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Nanowire Spin Torque Oscillator Driven by Spin Orbit Torques

Spin torque from spin current applied to a nanoscale region of a ferromagnet can act as negative magnetic damping and thereby excite self-oscillations of its magnetization. In contrast, spin torque uniformly applied to the magnetization of an extended ferromagnetic film does not generate self-oscillatory magnetic dynamics but leads to reduction of the saturation magnetization. Here we report studies of the effect of spin torque on a system of intermediate dimensionality - a ferromagnetic nanowire. We observe coherent self-oscillations of magnetization in a ferromagnetic nanowire serving as the active region of a spin torque oscillator driven by spin orbit torques. Our work demonstrates that magnetization self-oscillations can be excited in a one-dimensional magnetic system and that dimensions of the active region of spin torque oscillators can be extended beyond the nanometer length scale.

In all STOs studied to date, the active region where the negative ST damping exceeds the positive Gilbert damping of the ferromagnet was restricted to nanoscale dimensions. A recent study 33 of spatially uniform ST applied to an extended ferromagnetic film revealed that coherent self-oscillations of magnetization cannot be excited in this two-dimensional (2D) magnetic system. Instead, spin torque was shown to significantly reduce the saturation magnetization of the film 33 . The absence of ST-driven self-oscillations in a 2D ferromagnet was attributed to amplitude-dependent damping arising from nonlinear magnon scattering that prevents any of the multiple interacting spin wave modes of the system from reaching the state of large amplitude self-oscillations. As a result, the energy and angular momentum pumped by ST into the film is redistributed among a large number of spin wave modes leading to reduction of the saturation magnetization of the film. This study raises an important question about the role of the magnetic system dimensionality in ST-induced magnetization dynamics.
In this article, we demonstrate that spatially uniform ST can excite self-oscillations of magnetization in a one-dimensional (1D) magnetic system -a ferromagnetic nanowire. We report studies of ST-driven dynamics in a Pt/Py bilayer nanowire, in which SO torques excite self-oscillations of magnetization over a 1.8 µm long active region. This nanowire STO exhibits two types of self-oscillatory modes that arise directly from the edge and bulk spin wave eigenmodes of the Py nanowire 40 . Our work suggests that geometric confinement of the spin wave spectrum in the 1D nanowire geometry limits the phase space for nonlinear magnon scattering compared to the 2D film geometry and thereby enables STOs with a spatially extended active region. The bright areas on both sides of the image are Au/Cr leads. Scale bar: 1 µm. The inset shows resistance versus in-plane magnetic field applied perpendicular to the nanowire measured at the bath temperature T b = 4.2 K and a bias current of 0.5 mA. The black arrow shows the magnetic field sweep direction.

Sample description
The nanowire STO devices studied in this work are patterned from Pt(5 nm)/Py≡Ni 80 Fe 20 (5 nm)/AlO x (4 nm)/(GaAs substrate) multilayers deposited by magnetron sputtering. Multilayer nanowires that are 6 µm long and 190 nm wide are defined via e-beam lithography and Ar plasma etching as described in the Methods section. Two Au(35 nm)/Cr (7 nm) leads are attached to each nanowire with a 1.8 µm gap between the leads, which defines the active region of the device as shown in Fig. 1. The resistance of the device measured at bath temperature T b = 4.2 K versus magnetic field applied in the sample plane perpendicular to the nanowire is shown in the inset of Fig. 1. This plot reveals that the anisotropic magnetoresistance (AMR) of the Pt/Py bilayer is 1 %.

Electrical measurements
To study self-oscillatory magnetic dynamics excited by SO torques, we apply a saturating magnetic field (H > 0.5 kOe) in the plane of the sample in a direction nearly perpendicular to the nanowire axis. In this configuration, SO torques applied to the Py magnetization act as effective magnetic damping 25 . We apply a direct current bias I dc to the nanowire and measure the microwave signal emitted by the device using a spectrum analyzer 3 . The microwave signal V ac ∼ I dc δR ac is generated by the AMR resistance oscillations δR ac arising from the magnetization self-oscillations 18 . Microwave signal emission, shown in Fig.   2, is only observed above a critical current I c with the current polarity corresponding to SO torques acting as negative damping 25 . We measured the microwave emission for five nominally identical devices and found similar results for all these samples.
For all samples, the microwave emission spectra for I dc > I c exhibit two groups of closely spaced peaks with a frequency gap between the groups of ∼1 GHz. Each group consists of 1 to 4 distinct emission peaks separated from each other by tens to hundreds of MHz. The high and low frequency groups of peaks appear at similar critical currents. Both groups of peaks are observed in the entire range of magnetic fields (H = 0.5 -1.5 kOe) employed in this study. For the high frequency group of peaks, microwave emission is observed not only at the fundamental frequency shown in Fig. 2, but also at the second harmonic. As illustrated in the inset of Fig. 2b, the emitted power at the fundamental frequency is zero for magnetic field applied at an angle β = 90 • with respect to the nanowire axis and increases with decreasing β. In contrast, the emitted power in the second harmonic has a maximum at β = 90 • and decreases with decreasing β. Such angular dependence of the emitted power in the fundamental and second harmonic is expected for a microwave signal arising from AMR. For the low frequency group of peaks, no emission is seen at the second harmonic, which can be explained by the smaller amplitude of magnetization precession reached by these modes and an equilibrium magnetization direction within the mode excitation area being closer to the nanowire axis. Fig. 2c illustrates the dependence of the emission spectra on I dc for H = 890 Oe and β = 85 • .
In order to determine the origin of the microwave emission signals, we make measurements of the spectrum of spin wave eigenmodes of the nanowire using spin torque ferromagnetic resonance (ST-FMR) [34][35][36] . In this technique, a microwave current I ac applied to the nanowire excites magnetization dynamics in Py by the combined action of current-induced SO torques and the Oersted field from the current in Pt, and thereby generates AMR resistance oscillations at the frequency of the microwave drive 36 . Mixing of the current and resistance oscillations as well as variation of the time-averaged sample resistance in response to the microwave drive 37,38 give rise to a direct voltage V dc that is measured as a function of magnetic field H applied to the sample. Peaks in V dc (H) arise from resonant excitation of spin wave eigenmodes of the nanowire. An ST-FMR spectrum of spin wave eigenmodes measured at β = 85 • , drive frequency of 6 GHz and I dc = 2.0 mA < I c is shown in Fig. 3a.
Similar to the microwave emission spectra, we observe two groups of modes in the ST-FMR spectra. In Fig. 3b, we compare the field dependence of the eigenmode frequencies measured by ST-FMR at I dc < I c to the frequencies of self-oscillatory modes measured at I c .
This figure demonstrates that the frequencies of all self-oscillatory modes at I c coincide with the frequencies of spin wave eigenmodes of the system measured by ST-FMR. Therefore, all self-oscillatory modes of the system arise directly from spin wave eigenmodes of the nanowire.
This type of eigenmode self-oscillation is qualitatively different from the spin wave bullet mode excited by SO torques in a planar point contact to an extended ferromagnetic film 16 .
The bullet mode is a nonlinear type of oscillation, self-localized to a region with dimensions below 100 nm and with frequency below the spectrum of spin wave eigenmodes of the film 39 .
It is well known that spin wave eigenmodes of a transversely magnetized thin-film ferromagnetic nanowire can be classified as bulk and edge eigenmodes [40][41][42][43]  Using these values of the nanowire magnetic parameters and assuming translational invariance along the nanowire axis, we perform micromagnetic simulations to find the spectrum of spin wave eigenmodes for magnetic field applied in the sample plane at β = 85 • . We find that the two lowest frequency spin wave modes are the edge and bulk modes, whose spatial   measured at the current bias that maximizes the integrated power of the peak. The spectral linewidth of the edge peak decreases with increasing temperature, which cannot be explained by a single-mode STO theory 4 . However, such temperature dependence has been previously observed for multi-mode STOs and was successfully explained by an STO theory taking into account coupling between multiple self-oscillatory modes 48 . The linewidth of the dominant bulk peak is an order of magnitude smaller than that of the dominant edge peak for T b ≤100 K but it rises precipitously to a value similar to that of the edge peak for T b >100 K, which might result from enhanced coupling between the bulk and the edge modes for T b >100 K.

Brillouin Light Scattering measurements
In order to better understand the nature of the self-oscillatory dynamics induced by SO torques in the Pt/Py nanowire system and to directly confirm that the self-oscillatory modes occupy the entire active region of the nanowire, we make BLS measurements 49 of the current-driven magnetization dynamics in this system. Since a 5 nm thick layer of Pt is not sufficiently transparent for BLS studies, we make a separate batch of samples for BLS measurements with the reverse order of deposition of the AlO x , Py and Pt layers. These  conserving four-magnon scattering processes from the uniform (k = 0) mode into spin wave modes with a finite wave vector k 4 are allowed for the BVSW but not for the SSW modes as schematically illustrated in Fig. 6a. This nonlinear scattering channel is always present in the 2D film geometry and it contributes to the nonlinear damping of the uniform mode.
When such a film is patterned into a nanowire aligned perpendicular to the magnetization direction, the BVSW mode spectrum becomes quantized and the energy-conserving four- magnon scattering channel becomes suppressed for wires narrower than approximately πk −1 4 , which is ≈ 0.5 µm for our 5 nm thick Py nanowires. Fig. 6b shows the spin wave dispersion relation numerically calculated for our 190 nm wide nanowire sample as described in the Supplementary Note. The frequencies of the bulk and edge modes are shifted below the frequencies of spin waves in the extended film shown in Fig. 6a by the demagnetizing field of the nanowire. The dispersion branches of both the edge and the bulk spin wave modes do not have minima at k =0 and therefore do not support the energy-and momentumconserving four-magnon scattering within each of the branches.
Since some nonlinear magnon scattering processes (such as the four-magnon process) responsible for limiting the amplitude of low-frequency spin wave modes in the 2D film are eliminated in the nanowire, ST excitation of large-amplitude self-oscillatory spin wave modes may become possible above the critical current. At higher currents, SO torques increase the occupation numbers for all magnon modes, which results in enhanced scattering in the remaining nonlinear channels and the associated decrease of the amplitude of self-oscillations seen in Fig. 3c as well as in the reduction of the saturation magnetization and the associated decrease of the self-oscillatory mode frequency with current seen in Fig. 2c. Our data in Fig. 4 showing decreasing self-oscillation amplitude with increasing temperature suggest that enhanced population of thermal magnons and magnon-phonon scattering might also play a role in suppression of the self-oscillatory dynamics. Development of a detailed theory accounting for all magnon scattering channels in the nanowire in the presence of SO torques is required for quantitative explanation of our experimental data, and we hope that our work will stimulate the development of such a theory.
In conclusion, we demonstrated that spatially uniform spin torques can excite selfoscillations of magnetization in a 1D ferromagnetic system -a Py nanowire. The selfoscillatory modes induced by spin torque in this system directly arise from the bulk and edge spin wave eigenmodes of the nanowire. The 1D nanowire geometry offers unique advantages for studies of magnetization dynamics driven by spin torques over the 2D extended thin film system, in which spatially uniform spin torque does not excite magnetization self-oscillations. Our results suggest that the self-oscillatory dynamics in the nanowire geometry is enabled by geometric confinement of magnons that suppresses nonlinear magnon scattering. Our work demonstrates the feasibility of spin torque oscillators with the active region dimensions extended beyond the nanometer length scale. to obtain information about the BLS intensity proportional to the square of the amplitude of the dynamic magnetization at the location of the probing spot.

Estimate of the amplitude of resistance self-oscillations
In this section, we estimate the maximum amplitude of the resistance oscillations achieved by the bulk and edge modes in the self-oscillatory regime. The measured integrated power at the fundamental frequency emitted by the bulk group of modes P b (corrected for frequencydependent attenuation and amplification in the measurement circuit) is plotted as a function of direct bias current I dc in Fig. 3c. This power is directly related to the amplitude of the nanowire resistance oscillations δR ac at the fundamental frequency of magnetization selfoscillations. These resistance oscillations arise from anisotropic magneto-resistance (AMR) of the nanowire. The microwave voltage generated by the nanowire STO device at the fundamental frequency of the bulk mode I dc δR ac is detected by a 50 Ohm microwave spectrum analyzer as microwave power P b 3 : where R 50 ≡ 50 Ohm is the spectrum analyzer impedance and R is the nanowire resistance. From this equation, we can express the amplitude of resistance oscillations at the fundamental frequency as a function of the emitted power: Using Eq.
(2) and the maximum experimentally measured integrated power in the bulk group of modes P b = 10 pW at I dc = 2.45 mA as shown in Fig. 4a, we evaluate δR ac ≈ 0.15 Ohm. A similar maximum value of δR ac arising from the edge mode self-oscillations is calculated from the data in Fig. 4b. These values of δR ac are substantial fractions of the full AMR amplitude of 1 Ohm shown in the inset in Fig. 1b. This demonstrates that large amplitude of magnetization self-oscillations can be achieved by both the bulk and the edge modes of the nanowire STO.

Sample temperature
Direct current bias I dc can significantly increase the temperature of the nanowire due to Ohmic heating. In order to estimate the actual temperature of the nanowire at large I dc , we compare measurements of the wire resistance as a function of temperature at small I dc = 0.1 mA to measurements of the wire resistance versus I dc taken at the bath temperature T b = 4.2 K 18 . These data are plotted in Supplementary Fig. 4, which shows that resistance of the nanowire is approximately quadratic in both T b and I dc . This allows us to estimate the actual temperature of the nanowire directly. For example, at I dc = 2.0 mA, the actual temperature of the nanowire T ≈ 150 K.