Current driven properties and the associated magnetic domain walls manipulation in U-shaped magnetic nanowires

Based on the extended Landau–Lifshitz–Gilbert method, the properties of current driven domain wall movement in U-shaped magnetic nanowires and the effect of spin wave assistance on their properties have been investigated. The results show that changes of the curvature radius of magnetic nanowire can cause the additional pinning action and the pinning action will weaken the speed of current driven domain wall movement. For U-shaped magnetic nanowires, the changes of curvature radius can be represented by the radius R at the bend. The results show that the decline of its speed non-monotonically increases with the decrease of the bending radius of magnetic nanowires. On the other hand, the assistance of applying spin waves not only enhances the movement of magnetic domain walls but also weakens the pinning action. Further research has shown that applying the appropriate spin waves at the bend changing point can completely eliminate the influence induced by bend changing, in order to ensure uniform and stable movement of current driven magnetic domain walls in U-shaped magnetic nanowires, and achieve the current driven three-dimensional racetrack memory technology.


Introduction
The fundamental issue of Racetrack Memory technology is a research hotspot in the field of next-generation magnetic information storage [1,2].This technology breaks through the limitations of planar storage in information recording and has the advantages of fast read/write and easy miniaturization.The track in the memory is actually a magnetic nanowire, and the data encoding for information storage is achieved through magnetic domains separated by magnetic domain walls on the magnetic nanowires.The magnetic nanowires are folded multiple times and vertically arranged to grow on a silicon substrate as shown in figure 1(a) achieving three-dimensional storage [1].Meanwhile, the reading and writing of information in track memory can be achieved by driving the magnetic domain walls through spin polarized currents, without moving the magnetic head to process the information different from traditional magnetic hard disk drives, making it easy to miniaturize devices.So, it is important to study the movement properties of current driven magnetic domain wall along the bent magnetic nanowires.
In recent years, many experiments have observed the phenomenon of current driven magnetic domain wall motion in magnetic nanostructures [1][2][3][4][5][6][7], and there have also been many theoretical works to reveal the physical mechanisms of the related experimental phenomena [8][9][10][11][12][13][14][15][16][17][18][19][20][21].However, most of the research is limited to the movement properties of current driven the single magnetic domain walls in long straight magnetic nanowires.In fact, in order to achieve three-dimensional storage devices in track memory, it is usually necessary to fold magnetic nanowires multiple times and grow them vertically on the silicon substrate.The adjacent vertical magnetic nanowires can be connected by U-shaped nanowires as shown in figure 1(a) [1].Therefore, it is crucial to ensure that the current driven magnetic domain walls move evenly in the U-shaped magnetic nanowires.
Whereas, due to the boundary differences caused by the folding and bending of magnetic nanowires, there will be differences of dynamic behavior of magnetic domain walls between in the straight and in the bent parts of the magnetic nanowires.In fact, for small-scale magnetic nanowires, their shapes have a significant impact on the dynamic properties of their magnetic domain walls [22][23][24][25][26][27].As a result, for three-dimensional racetrack storage devices based on spin transfer torque, it is necessary to investigate the influence of different folding and bending of magnetic nanowires on the dynamic behavior of current driven magnetic domain walls.In this work, we plan to employ an external field at the bending point as assistance to eliminate influences by curvature radius changing to ensure the balanced movement of current driven magnetic domain walls in U-shaped magnetic nanowires.
To weaken the Joule heating caused by large driving currents [28][29][30], the spin wave assistance can be used to decrease the driving current then reduce Joule heating [31].Here, we try to use the spin wave assisting to weaken the influence by nanowire bending for current driving magnetic domain wall in U-shaped magnetic nanowires.The results show that, on the one hand, the changes of the curvature radius of magnetic nanowires do reduce the movement of current driven magnetic domain walls.On the other hand, the assistances of spin waves can weaken the influence of changes in the curvature radius of magnetic nanowires.Specifically, adding appropriate spin waves at the curvature radius changing spot of U-shaped magnetic nanowires can eliminate bending influence on the current driven domain walls completely, achieving uniform motion of the current driven domain walls in U-shaped magnetic nanowires.

Models and methods
The U-shaped magnetic nanowire is taken as 40 nm wide, 3000 nm long and 5 nm thick, and its bending degree is represented by the bending radius R. As shown in figure 1(b), the nanowire is placed in the x-y plane.The extended Landau-Lifshitz-Gilbert (LLG) equation which includes current and spin wave can be written as follows: where M is the magnetization vector of the lattice point, H eff an effective field that includes exchange energy and demagnetization energy, γ is the rotational magnetic ratio, α is the Gilbert dissipation coefficient, β is a dimensionless constant that describes the non adiabatic degree of spin and local magnetization of conducting electrons [8].u is a vector of electronic motion, its value is u = Jγµ B / (2eM S ) [8], J is the current intensity.H w is the spin wave field, when this item is 0, the equation is reduced to a current driven LLG equation.H w is formulated as, In this article, the widely used parameter values have been adopted.Namely, the lattice size a × a × a = 2 nm × 2 nm × 5 nm, Saturation magnetization M S = 8 × 10 5 A m −1 , exchange constant A = 1.0 × 10 −11 J m −1 .In addition, α = 0.02, β = 0.1 [8,10].The boundary effect by the bending of the nanowire can be equivalent to the effective field H eff through Stoner's criterion and magnetic dipole moment interaction.Based on the extended LLG equation, the related calculations have been obtained by using the lattice discretization.For simplicity, it is assumed that the current density is still uniform after the current transitioning from a straight portion to a curved portion, but its current direction is consistent with the boundary of the grid point.

Results and discussions
The influences of the bending changes of the U-shaped magnetic nanowires on their current driven magnetic domain wall movement have been investigated.As shown in figure 1(b), for simplicity, the changes in the curvature radius of the U-shaped magnetic nanowires can be represented by the radius R at the bend.The results show that changes of the curvature radius of magnetic nanowires can weaken the speed of current driven domain wall movement, and the decline of the speed (∆υ) decreases non-monotonically with the increase of the radius R as shown in figure 2. Specifically, the bending radius is smaller, the decreasing of the velocity is greater, but the increasing of the bending radius is bigger, the decreasing of the velocity is smaller.The decline in velocity gradually approaches zero in the range with a larger bending radius.
In order to understand the dynamic process of current driven domain wall movement in U-shaped magnetic nanowires, we present the micro-structural changes of domain walls over time in U-shaped magnetic nanowires under current driving, as shown in figure 3. The results show that the magnetic domain  is constrained when the magnetic domain moves to the bending point.Further research finds that the confinement of magnetic domains arises from the bending of magnetic nanowires, which generates additional magnetic dipole fields that hinder the movement of magnetic domain walls.Figure 4 shows the relationship between the speed of magnetic domain wall and its position in nanowire.The results show that the speed of magnetic domain wall decreases while domain wall transporting from the straight line to the curved point.That is, the decrease in the speed of the magnetic domain wall is due to the additional magnetic dipole field caused by the curvature radius changing in the magnetic nanowires.The additional magnetic dipole field induced by the different magnetic coupling between magnetic domains and curved or straight boundaries can be equivalent to a pinning field (H p ).The H p can be equivalent to the difference of magnetic coupling field (H c ) between the magnetic domain and the boundary when it in the curved and straight parts, and the H c can be calculated using the following formula, where S i and S j are the unit magnetization for cells i and j. r ij denotes the lattice vector between cells i and j.
As an approximation, it is assumed that the influence of boundaries on magnetic domains can be described by their magnetic dipole effect.The depending of the pinning field on the curvature radius (R) of the U-shaped magnetic nanowires has been calculated as shown in figure 5.The figure 5 shows that the variation of the equivalent pinning field with the bending radius of the magnetic nanowires is highly consistent with the variation of the magnetic domain wall moving speed in the U-shaped magnetic nanowires with its bending radius, indicating that the   bending of magnetic nanowires weakens the current driven movement of magnetic domain walls is due to the additional magnetic dipole field caused by changes in the curvature radius of magnetic nanowires.
The [31] shows that the spin wave assistance can enhance the current driven magnetic domain wall movement, so it can be used to reduce driving current.In addition, the previous research has shown that  spin wave driving the magnetic domain walls movement depends on its amplitude and frequency [32][33][34].In detail, the speed of spin wave driven domain wall movement is linearly dependent on the amplitude of the spin wave, but nonlinearly dependent on the frequency of the spin wave.Here, we try to apply spin wave assistance at the curvature radius changing point of U-shaped magnetic nanowires to achieve balanced movement of current driven magnetic domain walls along U-shaped magnetic nanowires.For ease of manipulation, we fix the frequency of the spin wave but adjust its amplitude to eliminate the influence of magnetic nanowire bending.Figure 6 shows that the increase of the movement speed (∆υ) of the magnetic domain wall under current driving linearly increases with the increase of the applied spin wave amplitude, and its slope decreases with the increase of driving current.The results show that we can completely eliminate the influence of the bending changing on the movement of current driven magnetic domain walls by applying the appropriate spin wave assistance at the curvature radius changing point in U-shaped magnetic nanowires.
Figure 7 shows the relationship between the amplitude of the assisting spin wave applied at curvature radius changing point and the bent radius R under the fixed current and frequency spin wave, to ensure domain wall uniform movement along in U-shaped magnetic nanowires.According to figure 7, we can use corresponding assisting spin wave for different curvature radius (R) to eliminate the effects caused by changes in curvature radius, and achieve uniform and stable movement of current driven magnetic domain walls in U-shaped magnetic nanowires.
In order to understand the physical reason why spin wave assistance at the curvature radius changing point can effectively increase the velocity of current driven domain wall movement, we specifically studied the effect of spin wave assistance on domain wall movement.The results show that the external spin wave not only weakens the pinning field (H p ) caused by changing of the bent radius, as shown in figure 8, but also enhances the vertical magnetization (mz) domain wall, as shown in figure 9.The increase in vertical   magnetization intensity can enhance the transfer of spin moment, thereby speed up the current driven domain wall movement [35].
The results of figures 8 and 9 also indicate that the increase in the velocity of current driven magnetic domain wall movement with spin wave assistance is non-monotonic and has multiple extremum with spin wave frequency, which is difficult to manipulate.However, the dependence with spin wave amplitude is monotonic and linear, making it easy to manipulate.Therefore, in practical applications, it is possible to fix the frequency of spin waves while adjusting their amplitude to assist current driving magnetic domain walls movement, in order to eliminate the influence of magnetic nanowires bending.
In addition, we have also calculated the microstructure of the magnetic domain and the H p at bent point as a function of the spin wave under current-driven as shown in the following figure 10 to understand the physical mechanism of spin wave assistance.Figures 10(a) and (b) represent the microstructure of the magnetic domain wall in the straight and curved parts driven only by current, while figures 10(c) and (d) represent the microstructure of the magnetic domain wall in the straight and curved parts driven by both current and spin waves, respectively.The results show that after applying spin waves, the microstructure of the magnetic domain undergoes slight changes as shown in figures 10(e)-(g), thereby weakening the pining field as shown in figure 10(h).So the assisting spin wave enhances current driven movement of magnetic domain walls.

Summary
Based on the extended LLG method, the relationship between the properties of spin wave assisted current driven domain wall movement and the bend radius of U-shaped magnetic nanowires has been systematically investigated.The results show that the U-shaped bending of magnetic nanowires hinders the movement of current driven magnetic domain walls, and the hindrance is due to the additional pinning field caused by the changes of curvature radius of the U-shaped magnetic nanowires.The decline of domain wall movement speed non-monotonically increases with the decrease of the bending radius of magnetic nanowires.On the other hand, the assistance of spin waves not only enhances the movement of current driven magnetic domain walls, but also weakens the pinning field caused by the changes of curvature radius.The reason is that spin waves can alter the microstructure of magnetic domain walls weakening their interaction with the boundary, and effectively increase the vertical magnetization of the magnetic domain walls enhancing the transfer of magnetic angular momentum.Among them, the speed of the spin wave assisting driven magnetic domain wall movement increases linearly with the of spin wave amplitude, but non-monotonic multiple peaks relationship with the increase of spin wave frequency.The research has shown that applying the appropriate spin waves assisting at the curvature radius changing point of U-shaped magnetic nanowires can completely eliminate the additional pinning field caused by the changes in curvature radius, in order to ensure the uniform and stable movement of the current driven magnetic domain wall along the curved magnetic nanowires, then achieve the uniform and stable movement of the current driven magnetic domain wall in the three-dimensional nanowires by the curved magnetic nanowires, and promote the reliable development of three-dimensional racetrack memory technology.

Figure 1 .
Figure 1.(a) Schematic diagram of the nanowire structure of a three-dimensional track memory device (see the figure 1 of [1]).(b) Schematic diagram of U-shaped nanowire and bending radius R.

Figure 2 .
Figure 2. The decline of magnetic domain wall moving speed caused by changes of curvature radius as a function of the bending radius R of U-shaped magnetic nanowires.

Figure 3 .
Figure 3. Simulation results of the position of magnetic domain walls in U-shaped magnetic nanowires over time under current driving.

Figure 4 .
Figure 4.The movement speed of magnetic domain wall vs. its position in nanowire.

Figure 5 .
Figure 5.The additional equivalent pinning field Hp caused by changes of curvature radius as a function of the bending radius R of magnetic nanowires.

Figure 6 .
Figure 6.The increase of domain wall movement speed as function of the amplitude of the assisting spin waves with its frequency f = 21 GHz under the different current driving.

Figure 7 .
Figure 7.The amplitude of the assisting spin wave applied as a function of the bending radius R with the current u = 10 m s −1 and the frequency f = 21 GHz to ensure domain wall uniform movement in the U-shaped magnetic nanowires.

Figure 8 .
Figure 8.The decline of pinning field by assisting spin waves applied as a function of its frequency (a) or its amplitude (b).

Figure
Figure The increase of vertical magnetization of magnetic domain walls caused by assisting spin waves applied as a function of its frequency (a) or its amplitude (b).

Figure 10 .
Figure 10.The changes in magnetic domain microstructure and pinning field under current drive with and without spin wave assistance.