Reversal of coupled vortices in advanced spintronics: A mechanistic study

TL;DR

This paper investigates how magnetic vortex cores in coupled ferromagnetic dots can be reversed using direct current without external magnetic fields, combining experiments and simulations to understand the underlying mechanisms.

Contribution

It provides new insights into vortex core reversal mechanisms and demonstrates current-induced manipulation of vortex orientation in coupled magnetic systems.

Findings

Vortex cores can be switched by applying a constant direct current.

Deformation of vortex profiles causes the reversal process.

Potential for vortex-based non-volatile memory applications.

Abstract

This study conducts a comprehensive investigation into the reversal mechanism of magnetic vortex cores in a nanopillar system composed of two coupled ferromagnetic dots under zero magnetic field conditions. The research employs a combination of experimental and simulation methods to gain a deeper understanding of the dynamics of magnetic vortex cores. The findings reveal that by applying a constant direct current, the orientation of the vortex cores can be manipulated, resulting in a switch in one of the dots at a specific current value. The micromagnetic simulations provide evidence that this switch is a consequence of a deformation in the vortex profile caused by the increasing velocity of the vortex cores resulting from the constant amplitude of the trajectory as frequency increases. These findings offer valuable new insights into the coupled dynamics of magnetic vortex cores and demonstrate the feasibility of manipulating their orientation using direct currents under zero magnetic field conditions. The results of this study have potential implications for the development of vortex-based non-volatile memory technologies. \end{abstract}

Full text

I Introduction

In a magnetic medium, a vortex state describes a region of in-plane curling magnetization distribution spiraling about a region with out of plane magnetization, namely the vortex core (VC). Magnetization patterns at the vortex state can be described by the vortex polarity and chirality. The vortex polarity describes the polarization of the magnetization at the VC, where a positive/negative polarity refers to an upward/downward direction of magnetization respectively. On the other hand, the chirality describes the direction of the curling magnetization about the VC, with a positive/negative chirality referring to a clockwise/counter clockwise configuration. In the past years, a number of theoretical, numerical and experimental studies have been devoted to understanding and manipulating the vortex dynamics [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. Magnetic vortices offer high density data storage application possibilities, such as non-volatile magnetic memories [11, 12]. These solitons also possess the ability to be exploited further as sources for spin wave generation [13, 14], which constitutes the backbone for beyond-CMOS-based technologies [15, 16, 17, 18].

In recent years, there has been increasing interest in studying vortex dynamics in the context of spin-transfer torque oscillators (STOs) [8, 9, 10, 19]. These non-linear devices have the ability to regulate their frequencies as to synchronize with external stimuli, such as current or field [20, 21, 22], noise in the surrounding environment [23], or other STOs [24, 25]. This nominates them as promising candidates for applications in vowel recognition [26], neural networks [27], and neuromorphic computing [28]. Vortices can be excited and driven in the magnetic layers of STOs by applying a DC current (IDC) through the multi-layered structure, a phenomenon known as spin-transfer torque (STT) [29, 30, 31]. Compared to traditional STOs, vortex-based STOs offer improved efficiency [32, 24], a wider range of tunability [33], and a narrower frequency linewidth [34, 35, 36]. One of the most significant applications of vortex-based STOs is their ability to synchronize more efficiently and achieve mutual phase locking [21, 32, 22, 37, 24, 25, 38, 39, 19].

In this work, we investigate the dynamics of two dipolarly coupled vortices for continuous applied DC current and zero field, while also aiming to reveal the origin of the reversal mechanism underwent by the VC based in the STO’s thick layer. Understanding the physics bound to the origin of the VC’s reversal mechanism in coupled systems is at the core of studies aiming at vortex engineering, control and manipulation. The concept of VC reversal itself is neither new nor surprising, as it has been previously approached and investigated via various methods [40, 41, 42, 43]. However, interpretations in regards of the VC’s dynamic reversal in coupled systems under zero magnetic field still lag as to vividly convey the nature of such mechanism. Here, we propose an alternative approach as to vortex coupled dynamics for applied DC current at zero field, thus presenting an efficiently convenient approach to controlling vortex dynamics in coupled ferromagnetic dots. After which, we go further to provide detailed understanding of the dynamic VC reversal mechanism in such coupled systems.

II Experimental Setup

To start, we present the geometry of the circular spin-valve nanopillar oscillator under investigation (see Figure 1(a)). The system is a multi-layered stack, consisting of Permalloy-based thick (vortex 1) and thin (vortex 2) layers with vortex magnetic configurations at remanence. The vortices’ configurations in these layers are prepared with opposite polarities and the same chiralities (as shown in Figure 1(a) for case 1 and case 2), where the polarities of the vortices in the thick layer are represented by $\Uparrow$ or $\Downarrow$, and those in the thin layer by $\uparrow$ or $\downarrow$. The thickness of the lower layer is 15 nm, while that of the upper layer is 4 nm. A non-magnetic copper spacer of 10 nm thickness is present between the two layers. The overall diameter of the multi-layered stack is 250 nm.

We initiate our study by experimentally investigating the power dependence of the thick layer’s frequency on DC current in the absence of applied magnetic field for the two above mentioned cases (see Fig. 1(b) and 1(c)). As IDC from 5 mA to 27 mA, the fundamental frequency of the thick layer’s VC starts increasing for both configurations $\Downarrow/\uparrow$ and $\Uparrow/\downarrow$ (see Fig. 1(b) and 1(c)). The increase of which is owed to increasing the Zeeman energy associated with the circumferential Oersted field generated by the injected current [35]. As such, the vortex exhibiting stronger confinement increases its frequency from 600 MHz until reaching a maximum value of 770 MHz for an applied current of 27 mA. Increasing the applied current IDC further, dynamics disappear and the signal’s power becomes null. Experimentally, hindered dynamics refer to coupled vortices with same polarities. For opposite core polarities, the thick layer polarizes electrons during IDC traversal, polarized spins thereafter transfer spin angular momentum onto magnetization in the thin layer resulting in the observed dynamics. However, this exerted torque by the polarized spins becomes absent as the thick layer’s VC switches its orientation. Thus, the region with no dynamics (see dotted switching region, Fig. 1(b) and 1(c)) represents the thick layer’s VC orientation reversal at which the signal’s power is null. This intriguing behavior of the thick layer’s VC’s coupled dynamics arises a fundamental question about the origin of the dynamic reversal itself. The origin of which has neither been explained nor revealed for a system based on coupled vortex dynamics.

III Micromagnetic Simulations: Results and Analysis

In the following we go further to reveal this origin, as well as explain vividly the dynamic reversal mechanism observed in such devices. Via micromagnetic simulations, we investigate the coupled dynamics of the thin and thick layers’ VCs for the vortex based oscillator stack introduced earlier in Fig. 1(a). To carry out the simulations and the necessary data analysis, we use FastMag [44] and the software platform Aithericon [45]. As such, we utilize the same oscillator dimensions presented earlier, in addition to structure parameters from the experimental work [46]. We monitor the dynamics of the two coupled vortices tracing the trajectories of their respective VCs (see Fig. 2). Clearly, it is noticeable that the radius of the trajectory followed by the thin layer’s VC is much smaller than that of the thick layer for both cases $\Downarrow/\uparrow$ (Fig. 2(b,d)) and $\Uparrow/\downarrow$ (Fig. 2(a,c)). Focusing on (Fig. 2(c,d)), a region on the inner side of the thick layer’s VC is observed. The region of which has a normalized amplitude $\delta M_{\text{z}}$, and exhibits a polarity opposing that of the thick layer’s VC.

To explain the appearance of $\delta M_{\text{z}}$, we study the thick layer’s VC frequency dependence on applied DC current in the absence of an external field (Fig. 3). The current traversing the structure downwards in both configurations ($\Downarrow/\uparrow$ and $\Uparrow/\downarrow$) excites a high-frequency orbital motion of the VC, and induces an Oersted field of clockwise direction. The Oersted field increases the tunable frequency range [47].

Upon increasing the current gradually above 5 mA, the system’s fundamental frequency starts increasing (see power spectrum Fig. 3(a)) for both $\Downarrow/\uparrow$ and $\Uparrow/\downarrow$) [48], raising the velocity and $\delta M_{\text{z}}$ at the VC. Here, it should be noted that the increase in velocity of the thick layer’s VC may be traced back to either an increase in the fundamental frequency, or a decrease in its trajectory’s radius. However, the thick layer’s radius is observed to vary only slightly with respect to that of the thin layer (see Fig. 3(b)). This goes back to the fact that the two VCs are coupled, and thus can’t change their trajectories’ radii freely, as the system consisting of the dipolarly coupled vortices exhibits a selection upon the radius to radius distance [49]. This implies that, on average R${}_{\text{VC}}$-r lies in the vicinity of a somewhat constant value. Whence, the increase in the system’s fundamental frequency is associated with an increase in the velocity of the thick layer’s VC (see Fig. 3(d)). Naturally, the VC being a topological soliton deforms upon a continuous increase in its velocity, this is made clear by the appearance of the previously mentioned region on the inner side of the thick layer’s VC (see Fig. 2(c,d) for $\delta M_{\text{z}}$). The respective dip and hump shaped regions represent the mentioned deformation at the inner side of the thick layer’s VC during its gyrotropic motion about its equilibrium position for both cases, each of which has an amplitude of $\delta M_{\text{z}}$ (see Fig. 2(c) dip, and Fig. 2(d) hump).

As the current reaches 22.5 mA, the vortex attains a critical velocity of 266 m/s (see Fig. 3(d)). Beyond this value, the VC reverses its orientation (see the switching region in Fig. 3(c) and Fig. 3(d) for both $\Downarrow/\uparrow$ and $\Uparrow/\downarrow$), leading to the anticipated velocity drop [50]. The results of the micromagnetic simulations are in qualitative agreement with our obtained experimental results. Quantitatively, the difference between the frequency measured experimentally and that measured via simulations is approximately $\absolutevalue{\Delta F}$ = 100 MHz. The respective difference is rather expected based on the fact that thermal fluctuations haven’t been accounted for in the simulations. The latter’s influence on frequency in such systems has been previously studied and demonstrated [51]. In addition to the mentioned thermal influence, the simulation’s accounted for Oersted field has been calculated analytically for the case of a DC current traversing an infinitely long wire, then plugged as part of the effective field.

Hereby, the value of 266 m/s reached constitutes a critical velocity of the VC, where any further increase in applied current implies a VC reversal. The obtained critical velocity is in good agreement with the analytical expression [52] V${}_{\text{c}}$= (1.66±0.18)$\times\gamma\times\sqrt{A_{\text{ex}}}$ (e.g V${}_{\text{c}}^{Py}$= 289$\pm$31 m/s), where $\gamma$= $5.51\times 10^{7}$ T${}^{-1}s^{-1}$ is the gyromagnetic ratio, and A${}_{\text{ex}}$= 1$\times 10^{-11}$ J/m is the exchange stiffness. The deformation of amplitude $\delta M_{\text{z}}$ initially stated in Fig. 2(c) and Fig. 2(d) opposes the polarity of the undeformed vortex core. For a positive VC polarity ($\Uparrow$) this deformation appeared as a dip, whereas it appeared as a hump for a negative VC polarity ($\Downarrow$). $\absolutevalue{\Delta M_{\text{z}}}$ increases in a similar manner to the velocity/frequency increase with current between 5 mA and 22.5 mA (see Fig. 3(c) for $\absolutevalue{\Delta M_{\text{z}}}$ vs. current for both $\Downarrow/\uparrow$ and $\Uparrow/\downarrow$). This dip’s/hump’s amplitude increase reaches a maximum at the critical velocity of 266 m/s, at which the deformation in the VC becomes large enough to induce an orientation reversal ($\Downarrow$ to $\Uparrow$ in 3(c-1) and $\Uparrow$ to $\Downarrow$ in 3(c-2)), leading to a drop in the VC’s velocity accompanied by spin waves’ emission [53].

IV Conclusion

The results of our experimental studies and micromagnetic simulations, along with the physical interpretations provided, offer a comprehensive understanding of the relationship between the dynamics of coupled vortices and applied DC current. Our research has clarified the dynamic reversal mechanism, which is traced to the appearance of a dip/hump region on the inner side of the vortex core in the thick layer due to increasing velocity of the vortex core. The continuously increasing velocity of the vortex motion in response to the applied DC current is shown by the emergence of deformation. Our findings provide significant insights into the engineering of vortex switching in nano-oscillators, enabling the manipulation of magnetization dynamics and control of vortex polarity. The vortex trajectories of coupled layers present many possibilities for engineering amplitude and velocity of VCs.This study provides a significant contribution to a deeper understanding of the physics behind vortex core reversal in coupled systems without a static magnetic field, which is crucial for the advancement of advanced spintronic devices.

Acknowledgments

This work has been supported by the European Research Council within the Starting Grant No. 101042439 ”CoSpiN” and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - TRR 173 - 268565370” (project B01).