Multiple modes of a single spin torque oscillator under the non-linear region

The spin-torque oscillator (STO)¹ is an auto-oscillator on the nanometer scale driven by the spin toque under the charge current. It has attracted considerable attention with prospects of a wide range of applications including the frequency signal generator,² signal modulation,³ spin-wave generation,⁴ and neuromorphic computing.⁵ 

The STO is normally governed by a single Kittel mode in a linear auto-oscillatory region, while the non-linear STO driven by the phase-locking⁶ enables us to engineer the frequency modulation. The sideband modulation^7–12 is a typical analog frequency modulator where equally spaced multiple modes are excited around the fundamental mode. The schemes of the sideband modulation are similar to multi-mode laser systems in terms of a delay-line oscillator. The delay in the latter case is due to the optical cavity, while the delay is brought by the external cavity in the case of a feedback STO, e.g., the external radio-frequency sources.^9,11 In addition, a recent experimental work reported that the reflection from the transmission line can also the trigger sideband-type spectrum through filtering effects.¹² However, these external cavities mostly require huge circuit components, including phase shifters and mixtures, and centimeter-sized cables for signal reflection. These device extensions prevent downsizing of the whole oscillatory system and abandon the great potential in the nanoscale communication system of the STO.

In this paper, we numerically show a different scheme to induce multiple modes only by using a single radio-frequency source without any special device structures. We have conducted micromagnetic simulations to investigate the non-linear self-sustained dynamics for a single STO setup based on the spin Hall effect.^12–18 The large angle precession under weak perpendicular magnetic anisotropy results in periodical perturbations in oscillation trajectories. This leads to the multiple modes via the non-linear frequency and amplitude modulation (NFAM).⁸ Importantly, this type of phenomenon is triggered only by intrinsic magnetization dynamics of a single STO without requiring any external sources to assist the synchronization. Further downsizing is possible down to the size limit of the single STO nanostructure, possibly several tens of nanometers.¹⁹ 

A schematic image of the present model system is shown in Fig. 1. We assume that the single STO is driven by the spin Hall effect asserted throughout a heavy metal electrode with strong spin–orbit interaction (Fig. 1). We focus on the magnetization dynamics of the spatially averaged free layers with typical material parameters of CoFeB²⁰ for simplicity. The micromagnetic simulations are carried out by using the open-source micromagnetic code, the finite difference package MuMax3.²¹ The motions of the spins are calculated by solving the Landau–Lifshitz–Gilbert equation^22,23 with spin-transfer and spin Hall torque contributions,

where m denotes the normalized free-layer magnetization and M_sm = M with the saturation magnetization M_s(=1.41 × 10⁴ Oe). γ(=1.76 × 10¹¹ Hz/T) is the gyromagnetic ratio, and α(=0.012) is the Gilbert damping constant. H_eff is the effective field including the external magnetic field H; the magnetic anisotropy field $Han=2KuMsu⋅mu$⁠, with the first order uniaxial anisotropy constant K_u and a unit vector indicating the anisotropy direction u; and the Heisenberg exchange field $Hex=2μ0AMsΔm$⁠, with the exchange stiffness constant A(=2.84×10⁻¹¹ J/m). T_I counts the spin-polarized current contribution given by T_I = (γℏPI_MTJ/2eM_sV)m × (m × p), and T_SH is the spin Hall torque given by $TSH=γℏθSHJSH/2eMsdm×m×y^−βm×y^$⁠, where p is the spin polarizer vector, ℏ is the Dirac constant, e is the elementary charge, P(=0.57) is the spin polarization of CoFeB,^20,I_MTJ(=50 μA) is the perpendicular DC current passing through the STO, V and d are the volume and thickness of the free layer, θ_SH(=−0.12) is the spin Hall angle of Ta,²⁴ and J_SH = I/(w_HMd_HM) is the current density in the heavy metal electrode with w_HM and d_HM as its width and thickness. The complex behavior of the field-like spin Hall torque coefficient β²⁵ will be crucial for investigating the magnetization switching process²⁶ but should not be critical for discussion about excitation modes. Here, β is set to be β = 1 for simplicity.²⁷ 

The STOs in the present calculation have an elliptical shape with the major and minor axes of 160 nm and 80 nm, respectively, with the free-layer thickness of d = 1.3 nm, allocated on the heavy metal electrode with (w_HM, d_HM) = (1 μm, 7 nm). These values are based on the experimental work by Iwakiri et al.¹² The discretization cell size is set at 2 × 2 × 1.3 nm³. The in-plane external magnetic field H = 200 Oe is applied along the ϕ = 60° direction, and p is fixed to be parallel to it.

Figure 2(a) shows the time evolution of the z component magnetization m_z in a linear oscillation region (top panel: I = 0.91 mA) and in a non-linear oscillation region (bottom panel: I = 1.06 mA). The magnetization continuously oscillates with a single frequency in the linear oscillation region, which is typical of STO. In the non-linear oscillation region, on the other hand, the precession angle gradually increases as a function of time. This transient state continues about 120 ns, and then, magnetization dynamics results in a non-linear steady state in which a steady oscillation with a slow beating of precession angle starts to appear.

The top and bottom panels of Fig. 2(b) show the spectral densities obtained by the fast Fourier transform (FFT) of m_z in the linear and non-linear oscillation regions, respectively. While the spectral density in the linear region shows a single peak at 695 MHz, the spectral density in the non-linear exhibits multiple peaks at 115 MHz, 490 MHz, 600 MHz, and 715 MHz. In addition, two tiny sideband peaks are observed at 230 MHz and 375 MHz, as discussed in detail below.

The time evolutions of the first and second fundamental modes (600 MHz and 115 MHz) are plotted in the top and middle panels of Fig. 3(b), accompanied by the full non-linear oscillation track in the bottom panel. As guided by the gray areas, the beating frequency coincides with the second fundamental mode 115 MHz. In addition, frequencies of other four peaks in the spectra (230 MHz, 375 MHz, 490 MHz, and 715 MHz) are constructed by making difference and/or doubling of these two fundamental frequencies, i.e., 230 MHz = 115 MHz × 2, 375 MHz ≅ 600 MHz − 115 MHz × 2, 490 MHz ≅ 600 MHz − 115 MHz, and 715 MHz = 600 MHz + 115 MHz. These sideband modes can be provided from the general solutions of the non-linear universal auto-oscillator model²⁸ and have a strong analogy with the difference frequency generation and the second harmonic generation in non-linear optics.²⁹ 

Let us introduce the analytical approach developed by Consolo et al.³⁰ to describe the multiple mode generation. The output signal of the STO can be denoted as c(t) = A_c cos(2πf_ct) with the oscillation amplitude A_c and the base frequency f_c = 600 MHz in the universal model. We can take into account the phenomenological beating frequency f_m = 115 MHz by writing A_c → A_c cos(2πf_mt + φ) with the phase difference φ. Then, the entire output wave can be written in the form of c(t) = A_c(t)cos[θ(t)] using the time dependent amplitude [A_c(t)] and phase [θ(t)] components, which can be generally categorized as the combined NFAM signal.⁸ The peak positions of such a NFAM spectrum can be obtained as $f=fcI±lfm$⁠, where $fcI$ is the central frequency and l is the positive integer labeling sideband order. This NFAM scenario can reproduce all non-linear modes of the spectrum in Fig. 3 (a) by using the central frequency as $fcI=600 MHz$ or 115 MHz, and f = 600 ± 115l₁ MHz (375 MHz, 490 MHz, 715 MHz) and f = 115 ± 115l₂ MHz (230 MHz) with l₁, l₂ = 0, 1, 2. In these manners, these multiple modes are regarded to be equivalent up to the second order of sideband modes of NFAM.

We next discuss the origin of the two different fundamental modes under the non-linear processes. The three-dimensional trajectories of the linear (I = 0.91 mA) and non-linear oscillations (I = 1.06 mA) are plotted in Fig. 4(a). Owning to the weak perpendicular anisotropy K_u = 7.55 × 10⁵ J/m³, the magnetization dynamics is no longer free from the finite shape-induced in-plane anisotropy of the thin film [K_i, a red vector in Fig. 4(a)]. The precession axis is then strongly tilted from the perpendicular axis (K_p, a black vector).^31,32 An application of the in-plane external field H (a blue vector) also tilts the azimuthal angle of the precession axis from the x-axis. As a consequence, the magnetization oscillates around the intermediate position where its azimuthal and elevation angles of the precession axis are located at (θ, ϕ) ∼ (30°, 30°).

Once the large-angle precessions are excited at such an intermediate position, the non-linear trajectories will be affected by the complicated triaxial-type anisotropy energy profile induced by (K_i, K_p, H). This leads to the fluctuation of the precession angle as plotted by the red lines in Fig. 4(a), in which the non-linear oscillation shows different trajectories for every period of the fundamental mode at 600 MHz.

Figure 4(b) shows time evolutions of the precession axis of the free-layer magnetization at the linear and non-linear oscillations obtained by averaging the trajectories for one period of the fundamental modes. While the precession axis settles on the identical position during the linear oscillation, it draws the elliptical orbits in the non-linear oscillation with ∼8.70 ns (=115 MHz) in its average periodicity. This fluctuation of the precession axis triggers the second fundamental mode. It should be noted that the present results indicate that such multiple modes are purely inherent in the magnetization dynamics of a common cylinder-shaped STO nanopillar and are free from any external circuits demonstrated so far.^9,11

The conditions of the multiple modes are addressed by calculating spectral densities as functions of electric current I, external field H, and perpendicular anisotropy K_u in Figs. 5(a)–5(c). The electric current I dependence of spectral density is presented in Fig. 5(a). The spectra show a linear single-mode of 700 ± 5 MHz under I ≤ 0.94 mA followed by the transient region with the nonlinear red shift (the red dotted lines) and then enter into the non-linear multiple modes above I ≥ 1.00 mA. The widths and amplitudes of the peaks of the multiple modes are barely affected by I under I < 1.20 mA. Interestingly, this result has a strong analogy with the recent experimental observation of the sideband-type spectrum,¹² which implies the intrinsic relevance of the NFAM. Once the current exceeds I > 1.20 mA, the magnetization starts continuous switching processes.

The external in-plane field H dependence under I = 1.06 mA is introduced in Fig. 5(b). The clear locking behavior is observed under 160 Oe ≤ H < 220 Oe. The first fundamental mode (∼600 MHz) shows a small dependence on H, while the second fundamental mode (∼115 MHz) is barely affected by H. These locking behaviors are easily corrupted under too small or too large field H since they are attributed to a competition between the perpendicular and in-plane anisotropy energies. As a result, single peak spectra are observed at H < 160 Oe and H > 220 Oe.

The perpendicular anisotropy K_u dependence under I = 1.06 mA is shown in Fig. 5(c). Among the three modulation parameters (I, H, K_u), the locking condition for K_u is confirmed to lie especially in the narrow region, 7.52 × 10⁵ J/m³ ≤ K_u ≤ 7.58 × 10⁵ J/m³, and around 0.5% deviation of K_u would break the locking. Under the stronger K_u(>7.58 × 10⁵ J/m³), the precession axis has been barely affected by in-plane anisotropy K_i and applied field H; hence, the free-layer magnetization would precess in the way of the usual perpendicular STO. While under the weaker K_u(<7.50 × 10⁵ J/m³), the STO property would be close to that of the in-plane oscillator, and the frequency of the fundamental peak decreases down to the 600 MHz range. In other words, fine-tuning of K_u may help in finding out these NFAM conditions experimentally, such as the voltage-controlled magnetic anisotropy.³³ 

Finally, we introduce these non-linear effects on the output power of the STO. The current dependence of the output power P_o can be written as⁹ 

where var(m_x) is the variance of the average value of m_x, and R_T, R_MTJ, and ΔR denote the termination resistance, magnetic tunneling junction resistance, and magnetoresistance between the parallel and antiparallel status, respectively. We set reference values as R_T = 50 Ω, R_MTJ = 3850 Ω, and ΔR = 2100 Ω after the experimental report about the non-linear multiple mode nucleation in a single STO.¹² Figure 5(d) shows the current I dependence of the output power P. The transient region from the linear precession to the non-linear precession is guided by the red dotted lines, same as in Fig. 5(a). The output power P shows a gradual increase in the linear region (I < 0.94 mA), and a notable increase is observed once the NFAM is conducted. In the NFAM region (I > 1.00 mA), P shows a further increase and eventually reaches almost ten times enhancement compared to the linear output power. These results indicate that the intrinsic sideband modulation possesses the potential advantage not only to the downsizing but also to improve an active device power following the typical phase-locking phenomena.

In summary, we have conducted numerical studies to induce the multiple modes of the STO inherent in its magnetization dynamics. By using the free layer with the weak perpendicular magnetic anisotropy, NFAM by the beating mode leads to the difference frequency generation and the second harmonic generation, which are accompanied with a sharp increase in the output power. Since these non-linear processes require neither extrinsic radio frequency sources nor complicated device structures, they would be potentially usable at any design of the circuit. These results may open a new route for developing a new class of non-linear STO systems.