Introduction

Spin-orbitronics combined with other sub-fields of spintronics, such as magnonics and spin-caloritronics, has created a novel paradigm in information processing which could become a viable alternative to Si-based electronics1.

Recent experimental and theoretical developments in spin-orbitronics have clearly shown a great potential in generation of spin-currents able to compensate damping in magnetic materials2,3,4,5,6. The spin-Hall effect (SHE) plays a dominant role in the above-mentioned experiments, as it converts the input charge current, flowing in a heavy metal, into a spin-current, diffusing perpendicularly into the adjacent ferromagnet, and creating a spin-transfer torque (STT) that acts on the ferromagnet magnetization7. Another interesting and highly non-trivial spin-orbital effect is the interfacial Dzyaloshinskii-Moriya interaction (i-DMI)8. Both SHE and i-DMI have been used to improve the performance of “racetrack” device prototypes in magnetic storage9,10, to add a new degree of freedom in the design of magnetoresistive memories3,11, to create nonreciprocity in the spin-wave (SW) propagation for signal processing applications12,13,14, to excite coherent magnetization self-oscillations4,5, and for the manipulation of skyrmions in ultrathin ferromagnetic materials15,16. However, to the best of our knowledge, the influence of i-DMI on the performance of a spin-Hall oscillator (SHO) has not been studied so far4,5.

Here, we present the magnetization dynamics induced by the SHE in a realistic SHO structure, taking into account the influence of the i-DMI8. We have chosen a state-of-the-art SHO geometry (Fig. 1a) where the charge current I flows in the Pt layer along the x-axis between the golden electrodes and, due to the SHE in Pt, a spin-current is locally injected into the ultrathin extended CoFe ferromagnet (SHO “free” layer). The CoFe layer has an in-plane easy axis at zero bias field, so when a sufficiently large out-of-plane bias field is applied at an oblique angle in the “yz” plane (Fig. 1b), the static magnetization M of the “free” layer also goes out-of-plane, making the angle θM with the vertical axis “z”. In such a geometry, the Slonczewski propagating spin waves17 can be excited in any in-plane direction18,19,20 and, due to the influence of the i-DMI, they have the maximum nonreciprocity when propagating along the x-axis, perpendicular to the in-plane projection of the bias magnetic field. Our numerical simulations have shown that the wave numbers of SWs excited at a particular frequency ω and propagating along the positive and negative directions of the x-axis are different. The difference is proportional to the magnitude of the i-DMI parameter D. This result, well reproduced by a simple one-dimensional analytical model, can be used to establish a novel procedure for the experimental measurements of D. Micromagnetic simulations have also demonstrated that (i) a novel propagating spin-wave mode, characterized by a spiral spatial profile, can be excited at sufficiently large magnitudes of D and I, and (ii) skyrmions can be efficiently nucleated by the SHE in the SHO geometry. Similarly to optics21,22, the excitation of spiral spin-waves in magnetism could be attractive for designing new information coding protocols. Recent experimental observations have demonstrated that skyrmions15,16,23,24,25 can be nucleated via conversion of domain walls in Ta/CoFeB/MgO26, or by applying an out-of-plane field in Ir/Co/Pt27 and Pt/Co/MgO28 multilayers. Although a single skyrmion can be nucleated by a spin-polarized scanning tunneling microscope29, the control of its room temperature nucleation is still an experimental challenge. Earlier achievements have shown the possibility to solve this problem15,23,30. Our results show an alternative method to control the nucleation of single skyrmions, based on the use of the SHE.

Figure 1
figure 1

Sketch of the SHO device under investigation and dynamical phase diagram of this device.

(a) Sketch of a bilayer composed of a CoFe ferromagnetic layer and a heavy metal layer (Pt) with a rectangular cross-section. The thick Au electrodes carry the charge current everywhere, except the inter-electrode gap of the width d, where the charge current flows inside the bilayer and excites perpendicular (vertical) spin current going into the CoFe ferromagnetic layer. A rectangular coordinate system for the above described SHO geometry is shown. (b) The angle θM characterizing the equilibrium direction of the static magnetization in the CoFe ferromagnetic layer as a function of the magnitude of the external bias magnetic field B. The vertical line at B = 200 mT separates the regions where it is possible to excite localized and propagating spin-wave modes, respectively. Inset: Cartesian coordinate reference system where the angles θM and θB are shown explicitly. (c) The phase diagram of the SHO excitations on the D vs. I plane. Seven different regions can be distinguished in this phase diagram: uniform states (US), Slonczewski linear modes (SLM), spiral modes (SpM), skyrmions (SKY), and the bistability regions uniform states/skyrmions (US/SKY), Slonczewski linear modes/spiral modes (SLM/SpM) and Slonczewski linear modes/skyrmions (SLM/SKY). The amplitude of the external field is B = 400 mT. The Oersted field is included in the model. DC is the critical value of the i-DMI parameter (see explanations below), Ith is the threshold current, Isky is the current needed to nucleate skyrmions (line between the points ‘A’ and ‘B’); (d) Comparison between the threshold current of the SLM excitation obtained by means of micromagnetic simulations (symbols) and using the analytical formula (3) (solid line).

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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 nm2 (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 θM, 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 θB = 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 modes31,32, for sufficiently large values of θM, and excitation of Slonczewski propagating spin-wave modes for sufficiently small values of θM18,33. In this study, numerical simulations showed that, for the bias field larger than 200 mT and θM < 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).

Figure 1c shows a phase diagram of dynamical excitations in the SHO on the plane D-vs-I. Seven different regions can be identified: (i) uniform states (US), (ii) Slonczewski linear modes (SLM), (iii) spiral modes (SpM), (iv) skyrmions (SKY), as well as the bistability regions (v) uniform states/skyrmions (US/SKY), (vi) Slonczewski linear modes/skyrmions (SLM/SKY) and (vii) Slonczewski linear/spiral modes (SLM/SpM). At small values of the driving current, the SHO is in the US, i.e. in a region characterized by a uniform magnetic configuration. SLMs are excited at a critical current Ith that slightly decreases as a function of D (see Fig. 1d). The excited modes in the SLM region exhibit a two-dimensional radiation pattern that changes from the isotropic (see Supplementary Movies 1 and 2 for the SLM dynamics at I = 4.22 mA and I = 5.28 mA respectively, B = 400 mT and D = 0.0 mJ/m2) to the anisotropic cylindrical profile with the increase of the i-DMI parameter D (see Supplementary Movies 3 and 4 for the SLM at I = 4.22 mA and I = 5.28 mA, respectively, B = 400 mT and D = 1.5 mJ/m2). The cylindrical profile of the spin-wave radiation evolves into a spiral-like profile for 1.5 mJ/m2<D<3.0 mJ/m2, and a sufficiently large I (SpM region) (see as an example Supplementary Movie 5). The identification of the scenario leading to the radiation of these spiral spin-wave modes is one of the most important results of this study.

The SKY region is observed starting from D values near the critical value of DC = 3.16 mJ/m2 ()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 Isky (solid line between the point ‘A’ and ‘B’). The Isky curve coincides with Ith for D values larger than 4 mJ/m2 (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 M0 (x-axis), while the propagation along the in-plane projection of M0 (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 measurements35 and the results of the analytical theory12. 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 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.

Figure 2
figure 2

Non-reciprocal Slonczewski linear modes.

(a,b) Example of the spatial profile of the reciprocal and nonreciprocal Slonczweski spin waves, respectively, calculated for I = 5.28 A and D = 0.0 and 1.5 mJ/m2. respectively. (c) Oscillation frequency of the excited SLM as a function of D for two values of the driving current. (d) Wave numbers along the −x and +x directions as functions of D for two values of current; (e) Difference between the wave numbers along the positive and negative x-directions as a function D for the same two values of current (solid lines) and the same difference calculated analytically from Eq. (2) (dashed line).

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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 Js. In the case of a nonreciprocal spectrum, the general quantization rule can be written as f(k+ − k−, Js(x)) = 036, or, equivalently k+ − k− = const = f1(Js(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 (see Methods), where ω0 is the angular frequency of the ferromagnetic resonance in the SHO, , , λ is the exchange length in the material of the SHO ferromagnetic layer, and Han = 2Ku/(μ0MS) 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:

where () 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 that gives the following dependence of the generation frequency on D: . 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:

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 Δk(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 scattering37 or time-resolved Kerr microscopy38. 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 μm, 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) in39),

where 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, is the spin-wave group velocity, ΓG = αGω is the spin-wave damping, ΓJ = σJ is the negative damping created by the spin-current, and σ = gμBαH sin θM/(2eMStCoFe) determines the spin-Hall efficiency (g is the Landè factor, μB the Bohr magneton, e the electronic charge and tCoFe 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 Jth found from Eq. (3) with the current Ith: Ith = CJth, which value was determined from the coincidence of calculated and micromagnetic threshold currents Ith = 3.7 mA at D = 0.0 mJ/m2. 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 ΓG = αGω.

The SLM in SHOs have not been observed experimentally, since the threshold current for their excitation is expected to be very large (>109A/cm2)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 × 108A/cm2) 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 excitation33, at a higher magnetization angle θM, which results in the higher spin-Hall efficiency, since it is proportional to sin θM. A further reduction of the current density can be achieved by including an additional Ta layer above the CoFe ferromagnet24.

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/m2 and D = 1.5 mJ/m2 (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/m2, 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).

Figure 3
figure 3

Excitation of a spin-wave with spiral profile.

(a) Oscillation frequency of the excited mode as a function of I without and with i-DMI, for d = 100 nm. (b) Same as (a) but with d = 200nm. (c) Example of a spatial profile of the spiral-type spin-wave for I = 6.33 mA and D = 1.5 mJ/m2. (d–g) Spatial distributions of the magnetization characterizing a topological-type magnetic soliton, the current-induced gyration of which causes the radiation of a spiral-type spin wave mode.

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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 soliton40,41,42. 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-current43 (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 source44,45. 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 bifurcation42. 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 DC, the skyrmions become energetically stable34 and, after the nucleation driven by the SHE (SKY region) (see Supplementary Movie 6 for the nucleation of a single skyrmion), 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 skyrmions16. For D below 4.0 mJ/m2 (point ‘A’ in Fig. 1c), Isky and Ith 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 skyrmions46 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/m2). 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 current15, 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 phase47. 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 in47.

Figure 4
figure 4

Skyrmion nucleation.

(a) Nucleation time of a single skyrmion as a function of the current amplitude for D = 3.5 and 4.0 mJ/m2. (b) Snapshot of a single skyrmion and zoom of the skyrmion nucleation region. (c) Snapshot of a skyrmion gas and zoom of the region indicated by a green square in the right frame.

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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.

Methods

Micromagnetic framework

Micromagnetic simulations were carried out by means of a state-of-the-art parallel micromagnetic solver, which numerically integrates the LLG equation including the Slonczewski-like torque due to SHE48,20:

where m and hEFE are the normalized magnetization and the effective field of the ferromagnet. The effective field includes the standard magnetic field contributions, as well as the i-DMI and Oersted field (see also Supplementary Note 1). τ is the dimensionless time τ = γ0MSt, where γ0 is the gyromagnetic ratio, and Ms is the saturation magnetization of the ferromagnet. αG is the Gilbert damping, g is the Landè factor, μB is the Bohr Magneton, e is the electron charge, tCoFe is the thickness of the ferromagnetic layer, αH is the spin-Hall angle obtained from the ratio between the spin current and the electrical current. is the unit vector of the out-of-plane direction and J is the in-plane current density injected via the heavy metal. The i-DMI energetic density expression, as derived considering the ultra-thin film hypothesis , is , D being the parameter taking into account the intensity of the DMI, and mz is the z-component of the normalized magnetization. By making the functional derivative of equation, the normalized i-DMI effective field is given by:

The boundary conditions related to the interfacial DMI are expressed by where n is the unit vector normal to the edge and (being A the exchange constant) is a characteristic length in the presence of i-DMI.

We have studied a bilayer system Pt(5 nm)/CoFe(1 nm) with a rectangular cross section of 1500 × 3000 nm2. The electric current was locally injected into the ferromagnet via a thick Au electrode (thickness of 150 nm) with two tips located at a distance d from each other. The charge current flowing in the Pt layer gives rise to the SHE and then, to flow of perpendicular (along the “z” axis) pure spin current at the Pt/CoFe interface creating an anti-damping Slonczewski-like torque in the ferromagnetic layer. At sufficiently large magnitudes of the charge current his torque compensates the Gilbert losses in the ferromagnetic layer and excites in it persistent magnetization oscillations. For the results discussed in the main text, we have considered the following physical parameters of the SHO (Fig. 1a): saturation magnetization MS = 1 × 106A/m49, exchange stiffness constant A = 2.0 × 10-11J/m, interfacial perpendicular anisotropy induced at the boundary between CoFe and Pt characterized by the anisotropy constant Ku = 5.5 × 105J/m3,50 damping constant αG = 0.0351, and the spin-Hall angle αH = 0.19. The ferromagnetic CoFe layer has an in-plane equilibrium magnetization at zero field which is directed along the y- in-plane direction due to the shape anisotropy of the ferromagnetic layer. The real spatial distributions of the density Je of the charge current, density Js of the spin current and the Oersted field were calculated numerically, as it is described in the Supplementary Note 1.

Derivation of analytical equations

The spin wave dispersion relation along the x-direction in the presence of i-DMI can be calculated analogously to ref. 12 and has the following form

where ωH = γBeff, Beff is the static effective field, ωM = γμ0Ms. In the range ωMλ2k2 ≪ ω0 it can be approximated as , where is the angular frequency of the ferromagnetic resonance in the ferromagnetic layer, and .

Making a formal substitution kx → −i(d/dx) in this dispersion equation, it is possible to obtain the following dynamical equation describing the spatial and temporal evolution of the spin wave complex amplitude a:

The spin wave damping is accounted for by the term αGω (spin wave ellipticity, which could modify the damping term52 in our case is small), while the influence of the spin current could be easily calculated from equation (4) within the framework of the perturbation theory52 and is given by the term σJ(x)a, with σ = gμBαH sin θM/(2eMStCoFe). Equation (3) for the threshold current can be obtained, analogously to ref. 13, by deriving general analytical solutions of equation (7) inside and outside the current-carrying region, and applying the boundary conditions of continuity for the spin wave complex amplitude a and its derivative. It is clear, that in the reciprocal case, when k-x = −k+x, Eq. (3) is reduced to Eq. (6) from ref. 39.

Additional Information

How to cite this article: Giordano, A. et al. Spin-Hall nano-oscillator with oblique magnetization and Dzyaloshinskii-Moriya interaction as generator of skyrmions and nonreciprocal spin-waves. Sci. Rep. 6, 36020; doi: 10.1038/srep36020 (2016).

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