High-frequency spin transfer nano-oscillator based on the motion of skyrmions in an annular groove

Magnetic skyrmion-based spin transfer nano-oscillators (STNOs) have been proposed as microwave signal generators and attracted enormous interest recently. However, the oscillation frequency of skyrmion-based STNOs is about 2 GHz, which is not so high for practical applications. In this paper, we create an annular groove in the surface of the free layer and put skyrmions in the annular groove. Due to the potential of the groove, skyrmions are confined to moving in the groove when driven by the spin-polarized currents. Through micromagnetic simulations, it is found that the frequency tunability of the STNO with the presence of the annular groove reaches to 15.63 GHz, which is more than 6 times higher than the case without the presence of the annular groove because of three reasons: the oscillation radius of skyrmions can be adjusted by the groove, the potential of the groove is larger than that of the edge and the groove can limit the diameter of skyrmions so that a larger number of skyrmions can be placed in the groove. Our results present the understanding of dynamic of skyrmions in an annular groove, which provides alternative possibilities for the design of skyrmion-based STNOs.


Introduction
Spin transfer nano-oscillators [1] (STNOs) are used to produce microwave signals, which have attracted large interest in recent years due to their outstanding advantages, such as the broad range of frequency, the lower working voltage, the small sizes and the wider working temperature. A simple structure of a STNO usually consists of two magnetic layers (a relatively thick fixed layer and a relatively thin free layer) that separated by a spacer layer. A DC current is polarized by flowing through the fixed layer which drives a steady precession of magnetization in the free layer. This precession can be detected as a microwave signal by either the giant magnetoresistance or the tunneling magnetoresistance effects. To date, STNOs are divided into various types according to the different configurations of free magnetic layers, such as a single domain [2,3], magnetic vortices [4][5][6], magnetic droplets [7,8] and magnetic skyrmions [9][10][11][12][13][14]. Different from these three-layered STNOs, there is an alternative single-layered magnetic oscillator that works based on the periodical motion (rotation [15][16][17] or translation [18]) of magnetic domain wall driven by spin currents or magnetic fields.
Magnetic skyrmions [19] are topologically protected chiral spin configurations. So far, different types of skyrmions including Bloch skyrmions, Néel skyrmions and Antiskyrmions have been observed in bulk lattices with B20 structure [20][21][22], heterostructures lacking interfacial inversion symmetry [23][24][25] and Heusler compounds [26] due to the presence of bulk Dzyaloshinskii-Moriya interaction [27] (DMI), interfacial DMI [28,29] and anisotropic DMI [30], respectively. Moreover, because of the translational symmetry breaking on the surface of bulk chiral material, Twisted skyrmions are observed in Cu 2 OSeO 3 below a critical thickness [31]. Recently, it is demonstrated that magnetic skyrmions can be excited into oscillation in a nanodisk by the Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. spin-polarized currents, which provides a possibility of skyrmion-based STNOs [9][10][11]. However, except that the oscillation frequency of antiferromagnetic skyrmion-based STNOs [12] can reach ten of GHz, ferromagnetic skyrmion-based STNOs only produce about 2GHz, which need a further improvement for applications. In this paper, we propose adding an annular groove to the free layer of ferromagnetic skyrmion-based STNOs. Then we investigate performance and oscillation frequency of the STNOs by using micromagnetic simulations. At last, we discuss the reasons for the increase in frequency of this type of STNOs by analyzing the Thiele's equation.

Model and simulation details
The skyrmion-based STNO is a sandwich structure, i.e, consisting a fixed layer, a spacer and a free layer, as shown in figure 1. The diameter of the nanodisk is 100 nm, and the thickness of the free layer is 1 nm with the mesh size of 1×1×0.5 nm 3 . We first create an annular groove with a depth of 0.5 nm in the free layer, and then set a single or multiple Néel skyrmions in the annular groove region. However, In addition to [32], which reported current-induced skyrmion motion guided by a groove. It is a challenge to create a groove experimentally. Here, we provide an idea for possible implementation of the groove: Néel skyrmions can be replaced by Bloch skyrmions, and therefore the thickness of the free layer, i.e. the cubic B20 chiral magnets can be slightly increased, which reduces the difficulty of digging the groove and possibly carried out by electronbeam lithography [33]. The creation of skyrmions can be achieved by a local perpendicular current injection, as shown in figure 2. A perpendicular spin-polarized current is injected into the annular groove at 0 ns. Under a continuous injection of current, skyrmion-liked structures are appeared as shown in the snapshots at 0.06 and 0.09 ns. The current is turned off at 0.1 ns and after a 4.9 ns long relaxation, a stable skyrmion is nucleated in the annular groove. The threshold current can be further reduced with the presence of a perpendicular magnetic field [34]. Therefore, a local perpendicular current injection combined with an external magnetic field may be easier to create the skyrmions in the annular groove experimentally. The skyrmions are confined in the annular groove due to the potential of the groove. With a continuous current vertically injecting into the sandwich  structure by a point contact electrode, the skyrmions begin to move around in the annular groove, and then a stable oscillation voltage signal can be detected when the skyrmions pass the region of detecting contact.
The micromagnetic simulation results are performed by using the Object Oriented MicroMagnetic Framework (OOMMF) public code [35] with the additional module for DMI. The Landau-Lifshitz-Gilbert (LLG) equation extended with the Slonczewski torque [36] is numerically solved for the dynamics of timedependent magnetization, as follow: where γ is the gyromagnetic ratio, m is the unit vector of the local magnetization, the Gilbert damping α is set to 0.5 and 0.02 for solving the ground state and the dynamic response procedure, respectively. t F is the thickness of the free magnetic layer, m p is the polarization direction of the fixed layer along the -z direction. u represents the spin-polarized electrons velocity with the form = g m  u J P eM 2 0 s , where J is the current density, e is the electron charge, P is the spin polarization, ÿ is the reduced Planck constant, μ 0 is the permeability of free space, and M s is the saturation magnetization. H eff is the effective field including the exchange field, perpendicular anisotropy field, demagnetization field and interfacial DMI effective field. The Oersted field induced by the current is neglected because of its little influence on the STNO [11]. The free layer is considered as a Co magnetic film on a heavy-metal substrate inducing the interfacial DMI. The material parameters of the free layer are from [11,34,37]

Results and discussion
3.1. A single skyrmion-based STNO First we investigate the dynamics of a single skyrmion in the annular groove, as shown in figure 3. Figure 3(a) shows that the initial position of a skyrmion is located in the annular groove, where the width and inner radius of the annular groove are defined as d and r with the value of 10 and 25 nm, respectively. The radius of the point contact electrode is set to be equal to r, when the nanodisk is continuously injected current with the density of -10 10 A m 11 2 , the skyrmion begin to move counterclockwise in the annular groove as shown in figure 3(b), which displays the guiding center (R x , R y ) of the skyrmion in the Cartesian coordinate system. The form of the guiding center is given by [38]: where q is the topological density. One can see that the initial oscillation radius of the skyrmion is 30 nm, i.e. the initial position of the skyrmion is in the center of the groove. After applying a spin current, the oscillation radius  figure 3(a)) and R x , respectively. It can be seen that m z =0.83 indicates the skyrmion moving pass the detection area, and the time dependence of m z is similar to that of R x . The corresponding power spectrum shows that the frequency of a single skyrmion-based STNO is 1.75 GHz, which is obtained by taking fast Fourier transform of the oscillation of R x , as shown in figure 3(e).
The different width and inner radius of the annular groove have an influence on the kerb force acting on the skyrmion and the oscillation radius of the skyrmion. Therefore, we then investigate the influence of d and r on the oscillation frequency of the STNO, as shown in figure 4. Figure 4(a) shows the oscillation frequency dependence of current density for different d with r fixed at 20 nm. For the case without the annular groove, the STNO starts to oscillate with the frequency of 440 MHz when current is applied with the density of -0.2 10 A m 11 2 , with the current density further increases to´-20 10 A m 11 2 , the frequency of the STNO gradually increases to 590 MHz and then remains unchanged. For the case with the annular groove, the oscillation frequency is much larger than the case without the annular groove and decreases with the increase in d from 15 to 25 nm under the same current density. To be noted, for d=10 nm, the frequency of the STNO drops from 2.68 to 1.1 GHz when the current density increases from´-22 10 A m 11 2 to´-23 10 A m 11 2 , this is because the gyrotropic force of the skyrmion exceeds the critical kerb force of the annular groove so that the skyrmion moves out of the annular groove. Figure 4(b) shows the frequency of the STNO versus the current density for different r with d fixed at 10 nm. It is found that the frequency of the STNO decreases with the increase in r from 25 to 35 nm under the same current density. The snapshot of the initial states of the STNO under different d and r are shown in the right side of figure 4. The presence of the annular groove not only increases the frequency of the STNOs but also limits the size of the skyrmion. Therefore, in the following simulations, in order to place more skyrmions in the annular groove and ensure that skyrmions do not move out of the annular groove, d and r are fixed at 10 and 25 nm, respectively.

Multiple skyrmions-based STNO
For the situation without the presence of the annular groove, there are at most 6 skyrmions with a centrosymmetric distribution that can stably exist in a single nanodisk, as reported in our previous work [9]. When the number of skyrmions (N) is larger than 6, the skyrmions are pushed to the middle of the nanodisk or even annihilation due to the large repulsive force between the skyrmions. For the situation with the presence of the annular groove, because the size of the skyrmions are limited by the potential of the groove, the maximum number of skyrmions with a centrosymmetric distribution increases to 12, as shown in figure 5. The excitation of multiple skyrmions in the annular groove can be achieved by repeating the process of the excitation of a single skyrmion, as shown in figure 2. When the nanodisk is continuously applied a point contact spin current, all the skyrmions in the groove move with the counterclockwise mode.
The dynamic of one of multiple skyrmions in the annular groove as shown in figure 6 can be described by the Thiele's equation as follow [9, 39]: is the gyrovector, S is the skyrmion number. For Néel skyrmions and Bloch skyrmions, S=1. While for Antiskyrmios, the skyrmion number S=−1, which leads the gyrovector G of the Antiskyrmions different from that of Néel skyrmions and Bloch skyrmions. Therefore, the rotation direction of the Antiskyrmions is changed to clockwise. v is the velocity of the skyrmion that can be decomposed into the tangential ( q e ) and radial , whose value is proportional to the current density J. = a F D v · is the damping force, where D is the damping tensor with the value of D α . U is the potential for the skyrmion due to the repulsive force (F ss ) from the nearest neighbor skyrmions and the kerb force of the annular groove (F Kerb ), i.e.
where F ss is inversely proportional to the distance between two skyrmions [40]. It should be noted that the potential U is also affected by the size of the nanodisk,we have demonstrated that the frequency of the STNO is inversely proportional to the radius of the nanodisk in our previous work [9]. When skyrmions moves steadily in the annular groove, which means v r =0. Therefore, equation (3) can be written as: The oscillation frequency of the multiple skyrmions-based STNOs can be expressed as: where r s is the oscillation radius of skyrmions. Following, we investigate the frequency dependence of current density for different N, as shown in figure 7(a). It is found that the frequency of the 12 types of STNOs shown in figure 5 all increase first and then keep unchanged with the increase in current density. Equations (4a) and (5) also indicate that v t is greatly affected by F ST , that is, f increases with the increase in the current density before it reaches saturation. Figure 7(b) compares the maximum frequency dependency of N with and without the annular groove. In the presence of the annular groove, with N increases from 1 to 12, the maximum frequency of the STNO first increases from 2.43 to 15.63 GHz (N=8), and then decreases to 10.93 GHz. Moreover, with α further decreases to 0.01, the skyrmions are annihilated due to the strong spin wave induced by the spin current. From equations (4a) and (5), it can be known that: when N < 6, the value of F ss is almost 0 due to the long distance between two skyrmions and therefore f increases linearly with the increase in N; when N>6, F ss increases sharply with the decrease in the distance between two skyrmions, which leads to the decrease of v t . That is the reason why, although the number of skyrmions has increased, the frequency of the STNO no longer increases linearly with the increase in N or even begins to decrease. In the absence of the annular groove, the maximum frequency of the STNO also first increases and then decreases with N increases from 1 to 6, and the extreme value of the maximum frequency is 2.47 GHz when N=5. It can be seen that the frequency tunability of STNOs with the annular groove is improved by 6.3 times than that of without the annular groove, and there are three reasons by analyzing equations (4a) and (5). First, the kerb force from the annular groove is far larger than the boundary-induced force; second, more number of skyrmions can stably locate in the annular groove because their size are restricted by the groove; third, the oscillation radius of those skyrmions are fixed in the annular groove and does not increase significantly with the increase in current density.

Conclusions
In summary, we proposed a skyrmion-based STNO with the presence of an annular groove. By micromagnetic simulations, we first set one skyrmion in the annular groove and demonstrate that the skyrmion can move around in the annular groove driven by a spin current. Following, we investigate the influence of the annular groove parameters including d and r on the frequency of the STNO, the results show that the oscillation frequency is inversely proportional to both d and r within a certain range under the same current density. Interestingly, the oscillation frequency with an annular is far larger than that of without an annular groove, the Thiele's equation indicates that this is because the potential of the kerb is larger than that of the boundary and the oscillation radius of the skyrmion can be controlled by the groove. Moreover, because the size of the skyrmion is restricted by the annular groove, up to 12 skyrmions can stably locate in the annular groove, which is much more than the case without the annular groove. On that basis, the frequency tunability of the STNO reaches to 15.63 GHz, which is about 6.3 times of the STNOs without the groove. Our results may provide a possibility for the design of high-frequency skyrmion-based STNOs.