Array of Synchronized Nano-Oscillators Based on Repulsion between

PHYSICAL REVIEW APPLIED 9, 044007 (2018)

Array of Synchronized Nano-Oscillators Based on Repulsion between Domain Wall and Skyrmion Chendong Jin,1 Jianbo Wang,1,2 Weiwei Wang,3 Chengkun Song,1 Jinshuai Wang,1 Haiyan Xia,1 and Qingfang Liu1,* 1

Key Laboratory for Magnetism and Magnetic Materials of the Ministry of Education, Lanzhou University, Lanzhou 730000, People’s Republic of China 2 Key Laboratory for Special Function Materials and Structural Design of the Ministry of the Education, Lanzhou University, Lanzhou 730000, People’s Republic of China 3 Department of Physics, Ningbo University, Ningbo 315211, People’s Republic of China (Received 26 July 2017; revised manuscript received 23 November 2017; published 5 April 2018) Spin-transfer nano-oscillators (STNOs) are nanosized microwave signal generators based on spintransfer torque and the magnetoresistance effect. So far, the low output power of STNOs is one of the key restrictive factors. Fabrication and synchronization of a multiple STNO array in one device is a promising way to increase the output power. However, previous studies have shown that only a limited number of STNOs achieve synchronization due to the complex coupling mechanism. In this work, we propose an alternative structure of STNOs based on the repulsion between the domain wall and the Skyrmion. It is found that the frequency tunability of this kind of STNO reaches up to 1.9 GHz. Moreover, we numerically demonstrate that the integrated arrays of STNOs can export synchronous signals, which is promising to potentially increase their total power. Our results provide alternatives for designing of Skyrmion-based devices and further improving the output power of STNOs. DOI: 10.1103/PhysRevApplied.9.044007

I. INTRODUCTION Spin-transfer nano-oscillators (STNOs) have attracted great interest in recent years due to their potential applications in microwave sources and memory devices [1–7]. However, the output power urgently needs to be improved because the microwave power emitted from a single STNO is typically on the order of nanowatts, which is too weak for practical applications [8–10]. Up to now, several promising solutions have been proposed to increase the output power such as synchronizing with the external signals [9,11–13], local synchronization of four magnetic vortex-based STNOs [14], and phase locking of two [8,15–19], three [20], and even five [21] nanocontacts. Although it has been predicted that the practical level of the output power can be further enhanced by means of integrating multiple STNOs into arrays [4,10], the complete synchronization in a larger number of STNOs has not been achieved yet because the synchronization mechanisms of the phase-locking process are more unpredictable and complex [9,17,20]. Therefore, the situation calls for alternative ideas and solutions to achieve the synchronized arrays of STNOs. Magnetic Skyrmions are stable particlelike magnetization configurations with topologically protected property which have already been discovered in bulk noncentrosymmetric B20-type transition metal compounds [22–25] *

Corresponding author. [email protected]

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and multilayered ultrathin films [26–29] lacking inversion symmetry with strong spin-orbit coupling due to the presence of the Dzyaloshinskii-Moriya interaction (DMI) [30–33]. Until now, Magnetic Skyrmions have shown huge potential in the applications of spintronic devices and high-density information storages due to their stable small size and low density of driving current [34–37]. In this paper, we utilize the topologically protected property of Skyrmions and review the problem of synchronized arrays of STNOs from the perspective of structural design. Through micromagnetic simulations, we devise a dumbbell-shaped STNO based on repulsion between the domain wall (DW) and Skyrmion. Then we analyze the oscillation principle of the STNO and investigate the factors that influence the oscillation frequency. In addition, we design four arrays of STNOs and demonstrate that the synchronization of the arrays is expected to be achieved with a large number of STNOs, which is promising to improve their output power. At last, we look forward to the practical applications of STNOs and also provide several challenges that need to be considered. II. MODEL AND MICROMAGNETIC SIMULATION DETAILS The dumbbell-shaped STNO consists of two sandwich structures, i.e., a thick fixed layer and a thin free layer and a spacer layer with a disk shape that connects by a bridge (same structure and material with left and right nanodisks),

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PHYS. REV. APPLIED 9, 044007 (2018)

FIG. 1. Schematic diagram of the dumbbell-shaped STNO which consists of a free layer, a spacer, and a fixed layer. In the free layer, the red, white, and blue represent where the z component of the magnetization is positive, zero, and negative, respectively. The yellow represents the spacer, and the purple represents the fixed layer with the polarization vector along the −z direction. The diameter of both nanodisks is 100 nm and the thickness of the free layer is 1 nm, the length and width of the bridge are defined as L and W, respectively. For the left nanodisk, the current is perpendicularly injected along the −z direction in a circle region with the diameter of 40 nm, while for the right nanodisk, the direction of the injected current is changed to the þz axis and with other parameters unchanged.

as shown in Fig. 1. Each sandwich nanodisk can work as a single STNO controlled by current injecting from the electrode with the diameter of 40 nm. The magnetization orientation of the left free layer (along the −z direction) is opposite to that of the right free layer (along the þz direction). Therefore, a N´eel DW is generated in the free layer of the bridge. Then we create two N´eel Skyrmions at the center of the two nanodisks in the free layer (the Skyrmion number in the left and right nanodisks are 1 and −1, respectively). With continuous current injecting perpendicularly into the two sandwich nanodisks by the point contact electrodes, a stable oscillation voltage signal is detected between the free layer and the fixed layer at the bridge region due to the periodical translation of DW. The Object Oriented MicroMagnetic Framework (OOMMF) public code [38] is performed for the simulations including the additional modules for DMI [33]. The dynamics of time-dependent magnetization is governed by numerically solving the Landau-Lifshitz-Gilbert (LLG) equation extended with a spin-transfer torque term, i.e., Slonczewski torque [39], as follows: d ⃗m ⃗ eff þ α ⃗m × d ⃗m þ u ⃗m × ð ⃗mp × ⃗mÞ; ¼ −γ ⃗m × H dt Ms dt tF ð1Þ where ⃗m is the unit vector of the local magnetization, γ is the gyromagnetic ratio, α is the Gilbert damping, which is set to 0.5 and 0.02 for the equilibrium state and the dynamic

response procedure, respectively, M s is the saturation magnetization, tF is the thickness of the free magnetic layer with the value of 1 nm, ⃗mp is the current polarization vector along the −z direction. The parameter u represents the spin-polarized electron velocity with the form u ¼ ½ðγℏPÞ=ð2μ0 eMs ÞJ, where J is the current density, the spin polarization P is set to 0.4 [34,40–42], e is the electron ⃗ eff is the effective field including the exchange charge. H field, anisotropy field, demagnetization field, and Oersted field induced by the current (in the Supplemental Material [43] we have demonstrated that the Oersted field has little effect on the STNO, the Oersted field is neglected in this work), and the DMI effective field. In this paper, we consider the interfacial DMI energy whose energy density can be expressed as [33,44] εInter-DMI ¼ D½mz ∇ · ⃗m − ð ⃗m · ∇Þmz ;

ð2Þ

where D is the DMI constant representing the DMI strength, mz is the components of the normalized magnetization with the form mz ¼ M z =Ms . The diameter of both nanodisks is 100 nm and the thickness of the free layer is 1 nm with a mesh size of 1 × 1 × 1 nm3 , the length and width of the bridge are defined as L and W, respectively. We have investigated the effect of the value of L and W on the frequency of the STNOs later. For micromagnetic simulations, the free magnetic layer is considered as a cobalt film with perpendicular magnetic anisotropy on a heavy-metal substrate and material parameters are chosen from Ref. [34]: saturation magnetization Ms ¼ 580 × 103 A=m, exchange constant A ¼ 1.5 × 10−11 J=m, perpendicular magnetic anisotropy constant K u ¼ 8 × 105 J=m3 , and DMI strength D ¼ 3.5 × 10−3 J=m2 . These parameters guarantee the existence of a single or even more Skyrmions in a nanodisk. III. WORK PRINCIPLE AND FINE-TUNING OF THE STNO Two Skyrmions are both initially located in the center of their own nanodisks. Then currents with the density of 10 × 1011 A=m2 are injected into both nanodisks continuously. Because of the torque from the current, two Skyrmions begin to move around the disk. The snapshot of the two Skyrmions in their own nanodisks is shown in Fig. 2(a). For the left nanodisk, the Skyrmion is driven out of the center of the nanodisk from the bottom-left direction and then moves counterclockwise in a circle as shown in Fig. 2(b), which displays the trajectory of the Skyrmion core in the left nanodisk. At the same time, the Skyrmion in the right nanodisk moves out of the center from the top-left direction and rotates clockwise as shown in Fig. 2(c). To understand the motion of the Skyrmion in the nanodisk, we use the Thiele equation to describe the dynamics of the Skyrmion by reducing the Landau-Lifshitz-Gilbert equation to the following equation [2,6,45]:

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FIG. 2. (a) Top views of the STNO with the length and width of the bridge fixed at 20 and 10 nm, respectively. The diameter of the two nanodisks is 100 nm. The current densities of both nanodisks are 10 × 1011 A=m2 , the trajectories of the two Skyrmion cores in the left and right nanodisk are shown in (b) and (c), respectively. The yellow arrows in (a) indicate the move direction of two Skyrmions. (d) The variation of mz in the bridge region as a function of time. We have calculated the variation for 500 ns with the step of 0.01 ns. The diagram just shows the part of the first 100 ns. (e) FFT power spectrum of mz in the bridge region with the x axis in logarithmic scale and y axis in linear scale. The inset is the corresponding FFT power spectrum with the x axis in the linear scale and the y axis in the logarithmic scale.

G×

dX ∂U ˆ dX − −D þ FST ¼ 0; dt ∂X dt

ð3Þ

where G is the gyrovector with the form G ¼ −½ð2πStF Ms Þ=ðγez Þ, with S the Skyrmion number and ez unit vector along the z direction. U is the potential for the Skyrmion rising from the boundary effects, X is the ˆ is the damping tensor, FST position of the Skyrmion core, D is the spin-transfer force. Our previous work has demonstrated that the Skyrmion can rotate around in the nanodisk [6], the rotation frequencies of the Skyrmion obtained from the Thiele equation are close to our simulation results. In addition, it is found that the rotation frequencies are mainly determined by the potential U, i.e., the size of the nanodisk. In this work, the diameter of the nanodisk is fixed as 100 nm, and later we investigate the influence of other parameters on the rotation frequency. Interestingly, as shown in Fig. 2(a), when the Skyrmion in the left nanodisk approaches the left end of the bridge, the DW in the bridge region moves to the right end of the bridge due to the repulsive force from the Skyrmion. Then, when the Skyrmion in the left nanodisk moves away from the bridge and the Skyrmion in the right nanodisk approaches the right end of the bridge, the DW is driven to the left end of the bridge by the force from the Skyrmion in the right nanodisk. In other words, the DW translates periodically between both ends of the bridge under the alternate repulsive forces from the two Skyrmions; see Supplemental Material, Video I [43]. Therefore, the oscillation frequency of the DW is equal to the rotation frequency of the Skyrmion. In the

following, we obtain the oscillation of mz (the out-of-plane components of the normalized magnetization) in the bridge region as shown in Fig. 2(d), mz ¼ 1 and −1 represent that the DW locates at the left and right end of the bridge, respectively. The excitation spectrum is obtained by using the fast Fourier transform (FFT) for the time-dependent mz of the bridge region as shown in Fig. 2(e). The fundamental frequency, i.e., the oscillation frequency with the value of 580 MHz and several higher harmonics appear in the FFT power spectrum. Then let us expound the original purpose of designing the dumbbell-shaped STNO. The amplitude of magnetization precession (mx , my , and mz ) in the whole nanodisk is extremely small, which limits the practical use of Skyrmion-based STNOs. We propose a solution using multiple pairs of local contact electrodes [6]. However, detecting the movement of Skyrmions is still too difficult. In this work, we translate the rotation of the Skyrmion to periodical translation of DW. Therefore, oscillation voltage signals of the STNO are obtained by detecting the periodical translation of DW, which greatly reduces the difficulty of detection technology. The different width and length of the bridge have an influence on the boundary conditions of the Skyrmion in the nanodisk. According to Eq. (3), both potential U and spin-transfer force FST determine the rotation frequency of the Skyrmion. Therefore, it is necessary to understand the influence of bridge parameters on the oscillation frequency of the STNO. Figure 3(a) shows the frequency of the STNO as a function of current density for different W with L fixed at 20 nm. The threshold current density

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FIG. 4. (a), (b), (c), and (d) are the stable states of 2, 3, 4, and 5 Skyrmions in a single nanodisk, respectively. The yellow arrows represent the move direction of Skyrmions with a continuous injection of current.

FIG. 3. (a) Frequency of the STNO as a function of current density for W ¼ 5, 10, 15, and 20 nm with L fixed at 20 nm. (b) The frequency of the STNO as a function of current density for L ¼ 15, 20, 30, and 35 nm with W fixed at 10 nm. The insets are enlarged views of small current density. It should be noted that the injection current densities of the two nanodisks are equal.

increases proportionally to W, and the maximum oscillation frequency decreases with the increase of W under the same current density. We also investigate the oscillation frequency as a function of current density for different L with W fixed at 10 nm as shown in Fig. 3(b). It is found that the threshold current density is proportional to L, and the maximum oscillation frequency keeps unchanged with different L. In the following simulations, W and L are fixed at 10 and 20 nm, respectively. IV. MULTIPLE SKYRMIONS IN EACH NANODISK Frequency tunability is an important feature of STNOs. However, we noticed that the maximum frequency tunability of the STNO is no larger than 200 MHz by adjusting the parameters of the bridge, which is too small for practical application. Besides, we consider the oscillation frequency

of DW to be equal to the repulsive frequency from the Skyrmion and, therefore, it is expected to be improved by putting multiple Skyrmions in one nanodisk. Figure 5 shows that multiple Skyrmions can stably exist in a single nanodisk with a centrosymmetric distribution due to the repulsions of Skyrmion-Skyrmion and Skyrmion-edge. A similar result has also been reported in our previous work [6]. When currents of the same density are continuously applied on both nanodisks by point contacts, Skyrmions in the left and right nanodisk move with counterclockwise and clockwise modes, respectively. Figure 5(a) shows the oscillation frequency as a function of current density for different numbers of Skyrmions in each nanodisk (N Sky ). It can be seen that all the frequencies increase with the increase of current density, and the growth rate of frequency is relatively large for a larger N Sky (except for N Sky ¼ 5). Taking N Sky ¼ 4 as an example, the minimum oscillation frequency is 260 MHz with the threshold current density of 0.4 × 1011 A=m2 . With the current density increasing to 30 × 1011 A=m2 , the frequency of the STNO gradually increases to 2.16 GHz. While for N Sky ¼ 5, Skyrmions are easy to annihilate at the connection of the nanodisk and bridge due to the low potential of the boundary driven by a large current. In contrast to the STNO with a single Skyrmion in each nanodisk, STNOs with multiple Skyrmions have evident advantages: (i) the maximum operating frequency of the STNO-based multiple Skyrmions is far larger than that of a single Skyrmion. The frequency tunability (the maximum frequency minus the minimum frequency) for N Sky ¼ 4 reaches to 1.9 GHz, which is 13 times larger than that of N Sky ¼ 1 (150 MHz) as shown in Fig. 5(b). (ii) Both the threshold current densities and the corresponding minimum frequencies for N Sky > 2 are smaller than that of N Sky ¼ 1. (iii) The onset time for N Sky > 1 is smaller than that of N Sky ¼ 1, because a single Skyrmion needs time to move out of the center of the nanodisk, while for N Sky > 1, the multiple Skyrmions can rotate directly due to their centrosymmetric distribution. Skyrmions are probably nucleated simultaneously on both sides of the bridge as shown in Fig. 4(c), which may

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PHYS. REV. APPLIED 9, 044007 (2018) FIG. 5. (a) Frequency of the STNO as a function of current density for the number of Skyrmions N Sky ¼ 1, 2, 3, 4, and 5. (b) Frequency tunability as a function of the number of Skyrmions. (c) The variation of mz in the bridge region as a function of time for the number of Skyrmions N Sky ¼ 4 with a current density of 10 × 1011 A=m2 . (d) and (e) are top views of the STNO at 3.1 and 70.3 ns, respectively.

lead to the DW stay at the bridge area without oscillating motion. After detailed simulations, we find that even in this case, the STNO can still output a stable oscillation signal. Taking N Sky ¼ 4 as an example again, when constant currents with the density of 10 × 1011 A=m2 are applied into both nanodisks, we obtain the continuous oscillation of mz in the bridge region, as shown in Fig. 5(c), see Supplemental Material, Video II [43]. In the beginning, both sides of the Skyrmions are almost simultaneously moving to the edge of the bridge as shown in Fig. 5(d), which displays the top view of the STNO at 3.1 ns. The DW only oscillates in a small range of the bridge region with the oscillation amplitude increasing gradually. After 8 ns, the DW can completely translate to the edge of the bridge. During the first 8 ns, the relative positions of the Skyrmions in two nanodisks are adjusted by a coupling interaction, i.e., the repulsive force from the DW. This is the reason why the DW can be finally driven alternately by the Skyrmions in the two nanodisks as shown in Fig. 5(e), which displays the top view of the STNO at 70.3 ns. Therefore, the initial positions of the Skyrmions do not affect the output signals of the multiple Skyrmions-based STNO. V. FOUR ARRAYS OF STNOS We first look at the factors that affect the output power of a single STNO. The output power transmitted from the single STNO to the load can be approximately expressed as [4,10]:

Pout ¼

I 2 ΔR2 RL ; 8 ðR þ RL Þ2

ð4Þ

where I and R are the dc current and resistance, respectively, RL is the impendence of the load, ΔR is the oscillation amplitude of the resistance with the form ΔR ¼ Am × MR, with Am the amplitude of magnetization oscillation, and MR the various resistances due to the magnetoresistive effect. Therefore, maximizing output power of a single STNO requires optimizing the impedance matching ratio R=RL and improving both MR and oscillation amplitude of magnetization. To further improve the output power, we design four arrays of STNOs with different numbers of Skyrmions in the free layer, as shown in Fig. 6. The first type of array for N Sky ¼ 1, as shown in Fig. 6(a). As mentioned in the legend of Fig. 1, the fixed layer with the magnetization vector along the −z direction, in the free layer, the bridge region in blue and red represent the magnetic components along the −z and þz directions, respectively. Figures 6(b) and 6(c) are the array elements, which show the relatively low (parallel) and high (antiparallel) resistance of the STNO, respectively. Therefore, this type of a single STNO has maximizing Pout due to the maximizing Am . Figures 6(d), 6(g), and 6(j) show the linear, hexagonal, and square array for N Sky ¼ 2, 3, and 4, respectively. Figures 6(e), 6(h), and 6(k) are the elements of the linear, hexagonal, and square array representing the relatively low resistances of the STNOs,

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FIG. 6. (a), (d), (g), and (j) display the four arrays of STNOs for N Sky ¼ 1, 2, 3, and 4, respectively. (b), (e), (h), and (k) are the array elements representing the low resistance (the bridge region in blue color) corresponding to (a), (d), (g), and (j), respectively. (c), (f), (i), and (l) give the corresponding high resistance (the bridge region in red color) of the four types of arrays.

respectively. The relatively high resistances of the STNOs are shown as Figs. 6(f), 6(i), and 6(l). It can be seen that Skyrmions exist stably in each nanodisk with centrosymmetric distribution. Each nanodisk connects with neighboring nanodisks by two, three, and four bridges with 180°, 120°, and 90° distribution for N Sky ¼ 2, 3, and 4, respectively. We emphasize that a single dumbbell in Fig. 6(a) is a fourterminal device including two injections (nanodisks), a detection (bridge), and grounding. When N Sky increases to 2 as shown in Fig. 6(d), each nanodisk connects with

neighboring nanodisks by two bridges, which can be understood that one nanodisk controls one bridge. In a similar way, a single nanodisk in a hexagonal [Fig. 6(g)] and a square [Fig. 6(j)] array controls 1.5 and 2 bridges, respectively. Therefore, the number of bridges controlled by a single nanodisk relatively increases with the increase of N Sky . In other words, on the premise of the same number of injection terminals, arrays can export more signals with the increase of N Sky . It is also important to note that increasing the number of STNOs in arrays may also lead to impedance mismatch problems in microwave circuits and the requirement for additional power to drive arrays, which may be a challenge that needs to be addressed in the future. In the following, let us take the fourth array as an example to discuss the principle of synchronization. During the simulation of the oscillation array, a 2D periodic boundary condition (PBC) is used in the xy plane [46] and the initial element is shown in Fig. 7(a). Here, we deliberately create Skyrmions on both sides of the bridge, which is the case we mention in Figs. 5(c)–5(e), to make sure that the array can work in all cases. When current with the density of 10 × 1011 A=m2 is perpendicularly injected into all nanodisks, Skyrmions in each of the disks begin to rotate, which leads to a periodical translation of DW in all bridges, see Supplemental Material, Video III [43]. The variations of mz in four bridge areas as a function of time are shown in Fig. 7(b). The mz in four bridges changes synchronously from the beginning of the DW movement, even within the first 0.5 ns, where the amplitudes of mz do not reach their maximum values. Each nanodisk in the array connects with four bridges, i.e., there are four repulsion forces from the DWs in the bridges affecting the relative positions of Skyrmions. This is the reason why the elapsed time before the approaching of the maximum amplitude in the array is much less than that of Fig. 5(c). From Video III FIG. 7. (a) Snapshot of the initial state of the square array element and the location of the four bridges have been marked out. The current densities of all the four nanodisks are 10 × 1011 A=m2 and the 2D PBC in the xy plane are used. (b) The variations of mz in the four bridge regions as a function of time.

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in the Supplemental Material, it can be seen that Skyrmions in one nanodisk exert repulsive forces to the DW in the nearest bridge at the same time and the DW is driven alternately by Skyrmions in the two nanodisks on the adjacent sides of the bridge. The above two conditions guarantee the operation of the oscillation array. Therefore, the synchronization of the oscillation array is achieved by the special structural designs. Hence, the total output power of the array (Ptot ) can be written as [8,20]

ACKNOWLEDGMENTS

Ptot ¼ N 2 Pindiv ;

ð5Þ

where N is the number of the bridge, and Pindiv is the power of individual STNOs. The total power of the array is expected to increase as N 2 because the output signals from N individual STNOs are zero phase shift. VI. CONCLUSIONS AND OUTLOOK In summary, we propose an alternative STNO based on repulsive force between DW and Skyrmion, which is promising to solve the challenges of STNOs for applications. By micromagnetic simulations, we analyze the working principle of the STNOs and investigate the influence of bridge parameters on the frequency of the STNO. However, the frequency tunability is only about 150 MHz after fine-tuning. In order to solve the problem of low-frequency tunability, we dope out a solution by creating multiple Skyrmions in each nanodisk. It is found that the performance of the STNO is greatly improved as follows: the frequency tunability reaches up to 1.9 GHz, which is 13 times larger than that of a single Skyrmion; the onset time, the critical current density, and the corresponding minimum frequencies are all smaller than that of a single Skyrmion. On that basis, we design four arrays of STNOs and demonstrate the synchronization of the array by using 2D PBC. Therefore, these kinds of STNOs can be synchronized with a large number which is expected to further improve their output power. We also note that there are several issues that need to be considered for further practical applications, see the Supplemental Material [43]. One of the first questions is how to achieve the dumbbell or even the checkerboard type of STNOs experimentally. We provide a possible idea to implement the special structure and also acknowledge that it is indeed a big challenge. Second, imperfections in the actual situations such as different densities of two applying currents would have an influence on the output signals. Third, the stray field of the fixed layer has an influence on the rotation frequency of Skyrmions. At last, the thermal noise broadens the linewidth and even results in a destruction of STNOs. Therefore, it is also necessary to consider how to reduce the effects of thermal noise. These results may present guidance for the design of Skyrmionbased microwave signal generators.

This work is supported by National Science Fund of China (Grants No. 11574121 and No. 51771086).

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