# Micromagnetic Design of Skyrmionic Materials and Chiral Magnetic Configurations in Patterned Nanostructures for Neuromorphic and Qubit Applications

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## Abstract

**:**

## 1. Introduction

^{21}. In the next 300 years, the power requested to sustain the digital production is expected to exceed 18.5 × 10

^{15}W, i.e., the total planetary power consumption, today. Therefore, aside from the existing global challenges related to the environment, climate, population, health, energy, food and security, it is obvious that the energetic efficiency of any device used in ICT represents a major request. New concepts in data storage and the manipulation area must emerge. Spintronics should subscribe to this paradigm and the magnetic skyrmions are particularly suitable for that. They have the advantage of being particularly small, can be manipulated with extremely low levels of energy when storing classic, neuromorphic or quantum information, and ultimately, bring additional functionality driven from magnetic topology issues into consumer-friendly, low-energy nanoscale electronics. Within this context, there is a strong motivation for the theoretical modeling of skyrmionic materials and devices. Our objective is to identify the critical magnetic parameters of materials for hosting and manipulating skyrmions.

## 2. Materials and Methods

#### 2.1. General Issues about Skyrmions

^{−2}times smaller than the direct exchange [5], the DMI was further found to be particularly important in ferromagnetic materials with a large spin-orbit coupling (SOC) ${H}_{SO}=\zeta \left(r\right)\overrightarrow{L}\xb7\overrightarrow{S}$ which also breaks the inversion symmetry. The corresponding interaction term in the Hamiltonian would be ${H}_{DMI}=\left({\overrightarrow{S}}_{1}\times {\overrightarrow{S}}_{2}\right)\xb7{\overrightarrow{D}}_{12}$, where ${\overrightarrow{D}}_{12}$ is a vector perpendicular to the plane of the spins ${\overrightarrow{S}}_{1\left(2\right)}$. This term tilts the magnetic moments away from the collinearity and its magnitude proportional to the SO interaction included in the SO coupling parameter $\zeta \left(r\right)$. The low dimensionality of magnetic nanostructures brings additional sizeable contributions to the DMI via the associated additional potential gradients and corresponding intrinsic electric fields. At the interface between magnetic and nonmagnetic materials (heavy metals, insulators, etc.) the lack of inversion symmetry leads to a gradient of the lattice potential. Moreover, in multilayered stacks, additive interface effects would provide enhanced DMI and other interesting features. For example, it is possible to tune the sign of the DMI. Because the vector ${\overrightarrow{D}}_{ij}$ is parallel to the film plane in magnetic multilayers with perpendicular magnetization, its orientation is defined by the chemical nature and electronic structure effects at the interface between the nonmagnetic and the ultrathin ferromagnet. This leads to an enhancement of the effective magnitude $\left|{\overrightarrow{D}}_{ij}\right|$ with decreasing ferromagnetic layer thickness.

_{n = 10}) were found to enhance the thermal stability of skyrmions, even above the room temperature [9]. This represents an extremely important issue for the technological implementation in RT working ICT devices. The skyrmions can be efficiently displaced either by spin-orbit torque [6] or by spin-transfer torque (STT) effects in skyrmion racetrack memories [16] and can be created and annihilated by electric fields [17].

- The topological charge or the winding number Q = |1| that describes how many times the magnetization vector
**m**can be mapped onto a sphere: $Q=\frac{1}{4\pi}\int {d}^{2}x\text{}\overrightarrow{m}\xb7\left(\frac{d\overrightarrow{m}}{dx}\times \frac{d\overrightarrow{m}}{dy}\right)$. In saturated magnetic states Q = 0, Q = |0.5|for magnetic vortices, and more complex structures that wrapping the Bloch sphere multiple times Q >|1|. - The vorticity (chirality) ω = +/−1; ω = ((end) − θ(start))/2π. The sign of ω and Q are directly related to the orientation of the skyrmion core: P = ω·Q = 1/−1 for up/down, respectively; topological charge calculated from vorticity $Q=\frac{m}{2}\left[\underset{r\to \infty}{\mathrm{lim}}\mathrm{cos}\theta \left(r\right)-\mathrm{cos}\theta \left(0\right)\right]$.
- The helicity is uniquely determined by the type of DMI that occurs in the energetic competition (bulk or surface): $\psi =\left[0,\pi \right]$ for a Néel skyrmion and $\psi =\left[\frac{\pi}{2},3\pi /2\right]$ for a Bloch skyrmion.

#### 2.2. Ab-Initio Calculations of Magnetic Properties

_{Fe})/MgO(001) configuration, where X = nonmagnetic metal (Au, V, Ag) or insulator (MgO). The thickness of the ferromagnetic Fe layer (t

_{Fe}) was chosen in a few monolayers (ML) range (e.g., 5ML), as required for promoting the perpendicular magnetization ground states driven by the interplay of both top and bottom interfacial PMA. The calculation of the magnetic anisotropy energy (MAE) was executed by means of a fully relativistic spin–orbit scheme using the total energy and force theorem approaches [24], both providing similar results. Using the total energy approach, including the spin-orbit, the perpendicular magnetic anisotropy (PMA) has been estimated as the total energy difference between the easy (magnetization perpendicular to the film plane) and the hard directions (in-plane magnetization). Considering the extreme sensitivity of the magnetic anisotropy energy to the k-space meshing, first, a convergence study of the total energy with respect to the total number of k-points has been thoroughly performed.

_{0}, we concluded that ${\alpha}_{R}$ could be calculated using the values of E and k corresponding to the minimum of the parabolic dispersion band: ${\alpha}_{R}=\frac{2{E}_{0}}{{k}_{0}}$. We used this expression to deduce the Rashba parameter from the Rashba parabolic offset of the bands calculated for different supercell architectures describing magnetic heterostructures. Knowing the Rashba parameter, one can calculate the Rashba contribution to the DMI as [25]: $DMI=2{k}_{R}A$, with A = the exchange stiffness of the ferromagnetic material (e.g., 21 pJ/m for Fe), ${k}_{R}=\frac{2{\alpha}_{R}{m}_{e}}{{\hslash}^{2}}$, with m

_{e}the electron effective mass of the Rashba shifted parabolic bands.

#### 2.3. Micromagnetic Modeling Tools

^{3}GPU accelerated code (Ghent University, Ghent, Belgium) [28], based on solving the Landau-Lifshitz-Gilbert equation:

^{7}s Oe

^{−1}) and μ

_{0}the vacuum permeability (4π × 10

^{−7}F/m).

**m**,t] that includes various contributions: Zeemann, demagnetizing, anisotropy, exchange: symmetric (J

_{0}) and asymmetric (DMI), STT effects, etc.: ${B}_{eff}=-\frac{1}{{M}_{s}}\frac{\xf0E}{\xf0m}$. In our calculations, we have employed an NVIDIA Geforce RTX 3070 graphic card with 5888 CUDA cores for parallel computing. The modeled systems are nanometric-size disks constituted from tri-layer stacks, like those modeled by ab-initio calculations, in which a ferromagnetic film is sandwiched between a top (T) and a bottom (B) nonmagnetic layer. Therefore, the PMA and DMI are controlled by additive interface effects (top panel of Figure 3). In the Mumax

^{3}code, the perpendicular magnetic anisotropy was introduced as a 1st order uniaxial anisotropy K

_{u}along the Oz (001) axis, and the DMI has been chosen to have interfacial origin (D

_{ind}), as expected for ultrathin films stacks. In the simulations, we included as variables: other specific material/multilayer stack parameters such as the α- Gilbert damping of FM, exchange stiffness A

_{ex}, the saturation magnetization of the FM, the shape/size of the patterned nanostructure (e.g., disk, track, …), the density of a spin-polarized current injected in a nano-disk for STT writing of skyrmions and the time length of current or gating electric field pulses. For dynamical simulations, the LLG equation has been integrated using the adaptive step RK45, the Dormand–Prince solver. For the calculations of the phase diagrams, the energy minimum of the states has been obtained using the mumax3 relax () function that disables the precession term of the LLG equation allowing the system to relax towards the minimum of energy. For including the temperature effects, a finite temperature has been set to the ground state after the relaxation. Then, the evolution/stability of the system has been studied within a finite time integration window. The classification of the final micromagnetic states has been performed by topological charge calculations: $Q=\frac{1}{4\pi}\int {d}^{2}x\text{}\overrightarrow{m}\xb7\left(\frac{d\overrightarrow{m}}{dx}\times \frac{d\overrightarrow{m}}{dy}\right)$ tested against either artificial intelligence (AI) algorithms (image recognition).

## 3. Results

#### 3.1. Ab-initio Calculations of Multilayer Stacks

**E**

_{B}and

**E**

_{T}, that would be responsible for the PMA and DMI. Moreover, one can qualitatively observe that a positive (negative) voltage bias has an effect, the decrease (enhancement) of the intrinsic field at the top Fe/MgO interface (Figure 4c) and, as expected, no effect for the metallic Au/Fe bottom interface (Figure 4d). It was demonstrated that this intrinsic interface electric field would be responsible for both PMA and DMI [25,29]. Moreover, as theoretically predicted by Barnes et Maekawa [30], the additive effect of the intrinsic electric fields of both top and bottom interfaces would lead to a net Rashba contribution to the PMA and DMI whose magnitude and signs can be modulated by the chemical nature of the underlayer and overlayer. Therefore, within this interface Rashba framework, one can easily understand the effect of an external gating electric field which, depending on its orientation will either enhance or decrease the value of the intrinsic interface fields and implicitly, the magnitude of the PMA and the DMI.

_{u}(J/m

^{3}), is calculated from the surface anisotropy energy K

_{s}(J/m

^{2}), using the equation K

_{u}(J/m

^{3}) = K

_{s}(J/m

^{2})/ t

_{Fe}, with t

_{Fe}being the thickness of Fe (0.715 nm for 5ML). The value of K

_{s}is obtained by dividing the total energy difference between the easy (magnetization perpendicular to the film plane) and the hard directions (in-plane magnetization) to the area of the supercell base (Figure 4a).

**M**fixed along (100) and (−100) directions (Figure 7).

^{3}, the DMI from 1.67 to 2.27 mJ/m

^{2}, and the variation of PMA with E-field roughly by a factor of 2. These results indicate promising strategies for tailoring the PMA, DMI and their response to an electric field, very important issues for micromagnetic properties and control, as we will illustrate later.

#### 3.2. Micromagnetic Design of Skyrmionic Materials

_{s}and disk size. We briefly extrapolated the phase diagrams to the case of antiferromagnetically coupled skyrmions that could be stabilized in synthetic antiferromagnetic materials with interesting perspectives for racetrack memories and coupled qubits applications. Furthermore, we discussed some skyrmion manipulation issues: in nano-oscillators and racetracks, we illustrate the possibility to generate multi-skyrmionic ground states interesting for memristor functions in neuromorphic devices. Finally, we address some issues related to the skyrmionic state manipulation (core polarization and chirality by external gating electric fields). The choice of the diameter size of the disk in most simulations (90 nm) has been optimized as a compromise between the following issues: (i) our initial fixed target, to have nanometric size skyrmions in materials with experimentally achievable DMI (having in view our observation concerning the inverse proportionality between the size of a skyrmion and the DMI required to stabilize it), (ii) the efficiency and versatility of the skyrmionic injection by STT in larger disks increases with the disk size.

#### 3.2.1. Injection of Skyrmions by STT in Patterned Nanodisks

_{u}(PMA) = 1.91 × 10

^{+6}J/m

^{3}and D

_{ind}(DMI) = 1.67 × 10

^{−}

^{3}J/m

^{2}; M

_{s}= 1714 × 10

^{+3}A/m and α = 0.01 corresponding to the bulk Fe. In our simulations, we also considered as a variable parameter the voltage modulation of the anisotropy ( ξ

_{VCMA}) and the DMI for the bottom free layer. During the current pulse, the MTJ gets polarized and the electric field across the insulator was chosen to reduce the PMA and the DMI: K

_{u}= Ku

^{ini}(1-ξ

_{VCMA}) and D

_{ind}= D

_{ind}

^{ini}(1-ξ

_{VCMA}); the initial Ku

^{ini}and DMI

^{ini}values being restored after the pulse. The size of the unit cell of the micromagnetic grid was 1 × 1 × 1 nm.

_{pulse}-J

_{c}-Q and T

_{pulse}-J

_{c}-m

_{z}phase diagrams, starting either from an initial antiparallel (AP) or parallel (P) state of the fixed polarizer and the free layer; T

_{pulse}is the length of the current pulse, J

_{c}the density of the polarized electric current, Q the topological charge and m

_{z}the normalized z component of the magnetization. The micromagnetic grid used is a unit cell of 1 × 1 × 1 nm and the diagrams are built from 10.000 relaxed dynamics. In Figure 10, we illustrate the skyrmion writing phase diagrams calculated considering a current polarization p = 0.8 and a voltage modulation of anisotropy and DMI, ξ = 0.1. Like in common experiments of magnetization switch by STT, as a function of the initial state, P or AP, the direction of the injected J

_{c}for writing the skyrmions by STT must be reversed (see Figure 10a,d). The T

_{pulse}-J

_{c}-Q diagrams (Figure 10b,d), corresponding to injection from AP(P) initial states, illustrate the J

_{c}and T

_{pulse}parameter range allowing to write a skyrmion: either with a positive core polarization (Figure 10a), when STT writing from an initial AP saturated state or a negative core polarization (Figure 10d), when injecting from an initial P state. Analyzing the sign and the value of the topological charge of the skyrmionic states range of the T

_{pulse}-J

_{c}-Q diagrams: (Q = 1, red color in (b) or Q = –1, blue color in (e)), one can see that, depending on the initial AP or P state, the core polarization, and the chirality of the written skyrmion will be opposite. Moreover, the T

_{pulse}-J

_{c}-m

_{z}phase diagrams (Figure 10c,d) bring complementary information, indistinguishable from the T

_{pulse}-J

_{c}-Q diagrams. They allow us to discriminate the parameter range J

_{c}–T

_{pulse}leading to the full reversal of the free-layer disk by STT, the STT writing experiments in STT-RAMs are the same.

_{c}and T

_{pulse}window for the (S) writing by STT is significantly larger for the 90 nm disks as compared to 60 nm ones.

_{pulse}-J

_{c}-Q and T

_{pulse}-J

_{c}-m

_{z}phase diagrams, we have chosen a specific point whose parameters (J

_{c}, T

_{pulse}) are suitable for writing a skyrmionic state by STT (white point designed by (WP) in Figure 10b,e). Corresponding to these (J

_{c}, T

_{pulse}) values that were kept fixed, we have calculated new phase diagrams in which the PMA and the DMI have been varied to potentially cover a broader range corresponding to other skyrmionic materials. The diagrams illustrated in Figure 12, correspond to a 90 nm diameter circular nanodisk, patterned from a 1 nm thick ferromagnetic film, and are built from 10.000 relaxed dynamics in which we chose as variables the anisotropy and the asymmetric exchange (DMI) and fixed the other magnetic parameters: the saturation magnetization M

_{s}= 1714 kA/m; the exchange stiffness A

_{ex}= 2.1 × 10

^{+11}J/m; the Gilbert damping constant α = 0.01, which are typical for a Fe thin film that can enter in a X/Fe(5ML)/MgO multilayer stack. The color scale represents the value of the topological charge: $Q=\frac{1}{4\pi}\int {d}^{2}x\text{}\overrightarrow{m}\xb7\left(\frac{d\overrightarrow{m}}{dx}\times \frac{d\overrightarrow{m}}{dy}\right)$ used to classify the micromagnetic configuration of the final relaxed chiral ground states. In our simulations, we start from an initial saturated P or AP (with respect to the hard reference layer) state, and after a current pulse (J

_{c}, T

_{pulse}) we cut the pulse and determine the final relaxed micromagnetic configuration of the disk.

#### 3.2.2. Skyrmionic Ground States in Patterned Nanodisks

^{+6}J/m

^{3}, DMI variable–line (1)) and DMI = 3 × 10

^{−}

^{3}J/m

^{2}, variable PMA–line (2). The resulting magnetic features are displayed in Figure 13 and Figure 14 and the following conclusions have been drawn:

- At a fixed DMI, at small PMA values, the ground state is vortex-like, with Q ≈ −0.5; when the PMA increases, within a certain range whose extension increases with the value of the DMI, the ground state becomes skyrmionic Q ≈ −1; above a threshold maximum value of the PMA, the ground state will be a saturated monodomain.
- Within the skyrmionic window, the skyrmion’s diameter progressively decreases with the PMA increase. The narrow transition from a skyrminonic to a saturated state implies some complex chiral states with $\left|Q\right|>1$.
- The skyrmionic states are stabilized by larger values of DMI but, above a critical value, at fixed PMA, large DMI will favour $\left|Q\right|>1$ configurations within a PMA window increasing with the increase in the DMI.
- In conclusion, an optimum window of PMA and DMI values is required to stabilize a skyrmionic ground state.

_{ind}= 4 mJ/m

^{2}) and variable PMA (K

_{u}= 0–2.5 × 10

^{+6}J/m

^{3}). We see that the skyrmion stability is a complex issue, being determined by the balance between the PMA, the DMI, the direct exchange and the demagnetizing (magnetostatic) energies. When the PMA increases during the evolution from a vortex state to a skyrmionic one, the total energy decreases. However, this evolution costs more and more in demagnetizing energy which is continuously increasing. At a certain critical value of PMA, an abrupt transition into a saturated perpendicular state will stop the increase in the magnetostatic energy and will accommodate both the anisotropy and the direct exchange energies, in detriment to the DMI energy (the abrupt transition from (S) to (P) is also illustrated in Figure 17).

_{s}and the size and the shape of the magnetic disks. Both have a direct impact on the DMI-PMA-Q ground state phase diagram, and, implicitly, on the skyrmion stabilization parameters window. This is illustrated by Figure 18 which depicts phase diagrams corresponding to three materials commonly used in spintronic devices, with different saturation magnetization values: Fe (M

_{s}= 1714 kA/m; A

_{ex}= 2.1 × 10

^{−}

^{11}J/m), Co (M

_{s}= 1200 kA/m; A

_{ex}= 1.8 × 10

^{−}

^{11}J/m) and CoFeB (M

_{s}= 580 kA/m; A

_{ex}= 1.5 × 10

^{−}

^{11}J/m). One can see that the decrease in M

_{s}and A

_{ex}promotes the skyrmion formation at larger DMI and PMA values and favor chiral structures with $\left|Q\right|>1$. From our simulations, we deduce that materials with a large M

_{s}value would promote skyrmion stabilization at lower DMI but, for that to happen, larger PMA values would be required. Moreover, large M

_{s}values are detrimental for complex chiral structures with $\left|Q\right|>1$.

_{s}= 1714 kA/m; A

_{ex}= 2.1 × 10

^{−11}J/m. One can clearly observe that the smaller ferromagnetic disks would need a larger DMI to promote skyrmionic states (because the core should be confined within a smaller area and spins are swirling within a narrow length–scale). A direct consequence would be the necessity to fabricate materials and/or profit from additive effects in complex multilayered architectures to gain access to the large DMI values necessary to host nanometric size skyrmions in nanometric disks, as required for high-density storage or qubit applications. As illustrated in Figure 19c, this issue can be partially overcome by using materials with a larger M

_{s}: the minimum value of DMI for skyrmionics ground state stabilization decreases with the increase in M

_{s}.

_{s}, A

_{ex}, PMA, DMI, to be able to fit within the parameter range required for skyrmion stabilization, in view of any skyrmionic application. These concepts are not only valid for nanostructures constituted from single magnetic layers but can be further extrapolated to more complex heterostructures.

_{s}, PMA, and DMI parameters (Figure 23). The simulations presented in this figure are performed considering a 250 nm diameter disk, large enough to be able to accommodate multiple skyrmions within a PMA-DMI window in the range of phase diagram parameters of Figure 14c; the simulations being for a CoFeB type material with M

_{s}= 580 kA/m; A

_{ex}= 1.5 × 10

^{−}

^{11}J/m.

#### 3.2.3. Skyrmionic Nano-Oscillators

_{pulse}-J

_{c}-Q and T

_{pulse}-J

_{c}-m

_{z}phase diagrams illustrate that for writing a skyrmion by STT, a spin-polarized current pulse must be applied, with properly chosen J

_{c}and T

_{pulse}. After cutting the pulse, the skyrmionic nucleated pattern will have a damped oscillation whose size and shape is evanescently converging towards the final diameter imposed by the magnetic material parameters (DMI, PMA, M

_{s}) and the patterned disk geometry. However, as if for a mechanical damped oscillator, we would supply energy to compensate the damping losses, e.g., by acting with an external driving torque provided by a spin polarized current that competes with the damping torque, the skyrmion can be driven in a steady self-sustained oscillation regime: in which the size of the skyrmion is cyclically increasing and decreasing (so-called “breathing” regime). In Figure 24, we illustrate micromagnetically simulated magnetization dynamics feature for the writing (damped oscillations from pulse nucleated pattern towards the final skyrmionic state) and the steady oscillation regime. They concern a skyrmion generated in a 90 nm diameter nano-disk, patterned from an Au/Fe(5ML)/MgO multilayer system with the magnetic parameters extracted from ab-initio calculations: K

_{u}(PMA) = 1.91 × 10

^{+6}J/m

^{3}and D

_{ind}(DMI) = 1.67 × 10

^{−3}J/m

^{2}; M

_{s}= 1714 × 10

^{+3}A/m and α = 0.01 are corresponding to bulk Fe.

_{c}= 1 × 10

^{+12}A/m

^{2}and T

_{pulse}= 1.5 ns for writing from an initial P state of the MTJ nanopillar, then J

_{c}= −1.5 × 10

^{+12}A/m

^{2}for STT driving torque driven oscillations. The spin-polarized current J

_{c}injected into the nanopillar has a double effect: the damping-like component will provide the negative torque for sustaining the steady oscillations and the field-like component will modulate the frequency of oscillation determining a strongly nonlinear variation with the current density J

_{c}. For maintaining the steady oscillation regime of the skyrmionic nano-oscillator, J

_{c}must be adjusted in a proper range. If it is too small, the damping will not be entirely compensated, and the oscillations will gradually vanish. J

_{c}values which are too large will provide anti-damping torques which are also too large. The skyrmion core will increase up to the limit of the disk size, and collapse in a stable saturated state. The skyrmionic nano-oscillators [32] gained a lot of interest in recent years. Their wide window of operating frequency, in the range of the microwaves (1–100 GHz), tunable by external stimuli (magnetic fields, STT of spin-polarized currents), open interesting application perspectives as nanoscale electric oscillators, and sensitive magnetic field sensors [33]. Their strongly nonlinear dynamics can be used in studies of chaotic phenomena [34] and for neuro-inspired devices suitable for neuromorphic applications, e.g., pattern recognition [35].

#### 3.2.4. Skyrmion Manipulation by Electric Fields

^{+6}J/m

^{3}and DMI = 2.4 × 10

^{−3}J/m

^{2}, chosen to be in the proximity of the transition zone between the skyrmionic (S) and the saturated (P) states. This initial skyrmion, being nucleated from a saturated state along −Oz m = (0,0,−1), has a negative core polarization, the other parameters used in the simulation are M

_{s}= 1714 × 10

^{+3}A/m, A

_{ex}= 2.1 × 10

^{−11}J/m and α = 0.01 (Fe).

## 4. Discussion

#### 4.1. Multiscale Modeling of Skyrmionic Nanomaterials

#### 4.2. Experimental Issues on Magnetic Skyrmionic Nano-Materials

_{s}parameter range of our calculated phase diagrams. A selection with recent data available from the literature, concerning selected values of the saturation magnetization, DMI and PMA is illustrated by Figure 29 (the corresponding references are indicated). Concerning the saturation magnetization of a ferromagnetic layer –Figure 29b–considering its ultrathin thickness range, the surface contributions will be significant. Therefore, the value of M

_{s}depends on the thin film thickness and the chemical nature of the top (X) and bottom (Y) interface layers X/FM/Y. From Figure 29a, one can see that the anisotropy range is well covered by available experimental systems. They not only provide a wide range of surface anisotropies, but one also has the possibility to tune the value of K

_{u}by the thickness of the ferromagnetic layer: K

_{u}= K

_{s}/t. On the other hand, the currently available DMI range, for the most part of the studied systems, is beyond 3 mJ/m

^{2}. As mentioned, the DMI can be enhanced by additive effects [9] or skillful engineering of multilayered sequences, use of synthetic antiferromagnets [37], etc. Indeed, recently, in the epitaxial W(110)/Fe/Co bilayer system a giant DMI, of 6.55 J/m

^{2}obtained by quantum engineering of the lattice symmetry has been demonstrated [38]. On the other hand, in this paper we show that ab-initio and micromagnetic complementary tools can be successfully used to calculate the magnetic properties of multilayer skyrmionic materials, the predicted DMI and PMA values being similar with those measured in experimental stacks. Moreover, we theoretically indicate a path for enhancing the DMI, the PMA and their response to external electric fields by the skillful engineering of interfaces: a monolayer of Pt inserted at the top Fe/MgO would enhance the PMA by almost an order of magnitude, from 1.91 to 15.77 MJ/m

^{3}, the DMI from 1.67 to 2.27 mJ/m

^{2}and their response to an external electric field roughly by a factor of 2.

^{2}, shows that there is still plenty of space for theoretical and experimental research concerning the topological spin texture materials. A promising issue would be to combine interfacial and bulk DMI mechanisms requiring magnetic materials with intrinsic large spin-orbit interactions. Rare-earth transition metal alloys are more and more considered promising candidates for small skyrmions and ultrafast chiral spin texture dynamics applications [39]. Within this emerging topic, skyrmionic bubbles have been recently obtained [40] in Rare Earth (RE)-based REMn

_{2}Ge

_{2}(RE = Ce, Pr, Nd) magnets, in a wide temperature range (220–320 K). Lattices of magnetic skyrmions have been observed in centrosymmetric rare earth compounds, such as Gd

_{2}PdSi

_{3}and GdRu

_{2}Si

_{2}[41]. Spontaneous topological magnetic transitions were identified in NdCo

_{5}RE magnets [42]. Compact ferrimagnetic skyrmions were observed in DyCo

_{3}films [43] and interfacial chiral magnetism and isolated skyrmions have been demonstrated in SmCo

_{5}-based magnetic multilayers featuring perpendicular magnetic anisotropy [44].

**Figure 29.**Selected data from the literature concerning (

**a**) PMA and DMI and (

**b**) saturation magnetization M

_{s}values, fitted within the parameters range used in our micromagnetic calculations. When only the surface contribution to the PMA (K

_{s}) was available, the anisotropy K

_{u}has been calculated considering a ferromagnetic thickness layer of t = 1 nm: K

_{u}= K

_{s}/t; when the anisotropy field ${B}_{k}={\mu}_{o}{H}_{k}$ (T) was available for a layer with given M

_{s}(A/m), K

_{u}(J/m

^{3}) was calculated as ${K}_{u}={M}_{s}{H}_{k}/2$[47,48].

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### The Influence of Thermal Fluctuations at Finite Temperature

**Figure A1.**(

**a**) DMI-PMA-Q phase diagrams calculated at different temperatures 0 K, 100 K, 300 K. (

**b**) Magnetization vector field representation of skyrmionic structures corresponding to the point (DMI = 5 mJ/m

^{2}, K

_{u}= 1.7 × 10

^{+6}J/m

^{3}) indicated with a white filled circle in the diagrams of the panel (

**a**). The calculation has been performed for a 90 nm disk with the following fixed parameters: M

_{s}= 1714 kA/m, A

_{ex}= 2.1 × 10

^{+11}J/m; the Gilbert damping constant α = 0.01.

_{z}maps, for each temperature.

**Figure A2.**Skyrmionic state at different temperatures: (

**a**) T = 0 K, (

**b**) T = 100 K, (

**c**) T = 300 K. The calculation has been performed for a 90 nm disk with the following parameters: DMI = 5 mJ/m

^{2}, K

_{u}= 1.7 × 10

^{+6}J/m

^{3}, M

_{s}= 1714 kA/m, A

_{ex}= 2.1 × 10

^{+11}J/m; the Gilbert damping constant α = 0.01. In each panel, the top image is a glyph representation of the magnetization vector field within the disk. The bottom image is a scalar field representation of the m

_{z}values.

**Figure A3.**Skyrmionic state at T = 300 K for: (

**a**) DMI = 5 mJ/m

^{2}and (

**b**) DMI = 2.5 mJ/m

^{2}. The calculation has been performed for a 90 nm disk with the following parameters: DMI = 5 mJ/m

^{2}, K

_{u}= 1.7 × 10

^{+6}J/m

^{3}, M

_{s}= 1714 kA/m, A

_{ex}= 2.1 × 10

^{+11}J/m; the Gilbert damping constant α = 0.01. In each panel, the top image is a glyph representation of the magnetization vector field within the disk. The bottom image is a scalar field representation of the m

_{z}values.

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**Figure 2.**Left: Hedgehog representation of a Néel skyrmion magnetization vector field projected on the Bloch sphere. We indicated the vorticity ω, the topological charge Q and the core polarization P. Right: Bloch sphere and the helicity qubit: encoding the quantum superposition of $\left|0\right.\u232a:{m}_{z}=+1$ and $\left|0\right.\u232a:{m}_{z}=-1$: $\left|\psi \right.\u232a=\alpha \left|0\right.\u232a$+ $\beta \left|1\right.\u232a$. The representation is issued following our micromagnetic simulations, see next paragraph.

**Figure 3.**Top: Multilayer stack in which the PMA and DMA result from additive effects of the top and bottom interfaces. Bottom: zig-zag potential model used for simulating the effect of an electric field on the electronic structure of a multilayer stack (adapted from [27]).

**Figure 4.**(

**a**) Supercell model used in the ab-initio calculation for X/Fe(t

_{Fe})/MgO stack. (

**b**) Valence charge distribution calculated for Au/Fe(5ML)/MgO at zero electric field, corresponding to (110) section of the supercell-complete stack. (

**c**) zoom at the top Fe/MgO and (

**d**) bottom Au/MgO interfaces illustrating the effect of an external electric field on the top and bottom interface intrinsic fields

**E**

_{T}and

**E**

_{B}(adapted from [28]).

**Figure 5.**Opposite sign variation of the perpendicular magnetic anisotropy energy (K

_{u}) with an applied electric field (E-field) calculated for (

**a**) V/Fe(5ML)/MgO and (

**b**) Au/Fe(5ML)/MgO supercells- (adapted from [28]).

**Figure 6.**Potential profiles and Rashba magnetic fields ${\overrightarrow{B}}_{Ri}=-{\alpha}_{R}\left(\overrightarrow{k}\times {\overrightarrow{E}}_{i}\right)$ related to the intrinsic electric fields ${\overrightarrow{E}}_{i}$, i = 1,2 the top (1) = Fe/MgO and the bottom (2) = X/MgO interfaces in (

**a**) V/Fe(5ML)/MgO and (

**b**) Au/Fe(5ML)/MgO systems–(adapted from [28]).

**Figure 7.**Energy bands oppositely shifted in k, and zooms in a narrow k and energy range (black box) - corresponding to the magnetization

**M**‖(100) and

**M**‖ ( (−100), respectively, calculated for (

**a**) Au/Fe/MgO and (

**b**) V/Fe/MgO systems demonstrate the opposite sign of the net Rashba field ${\overrightarrow{B}}_{R}^{net}$ in the two situations (adapted from [28]).

**Figure 8.**Electric field dependence of the surface anisotropy K

_{u}(PMA), ${\alpha}_{R}$ and DMI, here for the Au/Fe(5ML)/MgO system–(adapted from [28]). The zero field values: K

_{u}(E-field = 0) = 1.91 MJ/m

^{3}and DMI (E-field = 0) = 1.67 mJ/m

^{2}will be further used in micromagnetic simulations (see Section 3.2.2).

**Figure 9.**Simulation geometry for skyrmion injection in nano-disks as free magnetic electrodes of magnetic tunnel junction devices.

**Figure 10.**(

**a**,

**d**) Simulation geometry and final skyrmionic state magnetization vector field used in skyrmion writing by STT. T

_{pulse}-J

_{c}-Q and T

_{pulse}-J

_{c}-m

_{z}phase diagrams, starting either from an initial antiparallel (AP) state (

**b**) and (

**c**) or parallel (

**P**) state (

**e**) and (

**f**).

**Figure 11.**T

_{pulse}-J

_{c}-Q phase diagrams for a 60 nm disk (

**a**) and 90 nm disk (

**b**) starting from an initial parallel (

**P**) state.

**Figure 12.**DMI-PMA-Q phase diagrams corresponding to simulation experiments started either from an initial P state (

**a**) or an initial AP state (

**b**). The parameters of the current pulse used in the simulation were: T

_{pulse}= 0.5 ns and J

_{c}= −3 × 10

^{12}A/m

^{2}or J

_{c}= + 3 × 10

^{12}A/m

^{2}, depending on the initial P or AP state (WP in Figure 22a,d). The sign of the written skyrmion topological charge is opposite in the two situations.

**Figure 13.**(

**a**) Ground state diagram DMI-PMA-Q (90 nm circular disk) in which the micromagnetic states are classified after their topological charge Q. (

**b**) Micromagnetic configuration in vector field glyph representation corresponding to chosen zones from the phase diagram: (P) = perpendicularly magnetized (PM) states, (V) = magnetic vortex states (Q ≈ −0.5), (S) = magnetic skyrmionic states (Q ≈ −1), (M) = complex chiral magnetic states with $\left|Q\right|$ >1 for different vorticity chiral structures, determined by the sign of the DMI.

**Figure 14.**Micromagnetic configuration in vector field glyph representation (

**a**) and (

**b**) projections on the Bloch sphere of the corresponding magnetization vector for: (V) = magnetic vortex states (Q ≈ – 0.5), (S) = magnetic skyrmionic states (Q ≈ −1) and (P) = perpendicularly magnetized (PM) states; (

**c**) Line profile of the total energy density of the spins within the ferromagnetic disk.

**Figure 15.**(

**a**) Ground state diagram DMI-PMA-Q (90 nm circular disk) with two analysis paths: (1): fixed PMA = 1.72 × 10

^{+6}J/m

^{3}and (2): fixed DMI = 5 mJ/m

^{2}. (

**b**) Micromagnetic chiral ground states and (

**c**) Total energy density landscapes evolution when the PMA and the DMI increase along the paths (1) and (2). In (

**a**), the white dotted lines are guiding lines for the threshold from (S) to (P) state, and the black vertical line corresponds to the analysis path used in Figure 13: fixed DMI D

_{ind}= 4 mJ/m

^{2}and variable PMA K

_{u}= 0−2.5 × 10

^{+6}J/m

^{3}.

**Figure 17.**Variation with the anisotropy at fixed DMI for: (

**a**) different contributions to the magnetic free energy; here, the exchange energy includes both the DMI and the direct exchange (

**b**) topological charge (

**c**) total exchange and only the DMI contributions energies.

**Figure 18.**The influence of the saturation magnetization on the DMI-PMA-Q ground state phase diagram (

**a**) Fe type ferromagnetic material with M

_{s}= 1714 kA/m; A

_{ex}= 2.1 × 10

^{−11}J/m (

**b**) Co type material with M

_{s}= 1200 kA/m; A

_{ex}= 1.8 × 10

^{−11}J/m and (

**c**) CoFeB type material with M

_{s}= 580 kA/m; A

_{ex}= 1.5 × 10

^{−11}J/m.

**Figure 19.**The influence of the ferromagnetic disk size on the DMI-PMA-Q ground state phase diagram. (

**a**) 90 nm diameter disk (

**b**) 50 nm diameter disk (

**c**) M

_{s}-DMI-Q phase diagram indicating the maximum DMI required to promote skyrmionics (S) state instead of vortex (V) for a given M

_{s}.

**Figure 20.**The influence of the magnetic field applied along the +Oz axis on the DMI-PMA-Q ground state phase diagram for a 90 nm disk. (

**a**) B = 0 T (

**b**) B = 0.1 T (

**c**) B = 0.3 T (

**d**) B = 0.4 T. (

**e**) The variation of size of the skyrmion core with increasing magnetic field. The images correspond to a point (DMI = 5 mJ/m

^{2}, K

_{u}= 1.7 × 10

^{+6}J/m

^{3}). The calculation has been performed for a 90 nm disk with the following fixed parameters: M

_{s}= 1714 kA/m, A

_{ex}= 2.1 × 10

^{+11}J/m; the Gilbert damping constant α = 0.01.

**Figure 21.**Antiferromagnetic configuration DMI-PMA-Q ground state phase diagram, for the top (

**a**) and bottom (

**b**) disks antiferromagnetically coupled (e.g., in a Synthetic antiferromagnet) (

**c**) Representations of the magnetization vector fields in the two AF coupled layers illustrating opposite spins and sign of the topological charge (chirality). This configuration corresponds to a skyrmionic AF ground state corresponding to PMA = 1.74 × 10

^{+6}J/m

^{3}and DMI = −4 × 10

^{−3}J/m

^{2}.

**Figure 22.**Illustration of correction of the “Magnus” drift due to skyrmionics Hall effect in AF coupled skyrmions.

**Figure 24.**(

**a**) Time dynamics of the magnetization component m

_{z}during the STT writing event. (

**b**) Fast Fourier Transform (FFT) spectrum for the damped oscillations illustrated in (

**a**). (

**c**) Time dynamics of the skyrmionic STT oscillator for which the amplitude is kept constant by the anti-damping torque provided by STT of the spin-polarized current. (

**d**) FFT spectrum corresponding to the skyrmionic STT oscillator.

**Figure 25.**(

**a**) Zoom on the DMI-PMA-Q phase diagram. (

**b**) Micromagnetic states of skyrmion: (i) = initial state, (1)–(5) intermediate states during the E-field pulse, (f) final state, after cutting the pulse. (

**c**) Line profiles of the total magnetic energy density in the disk.

**Figure 26.**Magnetization dynamics of the m

_{z}component during two successive cycles of skyrmionic core reversal by electric field. Corresponding micromagnetic configurations of some representative magnetic states: skyrmion with up(down) core orientation [S- ↑(↓)], saturated perpendicular up(down) [P-↑(↓)], intermediate up(down) states [i-↑(↓)], are represented.

**Figure 27.**(

**a**) Ab-initio calculated DMI variation with the electric field. The variation rate is described by the parameter $\beta =\frac{\mathsf{\Delta}DMI}{\mathsf{\Delta}E-field}$ (fJ/Vm). (

**b**) DMI-PMA-Q phase diagram in which we indicated the path between the initial (i) and the final (f) states showing opposite core polarization and chirality. Top insert: sketch of E-field biasing geometry, the electric field at the surface of the ferromagnetic layer (FM) is applied across an insulator (I) using a top metallic contact electrode (MC).

**Table 1.**Selection of thin film multilayered materials illustrating Néel skyrmion stabilization. We indicate the material (multilayer system), the measured diameter of the skyrmion core, the magnitude of the DMI $\left|\mathit{D}\right|\left(\frac{mJ}{{m}^{2}}\right)$, the temperature of the skyrmion stability and the reference of the paper containing the study.

Multilayer System | Diameter of Skyrmion Core (nm) | $\left|\mathit{D}\right|\left(\frac{\mathit{m}\mathit{J}}{{\mathit{m}}^{2}}\right)$ | Temperature of Skyrmion Stability (K) | Reference |
---|---|---|---|---|

Pt/Co/Ta | 75–200 | 1.3 | $\lessgtr 300$ | [5] |

Pt/Co/MgO | 70–130 | 2.0 | $\lessgtr 300$ | [6] |

Ir/Co/Pt | 25–100 | N.A. | $\lessgtr 300$ | [7] |

[Ir/Co/Pt]_{10} | 100 | 2 | $>300$ | [8] |

Pt/CoFeB/MgO | <250 | 1.35 | $\lessgtr 300$ | [9,10,11] |

Pd/CoFeB/MgO | <200 | 0.78 | $\lessgtr 300$ | [12] |

W/CoFeB/MgO | 250 | 0.3–0.7 | $\lessgtr 300$ | [13] |

Ta/CoFeB/MgO | 300 | 0.33 | $\lessgtr 300$ | [14] |

Ta/CoFeB/Ta/MgO | 1000–2000 | 0.33 | >300 | [14] |

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One, R.-A.; Mican, S.; Cimpoeșu, A.-G.; Joldos, M.; Tetean, R.; Tiușan, C.V.
Micromagnetic Design of Skyrmionic Materials and Chiral Magnetic Configurations in Patterned Nanostructures for Neuromorphic and Qubit Applications. *Nanomaterials* **2022**, *12*, 4411.
https://doi.org/10.3390/nano12244411

**AMA Style**

One R-A, Mican S, Cimpoeșu A-G, Joldos M, Tetean R, Tiușan CV.
Micromagnetic Design of Skyrmionic Materials and Chiral Magnetic Configurations in Patterned Nanostructures for Neuromorphic and Qubit Applications. *Nanomaterials*. 2022; 12(24):4411.
https://doi.org/10.3390/nano12244411

**Chicago/Turabian Style**

One, Roxana-Alina, Sever Mican, Angela-Georgiana Cimpoeșu, Marius Joldos, Romulus Tetean, and Coriolan Viorel Tiușan.
2022. "Micromagnetic Design of Skyrmionic Materials and Chiral Magnetic Configurations in Patterned Nanostructures for Neuromorphic and Qubit Applications" *Nanomaterials* 12, no. 24: 4411.
https://doi.org/10.3390/nano12244411