Modeling of magnetization precession in spin-torque nano-oscillators with a tilted polarizer

The magnetization dynamics triggered by the spin-transfer torque^1,2 holds great promise in the applications of the non-volatile magnetic random-access memory (MRAM)³ and nano-oscillators.^4,5 The spin-torque nano-oscillator (STNO) is a prospective nano-sized radio frequency (rf) oscillator, which is promising in the potential microwave generation and wireless communication applications.^6–10 However, some challenges need to be solved before STNOs can be put into practical application, which includes the limitation of output power, improving the rf oscillation quality, and increasing the frequency tunability. Various solutions have recently been proposed to overcome these problems, e.g., the utilization of perpendicular magnetic anisotropy materials as the fixed layer or free layer,^11–15 vortex oscillators,^16–22 wavy-torque spin-torque oscillators,²³ a unique device structure such as a somebrero-shape,^24,25 dual spin-polarizers,^26–28 and tilted-polarizer STNOs (TP-STNOs).^29–34 

In the TP-STNOs device, the magnetization of the fixed layer is tilted with respect to the film plane, which can be achieved by using materials with strong titled perpendicular magneto crystalline anisotropy.³⁵ In such devices, the tilted spin polarization has both an out-of-plane (OP) component and an in-plane (IP) component. The OP component of spin polarized current can drive the magnetization of free layer into a steady precession state under zero applied magnetic field. In our previous study, we found that the precession trajectory of the free-layer magnetization demonstrates three distinct dynamics including “N” state (a direct way to the final state), “S” state (a spiral trajectory to the final sate) and “L” state (periodic precession state).²⁹ It was found that the magnetization precession frequency in the L state is dependent on the material parameters. In this work, macrospin simulation is employed to study the phase diagram of the magnetization precession frequency as a function of the applied current and the tilt angle of the spin polarizer magnetization. The simulations indicate that the precession frequency also depends on the damping factor and saturation magnetizations μ₀M_s of the free layer. In order to interpret the dependence of the precession frequency on different material parameters, we develop an analytic model to elaborate the features of the precession frequency.

The model structure of the TP-STNO device is illustrated in Fig. 1(a). The magnetization of the spin polarizer is supposed to be in the x-z plane with an angle of β with respect to the x-axis. The magnetization dynamics of the free layer is described by the Landau-Lifshitz-Gilbert (LLG) equation including the spin-torque term:^1,36

where M_s is the saturation magnetization of the free layer, α is the Gilbert damping factor and γ is the gyromagnetic ratio. H_eff is the total effective field acting on the free layer. The third term represents the spin-transfer torque, where p is the unit vector whose direction is along the spin polarizer magnetization. The spin-torque factor a_J = ℏPJ/(2μ₀|e|M_st), in which ℏ, P,J, e, and t are the Planck constant, spin polarization, current density, electron charge and free layer thickness, respectively. Here we define the positive current as the electrons flowing from the spin polarizer to the free layer direction. In this study, the current-induced heating and multiple reflection effects are not included.^37,38

The orientation of spin-torques generated from OP and IP component of the tilted spin polarizer are sketched in Fig. 1(b). The free layer magnetization can experience a periodic precession state at a proper combination of current density and tilt angle of β.²⁹ As shown in Fig. 1(b), in the presence of positive spin-polarized current case, the spin-torque generated by the OP component of the spin polarizer forces the free layer magnetization out of plane to the +z direction while the spin-torque generated by the IP component of the polarizer drives the free layer to +x direction. Thus, the free layer magnetization is tilted and possesses a nonzero z-component of magnetization, which results in a demagnetizing field of ∼4πM_z. Consequently, a precession state of the free layer magnetization around the z-axis is excited. Fig. 2(a) shows a typical precession trajectory of the free layer magnetization excited at J = 8.0 × 10¹¹ A/m². In this simulation, the parameters are taken as γ = 2.21 × 10⁵ m/(As), μ₀M_s = 1 T, α = 0.01, and β = 10°. The free layer starts from an IP magnetization state along the +x direction. Under the influence of the spin-torque effect, the magnetization starts to rotate along +z axis with a spiral orbit motion. As shown in Fig. 2(b), at ∼2 ns it reaches to its stable periodic precession state with a polar angle θ of 48.7° (with respective to +z axis). The frequency of this stable magnetization precession is 18.6 GHz from the Fast Fourier Transform (FFT) analysis. Fig. 2(c) shows the corresponding frequency spectrum with a sharp peak of frequency.

In order to depict the detailed magnetic precession dynamics of the TP-STNO and the frequency dependence, we carried out a series of simulations at different current and tilt angle of spin polarizer. The results are summarized in Figs. 3(a)-3(f), showing the phase diagram of the precession frequency as a function of the applied current and tilt angle (β ≤ 30°) for different damping constant α and saturation magnetization. Note that, at a given tilted angle β, the precession frequency increases from sub-gigahertz to 26 gigahertz with the increasing current density [see Fig. 3(a), 3(b), 3(c)]. Similarly, at a given current the frequency also increases with the increased tilt angle β of the spin polarizer. The frequency region becomes narrow slightly with the increasing tilt angle β, which is in consistent with our previous micromagnetic simulation.²⁹ In addition, the frequency region slightly becomes wide when the damping factor α increases. The interesting thing is that the frequency range is almost no change. The simulations with different saturation magnetizations M_s show that the frequency region can be broadened by increasing the M_s, as shown in Figs. 3(d)-3(f). In contrast to the case of damping changing, one significant difference is that the frequency range obviously changes with the increasing μ₀M_s, e.g. f = 41 GHz of frequency can be achieved when μ₀M_s = 1.5 T, see Fig. 3(f). We attribute this change to the counterbalancing of the damping torque (proportional to M_s) with the spin-transfer torque (proportional to J/M_s).

When the polarizer angle is increased further (β ≥ 30°), the stable periodic precession state (i.e. L state) is still observed. But due to the increased OP-STT strength, the free layer is more easily to be switched to the +z direction at the same current. Therefore, the current parameter range to excite the stable periodic magnetization precession will reduce for the larger beta, but the qualitative feature of the magnetization precession is almost unchanged. Therefore, for simplicity, the cut-off for the polarizer angle is set to 30 degree in this study. In addition, the white area of phase diagrams corresponds to the parameters of N or S state, in which the free layer goes to a stable final state. For example, if the current is increased up to J = 17.7 × 10¹¹ A/m² at β = 10° in Fig. 3(b), the magnetization turns into a single spiral state and eventually switches to be a steady state with m_z ≈ + 1.

Figs. 3(a′)-3(f′) show the corresponding phase diagram of polar angle θ of the free layer magnetization precession. In each plot, the polar angle θ decreases with the increase of current density at a fixed tilt angle β, and also decreases with the β increasing at a given current, indicating that the free layer magnetization component M_z increases with the two factors. This is attributed to the enhanced OP component of STT with the increased current as well as the tilt angle β of the spin-polarizer. Consider that the precession frequency is proportional to the z component of magnetization,³⁹ f ∝ M_z, thus the tunable frequency can be easily achieved by changing either the intrinsic or extrinsic parameters of devices, as shown in Fig. 3. It is notice that the θ could vary from 6 to 88 degree (almost from the out-of-plane to the in-plane), meaning that the large range of angles with stable periodic magnetization oscillations can be excited in such STNO devices with a tilted spin-polarizer.

To gain deep insights into the frequency characteristics of the stable periodic precessions, an analytical model is developed. For the TP-STNO device, the effective field acting on the free layer is given by

Where H_k is the intrinsic in-plane anisotropy field, H_x and H_z are the stray fields generated from the IP and OP component of the polarizer, respectively. The demagnetizing field is −4πM_z.

By setting $p=cosβ e ˆ x +sinβ e ˆ z$ and inserting Eq. (2) into Eq. (1), one finds the following equation for the z-component of free-layer magnetization (M_z),

Taking $M= M ⊥ cos ( 2 π f ⋅ t ) e x + M ⊥ sin ( 2 π f ⋅ t ) e y + M z e z$⁠, here M_⊥ is the IP component and f is the precession frequency, Eq. (3) can be rewritten as:

Consider that the change of M_z is very small in a precession period [Fig. 2(b)] and by time-integrating both sides of Eq. (4) in a period of 2π, the precession frequency can be derived:

Equation (5) quantitatively describes the dependence of precession frequency f on different material parameters and clearly indicates that the precession frequency is linearly proportional to the spin-torque from the OP component of the polarizer only (i.e. f ∼ a_Jsinβ) and is irrelevant to the spin-torque of the IP component (a_Jcosβ). According to the equation (5), the precession frequency f increases with the increasing current density J and the tilted angle β of the polarizer. At a given frequency value of f, the current density J will decrease to a smaller value with the decrease of the M_s or the damping constant α, which certainly results in the phase diagram of precession frequency, see Fig. 3. At a given M_s, the damping torque is approximately proportional to damping factor α. However the spin-transfer torque is proportional to J. In order to maintain the balance between the T_damp and T_OP, the current density J also needs to increase n times with α increasing n times. Therefore, the frequency range is almost no change according to the Eq. (5). In contrast, at a given α, the damping torque is proportional to M_s while the spin-transfer torque is proportional to J/M_s. Thus the current density J needs to increase n² times when M_s increases n times. According to the Eq. (5), the frequency range have a corresponding growth with the increasing μ₀M_s, exhibiting that the frequency range of Fig. 3(d)-3(f) is 1.8-13.6 GHz, 1.6-27.5 GHz and 1.8-41.2 GHz, respectively. Furthermore, taking the same material parameters using in Fig. 2, Eq. (5) yields f ≈ 18.8 GHz, which is quite close to the simulation value f ≈ 18.6 GHz. Therefore, this analytic result is in perfect agreement with our simulations.

In summary, the magnetization precession excited by the spin-torque effect in a spin valve with a tilted spin polarizer is investigated systematically through macrospin simulations. We have obtained the phase diagram of the precession frequency as a function of the applied current and the intrinsic parameters of the device as well. The precession frequency dependence can be quantitatively described through the analytical analysis, showing the frequency increases with the increase of the current density J and/or the tilted angle β, and with the decrease of the damping constant and saturation magnetizations. The analytic result is in perfect agreement with our simulations.

This work is supported by the NSFC (Grant No. 11274241, 51471118, 51201059 and 51302157).