Shape anisotropy effects on spin-torque oscillators

Since the discovery of spin-transfer torque (STT) effect,^1–4 spin-torque oscillators (STOs) have been extensively studied due to their numerous potential applications including tunable microwave oscillators,^5–8 hard disk drive read head sensors,⁹ neuromorphic computing,^10–12 parallel microwave signal processors,¹³ and wireless communications.^14–16 STO devices utilize the spin torque generated by a spin-polarized current to drive persistent magnetization precession. Spin-torque oscillations have been observed in giant magnetoresistive (GMR) spin valves^17–21 and MgO-based magnetic tunnel junctions (MTJs).^22–25 The precession of the free layer magnetization leads to a microwave oscillatory component of the device resistance, thus an oscillatory component of the output power. Compared to traditional semiconductor-based oscillators, the nanoscale spin-torque oscillators have their unique advantages. They are small, frequency tunable, with a wide working temperature range, and are also compatible with complementary metal-oxide- semiconductor (CMOS) processes.²⁶ 

For most applications, the desired microwave signal should be with high output power, narrow linewidth, and tunable frequency.²⁷ A narrow linewidth is required for a high quality factor (Q factor) $Q=f0Δf$⁠, where f₀ is the peak frequency of the power spectrum, and Δf is the linewidth. As the STO size is scaled down, the thermal effect plays an important role in the linewidth broadening of the output signal.²⁸ Theoretically, the STO linewidth is mainly affected by the thermal phase noise. Temperature dependence studies of the STO linewidth show that the linewidth is determined by phase fluctuations and two-state switching.^17,29 The linewidth discussed later in this paper is about the two-state switching contribution of the thermal noise only. The thermal noise in STOs is directly related to the energy barrier of the magnetic layer. The STO linewidth can be expressed by the formula

where E_b is an effective activation barrier.²⁹ Experiments have demonstrated that the spectral linewidth depends on the temperature,³⁰ current,^22,31,32 and magnetic field angle.^33,34 The effect of shape anisotropy on the threshold current has been studied theoretically.³⁵ In this paper, we investigate the relationship between the shape anisotropy and the threshold current experimentally, which influences the linewidth of spin-torque oscillations excited in nanopillar MgO-based MTJ samples.

The in-plane MTJ thin film has the following stack structure: seed layer/PtMn (15)/Co₇₀Fe₃₀ (2.3)/Ru (0.85)/Co₄₀Fe₄₀B₂₀ (2.4)/MgO (0.82)/Co₂₀Fe₆₀B₂₀ (2)/capping layer, where the numbers in parentheses are the layer thicknesses in nanometers. The thin film was deposited by a sputtering system and was post-annealed at 300°C for 2 hours in a 1 T in-plane magnetic field. The wafer was patterned by e-beam lithography into elliptical nanopillars with the long axis along the annealing direction. The nanopillars presented in this paper are 45 × 170 nm², 50 × 150 nm², and 60 × 130 nm² in lateral size with the corresponding cross-sectional areas of 6008, 5890, and 6126 nm² for devices A, B, and C, respectively. Since the volumes of the devices are close, the difference in aspect ratio is the main contributor to the energy barrier difference among the devices. The shape anisotropy field can be expressed by H_k = (N_y − N_x)M_s, where N_x and N_y are the demagnetization factors of the Co₂₀Fe₆₀B₂₀ free layer along the x and y direction, respectively. The analytic demagnetization factors can be obtained as a series expansion in the squat cylinder limit.^36,37 Here, we assume M_s = 1200 emu/cm³, and the anisotropy field of devices A, B, and C is H_kA = 1132 Oe, H_kB = 1033 Oe, and H_kC = 903 Oe, respectively. Figure 1(a) shows the magnetoresistance minor loops of the three MTJ samples. The resistance-area product (RA) is 3.5 Ω·m². The tunneling magnetoresistance (TMR) ratio is 109% for all three devices. Small loop shifts of 55 Oe, 30 Oe, and 60 Oe are observed for devices A-C due to uncompensated magnetostatic coupling between layers. The room temperature coercivities of devices A, B, and C are 170 Oe, 135 Oe, and 60 Oe, respectively. The coercivity trend of the three devices indicates that the energy barrier is high for the device with a large aspect ratio. Figure 1(b) presents the STO measurement setup with the sign convention for the applied current and magnetic field, which was used in other reports before.^38,39 The voltage signal was transmitted through a microwave probe, with the DC component blocked by a bias tee. The AC component was amplified to +30 dB by a low noise microwave amplifier and was monitored on a real-time oscilloscope. The amplification and the background noise have been subtracted from all the data, where the background noise was obtained at I_dc = 0 in the same field.

The thermal stability factor is proportional to the energy barrier, which is defined by $Δ=EkBT$⁠, where k_B is the Boltzmann constant, and T is the temperature. Therefore, in order to compare the energy barrier of the three devices, we obtained the thermal stability factors through the spin-transfer torque switching probability experiments.^40,41 In our experiment, the switching probability was measured by switching from the antiparallel state to the parallel state. The measurement process was started by setting the MTJ to the antiparallel state by a large magnetic field. Next, a small bias field was applied to compensate for the offset field of the free layer. By varying the pulse width and amplitude, the MTJ switching probability was obtained. In the experiment, 100 trials were carried out to calculate the switching probability for each pulse amplitude and pulse width. After each trial, the final resistance state of the MTJ was measured to estimate the switching probability.

In Figure 2(a), we plot the switching probability of device C as a function of the pulse amplitude at different pulse widths from 3 μs up to 1 ms. Taking the impedance mismatch into consideration, we define the pulse amplitude by 2VZ/(Z + Z₀), where V is the output voltage pulse of the pulse generator, Z is the impedance of the MTJ, and Z₀ is the impedance of the circuit.⁴⁰ Figure 2(a) shows that the increase of the pulse voltage reduces the switching time, which is consistent with the Macrospin model.^40,42 At long voltage pulses (>10 ns), the free layer is in the thermally activated spin-transfer torque switching regime. The pulse amplitude is described by $VC=VC01−1Δlntpτ0$⁠, where $VC0$ is the intrinsic switching voltage, t_p is the pulse width, and τ₀ is the attempt time, usually set to be 1 ns.⁴¹ This equation is strictly applicable when the switching probability is 63%. Therefore, we extract the pulse amplitudes that correspond to 63% of the switching probability and plot them in Figure 2(b). From the linear fits of this expression to the experimental results, we obtain that Δ = 39 and $VC0$ = 0.445 V for device A, Δ = 34 and $VC0$ = 0.374 V for device B, Δ = 30 and $VC0$ = 0.367 V for device C. Thus, the corresponding energy barriers of devices A, B, and C are 39k_BT, 34k_BT, 30k_BT, which demonstrates that among these three devices, the energy barrier of a high-aspect-ratio device is high.

Before comparing the three devices, we first study the STO power spectra of a single device. Figure 3(a) and (b) report quantitative data on the frequency and linewidth of device B as a function of current at 210, 250, and 290 Oe. The linewidth is determined by Lorentzian fits to the power spectra. In Figure 3(a), the frequency redshift can be divided into two regimes. At 210 Oe, the frequency drops slowly with the current in the low current regime (I < 0.32 mA), while it decreases fast in the high current regime (I > 0.32 mA), which is similar to the result in Ref. 31. However, at 250 and 290 Oe, the frequency remains constant at low current. The redshift of frequency is characteristic of in-plane magnetization precession in STOs.^5,43–45 The low current regime is mainly caused by thermally excited ferromagnetic resonance noise, and no frequency variation is expected. Here, the observed small redshift at 210 Oe may be ascribed to current dependent torques induced by the oersted field or field-like torque.³¹ In the high current regime, the precession cone angle becomes large with the increase of current, leading to the reduction of the oscillation frequency. The STO linewidth trend can also be divided into two regimes. At low current, the linewidth is almost flat at 210 Oe, while it decreases with the current at 250 and 290 Oe. At high current, the linewidth increases with current. This phenomenon may be explained by the following: at low current, the spin-transfer torque compensates for the damping torque gradually. The nonlinearities are ignored, and the linear reduction of linewidth is given by Δf = Γ_g − σ/2πI, where Γ_g ≈ αγμ₀M_eff/2π, α is the Gilbert damping constant, γ is the gyromagnetic ratio, μ₀M_eff is the effective magnetization, and σ is proportional to the spin-polarization efficiency.³¹ Above the threshold current, the magnetization undergoes a nonlinear quasi-uniform precession. The peak broadening can be explained by the strong impact of the nonlinearity associated with the spin-transfer torque.^46,47 The STO linewidth not only changes with the current but also with the field. Below the threshold current, the linewidth gets broad with increasing magnetic field, while above the threshold current, it becomes narrow with increasing field. This is because, below the threshold current, the magnetization is in thermally excited small-amplitude dynamics regime, where thermal fluctuations are crucial for the linewidth broadening and strongly influence the dynamics. In this ferromagnetic resonance regime, with the increase of the frequency, the linewidth becomes broad.⁴⁸ Based on Kittel’s formula, the frequency increases with the magnetic field, so below the threshold current, the linewidth increases with the magnetic field as well. Above the threshold current, the linewidth drop is due to the reduction of the magnetization precession cone angle at high bias fields. A small precession cone angle brings the magnetization oscillation from a nonuniform precession to a uniform precession mode. This result agrees with the simulation and theoretical results in Refs. 28 and 49.

To study the shape anisotropy effects on spin-torque oscillators, we compare the dependence of the oscillation linewidth and corresponding Q factor of devices A, B, and C on the current in Figure 4. Due to the different hysteresis loop shifts of the three devices, small bias fields of 55, 30, and 60 Oe are applied to devices A, B, and C respectively to compensate for the magnetostatic coupling between the free layer and fixed layer. Besides, an additional field of 210 Oe is applied to all the MTJs, so that each MTJ device is in the same bias field relative to the center of its hysteresis loop. The corresponding current density applied to each device at 0.4 mA is 7 × 10⁶ A/cm². Figure 4(a) shows the linewidth variation of devices A, B, and C with the current. Similar to Figure 3(b), the linewidth of each device shows two different dynamical regimes: a linewidth decrease at low current, followed by an increase at high current. The dashed lines are linear fits to the experimental results in the regime below the threshold currents I_th. From the extrapolation of the linear fits to the zero linewidth, we find the threshold currents of devices A, B, and C to be I_th ≈ 0.34, 0.29, and 0.22 mA, respectively. The results demonstrate that the increase in the energy barrier enhances the STO threshold current. This is because, based on the Macrospin model, the threshold current is linear with the anisotropy field,⁴² which is proportional to the energy barrier.⁵⁰ Due to different threshold currents, the linewidth is different in MTJs with different shape anisotropy. For the device with a small shape anisotropy, it enters the oscillation state at a smaller current. Therefore, to obtain the same linewidth, the desired current is small. In addition, above the threshold current of device A, the increase of the energy barrier narrows the STO linewidth. This can be tentatively understood by the different precession cone angles induced by different damping torques within the three devices. As the anisotropy field becomes large, the effective field is enhanced as well, and so is the amplitude of the damping torque, which leads to a small precession cone angle.^1,51 During this process, the magnetization turns from a nonuniform precession to a uniform precession, reflected in the linewidth drop. We also compare the linewidth of devices A, B, and C as a function of supercriticality (ζ = I/I_th) in the inset of Figure 4(a). The shape anisotropy effect is not very obvious. In Figure 4(b), we show the corresponding Q factors of the three devices. Upon increasing the applied current, all the devices undergo an increase in the Q factor, followed by a decrease. The critical current for the maximum Q factor is enhanced with increasing aspect ratio and energy barrier. Our results also show that when all the devices are in the quasi-uniform precession state, due to the enhanced critical current, the Q factor of spin-torque oscillators can be improved with an increasing energy barrier. It is important to note that the effective activation barrier for switching between dynamical states in Eq. (1) is not the static anisotropy barrier. If we take device A as an example, E_b ≈ 2k_BT, which is much smaller than the static anisotropy barrier. Our result agrees with the result in Ref. 29.

In summary, the shape anisotropy effects on the threshold current of spin-torque oscillators have been studied experimentally. The threshold current is large in devices with a large aspect ratio. Due to the large threshold current, the corresponding linewidth is dramatically reduced. Meanwhile, the quality factor is increased. Our results demonstrate that a narrow linewidth and a high quality factor could be achieved by tailoring the device anisotropy for HDD reader application.

This work was supported by Seagate Technology and National Science Foundation through the MRSEC Program under Award No. DMR-0819885. We thank Dr. Wonjoon Jung, Dr. Jae-Young Yi, Dr. Eric W. Singleton, and Dr. Mark Kief for useful discussions and suggestions. We gratefully acknowledge Professor Randall H. Victora and Dr. Tao Qu for their help on the micromagnetic simulations, which are not included in this manuscript.