Theory of isolated magnetic skyrmions: From fundamentals to room temperature applications

Abstract

Magnetic skyrmions are topological quasiparticles of great interest for data storage applications because of their small size, high stability, and ease of manipulation via electric current. However, although models exist for some limiting cases, there is no universal theory capable of accurately describing the structure and energetics of all skyrmions. The main barrier is the complexity of non-local stray field interactions, which are usually included through crude approximations. Here we present an accurate analytical framework to treat isolated skyrmions in any material, assuming only a circularly-symmetric 360° domain wall profile and a homogeneous magnetization profile in the out-of-plane direction. We establish the first rigorous criteria to distinguish stray field from DMI skyrmions, resolving a major dispute in the community. We discover new phases, such as bi-stability, a phenomenon unknown in magnetism so far. We predict materials for sub-10 nm zero field room temperature stable skyrmions suitable for applications. Finally, we derive analytical equations to describe current-driven dynamics, find a topological damping, and show how to engineer materials in which compact skyrmions can be driven at velocities >1000 m/s.

Figure 1

Skyrmion profile and stabilizing energy. (a) Illustration of the 360° domain wall model. The plot shows the normalized perpendicular magnetization m_z as a function of position x along the diameter of a skyrmion and defines the characteristic parameters R (radius), Δ (domain wall width), ψ (domain wall angle, inset), and N (polarity or skyrmion charge). The profile corresponds to the μ₀H_z  = −1 T data point in (b). The negative value of the field indicates that it is oriented antiparallel to the skyrmion core. (b) Radius and domain wall width as a function of applied magnetic field μ₀H_z. The small solid data points are predictions of our analytical model, the solid large data points are results from micromagnetic simulations, the open points are experimental results of Romming et al.. The insets show the relaxed spin structures obtained from micromagnetic simulations corresponding to the large solid data points. (c) Total energy, domain wall energy, and bulk energy as a function of skyrmion radius at a field of μ₀H_z = −2 T. At each value of R the energy has been minimized to determine Δ and ψ. R_eq is the equilibrium radius as plotted in (b). At the minimum, the domain wall energy has a negative slope, qualifying this skyrmion as a DMI skyrmion. (d) Decomposition of the total energy in c into individual components. Domain wall energies (yellow): DMI energy (inverted), exchange energy, effective anisotropy energy, and volume stray field energy (multiplied by 10). Bulk energies (pink): effective Zeeman energy and remaining surface stray field energies (multiplied by 10).

Figure 2

Fundamental predictions of our model. (a–c) Room temperature collapse diameter predicted by different models (always requiring 50k_BT for stability, but all diagrams would look very similar for 30k_BT). In all diagrams, d is the thickness of the magnetic material and the total film thickness (including non-magnetic spacer layers) is 4d. All three diagrams were derived with A = 10 pJ/m, Q = 1.4, M_s = 1.4 MA/m, and H_z such that R is minimum. (a) Wall energy model of bubble domains. (b) Bogdanov and Hubert’s model^,, which considers stray fields by effective anisotropies. (c) Our model including stray field in their full non-local nature. (d) Characteristic energy plots for different values of D_i at d = 31.6 nm. The yellow and pink solid lines are the domain wall and bulk energy corresponding to the red solid line, respectively. The bulk energy has been offset by 155Ad to present it in the given plot range. (e) Radial cross section of a simulated non-DMI skyrmion (green) and low DMI (stray field) skyrmion (light blue). The red and blue lines are reproduced from ref., where the attempt was made to distinguish proper skyrmions from stray-field stabilized bubbles based on their radial profile. The radial profile of the simulated stray field skyrmions precisely agrees with the suggested shape of a DMI skyrmion, demonstrating that radial profile is not a valid selection criterion. θ = arccos (m_z) is the out-of-plane spin angle. The inset shows a fit of the 360° domain wall model to the numerically obtained solutions of the Euler equation of a zero-M_s system. The following parameters were used for the simulations: d = 2.8 nm, A = 10 pJ/m, M_s = 1.4 MA/m, Q = 1.01, D_i = 0, μ₀H_z = −63 mT for the non-DMI skyrmion and d = 1 nm, A = 10 pJ/m, M_s = 1.4 MA/m, Q = 1.01, D_i = 0.8 mJ/m², and μ₀H_z = −50 mT for the low DMI skyrmion (and no non-magnetic spacer layers in both cases).

Figure 3

Multiple minima in systems with competing DMI and stray-field energies. (a) Phase diagram of the multi-stability of skyrmions as a function of applied field and DMI strength, where white, blue, and orange indicate the regions of instability, mono-stability, and bi-stability, respectively. Stability requirement is that all energy barriers are larger than 50k_BT. The red line is the collapse field of DMI skyrmions and the gray line is the collapse field of stray field skyrmions. The dashed horizontal line indicates the slice that is plotted in panel (b). The data was obtained for a magnetic layer thickness d = 31.6 nm, a total film thickness of 4d, and magnetic layer M_s = 1.4 MA/m and quality factor Q = 1.01. (b) Radius as a function of applied field for skyrmions in the bi-stability region in (a) at D_i = 1.4 mJ/m². The background color indicates the level of multi-stability as defined (a). The left inset shows a simulation of the stray field and DMI skyrmion which are simultaneously present in the same simulation area at μ₀H_z = −254 mT. The right insets show zoom-ins on the individual skyrmions (stray field skyrmion on top, DMI skrymion at the bottom). The coloured arrows indicate the domain wall spin orientation. (c) Energy as a function of radius for a zero stiffness skyrmion. The thermal energy available to the skyrmion at room temperature is indicated by the blue shading and the dotted lines. All states between R = 2 nm and R = 11 nm are accessible by thermal excitation on a sub-nanosecond time scale. The shown data corresponds to d = 4 nm (no spacer layers), D_i = 1.85 mJ/m², Q = 1.4, and μ₀H_z = −100 mT. The minimum at $R\approx 6\mathrm{nm}$ is an artifact within the 1% precision of our model. The model actually predicts at most two minima.

Figure 4

Zero field skyrmions. (a) Diameter of zero field skyrmions as a function of anisotropy and saturation magnetization, in a 2 nm thick film with A = 10 pJ/m and D_i = 2 mJ/m². The black solid line is a plot of $\kappa =\frac{1}{2}$ and the dashed line follows the solution of $\sqrt{8A(Q-\mathrm{1)}{\mu }_0{M}_s^2}-\pi D_i={\mu }_0{M}_s^{2}d$, where the latter has been imposed as a threshold because stripe domains are expected to nucleate spontaneously if K_u is smaller. (b) Stabilizing energy (in units of k_BT) for the skyrmions in (a). The boxes illustrate the range of anisotropy and saturation magnetization values reported in the literature for different material classes. (c,d) The same diagrams as in (a and b), but for an applied field of 100 mT. (e,f) Diameter of zero-M_s skyrmions in a d = 5 nm (g) and a d = 10 nm (h) thick film as a function of exchange constant and DMI strength. The anisotropy constant at each point is adjusted to obtain $E_a^{\mathrm{eff}}=50k_{B}T$.

Figure 5

Spin-orbit torque driven skyrmion motion. (a–c) Simulated skyrmion motion for different ferromagnetic skyrmions, moved by a pure damping-like SOT current j_HM = 10¹¹ A/m². (a) Intermediate skyrmion with ψ ≈ 45°, illustrating the different directions of current density j, force F, and velocity v as well as the difference between ξ and ξ′. Other parameters are M_s = 1.4 MA/m, Q = 1.4, A = 10 pJ/m, D_i = 0.7 mJ/m², $\mathcal{N}=10$ layers (affects the vertical spin current), d = 10 nm and total film thickness (including non-magnetic spacers) of 40 nm. (b) Motion of a 2R ≈ 200 nm diameter skyrmion at μ₀H_z = −4.6 mT. (c) The same skyrmion as in (b), but at a field of μ₀H_z = −90 mT and a corresponding diameter of 2R ≈ 20 nm. (d) Mobility (velocity per current density) and skyrmion Hall angle as a function of skyrmion diameter (the same skyrmion as in (b) and (c). The size is controlled by the out-of-plane field). Material parameters are M_s = 1MA/m, K_u = 765kJ/m³, A = 20 pJ/m, D_i = 2 mJ/m², $\mathcal{N}=1$ layer, d = 1 nm without non-magnetic spacers. Simulations in (d) were performed using current densities between 10¹¹A/m² and 3 × 10¹¹A/m². (e) Mobility and skyrmion Hall angle for a SAF bilayer with fixed M_s1 of the bottom layer and variable M_s2 of the top layer. The same spin current is assumed for both layers. Other simulation parameters are A_x,y = 10 pJ/m, A_z = −10 pJ/m, K_u = 500 kJ/m³, D_i = 2 mJ/m², and d₁ = d₂ = 2.5 nm. No field is applied. (f) 3D representation of a simulated 10 nm diameter skyrmion in a SAF bilayer, moving colinear with the current at a velocity of 1000 m/s at j_HM = 10¹² A/m², using M_s1 = M_s2 = 50 kA/m and other parameters as in (e). The two magnetic layers are colored black (top layer, predominantely magnetized down) and white (bottom layer, magnetized up). The skyrmion texture in the bottom layer can be seen in the reflection in mirror included in the ray-traced image. The small image shows a side view. Thermal stability of this skyrmion is 34k_BT. In all simulations, α = 0.2 and ${\theta }_{\mathrm{SH}}^{\mathrm{eff}}=0.15$.