-PDMS) diblock copolymer subject to solvent vapor annealing are described. The PS-b-PDMS has a double-gyroid morphol. in bulk, but as a thin film the morphol. can form spheres, cylinders, perforated lamellae, or gyroids, depending on the film thickness, its commensurability with the microdomain period, and the ratio of toluene:heptane vapors used for the solvent annealing process. The morphologies are described by SCF theory simulations. Thin film structures with excellent long-range order were produced, which are promising for nanopatterning applications.
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    34. 34
      Hsueh, H. Y.; Yao, C. T.; Ho, R. M. Well-ordered nanohybrids and nanoporous materials from gyroid block copolymer templates. Chem. Soc. Rev. 2015, 44, 1974– 2018,  DOI: 10.1039/C4CS00424H
      34
      Well-ordered nanohybrids and nanoporous materials from gyroid block copolymer templates
      Hsueh, Han-Yu; Yao, Cheng-Thai; Ho, Rong-Ming
      Chemical Society Reviews (2015), 44 (7), 1974-2018CODEN: CSRVBR; ISSN:0306-0012. (Royal Society of Chemistry)
      A review. The design of nanostructured materials and their corresponding morphologies has attracted intense attention because of their effectiveness in tuning electronic, optical, magnetic, and catalytic properties, as well as mech. properties. Although many technologies have been explored to fabricate nanostructured materials, templated synthesis is one of the most important approaches to fabricate nanostructured materials with precisely controlled structures and morphologies from their constituent components. In this review article, we aim to highlight the use of the self-assembly of block copolymers as an emerging and powerful tool to fabricate well-defined nanomaterials with precise control over the structural dimensions and shape, as well as over the compn. and corresponding spatial arrangement. After providing a brief introduction to the synthesis of regular porous materials, including silica- and carbon-based mesoporous materials, the review focuses on the fabrication of well-ordered nanoporous polymers from the selfassembly of degradable block copolymers, in particular with gyroid-forming network morphologies, as templates for the syntheses of various materials with different entities. We highlight the principles of different templated syntheses, from the fundamentals to their practical uses in the fabrication of nanohybrids and nanoporous materials; moreover, we provide an introduction to templates, precursors, solvents, and processing. Finally, some recent examples using block copolymer structure-directed nanomaterials for applications, such as solar cells, catalysis, and drug delivery, are presented. In particular, by taking advantage of the "well-ordered" structural characteristics of the gyroid texture, the properties and applications of 3D regular nanostructures, such as the photonic behavior and optical properties of gyroid-forming nanostructures, as well as of gyroid-forming metamaterials, will be emphasized. Special attention is also given to present new developments and future perspectives in this field.
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    35. 35
      Lich, L. V.; Hue, D. T. H.; Giang, D. T. H.; Duc, N. H.; Shimada, T.; Kitamura, T.; Dinh, V. H. Formation and switching of chiral magnetic field textures in three-dimensional gyroid nanostructures. Acta Mater. 2023, 249, 118802,  DOI: 10.1016/j.actamat.2023.118802
      There is no corresponding record for this reference.
    36. 36
      Hertel, R. Curvature-induced magnetochirality. SPIN 2013, 03, 1340009
      36
      Curvature-induced magnetochirality
      Hertel, Riccardo
      SPIN (2013), 3 (3), 1340009/1-1340009/9CODEN: SPINCC; ISSN:2010-3247. (World Scientific Publishing Co. Pte. Ltd.)
      Curved geometries like nanotubes and flexible membranes generally differ from flat films by internal strain, geodesic pathways for transport phenomena, and a break of the local inversion symmetry. In ferromagnetism, these characteristics can lead to surprising effects, esp. when the curvature radius reaches intrinsic length scales, like the domain wall width or the magnon wave length. Simulation studies demonstrate that curved ferromagnetic thin films display magnetochiral properties similar to the Dzyaloshinskii-Moriya interaction (DMI). In close analogy to the emerging field of flexoelectricity, it is suggested that the controlled bending of ferromagnetic membranes provides a new, reversible and universal method to manipulate their magnetic properties.
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    37. 37
      Gaididei, Y.; Kravchuk, V. P.; Sheka, D. D. Curvature Effects in Thin Magnetic Shells. Phys. Rev. Lett. 2014, 112, 257203,  DOI: 10.1103/PhysRevLett.112.257203
      37
      Curvature effects in thin magnetic shells
      Gaididei, Yuri; Kravchuk, Volodymyr P.; Sheka, Denis D.
      Physical Review Letters (2014), 112 (25), 257203/1-257203/5, 5 pp.CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
      A magnetic energy functional is derived for an arbitrary curved thin shell on the assumption that the magnetostatic effects can be reduced to an effective easy-surface anisotropy; it can be used for solving both static and dynamic problems. General static solns. are obtained in the limit of a strong anisotropy of both signs (easy-surface and easy-normal cases). It is shown that the effect of the curvature can be treated as the appearance of an effective magnetic field, which is aligned along the surface normal for the case of easy-surface anisotropy and is tangential to the surface for the case of easy-normal anisotropy. In general, the existence of such a field excludes the solns. that are strictly tangential or strictly normal to the surface. As an example, we consider static equil. solns. for a cone surface magnetization.
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    38. 38
      Streubel, R.; Fischer, P.; Kronast, F.; Kravchuk, V. P.; Sheka, D. D.; Gaididei, Y.; Schmidt, O. G.; Makarov, D. Magnetism in curved geometries. J. Phys. D: Appl. Phys. 2016, 49, 363001,  DOI: 10.1088/0022-3727/49/36/363001
      38
      Magnetism in curved geometries
      Streubel, Robert; Fischer, Peter; Kronast, Florian; Kravchuk, Volodymyr P.; Sheka, Denis D.; Gaididei, Yuri; Schmidt, Oliver G.; Makarov, Denys
      Journal of Physics D: Applied Physics (2016), 49 (36), 363001/1-363001/45CODEN: JPAPBE; ISSN:0022-3727. (IOP Publishing Ltd.)
      Extending planar two-dimensional structures into the three-dimensional space has become a general trend in multiple disciplines, including electronics, photonics, plasmonics and magnetics. This approach provides means to modify conventional or to launch novel functionalities by tailoring the geometry of an object, e.g. its local curvature. In a generic electronic system, curvature results in the appearance of scalar and vector geometric potentials inducing anisotropic and chiral effects. In the specific case of magnetism, even in the simplest case of a curved anisotropic Heisenberg magnet, the curvilinear geometry manifests two exchange-driven interactions, namely effective anisotropy and antisym. exchange, i.e. Dzyaloshinskii-Moriya-like interaction. As a consequence, a family of novel curvature-driven effects emerges, which includes magnetochiral effects and topol. induced magnetization patterning, resulting in theor. predicted unlimited domain wall velocities, chirality symmetry breaking and Cherenkov-like effects for magnons. The broad range of altered phys. properties makes these curved architectures appealing in view of fundamental research on e.g. skyrmionic systems, magnonic crystals or exotic spin configurations. In addn. to these rich physics, the application potential of three-dimensionally shaped objects is currently being explored as magnetic field sensorics for magnetofluidic applications, spin-wave filters, advanced magneto-encephalog. devices for diagnosis of epilepsy or for energy-efficient racetrack memory devices. These recent developments ranging from theor. predictions over fabrication of three-dimensionally curved magnetic thin films, hollow cylinders or wires, to their characterization using integral means as well as the development of advanced tomog. approaches are in the focus of this review.
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    39. 39
      Sander, D.; Valenzuela, S. O.; Makarov, D.; Marrows, C. H.; Fullerton, E. E.; Fischer, P.; McCord, J.; Vavassori, P.; Mangin, S.; Pirro, P. The 2017 Magnetism Roadmap. J. Phys. D: Appl. Phys. 2017, 50, 363001,  DOI: 10.1088/1361-6463/aa81a1
      39
      The 2017 Magnetism Roadmap
      Sander, D.; Valenzuela, S. O.; Makarov, D.; Marrows, C. H.; Fullerton, E. E.; Fischer, P.; McCord, J.; Vavassori, P.; Mangin, S.; Pirro, P.; Hillebrands, B.; Kent, A. D.; Jungwirth, T.; Gutfleisch, O.; Kim, C. G.; Berger, A.
      Journal of Physics D: Applied Physics (2017), 50 (36), 363001/1-363001/33CODEN: JPAPBE; ISSN:0022-3727. (IOP Publishing Ltd.)
      Building upon the success and relevance of the 2014 Magnetism Roadmap, this 2017 Magnetism Roadmap edition follows a similar general layout, even if its focus is naturally shifted, and a different group of experts and, thus, viewpoints are being collected and presented. More importantly, key developments have changed the research landscape in very relevant ways, so that a novel view onto some of the most crucial developments is warranted, and thus, this 2017 Magnetism Roadmap article is a timely endeavour. The change in landscape is hereby not exclusively scientific, but also reflects the magnetism related industrial application portfolio. Specifically, Hard Disk Drive technol., which still dominates digital storage and will continue to do so for many years, if not decades, has now limited its footprint in the scientific and research community, whereas significantly growing interest in magnetism and magnetic materials in relation to energy applications is noticeable, and other technol. fields are emerging as well. Also, more and more work is occurring in which complex topologies of magnetically ordered states are being explored, hereby aiming at a technol. utilization of the very theor. concepts that were recognized by the 2016 Nobel Prize in Physics. Given this somewhat shifted scenario, it seemed appropriate to select topics for this Roadmap article that represent the three core pillars of magnetism, namely magnetic materials, magnetic phenomena and assocd. characterization techniques, as well as applications of magnetism. While many of the contributions in this Roadmap have clearly overlapping relevance in all three fields, their relative focus is mostly assocd. to one of the three pillars. In this way, the interconnecting roles of having suitable magnetic materials, understanding (and being able to characterize) the underlying physics of their behavior and utilizing them for applications and devices is well illustrated, thus giving an accurate snapshot of the world of magnetism in 2017. The article consists of 14 sections, each written by an expert in the field and addressing a specific subject on two pages. Evidently, the depth at which each contribution can describe the subject matter is limited and a full review of their statuses, advances, challenges and perspectives cannot be fully accomplished. Also, magnetism, as a vibrant research field, is too diverse, so that a no. of areas will not be adequately represented here, leaving space for further Roadmap editions in the future. However, this 2017 Magnetism Roadmap article can provide a frame that will enable the reader to judge where each subject and magnetism research field stands overall today and which directions it might take in the foreseeable future. The first material focused pillar of the 2017 Magnetism Roadmap contains five articles, which address the questions of at. scale confinement, 2D, curved and topol. magnetic materials, as well as materials exhibiting unconventional magnetic phase transitions. The second pillar also has five contributions, which are devoted to advances in magnetic characterization, magneto-optics and magneto-plasmonics, ultrafast magnetization dynamics and magnonic transport. The final and application focused pillar has four contributions, which present non-volatile memory technol., antiferromagnetic spintronics, as well as magnet technol. for energy and bio-related applications. As a whole, the 2017 Magnetism Roadmap article, just as with its 2014 predecessor, is intended to act as a ref. point and guideline for emerging research directions in modern magnetism.
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    40. 40
      Sheka, D. D. A perspective on curvilinear magnetism. Appl. Phys. Lett. 2021, 118, 230502,  DOI: 10.1063/5.0048891
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      Shindou, R.; Matsumoto, R.; Murakami, S.; Ohe, J. I. Topological chiral magnonic edge mode in a magnonic crystal. Phys. Rev. B: Condens. Matter Mater. Phys. 2013, 87, 174427,  DOI: 10.1103/PhysRevB.87.174427
      41
      Topological chiral magnonic edge mode in a magnonic crystal
      Shindou, Ryuichi; Matsumoto, Ryo; Murakami, Shuichi; Ohe, Jun-ichiro
      Physical Review B: Condensed Matter and Materials Physics (2013), 87 (17), 174427/1-174427/11CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)
      Topol. phases have been explored in various fields in physics such as spintronics, photonics, liq. helium, correlated electron system, and cold-at. system. This leads to the recent foundation of emerging materials such as topol. band insulators, topol. photonic crystals, and topol. superconductors/superfluid. In this paper, we propose a topol. magnonic crystal which provides protected chiral edge modes for magnetostatic spin waves. Based on a linearized Landau-Lifshitz equation, we show that a magnonic crystal with the dipolar interaction acquires a spin-wave vol.-mode band with nonzero Chern integer. We argue that such magnonic systems are accompanied by the same integer nos. of chiral spin-wave edge modes within a band gap for the vol.-mode bands. In these edge modes, the spin wave propagates in a unidirectional manner without being scattered backward, which implements novel fault-tolerant spintronic devices.
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    42. 42
      McClarty, P. A. Topological Magnons: A Review. Annu. Rev. Condens. Matter Phys. 2022, 13, 171– 190,  DOI: 10.1146/annurev-conmatphys-031620-104715
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    43. 43
      May, A.; Saccone, M.; van den Berg, A.; Askey, J.; Hunt, M.; Ladak, S. Magnetic charge propagation upon a 3D artificial spin-ice. Nat. Commun. 2021, 12, 3217– 3310,  DOI: 10.1038/s41467-021-23480-7
      There is no corresponding record for this reference.
    44. 44
      Guo, H.; Deenen, A. J. M.; Xu, M.; Hamdi, M.; Grundler, D. Realization and Control of Bulk and Surface Modes in 3D Nanomagnonic Networks by Additive Manufacturing of Ferromagnets. Adv. Mater. 2023, 35, 2303292,  DOI: 10.1002/adma.202303292
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      Krawczyk, M.; Puszkarski, H. Magnonic crystal theory of the spin-wave frequency gap in low-doped manganites. J. Appl. Phys. 2006, 100, 073905,  DOI: 10.1063/1.2356082
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      Krawczyk, M.; Puszkarski, H. Plane-wave theory of three-dimensional magnonic crystals. Phys. Rev. B: Condens. Matter Mater. Phys. 2008, 77, 054437,  DOI: 10.1103/PhysRevB.77.054437
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    47. 47
      Volkov, O. M.; Rößler, U. K.; Fassbender, J.; Makarov, D. Concept of artificial magnetoelectric materials via geometrically controlling curvilinear helimagnets. J. Phys. D: Appl. Phys. 2019, 52, 345001,  DOI: 10.1088/1361-6463/ab2368
      47
      Concept of artificial magnetoelectric materials via geometrically controlling curvilinear helimagnets
      Volkov, O. M.; Roessler, U. K.; Fassbender, J.; Makarov, D.
      Journal of Physics D: Applied Physics (2019), 52 (34), 345001CODEN: JPAPBE; ISSN:0022-3727. (IOP Publishing Ltd.)
      A novel type of artificial magnetoelec. material, which allows an elec. field-induced deterministic switching between magnetic states without influencing intrinsic magnetic parameters, is proposed. It refers to three dimensional curvilinear helimagnets, e.g. torsion springs, embedded in a piezoelec. matrix. In contrast to conventional strain-coupled magnetoelec. heterostructures based on piezoelec.-magnetostrictive bilayers, we exploit the geometrical coupling of the matrix to the curvilinear helimagnet with intrinsic chiral Dzyaloshinskii-Moriya interactions. Namely, the magnetic state is modified due to the change of geometrical parameters of the curved nanomagnet. Theor., the essence of the proposal is analyzed for a deformable torsional spring made of helimagnetic material. In response to the geometrical change magnetic phase transition between the homogeneous and a periodically modulated state can be driven in a wide range of geometrical parameters. Resulting transformations of the av. magnetization from non-zero to zero value can be uniquely assigned to logical '1' and '0'. As the chiral magnetic properties are easier to control by mech. distortion than effective anisotropies, our concept should lead to a robust design of novel magnetoelec. devices.
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      Hopfield, J. J. Neural networks and physical systems with emergent collective computational abilities. Proc. Natl. Acad. Sci. U.S.A. 1982, 79, 2554– 2558,  DOI: 10.1073/pnas.79.8.2554
      48
      Neural networks and physical systems with emergent collective computational abilities
      Hopfield J J
      Proceedings of the National Academy of Sciences of the United States of America (1982), 79 (8), 2554-8 ISSN:0027-8424.
      Computational properties of use of biological organisms or to the construction of computers can emerge as collective properties of systems having a large number of simple equivalent components (or neurons). The physical meaning of content-addressable memory is described by an appropriate phase space flow of the state of a system. A model of such a system is given, based on aspects of neurobiology but readily adapted to integrated circuits. The collective properties of this model produce a content-addressable memory which correctly yields an entire memory from any subpart of sufficient size. The algorithm for the time evolution of the state of the system is based on asynchronous parallel processing. Additional emergent collective properties include some capacity for generalization, familiarity recognition, categorization, error correction, and time sequence retention. The collective properties are only weakly sensitive to details of the modeling or the failure of individual devices.
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    49. 49
      Karcher, H. The triply periodic minimal surfaces of Alan Schoen and their constant mean curvature companions. Manuscripta Math. 1989, 64, 291– 357,  DOI: 10.1007/BF01165824
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      Große-Brauckmann, K.; Wohlgemuth, M. The gyroid is embedded and has constant mean curvature companions. Calc. Var. Partial Differ. Equ. 1996, 4, 499– 523,  DOI: 10.1007/s005260050052
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      Große-Brauckmann, K. Gyroids of constant mean curvature. Exp. Math. 1997, 6, 33– 50,  DOI: 10.1080/10586458.1997.10504349
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      Sunada, T. Crystals That Nature Might Miss Creating. Not. AMS 2008, 55, 208– 215
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      Hyde, S. T.; O’Keeffe, M.; Proserpio, D. M. A short history of an elusive yet ubiquitous structure in chemistry, materials, and mathematics. Angew. Chem., Int. Ed. 2008, 47, 7996– 8000,  DOI: 10.1002/anie.200801519
      53
      A short history of an elusive yet ubiquitous structure in chemistry, materials, and mathematics
      Hyde, Stephen T.; O'Keeffe, Michael; Proserpio, Davide M.
      Angewandte Chemie, International Edition (2008), 47 (42), 7996-8000CODEN: ACIEF5; ISSN:1433-7851. (Wiley-VCH Verlag GmbH & Co. KGaA)
      Beauty in the sciences: The extraordinary history of a three-periodic net and its assocd. surface, the gyroid, is recounted. These structures appear in diverse contexts in mathematics, as the topol. for crystal structures in materials, which is the basis for liq. crystal phases and derived mesoporous materials, and in insect pigments.
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      Coey, J. M. Magnetism and Magnetic Materials; Cambridge University Press, 2010; pp 1– 617.
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      Hertel, R. tetmag. 2023, https://github.com/R-Hertel/tetmag.
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      Hertel, R.; Christophersen, S.; Börm, S. Large-scale magnetostatic field calculation in finite element micromagnetics with H 2 -matrices. J. Magn. Magn. Mater. 2019, 477, 118– 123,  DOI: 10.1016/j.jmmm.2018.12.103
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      d’Aquino, M.; Hertel, R. Micromagnetic frequency-domain simulation methods for magnonic systems. J. Appl. Phys. 2023, 133, 033902,  DOI: 10.1063/5.0131922
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      d’Aquino, M.; Serpico, C.; Miano, G.; Forestiere, C. A novel formulation for the numerical computation of magnetization modes in complex micromagnetic systems. J. Comput. Phys. 2009, 228, 6130– 6149,  DOI: 10.1016/j.jcp.2009.05.026
      There is no corresponding record for this reference.

      Number: 17.

    59. 59
      Heinrich, B. In Ultrathin Magnetic Structures II Heinrich, B., Bland, J., Eds.; Springer: Berlin, Heidelberg, 1994; pp 195– 296.
      There is no corresponding record for this reference.
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      Montoya, E.; McKinnon, T.; Zamani, A.; Girt, E.; Heinrich, B. Broadband ferromagnetic resonance system and methods for ultrathin magnetic films. J. Magn. Magn. Mater. 2014, 356, 12– 20,  DOI: 10.1016/j.jmmm.2013.12.032
      60
      Broadband ferromagnetic resonance system and methods for ultrathin magnetic films
      Montoya, Eric; McKinnon, Tommy; Zamani, Atieh; Girt, Erol; Heinrich, Bret
      Journal of Magnetism and Magnetic Materials (2014), 356 (), 12-20CODEN: JMMMDC; ISSN:0304-8853. (Elsevier B.V.)
      Spintronics requires the development of magnetic thin film structures having a wide range of magnetic properties. FMR is a well understood exptl. technique that has proven to be an invaluable tool to probe the static and dynamic magnetic properties of ultrathin films, multilayer nanostructures, and superlattices. In order to achieve a full characterization of thin film materials, one needs to carry out FMR measurements at a wide range of microwave frequencies. We show that one does not have to use a broadband vector network analyzer; similar performance can be achieved by a broadband microwave signal generator, a coplanar waveguide, and a broadband microwave detector. To obtain a good signal to noise ratio, one needs to employ a modulation technique to use lock-in detection;. we use low frequency external field modulation (105 Hz) and microwave power amplitude pulse modulation (10 kHz). The sensitivity and the performance of this broadband microwave system is demonstrated on 2 types of samples: MBE grown single crystal GaAs(001)/Fe/Au and sputter deposited textured Si(111)/Ta/Ru/Co/Ru superlattice structures. The samples were mounted on a coplanar waveguide, allowing one a broadband measurement, ∼0.1-50 GHz, of d.c. field swept FMR signals. The results are compared to traditional field swept, field modulated measurements in microwave cavity resonators. Despite the fact that the FMR signal can be very different from that obtained by std. microwave cavities, we show that the anal. of the FMR signal is fairly simple using an admixt. of the in-phase and out-of-phase components of r.f. susceptibility and that the resulting fitted magnetic parameters are in excellent agreement. Addnl., we demonstrate that microwave power amplitude pulse modulation can be used to greatly speed up data collection times, esp. for very weak and broad FMR signals.
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    61. 61
      Dubowik, J.; Głowiński, H. Broad-Band Ferromagnetic Resonance in Thin Magnetic Films and Nanostructures. Current Topics in Biophysics 2010, 33, 43– 45
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      Mikhaylovskiy, R. V.; Hendry, E.; Kruglyak, V. V. Negative permeability due to exchange spin-wave resonances in thin magnetic films with surface pinning. Phys. Rev. B: Condens. Matter Mater. Phys. 2010, 82, 195446,  DOI: 10.1103/PhysRevB.82.195446
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      Mruczkiewicz, M.; Krawczyk, M.; Mikhaylovskiy, R. V.; Kruglyak, V. V. Towards high-frequency negative permeability using magnonic crystals in metamaterial design. Phys. Rev. B: Condens. Matter Mater. Phys. 2012, 86, 024425,  DOI: 10.1103/PhysRevB.86.024425
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    64. 64
      Kruglyak, V.; In Metamaterial; Jiang, X.-Y., Ed.; IntechOpen: Rijeka, 2012; Chapter 14.
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      Zhuo, F.; Li, H.; Cheng, Z.; Manchon, A. Magnonic Metamaterials for Spin-Wave Control with Inhomogeneous Dzyaloshinskii-Moriya Interactions. Nanomaterials 2022, 12, 1159,  DOI: 10.3390/nano12071159
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      Haldar, A.; Adeyeye, A. O. Reconfigurable and self-biased magnonic metamaterials. J. Appl. Phys. 2020, 128, 240902,  DOI: 10.1063/5.0033254
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      Beaujour, J.-M.; Ravelosona, D.; Tudosa, I.; Fullerton, E. E.; Kent, A. D. Ferromagnetic resonance linewidth in ultrathin films with perpendicular magnetic anisotropy. Phys. Rev. B: Condens. Matter Mater. Phys. 2009, 80, 180415,  DOI: 10.1103/PhysRevB.80.180415
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      Walowski, J.; Kaufmann, M. D.; Lenk, B.; Hamann, C.; McCord, J.; Münzenberg, M. Intrinsic and non-local Gilbert damping in polycrystalline nickel studied by Ti: sapphire laser fs spectroscopy. J. Phys. D: Appl. Phys. 2008, 41, 164016,  DOI: 10.1088/0022-3727/41/16/164016
      68
      Intrinsic and non-local Gilbert damping in polycrystalline nickel studied by Ti: sapphire laser fs spectroscopy
      Walowski, J.; Kaufmann, M. Djordjevic; Lenk, B.; Hamann, C.; McCord, J.; Muenzenberg, M.
      Journal of Physics D: Applied Physics (2008), 41 (16), 164016/1-164016/10CODEN: JPAPBE; ISSN:0022-3727. (Institute of Physics Publishing)
      The use of femtosecond laser pulses generated by a Ti: sapphire laser system allows the authors to gain an insight into the magnetization dynamics on time scales from sub-picosecond up to 1 ns directly in the time domain. This exptl. technique was used to excite a polycryst. Ni film optically and probe the dynamics afterwards. Different spin-wave modes (the Kittel mode, perpendicular standing spin-wave modes and dipolar spin-wave modes (Damon-Eshbach modes)) are identified as the Ni thickness is increased. The Kittel mode allows detn. of the Gilbert damping parameter α extd. from the magnetization relaxation time τα. The nonlocal damping by spin currents emitted into a nonmagnetic metallic layer of V, Pd and the rare earth Dy were studied for wedge-shaped Ni films of 1-30 nm. The damping parameter increases from α 0.045 intrinsic for Ni to α > 0.10 for the heavy materials, such as Pd and Dy, for the thinnest Ni films <10 nm thickness. Also, for the thinnest ref. Ni film thickness, an increased magnetic damping <4 nm is obsd. The origin of this increase is discussed within the framework of line broadening by locally different precessional frequencies within the laser spot region.
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    69. 69
      Abdelrahman, D.; Iseli, R.; Musya, M.; Jinnai, B.; Fukami, S.; Yuasa, T.; Sai, H.; Wiesner, U. B.; Saba, M.; Wilts, B. D.; Steiner, U.; Llandro, J.; Gunkel, I. Directed Self-Assembly of Diamond Networks in Triblock Terpolymer Films on Patterned Substrates. ACS Appl. Mater. Interfaces 2023, 15, 57981– 57991,  DOI: 10.1021/acsami.3c10619
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      Gurevich, A.; Melkov, G. Magnetization Oscillations and Waves; Taylor & Francis, 1996.
      There is no corresponding record for this reference.

  • Supporting Information

    Supporting Information

    The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsami.4c02366.

    • BBFMR gyroid structure analysis under in-plane field rotation in 40° increments over 180° relative to the CPW; micromagnetic results of 4 × 4 × 4 gyroid spectra at 100 mT: detailed focus on satellite peaks; micromagnetic results of 4 × 4 × 4 gyroid spectra at 300 mT: detailed focus on satellite peaks; and FWHM vs frequency: comparison of BBFMR signals of gyroids and uniform Ni, under sample rotations on CPW, revealing the variation in the resulting effective parameters of damping and inhomogeneous contributions to the line width (PDF)


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