## Introduction

High-density arrays of mutually coupled nanoscale spintronic oscillators have the potential to closely mimic the behavior of the non-linear oscillatory neural networks of the human brain and thus open the way for efficient high-speed and low-power neuromorphic computing and signal processing1,2,3. Recently, a proof-of-principle demonstration of vowel recognition using a chain of four electrically coupled spin-torque nano-oscillators (STNOs) with individual drive current control was reported4. However, the current-based tunability is neither an energy-efficient approach nor an ideal arrangement to scale the neuromorphic computing to large dynamical neural networks.

Spin Hall nano-oscillators (SHNOs)5,6,7,8,9 have recently shown enormous potential as an energy-efficient alternative to conventional STNOs thanks to their easy fabrication process, flexible device geometry, reduced power dissipation, and CMOS compatibility10. SHNOs take advantage of a pure spin current, produced by the spin Hall effect in a non-magnetic heavy-metal layer, to excite the local magnetization of a magnetic thin film into steady-state auto-oscillating precession at microwave frequencies. Among a variety of SHNO device layouts presented so far, nano-constriction based SHNOs10,11,12,13,14 exhibit robust mutual synchronization both in linear chains15 and in two-dimensional arrays of as many as 64 SHNOs16. 4 × 4 arrays of 16 such SHNOs were also used to demonstrate16 the same type of synchronization maps as in ref. 4 employed for neuromorphic vowel recognition. However, a major challenge that has to be addressed, before more complex neuromorphic tasks can be attempted, is the individual frequency control of each SHNO inside the network. It is also of particular interest to be able to turn individual networked SHNOs on and off at will as this forms the basis of other recent oscillator network-based computing paradigms17,18.

Voltage-controlled magnetic anisotropy (VCMA)19,20,21,22,23,24,25,26,27,28,29 has recently attracted great interest as it offers a highly energy-efficient approach to drive magnetization switching with significantly lower energy dissipation compared to current control in, e.g., spin-transfer torque-MRAM22,23,30. Here, we demonstrate how VCMA can also provide strong individual control of the threshold current and the auto-oscillation frequency of nano-constriction based W(5 nm)/$${({{\rm{Co}}}_{0.75}{{\rm{Fe}}}_{0.25})}_{75}$$B25(1.7 nm)/MgO(2 nm)/AlOx(2 nm) SHNOs with relatively strong interfacial perpendicular magnetic anisotropy (PMA)9. We observe a giant 42% modulation of the total effective damping over four volts (10.5%/V), a very large 22% modulation in threshold current (5.5%/V), and a robust 50 MHz frequency tunability (12 MHz/V). Our detailed analysis, using spin-torque ferromagnetic resonance (ST-FMR) and micromagnetic simulations, shows that these large effects are caused by a moderate VCMA, which tunes the auto-oscillation frequency of the nano-constriction such that its mode volume and coupling to propagating spin waves in the extended CoFeB layer can be greatly varied. When the coupling to the surrounding spin waves increases, the auto-oscillating mode experiences an increasing load due to radiation of magnons into the magnetic leads, which is experimentally observed as strongly increased effective damping in the ST-FMR measurements and an increased threshold current of the SHNOs. The increased frequency tunability and the possibility to turn individual oscillators on/off within a large network will allow for recently suggested oscillator computing approaches to be implemented using voltage-controlled SHNOs.

## Results

### Nano-patterned gated SHNO device schematic and layout with individual layers

Figure 1a shows the optical microscope image of an electrically gated nano-constriction SHNO device with electrical signal pads and how these connect to the experimental set-up. The top signal-ground pads are utilized to apply different gate voltages via the gate electrode. The bottom signal-ground pads are used to locally inject the electrical drive current into the device and to pick up the microwave signals from the spin-orbit torque (SOT) driven auto-oscillations of the CoFeB magnetization. Figure 1b shows an enlarged zoomed-in area of the gated nano-constriction SHNO; a further zoomed-in scanning electron microscope (SEM) image of the nano-constriction is displayed in Fig. 1c. A positive direct current (d.c.) is injected along the y-direction while ϕ and θ define the in-plane (IP) and out-of-plane (OOP) field angles, respectively, for the applied magnetic field. The two different schematic cross-sectional views of the material stack with structure β-W(5 nm)/$${({{\rm{Co}}}_{0.75}{{\rm{Fe}}}_{0.25})}_{75}$$B25(1.7 nm)/MgO(2 nm)/AlOx(2 nm) grown on highly resistive Si substrates are shown in Fig. 1d, e. The β-phase of W is known to produce large SOT13,31,32 and the thinner CoFeB exposed to MgO layer facilitates the PMA in the CoFeB layer33.

### Voltage-controlled tunabilities of threshold operational current and auto-oscillation frequency

Figure 2a–c shows color plots of the current-dependent auto-oscillation power spectral density (PSD) excited at a fixed OOP field strength of 0.4 T and subject to three different gate voltages. Note that the leakage current does not exceed 5 nA at VG = ±2 V during auto-oscillation measurements. One can observe a dramatic reduction of the threshold current (Ith) as the gate voltage increases from −2 to +2 V. We extract Ith via a linear fit of 1/P vs. Idc (employing the method described in ref. 34), as shown in the inset of Fig. 2d for one representative gate voltage, VG =−1 V. The main panel of Fig. 2d displays the variation of Ith as a function of gate voltage exhibiting a linear dependence with a very large overall modulation of 22% with VG = ±2 V. In Fig. 2e, we show a comparison of the auto-oscillation frequencies at different gate voltages, exhibiting a remarkable 50 MHz frequency tunability with gate voltage. We analyze the tunability behavior by extracting the auto-oscillation frequency shift Δf(VG) = f(VG) − f(VG = 0) at 1.2 mA, exhibiting a linear dependence on gate voltage with a rate of 12 MHz/V, as shown in Fig. 2f. We note that this is twice as large as the frequency tunability demonstrated in nano-gap SHNOs using back-gated permalloy films35, corroborating the benefit of VCMA in CoFeB/MgO-based system for frequency tuning.

### Spin-torque ferromagnetic resonance measurements on a gated SHNO device

To understand the origin of the voltage-controlled tunability of threshold current and auto-oscillation frequency, we performed detailed ST-FMR measurements on a similar gated 150 nm nano-constriction SHNO device subject to different microwave frequencies ranging from 4 to 12 GHz at different gate voltages. In Fig. 3a, we show typical ST-FMR peaks recorded at a frequency of 7 GHz for six different gate voltages starting from VG = +1 V to VG = −3 V. The mixing voltages Vmix are offset along the y-axis to enable comparison of the line shapes. Figure 3b shows the variation of the effective magnetization μ0Meff, extracted from a Kittel fit (Eq. (1)) of field-dependent resonance peak positions at different microwave frequencies ranging from 4 to 12 GHz, as a function of gate voltage. We observe a linear dependence of μ0Meff with gate voltage, exhibiting an overall 14% modulation under electrostatic gating of ΔVG = 4 V. Using equation $${H}_{\,\text{k}}^{\perp }={M}_{\text{S}}-{M}_{\text{eff}}$$ with the saturation magnetization, μ0MS = 1.3 T, this translates into a minor tunability (<1%) of the interfacial PMA. Figure 3c displays the plot of line-widths extracted as half-width at half maximum (HWHMs) from the ST-FMR peaks as a function of different microwave frequencies under different gate voltages. The experimental data together with their respective linear fits to ΔH = ΔH0  + 2παf/γμ0 (solid lines) with a gyromagnetic ratio of γ/2π = 29.1 GHz/T show a remarkable variation in line-widths indicating a dramatically large modulation in effective damping constant, α. In Fig. 3d, we show that the modulation of the effective damping constant as a function of gate voltage is as large as 42%, exhibiting a linear dependence with a slope of 0.002/V.

Note that our ST-FMR measurements are sensitive only to the area in the vicinity of the gate, since ST-FMR peak shifts with the applied gate voltage. Such a result can be somewhat expected since the applied current density is only substantial in the nano-constriction region, i.e., under the gate. In addition, the strongly non-uniform nature of the spin-current density does not allow to excite the magnetization uniformly over the whole sample36. To gain deeper insight into the observed effective damping modulation, we carried out micromagnetic simulations to study the spatial amplitude profiles of the dynamical mode in the nano-constriction region under different gate voltages for a 150 nm SHNO.

Our simulations show that for the given device configuration with the large value of PMA, the demagnetizing field does not create substantial confinement for the magnons in the constriction region, which means the absence of complete localization of the linear-mode9. The mode consists of an oscillating core in the center, coupled with the magnons propagating out of the nano-constriction. The total energy loss of the excited mode hence consists of both the intrinsic damping of the core and the energy radiated away by the emitted magnons37. The latter is quite substantial in our case, due to the small volume of the excited core.

The applied voltage on the gate, through the above-mentioned tunability of the interfacial PMA, changes the potential for the magnons in the constriction region, which in turn changes the ratio of magnons in the core and emitted spin waves6,37. Thus, the negative voltage increases the PMA value, which raises the potential and leads to the further delocalization of the mode and, in turn, a larger value of the emission losses and total damping via the relaxation rate of the excited dynamical mode. On the other hand, a positive voltage leads to a more localized behavior of the excited mode with predominantly intrinsic losses of the core, which substantially reduces the total damping. The spatial profiles of the mode amplitudes $$| {\tilde{m}}_{x}|$$ normalized to their respective maximum amplitudes are shown in Fig. 3e for VG = −3 V and VG = +1 V. Note that the core of the mode in the center of the nano-constriction is substantially more pronounced for VG = +1 V whereas less localized for VG = −3 V. The extracted values of the total damping from simulation, as shown in Fig. 3d by red spheres with a linear fit, are in a good agreement with the experimental results. While our simulations reproduce quite well the experimentally observed effective damping modulation, we do not rule out the possibility of a minor change in intrinsic damping under the application of gate voltage25.

We also performed ST-FMR measurements at different levels of a device current under positive field direction at two fixed microwave frequencies of 6 and 12 GHz, respectively. The variations of extracted ST-FMR linewidth as a function of dc current at different gate voltages for the reference SHNO device are shown in Fig. 4a, b. We fit the linewidth variations to a linear function and the extracted slopes of lines are plotted as a function of gate voltages for both frequencies in Fig. 4c, d. Each value of the slope determines the efficiency of the linewidth change with dc current and is, therefore, a measure of the extent to which the excited mode is affected by the drive current. As the W/CoFeB interface is unlikely to get modified by applied gate voltage due to effective screening of electric field within the CoFeB(1.7 nm) layer, the impact of gating on the spin Hall efficiency (SHE) should be minimal and the slope, therefore, depends on the volume of the excited mode37. The observed variation of the slope with the gate voltage is also in a good agreement with our simulation results exhibiting stronger localization of the dynamical mode at +1 V and weaker localization at −3 V, as shown in Fig. 3e. This is because the strongly localized mode resides in the nano-constriction region and has a greater overlap with the current density profile.

Our ST-FMR results indicate that the large voltage-controlled modulation of the threshold current and auto-oscillation frequency of our SHNOs are produced by two distinct contributions. The former is predominantly caused by the observed strong voltage-controlled change in the effective damping constant, and the latter by the moderate change in effective magnetization. The linear decrease in μ0Meff with negative gate voltage leads to an effective increase in the FMR frequency as per the Kittel equation38 for the obliquely magnetized case. This, in turn, increases the auto-oscillation frequency, which essentially mimics the FMR frequency37.

## Discussion

There are in general two types of VCMA effects reported in the literature to tailor the magnetic properties of FM thin films. One arises due to a redistribution of electronic densities among different 3d orbitals in interface FM atoms20,35,39 or via modification of the Rashba spin-orbit coupling and the Dzyaloshinskii-Moriya interaction (DMI)24,40; all these effects have an electronic band structure origin. The other mechanism is of ionic origin, where the voltage-driven migration of O2− ions is exploited to engineer the interfacial oxidation states of the ferromagnetic layer26,29,41,42,43. Unlike engineering the electronic bands at the FM/oxide interface, the ionic mechanism can have a much larger impact on the interface PMA26,29,42. Given the observed minor tunability of interface PMA under applied gate voltage, which gives rise to an estimated VCMA value of about −103.4 fJ/Vm akin to earlier reported values for this material system23,44, we believe that the voltage-induced changes in the electronic band structure across the CoFeB/MgO interface have a predominant role in our case.

The damping parameter governs the magnetization dynamics in ferromagnets and therefore critically influences the performance of spintronics devices, such as the auto-oscillation threshold current. While we observed a rather modest VCMA effect of <1%, its substantial impact on the spin-wave mode volume in the nano-constriction region led to the giant 42% modulation of the effective damping. Recently, Nozaki et al. demonstrated a significant enhancement in the VCMA effect, as high as 350 fJ/Vm, via engineering of the FM/oxide interface in an ultrathin Ir-doped Fe/MgO structures with a CoFe termination layer.45 Employing an ionic mechanism to control the PMA could potentially result in even stronger modulation of the damping and possibly also non-volatile storage of different damping values.46 Further improvement in the damping modulation should also be possible via optimizing the device layout. For example, by tuning the gate width and constriction opening angle, one can change the ratio between the radiative and core part of the mode, thus control the SHNO damping and frequency tunability.

The demonstrated strong tunability of SHNO threshold current will allow individual oscillator control within long SHNO chains or large two-dimensional SHNO arrays. While a single global drive current is applied to all SHNOs in the chain/array, the voltage applied to each oscillator node determines whether that oscillator will be turned on or stay off. This specific combination of functionalities—i.e., individual on/off control and mutual synchronization—has very recently been suggested based on thermally coupled VO2 relaxation oscillators interacting in two-dimensional arrays17,18. This proposed approach will work equally well for two-dimensional SHNO arrays.

Our demonstration of  ~50 MHz frequency tunability is also sufficient to cover the frequency tuning used by Romera et al. (~10 MHz) to optimize the synchronization map for vowel recognition4. The SHNO synchronization map demonstrated by Zahedinejad et al. showed detrimental gaps of about 50 MHz, which may hence be possible to close using our demonstrated voltage control16.

The capability of energy-efficient and individual oscillator control in nano-constriction based SHNO networks has several important advantages over other device layouts for neuromorphic computing applications. First, our device structure is fully integrated without any additional external requirements for injection locking (antenna) and mutual coupling (bonding wires). Second, the nano-constriction based SHNO architecture is highly scalable accommodating as many as 100 partially synchronized oscillators while taking up an area of <1 μm2 with the possibility of further downscaling using already demonstrated 20 nm SHNOs12,16. Finally, strong non-linear interactions between neighboring oscillators, direct optical access to the active dynamical area, higher operating frequency10 (3–22 GHz) and a wide locking range (1 GHz)9,15, both being two orders of magnitude higher than that of vortex STNOs4, make nano-constriction SHNOs the most viable choice for oscillatory computing. However, in order to truly benefit from these important merits, output power needs to be increased by other means than synchronization. This could, e.g., be done by fabricating a W/CoFeB/MgO/CoFeB magnetic tunnel junction over part of the constriction region by adding a patterned CoFeB layer to the existing structure.

## Methods

### Gated nano-constriction SHNO fabrication

A trilayer stack of W(5)/$${({{\rm{Co}}}_{0.75}{{\rm{Fe}}}_{0.25})}_{75}$$B25(1.7)/MgO(2) (thicknesses in nm) was grown at room temperature on an intrinsic high-resistivity Si substrate (ρSi > 10 kΩ  cm) using an ultra-high vacuum sputtering system. dc and rf sputtering were sequentially employed for the depositions of metallic and insulating layers, respectively. The stack was covered with a 2 nm thin layer of sputtered AlOx at room temperature to protect the MgO layer from degradation due to exposure to the ambient conditions. The stack was subsequently annealed at 300 C for 1 h to induce PMA. The stack was then patterned into an array of 4 × 14 μm2 rectangular mesas and the nano-constriction SHNO devices with different widths were defined at the center of these mesas by a combination of electron beam lithography and Argon ion beam etching (IBE) using negative electron beam resist as the etching mask. After removing the electron resist, the sample was covered with 15 nm stoichiometric room–temperature sputtered Si3N4 to isolate the gate contacts from the nano-constriction metallic sidewalls. The gates were defined by sputtering a bilayer of Ti(2 nm)/Cu(40 nm) followed by EB lithography using negative electron resist to fabricate 100 nm wide gate pattern on top of nano-constriction. The pattern was then transferred to the Ti/Cu bilayer using IBE technique. After removing the remaining negative resist, the sample went through optical lithography using a positive resist to define vias in Si3N4, giving access to the SHNO metal layers for the contact pads. Finally, the electrical contacts, including two SG-CPWs for microwave and dc measurements were defined by optical lithography and lift-off for a bilayer of Cu(700 nm)/Pt(20 nm).

### Microwave measurements

Microwave measurements were carried out at room temperature using a custom-built probe station where the sample was mounted at a fixed IP angle of ϕ = 22 on an OOP rotatable sample holder between the pole pieces of an electromagnet producing a uniform magnetic field. Using a Keithley 6221 current source, a direct positive electric current, Idc, was injected to gated nano-constriction SHNO device through dc port of a high-frequency bias-T under different applied gate voltages sourced from Keithley 2400 sourcemeter. The resulting auto-oscillating signal was then amplified by a low-noise amplifier (LNA) with a gain of ≥32 dB and finally recorded using a spectrum analyzer (SA) from Rhode & Schwarz (10 Hz–40 GHz) comprising a resolution bandwidth of 1 MHz. We measured multiple gated SHNO devices and restricted the maximum current up to 1.4 mA in order to avoid irreversible changes in the output microwave characteristics due to device degradation.

### ST-FMR measurements

ST-FMR measurements were performed at room temperature on a 150 nm gated nano-constriction SHNO device under the identical conditions employed during auto-oscillation measurements. A radio-frequency (rf) current modulated at 98.76 Hz is supplied through a high-frequency bias-T at a fixed frequency (ranging from 4 to 12 GHz) with an input rf power of P = −18 dBm, generating spin-orbit torques as well as Oersted field under an externally applied OOP magnetic field. The resulting torques excite the magnetization precession in the CoFeB layer, leading to a time-dependent change in the resistance of the device due to the magnetoresistance (MR) of the CoFeB layer47. The oscillating MR mixes with the rf current, yielding a dc mixing voltage, Vmix, and was measured using a lock-in amplifier. All ST-FMR measurements shown in Fig. 3 were carried out by sweeping the applied field oriented at a fixed IP angle of ϕ = 22 and OOP angle of θ = 60 from 550 to 0 mT, while the frequency of the input rf signal is kept fixed. Unlike faster auto-oscillation measurements, the gate breakdown occurred during longer ST-FMR field sweeps at higher positive gate voltage exceeding 1 V. Therefore, we restricted our ST-FMR measurements to +1 V for a fair comparison. To estimate the degree of mode localization in Fig. 4, we injected small dc currents in addition to rf current through dc and rf ports, respectively, of a bias-T. The resonance feature in voltage response from each field sweep was fitted to a sum of one symmetric and one anti-symmetric Lorentzian sharing the same resonance field and linewidth.

The values of Meff were extracted through the fitting by the Kittel equation:

$$f=\frac{\gamma {\mu }_{0}}{2\pi }\sqrt{{H}_{\text{int}}({H}_{\text{int}}+{M}_{\text{eff}}\cos {\theta }_{\text{int}})},$$
(1)

where Hint and θint are the internal magnetic field magnitude and out-of-plane angle, respectively, which can be calculated from the magnitude Hex and angle θex of the applied magnetic field as:

$${H}_{\text{ex}}\ \cos {\theta }_{\text{ex}}={H}_{\text{int}}\ \cos {\theta }_{\text{int}},$$
(2)
$${{H}}_{\text{ex}}\ \sin {\theta }_{\text{ex}}=({H}_{\text{int}}+{M}_{\text{eff}})\ \sin {\theta }_{\text{int}},$$
(3)

### Micromagnetic simulations

The micromagnetic simulations were performed using the GPU-accelerated program mumax348. The geometry of a 150-nm nano-constriction width SHNO device was extracted from the CAD model, used for the fabrication, and discretized into 1024 × 1024 × 1 cells with an individual cell size of 4 × 4 × 1.7  nm3. The magnetization dynamics was excited by the rapid (σ = 10 ps) Gaussian pulse of the current-induced spin-transfer-torque. The in-plane current distribution was modeled in COMSOL. After the pre-relaxation to the linear behavior, the residual oscillations of magnetization, averaged over the under-gate region, was analyzed and fitted by the decay function $$\tilde{m}=A\sin (2\pi ft+{\phi }_{0})\exp (-2\pi \Delta ft)$$. Here A and ϕ0 are the initial amplitude and phase, f is the FMR frequency, and Δf is the decay rate. The damping parameter α, shown on Fig. 3d was calculated as $$\alpha =\frac{\gamma {\mu }_{0}}{2\pi }\frac{\Delta f}{f}\frac{dH}{df}$$.

In the simulation we used the following parameters: μ0Hext = 0.4 T—applied magnetic field, MS = 1050 kA/m—saturation magnetization, α0 = 0.009—Gilbert damping constant49, Aex = 19 pJ/m50. The intrinsic PMA value Ku(VG = 0) = 645 kJ/m3 (tCoFeBKu = 1096.5 μJ/m2) with VCMA value of −2.5 kJ/m3 per Volt (i.e., −103.4 fJ/Vm per area per electric field) were chosen to have a good agreement with the experimentally obtained frequencies by ST-FMR.