Voltage-controlled magnetic solitons motion in an anisotropic ferromagnetic nanowire

The precise manipulation of magnetic solitons remains a challenge and is considered a crucial process in magnetic storage. In this paper, we investigate the control of velocity and spatial manipulation of magnetic solitons using the voltage-controlled magnetic anisotropy effect. A long-wave model, known as the generalized derivative nonlinear Schrödinger (GDNLS) equation, is developed to describe the dynamics of magnetic solitons in an anisotropic ferromagnetic nanowire. By constructing the Lax pair for the GDNLS equation, we obtain the exact solutions including magnetic dark solitons, anti-dark solitons, and periodic solutions. Moreover, we propose two approaches to manipulate magnetic solitons: direct voltage application and inhomogeneous insulation layer design. Numerically results show the direct modulation of soliton velocity by a constant voltage, while time-varying voltage induces periodic oscillations. Investigation of Gaussian-type defects reveals soliton being trapped beyond a critical defect depth. These results provide a theoretical basis for future applications in magnetic soliton-based memory devices.


Introduction
Magnetic materials play a vital role in modern information storage, compared to traditional semiconductor memory, they possess higher stability and reliability characteristics [1][2][3][4][5][6].Magnetic solitons on the ferromagnetic nanoscale have been studied extensively as information carriers for memory storage device applications [7][8][9][10][11][12].However, the precise manipulation of magnetic solitons remains a challenge and is regarded as an important process in magnetic storage.Although many manipulation techniques have been developed over the past few decades, they have some drawbacks that could be unfavorable in certain case.For instance, approaches relying on electric current-based methods may consume excessive energy due to Joule heating.Meanwhile, high current densities are impractical for device applications [13][14][15].The characteristic dispersion of the magnetic field generally increases the difficulty of manipulate [5,16].Temperature gradient and spin wave-driven methods have been relatively neglected due to their experimental challenges [17][18][19][20].Therefore, exploring more efficient and well-controlled techniques for magnetic solitons manipulation is urgently necessary to the development of spintronic devices.
Comparing to current or magnetic fields manipulation, the voltage control scheme is more attractive due to its feasible features, such as simple manipulation techniques for nanoscale applications, low cost, and no magnetic hysteresis [21][22][23][24][25].In recent years, the voltage-controlled magnetic anisotropy (VCMA) effect has gained widespread attention as a promising technology [3,[26][27][28][29][30].This effect occurs when an interface voltage is applied between ferromagnetic metals and insulators, leading to a localized modulation of perpendicular magnetic anisotropy [27,31].More interestingly, it has been proven that the perpendicular anisotropy of magnetic film scales linearly with applied voltage [32,33].Moreover, the thickness of the insulation layer has a significant impact on the perpendicular magnetic anisotropy [34].These excellent properties provide a new toolbox for potentially more competitive magnetic applications.Based on the VCMA effect, the voltage-controlled motion of domain walls and skyrmions has been successfully demonstrated, indicating that the domain walls and skyrmions velocity can be significantly changed by applied voltage [35][36][37][38][39][40][41].In addition, it is found that designing the thickness of the insulating layer (anisotropy gradient can be obtained through a wedged heterostructures) could be also effectively control the domain walls and skyrmions motion [40,[42][43][44][45][46].
So far, the majority of research has relied on micromagnetic simulations, lacking analytical solutions for magnetic solitons in anisotropic ferromagnetic nanowires.Meanwhile, the current approach to regulating magnetic solitons based on the VCMA effect is relatively limited in its scope.Here, we develop a long-wave model to describe the dynamics of magnetic solitons in the anisotropic ferromagnetic nanowire.Interestingly, the 1D magnetic solitons we obtained are not the typical (kπ or 2kπ ) magnetic domain walls but rather small-amplitude excitations near the state of saturated magnetization.The exact solutions are obtained by employing the Darboux transformation (DT).Furthermore, we propose two schemes to manipulate the magnetic soliton using the VCMA effect.The paper is organized as follows.In section 2, we derive the generalized derivative nonlinear Schrödinger (GDNLS) equation under the long-wave approximation, which is used to describe the propagation of magnetic solitons in anisotropic ferromagnetic nanowire.In section 3, the Lax pair and DT of the GDNLS equation are constructed, and analytical solutions (including dark solitons, anti-dark solitons, and periodic solutions) are obtained.In section 4, the manipulation of magnetic dark solitons using the VCMA effect is numerically investigated.Moreover, we consider the influence of the depth and width of inhomogeneous defect (Gaussian type) on the magnetic solitons motion, and our findings indicate that the magnetic solitons will be trapped within the defect if the depth of the defect exceeds a critical value.A concise conclusion is provided in section 5.

Physical background 2.1. Basic equations
We consider a ferromagnetic sandwich structure shown in figure 1(a), which consists of an anisotropic ferromagnetic film (Co/Pt), a thin AlOx insulating layer, and an electrode layer.Under the applied gate voltage V g , a large perpendicular magnetic anisotropy can be generated at the interface between the ferromagnetic and the insulator films [27], which can be tuned by adjusting the applied voltage V g .
The ferromagnetic film (Co/Pt) is regarded as quasi-one-dimensional since its length is significantly greater than its width and thickness.The nanowire is magnetized to saturation by an external field H 0 along the z-axis, i.e. the initial magnetization density is M 0 = (0, 0, M 0 ), as shown in figure 1(b).In this case, the existence of domain walls is prohibited, and only nonlinear excitations near the equilibrium state are present.It is noteworthy that, as we are considering small amplitude excitations near the state of saturated magnetization, the resulting magnetic soliton solution will differ from the commonly observed one-dimensional domain wall solutions.The external magnetic field does not serve as the driving force but rather plays a supporting role in maintaining the saturation magnetization of the ferromagnetic nanowire.
The dynamics of nonlinear waves in the anisotropic ferromagnetic nanowire can be fully characterized by a system of coupled equations, in which the dynamics of magnetization is governed by the Landau-Lifshitz-Gilbert equation here, γ is the gyromagnetic ratio, A is the exchange interaction coefficient, M s is the saturation magnetization, s is the Gilbert-damping constant and n is the unit vector along the anisotropic axis.
In the present paper, we consider the following physical parameters, γ = 1.76 × 10 11 rad (s The evolution of the magnetic field H is governed by the Maxwell equation, which is simplified as where c = 1/( √ µ 0 ε) is the speed of light in the ferromagnetic medium, µ 0 is the magnetic permeability in the vacuum and ε is the dielectric constant of the ferromagnetic medium [47,48].

Long-wave approximation and reductive perturbation expansion
In order to describe the propagation of nonlinear waves in the ferromagnetic nanowire, we aim to solve the coupled nonlinear partial differential equations ( 1) and ( 2).The characteristic length of the nanowire are its length L and the exchange length l.We consider the following physical model: the typical magnitude of the parameters are L = 20 µm, and l = 3 nm for cobalt.The typical wavelength Λ of about 0.5 µm, thus the condition l ≪ Λ ≪ L is obviously satisfied.Setting exchange length l as the reference length, the corresponding wavelength Λ ∼ 1/χ (χ is a small perturbation parameter) is large, indicating the long-wave propagation.Then we introduce the following slow space and time variables, ξ = χ (z − vt) , τ = χ 2 t, where ξ describes the shape of the pulse propagating at the velocity v. Furthermore, assuming that the natural ferromagnetic nanowire has strong exchange interaction and weak anisotropy, thus A and K are rescaled by χ −1 A and χ K. Next we nonuniformly expand the magnetization M and magnetic field H in powers of χ since the naturally presence of anisotropic axis along the z-direction [49,50].
] , Where δ = x, y, the M 0 and H 0 characterize the unperturbed (initial) state of the system.Substituting the perturbation expansion (3) into equations ( 1) and ( 2) and collecting the coefficients of each order of χ, we obtain the following relations.At the order χ 0 : where . At the order χ 1 : and Equation ( 6) can be simplified by introducing a new complex field ψ = M x 1 − iM y 1 .Using the relation for conservation of length of the magnetization vector, M z 2 .Performing a single differentiation on equation ( 6), we derive a new nonlinear evolution equation using the complex field ψ: where , κ = 2γK/v and α = 2γA/v.Equation ( 7) is a GDNLS equation describing the dynamics of magnetization in the ferromagnetic nanowire at the first-order perturbation.

DT and soliton solutions of the GDNLS equation
In this section, we aim at solving the soliton solution for GDNLS equation by the DT method.In some particular cases, the GDNLS equation can be further simplified, the third-order dispersion term can be neglected as the order of α is significantly much lower compared to that of η and σ.When the magnetic soliton moves a small distance, the damping effect can also be disregarded.Rescaling the transformation Notably, the equation ( 8) reduces to the well-known Kaup-Newell equation when η = σ = 1 [51].When η and σ take on specific values, the corresponding equations have also been investigated [52,53].
We take the plane-wave solution of equation ( 8) as the seed solution, where ω = ( k 2 − ka 2 σ ) /η.The soliton solution is obtained by the DT with the spectral parameter taken as λ = iβ (see appendix for detailed calculations). with It is noteworthy that the dark solitons, anti-dark solitons, and periodic solutions can be obtained in different parameter space.The nature of the solutions, whether they are solitons or periodic solutions, is dictated by the sign of Γ 2 , where Γ 2 > 0 corresponds to solitons, and Γ 2 < 0 corresponds to periodic solutions.What's more, the dark solitons, anti-dark solitons, can be distinguished by comparing the value at infinity of the solution |ψ | 2 infinity = a 2 − 2k/σ with the extreme value |ψ | 2 extreme = (a − 2βη/σ) 2 of equation (10).It generates an anti-dark soliton when |ψ | 2 extreme > |ψ | 2 infinity , otherwise a dark soliton.Figure 2 shows the classification of exact solutions (10), which can be divided into four distinct situations according to the parameter conditions.Figure 2   figure 2(a), the region where Γ 2 < 0 represents the periodic solution, while the zero point of the orange line separates the region where Γ 2 > 0 into two parts, corresponding to anti-dark and dark solitons, respectively.It is notable that only periodic solutions exist when a 2 σ/2 ⩽ k (see figure 2(d)).Three typical density plots of anti-dark soliton, dark soliton, and periodic solution are depicted in figure 3, taken from different regions of figure 2(a).

Two schemes for voltage-controlled the magnetic solitons motion
After acquiring the soliton solutions, the subsequent objective is to investigate the manipulation of the magnetic solitons.Remember that the perpendicular magnetic anisotropy of the interface between AlOx and Co/Pt can be significant changed by an applied voltage V g , which contributes to the fourth term of the equation (7).After obtaining the analytical solution for magnetic solitons (equation ( 10)), the velocity of magnetic solitons can be written as v MS = −2γK/ηv − (kσ − a 2 σ − 2β 2 η 2 )/ησ.The soliton velocity is determined by its intrinsic velocity and the portion that modified by adjusting K. From the expression, it is evident that the soliton velocity is proportional to the gradient of K, i.e. ∆v MS ∝ ∆K.Based on the above relationship, it is possible to manipulate the propagation of magnetic solitons by changing the applied voltage.Therefore, in this section, we numerically explore the above possibilities based on the analytical solution.
Firstly, a constant voltage is applied to a dark soliton with an initial velocity.It is observed that, upon applying an appropriate voltage, the velocity of the dark soliton reaches zero.Further increasing the voltage causes the dark soliton to move in the reverse direction.These three processes are illustrated in figures 4(a)-(c).This result suggests that the magnetic soliton velocity can be directly controlled by applying different voltages.Secondly, we investigate the effects of applying a time-varying voltage, where the voltage amplitude periodically oscillates between 0 and V 0 over time, with an adjustable period.Remarkably, we observe that magnetic solitons exhibit serpentine motion and their velocity dynamically respond to the applied voltage in real-time.Furthermore, it is found that the group velocity of the magnetic solitons are correlated with the average value of the changing voltage.Figures 4(d When the insulation layer exhibits inhomogeneous thickness t AlOx = t AlOx (ξ) (or thickness gradient), the accumulation of charges leads to the development of a magnetic anisotropy gradient within the layer when subjected to a constant voltage.In practical materials, there may be defects inside the insulation layer.In the following investigation, we aim to explore the impact of Gaussian-type defects within the insulation layer, which induce the inhomogeneous magnetic anisotropy, on the transmission of magnetic soliton.
We consider a Gaussian shape defect (as shown in figure 5(a)), resulting in an inhomogeneous distribution of perpendicular magnetic anisotropy in space (κ = −A • exp[−(ξ/W) 2 ]), where A and W describe the depth and width of the defect respectively.To investigate the propagation of magnetic solitons at different depths of the defect, we set a specific width of W = 4.For a shallow defect depth (A = 0.4), the magnetic soliton experience a slight deceleration before escaping (see figure 5(b)).As the Gaussian defect become deeper, its limitations on magnetic soliton become increasingly apparent.As shown in figure 5(c), when A = 2, the magnetic soliton is trapped in the defect and oscillate periodically.With an increasing depth of the defect, the motion of magnetic soliton undergoes a transition from escaping to periodic oscillation in the defect.To demonstrate such a transition, we plot the amplitude of the magnetic solitons as a function of A in figure 5(d).It is evident that there exists a critical depth of the Gaussian defect at approximately 0.6.Once the defect depth exceeds this critical value, the magnetic solitons are unable to escape.We also investigate the impact of the width of Gaussian impurity potential on the critical behavior and find that it slightly affects the critical strength.We perform a series of numerical simulations on the critical behavior of solitons in Gaussian defects and plot the resulting phase diagram of depth versus width of the defects in figure 5(e).The pink region represents magnetic solitons trapped within defects, while the blue region signifies the rapid escape of magnetic solitons.Furthermore, other forms of defects can also induce similar effects, such as one-dimensional finite-depth potential wells.
The above results indicate another scheme for manipulating (especially trapping) magnetic solitons, which allows us to regulate the motion of magnetic solitons by designing the thickness of the insulation layer.

Conclusion
In conclusion, we have studied analytically and numerically the dynamics of the magnetic solitons motion in an anisotropic ferromagnetic nanowire controlled by the applied voltage.We derive the GDNLS equation under the long-wave approximation, which describes the evolution of magnetic solitons (small amplitude nonlinear excitation) in the anisotropic ferromagnetic nanowire under magnetization saturation.By constructing the Lax pair and employing DT, the exact solutions of GDNLS equation are obtained.Our results show that the dark solitons, anti-dark solitons and periodic solutions can be obtained in different parameter space.After that, numerical simulations are performed to investigate the magnetic solitons motion by the applied voltage.One noteworthy observation is that magnetic solitons can be directly manipulated by applying varying voltages, resulting in changes in their trajectory, cessation of motion, or even the induction of spatial oscillations.In addition, we show that the inhomogeneous of the insulation layer thickness is an important factor affecting the motion of magnetic solitons.Our research primarily focuses on Gaussian defects, and we discover that the depth and width of defects jointly determine whether the solitons can pass through unimpeded or be captured.
This work theoretically provides a low-power and high-efficiency approach for manipulating magnetic solitons in the anisotropic ferromagnetic nanowire.Our results may be useful for spintronic devices based on magnetic solitons.

Figure 1 .
Figure 1.(a) A schematic diagram of the device structure.The device consisted of a Co/Pt nanowire, a AlOx insulator and an electrode layer, Vg is applied voltage.(b) Ferromagnetic nanowire under consideration.The sample is saturated with a strong magnetic field H0 applied in the z-direction, while the nonlinear waves propagation also along the z-direction at the velocity v.
(a)  illustrates the distribution range of different solutions when the wave number k = 0.When the initial plane-wave solution satisfies a 2 σ/2 > k, figures 2(b) and (c) illustrate the cases when k > 0 and k < 0, respectively.On the other hand, figure2(d) corresponds to the situation where a 2 σ/2 ⩽ k.The existence ranges of the three types of solutions are indicated by different colors in figure2: the anti-dark solitons are marked in red, dark solitons in black, and periodic solutions in green.In each subfigure, two lines are employed to determine the partition between each solution.The orange line represents y = ( a 2 − 2k/σ ) − (a − 2βη/σ) 2 , while the blue line refers to y = Γ 2 .In particular, in
figures 4(a)-(c).This result suggests that the magnetic soliton velocity can be directly controlled by applying different voltages.Secondly, we investigate the effects of applying a time-varying voltage, where the voltage amplitude periodically oscillates between 0 and V 0 over time, with an adjustable period.Remarkably, we observe that magnetic solitons exhibit serpentine motion and their velocity dynamically respond to the applied voltage in real-time.Furthermore, it is found that the group velocity of the magnetic solitons are correlated with the average value of the changing voltage.Figures4(d) and (e) show the propagation of magnetic solitons under the time-varying voltage with different periods.When the insulation layer exhibits inhomogeneous thickness t AlOx = t AlOx (ξ) (or thickness gradient), the accumulation of charges leads to the development of a magnetic anisotropy gradient within the layer when subjected to a constant voltage.In practical materials, there may be defects inside the insulation layer.In the following investigation, we aim to explore the impact of Gaussian-type defects within the insulation layer, which induce the inhomogeneous magnetic anisotropy, on the transmission of magnetic soliton.We consider a Gaussian shape defect (as shown in figure5(a)), resulting in an inhomogeneous distribution of perpendicular magnetic anisotropy in space (κ = −A • exp[−(ξ/W) 2 ]), where A and W describe the depth and width of the defect respectively.To investigate the propagation of magnetic solitons at different depths of the defect, we set a specific width of W = 4.For a shallow defect depth (A = 0.4), the magnetic soliton experience a slight deceleration before escaping (see figure5(b)).As the Gaussian defect become deeper, its limitations on magnetic soliton become increasingly apparent.As shown in figure5(c), when A = 2, the magnetic soliton is trapped in the defect and oscillate periodically.With an increasing depth of the defect, the motion of magnetic soliton undergoes a transition from escaping to periodic oscillation in the defect.To demonstrate such a transition, we plot the amplitude of the magnetic solitons as a function of A in figure5(d).It is evident that there exists a critical depth of the Gaussian defect at approximately 0.6.Once the defect depth exceeds this critical value, the magnetic solitons are unable to escape.We also investigate the impact of the width of Gaussian impurity potential on the critical behavior and find that it slightly affects the critical strength.We perform a series of numerical simulations on the critical behavior of solitons in Gaussian defects and plot the resulting phase diagram of depth versus width of the defects in figure5(e).The pink region represents magnetic solitons trapped within defects, while the blue region signifies the rapid escape of magnetic solitons.Furthermore, other forms of defects can also induce similar effects, such as one-dimensional finite-depth potential wells.The above results indicate another scheme for manipulating (especially trapping) magnetic solitons, which allows us to regulate the motion of magnetic solitons by designing the thickness of the insulation layer.

Figure 5 .
Figure 5. (a) A sketch of Gaussian-type defect within the insulation layer.(b) and (c) The evolution of magnetic solitons for two different values of depth, A = 0.4 and A = 2.(d) The amplitude of the magnetic solitons as a function of A. (e) phase diagram plotted by the depth A versus width W of the defects.