Spin-Hall nano-oscillator with oblique magnetization and Dzyaloshinskii-Moriya interaction as generator of skyrmions and nonreciprocal spin-waves

Spin-Hall oscillators (SHO) are promising sources of spin-wave signals for magnonics applications, and can serve as building blocks for magnonic logic in ultralow power computation devices. Thin magnetic layers used as “free” layers in SHO are in contact with heavy metals having large spin-orbital interaction, and, therefore, could be subject to the spin-Hall effect (SHE) and the interfacial Dzyaloshinskii-Moriya interaction (i-DMI), which may lead to the nonreciprocity of the excited spin waves and other unusual effects. Here, we analytically and micromagnetically study magnetization dynamics excited in an SHO with oblique magnetization when the SHE and i-DMI act simultaneously. Our key results are: (i) excitation of nonreciprocal spin-waves propagating perpendicularly to the in-plane projection of the static magnetization; (ii) skyrmions generation by pure spin-current; (iii) excitation of a new spin-wave mode with a spiral spatial profile originating from a gyrotropic rotation of a dynamical skyrmion. These results demonstrate that SHOs can be used as generators of magnetic skyrmions and different types of propagating spin-waves for magnetic data storage and signal processing applications.

out-of-plane field in Ir/Co/Pt 27 and Pt/Co/MgO 28 multilayers. Although a single skyrmion can be nucleated by a spin-polarized scanning tunneling microscope 29 , the control of its room temperature nucleation is still an experimental challenge. Earlier achievements have shown the possibility to solve this problem 15,23,30 . Our results show an alternative method to control the nucleation of single skyrmions, based on the use of the SHE.

Results
Static characterization of the SHO structure and phase diagram of the SHO excitations. We have micromagnetically studied a Pt(5 nm)/CoFe(1 nm) SHO with a rectangular cross section of 1500 × 3000 nm 2 (see Fig. 1a for the sketch of the device, including a Cartesian coordinate system where x and y are the in-plane axes, while z is the out-of-plane axis, Methods and Supplementary Note 1 for the detailed description of the micromagnetic framework and simulation parameters). Figure 1b shows the angle θ M , characterizing the equilibrium orientation of the static magnetization in the SHO, as a function of the external bias magnetic field B. This field is applied at the tilting angle θ B = 15° with respect to the perpendicular of the SHO ferromagnetic layer in the y-z plane (see inset in Fig. 1b). As the bias field increases, the magnetization vector tends to align along the field direction.
Similarly to what is observed in STT oscillators based on the point-contact geometry, the type of the spin-wave mode excited by the SHE can be controlled by the direction of the bias magnetic field and the effective anisotropy. In particular, the materials with in-plane easy axis demonstrate excitation of self-localized spin-wave "bullets", or co-existence of bullets and Slonczewski modes 31,32 , for sufficiently large values of θ M, and excitation of Slonczewski propagating spin-wave modes for sufficiently small values of θ M 18,33 . In this study, numerical simulations showed that, for the bias field larger than 200 mT and θ M < 37°, the Slonczewski propagating spin-wave modes were excited.
As it will be discussed below, the additional degree of freedom of the i-DMI can introduce qualitative differences in the spatial profile of the Slonczewski-type cylindrical mode, compared to the case when i-DMI is ignored. Hereafter, we focus on the results obtained at the bias field of 400 mT and active region (distance between the Au electrodes in Fig. 1a) of d = 100 nm, however similar findings have been obtained at d = 200 nm and at larger bias fields (up to 800 mT).  34 .The skyrmion nucleation process, driven by the SHE, occurs together with the excitation of propagating spin-waves (see for an example Supplementary Movie 6), and at the current I sky (solid line between the point ' A' and 'B'). The I sky curve coincides with I th for D values larger than 4 mJ/m 2 (point ' A' in Fig. 1c). This fact constitutes the second key result of this study, i.e. the prediction that a pure spin-current with in-plane polarization can be used for the nucleation of skyrmions. The regions US/SKY, SKY/ SLM and SLM/SpM are the bistability regions, obtained sweeping the current back and forth and using in each simulation the final state at the previous current. In the first of these regions, we have either a uniform ground state or skyrmions, depending on the excitation history. In particular, the US is achieved when the current is not sufficiently large to obtain the SKY region. On the other hand, when the current is large enough to reach the SKY region, skyrmions are nucleated and remain stable also at zero current, therefore the SKY state is achieved in the US/SKY region. Concerning the second region SKY/SLM, a SLM is excited if the current is increased from the US/SKY region. Skyrmions and a SLM coexist if the current is decreased from the SKY region. In the last region, an SLM (SpM) is observed, if the current is increased (decreased) from SLM (SpM) region. The origin of this hysteretic behavior will be discussed in detail below. We have also investigated the role of the Oersted field, finding that it does not influence qualitatively the results of Fig. 1c (see Supplementary Note 2 for more details).

Excitation of Slonczewski linear spin-wave mode.
As it was pointed out earlier, the i-DMI leads to the excitation of nonreciprocal spin-waves. It can be observed qualitatively in the Supplementary Movies 3 and 4, and by comparing the mode profile in Fig. 2a,b. The largest nonreciprocal effect induced by the i-DMI occurs in the direction perpendicular to the in-plane projection of the static magnetization M 0 (x-axis), while the propagation along the in-plane projection of M 0 (y-axis) is reciprocal, and is characterized by the wave number that is the same as in the case of zero D (0.03 and 0.035 nm -1 at I = 4.22 mA and I = 5.28 mA, respectively, see Supplementary Note 2). Those results are consistent with the previous experimental measurements 35 and the results of the analytical theory 12 . The i-DMI-induced appearance of the nonreciprocal spin-waves leads to the decrease of the threshold current (Fig. 1d), and to a "red" shift of the generation frequency for increasing values of D, at a constant current (Fig. 2c). Figure 2d summarizes the dependence of the wave numbers (|k -x | and |k +x |) on D computed from the spatial distribution of the magnetization for I = 4.22 mA and I = 5.28 mA. The difference between the |k -x | and |k +x | is shown in Fig. 2e, and, as it can be noticed, is independent of I. All these numerically obtained features can Scientific RepoRts | 6:36020 | DOI: 10.1038/srep36020 be understood using a simple one-dimensional analytical model. In the framework of this model, we consider only the spin-waves propagating along the x-direction, where the spin-waves exhibit the largest nonreciprocity.
The frequency and wave vectors of the excited spin-waves are defined by the spatial quantization rule, which is determined by the spatial distribution of the spin-current J s . In the case of a nonreciprocal spectrum, the general quantization rule can be written as f(k + − k − , J s (x)) = 0 36 , or, equivalently k + − k − = const = f 1 (J s (x)), where k + and k − are the wave vectors of spin-waves propagating in opposite directions along the direction of maximum nonreciprocity and having the same frequency (for a reciprocal wave spectrum this rule is reduced to the condition |k| = const). The approximate spin-wave spectrum in the x-direction can be written as where ω 0 is the angular frequency of the ferromagnetic resonance in the SHO, 0 , λ is the exchange length in the material of the SHO ferromagnetic layer, and H an = 2K u /(μ 0 M S ) is the anisotropy field. From this equation, the wave vectors of counter-propagating nonreciprocal spin-waves, having the same frequency ω, can be computed as: is associated with the plus (minus) sign in the second term in the circular brackets in the equation (1). Substituting the wavenumber of equation (1) in the quantization rule, we get the condition . Thus, the generation frequency has a "red" shift with the increased D, as obtained from our micromagnetic simulations (see Fig. 2c). This effect could be easily understood by noting that the minimum spin-wave frequency in the spectrum becomes lower with the increase of D. From equation (1), it is easy to calculate the difference between the wave numbers of the excited waves: x x 2 and to verify that this difference is independent of the quantization constant and, therefore, of the spatial distribution of the spin-current. Hence, the condition (2) can be used for the experimental determination of the magnitude and sign of the i-DMI parameter. Equation (2) gives a reasonable description of the simulation data, considering the same physical parameters of the SHO (Fig. 2e). Small deviation of Eq. (2) from micromagnetic results are related to the usage of approximate SW spectrum which allows us to give simple and clear qualitative explanation of the observed effect and derive explicit expression for Δk. The fact that the dependence Δ k(D) is almost the same for different I is linked to a weak nonlinear variation of the spin-wave spectrum with driving current, due to a small difference in amplitudes of the excited spin-waves. Therefore, the difference of the spin-wave numbers is mainly determined by the linear spin-wave spectrum. From an experimental point of view, a direct determination of D can be achieved by measuring the wavelength of the emitted spin-waves along the + x and -x direction, using the phase-resolved micro-focused Brillouin light scattering 37 or time-resolved Kerr microscopy 38 . However, this method of determination of D may have practical limitations due to the fact that the wavelength of the excited spin-waves (see Fig. 2d) are in the range 0.13-0.63 μ m, i.e. being comparable with the lateral resolution of the above mentioned optical techniques.
Within the above described one-dimensional model, we can also calculate the dependence of the threshold current for spin-wave excitation on D. Assuming a rectangular profile of the charge current density in the active region (J(x) = J within x = [0,d], and J(x) = 0 otherwise), one can get the following implicit expression (similar to equation (6c) in 39 ), is the average wave number of excited nonreciprocal spin-waves (note, that in our notation k −x < 0), d is the distance between the SHO golden electrodes characterizing the spatial localization of the spin-current, damping, Γ J = σJ is the negative damping created by the spin-current, and σ = gμ B α H sin θ M /(2eM S t CoFe ) determines the spin-Hall efficiency (g is the Landè factor, μ B the Bohr magneton, e the electronic charge and t CoFe the CoFe layer thickness). The threshold current calculated from Eq. (3) is compared with numerical results in Fig. 1d. Here we use fitting coefficient C which relates threshold current density J th found from Eq. (3) with the current I th : I th = CJ th , which value was determined from the coincidence of calculated and micromagnetic threshold currents I th = 3.7 mA at D = 0.0 mJ/m 2 . One can see a good coincidence between the analytical and numerical results. Note that the decrease of the threshold current with D has the same nature, as a frequency "red" shift-lowering the bottom of the spin-wave spectrum with the increase of D and, consequently, the decrease of spin-wave damping Γ G = α G ω.
The SLM in SHOs have not been observed experimentally, since the threshold current for their excitation is expected to be very large (> 10 9 A/cm 2 ) 20 , around three times larger than the current necessary to excite a "bullet" spin-wave mode in an SHO with in-plane magnetization. In the SHO of this study, we were able to reduce the critical current density of one order of magnitude (< 4 × 10 8 A/cm 2 ) thanks to the additional perpendicular interface anisotropy in the CoFe ferromagnet. This additional anisotropy allows one to achieve the positive nonlinear frequency shift, required for the SLM excitation 33 , at a higher magnetization angle θ M , which results in the higher spin-Hall efficiency, since it is proportional to sin θ M . A further reduction of the current density can be achieved by including an additional Ta layer above the CoFe ferromagnet 24 .
Excitation of spin-wave modes with a spiral spatial profile. Figure 3a summarizes the spin-wave frequency as a function of I computed for D = 0.0 mJ/m 2 and D = 1.5 mJ/m 2 (d = 100 nm). In the absence of the i-DMI, the oscillation frequency shows a monotonic increase with current, or a "blue" frequency shift, typical for the Slonczewski linear propagating spin-wave mode. A different frequency behavior is seen for D = 1.5 mJ/m 2 , where the frequency tunability with current becomes non-monotonic. This behavior is robust under the variation of d, as seen from Fig. 3b where d = 200 nm. At sufficiently large I and D, the spin-wave is converted from the cylindrical to a spiral-like (SpM region in Fig. 1c). Figure 3c shows a spiral-type profile (the color is linked to the y-component of the magnetization).
In order to understand the origin of the spiral mode, we have performed a detailed analysis of the spatial distribution of the dynamic magnetization in the SHO ferromagnetic layer in this regime. Figure 3d-g illustrate four snapshots (I = 6.33 mA) which clearly reveal the physics of the spiral mode formation. In the SpM region, the SHE is able to nucleate a dynamical soliton [40][41][42] . It is characterized by a central core with the magnetization pointing along the negative out-of-plane direction (opposite to the equilibrium axis of the magnetization), and by the rotation of its boundary spins through 360° (see Fig. 3d-g). The dynamical skyrmion exhibits a rotational motion (gyration) along a circular trajectory within the region of the high current density, that is typical for solitons with nonzero topological charge under the influence of spin-current 43 (see Supplementary Movie 7). Dynamical skyrmion plays a role of a "source" for magnetization oscillations in the outer region, and, since the source is gyrating, the radiation acquires the form of a spiral wave, as it happens in many other fields with gyrating source 44,45 . Note, also, that once it has been excited the SpM is still stable at lower current magnitudes in the SpM/SLM region, because the excitation of the dynamical skyrmion is linked to a sub-critical Hopf bifurcation 42 . Spiral mode is strongly nonlinear because it is originated by the interaction between a dynamical skyrmion and propagating spin-waves, this is the reason of the non-monotonic behavior of the frequency of the excited mode as a function of the current. Generation of single skyrmions and "gas" of skyrmions. The last regions of the phase diagram of Fig. 1c are related to skyrmions. For the critical D C , the skyrmions become energetically stable 34 and, after the nucleation driven by the SHE (SKY region) (see Supplementary Movie 6 for the nucleation of a single skyrmion), Scientific RepoRts | 6:36020 | DOI: 10.1038/srep36020 they remain stable even when the driving current is switched off (US/SKY region). Once the skyrmion is nucleated, it is shifted along the spin-current direction, as expected for Néel skyrmions 16 . For D below 4.0 mJ/m 2 (point ' A' in Fig. 1c), I sky and I th split into different curves, and, hence, in the SKY/SLM region when the current increases from the uniform state, only the SLMs are excited. The presence of this region in the phase diagram is interesting from a fundamental point of view, as it identifies a scenario where the interaction between the spin-waves and skyrmions 46 can be studied. Figure 4a shows the nucleation time of a single skyrmion as a function of the current magnitude for two values of D (3.5 and 4.0 mJ/m 2 ). It can be seen from Fig. 4a that a sub-nanosecond skyrmion nucleation time can be achieved (see Fig. 4b for a single skyrmion snapshot). Our results predict a new scenario for a single skyrmion nucleation driven by a pure spin-current. This method can be used as an alternative to the method based on the STT from a perpendicular spin-polarized current 15 , with the possible advantage of  the simpler fabrication process of the device. If current pulses are applied consecutively or if the current is not switched off, more skyrmions are nucleated up to a saturation value that marks a transition to a skyrmion gas phase 47 . In detail, since the current is non-uniformly applied, the skyrmions tend to accumulate in one side of the ferromagnet until no more skyrmions can be hosted because of the skyrmion-skyrmion magnetostatic repulsion (see Fig. 4c for an example of the spatial distribution of the skyrmions). A skyrmion gas is, therefore, formed, and each skyrmion further nucleated is immediately annihilated (see Supplementary Movie 8). This result paves the way to study the magnetic properties of skyrmion gas described theoretically in 47 .

Discussion
In our study, we propose an SHO device geometry that, combining SHE and i-DMI, offers a unique opportunity to study nonreciprocal effects of spin-wave propagation in two dimensional systems and to observe a new type of dynamical spin-wave modes having a spiral spatial profile. This novel spin-wave mode originates from the gyrotropic rotation of a dynamical skyrmion. From the technological point of view, the proposed SHO geometry could be useful for the development of novel generators of short propagating spin-waves in future magnonic signal processing devices. From the fundamental point of view, it is also very interesting, as it allows to study the interaction of spin-wave and skyrmions, as well as to control the number of the nucleated skyrmions by applying a properly designed current pulse.