Spin-Orbit-Torque Driven Propagating Spin Waves

Spin-orbit torque (SOT) can drive sustained spin wave (SW) auto-oscillations in a class of emerging microwave devices known as spin Hall nano-oscillators (SHNOs), which have highly non-linear properties governing robust mutual synchronization at frequencies directly amenable to high-speed neuromorphic computing. However, all demonstrations have relied on localized SW modes interacting through dipolar coupling and/or direct exchange. As nanomagnonics requires propagating SWs for data transfer, and additional computational functionality can be achieved using SW interference, SOT driven propagating SWs would be highly advantageous. Here, we demonstrate how perpendicular magnetic anisotropy can raise the frequency of SOT driven auto-oscillations in magnetic nano-constrictions well above the SW gap, resulting in the efficient generation of field and current tunable propagating SWs. Our demonstration greatly extends the functionality and design freedom of SHNOs enabling long range SOT driven SW propagation for nanomagnonics, SW logic, and neuro-morphic computing, directly compatible with CMOS technology.


Introduction
The recent emergence of spin-orbit torque (SOT) 1-3 from pure spin currents, has opened a new avenue for the controlled manipulation of magnetic moments in spintronic devices resulting in dramatically improved efficiency 4,5 and much lower power dissipation 6,7 compared to conventional spin-transfer torque (STT) 8 . Thanks to the ability of SOT to compensate natural magnetic damping over spatially extended regions, there has been considerable interest in exploring the long range enhancement of spin wave (SW) propagation [9][10][11][12][13][14][15] in a variety of nanoscale devices with the aim of developing an energy-efficient and ultra-high-speed beyond-CMOS spin-wave-based technology for signal processing 16,17 and computation 11,[18][19][20] , so-called magnonics 21,22 .
A major limitation of nano-constriction SHNOs, however, is the localized nature of the SOT driven auto-oscillations. 37 The localization is a consequence of the easy-plane anisotropy and the geometry of the device, which lead to a negative magnetodynamic non-linearity, further exacerbated by the Oersted field from the drive current, and from the SOT itself. 37,38 It would be highly 2 advantageous if the localization could be mitigated so as to generate truly propagating SWs. Not only should this lead to mutual synchronization over much longer distances, it would also make SOT driven SWs directly applicable to additional non-conventional computing schemes such as wave based computing 18,[39][40][41] .
In a recent work 31 , Evelt and coworkers demonstrated SOT driven propagating SWs in extended Bi-substituted YIG films with perpendicular magnetic anisotropy 42 (PMA). While the autooscillations could only be observed optically and did not exhibit any frequency tunability via the drive current, the demonstration raises the question whether the addition of PMA to metal based nano-constriction SHNOs could potentially lead to SOT generated propagating SWs in more practical devices directly compatible with CMOS technology 29 . Here we show, using 150 nm and 200 nm nano-constrictions in W/CoFeB/MgO material stacks with substantial PMA, that it is indeed possible to generate strongly current-tunable propagating SWs over a very wide frequency range of about 3-23 GHz. The SWs are studied using electrical microwave spectroscopy and modelled using micromagnetic simulations. Auto-oscillations are observed at currents as low as 0.15 mA, where they are still localized and exhibit negative non-linearity. As the current is increased, the non-linearity changes sign and the localized SWs exhibit smooth transition into propagating SWs at about 0.5 mA. It is hence possible to seamlessly turn on and off the localization, which will allow the generation of ultra-short SW pulses driven by the current alone, which is much faster than using external fields 43 .
Perpendicular magnetic anisotropy controlling the magnetodynamic non-linearity. The rich non-linear magneto-dynamics in patterned magnetic thin films can be analytically described by a single non-linearity coefficient, N , the magnitude and sign of which determine the strength and nature of magnon-magnon interactions, with positive and negative values signifying magnon repulsion and attraction, respectively. 37,44,45 As spin transfer torque and SOT can generate very high SW amplitudes, the sign and magnitude of N leads to dinstinctly different behavior of the autooscillations. A negative non-linearity makes the auto-oscillation frequency decrease with amplitude, eventually moving it into the magnonic band gap, where it first leads to SW self-localization, and can further promote the nucleation of magnetodynamical solitons such as SW bullets [46][47][48] in easy-plane magnetic films and magnetic droplets 49,50 in films with very large PMA. A large positive non-linearity, on the other hand, makes the auto-oscillation frequency increase with amplitude to well above the ferromagnetic resonance (FMR) frequency leading to the propagation of spin waves with a finite real wave-vector. A prominent example is the so-called spin torque driven Slonczewski modes [51][52][53] , which can also form SW beams 54 in oblique fields, particularly useful for mutual synchronization 55 .
The non-linearity is generally governed by the applied field vector and/or an effective magnetic anisotropy tensor and is zero for an isotropic magnet regardless of the applied external field strength. The easy-plane shape anisotropy of a magnetic thin film holds the magnetization vector in the film plane and therefore the non-linearity strongly depends on the strength and orientation of the applied field. However, anisotropy induced by an interface to a different material can counteract the shape anisotropy and even pull the magnetization vector out of the film plane. Such One notes that N increases monotonically from negative (red regions) to positive (blue regions) values as a function of the OOP field strength and goes through zero at a certain field value (black dashed line) that depends on the PMA strength 56 , i.e. the PMA shifts the point of zero non-linearity towards lower values of applied field. Adding PMA hence makes it possible to reach positive non-linearity at much lower fields. As the auto-oscillation threshold current decreases with decreasing field, it should in principle be possible to drive propagating SWs at much lower currents, which the SHNO can sustain without degradation.
As a side note, any further increase of the PMA beyond the point where it completely compensates the shape anisotropy, i.e. where the magnetization equilibrium angle changes from inplane (IP) to perpendicular to the plane, again results in a negative N (the second red region at high PMA in Fig. 1b). As a consequence, the current dependence of the auto-oscillation frequency again becomes negative 57, 58 , the generated spin waves self-localize, and can eventually nucleate magnetic droplet solitons 49,50,[59][60][61][62] .

Results
Nano-patterned SHNO device schematic and magneto-dynamics. A schematic of a nanoconstriction SHNO is shown in Fig. 1a. The material stack consisted of sputtered β-W(5 nm)/Co 20 Fe 60 B 20 (1.4 nm)/MgO(2 nm). The β-phase of W has been shown to produce large SOT 63, 64 and a thinner CoFeB interfaced with MgO layer enhances PMA in the CoFeB layer 65 .
The stack was fabricated on highly resistive Si substrates to both dissipate the local heat generated during operation and to reduce microwave losses; the SHNOs are hence CMOS compatible 29 . A positive direct current (d.c.) is injected from the signal pad to ground along the y-direction while φ and θ define the IP and OOP field angles, respectively. Fig. 1c shows the in-plane angular depen-dence of the anisotropic magneto-resitance (AMR) measured for a 200 nm wide nano-constriction SHNO exhibiting a relatively large overall AMR value of 0.46% between the parallel and perpendicular orientation. Propagating spin waves Figure 3 shows color plots of the generated microwave power spectral density (PSD) as a function of OOP applied field strength measured for two different nanoconstriction widths with fixed direct currents of I dc = 1 and 2 mA, respectively. In both measurements, the IP field angle was fixed at φ = 22 • to ensure sufficient electrical sensitivity to the auto-oscillation signal while the OOP field angle θ = 80 • was chosen in a way to achieve large positive non-linearity in the active nano-constriction region. The orange circles, fitted with a solid 8 orange line using the Kittel equation (Eq. (2)), represent the FMR frequencies obtained from ST-FMR measurements on microstrips under identical applied field conditions. As the FMR frequency corresponds to a wave vector of k = 0, it allows us to distinguish between propagating and localized spin-wave modes in the SHNO, since spatially localized modes with a frequency well below FMR have no well-defined real wave vector, while propagating modes with frequency higher than the FMR have a finite real k.
It is noteworthy that in all previously investigated nano-constriction and nano-gap SHNOs 26, 66, 67 , the auto-oscillations remained localized with frequency lower than the FMR spectrum of the magnetic material. As can be seen in Fig. 3a, this is in our case only true for µ 0 H < 0.2 T and at all higher fields, the auto-oscillation frequency lies up to several GHz above the FMR frequency. This general behavior was observed in all devices as e.g. in Fig. 3b where a larger nano-constriction with w = 200 nm follows the same trend, but now with improved microwave characteristics. It can be noted that in addition to a higher output power, the auto-oscillations in the larger nano-constriction cross over into propagating spin waves already at µ 0 H ∼ 0.15 T, indicating a higher PMA than in the smaller nano-constriction. This is a general trend for different nano-constriction widths, which we believe is an effect of an etch induced reduction of PMA at the nano-constriction edges, which affects the smaller nano-constrictions in greater proportion.
The field dependence in Fig. 3 is entirely consistent with the expected behavior based on is interesting to note that we no longer observe any auto-oscillation localization in the wider nanoconstriction, even at the lowest field of 0.4 T, which is consistent with stronger PMA in wider nano-constrictions. We also emphasize that the spectral linewidth of auto-oscillations ∆f < 20 MHz, extracted using a Lorentzian fit, yields a quality factor Q = f /∆f of up to 1000, indicating a considerably higher degree of oscillation coherence of the generated propagating spin-waves in Micromagnetic simulations Finally, we present micromagnetic simulations performed using comparable conditions as in our electrical measurements to study the spatial profiles of the SW auto-oscillations for a 150 nm nano-constriction SHNO. All the magnetodynamical parameters used in the simulations are directly taken from the ST-FMR measurements discussed above. Fig. 5a-c show the current dependent PSD under three different fixed OOP field strengths indicating an excellent agreement with our experimental results in Fig. 4a-c. At 0.4 T, we observe a similar non-monotonic current dependence of the auto-oscillation frequency with simulated current, starting with a red-shifting frequency followed by a blue-shifted behavior (see Fig. 5a). It is interesting to note the appearance of multiple frequency steps at the lowest field. These are likely related to discreet changes in the wave vector and can also be observed in our experimental results at the same field (Fig. 4a), albeit as smoother transitions. At higher applied fields, these mode

Discussion
The capability to generate high frequency SOT-driven coherent propagating spin waves in metal based CMOS-compatible SHNO devices has particular potential for a number of reasons. First, the SW generation takes place already at very small operational currents with thresholds as low as 0.15 mA, with a straightforward development path towards even lower currents, making these oscillators the most amenable to adaptation in nanomagnonic circuits. Thanks to the large spin Hall angle provided by the β-W layer together with the strong PMA due to thinner CoFeB layer, the critical threshold current density required to excite propagating spin wave with SOT in metal devices has been reduced by about two orders of magnitude (10 7 A/cm 2 ) compared to theoretical predictions based on Pt and zero PMA 69 . Further reduction of the critical current density should also be possible by yet higher PMA in yet thinner CoFeB. Second, their current tunability should allow these SHNO to rapidly switch between localized and propagating spin waves, effectively acting as tunable nanoscale sources of ultra-short SW pulses and wave packets 43 where The FMR frequency, shown by a solid line on Figure 3, can be calculated as: The coefficients of Bogolyubov transformation are: In the above expressions we used the notations ω M = γµ 0 M S , ω k = γµ 0 H ⊥ k , ω H = γµ 0 H M , where M S is the saturation magnetization, H ⊥ k is the PMA field, H M -effective internal field. The latter can be defined together with the internal angle of magnetization θ M from the following equations: where H ex is the externally applied magnetic field applied at the out-of-plane angle θ ex . voltage, V mix , across the microstrip and is measured using the circuit displayed in inset of Fig. 2.
All ST-FMR measurements shown in Fig. 2 were carried out by sweeping an in-plane field (φ = 30 • ) from 350 to 0 mT, while the frequency of the input RF signal is kept fixed. To determine the SHA, 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.
In Fig. 3, ST-FMR measurements on the microstrip were performed by sweeping the applied field at a fixed IP angle of φ = 22 • and OOP angle of θ = 80 • to measure FMR frequency under the identical conditions employed during auto-oscillation measurements.
Microwave measurements. All microwave electrical measurements were carried out at room temperature using a custom built probe station with the sample mounted at a fixed in-plane angle on an out-of-plane rotatable sample holder between the pole pieces of an electromagnet capable of producing a uniform magnetic field. A direct positive electric current, I dc , was made to inject through dc port of a high frequency bias-T and the resulting auto-oscillating signal was then amplified by a low-noise amplifier with a gain of ≥ 32 dB and subsequently recorded using a spectrum analyzer from Rhode & Schwarz (10 Hz-40 GHz) comprising a low resolution bandwidth of 300 kHz. We measured multiple SHNO devices and restricted the maximum current up to 1 mA for 150 nm and 2 mA for 200 nm nano-constriction devices in order to avoid irreversible changes due to device degradation in the output microwave characteristics.