Influence of size and shape on key performance metrics in spin-torque oscillators

Spin-torque oscillators (STOs) are nanoscale, ferromagnetic devices capable of generating self-sustained, high frequency signals, which are caused by stable magnetization precessions induced by a spin polarized current via the spin transfer torque (STT) effect.^1–6 Implementing STOs in modern technologies is challenging since the output power and quality factors of their signals are several orders of magnitude lower than needed for modern applications.^1–6 Nevertheless, STOs exhibit many unique and novel properties such as nonlinearity, frequency tunability and synchronization⁷ which give them exciting prospects for next generation computation, communication, and sensor technologies.

Experimental work performed over the last decade has demonstrated that the frequency in STOs can be tuned over several GHz using external DC biases such as a magnetic field and an electrical current.⁸ This feature has led to various studies which propose STOs as the working principal in wireless on-chip and chip-to-chip communication technologies^9,10 as well as new sensor technologies such as STO-based read-head sensors in high density magnetic recording arrays^11,12 and bio detection systems.¹³ Additionally, STO frequencies can be synchronized to an external source, which includes either an RF field¹⁴ or an RF current through injection locking¹⁵ or to other STOs through mutual coupling. STO coupling can be done via spin wave propagation,¹⁶ dipole interactions,¹⁷ or electrical coupling, which is caused by self-modulation of the current through each STO.¹⁸ Not only is this feature is a promising solution for enhancing output power and quality of STO signals,^15–18 but it has also made STOs a promising solution in a novel next generation computing paradigm where large scale oscillator arrays mimic neural activities for bio-inspired functions.^19–22 

There are numerous metrics that can be used to quantify the performance of an STO, many of which are independent on one another. Determining which of the metrics is the most important is dependent on the desired application which is why it is important to understand how the intrinsic properties of the device influence each of these performance metrics so that STO designs can be optimized for the desired function. Simulations in OOMMF have shown that when the STO size decreases from 40 x 40 nm² to 10 x 10 nm², then the linewidth increased approximately three-fold and the frequency is reduced by almost 0.8 GHz.²³ X. Chao et al.²⁴ studied the influence of shape anisotropy on the STOs signal quality and demonstrated that the quality factor of the signals generated by STOs improved ∼1.5X as the STOs coercivity increased from 60 Oe to 170 Oe. However, it is not clear if the STOs signal quality has a stronger dependence on STO size or shape anisotropy as well as the effects that these two parameters have on other STO performance metrics.

In this work, we study spin-torque oscillations generated from 20 MgO-based magnetic tunnel junctions (MTJs) and their dependence on both device size and shape. We characterized the output signals generated from these MTJs by 12 different performance metrics which include output power, precession frequency, linewidth, quality factor, the bias field and current required to optimize STO performance, and the change in frequency with both bias field and bias current. The relative strength of each of these metrics was quantified with respect to STO size and shape using correlation coefficients. By studying all aspects of the STOs performance, our study provides a basis for application-specific optimization of future STO designs.

There were 20 MTJs tested with varying sizes and aspect ratios (long-axis/short-axis lengths). For the complete stack structure of the MTJs and descriptions of each device tested, see supplementary material, note 1 and supplementary material, Table I. However, some of our MTJs had similar nominal dimensions but had significantly different field switching behavior, which illustrates that the actual device dimensions may have varied from their nominal dimensions. To compensate for these variations in our analysis, we represented the STOs size and shape by their measured P-state resistance (R_P) and the coercivity (H_C) respectively, which were measured from field switching (R-H) hysteresis plots (see supplementary material, note 2 and supplementary material, figure 1).

Spin-torque oscillations were generated through the application of an applied bias field (H_bias) and a DC current bias (I_bias), which favor opposing states. The output RF signals were transmitted through a microwave probe which were isolated from the DC component using a bias tee. These signals were amplified by +27 dB then measured using a Tektronix DPO 72004C mixed signal oscilloscope with a sampling rate of 50 Gs/second. The STO waveforms acquired were analyzed using power spectral density (PSD) plots, an example of which is shown in Figure 1a. For each data set, we found precession frequency (f_p), and linewidth (Δf) by fitting the first harmonic peak to a Lorentzian curve, as illustrated in the example in Figure 1b. From these measurements, we calculated the quality factor (Q_f) of the signal, which is defined as f_p/Δf. Lastly, output power (P_out) was also be obtained by integrating the PSD curve through the entire frequency bandwidth.

In addition to P_out, f_p, Δf, and Q_f, we also investigated the influence of R_P and H_C on the H_bias and I_bias values required to generate the precession signals for each STO. Note that most of the STOs tested had an intrinsic stay field (H_stray), which is represented as the offset in the R-H hysteresis curve along the x-axis, an example of which is shown in Figure 1c. Note that H_stray = -35 Oe in Figure 1c, however, each STO had a different H_stray. For valid comparison between STOs, we defined H_bias as H_bias = H_appl + H_stray, where H_appl is the applied field.

In our analysis, we compared each of these performance metrics between all STOs tested in order to determine their dependence on device size and aspect ratio. Note that each STO tested contained between 30 and 50 data sets, each with different bias conditions, so to make valid comparisons between STOs, we defined the top 8 data sets for each STO and calculated averages for P_out, f_p, Δf, Q_f, H_bias and I_bias measurements among these sets. This way, we are only comparing these performance metrics in the sets with optimum performance. A challenge faced with this approach is that it is not clear how to define the ‘best’ sets because the criteria for ‘best’ is likely to be application dependent. To avoid this problem, we analyzed our results using two independent methods of defining the top data sets: 1) Sets ranked by P_out and 2) sets ranked by Q_f. To analyze the relation of each metric with STO size and shape, we plotted P_out, f_p, Δf, Q_f, H_bias and I_bias with R_P and H_C separately. The magnitude of the dependence of R_P and H_C on each performance metric was quantified using their linear correlation coefficients (ρ). Note that it is unlikely that all seven metrics studied actually have linear relationships with R_P and H_C, so ρ simply serves as a factor to quantify the relative dependencies on R_P and H_C but does not necessarily indicate linear relations. For further details regarding our error analysis of ρ as well as all plots used to calculate ρ values, see supplementary material, note 3 and supplementary material, Figures 2–4.

Figures 2a–b show the linear correlation coefficients (ρ) for P_out, f_p, Δf, Q_f, H_bias and I_bias with R_P and H_C, where data sets are ranked by P_out and Q_f in Figures 2a and 2b, respectively. This result shows that increasing R_P causes Δf to increase, has no influence on f_p, and thus causes Q_f to decrease at both maximum P_out and maximum Q_f. Our data also shows that increasing H_C causes Δf to decrease and f_p to increases, and therefore, causes Q_f to increase. The dependence of Δf on R_P is the result of an increase in phase noise due to thermal fluctuations and is a predicted trend based on the simulation results in Ref. 23. Additionally, the decrease in Δf and the increase in Q_f with H_C confirms the experiment results presented in Ref. 24 since a larger H_C indicates a larger shape anisotropy. The key observation in our results is that both Δf and Q_f had a much stronger correlation with H_C than with R_P, as seen from the relative magnitudes of ρ. This suggests that decreases in signal quality due to reduction in STO size can be easily mitigated if the aspect ratio is designed to maximize device coercivity.

Our analysis also shows that increasing R_P causes decreases in H_bias and I_bias at both maximum P_out and Q_f, as seen in Figures 2a and 2b, which implies a reduction in energy consumption when operating at maximum P_out and Q_f. On the other hand, these figures show that increasing H_C causes H_bias and I_bias to increase. Again, these results are not surprising since a larger coercivity indicates larger thermal stability. However, the important findings from our data are revealed through the correlation coefficients, which show that I_bias has a stronger correlation with R_P than with H_C, meaning that I_bias will likely decreases as the STO size decreases, regardless of the devices coercivity. Alternatively, H_bias has a slightly stronger correlation with H_C than with R_P, however, this is not necessarily a detrimental effect since larger H_bias does not necessarily mean larger applied field. Recall that H_bias also considered H_stray, which means that increases in H_bias do not need to be compensated by R_P, but rather by a stray field with the correct orientation. In fact, several of the STOs in our study generated precession signals at zero-applied fields (see supplementary material, note 4).

The correlation between R_P and P_out is quite insignificant, however H_C is noticeably correlated with reductions P_out. The cause of this effect is most likely due to device failure in our experiment and does not represent the influence of H_C on P_out. Since STOs with larger H_C require I_bias, they are more susceptible to device failure caused by breakdown in MgO tunneling barrier. In our experiment, many of our devices failed at I_bias ≈ 300 – 350 μA. This was not an issue for STOs with R_P ≥ 3kΩ and/or H_C ≤ 20 Oe since maximum P_out was achieved at bias currents well below 300 μA. However, those devices with H_C > 50 Oe may have failed at I_bias below its maximum P_out capability. Since it is questionable if maximum P_out was achieved in STOs with H_C > 50 Oe, any relation observed for H_C versus P_out are not conclusive. However, from a practical point of view, these correlations should not be disregarded since they illustrate that device failure is another factor that should be considered when increasing H_C.

The overall trends observed in Figures 2a and 2b are in good agreement, however, Figure 2c suggests that there are some noticeable discrepancies. Figure 2c shows the correlations in difference between the H_bias, I_bias, P_out, and Q_f values between maximum P_out and Q_f (represented as ΔH_bias, ΔI_bias, ΔP_out, and ΔQ_f respectively) with R_P and H_C (see supplementary material, note 5 and supplementary material, Figure 6). This plot shows that R_P causes both ΔH_bias and ΔI_bias to decrease and H_C causes both ΔH_bias and ΔI_bias to increase. The ρ values displayed in Figure 2c shows that ΔH_bias has a much stronger correlation with H_C than with R_P, whereas ΔI_bias has a stronger correlation with R_P. However, the correlation coefficients for ΔI_bias with both R_P and H_C are too small to be conclusive on their relative influence.

The data in Figure 2c shows that ΔQ_f decreases with R_P but has a stronger correlation to increases with H_C. Recall that increasing H_C leads to improved maximum Q_f; however, the correlation between ΔQ_f and H_C indicates that some signal quality will be sacrificed when operating maximum P_out for STOs with high H_C. It should be noted that H_C still has a relatively strong, positive correlation with Q_f at maximum P_out (ρ > 0.6), which suggests that STOs with high H_C will still produce signals with higher Q_f values at maximum P_out than STOs with lower H_C, despite the increase in ΔQ_f. The correlation coefficients for ΔP_out suggest that ΔP_out does not have a strong dependence on either R_P or H_C. The lack of dependence of ΔP_out on R_P is an important feature in terms of device size scaling since it suggests that reducing STO size will not cause further reductions in P_out when operating at maximum Q_f. Our data indicates that ΔP_out also has a weak correlation H_C, however, the actual dependence on H_C may be misleading since we suspect that maximum P_out was not achieved due to device failure for STOs with H_C > 50 Oe.

The final two STO performance metrics investigated are the frequency slopes with H_bias and I_bias (df_p/dH_bias and df_p/dI_bias, respectively). Note that large df_p/dH_bias is a crucial feature for STO read-head sensors and bio-sensors, as well as for components in neural oscillating networks, where STOs interact through dipole interactions.²¹ Figure 3a shows that R_P has no significant correlation with df_p/dI_bias while Figure 3b shows that H_C causes df_p/dI_bias to decrease with a noticeable correlation. Figures 3c–d show that R_P caused df_p/dH_bias to increase and H_C caused df_p/dH_bias to decrease with much stronger correlations with both R_P and H_C than df_p/dI_bias had. The correlation coefficients shown on these figures indicate that df_p/dH_bias has a much stronger correlation with H_C than with R_P.

An overview of the influence of R_P and H_C on a full spectrum of STO performance metrics based on our findings is shown in Table I. In this table, the effects of increasing R_P are listed in the “Effects of decreasing STO area” column and the effects of increasing H_C are listed in the “Effects of increasing STO aspect ratio” column. Here we list all of the key findings from our analysis on the influences of R_P and H_C and categorize them as either advantages, neutral effects, avoidable disadvantages, or unavoidable disadvantages. The difference between avoidable and unavoidable disadvantages is based on the relative correlation coefficients calculated between R_P and H_C for each performance metric studied.

In summary, we analyzed spin-torque oscillations generated from 20 MTJs with shape magnetic anisotropy. We then studied the influence device size and shape on multiple key STO performance metrics. Our results showed that there were multiple advantages as well as disadvantages of reducing the STO’s size and increasing its aspect ratio. This means that changing either size or shape may improve one aspect of the STO’s performance, however, this change will be accompanied by a reduction in another aspect of its performance. While the explanation for some of the results presented are unclear at this point, our results still demonstrate all of the trade-offs in STO performances with as well as illustrating the relative magnitude of their effects between two key device parameters. STOs are promising solutions in a variety of novel applications and our analysis could serve as a guide in designing STOs to optimize the performance metric most important for the desired functions.

See supplementary material for device fabrication methods and stack structure, a list of key device properties for our analysis, field switching hysteresis plots for each device tested, and data used to calculate all correlation coefficients and their uncertainties.

This work is supported in part by Seagate Technology and CAPSL, an SRC program sponsored by the NSF through 1739635. Portions of this work were conducted in the Minnesota Nano Center, which is supported by the National Science Foundation through the National Nano Coordinated Infrastructure Network, award number ECCS-2025124. The authors thank the useful discussion with Dr. Pavol Krivosik and Dr. Mark Kief from Seagate Technology.

The data that support the findings of this study are available from the corresponding author upon reasonable request.