Skyrmion Pair Racetrack Utilizing Hall Motion

Abstract

The skyrmion racetrack is a promising concept for future information technology. The primary issues with skyrmion racetrack memory are now error codes and Hall motion. Here, we propose a skyrmion pair racetrack memory. The Oersted fields generated by the non-contact current-carrying wire in the middle of the magnetic nanostrip stabilize the skyrmion pairs in the nanostrip, which are separated by a naturally formed domain wall. Through numerical models and micromagnetic simulations, we demonstrate that such a skyrmion pair can produce linear Hall motion along the nanostrip under the linear control of the Oersted field gradient. These findings offer a high-reliability method for skyrmion racetrack memory and a more efficient approach to designing devices that use the skyrmion Hall effect.

1. Introduction

Magnetic skyrmions are a type of particle-like spin texture with a unique topological charge [1,2,3,4,5,6,7,8,9], which are considered to be ideal information storage units due to their small size, non-volatility, and low energy consumption [10,11,12,13,14]. In particular, in 2013, Fert et al. proposed a skyrmion-based racetrack memory, encoding binary information by the presence or absence of skyrmions in a nanostrip [10]. The theoretically driven current density of skyrmion-based racetrack memory is only [15,16], which is five orders of magnitude lower than the current density required for traditional magnetic domain wall (DW) racetrack memory [17,18]. This study illustrated the benefits of skyrmions in terms of low energy usage and also made skyrmion-based racetrack memory a research hotspot in the past 10 years.

However, there are certain significant architectural obstacles that current research must overcome before skyrmion-based racetrack memory can be used in practice. On the one hand, it is unreliable to move and read lengthy data bit sequences in nanostrips, particularly when process changes and temperature fluctuations caused by current are taken into account. An electric field-controlled dual-track skyrmion racetrack memory is one such approach [19]. In these devices, the bit value (0 or 1) is no longer determined by the presence of magnetic skyrmions in one track, but by the presence of skyrmions in two tracks [20,21]. On the other hand, when the skyrmions are moved by the spin-polarized current, the Magnus force acting on the moving skyrmion leads to the transverse motion of a magnetic skyrmion in the nanostrip, which is called the magnetic skyrmion Hall effect (SkHE) [22,23,24]. One alternative solution is to construct antiferromagnetic skyrmions [25,26,27,28,29,30,31,32,33,34], compensated ferrimagnetic skyrmions [13,35,36,37,38], or skyrmioniums [39,40,41] with zero topological charge. Recently, researchers otherwise implemented intrinsic skyrmion Hall motion for building racetrack-like devices.

Here, we propose a skyrmion pair racetrack memory device. In the center of the nanostrip, a non-contact neighboring wire is built. As a current is passed through the wire, Oersted fields of equal magnitude and opposite direction are generated on the nanostrip, together with a corresponding Oersted field gradient. Through micromagnetic simulations, we demonstrate that such Oersted fields can stabilize a pair of opposite polarized skyrmions separated by a DW in the nanostrip. The Oersted field gradients generate a Magnus force in the direction perpendicular to the gradient, leading to the Hall motion of the skyrmion pair along the nanostrip. Due to the effect of the DW at the center of the nanostrip, the stability and movement speed of the skyrmion pair in the racetrack are higher than those of a single skyrmion. Furthermore, this racetrack storage may successfully avoid the local temperature effect caused by the current because it does not require current to pass across the nanostrip, enhancing the device’s dependability and longevity.

2. Results and Discussions

Figure 1a shows the device structure, which consists of a magnetic nanostrip and a non-contacting wire above it. When current is passed through the middle wire, the generated Oersted fields can be calculated by the Biot-Savart law:

$$\mathrm{d}{B}_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}}=-{\displaystyle \frac{{\mu }_{0}I}{4\pi x^3}}\left(\mathrm{d}l\times x\right)$$

(1)

where $I$ is the applied current, $l$ is the integration path, and x is the transverse distance from the measuring position to the wire. Since the length of the sample is much smaller than the length of the conducting wire, we can approximate that the length of the conducting wire is infinite. This result is a simplification of $B_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}}=-{\int }_{-{\displaystyle \frac{\pi }{2}}}^{{\displaystyle \frac{\pi }{2}}}{\displaystyle \frac{{\mu }_{0}I(dl\mathrm{s}\mathrm{i}\mathrm{n}\theta )}{4\pi x^2}}=-{\mu }_{0}I/(2\pi x)$. Considering a thin wire with a given width a, the generated Oersted field in the z-direction ($B_z$) can be expressed as follows:

$$B_z=-{\displaystyle \frac{{\mu }_{0}I}{2\pi a}}\ln {\displaystyle \frac{x+a}{x}}$$

(2)

where ${\mu }_0$ is the vacuum permeability. The corresponding Oersted field gradient $g={\displaystyle \frac{\partial }{{\partial }_x}}{B}_z$ can be determined by calculating the derivative of the Oersted field:

$$g={\displaystyle \frac{{\mu }_{0}I}{2\pi }}{\displaystyle \frac{1}{x(x+a)}}$$

(3)

This equation indicates a monotonic decrease of $g$ as a function of distance ($r)$.

For $a=32$ nm, $I=0.02$ A, we use the finite element method (FEM) in COMSOL Multi-physics software 6.1 to calculate the distribution of the generated Oersted fields on the nanostrip. The calculation results are shown in Figure 1b. The depth of red (blue) represents the magnitude of the Oersted field on the nanostrip. The intensity of the Oersted field gradually decays with the increase in distance $r$, consistent with the prediction of the Biot–Savart law.

The Oersted field gradient has been shown to effectively manipulate the movement of skyrmions [42,43,44,45]. All micromagnetic simulations were performed using Mumax 3.10 [46], based on the Landau-Lifshitz-Gilbert (LLG) equation, expressed as follows [47]:

$$\dot{\mathit{m}} = -{\displaystyle \frac{\gamma }{1+{\alpha }^2}}[\overrightarrow{\mathit{m}} \times {\overrightarrow{\mathit{H}}}_{\mathrm{eff}} + \alpha \overrightarrow{\mathit{m}} \times \left(\overrightarrow{\mathit{m}} \times {\overrightarrow{ \mathit{H}}}_{\mathrm{eff}}\right)]$$

(4)

where $\gamma$ is the gyromagnetic ratio, $\alpha$ is the damping coefficient, $\overrightarrow{\mathit{m}}$ is the normalized magnetic vector, $\dot{\mathit{m}}$ is the time derivative of $\overrightarrow{\mathit{m}}$. ${\overrightarrow{\mathit{H}}}_{\mathrm{eff}} = -{\displaystyle \frac{1}{{\mu }_0{\mathit{M}}_{\mathrm{s}}}}{\displaystyle \frac{\mathrm{d}\mathit{E}}{\mathrm{d}\overrightarrow{\mathit{m}}}}$ is the effective field where the total energy $\mathit{E}$ in our studied system is $\mathit{E} = {\mathit{E}}_{\mathrm{DMI}}+{\mathit{E}}_{\mathrm{ex}}+{\mathit{E}}_{\mathrm{de}}+{\mathit{E}}_{\mathrm{an}}+{\mathit{E}}_{\mathrm{Oe}}$. We consider the Neumann boundary conditions and the interfacial Dzyaloshinskii-Moriya interaction (i-DMI) energy as follows:

$${\mathit{E}}_{\mathrm{DMI}} = \mathit{D}[{\mathit{m}}_{\mathrm{z}}(\nabla ·\overrightarrow{\mathit{m}})-(\overrightarrow{\mathit{m}}·\nabla ){\mathit{m}}_{\mathrm{z}}]$$

(5)

where $\mathit{D}$ is the i-DMI strength. The exchange energy is given by the following:

$${\mathit{E}}_{\mathrm{ex}} = \mathit{A}{(\nabla \overrightarrow{\mathit{m}})}^2$$

(6)

where $\mathit{A}$ is the exchange constant. The demagnetization energy is as follows:

$${\mathit{E}}_{\mathrm{de} }= -{\displaystyle \frac{1}{2}}{\mu }_0{\mathit{M}}_{\mathrm{s}}\overrightarrow{\mathit{m}}·{\overrightarrow{\mathit{H}}}_{\mathrm{de}}$$

(7)

where ${\overrightarrow{\mathit{H}}}_{\mathrm{de}}$ is the demagnetization field, ${\mu }_0$ is the vacuum permeability, and ${\mathit{M}}_{\mathrm{s}}$ is the saturation magnetization. The uniaxial anisotropy energy ${\mathit{E}}_{\mathrm{an}}$ is as follows:

$${\mathit{E}}_{\mathrm{an}} = -{\mathit{K}}_{\mathrm{u}}{(\widehat{\mathit{z}}·\overrightarrow{\mathit{m}})}^2$$

(8)

where ${\mathit{K}}_{\mathrm{u}}$ is perpendicular magnetization anisotropy. The effect of Oersted fields is given by a Zeeman energy ${\mathit{E}}_{\mathrm{Oe}}$ as follows:

$${\mathit{E}}_{\mathrm{Oe}} = -{\displaystyle \frac{1}{2}}{\mathit{M}}_{\mathrm{s}}\overrightarrow{\mathit{m}}·\overrightarrow{\mathit{B}}$$

(9)

where $\overrightarrow{\mathit{B}}$ is the Oersted field. We study the case where the skyrmion pair moves under a spatially inhomogeneous magnetic field of the form $\overrightarrow{\mathit{B}} = [0, 0, {\mathit{B}}_{\mathrm{z}}(\mathit{x})]$, with a gradient along the x-direction (${\displaystyle \frac{\partial }{{\partial }_x}}{\mathit{B}}_{\mathit{z}}$).

Through micromagnetic simulations, Mumax3, we initialize two skyrmions with opposite topological charges on the left and right sides of the nanostrip, as shown in Figure 1a. In the micromagnetic simulation, the magnetic parameters are based on a known chiral magnet system, specifically the Ta/CoFeB/MgO material parameter system in reference [47], which exhibits finite DMI that can stabilize skyrmions: In the micromagnetic simulation, the mesh size in the simulation is $4 \mathrm{n}\mathrm{m} \times 4 \mathrm{n}\mathrm{m} \times 1 \mathrm{n}\mathrm{m}$, the diameter of the nanostrip is $512 \mathrm{n}\mathrm{m}$, the vertical magnetization anisotropy is $K_u= 470 \mathrm{k}\mathrm{J}{\mathrm{m}}^{-3}$, the saturation magnetization is $M_{\mathrm{s}}=7.7 \times {10}^5 \mathrm{A}/\mathrm{m}$, the exchange constant is $A=1.0 \times {10}^{-11} \mathrm{J}/\mathrm{m}$, the damping coefficient is $\alpha =0.01$, and the interfacial Dzyaloshinskii-Moriya interaction (iDMI) intensity is $D=1 \mathrm{m}\mathrm{J}/{\mathrm{m}}^2$. In the simulation, a periodic boundary condition was applied only along the x direction of the nanostrip. The Oersted fields obtained by the FEM are introduced into the simulations through external magnetic fields, and their magnetic field gradient is approximated as a uniform value.

Under the combined action of the Oersted fields and the DMI, a chiral DW is formed under the wire, forming a skyrmion–DW–skyrmion magnetic moment structure in the nanostrip, as shown in Figure 1c. To investigate the correlation between the current and the skyrmion–DW–skyrmion structure, the phase diagram as a function of the Oersted field, the corresponding current, and PMA is presented in Figure 1d. When the Oersted field exceeds the maximum tolerable threshold, it will provide sufficient driving force for skyrmions to surmount the energy barrier, thereby inducing their gradual annihilation and a subsequent phase transition to the ferromagnetic (FM) state. Notably, as the PMA increases, the magnetic system exhibits an enhanced tendency to transition to the FM state. Consequently, the maximum Oersted field that the system can tolerate decreases accordingly.

Assuming that the skyrmion is a rigid particle, the dynamics of skyrmions under an Oersted field gradient can be studied by using the Thiele equation [48]:

$$\overrightarrow{G}\times \overrightarrow{v}-\mathcal{D}\alpha \overrightarrow{v}=-\overrightarrow{F}$$

(10)

where $\overrightarrow{G}=4\pi Q{M}_{\mathrm{s}}\widehat{z}/\gamma$ is the gyromagnetic coupling vector, $\gamma$ is the gyromagnetic ratio, $\overrightarrow{v}$ is the skyrmion drift velocity, and $\mathcal{D}$ is the dissipation tensor. Here, the force $\overrightarrow{F}$ induced by the Oersted field gradient is given by $\overrightarrow{F}=\nabla E_{\mathrm{Z}\mathrm{e}\mathrm{e}}=-M_{\mathrm{s}}\int \nabla (\overrightarrow{m}\cdot \overrightarrow{B})dxdy$, where $E_{\mathrm{Z}\mathrm{e}\mathrm{e}}$ is the Zeeman energy, $\overrightarrow{m}$ is the magnetic vector, and $\overrightarrow{B}$ is the Oersted field. It can be written as follows:

$$F_i=-M_{\mathrm{s}}\int (\overrightarrow{B}{\cdot \partial }_i\overrightarrow{m}+\overrightarrow{m}{\cdot \partial }_i\overrightarrow{B})dxdy$$

(11)

By assuming that the magnetic moment of the ferromagnetic background is opposite to the topological charge, we can simplify Equation (11) to the following:

$$F_y=-S{M}_{\mathrm{s}}gQ$$

(12)

where we have applied a partial integration [41,49,50] and converted the integration interval to the area of skyrmion S, and the following:

$$F_x=-M_{\mathrm{s}}\int B\left(x\right){\displaystyle \frac{\partial }{{\partial }_y}}{m}_{z}dxdy$$

(13)

For a unit skyrmion, $F_x=0$. Under DW and boundary constraints, the motion of skyrmions in the y direction is restricted ($v_y=0$). Here, we can simplify Equation (10) to the following:

$$G{v}_x=-F_y$$

(14)

Therefore, the skyrmion velocity can be given by the following:

$$v_x=-{\displaystyle \frac{\gamma Sg}{4\pi }}$$

(15)

It can be predicted from the Thiele equation that (1) the velocity of the skyrmions is along the x direction, (2) the velocity of the skyrmions is independent of $Q$, and (3) the velocity of the skyrmions is proportional to $g$. Since the Oersted field gradient $g$ generated by the current-carrying wire on both sides has the same direction and symmetric magnitude, the skyrmions on both sides will move with the same velocity.

We performed micromagnetic simulations on the movement of the skyrmion pair in a nanostrip. The skyrmion-pair movement trajectories (marked by a blue-dotted box) are shown in Figure 2a. As expected, throughout its motion, the skyrmion pair does not cause any displacement in the y direction. Figure 2b illustrates how the displacement $L$ in the x direction varies with time $t$. The displacement of the skyrmion pair changes linearly with time, indicating a uniform speed. From the slope, we can read that the velocity of the skyrmion pair is 1.2 m/s.

In the movement of the skyrmion pair, the DW between the two skyrmions is responsible for restricting the y-directional deviation of the skyrmion pair. In order to analyze the effect of the DW on the movement of the skyrmion pair, we further compared the movement of the skyrmion pair and the single skyrmion under the Oersted field gradient. Under the action of the initial Oersted field $B$ of 10 mT and the gradient $g$ of 0.234 nm/mT, the trajectories of the skyrmion pair and the single skyrmion in 600 ns are shown in Figure 3a. Obviously, the movement speed of the skyrmion pair is much faster than that of the single skyrmion. Due to the presence of the skyrmion–DW–skyrmion coupling structure, a “push-pull” effect emerges between the intermediate DW and the skyrmions on both sides under the influence of the Oersted field gradient. Specifically, the repulsive forces between the intermediate DW and these skyrmions serve to enhance the y-direction driving force acting on them, resulting in a stronger SkHE and thus further leading to a higher motion velocity in the x direction.

In contrast, in the case of a single skyrmion, the current wire is fabricated on the right side of the entire nanostrip (its structure is shown in Supplementary Figure S2). Figure 3b displays the velocity of the single skyrmion and the skyrmion pair as a function of the Oersted field gradient. Since the skyrmion pair is influenced in the y direction under the action of the DW, its movement speed in the x direction $v_x$ is linearly regulated by $g$, which is also in line with the calculation conclusion of the Thiele equation (Equation (15)). Due to the absence of repulsive force between the skyrmion and the DW, the speed of the single skyrmion is slower than that of the skyrmion pair, and the Hall angle of the single skyrmion is slightly smaller than that of the skyrmion with the same topological charge in the skyrmion pair. The Hall angle comparison between the skyrmion pair ($Q=\pm 1$) and the single skyrmion is shown in Supplementary Figure S5.

In addition to the limitation of the DW, the boundary effect of the nanostrip also contributes to the relocation of skyrmions in the y direction. As illustrated in Figure 4, we determined the skyrmion pair velocity $v_x$ as a function of the nanostrip width $w$ under three Oersted field gradient magnitudes. This is because the boundary effect of the nanostrip decreases with the increase in width. At this time, the skyrmion will have a velocity component in the y direction, following a decrease in the velocity component in the x direction. However, the skyrmions only have the Hall motion induced by SkHE along the x direction due to the boundary effect. Furthermore, when the track length reaches 5 μm, snapshots of the skyrmion motion process (Supplementary Figure S1) show the the skyrmion Hall motion induced by the transverse Oersted field can be fully utilized for directional transport along the x direction, without reaching the width edge of the track and being pinned.

3. Conclusions

In this work, we propose a skyrmion pair racetrack memory. Through the Oersted fields generated by an on-chip non-contact current-carrying wire above the magnetic nanostrip, a pair of skyrmions with opposite topological charges can be stabilized on two sides of the wire, forming a skyrmion–DW–skyrmion spin structure. Through FEM, the Thiele equation, and micromagnetic simulations, we demonstrate that in this racetrack memory, the skyrmion pair can be linearly manipulated to move along the strip in a straight line through the Oersted field gradient. This method offers a dependable design concept for the skyrmion racetrack memory and could also be used for magnetic insulator materials.