The reduction in phase noise in electronic systems is of utmost importance in modern communication and signal processing applications and requires an understanding of the underlying physical processes. Here, we systematically study the phase noise in mutually synchronized chains of nano-constriction spin Hall nano-oscillators (SHNOs). We find that longer chains have improved phase noise figures at low offset frequencies (1/f noise), where chains of two and ten mutually synchronized SHNOs have 2.8 and 6.2 dB lower phase noise than single SHNOs. This is close to the theoretical values of 3 and 10 dB, and the deviation is ascribed to process variations between nano-constrictions. However, at higher offset frequencies (thermal noise), the phase noise unexpectedly increases with chain length, which we ascribe to process variations, a higher operating temperature in the long chains at the same drive current and phase delays in the coupling between nano-constrictions.
Spin transfer and spin–orbit torque provide a means to drive nanomagnetic systems into current tunable high-frequency precession.1–3 The resulting microwave voltage signal can be used for communication applications4–8 and spectral analysis,9,10 where the small footprint, ready integration with CMOS technology, and wide frequency tunability make these oscillators particularly interesting. While spin-torque nano-oscillators (STNOs), comprising ferromagnetic/non-magnetic/ferromagnetic structures, require a somewhat complex fabrication process due to the current flowing out-of-plane, spin–orbit torque-driven spin Hall nano-oscillators (SHNOs) utilize in-plane currents in simple ferromagnetic/heavy metal bilayer systems,11–20 where heavy metals (e.g., Pt,21 Ta,22,23 and W15,24,25) produce pure spin currents through the spin Hall effect.26 The simple geometry and in-plane current flow allow ease of fabrication, direct optical access, and the ability to synchronize multiple oscillators in chains and two-dimensional arrays,16,27,28 making such SHNOs promising candidates for emerging spintronic applications including Ising machines29–31 and neuromorphic computing.32–35 For communication applications, the phase noise plays a crucial role, as it directly determines the performance of the system. To evaluate the potential of nano-oscillators for conventional signal processing applications, it is, hence, essential to characterize their phase noise performance,36–42 understand its physical origin, and suggest methods43 for its improvement.
Here, we perform a comprehensive analysis of phase noise in single nanoconstriction (NC) SHNOs as well as short (two NCs) and longer (ten NCs) chains of mutually synchronized NC SHNOs and demonstrate that it can be significantly reduced compared to single SHNOs. We find that in the case of two NC SHNOs, the mutual synchronization leads to an improvement of 2.8 dB in the flicker frequency phase noise, which is very close to the theoretical prediction of 3 dB. In the longer chain of 10 NCs, the noise improves by 6.2 dB, which is substantial but further (3.8 dB) from the theoretical expectation of 10 dB. We argue that this deviation originates from process variations between individual NCs, since the theoretical value assumes identical intrinsic frequencies of all oscillators. Somewhat unexpectedly, the white (thermal) frequency phase noise at higher noise frequencies is found to increase with chain length, being 2.1 dB worse for two NCs and 3.1 dB worse for ten NCs, compared to single SHNOs. In addition to process variations, the longer chains also operate at a higher temperature, due to the higher power dissipation, which may further increase the phase noise, in particular in the thermal region. As the measured linewidth improves substantially with chain length, we conclude that it is governed primarily by the noise.
Single NC SHNOs and chains were fabricated from the DC/RF magnetron sputtered W(5 nm)/NiFe(5 nm)/Al2O3(4 nm) stacks. The large spin Hall angle of W ( 0.44) reduces the threshold current,15,25 and the anisotropic magnetoresistance (AMR) of NiFe (0.65%) provides a reasonable output power.21,44 The devices were patterned into 150 nm NCs with 200 nm center-to-center separation (in chains) using e-beam lithography, followed by Ar-ion etching (for details, see, e.g., Ref. 45). Figure 1(a) shows a schematic of a ten NC chain.
(a) Schematic of a SHNO chain with ten nano-constrictions in series. (b) The phase noise measurement setup, where LO is a local oscillator tuned around 18 GHz to downconvert the SHNO signal to 10 MHz. The mixer is a ZMDB-44H-K+ double-balanced mixer, and the LFP is a lumped-element lowpass filter with a cutoff frequency of 30 MHz and stop band attenuation of 40 dB. The sampling rate of the oscilloscope is 50 MS/s.
(a) Schematic of a SHNO chain with ten nano-constrictions in series. (b) The phase noise measurement setup, where LO is a local oscillator tuned around 18 GHz to downconvert the SHNO signal to 10 MHz. The mixer is a ZMDB-44H-K+ double-balanced mixer, and the LFP is a lumped-element lowpass filter with a cutoff frequency of 30 MHz and stop band attenuation of 40 dB. The sampling rate of the oscilloscope is 50 MS/s.
Phase noise measurements were performed at fixed current and magnetic field and analyzed using a Hilbert transform technique.8 The analysis of close-in phase noise at low offset frequencies of sub-Hz range requires that the experimental time traces are accumulated over seconds time scales, which would require terabytes of data to be processed with a direct signal sampling at a 40 GS/s rate. In order to reduce the processed amount of data for such a long time series, we performed SHNO signal down-sampling using a frequency mixer, as shown in Fig. 1(b). Prior to mixing the SHNO signal with the local oscillator (LO) signal, the SHNO signal gets amplified to improve the efficiency of frequency downconversion. Amplification does not affect the instantaneous phase dynamics, and no corrections are required. The NC SHNO signal is downconverted to a 10 MHz intermediate frequency by adjusting the LO frequency and captured with a real-time oscilloscope at a low sampling rate of 50 MS/s. The frequency downconversion using a mixer has an important advantage as it performs a parallel shift of the signal in the frequency domain down to the intermediate frequency, i.e., the instantaneous phase dynamics is preserved.46
We process the captured signal in several steps. First, the captured SHNOs signal gets filtered with a finite impulse response (FIR) digital bandpass filter with a central frequency of 10 MHz, bandwidth of 12 MHz, in-band ripple of 0.1 dB, and stop band attenuation of 60 dB to improve the signal-to-noise ratio (SNR) so that the amplitude of the SHNO signal is sufficiently higher (>3 dB) than the RMS amplitude of the thermal noise floor. It is important to emphasize that the use of a bandpass filter only affects the phase noise data above 6 MHz, which is half of the 12 MHz bandwidth of the used bandpass filter. The small 0.1 dB in-band ripple of the used bandpass filter has almost no effect on the phase noise data below 6 MHz offset frequency. Then, the instantaneous phase gets extracted with a Hilbert transform47–49 of signal time traces. At the next step, the instantaneous phase signal is detrended. Next, the power spectral density (PSD) is calculated from the detrended instantaneous phase signal using a FFT to obtain the spectral density of phase fluctuations . We present phase noise data as the spectral density of phase fluctuations expressed in the logarithmic scale dB rad2/Hz as it is consistent with the International System of Units (SI).50
The free-running auto-oscillations with varying DC (IDC) are shown in Figs. 2(a), 2(b), and 2(c) for single (1 NC), two (2 NC), and ten (10 NC) mutually synchronized SHNOs, respectively. Figure 2(d) summarizes their operating frequency, where it can be observed that the mutual synchronization of SHNOs leads to higher frequency tunability. This could be understood as an absolute increase in their magneto-dynamical region. Figures 2(e) and 2(f) show the linewidth and integrated output power for the oscillator chains. It is clear from the observed parameters that mutual synchronization leads to a larger output power and lower linewidth for a larger number of synchronized oscillators in a chain, consistent with earlier work.27,28
Free running properties of single and mutually synchronized 2 NC and 10 NC SHNO chains. Power spectral density (PSD) of the auto-oscillation for (a) single NC, (b) 2 NC, and (c) 10 NC, respectively. Extracted (d) auto-oscillation frequency, (e) linewidth, and (f) integrated power. The dashed yellow line represents the current used during phase noise measurements.
Free running properties of single and mutually synchronized 2 NC and 10 NC SHNO chains. Power spectral density (PSD) of the auto-oscillation for (a) single NC, (b) 2 NC, and (c) 10 NC, respectively. Extracted (d) auto-oscillation frequency, (e) linewidth, and (f) integrated power. The dashed yellow line represents the current used during phase noise measurements.
The results of phase noise measurement are presented in Figs. 3(a)–3(e) for a single and mutually synchronized 2 NC and 10 NC oscillators in a chain. Phase noise measurements are performed at IDC = 2.53, 2.53, and 2.35 mA [also indicated by the yellow dashed line in Figs. 2(a)–2(c)] for single and mutually synchronized 2 NC and 10 NC oscillators, respectively. As can be seen from Fig. 3(a), all devices demonstrate regions with and white (thermal) frequency phase noise. Interestingly, in our SHNOs, the phase noise corner appears at a much lower offset frequency of 50 kHz compared to the GHz frequencies for magnetic tunnel junction (MTJ) STNOs.40 A lower value of the corner leads to an improved linewidth at laboratory time scales as the noise has a steep slope and a much higher contribution to the integrated power of phase noise.
Phase noise as a spectral density of phase fluctuations for a single and mutually synchronized 2 NC and 10 NC SHNOs in a chain. The dashed vertical line represents the -corner frequency of 50 kHz and separates regions with flicker frequency noise and white frequency phase noise. The steep reduction in phase noise above 6 MHz is associated with the applied bandpass filter used to improve SNR.
Phase noise as a spectral density of phase fluctuations for a single and mutually synchronized 2 NC and 10 NC SHNOs in a chain. The dashed vertical line represents the -corner frequency of 50 kHz and separates regions with flicker frequency noise and white frequency phase noise. The steep reduction in phase noise above 6 MHz is associated with the applied bandpass filter used to improve SNR.
A single SHNO exhibits a phase noise of 0, −17, and −67 dB rad2/Hz at the offset frequencies 100 Hz, 10 kHz, and 1 MHz, respectively. In Fig. 3(b), it can be seen that two mutually synchronized SHNOs demonstrate a 2.8 dB improvement in the region, which is in good agreement with the theoretical expectation of 3 dB improvement for each doubling in the number of synchronized identical oscillators.51–53 In the case of 10 NC SHNOs, we expect to see a 10 dB improvement, but the experiment only showed a reduction of 6.2 dB. This may be attributed to process variations in the NC width, which lead to variations in the intrinsic frequency of the nano-constrictions in the chain. Process variations naturally become more noticeable in longer chains as the probability to find N identical oscillators decreases rapidly with N. Another factor could be attributed to the geometry of the chain and its associated thermal effects. A chain of two identical nano-constrictions will retain the same zero difference in their relative frequencies even when the temperature and its gradient increase. However, in the case of more than two coupled NC-SHNOs, the inner and outer NCs will heat up differently, leading to a varying intrinsic frequency as a function of position in the chain.
Unexpectedly, in the region of white frequency phase noise, the 2 NC and 10 NC SHNO, instead of an improvement, show an increase in the phase noise by 2.1 and 3.1 dB, respectively, as compared to a single NC SHNO [see Fig. 3(d)]. A possible explanation could be that process variations affect the thermal noise much more than the noise. From Fig. 2(d), we can deduce from the increase in the frequency variation with current that nonlinearity of NC SHNO chains increases with the number of oscillators. It may lead to a sufficient shift in the corner of white frequency phase noise. Additionally, the temperature of the NC SHNO is higher for longer chains, which contributes to the region of up-converted thermal noise. From the inset of Fig. 3(a) where we plot frequency noise, it is more evident that for 2 and 10 NC, the level of white frequency noise, which corresponds to the flicker frequency type of phase noise, increases with chain length.
To understand the extent of the temperature gradient in long chains, we performed COMSOL simulations of a 10 NC SHNO chain. We used the COMSOL module Electric Currents (ec) to simulate the current density variation in the nanoconstrictions together with the Heat Transfer in Solids (ht) module. Multiphysics simulations were performed using the Electromagnetic Heating (emh1) module. In our simulation, we took into account the 2 nm silicon oxide layer on top of the silicon wafer, which has a significantly lower thermal conductivity of 1.4 W/(m K). The base silicon wafer has a thermal conductivity of 34 W/(m K). The simulations are performed using the measured resistivity for the thin films, i.e., W (300 cm) and NiFe (40 cm). In order to reduce the simulation time and resources, we simulate a limited chip area of 1.5 × 1.5 × 0.5 mm3. Temperature boundary conditions of 293.15 K are applied at the edges of the simulated area. The top panel in Fig. 4 shows a thermal map for an applied DC of 2.35 mA flowing through the chain. In order to visualize the temperature gradients, we have plotted a temperature profile along the x axis in the bottom panel of Fig. 4. It can be seen that the temperature gradient exponentially increases to the edge of an array. The temperature deviation ΔT between the central and the outer NC SHNOs is 13 K. In our previous studies,54 we have experimentally observed a large change in the operating frequency due to thermal effects. In our present work, we estimate that the temperature gradient contributes to a 20 MHz change in the intrinsic frequency between the oscillators. However, since a deviation of 20 MHz unequivocally falls within the broad locking range of SHNOs,27 it cannot be the main factor for the sufficient increase in phase noise. Another reason that can lead to an increase in the phase noise in mutually synchronized chains of oscillators with primarily nearest-neighbor coupling is the phase delay in the coupling. In Ref. 55, it has been shown that the total phase noise can sufficiently increase in a chain of oscillators with nearest-neighbor coupling. Since NC SHNO chains demonstrate positive nonlinearity, the coupling between oscillators most likely happens through propagating spinwaves, which may lead to a large delay. The phase delay of the coupling between NC SHNOs has to be explored further in order to fully understand its contribution to the phase noise increase in both flicker frequency and white frequency regions of the phase noise.
COMSOL simulation of 10 NC SHNOs in a chain. Top panel: a thermal map for an applied DC of 2.35 mA. Bottom panel: a temperature profile along x axis depicted as a dashed green line in the top panel. The temperature difference ΔT between the central and the edge NC SHNOs is 13 K.
COMSOL simulation of 10 NC SHNOs in a chain. Top panel: a thermal map for an applied DC of 2.35 mA. Bottom panel: a temperature profile along x axis depicted as a dashed green line in the top panel. The temperature difference ΔT between the central and the edge NC SHNOs is 13 K.
In summary, we have analyzed the phase noise for single, double, and ten nano-constriction SHNOs. Two mutually synchronized SHNOs demonstrate a 2.8 dB reduction in the phase noise, which corresponds well with the theoretical estimation of 3 dB. The longer chains of ten nano-constrictions demonstrate an improvement of 6.2 dB, which is further from the theoretical value of 10 dB and can be associated with several factors, such as (i) process variation of the nano-constrictions, (ii) temperature gradients within the chain making the NC SHNOs nonidentical and increasing the overall temperature, and (iii) phase delays in the coupling between nano-constrictions, which may lead to decoherence in the chain and elevated noise levels. Further phase noise measurements and analysis will be required for a more complete understanding of these different mechanisms and ways to mitigate their impact.
This work was partially supported by the Horizon 2020 Research and Innovation Program (ERC Advanced Grant No. 835068 “TOPSPIN” and Grant No. 899559 “SpinAge,” DOI 10.3030/899559) and the Swedish Research Council (VR; Dnr 2016-05980).
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
Artem Litvinenko and Akash Kumar contributed equally to this work.
Artem Litvinenko: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Writing – original draft (equal); Writing – review & editing (equal). Akash Kumar: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Writing – original draft (equal); Writing – review & editing (equal). Mona Rajabali: Data curation (supporting); Writing – review & editing (supporting). Ahmad A. AWAD: Data curation (supporting); Formal analysis (supporting); Project administration (equal); Writing – review & editing (supporting). Roman Khymyn: Conceptualization (equal); Data curation (supporting); Formal analysis (supporting); Project administration (equal); Writing – review & editing (supporting). Johan Akerman: Conceptualization (equal); Funding acquisition (equal); Project administration (equal); Writing – review & editing (equal).
DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding author upon reasonable request.