Detailed examination of domain wall types, their widths and critical diameters in cylindrical magnetic nanowires

Highlights

• A phase diagram determining domain walls in cylindrical nanowires is presented.

• A large region of co-existence of the Bloch point with other domain walls is shown.

• The vortex/antivortex is the most probable domain wall rather than the Bloch point.

• An expression delimiting the domain walls in cylindrical nanowires is shown.

Abstract

Understanding and mastering the magnetic properties of domain walls in cylindrical nanowires is a fundamental pillar for developing novel 3D information technologies. For this purpose, a good comprehension of domain wall (DW) dynamical properties, which are strongly dependent on its type and size, is needed. In this work, by means of micromagnetic simulations, we focus on the accurate determination of DW types (transverse, vortex-antivortex, and the Bloch point) and DW widths and present a state diagram as a function of the diameter and the nanowire material. Using different initial states and trajectories we find a large region of metastability where the Bloch-point domain wall co-exists with either transverse or a more energetical vortex-antivortex one. We determine the domain wall width and its dependence on the nanowire diameter either for the transverse or the vortex-antivortex domain wall showing that it is always larger than the nanowire diameter. We also find simple expressions for the DW widths and the critical diameter for the transition between different DW types which agree well with direct simulations. Our results are useful for the experimental design of cylindrical nanowires for multiple applications.

Introduction

Cylindrical magnetic nanowires (NWs) are nanostructured materials with a large aspect ratio (length to other two dimensions) [1], [2], [3]. This feature provides important properties such as a strong uniaxial anisotropy parallel to the nanowire axis [4] or the possibility of creating and driving magnetic domain walls along the long dimension by means of external magnetic fields and/or currents [5], [6], [7], [8]. The circular cross-section comes up with many intriguing properties. First, the so-called Walker breakdown [9], [10], a complex oscillatory domain wall (DW) motion, limiting the DW velocity in magnetic stripes, is suppressed in cylindrical geometries [11], thus, high DW velocities (limited only by the spin-Cherenkov effect) can be obtained [10], [12], [8], [13]. Secondly, cylindrical magnetic nanowires are intrinsically magneto-chiral systems [5], leading to non-reciprocity of DWs [14], [10] and spin-wave dynamics [15], [10]. Third, the cylindrical geometry leads to the appearance of a curvature-induced Dzyalonshinki-Moriya interaction [14], and cylindrical magnetic NWs can also host skyrmions (e.g. the Bloch point) [16] and skyrmion tubes [17] with no need of special materials. For applications, magnetic cylindrical nanowires are considered as the building blocks for future emerging 3D internet-of-thing nanoscale platforms. [18], [19], [20], [21].

Magnetic cylindrical nanowires can be easily produced using different routes such as chemical methods [22], electrodeposition in alumina templates [23], [6], [24], or FEBID technique [25] using several magnetic materials with tunable geometric dimensions, aiming to accomplish a specific technological requirement. In particular, the material composition determines intrinsic magnetic properties of the nanowire, such as the saturation magnetization ($M_S$), the exchange stiffness (A), or the magnetocrystalline anisotropy (K), while geometric dimensions strongly influence the role that the magnetostatic energy plays. The combination of composition and geometry dictates the cylindrical nanowires magnetic properties, e.g., coercive fields ($H_C$), remanence ($M_R$), as well as the type of the main reversal mode, consisting in DW propagation [26].

Consequently, controlling the aforementioned magnetic properties in terms of the geometry and the composition is essential to develop novel applications for which a precise magnetic response is required. Amid them, having an insight into DW properties becomes even more crucial for designing the next generation of magnetic nanodevices since it should be based on a solid control of DW dynamics under the effect of external stimuli [27], [28]. The DW type and the DW width (${\delta }_W$) are the most fundamental intrinsic aspects to focus on, due to the key role they play in the DW dynamics [7], [8].

It is well known that cylindrical nanowires exhibit different kinds of DWs in terms of the wire diameter, being possible their coexistence for a certain range of values [29], [30], [7]. In fact, very narrow nanowires ($D<D_{\mathit{BP}}$) exhibit only transverse DWs ($T_{\mathit{dw}}$). Nanowires of larger diameter present a Bloch-point DW (${\mathit{BP}}_{\mathit{dw}}$), a special configuration with cylindrical symmetry, a singularity in its center [31], [32], [33], [34] and very intriguing dynamical properties. Notice that historically in many publications, this DW type was called vortex DW [35], [26]. ${\mathit{BP}}_{\mathit{dw}}$ and $T_{\mathit{dw}}$ coexist in some interval of diameters ($D_{\mathit{BP}}<D<D_{\mathit{cr}}$) being the $T_{\mathit{dw}}$ the minimum energy state. For wider NWs ($D>D_{\mathit{cr}}$), the ${\mathit{BP}}_{\mathit{dw}}$ becomes the stable state, while the $T_{\mathit{dw}}$ deforms into a third type of DW in which the magnetization curls to form a vortex on the surface of the cylinder. In this case, an antivortex appears on the opposite surface to conserve the topological charge. This structure is similar but not equal that the Vortex/Antivortex pairs presented in thin films[36], [37]. Dubbed in the literature in several ways [7], [29], [30], [38], including “the vortex DW” and the “transverse vortex DW”, which produces confusion with the historical name for the Bloch point DW, in this work, we call this third type of wall a “vortex/antivortex pair” wall (${\mathit{VAV}}_{\mathit{dw}}$). The critical diameters separating the different DW regimes, $D_{\mathit{BP}}$ and $D_{\mathit{cr}}$, are material-dependent parameters [39], [30], [40], [35], [38], [29]. Indeed, previous studies on permalloy nanowires have established that the $D_{\mathit{cr}}$ is proportional to the exchange length for soft magnetic materials, $\mathrm{\Delta }$ [41], [29], extending its validity to this set of materials. Nevertheless, a rigorous corroboration of this statement with micromagnetic simulations is a subject still pending. Besides, $D_{\mathit{BP}}$ has been suggested to be smaller than the exchange length [39].

Below $D_{\mathit{cr}}$, it has been widely demonstrated for $T_{\mathit{dw}}$ that the DW width (${\delta }_W$) linearly increases with D, deviating from this relationship only for very narrow nanowires. In this case, it reaches the analytical value derived for an infinitely long and narrow nanowire ${\delta }_{0W}=\pi \sqrt{\frac{2A}{{\mu }_0{M}_S^2{N}_X}}$, where $N_X=1/2$ is the demagnetization factor of a nanowire [29], [42]. This result predicts that in the absence of a strong magnetocrystalline anisotropy, the larger is the material magnetization, the narrower is ${\delta }_W$ for a given nanowire diameter. Contrarily, what happens to the DW width once the $D_{\mathit{cr}}$ limit is crossed and if there is a simple relation between ${\delta }_W$ and $M_S$ is an open question. Furthermore, above $D_{\mathit{cr}}$, the relation between ${\delta }_W$ and D is also unknown since there exist two analytical results predicting completely opposite behaviours [29], [43].

Thus, while there exists partial state diagrams for DW types in terms of nanowire diameter [30], [29], a systematic study of the different stability regions of the three DW types mentioned before concerning different materials has not been performed yet. Unlike previous existing studies in cylindrical nanowires, mainly interested in materials such as Py or Ni, in this article, we vary the magnetization saturation value from a very small value for soft magnetic nanowires (useful for sensor applications) up to high values for hard magnetic nanowires such as FeCo, suitable for permanent magnets. [44]

The knowledge of DW type is essential for many applications. For example, the $T_{\mathit{dw}}$ is associated with a higher coercive value ($H_C$) [40], [45] than the ${\mathit{BP}}_{\mathit{dw}}$, which is very useful for the design of the above applications. ${\mathit{BP}}_{\mathit{dw}}$ has a higher mobility [39] than the $T_{\mathit{dw}}$, a property useful for high-frequency applications [46]. Therefore, in this article, we perform a systematic micromagnetic study in cylindrical nanowires to determine the different possible types for DWs and the transitions between these DWs in terms of the saturation magnetization and the nanowire diameter. We present a state diagram which will be very useful for future nanowires design. We found a very simple expression for $D_{\mathit{cr}}$ as a function of the demagnetising factor and magnetisation correlation length. We also checked the validity of this expression for the existing results in spherical geometries. Finally, we studied the corresponding widths for each DW and found their dependence on the nanowire diameter and saturation magnetisation value.