Grain structure influence on synchronized two-dimensional spin-Hall nano-oscillators

In recent years, an increasing interest in neuromorphic computing¹ has motivated research in concepts for neuron-like devices,^2–4 capable of being programmable, non-linear, and scalable into high-density networks. Current and frequency tunable spintronic oscillators are of special interest, thanks to their intrinsic characteristics of scalability,⁵ ease of CMOS integration,^6–8 low power consumption,⁹ and GHz bandwidth.¹⁰ 

An alternative to the widely investigated spin-torque nano-oscillators (STNOs)^11–13 are spin-Hall nano-oscillators,^14–17 based on the spin-Hall effect.^18,19 Nanoconstriction-based devices are especially suitable to generate a high current local density that drives the oscillations.²⁰ As other spin-torque based devices, e.g., STNOs,^21–27 spin-Hall nano-oscillators (SHNOs) can be easily synchronized to an external field,²⁸ and mutual synchronization has been demonstrated in chains^29–32 as well as in two-dimensional arrays.^33,34

Two-dimensional synchronized arrays of nanoconstriction spin-Hall nano-oscillators have several promising features such as higher output power, better frequency stability (in the sense of a reduced phase noise), and applications in neuromorphic computing. For all these applications, the synchronization of the oscillator array is crucial, in both amplitude and phase. Oscillators are synchronized with the neighboring ones via dipolar and exchange coupling. For larger oscillator separation or nanoconstriction dimensions on the order of several 100 nm, mutual coupling is also mediated by propagating spin waves .³⁰ Both exchange coupling and spin wave propagation would depend on local properties of the free magnetic layer. To the best of our knowledge, no extensive investigation has been performed on the influence of the layer properties on the synchronization quality.

Our previous study³⁵ demonstrated how local variations in the exchange coupling can induce multiple oscillation modes in single devices and cause variability in the oscillation frequency. In this study, we expand the modeling using the MuMax3 micromagnetic simulator^36,37 to free-running 2 × 2 and 4 × 4 synchronized NC-SHNOs arrays, arranged with different pitches and widths. Moreover, we investigate the stability and synchronization vs temperature—from low temperature (4 K) to room temperature (300 K) and more—including self-heating by the DC-drive current. Finally, the influence of free layer local exchange coupling variations is investigated.

The paper is structured into three main sections. The simulation methods are illustrated in Sec. II, the results of the influence of the array dimensions, temperature, and free layer properties on the synchronization of arrays are presented and discussed in Sec. III, and the paper is concluded in Sec. IV.

Our simulation setup is based on a recent experimental work³⁴ using the same material stack of platinum (Pt), hafnium (Hf), and permalloy (Py), Ni₈₀Fe₂₀, as shown in inset of Fig. 1. Similar to that study, we considered various array configurations, from the single NC-SHNO, shown in Fig. 1(a), to have a reference point, up to a 16 = 4 × 4 oscillator two-dimensional array, shown in Fig. 1(d). The current density and Oersted field were calculated in COMSOL Multiphysics and then imported to the micromagnetic simulations performed in MuMax3.

For every array, the pitch (p) and the width (w) were parameterized, as shown in Fig. 1(d). The width represents the dimension of the nanoconstriction, while the pitch is the separation between adjacent oscillators. For the 2 × 2 case, we tested a complete set of dimensions, while for the 4 × 4 case, we only considered the two extremes, shown in Table I. As explained in Ref. 34, we expect the cases of p = 300 nm to have a weaker mutual synchronization and a higher likelihood of displaying, e.g., column-wise synchronization where oscillator columns possibly operate at slightly different frequencies.

The core method of the study was the technique developed in our previous work,³⁵ where oscillator stability in the presence of local reduction in the exchange coupling was studied. To achieve a deeper understanding of the influence of grains on array synchronization, a single grain boundary case was explored. This is what we refer to as a cut in Fig. 1(c). Focusing on the horizontal and the tilted cuts allowed the exploration of row-wise or column-wise synchronization. Even with a reduced exchange coupling, there is still dipolar interaction between individual oscillators in the array, and the relative influence of these two coupling mechanisms depends on the oscillator pitch. A wide range of exchange reduction, 10–50% of the original exchange value (10⁻¹¹ J m⁻¹), was explored. It was found that exchange values of less than 10% were effective and resulted in disturbed oscillator behavior. For larger exchange values, the arrays maintained their original behavior and stability. This is reasonable since the stability is also governed by dipolar coupling and a relatively small direct exchange interaction for the larger considered pitches.

The natural expansion of the cut study adds a full grain map to the device, as shown in Fig. 1(b). Each grain boundary is assigned a locally reduced exchange coupling value. The map was created by first analyzing a full layer stack with an atomic force microscope (AFM), the result was then post-processed in MATLAB to extract single grains, and finally, these grains were imported as individual regions in MuMax3. This technique, in principle, allows full control over the properties of the grains, such as saturation magnetization and damping constant, while here, only the exchange at grain boundaries was considered. The value of the exchange was randomly assigned in a range of 10%–30% to each grain boundary. In the simulations, three grain cases were defined that differ in the randomly assigned inter-grain exchange, while the grain map itself was kept fixed. Changing the grain map is also a possibility, but it is outside the purpose of this study.

In order to allow closely spaced modes to be resolved, all the simulations were run for a minimum of 500 ns, resulting in 2 MHz resolution. For low-temperature simulations, with small thermal broadening, the linewidth of the frequency peaks is well below this limit. The cell size was kept fixed at 3.9 × 3.9 × 3 nm³ over a domain size of 2000 × 2000 × 3 nm³. The magnetic field was kept constant at 6.8 kOe, while the in-plane (IP) angle was changed in the range of 29°–31° and the out-of-plane (OOP) angle was changed in the range of 75°–77°. The effect of the IP-angle is only visible at very high currents and very marginal. Any experimental variation in this angle causes a minor change in the observed frequency. On the other hand, the effect of the OOP-angle is considerable since a 1° increase corresponds to a 500 MHz decrease in the frequency of oscillation. Therefore, this angle is very effective in tuning the oscillator frequency, which is both a benefit and a drawback.

The most relevant properties of a tunable spintronic microwave oscillator array are frequency stability, i.e., linewidth, the number of simultaneously excited frequency modes, and the total output power. For an in-phase synchronization of multiple oscillators in an array, the power should be proportional to the number of oscillators or more precisely to the energy of the coherent oscillation volume. Such synchronization has already been experimentally confirmed by Brillouin light scattering (BLS) in large arrays.^9,34 On the other hand, out-of-phase synchronization, or non-coherent operation at multiple frequencies, e.g., along columns, results in significantly lower total power. Finally, it should be remembered that the spin precession is converted into electrical power by the anisotropic magnetoresistance effect with relatively low efficiency. In order to compare different array sizes, the DC driving current can be normalized by width, as shown in Fig. 2. All investigated oscillator arrays display a characteristic frequency tuning by the DC driving current. The 2 × 2 case has a local frequency minimum at $∼25×10−3$ A μm⁻¹ followed by a steep increase and a plateau, as shown in Fig. 2(a). The regions with a small curvature indicate good synchronization, as evidenced by high output power and the absence of additional frequency modes, which is also shown in Sec. III C. A larger oscillator array 4 × 4, shown in Fig. 2(b), shows the same general behavior for the frequency vs current tuning. However, the preferred frequency, i.e., the local minima, is typically shifted to a lower frequency value, which is attributed to more disturbing interaction within the array. In essence, the array cannot sustain the free running frequency of an undisturbed individual oscillator for the same DC driving current.

For the case of the 2 × 2 array, a relatively large pitch, 300 nm, was considered first. In Fig. 3(a), two dominant frequency modes were extracted from the analysis of the power spectral density. These two coexisting modes are separated by 20–30 MHz, from the onset of oscillation to about 3.5 mA. As demonstrated below, the modes are temporally coexisting but spatially separated in their origin. The dual mode operation persists at higher current, but the modes become harder to distinguish due to small spacing and lower power, as shown in Fig. 3(b).

To further investigate the dual mode origin, we considered the average power and phase spatial maps obtained from a fast Fourier transform (FFT) of the y-component of the magnetization. Figures 3(c)–3(f) show these results at 3.2 mA. This operating point represents one of the most stable conditions since the slope of frequency vs current is zero. Here, the two modes are oscillating at 9.73 and 9.76 GHz, respectively. Moreover, Figs. 3(c) and 3(d) also show that the oscillator can be locked at a non-zero phase difference, 1.27 and 1.09 rad for the lower and higher frequency modes, respectively.³⁰ 

These results show that column-wise synchronization is favored. There is no strong synchronization along rows since there is no direct exchange coupling in the lateral direction. The columns can sustain a different oscillating frequency, given by the local current density in each column, as shown in Fig. 4(a). The current density is higher in the left column due to the tilt of the array. Without the tilt angle, two distinct frequency modes should not appear. In some 4 × 4 array configurations, four frequencies have been detected. Here, the current density pattern for each column is more complicated, and bottom left and top right oscillators are the extreme cases, as shown in Fig. 4(b).

Operating the oscillator arrays at higher temperatures results in a small parallel blue-shift of the frequency vs current characteristic [see Fig. 5(a)]. The thermal field adds energy to the system, effectively increasing the frequency of oscillation. Similar magnitude of frequency blue-shifts has been previously observed in STNOs.³⁸ For the general conclusion of this paper, it is a minor effect but still representative of experimental conditions. In this study, the interplay between the random thermal field and exchange-mediated coupling between oscillators is of importance. The thermal fluctuations will clearly reduce the local spin coherence of the processing volume and also effectively smear or damp any propagating spin wave modes. This type of change in the effective coupling between oscillators could result in different relative phases, which in turn affects the total power of the array. For the effects of room temperature, we considered both the 2 × 2 and the 4 × 4 arrays. At both LT and RT, the power vs current behavior, shown in Fig. 5(b), shows that the power peaks for current regions with a low frequency curvature. These current regions correspond to a favorable phase difference in the coupling configuration. As the frequency increases further with current, the total power falls off.

The phase and power spatial maps at 6.7 mA at RT, shown in Figs. 6(a)–6(d), are similar to the situation of the 2 × 2 array at p = 300 nm, shown in Fig. 3. However, here, we have a diagonal synchronization at 45°, where oscillators above the main diagonal favor a slightly lower frequency than the ones in the lower left part of the array. Overall, the latter case represents a locally higher current density, shown in Fig. 4(b), and hence, a higher frequency is induced.

In order to study stability in the presence of grains and reduced exchange along lines, i.e., cuts, we considered 2 × 2 arrays since they exhibit good phase coherence or almost complete synchronization at pitches of 200 nm or less. In Figs. 7 and 9, the frequency and power at each current step are compared for the ideal, cut (exchange reduced to 10%), and grain cases.

Focusing on the smaller dimension shown in Fig. 7, it is observed that modifying the exchange coupling inside the array, by the addition of a cut, results in a minor shift of the synchronized frequency at low and medium currents, while the array is much more disturbed at high currents (I > 3 mA). Two cut cases were tried, horizontal and slanted. The horizontal cut divides the array along rows, while the slant is tilted at the same angle as the natural angle of the array, reducing the exchange coupling between neighboring columns. In the simulations, a relatively minor effect on frequency is found for either cut, along columns or rows. On the other hand, the power in the synchronized state is a more sensitive parameter since it reflects the obtained phase coupling. The effects of the introduction of a full grain map start to appear at lower currents (I > 2.5 mA). The three grain cases differ in the exchange, which is randomly assigned, while the map is kept fixed. The opposite is also a possibility, but it was not explored in this study. A divergence of the normal stable mode occurs toward higher currents, and a decrease in power output of almost 15 dB is seen. Figure 8 presents the spatial maps of phase and power at 2 mA. The ideal case illustrates a coherent oscillation volume that includes all constriction and extends also into the surrounding part of the structure, as shown in Fig. 8(b). A closer inspection of the power map shows that the power peaks at the center of the constriction, meaning that the two characteristic edge modes have joined. High power is also detected above and below the constrictions. Effectively, the oscillation volume is not limited to the constrictions. The slant cut case has some minor fluctuations in the phase coherence, but contrary to the cases discussed above, no non-zero phase shift can be found in this system, as shown in Fig. 8(c). The power is higher, and it is observed that a dB scale is used, so a comparison to the ideal case shows a significant difference, as shown in Fig. 8(d). Again, the maximum power appears centered in the constriction, and high power is generated beyond the constrictions. The actual cut is resolved in both phase and power representations, albeit very faintly, on this color scale. Finally, the grain case shows a phase separation between the constriction regions and the rest of the structures, as shown in Fig. 8(e). Here, the power maps reflect the presence of individual grains, as shown in Fig. 8(f). The power in each grain region differs slightly, but overall, the behavior discussed for the former cases is reflected here as well.

In the case of an oscillator with a wider pitch and width, shown in Fig. 9, the situation is similar. Above a critical current, the normal stable mode is disturbed. Here, only one grain map case was considered, as comparable effects were detected among the three different gain cases in Fig. 7. The behavior of the frequency vs current shows that cuts do not have any noticeable effects for the considered current range, as shown in Fig. 10(a), while the usage of the grain map clearly limits the stable current region and reduces the power, as shown in Fig. 10(b). The oscillator mode has a blue shift of ∼100 MHz.

The power and phase spatial maps, shown in Fig. 10, show the effects of a cut and the grains at a current of I = 3 mA. All cases show relatively good phase coherence. However, the grain map results in some phase separation in the middle region, as shown in Fig. 10(e). Since the constrictions are wider, they can now fit two power maxima, meaning that the two edge modes have not completely joined. The cut creates an asymmetry between the left and right columns of the oscillator, where stronger power is delivered from the left side, as shown in Fig. 10(d). The grains created a much more chaotic situation, where the bottom left oscillator shows lower power at this frequency as well as a phase separation, as shown in Figs. 10(e) and 10(f). As discussed above, this oscillator is located in the region of the highest current density, so, in fact, its peak power appears shifted to a slightly higher frequency.

A micromagnetic simulation setup, based on realistic local variations in exchange coupling, has been used to investigate 2 × 2 and 4 × 4 NC-SHNO arrays. It was demonstrated with microspin spatial representations that the modal stability of an NC-SHNO can be disturbed by the introduction of single or multiple grain boundaries with reduced exchange coupling within the active areas of the arrays.

The stability of 2 × 2 and 4 × 4 arrays was tested at room temperature and compared to the low-temperature results. The effect is mostly a small shift toward higher frequencies of oscillation and slightly higher power peaks independent of the temperature. The arrays synchronize mainly along columns, and local variations in the current density induce multiple simultaneous oscillation modes.

Our study also suggests that larger array configurations are synchronized at a frequency that differs from that of a single free-running oscillator.

A logical continuation for this study is to expand the array dimensions to 6 × 6 and beyond. For arrays of these sizes, use of smaller pitch and width dimensions may be advised to reach a phase-coherent state with large output power.

This work was partially supported by the Swedish Research Council (VR), Project Fundamental Fluctuations in Spintronics, under Grant No. 2017-04196.

The computations were enabled by resources provided by the Swedish National Infrastructure for Computing (SNIC) at High Performance Computing Center North (HPC2N) partially funded by the Swedish Research Council, through Grant Agreement No. 2018-05973.

We thank Mykola Dvornik and the Åkerman Applied Spintronics group at Gothenburg University for the helpful discussions and the support with the Ragnarok cluster and Ari Sigurdsson for the help with COMSOL simulations.

The authors have no conflicts to disclose.

Corrado Carlo Maria Capriata: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Bengt Gunnar Malm: Conceptualization (equal); Funding acquisition (equal); Methodology (equal); Project administration (equal); Resources (equal); Supervision (equal); Validation (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.